LMFDB - Embedded newform 2912.2.i.a.337.13

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2912,2,Mod(337,2912)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2912, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0, 1])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2912.337"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 2912 = 2^{5} \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2912.i (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$23.2524370686$
Analytic rank:	$0$
Dimension:	$84$
Twist minimal:	no (minimal twist has level 728)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			337.13
Character	$\chi$	$=$	2912.337
Dual form			2912.2.i.a.337.14

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-0.208275i q^{3} -3.52272 q^{5} +1.00000i q^{7} +2.95662 q^{9} +1.92616 q^{11} +(-3.39827 - 1.20488i) q^{13} +0.733695i q^{15} -3.54436 q^{17} +0.129878 q^{19} +0.208275 q^{21} +5.09962 q^{23} +7.40954 q^{25} -1.24062i q^{27} -6.31068i q^{29} +1.15279i q^{31} -0.401171i q^{33} -3.52272i q^{35} +0.300461 q^{37} +(-0.250947 + 0.707776i) q^{39} +2.93462i q^{41} +5.57927i q^{43} -10.4153 q^{45} +2.30877i q^{47} -1.00000 q^{49} +0.738202i q^{51} +11.5827i q^{53} -6.78531 q^{55} -0.0270504i q^{57} -7.08922 q^{59} -6.22897i q^{61} +2.95662i q^{63} +(11.9712 + 4.24446i) q^{65} -8.04110 q^{67} -1.06212i q^{69} +16.5533i q^{71} -11.4665i q^{73} -1.54322i q^{75} +1.92616i q^{77} -14.6896 q^{79} +8.61147 q^{81} +3.76948 q^{83} +12.4858 q^{85} -1.31436 q^{87} -0.982458i q^{89} +(1.20488 - 3.39827i) q^{91} +0.240098 q^{93} -0.457523 q^{95} +9.30657i q^{97} +5.69492 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 84 q - 84 q^{9} + 8 q^{17} + 24 q^{23} + 92 q^{25} + 24 q^{39} - 84 q^{49} - 32 q^{55} - 24 q^{65} + 40 q^{79} + 84 q^{81} + 48 q^{87} + 16 q^{95}+O(q^{100}) $ 84 * q - 84 * q^9 + 8 * q^17 + 24 * q^23 + 92 * q^25 + 24 * q^39 - 84 * q^49 - 32 * q^55 - 24 * q^65 + 40 * q^79 + 84 * q^81 + 48 * q^87 + 16 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2912\mathbb{Z}\right)^\times$.

$n$	$1093$	$1249$	$2017$	$2367$
$\chi(n)$	$-1$	$1$	$-1$	$1$

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
$3$		−	0.208275i		−	0.120248i	−0.998191	−	0.0601239i	$-0.980850\pi$
$3$		−	0.208275i		−	0.120248i	0.998191	−	0.0601239i	$-0.0191496\pi$
$4$		0			0
$5$	−3.52272			−1.57541			−0.787704	−	0.616055i	$-0.788730\pi$
$5$	−3.52272			−1.57541			−0.787704	+	0.616055i	$0.788730\pi$
$6$		0			0
$7$			1.00000i			0.377964i
$7$			1.00000i			0.377964i
$8$		0			0
$9$	2.95662			0.985540
$10$		0			0
$11$	1.92616			0.580758			0.290379	−	0.956912i	$-0.406219\pi$
$11$	1.92616			0.580758			0.290379	+	0.956912i	$0.406219\pi$
$12$		0			0
$13$	−3.39827	−	1.20488i	−0.942511	−	0.334174i
$13$	−3.39827	−	1.20488i	−0.942511	−	0.334174i
$14$		0			0
$15$			0.733695i			0.189439i
$16$		0			0
$17$	−3.54436			−0.859633			−0.429816	−	0.902916i	$-0.641422\pi$
$17$	−3.54436			−0.859633			−0.429816	+	0.902916i	$0.641422\pi$
$18$		0			0
$19$	0.129878			0.0297960			0.0148980	−	0.999889i	$-0.495258\pi$
$19$	0.129878			0.0297960			0.0148980	+	0.999889i	$0.495258\pi$
$20$		0			0
$21$	0.208275			0.0454494
$22$		0			0
$23$	5.09962			1.06334			0.531672	−	0.846950i	$-0.321564\pi$
$23$	5.09962			1.06334			0.531672	+	0.846950i	$0.321564\pi$
$24$		0			0
$25$	7.40954			1.48191
$26$		0			0
$27$		−	1.24062i		−	0.238757i
$28$		0			0
$29$		−	6.31068i		−	1.17186i	−0.810360	−	0.585932i	$-0.800729\pi$
$29$		−	6.31068i		−	1.17186i	0.810360	−	0.585932i	$-0.199271\pi$
$30$		0			0
$31$			1.15279i			0.207048i	0.994627	+	0.103524i	$0.0330118\pi$
$31$			1.15279i			0.207048i	−0.994627	+	0.103524i	$0.966988\pi$
$32$		0			0
$33$		−	0.401171i		−	0.0698349i
$34$		0			0
$35$		−	3.52272i		−	0.595448i
$36$		0			0
$37$	0.300461			0.0493954			0.0246977	−	0.999695i	$-0.492138\pi$
$37$	0.300461			0.0493954			0.0246977	+	0.999695i	$0.492138\pi$
$38$		0			0
$39$	−0.250947	+	0.707776i	−0.0401837	+	0.113335i
$40$		0			0
$41$			2.93462i			0.458311i	0.973390	+	0.229156i	$0.0735964\pi$
$41$			2.93462i			0.458311i	−0.973390	+	0.229156i	$0.926404\pi$
$42$		0			0
$43$			5.57927i			0.850830i	0.904998	+	0.425415i	$0.139872\pi$
$43$			5.57927i			0.850830i	−0.904998	+	0.425415i	$0.860128\pi$
$44$		0			0
$45$	−10.4153			−1.55263
$46$		0			0
$47$			2.30877i			0.336769i	0.985721	+	0.168384i	$0.0538549\pi$
$47$			2.30877i			0.336769i	−0.985721	+	0.168384i	$0.946145\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$		0			0
$51$			0.738202i			0.103369i
$52$		0			0
$53$			11.5827i			1.59101i	0.605946	+	0.795506i	$0.292795\pi$
$53$			11.5827i			1.59101i	−0.605946	+	0.795506i	$0.707205\pi$
$54$		0			0
$55$	−6.78531			−0.914930
$56$		0			0
$57$		−	0.0270504i		−	0.00358291i
$58$		0			0
$59$	−7.08922			−0.922938			−0.461469	−	0.887156i	$-0.652677\pi$
$59$	−7.08922			−0.922938			−0.461469	+	0.887156i	$0.652677\pi$
$60$		0			0
$61$		−	6.22897i		−	0.797538i	−0.917051	−	0.398769i	$-0.869438\pi$
$61$		−	6.22897i		−	0.797538i	0.917051	−	0.398769i	$-0.130562\pi$
$62$		0			0
$63$			2.95662i			0.372499i
$64$		0			0
$65$	11.9712	+	4.24446i	1.48484	+	0.526460i
$66$		0			0
$67$	−8.04110			−0.982376			−0.491188	−	0.871053i	$-0.663437\pi$
$67$	−8.04110			−0.982376			−0.491188	+	0.871053i	$0.663437\pi$
$68$		0			0
$69$		−	1.06212i		−	0.127865i
$70$		0			0
$71$			16.5533i			1.96452i	0.187535	+	0.982258i	$0.439950\pi$
