LMFDB - Embedded newform 4004.2.m.c.2157.15

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4004,2,Mod(2157,4004)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4004, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4004.2157");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4004 = 2^{2} \cdot 7 \cdot 11 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4004.m (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:	$31.9721009693$
Analytic rank:	$0$
Dimension:	$36$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2157.15
Character	$\chi$	$=$	4004.2157
Dual form			4004.2.m.c.2157.16

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.819032 q^{3} +1.69723i q^{5} +1.00000i q^{7} -2.32919 q^{9} +O(q^{10})$	$q-0.819032 q^{3} +1.69723i q^{5} +1.00000i q^{7} -2.32919 q^{9} -1.00000i q^{11} +(-1.93282 - 3.04372i) q^{13} -1.39008i q^{15} -2.32085 q^{17} +2.77344i q^{19} -0.819032i q^{21} -0.366187 q^{23} +2.11942 q^{25} +4.36478 q^{27} +2.95197 q^{29} +9.83529i q^{31} +0.819032i q^{33} -1.69723 q^{35} -7.39804i q^{37} +(1.58304 + 2.49290i) q^{39} +9.67457i q^{41} -10.1237 q^{43} -3.95316i q^{45} -10.4378i q^{47} -1.00000 q^{49} +1.90085 q^{51} -7.02888 q^{53} +1.69723 q^{55} -2.27154i q^{57} -11.9025i q^{59} -8.21477 q^{61} -2.32919i q^{63} +(5.16588 - 3.28043i) q^{65} -10.3012i q^{67} +0.299919 q^{69} +0.127032i q^{71} -11.1281i q^{73} -1.73587 q^{75} +1.00000 q^{77} +14.2135 q^{79} +3.41266 q^{81} +4.44302i q^{83} -3.93902i q^{85} -2.41776 q^{87} -5.15887i q^{89} +(3.04372 - 1.93282i) q^{91} -8.05542i q^{93} -4.70716 q^{95} +4.55893i q^{97} +2.32919i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 36 q - 4 q^{3} + 40 q^{9}+O(q^{10}) $ 36 * q - 4 * q^3 + 40 * q^9	$ 36 q - 4 q^{3} + 40 q^{9} - 4 q^{17} + 8 q^{23} - 80 q^{25} + 8 q^{27} + 8 q^{29} - 24 q^{39} + 32 q^{43} - 36 q^{49} - 20 q^{51} + 12 q^{53} + 32 q^{61} - 24 q^{65} + 80 q^{69} - 36 q^{75} + 36 q^{77} + 16 q^{79} + 132 q^{81} + 8 q^{91} + 56 q^{95}+O(q^{100}) $ 36 * q - 4 * q^3 + 40 * q^9 - 4 * q^17 + 8 * q^23 - 80 * q^25 + 8 * q^27 + 8 * q^29 - 24 * q^39 + 32 * q^43 - 36 * q^49 - 20 * q^51 + 12 * q^53 + 32 * q^61 - 24 * q^65 + 80 * q^69 - 36 * q^75 + 36 * q^77 + 16 * q^79 + 132 * q^81 + 8 * q^91 + 56 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4004\mathbb{Z}\right)^\times$.

$n$	$365$	$925$	$2003$	$3433$
$\chi(n)$	$1$	$-1$	$1$	$1$

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 $

$2$		0			0
$2$		0			0
$3$	−0.819032			−0.472869			−0.236434	−	0.971647i	$-0.575979\pi$
$3$	−0.819032			−0.472869			−0.236434	+	0.971647i	$0.575979\pi$
$4$		0			0
$5$			1.69723i			0.759023i	0.925187	+	0.379511i	$0.123908\pi$
$5$			1.69723i			0.759023i	−0.925187	+	0.379511i	$0.876092\pi$
$6$		0			0
$7$			1.00000i			0.377964i
$7$			1.00000i			0.377964i
$8$		0			0
$9$	−2.32919			−0.776395
$10$		0			0
$11$		−	1.00000i		−	0.301511i
$11$		−	1.00000i		−	0.301511i
$12$		0			0
$13$	−1.93282	−	3.04372i	−0.536067	−	0.844175i
$13$	−1.93282	−	3.04372i	−0.536067	−	0.844175i
$14$		0			0
$15$		−	1.39008i		−	0.358918i
$16$		0			0
$17$	−2.32085			−0.562890			−0.281445	−	0.959577i	$-0.590814\pi$
$17$	−2.32085			−0.562890			−0.281445	+	0.959577i	$0.590814\pi$
$18$		0			0
$19$			2.77344i			0.636271i	0.948045	+	0.318136i	$0.103057\pi$
$19$			2.77344i			0.636271i	−0.948045	+	0.318136i	$0.896943\pi$
$20$		0			0
$21$		−	0.819032i		−	0.178728i
$22$		0			0
$23$	−0.366187			−0.0763553			−0.0381776	−	0.999271i	$-0.512155\pi$
$23$	−0.366187			−0.0763553			−0.0381776	+	0.999271i	$0.512155\pi$
$24$		0			0
$25$	2.11942			0.423884
$26$		0			0
$27$	4.36478			0.840002
$28$		0			0
$29$	2.95197			0.548168			0.274084	−	0.961706i	$-0.411625\pi$
$29$	2.95197			0.548168			0.274084	+	0.961706i	$0.411625\pi$
$30$		0			0
$31$			9.83529i			1.76647i	0.468931	+	0.883235i	$0.344639\pi$
$31$			9.83529i			1.76647i	−0.468931	+	0.883235i	$0.655361\pi$
$32$		0			0
$33$			0.819032i			0.142575i
$34$		0			0
$35$	−1.69723			−0.286884
$36$		0			0
$37$		−	7.39804i		−	1.21623i	−0.793849	−	0.608115i	$-0.791926\pi$
$37$		−	7.39804i		−	1.21623i	0.793849	−	0.608115i	$-0.208074\pi$
$38$		0			0
$39$	1.58304	+	2.49290i	0.253489	+	0.399184i
$40$		0			0
$41$			9.67457i			1.51091i	0.655199	+	0.755457i	$0.272585\pi$
$41$			9.67457i			1.51091i	−0.655199	+	0.755457i	$0.727415\pi$
$42$		0			0
$43$	−10.1237			−1.54385			−0.771924	−	0.635715i	$-0.780705\pi$
$43$	−10.1237			−1.54385			−0.771924	+	0.635715i	$0.780705\pi$
$44$		0			0
$45$		−	3.95316i		−	0.589302i
$46$		0			0
$47$		−	10.4378i		−	1.52250i	−0.648457	−	0.761251i	$-0.724585\pi$
$47$		−	10.4378i		−	1.52250i	0.648457	−	0.761251i	$-0.275415\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$		0			0
$51$	1.90085			0.266173
$52$		0			0
$53$	−7.02888			−0.965490			−0.482745	−	0.875761i	$-0.660360\pi$
$53$	−7.02888			−0.965490			−0.482745	+	0.875761i	$0.660360\pi$
$54$		0			0
$55$	1.69723			0.228854
$56$		0			0
$57$		−	2.27154i		−	0.300873i
$58$		0			0
$59$		−	11.9025i		−	1.54958i	−0.632219	−	0.774790i	$-0.717856\pi$
$59$		−	11.9025i		−	1.54958i	0.632219	−	0.774790i	$-0.282144\pi$
$60$		0			0
$61$	−8.21477			−1.05179			−0.525897	−	0.850548i	$-0.676270\pi$
$61$	−8.21477			−1.05179			−0.525897	+	0.850548i	$0.676270\pi$
$62$		0			0
$63$		−	2.32919i		−	0.293450i
$64$		0			0
$65$	5.16588	−	3.28043i	0.640748	−	0.406887i
$66$		0			0
$67$		−	10.3012i		−	1.25849i	−0.777208	−	0.629244i	$-0.783365\pi$
