LMFDB - Embedded newform 76.6.e.a.45.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [76,6,Mod(45,76)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(76, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 2]))

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("76.45");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 76 = 2^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	76.e (of order $3$, degree $2$, 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:	$12.1891703058$
Analytic rank:	$0$
Dimension:	$18$
Relative dimension:	$9$ over $\Q(\zeta_{3})$
Coefficient field:	$\mathbb{Q}[x]/(x^{18} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{18} - 2 x^{17} + 1540 x^{16} - 768 x^{15} + 1608492 x^{14} - 1027368 x^{13} + 897054160 x^{12} + \cdots + 80\!\cdots\!36 $ x^18 - 2x^17 + 1540x^16 - 768x^15 + 1608492x^14 - 1027368x^13 + 897054160x^12 - 1275481376x^11 + 361098181456x^10 - 863969476320x^9 + 79755165392064x^8 - 375077568148992x^7 + 12736924096193536x^6 - 57314532742553600x^5 + 977121800205220864x^4 - 4977732006498379776x^3 + 53672321824823513088x^2 - 185653809995679793152*x + 804303742853852430336
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{16}\cdot 3^{3}\cdot 5^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			45.8
Root			$-10.9539 + 18.9727i$ of defining polynomial
Character	$\chi$	$=$	76.45
Dual form			76.6.e.a.49.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(10.4539 + 18.1067i) q^{3} +(50.5928 + 87.6293i) q^{5} +95.5451 q^{7} +(-97.0688 + 168.128i) q^{9} +O(q^{10})$	\(q+(10.4539 + 18.1067i) q^{3} +(50.5928 + 87.6293i) q^{5} +95.5451 q^{7} +(-97.0688 + 168.128i) q^{9} -119.504 q^{11} +(297.428 - 515.161i) q^{13} +(-1057.79 + 1832.14i) q^{15} +(-459.911 - 796.589i) q^{17} +(-806.703 + 1351.05i) q^{19} +(998.821 + 1730.01i) q^{21} +(2124.58 - 3679.89i) q^{23} +(-3556.76 + 6160.48i) q^{25} +1021.61 q^{27} +(-2214.69 + 3835.95i) q^{29} -4955.30 q^{31} +(-1249.29 - 2163.83i) q^{33} +(4833.89 + 8372.55i) q^{35} +7651.95 q^{37} +12437.2 q^{39} +(2006.97 + 3476.18i) q^{41} +(-5228.80 - 9056.55i) q^{43} -19643.9 q^{45} +(4318.48 - 7479.83i) q^{47} -7678.13 q^{49} +(9615.75 - 16655.0i) q^{51} +(18534.0 - 32101.9i) q^{53} +(-6046.06 - 10472.1i) q^{55} +(-32896.2 - 482.998i) q^{57} +(-20727.5 - 35901.1i) q^{59} +(-2244.12 + 3886.93i) q^{61} +(-9274.45 + 16063.8i) q^{63} +60190.9 q^{65} +(18484.7 - 32016.5i) q^{67} +88840.9 q^{69} +(23408.1 + 40543.9i) q^{71} +(40666.6 + 70436.6i) q^{73} -148728. q^{75} -11418.1 q^{77} +(6656.35 + 11529.1i) q^{79} +(34267.5 + 59353.1i) q^{81} -18380.7 q^{83} +(46536.4 - 80603.3i) q^{85} -92608.6 q^{87} +(-7307.86 + 12657.6i) q^{89} +(28417.8 - 49221.1i) q^{91} +(-51802.3 - 89724.2i) q^{93} +(-159205. - 2337.52i) q^{95} +(61844.5 + 107118. i) q^{97} +(11600.1 - 20092.0i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q - 11 q^{3} + 11 q^{5} + 336 q^{7} - 902 q^{9}+O(q^{10}) $ 18 * q - 11 * q^3 + 11 * q^5 + 336 * q^7 - 902 * q^9	$ 18 q - 11 q^{3} + 11 q^{5} + 336 q^{7} - 902 q^{9} - 320 q^{11} + 227 q^{13} - 101 q^{15} + 179 q^{17} - 868 q^{19} - 5700 q^{21} - 3425 q^{23} - 7054 q^{25} + 14722 q^{27} - 7349 q^{29} - 9960 q^{31} - 2998 q^{33} + 15888 q^{35} + 26444 q^{37} - 30246 q^{39} - 7311 q^{41} - 8283 q^{43} - 62164 q^{45} + 37603 q^{47} + 124738 q^{49} + 47227 q^{51} - 20337 q^{53} + 716 q^{55} - 57555 q^{57} - 74455 q^{59} - 7569 q^{61} - 52544 q^{63} + 188998 q^{65} - 26177 q^{67} + 116282 q^{69} - 53463 q^{71} - 14103 q^{73} + 120912 q^{75} - 31960 q^{77} + 31825 q^{79} - 21137 q^{81} + 82600 q^{83} - 50787 q^{85} - 339766 q^{87} - 155197 q^{89} - 2800 q^{91} - 46460 q^{93} + 49315 q^{95} + 111241 q^{97} - 193544 q^{99}+O(q^{100}) $ 18 * q - 11 * q^3 + 11 * q^5 + 336 * q^7 - 902 * q^9 - 320 * q^11 + 227 * q^13 - 101 * q^15 + 179 * q^17 - 868 * q^19 - 5700 * q^21 - 3425 * q^23 - 7054 * q^25 + 14722 * q^27 - 7349 * q^29 - 9960 * q^31 - 2998 * q^33 + 15888 * q^35 + 26444 * q^37 - 30246 * q^39 - 7311 * q^41 - 8283 * q^43 - 62164 * q^45 + 37603 * q^47 + 124738 * q^49 + 47227 * q^51 - 20337 * q^53 + 716 * q^55 - 57555 * q^57 - 74455 * q^59 - 7569 * q^61 - 52544 * q^63 + 188998 * q^65 - 26177 * q^67 + 116282 * q^69 - 53463 * q^71 - 14103 * q^73 + 120912 * q^75 - 31960 * q^77 + 31825 * q^79 - 21137 * q^81 + 82600 * q^83 - 50787 * q^85 - 339766 * q^87 - 155197 * q^89 - 2800 * q^91 - 46460 * q^93 + 49315 * q^95 + 111241 * q^97 - 193544 * q^99

Display 100 coefficients 10 coefficients

Character values

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

$n$	$21$	$39$
$\chi(n)$	$e\left(\frac{1}{3}\right)$	$1$

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 $

$2$		0			0
$2$		0			0
$3$	10.4539	+	18.1067i	0.670619	+	1.16155i	0.977729	+	0.209872i	$0.0673048\pi$
$3$	10.4539	+	18.1067i	0.670619	+	1.16155i	−0.307110	+	0.951674i	$0.599362\pi$
$4$		0			0
$5$	50.5928	+	87.6293i	0.905031	+	1.56756i	0.820876	+	0.571107i	$0.193486\pi$
$5$	50.5928	+	87.6293i	0.905031	+	1.56756i	0.0841554	+	0.996453i	$0.473181\pi$
$6$		0			0
$7$	95.5451			0.736993			0.368497	−	0.929629i	$-0.379873\pi$
$7$	95.5451			0.736993			0.368497	+	0.929629i	$0.379873\pi$
$8$		0			0
$9$	−97.0688	+	168.128i	−0.399460	+	0.691885i
$10$		0			0
$11$	−119.504			−0.297784			−0.148892	−	0.988853i	$-0.547571\pi$
$11$	−119.504			−0.297784			−0.148892	+	0.988853i	$0.547571\pi$
$12$		0			0
$13$	297.428	−	515.161i	0.488117	−	0.845443i	−0.511790	−	0.859111i	$-0.671017\pi$
$13$	297.428	−	515.161i	0.488117	−	0.845443i	0.999907	+	0.0136677i	$0.00435069\pi$
$14$		0			0
$15$	−1057.79	+	1832.14i	−1.21386	+	2.10247i
$16$		0			0
$17$	−459.911	−	796.589i	−0.385968	−	0.668517i	0.605935	−	0.795514i	$-0.292799\pi$
$17$	−459.911	−	796.589i	−0.385968	−	0.668517i	−0.991903	+	0.126998i	$0.959466\pi$
$18$		0			0
$19$	−806.703	+	1351.05i	−0.512660	+	0.858592i
$19$	−806.703	+	1351.05i	−0.512660	+	0.858592i
$20$		0			0
$21$	998.821	+	1730.01i	0.494242	+	0.856052i
$22$		0			0
$23$	2124.58	−	3679.89i	0.837441	−	1.45049i	−0.0545863	−	0.998509i	$-0.517384\pi$
$23$	2124.58	−	3679.89i	0.837441	−	1.45049i	0.892027	−	0.451981i	$-0.149283\pi$
$24$		0			0
$25$	−3556.76	+	6160.48i	−1.13816	+	1.97135i
$26$		0			0
$27$	1021.61			0.269696
$28$		0			0
$29$	−2214.69	+	3835.95i	−0.489009	+	0.846989i	−0.999920	−	0.0126447i	$-0.995975\pi$
$29$	−2214.69	+	3835.95i	−0.489009	+	0.846989i	0.510911	+	0.859634i	$0.329308\pi$
$30$		0			0
$31$	−4955.30			−0.926116			−0.463058	−	0.886328i	$-0.653248\pi$
$31$	−4955.30			−0.926116			−0.463058	+	0.886328i	$0.653248\pi$
$32$		0			0
$33$	−1249.29	−	2163.83i	−0.199700	−	0.345890i
$34$		0			0
$35$	4833.89	+	8372.55i	0.667002	+	1.15528i
$36$		0			0
$37$	7651.95			0.918900			0.459450	−	0.888204i	$-0.348047\pi$
$37$	7651.95			0.918900			0.459450	+	0.888204i	$0.348047\pi$
$38$		0			0
$39$	12437.2			1.30936
$40$		0			0
$41$	2006.97	+	3476.18i	0.186458	+	0.322955i	0.944067	−	0.329754i	$-0.106966\pi$