$71$			16.5533i			1.96452i	−0.187535	+	0.982258i	$0.560050\pi$
$72$		0			0
$73$		−	11.4665i		−	1.34205i	−0.741436	−	0.671024i	$-0.765855\pi$
$73$		−	11.4665i		−	1.34205i	0.741436	−	0.671024i	$-0.234145\pi$
$74$		0			0
$75$		−	1.54322i		−	0.178196i
$76$		0			0
$77$			1.92616i			0.219506i
$78$		0			0
$79$	−14.6896			−1.65271			−0.826357	−	0.563147i	$-0.809591\pi$
$79$	−14.6896			−1.65271			−0.826357	+	0.563147i	$0.809591\pi$
$80$		0			0
$81$	8.61147			0.956830
$82$		0			0
$83$	3.76948			0.413754			0.206877	−	0.978367i	$-0.433670\pi$
$83$	3.76948			0.413754			0.206877	+	0.978367i	$0.433670\pi$
$84$		0			0
$85$	12.4858			1.35427
$86$		0			0
$87$	−1.31436			−0.140914
$88$		0			0
$89$		−	0.982458i		−	0.104140i	−0.998643	−	0.0520702i	$-0.983418\pi$
$89$		−	0.982458i		−	0.104140i	0.998643	−	0.0520702i	$-0.0165820\pi$
$90$		0			0
$91$	1.20488	−	3.39827i	0.126306	−	0.356236i
$92$		0			0
$93$	0.240098			0.0248970
$94$		0			0
$95$	−0.457523			−0.0469409
$96$		0			0
$97$			9.30657i			0.944939i	0.881347	+	0.472469i	$0.156637\pi$
$97$			9.30657i			0.944939i	−0.881347	+	0.472469i	$0.843363\pi$
$98$		0			0
$99$	5.69492			0.572361
$100$		0			0
$101$			9.15160i			0.910618i	0.890333	+	0.455309i	$0.150471\pi$
$101$			9.15160i			0.910618i	−0.890333	+	0.455309i	$0.849529\pi$
$102$		0			0
$103$	−5.49996			−0.541927			−0.270964	−	0.962590i	$-0.587342\pi$
$103$	−5.49996			−0.541927			−0.270964	+	0.962590i	$0.587342\pi$
$104$		0			0
$105$	−0.733695			−0.0716013
$106$		0			0
$107$			6.53885i			0.632134i	0.948737	+	0.316067i	$0.102363\pi$
$107$			6.53885i			0.632134i	−0.948737	+	0.316067i	$0.897637\pi$
$108$		0			0
$109$	12.0875			1.15778			0.578888	−	0.815407i	$-0.303487\pi$
$109$	12.0875			1.15778			0.578888	+	0.815407i	$0.303487\pi$
$110$		0			0
$111$		−	0.0625785i		−	0.00593969i
$112$		0			0
$113$	−4.68304			−0.440543			−0.220272	−	0.975439i	$-0.570694\pi$
$113$	−4.68304			−0.440543			−0.220272	+	0.975439i	$0.570694\pi$
$114$		0			0
$115$	−17.9645			−1.67520
$116$		0			0
$117$	−10.0474	−	3.56238i	−0.928883	−	0.329342i
$118$		0			0
$119$		−	3.54436i		−	0.324911i
$120$		0			0
$121$	−7.28992			−0.662720
$122$		0			0
$123$	0.611209			0.0551109
$124$		0			0
$125$	−8.48812			−0.759200
$126$		0			0
$127$	14.7640			1.31009			0.655045	−	0.755590i	$-0.272650\pi$
$127$	14.7640			1.31009			0.655045	+	0.755590i	$0.272650\pi$
$128$		0			0
$129$	1.16202			0.102310
$130$		0			0
$131$			15.8504i			1.38486i	0.721487	+	0.692428i	$0.243459\pi$
$131$			15.8504i			1.38486i	−0.721487	+	0.692428i	$0.756541\pi$
$132$		0			0
$133$			0.129878i			0.0112618i
$134$		0			0
$135$			4.37034i			0.376139i
$136$		0			0
$137$			8.39453i			0.717193i	0.933493	+	0.358596i	$0.116745\pi$
$137$			8.39453i			0.717193i	−0.933493	+	0.358596i	$0.883255\pi$
$138$		0			0
$139$			13.5249i			1.14716i	0.819149	+	0.573581i	$0.194446\pi$
$139$			13.5249i			1.14716i	−0.819149	+	0.573581i	$0.805554\pi$
$140$		0			0
$141$	0.480859			0.0404957
$142$		0			0
$143$	−6.54561	−	2.32079i	−0.547371	−	0.194074i
$144$		0			0
$145$			22.2307i			1.84616i
$146$		0			0
$147$			0.208275i			0.0171783i
$148$		0			0
$149$	−19.3528			−1.58545			−0.792723	−	0.609582i	$-0.791337\pi$
$149$	−19.3528			−1.58545			−0.792723	+	0.609582i	$0.791337\pi$
$150$		0			0
$151$		−	5.55900i		−	0.452385i	−0.974083	−	0.226193i	$-0.927372\pi$
$151$		−	5.55900i		−	0.452385i	0.974083	−	0.226193i	$-0.0726279\pi$
$152$		0			0
$153$	−10.4793			−0.847203
$154$		0			0
$155$		−	4.06096i		−	0.326184i
$156$		0			0
$157$			16.1401i			1.28812i	0.764974	+	0.644061i	$0.222752\pi$
$157$			16.1401i			1.28812i	−0.764974	+	0.644061i	$0.777248\pi$
$158$		0			0
$159$	2.41240			0.191316
$160$		0			0
$161$			5.09962i			0.401906i
$162$		0			0
$163$	−3.76604			−0.294979			−0.147490	−	0.989064i	$-0.547119\pi$
$163$	−3.76604			−0.294979			−0.147490	+	0.989064i	$0.547119\pi$
$164$		0			0
$165$			1.41321i			0.110018i
$166$		0			0
$167$			23.2916i			1.80236i	0.433449	+	0.901178i	$0.357296\pi$
$167$			23.2916i			1.80236i	−0.433449	+	0.901178i	$0.642704\pi$
$168$		0			0
$169$	10.0965	+	8.18903i	0.776656	+	0.629925i
$170$		0			0
$171$	0.384000			0.0293652
$172$		0			0
$173$			19.8402i			1.50842i	0.656633	+	0.754210i	$0.271980\pi$
$173$			19.8402i			1.50842i	−0.656633	+	0.754210i	$0.728020\pi$
$174$		0			0
$175$			7.40954i			0.560108i
$176$		0			0
$177$			1.47651i			0.110981i
$178$		0			0
$179$			0.396420i			0.0296298i	0.999890	+	0.0148149i	$0.00471591\pi$
$179$			0.396420i			0.0296298i	−0.999890	+	0.0148149i	$0.995284\pi$
$180$		0			0
$181$		−	21.0449i		−	1.56425i	−0.623121	−	0.782126i	$-0.714135\pi$
$181$		−	21.0449i		−	1.56425i	0.623121	−	0.782126i	$-0.285865\pi$
$182$		0			0
$183$	−1.29734			−0.0959022
$184$		0			0
$185$	−1.05844			−0.0778179
$186$		0			0
$187$	−6.82699			−0.499239
$188$		0			0
$189$	1.24062			0.0902416
$190$		0			0
$191$	−5.37919			−0.389225			−0.194612	−	0.980880i	$-0.562345\pi$
$191$	−5.37919			−0.389225			−0.194612	+	0.980880i	$0.562345\pi$
$192$		0			0
$193$			23.0841i			1.66163i	0.556548	+	0.830816i	$0.312126\pi$
$193$			23.0841i			1.66163i	−0.556548	+	0.830816i	$0.687874\pi$
$194$		0			0
$195$	0.884015	−	2.49330i	0.0633056	−	0.178549i
$196$		0			0
$197$	−8.31831			−0.592655			−0.296328	−	0.955086i	$-0.595762\pi$
$197$	−8.31831			−0.592655			−0.296328	+	0.955086i	$0.595762\pi$
$198$		0			0