$67$		−	10.3012i		−	1.25849i	0.777208	−	0.629244i	$-0.216635\pi$
$68$		0			0
$69$	0.299919			0.0361060
$70$		0			0
$71$			0.127032i			0.0150759i	0.999972	+	0.00753794i	$0.00239942\pi$
$71$			0.127032i			0.0150759i	−0.999972	+	0.00753794i	$0.997601\pi$
$72$		0			0
$73$		−	11.1281i		−	1.30244i	−0.758888	−	0.651221i	$-0.774257\pi$
$73$		−	11.1281i		−	1.30244i	0.758888	−	0.651221i	$-0.225743\pi$
$74$		0			0
$75$	−1.73587			−0.200441
$76$		0			0
$77$	1.00000			0.113961
$78$		0			0
$79$	14.2135			1.59915			0.799574	−	0.600567i	$-0.205058\pi$
$79$	14.2135			1.59915			0.799574	+	0.600567i	$0.205058\pi$
$80$		0			0
$81$	3.41266			0.379185
$82$		0			0
$83$			4.44302i			0.487684i	0.969815	+	0.243842i	$0.0784079\pi$
$83$			4.44302i			0.487684i	−0.969815	+	0.243842i	$0.921592\pi$
$84$		0			0
$85$		−	3.93902i		−	0.427246i
$86$		0			0
$87$	−2.41776			−0.259211
$88$		0			0
$89$		−	5.15887i		−	0.546839i	−0.961895	−	0.273419i	$-0.911845\pi$
$89$		−	5.15887i		−	0.546839i	0.961895	−	0.273419i	$-0.0881547\pi$
$90$		0			0
$91$	3.04372	−	1.93282i	0.319068	−	0.202614i
$92$		0			0
$93$		−	8.05542i		−	0.835308i
$94$		0			0
$95$	−4.70716			−0.482945
$96$		0			0
$97$			4.55893i			0.462890i	0.972848	+	0.231445i	$0.0743453\pi$
$97$			4.55893i			0.462890i	−0.972848	+	0.231445i	$0.925655\pi$
$98$		0			0
$99$			2.32919i			0.234092i
$100$		0			0
$101$	16.4229			1.63414			0.817068	−	0.576541i	$-0.195598\pi$
$101$	16.4229			1.63414			0.817068	+	0.576541i	$0.195598\pi$
$102$		0			0
$103$	−5.61320			−0.553085			−0.276543	−	0.961002i	$-0.589189\pi$
$103$	−5.61320			−0.553085			−0.276543	+	0.961002i	$0.589189\pi$
$104$		0			0
$105$	1.39008			0.135658
$106$		0			0
$107$	6.32562			0.611520			0.305760	−	0.952109i	$-0.401089\pi$
$107$	6.32562			0.611520			0.305760	+	0.952109i	$0.401089\pi$
$108$		0			0
$109$			0.445333i			0.0426552i	0.999773	+	0.0213276i	$0.00678930\pi$
$109$			0.445333i			0.0426552i	−0.999773	+	0.0213276i	$0.993211\pi$
$110$		0			0
$111$			6.05924i			0.575117i
$112$		0			0
$113$	1.69007			0.158988			0.0794940	−	0.996835i	$-0.474670\pi$
$113$	1.69007			0.158988			0.0794940	+	0.996835i	$0.474670\pi$
$114$		0			0
$115$		−	0.621502i		−	0.0579554i
$116$		0			0
$117$	4.50189	+	7.08938i	0.416200	+	0.655414i
$118$		0			0
$119$		−	2.32085i		−	0.212752i
$120$		0			0
$121$	−1.00000			−0.0909091
$122$		0			0
$123$		−	7.92378i		−	0.714463i
$124$		0			0
$125$			12.0833i			1.08076i
$126$		0			0
$127$	13.3288			1.18274			0.591369	−	0.806401i	$-0.298588\pi$
$127$	13.3288			1.18274			0.591369	+	0.806401i	$0.298588\pi$
$128$		0			0
$129$	8.29163			0.730037
$130$		0			0
$131$	−15.1942			−1.32752			−0.663761	−	0.747944i	$-0.731041\pi$
$131$	−15.1942			−1.32752			−0.663761	+	0.747944i	$0.731041\pi$
$132$		0			0
$133$	−2.77344			−0.240488
$134$		0			0
$135$			7.40802i			0.637581i
$136$		0			0
$137$			3.05099i			0.260663i	0.991470	+	0.130332i	$0.0416042\pi$
$137$			3.05099i			0.260663i	−0.991470	+	0.130332i	$0.958396\pi$
$138$		0			0
$139$	15.1151			1.28205			0.641023	−	0.767522i	$-0.278510\pi$
$139$	15.1151			1.28205			0.641023	+	0.767522i	$0.278510\pi$
$140$		0			0
$141$			8.54886i			0.719944i
$142$		0			0
$143$	−3.04372	+	1.93282i	−0.254528	+	0.161630i
$144$		0			0
$145$			5.01017i			0.416072i
$146$		0			0
$147$	0.819032			0.0675527
$148$		0			0
$149$		−	4.07402i		−	0.333757i	−0.985977	−	0.166878i	$-0.946631\pi$
$149$		−	4.07402i		−	0.333757i	0.985977	−	0.166878i	$-0.0533688\pi$
$150$		0			0
$151$		−	17.4602i		−	1.42089i	−0.703750	−	0.710447i	$-0.748493\pi$
$151$		−	17.4602i		−	1.42089i	0.703750	−	0.710447i	$-0.251507\pi$
$152$		0			0
$153$	5.40570			0.437025
$154$		0			0
$155$	−16.6927			−1.34079
$156$		0			0
$157$	14.0108			1.11818			0.559091	−	0.829106i	$-0.311150\pi$
$157$	14.0108			1.11818			0.559091	+	0.829106i	$0.311150\pi$
$158$		0			0
$159$	5.75688			0.456550
$160$		0			0
$161$		−	0.366187i		−	0.0288596i
$162$		0			0
$163$		−	12.1130i		−	0.948766i	−0.880319	−	0.474383i	$-0.842671\pi$
$163$		−	12.1130i		−	0.948766i	0.880319	−	0.474383i	$-0.157329\pi$
$164$		0			0
$165$	−1.39008			−0.108218
$166$		0			0
$167$		−	11.6051i		−	0.898033i	−0.893523	−	0.449016i	$-0.851774\pi$
$167$		−	11.6051i		−	0.898033i	0.893523	−	0.449016i	$-0.148226\pi$
$168$		0			0
$169$	−5.52843	+	11.7659i	−0.425264	+	0.905069i
$170$		0			0
$171$		−	6.45986i		−	0.493998i
$172$		0			0
$173$	14.4572			1.09916			0.549582	−	0.835440i	$-0.314787\pi$
$173$	14.4572			1.09916			0.549582	+	0.835440i	$0.314787\pi$
$174$		0			0
$175$			2.11942i			0.160213i
$176$		0			0
$177$			9.74857i			0.732747i
$178$		0			0
$179$	−2.06658			−0.154464			−0.0772318	−	0.997013i	$-0.524608\pi$
$179$	−2.06658			−0.154464			−0.0772318	+	0.997013i	$0.524608\pi$
$180$		0			0
$181$	9.45225			0.702580			0.351290	−	0.936267i	$-0.385743\pi$
$181$	9.45225			0.702580			0.351290	+	0.936267i	$0.385743\pi$
$182$		0			0
$183$	6.72817			0.497360
$184$		0			0
$185$	12.5562			0.923147
$186$		0			0
$187$			2.32085i			0.169718i
$188$		0			0
$189$			4.36478i			0.317491i
$190$		0			0
$191$	−0.193160			−0.0139765			−0.00698827	−	0.999976i	$-0.502224\pi$
$191$	−0.193160			−0.0139765			−0.00698827	+	0.999976i	$0.502224\pi$
$192$		0			0
$193$		−	21.0349i		−	1.51413i	−0.653340	−	0.757064i	$-0.726633\pi$
$193$		−	21.0349i		−	1.51413i	0.653340	−	0.757064i	$-0.273367\pi$
$194$		0			0
$195$	−4.23102	+	2.68678i	−0.302990	+	0.192404i
