LMFDB - Embedded newform 76.3.b.b.39.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [76,3,Mod(39,76)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(76, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0]))

N = Newforms(chi, 3, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("76.39");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 76 = 2^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	76.b (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$2.07085000914$
Analytic rank:	$0$
Dimension:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} - 2 x^{13} + x^{12} + 14 x^{11} - 42 x^{10} + 28 x^{9} + 132 x^{8} - 440 x^{7} + 528 x^{6} + \cdots + 16384 $ x^14 - 2x^13 + x^12 + 14x^11 - 42x^10 + 28x^9 + 132x^8 - 440x^7 + 528x^6 + 448x^5 - 2688x^4 + 3584x^3 + 1024x^2 - 8192x + 16384
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{12} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			39.1
Root			$-1.92254 + 0.551226i$ of defining polynomial
Character	$\chi$	$=$	76.39
Dual form			76.3.b.b.39.2

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.92254 - 0.551226i) q^{2} +0.644704i q^{3} +(3.39230 + 2.11950i) q^{4} -2.32715 q^{5} +(0.355377 - 1.23947i) q^{6} +8.62924i q^{7} +(-5.35350 - 5.94475i) q^{8} +8.58436 q^{9} +O(q^{10})$	\(q+(-1.92254 - 0.551226i) q^{2} +0.644704i q^{3} +(3.39230 + 2.11950i) q^{4} -2.32715 q^{5} +(0.355377 - 1.23947i) q^{6} +8.62924i q^{7} +(-5.35350 - 5.94475i) q^{8} +8.58436 q^{9} +(4.47404 + 1.28279i) q^{10} +19.2717i q^{11} +(-1.36645 + 2.18703i) q^{12} +13.8067 q^{13} +(4.75666 - 16.5900i) q^{14} -1.50032i q^{15} +(7.01540 + 14.3800i) q^{16} -12.5180 q^{17} +(-16.5037 - 4.73192i) q^{18} +4.35890i q^{19} +(-7.89440 - 4.93241i) q^{20} -5.56330 q^{21} +(10.6231 - 37.0506i) q^{22} -37.4981i q^{23} +(3.83260 - 3.45142i) q^{24} -19.5844 q^{25} +(-26.5438 - 7.61058i) q^{26} +11.3367i q^{27} +(-18.2897 + 29.2730i) q^{28} -6.36930 q^{29} +(-0.827018 + 2.88443i) q^{30} -5.44851i q^{31} +(-5.56075 - 31.5131i) q^{32} -12.4245 q^{33} +(24.0663 + 6.90025i) q^{34} -20.0816i q^{35} +(29.1207 + 18.1946i) q^{36} +20.9250 q^{37} +(2.40274 - 8.38015i) q^{38} +8.90120i q^{39} +(12.4584 + 13.8343i) q^{40} +72.8337 q^{41} +(10.6957 + 3.06664i) q^{42} +10.1553i q^{43} +(-40.8465 + 65.3755i) q^{44} -19.9771 q^{45} +(-20.6699 + 72.0914i) q^{46} -32.4450i q^{47} +(-9.27083 + 4.52286i) q^{48} -25.4638 q^{49} +(37.6517 + 10.7954i) q^{50} -8.07040i q^{51} +(46.8363 + 29.2633i) q^{52} -42.9339 q^{53} +(6.24908 - 21.7952i) q^{54} -44.8482i q^{55} +(51.2987 - 46.1966i) q^{56} -2.81020 q^{57} +(12.2452 + 3.51092i) q^{58} +38.9728i q^{59} +(3.17994 - 5.08955i) q^{60} +25.5611 q^{61} +(-3.00336 + 10.4750i) q^{62} +74.0765i q^{63} +(-6.68010 + 63.6504i) q^{64} -32.1302 q^{65} +(23.8867 + 6.84873i) q^{66} -65.3183i q^{67} +(-42.4648 - 26.5320i) q^{68} +24.1751 q^{69} +(-11.0695 + 38.6076i) q^{70} -18.7557i q^{71} +(-45.9563 - 51.0319i) q^{72} +72.8251 q^{73} +(-40.2292 - 11.5344i) q^{74} -12.6261i q^{75} +(-9.23871 + 14.7867i) q^{76} -166.300 q^{77} +(4.90657 - 17.1129i) q^{78} -139.874i q^{79} +(-16.3259 - 33.4644i) q^{80} +69.9504 q^{81} +(-140.026 - 40.1478i) q^{82} +94.7116i q^{83} +(-18.8724 - 11.7914i) q^{84} +29.1313 q^{85} +(5.59789 - 19.5240i) q^{86} -4.10631i q^{87} +(114.566 - 103.171i) q^{88} +33.3546 q^{89} +(38.4068 + 11.0119i) q^{90} +119.141i q^{91} +(79.4773 - 127.205i) q^{92} +3.51267 q^{93} +(-17.8845 + 62.3766i) q^{94} -10.1438i q^{95} +(20.3166 - 3.58504i) q^{96} -150.817 q^{97} +(48.9551 + 14.0363i) q^{98} +165.435i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q + 2 q^{2} + 2 q^{4} - 40 q^{8} - 68 q^{9}+O(q^{10}) $ 14 * q + 2 * q^2 + 2 * q^4 - 40 * q^8 - 68 * q^9	$ 14 q + 2 q^{2} + 2 q^{4} - 40 q^{8} - 68 q^{9} - 12 q^{10} + 4 q^{12} + 54 q^{13} + 30 q^{14} + 58 q^{16} + 34 q^{17} + 36 q^{18} + 32 q^{20} - 38 q^{21} + 36 q^{22} - 98 q^{24} - 86 q^{25} - 16 q^{26} + 18 q^{28} + 54 q^{29} - 204 q^{30} + 72 q^{32} + 20 q^{33} - 82 q^{34} + 96 q^{36} + 100 q^{37} - 148 q^{40} + 224 q^{41} + 224 q^{42} - 96 q^{44} - 168 q^{45} + 46 q^{46} + 296 q^{48} - 220 q^{49} - 58 q^{50} - 288 q^{52} + 14 q^{53} - 128 q^{54} + 12 q^{56} + 38 q^{57} - 72 q^{58} + 188 q^{60} + 28 q^{61} + 396 q^{62} - 118 q^{64} - 472 q^{65} - 32 q^{66} + 30 q^{68} + 122 q^{69} + 156 q^{70} + 80 q^{72} + 70 q^{73} - 224 q^{74} + 228 q^{77} + 274 q^{78} - 348 q^{80} + 334 q^{81} - 400 q^{82} - 216 q^{84} + 48 q^{85} - 124 q^{86} + 472 q^{88} + 416 q^{90} + 126 q^{92} - 176 q^{93} - 88 q^{94} - 106 q^{96} + 308 q^{97} + 68 q^{98}+O(q^{100}) $ 14 * q + 2 * q^2 + 2 * q^4 - 40 * q^8 - 68 * q^9 - 12 * q^10 + 4 * q^12 + 54 * q^13 + 30 * q^14 + 58 * q^16 + 34 * q^17 + 36 * q^18 + 32 * q^20 - 38 * q^21 + 36 * q^22 - 98 * q^24 - 86 * q^25 - 16 * q^26 + 18 * q^28 + 54 * q^29 - 204 * q^30 + 72 * q^32 + 20 * q^33 - 82 * q^34 + 96 * q^36 + 100 * q^37 - 148 * q^40 + 224 * q^41 + 224 * q^42 - 96 * q^44 - 168 * q^45 + 46 * q^46 + 296 * q^48 - 220 * q^49 - 58 * q^50 - 288 * q^52 + 14 * q^53 - 128 * q^54 + 12 * q^56 + 38 * q^57 - 72 * q^58 + 188 * q^60 + 28 * q^61 + 396 * q^62 - 118 * q^64 - 472 * q^65 - 32 * q^66 + 30 * q^68 + 122 * q^69 + 156 * q^70 + 80 * q^72 + 70 * q^73 - 224 * q^74 + 228 * q^77 + 274 * q^78 - 348 * q^80 + 334 * q^81 - 400 * q^82 - 216 * q^84 + 48 * q^85 - 124 * q^86 + 472 * q^88 + 416 * q^90 + 126 * q^92 - 176 * q^93 - 88 * q^94 - 106 * q^96 + 308 * q^97 + 68 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/76\mathbb{Z}\right)^\times$.

$n$	$21$	$39$
$\chi(n)$	$1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$	−1.92254	−	0.551226i	−0.961269	−	0.275613i
$2$	−1.92254	−	0.551226i	−0.961269	−	0.275613i
$3$			0.644704i			0.214901i	0.994210	+	0.107451i	$0.0342688\pi$
$3$			0.644704i			0.214901i	−0.994210	+	0.107451i	$0.965731\pi$
$4$	3.39230	+	2.11950i	0.848075	+	0.529876i
$5$	−2.32715			−0.465431			−0.232715	−	0.972545i	$-0.574761\pi$
$5$	−2.32715			−0.465431			−0.232715	+	0.972545i	$0.574761\pi$
$6$	0.355377	−	1.23947i	0.0592296	−	0.206578i
$7$			8.62924i			1.23275i	0.787453	+	0.616374i	$0.211399\pi$
$7$			8.62924i			1.23275i	−0.787453	+	0.616374i	$0.788601\pi$
$8$	−5.35350	−	5.94475i	−0.669187	−	0.743094i
$9$	8.58436			0.953817
$10$	4.47404	+	1.28279i	0.447404	+	0.128279i
$11$			19.2717i			1.75197i	0.482334	+	0.875987i	$0.339789\pi$
$11$			19.2717i			1.75197i	−0.482334	+	0.875987i	$0.660211\pi$
$12$	−1.36645	+	2.18703i	−0.113871	+	0.182252i
$13$	13.8067			1.06205			0.531025	−	0.847356i	$-0.321807\pi$
$13$	13.8067			1.06205			0.531025	+	0.847356i	$0.321807\pi$
$14$	4.75666	−	16.5900i	0.339761	−	1.18500i
$15$		−	1.50032i		−	0.100022i
$16$	7.01540	+	14.3800i	0.438463	+	0.898749i
$17$	−12.5180			−0.736353			−0.368177	−	0.929756i	$-0.620018\pi$
$17$	−12.5180			−0.736353			−0.368177	+	0.929756i	$0.620018\pi$
$18$	−16.5037	−	4.73192i	−0.916875	−	0.262884i
$19$			4.35890i			0.229416i
$19$			4.35890i			0.229416i
$20$	−7.89440	−	4.93241i	−0.394720	−	0.246621i
$21$	−5.56330			−0.264919
$22$	10.6231	−	37.0506i	0.482867	−	1.68412i
$23$		−	37.4981i		−	1.63035i	−0.579214	−	0.815175i	$-0.696640\pi$
$23$		−	37.4981i		−	1.63035i	0.579214	−	0.815175i	$-0.303360\pi$
$24$	3.83260	−	3.45142i	0.159692	−	0.143809i
$25$	−19.5844			−0.783374
$26$	−26.5438	−	7.61058i	−1.02092	−	0.292715i
$27$			11.3367i			0.419878i
$28$	−18.2897	+	29.2730i	−0.653204	+	1.04546i
$29$	−6.36930			−0.219631			−0.109816	−	0.993952i	$-0.535026\pi$
$29$	−6.36930			−0.219631			−0.109816	+	0.993952i	$0.535026\pi$
$30$	−0.827018	+	2.88443i	−0.0275673	+	0.0961477i
$31$		−	5.44851i		−	0.175758i	−0.996131	−	0.0878791i	$-0.971991\pi$
$31$		−	5.44851i		−	0.175758i	0.996131	−	0.0878791i	$-0.0280089\pi$
$32$	−5.56075	−	31.5131i	−0.173774	−	0.984786i
$33$	−12.4245			−0.376502
$34$	24.0663	+	6.90025i	0.707833	+	0.202948i
$35$		−	20.0816i		−	0.573759i
$36$	29.1207	+	18.1946i	0.808909	+	0.505405i
$37$	20.9250			0.565542			0.282771	−	0.959187i	$-0.408746\pi$
$37$	20.9250			0.565542			0.282771	+	0.959187i	$0.408746\pi$
$38$	2.40274	−	8.38015i	0.0632299	−	0.220530i
$39$			8.90120i			0.228236i
$40$	12.4584	+	13.8343i	0.311460	+	0.345859i
$41$	72.8337			1.77643			0.888216	−	0.459425i	$-0.151945\pi$