$199$	18.8463			1.33598			0.667989	−	0.744171i	$-0.267155\pi$
$199$	18.8463			1.33598			0.667989	+	0.744171i	$0.267155\pi$
$200$		0			0
$201$			1.67476i			0.118129i
$202$		0			0
$203$	6.31068			0.442923
$204$		0			0
$205$		−	10.3378i		−	0.722026i
$206$		0			0
$207$	15.0776			1.04797
$208$		0			0
$209$	0.250165			0.0173043
$210$		0			0
$211$			19.4654i			1.34005i	0.742337	+	0.670027i	$0.233717\pi$
$211$			19.4654i			1.34005i	−0.742337	+	0.670027i	$0.766283\pi$
$212$		0			0
$213$	3.44764			0.236229
$214$		0			0
$215$		−	19.6542i		−	1.34040i
$216$		0			0
$217$	−1.15279			−0.0782567
$218$		0			0
$219$	−2.38818			−0.161378
$220$		0			0
$221$	12.0447	+	4.27053i	0.810214	+	0.287267i
$222$		0			0
$223$			17.6477i			1.18178i	0.806753	+	0.590889i	$0.201223\pi$
$223$			17.6477i			1.18178i	−0.806753	+	0.590889i	$0.798777\pi$
$224$		0			0
$225$	21.9072			1.46048
$226$		0			0
$227$	−18.0842			−1.20029			−0.600144	−	0.799892i	$-0.704890\pi$
$227$	−18.0842			−1.20029			−0.600144	+	0.799892i	$0.704890\pi$
$228$		0			0
$229$	4.25750			0.281343			0.140672	−	0.990056i	$-0.455074\pi$
$229$	4.25750			0.281343			0.140672	+	0.990056i	$0.455074\pi$
$230$		0			0
$231$	0.401171			0.0263951
$232$		0			0
$233$	−12.0468			−0.789210			−0.394605	−	0.918851i	$-0.629119\pi$
$233$	−12.0468			−0.789210			−0.394605	+	0.918851i	$0.629119\pi$
$234$		0			0
$235$		−	8.13314i		−	0.530547i
$236$		0			0
$237$			3.05949i			0.198735i
$238$		0			0
$239$			1.26186i			0.0816230i	0.999167	+	0.0408115i	$0.0129943\pi$
$239$			1.26186i			0.0816230i	−0.999167	+	0.0408115i	$0.987006\pi$
$240$		0			0
$241$		−	16.9944i		−	1.09471i	−0.836902	−	0.547353i	$-0.815636\pi$
$241$		−	16.9944i		−	1.09471i	0.836902	−	0.547353i	$-0.184364\pi$
$242$		0			0
$243$		−	5.51541i		−	0.353814i
$244$		0			0
$245$	3.52272			0.225058
$246$		0			0
$247$	−0.441360	−	0.156487i	−0.0280831	−	0.00995705i
$248$		0			0
$249$		−	0.785089i		−	0.0497530i
$250$		0			0
$251$		−	16.4332i		−	1.03725i	−0.855001	−	0.518626i	$-0.826444\pi$
$251$		−	16.4332i		−	1.03725i	0.855001	−	0.518626i	$-0.173556\pi$
$252$		0			0
$253$	9.82267			0.617546
$254$		0			0
$255$		−	2.60048i		−	0.162848i
$256$		0			0
$257$	−18.6782			−1.16511			−0.582557	−	0.812790i	$-0.697948\pi$
$257$	−18.6782			−1.16511			−0.582557	+	0.812790i	$0.697948\pi$
$258$		0			0
$259$			0.300461i			0.0186697i
$260$		0			0
$261$		−	18.6583i		−	1.15492i
$262$		0			0
$263$	0.495204			0.0305356			0.0152678	−	0.999883i	$-0.495140\pi$
$263$	0.495204			0.0305356			0.0152678	+	0.999883i	$0.495140\pi$
$264$		0			0
$265$		−	40.8027i		−	2.50649i
$266$		0			0
$267$	−0.204622			−0.0125226
$268$		0			0
$269$			10.0263i			0.611314i	0.952142	+	0.305657i	$0.0988760\pi$
$269$			10.0263i			0.611314i	−0.952142	+	0.305657i	$0.901124\pi$
$270$		0			0
$271$		−	19.8983i		−	1.20874i	−0.796705	−	0.604369i	$-0.793425\pi$
$271$		−	19.8983i		−	1.20874i	0.796705	−	0.604369i	$-0.206575\pi$
$272$		0			0
$273$	−0.707776	−	0.250947i	−0.0428366	−	0.0151880i
$274$		0			0
$275$	14.2719			0.860630
$276$		0			0
$277$		−	29.0585i		−	1.74596i	−0.487759	−	0.872978i	$-0.662186\pi$
$277$		−	29.0585i		−	1.74596i	0.487759	−	0.872978i	$-0.337814\pi$
$278$		0			0
$279$			3.40837i			0.204054i
$280$		0			0
$281$		−	14.5021i		−	0.865120i	−0.901605	−	0.432560i	$-0.857610\pi$
$281$		−	14.5021i		−	0.865120i	0.901605	−	0.432560i	$-0.142390\pi$
$282$		0			0
$283$		−	5.54760i		−	0.329771i	−0.986313	−	0.164885i	$-0.947275\pi$
$283$		−	5.54760i		−	0.329771i	0.986313	−	0.164885i	$-0.0527254\pi$
$284$		0			0
$285$			0.0952907i			0.00564454i
$286$		0			0
$287$	−2.93462			−0.173225
$288$		0			0
$289$	−4.43753			−0.261031
$290$		0			0
$291$	1.93833			0.113627
$292$		0			0
$293$	24.7162			1.44393			0.721967	−	0.691928i	$-0.243238\pi$
$293$	24.7162			1.44393			0.721967	+	0.691928i	$0.243238\pi$
$294$		0			0
$295$	24.9733			1.45400
$296$		0			0
$297$		−	2.38962i		−	0.138660i
$298$		0			0
$299$	−17.3299	−	6.14444i	−1.00221	−	0.355342i
$300$		0			0
$301$	−5.57927			−0.321584
$302$		0			0
$303$	1.90605			0.109500
$304$		0			0
$305$			21.9429i			1.25645i
$306$		0			0
$307$	14.9768			0.854770			0.427385	−	0.904070i	$-0.359435\pi$
$307$	14.9768			0.854770			0.427385	+	0.904070i	$0.359435\pi$
$308$		0			0
$309$			1.14551i			0.0651656i
$310$		0			0
$311$	22.2304			1.26057			0.630284	−	0.776365i	$-0.282939\pi$
$311$	22.2304			1.26057			0.630284	+	0.776365i	$0.282939\pi$
$312$		0			0
$313$	8.34646			0.471770			0.235885	−	0.971781i	$-0.424201\pi$
$313$	8.34646			0.471770			0.235885	+	0.971781i	$0.424201\pi$
$314$		0			0
$315$		−	10.4153i		−	0.586838i
$316$		0			0
$317$	−19.6311			−1.10260			−0.551298	−	0.834309i	$-0.685867\pi$
$317$	−19.6311			−1.10260			−0.551298	+	0.834309i	$0.685867\pi$
$318$		0			0
$319$		−	12.1554i		−	0.680570i
$320$		0			0
$321$	1.36188			0.0760128
$322$		0			0
$323$	−0.460334			−0.0256136
$324$		0			0
$325$	−25.1796	−	8.92761i	−1.39671	−	0.495215i
$326$		0			0
$327$		−	2.51754i		−	0.139220i
$328$		0			0
$329$	−2.30877			−0.127287
$330$		0			0
$331$	−33.6438			−1.84923			−0.924616	−	0.380900i	$-0.875614\pi$
$331$	−33.6438			−1.84923			−0.924616	+	0.380900i	$0.875614\pi$
$332$		0			0
$333$	0.888348			0.0486812
$334$		0			0
$335$	28.3265			1.54764
$336$		0			0
$337$	0.0143341			0.000780827			0.000390413	−	1.00000i	$-0.499876\pi$
$337$	0.0143341			0.000780827			0.000390413		1.00000i	$0.499876\pi$
$338$		0			0
