LMFDB - Embedded newform 15.5.c.a.11.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [15,5,Mod(11,15)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(15, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 5, names="a")

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

chi := DirichletCharacter("15.11");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 15 = 3 \cdot 5 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	15.c (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:	$1.55054944626$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 73x^{4} + 1096x^{2} + 180 $ x^6 + 73x^4 + 1096x^2 + 180
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 3\cdot 5^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			11.3
Root			$-0.407512i$ of defining polynomial
Character	$\chi$	$=$	15.11
Dual form			15.5.c.a.11.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.407512i q^{2} +(3.21297 + 8.40695i) q^{3} +15.8339 q^{4} -11.1803i q^{5} +(3.42594 - 1.30932i) q^{6} -46.9457 q^{7} -12.9727i q^{8} +(-60.3537 + 54.0225i) q^{9} +O(q^{10})$	\(q-0.407512i q^{2} +(3.21297 + 8.40695i) q^{3} +15.8339 q^{4} -11.1803i q^{5} +(3.42594 - 1.30932i) q^{6} -46.9457 q^{7} -12.9727i q^{8} +(-60.3537 + 54.0225i) q^{9} -4.55613 q^{10} -200.952i q^{11} +(50.8739 + 133.115i) q^{12} -22.3321 q^{13} +19.1309i q^{14} +(93.9926 - 35.9221i) q^{15} +248.056 q^{16} +344.602i q^{17} +(22.0148 + 24.5949i) q^{18} -59.9990 q^{19} -177.029i q^{20} +(-150.835 - 394.670i) q^{21} -81.8903 q^{22} +212.905i q^{23} +(109.061 - 41.6809i) q^{24} -125.000 q^{25} +9.10062i q^{26} +(-648.079 - 333.818i) q^{27} -743.335 q^{28} +578.043i q^{29} +(-14.6387 - 38.3031i) q^{30} +490.110 q^{31} -308.650i q^{32} +(1689.39 - 645.652i) q^{33} +140.429 q^{34} +524.869i q^{35} +(-955.636 + 855.389i) q^{36} +1936.81 q^{37} +24.4503i q^{38} +(-71.7524 - 187.745i) q^{39} -145.039 q^{40} +1638.25i q^{41} +(-160.833 + 61.4671i) q^{42} -2160.31 q^{43} -3181.86i q^{44} +(603.990 + 674.775i) q^{45} +86.7615 q^{46} -2282.28i q^{47} +(796.997 + 2085.40i) q^{48} -197.104 q^{49} +50.9390i q^{50} +(-2897.05 + 1107.19i) q^{51} -353.606 q^{52} -2626.84i q^{53} +(-136.035 + 264.100i) q^{54} -2246.71 q^{55} +609.013i q^{56} +(-192.775 - 504.409i) q^{57} +235.560 q^{58} +4106.06i q^{59} +(1488.27 - 568.788i) q^{60} +4790.63 q^{61} -199.726i q^{62} +(2833.34 - 2536.12i) q^{63} +3843.12 q^{64} +249.681i q^{65} +(-263.111 - 688.448i) q^{66} -1565.70 q^{67} +5456.40i q^{68} +(-1789.88 + 684.058i) q^{69} +213.890 q^{70} -5372.27i q^{71} +(700.819 + 782.951i) q^{72} -4321.67 q^{73} -789.274i q^{74} +(-401.621 - 1050.87i) q^{75} -950.020 q^{76} +9433.82i q^{77} +(-76.5085 + 29.2400i) q^{78} -4801.14 q^{79} -2773.35i q^{80} +(724.132 - 6520.92i) q^{81} +667.606 q^{82} -3384.94i q^{83} +(-2388.31 - 6249.18i) q^{84} +3852.77 q^{85} +880.352i q^{86} +(-4859.58 + 1857.23i) q^{87} -2606.89 q^{88} +3446.38i q^{89} +(274.979 - 246.133i) q^{90} +1048.40 q^{91} +3371.13i q^{92} +(1574.71 + 4120.33i) q^{93} -930.059 q^{94} +670.809i q^{95} +(2594.80 - 991.681i) q^{96} +5447.68 q^{97} +80.3221i q^{98} +(10855.9 + 12128.2i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 8 q^{3} - 50 q^{4} - 2 q^{6} + 76 q^{7} + 118 q^{9}+O(q^{10}) $ 6 * q + 8 * q^3 - 50 * q^4 - 2 * q^6 + 76 * q^7 + 118 * q^9	$ 6 q + 8 q^{3} - 50 q^{4} - 2 q^{6} + 76 q^{7} + 118 q^{9} + 50 q^{10} - 452 q^{12} - 424 q^{13} + 50 q^{15} + 802 q^{16} + 1160 q^{18} - 244 q^{19} - 876 q^{21} + 340 q^{22} - 786 q^{24} - 750 q^{25} - 352 q^{27} - 3764 q^{28} + 2200 q^{30} + 3772 q^{31} + 4420 q^{33} + 3124 q^{34} - 7606 q^{36} + 1896 q^{37} - 1336 q^{39} - 4650 q^{40} - 1980 q^{42} - 7384 q^{43} + 1900 q^{45} + 8196 q^{46} + 14668 q^{48} - 1318 q^{49} - 8492 q^{51} + 8976 q^{52} - 278 q^{54} - 1300 q^{55} - 11584 q^{57} - 23740 q^{58} + 5050 q^{60} + 6452 q^{61} + 14796 q^{63} + 3174 q^{64} - 12760 q^{66} + 13816 q^{67} + 5472 q^{69} - 2100 q^{70} - 2040 q^{72} + 596 q^{73} - 1000 q^{75} + 21348 q^{76} - 1400 q^{78} - 16124 q^{79} + 5086 q^{81} - 31240 q^{82} - 14736 q^{84} - 3100 q^{85} - 4900 q^{87} + 15660 q^{88} + 7550 q^{90} - 11632 q^{91} - 8184 q^{93} + 34924 q^{94} + 14354 q^{96} + 9756 q^{97} + 9680 q^{99}+O(q^{100}) $ 6 * q + 8 * q^3 - 50 * q^4 - 2 * q^6 + 76 * q^7 + 118 * q^9 + 50 * q^10 - 452 * q^12 - 424 * q^13 + 50 * q^15 + 802 * q^16 + 1160 * q^18 - 244 * q^19 - 876 * q^21 + 340 * q^22 - 786 * q^24 - 750 * q^25 - 352 * q^27 - 3764 * q^28 + 2200 * q^30 + 3772 * q^31 + 4420 * q^33 + 3124 * q^34 - 7606 * q^36 + 1896 * q^37 - 1336 * q^39 - 4650 * q^40 - 1980 * q^42 - 7384 * q^43 + 1900 * q^45 + 8196 * q^46 + 14668 * q^48 - 1318 * q^49 - 8492 * q^51 + 8976 * q^52 - 278 * q^54 - 1300 * q^55 - 11584 * q^57 - 23740 * q^58 + 5050 * q^60 + 6452 * q^61 + 14796 * q^63 + 3174 * q^64 - 12760 * q^66 + 13816 * q^67 + 5472 * q^69 - 2100 * q^70 - 2040 * q^72 + 596 * q^73 - 1000 * q^75 + 21348 * q^76 - 1400 * q^78 - 16124 * q^79 + 5086 * q^81 - 31240 * q^82 - 14736 * q^84 - 3100 * q^85 - 4900 * q^87 + 15660 * q^88 + 7550 * q^90 - 11632 * q^91 - 8184 * q^93 + 34924 * q^94 + 14354 * q^96 + 9756 * q^97 + 9680 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/15\mathbb{Z}\right)^\times$.

$n$	$7$	$11$
$\chi(n)$	$1$	$-1$

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 $

$2$		−	0.407512i		−	0.101878i	−0.998702	−	0.0509390i	$-0.983779\pi$
$2$		−	0.407512i		−	0.101878i	0.998702	−	0.0509390i	$-0.0162214\pi$
$3$	3.21297	+	8.40695i	0.356996	+	0.934106i
$3$	3.21297	+	8.40695i	0.356996	+	0.934106i
$4$	15.8339			0.989621
$5$		−	11.1803i		−	0.447214i
$5$		−	11.1803i		−	0.447214i
$6$	3.42594	−	1.30932i	0.0951649	−	0.0363701i
$7$	−46.9457			−0.958075			−0.479038	−	0.877794i	$-0.659014\pi$
$7$	−46.9457			−0.958075			−0.479038	+	0.877794i	$0.659014\pi$
$8$		−	12.9727i		−	0.202699i
$9$	−60.3537	+	54.0225i	−0.745107	+	0.666945i
$10$	−4.55613			−0.0455613
$11$		−	200.952i		−	1.66076i	−0.557198	−	0.830379i	$-0.688124\pi$
$11$		−	200.952i		−	1.66076i	0.557198	−	0.830379i	$-0.311876\pi$
$12$	50.8739	+	133.115i	0.353291	+	0.924411i
$13$	−22.3321			−0.132143			−0.0660714	−	0.997815i	$-0.521047\pi$
$13$	−22.3321			−0.132143			−0.0660714	+	0.997815i	$0.521047\pi$
$14$			19.1309i			0.0976068i
$15$	93.9926	−	35.9221i	0.417745	−	0.159654i
$16$	248.056			0.968970
$17$			344.602i			1.19239i	0.802838	+	0.596197i	$0.203322\pi$
$17$			344.602i			1.19239i	−0.802838	+	0.596197i	$0.796678\pi$
$18$	22.0148	+	24.5949i	0.0679471	+	0.0759101i
$19$	−59.9990			−0.166202			−0.0831011	−	0.996541i	$-0.526482\pi$
$19$	−59.9990			−0.166202			−0.0831011	+	0.996541i	$0.526482\pi$
$20$		−	177.029i		−	0.442572i
$21$	−150.835	−	394.670i	−0.342029	−	0.894943i
$22$	−81.8903			−0.169195
$23$			212.905i			0.402467i	0.979543	+	0.201234i	$0.0644951\pi$
$23$			212.905i			0.402467i	−0.979543	+	0.201234i	$0.935505\pi$
$24$	109.061	−	41.6809i	0.189342	−	0.0723627i
$25$	−125.000			−0.200000
$26$			9.10062i			0.0134625i
$27$	−648.079	−	333.818i	−0.888998	−	0.457912i
$28$	−743.335			−0.948131
$29$			578.043i			0.687328i	0.939093	+	0.343664i	$0.111668\pi$
$29$			578.043i			0.687328i	−0.939093	+	0.343664i	$0.888332\pi$
$30$	−14.6387	−	38.3031i	−0.0162652	−	0.0425590i
$31$	490.110			0.510000			0.255000	−	0.966941i	$-0.417925\pi$
$31$	490.110			0.510000			0.255000	+	0.966941i	$0.417925\pi$
$32$		−	308.650i		−	0.301416i
$33$	1689.39	−	645.652i	1.55132	−	0.592885i
$34$	140.429			0.121479
$35$			524.869i			0.428464i
$36$	−955.636	+	855.389i	−0.737374	+	0.660023i
$37$	1936.81			1.41476			0.707382	−	0.706832i	$-0.249876\pi$
$37$	1936.81			1.41476			0.707382	+	0.706832i	$0.249876\pi$
$38$			24.4503i			0.0169324i
$39$	−71.7524	−	187.745i	−0.0471745	−	0.123435i
$40$	−145.039			−0.0906496
$41$			1638.25i			0.974567i	0.873244	+	0.487284i	$0.162012\pi$