$41$	2006.97	+	3476.18i	0.186458	+	0.322955i	−0.757609	+	0.652709i	$0.773632\pi$
$42$		0			0
$43$	−5228.80	−	9056.55i	−0.431252	−	0.746950i	0.565729	−	0.824591i	$-0.308595\pi$
$43$	−5228.80	−	9056.55i	−0.431252	−	0.746950i	−0.996981	+	0.0776405i	$0.975261\pi$
$44$		0			0
$45$	−19643.9			−1.44610
$46$		0			0
$47$	4318.48	−	7479.83i	0.285159	−	0.493909i	−0.687489	−	0.726195i	$-0.741287\pi$
$47$	4318.48	−	7479.83i	0.285159	−	0.493909i	0.972648	+	0.232286i	$0.0746204\pi$
$48$		0			0
$49$	−7678.13			−0.456841
$50$		0			0
$51$	9615.75	−	16655.0i	0.517675	−	0.896640i
$52$		0			0
$53$	18534.0	−	32101.9i	0.906318	−	1.56979i	0.0871791	−	0.996193i	$-0.472215\pi$
$53$	18534.0	−	32101.9i	0.906318	−	1.56979i	0.819139	−	0.573596i	$-0.194452\pi$
$54$		0			0
$55$	−6046.06	−	10472.1i	−0.269504	−	0.466795i
$56$		0			0
$57$	−32896.2	−	482.998i	−1.34109	−	0.0196906i
$58$		0			0
$59$	−20727.5	−	35901.1i	−0.775207	−	1.34270i	−0.934678	−	0.355495i	$-0.884312\pi$
$59$	−20727.5	−	35901.1i	−0.775207	−	1.34270i	0.159472	−	0.987203i	$-0.449021\pi$
$60$		0			0
$61$	−2244.12	+	3886.93i	−0.0772186	+	0.133747i	−0.902049	−	0.431634i	$-0.857937\pi$
$61$	−2244.12	+	3886.93i	−0.0772186	+	0.133747i	0.824830	+	0.565380i	$0.191271\pi$
$62$		0			0
$63$	−9274.45	+	16063.8i	−0.294399	+	0.509915i
$64$		0			0
$65$	60190.9			1.76704
$66$		0			0
$67$	18484.7	−	32016.5i	0.503067	−	0.871338i	−0.496927	−	0.867793i	$-0.665538\pi$
$67$	18484.7	−	32016.5i	0.503067	−	0.871338i	0.999994	−	0.00354526i	$-0.00112849\pi$
$68$		0			0
$69$	88840.9			2.24642
$70$		0			0
$71$	23408.1	+	40543.9i	0.551086	+	0.954509i	0.998197	+	0.0600303i	$0.0191197\pi$
$71$	23408.1	+	40543.9i	0.551086	+	0.954509i	−0.447110	+	0.894479i	$0.647547\pi$
$72$		0			0
$73$	40666.6	+	70436.6i	0.893163	+	1.54700i	0.836062	+	0.548635i	$0.184852\pi$
$73$	40666.6	+	70436.6i	0.893163	+	1.54700i	0.0571012	+	0.998368i	$0.481814\pi$
$74$		0			0
$75$	−148728.			−3.05309
$76$		0			0
$77$	−11418.1			−0.219465
$78$		0			0
$79$	6656.35	+	11529.1i	0.119996	+	0.207840i	0.919766	−	0.392467i	$-0.128378\pi$
$79$	6656.35	+	11529.1i	0.119996	+	0.207840i	−0.799770	+	0.600307i	$0.795045\pi$
$80$		0			0
$81$	34267.5	+	59353.1i	0.580323	+	1.00515i
$82$		0			0
$83$	−18380.7			−0.292864			−0.146432	−	0.989221i	$-0.546779\pi$
$83$	−18380.7			−0.292864			−0.146432	+	0.989221i	$0.546779\pi$
$84$		0			0
$85$	46536.4	−	80603.3i	0.698627	−	1.21006i
$86$		0			0
$87$	−92608.6			−1.31176
$88$		0			0
$89$	−7307.86	+	12657.6i	−0.0977946	+	0.169385i	−0.910772	−	0.412911i	$-0.864512\pi$
$89$	−7307.86	+	12657.6i	−0.0977946	+	0.169385i	0.812977	+	0.582296i	$0.197846\pi$
$90$		0			0
$91$	28417.8	−	49221.1i	0.359739	−	0.623086i
$92$		0			0
$93$	−51802.3	−	89724.2i	−0.621071	−	1.07573i
$94$		0			0
$95$	−159205.	−	2337.52i	−1.80987	−	0.0265733i
$96$		0			0
$97$	61844.5	+	107118.i	0.667378	+	1.15593i	0.978635	+	0.205606i	$0.0659166\pi$
$97$	61844.5	+	107118.i	0.667378	+	1.15593i	−0.311257	+	0.950326i	$0.600750\pi$
$98$		0			0
$99$	11600.1	−	20092.0i	0.118953	−	0.206033i
$100$		0			0
$101$	−24891.8	+	43113.9i	−0.242803	+	0.420546i	−0.961511	−	0.274765i	$-0.911400\pi$
$101$	−24891.8	+	43113.9i	−0.242803	+	0.420546i	0.718709	+	0.695311i	$0.244733\pi$
$102$		0			0
$103$	−102447.			−0.951496			−0.475748	−	0.879582i	$-0.657823\pi$
$103$	−102447.			−0.951496			−0.475748	+	0.879582i	$0.657823\pi$
$104$		0			0
$105$	−101066.	+	175052.i	−0.894608	+	1.54951i
$106$		0			0
$107$	45310.9			0.382598			0.191299	−	0.981532i	$-0.438730\pi$
$107$	45310.9			0.382598			0.191299	+	0.981532i	$0.438730\pi$
$108$		0			0
$109$	75259.9	+	130354.i	0.606733	+	1.05089i	0.991775	+	0.127993i	$0.0408535\pi$
$109$	75259.9	+	130354.i	0.606733	+	1.05089i	−0.385042	+	0.922899i	$0.625813\pi$
$110$		0			0
$111$	79992.9	+	138552.i	0.616232	+	1.06734i
$112$		0			0
$113$	71068.5			0.523577			0.261789	−	0.965125i	$-0.415688\pi$
$113$	71068.5			0.523577			0.261789	+	0.965125i	$0.415688\pi$
$114$		0			0
$115$	429954.			3.03164
$116$		0			0
$117$	57742.0	+	100012.i	0.389966	+	0.675442i
$118$		0			0
$119$	−43942.3	−	76110.2i	−0.284456	−	0.492692i
$120$		0			0
$121$	−146770.			−0.911324
$122$		0			0
$123$	−41961.5	+	72679.4i	−0.250085	+	0.433160i
$124$		0			0
$125$	−403580.			−2.31023
$126$		0			0
$127$	65658.5	−	113724.i	0.361228	−	0.625666i	−0.626935	−	0.779071i	$-0.715691\pi$
$127$	65658.5	−	113724.i	0.361228	−	0.625666i	0.988163	+	0.153406i	$0.0490242\pi$
$128$		0			0
$129$	109323.	−	189353.i	0.578412	−	1.00184i
$130$		0			0
$131$	−100412.	−	173920.i	−0.511222	−	0.885462i	−0.999915	−	0.0130067i	$-0.995860\pi$
$131$	−100412.	−	173920.i	−0.511222	−	0.885462i	0.488694	−	0.872456i	$-0.337474\pi$
$132$		0			0
$133$	−77076.5	+	129086.i	−0.377827	+	0.632776i
$134$		0			0
$135$	51685.9	+	89522.6i	0.244083	+	0.422764i
$136$		0			0
$137$	181520.	−	314402.i	0.826272	−	1.43114i	−0.0746720	−	0.997208i	$-0.523791\pi$
$137$	181520.	−	314402.i	0.826272	−	1.43114i	0.900944	−	0.433936i	$-0.142876\pi$
$138$		0			0
$139$	190140.	−	329331.i	0.834710	−	1.44576i	−0.0595567	−	0.998225i	$-0.518969\pi$
$139$	190140.	−	329331.i	0.834710	−	1.44576i	0.894266	−	0.447535i	$-0.147698\pi$
$140$		0			0
$141$	180580.			0.764931
$142$		0			0
$143$	−35544.0	+	61563.9i	−0.145354	+	0.251760i
$144$		0			0
$145$	−448188.			−1.77027
$146$		0			0
$147$	−80266.5	−	139026.i	−0.306366	−	0.530642i
$148$		0			0
$149$	−21911.4	−	37951.7i	−0.0808546	−	0.140044i	0.822763	−	0.568385i	$-0.192432\pi$
$149$	−21911.4	−	37951.7i	−0.0808546	−	0.140044i	−0.903617	+	0.428341i	$0.859098\pi$
$150$		0			0
$151$	−131110.			−0.467942			−0.233971	−	0.972244i	$-0.575172\pi$
$151$	−131110.			−0.467942			−0.233971	+	0.972244i	$0.575172\pi$
$152$		0			0
$153$	178572.			0.616716
$154$		0			0
$155$	−250702.	−	434229.i	−0.838164	−	1.45174i
$156$		0			0
$157$	239102.	+	414137.i	0.774166	+	1.34089i	0.935262	+	0.353956i	$0.115164\pi$
$157$	239102.	+	414137.i	0.774166	+	1.34089i	−0.161096	+	0.986939i	$0.551503\pi$
$158$		0			0
$159$	775013.			2.43118
$160$		0			0
$161$	202994.	−	351595.i	0.617188	−	1.06900i
$162$		0			0
$163$	−389081.			−1.14702			−0.573509	−	0.819199i	$-0.694418\pi$
$163$	−389081.			−1.14702			−0.573509	+	0.819199i	$0.694418\pi$
$164$		0			0
$165$	126410.	−	218948.i	0.361469	−	0.626083i
$166$		0			0
$167$	−261490.	+	452914.i	−0.725544	+	1.25668i	0.233206	+	0.972427i	$0.425078\pi$
$167$	−261490.	+	452914.i	−0.725544	+	1.25668i	−0.958750	+	0.284252i	$0.908255\pi$
$168$		0			0
$169$	8719.47	+	15102.6i	0.0234841	+	0.0406756i
$170$		0			0
$171$	−148843.	−	266774.i	−0.389260	−	0.697675i
$172$		0			0
$173$	−228433.	−	395658.i	−0.580289	−	1.00509i	−0.995445	−	0.0953400i	$-0.969606\pi$
$173$	−228433.	−	395658.i	−0.580289	−	1.00509i	0.415156	−	0.909750i	$-0.363727\pi$
$174$		0			0
$175$	−339831.	+	588604.i	−0.838818	+	1.45288i
$176$		0			0
$177$	433368.	−	750615.i	1.03974	−	1.80088i
$178$		0			0
$179$	380329.			0.887211			0.443606	−	0.896222i	$-0.353699\pi$
$179$	380329.			0.887211			0.443606	+	0.896222i	$0.353699\pi$