$339$			0.975362i			0.0529744i
$340$		0			0
$341$			2.22046i			0.120245i
$342$		0			0
$343$		−	1.00000i		−	0.0539949i
$344$		0			0
$345$			3.74157i			0.201439i
$346$		0			0
$347$		−	3.13277i		−	0.168176i	−0.996458	−	0.0840880i	$-0.973202\pi$
$347$		−	3.13277i		−	0.168176i	0.996458	−	0.0840880i	$-0.0267977\pi$
$348$		0			0
$349$	−1.24421			−0.0666011			−0.0333006	−	0.999445i	$-0.510602\pi$
$349$	−1.24421			−0.0666011			−0.0333006	+	0.999445i	$0.510602\pi$
$350$		0			0
$351$	−1.49480	+	4.21596i	−0.0797863	+	0.225031i
$352$		0			0
$353$			7.40604i			0.394184i	0.980385	+	0.197092i	$0.0631497\pi$
$353$			7.40604i			0.394184i	−0.980385	+	0.197092i	$0.936850\pi$
$354$		0			0
$355$		−	58.3126i		−	3.09491i
$356$		0			0
$357$	−0.738202			−0.0390698
$358$		0			0
$359$		−	36.1561i		−	1.90824i	−0.299418	−	0.954122i	$-0.596792\pi$
$359$		−	36.1561i		−	1.90824i	0.299418	−	0.954122i	$-0.403208\pi$
$360$		0			0
$361$	−18.9831			−0.999112
$362$		0			0
$363$			1.51831i			0.0796906i
$364$		0			0
$365$			40.3931i			2.11427i
$366$		0			0
$367$	−8.10201			−0.422922			−0.211461	−	0.977386i	$-0.567822\pi$
$367$	−8.10201			−0.422922			−0.211461	+	0.977386i	$0.567822\pi$
$368$		0			0
$369$			8.67657i			0.451684i
$370$		0			0
$371$	−11.5827			−0.601346
$372$		0			0
$373$			7.73552i			0.400530i	0.979742	+	0.200265i	$0.0641803\pi$
$373$			7.73552i			0.400530i	−0.979742	+	0.200265i	$0.935820\pi$
$374$		0			0
$375$			1.76787i			0.0912922i
$376$		0			0
$377$	−7.60362	+	21.4454i	−0.391606	+	1.10450i
$378$		0			0
$379$	32.4599			1.66735			0.833675	−	0.552255i	$-0.186232\pi$
$379$	32.4599			1.66735			0.833675	+	0.552255i	$0.186232\pi$
$380$		0			0
$381$		−	3.07497i		−	0.157535i
$382$		0			0
$383$			35.7865i			1.82860i	0.405032	+	0.914302i	$0.367260\pi$
$383$			35.7865i			1.82860i	−0.405032	+	0.914302i	$0.632740\pi$
$384$		0			0
$385$		−	6.78531i		−	0.345811i
$386$		0			0
$387$			16.4958i			0.838528i
$388$		0			0
$389$			6.32180i			0.320528i	0.987074	+	0.160264i	$0.0512345\pi$
$389$			6.32180i			0.320528i	−0.987074	+	0.160264i	$0.948765\pi$
$390$		0			0
$391$	−18.0749			−0.914086
$392$		0			0
$393$	3.30124			0.166526
$394$		0			0
$395$	51.7474			2.60370
$396$		0			0
$397$	10.2742			0.515649			0.257824	−	0.966192i	$-0.416994\pi$
$397$	10.2742			0.515649			0.257824	+	0.966192i	$0.416994\pi$
$398$		0			0
$399$	0.0270504			0.00135421
$400$		0			0
$401$			1.79881i			0.0898285i	0.998991	+	0.0449142i	$0.0143015\pi$
$401$			1.79881i			0.0898285i	−0.998991	+	0.0449142i	$0.985699\pi$
$402$		0			0
$403$	1.38898	−	3.91751i	0.0691899	−	0.195145i
$404$		0			0
$405$	−30.3358			−1.50740
$406$		0			0
$407$	0.578734			0.0286868
$408$		0			0
$409$		−	21.9102i		−	1.08339i	−0.840575	−	0.541696i	$-0.817783\pi$
$409$		−	21.9102i		−	1.08339i	0.840575	−	0.541696i	$-0.182217\pi$
$410$		0			0
$411$	1.74837			0.0862409
$412$		0			0
$413$		−	7.08922i		−	0.348838i
$414$		0			0
$415$	−13.2788			−0.651831
$416$		0			0
$417$	2.81689			0.137944
$418$		0			0
$419$		−	18.8887i		−	0.922773i	−0.887199	−	0.461386i	$-0.847352\pi$
$419$		−	18.8887i		−	0.922773i	0.887199	−	0.461386i	$-0.152648\pi$
$420$		0			0
$421$	−6.34070			−0.309027			−0.154513	−	0.987991i	$-0.549381\pi$
$421$	−6.34070			−0.309027			−0.154513	+	0.987991i	$0.549381\pi$
$422$		0			0
$423$			6.82615i			0.331899i
$424$		0			0
$425$	−26.2620			−1.27390
$426$		0			0
$427$	6.22897			0.301441
$428$		0			0
$429$	−0.483363	+	1.36329i	−0.0233370	+	0.0658202i
$430$		0			0
$431$		−	27.2674i		−	1.31342i	−0.754141	−	0.656712i	$-0.771947\pi$
$431$		−	27.2674i		−	1.31342i	0.754141	−	0.656712i	$-0.228053\pi$
$432$		0			0
$433$	17.8959			0.860023			0.430012	−	0.902823i	$-0.358509\pi$
$433$	17.8959			0.860023			0.430012	+	0.902823i	$0.358509\pi$
$434$		0			0
$435$	4.63012			0.221997
$436$		0			0
$437$	0.662328			0.0316834
$438$		0			0
$439$	−16.4464			−0.784944			−0.392472	−	0.919764i	$-0.628380\pi$
$439$	−16.4464			−0.784944			−0.392472	+	0.919764i	$0.628380\pi$
$440$		0			0
$441$	−2.95662			−0.140791
$442$		0			0
$443$			21.4873i			1.02089i	0.859910	+	0.510446i	$0.170520\pi$
$443$			21.4873i			1.02089i	−0.859910	+	0.510446i	$0.829480\pi$
$444$		0			0
$445$			3.46092i			0.164063i
$446$		0			0
$447$			4.03072i			0.190646i
$448$		0			0
$449$		−	11.6297i		−	0.548840i	−0.961610	−	0.274420i	$-0.911514\pi$
$449$		−	11.6297i		−	0.548840i	0.961610	−	0.274420i	$-0.0884858\pi$
$450$		0			0
$451$			5.65254i			0.266168i
$452$		0			0
$453$	−1.15780			−0.0543983
$454$		0			0
$455$	−4.24446	+	11.9712i	−0.198983	+	0.561216i
$456$		0			0
$457$		−	20.1702i		−	0.943524i	−0.881726	−	0.471762i	$-0.843618\pi$
$457$		−	20.1702i		−	0.943524i	0.881726	−	0.471762i	$-0.156382\pi$
$458$		0			0
$459$			4.39719i			0.205243i
$460$		0			0
$461$	15.3054			0.712845			0.356423	−	0.934325i	$-0.383996\pi$
$461$	15.3054			0.712845			0.356423	+	0.934325i	$0.383996\pi$
$462$		0			0
$463$		−	22.3957i		−	1.04082i	−0.853917	−	0.520409i	$-0.825780\pi$
$463$		−	22.3957i		−	1.04082i	0.853917	−	0.520409i	$-0.174220\pi$
$464$		0			0
$465$	−0.845798			−0.0392230
$466$		0			0
$467$			10.1198i			0.468287i	0.972202	+	0.234143i	$0.0752285\pi$
$467$			10.1198i			0.468287i	−0.972202	+	0.234143i	$0.924771\pi$
$468$		0			0
$469$		−	8.04110i		−	0.371303i
$470$		0			0
$471$	3.36159			0.154894
$472$		0			0
$473$			10.7465i			0.494127i
$474$		0			0
$475$	0.962335			0.0441550
$476$		0			0