$41$			1638.25i			0.974567i	−0.873244	+	0.487284i	$0.837988\pi$
$42$	−160.833	+	61.4671i	−0.0911751	+	0.0348453i
$43$	−2160.31			−1.16837			−0.584183	−	0.811622i	$-0.698585\pi$
$43$	−2160.31			−1.16837			−0.584183	+	0.811622i	$0.698585\pi$
$44$		−	3181.86i		−	1.64352i
$45$	603.990	+	674.775i	0.298267	+	0.333222i
$46$	86.7615			0.0410026
$47$		−	2282.28i		−	1.03318i	−0.856234	−	0.516588i	$-0.827202\pi$
$47$		−	2282.28i		−	1.03318i	0.856234	−	0.516588i	$-0.172798\pi$
$48$	796.997	+	2085.40i	0.345919	+	0.905121i
$49$	−197.104			−0.0820923
$50$			50.9390i			0.0203756i
$51$	−2897.05	+	1107.19i	−1.11382	+	0.425680i
$52$	−353.606			−0.130771
$53$		−	2626.84i		−	0.935151i	−0.883953	−	0.467575i	$-0.845128\pi$
$53$		−	2626.84i		−	0.935151i	0.883953	−	0.467575i	$-0.154872\pi$
$54$	−136.035	+	264.100i	−0.0466512	+	0.0905694i
$55$	−2246.71			−0.742714
$56$			609.013i			0.194201i
$57$	−192.775	−	504.409i	−0.0593336	−	0.155250i
$58$	235.560			0.0700237
$59$			4106.06i			1.17956i	0.807563	+	0.589782i	$0.200786\pi$
$59$			4106.06i			1.17956i	−0.807563	+	0.589782i	$0.799214\pi$
$60$	1488.27	−	568.788i	0.413409	−	0.157997i
$61$	4790.63			1.28746			0.643729	−	0.765254i	$-0.277386\pi$
$61$	4790.63			1.28746			0.643729	+	0.765254i	$0.277386\pi$
$62$		−	199.726i		−	0.0519578i
$63$	2833.34	−	2536.12i	0.713868	−	0.638983i
$64$	3843.12			0.938263
$65$			249.681i			0.0590961i
$66$	−263.111	−	688.448i	−0.0604020	−	0.158046i
$67$	−1565.70			−0.348785			−0.174393	−	0.984676i	$-0.555796\pi$
$67$	−1565.70			−0.348785			−0.174393	+	0.984676i	$0.555796\pi$
$68$			5456.40i			1.18002i
$69$	−1789.88	+	684.058i	−0.375947	+	0.143679i
$70$	213.890			0.0436511
$71$		−	5372.27i		−	1.06571i	−0.846205	−	0.532857i	$-0.821118\pi$
$71$		−	5372.27i		−	1.06571i	0.846205	−	0.532857i	$-0.178882\pi$
$72$	700.819	+	782.951i	0.135189	+	0.151032i
$73$	−4321.67			−0.810972			−0.405486	−	0.914101i	$-0.632898\pi$
$73$	−4321.67			−0.810972			−0.405486	+	0.914101i	$0.632898\pi$
$74$		−	789.274i		−	0.144133i
$75$	−401.621	−	1050.87i	−0.0713993	−	0.186821i
$76$	−950.020			−0.164477
$77$			9433.82i			1.59113i
$78$	−76.5085	+	29.2400i	−0.0125754	+	0.00480605i
$79$	−4801.14			−0.769290			−0.384645	−	0.923065i	$-0.625676\pi$
$79$	−4801.14			−0.769290			−0.384645	+	0.923065i	$0.625676\pi$
$80$		−	2773.35i		−	0.433337i
$81$	724.132	−	6520.92i	0.110369	−	0.993891i
$82$	667.606			0.0992870
$83$		−	3384.94i		−	0.491354i	−0.969352	−	0.245677i	$-0.920990\pi$
$83$		−	3384.94i		−	0.491354i	0.969352	−	0.245677i	$-0.0790103\pi$
$84$	−2388.31	−	6249.18i	−0.338479	−	0.885655i
$85$	3852.77			0.533255
$86$			880.352i			0.119031i
$87$	−4859.58	+	1857.23i	−0.642037	+	0.245374i
$88$	−2606.89			−0.336634
$89$			3446.38i			0.435094i	0.976050	+	0.217547i	$0.0698056\pi$
$89$			3446.38i			0.435094i	−0.976050	+	0.217547i	$0.930194\pi$
$90$	274.979	−	246.133i	0.0339480	−	0.0303868i
$91$	1048.40			0.126603
$92$			3371.13i			0.398290i
$93$	1574.71	+	4120.33i	0.182068	+	0.476394i
$94$	−930.059			−0.105258
$95$			670.809i			0.0743279i
$96$	2594.80	−	991.681i	0.281554	−	0.107604i
$97$	5447.68			0.578986			0.289493	−	0.957180i	$-0.406513\pi$
$97$	5447.68			0.578986			0.289493	+	0.957180i	$0.406513\pi$
$98$			80.3221i			0.00836340i
$99$	10855.9	+	12128.2i	1.10763	+	1.23744i
$100$	−1979.24			−0.197924
$101$		−	7655.22i		−	0.750438i	−0.926936	−	0.375219i	$-0.877567\pi$
$101$		−	7655.22i		−	0.750438i	0.926936	−	0.375219i	$-0.122433\pi$
$102$	451.195	+	1180.58i	0.0433675	+	0.113474i
$103$	16163.7			1.52358			0.761790	−	0.647824i	$-0.224321\pi$
$103$	16163.7			1.52358			0.761790	+	0.647824i	$0.224321\pi$
$104$			289.708i			0.0267852i
$105$	−4412.55	+	1686.39i	−0.400231	+	0.152960i
$106$	−1070.47			−0.0952713
$107$		−	18944.3i		−	1.65466i	−0.561713	−	0.827332i	$-0.689857\pi$
$107$		−	18944.3i		−	1.65466i	0.561713	−	0.827332i	$-0.310143\pi$
$108$	−10261.6	−	5285.65i	−0.879771	−	0.453159i
$109$	−4373.48			−0.368107			−0.184053	−	0.982916i	$-0.558922\pi$
$109$	−4373.48			−0.368107			−0.184053	+	0.982916i	$0.558922\pi$
$110$			915.562i			0.0756663i
$111$	6222.91	+	16282.7i	0.505065	+	1.32154i
$112$	−11645.2			−0.928346
$113$			1242.99i			0.0973443i	0.998815	+	0.0486722i	$0.0154990\pi$
$113$			1242.99i			0.0973443i	−0.998815	+	0.0486722i	$0.984501\pi$
$114$	−205.553	+	78.5581i	−0.0158166	+	0.00604479i
$115$	2380.35			0.179989
$116$			9152.69i			0.680194i
$117$	1347.83	−	1206.44i	0.0984605	−	0.0881320i
$118$	1673.27			0.120172
$119$		−	16177.6i		−	1.14240i
$120$	−466.007	−	1219.34i	−0.0323616	−	0.0846763i
$121$	−25740.6			−1.75812
$122$		−	1952.24i		−	0.131164i
$123$	−13772.7	+	5263.64i	−0.910349	+	0.347917i
$124$	7760.36			0.504706
$125$			1397.54i			0.0894427i
$126$	−1033.50	−	1154.62i	−0.0650984	−	0.0727275i
$127$	4143.92			0.256924			0.128462	−	0.991714i	$-0.458996\pi$
$127$	4143.92			0.256924			0.128462	+	0.991714i	$0.458996\pi$
$128$		−	6504.51i		−	0.397004i
$129$	−6941.00	−	18161.6i	−0.417102	−	1.09138i
$130$	101.748			0.00602059
$131$			23414.5i			1.36440i	0.731166	+	0.682200i	$0.238977\pi$
$131$			23414.5i			1.36440i	−0.731166	+	0.682200i	$0.761023\pi$
$132$	26749.7	−	10223.2i	1.53522	−	0.586731i
$133$	2816.69			0.159234
$134$			638.041i			0.0355336i
$135$	−3732.20	+	7245.75i	−0.204784	+	0.397572i
$136$	4470.42			0.241697
$137$			4168.41i			0.222090i	0.993815	+	0.111045i	$0.0354198\pi$
$137$			4168.41i			0.222090i	−0.993815	+	0.111045i	$0.964580\pi$
$138$	278.762	+	729.400i	0.0146378	+	0.0383008i
$139$	6702.86			0.346921			0.173460	−	0.984841i	$-0.444505\pi$
$139$	6702.86			0.346921			0.173460	+	0.984841i	$0.444505\pi$
$140$			8310.73i			0.424017i
$141$	19187.1	−	7332.91i	0.965095	−	0.368840i
$142$	−2189.26			−0.108573
$143$			4487.68i			0.219457i
$144$	−14971.1	+	13400.6i	−0.721987	+	0.646250i
$145$	6462.72			0.307382
$146$			1761.13i			0.0826203i
$147$	−633.287	−	1657.04i	−0.0293067	−	0.0766829i
$148$	30667.3			1.40008
$149$			7267.97i			0.327371i	0.986513	+	0.163686i	$0.0523383\pi$
$149$			7267.97i			0.327371i	−0.986513	+	0.163686i	$0.947662\pi$
$150$	−428.242	+	163.665i	−0.0190330	+	0.00727402i
$151$	−17022.9			−0.746584			−0.373292	−	0.927714i	$-0.621771\pi$
$151$	−17022.9			−0.746584			−0.373292	+	0.927714i	$0.621771\pi$
$152$			778.350i			0.0336890i
$153$	−18616.3	−	20798.0i	−0.795261	−	0.888461i
$154$	3844.40			0.162101
$155$		−	5479.59i		−	0.228079i
$156$	−1136.12	−	2972.74i	−0.0466849	−	0.122154i
$157$	−11560.8			−0.469017			−0.234509	−	0.972114i	$-0.575348\pi$
$157$	−11560.8			−0.469017			−0.234509	+	0.972114i	$0.575348\pi$
$158$			1956.52i			0.0783738i
$159$	22083.7	−	8439.95i	0.873530	−	0.333845i
$160$	−3450.81			−0.134797
$161$		−	9994.98i		−	0.385594i
$162$	−2657.35	−	295.093i	−0.101256	−	0.0112442i
$163$	−6066.12			−0.228316			−0.114158	−	0.993463i	$-0.536417\pi$
$163$	−6066.12			−0.228316			−0.114158	+	0.993463i	$0.536417\pi$
$164$			25939.9i			0.964452i
$165$	−7218.61	−	18888.0i	−0.265146	−	0.693773i
$166$	−1379.40			−0.0500582
$167$			6974.71i			0.250088i	0.992151	+	0.125044i	$0.0399073\pi$
$167$			6974.71i			0.250088i	−0.992151	+	0.125044i	$0.960093\pi$
$168$	−5119.94	+	1956.74i	−0.181404	+	0.0693289i
$169$	−28062.3			−0.982538
$170$		−	1570.05i		−	0.0543270i
$171$	3621.16	−	3241.30i	0.123838	−	0.110848i
$172$	−34206.2			−1.15624
$173$			27868.0i			0.931135i	0.885012	+	0.465568i	$0.154150\pi$
$173$			27868.0i			0.931135i	−0.885012	+	0.465568i	$0.845850\pi$
$174$	756.845	+	1980.34i	0.0249982	+	0.0654095i
$175$	5868.21			0.191615
$176$		−	49847.4i		−	1.60923i
$177$	−34519.4	+	13192.6i	−1.10184	+	0.421100i
$178$	1404.44			0.0443266
$179$		−	22242.5i		−	0.694189i	−0.937830	−	0.347095i	$-0.887168\pi$
$179$		−	22242.5i		−	0.694189i	0.937830	−	0.347095i	$-0.112832\pi$
$180$	9563.54	+	10684.3i	0.295171	+	0.329763i
$181$	33471.7			1.02169			0.510847	−	0.859672i	$-0.329332\pi$
