LMFDB - Embedded newform 4004.2.m.c.2157.30

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			2157.30
Character	$\chi$	$=$	4004.2157
Dual form			4004.2.m.c.2157.29

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.18185 q^{3} +0.656342i q^{5} -1.00000i q^{7} +1.76049 q^{9} +O(q^{10})$	$q+2.18185 q^{3} +0.656342i q^{5} -1.00000i q^{7} +1.76049 q^{9} +1.00000i q^{11} +(-0.945058 + 3.47949i) q^{13} +1.43204i q^{15} +2.46142 q^{17} +5.98084i q^{19} -2.18185i q^{21} -1.56474 q^{23} +4.56922 q^{25} -2.70444 q^{27} +1.39681 q^{29} +7.01087i q^{31} +2.18185i q^{33} +0.656342 q^{35} -6.51691i q^{37} +(-2.06198 + 7.59174i) q^{39} +1.43526i q^{41} -2.49124 q^{43} +1.15548i q^{45} +2.52117i q^{47} -1.00000 q^{49} +5.37045 q^{51} -7.93016 q^{53} -0.656342 q^{55} +13.0493i q^{57} +6.65406i q^{59} +9.05764 q^{61} -1.76049i q^{63} +(-2.28374 - 0.620281i) q^{65} +1.47785i q^{67} -3.41402 q^{69} -4.89962i q^{71} +0.638688i q^{73} +9.96936 q^{75} +1.00000 q^{77} -1.86672 q^{79} -11.1821 q^{81} +9.14684i q^{83} +1.61553i q^{85} +3.04763 q^{87} +8.47760i q^{89} +(3.47949 + 0.945058i) q^{91} +15.2967i q^{93} -3.92547 q^{95} +9.88002i q^{97} +1.76049i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	2.18185			1.25969			0.629847	−	0.776719i	$-0.283118\pi$
$3$	2.18185			1.25969			0.629847	+	0.776719i	$0.283118\pi$
$4$		0			0
$5$			0.656342i			0.293525i	0.989172	+	0.146762i	$0.0468853\pi$
$5$			0.656342i			0.293525i	−0.989172	+	0.146762i	$0.953115\pi$
$6$		0			0
$7$		−	1.00000i		−	0.377964i
$7$		−	1.00000i		−	0.377964i
$8$		0			0
$9$	1.76049			0.586828
$10$		0			0
$11$			1.00000i			0.301511i
$11$			1.00000i			0.301511i
$12$		0			0
$13$	−0.945058	+	3.47949i	−0.262112	+	0.965037i
$13$	−0.945058	+	3.47949i	−0.262112	+	0.965037i
$14$		0			0
$15$			1.43204i			0.369751i
$16$		0			0
$17$	2.46142			0.596981			0.298491	−	0.954413i	$-0.403517\pi$
$17$	2.46142			0.596981			0.298491	+	0.954413i	$0.403517\pi$
$18$		0			0
$19$			5.98084i			1.37210i	0.727555	+	0.686049i	$0.240657\pi$
$19$			5.98084i			1.37210i	−0.727555	+	0.686049i	$0.759343\pi$
$20$		0			0
$21$		−	2.18185i		−	0.476119i
$22$		0			0
$23$	−1.56474			−0.326270			−0.163135	−	0.986604i	$-0.552161\pi$
$23$	−1.56474			−0.326270			−0.163135	+	0.986604i	$0.552161\pi$
$24$		0			0
$25$	4.56922			0.913843
$26$		0			0
$27$	−2.70444			−0.520470
$28$		0			0
$29$	1.39681			0.259381			0.129690	−	0.991555i	$-0.458602\pi$
$29$	1.39681			0.259381			0.129690	+	0.991555i	$0.458602\pi$
$30$		0			0
$31$			7.01087i			1.25919i	0.776924	+	0.629595i	$0.216779\pi$
$31$			7.01087i			1.25919i	−0.776924	+	0.629595i	$0.783221\pi$
$32$		0			0
$33$			2.18185i			0.379812i
$34$		0			0
$35$	0.656342			0.110942
$36$		0			0
$37$		−	6.51691i		−	1.07137i	−0.844417	−	0.535687i	$-0.820053\pi$
$37$		−	6.51691i		−	1.07137i	0.844417	−	0.535687i	$-0.179947\pi$
$38$		0			0
$39$	−2.06198	+	7.59174i	−0.330181	+	1.21565i
$40$		0			0
$41$			1.43526i			0.224151i	0.993700	+	0.112075i	$0.0357498\pi$
$41$			1.43526i			0.224151i	−0.993700	+	0.112075i	$0.964250\pi$
$42$		0			0
$43$	−2.49124			−0.379911			−0.189955	−	0.981793i	$-0.560834\pi$
$43$	−2.49124			−0.379911			−0.189955	+	0.981793i	$0.560834\pi$
$44$		0			0
$45$			1.15548i			0.172249i
$46$		0			0
$47$			2.52117i			0.367750i	0.982950	+	0.183875i	$0.0588642\pi$
$47$			2.52117i			0.367750i	−0.982950	+	0.183875i	$0.941136\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$		0			0
$51$	5.37045			0.752014
$52$		0			0
$53$	−7.93016			−1.08929			−0.544646	−	0.838666i	$-0.683336\pi$
$53$	−7.93016			−1.08929			−0.544646	+	0.838666i	$0.683336\pi$
$54$		0			0
$55$	−0.656342			−0.0885011
$56$		0			0
$57$			13.0493i			1.72842i
$58$		0			0
$59$			6.65406i			0.866284i	0.901326	+	0.433142i	$0.142595\pi$
$59$			6.65406i			0.866284i	−0.901326	+	0.433142i	$0.857405\pi$
$60$		0			0
$61$	9.05764			1.15971			0.579856	−	0.814719i	$-0.303109\pi$
$61$	9.05764			1.15971			0.579856	+	0.814719i	$0.303109\pi$
$62$		0			0
$63$		−	1.76049i		−	0.221800i
$64$		0			0
$65$	−2.28374	−	0.620281i	−0.283263	−	0.0769364i
$66$		0			0
$67$			1.47785i			0.180548i	0.995917	+	0.0902742i	$0.0287743\pi$
$67$			1.47785i			0.180548i	−0.995917	+	0.0902742i	$0.971226\pi$
$68$		0			0
$69$	−3.41402			−0.411000
$70$		0			0
$71$		−	4.89962i		−	0.581478i	−0.956802	−	0.290739i	$-0.906099\pi$
$71$		−	4.89962i		−	0.581478i	0.956802	−	0.290739i	$-0.0939011\pi$
$72$		0			0
$73$			0.638688i			0.0747528i	0.999301	+	0.0373764i	$0.0119001\pi$
$73$			0.638688i			0.0747528i	−0.999301	+	0.0373764i	$0.988100\pi$
$74$		0			0
$75$	9.96936			1.15116
$76$		0			0
$77$	1.00000			0.113961
$78$		0			0
$79$	−1.86672			−0.210022			−0.105011	−	0.994471i	$-0.533488\pi$
$79$	−1.86672			−0.210022			−0.105011	+	0.994471i	$0.533488\pi$
$80$		0			0
$81$	−11.1821			−1.24246
$82$		0			0
$83$			9.14684i			1.00400i	0.864869	+	0.501998i	$0.167402\pi$
$83$			9.14684i			1.00400i	−0.864869	+	0.501998i	$0.832598\pi$
$84$		0			0
$85$			1.61553i			0.175229i
$86$		0			0
$87$	3.04763			0.326741
$88$		0			0
$89$			8.47760i			0.898624i	0.893375	+	0.449312i	$0.148331\pi$
$89$			8.47760i			0.898624i	−0.893375	+	0.449312i	$0.851669\pi$
