LMFDB - Embedded newform 429.2.e.d.428.15

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [429,2,Mod(428,429)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(429, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("429.428");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 429 = 3 \cdot 11 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	429.e (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:	$3.42558224671$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 53x^{10} + 1024 $ x^20 - 53*x^10 + 1024
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{12}\cdot 3^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			428.15
Root			$-1.09266 - 0.897823i$ of defining polynomial
Character	$\chi$	$=$	429.428
Dual form			429.2.e.d.428.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.79565i q^{2} +(-0.240479 - 1.71528i) q^{3} -1.22434 q^{4} +(3.08003 - 0.431815i) q^{6} -5.23009 q^{7} +1.39281i q^{8} +(-2.88434 + 0.824975i) q^{9} +O(q^{10})$	\(q+1.79565i q^{2} +(-0.240479 - 1.71528i) q^{3} -1.22434 q^{4} +(3.08003 - 0.431815i) q^{6} -5.23009 q^{7} +1.39281i q^{8} +(-2.88434 + 0.824975i) q^{9} +3.31662i q^{11} +(0.294428 + 2.10008i) q^{12} +3.60555 q^{13} -9.39139i q^{14} -4.94967 q^{16} +(-1.48136 - 5.17925i) q^{18} -7.00719 q^{19} +(1.25773 + 8.97104i) q^{21} -5.95548 q^{22} +6.19820i q^{23} +(2.38905 - 0.334941i) q^{24} -5.00000 q^{25} +6.47429i q^{26} +(2.10868 + 4.74905i) q^{27} +6.40342 q^{28} -6.10224i q^{32} +(5.68893 - 0.797578i) q^{33} +(3.53142 - 1.01005i) q^{36} -12.5824i q^{38} +(-0.867059 - 6.18451i) q^{39} -7.55666i q^{41} +(-16.1088 + 2.25843i) q^{42} -4.06068i q^{44} -11.1298 q^{46} +(1.19029 + 8.49005i) q^{48} +20.3538 q^{49} -8.97823i q^{50} -4.41443 q^{52} -2.65693i q^{53} +(-8.52761 + 3.78644i) q^{54} -7.28451i q^{56} +(1.68508 + 12.0193i) q^{57} +(15.0854 - 4.31469i) q^{63} +1.05811 q^{64} +(1.43217 + 10.2153i) q^{66} +(10.6316 - 1.49054i) q^{69} +(-1.14903 - 4.01733i) q^{72} +6.10477 q^{73} +(1.20239 + 8.57638i) q^{75} +8.57920 q^{76} -17.3462i q^{77} +(11.1052 - 1.55693i) q^{78} +(7.63883 - 4.75902i) q^{81} +13.5691 q^{82} +17.5249i q^{83} +(-1.53989 - 10.9836i) q^{84} -4.61942 q^{88} -18.8574 q^{91} -7.58871i q^{92} +(-10.4670 + 1.46746i) q^{96} +36.5483i q^{98} +(-2.73613 - 9.56627i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 40 q^{4}+O(q^{10}) $ 20 * q - 40 * q^4	$ 20 q - 40 q^{4} + 80 q^{16} - 100 q^{25} + 10 q^{36} - 50 q^{42} + 70 q^{48} + 140 q^{49} - 160 q^{64} - 110 q^{66} + 130 q^{78}+O(q^{100}) $ 20 * q - 40 * q^4 + 80 * q^16 - 100 * q^25 + 10 * q^36 - 50 * q^42 + 70 * q^48 + 140 * q^49 - 160 * q^64 - 110 * q^66 + 130 * q^78

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/429\mathbb{Z}\right)^\times$.

$n$	$67$	$79$	$287$
$\chi(n)$	$-1$	$-1$	$-1$

Coefficient data

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

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

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

$2$			1.79565i			1.26971i	0.772630	+	0.634856i	$0.218941\pi$
$2$			1.79565i			1.26971i	−0.772630	+	0.634856i	$0.781059\pi$
$3$	−0.240479	−	1.71528i	−0.138841	−	0.990315i
$3$	−0.240479	−	1.71528i	−0.138841	−	0.990315i
$4$	−1.22434			−0.612171
$5$		0			0			−	1.00000i	$-0.5\pi$
$5$		0			0				1.00000i	$0.5\pi$
$6$	3.08003	−	0.431815i	1.25742	−	0.176288i
$7$	−5.23009			−1.97679			−0.988394	−	0.151912i	$-0.951457\pi$
$7$	−5.23009			−1.97679			−0.988394	+	0.151912i	$0.951457\pi$
$8$			1.39281i			0.492432i
$9$	−2.88434	+	0.824975i	−0.961447	+	0.274992i
$10$		0			0
$11$			3.31662i			1.00000i
$11$			3.31662i			1.00000i
$12$	0.294428	+	2.10008i	0.0849941	+	0.606242i
$13$	3.60555			1.00000
$13$	3.60555			1.00000
$14$		−	9.39139i		−	2.50995i
$15$		0			0
$16$	−4.94967			−1.23742
$17$		0			0			−	1.00000i	$-0.5\pi$
$17$		0			0				1.00000i	$0.5\pi$
$18$	−1.48136	−	5.17925i	−0.349160	−	1.22076i
$19$	−7.00719			−1.60756			−0.803780	−	0.594926i	$-0.797181\pi$
$19$	−7.00719			−1.60756			−0.803780	+	0.594926i	$0.797181\pi$
$20$		0			0
$21$	1.25773	+	8.97104i	0.274458	+	1.95764i
$22$	−5.95548			−1.26971
$23$			6.19820i			1.29241i	0.763162	+	0.646207i	$0.223646\pi$
$23$			6.19820i			1.29241i	−0.763162	+	0.646207i	$0.776354\pi$
$24$	2.38905	−	0.334941i	0.487662	−	0.0683695i
$25$	−5.00000			−1.00000
$26$			6.47429i			1.26971i
$27$	2.10868	+	4.74905i	0.405816	+	0.913955i
$28$	6.40342			1.21013
$29$		0			0			−	1.00000i	$-0.5\pi$
$29$		0			0				1.00000i	$0.5\pi$
$30$		0			0
$31$		0			0		1.00000			$0$
$31$		0			0		−1.00000			$\pi$
$32$		−	6.10224i		−	1.07873i
$33$	5.68893	−	0.797578i	0.990315	−	0.138841i
$34$		0			0
$35$		0			0
$36$	3.53142	−	1.01005i	0.588570	−	0.168342i
$37$		0			0		1.00000			$0$
$37$		0			0		−1.00000			$\pi$
$38$		−	12.5824i		−	2.04114i
$39$	−0.867059	−	6.18451i	−0.138841	−	0.990315i
$40$		0			0
$41$		−	7.55666i		−	1.18015i	−0.807347	−	0.590076i	$-0.799098\pi$
$41$		−	7.55666i		−	1.18015i	0.807347	−	0.590076i	$-0.200902\pi$
$42$	−16.1088	+	2.25843i	−2.48564	+	0.348483i
$43$		0			0		1.00000			$0$
$43$		0			0		−1.00000			$\pi$
$44$		−	4.06068i		−	0.612171i
$45$		0			0
$46$	−11.1298			−1.64099
$47$		0			0			−	1.00000i	$-0.5\pi$
$47$		0			0				1.00000i	$0.5\pi$