$41$	72.8337			1.77643			0.888216	+	0.459425i	$0.151945\pi$
$42$	10.6957	+	3.06664i	0.254659	+	0.0730152i
$43$			10.1553i			0.236171i	0.993003	+	0.118085i	$0.0376757\pi$
$43$			10.1553i			0.236171i	−0.993003	+	0.118085i	$0.962324\pi$
$44$	−40.8465	+	65.3755i	−0.928329	+	1.48581i
$45$	−19.9771			−0.443936
$46$	−20.6699	+	72.0914i	−0.449346	+	1.56721i
$47$		−	32.4450i		−	0.690318i	−0.938544	−	0.345159i	$-0.887825\pi$
$47$		−	32.4450i		−	0.690318i	0.938544	−	0.345159i	$-0.112175\pi$
$48$	−9.27083	+	4.52286i	−0.193142	+	0.0942262i
$49$	−25.4638			−0.519669
$50$	37.6517	+	10.7954i	0.753033	+	0.215908i
$51$		−	8.07040i		−	0.158243i
$52$	46.8363	+	29.2633i	0.900698	+	0.562755i
$53$	−42.9339			−0.810074			−0.405037	−	0.914300i	$-0.632741\pi$
$53$	−42.9339			−0.810074			−0.405037	+	0.914300i	$0.632741\pi$
$54$	6.24908	−	21.7952i	0.115724	−	0.403615i
$55$		−	44.8482i		−	0.815423i
$56$	51.2987	−	46.1966i	0.916048	−	0.824940i
$57$	−2.81020			−0.0493017
$58$	12.2452	+	3.51092i	0.211125	+	0.0605332i
$59$			38.9728i			0.660556i	0.943884	+	0.330278i	$0.107142\pi$
$59$			38.9728i			0.660556i	−0.943884	+	0.330278i	$0.892858\pi$
$60$	3.17994	−	5.08955i	0.0529991	−	0.0848258i
$61$	25.5611			0.419034			0.209517	−	0.977805i	$-0.432811\pi$
$61$	25.5611			0.419034			0.209517	+	0.977805i	$0.432811\pi$
$62$	−3.00336	+	10.4750i	−0.0484412	+	0.168951i
$63$			74.0765i			1.17582i
$64$	−6.68010	+	63.6504i	−0.104377	+	0.994538i
$65$	−32.1302			−0.494311
$66$	23.8867	+	6.84873i	0.361919	+	0.103769i
$67$		−	65.3183i		−	0.974900i	−0.873151	−	0.487450i	$-0.837927\pi$
$67$		−	65.3183i		−	0.974900i	0.873151	−	0.487450i	$-0.162073\pi$
$68$	−42.4648	−	26.5320i	−0.624483	−	0.390176i
$69$	24.1751			0.350364
$70$	−11.0695	+	38.6076i	−0.158135	+	0.551537i
$71$		−	18.7557i		−	0.264165i	−0.991239	−	0.132083i	$-0.957834\pi$
$71$		−	18.7557i		−	0.264165i	0.991239	−	0.132083i	$-0.0421665\pi$
$72$	−45.9563	−	51.0319i	−0.638283	−	0.708776i
$73$	72.8251			0.997604			0.498802	−	0.866716i	$-0.333773\pi$
$73$	72.8251			0.997604			0.498802	+	0.866716i	$0.333773\pi$
$74$	−40.2292	−	11.5344i	−0.543638	−	0.155871i
$75$		−	12.6261i		−	0.168348i
$76$	−9.23871	+	14.7867i	−0.121562	+	0.194562i
$77$	−166.300			−2.15974
$78$	4.90657	−	17.1129i	0.0629048	−	0.219396i
$79$		−	139.874i		−	1.77056i	−0.465060	−	0.885279i	$-0.653967\pi$
$79$		−	139.874i		−	1.77056i	0.465060	−	0.885279i	$-0.346033\pi$
$80$	−16.3259	−	33.4644i	−0.204074	−	0.418306i
$81$	69.9504			0.863585
$82$	−140.026	−	40.1478i	−1.70763	−	0.489608i
$83$			94.7116i			1.14110i	0.821262	+	0.570552i	$0.193271\pi$
$83$			94.7116i			1.14110i	−0.821262	+	0.570552i	$0.806729\pi$
$84$	−18.8724	−	11.7914i	−0.224671	−	0.140374i
$85$	29.1313			0.342721
$86$	5.59789	−	19.5240i	0.0650917	−	0.227024i
$87$		−	4.10631i		−	0.0471990i
$88$	114.566	−	103.171i	1.30188	−	1.17240i
$89$	33.3546			0.374770			0.187385	−	0.982287i	$-0.439999\pi$
$89$	33.3546			0.374770			0.187385	+	0.982287i	$0.439999\pi$
$90$	38.4068	+	11.0119i	0.426742	+	0.122354i
$91$			119.141i			1.30924i
$92$	79.4773	−	127.205i	0.863884	−	1.38266i
$93$	3.51267			0.0377707
$94$	−17.8845	+	62.3766i	−0.190261	+	0.663581i
$95$		−	10.1438i		−	0.106777i
$96$	20.3166	−	3.58504i	0.211632	−	0.0373441i
$97$	−150.817			−1.55482			−0.777408	−	0.628996i	$-0.783466\pi$
$97$	−150.817			−1.55482			−0.777408	+	0.628996i	$0.783466\pi$
$98$	48.9551	+	14.0363i	0.499542	+	0.143228i
$99$			165.435i			1.67106i
$100$	−66.4360	−	41.5091i	−0.664360	−	0.415091i
$101$	77.7745			0.770045			0.385022	−	0.922907i	$-0.374194\pi$
$101$	77.7745			0.770045			0.385022	+	0.922907i	$0.374194\pi$
$102$	−4.44862	+	15.5157i	−0.0436139	+	0.152114i
$103$		−	14.9481i		−	0.145127i	−0.997364	−	0.0725637i	$-0.976882\pi$
$103$		−	14.9481i		−	0.145127i	0.997364	−	0.0725637i	$-0.0231180\pi$
$104$	−73.9139	−	82.0771i	−0.710711	−	0.789203i
$105$	12.9467			0.123302
$106$	82.5420	+	23.6663i	0.778698	+	0.223267i
$107$		−	80.5205i		−	0.752528i	−0.926512	−	0.376264i	$-0.877208\pi$
$107$		−	80.5205i		−	0.752528i	0.926512	−	0.376264i	$-0.122792\pi$
$108$	−24.0282	+	38.4575i	−0.222483	+	0.356088i
$109$	53.8512			0.494048			0.247024	−	0.969009i	$-0.420547\pi$
$109$	53.8512			0.494048			0.247024	+	0.969009i	$0.420547\pi$
$110$	−24.7215	+	86.2224i	−0.224741	+	0.783840i
$111$			13.4905i			0.121536i
$112$	−124.088	+	60.5376i	−1.10793	+	0.540514i
$113$	105.921			0.937355			0.468677	−	0.883369i	$-0.344731\pi$
$113$	105.921			0.937355			0.468677	+	0.883369i	$0.344731\pi$
$114$	5.40271	+	1.54905i	0.0473922	+	0.0135882i
$115$			87.2638i			0.758815i
$116$	−21.6066	−	13.4998i	−0.186264	−	0.116377i
$117$	118.521			1.01300
$118$	21.4828	−	74.9266i	0.182058	−	0.634972i
$119$		−	108.021i		−	0.907739i
$120$	−8.91905	+	8.03199i	−0.0743254	+	0.0669332i
$121$	−250.399			−2.06941
$122$	−49.1421	−	14.0899i	−0.402804	−	0.115491i
$123$			46.9562i			0.381758i
$124$	11.5481	−	18.4830i	0.0931301	−	0.149056i
$125$	103.755			0.830037
$126$	40.8329	−	142.415i	0.324070	−	1.13028i
$127$			17.1409i			0.134968i	0.997720	+	0.0674839i	$0.0214971\pi$
$127$			17.1409i			0.134968i	−0.997720	+	0.0674839i	$0.978503\pi$
$128$	47.9285	−	118.688i	0.374441	−	0.927251i
$129$	−6.54719			−0.0507534
$130$	61.7715	+	17.7110i	0.475166	+	0.136238i
$131$			229.336i			1.75066i	0.483526	+	0.875330i	$0.339356\pi$
$131$			229.336i			1.75066i	−0.483526	+	0.875330i	$0.660644\pi$
$132$	−42.1478	−	26.3339i	−0.319302	−	0.199499i
$133$	−37.6140			−0.282812
$134$	−36.0051	+	125.577i	−0.268695	+	0.937141i
$135$		−	26.3822i		−	0.195424i
$136$	67.0151	+	74.4164i	0.492758	+	0.547179i
$137$	−120.567			−0.880054			−0.440027	−	0.897985i	$-0.645031\pi$
$137$	−120.567			−0.880054			−0.440027	+	0.897985i	$0.645031\pi$
$138$	−46.4776	−	13.3260i	−0.336794	−	0.0965650i
$139$		−	145.864i		−	1.04938i	−0.851292	−	0.524692i	$-0.824181\pi$
$139$		−	145.864i		−	1.04938i	0.851292	−	0.524692i	$-0.175819\pi$
$140$	42.5630	−	68.1227i	0.304021	−	0.486591i
$141$	20.9174			0.148350
$142$	−10.3387	+	36.0586i	−0.0728074	+	0.253934i
$143$			266.078i			1.86069i
$144$	60.2227	+	123.443i	0.418213	+	0.857243i
$145$	14.8223			0.102223
$146$	−140.009	−	40.1431i	−0.958966	−	0.274953i
$147$		−	16.4166i		−	0.111678i
$148$	70.9840	+	44.3507i	0.479622	+	0.299667i
$149$	19.2547			0.129226			0.0646132	−	0.997910i	$-0.479419\pi$
$149$	19.2547			0.129226			0.0646132	+	0.997910i	$0.479419\pi$
$150$	−6.95984	+	24.2742i	−0.0463989	+	0.161828i
$151$		−	165.526i		−	1.09620i	−0.836414	−	0.548099i	$-0.815352\pi$
$151$		−	165.526i		−	1.09620i	0.836414	−	0.548099i	$-0.184648\pi$
$152$	25.9126	−	23.3354i	0.170477	−	0.153522i
$153$	−107.459			−0.702347
$154$	319.719	+	91.6690i	2.07609	+	0.595253i
$155$			12.6795i			0.0818033i
$156$	−18.8661	+	30.1956i	−0.120937	+	0.193561i
$157$	149.449			0.951907			0.475954	−	0.879470i	$-0.342103\pi$
$157$	149.449			0.951907			0.475954	+	0.879470i	$0.342103\pi$
$158$	−77.1022	+	268.913i	−0.487989	+	1.70198i
$159$		−	27.6796i		−	0.174086i
$160$	12.9407	+	73.3359i	0.0808795	+	0.458349i
$161$	323.580			2.00981
$162$	−134.482	−	38.5585i	−0.830137	−	0.238015i
$163$		−	146.763i		−	0.900384i	−0.892932	−	0.450192i	$-0.851356\pi$
$163$		−	146.763i		−	0.900384i	0.892932	−	0.450192i	$-0.148644\pi$
$164$	247.074	+	154.371i	1.50655	+	0.941289i
$165$	28.9138			0.175235
$166$	52.2075	−	182.087i	0.314503	−	1.09691i
$167$		−	213.493i		−	1.27840i	−0.769040	−	0.639201i	$-0.779265\pi$
$167$		−	213.493i		−	1.27840i	0.769040	−	0.639201i	$-0.220735\pi$
$168$	29.7831	+	33.0724i	0.177281	+	0.196860i
$169$	21.6237			0.127951
$170$	−56.0061	−	16.0579i	−0.329447	−	0.0944584i
$171$			37.4183i			0.218821i
$172$	−21.5243	+	34.4500i	−0.125141	+	0.200291i
$173$	−280.715			−1.62263			−0.811314	−	0.584610i	$-0.801247\pi$
$173$	−280.715			−1.62263			−0.811314	+	0.584610i	$0.801247\pi$
$174$	−2.26351	+	7.89454i	−0.0130087	+	0.0453709i
$175$		−	168.998i		−	0.965704i
$176$	−277.127	+	135.199i	−1.57459	+	0.768175i
$177$	−25.1259			−0.141954
$178$	−64.1254	−	18.3859i	−0.360255	−	0.103292i
$179$			35.1878i			0.196580i	0.995158	+	0.0982899i	$0.0313372\pi$
$179$			35.1878i			0.196580i	−0.995158	+	0.0982899i	$0.968663\pi$
$180$	−67.7684	−	42.3416i	−0.376491	−	0.235231i
$181$	−345.539			−1.90906			−0.954529	−	0.298120i	$-0.903641\pi$