$180$		0			0
$181$	−172037.	+	297977.i	−0.390325	+	0.676062i	−0.992492	−	0.122308i	$-0.960971\pi$
$181$	−172037.	+	297977.i	−0.390325	+	0.676062i	0.602168	+	0.798370i	$0.294304\pi$
$182$		0			0
$183$	−93839.5			−0.207137
$184$		0			0
$185$	387134.	+	670535.i	0.831633	+	1.44043i
$186$		0			0
$187$	54961.4	+	95195.9i	0.114935	+	0.199074i
$188$		0			0
$189$	97609.5			0.198764
$190$		0			0
$191$	−699156.			−1.38673			−0.693363	−	0.720588i	$-0.743872\pi$
$191$	−699156.			−1.38673			−0.693363	+	0.720588i	$0.743872\pi$
$192$		0			0
$193$	27621.7	+	47842.2i	0.0533774	+	0.0924524i	0.891480	−	0.453061i	$-0.149668\pi$
$193$	27621.7	+	47842.2i	0.0533774	+	0.0924524i	−0.838102	+	0.545513i	$0.816335\pi$
$194$		0			0
$195$	629230.	+	1.08986e6i	1.18501	+	2.05250i
$196$		0			0
$197$	−1.00021e6			−1.83623			−0.918113	−	0.396318i	$-0.870288\pi$
$197$	−1.00021e6			−1.83623			−0.918113	+	0.396318i	$0.870288\pi$
$198$		0			0
$199$	246676.	−	427255.i	0.441564	−	0.764811i	−0.556242	−	0.831020i	$-0.687757\pi$
$199$	246676.	−	427255.i	0.441564	−	0.764811i	0.997806	+	0.0662094i	$0.0210905\pi$
$200$		0			0
$201$	772951.			1.34947
$202$		0			0
$203$	−211602.	+	366506.i	−0.360397	+	0.624225i
$204$		0			0
$205$	−203077.	+	351739.i	−0.337501	+	0.584569i
$206$		0			0
$207$	412462.	+	714405.i	0.669049	+	1.15883i
$208$		0			0
$209$	96404.5	−	161456.i	0.152662	−	0.255675i
$210$		0			0
$211$	−309293.	−	535711.i	−0.478259	−	0.828369i	0.521430	−	0.853294i	$-0.325399\pi$
$211$	−309293.	−	535711.i	−0.478259	−	0.828369i	−0.999689	+	0.0249246i	$0.992065\pi$
$212$		0			0
$213$	−489412.	+	847686.i	−0.739138	+	1.28022i
$214$		0			0
$215$	529079.	−	916392.i	0.780593	−	1.35203i
$216$		0			0
$217$	−473455.			−0.682541
$218$		0			0
$219$	−850251.	+	1.47268e6i	−1.19794	+	2.07490i
$220$		0			0
$221$	−547162.			−0.753590
$222$		0			0
$223$	−75025.2	−	129947.i	−0.101029	−	0.174987i	0.811080	−	0.584935i	$-0.198880\pi$
$223$	−75025.2	−	129947.i	−0.101029	−	0.174987i	−0.912109	+	0.409948i	$0.865547\pi$
$224$		0			0
$225$	−690500.	−	1.19598e6i	−0.909301	−	1.57496i
$226$		0			0
$227$	1.05214e6			1.35522			0.677610	−	0.735421i	$-0.263016\pi$
$227$	1.05214e6			1.35522			0.677610	+	0.735421i	$0.263016\pi$
$228$		0			0
$229$	766814.			0.966276			0.483138	−	0.875544i	$-0.339497\pi$
$229$	766814.			0.966276			0.483138	+	0.875544i	$0.339497\pi$
$230$		0			0
$231$	−119363.	−	206744.i	−0.147177	−	0.254919i
$232$		0			0
$233$	−722964.	−	1.25221e6i	−0.872423	−	1.51108i	−0.859483	−	0.511165i	$-0.829214\pi$
$233$	−722964.	−	1.25221e6i	−0.872423	−	1.51108i	−0.0129403	−	0.999916i	$-0.504119\pi$
$234$		0			0
$235$	873936.			1.03231
$236$		0			0
$237$	−139170.	+	241049.i	−0.160944	+	0.278763i
$238$		0			0
$239$	−749353.			−0.848578			−0.424289	−	0.905527i	$-0.639476\pi$
$239$	−749353.			−0.848578			−0.424289	+	0.905527i	$0.639476\pi$
$240$		0			0
$241$	−459395.	+	795696.i	−0.509499	+	0.882479i	0.490440	+	0.871475i	$0.336836\pi$
$241$	−459395.	+	795696.i	−0.509499	+	0.882479i	−0.999939	+	0.0110040i	$0.996497\pi$
$242$		0			0
$243$	−592334.	+	1.02595e6i	−0.643504	+	1.11458i
$244$		0			0
$245$	−388458.	−	672829.i	−0.413455	−	0.716126i
$246$		0			0
$247$	456071.	+	817421.i	0.475652	+	0.852518i
$248$		0			0
$249$	−192150.	−	332814.i	−0.196400	−	0.340176i
$250$		0			0
$251$	158549.	−	274616.i	0.158848	−	0.275132i	−0.775606	−	0.631218i	$-0.782556\pi$
$251$	158549.	−	274616.i	0.158848	−	0.275132i	0.934453	+	0.356086i	$0.115889\pi$
$252$		0			0
$253$	−253897.	+	439762.i	−0.249377	+	0.431933i
$254$		0			0
$255$	1.94595e6			1.87405
$256$		0			0
$257$	−293395.	+	508174.i	−0.277089	+	0.479932i	−0.970660	−	0.240456i	$-0.922703\pi$
$257$	−293395.	+	508174.i	−0.277089	+	0.479932i	0.693571	+	0.720388i	$0.256036\pi$
$258$		0			0
$259$	731107.			0.677223
$260$		0			0
$261$	−429954.	−	744702.i	−0.390680	−	0.676677i
$262$		0			0
$263$	−602410.	−	1.04341e6i	−0.537036	−	0.930173i	−0.999062	−	0.0433068i	$-0.986211\pi$
$263$	−602410.	−	1.04341e6i	−0.537036	−	0.930173i	0.462026	−	0.886866i	$-0.347123\pi$
$264$		0			0
$265$	3.75075e6			3.28098
$266$		0			0
$267$	−305583.			−0.262332
$268$		0			0
$269$	154072.	+	266860.i	0.129820	+	0.224855i	0.923607	−	0.383341i	$-0.125227\pi$
$269$	154072.	+	266860.i	0.129820	+	0.224855i	−0.793787	+	0.608196i	$0.791893\pi$
$270$		0			0
$271$	476222.	+	824840.i	0.393900	+	0.682255i	0.992960	−	0.118449i	$-0.0377923\pi$
$271$	476222.	+	824840.i	0.393900	+	0.682255i	−0.599060	+	0.800704i	$0.704459\pi$
$272$		0			0
$273$	1.18831e6			0.964991
$274$		0			0
$275$	425048.	−	736205.i	0.338927	−	0.587039i
$276$		0			0
$277$	−575120.			−0.450359			−0.225179	−	0.974317i	$-0.572297\pi$
$277$	−575120.			−0.450359			−0.225179	+	0.974317i	$0.572297\pi$
$278$		0			0
$279$	481005.	−	833125.i	0.369947	−	0.640766i
$280$		0			0
$281$	−917605.	+	1.58934e6i	−0.693250	+	1.20074i	0.277517	+	0.960721i	$0.410488\pi$
$281$	−917605.	+	1.58934e6i	−0.693250	+	1.20074i	−0.970767	+	0.240023i	$0.922845\pi$
$282$		0			0
$283$	−615501.	−	1.06608e6i	−0.456838	−	0.791267i	0.541954	−	0.840408i	$-0.317685\pi$
$283$	−615501.	−	1.06608e6i	−0.456838	−	0.791267i	−0.998792	+	0.0491413i	$0.984352\pi$
$284$		0			0
$285$	−1.62199e6	−	2.90711e6i	−1.18287	−	2.12007i
$286$		0			0
$287$	191757.	+	332132.i	0.137419	+	0.238016i
$288$		0			0
$289$	286892.	−	496912.i	0.202057	−	0.349973i
$290$		0			0
$291$	−1.29303e6	+	2.23960e6i	−0.895113	+	1.55038i
$292$		0			0
$293$	−923293.			−0.628305			−0.314153	−	0.949373i	$-0.601720\pi$
$293$	−923293.			−0.628305			−0.314153	+	0.949373i	$0.601720\pi$
$294$		0			0
$295$	2.09733e6	−	3.63268e6i	1.40317	−	2.43037i
$296$		0			0
$297$	−122086.			−0.0803112
$298$		0			0
$299$	−1.26382e6	−	2.18900e6i	−0.817538	−	1.41602i
$300$		0			0
$301$	−499587.	−	865310.i	−0.317830	−	0.550497i
$302$		0			0
$303$	−1.04087e6			−0.651312
$304$		0			0
$305$	−454145.			−0.279541
$306$		0			0
$307$	843091.	+	1.46028e6i	0.510539	+	0.884279i	0.999925	+	0.0122119i	$0.00388726\pi$
$307$	843091.	+	1.46028e6i	0.510539	+	0.884279i	−0.489387	+	0.872067i	$0.662779\pi$
$308$		0			0
$309$	−1.07097e6	−	1.85498e6i	−0.638091	−	1.10521i
$310$		0			0
$311$	−1.06080e6			−0.621915			−0.310957	−	0.950424i	$-0.600650\pi$
$311$	−1.06080e6			−0.621915			−0.310957	+	0.950424i	$0.600650\pi$
$312$		0			0
$313$	8090.23	−	14012.7i	0.00466767	−	0.00808464i	−0.863682	−	0.504037i	$-0.831848\pi$
$313$	8090.23	−	14012.7i	0.00466767	−	0.00808464i	0.868350	+	0.495952i	$0.165181\pi$
$314$		0			0
$315$	−1.87688e6			−1.06576
$316$		0			0
$317$	1.27571e6	−	2.20960e6i	0.713024	−	1.23499i	−0.250693	−	0.968067i	$-0.580658\pi$
$317$	1.27571e6	−	2.20960e6i	0.713024	−	1.23499i	0.963717	−	0.266927i	$-0.0860082\pi$
$318$		0			0
$319$	264665.	−	458413.i	0.145619	−	0.252220i
$320$		0			0
$321$	473676.	+	820431.i	0.256578	+	0.444406i
$322$		0			0
$323$	1.44724e6	+	21249.1i	0.771853	+	0.0113327i
$324$		0			0
$325$	2.11576e6	+	3.66460e6i	1.11111	+	1.92450i
$326$		0			0
$327$	−1.57352e6	+	2.72542e6i	−0.813773	+	1.40950i