$48$	1.19029	+	8.49005i	0.171804	+	1.22543i
$49$	20.3538			2.90769
$50$		−	8.97823i		−	1.26971i
$51$		0			0
$52$	−4.41443			−0.612171
$53$		−	2.65693i		−	0.364957i	−0.983210	−	0.182479i	$-0.941588\pi$
$53$		−	2.65693i		−	0.364957i	0.983210	−	0.182479i	$-0.0584120\pi$
$54$	−8.52761	+	3.78644i	−1.16046	+	0.515270i
$55$		0			0
$56$		−	7.28451i		−	0.973433i
$57$	1.68508	+	12.0193i	0.223195	+	1.59199i
$58$		0			0
$59$		0			0			−	1.00000i	$-0.5\pi$
$59$		0			0				1.00000i	$0.5\pi$
$60$		0			0
$61$		0			0		1.00000			$0$
$61$		0			0		−1.00000			$\pi$
$62$		0			0
$63$	15.0854	−	4.31469i	1.90058	−	0.543600i
$64$	1.05811			0.132264
$65$		0			0
$66$	1.43217	+	10.2153i	0.176288	+	1.25742i
$67$		0			0		1.00000			$0$
$67$		0			0		−1.00000			$\pi$
$68$		0			0
$69$	10.6316	−	1.49054i	1.27990	−	0.179439i
$70$		0			0
$71$		0			0			−	1.00000i	$-0.5\pi$
$71$		0			0				1.00000i	$0.5\pi$
$72$	−1.14903	−	4.01733i	−0.135415	−	0.473447i
$73$	6.10477			0.714510			0.357255	−	0.934007i	$-0.383713\pi$
$73$	6.10477			0.714510			0.357255	+	0.934007i	$0.383713\pi$
$74$		0			0
$75$	1.20239	+	8.57638i	0.138841	+	0.990315i
$76$	8.57920			0.984102
$77$		−	17.3462i		−	1.97679i
$78$	11.1052	−	1.55693i	1.25742	−	0.176288i
$79$		0			0		1.00000			$0$
$79$		0			0		−1.00000			$\pi$
$80$		0			0
$81$	7.63883	−	4.75902i	0.848759	−	0.528779i
$82$	13.5691			1.49846
$83$			17.5249i			1.92360i	0.273747	+	0.961802i	$0.411737\pi$
$83$			17.5249i			1.92360i	−0.273747	+	0.961802i	$0.588263\pi$
$84$	−1.53989	−	10.9836i	−0.168015	−	1.19841i
$85$		0			0
$86$		0			0
$87$		0			0
$88$	−4.61942			−0.492432
$89$		0			0			−	1.00000i	$-0.5\pi$
$89$		0			0				1.00000i	$0.5\pi$
$90$		0			0
$91$	−18.8574			−1.97679
$92$		−	7.58871i		−	0.791178i
$93$		0			0
$94$		0			0
$95$		0			0
$96$	−10.4670	+	1.46746i	−1.06829	+	0.149772i
$97$		0			0		1.00000			$0$
$97$		0			0		−1.00000			$\pi$
$98$			36.5483i			3.69193i
$99$	−2.73613	−	9.56627i	−0.274992	−	0.961447i
$100$	6.12171			0.612171
$101$		0			0			−	1.00000i	$-0.5\pi$
$101$		0			0				1.00000i	$0.5\pi$
$102$		0			0
$103$	15.7547			1.55236			0.776179	−	0.630512i	$-0.217155\pi$
$103$	15.7547			1.55236			0.776179	+	0.630512i	$0.217155\pi$
$104$			5.02184i			0.492432i
$105$		0			0
$106$	4.77090			0.463391
$107$		0			0			−	1.00000i	$-0.5\pi$
$107$		0			0				1.00000i	$0.5\pi$
$108$	−2.58175	−	5.81446i	−0.248429	−	0.559497i
$109$	−20.1192			−1.92707			−0.963533	−	0.267589i	$-0.913773\pi$
$109$	−20.1192			−1.92707			−0.963533	+	0.267589i	$0.913773\pi$
$110$		0			0
$111$		0			0
$112$	25.8872			2.44611
$113$			17.9264i			1.68637i	0.537622	+	0.843186i	$0.319323\pi$
$113$			17.9264i			1.68637i	−0.537622	+	0.843186i	$0.680677\pi$
$114$	−21.5823	+	3.02581i	−2.02137	+	0.283393i
$115$		0			0
$116$		0			0
$117$	−10.3996	+	2.97449i	−0.961447	+	0.274992i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−11.0000			−1.00000
$122$		0			0
$123$	−12.9618	+	1.81722i	−1.16872	+	0.163853i
$124$		0			0
$125$		0			0
$126$	7.74766	+	27.0879i	0.690216	+	2.41319i
$127$		0			0		1.00000			$0$
$127$		0			0		−1.00000			$\pi$
$128$		−	10.3045i		−	0.910796i
$129$		0			0
$130$		0			0
$131$		0			0			−	1.00000i	$-0.5\pi$
$131$		0			0				1.00000i	$0.5\pi$
$132$	−6.96519	+	0.976508i	−0.606242	+	0.0849941i
$133$	36.6483			3.17781
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0			−	1.00000i	$-0.5\pi$
$137$		0			0				1.00000i	$0.5\pi$
$138$	2.67647	+	19.0906i	0.227836	+	1.62510i
$139$		0			0		1.00000			$0$
$139$		0			0		−1.00000			$\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			11.9583i			1.00000i
$144$	14.2765	−	4.08335i	1.18971	−	0.340279i
$145$		0			0
$146$			10.9620i			0.907222i
$147$	−4.89467	−	34.9124i	−0.403705	−	2.87953i
$148$		0			0
$149$		−	12.3797i		−	1.01419i	−0.861891	−	0.507093i	$-0.830720\pi$
$149$		−	12.3797i		−	1.01419i	0.861891	−	0.507093i	$-0.169280\pi$
$150$	−15.4001	+	2.15907i	−1.25742	+	0.176288i
$151$	−14.4222			−1.17366			−0.586831	−	0.809709i	$-0.699625\pi$
$151$	−14.4222			−1.17366			−0.586831	+	0.809709i	$0.699625\pi$
$152$		−	9.75967i		−	0.791614i
$153$		0			0
$154$	31.1477			2.50995
$155$		0			0
$156$	1.06158	+	7.57196i	0.0849941	+	0.606242i
$157$	−14.2034			−1.13356			−0.566778	−	0.823871i	$-0.691810\pi$
$157$	−14.2034			−1.13356			−0.566778	+	0.823871i	$0.691810\pi$
$158$		0			0
$159$	−4.55737	+	0.638935i	−0.361423	+	0.0506709i
$160$		0			0
$161$		−	32.4171i		−	2.55483i
$162$	8.54550	+	13.7166i	0.671398	+	1.07768i
$163$		0			0		1.00000			$0$
$163$		0			0		−1.00000			$\pi$
$164$			9.25194i			0.722455i
$165$		0			0
$166$	−31.4684			−2.44242
$167$			22.7419i			1.75982i	0.475138	+	0.879911i	$0.342398\pi$
$167$			22.7419i			1.75982i	−0.475138	+	0.879911i	$0.657602\pi$
$168$	−12.4949	+	1.75177i	−0.964005	+	0.135152i
$169$	13.0000			1.00000
$170$		0			0
$171$	20.2111	−	5.78076i	1.54558	−	0.442066i
$172$		0			0
$173$		0			0			−	1.00000i	$-0.5\pi$
$173$		0			0				1.00000i	$0.5\pi$
$174$		0			0
$175$	26.1504			1.97679
$176$		−	16.4162i		−	1.23742i
$177$		0			0
$178$		0			0
$179$			23.9165i			1.78760i	0.448461	+	0.893802i	$0.351972\pi$
$179$			23.9165i			1.78760i	−0.448461	+	0.893802i	$0.648028\pi$
$180$		0			0
$181$	10.4681			0.778087			0.389044	−	0.921219i	$-0.372806\pi$