$90$		0			0
$91$	3.47949	+	0.945058i	0.364750	+	0.0990690i
$92$		0			0
$93$			15.2967i			1.58619i
$94$		0			0
$95$	−3.92547			−0.402745
$96$		0			0
$97$			9.88002i			1.00316i	0.865110	+	0.501582i	$0.167248\pi$
$97$			9.88002i			1.00316i	−0.865110	+	0.501582i	$0.832752\pi$
$98$		0			0
$99$			1.76049i			0.176935i
$100$		0			0
$101$	−3.36765			−0.335093			−0.167547	−	0.985864i	$-0.553584\pi$
$101$	−3.36765			−0.335093			−0.167547	+	0.985864i	$0.553584\pi$
$102$		0			0
$103$	3.53773			0.348582			0.174291	−	0.984694i	$-0.444237\pi$
$103$	3.53773			0.348582			0.174291	+	0.984694i	$0.444237\pi$
$104$		0			0
$105$	1.43204			0.139753
$106$		0			0
$107$	10.6001			1.02475			0.512375	−	0.858762i	$-0.328766\pi$
$107$	10.6001			1.02475			0.512375	+	0.858762i	$0.328766\pi$
$108$		0			0
$109$		−	3.36087i		−	0.321913i	−0.986961	−	0.160957i	$-0.948542\pi$
$109$		−	3.36087i		−	0.321913i	0.986961	−	0.160957i	$-0.0514579\pi$
$110$		0			0
$111$		−	14.2189i		−	1.34960i
$112$		0			0
$113$	6.15074			0.578613			0.289306	−	0.957237i	$-0.406575\pi$
$113$	6.15074			0.578613			0.289306	+	0.957237i	$0.406575\pi$
$114$		0			0
$115$		−	1.02700i		−	0.0957684i
$116$		0			0
$117$	−1.66376	+	6.12559i	−0.153815	+	0.566311i
$118$		0			0
$119$		−	2.46142i		−	0.225638i
$120$		0			0
$121$	−1.00000			−0.0909091
$122$		0			0
$123$			3.13154i			0.282361i
$124$		0			0
$125$			6.28067i			0.561761i
$126$		0			0
$127$	13.0011			1.15366			0.576829	−	0.816865i	$-0.304290\pi$
$127$	13.0011			1.15366			0.576829	+	0.816865i	$0.304290\pi$
$128$		0			0
$129$	−5.43553			−0.478571
$130$		0			0
$131$	8.94054			0.781138			0.390569	−	0.920574i	$-0.372278\pi$
$131$	8.94054			0.781138			0.390569	+	0.920574i	$0.372278\pi$
$132$		0			0
$133$	5.98084			0.518605
$134$		0			0
$135$		−	1.77504i		−	0.152771i
$136$		0			0
$137$		−	3.41337i		−	0.291624i	−0.989312	−	0.145812i	$-0.953421\pi$
$137$		−	3.41337i		−	0.291624i	0.989312	−	0.145812i	$-0.0465795\pi$
$138$		0			0
$139$	3.60469			0.305746			0.152873	−	0.988246i	$-0.451147\pi$
$139$	3.60469			0.305746			0.152873	+	0.988246i	$0.451147\pi$
$140$		0			0
$141$			5.50082i			0.463253i
$142$		0			0
$143$	−3.47949	−	0.945058i	−0.290970	−	0.0790297i
$144$		0			0
$145$			0.916784i			0.0761348i
$146$		0			0
$147$	−2.18185			−0.179956
$148$		0			0
$149$			1.57686i			0.129181i	0.997912	+	0.0645905i	$0.0205741\pi$
$149$			1.57686i			0.129181i	−0.997912	+	0.0645905i	$0.979426\pi$
$150$		0			0
$151$		−	3.78274i		−	0.307835i	−0.988084	−	0.153918i	$-0.950811\pi$
$151$		−	3.78274i		−	0.307835i	0.988084	−	0.153918i	$-0.0491890\pi$
$152$		0			0
$153$	4.33329			0.350326
$154$		0			0
$155$	−4.60152			−0.369603
$156$		0			0
$157$	20.0791			1.60248			0.801242	−	0.598340i	$-0.204173\pi$
$157$	20.0791			1.60248			0.801242	+	0.598340i	$0.204173\pi$
$158$		0			0
$159$	−17.3024			−1.37217
$160$		0			0
$161$			1.56474i			0.123318i
$162$		0			0
$163$			11.4567i			0.897359i	0.893693	+	0.448680i	$0.148106\pi$
$163$			11.4567i			0.897359i	−0.893693	+	0.448680i	$0.851894\pi$
$164$		0			0
$165$	−1.43204			−0.111484
$166$		0			0
$167$		−	4.52433i		−	0.350103i	−0.984559	−	0.175051i	$-0.943991\pi$
$167$		−	4.52433i		−	0.350103i	0.984559	−	0.175051i	$-0.0560092\pi$
$168$		0			0
$169$	−11.2137	−	6.57664i	−0.862595	−	0.505896i
$170$		0			0
$171$			10.5292i			0.805186i
$172$		0			0
$173$	5.96019			0.453145			0.226573	−	0.973994i	$-0.427248\pi$
$173$	5.96019			0.453145			0.226573	+	0.973994i	$0.427248\pi$
$174$		0			0
$175$		−	4.56922i		−	0.345400i
$176$		0			0
$177$			14.5182i			1.09125i
$178$		0			0
$179$	−7.26747			−0.543196			−0.271598	−	0.962411i	$-0.587552\pi$
$179$	−7.26747			−0.543196			−0.271598	+	0.962411i	$0.587552\pi$
$180$		0			0
$181$	−22.4943			−1.67199			−0.835995	−	0.548737i	$-0.815109\pi$
$181$	−22.4943			−1.67199			−0.835995	+	0.548737i	$0.815109\pi$
$182$		0			0
$183$	19.7624			1.46088
$184$		0			0
$185$	4.27732			0.314475
$186$		0			0
$187$			2.46142i			0.179997i
$188$		0			0
$189$			2.70444i			0.196719i
$190$		0			0
$191$	12.6293			0.913825			0.456913	−	0.889512i	$-0.348955\pi$
$191$	12.6293			0.913825			0.456913	+	0.889512i	$0.348955\pi$
$192$		0			0
$193$		−	8.95703i		−	0.644741i	−0.946614	−	0.322371i	$-0.895520\pi$
$193$		−	8.95703i		−	0.644741i	0.946614	−	0.322371i	$-0.104480\pi$
$194$		0			0
$195$	−4.98278	−	1.35336i	−0.356824	−	0.0969162i
$196$		0			0
$197$			5.77052i			0.411133i	0.978643	+	0.205566i	$0.0659037\pi$
$197$			5.77052i			0.411133i	−0.978643	+	0.205566i	$0.934096\pi$
$198$		0			0
$199$	8.41221			0.596326			0.298163	−	0.954515i	$-0.403626\pi$
$199$	8.41221			0.596326			0.298163	+	0.954515i	$0.403626\pi$
$200$		0			0
$201$			3.22446i			0.227436i
$202$		0			0
$203$		−	1.39681i		−	0.0980368i
$204$		0			0
$205$	−0.942024			−0.0657938
$206$		0			0
$207$	−2.75469			−0.191464
$208$		0			0
$209$	−5.98084			−0.413703
$210$		0			0
$211$	9.86340			0.679025			0.339512	−	0.940602i	$-0.389738\pi$
$211$	9.86340			0.679025			0.339512	+	0.940602i	$0.389738\pi$
$212$		0			0
$213$		−	10.6902i		−	0.732484i
$214$		0			0
$215$		−	1.63511i		−	0.111513i
$216$		0			0
$217$	7.01087			0.475929
$218$		0			0
$219$			1.39352i			0.0941657i