$181$	33471.7			1.02169			0.510847	+	0.859672i	$0.329332\pi$
$182$		−	427.235i		−	0.0128980i
$183$	15392.1	+	40274.6i	0.459618	+	1.20262i
$184$	2761.96			0.0815796
$185$		−	21654.2i		−	0.632701i
$186$	1679.08	−	641.712i	0.0485341	−	0.0185487i
$187$	69248.4			1.98028
$188$		−	36137.5i		−	1.02245i
$189$	30424.5	+	15671.3i	0.851726	+	0.438714i
$190$	273.363			0.00757238
$191$			602.154i			0.0165060i	0.999966	+	0.00825298i	$0.00262704\pi$
$191$			602.154i			0.0165060i	−0.999966	+	0.00825298i	$0.997373\pi$
$192$	12347.8	+	32309.0i	0.334956	+	0.876437i
$193$	−27201.4			−0.730259			−0.365130	−	0.930957i	$-0.618975\pi$
$193$	−27201.4			−0.730259			−0.365130	+	0.930957i	$0.618975\pi$
$194$		−	2220.00i		−	0.0589860i
$195$	−2099.05	+	802.217i	−0.0552020	+	0.0210971i
$196$	−3120.92			−0.0812402
$197$			61640.6i			1.58831i	0.607717	+	0.794154i	$0.292085\pi$
$197$			61640.6i			1.58831i	−0.607717	+	0.794154i	$0.707915\pi$
$198$	4942.38	−	4423.92i	0.126068	−	0.112844i
$199$	−10712.4			−0.270507			−0.135254	−	0.990811i	$-0.543185\pi$
$199$	−10712.4			−0.270507			−0.135254	+	0.990811i	$0.543185\pi$
$200$			1621.59i			0.0405397i
$201$	−5030.53	−	13162.7i	−0.124515	−	0.325802i
$202$	−3119.60			−0.0764532
$203$		−	27136.6i		−	0.658512i
$204$	−45871.7	+	17531.2i	−1.10226	+	0.421262i
$205$	18316.2			0.435840
$206$		−	6586.89i		−	0.155219i
$207$	−11501.7	−	12849.6i	−0.268424	−	0.299881i
$208$	−5539.63			−0.128042
$209$			12056.9i			0.276022i
$210$	687.223	+	1798.17i	0.0155833	+	0.0407747i
$211$	19405.1			0.435865			0.217933	−	0.975964i	$-0.430069\pi$
$211$	19405.1			0.435865			0.217933	+	0.975964i	$0.430069\pi$
$212$		−	41593.2i		−	0.925445i
$213$	45164.4	−	17260.9i	0.995490	−	0.380456i
$214$	−7720.02			−0.168574
$215$			24153.0i			0.522509i
$216$	−4330.52	+	8407.35i	−0.0928182	+	0.180199i
$217$	−23008.5			−0.488618
$218$			1782.25i			0.0375020i
$219$	−13885.4	−	36332.1i	−0.289514	−	0.757534i
$220$	−35574.3			−0.735005
$221$		−	7695.70i		−	0.157566i
$222$	6635.39	−	2535.91i	0.134636	−	0.0514551i
$223$	−60220.2			−1.21097			−0.605484	−	0.795857i	$-0.707021\pi$
$223$	−60220.2			−1.21097			−0.605484	+	0.795857i	$0.707021\pi$
$224$			14489.8i			0.288779i
$225$	7544.21	−	6752.82i	0.149021	−	0.133389i
$226$	506.534			0.00991725
$227$		−	52638.3i		−	1.02153i	−0.859721	−	0.510764i	$-0.829363\pi$
$227$		−	52638.3i		−	1.02153i	0.859721	−	0.510764i	$-0.170637\pi$
$228$	−3052.38	−	7986.77i	−0.0587177	−	0.153639i
$229$	54102.0			1.03167			0.515837	−	0.856687i	$-0.327481\pi$
$229$	54102.0			1.03167			0.515837	+	0.856687i	$0.327481\pi$
$230$		−	970.023i		−	0.0183369i
$231$	−79309.7	+	30310.6i	−1.48629	+	0.568028i
$232$	7498.79			0.139321
$233$			82178.1i			1.51372i	0.653580	+	0.756858i	$0.273266\pi$
$233$			82178.1i			1.51372i	−0.653580	+	0.756858i	$0.726734\pi$
$234$	−491.638	−	549.256i	−0.00897871	−	0.0100310i
$235$	−25516.7			−0.462050
$236$			65015.1i			1.16732i
$237$	−15425.9	−	40362.9i	−0.274634	−	0.718598i
$238$	−6592.56			−0.116386
$239$		−	44047.8i		−	0.771132i	−0.922680	−	0.385566i	$-0.874006\pi$
$239$		−	44047.8i		−	0.771132i	0.922680	−	0.385566i	$-0.125994\pi$
$240$	23315.5	−	8910.70i	0.404782	−	0.154700i
$241$	11775.6			0.202744			0.101372	−	0.994849i	$-0.467677\pi$
$241$	11775.6			0.202744			0.101372	+	0.994849i	$0.467677\pi$
$242$			10489.6i			0.179114i
$243$	57147.6	−	14863.8i	0.967800	−	0.251719i
$244$	75854.5			1.27410
$245$			2203.68i			0.0367128i
$246$	2145.00	+	5612.53i	0.0354451	+	0.0927446i
$247$	1339.91			0.0219624
$248$		−	6358.05i		−	0.103376i
$249$	28457.0	−	10875.7i	0.458977	−	0.175412i
$250$	569.516			0.00911225
$251$			114693.i			1.82050i	0.414061	+	0.910249i	$0.364110\pi$
$251$			114693.i			1.82050i	−0.414061	+	0.910249i	$0.635890\pi$
$252$	44863.0	−	40156.8i	0.706459	−	0.632351i
$253$	42783.7			0.668401
$254$		−	1688.70i		−	0.0261749i
$255$	12378.8	+	32390.0i	0.190370	+	0.498116i
$256$	58839.3			0.897817
$257$		−	44360.2i		−	0.671626i	−0.941929	−	0.335813i	$-0.890989\pi$
$257$		−	44360.2i		−	0.671626i	0.941929	−	0.335813i	$-0.109011\pi$
$258$	−7401.08	+	2828.54i	−0.111187	+	0.0424936i
$259$	−90924.9			−1.35545
$260$			3953.43i			0.0584827i
$261$	−31227.3	−	34887.0i	−0.458410	−	0.512133i
$262$	9541.68			0.139002
$263$		−	61162.6i		−	0.884249i	−0.896954	−	0.442124i	$-0.854225\pi$
$263$		−	61162.6i		−	0.884249i	0.896954	−	0.442124i	$-0.145775\pi$
$264$	−8375.86	−	21916.0i	−0.120177	−	0.314451i
$265$	−29368.9			−0.418212
$266$		−	1147.84i		−	0.0162225i
$267$	−28973.6	+	11073.1i	−0.406424	+	0.155327i
$268$	−24791.1			−0.345165
$269$			54930.0i			0.759110i	0.925169	+	0.379555i	$0.123923\pi$
$269$			54930.0i			0.759110i	−0.925169	+	0.379555i	$0.876077\pi$
$270$	2952.73	+	1520.92i	0.0405038	+	0.0208630i
$271$	−48199.5			−0.656302			−0.328151	−	0.944625i	$-0.606425\pi$
$271$	−48199.5			−0.656302			−0.328151	+	0.944625i	$0.606425\pi$
$272$			85480.7i			1.15539i
$273$	3368.47	+	8813.82i	0.0451967	+	0.118260i
$274$	1698.68			0.0226261
$275$			25119.0i			0.332152i
$276$	−28340.9	+	10831.3i	−0.372045	+	0.142188i
$277$	101493.			1.32275			0.661375	−	0.750056i	$-0.269973\pi$
$277$	101493.			1.32275			0.661375	+	0.750056i	$0.269973\pi$
$278$		−	2731.50i		−	0.0353436i
$279$	−29579.9	+	26477.0i	−0.380004	+	0.340142i
$280$	6808.97			0.0868491
$281$		−	120902.i		−	1.53117i	−0.643337	−	0.765583i	$-0.722451\pi$
$281$		−	120902.i		−	1.53117i	0.643337	−	0.765583i	$-0.277549\pi$
$282$	−2988.25	−	7818.96i	−0.0375767	−	0.0983220i
$283$	−111286.			−1.38953			−0.694767	−	0.719235i	$-0.744493\pi$
$283$	−111286.			−1.38953			−0.694767	+	0.719235i	$0.744493\pi$
$284$		−	85064.1i		−	1.05465i
$285$	−5639.46	+	2155.29i	−0.0694301	+	0.0265348i
$286$	1828.79			0.0223579
$287$		−	76908.6i		−	0.933708i
$288$	16674.0	+	18628.1i	0.201028	+	0.224587i
$289$	−35229.5			−0.421804
$290$		−	2633.64i		−	0.0313155i
$291$	17503.2	+	45798.4i	0.206696	+	0.540834i
$292$	−68429.1			−0.802555
$293$		−	64072.7i		−	0.746342i	−0.927763	−	0.373171i	$-0.878270\pi$
$293$		−	64072.7i		−	0.746342i	0.927763	−	0.373171i	$-0.121730\pi$
$294$	−675.264	+	258.072i	−0.00781230	+	0.00298570i
$295$	45907.1			0.527517
$296$		−	25125.7i		−	0.286771i
$297$	−67081.3	+	130233.i	−0.760481	+	1.47641i
$298$	2961.79			0.0333520
$299$		−	4754.63i		−	0.0531832i
$300$	−6359.24	−	16639.4i	−0.0706582	−	0.184882i
$301$	101417.			1.11938
$302$			6937.02i			0.0760605i
$303$	64357.1	−	24596.0i	0.700989	−	0.267904i
$304$	−14883.1			−0.161045
$305$		−	53560.9i		−	0.575769i
$306$	−8475.44	+	7586.36i	−0.0905147	+	0.0810197i
$307$	33666.4			0.357207			0.178604	−	0.983921i	$-0.442842\pi$
$307$	33666.4			0.357207			0.178604	+	0.983921i	$0.442842\pi$
$308$			149374.i			1.57462i
$309$	51933.3	+	135887.i	0.543913	+	1.42318i
$310$	−2233.00			−0.0232362
$311$		−	58424.9i		−	0.604056i	−0.953299	−	0.302028i	$-0.902336\pi$
$311$		−	58424.9i		−	0.604056i	0.953299	−	0.302028i	$-0.0976636\pi$
$312$	−2435.57	+	930.824i	−0.0250202	+	0.00956221i
$313$	40228.4			0.410624			0.205312	−	0.978697i	$-0.434179\pi$
$313$	40228.4			0.410624			0.205312	+	0.978697i	$0.434179\pi$
$314$			4711.17i			0.0477826i
$315$	−28354.7	−	31677.7i	−0.285762	−	0.319252i
$316$	−76020.9			−0.761305
$317$			114181.i			1.13625i	0.822943	+	0.568124i	$0.192331\pi$
$317$			114181.i			1.13625i	−0.822943	+	0.568124i	$0.807669\pi$
$318$	−3439.38	−	8999.38i	−0.0340115	−	0.0889935i
$319$	116159.			1.14149
$320$		−	42967.4i		−	0.419604i
$321$	159263.	−	60867.3i	1.54563	−	0.590709i
$322$	−4073.08			−0.0392836
$323$		−	20675.8i		−	0.198178i
$324$	11465.9	−	103252.i	0.109224	−	0.983575i
$325$	2791.52			0.0264286
$326$			2472.02i			0.0232604i
$327$	−14051.8	−	36767.6i	−0.131413	−	0.343851i
$328$	21252.5			0.197544
$329$			107143.i			0.989860i
$330$	−7697.08	+	2941.67i	−0.0706803	+	0.0270126i
$331$	479.214			0.00437395			0.00218698	−	0.999998i	$-0.499304\pi$