$181$	−345.539			−1.90906			−0.954529	+	0.298120i	$0.903641\pi$
$182$	65.6736	−	229.053i	0.360844	−	1.25853i
$183$			16.4793i			0.0900509i
$184$	−222.917	+	200.746i	−1.21150	+	1.09101i
$185$	−48.6958			−0.263220
$186$	−6.75324	−	1.93628i	−0.0363078	−	0.0104101i
$187$		−	241.244i		−	1.29007i
$188$	68.7672	−	110.063i	0.365783	−	0.585442i
$189$	−97.8271			−0.517604
$190$	−5.59154	+	19.5019i	−0.0294292	+	0.102642i
$191$			100.233i			0.524782i	0.964962	+	0.262391i	$0.0845110\pi$
$191$			100.233i			0.524782i	−0.964962	+	0.262391i	$0.915489\pi$
$192$	−41.0357	−	4.30669i	−0.213727	−	0.0224307i
$193$	168.119			0.871085			0.435543	−	0.900168i	$-0.356557\pi$
$193$	168.119			0.871085			0.435543	+	0.900168i	$0.356557\pi$
$194$	289.952	+	83.1343i	1.49460	+	0.428528i
$195$		−	20.7145i		−	0.106228i
$196$	−86.3808	−	53.9706i	−0.440719	−	0.275360i
$197$	216.705			1.10002			0.550012	−	0.835157i	$-0.314623\pi$
$197$	216.705			1.10002			0.550012	+	0.835157i	$0.314623\pi$
$198$	91.1922	−	318.056i	0.460567	−	1.60634i
$199$		−	82.2545i		−	0.413339i	−0.978411	−	0.206670i	$-0.933738\pi$
$199$		−	82.2545i		−	0.413339i	0.978411	−	0.206670i	$-0.0662625\pi$
$200$	104.845	+	116.424i	0.524224	+	0.582121i
$201$	42.1109			0.209507
$202$	−149.524	−	42.8713i	−0.740220	−	0.212234i
$203$		−	54.9623i		−	0.270750i
$204$	17.1053	−	27.3772i	0.0838493	−	0.134202i
$205$	−169.495			−0.826806
$206$	−8.23978	+	28.7383i	−0.0399990	+	0.139506i
$207$		−	321.897i		−	1.55506i
$208$	96.8592	+	198.540i	0.465669	+	0.954517i
$209$	−84.0035			−0.401931
$210$	−24.8904	−	7.13653i	−0.118526	−	0.0339835i
$211$		−	16.4195i		−	0.0778173i	−0.999243	−	0.0389087i	$-0.987612\pi$
$211$		−	16.4195i		−	0.0778173i	0.999243	−	0.0389087i	$-0.0123881\pi$
$212$	−145.645	−	90.9986i	−0.687003	−	0.429239i
$213$	12.0919			0.0567695
$214$	−44.3850	+	154.804i	−0.207407	+	0.723382i
$215$		−	23.6331i		−	0.109921i
$216$	67.3938	−	60.6910i	0.312009	−	0.280977i
$217$	47.0165			0.216666
$218$	−103.531	−	29.6842i	−0.474913	−	0.136166i
$219$			46.9506i			0.214386i
$220$	95.0561	−	152.139i	0.432073	−	0.691540i
$221$	−172.832			−0.782044
$222$	7.43629	−	25.9359i	0.0334968	−	0.116828i
$223$			195.225i			0.875450i	0.899109	+	0.437725i	$0.144216\pi$
$223$			195.225i			0.875450i	−0.899109	+	0.437725i	$0.855784\pi$
$224$	271.934	−	47.9851i	1.21399	−	0.214219i
$225$	−168.119			−0.747196
$226$	−203.637	−	58.3864i	−0.901050	−	0.258347i
$227$			305.403i			1.34539i	0.739920	+	0.672695i	$0.234863\pi$
$227$			305.403i			1.34539i	−0.739920	+	0.672695i	$0.765137\pi$
$228$	−9.53304	−	5.95623i	−0.0418116	−	0.0261238i
$229$	−229.686			−1.00300			−0.501499	−	0.865158i	$-0.667218\pi$
$229$	−229.686			−1.00300			−0.501499	+	0.865158i	$0.667218\pi$
$230$	48.1020	−	167.768i	0.209139	−	0.729425i
$231$		−	107.214i		−	0.464132i
$232$	34.0981	+	37.8639i	0.146974	+	0.163207i
$233$	187.667			0.805436			0.402718	−	0.915324i	$-0.368065\pi$
$233$	187.667			0.805436			0.402718	+	0.915324i	$0.368065\pi$
$234$	−227.862	−	65.3320i	−0.973767	−	0.279196i
$235$			75.5044i			0.321295i
$236$	−82.6030	+	132.207i	−0.350013	+	0.560201i
$237$	90.1774			0.380495
$238$	−59.5439	+	207.674i	−0.250184	+	0.872581i
$239$			69.8504i			0.292261i	0.989265	+	0.146130i	$0.0466819\pi$
$239$			69.8504i			0.292261i	−0.989265	+	0.146130i	$0.953318\pi$
$240$	21.5747	−	10.5254i	0.0898944	−	0.0438557i
$241$	297.561			1.23469			0.617346	−	0.786691i	$-0.288208\pi$
$241$	297.561			1.23469			0.617346	+	0.786691i	$0.288208\pi$
$242$	481.402	+	138.026i	1.98926	+	0.570357i
$243$			147.128i			0.605463i
$244$	86.7109	+	54.1768i	0.355372	+	0.222036i
$245$	59.2582			0.241870
$246$	25.8835	−	90.2750i	0.105217	−	0.366972i
$247$			60.1818i			0.243651i
$248$	−32.3900	+	29.1686i	−0.130605	+	0.117615i
$249$	−61.0609			−0.245225
$250$	−199.472	−	57.1922i	−0.797889	−	0.228769i
$251$			36.4667i			0.145286i	0.997358	+	0.0726429i	$0.0231433\pi$
$251$			36.4667i			0.145286i	−0.997358	+	0.0726429i	$0.976857\pi$
$252$	−157.005	+	251.290i	−0.623037	+	0.997181i
$253$	722.652			2.85633
$254$	9.44851	−	32.9540i	0.0371988	−	0.129740i
$255$			18.7811i			0.0736513i
$256$	−157.568	+	201.763i	−0.615501	+	0.788136i
$257$	−175.943			−0.684604			−0.342302	−	0.939590i	$-0.611207\pi$
$257$	−175.943			−0.684604			−0.342302	+	0.939590i	$0.611207\pi$
$258$	12.5872	+	3.60898i	0.0487877	+	0.0139883i
$259$			180.567i			0.697171i
$260$	−108.995	−	68.1001i	−0.419213	−	0.261924i
$261$	−54.6764			−0.209488
$262$	126.416	−	440.908i	0.482504	−	1.68285i
$263$			444.887i			1.69158i	0.533512	+	0.845792i	$0.320872\pi$
$263$			444.887i			1.69158i	−0.533512	+	0.845792i	$0.679128\pi$
$264$	66.5148	+	73.8608i	0.251950	+	0.279776i
$265$	99.9138			0.377033
$266$	72.3143	+	20.7338i	0.271858	+	0.0779466i
$267$			21.5038i			0.0805386i
$268$	138.442	−	221.579i	0.516576	−	0.826788i
$269$	149.744			0.556670			0.278335	−	0.960484i	$-0.410217\pi$
$269$	149.744			0.556670			0.278335	+	0.960484i	$0.410217\pi$
$270$	−14.5426	+	50.7208i	−0.0538614	+	0.187855i
$271$			439.685i			1.62245i	0.584731	+	0.811227i	$0.301200\pi$
$271$			439.685i			1.62245i	−0.584731	+	0.811227i	$0.698800\pi$
$272$	−87.8188	−	180.009i	−0.322863	−	0.661797i
$273$	−76.8106			−0.281358
$274$	231.795	+	66.4598i	0.845968	+	0.242554i
$275$		−	377.424i		−	1.37245i
$276$	82.0094	+	51.2393i	0.297135	+	0.185650i
$277$	83.2452			0.300524			0.150262	−	0.988646i	$-0.451988\pi$
$277$	83.2452			0.300524			0.150262	+	0.988646i	$0.451988\pi$
$278$	−80.4042	+	280.430i	−0.289224	+	1.00874i
$279$		−	46.7719i		−	0.167641i
$280$	−119.380	+	107.507i	−0.426357	+	0.383952i
$281$	−12.9851			−0.0462105			−0.0231052	−	0.999733i	$-0.507355\pi$
$281$	−12.9851			−0.0462105			−0.0231052	+	0.999733i	$0.507355\pi$
$282$	−40.2144	−	11.5302i	−0.142604	−	0.0408872i
$283$			111.022i			0.392303i	0.980574	+	0.196152i	$0.0628445\pi$
$283$			111.022i			0.392303i	−0.980574	+	0.196152i	$0.937156\pi$
$284$	39.7529	−	63.6251i	0.139975	−	0.224032i
$285$	6.53976			0.0229465
$286$	146.669	−	511.545i	0.512829	−	1.78862i
$287$			628.500i			2.18990i
$288$	−47.7355	−	270.520i	−0.165748	−	0.939306i
$289$	−132.300			−0.457784
$290$	−28.4965	−	8.17046i	−0.0982639	−	0.0281740i
$291$		−	97.2324i		−	0.334132i
$292$	247.045	+	154.353i	0.846043	+	0.528607i
$293$	−141.572			−0.483179			−0.241590	−	0.970378i	$-0.577669\pi$
$293$	−141.572			−0.483179			−0.241590	+	0.970378i	$0.577669\pi$
$294$	−9.04926	+	31.5615i	−0.0307798	+	0.107352i
$295$		−	90.6956i		−	0.307443i
$296$	−112.022	−	124.394i	−0.378453	−	0.420251i
$297$	−218.478			−0.735615
$298$	−37.0179	−	10.6137i	−0.124221	−	0.0356165i
$299$		−	517.723i		−	1.73151i
$300$	26.7611	−	42.8315i	0.0892036	−	0.142772i
$301$	−87.6329			−0.291139
$302$	−91.2421	+	318.229i	−0.302126	+	1.05374i
$303$			50.1415i			0.165484i
$304$	−62.6809	+	30.5794i	−0.206187	+	0.100590i
$305$	−59.4846			−0.195031
$306$	206.594	+	59.2342i	0.675144	+	0.193576i
$307$			357.744i			1.16529i	0.812727	+	0.582645i	$0.197982\pi$
$307$			357.744i			1.16529i	−0.812727	+	0.582645i	$0.802018\pi$
$308$	−564.141	−	352.474i	−1.83163	−	1.14440i
$309$	9.63710			0.0311880
$310$	6.98927	−	24.3768i	0.0225460	−	0.0786349i
$311$		−	472.383i		−	1.51892i	−0.650557	−	0.759458i	$-0.725464\pi$
$311$		−	472.383i		−	1.51892i	0.650557	−	0.759458i	$-0.274536\pi$
$312$	52.9154	−	47.6526i	0.169601	−	0.152733i
$313$	63.7596			0.203705			0.101852	−	0.994800i	$-0.467523\pi$
$313$	63.7596			0.203705			0.101852	+	0.994800i	$0.467523\pi$
$314$	−287.322	−	82.3804i	−0.915039	−	0.262358i
$315$		−	172.387i		−	0.547261i
$316$	296.464	−	474.495i	0.938177	−	1.50157i
$317$	198.721			0.626880			0.313440	−	0.949608i	$-0.398519\pi$
$317$	198.721			0.626880			0.313440	+	0.949608i	$0.398519\pi$
$318$	−15.2577	+	53.2151i	−0.0479803	+	0.167343i
$319$		−	122.747i		−	0.384788i
$320$	15.5456	−	148.124i	0.0485801	−	0.462888i
$321$	51.9119			0.161719
$322$	−622.094	−	178.366i	−1.93197	−	0.553930i
$323$		−	54.5647i		−	0.168931i
$324$	237.293	+	148.260i	0.732385	+	0.457593i
$325$	−270.394			−0.831983
$326$	−80.8993	+	282.156i	−0.248157	+	0.865511i
$327$			34.7181i			0.106172i
$328$	−389.915	−	432.978i	−1.18877	−	1.32006i
$329$	279.975			0.850989
$330$	−55.5879	−	15.9380i	−0.168448	−	0.0482971i