$220$		0			0
$221$	−2.32618	+	8.56448i	−0.156476	+	0.576109i
$222$		0			0
$223$			18.2538i			1.22236i	0.791491	+	0.611181i	$0.209305\pi$
$223$			18.2538i			1.22236i	−0.791491	+	0.611181i	$0.790695\pi$
$224$		0			0
$225$	8.04404			0.536269
$226$		0			0
$227$		−	6.10395i		−	0.405133i	−0.979268	−	0.202567i	$-0.935072\pi$
$227$		−	6.10395i		−	0.405133i	0.979268	−	0.202567i	$-0.0649283\pi$
$228$		0			0
$229$		−	2.63647i		−	0.174223i	−0.996199	−	0.0871114i	$-0.972236\pi$
$229$		−	2.63647i		−	0.174223i	0.996199	−	0.0871114i	$-0.0277636\pi$
$230$		0			0
$231$	2.18185			0.143555
$232$		0			0
$233$	11.9511			0.782941			0.391471	−	0.920191i	$-0.371966\pi$
$233$	11.9511			0.782941			0.391471	+	0.920191i	$0.371966\pi$
$234$		0			0
$235$	−1.65475			−0.107944
$236$		0			0
$237$	−4.07290			−0.264563
$238$		0			0
$239$		−	27.7702i		−	1.79631i	−0.439682	−	0.898153i	$-0.644909\pi$
$239$		−	27.7702i		−	1.79631i	0.439682	−	0.898153i	$-0.355091\pi$
$240$		0			0
$241$			1.79208i			0.115438i	0.998333	+	0.0577191i	$0.0183828\pi$
$241$			1.79208i			0.115438i	−0.998333	+	0.0577191i	$0.981617\pi$
$242$		0			0
$243$	−16.2845			−1.04465
$244$		0			0
$245$		−	0.656342i		−	0.0419321i
$246$		0			0
$247$	−20.8103	−	5.65224i	−1.32413	−	0.359643i
$248$		0			0
$249$			19.9571i			1.26473i
$250$		0			0
$251$	−28.3399			−1.78880			−0.894399	−	0.447271i	$-0.852396\pi$
$251$	−28.3399			−1.78880			−0.894399	+	0.447271i	$0.852396\pi$
$252$		0			0
$253$		−	1.56474i		−	0.0983741i
$254$		0			0
$255$			3.52485i			0.220735i
$256$		0			0
$257$	−2.79250			−0.174191			−0.0870956	−	0.996200i	$-0.527759\pi$
$257$	−2.79250			−0.174191			−0.0870956	+	0.996200i	$0.527759\pi$
$258$		0			0
$259$	−6.51691			−0.404941
$260$		0			0
$261$	2.45906			0.152212
$262$		0			0
$263$	15.7603			0.971823			0.485911	−	0.874008i	$-0.338488\pi$
$263$	15.7603			0.971823			0.485911	+	0.874008i	$0.338488\pi$
$264$		0			0
$265$		−	5.20489i		−	0.319734i
$266$		0			0
$267$			18.4969i			1.13199i
$268$		0			0
$269$	−12.6893			−0.773681			−0.386841	−	0.922147i	$-0.626434\pi$
$269$	−12.6893			−0.773681			−0.386841	+	0.922147i	$0.626434\pi$
$270$		0			0
$271$		−	6.82466i		−	0.414569i	−0.978281	−	0.207284i	$-0.933537\pi$
$271$		−	6.82466i		−	0.414569i	0.978281	−	0.207284i	$-0.0664625\pi$
$272$		0			0
$273$	7.59174	+	2.06198i	0.459473	+	0.124797i
$274$		0			0
$275$			4.56922i			0.275534i
$276$		0			0
$277$	−21.4740			−1.29025			−0.645124	−	0.764078i	$-0.723194\pi$
$277$	−21.4740			−1.29025			−0.645124	+	0.764078i	$0.723194\pi$
$278$		0			0
$279$			12.3425i			0.738928i
$280$		0			0
$281$		−	7.76841i		−	0.463425i	−0.972784	−	0.231712i	$-0.925567\pi$
$281$		−	7.76841i		−	0.463425i	0.972784	−	0.231712i	$-0.0744328\pi$
$282$		0			0
$283$	−18.5355			−1.10182			−0.550911	−	0.834564i	$-0.685720\pi$
$283$	−18.5355			−1.10182			−0.550911	+	0.834564i	$0.685720\pi$
$284$		0			0
$285$	−8.56481			−0.507336
$286$		0			0
$287$	1.43526			0.0847210
$288$		0			0
$289$	−10.9414			−0.643613
$290$		0			0
$291$			21.5568i			1.26368i
$292$		0			0
$293$		−	18.9627i		−	1.10781i	−0.832579	−	0.553906i	$-0.813137\pi$
$293$		−	18.9627i		−	1.10781i	0.832579	−	0.553906i	$-0.186863\pi$
$294$		0			0
$295$	−4.36733			−0.254276
$296$		0			0
$297$		−	2.70444i		−	0.156928i
$298$		0			0
$299$	1.47877	−	5.44449i	0.0855192	−	0.314863i
$300$		0			0
$301$			2.49124i			0.143593i
$302$		0			0
$303$	−7.34771			−0.422115
$304$		0			0
$305$			5.94490i			0.340404i
$306$		0			0
$307$		−	5.83629i		−	0.333094i	−0.986033	−	0.166547i	$-0.946738\pi$
$307$		−	5.83629i		−	0.333094i	0.986033	−	0.166547i	$-0.0532618\pi$
$308$		0			0
$309$	7.71880			0.439107
$310$		0			0
$311$	17.6603			1.00142			0.500711	−	0.865614i	$-0.333072\pi$
$311$	17.6603			1.00142			0.500711	+	0.865614i	$0.333072\pi$
$312$		0			0
$313$	15.5314			0.877886			0.438943	−	0.898515i	$-0.355353\pi$
$313$	15.5314			0.877886			0.438943	+	0.898515i	$0.355353\pi$
$314$		0			0
$315$	1.15548			0.0651039
$316$		0			0
$317$		−	34.9635i		−	1.96375i	−0.189539	−	0.981873i	$-0.560699\pi$
$317$		−	34.9635i		−	1.96375i	0.189539	−	0.981873i	$-0.439301\pi$
$318$		0			0
$319$			1.39681i			0.0782063i
$320$		0			0
$321$	23.1278			1.29087
$322$		0			0
$323$			14.7213i			0.819117i
$324$		0			0
$325$	−4.31817	+	15.8986i	−0.239529	+	0.881893i
$326$		0			0
$327$		−	7.33293i		−	0.405512i
$328$		0			0
$329$	2.52117			0.138997
$330$		0			0
$331$			5.89561i			0.324052i	0.986787	+	0.162026i	$0.0518028\pi$
$331$			5.89561i			0.324052i	−0.986787	+	0.162026i	$0.948197\pi$
$332$		0			0
$333$		−	11.4729i		−	0.628712i
$334$		0			0
$335$	−0.969976			−0.0529954
$336$		0			0
$337$	17.4845			0.952439			0.476220	−	0.879326i	$-0.342007\pi$
$337$	17.4845			0.952439			0.476220	+	0.879326i	$0.342007\pi$
$338$		0			0
$339$	13.4200			0.728875
$340$		0			0
$341$	−7.01087			−0.379660
$342$		0			0
$343$			1.00000i			0.0539949i
$344$		0			0
$345$		−	2.24077i		−	0.120639i
$346$		0			0
$347$	10.5389			0.565757			0.282879	−	0.959156i	$-0.408711\pi$
$347$	10.5389			0.565757			0.282879	+	0.959156i	$0.408711\pi$
$348$		0			0
$349$		−	0.989807i		−	0.0529832i	−0.999649	−	0.0264916i	$-0.991566\pi$