$181$	10.4681			0.778087			0.389044	+	0.921219i	$0.372806\pi$
$182$		−	33.8611i		−	2.50995i
$183$		0			0
$184$	−8.63289			−0.636425
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	−11.0286	−	24.8379i	−0.802212	−	1.80669i
$190$		0			0
$191$		−	4.97779i		−	0.360181i	−0.983650	−	0.180090i	$-0.942361\pi$
$191$		−	4.97779i		−	0.360181i	0.983650	−	0.180090i	$-0.0576390\pi$
$192$	−0.254454	−	1.81496i	−0.0183636	−	0.130983i
$193$	10.0969			0.726792			0.363396	−	0.931635i	$-0.381617\pi$
$193$	10.0969			0.726792			0.363396	+	0.931635i	$0.381617\pi$
$194$		0			0
$195$		0			0
$196$	−24.9201			−1.78000
$197$			11.9680i			0.852688i	0.904561	+	0.426344i	$0.140199\pi$
$197$			11.9680i			0.852688i	−0.904561	+	0.426344i	$0.859801\pi$
$198$	17.1776	−	4.91312i	1.22076	−	0.349160i
$199$	−28.1505			−1.99554			−0.997770	−	0.0667521i	$-0.978736\pi$
$199$	−28.1505			−1.99554			−0.997770	+	0.0667521i	$0.978736\pi$
$200$		−	6.96403i		−	0.492432i
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$			28.2899i			1.97105i
$207$	−5.11336	−	17.8777i	−0.355403	−	1.24259i
$208$	−17.8463			−1.23742
$209$		−	23.2402i		−	1.60756i
$210$		0			0
$211$		0			0		1.00000			$0$
$211$		0			0		−1.00000			$\pi$
$212$			3.25299i			0.223416i
$213$		0			0
$214$		0			0
$215$		0			0
$216$	−6.61451	+	2.93699i	−0.450060	+	0.199837i
$217$		0			0
$218$		−	36.1269i		−	2.44682i
$219$	−1.46807	−	10.4714i	−0.0992029	−	0.707589i
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		1.00000			$0$
$223$		0			0		−1.00000			$\pi$
$224$			31.9153i			2.13243i
$225$	14.4217	−	4.12487i	0.961447	−	0.274992i
$226$	−32.1894			−2.14121
$227$		−	29.9245i		−	1.98616i	−0.117445	−	0.993079i	$-0.537470\pi$
$227$		−	29.9245i		−	1.98616i	0.117445	−	0.993079i	$-0.462530\pi$
$228$	−2.06312	−	14.7157i	−0.136633	−	0.974571i
$229$		0			0		1.00000			$0$
$229$		0			0		−1.00000			$\pi$
$230$		0			0
$231$	−29.7536	+	4.17140i	−1.95764	+	0.274458i
$232$		0			0
$233$		0			0			−	1.00000i	$-0.5\pi$
$233$		0			0				1.00000i	$0.5\pi$
$234$	−5.34113	−	18.6741i	−0.349160	−	1.22076i
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	8.37676i		−	0.541847i	−0.962601	−	0.270924i	$-0.912671\pi$
$239$		−	8.37676i		−	0.541847i	0.962601	−	0.270924i	$-0.0873291\pi$
$240$		0			0
$241$	10.8234			0.697199			0.348600	−	0.937272i	$-0.386657\pi$
$241$	10.8234			0.697199			0.348600	+	0.937272i	$0.386657\pi$
$242$		−	19.7521i		−	1.26971i
$243$	−10.0000	−	11.9583i	−0.641500	−	0.767123i
$244$		0			0
$245$		0			0
$246$	−3.26308	−	23.2747i	−0.208046	−	1.48394i
$247$	−25.2648			−1.60756
$248$		0			0
$249$	30.0600	−	4.21436i	1.90497	−	0.267074i
$250$		0			0
$251$			16.0979i			1.01609i	0.861330	+	0.508045i	$0.169632\pi$
$251$			16.0979i			1.01609i	−0.861330	+	0.508045i	$0.830368\pi$
$252$	−18.4696	+	5.28266i	−1.16348	+	0.332776i
$253$	−20.5571			−1.29241
$254$		0			0
$255$		0			0
$256$	20.6194			1.28871
$257$			29.7426i			1.85529i	0.373457	+	0.927647i	$0.378172\pi$
$257$			29.7426i			1.85529i	−0.373457	+	0.927647i	$0.621828\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0			−	1.00000i	$-0.5\pi$
$263$		0			0				1.00000i	$0.5\pi$
$264$	1.11087	+	7.92357i	0.0683695	+	0.487662i
$265$		0			0
$266$			65.8073i			4.03490i
$267$		0			0
$268$		0			0
$269$		−	7.44655i		−	0.454024i	−0.973892	−	0.227012i	$-0.927104\pi$
$269$		−	7.44655i		−	0.454024i	0.973892	−	0.227012i	$-0.0728957\pi$
$270$		0			0
$271$	19.2167			1.16733			0.583667	−	0.811993i	$-0.301617\pi$
$271$	19.2167			1.16733			0.583667	+	0.811993i	$0.301617\pi$
$272$		0			0
$273$	4.53479	+	32.3456i	0.274458	+	1.95764i
$274$		0			0
$275$		−	16.5831i		−	1.00000i
$276$	−13.0167	+	1.82492i	−0.783515	+	0.109848i
$277$		0			0		1.00000			$0$
$277$		0			0		−1.00000			$\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			33.5158i			1.99938i	0.0248074	+	0.999692i	$0.492103\pi$
$281$			33.5158i			1.99938i	−0.0248074	+	0.999692i	$0.507897\pi$
$282$		0			0
$283$		0			0		1.00000			$0$
$283$		0			0		−1.00000			$\pi$
$284$		0			0
$285$		0			0
$286$	−21.4728			−1.26971
$287$			39.5220i			2.33291i
$288$	5.03419	+	17.6009i	0.296643	+	1.03714i
$289$	−17.0000			−1.00000
$290$		0			0
$291$		0			0
$292$	−7.47433			−0.437402
$293$			26.5330i			1.55007i	0.631916	+	0.775037i	$0.282269\pi$
$293$			26.5330i			1.55007i	−0.631916	+	0.775037i	$0.717731\pi$
$294$	62.6903	−	8.78908i	3.65618	−	0.512590i
$295$		0			0
$296$		0			0
$297$	−15.7508	+	6.99371i	−0.913955	+	0.405816i
$298$	22.2296			1.28773
$299$			22.3479i			1.29241i
$300$	−1.47214	−	10.5004i	−0.0849941	−	0.606242i
$301$		0			0
$302$		−	25.8972i		−	1.49021i
$303$		0			0
$304$	34.6833			1.98922
$305$		0			0
$306$		0			0
$307$	28.8444			1.64624			0.823119	−	0.567869i	$-0.192232\pi$
$307$	28.8444			1.64624			0.823119	+	0.567869i	$0.192232\pi$
$308$			21.2377i			1.21013i
$309$	−3.78868	−	27.0237i	−0.215530	−	1.53732i
$310$		0			0
$311$		−	25.5611i		−	1.44944i	−0.689045	−	0.724719i	$-0.741970\pi$
$311$		−	25.5611i		−	1.44944i	0.689045	−	0.724719i	$-0.258030\pi$
$312$	8.61383	−	1.20765i	0.487662	−	0.0683695i
$313$	−23.5113			−1.32894			−0.664469	−	0.747316i	$-0.731342\pi$
$313$	−23.5113			−1.32894			−0.664469	+	0.747316i	$0.731342\pi$
$314$		−	25.5043i		−	1.43929i