$331$		−	438.736i		−	1.32549i	−0.748846	−	0.662744i	$-0.769392\pi$
$331$		−	438.736i		−	1.32549i	0.748846	−	0.662744i	$-0.230608\pi$
$332$	−200.742	+	321.290i	−0.604644	+	0.967742i
$333$	179.628			0.539424
$334$	−117.683	+	410.449i	−0.352344	+	1.22889i
$335$			152.006i			0.453748i
$336$	−39.0288	−	80.0003i	−0.116157	−	0.238096i
$337$	−538.699			−1.59851			−0.799257	−	0.600989i	$-0.794773\pi$
$337$	−538.699			−1.59851			−0.799257	+	0.600989i	$0.794773\pi$
$338$	−41.5724	−	11.9196i	−0.122995	−	0.0352650i
$339$			68.2877i			0.201439i
$340$	98.8222	+	61.7440i	0.290653	+	0.181600i
$341$	105.002			0.307924
$342$	20.6260	−	71.9382i	0.0603098	−	0.210346i
$343$			203.100i			0.592127i
$344$	60.3710	−	54.3666i	0.175497	−	0.158043i
$345$	−56.2593			−0.163070
$346$	539.685	+	154.737i	1.55978	+	0.447217i
$347$		−	584.405i		−	1.68417i	−0.539349	−	0.842083i	$-0.681329\pi$
$347$		−	584.405i		−	1.68417i	0.539349	−	0.842083i	$-0.318671\pi$
$348$	8.70335	−	13.9299i	0.0250096	−	0.0400283i
$349$	−300.832			−0.861983			−0.430991	−	0.902356i	$-0.641836\pi$
$349$	−300.832			−0.861983			−0.430991	+	0.902356i	$0.641836\pi$
$350$	−93.1561	+	324.905i	−0.266160	+	0.928301i
$351$			156.522i			0.445931i
$352$	607.312	−	107.165i	1.72532	−	0.304447i
$353$	−316.023			−0.895250			−0.447625	−	0.894221i	$-0.647730\pi$
$353$	−316.023			−0.895250			−0.447625	+	0.894221i	$0.647730\pi$
$354$	48.3055	+	13.8500i	0.136456	+	0.0391244i
$355$			43.6475i			0.122951i
$356$	113.149	+	70.6951i	0.317833	+	0.198582i
$357$	69.6415			0.195074
$358$	19.3964	−	67.6498i	0.0541799	−	0.188966i
$359$		−	116.055i		−	0.323272i	−0.986850	−	0.161636i	$-0.948323\pi$
$359$		−	116.055i		−	0.323272i	0.986850	−	0.161636i	$-0.0516770\pi$
$360$	106.947	+	118.759i	0.297076	+	0.329886i
$361$	−19.0000			−0.0526316
$362$	664.312	+	190.470i	1.83512	+	0.526161i
$363$		−	161.433i		−	0.444720i
$364$	−252.520	+	404.162i	−0.693736	+	1.11033i
$365$	−169.475			−0.464316
$366$	9.08383	−	31.6821i	0.0248192	−	0.0865632i
$367$		−	67.8773i		−	0.184952i	−0.995715	−	0.0924759i	$-0.970522\pi$
$367$		−	67.8773i		−	0.184952i	0.995715	−	0.0924759i	$-0.0294781\pi$
$368$	539.222	−	263.064i	1.46528	−	0.714848i
$369$	625.231			1.69439
$370$	93.6195	+	26.8424i	0.253026	+	0.0725470i
$371$		−	370.487i		−	0.998617i
$372$	11.9160	+	7.44512i	0.0320324	+	0.0200138i
$373$	−267.337			−0.716721			−0.358361	−	0.933583i	$-0.616664\pi$
$373$	−267.337			−0.716721			−0.358361	+	0.933583i	$0.616664\pi$
$374$	−132.980	+	463.800i	−0.355561	+	1.24011i
$375$			66.8910i			0.178376i
$376$	−192.877	+	173.694i	−0.512971	+	0.461952i
$377$	−87.9388			−0.233259
$378$	188.076	+	53.9248i	0.497556	+	0.142658i
$379$		−	372.834i		−	0.983731i	−0.870671	−	0.491866i	$-0.836315\pi$
$379$		−	372.834i		−	0.983731i	0.870671	−	0.491866i	$-0.163685\pi$
$380$	21.4999	−	34.4109i	0.0565786	−	0.0905550i
$381$	−11.0508			−0.0290047
$382$	55.2512	−	192.702i	0.144637	−	0.504457i
$383$			19.6246i			0.0512392i	0.999672	+	0.0256196i	$0.00815587\pi$
$383$			19.6246i			0.0512392i	−0.999672	+	0.0256196i	$0.991844\pi$
$384$	76.5186	+	30.8997i	0.199267	+	0.0804679i
$385$	387.006			1.00521
$386$	−323.216	−	92.6718i	−0.837347	−	0.240082i
$387$			87.1771i			0.225264i
$388$	−511.617	−	319.658i	−1.31860	−	0.823860i
$389$	198.727			0.510867			0.255434	−	0.966827i	$-0.417782\pi$
$389$	198.727			0.510867			0.255434	+	0.966827i	$0.417782\pi$
$390$	−11.4183	+	39.8243i	−0.0292778	+	0.102114i
$391$			469.401i			1.20051i
$392$	136.320	+	151.376i	0.347756	+	0.386163i
$393$	−147.854			−0.376219
$394$	−416.623	−	119.453i	−1.05742	−	0.303181i
$395$			325.509i			0.824072i
$396$	−350.641	+	561.206i	−0.885457	+	1.41719i
$397$	295.448			0.744203			0.372101	−	0.928192i	$-0.378637\pi$
$397$	295.448			0.744203			0.372101	+	0.928192i	$0.378637\pi$
$398$	−45.3408	+	158.137i	−0.113922	+	0.397330i
$399$		−	24.2499i		−	0.0607766i
$400$	−137.392	−	281.623i	−0.343480	−	0.704057i
$401$	−63.9986			−0.159597			−0.0797987	−	0.996811i	$-0.525428\pi$
$401$	−63.9986			−0.159597			−0.0797987	+	0.996811i	$0.525428\pi$
$402$	−80.9599	−	23.2126i	−0.201393	−	0.0577429i
$403$		−	75.2256i		−	0.186664i
$404$	263.835	+	164.843i	0.653056	+	0.408028i
$405$	−162.785			−0.401939
$406$	−30.2966	+	105.667i	−0.0746222	+	0.260264i
$407$			403.262i			0.990815i
$408$	−47.9765	+	43.2049i	−0.117590	+	0.105894i
$409$	504.473			1.23343			0.616715	−	0.787186i	$-0.288463\pi$
$409$	504.473			1.23343			0.616715	+	0.787186i	$0.288463\pi$
$410$	325.861	+	93.4302i	0.794783	+	0.227878i
$411$		−	77.7302i		−	0.189125i
$412$	31.6826	−	50.7085i	0.0768995	−	0.123079i
$413$	−336.306			−0.814299
$414$	−177.438	+	618.859i	−0.428594	+	1.49483i
$415$		−	220.408i		−	0.531105i
$416$	−76.7754	−	435.091i	−0.184556	−	1.04589i
$417$	94.0393			0.225514
$418$	161.500	+	46.3049i	0.386363	+	0.110777i
$419$		−	470.299i		−	1.12243i	−0.827670	−	0.561215i	$-0.810334\pi$
$419$		−	470.299i		−	1.12243i	0.827670	−	0.561215i	$-0.189666\pi$
$420$	43.9190	+	27.4405i	0.104569	+	0.0653345i
$421$	−470.034			−1.11647			−0.558235	−	0.829683i	$-0.688521\pi$
$421$	−470.034			−1.11647			−0.558235	+	0.829683i	$0.688521\pi$
$422$	−9.05083	+	31.5670i	−0.0214475	+	0.0748033i
$423$		−	278.519i		−	0.658437i
$424$	229.847	+	255.231i	0.542091	+	0.601961i
$425$	245.157			0.576840
$426$	−23.2471	−	6.66537i	−0.0545707	−	0.0156464i
$427$			220.573i			0.516564i
$428$	170.664	−	273.150i	0.398747	−	0.638201i
$429$	−171.541			−0.399864
$430$	−13.0271	+	45.4354i	−0.0302957	+	0.105664i
$431$		−	224.005i		−	0.519733i	−0.965645	−	0.259866i	$-0.916321\pi$
$431$		−	224.005i		−	0.519733i	0.965645	−	0.259866i	$-0.0836785\pi$
$432$	−163.022	+	79.5315i	−0.377365	+	0.184101i
$433$	180.128			0.415999			0.207999	−	0.978129i	$-0.433305\pi$
$433$	180.128			0.415999			0.207999	+	0.978129i	$0.433305\pi$
$434$	−90.3909	−	25.9167i	−0.208274	−	0.0597159i
$435$			9.55602i			0.0219679i
$436$	182.680	+	114.138i	0.418990	+	0.261784i
$437$	163.450			0.374028
$438$	25.8804	−	90.2643i	0.0590877	−	0.206083i
$439$		−	590.004i		−	1.34397i	−0.740563	−	0.671987i	$-0.765441\pi$
$439$		−	590.004i		−	1.34397i	0.740563	−	0.671987i	$-0.234559\pi$
$440$	−266.612	+	240.095i	−0.605936	+	0.545671i
$441$	−218.590			−0.495670
$442$	332.276	+	95.2693i	0.751755	+	0.215541i
$443$		−	157.459i		−	0.355439i	−0.984081	−	0.177719i	$-0.943128\pi$
$443$		−	157.459i		−	0.355439i	0.984081	−	0.177719i	$-0.0568720\pi$
$444$	−28.5931	+	45.7637i	−0.0643988	+	0.103071i
$445$	−77.6212			−0.174430
$446$	107.613	−	375.328i	0.241285	−	0.841543i
$447$			12.4136i			0.0277709i
$448$	−549.255	−	57.6442i	−1.22602	−	0.128670i
$449$	−715.434			−1.59339			−0.796697	−	0.604379i	$-0.793421\pi$
$449$	−715.434			−1.59339			−0.796697	+	0.604379i	$0.793421\pi$
$450$	323.215	+	92.6716i	0.718256	+	0.205937i
$451$			1403.63i			3.11226i
$452$	359.316	+	224.500i	0.794947	+	0.496682i
$453$	106.715			0.235574
$454$	168.346	−	587.149i	0.370807	−	1.29328i
$455$		−	277.259i		−	0.609361i
$456$	15.0444	+	16.7059i	0.0329921	+	0.0366358i
$457$	31.3497			0.0685988			0.0342994	−	0.999412i	$-0.489080\pi$
$457$	31.3497			0.0685988			0.0342994	+	0.999412i	$0.489080\pi$
$458$	441.581	+	126.609i	0.964150	+	0.276439i
$459$		−	141.913i		−	0.309178i
$460$	−184.956	+	296.025i	−0.402078	+	0.643532i
$461$	368.719			0.799824			0.399912	−	0.916554i	$-0.369041\pi$
$461$	368.719			0.799824			0.399912	+	0.916554i	$0.369041\pi$
$462$	−59.0994	+	206.124i	−0.127921	+	0.446155i
$463$			817.405i			1.76545i	0.469887	+	0.882726i	$0.344295\pi$
$463$			817.405i			1.76545i	−0.469887	+	0.882726i	$0.655705\pi$
$464$	−44.6832	−	91.5905i	−0.0963001	−	0.197393i
$465$	−8.17453			−0.0175796
$466$	−360.796	−	103.447i	−0.774241	−	0.221989i
$467$		−	86.9194i		−	0.186123i	−0.995660	−	0.0930614i	$-0.970335\pi$
$467$		−	86.9194i		−	0.186123i	0.995660	−	0.0930614i	$-0.0296653\pi$
$468$	402.060	+	251.206i	0.859102	+	0.536766i
$469$	563.647			1.20181
$470$	41.6200	−	145.160i	0.0885531	−	0.308851i
$471$			96.3506i			0.204566i
$472$	231.683	−	208.641i	0.490855	−	0.442036i
$473$	−195.711			−0.413765
$474$	−173.369	−	49.7081i	−0.365758	−	0.104869i
$475$		−	85.3662i		−	0.179718i
$476$	228.951	−	366.439i	0.480989	−	0.769830i
$477$	−368.560			−0.772662
$478$	38.5033	−	134.290i	0.0805509	−	0.280941i