$331$	479.214			0.00437395			0.00218698	+	0.999998i	$0.499304\pi$
$332$		−	53596.9i		−	0.486254i
$333$	−116894.	+	104631.i	−1.05415	+	0.943569i
$334$	2842.28			0.0254785
$335$			17505.0i			0.155981i
$336$	−37415.6	−	97900.4i	−0.331416	−	0.867174i
$337$	−139642.			−1.22958			−0.614790	−	0.788691i	$-0.710759\pi$
$337$	−139642.			−1.22958			−0.614790	+	0.788691i	$0.710759\pi$
$338$			11435.7i			0.100099i
$339$	−10449.8	+	3993.69i	−0.0909299	+	0.0347516i
$340$	61004.4			0.527720
$341$		−	98488.4i		−	0.846986i
$342$	−1320.87	−	1475.67i	−0.0112929	−	0.0126164i
$343$	121970.			1.03673
$344$			28025.1i			0.236826i
$345$	7648.00	+	20011.5i	0.0642554	+	0.168129i
$346$	11356.5			0.0948623
$347$			131835.i			1.09489i	0.836842	+	0.547445i	$0.184400\pi$
$347$			131835.i			1.09489i	−0.836842	+	0.547445i	$0.815600\pi$
$348$	−76946.3	+	29407.3i	−0.635373	+	0.242827i
$349$	−238053.			−1.95444			−0.977221	−	0.212222i	$-0.931930\pi$
$349$	−238053.			−1.95444			−0.977221	+	0.212222i	$0.931930\pi$
$350$		−	2391.37i		−	0.0195214i
$351$	14473.0	+	7454.86i	0.117475	+	0.0605098i
$352$	−62023.7			−0.500579
$353$		−	181533.i		−	1.45682i	−0.685142	−	0.728410i	$-0.740260\pi$
$353$		−	181533.i		−	1.45682i	0.685142	−	0.728410i	$-0.259740\pi$
$354$	5376.16	+	14067.1i	0.0429008	+	0.112253i
$355$	−60063.8			−0.476602
$356$			54569.8i			0.430578i
$357$	136004.	−	51978.0i	1.06713	−	0.407834i
$358$	−9064.10			−0.0707227
$359$			32567.2i			0.252692i	0.991986	+	0.126346i	$0.0403249\pi$
$359$			32567.2i			0.252692i	−0.991986	+	0.126346i	$0.959675\pi$
$360$	8753.66	−	7835.40i	0.0675437	−	0.0604583i
$361$	−126721.			−0.972377
$362$		−	13640.1i		−	0.104088i
$363$	−82703.8	−	216400.i	−0.627643	−	1.64227i
$364$	16600.2			0.125289
$365$			48317.8i			0.362678i
$366$	16412.4	−	6272.49i	0.122521	−	0.0468250i
$367$	117190.			0.870076			0.435038	−	0.900412i	$-0.356735\pi$
$367$	117190.			0.870076			0.435038	+	0.900412i	$0.356735\pi$
$368$			52812.5i			0.389979i
$369$	−88502.3	−	98874.2i	−0.649983	−	0.726157i
$370$	−8824.35			−0.0644584
$371$			123319.i			0.895945i
$372$	24933.8	+	65241.0i	0.180178	+	0.471449i
$373$	155803.			1.11984			0.559922	−	0.828546i	$-0.310831\pi$
$373$	155803.			1.11984			0.559922	+	0.828546i	$0.310831\pi$
$374$		−	28219.6i		−	0.201747i
$375$	−11749.1	+	4490.26i	−0.0835490	+	0.0319307i
$376$	−29607.4			−0.209423
$377$		−	12908.9i		−	0.0908255i
$378$	6386.25	−	12398.4i	0.0446953	−	0.0867722i
$379$	174237.			1.21301			0.606503	−	0.795082i	$-0.292572\pi$
$379$	174237.			1.21301			0.606503	+	0.795082i	$0.292572\pi$
$380$			10621.5i			0.0735564i
$381$	13314.3	+	34837.8i	0.0917209	+	0.239994i
$382$	245.385			0.00168160
$383$			252899.i			1.72405i	0.506869	+	0.862023i	$0.330803\pi$
$383$			252899.i			1.72405i	−0.506869	+	0.862023i	$0.669197\pi$
$384$	54683.1	−	20898.8i	0.370844	−	0.141729i
$385$	105473.			0.711576
$386$			11084.9i			0.0743974i
$387$	130383.	−	116705.i	0.870558	−	0.779235i
$388$	86258.2			0.572977
$389$		−	125454.i		−	0.829058i	−0.910036	−	0.414529i	$-0.863946\pi$
$389$		−	125454.i		−	0.829058i	0.910036	−	0.414529i	$-0.136054\pi$
$390$	326.913	+	855.391i	0.00214933	+	0.00562387i
$391$	−73367.5			−0.479900
$392$			2556.97i			0.0166400i
$393$	−196844.	+	75229.9i	−1.27449	+	0.487086i
$394$	25119.3			0.161814
$395$			53678.3i			0.344037i
$396$	171892.	+	192037.i	1.09614	+	1.22460i
$397$	−225399.			−1.43012			−0.715059	−	0.699064i	$-0.753600\pi$
$397$	−225399.			−1.43012			−0.715059	+	0.699064i	$0.753600\pi$
$398$			4365.42i			0.0275588i
$399$	9049.94	+	23679.8i	0.0568460	+	0.148742i
$400$	−31007.0			−0.193794
$401$			128670.i			0.800184i	0.916475	+	0.400092i	$0.131022\pi$
$401$			128670.i			0.800184i	−0.916475	+	0.400092i	$0.868978\pi$
$402$	−5363.98	+	2050.00i	−0.0331921	+	0.0126854i
$403$	−10945.2			−0.0673928
$404$		−	121212.i		−	0.742649i
$405$	−72906.1	−	8096.04i	−0.444481	−	0.0493586i
$406$	−11058.5			−0.0670879
$407$		−	389206.i		−	2.34958i
$408$	14363.3	+	37582.6i	0.0862849	+	0.225770i
$409$	105708.			0.631917			0.315958	−	0.948773i	$-0.397674\pi$
$409$	105708.			0.631917			0.315958	+	0.948773i	$0.397674\pi$
$410$		−	7464.06i		−	0.0444025i
$411$	−35043.6	+	13393.0i	−0.207456	+	0.0792854i
$412$	255934.			1.50777
$413$		−	192762.i		−	1.13011i
$414$	−5236.37	+	4687.08i	−0.0305513	+	0.0273465i
$415$	−37844.8			−0.219740
$416$			6892.80i			0.0398299i
$417$	21536.1	+	56350.6i	0.123850	+	0.324061i
$418$	4913.34			0.0281206
$419$			157919.i			0.899512i	0.893151	+	0.449756i	$0.148489\pi$
$419$			157919.i			0.899512i	−0.893151	+	0.449756i	$0.851511\pi$
$420$	−69867.9	+	26702.1i	−0.396077	+	0.151373i
$421$	−245033.			−1.38248			−0.691242	−	0.722623i	$-0.742936\pi$
$421$	−245033.			−1.38248			−0.691242	+	0.722623i	$0.742936\pi$
$422$		−	7907.84i		−	0.0444051i
$423$	123295.	+	137744.i	0.689071	+	0.769826i
$424$	−34077.2			−0.189554
$425$		−	43075.2i		−	0.238479i
$426$	−7034.04	−	18405.0i	−0.0387602	−	0.101419i
$427$	−224899.			−1.23348
$428$		−	299962.i		−	1.63749i
$429$	−37727.7	+	14418.8i	−0.204996	+	0.0783455i
$430$	9842.64			0.0532322
$431$			61385.4i			0.330454i	0.986256	+	0.165227i	$0.0528356\pi$
$431$			61385.4i			0.330454i	−0.986256	+	0.165227i	$0.947164\pi$
$432$	−160760.	−	82805.6i	−0.861412	−	0.443703i
$433$	25262.9			0.134743			0.0673716	−	0.997728i	$-0.478539\pi$
$433$	25262.9			0.134743			0.0673716	+	0.997728i	$0.478539\pi$
$434$			9376.26i			0.0497795i
$435$	20764.5	+	54331.7i	0.109734	+	0.287128i
$436$	−69249.3			−0.364286
$437$		−	12774.1i		−	0.0668909i
$438$	−14805.8	+	5658.47i	−0.0771761	+	0.0294952i
$439$	−207087.			−1.07455			−0.537273	−	0.843409i	$-0.680545\pi$
$439$	−207087.			−1.07455			−0.537273	+	0.843409i	$0.680545\pi$
$440$			29145.9i			0.150547i
$441$	11895.9	−	10648.0i	0.0611675	−	0.0547510i
$442$	−3136.09			−0.0160525
$443$			132568.i			0.675508i	0.941234	+	0.337754i	$0.109667\pi$
$443$			132568.i			0.675508i	−0.941234	+	0.337754i	$0.890333\pi$
$444$	98533.2	+	257819.i	0.499823	+	1.30782i
$445$	38531.7			0.194580
$446$			24540.5i			0.123371i
$447$	−61101.5	+	23351.8i	−0.305799	+	0.116870i
$448$	−180418.			−0.898926
$449$		−	200739.i		−	0.995726i	−0.867256	−	0.497863i	$-0.834118\pi$
$449$		−	200739.i		−	0.995726i	0.867256	−	0.497863i	$-0.165882\pi$
$450$	−2751.86	−	3074.36i	−0.0135894	−	0.0151820i
$451$	329209.			1.61852
$452$			19681.4i			0.0963340i
$453$	−54693.9	−	143110.i	−0.266528	−	0.697388i
$454$	−21450.8			−0.104071
$455$		−	11721.4i		−	0.0566185i
$456$	−6543.55	+	2500.81i	−0.0314691	+	0.0120268i
$457$	396196.			1.89705			0.948523	−	0.316709i	$-0.102578\pi$
$457$	396196.			1.89705			0.948523	+	0.316709i	$0.102578\pi$
$458$		−	22047.2i		−	0.105105i
$459$	115034.	−	223329.i	0.546011	−	1.06004i
$460$	37690.3			0.178121
$461$		−	182217.i		−	0.857405i	−0.903446	−	0.428702i	$-0.858971\pi$
$461$		−	182217.i		−	0.857405i	0.903446	−	0.428702i	$-0.141029\pi$
$462$	12351.9	+	32319.7i	0.0578696	+	0.151420i
$463$	181212.			0.845329			0.422665	−	0.906286i	$-0.361095\pi$
$463$	181212.			0.845329			0.422665	+	0.906286i	$0.361095\pi$
$464$			143387.i			0.666001i
$465$	46066.7	−	17605.8i	0.213050	−	0.0814233i
$466$	33488.6			0.154214
$467$			370180.i			1.69738i	0.528889	+	0.848691i	$0.322609\pi$
$467$			370180.i			1.69738i	−0.528889	+	0.848691i	$0.677391\pi$
$468$	21341.4	−	19102.7i	0.0974386	−	0.0872172i
$469$	73502.7			0.334162
$470$			10398.4i			0.0470728i
$471$	−37144.5	−	97191.1i	−0.167437	−	0.438112i
$472$	53266.8			0.239096
$473$			434118.i			1.94037i
$474$	−16448.4	+	6286.24i	−0.0732094	+	0.0279792i
$475$	7499.87			0.0332404
$476$		−	256155.i		−	1.13055i
$477$	141908.	+	158539.i	0.623694	+	0.696787i
$478$	−17950.0			−0.0785615
$479$			13652.2i			0.0595020i	0.999557	+	0.0297510i	$0.00947143\pi$
$479$			13652.2i			0.0595020i	−0.999557	+	0.0297510i	$0.990529\pi$
$480$	−11087.3	−	29010.8i	−0.0481221	−	0.125915i