$328$		0			0
$329$	412610.	−	714661.i	0.210160	−	0.364008i
$330$		0			0
$331$	1.14830e6			0.576082			0.288041	−	0.957618i	$-0.406996\pi$
$331$	1.14830e6			0.576082			0.288041	+	0.957618i	$0.406996\pi$
$332$		0			0
$333$	−742766.	+	1.28651e6i	−0.367064	+	0.635773i
$334$		0			0
$335$	3.74077e6			1.82117
$336$		0			0
$337$	−742181.	−	1.28549e6i	−0.355988	−	0.616589i	0.631299	−	0.775540i	$-0.282522\pi$
$337$	−742181.	−	1.28549e6i	−0.355988	−	0.616589i	−0.987286	+	0.158951i	$0.949189\pi$
$338$		0			0
$339$	742944.	+	1.28682e6i	0.351121	+	0.608160i
$340$		0			0
$341$	592180.			0.275783
$342$		0			0
$343$	−2.33943e6			−1.07368
$344$		0			0
$345$	4.49471e6	+	7.78506e6i	2.03308	+	3.52139i
$346$		0			0
$347$	1.09955e6	+	1.90448e6i	0.490222	+	0.849089i	0.999937	−	0.0112544i	$-0.00358245\pi$
$347$	1.09955e6	+	1.90448e6i	0.490222	+	0.849089i	−0.509715	+	0.860343i	$0.670249\pi$
$348$		0			0
$349$	−3.81500e6			−1.67660			−0.838302	−	0.545206i	$-0.816451\pi$
$349$	−3.81500e6			−1.67660			−0.838302	+	0.545206i	$0.816451\pi$
$350$		0			0
$351$	303855.	−	526292.i	0.131643	−	0.228012i
$352$		0			0
$353$	550982.			0.235342			0.117671	−	0.993053i	$-0.462457\pi$
$353$	550982.			0.235342			0.117671	+	0.993053i	$0.462457\pi$
$354$		0			0
$355$	−2.36856e6	+	4.10246e6i	−0.997500	+	1.72772i
$356$		0			0
$357$	918738.	−	1.59130e6i	0.381523	−	0.660818i
$358$		0			0
$359$	−110651.	−	191653.i	−0.0453126	−	0.0784838i	0.842480	−	0.538728i	$-0.181095\pi$
$359$	−110651.	−	191653.i	−0.0453126	−	0.0784838i	−0.887792	+	0.460245i	$0.847762\pi$
$360$		0			0
$361$	−1.17456e6	−	2.17979e6i	−0.474359	−	0.880331i
$362$		0			0
$363$	−1.53432e6	−	2.65752e6i	−0.611152	−	1.05855i
$364$		0			0
$365$	−4.11487e6	+	7.12717e6i	−1.61668	+	2.80017i
$366$		0			0
$367$	−680360.	+	1.17842e6i	−0.263678	+	0.456704i	−0.967216	−	0.253954i	$-0.918269\pi$
$367$	−680360.	+	1.17842e6i	−0.263678	+	0.456704i	0.703538	+	0.710657i	$0.251602\pi$
$368$		0			0
$369$	−779258.			−0.297931
$370$		0			0
$371$	1.77084e6	−	3.06718e6i	0.667950	−	1.15692i
$372$		0			0
$373$	4.37071e6			1.62660			0.813298	−	0.581848i	$-0.197670\pi$
$373$	4.37071e6			1.62660			0.813298	+	0.581848i	$0.197670\pi$
$374$		0			0
$375$	−4.21899e6	−	7.30751e6i	−1.54928	−	2.68344i
$376$		0			0
$377$	1.31742e6	+	2.28184e6i	0.477387	+	0.826859i
$378$		0			0
$379$	−3.26062e6			−1.16601			−0.583005	−	0.812469i	$-0.698123\pi$
$379$	−3.26062e6			−1.16601			−0.583005	+	0.812469i	$0.698123\pi$
$380$		0			0
$381$	2.74555e6			0.968986
$382$		0			0
$383$	1.11542e6	+	1.93197e6i	0.388546	+	0.672981i	0.992254	−	0.124224i	$-0.0396441\pi$
$383$	1.11542e6	+	1.93197e6i	0.388546	+	0.672981i	−0.603708	+	0.797205i	$0.706311\pi$
$384$		0			0
$385$	−577671.	−	1.00056e6i	−0.198623	−	0.344025i
$386$		0			0
$387$	2.03021e6			0.689072
$388$		0			0
$389$	520809.	−	902068.i	0.174504	−	0.302249i	−0.765486	−	0.643453i	$-0.777501\pi$
$389$	520809.	−	902068.i	0.174504	−	0.302249i	0.939989	+	0.341204i	$0.110835\pi$
$390$		0			0
$391$	−3.90848e6			−1.29290
$392$		0			0
$393$	2.09941e6	−	3.63628e6i	0.685670	−	1.18762i
$394$		0			0
$395$	−673527.	+	1.16658e6i	−0.217201	+	0.376203i
$396$		0			0
$397$	−2.19687e6	−	3.80509e6i	−0.699565	−	1.21168i	−0.968618	−	0.248556i	$-0.920044\pi$
$397$	−2.19687e6	−	3.80509e6i	−0.699565	−	1.21168i	0.269053	−	0.963125i	$-0.413289\pi$
$398$		0			0
$399$	−3.14308e6	−	46148.1i	−0.988377	−	0.0145118i
$400$		0			0
$401$	393217.	+	681071.i	0.122116	+	0.211510i	0.920602	−	0.390503i	$-0.127699\pi$
$401$	393217.	+	681071.i	0.122116	+	0.211510i	−0.798486	+	0.602013i	$0.794365\pi$
$402$		0			0
$403$	−1.47385e6	+	2.55277e6i	−0.452053	+	0.782979i
$404$		0			0
$405$	−3.46738e6	+	6.00567e6i	−1.05042	+	1.81938i
$406$		0			0
$407$	−914442.			−0.273634
$408$		0			0
$409$	−584641.	+	1.01263e6i	−0.172815	+	0.299324i	−0.939403	−	0.342815i	$-0.888620\pi$
$409$	−584641.	+	1.01263e6i	−0.172815	+	0.299324i	0.766588	+	0.642139i	$0.221953\pi$
$410$		0			0
$411$	7.59038e6			2.21645
$412$		0			0
$413$	−1.98041e6	−	3.43018e6i	−0.571322	−	0.989559i
$414$		0			0
$415$	−929930.	−	1.61069e6i	−0.265051	−	0.459082i
$416$		0			0
$417$	7.95081e6			2.23909
$418$		0			0
$419$	−4.14739e6			−1.15409			−0.577045	−	0.816712i	$-0.695794\pi$
$419$	−4.14739e6			−1.15409			−0.577045	+	0.816712i	$0.695794\pi$
$420$		0			0
$421$	1.59962e6	+	2.77063e6i	0.439858	+	0.761857i	0.997678	−	0.0681052i	$-0.0216954\pi$
$421$	1.59962e6	+	2.77063e6i	0.439858	+	0.761857i	−0.557820	+	0.829962i	$0.688362\pi$
$422$		0			0
$423$	838380.	+	1.45212e6i	0.227819	+	0.394594i
$424$		0			0
$425$	6.54317e6			1.75718
$426$		0			0
$427$	−214415.	+	371378.i	−0.0569096	+	0.0985703i
$428$		0			0
$429$	−1.48629e6			−0.389908
$430$		0			0
$431$	2.91306e6	−	5.04557e6i	0.755364	−	1.30833i	−0.189830	−	0.981817i	$-0.560794\pi$
$431$	2.91306e6	−	5.04557e6i	0.755364	−	1.30833i	0.945193	−	0.326511i	$-0.105873\pi$
$432$		0			0
$433$	1.29561e6	−	2.24407e6i	0.332090	−	0.575196i	−0.650832	−	0.759222i	$-0.725580\pi$
$433$	1.29561e6	−	2.24407e6i	0.332090	−	0.575196i	0.982921	+	0.184026i	$0.0589130\pi$
$434$		0			0
$435$	−4.68533e6	−	8.11522e6i	−1.18718	−	2.05626i
$436$		0			0
$437$	3.25779e6	+	5.83899e6i	0.816056	+	1.46263i
$438$		0			0
$439$	986761.	+	1.70912e6i	0.244372	+	0.423264i	0.961955	−	0.273209i	$-0.0880850\pi$
$439$	986761.	+	1.70912e6i	0.244372	+	0.423264i	−0.717583	+	0.696473i	$0.754752\pi$
$440$		0			0
$441$	745307.	−	1.29091e6i	0.182490	−	0.316082i
$442$		0			0
$443$	73870.7	−	127948.i	0.0178839	−	0.0309759i	−0.856945	−	0.515408i	$-0.827640\pi$
$443$	73870.7	−	127948.i	0.0178839	−	0.0309759i	0.874829	+	0.484432i	$0.160974\pi$
$444$		0			0
$445$	−1.47890e6			−0.354029
$446$		0			0
$447$	458120.	−	793487.i	0.108445	−	0.187833i
$448$		0			0
$449$	−6.33075e6			−1.48197			−0.740985	−	0.671522i	$-0.765641\pi$
$449$	−6.33075e6			−1.48197			−0.740985	+	0.671522i	$0.765641\pi$
$450$		0			0
$451$	−239842.	−	415419.i	−0.0555244	−	0.0961711i
$452$		0			0
$453$	−1.37061e6	−	2.37396e6i	−0.313811	−	0.543536i
$454$		0			0
$455$	5.75094e6			1.30230
$456$		0			0
$457$	−175313.			−0.0392667			−0.0196334	−	0.999807i	$-0.506250\pi$
$457$	−175313.			−0.0392667			−0.0196334	+	0.999807i	$0.506250\pi$
$458$		0			0
$459$	−469848.	−	813801.i	−0.104094	−	0.180296i
$460$		0			0
$461$	−3.20957e6	−	5.55914e6i	−0.703388	−	1.21830i	−0.967270	−	0.253749i	$-0.918336\pi$
$461$	−3.20957e6	−	5.55914e6i	−0.703388	−	1.21830i	0.263882	−	0.964555i	$-0.414997\pi$
$462$		0			0
$463$	2.17798e6			0.472174			0.236087	−	0.971732i	$-0.424135\pi$
$463$	2.17798e6			0.472174			0.236087	+	0.971732i	$0.424135\pi$
$464$		0			0
$465$	5.24164e6	−	9.07879e6i	1.12418	−	1.94713i
$466$		0			0
$467$	742392.			0.157522			0.0787610	−	0.996894i	$-0.474904\pi$
$467$	742392.			0.157522			0.0787610	+	0.996894i	$0.474904\pi$
$468$		0			0
$469$	1.76613e6	−	3.05902e6i	0.370757	−	0.642170i
$470$		0			0
$471$	−4.99910e6	+	8.65870e6i	−1.03834	+	1.79846i
$472$		0			0
$473$	624865.	+	1.08230e6i	0.128420	+	0.222430i