$349$		−	0.989807i		−	0.0529832i	0.999649	−	0.0264916i	$-0.00843352\pi$
$350$		0			0
$351$	2.55585	−	9.41008i	0.136421	−	0.502273i
$352$		0			0
$353$		−	19.8717i		−	1.05766i	−0.848727	−	0.528832i	$-0.822630\pi$
$353$		−	19.8717i		−	1.05766i	0.848727	−	0.528832i	$-0.177370\pi$
$354$		0			0
$355$	3.21582			0.170678
$356$		0			0
$357$		−	5.37045i		−	0.284234i
$358$		0			0
$359$		−	4.52227i		−	0.238676i	−0.992854	−	0.119338i	$-0.961923\pi$
$359$		−	4.52227i		−	0.238676i	0.992854	−	0.119338i	$-0.0380773\pi$
$360$		0			0
$361$	−16.7704			−0.882655
$362$		0			0
$363$	−2.18185			−0.114518
$364$		0			0
$365$	−0.419198			−0.0219418
$366$		0			0
$367$	−14.5331			−0.758620			−0.379310	−	0.925270i	$-0.623839\pi$
$367$	−14.5331			−0.758620			−0.379310	+	0.925270i	$0.623839\pi$
$368$		0			0
$369$			2.52676i			0.131538i
$370$		0			0
$371$			7.93016i			0.411713i
$372$		0			0
$373$	−5.01263			−0.259544			−0.129772	−	0.991544i	$-0.541425\pi$
$373$	−5.01263			−0.259544			−0.129772	+	0.991544i	$0.541425\pi$
$374$		0			0
$375$			13.7035i			0.707646i
$376$		0			0
$377$	−1.32007	+	4.86019i	−0.0679868	+	0.250312i
$378$		0			0
$379$		−	7.00903i		−	0.360030i	−0.983664	−	0.180015i	$-0.942385\pi$
$379$		−	7.00903i		−	0.360030i	0.983664	−	0.180015i	$-0.0576146\pi$
$380$		0			0
$381$	28.3664			1.45326
$382$		0			0
$383$		−	18.6150i		−	0.951185i	−0.879666	−	0.475592i	$-0.842234\pi$
$383$		−	18.6150i		−	0.951185i	0.879666	−	0.475592i	$-0.157766\pi$
$384$		0			0
$385$			0.656342i			0.0334503i
$386$		0			0
$387$	−4.38580			−0.222943
$388$		0			0
$389$	−32.5643			−1.65107			−0.825537	−	0.564347i	$-0.809128\pi$
$389$	−32.5643			−1.65107			−0.825537	+	0.564347i	$0.809128\pi$
$390$		0			0
$391$	−3.85147			−0.194777
$392$		0			0
$393$	19.5069			0.983995
$394$		0			0
$395$		−	1.22520i		−	0.0616466i
$396$		0			0
$397$			14.3648i			0.720949i	0.932769	+	0.360475i	$0.117385\pi$
$397$			14.3648i			0.720949i	−0.932769	+	0.360475i	$0.882615\pi$
$398$		0			0
$399$	13.0493			0.653283
$400$		0			0
$401$			0.199184i			0.00994677i	0.999988	+	0.00497339i	$0.00158308\pi$
$401$			0.199184i			0.00994677i	−0.999988	+	0.00497339i	$0.998417\pi$
$402$		0			0
$403$	−24.3943	−	6.62568i	−1.21516	−	0.330048i
$404$		0			0
$405$		−	7.33931i		−	0.364693i
$406$		0			0
$407$	6.51691			0.323031
$408$		0			0
$409$			4.93924i			0.244230i	0.992516	+	0.122115i	$0.0389676\pi$
$409$			4.93924i			0.244230i	−0.992516	+	0.122115i	$0.961032\pi$
$410$		0			0
$411$		−	7.44747i		−	0.367357i
$412$		0			0
$413$	6.65406			0.327425
$414$		0			0
$415$	−6.00345			−0.294698
$416$		0			0
$417$	7.86491			0.385146
$418$		0			0
$419$	12.4639			0.608904			0.304452	−	0.952528i	$-0.401527\pi$
$419$	12.4639			0.608904			0.304452	+	0.952528i	$0.401527\pi$
$420$		0			0
$421$			9.88924i			0.481972i	0.970529	+	0.240986i	$0.0774708\pi$
$421$			9.88924i			0.481972i	−0.970529	+	0.240986i	$0.922529\pi$
$422$		0			0
$423$			4.43848i			0.215806i
$424$		0			0
$425$	11.2467			0.545547
$426$		0			0
$427$		−	9.05764i		−	0.438330i
$428$		0			0
$429$	−7.59174	−	2.06198i	−0.366533	−	0.0995532i
$430$		0			0
$431$		−	12.8696i		−	0.619904i	−0.950752	−	0.309952i	$-0.899687\pi$
$431$		−	12.8696i		−	0.619904i	0.950752	−	0.309952i	$-0.100313\pi$
$432$		0			0
$433$	−1.60281			−0.0770261			−0.0385130	−	0.999258i	$-0.512262\pi$
$433$	−1.60281			−0.0770261			−0.0385130	+	0.999258i	$0.512262\pi$
$434$		0			0
$435$			2.00029i			0.0959065i
$436$		0			0
$437$		−	9.35843i		−	0.447675i
$438$		0			0
$439$	−5.05805			−0.241408			−0.120704	−	0.992689i	$-0.538515\pi$
$439$	−5.05805			−0.241408			−0.120704	+	0.992689i	$0.538515\pi$
$440$		0			0
$441$	−1.76049			−0.0838326
$442$		0			0
$443$	17.8159			0.846460			0.423230	−	0.906022i	$-0.360896\pi$
$443$	17.8159			0.846460			0.423230	+	0.906022i	$0.360896\pi$
$444$		0			0
$445$	−5.56420			−0.263769
$446$		0			0
$447$			3.44047i			0.162729i
$448$		0			0
$449$		−	1.07772i		−	0.0508610i	−0.999677	−	0.0254305i	$-0.991904\pi$
$449$		−	1.07772i		−	0.0508610i	0.999677	−	0.0254305i	$-0.00809565\pi$
$450$		0			0
$451$	−1.43526			−0.0675840
$452$		0			0
$453$		−	8.25339i		−	0.387778i
$454$		0			0
$455$	−0.620281	+	2.28374i	−0.0290792	+	0.107063i
$456$		0			0
$457$		−	27.5500i		−	1.28873i	−0.764717	−	0.644367i	$-0.777121\pi$
$457$		−	27.5500i		−	1.28873i	0.764717	−	0.644367i	$-0.222879\pi$
$458$		0			0
$459$	−6.65675			−0.310711
$460$		0			0
$461$		−	6.23110i		−	0.290211i	−0.989416	−	0.145106i	$-0.953648\pi$
$461$		−	6.23110i		−	0.290211i	0.989416	−	0.145106i	$-0.0463522\pi$
$462$		0			0
$463$		−	15.3761i		−	0.714587i	−0.933992	−	0.357293i	$-0.883700\pi$
$463$		−	15.3761i		−	0.714587i	0.933992	−	0.357293i	$-0.116300\pi$
$464$		0			0
$465$	−10.0399			−0.465587
$466$		0			0
$467$	−13.4541			−0.622583			−0.311292	−	0.950314i	$-0.600762\pi$
$467$	−13.4541			−0.622583			−0.311292	+	0.950314i	$0.600762\pi$
$468$		0			0
$469$	1.47785			0.0682409
$470$		0			0
$471$	43.8096			2.01864
$472$		0			0
$473$		−	2.49124i		−	0.114547i
$474$		0			0
$475$			27.3277i			1.25388i
$476$		0			0
$477$	−13.9609			−0.639227
$478$		0			0
$479$		−	12.7525i		−	0.582675i	−0.956620	−	0.291337i	$-0.905900\pi$