$315$		0			0
$316$		0			0
$317$		0			0			−	1.00000i	$-0.5\pi$
$317$		0			0				1.00000i	$0.5\pi$
$318$	−1.14730	−	8.18341i	−0.0643374	−	0.458903i
$319$		0			0
$320$		0			0
$321$		0			0
$322$	58.2097			3.24390
$323$		0			0
$324$	−9.35254	+	5.82666i	−0.519586	+	0.323703i
$325$	−18.0278			−1.00000
$326$		0			0
$327$	4.83823	+	34.5099i	0.267555	+	1.90840i
$328$	10.5250			0.581145
$329$		0			0
$330$		0			0
$331$		0			0		1.00000			$0$
$331$		0			0		−1.00000			$\pi$
$332$		−	21.4564i		−	1.17757i
$333$		0			0
$334$	−40.8364			−2.23447
$335$		0			0
$336$	−6.22533	−	44.4037i	−0.339619	−	2.42242i
$337$		0			0		1.00000			$0$
$337$		0			0		−1.00000			$\pi$
$338$			23.3434i			1.26971i
$339$	30.7487	−	4.31091i	1.67004	−	0.234137i
$340$		0			0
$341$		0			0
$342$	10.3802	+	36.2920i	0.561297	+	1.96245i
$343$	−69.8418			−3.77110
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0			−	1.00000i	$-0.5\pi$
$347$		0			0				1.00000i	$0.5\pi$
$348$		0			0
$349$	−21.2836			−1.13929			−0.569643	−	0.821892i	$-0.692919\pi$
$349$	−21.2836			−1.13929			−0.569643	+	0.821892i	$0.692919\pi$
$350$			46.9569i			2.50995i
$351$	7.60296	+	17.1229i	0.405816	+	0.913955i
$352$	20.2388			1.07873
$353$		0			0			−	1.00000i	$-0.5\pi$
$353$		0			0				1.00000i	$0.5\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$	−42.9456			−2.26974
$359$		−	37.4612i		−	1.97713i	−0.150798	−	0.988565i	$-0.548184\pi$
$359$		−	37.4612i		−	1.97713i	0.150798	−	0.988565i	$-0.451816\pi$
$360$		0			0
$361$	30.1008			1.58425
$362$			18.7970i			0.987947i
$363$	2.64527	+	18.8680i	0.138841	+	0.990315i
$364$	23.0878			1.21013
$365$		0			0
$366$		0			0
$367$	−31.0363			−1.62008			−0.810041	−	0.586374i	$-0.800555\pi$
$367$	−31.0363			−1.62008			−0.810041	+	0.586374i	$0.800555\pi$
$368$		−	30.6790i		−	1.59926i
$369$	6.23406	+	21.7960i	0.324532	+	1.13465i
$370$		0			0
$371$			13.8960i			0.721443i
$372$		0			0
$373$		0			0		1.00000			$0$
$373$		0			0		−1.00000			$\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$	44.6001	−	19.8034i	2.29398	−	1.01858i
$379$		0			0		1.00000			$0$
$379$		0			0		−1.00000			$\pi$
$380$		0			0
$381$		0			0
$382$	8.93835			0.457326
$383$		0			0			−	1.00000i	$-0.5\pi$
$383$		0			0				1.00000i	$0.5\pi$
$384$	−17.6750	+	2.47801i	−0.901975	+	0.126455i
$385$		0			0
$386$			18.1305i			0.922817i
$387$		0			0
$388$		0			0
$389$			38.5097i			1.95252i	0.216606	+	0.976259i	$0.430501\pi$
$389$			38.5097i			1.95252i	−0.216606	+	0.976259i	$0.569499\pi$
$390$		0			0
$391$		0			0
$392$			28.3490i			1.43184i
$393$		0			0
$394$	−21.4904			−1.08267
$395$		0			0
$396$	3.34996	+	11.7124i	0.168342	+	0.588570i
$397$		0			0		1.00000			$0$
$397$		0			0		−1.00000			$\pi$
$398$		−	50.5484i		−	2.53376i
$399$	−8.81313	−	62.8619i	−0.441208	−	3.14703i
$400$	24.7484			1.23742
$401$		0			0			−	1.00000i	$-0.5\pi$
$401$		0			0				1.00000i	$0.5\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	7.21110			0.356566			0.178283	−	0.983979i	$-0.442946\pi$
$409$	7.21110			0.356566			0.178283	+	0.983979i	$0.442946\pi$
$410$		0			0
$411$		0			0
$412$	−19.2892			−0.950309
$413$		0			0
$414$	32.1020	−	9.18178i	1.57773	−	0.451260i
$415$		0			0
$416$		−	22.0019i		−	1.07873i
$417$		0			0
$418$	41.7312			2.04114
$419$		−	40.8907i		−	1.99764i	−0.0485577	−	0.998820i	$-0.515462\pi$
$419$		−	40.8907i		−	1.99764i	0.0485577	−	0.998820i	$-0.484538\pi$
$420$		0			0
$421$		0			0		1.00000			$0$
$421$		0			0		−1.00000			$\pi$
$422$		0			0
$423$		0			0
$424$	3.70059			0.179717
$425$		0			0
$426$		0			0
$427$		0			0
$428$		0			0
$429$	20.5117	−	2.87571i	0.990315	−	0.138841i
$430$		0			0
$431$		−	32.6382i		−	1.57213i	−0.618146	−	0.786063i	$-0.712116\pi$
$431$		−	32.6382i		−	1.57213i	0.618146	−	0.786063i	$-0.287884\pi$
$432$	−10.4373	−	23.5062i	−0.502164	−	1.13094i
$433$	19.1253			0.919105			0.459552	−	0.888151i	$-0.348010\pi$
$433$	19.1253			0.919105			0.459552	+	0.888151i	$0.348010\pi$
$434$		0			0
$435$		0			0
$436$	24.6327			1.17969
$437$		−	43.4320i		−	2.07763i
$438$	18.8029	−	2.63613i	0.898435	−	0.125959i
$439$		0			0		1.00000			$0$
$439$		0			0		−1.00000			$\pi$
$440$		0			0
$441$	−58.7074	+	16.7914i	−2.79559	+	0.799591i
$442$		0			0
$443$		−	3.70150i		−	0.175864i	−0.996126	−	0.0879318i	$-0.971974\pi$
$443$		−	3.70150i		−	0.175864i	0.996126	−	0.0879318i	$-0.0280258\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	−21.2346	+	2.97706i	−1.00436	+	0.140810i
$448$	−5.53403			−0.261458
$449$		0			0			−	1.00000i	$-0.5\pi$
$449$		0			0				1.00000i	$0.5\pi$
$450$	7.40681	+	25.8963i	0.349160	+	1.22076i
$451$	25.0626			1.18015
$452$		−	21.9480i		−	1.03235i
$453$	3.46823	+	24.7381i	0.162952	+	1.16230i
$454$	53.7338			2.52185
$455$		0			0
$456$	−16.7405	+	2.34699i	−0.783947	+	0.109908i
$457$	21.9240			1.02556			0.512781	−	0.858520i	$-0.328615\pi$
$457$	21.9240			1.02556			0.512781	+	0.858520i	$0.328615\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			2.39711i			0.111645i	0.998441	+	0.0558223i	$0.0177780\pi$
$461$			2.39711i			0.111645i	−0.998441	+	0.0558223i	$0.982222\pi$
$462$	−7.49036	−	53.4269i	−0.348483	−	2.48564i
$463$		0			0		1.00000			$0$
$463$		0			0		−1.00000			$\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			23.9165i			1.10672i	0.832941	+	0.553362i	$0.186655\pi$