$474$		0			0
$475$	−5.45386e6	−	9.77503e6i	−1.10910	−	1.98785i
$476$		0			0
$477$	3.59815e6	+	6.23219e6i	0.724076	+	1.25414i
$478$		0			0
$479$	−2.42943e6	+	4.20789e6i	−0.483799	+	0.837965i	−0.999827	−	0.0186069i	$-0.994077\pi$
$479$	−2.42943e6	+	4.20789e6i	−0.483799	+	0.837965i	0.516028	+	0.856572i	$0.327410\pi$
$480$		0			0
$481$	2.27591e6	−	3.94199e6i	0.448530	−	0.776877i
$482$		0			0
$483$	8.48832e6			1.65559
$484$		0			0
$485$	−6.25777e6	+	1.08388e7i	−1.20800	+	2.09231i
$486$		0			0
$487$	3.50290e6			0.669275			0.334638	−	0.942347i	$-0.391386\pi$
$487$	3.50290e6			0.669275			0.334638	+	0.942347i	$0.391386\pi$
$488$		0			0
$489$	−4.06742e6	−	7.04497e6i	−0.769213	−	1.33232i
$490$		0			0
$491$	368324.	+	637956.i	0.0689488	+	0.119423i	0.898439	−	0.439099i	$-0.144702\pi$
$491$	368324.	+	637956.i	0.0689488	+	0.119423i	−0.829490	+	0.558521i	$0.811369\pi$
$492$		0			0
$493$	4.07424e6			0.754968
$494$		0			0
$495$	2.34753e6			0.430625
$496$		0			0
$497$	2.23653e6	+	3.87378e6i	0.406147	+	0.703467i
$498$		0			0
$499$	9216.83	+	15964.0i	0.00165703	+	0.00287006i	0.866853	−	0.498564i	$-0.166139\pi$
$499$	9216.83	+	15964.0i	0.00165703	+	0.00287006i	−0.865196	+	0.501434i	$0.832806\pi$
$500$		0			0
$501$	−1.09344e7			−1.94625
$502$		0			0
$503$	249975.	−	432970.i	0.0440532	−	0.0763024i	−0.843158	−	0.537666i	$-0.819306\pi$
$503$	249975.	−	432970.i	0.0440532	−	0.0763024i	0.887211	+	0.461363i	$0.152640\pi$
$504$		0			0
$505$	−5.03738e6			−0.878975
$506$		0			0
$507$	−182305.	+	315762.i	−0.0314977	+	0.0545557i
$508$		0			0
$509$	−2.32529e6	+	4.02752e6i	−0.397816	+	0.689038i	−0.993456	−	0.114213i	$-0.963565\pi$
$509$	−2.32529e6	+	4.02752e6i	−0.397816	+	0.689038i	0.595640	+	0.803252i	$0.296899\pi$
$510$		0			0
$511$	3.88550e6	+	6.72988e6i	0.658255	+	1.14013i
$512$		0			0
$513$	−824133.	+	1.38024e6i	−0.138262	+	0.231559i
$514$		0			0
$515$	−5.18309e6	−	8.97737e6i	−0.861133	−	1.49153i
$516$		0			0
$517$	−516077.	+	893872.i	−0.0849158	+	0.147078i
$518$		0			0
$519$	4.77605e6	−	8.27236e6i	0.778306	−	1.34807i
$520$		0			0
$521$	7.64507e6			1.23392			0.616960	−	0.786994i	$-0.288364\pi$
$521$	7.64507e6			1.23392			0.616960	+	0.786994i	$0.288364\pi$
$522$		0			0
$523$	794941.	−	1.37688e6i	0.127081	−	0.220111i	−0.795463	−	0.606002i	$-0.792773\pi$
$523$	794941.	−	1.37688e6i	0.127081	−	0.220111i	0.922544	+	0.385891i	$0.126106\pi$
$524$		0			0
$525$	−1.42103e7			−2.25011
$526$		0			0
$527$	2.27900e6	+	3.94734e6i	0.357451	+	0.619124i
$528$		0			0
$529$	−5.80954e6	−	1.00624e7i	−0.902615	−	1.56338i
$530$		0			0
$531$	8.04799e6			1.23866
$532$		0			0
$533$	2.38772e6			0.364054
$534$		0			0
$535$	2.29240e6	+	3.97056e6i	0.346263	+	0.599746i
$536$		0			0
$537$	3.97593e6	+	6.88651e6i	0.594981	+	1.03054i
$538$		0			0
$539$	917570.			0.136040
$540$		0			0
$541$	−2.63644e6	+	4.56645e6i	−0.387280	+	0.670789i	−0.992083	−	0.125587i	$-0.959919\pi$
$541$	−2.63644e6	+	4.56645e6i	−0.387280	+	0.670789i	0.604803	+	0.796375i	$0.293252\pi$
$542$		0			0
$543$	−7.19385e6			−1.04704
$544$		0			0
$545$	−7.61521e6	+	1.31899e7i	−1.09822	+	1.90218i
$546$		0			0
$547$	−1.94792e6	+	3.37389e6i	−0.278357	+	0.482128i	−0.970977	−	0.239175i	$-0.923123\pi$
$547$	−1.94792e6	+	3.37389e6i	−0.278357	+	0.482128i	0.692620	+	0.721303i	$0.256456\pi$
$548$		0			0
$549$	−435669.	−	754600.i	−0.0616915	−	0.106853i
$550$		0			0
$551$	−3.39596e6	−	6.08662e6i	−0.476522	−	0.854077i
$552$		0			0
$553$	635982.	+	1.10155e6i	0.0884366	+	0.153177i
$554$		0			0
$555$	−8.09413e6	+	1.40194e7i	−1.11542	+	1.93196i
$556$		0			0
$557$	3.90618e6	−	6.76570e6i	0.533475	−	0.924006i	−0.465761	−	0.884911i	$-0.654219\pi$
$557$	3.90618e6	−	6.76570e6i	0.533475	−	0.924006i	0.999235	−	0.0390950i	$-0.0124475\pi$
$558$		0			0
$559$	−6.22077e6			−0.842005
$560$		0			0
$561$	−1.14912e6	+	1.99034e6i	−0.154156	+	0.267005i
$562$		0			0
$563$	7.01256e6			0.932407			0.466204	−	0.884677i	$-0.345621\pi$
$563$	7.01256e6			0.932407			0.466204	+	0.884677i	$0.345621\pi$
$564$		0			0
$565$	3.59555e6	+	6.22768e6i	0.473854	+	0.820739i
$566$		0			0
$567$	3.27409e6	+	5.67090e6i	0.427694	+	0.740788i
$568$		0			0
$569$	6.70622e6			0.868355			0.434177	−	0.900827i	$-0.357039\pi$
$569$	6.70622e6			0.868355			0.434177	+	0.900827i	$0.357039\pi$
$570$		0			0
$571$	5.59306e6			0.717893			0.358946	−	0.933358i	$-0.383136\pi$
$571$	5.59306e6			0.717893			0.358946	+	0.933358i	$0.383136\pi$
$572$		0			0
$573$	−7.30892e6	−	1.26594e7i	−0.929965	−	1.61075i
$574$		0			0
$575$	1.51133e7	+	2.61769e7i	1.90629	+	3.30179i
$576$		0			0
$577$	1.19869e7			1.49888			0.749442	−	0.662070i	$-0.230322\pi$
$577$	1.19869e7			1.49888			0.749442	+	0.662070i	$0.230322\pi$
$578$		0			0
$579$	−577510.	+	1.00028e6i	−0.0715918	+	0.124001i
$580$		0			0
$581$	−1.75619e6			−0.215839
$582$		0			0
$583$	−2.21490e6	+	3.83632e6i	−0.269887	+	0.467458i
$584$		0			0
$585$	−5.84266e6	+	1.01198e7i	−0.705863	+	1.22259i
$586$		0			0
$587$	7.13321e6	+	1.23551e7i	0.854456	+	1.47996i	0.877149	+	0.480219i	$0.159443\pi$
$587$	7.13321e6	+	1.23551e7i	0.854456	+	1.47996i	−0.0226925	+	0.999742i	$0.507224\pi$
$588$		0			0
$589$	3.99745e6	−	6.69484e6i	0.474783	−	0.795156i
$590$		0			0
$591$	−1.04561e7	−	1.81105e7i	−1.23141	−	2.13286i
$592$		0			0
$593$	3.81201e6	−	6.60260e6i	0.445161	−	0.771042i	−0.552902	−	0.833246i	$-0.686480\pi$
$593$	3.81201e6	−	6.60260e6i	0.445161	−	0.771042i	0.998063	+	0.0622043i	$0.0198130\pi$
$594$		0			0
$595$	4.44632e6	−	7.70126e6i	0.514883	−	0.891803i
$596$		0			0
$597$	1.03149e7			1.18448
$598$		0			0
$599$	−5.20469e6	+	9.01479e6i	−0.592691	+	1.02657i	0.401178	+	0.916000i	$0.368601\pi$
$599$	−5.20469e6	+	9.01479e6i	−0.592691	+	1.02657i	−0.993868	+	0.110570i	$0.964732\pi$
$600$		0			0
$601$	1.96843e6			0.222297			0.111149	−	0.993804i	$-0.464547\pi$
$601$	1.96843e6			0.222297			0.111149	+	0.993804i	$0.464547\pi$
$602$		0			0
$603$	3.58858e6	+	6.21560e6i	0.401911	+	0.696129i
$604$		0			0
$605$	−7.42549e6	−	1.28613e7i	−0.824777	−	1.42856i
$606$		0			0
$607$	−1.79114e7			−1.97314			−0.986569	−	0.163347i	$-0.947771\pi$
$607$	−1.79114e7			−1.97314			−0.986569	+	0.163347i	$0.947771\pi$
$608$		0			0
$609$	−8.84830e6			−0.966755
$610$		0			0
$611$	−2.56888e6	−	4.44942e6i	−0.278381	−	0.482171i
$612$		0			0
$613$	−206107.	−	356988.i	−0.0221535	−	0.0383709i	0.854736	−	0.519063i	$-0.173719\pi$
$613$	−206107.	−	356988.i	−0.0221535	−	0.0383709i	−0.876890	+	0.480692i	$0.840386\pi$
$614$		0			0
$615$	−8.49179e6			−0.905339
$616$		0			0
$617$	6.97181e6	−	1.20755e7i	0.737281	−	1.27701i	−0.216435	−	0.976297i	$-0.569443\pi$
$617$	6.97181e6	−	1.20755e7i	0.737281	−	1.27701i	0.953715	−	0.300711i	$-0.0972238\pi$
$618$		0			0
$619$	−1.59596e7			−1.67415			−0.837075	−	0.547088i	$-0.815736\pi$
$619$	−1.59596e7			−1.67415			−0.837075	+	0.547088i	$0.815736\pi$
$620$		0			0
$621$	2.17049e6	−	3.75940e6i	0.225854	−	0.391191i
$622$		0			0
$623$	−698230.	+	1.20937e6i	−0.0720740	+	0.124836i
$624$		0			0
$625$	−9.30336e6	−	1.61139e7i	−0.952664	−	1.65006i