$477$			34.2458i			1.56801i
$478$		0			0
$479$			20.1092i			0.918811i	0.888227	+	0.459405i	$0.151938\pi$
$479$			20.1092i			0.918811i	−0.888227	+	0.459405i	$0.848062\pi$
$480$		0			0
$481$	−1.02105	−	0.362019i	−0.0465557	−	0.0165067i
$482$		0			0
$483$	1.06212			0.0483284
$484$		0			0
$485$		−	32.7844i		−	1.48866i
$486$		0			0
$487$		−	14.2922i		−	0.647643i	−0.946118	−	0.323822i	$-0.895032\pi$
$487$		−	14.2922i		−	0.647643i	0.946118	−	0.323822i	$-0.104968\pi$
$488$		0			0
$489$			0.784373i			0.0354706i
$490$		0			0
$491$			20.8674i			0.941734i	0.882204	+	0.470867i	$0.156059\pi$
$491$			20.8674i			0.941734i	−0.882204	+	0.470867i	$0.843941\pi$
$492$		0			0
$493$			22.3673i			1.00737i
$494$		0			0
$495$	−20.0616			−0.901701
$496$		0			0
$497$	−16.5533			−0.742517
$498$		0			0
$499$	19.1219			0.856016			0.428008	−	0.903775i	$-0.359216\pi$
$499$	19.1219			0.856016			0.428008	+	0.903775i	$0.359216\pi$
$500$		0			0
$501$	4.85106			0.216729
$502$		0			0
$503$	−27.5342			−1.22769			−0.613844	−	0.789427i	$-0.710378\pi$
$503$	−27.5342			−1.22769			−0.613844	+	0.789427i	$0.710378\pi$
$504$		0			0
$505$		−	32.2385i		−	1.43459i
$506$		0			0
$507$	1.70557	−	2.10286i	0.0757471	−	0.0933911i
$508$		0			0
$509$	13.8266			0.612851			0.306426	−	0.951895i	$-0.400867\pi$
$509$	13.8266			0.612851			0.306426	+	0.951895i	$0.400867\pi$
$510$		0			0
$511$	11.4665			0.507246
$512$		0			0
$513$		−	0.161129i		−	0.00711401i
$514$		0			0
$515$	19.3748			0.853756
$516$		0			0
$517$			4.44705i			0.195581i
$518$		0			0
$519$	4.13222			0.181384
$520$		0			0
$521$	−4.40236			−0.192871			−0.0964355	−	0.995339i	$-0.530744\pi$
$521$	−4.40236			−0.192871			−0.0964355	+	0.995339i	$0.530744\pi$
$522$		0			0
$523$		−	5.70151i		−	0.249309i	−0.992200	−	0.124655i	$-0.960218\pi$
$523$		−	5.70151i		−	0.249309i	0.992200	−	0.124655i	$-0.0397823\pi$
$524$		0			0
$525$	1.54322			0.0673518
$526$		0			0
$527$		−	4.08591i		−	0.177985i
$528$		0			0
$529$	3.00612			0.130701
$530$		0			0
$531$	−20.9601			−0.909593
$532$		0			0
$533$	3.53587	−	9.97265i	0.153156	−	0.431963i
$534$		0			0
$535$		−	23.0345i		−	0.995869i
$536$		0			0
$537$	0.0825646			0.00356292
$538$		0			0
$539$	−1.92616			−0.0829654
$540$		0			0
$541$	21.6901			0.932529			0.466265	−	0.884645i	$-0.345600\pi$
$541$	21.6901			0.932529			0.466265	+	0.884645i	$0.345600\pi$
$542$		0			0
$543$	−4.38312			−0.188098
$544$		0			0
$545$	−42.5810			−1.82397
$546$		0			0
$547$		−	8.57481i		−	0.366632i	−0.983054	−	0.183316i	$-0.941317\pi$
$547$		−	8.57481i		−	0.366632i	0.983054	−	0.183316i	$-0.0586832\pi$
$548$		0			0
$549$		−	18.4167i		−	0.786006i
$550$		0			0
$551$		−	0.819618i		−	0.0349169i
$552$		0			0
$553$		−	14.6896i		−	0.624667i
$554$		0			0
$555$			0.220446i			0.00935743i
$556$		0			0
$557$	2.70216			0.114494			0.0572471	−	0.998360i	$-0.481768\pi$
$557$	2.70216			0.114494			0.0572471	+	0.998360i	$0.481768\pi$
$558$		0			0
$559$	6.72235	−	18.9599i	0.284325	−	0.801917i
$560$		0			0
$561$			1.42189i			0.0600324i
$562$		0			0
$563$			39.2914i			1.65593i	0.560777	+	0.827967i	$0.310503\pi$
$563$			39.2914i			1.65593i	−0.560777	+	0.827967i	$0.689497\pi$
$564$		0			0
$565$	16.4970			0.694035
$566$		0			0
$567$			8.61147i			0.361648i
$568$		0			0
$569$	1.67637			0.0702771			0.0351386	−	0.999382i	$-0.488813\pi$
$569$	1.67637			0.0702771			0.0351386	+	0.999382i	$0.488813\pi$
$570$		0			0
$571$			34.5286i			1.44498i	0.691384	+	0.722488i	$0.257001\pi$
$571$			34.5286i			1.44498i	−0.691384	+	0.722488i	$0.742999\pi$
$572$		0			0
$573$			1.12035i			0.0468034i
$574$		0			0
$575$	37.7858			1.57578
$576$		0			0
$577$			22.3246i			0.929384i	0.885472	+	0.464692i	$0.153835\pi$
$577$			22.3246i			0.929384i	−0.885472	+	0.464692i	$0.846165\pi$
$578$		0			0
$579$	4.80785			0.199808
$580$		0			0
$581$			3.76948i			0.156384i
$582$		0			0
$583$			22.3102i			0.923993i
$584$		0			0
$585$	35.3942	+	12.5492i	1.46337	+	0.518848i
$586$		0			0
$587$	13.2895			0.548517			0.274259	−	0.961656i	$-0.411568\pi$
$587$	13.2895			0.548517			0.274259	+	0.961656i	$0.411568\pi$
$588$		0			0
$589$			0.149722i			0.00616920i
$590$		0			0
$591$			1.73250i			0.0712655i
$592$		0			0
$593$		−	22.1464i		−	0.909444i	−0.890634	−	0.454722i	$-0.849739\pi$
$593$		−	22.1464i		−	0.909444i	0.890634	−	0.454722i	$-0.150261\pi$
$594$		0			0
$595$			12.4858i			0.511867i
$596$		0			0
$597$		−	3.92522i		−	0.160649i
$598$		0			0
$599$	−23.3029			−0.952129			−0.476065	−	0.879410i	$-0.657937\pi$
$599$	−23.3029			−0.952129			−0.476065	+	0.879410i	$0.657937\pi$
$600$		0			0
$601$	−14.8316			−0.604993			−0.302496	−	0.953151i	$-0.597820\pi$
$601$	−14.8316			−0.604993			−0.302496	+	0.953151i	$0.597820\pi$
$602$		0			0
$603$	−23.7745			−0.968172
$604$		0			0
$605$	25.6803			1.04405
$606$		0			0
$607$	−38.0443			−1.54417			−0.772085	−	0.635520i	$-0.780786\pi$
$607$	−38.0443			−1.54417			−0.772085	+	0.635520i	$0.780786\pi$
$608$		0			0
$609$		−	1.31436i		−	0.0532605i
$610$		0			0
$611$	2.78179	−	7.84583i	0.112539	−	0.317408i
$612$		0			0
$613$	−17.0574			−0.688941			−0.344470	−	0.938797i	$-0.611941\pi$
$613$	−17.0574			−0.688941			−0.344470	+	0.938797i	$0.611941\pi$
$614$		0			0
$615$	−2.15312			−0.0868221
$616$		0			0
$617$			7.95481i			0.320249i	0.987097	+	0.160124i	$0.0511895\pi$
$617$			7.95481i			0.320249i	−0.987097	+	0.160124i	$0.948810\pi$
$618$		0			0