$467$			23.9165i			1.10672i	−0.832941	+	0.553362i	$0.813345\pi$
$468$	12.7327	−	3.64179i	0.588570	−	0.168342i
$469$		0			0
$470$		0			0
$471$	3.41562	+	24.3628i	0.157383	+	1.12258i
$472$		0			0
$473$		0			0
$474$		0			0
$475$	35.0360			1.60756
$476$		0			0
$477$	2.19190	+	7.66349i	0.100360	+	0.350887i
$478$	15.0417			0.687991
$479$			6.63325i			0.303081i	0.988451	+	0.151540i	$0.0484234\pi$
$479$			6.63325i			0.303081i	−0.988451	+	0.151540i	$0.951577\pi$
$480$		0			0
$481$		0			0
$482$			19.4351i			0.885243i
$483$	−55.6043	+	7.79563i	−2.53008	+	0.354714i
$484$	13.4678			0.612171
$485$		0			0
$486$	21.4728	−	17.9565i	0.974026	−	0.814521i
$487$		0			0		1.00000			$0$
$487$		0			0		−1.00000			$\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		0			0			−	1.00000i	$-0.5\pi$
$491$		0			0				1.00000i	$0.5\pi$
$492$	15.8696	−	2.22490i	0.715458	−	0.100306i
$493$		0			0
$494$		−	45.3666i		−	2.04114i
$495$		0			0
$496$		0			0
$497$		0			0
$498$	7.56749	+	53.9770i	0.339107	+	2.41877i
$499$		0			0		1.00000			$0$
$499$		0			0		−1.00000			$\pi$
$500$		0			0
$501$	39.0087	−	5.46895i	1.74278	−	0.244335i
$502$	−28.9061			−1.29014
$503$		0			0			−	1.00000i	$-0.5\pi$
$503$		0			0				1.00000i	$0.5\pi$
$504$	6.00953	+	21.0110i	0.267686	+	0.935904i
$505$		0			0
$506$		−	36.9133i		−	1.64099i
$507$	−3.12622	−	22.2986i	−0.138841	−	0.990315i
$508$		0			0
$509$		0			0			−	1.00000i	$-0.5\pi$
$509$		0			0				1.00000i	$0.5\pi$
$510$		0			0
$511$	−31.9285			−1.41243
$512$			16.4162i			0.725500i
$513$	−14.7759	−	33.2775i	−0.652374	−	1.46924i
$514$	−53.4072			−2.35569
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$			39.6423i			1.73676i	0.495897	+	0.868381i	$0.334839\pi$
$521$			39.6423i			1.73676i	−0.495897	+	0.868381i	$0.665161\pi$
$522$		0			0
$523$		0			0		1.00000			$0$
$523$		0			0		−1.00000			$\pi$
$524$		0			0
$525$	−6.28863	−	44.8552i	−0.274458	−	1.95764i
$526$		0			0
$527$		0			0
$528$	−28.1583	+	3.94775i	−1.22543	+	0.171804i
$529$	−15.4177			−0.670333
$530$		0			0
$531$		0			0
$532$	−44.8700			−1.94536
$533$		−	27.2459i		−	1.18015i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	41.0234	−	5.75142i	1.77029	−	0.248192i
$538$	13.3714			0.576480
$539$			67.5060i			2.90769i
$540$		0			0
$541$	41.4775			1.78326			0.891628	−	0.452770i	$-0.149564\pi$
$541$	41.4775			1.78326			0.891628	+	0.452770i	$0.149564\pi$
$542$			34.5064i			1.48218i
$543$	−2.51735	−	17.9557i	−0.108030	−	0.770551i
$544$		0			0
$545$		0			0
$546$	−58.0812	+	8.14288i	−2.48564	+	0.348483i
$547$		0			0		1.00000			$0$
$547$		0			0		−1.00000			$\pi$
$548$		0			0
$549$		0			0
$550$	29.7774			1.26971
$551$		0			0
$552$	2.07603	+	14.8078i	0.0883616	+	0.630261i
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$			42.6064i			1.80529i	0.430385	+	0.902645i	$0.358378\pi$
$557$			42.6064i			1.80529i	−0.430385	+	0.902645i	$0.641622\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	−60.1825			−2.53864
$563$		0			0			−	1.00000i	$-0.5\pi$
$563$		0			0				1.00000i	$0.5\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−39.9518	+	24.8901i	−1.67782	+	1.04528i
$568$		0			0
$569$		0			0			−	1.00000i	$-0.5\pi$
$569$		0			0				1.00000i	$0.5\pi$
$570$		0			0
$571$		0			0		1.00000			$0$
$571$		0			0		−1.00000			$\pi$
$572$		−	14.6410i		−	0.612171i
$573$	−8.53829	+	1.19705i	−0.356692	+	0.0500077i
$574$	−70.9675			−2.96213
$575$		−	30.9910i		−	1.29241i
$576$	−3.05196	+	0.872918i	−0.127165	+	0.0363716i
$577$		0			0		1.00000			$0$
$577$		0			0		−1.00000			$\pi$
$578$		−	30.5260i		−	1.26971i
$579$	−2.42809	−	17.3190i	−0.100908	−	0.719753i
$580$		0			0
$581$		−	91.6566i		−	3.80256i
$582$		0			0
$583$	8.81204			0.364957
$584$			8.50277i			0.351847i
$585$		0			0
$586$	−47.6439			−1.96815
$587$		0			0			−	1.00000i	$-0.5\pi$
$587$		0			0				1.00000i	$0.5\pi$
$588$	5.99274	+	42.7448i	0.247137	+	1.76276i
$589$		0			0
$590$		0			0
$591$	20.5285	−	2.87806i	0.844430	−	0.118388i
$592$		0			0
$593$			47.4294i			1.94769i	0.227205	+	0.973847i	$0.427041\pi$
$593$			47.4294i			1.94769i	−0.227205	+	0.973847i	$0.572959\pi$
$594$	−12.5582	−	28.2829i	−0.515270	−	1.16046i
$595$		0			0
$596$			15.1570i			0.620856i
$597$	6.76961	+	48.2859i	0.277062	+	1.97621i
$598$	−40.1289			−1.64099
$599$			36.1888i			1.47863i	0.673357	+	0.739317i	$0.264852\pi$
$599$			36.1888i			1.47863i	−0.673357	+	0.739317i	$0.735148\pi$
$600$	−11.9452	+	1.67470i	−0.487662	+	0.0683695i
$601$		0			0		1.00000			$0$
$601$		0			0		−1.00000			$\pi$
$602$		0			0
$603$		0			0
$604$	17.6577			0.718482
$605$		0			0
$606$		0			0
$607$		0			0		1.00000			$0$
$607$		0			0		−1.00000			$\pi$
$608$			42.7596i			1.73413i
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	32.3287			1.30574			0.652872	−	0.757468i	$-0.273564\pi$
$613$	32.3287			1.30574			0.652872	+	0.757468i	$0.273564\pi$
$614$			51.7943i			2.09025i
$615$		0			0
$616$	24.1600			0.973433
$617$		0			0			−	1.00000i	$-0.5\pi$
$617$		0			0				1.00000i	$0.5\pi$
$618$	48.5250	−	6.80312i	1.95196	−	0.273662i
$619$		0			0		1.00000			$0$
$619$		0			0		−1.00000			$\pi$
$620$		0			0
$621$	−29.4355	+	13.0700i	−1.18121	+	0.524482i
$622$	45.8987			1.84037
$623$		0			0
$624$	4.29165	+	30.6113i	0.171804	+	1.22543i