$626$		0			0
$627$	3.93124e6	+	57720.4i	0.399357	+	0.00586354i
$628$		0			0
$629$	−3.51922e6	−	6.09547e6i	−0.354666	−	0.614300i
$630$		0			0
$631$	3.10790e6	−	5.38304e6i	0.310737	−	0.538213i	−0.667785	−	0.744354i	$-0.732757\pi$
$631$	3.10790e6	−	5.38304e6i	0.310737	−	0.538213i	0.978522	+	0.206141i	$0.0660907\pi$
$632$		0			0
$633$	6.46664e6	−	1.12005e7i	0.641460	−	1.11104i
$634$		0			0
$635$	1.32874e7			1.30769
$636$		0			0
$637$	−2.28369e6	+	3.95547e6i	−0.222992	+	0.386233i
$638$		0			0
$639$	−9.08877e6			−0.880548
$640$		0			0
$641$	4.32033e6	+	7.48302e6i	0.415309	+	0.719337i	0.995461	−	0.0951719i	$-0.0303401\pi$
$641$	4.32033e6	+	7.48302e6i	0.415309	+	0.719337i	−0.580152	+	0.814508i	$0.697007\pi$
$642$		0			0
$643$	−8.07254e6	−	1.39820e7i	−0.769985	−	1.33365i	−0.937570	−	0.347796i	$-0.886930\pi$
$643$	−8.07254e6	−	1.39820e7i	−0.769985	−	1.33365i	0.167585	−	0.985858i	$-0.446403\pi$
$644$		0			0
$645$	2.21238e7			2.09392
$646$		0			0
$647$	8.16344e6			0.766677			0.383338	−	0.923608i	$-0.374774\pi$
$647$	8.16344e6			0.766677			0.383338	+	0.923608i	$0.374774\pi$
$648$		0			0
$649$	2.47703e6	+	4.29034e6i	0.230844	+	0.399834i
$650$		0			0
$651$	−4.94946e6	−	8.57271e6i	−0.457725	−	0.792803i
$652$		0			0
$653$	1.08567e7			0.996359			0.498179	−	0.867074i	$-0.334002\pi$
$653$	1.08567e7			0.996359			0.498179	+	0.867074i	$0.334002\pi$
$654$		0			0
$655$	1.01603e7	−	1.75981e7i	0.925343	−	1.60274i
$656$		0			0
$657$	−1.57898e7			−1.42713
$658$		0			0
$659$	2.49623e6	−	4.32359e6i	0.223908	−	0.387821i	−0.732083	−	0.681215i	$-0.761452\pi$
$659$	2.49623e6	−	4.32359e6i	0.223908	−	0.387821i	0.955991	+	0.293395i	$0.0947850\pi$
$660$		0			0
$661$	−7.54002e6	+	1.30597e7i	−0.671226	+	1.16260i	0.306330	+	0.951925i	$0.400899\pi$
$661$	−7.54002e6	+	1.30597e7i	−0.671226	+	1.16260i	−0.977557	+	0.210673i	$0.932435\pi$
$662$		0			0
$663$	−5.71999e6	−	9.90731e6i	−0.505372	−	0.875330i
$664$		0			0
$665$	−1.52112e7	−	223338.i	−1.33386	−	0.0195844i
$666$		0			0
$667$	9.41057e6	+	1.62996e7i	0.819033	+	1.41861i
$668$		0			0
$669$	1.56862e6	−	2.71692e6i	0.135504	−	0.234699i
$670$		0			0
$671$	268182.	−	464505.i	0.0229945	−	0.0398276i
$672$		0			0
$673$	−1.89452e7			−1.61236			−0.806180	−	0.591671i	$-0.798469\pi$
$673$	−1.89452e7			−1.61236			−0.806180	+	0.591671i	$0.798469\pi$
$674$		0			0
$675$	−3.63361e6	+	6.29359e6i	−0.306958	+	0.531666i
$676$		0			0
$677$	−4.05825e6			−0.340304			−0.170152	−	0.985418i	$-0.554426\pi$
$677$	−4.05825e6			−0.340304			−0.170152	+	0.985418i	$0.554426\pi$
$678$		0			0
$679$	5.90894e6	+	1.02346e7i	0.491853	+	0.851914i
$680$		0			0
$681$	1.09990e7	+	1.90508e7i	0.908836	+	1.57415i
$682$		0			0
$683$	2.85503e6			0.234185			0.117092	−	0.993121i	$-0.462643\pi$
$683$	2.85503e6			0.234185			0.117092	+	0.993121i	$0.462643\pi$
$684$		0			0
$685$	3.67344e7			2.99121
$686$		0			0
$687$	8.01621e6	+	1.38845e7i	0.648003	+	1.12237i
$688$		0			0
$689$	−1.10251e7	−	1.90960e7i	−0.884778	−	1.53248i
$690$		0			0
$691$	4.38220e6			0.349138			0.174569	−	0.984645i	$-0.444147\pi$
$691$	4.38220e6			0.349138			0.174569	+	0.984645i	$0.444147\pi$
$692$		0			0
$693$	1.10834e6	−	1.91970e6i	0.0876675	−	0.151845i
$694$		0			0
$695$	3.84788e7			3.02175
$696$		0			0
$697$	1.84606e6	−	3.19747e6i	0.143934	−	0.249301i
$698$		0			0
$699$	1.51156e7	−	2.61810e7i	1.17013	−	2.02672i
$700$		0			0
$701$	−2.33852e6	−	4.05044e6i	−0.179741	−	0.311320i	0.762051	−	0.647517i	$-0.224193\pi$
$701$	−2.33852e6	−	4.05044e6i	−0.179741	−	0.311320i	−0.941792	+	0.336197i	$0.890859\pi$
$702$		0			0
$703$	−6.17285e6	+	1.03382e7i	−0.471083	+	0.788959i
$704$		0			0
$705$	9.13605e6	+	1.58241e7i	0.692287	+	1.19908i
$706$		0			0
$707$	−2.37829e6	+	4.11932e6i	−0.178944	+	0.309940i
$708$		0			0
$709$	−177992.	+	308291.i	−0.0132980	+	0.0230328i	−0.872598	−	0.488439i	$-0.837566\pi$
$709$	−177992.	+	308291.i	−0.0132980	+	0.0230328i	0.859300	+	0.511472i	$0.170900\pi$
$710$		0			0
$711$	−2.58450e6			−0.191735
$712$		0			0
$713$	−1.05279e7	+	1.82349e7i	−0.775568	+	1.34332i
$714$		0			0
$715$	−7.19307e6			−0.526198
$716$		0			0
$717$	−7.83368e6	−	1.35683e7i	−0.569073	−	0.985663i
$718$		0			0
$719$	−705747.	−	1.22239e6i	−0.0509128	−	0.0881835i	0.839446	−	0.543443i	$-0.182880\pi$
$719$	−705747.	−	1.22239e6i	−0.0509128	−	0.0881835i	−0.890359	+	0.455260i	$0.849546\pi$
$720$		0			0
$721$	−9.78833e6			−0.701246
$722$		0			0
$723$	−1.92099e7			−1.36672
$724$		0			0
$725$	−1.57542e7	−	2.72871e7i	−1.11314	−	1.92802i
$726$		0			0
$727$	6.82049e6	+	1.18134e7i	0.478608	+	0.828973i	0.999699	−	0.0245278i	$-0.00780822\pi$
$727$	6.82049e6	+	1.18134e7i	0.478608	+	0.828973i	−0.521091	+	0.853501i	$0.674475\pi$
$728$		0			0
$729$	−8.11485e6			−0.565538
$730$		0			0
$731$	−4.80957e6	+	8.33042e6i	−0.332899	+	0.576598i
$732$		0			0
$733$	−2.66079e7			−1.82915			−0.914577	−	0.404412i	$-0.867476\pi$
$733$	−2.66079e7			−1.82915			−0.914577	+	0.404412i	$0.867476\pi$
$734$		0			0
$735$	8.12181e6	−	1.40674e7i	0.554542	−	0.960495i
$736$		0			0
$737$	−2.20900e6	+	3.82611e6i	−0.149806	+	0.259471i
$738$		0			0
$739$	4.22169e6	+	7.31218e6i	0.284364	+	0.492533i	0.972455	−	0.233091i	$-0.0748842\pi$
$739$	4.22169e6	+	7.31218e6i	0.284364	+	0.492533i	−0.688091	+	0.725625i	$0.741551\pi$
$740$		0			0
$741$	−1.00331e7	+	1.68032e7i	−0.671258	+	1.12421i
$742$		0			0
$743$	−2.70095e6	−	4.67819e6i	−0.179492	−	0.310889i	0.762215	−	0.647324i	$-0.224112\pi$
$743$	−2.70095e6	−	4.67819e6i	−0.179492	−	0.310889i	−0.941707	+	0.336435i	$0.890779\pi$
$744$		0			0
$745$	2.21712e6	−	3.84016e6i	0.146352	−	0.253489i
$746$		0			0
$747$	1.78419e6	−	3.09031e6i	0.116988	−	0.202629i
$748$		0			0
$749$	4.32923e6			0.281972
$750$		0			0
$751$	6.66101e6	−	1.15372e7i	0.430963	−	0.746450i	−0.565994	−	0.824410i	$-0.691507\pi$
$751$	6.66101e6	−	1.15372e7i	0.430963	−	0.746450i	0.996956	+	0.0779599i	$0.0248406\pi$
$752$		0			0
$753$	6.62985e6			0.426105
$754$		0			0
$755$	−6.63319e6	−	1.14890e7i	−0.423502	−	0.733527i
$756$		0			0
$757$	1.61706e6	+	2.80083e6i	0.102562	+	0.177642i	0.912739	−	0.408542i	$-0.133963\pi$
$757$	1.61706e6	+	2.80083e6i	0.102562	+	0.177642i	−0.810178	+	0.586185i	$0.800629\pi$
$758$		0			0
$759$	−1.06169e7			−0.668948
$760$		0			0
$761$	602035.			0.0376843			0.0188421	−	0.999822i	$-0.494002\pi$
$761$	602035.			0.0376843			0.0188421	+	0.999822i	$0.494002\pi$
$762$		0			0
$763$	7.19072e6	+	1.24547e7i	0.447158	+	0.774500i
$764$		0			0
$765$	9.03446e6	+	1.56481e7i	0.558147	+	0.966739i
$766$		0			0
$767$	−2.46598e7			−1.51357
$768$		0			0
$769$	−9.81675e6	+	1.70031e7i	−0.598621	+	1.03684i	0.394404	+	0.918937i	$0.370951\pi$
$769$	−9.81675e6	+	1.70031e7i	−0.598621	+	1.03684i	−0.993025	+	0.117904i	$0.962382\pi$
$770$		0			0
$771$	−1.22685e7			−0.743285
$772$		0			0
$773$	2.03679e6	−	3.52782e6i	0.122602	−	0.212353i	−0.798191	−	0.602404i	$-0.794210\pi$
$773$	2.03679e6	−	3.52782e6i	0.122602	−	0.212353i	0.920793	+	0.390052i	$0.127543\pi$
$774$		0			0