$619$	27.3721			1.10018			0.550088	−	0.835106i	$-0.314594\pi$
$619$	27.3721			1.10018			0.550088	+	0.835106i	$0.314594\pi$
$620$		0			0
$621$		−	6.32668i		−	0.253881i
$622$		0			0
$623$	0.982458			0.0393614
$624$		0			0
$625$	−7.14645			−0.285858
$626$		0			0
$627$		−	0.0521032i		−	0.00208080i
$628$		0			0
$629$	−1.06494			−0.0424619
$630$		0			0
$631$		−	21.1834i		−	0.843298i	−0.906759	−	0.421649i	$-0.861452\pi$
$631$		−	21.1834i		−	0.843298i	0.906759	−	0.421649i	$-0.138548\pi$
$632$		0			0
$633$	4.05416			0.161138
$634$		0			0
$635$	−52.0093			−2.06393
$636$		0			0
$637$	3.39827	+	1.20488i	0.134644	+	0.0477391i
$638$		0			0
$639$			48.9419i			1.93611i
$640$		0			0
$641$	0.478600			0.0189036			0.00945179	−	0.999955i	$-0.496991\pi$
$641$	0.478600			0.0189036			0.00945179	+	0.999955i	$0.496991\pi$
$642$		0			0
$643$	−5.62114			−0.221676			−0.110838	−	0.993838i	$-0.535353\pi$
$643$	−5.62114			−0.221676			−0.110838	+	0.993838i	$0.535353\pi$
$644$		0			0
$645$	−4.09348			−0.161181
$646$		0			0
$647$	44.2220			1.73854			0.869272	−	0.494333i	$-0.164588\pi$
$647$	44.2220			1.73854			0.869272	+	0.494333i	$0.164588\pi$
$648$		0			0
$649$	−13.6550			−0.536004
$650$		0			0
$651$			0.240098i			0.00941019i
$652$		0			0
$653$			4.90631i			0.191999i	0.995381	+	0.0959994i	$0.0306047\pi$
$653$			4.90631i			0.191999i	−0.995381	+	0.0959994i	$0.969395\pi$
$654$		0			0
$655$		−	55.8364i		−	2.18171i
$656$		0			0
$657$		−	33.9020i		−	1.32264i
$658$		0			0
$659$			2.44724i			0.0953310i	0.998863	+	0.0476655i	$0.0151782\pi$
$659$			2.44724i			0.0953310i	−0.998863	+	0.0476655i	$0.984822\pi$
$660$		0			0
$661$	46.9632			1.82666			0.913328	−	0.407225i	$-0.133503\pi$
$661$	46.9632			1.82666			0.913328	+	0.407225i	$0.133503\pi$
$662$		0			0
$663$	0.889446	−	2.50861i	0.0345432	−	0.0974264i
$664$		0			0
$665$		−	0.457523i		−	0.0177420i
$666$		0			0
$667$		−	32.1821i		−	1.24609i
$668$		0			0
$669$	3.67558			0.142106
$670$		0			0
$671$		−	11.9980i		−	0.463177i
$672$		0			0
$673$	42.5580			1.64049			0.820245	−	0.572013i	$-0.193837\pi$
$673$	42.5580			1.64049			0.820245	+	0.572013i	$0.193837\pi$
$674$		0			0
$675$		−	9.19240i		−	0.353816i
$676$		0			0
$677$		−	27.1383i		−	1.04301i	−0.853249	−	0.521504i	$-0.825371\pi$
$677$		−	27.1383i		−	1.04301i	0.853249	−	0.521504i	$-0.174629\pi$
$678$		0			0
$679$	−9.30657			−0.357153
$680$		0			0
$681$			3.76648i			0.144332i
$682$		0			0
$683$	−34.5690			−1.32275			−0.661373	−	0.750057i	$-0.730026\pi$
$683$	−34.5690			−1.32275			−0.661373	+	0.750057i	$0.730026\pi$
$684$		0			0
$685$		−	29.5715i		−	1.12987i
$686$		0			0
$687$		−	0.886732i		−	0.0338309i
$688$		0			0
$689$	13.9558	−	39.3613i	0.531675	−	1.49955i
$690$		0			0
$691$	26.6922			1.01542			0.507709	−	0.861529i	$-0.330492\pi$
$691$	26.6922			1.01542			0.507709	+	0.861529i	$0.330492\pi$
$692$		0			0
$693$			5.69492i			0.216332i
$694$		0			0
$695$		−	47.6442i		−	1.80725i
$696$		0			0
$697$		−	10.4014i		−	0.393979i
$698$		0			0
$699$			2.50904i			0.0949007i
$700$		0			0
$701$			21.2235i			0.801602i	0.916165	+	0.400801i	$0.131268\pi$
$701$			21.2235i			0.801602i	−0.916165	+	0.400801i	$0.868732\pi$
$702$		0			0
$703$	0.0390232			0.00147179
$704$		0			0
$705$	−1.69393			−0.0637972
$706$		0			0
$707$	−9.15160			−0.344181
$708$		0			0
$709$	38.7261			1.45439			0.727194	−	0.686432i	$-0.240824\pi$
$709$	38.7261			1.45439			0.727194	+	0.686432i	$0.240824\pi$
$710$		0			0
$711$	−43.4317			−1.62882
$712$		0			0
$713$			5.87881i			0.220163i
$714$		0			0
$715$	23.0583	+	8.17549i	0.862332	+	0.305746i
$716$		0			0
$717$	0.262814			0.00981499
$718$		0			0
$719$	−44.9233			−1.67536			−0.837679	−	0.546163i	$-0.816088\pi$
$719$	−44.9233			−1.67536			−0.837679	+	0.546163i	$0.816088\pi$
$720$		0			0
$721$		−	5.49996i		−	0.204829i
$722$		0			0
$723$	−3.53951			−0.131636
$724$		0			0
$725$		−	46.7592i		−	1.73659i
$726$		0			0
$727$	−24.2750			−0.900310			−0.450155	−	0.892950i	$-0.648631\pi$
$727$	−24.2750			−0.900310			−0.450155	+	0.892950i	$0.648631\pi$
$728$		0			0
$729$	24.6857			0.914285
$730$		0			0
$731$		−	19.7749i		−	0.731402i
$732$		0			0
$733$	21.7151			0.802067			0.401033	−	0.916063i	$-0.368651\pi$
$733$	21.7151			0.802067			0.401033	+	0.916063i	$0.368651\pi$
$734$		0			0
$735$		−	0.733695i		−	0.0270627i
$736$		0			0
$737$	−15.4884			−0.570523
$738$		0			0
$739$	4.13272			0.152025			0.0760123	−	0.997107i	$-0.475781\pi$
$739$	4.13272			0.152025			0.0760123	+	0.997107i	$0.475781\pi$
$740$		0			0
$741$	−0.0325925	+	0.0919245i	−0.00119731	+	0.00337693i
$742$		0			0
$743$		−	20.2682i		−	0.743567i	−0.928319	−	0.371784i	$-0.878746\pi$
$743$		−	20.2682i		−	0.743567i	0.928319	−	0.371784i	$-0.121254\pi$
$744$		0			0
$745$	68.1746			2.49772
$746$		0			0
$747$	11.1449			0.407771
$748$		0			0
$749$	−6.53885			−0.238924
$750$		0			0
$751$	−50.2644			−1.83417			−0.917087	−	0.398687i	$-0.869466\pi$
$751$	−50.2644			−1.83417			−0.917087	+	0.398687i	$0.869466\pi$
$752$		0			0
$753$	−3.42262			−0.124727
$754$		0			0
$755$			19.5828i			0.712691i
$756$		0			0
$757$		−	20.4735i		−	0.744122i	−0.928208	−	0.372061i	$-0.878651\pi$
$757$		−	20.4735i		−	0.744122i	0.928208	−	0.372061i	$-0.121349\pi$
$758$		0			0
$759$		−	2.04582i		−	0.0742585i
$760$		0			0
$761$			31.2031i			1.13111i	0.824710	+	0.565556i	$0.191338\pi$
$761$			31.2031i			1.13111i	−0.824710	+	0.565556i	$0.808662\pi$
$762$		0			0