$479$		−	12.7525i		−	0.582675i	0.956620	−	0.291337i	$-0.0941002\pi$
$480$		0			0
$481$	22.6755	+	6.15885i	1.03392	+	0.280820i
$482$		0			0
$483$			3.41402i			0.155343i
$484$		0			0
$485$	−6.48467			−0.294454
$486$		0			0
$487$			14.6236i			0.662659i	0.943515	+	0.331329i	$0.107497\pi$
$487$			14.6236i			0.662659i	−0.943515	+	0.331329i	$0.892503\pi$
$488$		0			0
$489$			24.9969i			1.13040i
$490$		0			0
$491$	14.5194			0.655253			0.327626	−	0.944807i	$-0.393751\pi$
$491$	14.5194			0.655253			0.327626	+	0.944807i	$0.393751\pi$
$492$		0			0
$493$	3.43813			0.154846
$494$		0			0
$495$	−1.15548			−0.0519349
$496$		0			0
$497$	−4.89962			−0.219778
$498$		0			0
$499$			9.05835i			0.405507i	0.979230	+	0.202754i	$0.0649891\pi$
$499$			9.05835i			0.405507i	−0.979230	+	0.202754i	$0.935011\pi$
$500$		0			0
$501$		−	9.87142i		−	0.441022i
$502$		0			0
$503$	0.990698			0.0441730			0.0220865	−	0.999756i	$-0.492969\pi$
$503$	0.990698			0.0441730			0.0220865	+	0.999756i	$0.492969\pi$
$504$		0			0
$505$		−	2.21033i		−	0.0983582i
$506$		0			0
$507$	−24.4667	−	14.3493i	−1.08661	−	0.637273i
$508$		0			0
$509$		−	30.5013i		−	1.35195i	−0.736927	−	0.675973i	$-0.763724\pi$
$509$		−	30.5013i		−	1.35195i	0.736927	−	0.675973i	$-0.236276\pi$
$510$		0			0
$511$	0.638688			0.0282539
$512$		0			0
$513$		−	16.1748i		−	0.714136i
$514$		0			0
$515$			2.32196i			0.102318i
$516$		0			0
$517$	−2.52117			−0.110881
$518$		0			0
$519$	13.0043			0.570824
$520$		0			0
$521$	7.74677			0.339392			0.169696	−	0.985496i	$-0.445721\pi$
$521$	7.74677			0.339392			0.169696	+	0.985496i	$0.445721\pi$
$522$		0			0
$523$	−9.24815			−0.404393			−0.202197	−	0.979345i	$-0.564808\pi$
$523$	−9.24815			−0.404393			−0.202197	+	0.979345i	$0.564808\pi$
$524$		0			0
$525$		−	9.96936i		−	0.435099i
$526$		0			0
$527$			17.2567i			0.751712i
$528$		0			0
$529$	−20.5516			−0.893548
$530$		0			0
$531$			11.7144i			0.508360i
$532$		0			0
$533$	−4.99399	−	1.35641i	−0.216314	−	0.0587526i
$534$		0			0
$535$			6.95728i			0.300789i
$536$		0			0
$537$	−15.8566			−0.684261
$538$		0			0
$539$		−	1.00000i		−	0.0430730i
$540$		0			0
$541$		−	13.9413i		−	0.599385i	−0.954036	−	0.299692i	$-0.903116\pi$
$541$		−	13.9413i		−	0.599385i	0.954036	−	0.299692i	$-0.0968841\pi$
$542$		0			0
$543$	−49.0793			−2.10620
$544$		0			0
$545$	2.20588			0.0944896
$546$		0			0
$547$	44.6039			1.90713			0.953563	−	0.301192i	$-0.0973846\pi$
$547$	44.6039			1.90713			0.953563	+	0.301192i	$0.0973846\pi$
$548$		0			0
$549$	15.9458			0.680552
$550$		0			0
$551$			8.35409i			0.355896i
$552$		0			0
$553$			1.86672i			0.0793808i
$554$		0			0
$555$	9.33248			0.396142
$556$		0			0
$557$		−	4.70184i		−	0.199223i	−0.995026	−	0.0996117i	$-0.968240\pi$
$557$		−	4.70184i		−	0.199223i	0.995026	−	0.0996117i	$-0.0317601\pi$
$558$		0			0
$559$	2.35437	−	8.66826i	0.0995792	−	0.366628i
$560$		0			0
$561$			5.37045i			0.226741i
$562$		0			0
$563$	−23.6609			−0.997190			−0.498595	−	0.866835i	$-0.666150\pi$
$563$	−23.6609			−0.997190			−0.498595	+	0.866835i	$0.666150\pi$
$564$		0			0
$565$			4.03699i			0.169837i
$566$		0			0
$567$			11.1821i			0.469606i
$568$		0			0
$569$	40.4744			1.69677			0.848387	−	0.529376i	$-0.177574\pi$
$569$	40.4744			1.69677			0.848387	+	0.529376i	$0.177574\pi$
$570$		0			0
$571$	−4.40484			−0.184337			−0.0921684	−	0.995743i	$-0.529380\pi$
$571$	−4.40484			−0.184337			−0.0921684	+	0.995743i	$0.529380\pi$
$572$		0			0
$573$	27.5553			1.15114
$574$		0			0
$575$	−7.14961			−0.298160
$576$		0			0
$577$			31.0780i			1.29379i	0.762578	+	0.646896i	$0.223933\pi$
$577$			31.0780i			1.29379i	−0.762578	+	0.646896i	$0.776067\pi$
$578$		0			0
$579$		−	19.5429i		−	0.812177i
$580$		0			0
$581$	9.14684			0.379475
$582$		0			0
$583$		−	7.93016i		−	0.328434i
$584$		0			0
$585$	−4.02048	−	1.09199i	−0.166226	−	0.0451484i
$586$		0			0
$587$		−	12.9431i		−	0.534217i	−0.963666	−	0.267109i	$-0.913932\pi$
$587$		−	12.9431i		−	0.534217i	0.963666	−	0.267109i	$-0.0860683\pi$
$588$		0			0
$589$	−41.9309			−1.72773
$590$		0			0
$591$			12.5904i			0.517901i
$592$		0			0
$593$			43.4579i			1.78460i	0.451441	+	0.892301i	$0.350910\pi$
$593$			43.4579i			1.78460i	−0.451441	+	0.892301i	$0.649090\pi$
$594$		0			0
$595$	1.61553			0.0662303
$596$		0			0
$597$	18.3542			0.751188
$598$		0			0
$599$	−5.14135			−0.210070			−0.105035	−	0.994469i	$-0.533495\pi$
$599$	−5.14135			−0.210070			−0.105035	+	0.994469i	$0.533495\pi$
$600$		0			0
$601$	4.31253			0.175912			0.0879559	−	0.996124i	$-0.471967\pi$
$601$	4.31253			0.175912			0.0879559	+	0.996124i	$0.471967\pi$
$602$		0			0
$603$			2.60174i			0.105951i
$604$		0			0
$605$		−	0.656342i		−	0.0266841i
$606$		0			0
$607$	−7.96984			−0.323486			−0.161743	−	0.986833i	$-0.551712\pi$
$607$	−7.96984			−0.323486			−0.161743	+	0.986833i	$0.551712\pi$
$608$		0			0
$609$		−	3.04763i		−	0.123496i
$610$		0			0
$611$	−8.77239	−	2.38265i	−0.354893	−	0.0963917i
$612$		0			0
$613$			1.27544i			0.0515143i	0.999668	+	0.0257572i	$0.00819967\pi$
$613$			1.27544i			0.0515143i	−0.999668	+	0.0257572i	$0.991800\pi$
$614$		0			0
$615$	−2.05536			−0.0828801
$616$		0			0
$617$		−	12.2740i		−	0.494134i	−0.968998	−	0.247067i	$-0.920533\pi$