$481$	−43253.1			−0.186951
$482$		−	4798.68i		−	0.0206551i
$483$	84027.3	−	32113.6i	0.360185	−	0.137656i
$484$	−407575.			−1.73987
$485$		−	60906.9i		−	0.258931i
$486$	−6057.16	−	23288.4i	−0.0256446	−	0.0985976i
$487$	−193653.			−0.816521			−0.408260	−	0.912865i	$-0.633864\pi$
$487$	−193653.			−0.816521			−0.408260	+	0.912865i	$0.633864\pi$
$488$		−	62147.5i		−	0.260966i
$489$	−19490.2	−	50997.6i	−0.0815079	−	0.213271i
$490$	898.029			0.00374023
$491$			26447.9i			0.109705i	0.998494	+	0.0548527i	$0.0174689\pi$
$491$			26447.9i			0.109705i	−0.998494	+	0.0548527i	$0.982531\pi$
$492$	−218075.	+	83344.1i	−0.900900	+	0.344306i
$493$	−199195.			−0.819566
$494$		−	546.028i		−	0.00223749i
$495$	135597.	−	121373.i	0.553401	−	0.495349i
$496$	121575.			0.494175
$497$			252205.i			1.02103i
$498$	−4431.98	−	11596.6i	−0.0178706	−	0.0467597i
$499$	365043.			1.46603			0.733015	−	0.680213i	$-0.238113\pi$
$499$	365043.			1.46603			0.733015	+	0.680213i	$0.238113\pi$
$500$			22128.6i			0.0885144i
$501$	−58636.1	+	22409.5i	−0.233609	+	0.0892806i
$502$	46738.9			0.185469
$503$		−	85439.2i		−	0.337693i	−0.985642	−	0.168846i	$-0.945996\pi$
$503$		−	85439.2i		−	0.337693i	0.985642	−	0.168846i	$-0.0540041\pi$
$504$	−32900.4	−	36756.2i	−0.129521	−	0.144700i
$505$	−85588.0			−0.335606
$506$		−	17434.9i		−	0.0680954i
$507$	−90163.2	−	235918.i	−0.350763	−	0.917795i
$508$	65614.6			0.254257
$509$			28412.1i			0.109665i	0.998496	+	0.0548325i	$0.0174625\pi$
$509$			28412.1i			0.109665i	−0.998496	+	0.0548325i	$0.982538\pi$
$510$	13199.3	−	5044.52i	0.0507471	−	0.0193945i
$511$	202884.			0.776972
$512$		−	128050.i		−	0.488472i
$513$	38884.1	+	20028.7i	0.147753	+	0.0761059i
$514$	−18077.3			−0.0684239
$515$		−	180715.i		−	0.681366i
$516$	−109903.	−	287570.i	−0.412773	−	1.08005i
$517$	−458629.			−1.71586
$518$			37053.0i			0.138091i
$519$	−234285.	+	89538.8i	−0.869779	+	0.332412i
$520$	3239.04			0.0119787
$521$			2801.47i			0.0103208i	0.999987	+	0.00516038i	$0.00164261\pi$
$521$			2801.47i			0.0103208i	−0.999987	+	0.00516038i	$0.998357\pi$
$522$	−14216.9	+	12725.5i	−0.0521751	+	0.0467019i
$523$	−524249.			−1.91661			−0.958306	−	0.285743i	$-0.907760\pi$
$523$	−524249.			−1.91661			−0.958306	+	0.285743i	$0.907760\pi$
$524$			370743.i			1.35024i
$525$	18854.4	+	49333.8i	0.0684059	+	0.178989i
$526$	−24924.5			−0.0900855
$527$			168893.i			0.608121i
$528$	419065.	−	160158.i	1.50319	−	0.574488i
$529$	234512.			0.838020
$530$			11968.2i			0.0426066i
$531$	−221820.	−	247816.i	−0.786704	−	0.878901i
$532$	44599.3			0.157581
$533$		−	36585.6i		−	0.128782i
$534$	4512.43	+	11807.1i	0.0158244	+	0.0414057i
$535$	−211803.			−0.739988
$536$			20311.3i			0.0706983i
$537$	186992.	−	71464.5i	0.648446	−	0.247823i
$538$	22384.6			0.0773367
$539$			39608.3i			0.136335i
$540$	−59095.3	+	114729.i	−0.202659	+	0.393445i
$541$	300315.			1.02608			0.513042	−	0.858364i	$-0.328519\pi$
$541$	300315.			1.02608			0.513042	+	0.858364i	$0.328519\pi$
$542$			19641.9i			0.0668628i
$543$	107544.	+	281395.i	0.364741	+	0.954370i
$544$	106361.			0.359406
$545$			48897.0i			0.164622i
$546$	3591.74	−	1372.69i	0.0120481	−	0.00460455i
$547$	255317.			0.853307			0.426653	−	0.904415i	$-0.359692\pi$
$547$	255317.			0.853307			0.426653	+	0.904415i	$0.359692\pi$
$548$			66002.4i			0.219785i
$549$	−289132.	+	258802.i	−0.959294	+	0.858663i
$550$	10236.3			0.0338390
$551$		−	34682.0i		−	0.114235i
$552$	8874.09	+	23219.7i	0.0291236	+	0.0762040i
$553$	225393.			0.737037
$554$		−	41359.7i		−	0.134759i
$555$	182046.	−	69574.3i	0.591010	−	0.225872i
$556$	106133.			0.343320
$557$		−	517752.i		−	1.66883i	−0.551138	−	0.834414i	$-0.685806\pi$
$557$		−	517752.i		−	1.66883i	0.551138	−	0.834414i	$-0.314194\pi$
$558$	10789.7	+	12054.2i	0.0346530	+	0.0387141i
$559$	48244.3			0.154391
$560$			130197.i			0.415169i
$561$	222493.	+	582168.i	0.706953	+	1.84979i
$562$	−49269.2			−0.155992
$563$		−	353826.i		−	1.11628i	−0.829747	−	0.558139i	$-0.811515\pi$
$563$		−	353826.i		−	1.11628i	0.829747	−	0.558139i	$-0.188485\pi$
$564$	303807.	−	116109.i	0.955078	−	0.365012i
$565$	13897.1			0.0435337
$566$			45350.6i			0.141563i
$567$	−33994.9	+	306129.i	−0.105742	+	0.952222i
$568$	−69692.9			−0.216019
$569$		−	570557.i		−	1.76228i	−0.472857	−	0.881139i	$-0.656777\pi$
$569$		−	570557.i		−	1.76228i	0.472857	−	0.881139i	$-0.343223\pi$
$570$	878.306	+	2298.15i	0.00270331	+	0.00707340i
$571$	−108185.			−0.331815			−0.165908	−	0.986141i	$-0.553055\pi$
$571$	−108185.			−0.331815			−0.165908	+	0.986141i	$0.553055\pi$
$572$			71057.7i			0.217180i
$573$	−5062.28	+	1934.70i	−0.0154183	+	0.00589257i
$574$	−31341.2			−0.0951244
$575$		−	26613.2i		−	0.0804935i
$576$	−231947.	+	207615.i	−0.699106	+	0.625769i
$577$	161728.			0.485772			0.242886	−	0.970055i	$-0.421906\pi$
$577$	161728.			0.485772			0.242886	+	0.970055i	$0.421906\pi$
$578$			14356.4i			0.0429725i
$579$	−87397.3	−	228681.i	−0.260700	−	0.682139i
$580$	102330.			0.304192
$581$			158908.i			0.470754i
$582$	18663.4	−	7132.78i	0.0550992	−	0.0210578i
$583$	−527868.			−1.55306
$584$			56063.8i			0.164383i
$585$	−13488.4	−	15069.2i	−0.0394138	−	0.0440329i
$586$	−26110.4			−0.0760359
$587$		−	61507.6i		−	0.178506i	−0.996009	−	0.0892529i	$-0.971552\pi$
$587$		−	61507.6i		−	0.178506i	0.996009	−	0.0892529i	$-0.0284479\pi$
$588$	−10027.4	−	26237.5i	−0.0290025	−	0.0758870i
$589$	−29406.1			−0.0847630
$590$		−	18707.7i		−	0.0537424i
$591$	−518210.	+	198049.i	−1.48365	+	0.567020i
$592$	480438.			1.37086
$593$			204770.i			0.582313i	0.956675	+	0.291157i	$0.0940400\pi$
$593$			204770.i			0.582313i	−0.956675	+	0.291157i	$0.905960\pi$
$594$	53071.4	+	27336.4i	0.150414	+	0.0774764i
$595$	−180871.			−0.510898
$596$			115081.i			0.323974i
$597$	−34418.5	−	90058.3i	−0.0965702	−	0.252683i
$598$	−1937.57			−0.00541820
$599$		−	460558.i		−	1.28360i	−0.766871	−	0.641801i	$-0.778188\pi$
$599$		−	460558.i		−	1.28360i	0.766871	−	0.641801i	$-0.221812\pi$
$600$	−13632.6	+	5210.12i	−0.0378684	+	0.0144725i
$601$	−273581.			−0.757419			−0.378710	−	0.925516i	$-0.623632\pi$
$601$	−273581.			−0.757419			−0.378710	+	0.925516i	$0.623632\pi$
$602$		−	41328.7i		−	0.114040i
$603$	94495.5	−	84582.9i	0.259882	−	0.232620i
$604$	−269539.			−0.738835
$605$			287789.i			0.786255i
$606$	−10023.2	−	26226.3i	−0.0272935	−	0.0714154i
$607$	6868.44			0.0186415			0.00932074	−	0.999957i	$-0.497033\pi$
$607$	6868.44			0.0186415			0.00932074	+	0.999957i	$0.497033\pi$
$608$			18518.7i			0.0500959i
$609$	228136.	−	87189.1i	0.615120	−	0.235086i
$610$	−21826.7			−0.0586582
$611$			50968.3i			0.136527i
$612$	−294769.	−	329314.i	−0.787007	−	0.879240i
$613$	189461.			0.504196			0.252098	−	0.967702i	$-0.418879\pi$
$613$	189461.			0.504196			0.252098	+	0.967702i	$0.418879\pi$
$614$		−	13719.5i		−	0.0363916i
$615$	58849.2	+	153983.i	0.155593	+	0.407120i
$616$	122382.			0.322520
$617$		−	510628.i		−	1.34132i	−0.741763	−	0.670662i	$-0.766010\pi$
$617$		−	510628.i		−	1.34132i	0.741763	−	0.670662i	$-0.233990\pi$
$618$	55375.7	−	21163.5i	0.144991	−	0.0554128i
$619$	−466167.			−1.21663			−0.608317	−	0.793694i	$-0.708155\pi$
$619$	−466167.			−1.21663			−0.608317	+	0.793694i	$0.708155\pi$
$620$		−	86763.5i		−	0.225712i
$621$	71071.5	−	137979.i	0.184295	−	0.357793i
$622$	−23808.9			−0.0615400
$623$		−	161793.i		−	0.416853i
$624$	−17798.6	−	46571.4i	−0.0457107	−	0.119605i
$625$	15625.0			0.0400000
$626$		−	16393.6i		−	0.0418336i
$627$	−101362.	+	38738.4i	−0.257833	+	0.0985388i
$628$	−183053.			−0.464149
$629$			667429.i			1.68696i
$630$	−12909.1	+	11554.9i	−0.0325247	+	0.0291129i
$631$	144471.			0.362847			0.181423	−	0.983405i	$-0.441930\pi$
$631$	144471.			0.362847			0.181423	+	0.983405i	$0.441930\pi$
$632$			62283.8i			0.155934i
$633$	62348.1	+	163138.i	0.155602	+	0.407144i
$634$	46530.0			0.115759
$635$		−	46330.5i		−	0.114900i