$763$			12.0875i			0.437599i
$764$		0			0
$765$	36.9157			1.33469
$766$		0			0
$767$	24.0911	+	8.54167i	0.869880	+	0.308422i
$768$		0			0
$769$		−	32.8772i		−	1.18558i	−0.805356	−	0.592791i	$-0.798026\pi$
$769$		−	32.8772i		−	1.18558i	0.805356	−	0.592791i	$-0.201974\pi$
$770$		0			0
$771$			3.89021i			0.140103i
$772$		0			0
$773$	−12.3295			−0.443463			−0.221731	−	0.975108i	$-0.571171\pi$
$773$	−12.3295			−0.443463			−0.221731	+	0.975108i	$0.571171\pi$
$774$		0			0
$775$			8.54166i			0.306826i
$776$		0			0
$777$	0.0625785			0.00224499
$778$		0			0
$779$			0.381143i			0.0136558i
$780$		0			0
$781$			31.8843i			1.14091i
$782$		0			0
$783$	−7.82914			−0.279791
$784$		0			0
$785$		−	56.8571i		−	2.02932i
$786$		0			0
$787$	34.8281			1.24149			0.620743	−	0.784014i	$-0.286831\pi$
$787$	34.8281			1.24149			0.620743	+	0.784014i	$0.286831\pi$
$788$		0			0
$789$		−	0.103139i		−	0.00367184i
$790$		0			0
$791$		−	4.68304i		−	0.166510i
$792$		0			0
$793$	−7.50517	+	21.1677i	−0.266516	+	0.751689i
$794$		0			0
$795$	−8.49820			−0.301400
$796$		0			0
$797$			37.3680i			1.32364i	0.749661	+	0.661821i	$0.230216\pi$
$797$			37.3680i			1.32364i	−0.749661	+	0.661821i	$0.769784\pi$
$798$		0			0
$799$		−	8.18310i		−	0.289497i
$800$		0			0
$801$		−	2.90476i		−	0.102635i
$802$		0			0
$803$		−	22.0862i		−	0.779405i
$804$		0			0
$805$		−	17.9645i		−	0.633166i
$806$		0			0
$807$	2.08823			0.0735091
$808$		0			0
$809$	36.3556			1.27819			0.639097	−	0.769126i	$-0.279308\pi$
$809$	36.3556			1.27819			0.639097	+	0.769126i	$0.279308\pi$
$810$		0			0
$811$	−29.4704			−1.03485			−0.517423	−	0.855730i	$-0.673109\pi$
$811$	−29.4704			−1.03485			−0.517423	+	0.855730i	$0.673109\pi$
$812$		0			0
$813$	−4.14433			−0.145348
$814$		0			0
$815$	13.2667			0.464712
$816$		0			0
$817$			0.724623i			0.0253514i
$818$		0			0
$819$	3.56238	−	10.0474i	0.124480	−	0.351085i
$820$		0			0
$821$	−8.34090			−0.291100			−0.145550	−	0.989351i	$-0.546495\pi$
$821$	−8.34090			−0.291100			−0.145550	+	0.989351i	$0.546495\pi$
$822$		0			0
$823$	−46.3329			−1.61506			−0.807531	−	0.589825i	$-0.799197\pi$
$823$	−46.3329			−1.61506			−0.807531	+	0.589825i	$0.799197\pi$
$824$		0			0
$825$		−	2.97249i		−	0.103489i
$826$		0			0
$827$	36.3252			1.26315			0.631575	−	0.775315i	$-0.282409\pi$
$827$	36.3252			1.26315			0.631575	+	0.775315i	$0.282409\pi$
$828$		0			0
$829$			12.3282i			0.428175i	0.976814	+	0.214088i	$0.0686778\pi$
$829$			12.3282i			0.428175i	−0.976814	+	0.214088i	$0.931322\pi$
$830$		0			0
$831$	−6.05217			−0.209947
$832$		0			0
$833$	3.54436			0.122805
$834$		0			0
$835$		−	82.0496i		−	2.83944i
$836$		0			0
$837$	1.43017			0.0494341
$838$		0			0
$839$			33.9276i			1.17131i	0.810560	+	0.585656i	$0.199163\pi$
$839$			33.9276i			1.17131i	−0.810560	+	0.585656i	$0.800837\pi$
$840$		0			0
$841$	−10.8247			−0.373266
$842$		0			0
$843$	−3.02042			−0.104029
$844$		0			0
$845$	−35.5672	−	28.8476i	−1.22355	−	0.992389i
$846$		0			0
$847$		−	7.28992i		−	0.250485i
$848$		0			0
$849$	−1.15543			−0.0396542
$850$		0			0
$851$	1.53223			0.0525243
$852$		0			0
$853$	−38.7697			−1.32745			−0.663725	−	0.747977i	$-0.731026\pi$
$853$	−38.7697			−1.32745			−0.663725	+	0.747977i	$0.731026\pi$
$854$		0			0
$855$	−1.35272			−0.0462621
$856$		0			0
$857$	−26.4866			−0.904764			−0.452382	−	0.891824i	$-0.649426\pi$
$857$	−26.4866			−0.904764			−0.452382	+	0.891824i	$0.649426\pi$
$858$		0			0
$859$			17.8500i			0.609033i	0.952507	+	0.304517i	$0.0984949\pi$
$859$			17.8500i			0.609033i	−0.952507	+	0.304517i	$0.901505\pi$
$860$		0			0
$861$			0.611209i			0.0208300i
$862$		0			0
$863$			13.6999i			0.466350i	0.972435	+	0.233175i	$0.0749115\pi$
$863$			13.6999i			0.466350i	−0.972435	+	0.233175i	$0.925088\pi$
$864$		0			0
$865$		−	69.8913i		−	2.37638i
$866$		0			0
$867$			0.924228i			0.0313884i
$868$		0			0
$869$	−28.2945			−0.959827
$870$		0			0
$871$	27.3258	+	9.68857i	0.925901	+	0.328285i
$872$		0			0
$873$			27.5160i			0.931276i
$874$		0			0
$875$		−	8.48812i		−	0.286951i
$876$		0			0
$877$	−1.66862			−0.0563452			−0.0281726	−	0.999603i	$-0.508969\pi$
$877$	−1.66862			−0.0563452			−0.0281726	+	0.999603i	$0.508969\pi$
$878$		0			0
$879$		−	5.14777i		−	0.173630i
$880$		0			0
$881$	−0.959187			−0.0323158			−0.0161579	−	0.999869i	$-0.505143\pi$
$881$	−0.959187			−0.0323158			−0.0161579	+	0.999869i	$0.505143\pi$
$882$		0			0
$883$			40.6249i			1.36714i	0.729886	+	0.683569i	$0.239573\pi$
$883$			40.6249i			1.36714i	−0.729886	+	0.683569i	$0.760427\pi$
$884$		0			0
$885$		−	5.20133i		−	0.174841i
$886$		0			0
$887$	58.6312			1.96864			0.984322	−	0.176380i	$-0.0564388\pi$
$887$	58.6312			1.96864			0.984322	+	0.176380i	$0.0564388\pi$
$888$		0			0
$889$			14.7640i			0.495168i
$890$		0			0
$891$	16.5870			0.555687
$892$		0			0
$893$			0.299858i			0.0100344i
$894$		0			0
$895$		−	1.39648i		−	0.0466791i
$896$		0			0
$897$	−1.27973	+	3.60939i	−0.0427291	+	0.120514i
$898$		0			0
$899$	7.27491			0.242632
$900$		0			0
$901$		−	41.0534i		−	1.36769i
$902$		0			0
$903$			1.16202i			0.0386697i
$904$		0			0
$905$			74.1351i			2.46433i
$906$		0			0
$907$		−	16.8426i		−	0.559251i	−0.960109	−	0.279625i	$-0.909790\pi$
$907$		−	16.8426i		−	0.559251i	0.960109	−	0.279625i	$-0.0902103\pi$
$908$		0			0
$909$			27.0578i			0.897451i
$910$		0			0
$911$	19.4339			0.643874			0.321937	−	0.946761i	$-0.395666\pi$
$911$	19.4339			0.643874			0.321937	+	0.946761i	$0.395666\pi$