$617$		−	12.2740i		−	0.494134i	0.968998	−	0.247067i	$-0.0794668\pi$
$618$		0			0
$619$		−	33.1647i		−	1.33300i	−0.745504	−	0.666501i	$-0.767791\pi$
$619$		−	33.1647i		−	1.33300i	0.745504	−	0.666501i	$-0.232209\pi$
$620$		0			0
$621$	4.23173			0.169814
$622$		0			0
$623$	8.47760			0.339648
$624$		0			0
$625$	18.7238			0.748952
$626$		0			0
$627$	−13.0493			−0.521139
$628$		0			0
$629$		−	16.0408i		−	0.639590i
$630$		0			0
$631$			2.67332i			0.106423i	0.998583	+	0.0532116i	$0.0169458\pi$
$631$			2.67332i			0.106423i	−0.998583	+	0.0532116i	$0.983054\pi$
$632$		0			0
$633$	21.5205			0.855363
$634$		0			0
$635$			8.53314i			0.338627i
$636$		0			0
$637$	0.945058	−	3.47949i	0.0374446	−	0.137862i
$638$		0			0
$639$		−	8.62570i		−	0.341228i
$640$		0			0
$641$	−14.7907			−0.584196			−0.292098	−	0.956388i	$-0.594353\pi$
$641$	−14.7907			−0.584196			−0.292098	+	0.956388i	$0.594353\pi$
$642$		0			0
$643$			21.3911i			0.843583i	0.906693	+	0.421791i	$0.138599\pi$
$643$			21.3911i			0.843583i	−0.906693	+	0.421791i	$0.861401\pi$
$644$		0			0
$645$		−	3.56756i		−	0.140473i
$646$		0			0
$647$	−19.2641			−0.757348			−0.378674	−	0.925530i	$-0.623620\pi$
$647$	−19.2641			−0.757348			−0.378674	+	0.925530i	$0.623620\pi$
$648$		0			0
$649$	−6.65406			−0.261195
$650$		0			0
$651$	15.2967			0.599524
$652$		0			0
$653$	−20.4144			−0.798877			−0.399438	−	0.916760i	$-0.630795\pi$
$653$	−20.4144			−0.798877			−0.399438	+	0.916760i	$0.630795\pi$
$654$		0			0
$655$			5.86805i			0.229284i
$656$		0			0
$657$			1.12440i			0.0438671i
$658$		0			0
$659$	21.0394			0.819578			0.409789	−	0.912180i	$-0.365602\pi$
$659$	21.0394			0.819578			0.409789	+	0.912180i	$0.365602\pi$
$660$		0			0
$661$			9.96053i			0.387420i	0.981059	+	0.193710i	$0.0620521\pi$
$661$			9.96053i			0.387420i	−0.981059	+	0.193710i	$0.937948\pi$
$662$		0			0
$663$	−5.07539	+	18.6864i	−0.197112	+	0.725721i
$664$		0			0
$665$			3.92547i			0.152223i
$666$		0			0
$667$	−2.18564			−0.0846282
$668$		0			0
$669$			39.8270i			1.53980i
$670$		0			0
$671$			9.05764i			0.349666i
$672$		0			0
$673$	−17.2320			−0.664244			−0.332122	−	0.943236i	$-0.607765\pi$
$673$	−17.2320			−0.664244			−0.332122	+	0.943236i	$0.607765\pi$
$674$		0			0
$675$	−12.3572			−0.475628
$676$		0			0
$677$	−1.55745			−0.0598578			−0.0299289	−	0.999552i	$-0.509528\pi$
$677$	−1.55745			−0.0598578			−0.0299289	+	0.999552i	$0.509528\pi$
$678$		0			0
$679$	9.88002			0.379160
$680$		0			0
$681$		−	13.3179i		−	0.510344i
$682$		0			0
$683$			22.7575i			0.870791i	0.900239	+	0.435395i	$0.143391\pi$
$683$			22.7575i			0.870791i	−0.900239	+	0.435395i	$0.856609\pi$
$684$		0			0
$685$	2.24034			0.0855989
$686$		0			0
$687$		−	5.75239i		−	0.219467i
$688$		0			0
$689$	7.49446	−	27.5929i	0.285516	−	1.05121i
$690$		0			0
$691$			50.1165i			1.90652i	0.302149	+	0.953261i	$0.402296\pi$
$691$			50.1165i			1.90652i	−0.302149	+	0.953261i	$0.597704\pi$
$692$		0			0
$693$	1.76049			0.0668753
$694$		0			0
$695$			2.36591i			0.0897441i
$696$		0			0
$697$			3.53278i			0.133814i
$698$		0			0
$699$	26.0755			0.986266
$700$		0			0
$701$	28.5023			1.07652			0.538259	−	0.842780i	$-0.319082\pi$
$701$	28.5023			1.07652			0.538259	+	0.842780i	$0.319082\pi$
$702$		0			0
$703$	38.9766			1.47003
$704$		0			0
$705$	−3.61042			−0.135976
$706$		0			0
$707$			3.36765i			0.126653i
$708$		0			0
$709$			16.9600i			0.636946i	0.947932	+	0.318473i	$0.103170\pi$
$709$			16.9600i			0.636946i	−0.947932	+	0.318473i	$0.896830\pi$
$710$		0			0
$711$	−3.28632			−0.123247
$712$		0			0
$713$		−	10.9702i		−	0.410836i
$714$		0			0
$715$	0.620281	−	2.28374i	0.0231972	−	0.0854069i
$716$		0			0
$717$		−	60.5906i		−	2.26280i
$718$		0			0
$719$	−35.7864			−1.33461			−0.667304	−	0.744785i	$-0.732552\pi$
$719$	−35.7864			−1.33461			−0.667304	+	0.744785i	$0.732552\pi$
$720$		0			0
$721$		−	3.53773i		−	0.131752i
$722$		0			0
$723$			3.91006i			0.145417i
$724$		0			0
$725$	6.38232			0.237033
$726$		0			0
$727$	35.0772			1.30094			0.650471	−	0.759531i	$-0.274572\pi$
$727$	35.0772			1.30094			0.650471	+	0.759531i	$0.274572\pi$
$728$		0			0
$729$	−1.98393			−0.0734787
$730$		0			0
$731$	−6.13199			−0.226800
$732$		0			0
$733$		−	24.3420i		−	0.899094i	−0.893257	−	0.449547i	$-0.851585\pi$
$733$		−	24.3420i		−	0.899094i	0.893257	−	0.449547i	$-0.148415\pi$
$734$		0			0
$735$		−	1.43204i		−	0.0528216i
$736$		0			0
$737$	−1.47785			−0.0544374
$738$		0			0
$739$			12.0429i			0.443004i	0.975160	+	0.221502i	$0.0710960\pi$
$739$			12.0429i			0.443004i	−0.975160	+	0.221502i	$0.928904\pi$
$740$		0			0
$741$	−45.4050	−	12.3324i	−1.66799	−	0.453040i
$742$		0			0
$743$			8.84351i			0.324437i	0.986755	+	0.162218i	$0.0518649\pi$
$743$			8.84351i			0.324437i	−0.986755	+	0.162218i	$0.948135\pi$
$744$		0			0
$745$	−1.03496			−0.0379179
$746$		0			0
$747$			16.1029i			0.589173i
$748$		0			0
$749$		−	10.6001i		−	0.387319i
$750$		0			0
$751$	34.9949			1.27698			0.638491	−	0.769629i	$-0.279559\pi$
$751$	34.9949			1.27698			0.638491	+	0.769629i	$0.279559\pi$
$752$		0			0
$753$	−61.8335			−2.25334
$754$		0			0
$755$	2.48277			0.0903573
$756$		0			0