$196$		0			0
$197$		−	6.65046i		−	0.473826i	−0.971531	−	0.236913i	$-0.923864\pi$
$197$		−	6.65046i		−	0.473826i	0.971531	−	0.236913i	$-0.0761356\pi$
$198$		0			0
$199$	19.4598			1.37947			0.689735	−	0.724062i	$-0.257727\pi$
$199$	19.4598			1.37947			0.689735	+	0.724062i	$0.257727\pi$
$200$		0			0
$201$			8.43699i			0.595099i
$202$		0			0
$203$			2.95197i			0.207188i
$204$		0			0
$205$	−16.4199			−1.14682
$206$		0			0
$207$	0.852917			0.0592819
$208$		0			0
$209$	2.77344			0.191843
$210$		0			0
$211$	10.1153			0.696367			0.348183	−	0.937426i	$-0.386799\pi$
$211$	10.1153			0.696367			0.348183	+	0.937426i	$0.386799\pi$
$212$		0			0
$213$		−	0.104043i		−	0.00712891i
$214$		0			0
$215$		−	17.1822i		−	1.17182i
$216$		0			0
$217$	−9.83529			−0.667663
$218$		0			0
$219$			9.11425i			0.615884i
$220$		0			0
$221$	4.48579	+	7.06402i	0.301747	+	0.475178i
$222$		0			0
$223$		−	20.6441i		−	1.38243i	−0.722647	−	0.691217i	$-0.757075\pi$
$223$		−	20.6441i		−	1.38243i	0.722647	−	0.691217i	$-0.242925\pi$
$224$		0			0
$225$	−4.93652			−0.329102
$226$		0			0
$227$		−	29.0731i		−	1.92965i	−0.262896	−	0.964824i	$-0.584678\pi$
$227$		−	29.0731i		−	1.92965i	0.262896	−	0.964824i	$-0.415322\pi$
$228$		0			0
$229$		−	1.72489i		−	0.113984i	−0.998375	−	0.0569921i	$-0.981849\pi$
$229$		−	1.72489i		−	0.113984i	0.998375	−	0.0569921i	$-0.0181510\pi$
$230$		0			0
$231$	−0.819032			−0.0538884
$232$		0			0
$233$	9.70662			0.635902			0.317951	−	0.948107i	$-0.397005\pi$
$233$	9.70662			0.635902			0.317951	+	0.948107i	$0.397005\pi$
$234$		0			0
$235$	17.7152			1.15561
$236$		0			0
$237$	−11.6414			−0.756187
$238$		0			0
$239$		−	9.18724i		−	0.594273i	−0.954835	−	0.297137i	$-0.903968\pi$
$239$		−	9.18724i		−	0.594273i	0.954835	−	0.297137i	$-0.0960317\pi$
$240$		0			0
$241$			14.7453i			0.949826i	0.880033	+	0.474913i	$0.157520\pi$
$241$			14.7453i			0.949826i	−0.880033	+	0.474913i	$0.842480\pi$
$242$		0			0
$243$	−15.8894			−1.01931
$244$		0			0
$245$		−	1.69723i		−	0.108432i
$246$		0			0
$247$	8.44158	−	5.36056i	0.537125	−	0.341084i
$248$		0			0
$249$		−	3.63897i		−	0.230611i
$250$		0			0
$251$	29.9720			1.89182			0.945909	−	0.324432i	$-0.105173\pi$
$251$	29.9720			1.89182			0.945909	+	0.324432i	$0.105173\pi$
$252$		0			0
$253$			0.366187i			0.0230220i
$254$		0			0
$255$			3.22618i			0.202031i
$256$		0			0
$257$	−14.6901			−0.916345			−0.458173	−	0.888863i	$-0.651496\pi$
$257$	−14.6901			−0.916345			−0.458173	+	0.888863i	$0.651496\pi$
$258$		0			0
$259$	7.39804			0.459692
$260$		0			0
$261$	−6.87570			−0.425595
$262$		0			0
$263$	8.47278			0.522454			0.261227	−	0.965277i	$-0.415873\pi$
$263$	8.47278			0.522454			0.261227	+	0.965277i	$0.415873\pi$
$264$		0			0
$265$		−	11.9296i		−	0.732829i
$266$		0			0
$267$			4.22528i			0.258583i
$268$		0			0
$269$	−16.9317			−1.03234			−0.516171	−	0.856486i	$-0.672643\pi$
$269$	−16.9317			−1.03234			−0.516171	+	0.856486i	$0.672643\pi$
$270$		0			0
$271$			1.06282i			0.0645616i	0.999479	+	0.0322808i	$0.0102771\pi$
$271$			1.06282i			0.0645616i	−0.999479	+	0.0322808i	$0.989723\pi$
$272$		0			0
$273$	−2.49290	+	1.58304i	−0.150877	+	0.0958100i
$274$		0			0
$275$		−	2.11942i		−	0.127806i
$276$		0			0
$277$	−16.9601			−1.01904			−0.509518	−	0.860460i	$-0.670176\pi$
$277$	−16.9601			−1.01904			−0.509518	+	0.860460i	$0.670176\pi$
$278$		0			0
$279$		−	22.9082i		−	1.37148i
$280$		0			0
$281$			1.53421i			0.0915236i	0.998952	+	0.0457618i	$0.0145715\pi$
$281$			1.53421i			0.0915236i	−0.998952	+	0.0457618i	$0.985428\pi$
$282$		0			0
$283$	24.4847			1.45546			0.727732	−	0.685861i	$-0.240574\pi$
$283$	24.4847			1.45546			0.727732	+	0.685861i	$0.240574\pi$
$284$		0			0
$285$	3.85532			0.228369
$286$		0			0
$287$	−9.67457			−0.571072
$288$		0			0
$289$	−11.6136			−0.683155
$290$		0			0
$291$		−	3.73392i		−	0.218886i
$292$		0			0
$293$			11.4585i			0.669414i	0.942322	+	0.334707i	$0.108637\pi$
$293$			11.4585i			0.669414i	−0.942322	+	0.334707i	$0.891363\pi$
$294$		0			0
$295$	20.2013			1.17617
$296$		0			0
$297$		−	4.36478i		−	0.253270i
$298$		0			0
$299$	0.707773	+	1.11457i	0.0409315	+	0.0644572i
$300$		0			0
$301$		−	10.1237i		−	0.583519i
$302$		0			0
$303$	−13.4509			−0.772732
$304$		0			0
$305$		−	13.9423i		−	0.798336i
$306$		0			0
$307$			16.2231i			0.925902i	0.886384	+	0.462951i	$0.153209\pi$
$307$			16.2231i			0.925902i	−0.886384	+	0.462951i	$0.846791\pi$
$308$		0			0
$309$	4.59740			0.261537
$310$		0			0
$311$	−32.9393			−1.86782			−0.933908	−	0.357514i	$-0.883625\pi$
$311$	−32.9393			−1.86782			−0.933908	+	0.357514i	$0.883625\pi$
$312$		0			0
$313$	−6.11898			−0.345865			−0.172933	−	0.984934i	$-0.555324\pi$
$313$	−6.11898			−0.345865			−0.172933	+	0.984934i	$0.555324\pi$
$314$		0			0
$315$	3.95316			0.222735
$316$		0			0
$317$			5.56290i			0.312444i	0.987722	+	0.156222i	$0.0499315\pi$
$317$			5.56290i			0.312444i	−0.987722	+	0.156222i	$0.950069\pi$
$318$		0			0
$319$		−	2.95197i		−	0.165279i
$320$		0			0
$321$	−5.18089			−0.289169
$322$		0			0
$323$		−	6.43675i		−	0.358151i
$324$		0			0
$325$	−4.09645	−	6.45092i	−0.227230	−	0.357832i
$326$		0			0
$327$		−	0.364743i		−	0.0201703i
$328$		0			0
$329$	10.4378			0.575452
$330$		0			0
$331$		−	6.30178i		−	0.346377i	−0.984889	−	0.173188i	$-0.944593\pi$
$331$		−	6.30178i		−	0.346377i	0.984889	−	0.173188i	$-0.0554070\pi$
$332$		0			0
$333$			17.2314i			0.944276i
$334$		0			0
$335$	17.4834			0.955221