$636$	349672.	−	133638.i	0.864463	−	0.330380i
$637$	4401.74			0.0108479
$638$		−	47336.1i		−	0.116292i
$639$	290223.	+	324236.i	0.710773	+	0.794071i
$640$	−72722.7			−0.177546
$641$			536191.i			1.30498i	0.757798	+	0.652490i	$0.226275\pi$
$641$			536191.i			1.30498i	−0.757798	+	0.652490i	$0.773725\pi$
$642$	−24804.2	−	64901.8i	−0.0601803	−	0.157466i
$643$	382957.			0.926250			0.463125	−	0.886293i	$-0.346728\pi$
$643$	382957.			0.926250			0.463125	+	0.886293i	$0.346728\pi$
$644$		−	158260.i		−	0.381592i
$645$	−203053.	+	77602.7i	−0.488079	+	0.186534i
$646$	−8425.63			−0.0201900
$647$			320474.i			0.765570i	0.923838	+	0.382785i	$0.125035\pi$
$647$			320474.i			0.765570i	−0.923838	+	0.382785i	$0.874965\pi$
$648$	−84594.0	−	9393.96i	−0.201460	−	0.0223717i
$649$	825120.			1.95897
$650$		−	1137.58i		−	0.00269249i
$651$	−73925.7	−	193432.i	−0.174435	−	0.456421i
$652$	−96050.5			−0.225946
$653$		−	227576.i		−	0.533703i	−0.963738	−	0.266851i	$-0.914017\pi$
$653$		−	227576.i		−	0.533703i	0.963738	−	0.266851i	$-0.0859833\pi$
$654$	−14983.2	+	5726.30i	−0.0350308	+	0.0133881i
$655$	261782.			0.610178
$656$			406378.i			0.944327i
$657$	260829.	−	233468.i	0.604261	−	0.540874i
$658$	43662.3			0.100845
$659$			531408.i			1.22365i	0.790993	+	0.611825i	$0.209564\pi$
$659$			531408.i			1.22365i	−0.790993	+	0.611825i	$0.790436\pi$
$660$	−114299.	−	299071.i	−0.262394	−	0.686573i
$661$	−251100.			−0.574704			−0.287352	−	0.957825i	$-0.592775\pi$
$661$	−251100.			−0.574704			−0.287352	+	0.957825i	$0.592775\pi$
$662$		−	195.286i		−	0.000445610i
$663$	64697.3	−	24726.0i	0.147184	−	0.0562506i
$664$	−43911.9			−0.0995969
$665$		−	31491.6i		−	0.0712117i
$666$	42638.6	+	47635.6i	0.0961290	+	0.107395i
$667$	−123068.			−0.276627
$668$			110437.i			0.247493i
$669$	−193486.	−	506269.i	−0.432311	−	1.13117i
$670$	7133.51			0.0158911
$671$		−	962686.i		−	2.13816i
$672$	−121815.	+	46555.1i	−0.269750	+	0.103093i
$673$	532133.			1.17487			0.587435	−	0.809271i	$-0.300138\pi$
$673$	532133.			1.17487			0.587435	+	0.809271i	$0.300138\pi$
$674$			56905.9i			0.125267i
$675$	81009.9	+	41727.2i	0.177800	+	0.0915824i
$676$	−444336.			−0.972340
$677$			73112.8i			0.159520i	0.996814	+	0.0797602i	$0.0254154\pi$
$677$			73112.8i			0.159520i	−0.996814	+	0.0797602i	$0.974585\pi$
$678$	1627.48	+	4258.40i	0.00354042	+	0.00926376i
$679$	−255745.			−0.554712
$680$		−	49980.9i		−	0.108090i
$681$	442528.	−	169125.i	0.954216	−	0.364682i
$682$	−40135.2			−0.0862893
$683$			674426.i			1.44575i	0.690979	+	0.722875i	$0.257180\pi$
$683$			674426.i			1.44575i	−0.690979	+	0.722875i	$0.742820\pi$
$684$	57337.2	−	51322.5i	0.122553	−	0.109697i
$685$	46604.3			0.0993218
$686$		−	49704.2i		−	0.105620i
$687$	173828.	+	454833.i	0.368304	+	0.963692i
$688$	−535878.			−1.13211
$689$			58662.9i			0.123573i
$690$	8154.94	−	3116.65i	0.0171286	−	0.00654621i
$691$	617153.			1.29252			0.646259	−	0.763118i	$-0.276333\pi$
$691$	617153.			1.29252			0.646259	+	0.763118i	$0.276333\pi$
$692$			441259.i			0.921471i
$693$	−509639.	−	569366.i	−1.06120	−	1.18556i
$694$	53724.2			0.111545
$695$		−	74940.3i		−	0.155148i
$696$	24093.4	+	63042.0i	0.0497369	+	0.130140i
$697$	−564543.			−1.16207
$698$			97009.6i			0.199115i
$699$	−690867.	+	264036.i	−1.41397	+	0.540391i
$700$	92916.8			0.189626
$701$		−	154788.i		−	0.314993i	−0.987520	−	0.157496i	$-0.949658\pi$
$701$		−	154788.i		−	0.314993i	0.987520	−	0.157496i	$-0.0503422\pi$
$702$	3037.95	−	5897.92i	0.00616462	−	0.0119681i
$703$	−116207.			−0.235137
$704$		−	772283.i		−	1.55823i
$705$	−81984.4	−	214518.i	−0.164950	−	0.431604i
$706$	−73976.9			−0.148418
$707$			359379.i			0.718976i
$708$	−546579.	+	208891.i	−1.09040	+	0.416729i
$709$	−480282.			−0.955440			−0.477720	−	0.878512i	$-0.658537\pi$
$709$	−480282.			−0.955440			−0.477720	+	0.878512i	$0.658537\pi$
$710$			24476.7i			0.0485553i
$711$	289766.	−	259370.i	0.573203	−	0.513074i
$712$	44708.9			0.0881931
$713$			104347.i			0.205258i
$714$	−21181.7	−	55423.3i	−0.0415493	−	0.108717i
$715$	50173.8			0.0981443
$716$		−	352187.i		−	0.686984i
$717$	370308.	−	141524.i	0.720319	−	0.275291i
$718$	13271.5			0.0257437
$719$		−	99977.2i		−	0.193394i	−0.995314	−	0.0966971i	$-0.969172\pi$
$719$		−	99977.2i		−	0.193394i	0.995314	−	0.0966971i	$-0.0308278\pi$
$720$	149824.	+	167382.i	0.289012	+	0.322882i
$721$	−758814.			−1.45970
$722$			51640.4i			0.0990639i
$723$	37834.5	+	98996.5i	0.0723788	+	0.189384i
$724$	529989.			1.01109
$725$		−	72255.4i		−	0.137466i
$726$	−88185.8	+	33702.8i	−0.167311	+	0.0639430i
$727$	−182930.			−0.346112			−0.173056	−	0.984912i	$-0.555364\pi$
$727$	−182930.			−0.346112			−0.173056	+	0.984912i	$0.555364\pi$
$728$		−	13600.6i		−	0.0256622i
$729$	308572.	+	432681.i	0.580633	+	0.814165i
$730$	19690.1			0.0369489
$731$		−	744446.i		−	1.39315i
$732$	243718.	+	637705.i	0.454847	+	1.19014i
$733$	671761.			1.25028			0.625139	−	0.780513i	$-0.285042\pi$
$733$	671761.			1.25028			0.625139	+	0.780513i	$0.285042\pi$
$734$		−	47756.2i		−	0.0886417i
$735$	−18526.3	+	7080.37i	−0.0342936	+	0.0131063i
$736$	65713.1			0.121310
$737$			314630.i			0.579248i
$738$	−40292.5	+	36065.8i	−0.0739795	+	0.0662190i
$739$	−209903.			−0.384353			−0.192176	−	0.981360i	$-0.561555\pi$
$739$	−209903.			−0.384353			−0.192176	+	0.981360i	$0.561555\pi$
$740$		−	342871.i		−	0.626135i
$741$	4305.07	+	11264.5i	0.00784051	+	0.0205152i
$742$	50253.9			0.0912771
$743$		−	980505.i		−	1.77612i	−0.459728	−	0.888060i	$-0.652053\pi$
$743$		−	980505.i		−	1.77612i	0.459728	−	0.888060i	$-0.347947\pi$
$744$	53451.9	−	20428.2i	0.0965644	−	0.0369050i
$745$	81258.4			0.146405
$746$		−	63491.5i		−	0.114087i
$747$	182863.	+	204294.i	0.327706	+	0.366112i
$748$	1.09647e6			1.95973
$749$			889351.i			1.58529i
$750$	1829.84	+	4787.89i	0.00325304	+	0.00851181i
$751$	642018.			1.13833			0.569164	−	0.822224i	$-0.307267\pi$
$751$	642018.			1.13833			0.569164	+	0.822224i	$0.307267\pi$
$752$		−	566135.i		−	1.00112i
$753$	−964220.	+	368506.i	−1.70054	+	0.649911i
$754$	−5260.55			−0.00925312
$755$			190321.i			0.333883i
$756$	481740.	+	248138.i	0.842886	+	0.434160i
$757$	−315426.			−0.550434			−0.275217	−	0.961382i	$-0.588750\pi$
$757$	−315426.			−0.550434			−0.275217	+	0.961382i	$0.588750\pi$
$758$		−	71003.8i		−	0.123579i
$759$	137463.	+	359680.i	0.238617	+	0.624357i
$760$	8702.22			0.0150662
$761$			843657.i			1.45679i	0.685158	+	0.728394i	$0.259733\pi$
$761$			843657.i			1.45679i	−0.685158	+	0.728394i	$0.740267\pi$
$762$	14196.8	−	5425.74i	0.0244501	−	0.00934435i
$763$	205316.			0.352674
$764$			9534.47i			0.0163347i
$765$	−232529.	+	208136.i	−0.397332	+	0.355652i
$766$	103059.			0.175643
$767$		−	91697.1i		−	0.155871i
$768$	189049.	+	494659.i	0.320517	+	0.838656i
$769$	−326397.			−0.551942			−0.275971	−	0.961166i	$-0.588999\pi$
$769$	−326397.			−0.551942			−0.275971	+	0.961166i	$0.588999\pi$
$770$		−	42981.7i		−	0.0724940i
$771$	372934.	−	142528.i	0.627369	−	0.239768i
$772$	−430706.			−0.722680
$773$		−	702706.i		−	1.17602i	−0.808853	−	0.588010i	$-0.799911\pi$
$773$		−	702706.i		−	1.17602i	0.808853	−	0.588010i	$-0.200089\pi$
$774$	−47558.8	−	53132.5i	−0.0793870	−	0.0886907i
$775$	−61263.7			−0.102000
$776$		−	70671.2i		−	0.117360i
$777$	−292139.	−	764401.i	−0.483891	−	1.26613i
$778$	−51124.0			−0.0844628
$779$		−	98293.2i		−	0.161975i
$780$	−33236.3	+	12702.2i	−0.0546290	+	0.0208781i
$781$	−1.07957e6			−1.76989
$782$			29898.2i			0.0488913i
$783$	192961.	−	374618.i	0.314736	−	0.611033i
$784$	−48892.8			−0.0795450
$785$			129254.i			0.209751i
$786$	30657.1	+	80216.5i	0.0496234	+	0.129843i
$787$	447743.			0.722902			0.361451	−	0.932391i	$-0.382282\pi$
$787$	447743.			0.722902			0.361451	+	0.932391i	$0.382282\pi$
$788$			976013.i			1.57182i
$789$	514191.	−	196513.i	0.825982	−	0.315674i
$790$	21874.6			0.0350498
$791$		−	58353.0i		−	0.0932632i