$775$	1.76248e7	−	3.05270e7i	1.05407	−	1.82570i
$776$		0			0
$777$	7.64293e6	+	1.32379e7i	0.454159	+	0.786626i
$778$		0			0
$779$	−6.31552e6	−	92727.4i	−0.372877	−	0.00547475i
$780$		0			0
$781$	−2.79736e6	−	4.84518e6i	−0.164105	−	0.284238i
$782$		0			0
$783$	−2.26254e6	+	3.91883e6i	−0.131884	+	0.228429i
$784$		0			0
$785$	−2.41937e7	+	4.19047e7i	−1.40129	+	2.42710i
$786$		0			0
$787$	4.13033e6			0.237710			0.118855	−	0.992912i	$-0.462078\pi$
$787$	4.13033e6			0.237710			0.118855	+	0.992912i	$0.462078\pi$
$788$		0			0
$789$	1.25951e7	−	2.18154e7i	0.720293	−	1.24758i
$790$		0			0
$791$	6.79025e6			0.385873
$792$		0			0
$793$	1.33493e6	+	2.31217e6i	0.0753834	+	0.130568i
$794$		0			0
$795$	3.92101e7	+	6.79138e7i	2.20029	+	3.81101i
$796$		0			0
$797$	−3.62791e6			−0.202307			−0.101154	−	0.994871i	$-0.532253\pi$
$797$	−3.62791e6			−0.202307			−0.101154	+	0.994871i	$0.532253\pi$
$798$		0			0
$799$	−7.94447e6			−0.440249
$800$		0			0
$801$	−1.41873e6	−	2.45731e6i	−0.0781301	−	0.135325i
$802$		0			0
$803$	−4.85983e6	−	8.41748e6i	−0.265970	−	0.460674i
$804$		0			0
$805$	4.10800e7			2.23430
$806$		0			0
$807$	−3.22131e6	+	5.57947e6i	−0.174120	+	0.301585i
$808$		0			0
$809$	5.98904e6			0.321726			0.160863	−	0.986977i	$-0.448572\pi$
$809$	5.98904e6			0.321726			0.160863	+	0.986977i	$0.448572\pi$
$810$		0			0
$811$	−8.65166e6	+	1.49851e7i	−0.461899	+	0.800033i	−0.999056	−	0.0434495i	$-0.986165\pi$
$811$	−8.65166e6	+	1.49851e7i	−0.461899	+	0.800033i	0.537156	+	0.843483i	$0.319499\pi$
$812$		0			0
$813$	−9.95677e6	+	1.72456e7i	−0.528314	+	0.915066i
$814$		0			0
$815$	−1.96847e7	−	3.40948e7i	−1.03809	−	1.79802i
$816$		0			0
$817$	1.64539e7	+	241584.i	0.862411	+	0.0126623i
$818$		0			0
$819$	5.51697e6	+	9.55567e6i	0.287403	+	0.497796i
$820$		0			0
$821$	−7.04373e6	+	1.22001e7i	−0.364708	+	0.631692i	−0.988729	−	0.149715i	$-0.952164\pi$
$821$	−7.04373e6	+	1.22001e7i	−0.364708	+	0.631692i	0.624022	+	0.781407i	$0.285498\pi$
$822$		0			0
$823$	1.76700e6	−	3.06053e6i	0.0909361	−	0.157506i	−0.816969	−	0.576681i	$-0.804347\pi$
$823$	1.76700e6	−	3.06053e6i	0.0909361	−	0.157506i	0.907905	+	0.419175i	$0.137681\pi$
$824$		0			0
$825$	1.77737e7			0.909164
$826$		0			0
$827$	−5.25927e6	+	9.10933e6i	−0.267400	+	0.463151i	−0.968190	−	0.250217i	$-0.919498\pi$
$827$	−5.25927e6	+	9.10933e6i	−0.267400	+	0.463151i	0.700789	+	0.713368i	$0.252831\pi$
$828$		0			0
$829$	3.82603e7			1.93358			0.966790	−	0.255571i	$-0.0822636\pi$
$829$	3.82603e7			1.93358			0.966790	+	0.255571i	$0.0822636\pi$
$830$		0			0
$831$	−6.01225e6	−	1.04135e7i	−0.302019	−	0.523113i
$832$		0			0
$833$	3.53126e6	+	6.11632e6i	0.176326	+	0.305406i
$834$		0			0
$835$	−5.29180e7			−2.62656
$836$		0			0
$837$	−5.06236e6			−0.249770
$838$		0			0
$839$	1.04200e7	+	1.80480e7i	0.511050	+	0.885164i	0.999918	+	0.0128063i	$0.00407650\pi$
$839$	1.04200e7	+	1.80480e7i	0.511050	+	0.885164i	−0.488868	+	0.872358i	$0.662590\pi$
$840$		0			0
$841$	445904.	+	772328.i	0.0217396	+	0.0376541i
$842$		0			0
$843$	−3.83703e7			−1.85963
$844$		0			0
$845$	−882284.	+	1.52816e6i	−0.0425076	+	0.0736253i
$846$		0			0
$847$	−1.40231e7			−0.671640
$848$		0			0
$849$	1.28688e7	−	2.22894e7i	0.612729	−	1.06128i
$850$		0			0
$851$	1.62572e7	−	2.81583e7i	0.769524	−	1.33286i
$852$		0			0
$853$	1.06809e7	+	1.84999e7i	0.502617	+	0.870558i	0.999995	+	0.00302417i	$0.000962625\pi$
$853$	1.06809e7	+	1.84999e7i	0.502617	+	0.870558i	−0.497379	+	0.867534i	$0.665704\pi$
$854$		0			0
$855$	1.58468e7	−	2.65399e7i	0.741355	−	1.24161i
$856$		0			0
$857$	−1.36791e7	−	2.36929e7i	−0.636216	−	1.10196i	−0.986256	−	0.165224i	$-0.947165\pi$
$857$	−1.36791e7	−	2.36929e7i	−0.636216	−	1.10196i	0.350040	−	0.936735i	$-0.386168\pi$
$858$		0			0
$859$	3.27009e6	−	5.66397e6i	0.151209	−	0.261901i	−0.780463	−	0.625202i	$-0.785017\pi$
$859$	3.27009e6	−	5.66397e6i	0.151209	−	0.261901i	0.931672	+	0.363300i	$0.118350\pi$
$860$		0			0
$861$	−4.00921e6	+	6.94416e6i	−0.184311	+	0.319236i
$862$		0			0
$863$	−3.18151e7			−1.45414			−0.727069	−	0.686564i	$-0.759118\pi$
$863$	−3.18151e7			−1.45414			−0.727069	+	0.686564i	$0.759118\pi$
$864$		0			0
$865$	2.31142e7	−	4.00349e7i	1.05036	−	1.81928i
$866$		0			0
$867$	1.19966e7			0.542013
$868$		0			0
$869$	−795463.	−	1.37778e6i	−0.0357331	−	0.0618915i
$870$		0			0
$871$	−1.09958e7	−	1.90452e7i	−0.491111	−	0.850629i
$872$		0			0
$873$	−2.40127e7			−1.06636
$874$		0			0
$875$	−3.85601e7			−1.70262
$876$		0			0
$877$	1.41774e7	+	2.45560e7i	0.622442	+	1.07810i	0.989030	+	0.147717i	$0.0471925\pi$
$877$	1.41774e7	+	2.45560e7i	0.622442	+	1.07810i	−0.366588	+	0.930383i	$0.619474\pi$
$878$		0			0
$879$	−9.65203e6	−	1.67178e7i	−0.421353	−	0.729806i
$880$		0			0
$881$	4.86162e6			0.211029			0.105514	−	0.994418i	$-0.466351\pi$
$881$	4.86162e6			0.211029			0.105514	+	0.994418i	$0.466351\pi$
$882$		0			0
$883$	−412959.	+	715267.i	−0.0178240	+	0.0308721i	−0.874800	−	0.484485i	$-0.839007\pi$
$883$	−412959.	+	715267.i	−0.0178240	+	0.0308721i	0.856976	+	0.515357i	$0.172341\pi$
$884$		0			0
$885$	8.77011e7			3.76398
$886$		0			0
$887$	6.55015e6	−	1.13452e7i	0.279539	−	0.484176i	−0.691731	−	0.722155i	$-0.743152\pi$
$887$	6.55015e6	−	1.13452e7i	0.279539	−	0.484176i	0.971270	+	0.237979i	$0.0764850\pi$
$888$		0			0
$889$	6.27335e6	−	1.08658e7i	0.266223	−	0.461111i
$890$		0			0
$891$	−4.09512e6	−	7.09295e6i	−0.172811	−	0.299318i
$892$		0			0
$893$	6.62188e6	+	1.18685e7i	0.277877	+	0.498042i
$894$		0			0
$895$	1.92419e7	+	3.33280e7i	0.802954	+	1.39076i
$896$		0			0
$897$	2.64238e7	−	4.57673e7i	1.09651	−	1.89922i
$898$		0			0
$899$	1.09744e7	−	1.90083e7i	0.452880	−	0.784410i
$900$		0			0
$901$	−3.40960e7			−1.39924
$902$		0			0
$903$	1.04453e7	−	1.80918e7i	0.426286	−	0.738348i
$904$		0			0
$905$	−3.48154e7			−1.41302
$906$		0			0
$907$	4.54799e6	+	7.87734e6i	0.183570	+	0.317952i	0.943094	−	0.332527i	$-0.107901\pi$
$907$	4.54799e6	+	7.87734e6i	0.183570	+	0.317952i	−0.759524	+	0.650479i	$0.774568\pi$
$908$		0			0
$909$	−4.83244e6	−	8.37003e6i	−0.193980	−	0.335983i
$910$		0			0
$911$	4.66286e6			0.186147			0.0930737	−	0.995659i	$-0.470331\pi$
$911$	4.66286e6			0.186147			0.0930737	+	0.995659i	$0.470331\pi$
$912$		0			0
$913$	2.19657e6			0.0872105
$914$		0			0
$915$	−4.74760e6	−	8.22308e6i	−0.187466	−	0.324700i
$916$		0			0
$917$	−9.59392e6	−	1.66172e7i	−0.376767	−	0.652580i
$918$		0			0
$919$	−3.34522e7			−1.30658			−0.653291	−	0.757107i	$-0.726612\pi$
$919$	−3.34522e7			−1.30658			−0.653291	+	0.757107i	$0.726612\pi$
$920$		0			0
$921$	−1.76272e7	+	3.05312e7i	−0.684754	+	1.18603i
$922$		0			0
$923$	2.78489e7			1.07598
$924$		0			0
$925$	−2.72161e7	+	4.71397e7i	−1.04586	+	1.81148i
$926$		0			0
$927$	9.94443e6	−	1.72243e7i	0.380085	−	0.658326i
$928$		0			0
$929$	−4.29234e6	−	7.43456e6i	−0.163176	−	0.282628i	0.772830	−	0.634613i	$-0.218840\pi$
$929$	−4.29234e6	−	7.43456e6i	−0.163176	−	0.282628i	−0.936006	+	0.351984i	$0.885507\pi$
$930$		0			0
$931$	6.19397e6	−	1.03735e7i	0.234204	−	0.392240i