$479$			333.107i			0.695421i	0.937602	+	0.347710i	$0.113041\pi$
$479$			333.107i			0.695421i	−0.937602	+	0.347710i	$0.886959\pi$
$480$	−47.2799	+	8.34293i	−0.0984999	+	0.0173811i
$481$	288.905			0.600634
$482$	−572.072	−	164.023i	−1.18687	−	0.340297i
$483$			208.613i			0.431911i
$484$	−849.429	−	530.722i	−1.75502	−	1.09653i
$485$	350.975			0.723659
$486$	81.1005	−	282.858i	0.166873	−	0.582013i
$487$			264.690i			0.543511i	0.962366	+	0.271755i	$0.0876042\pi$
$487$			264.690i			0.543511i	−0.962366	+	0.271755i	$0.912396\pi$
$488$	−136.841	−	151.954i	−0.280412	−	0.311382i
$489$	94.6183			0.193494
$490$	−113.926	−	32.6646i	−0.232502	−	0.0666625i
$491$		−	402.874i		−	0.820518i	−0.911969	−	0.410259i	$-0.865438\pi$
$491$		−	402.874i		−	0.820518i	0.911969	−	0.410259i	$-0.134562\pi$
$492$	−99.5238	+	159.289i	−0.202284	+	0.323759i
$493$	79.7310			0.161726
$494$	33.1738	−	115.702i	0.0671534	−	0.234214i
$495$		−	384.993i		−	0.777764i
$496$	78.3495	−	38.2235i	0.157963	−	0.0770634i
$497$	161.848			0.325650
$498$	117.392	+	33.6584i	0.235727	+	0.0675871i
$499$		−	350.536i		−	0.702477i	−0.936286	−	0.351238i	$-0.885761\pi$
$499$		−	350.536i		−	0.702477i	0.936286	−	0.351238i	$-0.114239\pi$
$500$	351.967	+	219.908i	0.703934	+	0.439817i
$501$	137.640			0.274730
$502$	20.1014	−	70.1086i	0.0400426	−	0.139659i
$503$			339.615i			0.675179i	0.941293	+	0.337589i	$0.109612\pi$
$503$			339.615i			0.675179i	−0.941293	+	0.337589i	$0.890388\pi$
$504$	440.366	−	396.568i	0.873742	−	0.786842i
$505$	−180.993			−0.358403
$506$	−1389.33	−	398.345i	−2.74570	−	0.787242i
$507$			13.9409i			0.0274969i
$508$	−36.3302	+	58.1471i	−0.0715162	+	0.114463i
$509$	−933.003			−1.83301			−0.916506	−	0.400020i	$-0.869003\pi$
$509$	−933.003			−1.83301			−0.916506	+	0.400020i	$0.869003\pi$
$510$	10.3526	−	36.1073i	0.0202992	−	0.0707986i
$511$			628.426i			1.22980i
$512$	414.148	−	301.041i	0.808882	−	0.587971i
$513$	−49.4155			−0.0963266
$514$	338.257	+	96.9844i	0.658088	+	0.188686i
$515$			34.7866i			0.0675467i
$516$	−22.2100	−	13.8768i	−0.0430427	−	0.0268930i
$517$	625.270			1.20942
$518$	99.5333	−	347.147i	0.192149	−	0.670169i
$519$		−	180.978i		−	0.348705i
$520$	172.009	+	191.006i	0.330787	+	0.367319i
$521$	−415.284			−0.797091			−0.398545	−	0.917149i	$-0.630485\pi$
$521$	−415.284			−0.797091			−0.398545	+	0.917149i	$0.630485\pi$
$522$	105.117	+	30.1390i	0.201374	+	0.0577376i
$523$			241.598i			0.461947i	0.972960	+	0.230973i	$0.0741911\pi$
$523$			241.598i			0.461947i	−0.972960	+	0.230973i	$0.925809\pi$
$524$	−486.080	+	777.978i	−0.927633	+	1.48469i
$525$	108.954			0.207531
$526$	245.233	−	855.311i	0.466223	−	1.62607i
$527$			68.2044i			0.129420i
$528$	−87.1632	−	178.665i	−0.165082	−	0.338381i
$529$	−877.105			−1.65804
$530$	−192.088	−	55.0750i	−0.362430	−	0.103915i
$531$			334.556i			0.630050i
$532$	−127.598	−	79.7230i	−0.239846	−	0.149855i
$533$	1005.59			1.88666
$534$	11.8535	−	41.3419i	0.0221975	−	0.0774192i
$535$			187.384i			0.350250i
$536$	−388.301	+	349.681i	−0.724442	+	0.652391i
$537$	−22.6857			−0.0422452
$538$	−287.889	−	82.5428i	−0.535109	−	0.153425i
$539$		−	490.731i		−	0.910447i
$540$	55.9173	−	89.4965i	0.103551	−	0.165734i
$541$	−304.787			−0.563377			−0.281689	−	0.959506i	$-0.590894\pi$
$541$	−304.787			−0.563377			−0.281689	+	0.959506i	$0.590894\pi$
$542$	242.366	−	845.311i	0.447169	−	1.55961i
$543$		−	222.770i		−	0.410259i
$544$	69.6095	+	394.482i	0.127959	+	0.725150i
$545$	−125.320			−0.229945
$546$	147.671	+	42.3400i	0.270460	+	0.0775458i
$547$		−	555.915i		−	1.01630i	−0.861269	−	0.508149i	$-0.830330\pi$
$547$		−	555.915i		−	1.01630i	0.861269	−	0.508149i	$-0.169670\pi$
$548$	−409.001	−	255.543i	−0.746352	−	0.466319i
$549$	219.425			0.399682
$550$	−208.046	+	725.612i	−0.378265	+	1.31930i
$551$		−	27.7632i		−	0.0503869i
$552$	−129.422	−	143.715i	−0.234459	−	0.260354i
$553$	1207.01			2.18265
$554$	−160.042	−	45.8869i	−0.288885	−	0.0828283i
$555$		−	31.3944i		−	0.0565664i
$556$	309.160	−	494.816i	0.556043	−	0.889956i
$557$	−192.055			−0.344802			−0.172401	−	0.985027i	$-0.555152\pi$
$557$	−192.055			−0.344802			−0.172401	+	0.985027i	$0.555152\pi$
$558$	−25.7819	+	89.9208i	−0.0462041	+	0.161148i
$559$			140.211i			0.250825i
$560$	288.773	−	140.880i	0.515666	−	0.251572i
$561$	155.531			0.277238
$562$	24.9644	+	7.15775i	0.0444207	+	0.0127362i
$563$			107.771i			0.191423i	0.995409	+	0.0957114i	$0.0305126\pi$
$563$			107.771i			0.191423i	−0.995409	+	0.0957114i	$0.969487\pi$
$564$	70.9580	+	44.3345i	0.125812	+	0.0786072i
$565$	−246.495			−0.436274
$566$	61.1981	−	213.444i	0.108124	−	0.377109i
$567$			603.619i			1.06458i
$568$	−111.498	+	100.409i	−0.196300	+	0.176776i
$569$	−647.934			−1.13872			−0.569362	−	0.822087i	$-0.692810\pi$
$569$	−647.934			−1.13872			−0.569362	+	0.822087i	$0.692810\pi$
$570$	−12.5729	−	3.60489i	−0.0220578	−	0.00632436i
$571$			285.402i			0.499828i	0.968268	+	0.249914i	$0.0804023\pi$
$571$			285.402i			0.499828i	−0.968268	+	0.249914i	$0.919598\pi$
$572$	−563.953	+	902.616i	−0.985933	+	1.57800i
$573$	−64.6208			−0.112776
$574$	346.445	−	1208.31i	0.603563	−	2.10508i
$575$			734.376i			1.27717i
$576$	−57.3444	+	546.398i	−0.0995562	+	0.948608i
$577$	−92.4624			−0.160247			−0.0801234	−	0.996785i	$-0.525531\pi$
$577$	−92.4624			−0.160247			−0.0801234	+	0.996785i	$0.525531\pi$
$578$	254.351	+	72.9269i	0.440053	+	0.126171i
$579$			108.387i			0.187197i
$580$	50.2819	+	31.4160i	0.0866929	+	0.0541656i
$581$	−817.289			−1.40669
$582$	−53.5970	+	186.933i	−0.0920911	+	0.321191i
$583$		−	827.410i		−	1.41923i
$584$	−389.869	−	432.927i	−0.667584	−	0.741314i
$585$	−275.817			−0.471482
$586$	272.177	+	78.0379i	0.464465	+	0.133170i
$587$		−	654.717i		−	1.11536i	−0.830056	−	0.557681i	$-0.811691\pi$
$587$		−	654.717i		−	1.11536i	0.830056	−	0.557681i	$-0.188309\pi$
$588$	34.7951	−	55.6901i	0.0591753	−	0.0947110i
$589$	23.7495			0.0403217
$590$	−49.9938	+	174.366i	−0.0847352	+	0.295535i
$591$			139.710i			0.236396i
$592$	146.798	+	300.902i	0.247969	+	0.508280i
$593$	106.297			0.179253			0.0896264	−	0.995975i	$-0.471433\pi$
$593$	106.297			0.179253			0.0896264	+	0.995975i	$0.471433\pi$
$594$	420.032	+	120.431i	0.707124	+	0.202745i
$595$			251.381i			0.422489i
$596$	65.3178	+	40.8105i	0.109594	+	0.0684740i
$597$	53.0298			0.0888271
$598$	−285.382	+	995.342i	−0.477228	+	1.66445i
$599$		−	478.675i		−	0.799124i	−0.916706	−	0.399562i	$-0.869162\pi$
$599$		−	478.675i		−	0.799124i	0.916706	−	0.399562i	$-0.130838\pi$
$600$	−75.0591	+	67.5939i	−0.125098	+	0.112656i
$601$	808.024			1.34447			0.672233	−	0.740340i	$-0.265335\pi$
$601$	808.024			1.34447			0.672233	+	0.740340i	$0.265335\pi$
$602$	168.478	+	48.3055i	0.279863	+	0.0802418i
$603$		−	560.715i		−	0.929876i
$604$	350.833	−	561.513i	0.580849	−	0.929657i
$605$	582.717			0.963169
$606$	27.6393	−	96.3990i	0.0456094	−	0.159074i
$607$			694.694i			1.14447i	0.820089	+	0.572236i	$0.193924\pi$
$607$			694.694i			1.14447i	−0.820089	+	0.572236i	$0.806076\pi$
$608$	137.363	−	24.2388i	0.225925	−	0.0398664i
$609$	35.4344			0.0581845
$610$	114.361	+	32.7894i	0.187478	+	0.0537531i
$611$		−	447.956i		−	0.733153i
$612$	−364.533	−	227.760i	−0.595643	−	0.372157i
$613$	−496.830			−0.810489			−0.405245	−	0.914208i	$-0.632814\pi$
$613$	−496.830			−0.810489			−0.405245	+	0.914208i	$0.632814\pi$
$614$	197.198	−	687.776i	0.321169	−	1.12016i
$615$		−	109.274i		−	0.177682i
$616$	890.289	+	988.614i	1.44527	+	1.60489i
$617$	387.951			0.628770			0.314385	−	0.949296i	$-0.398202\pi$
$617$	387.951			0.628770			0.314385	+	0.949296i	$0.398202\pi$
$618$	−18.5277	−	5.31222i	−0.0299801	−	0.00859583i
$619$			882.105i			1.42505i	0.701647	+	0.712525i	$0.252448\pi$
$619$			882.105i			1.42505i	−0.701647	+	0.712525i	$0.747552\pi$
$620$	−26.8743	+	43.0127i	−0.0433456	+	0.0693753i
$621$	425.104			0.684548
$622$	−260.389	+	908.173i	−0.418633	+	1.46009i
$623$			287.824i			0.461998i
$624$	−127.999	+	62.4455i	−0.205127	+	0.100073i
$625$	248.156			0.397050
$626$	−122.580	−	35.1459i	−0.195815	−	0.0561437i
$627$		−	54.1574i		−	0.0863754i
$628$	506.977	+	316.759i	0.807289	+	0.504393i
$629$	−261.940			−0.416439
$630$	−95.0243	+	331.421i	−0.150832	+	0.526065i
$631$		−	502.601i		−	0.796515i	−0.917274	−	0.398258i	$-0.869615\pi$