$336$		0			0
$337$	−13.2305			−0.720709			−0.360355	−	0.932815i	$-0.617344\pi$
$337$	−13.2305			−0.720709			−0.360355	+	0.932815i	$0.617344\pi$
$338$		0			0
$339$	−1.38422			−0.0751805
$340$		0			0
$341$	9.83529			0.532611
$342$		0			0
$343$		−	1.00000i		−	0.0539949i
$344$		0			0
$345$			0.509031i			0.0274053i
$346$		0			0
$347$	−33.1097			−1.77742			−0.888710	−	0.458470i	$-0.848398\pi$
$347$	−33.1097			−1.77742			−0.888710	+	0.458470i	$0.848398\pi$
$348$		0			0
$349$		−	3.39507i		−	0.181734i	−0.995863	−	0.0908670i	$-0.971036\pi$
$349$		−	3.39507i		−	0.181734i	0.995863	−	0.0908670i	$-0.0289638\pi$
$350$		0			0
$351$	−8.43632	−	13.2851i	−0.450297	−	0.709109i
$352$		0			0
$353$			26.9734i			1.43565i	0.696224	+	0.717824i	$0.254862\pi$
$353$			26.9734i			1.43565i	−0.696224	+	0.717824i	$0.745138\pi$
$354$		0			0
$355$	−0.215601			−0.0114429
$356$		0			0
$357$			1.90085i			0.100604i
$358$		0			0
$359$		−	22.2390i		−	1.17373i	−0.809684	−	0.586866i	$-0.800362\pi$
$359$		−	22.2390i		−	1.17373i	0.809684	−	0.586866i	$-0.199638\pi$
$360$		0			0
$361$	11.3080			0.595159
$362$		0			0
$363$	0.819032			0.0429881
$364$		0			0
$365$	18.8869			0.988584
$366$		0			0
$367$	−9.31898			−0.486447			−0.243223	−	0.969970i	$-0.578205\pi$
$367$	−9.31898			−0.486447			−0.243223	+	0.969970i	$0.578205\pi$
$368$		0			0
$369$		−	22.5339i		−	1.17307i
$370$		0			0
$371$		−	7.02888i		−	0.364921i
$372$		0			0
$373$	−5.75842			−0.298160			−0.149080	−	0.988825i	$-0.547631\pi$
$373$	−5.75842			−0.298160			−0.149080	+	0.988825i	$0.547631\pi$
$374$		0			0
$375$		−	9.89659i		−	0.511058i
$376$		0			0
$377$	−5.70563	−	8.98498i	−0.293855	−	0.462750i
$378$		0			0
$379$		−	11.5494i		−	0.593251i	−0.954994	−	0.296625i	$-0.904139\pi$
$379$		−	11.5494i		−	0.593251i	0.954994	−	0.296625i	$-0.0958613\pi$
$380$		0			0
$381$	−10.9167			−0.559280
$382$		0			0
$383$			1.63226i			0.0834043i	0.999130	+	0.0417022i	$0.0132781\pi$
$383$			1.63226i			0.0834043i	−0.999130	+	0.0417022i	$0.986722\pi$
$384$		0			0
$385$			1.69723i			0.0864987i
$386$		0			0
$387$	23.5799			1.19864
$388$		0			0
$389$	−34.1493			−1.73144			−0.865720	−	0.500528i	$-0.833139\pi$
$389$	−34.1493			−1.73144			−0.865720	+	0.500528i	$0.833139\pi$
$390$		0			0
$391$	0.849866			0.0429796
$392$		0			0
$393$	12.4445			0.627744
$394$		0			0
$395$			24.1236i			1.21379i
$396$		0			0
$397$		−	22.1368i		−	1.11102i	−0.831511	−	0.555508i	$-0.812524\pi$
$397$		−	22.1368i		−	1.11102i	0.831511	−	0.555508i	$-0.187476\pi$
$398$		0			0
$399$	2.27154			0.113719
$400$		0			0
$401$		−	17.3380i		−	0.865819i	−0.901437	−	0.432910i	$-0.857487\pi$
$401$		−	17.3380i		−	0.865819i	0.901437	−	0.432910i	$-0.142513\pi$
$402$		0			0
$403$	29.9358	−	19.0098i	1.49121	−	0.946946i
$404$		0			0
$405$			5.79206i			0.287810i
$406$		0			0
$407$	−7.39804			−0.366707
$408$		0			0
$409$			5.96973i			0.295184i	0.989048	+	0.147592i	$0.0471523\pi$
$409$			5.96973i			0.295184i	−0.989048	+	0.147592i	$0.952848\pi$
$410$		0			0
$411$		−	2.49886i		−	0.123260i
$412$		0			0
$413$	11.9025			0.585686
$414$		0			0
$415$	−7.54081			−0.370164
$416$		0			0
$417$	−12.3798			−0.606239
$418$		0			0
$419$	−17.1740			−0.839006			−0.419503	−	0.907754i	$-0.637796\pi$
$419$	−17.1740			−0.839006			−0.419503	+	0.907754i	$0.637796\pi$
$420$		0			0
$421$			14.8026i			0.721436i	0.932675	+	0.360718i	$0.117468\pi$
$421$			14.8026i			0.721436i	−0.932675	+	0.360718i	$0.882532\pi$
$422$		0			0
$423$			24.3115i			1.18206i
$424$		0			0
$425$	−4.91887			−0.238600
$426$		0			0
$427$		−	8.21477i		−	0.397541i
$428$		0			0
$429$	2.49290	−	1.58304i	0.120359	−	0.0764299i
$430$		0			0
$431$		−	1.17641i		−	0.0566655i	−0.999599	−	0.0283327i	$-0.990980\pi$
$431$		−	1.17641i		−	0.0566655i	0.999599	−	0.0283327i	$-0.00901980\pi$
$432$		0			0
$433$	33.7302			1.62097			0.810485	−	0.585760i	$-0.199204\pi$
$433$	33.7302			1.62097			0.810485	+	0.585760i	$0.199204\pi$
$434$		0			0
$435$		−	4.10349i		−	0.196747i
$436$		0			0
$437$		−	1.01560i		−	0.0485827i
$438$		0			0
$439$	−33.0484			−1.57731			−0.788656	−	0.614835i	$-0.789223\pi$
$439$	−33.0484			−1.57731			−0.788656	+	0.614835i	$0.789223\pi$
$440$		0			0
$441$	2.32919			0.110914
$442$		0			0
$443$	28.5853			1.35813			0.679064	−	0.734079i	$-0.262386\pi$
$443$	28.5853			1.35813			0.679064	+	0.734079i	$0.262386\pi$
$444$		0			0
$445$	8.75577			0.415063
$446$		0			0
$447$			3.33676i			0.157823i
$448$		0			0
$449$		−	0.750950i		−	0.0354395i	−0.999843	−	0.0177198i	$-0.994359\pi$
$449$		−	0.750950i		−	0.0354395i	0.999843	−	0.0177198i	$-0.00564067\pi$
$450$		0			0
$451$	9.67457			0.455557
$452$		0			0
$453$			14.3005i			0.671897i
$454$		0			0
$455$	3.28043	+	5.16588i	0.153789	+	0.242180i
$456$		0			0
$457$			22.9719i			1.07458i	0.843398	+	0.537290i	$0.180552\pi$
$457$			22.9719i			1.07458i	−0.843398	+	0.537290i	$0.819448\pi$
$458$		0			0
$459$	−10.1300			−0.472828
$460$		0			0
$461$			13.4739i			0.627541i	0.949499	+	0.313770i	$0.101592\pi$
$461$			13.4739i			0.627541i	−0.949499	+	0.313770i	$0.898408\pi$
$462$		0			0
$463$		−	23.5879i		−	1.09622i	−0.836405	−	0.548112i	$-0.815347\pi$
$463$		−	23.5879i		−	1.09622i	0.836405	−	0.548112i	$-0.184653\pi$
$464$		0			0
$465$	13.6719			0.634018
$466$		0			0
$467$	27.0362			1.25109			0.625544	−	0.780189i	$-0.284877\pi$
$467$	27.0362			1.25109			0.625544	+	0.780189i	$0.284877\pi$
$468$		0			0
$469$	10.3012			0.475664
$470$		0			0
$471$	−11.4753			−0.528753