$625$	25.0000			1.00000
$626$		−	42.2180i		−	1.68737i
$627$	−39.8634	+	5.58878i	−1.59199	+	0.223195i
$628$	17.3898			0.693930
$629$		0			0
$630$		0			0
$631$		0			0		1.00000			$0$
$631$		0			0		−1.00000			$\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$	5.57977	−	0.782275i	0.221252	−	0.0310192i
$637$	73.3868			2.90769
$638$		0			0
$639$		0			0
$640$		0			0
$641$			19.8429i			0.783749i	0.920019	+	0.391875i	$0.128173\pi$
$641$			19.8429i			0.783749i	−0.920019	+	0.391875i	$0.871827\pi$
$642$		0			0
$643$		0			0		1.00000			$0$
$643$		0			0		−1.00000			$\pi$
$644$			39.6896i			1.56399i
$645$		0			0
$646$		0			0
$647$		−	46.1444i		−	1.81412i	−0.420997	−	0.907062i	$-0.638320\pi$
$647$		−	46.1444i		−	1.81412i	0.420997	−	0.907062i	$-0.361680\pi$
$648$	6.62839	+	10.6394i	0.260388	+	0.417956i
$649$		0			0
$650$		−	32.3715i		−	1.26971i
$651$		0			0
$652$		0			0
$653$		−	47.8330i		−	1.87185i	−0.352197	−	0.935926i	$-0.614565\pi$
$653$		−	47.8330i		−	1.87185i	0.352197	−	0.935926i	$-0.385435\pi$
$654$	−61.9675	+	8.68775i	−2.42312	+	0.339718i
$655$		0			0
$656$			37.4030i			1.46034i
$657$	−17.6082	+	5.03628i	−0.686963	+	0.196484i
$658$		0			0
$659$		0			0			−	1.00000i	$-0.5\pi$
$659$		0			0				1.00000i	$0.5\pi$
$660$		0			0
$661$		0			0		1.00000			$0$
$661$		0			0		−1.00000			$\pi$
$662$		0			0
$663$		0			0
$664$	−24.4087			−0.947243
$665$		0			0
$666$		0			0
$667$		0			0
$668$		−	27.8439i		−	1.07731i
$669$		0			0
$670$		0			0
$671$		0			0
$672$	54.7435	−	7.67494i	2.11177	−	0.296067i
$673$		0			0		1.00000			$0$
$673$		0			0		−1.00000			$\pi$
$674$		0			0
$675$	−10.5434	−	23.7452i	−0.405816	−	0.913955i
$676$	−15.9164			−0.612171
$677$		0			0			−	1.00000i	$-0.5\pi$
$677$		0			0				1.00000i	$0.5\pi$
$678$	7.74087	+	55.2137i	0.297286	+	2.12047i
$679$		0			0
$680$		0			0
$681$	−51.3288	+	7.19621i	−1.96692	+	0.275759i
$682$		0			0
$683$		0			0			−	1.00000i	$-0.5\pi$
$683$		0			0				1.00000i	$0.5\pi$
$684$	−24.7453	+	7.07762i	−0.946161	+	0.270620i
$685$		0			0
$686$		−	125.411i		−	4.78821i
$687$		0			0
$688$		0			0
$689$		−	9.57969i		−	0.364957i
$690$		0			0
$691$		0			0		1.00000			$0$
$691$		0			0		−1.00000			$\pi$
$692$		0			0
$693$	14.3102	+	50.0325i	0.543600	+	1.90058i
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		−	38.2178i		−	1.44657i
$699$		0			0
$700$	−32.0171			−1.21013
$701$		0			0			−	1.00000i	$-0.5\pi$
$701$		0			0				1.00000i	$0.5\pi$
$702$	−30.7467	+	13.6522i	−1.16046	+	0.515270i
$703$		0			0
$704$			3.50937i			0.132264i
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$		0			0		1.00000			$0$
$709$		0			0		−1.00000			$\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		−	29.2820i		−	1.09432i
$717$	−14.3684	+	2.01443i	−0.536600	+	0.0752304i
$718$	67.2671			2.51039
$719$			23.9165i			0.891936i	0.895049	+	0.445968i	$0.147140\pi$
$719$			23.9165i			0.891936i	−0.895049	+	0.445968i	$0.852860\pi$
$720$		0			0
$721$	−82.3986			−3.06868
$722$			54.0503i			2.01154i
$723$	−2.60281	−	18.5652i	−0.0967995	−	0.690447i
$724$	−12.8165			−0.476322
$725$		0			0
$726$	−33.8803	+	4.74996i	−1.25742	+	0.176288i
$727$	−53.4694			−1.98307			−0.991536	−	0.129834i	$-0.958556\pi$
$727$	−53.4694			−1.98307			−0.991536	+	0.129834i	$0.958556\pi$
$728$		−	26.2647i		−	0.973433i
$729$	−18.1069	+	20.0285i	−0.670627	+	0.741795i
$730$		0			0
$731$		0			0
$732$		0			0
$733$	42.2040			1.55884			0.779420	−	0.626502i	$-0.215514\pi$
$733$	42.2040			1.55884			0.779420	+	0.626502i	$0.215514\pi$
$734$		−	55.7302i		−	2.05704i
$735$		0			0
$736$	37.8229			1.39417
$737$		0			0
$738$	−39.1379	+	11.1942i	−1.44068	+	0.412063i
$739$	−4.50357			−0.165666			−0.0828332	−	0.996563i	$-0.526397\pi$
$739$	−4.50357			−0.165666			−0.0828332	+	0.996563i	$0.526397\pi$
$740$		0			0
$741$	6.07565	+	43.3361i	0.223195	+	1.59199i
$742$	−24.9522			−0.916026
$743$		−	33.1662i		−	1.21675i	−0.793649	−	0.608376i	$-0.791821\pi$
$743$		−	33.1662i		−	1.21675i	0.793649	−	0.608376i	$-0.208179\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	−14.4576	−	50.5477i	−0.528975	−	1.84944i
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−33.9220			−1.23783			−0.618916	−	0.785457i	$-0.712428\pi$
$751$	−33.9220			−1.23783			−0.618916	+	0.785457i	$0.712428\pi$
$752$		0			0
$753$	27.6123	−	3.87120i	1.00625	−	0.141074i
$754$		0			0
$755$		0			0
$756$	13.5028	+	30.4101i	0.491091	+	1.10601i
$757$	55.0208			1.99976			0.999882	−	0.0153878i	$-0.00489828\pi$
$757$	55.0208			1.99976			0.999882	+	0.0153878i	$0.00489828\pi$
$758$		0			0
$759$	4.94355	+	35.2611i	0.179439	+	1.27990i
$760$		0			0
$761$		−	17.2028i		−	0.623600i	−0.950148	−	0.311800i	$-0.899068\pi$
$761$		−	17.2028i		−	0.623600i	0.950148	−	0.311800i	$-0.100932\pi$
$762$		0			0
$763$	105.225			3.80940
$764$			6.09452i			0.220492i
$765$		0			0
$766$		0			0
$767$		0			0
$768$	−4.95853	−	35.3680i	−0.178926	−	1.27623i
$769$	−51.9376			−1.87292			−0.936459	−	0.350776i	$-0.885918\pi$
$769$	−51.9376			−1.87292			−0.936459	+	0.350776i	$0.885918\pi$
$770$		0			0
$771$	51.0168	−	7.15247i	1.83733	−	0.257590i
$772$	−12.3621			−0.444921
$773$		0			0			−	1.00000i	$-0.5\pi$
$773$		0			0				1.00000i	$0.5\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$	−69.1497			−2.47914