$631$		−	502.601i		−	0.796515i	0.917274	−	0.398258i	$-0.130385\pi$
$632$	−831.517	+	748.816i	−1.31569	+	1.18484i
$633$	10.5857			0.0167230
$634$	−382.048	−	109.540i	−0.602600	−	0.172776i
$635$		−	39.8895i		−	0.0628181i
$636$	58.6671	−	93.8977i	0.0922439	−	0.147638i
$637$	−351.570			−0.551915
$638$	−67.6616	+	235.987i	−0.106053	+	0.369885i
$639$		−	161.006i		−	0.251966i
$640$	−111.537	+	276.205i	−0.174277	+	0.431571i
$641$	385.625			0.601600			0.300800	−	0.953687i	$-0.402746\pi$
$641$	385.625			0.601600			0.300800	+	0.953687i	$0.402746\pi$
$642$	−99.8026	−	28.6152i	−0.155456	−	0.0445719i
$643$			925.293i			1.43902i	0.694480	+	0.719512i	$0.255635\pi$
$643$			925.293i			1.43902i	−0.694480	+	0.719512i	$0.744365\pi$
$644$	1097.68	+	685.829i	1.70447	+	1.06495i
$645$	15.2363			0.0236222
$646$	−30.0775	+	104.903i	−0.0465596	+	0.162388i
$647$		−	656.956i		−	1.01539i	−0.861538	−	0.507694i	$-0.830498\pi$
$647$		−	656.956i		−	1.01539i	0.861538	−	0.507694i	$-0.169502\pi$
$648$	−374.479	−	415.838i	−0.577900	−	0.641725i
$649$	−751.073			−1.15728
$650$	519.843	+	149.048i	0.799759	+	0.229305i
$651$			30.3117i			0.0465617i
$652$	311.064	−	497.863i	0.477092	−	0.763593i
$653$	392.897			0.601680			0.300840	−	0.953675i	$-0.402733\pi$
$653$	392.897			0.601680			0.300840	+	0.953675i	$0.402733\pi$
$654$	19.1375	−	66.7468i	0.0292622	−	0.102059i
$655$		−	533.701i		−	0.814811i
$656$	510.958	+	1047.35i	0.778899	+	1.59657i
$657$	625.157			0.951533
$658$	−538.263	−	154.330i	−0.818029	−	0.234543i
$659$			695.576i			1.05550i	0.849399	+	0.527751i	$0.176965\pi$
$659$			695.576i			1.05550i	−0.849399	+	0.527751i	$0.823035\pi$
$660$	98.0844	+	61.2830i	0.148613	+	0.0928530i
$661$	1179.05			1.78373			0.891866	−	0.452299i	$-0.149396\pi$
$661$	1179.05			1.78373			0.891866	+	0.452299i	$0.149396\pi$
$662$	−241.843	+	843.487i	−0.365322	+	1.27415i
$663$		−	111.425i		−	0.168062i
$664$	563.037	−	507.038i	0.847947	−	0.763612i
$665$	87.5335			0.131629
$666$	−345.342	−	99.0156i	−0.518531	−	0.148672i
$667$			238.837i			0.358076i
$668$	452.500	−	724.233i	0.677395	−	1.08418i
$669$	−125.863			−0.188135
$670$	83.7894	−	292.237i	0.125059	−	0.436174i
$671$			492.606i			0.734137i
$672$	30.9362	+	175.317i	0.0460359	+	0.260889i
$673$	−881.170			−1.30932			−0.654658	−	0.755925i	$-0.727187\pi$
$673$	−881.170			−1.30932			−0.654658	+	0.755925i	$0.727187\pi$
$674$	1035.67	+	296.945i	1.53660	+	0.440571i
$675$		−	222.022i		−	0.328921i
$676$	73.3542	+	45.8316i	0.108512	+	0.0677982i
$677$	−126.536			−0.186907			−0.0934533	−	0.995624i	$-0.529791\pi$
$677$	−126.536			−0.186907			−0.0934533	+	0.995624i	$0.529791\pi$
$678$	37.6420	−	131.286i	0.0555191	−	0.193637i
$679$		−	1301.44i		−	1.91670i
$680$	−155.954	−	173.178i	−0.229345	−	0.254674i
$681$	−196.895			−0.289126
$682$	−201.870	−	57.8798i	−0.295998	−	0.0848678i
$683$		−	873.042i		−	1.27825i	−0.769105	−	0.639123i	$-0.779298\pi$
$683$		−	873.042i		−	1.27825i	0.769105	−	0.639123i	$-0.220702\pi$
$684$	−79.3083	+	126.934i	−0.115948	+	0.185576i
$685$	280.579			0.409604
$686$	111.954	−	390.467i	0.163198	−	0.569193i
$687$		−	148.080i		−	0.215545i
$688$	−146.034	+	71.2439i	−0.212258	+	0.103552i
$689$	−592.774			−0.860339
$690$	108.161	+	31.0116i	0.156754	+	0.0449443i
$691$		−	610.841i		−	0.883996i	−0.897016	−	0.441998i	$-0.854270\pi$
$691$		−	610.841i		−	0.883996i	0.897016	−	0.441998i	$-0.145730\pi$
$692$	−952.269	−	594.976i	−1.37611	−	0.859792i
$693$	−1427.58			−2.06000
$694$	−322.139	+	1123.54i	−0.464178	+	1.61894i
$695$			339.449i			0.488415i
$696$	−24.4110	+	21.9831i	−0.0350733	+	0.0315850i
$697$	−911.733			−1.30808
$698$	578.361	+	165.826i	0.828597	+	0.237574i
$699$			120.989i			0.173089i
$700$	358.192	−	573.292i	0.511703	−	0.818989i
$701$	−506.393			−0.722387			−0.361193	−	0.932491i	$-0.617631\pi$
$701$	−506.393			−0.722387			−0.361193	+	0.932491i	$0.617631\pi$
$702$	86.2789	−	300.919i	0.122904	−	0.428660i
$703$			91.2102i			0.129744i
$704$	−1226.65	−	128.737i	−1.74241	−	0.182865i
$705$	−48.6780			−0.0690467
$706$	607.566	+	174.200i	0.860576	+	0.246742i
$707$			671.135i			0.949272i
$708$	−85.2346	−	53.2545i	−0.120388	−	0.0752182i
$709$	−1036.42			−1.46180			−0.730902	−	0.682483i	$-0.760900\pi$
$709$	−1036.42			−1.46180			−0.730902	+	0.682483i	$0.760900\pi$
$710$	24.0596	−	83.9140i	0.0338868	−	0.118189i
$711$		−	1200.73i		−	1.68879i
$712$	−178.564	−	198.284i	−0.250792	−	0.278489i
$713$	−204.308			−0.286548
$714$	−133.888	−	38.3882i	−0.187519	−	0.0537649i
$715$		−	619.204i		−	0.866020i
$716$	−74.5806	+	119.367i	−0.104163	+	0.166714i
$717$	−45.0328			−0.0628072
$718$	−63.9722	+	223.119i	−0.0890978	+	0.310751i
$719$		−	834.943i		−	1.16126i	−0.814169	−	0.580628i	$-0.802807\pi$
$719$		−	834.943i		−	1.16126i	0.814169	−	0.580628i	$-0.197193\pi$
$720$	−140.148	−	287.271i	−0.194649	−	0.398987i
$721$	128.991			0.178905
$722$	36.5282	+	10.4733i	0.0505931	+	0.0145059i
$723$			191.839i			0.265337i
$724$	−1172.17	−	732.372i	−1.61902	−	1.01156i
$725$	124.739			0.172053
$726$	−88.9862	+	310.362i	−0.122571	+	0.427495i
$727$		−	225.252i		−	0.309838i	−0.987927	−	0.154919i	$-0.950488\pi$
$727$		−	225.252i		−	0.309838i	0.987927	−	0.154919i	$-0.0495116\pi$
$728$	708.263	−	637.821i	0.972889	−	0.876128i
$729$	534.700			0.733470
$730$	325.822	+	93.4191i	0.446332	+	0.127971i
$731$		−	127.125i		−	0.173905i
$732$	−34.9280	+	55.9028i	−0.0477158	+	0.0763700i
$733$	697.401			0.951434			0.475717	−	0.879598i	$-0.342189\pi$
$733$	697.401			0.951434			0.475717	+	0.879598i	$0.342189\pi$
$734$	−37.4157	+	130.497i	−0.0509751	+	0.177788i
$735$			38.2040i			0.0519782i
$736$	−1181.68	+	208.517i	−1.60555	+	0.283312i
$737$	1258.80			1.70800
$738$	−1202.03	−	344.643i	−1.62877	−	0.466996i
$739$		−	1065.87i		−	1.44231i	−0.692772	−	0.721157i	$-0.743611\pi$
$739$		−	1065.87i		−	1.44231i	0.692772	−	0.721157i	$-0.256389\pi$
$740$	−165.191	−	103.211i	−0.223231	−	0.139474i
$741$	−38.7994			−0.0523609
$742$	−204.222	+	712.275i	−0.275232	+	0.959939i
$743$		−	1050.43i		−	1.41376i	−0.707332	−	0.706881i	$-0.750102\pi$
$743$		−	1050.43i		−	1.41376i	0.707332	−	0.706881i	$-0.249898\pi$
$744$	−18.8051	−	20.8820i	−0.0252756	−	0.0280671i
$745$	−44.8087			−0.0601459
$746$	513.966	+	147.363i	0.688962	+	0.197538i
$747$			813.038i			1.08840i
$748$	511.317	−	818.370i	0.683578	−	1.09408i
$749$	694.831			0.927678
$750$	36.8720	−	128.600i	0.0491627	−	0.171467i
$751$		−	26.6606i		−	0.0355002i	−0.999842	−	0.0177501i	$-0.994350\pi$
$751$		−	26.6606i		−	0.0355002i	0.999842	−	0.0177501i	$-0.00565033\pi$
$752$	466.558	−	227.614i	0.620423	−	0.302679i
$753$	−23.5102			−0.0312221
$754$	169.066	+	48.4741i	0.224225	+	0.0642893i
$755$			385.204i			0.510204i
$756$	−331.859	−	207.345i	−0.438967	−	0.274266i
$757$	445.989			0.589153			0.294577	−	0.955628i	$-0.404821\pi$
$757$	445.989			0.589153			0.294577	+	0.955628i	$0.404821\pi$
$758$	−205.516	+	716.788i	−0.271129	+	0.945630i
$759$			465.897i			0.613830i
$760$	−60.3025	+	54.3050i	−0.0793454	+	0.0714539i
$761$	294.799			0.387384			0.193692	−	0.981062i	$-0.437954\pi$
$761$	294.799			0.387384			0.193692	+	0.981062i	$0.437954\pi$
$762$	21.2456	+	6.09149i	0.0278813	+	0.00799408i
$763$			464.695i			0.609037i
$764$	−212.445	+	340.022i	−0.278070	+	0.445055i
$765$	250.074			0.326894
$766$	10.8176	−	37.7291i	0.0141222	−	0.0492547i
$767$			538.084i			0.701543i
$768$	−130.077	−	101.585i	−0.169371	−	0.132272i
$769$	−279.363			−0.363281			−0.181641	−	0.983365i	$-0.558141\pi$
$769$	−279.363			−0.363281			−0.181641	+	0.983365i	$0.558141\pi$
$770$	−744.034	−	213.328i	−0.966278	−	0.277049i
$771$		−	113.431i		−	0.147122i
$772$	570.312	+	356.330i	0.738746	+	0.461567i
$773$	−1092.48			−1.41329			−0.706647	−	0.707566i	$-0.749793\pi$
$773$	−1092.48			−1.41329			−0.706647	+	0.707566i	$0.749793\pi$
$774$	48.0543	−	167.601i	0.0620856	−	0.216539i
$775$			106.705i			0.137684i
$776$	807.400	+	896.571i	1.04046	+	1.15537i
$777$	−116.412			−0.149823
$778$	−382.061	−	109.544i	−0.491081	−	0.140802i
$779$			317.475i			0.407542i
$780$	43.9044	−	70.2697i	0.0562877	−	0.0900893i
$781$	361.456			0.462811
$782$	258.746	−	902.441i	0.330877	−	1.15402i
$783$		−	72.2069i		−	0.0922183i
$784$	−178.639	−	366.169i	−0.227856	−	0.467052i
$785$	−347.792			−0.443047