$932$		0			0
$933$	−1.10895e7	−	1.92075e7i	−0.417068	−	0.722383i
$934$		0			0
$935$	−5.56130e6	+	9.63245e6i	−0.208040	+	0.360336i
$936$		0			0
$937$	−2.35809e7	+	4.08434e7i	−0.877429	+	1.51975i	−0.0232757	+	0.999729i	$0.507410\pi$
$937$	−2.35809e7	+	4.08434e7i	−0.877429	+	1.51975i	−0.854153	+	0.520022i	$0.825924\pi$
$938$		0			0
$939$	338298.			0.0125209
$940$		0			0
$941$	1.05661e7	−	1.83010e7i	0.388992	−	0.673754i	−0.603322	−	0.797498i	$-0.706157\pi$
$941$	1.05661e7	−	1.83010e7i	0.388992	−	0.673754i	0.992314	+	0.123744i	$0.0394900\pi$
$942$		0			0
$943$	1.70559e7			0.624592
$944$		0			0
$945$	4.93834e6	+	8.55345e6i	0.179888	+	0.311574i
$946$		0			0
$947$	−1.04875e7	−	1.81649e7i	−0.380012	−	0.658199i	0.611052	−	0.791591i	$-0.290747\pi$
$947$	−1.04875e7	−	1.81649e7i	−0.380012	−	0.658199i	−0.991063	+	0.133391i	$0.957413\pi$
$948$		0			0
$949$	4.83816e7			1.74387
$950$		0			0
$951$	5.33447e7			1.91267
$952$		0			0
$953$	−2.41521e7	−	4.18327e7i	−0.861435	−	1.49205i	−0.870544	−	0.492091i	$-0.836233\pi$
$953$	−2.41521e7	−	4.18327e7i	−0.861435	−	1.49205i	0.00910824	−	0.999959i	$-0.497101\pi$
$954$		0			0
$955$	−3.53723e7	−	6.12665e7i	−1.25503	−	2.17378i
$956$		0			0
$957$	1.10671e7			0.390621
$958$		0			0
$959$	1.73433e7	−	3.00396e7i	0.608957	−	1.05474i
$960$		0			0
$961$	−4.07418e6			−0.142309
$962$		0			0
$963$	−4.39827e6	+	7.61803e6i	−0.152833	+	0.264714i
$964$		0			0
$965$	−2.79492e6	+	4.84094e6i	−0.0966164	+	0.167345i
$966$		0			0
$967$	2.71370e7	+	4.70027e7i	0.933247	+	1.61643i	0.777731	+	0.628597i	$0.216371\pi$
$967$	2.71370e7	+	4.70027e7i	0.933247	+	1.61643i	0.155516	+	0.987833i	$0.450296\pi$
$968$		0			0
$969$	1.47446e7	+	2.64269e7i	0.504456	+	0.904143i
$970$		0			0
$971$	1.62480e7	+	2.81423e7i	0.553033	+	0.957882i	0.998054	+	0.0623611i	$0.0198631\pi$
$971$	1.62480e7	+	2.81423e7i	0.553033	+	0.957882i	−0.445021	+	0.895520i	$0.646804\pi$
$972$		0			0
$973$	1.81669e7	−	3.14660e7i	0.615175	−	1.06552i
$974$		0			0
$975$	−4.42360e7	+	7.66189e7i	−1.49027	+	2.58122i
$976$		0			0
$977$	−6.98320e6			−0.234055			−0.117028	−	0.993129i	$-0.537337\pi$
$977$	−6.98320e6			−0.234055			−0.117028	+	0.993129i	$0.537337\pi$
$978$		0			0
$979$	873321.	−	1.51264e6i	0.0291217	−	0.0504403i
$980$		0			0
$981$	−2.92216e7			−0.969462
$982$		0			0
$983$	−2.88200e6	−	4.99177e6i	−0.0951285	−	0.164767i	0.814534	−	0.580116i	$-0.196993\pi$
$983$	−2.88200e6	−	4.99177e6i	−0.0951285	−	0.164767i	−0.909662	+	0.415349i	$0.863660\pi$
$984$		0			0
$985$	−5.06035e7	−	8.76478e7i	−1.66184	−	2.87839i
$986$		0			0
$987$	1.72536e7			0.563749
$988$		0			0
$989$	−4.44361e7			−1.44459
$990$		0			0
$991$	3.74277e6	+	6.48266e6i	0.121062	+	0.209686i	0.920187	−	0.391480i	$-0.128037\pi$
$991$	3.74277e6	+	6.48266e6i	0.121062	+	0.209686i	−0.799125	+	0.601165i	$0.794703\pi$
$992$		0			0
$993$	1.20042e7	+	2.07919e7i	0.386331	+	0.669146i
$994$		0			0
$995$	4.99200e7			1.59852
$996$		0			0
$997$	−1.00119e7	+	1.73412e7i	−0.318992	+	0.552510i	−0.980278	−	0.197624i	$-0.936678\pi$
$997$	−1.00119e7	+	1.73412e7i	−0.318992	+	0.552510i	0.661286	+	0.750134i	$0.270011\pi$
$998$		0			0
$999$	7.81729e6			0.247823

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	76.6.e.a.45.8	✓	18
3.2	odd	2		684.6.k.f.577.1		18
4.3	odd	2		304.6.i.d.273.2		18
19.11	even	3	inner	76.6.e.a.49.8	yes	18
57.11	odd	6		684.6.k.f.505.1		18
76.11	odd	6		304.6.i.d.49.2		18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
76.6.e.a.45.8	✓	18	1.1	even	1	trivial
76.6.e.a.49.8	yes	18	19.11	even	3	inner
304.6.i.d.49.2		18	76.11	odd	6
304.6.i.d.273.2		18	4.3	odd	2
684.6.k.f.505.1		18	57.11	odd	6
684.6.k.f.577.1		18	3.2	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:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	76.e (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(10.4539 + 18.1067i) q^{3} +(50.5928 + 87.6293i) q^{5} +95.5451 q^{7} +(-97.0688 + 168.128i) q^{9} +O(q^{10})\)	\(q+(10.4539 + 18.1067i) q^{3} +(50.5928 + 87.6293i) q^{5} +95.5451 q^{7} +(-97.0688 + 168.128i) q^{9} -119.504 q^{11} +(297.428 - 515.161i) q^{13} +(-1057.79 + 1832.14i) q^{15} +(-459.911 - 796.589i) q^{17} +(-806.703 + 1351.05i) q^{19} +(998.821 + 1730.01i) q^{21} +(2124.58 - 3679.89i) q^{23} +(-3556.76 + 6160.48i) q^{25} +1021.61 q^{27} +(-2214.69 + 3835.95i) q^{29} -4955.30 q^{31} +(-1249.29 - 2163.83i) q^{33} +(4833.89 + 8372.55i) q^{35} +7651.95 q^{37} +12437.2 q^{39} +(2006.97 + 3476.18i) q^{41} +(-5228.80 - 9056.55i) q^{43} -19643.9 q^{45} +(4318.48 - 7479.83i) q^{47} -7678.13 q^{49} +(9615.75 - 16655.0i) q^{51} +(18534.0 - 32101.9i) q^{53} +(-6046.06 - 10472.1i) q^{55} +(-32896.2 - 482.998i) q^{57} +(-20727.5 - 35901.1i) q^{59} +(-2244.12 + 3886.93i) q^{61} +(-9274.45 + 16063.8i) q^{63} +60190.9 q^{65} +(18484.7 - 32016.5i) q^{67} +88840.9 q^{69} +(23408.1 + 40543.9i) q^{71} +(40666.6 + 70436.6i) q^{73} -148728. q^{75} -11418.1 q^{77} +(6656.35 + 11529.1i) q^{79} +(34267.5 + 59353.1i) q^{81} -18380.7 q^{83} +(46536.4 - 80603.3i) q^{85} -92608.6 q^{87} +(-7307.86 + 12657.6i) q^{89} +(28417.8 - 49221.1i) q^{91} +(-51802.3 - 89724.2i) q^{93} +(-159205. - 2337.52i) q^{95} +(61844.5 + 107118. i) q^{97} +(11600.1 - 20092.0i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q - 11 q^{3} + 11 q^{5} + 336 q^{7} - 902 q^{9}+O(q^{10}) \) 18 * q - 11 * q^3 + 11 * q^5 + 336 * q^7 - 902 * q^9	\( 18 q - 11 q^{3} + 11 q^{5} + 336 q^{7} - 902 q^{9} - 320 q^{11} + 227 q^{13} - 101 q^{15} + 179 q^{17} - 868 q^{19} - 5700 q^{21} - 3425 q^{23} - 7054 q^{25} + 14722 q^{27} - 7349 q^{29} - 9960 q^{31} - 2998 q^{33} + 15888 q^{35} + 26444 q^{37} - 30246 q^{39} - 7311 q^{41} - 8283 q^{43} - 62164 q^{45} + 37603 q^{47} + 124738 q^{49} + 47227 q^{51} - 20337 q^{53} + 716 q^{55} - 57555 q^{57} - 74455 q^{59} - 7569 q^{61} - 52544 q^{63} + 188998 q^{65} - 26177 q^{67} + 116282 q^{69} - 53463 q^{71} - 14103 q^{73} + 120912 q^{75} - 31960 q^{77} + 31825 q^{79} - 21137 q^{81} + 82600 q^{83} - 50787 q^{85} - 339766 q^{87} - 155197 q^{89} - 2800 q^{91} - 46460 q^{93} + 49315 q^{95} + 111241 q^{97} - 193544 q^{99}+O(q^{100}) \) 18 * q - 11 * q^3 + 11 * q^5 + 336 * q^7 - 902 * q^9 - 320 * q^11 + 227 * q^13 - 101 * q^15 + 179 * q^17 - 868 * q^19 - 5700 * q^21 - 3425 * q^23 - 7054 * q^25 + 14722 * q^27 - 7349 * q^29 - 9960 * q^31 - 2998 * q^33 + 15888 * q^35 + 26444 * q^37 - 30246 * q^39 - 7311 * q^41 - 8283 * q^43 - 62164 * q^45 + 37603 * q^47 + 124738 * q^49 + 47227 * q^51 - 20337 * q^53 + 716 * q^55 - 57555 * q^57 - 74455 * q^59 - 7569 * q^61 - 52544 * q^63 + 188998 * q^65 - 26177 * q^67 + 116282 * q^69 - 53463 * q^71 - 14103 * q^73 + 120912 * q^75 - 31960 * q^77 + 31825 * q^79 - 21137 * q^81 + 82600 * q^83 - 50787 * q^85 - 339766 * q^87 - 155197 * q^89 - 2800 * q^91 - 46460 * q^93 + 49315 * q^95 + 111241 * q^97 - 193544 * q^99

Introduction

L-functions

Modular forms

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Database

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