$472$		0			0
$473$			10.1237i			0.465487i
$474$		0			0
$475$			5.87809i			0.269705i
$476$		0			0
$477$	16.3716			0.749602
$478$		0			0
$479$		−	7.28707i		−	0.332955i	−0.986045	−	0.166477i	$-0.946761\pi$
$479$		−	7.28707i		−	0.332955i	0.986045	−	0.166477i	$-0.0532393\pi$
$480$		0			0
$481$	−22.5175	+	14.2991i	−1.02671	+	0.651981i
$482$		0			0
$483$			0.299919i			0.0136468i
$484$		0			0
$485$	−7.73755			−0.351344
$486$		0			0
$487$			19.3351i			0.876155i	0.898937	+	0.438077i	$0.144340\pi$
$487$			19.3351i			0.876155i	−0.898937	+	0.438077i	$0.855660\pi$
$488$		0			0
$489$			9.92096i			0.448642i
$490$		0			0
$491$	2.36812			0.106872			0.0534358	−	0.998571i	$-0.482983\pi$
$491$	2.36812			0.106872			0.0534358	+	0.998571i	$0.482983\pi$
$492$		0			0
$493$	−6.85110			−0.308558
$494$		0			0
$495$	−3.95316			−0.177681
$496$		0			0
$497$	−0.127032			−0.00569814
$498$		0			0
$499$		−	39.6681i		−	1.77579i	−0.460048	−	0.887894i	$-0.652168\pi$
$499$		−	39.6681i		−	1.77579i	0.460048	−	0.887894i	$-0.347832\pi$
$500$		0			0
$501$			9.50499i			0.424652i
$502$		0			0
$503$	4.75868			0.212179			0.106089	−	0.994357i	$-0.466167\pi$
$503$	4.75868			0.212179			0.106089	+	0.994357i	$0.466167\pi$
$504$		0			0
$505$			27.8733i			1.24035i
$506$		0			0
$507$	4.52797	−	9.63666i	0.201094	−	0.427979i
$508$		0			0
$509$		−	4.79819i		−	0.212676i	−0.994330	−	0.106338i	$-0.966087\pi$
$509$		−	4.79819i		−	0.212676i	0.994330	−	0.106338i	$-0.0339126\pi$
$510$		0			0
$511$	11.1281			0.492277
$512$		0			0
$513$			12.1055i			0.534469i
$514$		0			0
$515$		−	9.52688i		−	0.419804i
$516$		0			0
$517$	−10.4378			−0.459052
$518$		0			0
$519$	−11.8410			−0.519760
$520$		0			0
$521$	−32.9554			−1.44380			−0.721902	−	0.691996i	$-0.756732\pi$
$521$	−32.9554			−1.44380			−0.721902	+	0.691996i	$0.756732\pi$
$522$		0			0
$523$	−0.616555			−0.0269601			−0.0134800	−	0.999909i	$-0.504291\pi$
$523$	−0.616555			−0.0269601			−0.0134800	+	0.999909i	$0.504291\pi$
$524$		0			0
$525$		−	1.73587i		−	0.0757598i
$526$		0			0
$527$		−	22.8263i		−	0.994328i
$528$		0			0
$529$	−22.8659			−0.994170
$530$		0			0
$531$			27.7232i			1.20309i
$532$		0			0
$533$	29.4466	−	18.6992i	1.27548	−	0.809951i
$534$		0			0
$535$			10.7360i			0.464158i
$536$		0			0
$537$	1.69260			0.0730410
$538$		0			0
$539$			1.00000i			0.0430730i
$540$		0			0
$541$		−	35.2915i		−	1.51730i	−0.651499	−	0.758650i	$-0.725859\pi$
$541$		−	35.2915i		−	1.51730i	0.651499	−	0.758650i	$-0.274141\pi$
$542$		0			0
$543$	−7.74170			−0.332228
$544$		0			0
$545$	−0.755832			−0.0323763
$546$		0			0
$547$	−27.8694			−1.19161			−0.595804	−	0.803130i	$-0.703166\pi$
$547$	−27.8694			−1.19161			−0.595804	+	0.803130i	$0.703166\pi$
$548$		0			0
$549$	19.1337			0.816608
$550$		0			0
$551$			8.18713i			0.348784i
$552$		0			0
$553$			14.2135i			0.604421i
$554$		0			0
$555$	−10.2839			−0.436527
$556$		0			0
$557$			20.7880i			0.880815i	0.897798	+	0.440408i	$0.145166\pi$
$557$			20.7880i			0.880815i	−0.897798	+	0.440408i	$0.854834\pi$
$558$		0			0
$559$	19.5672	+	30.8136i	0.827606	+	1.30328i
$560$		0			0
$561$		−	1.90085i		−	0.0802542i
$562$		0			0
$563$	36.3083			1.53021			0.765106	−	0.643905i	$-0.222687\pi$
$563$	36.3083			1.53021			0.765106	+	0.643905i	$0.222687\pi$
$564$		0			0
$565$			2.86843i			0.120676i
$566$		0			0
$567$			3.41266i			0.143318i
$568$		0			0
$569$	−30.4117			−1.27493			−0.637463	−	0.770481i	$-0.720016\pi$
$569$	−30.4117			−1.27493			−0.637463	+	0.770481i	$0.720016\pi$
$570$		0			0
$571$	27.8569			1.16578			0.582888	−	0.812553i	$-0.301923\pi$
$571$	27.8569			1.16578			0.582888	+	0.812553i	$0.301923\pi$
$572$		0			0
$573$	0.158204			0.00660906
$574$		0			0
$575$	−0.776104			−0.0323658
$576$		0			0
$577$			21.5422i			0.896812i	0.893830	+	0.448406i	$0.148008\pi$
$577$			21.5422i			0.896812i	−0.893830	+	0.448406i	$0.851992\pi$
$578$		0			0
$579$			17.2283i			0.715984i
$580$		0			0
$581$	−4.44302			−0.184327
$582$		0			0
$583$			7.02888i			0.291106i
$584$		0			0
$585$	−12.0323	+	7.64073i	−0.497474	+	0.315905i
$586$		0			0
$587$			8.25568i			0.340748i	0.985379	+	0.170374i	$0.0544977\pi$
$587$			8.25568i			0.340748i	−0.985379	+	0.170374i	$0.945502\pi$
$588$		0			0
$589$	−27.2776			−1.12395
$590$		0			0
$591$			5.44694i			0.224057i
$592$		0			0
$593$			0.692417i			0.0284342i	0.999899	+	0.0142171i	$0.00452559\pi$
$593$			0.692417i			0.0284342i	−0.999899	+	0.0142171i	$0.995474\pi$
$594$		0			0
$595$	3.93902			0.161484
$596$		0			0
$597$	−15.9382			−0.652308
$598$		0			0
$599$	−13.4964			−0.551450			−0.275725	−	0.961237i	$-0.588918\pi$
$599$	−13.4964			−0.551450			−0.275725	+	0.961237i	$0.588918\pi$
$600$		0			0
$601$	40.9950			1.67222			0.836111	−	0.548560i	$-0.184824\pi$
$601$	40.9950			1.67222			0.836111	+	0.548560i	$0.184824\pi$
$602$		0			0
$603$			23.9933i			0.977084i
$604$		0			0
$605$		−	1.69723i		−	0.0690021i
$606$		0			0
$607$	−9.68506			−0.393105			−0.196552	−	0.980493i	$-0.562975\pi$
$607$	−9.68506			−0.393105			−0.196552	+	0.980493i	$0.562975\pi$
$608$		0			0
$609$		−	2.41776i		−	0.0979727i
$610$		0			0
$611$	−31.7696	+	20.1743i	−1.28526	+	0.816164i
$612$		0			0
$613$		−	47.2799i		−	1.90962i	−0.297223	−	0.954808i	$-0.596060\pi$
$613$		−	47.2799i		−	1.90962i	0.297223	−	0.954808i	$-0.403940\pi$
$614$		0			0
$615$	13.4485			0.542294
$616$		0			0
$617$			12.4585i			0.501562i	0.968044	+	0.250781i	$0.0806873\pi$