$757$	9.83780			0.357561			0.178780	−	0.983889i	$-0.442785\pi$
$757$	9.83780			0.357561			0.178780	+	0.983889i	$0.442785\pi$
$758$		0			0
$759$		−	3.41402i		−	0.123921i
$760$		0			0
$761$		−	37.4066i		−	1.35599i	−0.735067	−	0.677994i	$-0.762849\pi$
$761$		−	37.4066i		−	1.35599i	0.735067	−	0.677994i	$-0.237151\pi$
$762$		0			0
$763$	−3.36087			−0.121672
$764$		0			0
$765$			2.84412i			0.102829i
$766$		0			0
$767$	−23.1527	−	6.28847i	−0.835997	−	0.227063i
$768$		0			0
$769$			15.9292i			0.574423i	0.957867	+	0.287212i	$0.0927283\pi$
$769$			15.9292i			0.574423i	−0.957867	+	0.287212i	$0.907272\pi$
$770$		0			0
$771$	−6.09282			−0.219428
$772$		0			0
$773$		−	33.2767i		−	1.19688i	−0.801167	−	0.598441i	$-0.795787\pi$
$773$		−	33.2767i		−	1.19688i	0.801167	−	0.598441i	$-0.204213\pi$
$774$		0			0
$775$			32.0342i			1.15070i
$776$		0			0
$777$	−14.2189			−0.510102
$778$		0			0
$779$	−8.58409			−0.307557
$780$		0			0
$781$	4.89962			0.175322
$782$		0			0
$783$	−3.77759			−0.135000
$784$		0			0
$785$			13.1787i			0.470369i
$786$		0			0
$787$			18.2175i			0.649382i	0.945820	+	0.324691i	$0.105260\pi$
$787$			18.2175i			0.649382i	−0.945820	+	0.324691i	$0.894740\pi$
$788$		0			0
$789$	34.3867			1.22420
$790$		0			0
$791$		−	6.15074i		−	0.218695i
$792$		0			0
$793$	−8.55999	+	31.5160i	−0.303974	+	1.11917i
$794$		0			0
$795$		−	11.3563i		−	0.402767i
$796$		0			0
$797$	−33.4861			−1.18614			−0.593070	−	0.805151i	$-0.702084\pi$
$797$	−33.4861			−1.18614			−0.593070	+	0.805151i	$0.702084\pi$
$798$		0			0
$799$			6.20565i			0.219540i
$800$		0			0
$801$			14.9247i			0.527338i
$802$		0			0
$803$	−0.638688			−0.0225388
$804$		0			0
$805$	−1.02700			−0.0361970
$806$		0			0
$807$	−27.6862			−0.974601
$808$		0			0
$809$	−19.1093			−0.671846			−0.335923	−	0.941889i	$-0.609048\pi$
$809$	−19.1093			−0.671846			−0.335923	+	0.941889i	$0.609048\pi$
$810$		0			0
$811$			29.6536i			1.04128i	0.853776	+	0.520640i	$0.174307\pi$
$811$			29.6536i			1.04128i	−0.853776	+	0.520640i	$0.825693\pi$
$812$		0			0
$813$		−	14.8904i		−	0.522230i
$814$		0			0
$815$	−7.51952			−0.263397
$816$		0			0
$817$		−	14.8997i		−	0.521275i
$818$		0			0
$819$	6.12559	+	1.66376i	0.214046	+	0.0581365i
$820$		0			0
$821$		−	0.505554i		−	0.0176440i	−0.999961	−	0.00882198i	$-0.997192\pi$
$821$		−	0.505554i		−	0.0176440i	0.999961	−	0.00882198i	$-0.00280816\pi$
$822$		0			0
$823$	3.21536			0.112080			0.0560402	−	0.998429i	$-0.482153\pi$
$823$	3.21536			0.112080			0.0560402	+	0.998429i	$0.482153\pi$
$824$		0			0
$825$			9.96936i			0.347089i
$826$		0			0
$827$		−	15.5749i		−	0.541593i	−0.962637	−	0.270797i	$-0.912713\pi$
$827$		−	15.5749i		−	0.541593i	0.962637	−	0.270797i	$-0.0872871\pi$
$828$		0			0
$829$	13.4136			0.465873			0.232936	−	0.972492i	$-0.425167\pi$
$829$	13.4136			0.465873			0.232936	+	0.972492i	$0.425167\pi$
$830$		0			0
$831$	−46.8532			−1.62532
$832$		0			0
$833$	−2.46142			−0.0852830
$834$		0			0
$835$	2.96950			0.102764
$836$		0			0
$837$		−	18.9605i		−	0.655370i
$838$		0			0
$839$		−	27.9840i		−	0.966116i	−0.875588	−	0.483058i	$-0.839526\pi$
$839$		−	27.9840i		−	0.966116i	0.875588	−	0.483058i	$-0.160474\pi$
$840$		0			0
$841$	−27.0489			−0.932722
$842$		0			0
$843$		−	16.9495i		−	0.583773i
$844$		0			0
$845$	4.31652	−	7.36004i	0.148493	−	0.253193i
$846$		0			0
$847$			1.00000i			0.0343604i
$848$		0			0
$849$	−40.4418			−1.38796
$850$		0			0
$851$			10.1972i			0.349557i
$852$		0			0
$853$		−	30.4541i		−	1.04273i	−0.853334	−	0.521364i	$-0.825423\pi$
$853$		−	30.4541i		−	1.04273i	0.853334	−	0.521364i	$-0.174577\pi$
$854$		0			0
$855$	−6.91074			−0.236342
$856$		0			0
$857$	−32.8623			−1.12255			−0.561277	−	0.827628i	$-0.689690\pi$
$857$	−32.8623			−1.12255			−0.561277	+	0.827628i	$0.689690\pi$
$858$		0			0
$859$	49.2975			1.68201			0.841005	−	0.541028i	$-0.181965\pi$
$859$	49.2975			1.68201			0.841005	+	0.541028i	$0.181965\pi$
$860$		0			0
$861$	3.13154			0.106723
$862$		0			0
$863$			38.2755i			1.30291i	0.758686	+	0.651457i	$0.225842\pi$
$863$			38.2755i			1.30291i	−0.758686	+	0.651457i	$0.774158\pi$
$864$		0			0
$865$			3.91192i			0.133009i
$866$		0			0
$867$	−23.8726			−0.810756
$868$		0			0
$869$		−	1.86672i		−	0.0633240i
$870$		0			0
$871$	−5.14218	−	1.39666i	−0.174236	−	0.0473239i
$872$		0			0
$873$			17.3936i			0.588685i
$874$		0			0
$875$	6.28067			0.212326
$876$		0			0
$877$		−	9.05953i		−	0.305919i	−0.988232	−	0.152959i	$-0.951120\pi$
$877$		−	9.05953i		−	0.305919i	0.988232	−	0.152959i	$-0.0488804\pi$
$878$		0			0
$879$		−	41.3738i		−	1.39550i
$880$		0			0
$881$	−20.0130			−0.674254			−0.337127	−	0.941459i	$-0.609455\pi$
$881$	−20.0130			−0.674254			−0.337127	+	0.941459i	$0.609455\pi$
$882$		0			0
$883$	40.7800			1.37236			0.686179	−	0.727433i	$-0.259287\pi$
$883$	40.7800			1.37236			0.686179	+	0.727433i	$0.259287\pi$
$884$		0			0
$885$	−9.52888			−0.320310
$886$		0			0
$887$	34.4779			1.15765			0.578827	−	0.815450i	$-0.303511\pi$
$887$	34.4779			1.15765			0.578827	+	0.815450i	$0.303511\pi$
$888$		0			0
$889$		−	13.0011i		−	0.436042i
$890$		0			0
$891$		−	11.1821i		−	0.374616i
$892$		0			0
$893$	−15.0787			−0.504590
$894$		0			0
$895$		−	4.76994i		−	0.159442i