$912$		0			0
$913$	7.26061			0.240291
$914$		0			0
$915$	4.57016			0.151085
$916$		0			0
$917$	−15.8504			−0.523426
$918$		0			0
$919$	4.84960			0.159974			0.0799868	−	0.996796i	$-0.474512\pi$
$919$	4.84960			0.159974			0.0799868	+	0.996796i	$0.474512\pi$
$920$		0			0
$921$		−	3.11929i		−	0.102784i
$922$		0			0
$923$	19.9448	−	56.2527i	0.656490	−	1.85158i
$924$		0			0
$925$	2.22627			0.0731994
$926$		0			0
$927$	−16.2613			−0.534091
$928$		0			0
$929$		−	15.2435i		−	0.500124i	−0.968230	−	0.250062i	$-0.919549\pi$
$929$		−	15.2435i		−	0.500124i	0.968230	−	0.250062i	$-0.0804509\pi$
$930$		0			0
$931$	−0.129878			−0.00425658
$932$		0			0
$933$		−	4.63003i		−	0.151580i
$934$		0			0
$935$	24.0495			0.786504
$936$		0			0
$937$	−58.4300			−1.90883			−0.954413	−	0.298490i	$-0.903517\pi$
$937$	−58.4300			−1.90883			−0.954413	+	0.298490i	$0.903517\pi$
$938$		0			0
$939$		−	1.73836i		−	0.0567293i
$940$		0			0
$941$	22.9904			0.749467			0.374733	−	0.927133i	$-0.377734\pi$
$941$	22.9904			0.749467			0.374733	+	0.927133i	$0.377734\pi$
$942$		0			0
$943$			14.9655i			0.487342i
$944$		0			0
$945$	−4.37034			−0.142167
$946$		0			0
$947$	−19.1597			−0.622608			−0.311304	−	0.950310i	$-0.600766\pi$
$947$	−19.1597			−0.622608			−0.311304	+	0.950310i	$0.600766\pi$
$948$		0			0
$949$	−13.8157	+	38.9662i	−0.448477	+	1.26490i
$950$		0			0
$951$			4.08868i			0.132585i
$952$		0			0
$953$	40.8834			1.32434			0.662171	−	0.749353i	$-0.269635\pi$
$953$	40.8834			1.32434			0.662171	+	0.749353i	$0.269635\pi$
$954$		0			0
$955$	18.9494			0.613188
$956$		0			0
$957$	−2.53166			−0.0818370
$958$		0			0
$959$	−8.39453			−0.271073
$960$		0			0
$961$	29.6711			0.957131
$962$		0			0
$963$			19.3329i			0.622994i
$964$		0			0
$965$		−	81.3188i		−	2.61775i
$966$		0			0
$967$		−	1.86092i		−	0.0598431i	−0.999552	−	0.0299216i	$-0.990474\pi$
$967$		−	1.86092i		−	0.0598431i	0.999552	−	0.0299216i	$-0.00952575\pi$
$968$		0			0
$969$			0.0958761i			0.00307998i
$970$		0			0
$971$			22.8850i			0.734416i	0.930139	+	0.367208i	$0.119686\pi$
$971$			22.8850i			0.734416i	−0.930139	+	0.367208i	$0.880314\pi$
$972$		0			0
$973$	−13.5249			−0.433587
$974$		0			0
$975$	−1.85940	+	5.24430i	−0.0595485	+	0.167952i
$976$		0			0
$977$			3.98677i			0.127548i	0.997964	+	0.0637740i	$0.0203137\pi$
$977$			3.98677i			0.127548i	−0.997964	+	0.0637740i	$0.979686\pi$
$978$		0			0
$979$		−	1.89237i		−	0.0604804i
$980$		0			0
$981$	35.7383			1.14104
$982$		0			0
$983$			50.8782i			1.62276i	0.584518	+	0.811381i	$0.301284\pi$
$983$			50.8782i			1.62276i	−0.584518	+	0.811381i	$0.698716\pi$
$984$		0			0
$985$	29.3031			0.933673
$986$		0			0
$987$			0.480859i			0.0153059i
$988$		0			0
$989$			28.4521i			0.904726i
$990$		0			0
$991$	54.2500			1.72331			0.861653	−	0.507497i	$-0.169429\pi$
$991$	54.2500			1.72331			0.861653	+	0.507497i	$0.169429\pi$
$992$		0			0
$993$			7.00718i			0.222366i
$994$		0			0
$995$	−66.3902			−2.10471
$996$		0			0
$997$			25.8118i			0.817467i	0.912654	+	0.408734i	$0.134029\pi$
$997$			25.8118i			0.817467i	−0.912654	+	0.408734i	$0.865971\pi$
$998$		0			0
$999$		−	0.372756i		−	0.0117935i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2912.2.i.a.337.13		84
4.3	odd	2		728.2.i.a.701.69	yes	84
8.3	odd	2		728.2.i.a.701.15	✓	84
8.5	even	2	inner	2912.2.i.a.337.72		84
13.12	even	2	inner	2912.2.i.a.337.71		84
52.51	odd	2		728.2.i.a.701.16	yes	84
104.51	odd	2		728.2.i.a.701.70	yes	84
104.77	even	2	inner	2912.2.i.a.337.14		84

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
728.2.i.a.701.15	✓	84	8.3	odd	2
728.2.i.a.701.16	yes	84	52.51	odd	2
728.2.i.a.701.69	yes	84	4.3	odd	2
728.2.i.a.701.70	yes	84	104.51	odd	2
2912.2.i.a.337.13		84	1.1	even	1	trivial
2912.2.i.a.337.14		84	104.77	even	2	inner
2912.2.i.a.337.71		84	13.12	even	2	inner
2912.2.i.a.337.72		84	8.5	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2912 = 2^{5} \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2912.i (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-0.208275i q^{3} -3.52272 q^{5} +1.00000i q^{7} +2.95662 q^{9} +1.92616 q^{11} +(-3.39827 - 1.20488i) q^{13} +0.733695i q^{15} -3.54436 q^{17} +0.129878 q^{19} +0.208275 q^{21} +5.09962 q^{23} +7.40954 q^{25} -1.24062i q^{27} -6.31068i q^{29} +1.15279i q^{31} -0.401171i q^{33} -3.52272i q^{35} +0.300461 q^{37} +(-0.250947 + 0.707776i) q^{39} +2.93462i q^{41} +5.57927i q^{43} -10.4153 q^{45} +2.30877i q^{47} -1.00000 q^{49} +0.738202i q^{51} +11.5827i q^{53} -6.78531 q^{55} -0.0270504i q^{57} -7.08922 q^{59} -6.22897i q^{61} +2.95662i q^{63} +(11.9712 + 4.24446i) q^{65} -8.04110 q^{67} -1.06212i q^{69} +16.5533i q^{71} -11.4665i q^{73} -1.54322i q^{75} +1.92616i q^{77} -14.6896 q^{79} +8.61147 q^{81} +3.76948 q^{83} +12.4858 q^{85} -1.31436 q^{87} -0.982458i q^{89} +(1.20488 - 3.39827i) q^{91} +0.240098 q^{93} -0.457523 q^{95} +9.30657i q^{97} +5.69492 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 84 q - 84 q^{9} + 8 q^{17} + 24 q^{23} + 92 q^{25} + 24 q^{39} - 84 q^{49} - 32 q^{55} - 24 q^{65} + 40 q^{79} + 84 q^{81} + 48 q^{87} + 16 q^{95}+O(q^{100}) \) 84 * q - 84 * q^9 + 8 * q^17 + 24 * q^23 + 92 * q^25 + 24 * q^39 - 84 * q^49 - 32 * q^55 - 24 * q^65 + 40 * q^79 + 84 * q^81 + 48 * q^87 + 16 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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