$617$			12.4585i			0.501562i	−0.968044	+	0.250781i	$0.919313\pi$
$618$		0			0
$619$		−	34.8410i		−	1.40038i	−0.713958	−	0.700188i	$-0.753099\pi$
$619$		−	34.8410i		−	1.40038i	0.713958	−	0.700188i	$-0.246901\pi$
$620$		0			0
$621$	−1.59832			−0.0641385
$622$		0			0
$623$	5.15887			0.206686
$624$		0			0
$625$	−9.91095			−0.396438
$626$		0			0
$627$	−2.27154			−0.0907165
$628$		0			0
$629$			17.1698i			0.684604i
$630$		0			0
$631$			30.9578i			1.23241i	0.787586	+	0.616205i	$0.211331\pi$
$631$			30.9578i			1.23241i	−0.787586	+	0.616205i	$0.788669\pi$
$632$		0			0
$633$	−8.28477			−0.329290
$634$		0			0
$635$			22.6220i			0.897726i
$636$		0			0
$637$	1.93282	+	3.04372i	0.0765810	+	0.120596i
$638$		0			0
$639$		−	0.295880i		−	0.0117048i
$640$		0			0
$641$	−37.1952			−1.46912			−0.734560	−	0.678543i	$-0.762612\pi$
$641$	−37.1952			−1.46912			−0.734560	+	0.678543i	$0.762612\pi$
$642$		0			0
$643$		−	0.449713i		−	0.0177349i	−0.999961	−	0.00886747i	$-0.997177\pi$
$643$		−	0.449713i		−	0.0177349i	0.999961	−	0.00886747i	$-0.00282264\pi$
$644$		0			0
$645$			14.0728i			0.554115i
$646$		0			0
$647$	−8.79899			−0.345924			−0.172962	−	0.984928i	$-0.555334\pi$
$647$	−8.79899			−0.345924			−0.172962	+	0.984928i	$0.555334\pi$
$648$		0			0
$649$	−11.9025			−0.467216
$650$		0			0
$651$	8.05542			0.315717
$652$		0			0
$653$	−36.0064			−1.40904			−0.704519	−	0.709685i	$-0.748837\pi$
$653$	−36.0064			−1.40904			−0.704519	+	0.709685i	$0.748837\pi$
$654$		0			0
$655$		−	25.7880i		−	1.00762i
$656$		0			0
$657$			25.9194i			1.01121i
$658$		0			0
$659$	0.969131			0.0377520			0.0188760	−	0.999822i	$-0.493991\pi$
$659$	0.969131			0.0377520			0.0188760	+	0.999822i	$0.493991\pi$
$660$		0			0
$661$		−	2.92011i		−	0.113579i	−0.998386	−	0.0567896i	$-0.981914\pi$
$661$		−	2.92011i		−	0.113579i	0.998386	−	0.0567896i	$-0.0180864\pi$
$662$		0			0
$663$	−3.67401	−	5.78567i	−0.142687	−	0.224697i
$664$		0			0
$665$		−	4.70716i		−	0.182536i
$666$		0			0
$667$	−1.08097			−0.0418555
$668$		0			0
$669$			16.9082i			0.653710i
$670$		0			0
$671$			8.21477i			0.317128i
$672$		0			0
$673$	18.8429			0.726342			0.363171	−	0.931723i	$-0.381694\pi$
$673$	18.8429			0.726342			0.363171	+	0.931723i	$0.381694\pi$
$674$		0			0
$675$	9.25080			0.356063
$676$		0			0
$677$	−19.6292			−0.754410			−0.377205	−	0.926130i	$-0.623115\pi$
$677$	−19.6292			−0.754410			−0.377205	+	0.926130i	$0.623115\pi$
$678$		0			0
$679$	−4.55893			−0.174956
$680$		0			0
$681$			23.8118i			0.912470i
$682$		0			0
$683$		−	42.0865i		−	1.61040i	−0.593006	−	0.805198i	$-0.702059\pi$
$683$		−	42.0865i		−	1.61040i	0.593006	−	0.805198i	$-0.297941\pi$
$684$		0			0
$685$	−5.17822			−0.197849
$686$		0			0
$687$			1.41274i			0.0538995i
$688$		0			0
$689$	13.5855	+	21.3939i	0.517568	+	0.815043i
$690$		0			0
$691$		−	0.595349i		−	0.0226481i	−0.999936	−	0.0113241i	$-0.996395\pi$
$691$		−	0.595349i		−	0.0226481i	0.999936	−	0.0113241i	$-0.00360464\pi$
$692$		0			0
$693$	−2.32919			−0.0884785
$694$		0			0
$695$			25.6537i			0.973102i
$696$		0			0
$697$		−	22.4533i		−	0.850478i
$698$		0			0
$699$	−7.95004			−0.300698
$700$		0			0
$701$	−8.54504			−0.322742			−0.161371	−	0.986894i	$-0.551592\pi$
$701$	−8.54504			−0.322742			−0.161371	+	0.986894i	$0.551592\pi$
$702$		0			0
$703$	20.5180			0.773853
$704$		0			0
$705$	−14.5094			−0.546454
$706$		0			0
$707$			16.4229i			0.617646i
$708$		0			0
$709$			2.69377i			0.101167i	0.998720	+	0.0505833i	$0.0161080\pi$
$709$			2.69377i			0.101167i	−0.998720	+	0.0505833i	$0.983892\pi$
$710$		0			0
$711$	−33.1060			−1.24157
$712$		0			0
$713$		−	3.60155i		−	0.134879i
$714$		0			0
$715$	−3.28043	−	5.16588i	−0.122681	−	0.193193i
$716$		0			0
$717$			7.52465i			0.281013i
$718$		0			0
$719$	−7.17053			−0.267416			−0.133708	−	0.991021i	$-0.542688\pi$
$719$	−7.17053			−0.267416			−0.133708	+	0.991021i	$0.542688\pi$
$720$		0			0
$721$		−	5.61320i		−	0.209047i
$722$		0			0
$723$		−	12.0769i		−	0.449143i
$724$		0			0
$725$	6.25648			0.232360
$726$		0			0
$727$	−11.7863			−0.437128			−0.218564	−	0.975823i	$-0.570137\pi$
$727$	−11.7863			−0.437128			−0.218564	+	0.975823i	$0.570137\pi$
$728$		0			0
$729$	2.77595			0.102813
$730$		0			0
$731$	23.4956			0.869016
$732$		0			0
$733$		−	16.2355i		−	0.599673i	−0.953991	−	0.299837i	$-0.903068\pi$
$733$		−	16.2355i		−	0.599673i	0.953991	−	0.299837i	$-0.0969321\pi$
$734$		0			0
$735$			1.39008i			0.0512740i
$736$		0			0
$737$	−10.3012			−0.379448
$738$		0			0
$739$			4.53787i			0.166928i	0.996511	+	0.0834642i	$0.0265984\pi$
$739$			4.53787i			0.166928i	−0.996511	+	0.0834642i	$0.973402\pi$
$740$		0			0
$741$	−6.91392	+	4.39047i	−0.253989	+	0.161288i
$742$		0			0
$743$			7.32658i			0.268786i	0.990928	+	0.134393i	$0.0429085\pi$
$743$			7.32658i			0.268786i	−0.990928	+	0.134393i	$0.957092\pi$
$744$		0			0
$745$	6.91454			0.253329
$746$		0			0
$747$		−	10.3486i		−	0.378636i
$748$		0			0
$749$			6.32562i			0.231133i
$750$		0			0
$751$	−7.03656			−0.256768			−0.128384	−	0.991725i	$-0.540979\pi$
$751$	−7.03656			−0.256768			−0.128384	+	0.991725i	$0.540979\pi$
$752$		0			0
$753$	−24.5481			−0.894582
$754$		0			0
$755$	29.6340			1.07849
$756$		0			0
$757$	13.4980			0.490594			0.245297	−	0.969448i	$-0.421114\pi$
$757$	13.4980			0.490594			0.245297	+	0.969448i	$0.421114\pi$
$758$		0			0
$759$		−	0.299919i		−	0.0108864i
$760$		0			0
$761$			13.4622i			0.488005i	0.969775	+	0.244002i	$0.0784605\pi$