$779$			52.9510i			1.89717i
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−100.745			−3.59803
$785$		0			0
$786$		0			0
$787$	−5.95661			−0.212330			−0.106165	−	0.994349i	$-0.533857\pi$
$787$	−5.95661			−0.212330			−0.106165	+	0.994349i	$0.533857\pi$
$788$		−	14.6530i		−	0.521991i
$789$		0			0
$790$		0			0
$791$		−	93.7566i		−	3.33360i
$792$	13.3240	−	3.81090i	0.473447	−	0.135415i
$793$		0			0
$794$		0			0
$795$		0			0
$796$	34.4659			1.22161
$797$		−	47.8330i		−	1.69433i	−0.531327	−	0.847167i	$-0.678307\pi$
$797$		−	47.8330i		−	1.69433i	0.531327	−	0.847167i	$-0.321693\pi$
$798$	112.878	−	15.8253i	3.99582	−	0.560208i
$799$		0			0
$800$			30.5112i			1.07873i
$801$		0			0
$802$		0			0
$803$			20.2472i			0.714510i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	−12.7729	+	1.79074i	−0.449627	+	0.0630369i
$808$		0			0
$809$		0			0			−	1.00000i	$-0.5\pi$
$809$		0			0				1.00000i	$0.5\pi$
$810$		0			0
$811$	25.4239			0.892755			0.446377	−	0.894845i	$-0.352714\pi$
$811$	25.4239			0.892755			0.446377	+	0.894845i	$0.352714\pi$
$812$		0			0
$813$	−4.62122	−	32.9620i	−0.162073	−	1.15603i
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$			12.9486i			0.452737i
$819$	54.3910	−	15.5568i	1.90058	−	0.543600i
$820$		0			0
$821$		−	53.0660i		−	1.85202i	−0.377504	−	0.926008i	$-0.623217\pi$
$821$		−	53.0660i		−	1.85202i	0.377504	−	0.926008i	$-0.376783\pi$
$822$		0			0
$823$	−28.1653			−0.981779			−0.490890	−	0.871222i	$-0.663328\pi$
$823$	−28.1653			−0.981779			−0.490890	+	0.871222i	$0.663328\pi$
$824$			21.9433i			0.764431i
$825$	−28.4446	+	3.98789i	−0.990315	+	0.138841i
$826$		0			0
$827$			13.1710i			0.458000i	0.973426	+	0.229000i	$0.0735456\pi$
$827$			13.1710i			0.458000i	−0.973426	+	0.229000i	$0.926454\pi$
$828$	6.26050	+	21.8884i	0.217567	+	0.760675i
$829$	−48.8155			−1.69543			−0.847716	−	0.530450i	$-0.822023\pi$
$829$	−48.8155			−1.69543			−0.847716	+	0.530450i	$0.822023\pi$
$830$		0			0
$831$		0			0
$832$	3.81509			0.132264
$833$		0			0
$834$		0			0
$835$		0			0
$836$			28.4540i			0.984102i
$837$		0			0
$838$	73.4252			2.53643
$839$		0			0			−	1.00000i	$-0.5\pi$
$839$		0			0				1.00000i	$0.5\pi$
$840$		0			0
$841$	−29.0000			−1.00000
$842$		0			0
$843$	57.4888	−	8.05984i	1.98002	−	0.277596i
$844$		0			0
$845$		0			0
$846$		0			0
$847$	57.5310			1.97679
$848$			13.1509i			0.451605i
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$	−4.29993			−0.147227			−0.0736134	−	0.997287i	$-0.523453\pi$
$853$	−4.29993			−0.147227			−0.0736134	+	0.997287i	$0.523453\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0			−	1.00000i	$-0.5\pi$
$857$		0			0				1.00000i	$0.5\pi$
$858$	5.16375	+	36.8318i	0.176288	+	1.25742i
$859$	51.9725			1.77328			0.886639	−	0.462462i	$-0.153034\pi$
$859$	51.9725			1.77328			0.886639	+	0.462462i	$0.153034\pi$
$860$		0			0
$861$	67.7912	−	9.50421i	2.31032	−	0.323903i
$862$	58.6066			1.99615
$863$		0			0			−	1.00000i	$-0.5\pi$
$863$		0			0				1.00000i	$0.5\pi$
$864$	28.9798	−	12.8677i	0.985914	−	0.437767i
$865$		0			0
$866$			34.3423i			1.16700i
$867$	4.08814	+	29.1597i	0.138841	+	0.990315i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		−	28.0221i		−	0.948948i
$873$		0			0
$874$	77.9884			2.63800
$875$		0			0
$876$	1.79742	+	12.8205i	0.0607291	+	0.433166i
$877$	−35.9384			−1.21355			−0.606777	−	0.794872i	$-0.707538\pi$
$877$	−35.9384			−1.21355			−0.606777	+	0.794872i	$0.707538\pi$
$878$		0			0
$879$	45.5114	−	6.38062i	1.53506	−	0.215213i
$880$		0			0
$881$			59.0930i			1.99089i	0.0953201	+	0.995447i	$0.469613\pi$
$881$			59.0930i			1.99089i	−0.0953201	+	0.995447i	$0.530387\pi$
$882$	−30.1514	−	105.418i	−1.01525	−	3.54960i
$883$	57.7440			1.94324			0.971620	−	0.236549i	$-0.0760164\pi$
$883$	57.7440			1.94324			0.971620	+	0.236549i	$0.0760164\pi$
$884$		0			0
$885$		0			0
$886$	6.64658			0.223296
$887$		0			0			−	1.00000i	$-0.5\pi$
$887$		0			0				1.00000i	$0.5\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	15.7839	+	25.3351i	0.528779	+	0.848759i
$892$		0			0
$893$		0			0
$894$	−5.34575	−	38.1299i	−0.178789	−	1.27525i
$895$		0			0
$896$			53.8933i			1.80045i
$897$	38.3328	−	5.37420i	1.27990	−	0.179439i
$898$		0			0
$899$		0			0
$900$	−17.6571	+	5.05026i	−0.588570	+	0.168342i
$901$		0			0
$902$			45.0036i			1.49846i
$903$		0			0
$904$	−24.9680			−0.830423
$905$		0			0
$906$	−44.4208	+	6.22772i	−1.47578	+	0.206902i
$907$	6.44682			0.214063			0.107032	−	0.994256i	$-0.465865\pi$
$907$	6.44682			0.214063			0.107032	+	0.994256i	$0.465865\pi$
$908$			36.6378i			1.21587i
$909$		0			0
$910$		0			0
$911$		−	50.7904i		−	1.68276i	−0.540444	−	0.841380i	$-0.681744\pi$
$911$		−	50.7904i		−	1.68276i	0.540444	−	0.841380i	$-0.318256\pi$
$912$	−8.34060	−	59.4914i	−0.276185	−	1.96996i
$913$	−58.1234			−1.92360
$914$			39.3677i			1.30217i
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$		0			0		1.00000			$0$
$919$		0			0		−1.00000			$\pi$
$920$		0			0
$921$	−6.93647	−	49.4761i	−0.228564	−	1.63029i
$922$	−4.30436			−0.141757
$923$		0			0
$924$	36.4286	−	5.10722i	1.19841	−	0.168015i
$925$		0			0
$926$		0			0
$927$	−45.4420	+	12.9972i	−1.49251	+	0.426886i
$928$		0			0
$929$		0			0			−	1.00000i	$-0.5\pi$