$786$	284.255	+	81.5010i	0.361648	+	0.103691i
$787$			231.147i			0.293706i	0.989158	+	0.146853i	$0.0469145\pi$
$787$			231.147i			0.293706i	−0.989158	+	0.146853i	$0.953086\pi$
$788$	735.127	+	459.306i	0.932903	+	0.582876i
$789$	−286.820			−0.363524
$790$	179.429	−	625.802i	0.227125	−	0.792155i
$791$			914.019i			1.15552i
$792$	983.472	−	885.658i	1.24176	−	1.11825i
$793$	352.913			0.445035
$794$	−568.011	−	162.859i	−0.715379	−	0.205112i
$795$			64.4148i			0.0810249i
$796$	174.339	−	279.032i	0.219018	−	0.350543i
$797$	453.389			0.568869			0.284435	−	0.958695i	$-0.408194\pi$
$797$	453.389			0.568869			0.284435	+	0.958695i	$0.408194\pi$
$798$	−13.3672	+	46.6213i	−0.0167508	+	0.0584227i
$799$			406.146i			0.508318i
$800$	108.904	+	617.165i	0.136130	+	0.771456i
$801$	286.327			0.357462
$802$	123.040	+	35.2777i	0.153416	+	0.0439871i
$803$			1403.47i			1.74778i
$804$	142.853	+	89.2543i	0.177678	+	0.111013i
$805$	−753.020			−0.935429
$806$	−41.4663	+	144.624i	−0.0514470	+	0.179434i
$807$			96.5406i			0.119629i
$808$	−416.366	−	462.350i	−0.515304	−	0.572216i
$809$	528.874			0.653738			0.326869	−	0.945070i	$-0.394006\pi$
$809$	528.874			0.653738			0.326869	+	0.945070i	$0.394006\pi$
$810$	312.961	+	89.7315i	0.386371	+	0.110780i
$811$			1185.82i			1.46217i	0.682287	+	0.731084i	$0.260985\pi$
$811$			1185.82i			1.46217i	−0.682287	+	0.731084i	$0.739015\pi$
$812$	116.493	−	186.449i	0.143464	−	0.229616i
$813$	−283.467			−0.348668
$814$	222.288	−	775.286i	0.273081	−	0.952439i
$815$			341.539i			0.419066i
$816$	116.052	−	56.6171i	0.142221	−	0.0693837i
$817$	−44.2661			−0.0541813
$818$	−969.868	−	278.079i	−1.18566	−	0.339949i
$819$			1022.75i			1.24878i
$820$	−574.979	−	359.246i	−0.701194	−	0.438105i
$821$	−570.471			−0.694849			−0.347424	−	0.937708i	$-0.612944\pi$
$821$	−570.471			−0.694849			−0.347424	+	0.937708i	$0.612944\pi$
$822$	−42.8469	+	149.439i	−0.0521252	+	0.181800i
$823$		−	1386.72i		−	1.68496i	−0.538728	−	0.842480i	$-0.681095\pi$
$823$		−	1386.72i		−	1.68496i	0.538728	−	0.842480i	$-0.318905\pi$
$824$	−88.8628	+	80.0247i	−0.107843	+	0.0971174i
$825$	243.327			0.294942
$826$	646.560	+	185.380i	0.782760	+	0.224431i
$827$			1073.59i			1.29818i	0.760712	+	0.649089i	$0.224850\pi$
$827$			1073.59i			1.29818i	−0.760712	+	0.649089i	$0.775150\pi$
$828$	682.262	−	1091.97i	0.823988	−	1.31881i
$829$	1109.24			1.33804			0.669021	−	0.743243i	$-0.266713\pi$
$829$	1109.24			1.33804			0.669021	+	0.743243i	$0.266713\pi$
$830$	−121.495	+	423.743i	−0.146379	+	0.510534i
$831$			53.6685i			0.0645830i
$832$	−92.2299	+	878.799i	−0.110853	+	1.05625i
$833$	318.756			0.382660
$834$	−180.794	−	51.8369i	−0.216779	−	0.0621545i
$835$			496.831i			0.595007i
$836$	−284.965	−	178.046i	−0.340867	−	0.212973i
$837$	61.7681			0.0737970
$838$	−259.241	+	904.167i	−0.309356	+	1.07896i
$839$			1547.29i			1.84421i	0.386943	+	0.922104i	$0.373531\pi$
$839$			1547.29i			1.84421i	−0.386943	+	0.922104i	$0.626469\pi$
$840$	−69.3099	−	76.9647i	−0.0825118	−	0.0916246i
$841$	−800.432			−0.951762
$842$	903.658	+	259.095i	1.07323	+	0.307714i
$843$		−	8.37157i		−	0.00993069i
$844$	34.8011	−	55.6997i	0.0412335	−	0.0659949i
$845$	−50.3218			−0.0595524
$846$	−153.527	+	535.463i	−0.181474	+	0.632935i
$847$		−	2160.76i		−	2.55107i
$848$	−301.199	−	617.389i	−0.355187	−	0.728053i
$849$	−71.5762			−0.0843065
$850$	−471.324	−	135.137i	−0.554498	−	0.158985i
$851$		−	784.649i		−	0.922031i
$852$	41.0194	+	25.6288i	0.0481448	+	0.0300808i
$853$	−341.108			−0.399892			−0.199946	−	0.979807i	$-0.564077\pi$
$853$	−341.108			−0.399892			−0.199946	+	0.979807i	$0.564077\pi$
$854$	121.585	−	424.059i	0.142372	−	0.496557i
$855$		−	87.0782i		−	0.101846i
$856$	−478.674	+	431.067i	−0.559199	+	0.503583i
$857$	1535.74			1.79200			0.896000	−	0.444054i	$-0.146460\pi$
$857$	1535.74			1.79200			0.896000	+	0.444054i	$0.146460\pi$
$858$	329.795	+	94.5581i	0.384376	+	0.110208i
$859$		−	1223.16i		−	1.42393i	−0.702213	−	0.711967i	$-0.747804\pi$
$859$		−	1223.16i		−	1.42393i	0.702213	−	0.711967i	$-0.252196\pi$
$860$	50.0904	−	80.1704i	0.0582446	−	0.0932214i
$861$	−405.196			−0.470611
$862$	−123.477	+	430.658i	−0.143245	+	0.499603i
$863$			672.603i			0.779377i	0.920947	+	0.389689i	$0.127417\pi$
$863$			672.603i			0.779377i	−0.920947	+	0.389689i	$0.872583\pi$
$864$	357.255	−	63.0406i	0.413490	−	0.0729636i
$865$	653.266			0.755221
$866$	−346.302	−	99.2909i	−0.399887	−	0.114655i
$867$		−	85.2940i		−	0.0983783i
$868$	159.494	+	99.6516i	0.183749	+	0.114806i
$869$	2695.62			3.10197
$870$	5.26753	−	18.3718i	0.00605463	−	0.0211170i
$871$		−	901.827i		−	1.03539i
$872$	−288.293	−	320.132i	−0.330611	−	0.367124i
$873$	−1294.67			−1.48301
$874$	−314.239	−	90.0980i	−0.359542	−	0.103087i
$875$			895.324i			1.02323i
$876$	−99.5121	+	159.271i	−0.113598	+	0.181816i
$877$	−1328.68			−1.51503			−0.757514	−	0.652819i	$-0.773586\pi$
$877$	−1328.68			−1.51503			−0.757514	+	0.652819i	$0.773586\pi$
$878$	−325.226	+	1134.31i	−0.370416	+	1.29192i
$879$		−	91.2717i		−	0.103836i
$880$	644.917	−	314.629i	0.732861	−	0.357532i
$881$	372.924			0.423296			0.211648	−	0.977346i	$-0.432117\pi$
$881$	372.924			0.423296			0.211648	+	0.977346i	$0.432117\pi$
$882$	420.248	+	120.493i	0.476472	+	0.136613i
$883$			669.314i			0.758000i	0.925397	+	0.379000i	$0.123732\pi$
$883$			669.314i			0.758000i	−0.925397	+	0.379000i	$0.876268\pi$
$884$	−586.297	−	366.318i	−0.663232	−	0.414387i
$885$	58.4718			0.0660699
$886$	−86.7957	+	302.722i	−0.0979635	+	0.341672i
$887$			874.225i			0.985598i	0.870143	+	0.492799i	$0.164026\pi$
$887$			874.225i			0.985598i	−0.870143	+	0.492799i	$0.835974\pi$
$888$	80.1974	−	72.2211i	0.0903124	−	0.0813301i
$889$	−147.913			−0.166381
$890$	149.230	+	42.7868i	0.167674	+	0.0480750i
$891$			1348.06i			1.51298i
$892$	−413.781	+	662.263i	−0.463880	+	0.742448i
$893$	141.424			0.158370
$894$	6.84270	−	23.8656i	0.00765402	−	0.0266953i
$895$		−	81.8873i		−	0.0914942i
$896$	1024.19	+	413.587i	1.14307	+	0.461592i
$897$	333.778			0.372105
$898$	1375.45	+	394.366i	1.53168	+	0.439160i
$899$			34.7032i			0.0386020i
$900$	−570.311	−	356.329i	−0.633678	−	0.395921i
$901$	537.447			0.596500
$902$	773.718	−	2698.53i	0.857780	−	2.99172i
$903$		−	56.4973i		−	0.0625662i
$904$	−567.048	−	629.674i	−0.627266	−	0.696543i
$905$	804.123			0.888534
$906$	−205.164	−	58.8241i	−0.226450	−	0.0649273i
$907$			33.1952i			0.0365989i	0.999833	+	0.0182994i	$0.00582522\pi$
$907$			33.1952i			0.0365989i	−0.999833	+	0.0182994i	$0.994175\pi$
$908$	−647.304	+	1036.02i	−0.712890	+	1.14099i
$909$	667.644			0.734482
$910$	−152.832	+	533.041i	−0.167948	+	0.585760i
$911$		−	25.4273i		−	0.0279114i	−0.999903	−	0.0139557i	$-0.995558\pi$
$911$		−	25.4273i		−	0.0279114i	0.999903	−	0.0139557i	$-0.00444238\pi$
$912$	−19.7147	−	40.4106i	−0.0216170	−	0.0443099i
$913$	−1825.26			−1.99918
$914$	−60.2709	−	17.2807i	−0.0659419	−	0.0189067i
$915$		−	38.3499i		−	0.0419125i
$916$	−779.165	−	486.821i	−0.850617	−	0.531464i
$917$	−1979.00			−2.15812
$918$	−78.2260	+	272.833i	−0.0852135	+	0.297204i
$919$			445.408i			0.484666i	0.970193	+	0.242333i	$0.0779126\pi$
$919$			445.408i			0.484666i	−0.970193	+	0.242333i	$0.922087\pi$
$920$	518.761	−	467.166i	0.563871	−	0.507790i
$921$	−230.639			−0.250422
$922$	−708.875	−	203.247i	−0.768845	−	0.220442i
$923$		−	258.954i		−	0.280557i
$924$	227.241	−	363.704i	0.245932	−	0.393619i
$925$	−409.804			−0.443031
$926$	450.575	−	1571.49i	0.486582	−	1.69707i
$927$		−	128.320i		−	0.138425i
$928$	35.4181	+	200.717i	0.0381661	+	0.216290i
$929$	−408.473			−0.439691			−0.219846	−	0.975535i	$-0.570555\pi$
$929$	−408.473			−0.439691			−0.219846	+	0.975535i	$0.570555\pi$
$930$	15.7158	+	4.50601i	0.0168987	+	0.00484517i
$931$		−	110.994i		−	0.119220i
$932$	636.622	+	397.760i	0.683070	+	0.426781i
$933$	304.547			0.326417
$934$	−47.9122	+	167.106i	−0.0512979	+	0.178914i
$935$			561.411i			0.600439i
$936$	−634.503	−	704.579i	−0.677888	−	0.752756i
$937$	−1101.68			−1.17576			−0.587878	−	0.808950i	$-0.700037\pi$
$937$	−1101.68			−1.17576			−0.587878	+	0.808950i	$0.700037\pi$
$938$	−1083.63	−	310.697i	−1.15526	−	0.331233i
$939$			41.1060i			0.0437764i
$940$	−160.032	+	256.134i	−0.170247	+	0.272482i
$941$	−1342.45			−1.42662			−0.713309	−	0.700850i	$-0.752804\pi$