$761$			13.4622i			0.488005i	−0.969775	+	0.244002i	$0.921540\pi$
$762$		0			0
$763$	−0.445333			−0.0161222
$764$		0			0
$765$			9.17470i			0.331712i
$766$		0			0
$767$	−36.2280	+	23.0054i	−1.30812	+	0.830678i
$768$		0			0
$769$		−	11.3243i		−	0.408366i	−0.978933	−	0.204183i	$-0.934546\pi$
$769$		−	11.3243i		−	0.408366i	0.978933	−	0.204183i	$-0.0654538\pi$
$770$		0			0
$771$	12.0317			0.433311
$772$		0			0
$773$			39.8536i			1.43343i	0.697364	+	0.716717i	$0.254356\pi$
$773$			39.8536i			1.43343i	−0.697364	+	0.716717i	$0.745644\pi$
$774$		0			0
$775$			20.8451i			0.748778i
$776$		0			0
$777$	−6.05924			−0.217374
$778$		0			0
$779$	−26.8318			−0.961351
$780$		0			0
$781$	0.127032			0.00454555
$782$		0			0
$783$	12.8847			0.460462
$784$		0			0
$785$			23.7795i			0.848726i
$786$		0			0
$787$		−	18.8270i		−	0.671109i	−0.942021	−	0.335555i	$-0.891076\pi$
$787$		−	18.8270i		−	0.671109i	0.942021	−	0.335555i	$-0.108924\pi$
$788$		0			0
$789$	−6.93948			−0.247052
$790$		0			0
$791$			1.69007i			0.0600918i
$792$		0			0
$793$	15.8777	+	25.0035i	0.563832	+	0.887899i
$794$		0			0
$795$			9.77073i			0.346532i
$796$		0			0
$797$	−9.21225			−0.326315			−0.163157	−	0.986600i	$-0.552168\pi$
$797$	−9.21225			−0.326315			−0.163157	+	0.986600i	$0.552168\pi$
$798$		0			0
$799$			24.2245i			0.857001i
$800$		0			0
$801$			12.0160i			0.424563i
$802$		0			0
$803$	−11.1281			−0.392701
$804$		0			0
$805$	0.621502			0.0219051
$806$		0			0
$807$	13.8676			0.488162
$808$		0			0
$809$	44.3347			1.55873			0.779363	−	0.626573i	$-0.215543\pi$
$809$	44.3347			1.55873			0.779363	+	0.626573i	$0.215543\pi$
$810$		0			0
$811$		−	2.16307i		−	0.0759558i	−0.999279	−	0.0379779i	$-0.987908\pi$
$811$		−	2.16307i		−	0.0759558i	0.999279	−	0.0379779i	$-0.0120916\pi$
$812$		0			0
$813$		−	0.870483i		−	0.0305292i
$814$		0			0
$815$	20.5586			0.720135
$816$		0			0
$817$		−	28.0775i		−	0.982306i
$818$		0			0
$819$	−7.08938	+	4.50189i	−0.247723	+	0.157309i
$820$		0			0
$821$		−	35.7421i		−	1.24741i	−0.781661	−	0.623704i	$-0.785627\pi$
$821$		−	35.7421i		−	1.24741i	0.781661	−	0.623704i	$-0.214373\pi$
$822$		0			0
$823$	27.2244			0.948982			0.474491	−	0.880260i	$-0.342632\pi$
$823$	27.2244			0.948982			0.474491	+	0.880260i	$0.342632\pi$
$824$		0			0
$825$			1.73587i			0.0604354i
$826$		0			0
$827$			39.2293i			1.36414i	0.731289	+	0.682068i	$0.238919\pi$
$827$			39.2293i			1.36414i	−0.731289	+	0.682068i	$0.761081\pi$
$828$		0			0
$829$	22.0601			0.766179			0.383090	−	0.923711i	$-0.374860\pi$
$829$	22.0601			0.766179			0.383090	+	0.923711i	$0.374860\pi$
$830$		0			0
$831$	13.8909			0.481870
$832$		0			0
$833$	2.32085			0.0804128
$834$		0			0
$835$	19.6966			0.681628
$836$		0			0
$837$			42.9288i			1.48384i
$838$		0			0
$839$			7.55390i			0.260790i	0.991462	+	0.130395i	$0.0416245\pi$
$839$			7.55390i			0.260790i	−0.991462	+	0.130395i	$0.958375\pi$
$840$		0			0
$841$	−20.2858			−0.699512
$842$		0			0
$843$		−	1.25657i		−	0.0432786i
$844$		0			0
$845$	−19.9694	−	9.38300i	−0.686968	−	0.322785i
$846$		0			0
$847$		−	1.00000i		−	0.0343604i
$848$		0			0
$849$	−20.0538			−0.688244
$850$		0			0
$851$			2.70907i			0.0928656i
$852$		0			0
$853$		−	29.6952i		−	1.01674i	−0.861138	−	0.508371i	$-0.830248\pi$
$853$		−	29.6952i		−	1.01674i	0.861138	−	0.508371i	$-0.169752\pi$
$854$		0			0
$855$	10.9639			0.374956
$856$		0			0
$857$	−0.967791			−0.0330591			−0.0165296	−	0.999863i	$-0.505262\pi$
$857$	−0.967791			−0.0330591			−0.0165296	+	0.999863i	$0.505262\pi$
$858$		0			0
$859$	39.2839			1.34035			0.670175	−	0.742204i	$-0.266219\pi$
$859$	39.2839			1.34035			0.670175	+	0.742204i	$0.266219\pi$
$860$		0			0
$861$	7.92378			0.270042
$862$		0			0
$863$			42.0058i			1.42989i	0.699179	+	0.714947i	$0.253549\pi$
$863$			42.0058i			1.42989i	−0.699179	+	0.714947i	$0.746451\pi$
$864$		0			0
$865$			24.5372i			0.834291i
$866$		0			0
$867$	9.51195			0.323043
$868$		0			0
$869$		−	14.2135i		−	0.482161i
$870$		0			0
$871$	−31.3538	+	19.9103i	−1.06238	+	0.674634i
$872$		0			0
$873$		−	10.6186i		−	0.359385i
$874$		0			0
$875$	−12.0833			−0.408489
$876$		0			0
$877$		−	7.72454i		−	0.260839i	−0.991459	−	0.130420i	$-0.958368\pi$
$877$		−	7.72454i		−	0.260839i	0.991459	−	0.130420i	$-0.0416325\pi$
$878$		0			0
$879$		−	9.38490i		−	0.316545i
$880$		0			0
$881$	−9.91245			−0.333959			−0.166979	−	0.985960i	$-0.553401\pi$
$881$	−9.91245			−0.333959			−0.166979	+	0.985960i	$0.553401\pi$
$882$		0			0
$883$	−40.8090			−1.37333			−0.686667	−	0.726972i	$-0.740927\pi$
$883$	−40.8090			−1.37333			−0.686667	+	0.726972i	$0.740927\pi$
$884$		0			0
$885$	−16.5455			−0.556172
$886$		0			0
$887$	38.0574			1.27784			0.638922	−	0.769272i	$-0.279381\pi$
$887$	38.0574			1.27784			0.638922	+	0.769272i	$0.279381\pi$
$888$		0			0
$889$			13.3288i			0.447033i
$890$		0			0
$891$		−	3.41266i		−	0.114329i
$892$		0			0
$893$	28.9485			0.968725
$894$		0			0
$895$		−	3.50746i		−	0.117241i
$896$		0			0
$897$	−0.579689	−	0.912869i	−0.0193552	−	0.0304798i
$898$		0			0
$899$			29.0335i			0.968322i
$900$		0			0
$901$	16.3130			0.543465
$902$		0			0
$903$			8.29163i			0.275928i
$904$		0			0
$905$			16.0426i			0.533274i
$906$		0			0
$907$	−17.2584			−0.573057			−0.286529	−	0.958072i	$-0.592501\pi$
$907$	−17.2584			−0.573057			−0.286529	+	0.958072i	$0.592501\pi$
$908$		0			0
$909$	−38.2519			−1.26874
$910$		0			0
$911$	−7.99574			−0.264911			−0.132455	−	0.991189i	$-0.542286\pi$