$792$	157335.	−	140831.i	0.250828	−	0.224516i
$793$	−106985.			−0.170128
$794$			91853.0i			0.145698i
$795$	−94361.5	−	246903.i	−0.149300	−	0.390654i
$796$	−169619.			−0.267700
$797$		−	662789.i		−	1.04342i	−0.853123	−	0.521710i	$-0.825294\pi$
$797$		−	662789.i		−	1.04342i	0.853123	−	0.521710i	$-0.174706\pi$
$798$	9649.81	−	3687.96i	0.0151535	−	0.00579136i
$799$	786480.			1.23195
$800$			38581.2i			0.0602831i
$801$	−186182.	−	208002.i	−0.290184	−	0.324192i
$802$	52434.7			0.0815212
$803$			868448.i			1.34683i
$804$	−79653.1	−	208418.i	−0.123223	−	0.322421i
$805$	−111747.			−0.172443
$806$			4460.30i			0.00686585i
$807$	−461794.	+	176488.i	−0.709089	+	0.271000i
$808$	−99309.0			−0.152113
$809$		−	1.20479e6i		−	1.84084i	−0.390936	−	0.920418i	$-0.627848\pi$
$809$		−	1.20479e6i		−	1.84084i	0.390936	−	0.920418i	$-0.372152\pi$
$810$	−3299.24	+	29710.1i	−0.00502856	+	0.0452829i
$811$	−380108.			−0.577917			−0.288958	−	0.957342i	$-0.593309\pi$
$811$	−380108.			−0.577917			−0.288958	+	0.957342i	$0.593309\pi$
$812$		−	429679.i		−	0.651677i
$813$	−154863.	−	405210.i	−0.234297	−	0.613055i
$814$	−158606.			−0.239371
$815$			67821.3i			0.102106i
$816$	−718632.	+	274647.i	−1.07926	+	0.412472i
$817$	129616.			0.194185
$818$		−	43077.2i		−	0.0643785i
$819$	−63274.6	+	56637.1i	−0.0943326	+	0.0844370i
$820$	290017.			0.431316
$821$			1.16909e6i			1.73444i	0.497923	+	0.867221i	$0.334096\pi$
$821$			1.16909e6i			1.73444i	−0.497923	+	0.867221i	$0.665904\pi$
$822$	5457.80	+	14280.7i	0.00807745	+	0.0211352i
$823$	−1.04796e6			−1.54719			−0.773594	−	0.633681i	$-0.781543\pi$
$823$	−1.04796e6			−1.54719			−0.773594	+	0.633681i	$0.781543\pi$
$824$		−	209687.i		−	0.308828i
$825$	−211174.	+	80706.5i	−0.310265	+	0.118577i
$826$	−78552.8			−0.115133
$827$		−	9106.01i		−	0.0133143i	−0.999978	−	0.00665713i	$-0.997881\pi$
$827$		−	9106.01i		−	0.0133143i	0.999978	−	0.00665713i	$-0.00211905\pi$
$828$	−182117.	−	203460.i	−0.265638	−	0.296769i
$829$	−715146.			−1.04060			−0.520302	−	0.853982i	$-0.674181\pi$
$829$	−715146.			−1.04060			−0.520302	+	0.853982i	$0.674181\pi$
$830$			15422.2i			0.0223867i
$831$	326094.	+	853249.i	0.472217	+	1.23559i
$832$	−85825.2			−0.123985
$833$		−	67922.3i		−	0.0978863i
$834$	22963.6	−	8776.22i	0.0330147	−	0.0126176i
$835$	77979.7			0.111843
$836$			190908.i			0.273157i
$837$	−317630.	−	163607.i	−0.453388	−	0.233535i
$838$	64354.1			0.0916406
$839$			242894.i			0.345059i	0.985004	+	0.172529i	$0.0551940\pi$
$839$			242894.i			0.345059i	−0.985004	+	0.172529i	$0.944806\pi$
$840$	21877.0	+	57242.7i	0.0310048	+	0.0811263i
$841$	373147.			0.527580
$842$			99853.9i			0.140845i
$843$	1.01642e6	−	388456.i	1.43027	−	0.546621i
$844$	307260.			0.431341
$845$			313746.i			0.439404i
$846$	56132.5	−	50244.1i	0.0784284	−	0.0702012i
$847$	1.20841e6			1.68441
$848$		−	651604.i		−	0.906133i
$849$	−357560.	−	935580.i	−0.496059	−	1.29797i
$850$	−17553.7			−0.0242958
$851$			412357.i			0.569396i
$852$	715130.	−	273308.i	0.985158	−	0.376507i
$853$	861767.			1.18438			0.592191	−	0.805797i	$-0.298263\pi$
$853$	861767.			1.18438			0.592191	+	0.805797i	$0.298263\pi$
$854$			91649.3i			0.125665i
$855$	−36238.8	−	40485.8i	−0.0495726	−	0.0553822i
$856$	−245758.			−0.335398
$857$			525214.i			0.715113i	0.933891	+	0.357557i	$0.116390\pi$
$857$			525214.i			0.715113i	−0.933891	+	0.357557i	$0.883610\pi$
$858$	5875.83	+	15374.5i	0.00798169	+	0.0208846i
$859$	66206.8			0.0897256			0.0448628	−	0.998993i	$-0.485715\pi$
$859$	66206.8			0.0897256			0.0448628	+	0.998993i	$0.485715\pi$
$860$			382437.i			0.517086i
$861$	646567.	−	247105.i	0.872182	−	0.333331i
$862$	25015.3			0.0336660
$863$			5062.88i			0.00679792i	0.999994	+	0.00339896i	$0.00108192\pi$
$863$			5062.88i			0.00679792i	−0.999994	+	0.00339896i	$0.998918\pi$
$864$	−103033.	+	200029.i	−0.138022	+	0.267958i
$865$	311573.			0.416416
$866$		−	10294.9i		−	0.0137274i
$867$	−113191.	−	296172.i	−0.150582	−	0.394009i
$868$	−364316.			−0.483546
$869$			964797.i			1.27760i
$870$	22140.9	−	8461.79i	0.0292520	−	0.0111795i
$871$	34965.3			0.0460894
$872$			56735.9i			0.0746148i
$873$	−328788.	+	294298.i	−0.431407	+	0.386152i
$874$	−5205.60			−0.00681472
$875$		−	65608.6i		−	0.0856928i
$876$	−219860.	−	575280.i	−0.286509	−	0.749672i
$877$	1.24871e6			1.62354			0.811772	−	0.583975i	$-0.198503\pi$
$877$	1.24871e6			1.62354			0.811772	+	0.583975i	$0.198503\pi$
$878$			84390.7i			0.109473i
$879$	538656.	−	205864.i	0.697162	−	0.266441i
$880$	−557311.			−0.719668
$881$		−	112260.i		−	0.144635i	−0.997382	−	0.0723173i	$-0.976961\pi$
$881$		−	112260.i		−	0.144635i	0.997382	−	0.0723173i	$-0.0230394\pi$
$882$	−4339.20	−	4847.73i	−0.00557793	−	0.00623163i
$883$	−329416.			−0.422496			−0.211248	−	0.977432i	$-0.567753\pi$
$883$	−329416.			−0.422496			−0.211248	+	0.977432i	$0.567753\pi$
$884$		−	121853.i		−	0.155931i
$885$	147498.	+	385939.i	0.188322	+	0.492756i
$886$	54023.0			0.0688194
$887$			152778.i			0.194185i	0.995275	+	0.0970923i	$0.0309542\pi$
$887$			152778.i			0.194185i	−0.995275	+	0.0970923i	$0.969046\pi$
$888$	211231.	−	80728.1i	0.267874	−	0.102376i
$889$	−194539.			−0.246152
$890$		−	15702.2i		−	0.0198234i
$891$	−1.31039e6	−	145516.i	−1.65061	−	0.183296i
$892$	−953523.			−1.19840
$893$			136935.i			0.171716i
$894$	9516.13	+	24899.6i	0.0119065	+	0.0311543i
$895$	−248679.			−0.310451
$896$			305359.i			0.380360i
$897$	39971.9	−	15276.5i	0.0496787	−	0.0189862i
$898$	−81803.8			−0.101443
$899$			283304.i			0.350537i
$900$	119455.	−	106924.i	0.147475	−	0.132005i
$901$	905213.			1.11507
$902$		−	134157.i		−	0.164892i
$903$	325850.	+	852609.i	0.399615	+	1.04562i
$904$	16125.0			0.0197316
$905$		−	374225.i		−	0.456916i
$906$	−58319.2	+	22288.4i	−0.0710486	+	0.0271533i
$907$	−1.26534e6			−1.53813			−0.769067	−	0.639169i	$-0.779279\pi$
$907$	−1.26534e6			−1.53813			−0.769067	+	0.639169i	$0.779279\pi$
$908$		−	833472.i		−	1.01093i
$909$	413554.	+	462021.i	0.500501	+	0.559157i
$910$	−4776.63			−0.00576818
$911$		−	702340.i		−	0.846274i	−0.906066	−	0.423137i	$-0.860929\pi$
$911$		−	702340.i		−	0.846274i	0.906066	−	0.423137i	$-0.139071\pi$
$912$	−47819.0	−	125122.i	−0.0574925	−	0.150433i
$913$	−680210.			−0.816021
$914$		−	161455.i		−	0.193267i
$915$	450284.	−	172089.i	0.537829	−	0.205547i
$916$	856648.			1.02097
$917$		−	1.09921e6i		−	1.30720i
$918$	−91009.4	−	46877.9i	−0.107994	−	0.0556266i
$919$	266756.			0.315851			0.157926	−	0.987451i	$-0.449519\pi$
$919$	266756.			0.315851			0.157926	+	0.987451i	$0.449519\pi$
$920$		−	30879.6i		−	0.0364835i
$921$	108169.	+	283032.i	0.127522	+	0.333670i
$922$	−74255.5			−0.0873507
$923$			119974.i			0.140827i
$924$	−1.25578e6	+	479935.i	−1.47086	+	0.562133i
$925$	−242101.			−0.282953
$926$		−	73846.3i		−	0.0861205i
$927$	−975536.	+	873202.i	−1.13523	+	1.01614i
$928$	178413.			0.207171
$929$		−	1.29140e6i		−	1.49634i	−0.663509	−	0.748169i	$-0.730933\pi$
$929$		−	1.29140e6i		−	1.49634i	0.663509	−	0.748169i	$-0.269067\pi$
$930$	−7174.56	−	18772.7i	−0.00829525	−	0.0217051i
$931$	11826.0			0.0136439
$932$			1.30120e6i			1.49800i
$933$	491175.	−	187717.i	0.564252	−	0.215646i
$934$	150853.			0.172926
$935$		−	774220.i		−	0.885608i
$936$	−15650.8	−	17485.0i	−0.0178642	−	0.0199578i
$937$	−399820.			−0.455392			−0.227696	−	0.973732i	$-0.573119\pi$
$937$	−399820.			−0.455392			−0.227696	+	0.973732i	$0.573119\pi$
$938$		−	29953.2i		−	0.0340438i
$939$	129253.	+	338199.i	0.146591	+	0.383566i
$940$	−404030.			−0.457254
$941$		−	1.22439e6i		−	1.38274i	−0.722499	−	0.691372i	$-0.757007\pi$
$941$		−	1.22439e6i		−	1.38274i	0.722499	−	0.691372i	$-0.242993\pi$
$942$	−39606.6	+	15136.8i	−0.0446340	+	0.0170582i
$943$	−348791.			−0.392231
$944$			1.01853e6i			1.14296i
$945$	175210.	−	340156.i	0.196199	−	0.380904i
$946$	176908.			0.197682
$947$		−	699832.i		−	0.780358i	−0.920739	−	0.390179i	$-0.872413\pi$