$929$		0			0				1.00000i	$0.5\pi$
$930$		0			0
$931$	−142.623			−4.67429
$932$		0			0
$933$	−43.8443	+	6.14690i	−1.43540	+	0.201241i
$934$	−42.9456			−1.40522
$935$		0			0
$936$	−4.14289	−	14.4847i	−0.135415	−	0.473447i
$937$		0			0		1.00000			$0$
$937$		0			0		−1.00000			$\pi$
$938$		0			0
$939$	5.65397	+	40.3284i	0.184510	+	1.31607i
$940$		0			0
$941$			55.0635i			1.79502i	0.440994	+	0.897510i	$0.354626\pi$
$941$			55.0635i			1.79502i	−0.440994	+	0.897510i	$0.645374\pi$
$942$	−43.7469	+	6.13324i	−1.42535	+	0.199832i
$943$	46.8377			1.52525
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0			−	1.00000i	$-0.5\pi$
$947$		0			0				1.00000i	$0.5\pi$
$948$		0			0
$949$	22.0111			0.714510
$950$			62.9122i			2.04114i
$951$		0			0
$952$		0			0
$953$		0			0			−	1.00000i	$-0.5\pi$
$953$		0			0				1.00000i	$0.5\pi$
$954$	−13.7609	+	3.93587i	−0.445526	+	0.127429i
$955$		0			0
$956$			10.2560i			0.331703i
$957$		0			0
$958$	−11.9110			−0.384826
$959$		0			0
$960$		0			0
$961$	31.0000			1.00000
$962$		0			0
$963$		0			0
$964$	−13.2516			−0.426805
$965$		0			0
$966$	−13.9982	−	99.8456i	−0.450384	−	3.21248i
$967$	−22.8264			−0.734048			−0.367024	−	0.930211i	$-0.619623\pi$
$967$	−22.8264			−0.734048			−0.367024	+	0.930211i	$0.619623\pi$
$968$		−	15.3209i		−	0.492432i
$969$		0			0
$970$		0			0
$971$		−	20.9194i		−	0.671335i	−0.941981	−	0.335667i	$-0.891038\pi$
$971$		−	20.9194i		−	0.671335i	0.941981	−	0.335667i	$-0.108962\pi$
$972$	12.2434	+	14.6410i	0.392708	+	0.469610i
$973$		0			0
$974$		0			0
$975$	4.33529	+	30.9226i	0.138841	+	0.990315i
$976$		0			0
$977$		0			0			−	1.00000i	$-0.5\pi$
$977$		0			0				1.00000i	$0.5\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	58.0305	−	16.5978i	1.85277	−	0.529927i
$982$		0			0
$983$		0			0			−	1.00000i	$-0.5\pi$
$983$		0			0				1.00000i	$0.5\pi$
$984$	−2.53103	−	18.0532i	−0.0806864	−	0.575516i
$985$		0			0
$986$		0			0
$987$		0			0
$988$	30.9327			0.984102
$989$		0			0
$990$		0			0
$991$	−19.4933			−0.619225			−0.309613	−	0.950863i	$-0.600199\pi$
$991$	−19.4933			−0.619225			−0.309613	+	0.950863i	$0.600199\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$	−36.8037	+	5.15981i	−1.16617	+	0.163495i
$997$		0			0		1.00000			$0$
$997$		0			0		−1.00000			$\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	429.2.e.d.428.15	yes	20
3.2	odd	2	inner	429.2.e.d.428.6	yes	20
11.10	odd	2	inner	429.2.e.d.428.5	✓	20
13.12	even	2	inner	429.2.e.d.428.5	✓	20
33.32	even	2	inner	429.2.e.d.428.16	yes	20
39.38	odd	2	inner	429.2.e.d.428.16	yes	20
143.142	odd	2	CM	429.2.e.d.428.15	yes	20
429.428	even	2	inner	429.2.e.d.428.6	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
429.2.e.d.428.5	✓	20	11.10	odd	2	inner
429.2.e.d.428.5	✓	20	13.12	even	2	inner
429.2.e.d.428.6	yes	20	3.2	odd	2	inner
429.2.e.d.428.6	yes	20	429.428	even	2	inner
429.2.e.d.428.15	yes	20	1.1	even	1	trivial
429.2.e.d.428.15	yes	20	143.142	odd	2	CM
429.2.e.d.428.16	yes	20	33.32	even	2	inner
429.2.e.d.428.16	yes	20	39.38	odd	2	inner

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:	\( N \)	\(=\)	\( 429 = 3 \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	429.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.79565i q^{2} +(-0.240479 - 1.71528i) q^{3} -1.22434 q^{4} +(3.08003 - 0.431815i) q^{6} -5.23009 q^{7} +1.39281i q^{8} +(-2.88434 + 0.824975i) q^{9} +O(q^{10})\)	\(q+1.79565i q^{2} +(-0.240479 - 1.71528i) q^{3} -1.22434 q^{4} +(3.08003 - 0.431815i) q^{6} -5.23009 q^{7} +1.39281i q^{8} +(-2.88434 + 0.824975i) q^{9} +3.31662i q^{11} +(0.294428 + 2.10008i) q^{12} +3.60555 q^{13} -9.39139i q^{14} -4.94967 q^{16} +(-1.48136 - 5.17925i) q^{18} -7.00719 q^{19} +(1.25773 + 8.97104i) q^{21} -5.95548 q^{22} +6.19820i q^{23} +(2.38905 - 0.334941i) q^{24} -5.00000 q^{25} +6.47429i q^{26} +(2.10868 + 4.74905i) q^{27} +6.40342 q^{28} -6.10224i q^{32} +(5.68893 - 0.797578i) q^{33} +(3.53142 - 1.01005i) q^{36} -12.5824i q^{38} +(-0.867059 - 6.18451i) q^{39} -7.55666i q^{41} +(-16.1088 + 2.25843i) q^{42} -4.06068i q^{44} -11.1298 q^{46} +(1.19029 + 8.49005i) q^{48} +20.3538 q^{49} -8.97823i q^{50} -4.41443 q^{52} -2.65693i q^{53} +(-8.52761 + 3.78644i) q^{54} -7.28451i q^{56} +(1.68508 + 12.0193i) q^{57} +(15.0854 - 4.31469i) q^{63} +1.05811 q^{64} +(1.43217 + 10.2153i) q^{66} +(10.6316 - 1.49054i) q^{69} +(-1.14903 - 4.01733i) q^{72} +6.10477 q^{73} +(1.20239 + 8.57638i) q^{75} +8.57920 q^{76} -17.3462i q^{77} +(11.1052 - 1.55693i) q^{78} +(7.63883 - 4.75902i) q^{81} +13.5691 q^{82} +17.5249i q^{83} +(-1.53989 - 10.9836i) q^{84} -4.61942 q^{88} -18.8574 q^{91} -7.58871i q^{92} +(-10.4670 + 1.46746i) q^{96} +36.5483i q^{98} +(-2.73613 - 9.56627i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 40 q^{4}+O(q^{10}) \) 20 * q - 40 * q^4	\( 20 q - 40 q^{4} + 80 q^{16} - 100 q^{25} + 10 q^{36} - 50 q^{42} + 70 q^{48} + 140 q^{49} - 160 q^{64} - 110 q^{66} + 130 q^{78}+O(q^{100}) \) 20 * q - 40 * q^4 + 80 * q^16 - 100 * q^25 + 10 * q^36 - 50 * q^42 + 70 * q^48 + 140 * q^49 - 160 * q^64 - 110 * q^66 + 130 * q^78

Introduction

L-functions

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Database

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