$911$	−7.99574			−0.264911			−0.132455	+	0.991189i	$0.542286\pi$
$912$		0			0
$913$	4.44302			0.147042
$914$		0			0
$915$			11.4192i			0.377508i
$916$		0			0
$917$		−	15.1942i		−	0.501756i
$918$		0			0
$919$	−11.5336			−0.380458			−0.190229	−	0.981740i	$-0.560923\pi$
$919$	−11.5336			−0.380458			−0.190229	+	0.981740i	$0.560923\pi$
$920$		0			0
$921$		−	13.2873i		−	0.437830i
$922$		0			0
$923$	0.386648	−	0.245529i	0.0127267	−	0.00808168i
$924$		0			0
$925$		−	15.6796i		−	0.515541i
$926$		0			0
$927$	13.0742			0.429413
$928$		0			0
$929$		−	8.35914i		−	0.274255i	−0.990553	−	0.137127i	$-0.956213\pi$
$929$		−	8.35914i		−	0.274255i	0.990553	−	0.137127i	$-0.0437869\pi$
$930$		0			0
$931$		−	2.77344i		−	0.0908959i
$932$		0			0
$933$	26.9784			0.883232
$934$		0			0
$935$	−3.93902			−0.128820
$936$		0			0
$937$	47.2457			1.54345			0.771726	−	0.635956i	$-0.219394\pi$
$937$	47.2457			1.54345			0.771726	+	0.635956i	$0.219394\pi$
$938$		0			0
$939$	5.01164			0.163549
$940$		0			0
$941$			43.7421i			1.42595i	0.701189	+	0.712975i	$0.252653\pi$
$941$			43.7421i			1.42595i	−0.701189	+	0.712975i	$0.747347\pi$
$942$		0			0
$943$		−	3.54270i		−	0.115366i
$944$		0			0
$945$	−7.40802			−0.240983
$946$		0			0
$947$		−	1.25908i		−	0.0409145i	−0.999791	−	0.0204572i	$-0.993488\pi$
$947$		−	1.25908i		−	0.0409145i	0.999791	−	0.0204572i	$-0.00651220\pi$
$948$		0			0
$949$	−33.8707	+	21.5085i	−1.09949	+	0.698197i
$950$		0			0
$951$		−	4.55620i		−	0.147745i
$952$		0			0
$953$	−13.1190			−0.424965			−0.212483	−	0.977165i	$-0.568155\pi$
$953$	−13.1190			−0.424965			−0.212483	+	0.977165i	$0.568155\pi$
$954$		0			0
$955$		−	0.327836i		−	0.0106085i
$956$		0			0
$957$			2.41776i			0.0781552i
$958$		0			0
$959$	−3.05099			−0.0985215
$960$		0			0
$961$	−65.7329			−2.12041
$962$		0			0
$963$	−14.7335			−0.474782
$964$		0			0
$965$	35.7011			1.14926
$966$		0			0
$967$		−	36.7342i		−	1.18129i	−0.806931	−	0.590646i	$-0.798873\pi$
$967$		−	36.7342i		−	1.18129i	0.806931	−	0.590646i	$-0.201127\pi$
$968$		0			0
$969$			5.27191i			0.169358i
$970$		0			0
$971$	−53.5486			−1.71846			−0.859228	−	0.511593i	$-0.829056\pi$
$971$	−53.5486			−1.71846			−0.859228	+	0.511593i	$0.829056\pi$
$972$		0			0
$973$			15.1151i			0.484568i
$974$		0			0
$975$	3.35513	+	5.28351i	0.107450	+	0.169208i
$976$		0			0
$977$			23.1365i			0.740203i	0.928991	+	0.370102i	$0.120677\pi$
$977$			23.1365i			0.740203i	−0.928991	+	0.370102i	$0.879323\pi$
$978$		0			0
$979$	−5.15887			−0.164878
$980$		0			0
$981$		−	1.03726i		−	0.0331173i
$982$		0			0
$983$		−	1.48080i		−	0.0472303i	−0.999721	−	0.0236151i	$-0.992482\pi$
$983$		−	1.48080i		−	0.0472303i	0.999721	−	0.0236151i	$-0.00751763\pi$
$984$		0			0
$985$	11.2873			0.359644
$986$		0			0
$987$	−8.54886			−0.272113
$988$		0			0
$989$	3.70716			0.117881
$990$		0			0
$991$	−42.6329			−1.35428			−0.677139	−	0.735855i	$-0.736780\pi$
$991$	−42.6329			−1.35428			−0.677139	+	0.735855i	$0.736780\pi$
$992$		0			0
$993$			5.16136i			0.163791i
$994$		0			0
$995$			33.0277i			1.04705i
$996$		0			0
$997$	−3.29043			−0.104209			−0.0521045	−	0.998642i	$-0.516593\pi$
$997$	−3.29043			−0.104209			−0.0521045	+	0.998642i	$0.516593\pi$
$998$		0			0
$999$		−	32.2908i		−	1.02164i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4004.2.m.c.2157.15	✓	36
13.12	even	2	inner	4004.2.m.c.2157.16	yes	36

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4004.2.m.c.2157.15	✓	36	1.1	even	1	trivial
4004.2.m.c.2157.16	yes	36	13.12	even	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:	\( N \)	\(=\)	\( 4004 = 2^{2} \cdot 7 \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4004.m (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-0.819032 q^{3} +1.69723i q^{5} +1.00000i q^{7} -2.32919 q^{9} +O(q^{10})\)	\(q-0.819032 q^{3} +1.69723i q^{5} +1.00000i q^{7} -2.32919 q^{9} -1.00000i q^{11} +(-1.93282 - 3.04372i) q^{13} -1.39008i q^{15} -2.32085 q^{17} +2.77344i q^{19} -0.819032i q^{21} -0.366187 q^{23} +2.11942 q^{25} +4.36478 q^{27} +2.95197 q^{29} +9.83529i q^{31} +0.819032i q^{33} -1.69723 q^{35} -7.39804i q^{37} +(1.58304 + 2.49290i) q^{39} +9.67457i q^{41} -10.1237 q^{43} -3.95316i q^{45} -10.4378i q^{47} -1.00000 q^{49} +1.90085 q^{51} -7.02888 q^{53} +1.69723 q^{55} -2.27154i q^{57} -11.9025i q^{59} -8.21477 q^{61} -2.32919i q^{63} +(5.16588 - 3.28043i) q^{65} -10.3012i q^{67} +0.299919 q^{69} +0.127032i q^{71} -11.1281i q^{73} -1.73587 q^{75} +1.00000 q^{77} +14.2135 q^{79} +3.41266 q^{81} +4.44302i q^{83} -3.93902i q^{85} -2.41776 q^{87} -5.15887i q^{89} +(3.04372 - 1.93282i) q^{91} -8.05542i q^{93} -4.70716 q^{95} +4.55893i q^{97} +2.32919i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q - 4 q^{3} + 40 q^{9}+O(q^{10}) \) 36 * q - 4 * q^3 + 40 * q^9	\( 36 q - 4 q^{3} + 40 q^{9} - 4 q^{17} + 8 q^{23} - 80 q^{25} + 8 q^{27} + 8 q^{29} - 24 q^{39} + 32 q^{43} - 36 q^{49} - 20 q^{51} + 12 q^{53} + 32 q^{61} - 24 q^{65} + 80 q^{69} - 36 q^{75} + 36 q^{77} + 16 q^{79} + 132 q^{81} + 8 q^{91} + 56 q^{95}+O(q^{100}) \) 36 * q - 4 * q^3 + 40 * q^9 - 4 * q^17 + 8 * q^23 - 80 * q^25 + 8 * q^27 + 8 * q^29 - 24 * q^39 + 32 * q^43 - 36 * q^49 - 20 * q^51 + 12 * q^53 + 32 * q^61 - 24 * q^65 + 80 * q^69 - 36 * q^75 + 36 * q^77 + 16 * q^79 + 132 * q^81 + 8 * q^91 + 56 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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