$947$		−	699832.i		−	0.780358i	0.920739	−	0.390179i	$-0.127587\pi$
$948$	−244253.	−	639104.i	−0.271783	−	0.711140i
$949$	96512.2			0.107164
$950$		−	3056.29i		−	0.00338647i
$951$	−959910.	+	366858.i	−1.06138	+	0.405637i
$952$	−209867.			−0.231564
$953$			1.25288e6i			1.37950i	0.724045	+	0.689752i	$0.242281\pi$
$953$			1.25288e6i			1.37950i	−0.724045	+	0.689752i	$0.757719\pi$
$954$	64606.7	−	57829.4i	0.0709874	−	0.0635407i
$955$	6732.29			0.00738169
$956$		−	697451.i		−	0.763128i
$957$	373214.	+	976541.i	0.407507	+	1.06627i
$958$	5563.43			0.00606194
$959$		−	195689.i		−	0.212779i
$960$	361225.	−	138053.i	0.391954	−	0.149797i
$961$	−683314.			−0.739900
$962$			17626.2i			0.0190462i
$963$	1.02342e6	+	1.14336e6i	1.10357	+	1.23290i
$964$	186453.			0.200639
$965$			304121.i			0.326582i
$966$	−13086.7	−	34242.2i	−0.0140241	−	0.0366950i
$967$	580972.			0.621301			0.310651	−	0.950524i	$-0.399453\pi$
$967$	580972.			0.621301			0.310651	+	0.950524i	$0.399453\pi$
$968$			333926.i			0.356369i
$969$	173820.	−	66430.6i	0.185120	−	0.0707490i
$970$	−24820.3			−0.0263793
$971$		−	648131.i		−	0.687423i	−0.939075	−	0.343712i	$-0.888316\pi$
$971$		−	648131.i		−	0.687423i	0.939075	−	0.343712i	$-0.111684\pi$
$972$	904872.	−	235352.i	0.957755	−	0.249106i
$973$	−314670.			−0.332376
$974$			78916.2i			0.0831856i
$975$	8969.05	+	23468.1i	0.00943490	+	0.0246871i
$976$	1.18835e6			1.24751
$977$			1.48636e6i			1.55717i	0.627539	+	0.778585i	$0.284062\pi$
$977$			1.48636e6i			1.55717i	−0.627539	+	0.778585i	$0.715938\pi$
$978$	−20782.1	+	7942.51i	−0.0217276	+	0.00830386i
$979$	692557.			0.722587
$980$			34893.0i			0.0363317i
$981$	263955.	−	236266.i	0.274279	−	0.245507i
$982$	10777.8			0.0111766
$983$		−	992676.i		−	1.02731i	−0.857998	−	0.513654i	$-0.828292\pi$
$983$		−	992676.i		−	1.02731i	0.857998	−	0.513654i	$-0.171708\pi$
$984$	68283.7	+	178669.i	0.0705223	+	0.184527i
$985$	689163.			0.710313
$986$			81174.3i			0.0834958i
$987$	−900749.	+	344248.i	−0.924634	+	0.353376i
$988$	21216.0			0.0217345
$989$		−	459941.i		−	0.470229i
$990$	−49461.0	−	55257.5i	−0.0504652	−	0.0563795i
$991$	1.60928e6			1.63864			0.819322	−	0.573334i	$-0.194350\pi$
$991$	1.60928e6			1.63864			0.819322	+	0.573334i	$0.194350\pi$
$992$		−	151272.i		−	0.153722i
$993$	1539.70	+	4028.73i	0.00156148	+	0.00408573i
$994$	102777.			0.104021
$995$			119768.i			0.120975i
$996$	450587.	−	172205.i	0.454213	−	0.173591i
$997$	375625.			0.377889			0.188945	−	0.981988i	$-0.439493\pi$
$997$	375625.			0.377889			0.188945	+	0.981988i	$0.439493\pi$
$998$		−	148759.i		−	0.149356i
$999$	−1.25521e6	−	646542.i	−1.25772	−	0.647837i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	15.5.c.a.11.3	✓	6
3.2	odd	2	inner	15.5.c.a.11.4	yes	6
4.3	odd	2		240.5.l.d.161.3		6
5.2	odd	4		75.5.d.d.74.7		12
5.3	odd	4		75.5.d.d.74.6		12
5.4	even	2		75.5.c.i.26.4		6
12.11	even	2		240.5.l.d.161.4		6
15.2	even	4		75.5.d.d.74.5		12
15.8	even	4		75.5.d.d.74.8		12
15.14	odd	2		75.5.c.i.26.3		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
15.5.c.a.11.3	✓	6	1.1	even	1	trivial
15.5.c.a.11.4	yes	6	3.2	odd	2	inner
75.5.c.i.26.3		6	15.14	odd	2
75.5.c.i.26.4		6	5.4	even	2
75.5.d.d.74.5		12	15.2	even	4
75.5.d.d.74.6		12	5.3	odd	4
75.5.d.d.74.7		12	5.2	odd	4
75.5.d.d.74.8		12	15.8	even	4
240.5.l.d.161.3		6	4.3	odd	2
240.5.l.d.161.4		6	12.11	even	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:	\( N \)	\(=\)	\( 15 = 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	15.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-0.407512i q^{2} +(3.21297 + 8.40695i) q^{3} +15.8339 q^{4} -11.1803i q^{5} +(3.42594 - 1.30932i) q^{6} -46.9457 q^{7} -12.9727i q^{8} +(-60.3537 + 54.0225i) q^{9} +O(q^{10})\)	\(q-0.407512i q^{2} +(3.21297 + 8.40695i) q^{3} +15.8339 q^{4} -11.1803i q^{5} +(3.42594 - 1.30932i) q^{6} -46.9457 q^{7} -12.9727i q^{8} +(-60.3537 + 54.0225i) q^{9} -4.55613 q^{10} -200.952i q^{11} +(50.8739 + 133.115i) q^{12} -22.3321 q^{13} +19.1309i q^{14} +(93.9926 - 35.9221i) q^{15} +248.056 q^{16} +344.602i q^{17} +(22.0148 + 24.5949i) q^{18} -59.9990 q^{19} -177.029i q^{20} +(-150.835 - 394.670i) q^{21} -81.8903 q^{22} +212.905i q^{23} +(109.061 - 41.6809i) q^{24} -125.000 q^{25} +9.10062i q^{26} +(-648.079 - 333.818i) q^{27} -743.335 q^{28} +578.043i q^{29} +(-14.6387 - 38.3031i) q^{30} +490.110 q^{31} -308.650i q^{32} +(1689.39 - 645.652i) q^{33} +140.429 q^{34} +524.869i q^{35} +(-955.636 + 855.389i) q^{36} +1936.81 q^{37} +24.4503i q^{38} +(-71.7524 - 187.745i) q^{39} -145.039 q^{40} +1638.25i q^{41} +(-160.833 + 61.4671i) q^{42} -2160.31 q^{43} -3181.86i q^{44} +(603.990 + 674.775i) q^{45} +86.7615 q^{46} -2282.28i q^{47} +(796.997 + 2085.40i) q^{48} -197.104 q^{49} +50.9390i q^{50} +(-2897.05 + 1107.19i) q^{51} -353.606 q^{52} -2626.84i q^{53} +(-136.035 + 264.100i) q^{54} -2246.71 q^{55} +609.013i q^{56} +(-192.775 - 504.409i) q^{57} +235.560 q^{58} +4106.06i q^{59} +(1488.27 - 568.788i) q^{60} +4790.63 q^{61} -199.726i q^{62} +(2833.34 - 2536.12i) q^{63} +3843.12 q^{64} +249.681i q^{65} +(-263.111 - 688.448i) q^{66} -1565.70 q^{67} +5456.40i q^{68} +(-1789.88 + 684.058i) q^{69} +213.890 q^{70} -5372.27i q^{71} +(700.819 + 782.951i) q^{72} -4321.67 q^{73} -789.274i q^{74} +(-401.621 - 1050.87i) q^{75} -950.020 q^{76} +9433.82i q^{77} +(-76.5085 + 29.2400i) q^{78} -4801.14 q^{79} -2773.35i q^{80} +(724.132 - 6520.92i) q^{81} +667.606 q^{82} -3384.94i q^{83} +(-2388.31 - 6249.18i) q^{84} +3852.77 q^{85} +880.352i q^{86} +(-4859.58 + 1857.23i) q^{87} -2606.89 q^{88} +3446.38i q^{89} +(274.979 - 246.133i) q^{90} +1048.40 q^{91} +3371.13i q^{92} +(1574.71 + 4120.33i) q^{93} -930.059 q^{94} +670.809i q^{95} +(2594.80 - 991.681i) q^{96} +5447.68 q^{97} +80.3221i q^{98} +(10855.9 + 12128.2i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 8 q^{3} - 50 q^{4} - 2 q^{6} + 76 q^{7} + 118 q^{9}+O(q^{10}) \) 6 * q + 8 * q^3 - 50 * q^4 - 2 * q^6 + 76 * q^7 + 118 * q^9	\( 6 q + 8 q^{3} - 50 q^{4} - 2 q^{6} + 76 q^{7} + 118 q^{9} + 50 q^{10} - 452 q^{12} - 424 q^{13} + 50 q^{15} + 802 q^{16} + 1160 q^{18} - 244 q^{19} - 876 q^{21} + 340 q^{22} - 786 q^{24} - 750 q^{25} - 352 q^{27} - 3764 q^{28} + 2200 q^{30} + 3772 q^{31} + 4420 q^{33} + 3124 q^{34} - 7606 q^{36} + 1896 q^{37} - 1336 q^{39} - 4650 q^{40} - 1980 q^{42} - 7384 q^{43} + 1900 q^{45} + 8196 q^{46} + 14668 q^{48} - 1318 q^{49} - 8492 q^{51} + 8976 q^{52} - 278 q^{54} - 1300 q^{55} - 11584 q^{57} - 23740 q^{58} + 5050 q^{60} + 6452 q^{61} + 14796 q^{63} + 3174 q^{64} - 12760 q^{66} + 13816 q^{67} + 5472 q^{69} - 2100 q^{70} - 2040 q^{72} + 596 q^{73} - 1000 q^{75} + 21348 q^{76} - 1400 q^{78} - 16124 q^{79} + 5086 q^{81} - 31240 q^{82} - 14736 q^{84} - 3100 q^{85} - 4900 q^{87} + 15660 q^{88} + 7550 q^{90} - 11632 q^{91} - 8184 q^{93} + 34924 q^{94} + 14354 q^{96} + 9756 q^{97} + 9680 q^{99}+O(q^{100}) \) 6 * q + 8 * q^3 - 50 * q^4 - 2 * q^6 + 76 * q^7 + 118 * q^9 + 50 * q^10 - 452 * q^12 - 424 * q^13 + 50 * q^15 + 802 * q^16 + 1160 * q^18 - 244 * q^19 - 876 * q^21 + 340 * q^22 - 786 * q^24 - 750 * q^25 - 352 * q^27 - 3764 * q^28 + 2200 * q^30 + 3772 * q^31 + 4420 * q^33 + 3124 * q^34 - 7606 * q^36 + 1896 * q^37 - 1336 * q^39 - 4650 * q^40 - 1980 * q^42 - 7384 * q^43 + 1900 * q^45 + 8196 * q^46 + 14668 * q^48 - 1318 * q^49 - 8492 * q^51 + 8976 * q^52 - 278 * q^54 - 1300 * q^55 - 11584 * q^57 - 23740 * q^58 + 5050 * q^60 + 6452 * q^61 + 14796 * q^63 + 3174 * q^64 - 12760 * q^66 + 13816 * q^67 + 5472 * q^69 - 2100 * q^70 - 2040 * q^72 + 596 * q^73 - 1000 * q^75 + 21348 * q^76 - 1400 * q^78 - 16124 * q^79 + 5086 * q^81 - 31240 * q^82 - 14736 * q^84 - 3100 * q^85 - 4900 * q^87 + 15660 * q^88 + 7550 * q^90 - 11632 * q^91 - 8184 * q^93 + 34924 * q^94 + 14354 * q^96 + 9756 * q^97 + 9680 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more