$896$		0			0
$897$	3.22645	−	11.8791i	0.107728	−	0.396631i
$898$		0			0
$899$			9.79284i			0.326610i
$900$		0			0
$901$	−19.5194			−0.650286
$902$		0			0
$903$			5.43553i			0.180883i
$904$		0			0
$905$		−	14.7640i		−	0.490771i
$906$		0			0
$907$	−21.8847			−0.726670			−0.363335	−	0.931659i	$-0.618362\pi$
$907$	−21.8847			−0.726670			−0.363335	+	0.931659i	$0.618362\pi$
$908$		0			0
$909$	−5.92869			−0.196642
$910$		0			0
$911$	41.2180			1.36561			0.682807	−	0.730599i	$-0.260759\pi$
$911$	41.2180			1.36561			0.682807	+	0.730599i	$0.260759\pi$
$912$		0			0
$913$	−9.14684			−0.302716
$914$		0			0
$915$			12.9709i			0.428805i
$916$		0			0
$917$		−	8.94054i		−	0.295242i
$918$		0			0
$919$	5.36922			0.177114			0.0885572	−	0.996071i	$-0.471774\pi$
$919$	5.36922			0.177114			0.0885572	+	0.996071i	$0.471774\pi$
$920$		0			0
$921$		−	12.7339i		−	0.419597i
$922$		0			0
$923$	17.0482	+	4.63042i	0.561148	+	0.152412i
$924$		0			0
$925$		−	29.7772i		−	0.979067i
$926$		0			0
$927$	6.22811			0.204558
$928$		0			0
$929$		−	37.9313i		−	1.24449i	−0.782825	−	0.622243i	$-0.786222\pi$
$929$		−	37.9313i		−	1.24449i	0.782825	−	0.622243i	$-0.213778\pi$
$930$		0			0
$931$		−	5.98084i		−	0.196014i
$932$		0			0
$933$	38.5321			1.26149
$934$		0			0
$935$	−1.61553			−0.0528335
$936$		0			0
$937$	28.4368			0.928989			0.464495	−	0.885576i	$-0.346236\pi$
$937$	28.4368			0.928989			0.464495	+	0.885576i	$0.346236\pi$
$938$		0			0
$939$	33.8872			1.10587
$940$		0			0
$941$		−	10.8106i		−	0.352417i	−0.984353	−	0.176208i	$-0.943617\pi$
$941$		−	10.8106i		−	0.352417i	0.984353	−	0.176208i	$-0.0563832\pi$
$942$		0			0
$943$		−	2.24581i		−	0.0731336i
$944$		0			0
$945$	−1.77504			−0.0577419
$946$		0			0
$947$		−	34.6788i		−	1.12691i	−0.826147	−	0.563455i	$-0.809472\pi$
$947$		−	34.6788i		−	1.12691i	0.826147	−	0.563455i	$-0.190528\pi$
$948$		0			0
$949$	−2.22231	−	0.603598i	−0.0721393	−	0.0195936i
$950$		0			0
$951$		−	76.2853i		−	2.47372i
$952$		0			0
$953$	38.7035			1.25373			0.626865	−	0.779128i	$-0.284338\pi$
$953$	38.7035			1.25373			0.626865	+	0.779128i	$0.284338\pi$
$954$		0			0
$955$			8.28915i			0.268231i
$956$		0			0
$957$			3.04763i			0.0985160i
$958$		0			0
$959$	−3.41337			−0.110223
$960$		0			0
$961$	−18.1523			−0.585557
$962$		0			0
$963$	18.6613			0.601352
$964$		0			0
$965$	5.87887			0.189248
$966$		0			0
$967$		−	16.9356i		−	0.544611i	−0.962211	−	0.272306i	$-0.912214\pi$
$967$		−	16.9356i		−	0.544611i	0.962211	−	0.272306i	$-0.0877861\pi$
$968$		0			0
$969$			32.1198i			1.03184i
$970$		0			0
$971$	−59.1761			−1.89905			−0.949526	−	0.313689i	$-0.898435\pi$
$971$	−59.1761			−1.89905			−0.949526	+	0.313689i	$0.898435\pi$
$972$		0			0
$973$		−	3.60469i		−	0.115561i
$974$		0			0
$975$	−9.42162	+	34.6883i	−0.301733	+	1.11091i
$976$		0			0
$977$		−	34.8626i		−	1.11535i	−0.830058	−	0.557677i	$-0.811693\pi$
$977$		−	34.8626i		−	1.11535i	0.830058	−	0.557677i	$-0.188307\pi$
$978$		0			0
$979$	−8.47760			−0.270945
$980$		0			0
$981$		−	5.91677i		−	0.188908i
$982$		0			0
$983$			19.6331i			0.626200i	0.949720	+	0.313100i	$0.101367\pi$
$983$			19.6331i			0.626200i	−0.949720	+	0.313100i	$0.898633\pi$
$984$		0			0
$985$	−3.78744			−0.120678
$986$		0			0
$987$	5.50082			0.175093
$988$		0			0
$989$	3.89814			0.123954
$990$		0			0
$991$	−12.0971			−0.384278			−0.192139	−	0.981368i	$-0.561542\pi$
$991$	−12.0971			−0.384278			−0.192139	+	0.981368i	$0.561542\pi$
$992$		0			0
$993$			12.8633i			0.408206i
$994$		0			0
$995$			5.52128i			0.175036i
$996$		0			0
$997$	48.1278			1.52422			0.762112	−	0.647446i	$-0.224163\pi$
$997$	48.1278			1.52422			0.762112	+	0.647446i	$0.224163\pi$
$998$		0			0
$999$			17.6246i			0.557617i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4004.2.m.c.2157.30	yes	36
13.12	even	2	inner	4004.2.m.c.2157.29	✓	36

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4004.2.m.c.2157.29	✓	36	13.12	even	2	inner
4004.2.m.c.2157.30	yes	36	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q+2.18185 q^{3} +0.656342i q^{5} -1.00000i q^{7} +1.76049 q^{9} +O(q^{10})\)	\(q+2.18185 q^{3} +0.656342i q^{5} -1.00000i q^{7} +1.76049 q^{9} +1.00000i q^{11} +(-0.945058 + 3.47949i) q^{13} +1.43204i q^{15} +2.46142 q^{17} +5.98084i q^{19} -2.18185i q^{21} -1.56474 q^{23} +4.56922 q^{25} -2.70444 q^{27} +1.39681 q^{29} +7.01087i q^{31} +2.18185i q^{33} +0.656342 q^{35} -6.51691i q^{37} +(-2.06198 + 7.59174i) q^{39} +1.43526i q^{41} -2.49124 q^{43} +1.15548i q^{45} +2.52117i q^{47} -1.00000 q^{49} +5.37045 q^{51} -7.93016 q^{53} -0.656342 q^{55} +13.0493i q^{57} +6.65406i q^{59} +9.05764 q^{61} -1.76049i q^{63} +(-2.28374 - 0.620281i) q^{65} +1.47785i q^{67} -3.41402 q^{69} -4.89962i q^{71} +0.638688i q^{73} +9.96936 q^{75} +1.00000 q^{77} -1.86672 q^{79} -11.1821 q^{81} +9.14684i q^{83} +1.61553i q^{85} +3.04763 q^{87} +8.47760i q^{89} +(3.47949 + 0.945058i) q^{91} +15.2967i q^{93} -3.92547 q^{95} +9.88002i q^{97} +1.76049i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q - 4 q^{3} + 40 q^{9}+O(q^{10}) \) 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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