$941$	−1342.45			−1.42662			−0.713309	+	0.700850i	$0.752804\pi$
$942$	53.1109	−	185.238i	0.0563810	−	0.196643i
$943$		−	2731.12i		−	2.89621i
$944$	−560.428	+	273.410i	−0.593674	+	0.289629i
$945$	227.659			0.240909
$946$	376.262	+	107.881i	0.397740	+	0.114039i
$947$		−	786.294i		−	0.830300i	−0.909753	−	0.415150i	$-0.863729\pi$
$947$		−	786.294i		−	0.830300i	0.909753	−	0.415150i	$-0.136271\pi$
$948$	305.909	+	191.131i	0.322689	+	0.201615i
$949$	1005.47			1.05951
$950$	−47.0561	+	164.120i	−0.0495327	+	0.172758i
$951$			128.116i			0.134717i
$952$	−642.157	+	578.290i	−0.674535	+	0.607447i
$953$	1134.64			1.19060			0.595301	−	0.803503i	$-0.297033\pi$
$953$	1134.64			1.19060			0.595301	+	0.803503i	$0.297033\pi$
$954$	708.570	+	203.160i	0.742736	+	0.212956i
$955$		−	233.258i		−	0.244250i
$956$	−148.048	+	236.953i	−0.154862	+	0.247859i
$957$	79.1357			0.0826915
$958$	183.617	−	640.410i	0.191667	−	0.668486i
$959$		−	1040.40i		−	1.08488i
$960$	95.4963	+	10.0223i	0.0994753	+	0.0104399i
$961$	931.314			0.969109
$962$	−555.430	−	159.252i	−0.577371	−	0.165542i
$963$		−	691.217i		−	0.717775i
$964$	1009.42	+	630.682i	1.04711	+	0.654234i
$965$	−391.240			−0.405430
$966$	114.993	−	401.067i	0.119040	−	0.415183i
$967$			963.333i			0.996208i	0.867117	+	0.498104i	$0.165970\pi$
$967$			963.333i			0.996208i	−0.867117	+	0.498104i	$0.834030\pi$
$968$	1340.51	+	1488.56i	1.38483	+	1.53777i
$969$	35.1781			0.0363035
$970$	−674.762	−	193.466i	−0.695631	−	0.199450i
$971$		−	1320.90i		−	1.36035i	−0.733050	−	0.680174i	$-0.761904\pi$
$971$		−	1320.90i		−	1.36035i	0.733050	−	0.680174i	$-0.238096\pi$
$972$	−311.838	+	499.101i	−0.320821	+	0.513478i
$973$	1258.70			1.29363
$974$	145.904	−	508.876i	0.149799	−	0.522460i
$975$		−	174.324i		−	0.178794i
$976$	179.321	+	367.568i	0.183731	+	0.376607i
$977$	−1159.80			−1.18710			−0.593550	−	0.804797i	$-0.702274\pi$
$977$	−1159.80			−1.18710			−0.593550	+	0.804797i	$0.702274\pi$
$978$	−181.907	−	52.1561i	−0.185999	−	0.0533293i
$979$			642.800i			0.656588i
$980$	201.021	+	125.598i	0.205124	+	0.128161i
$981$	462.278			0.471232
$982$	−222.075	+	774.541i	−0.226145	+	0.788738i
$983$			8.37645i			0.00852131i	0.999991	+	0.00426065i	$0.00135621\pi$
$983$			8.37645i			0.00852131i	−0.999991	+	0.00426065i	$0.998644\pi$
$984$	279.143	−	251.380i	0.283682	−	0.255467i
$985$	−504.305			−0.511985
$986$	−153.286	−	43.9498i	−0.155462	−	0.0445738i
$987$			180.501i			0.182879i
$988$	−127.556	+	204.155i	−0.129105	+	0.206634i
$989$	380.806			0.385041
$990$	−212.218	+	740.164i	−0.214362	+	0.747641i
$991$		−	854.355i		−	0.862114i	−0.902325	−	0.431057i	$-0.858141\pi$
$991$		−	854.355i		−	0.862114i	0.902325	−	0.431057i	$-0.141859\pi$
$992$	−171.700	+	30.2978i	−0.173084	+	0.0305421i
$993$	282.855			0.284849
$994$	−311.159	−	89.2147i	−0.313037	−	0.0897532i
$995$			191.419i			0.192381i
$996$	−207.137	−	129.419i	−0.207969	−	0.129939i
$997$	−77.4664			−0.0776995			−0.0388497	−	0.999245i	$-0.512369\pi$
$997$	−77.4664			−0.0776995			−0.0388497	+	0.999245i	$0.512369\pi$
$998$	−193.224	+	673.918i	−0.193612	+	0.675269i
$999$			237.221i			0.237458i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	76.3.b.b.39.1	✓	14
3.2	odd	2		684.3.g.b.343.14		14
4.3	odd	2	inner	76.3.b.b.39.2	yes	14
8.3	odd	2		1216.3.d.d.191.8		14
8.5	even	2		1216.3.d.d.191.7		14
12.11	even	2		684.3.g.b.343.13		14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
76.3.b.b.39.1	✓	14	1.1	even	1	trivial
76.3.b.b.39.2	yes	14	4.3	odd	2	inner
684.3.g.b.343.13		14	12.11	even	2
684.3.g.b.343.14		14	3.2	odd	2
1216.3.d.d.191.7		14	8.5	even	2
1216.3.d.d.191.8		14	8.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 76 = 2^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	76.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.92254 - 0.551226i) q^{2} +0.644704i q^{3} +(3.39230 + 2.11950i) q^{4} -2.32715 q^{5} +(0.355377 - 1.23947i) q^{6} +8.62924i q^{7} +(-5.35350 - 5.94475i) q^{8} +8.58436 q^{9} +O(q^{10})\)	\(q+(-1.92254 - 0.551226i) q^{2} +0.644704i q^{3} +(3.39230 + 2.11950i) q^{4} -2.32715 q^{5} +(0.355377 - 1.23947i) q^{6} +8.62924i q^{7} +(-5.35350 - 5.94475i) q^{8} +8.58436 q^{9} +(4.47404 + 1.28279i) q^{10} +19.2717i q^{11} +(-1.36645 + 2.18703i) q^{12} +13.8067 q^{13} +(4.75666 - 16.5900i) q^{14} -1.50032i q^{15} +(7.01540 + 14.3800i) q^{16} -12.5180 q^{17} +(-16.5037 - 4.73192i) q^{18} +4.35890i q^{19} +(-7.89440 - 4.93241i) q^{20} -5.56330 q^{21} +(10.6231 - 37.0506i) q^{22} -37.4981i q^{23} +(3.83260 - 3.45142i) q^{24} -19.5844 q^{25} +(-26.5438 - 7.61058i) q^{26} +11.3367i q^{27} +(-18.2897 + 29.2730i) q^{28} -6.36930 q^{29} +(-0.827018 + 2.88443i) q^{30} -5.44851i q^{31} +(-5.56075 - 31.5131i) q^{32} -12.4245 q^{33} +(24.0663 + 6.90025i) q^{34} -20.0816i q^{35} +(29.1207 + 18.1946i) q^{36} +20.9250 q^{37} +(2.40274 - 8.38015i) q^{38} +8.90120i q^{39} +(12.4584 + 13.8343i) q^{40} +72.8337 q^{41} +(10.6957 + 3.06664i) q^{42} +10.1553i q^{43} +(-40.8465 + 65.3755i) q^{44} -19.9771 q^{45} +(-20.6699 + 72.0914i) q^{46} -32.4450i q^{47} +(-9.27083 + 4.52286i) q^{48} -25.4638 q^{49} +(37.6517 + 10.7954i) q^{50} -8.07040i q^{51} +(46.8363 + 29.2633i) q^{52} -42.9339 q^{53} +(6.24908 - 21.7952i) q^{54} -44.8482i q^{55} +(51.2987 - 46.1966i) q^{56} -2.81020 q^{57} +(12.2452 + 3.51092i) q^{58} +38.9728i q^{59} +(3.17994 - 5.08955i) q^{60} +25.5611 q^{61} +(-3.00336 + 10.4750i) q^{62} +74.0765i q^{63} +(-6.68010 + 63.6504i) q^{64} -32.1302 q^{65} +(23.8867 + 6.84873i) q^{66} -65.3183i q^{67} +(-42.4648 - 26.5320i) q^{68} +24.1751 q^{69} +(-11.0695 + 38.6076i) q^{70} -18.7557i q^{71} +(-45.9563 - 51.0319i) q^{72} +72.8251 q^{73} +(-40.2292 - 11.5344i) q^{74} -12.6261i q^{75} +(-9.23871 + 14.7867i) q^{76} -166.300 q^{77} +(4.90657 - 17.1129i) q^{78} -139.874i q^{79} +(-16.3259 - 33.4644i) q^{80} +69.9504 q^{81} +(-140.026 - 40.1478i) q^{82} +94.7116i q^{83} +(-18.8724 - 11.7914i) q^{84} +29.1313 q^{85} +(5.59789 - 19.5240i) q^{86} -4.10631i q^{87} +(114.566 - 103.171i) q^{88} +33.3546 q^{89} +(38.4068 + 11.0119i) q^{90} +119.141i q^{91} +(79.4773 - 127.205i) q^{92} +3.51267 q^{93} +(-17.8845 + 62.3766i) q^{94} -10.1438i q^{95} +(20.3166 - 3.58504i) q^{96} -150.817 q^{97} +(48.9551 + 14.0363i) q^{98} +165.435i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q + 2 q^{2} + 2 q^{4} - 40 q^{8} - 68 q^{9}+O(q^{10}) \) 14 * q + 2 * q^2 + 2 * q^4 - 40 * q^8 - 68 * q^9	\( 14 q + 2 q^{2} + 2 q^{4} - 40 q^{8} - 68 q^{9} - 12 q^{10} + 4 q^{12} + 54 q^{13} + 30 q^{14} + 58 q^{16} + 34 q^{17} + 36 q^{18} + 32 q^{20} - 38 q^{21} + 36 q^{22} - 98 q^{24} - 86 q^{25} - 16 q^{26} + 18 q^{28} + 54 q^{29} - 204 q^{30} + 72 q^{32} + 20 q^{33} - 82 q^{34} + 96 q^{36} + 100 q^{37} - 148 q^{40} + 224 q^{41} + 224 q^{42} - 96 q^{44} - 168 q^{45} + 46 q^{46} + 296 q^{48} - 220 q^{49} - 58 q^{50} - 288 q^{52} + 14 q^{53} - 128 q^{54} + 12 q^{56} + 38 q^{57} - 72 q^{58} + 188 q^{60} + 28 q^{61} + 396 q^{62} - 118 q^{64} - 472 q^{65} - 32 q^{66} + 30 q^{68} + 122 q^{69} + 156 q^{70} + 80 q^{72} + 70 q^{73} - 224 q^{74} + 228 q^{77} + 274 q^{78} - 348 q^{80} + 334 q^{81} - 400 q^{82} - 216 q^{84} + 48 q^{85} - 124 q^{86} + 472 q^{88} + 416 q^{90} + 126 q^{92} - 176 q^{93} - 88 q^{94} - 106 q^{96} + 308 q^{97} + 68 q^{98}+O(q^{100}) \) 14 * q + 2 * q^2 + 2 * q^4 - 40 * q^8 - 68 * q^9 - 12 * q^10 + 4 * q^12 + 54 * q^13 + 30 * q^14 + 58 * q^16 + 34 * q^17 + 36 * q^18 + 32 * q^20 - 38 * q^21 + 36 * q^22 - 98 * q^24 - 86 * q^25 - 16 * q^26 + 18 * q^28 + 54 * q^29 - 204 * q^30 + 72 * q^32 + 20 * q^33 - 82 * q^34 + 96 * q^36 + 100 * q^37 - 148 * q^40 + 224 * q^41 + 224 * q^42 - 96 * q^44 - 168 * q^45 + 46 * q^46 + 296 * q^48 - 220 * q^49 - 58 * q^50 - 288 * q^52 + 14 * q^53 - 128 * q^54 + 12 * q^56 + 38 * q^57 - 72 * q^58 + 188 * q^60 + 28 * q^61 + 396 * q^62 - 118 * q^64 - 472 * q^65 - 32 * q^66 + 30 * q^68 + 122 * q^69 + 156 * q^70 + 80 * q^72 + 70 * q^73 - 224 * q^74 + 228 * q^77 + 274 * q^78 - 348 * q^80 + 334 * q^81 - 400 * q^82 - 216 * q^84 + 48 * q^85 - 124 * q^86 + 472 * q^88 + 416 * q^90 + 126 * q^92 - 176 * q^93 - 88 * q^94 - 106 * q^96 + 308 * q^97 + 68 * q^98

Introduction

L-functions

Modular forms

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