LMFDB - Embedded newform 4680.2.g.h.2521.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4680,2,Mod(2521,4680)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("4680.2521"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4680, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 4680 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4680.g (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,0,0,0,0,0,0,0,0,8,0,0,0,-14] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

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

Self dual:	no
Analytic conductor:	$37.3699881460$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{41})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 21x^{2} + 100 $ x^4 + 21*x^2 + 100
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1560)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2521.4
Root			$2.70156i$ of defining polynomial
Character	$\chi$	$=$	4680.2521
Dual form			4680.2.g.h.2521.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{5} +1.70156i q^{7} +3.70156i q^{11} +(2.00000 + 3.00000i) q^{13} -0.298438 q^{17} -5.40312i q^{19} +3.70156 q^{23} -1.00000 q^{25} -2.00000 q^{29} +3.40312i q^{31} -1.70156 q^{35} +3.70156i q^{37} +2.29844i q^{41} +11.4031 q^{43} +8.00000i q^{47} +4.10469 q^{49} -2.29844 q^{53} -3.70156 q^{55} +13.4031i q^{59} +0.298438 q^{61} +(-3.00000 + 2.00000i) q^{65} -8.00000i q^{67} +5.70156i q^{71} -12.8062i q^{73} -6.29844 q^{77} -9.70156 q^{79} -3.40312i q^{83} -0.298438i q^{85} -1.70156i q^{89} +(-5.10469 + 3.40312i) q^{91} +5.40312 q^{95} -3.10469i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 8 q^{13} - 14 q^{17} + 2 q^{23} - 4 q^{25} - 8 q^{29} + 6 q^{35} + 20 q^{43} - 22 q^{49} - 22 q^{53} - 2 q^{55} + 14 q^{61} - 12 q^{65} - 38 q^{77} - 26 q^{79} + 18 q^{91} - 4 q^{95}+O(q^{100}) $ 4 * q + 8 * q^13 - 14 * q^17 + 2 * q^23 - 4 * q^25 - 8 * q^29 + 6 * q^35 + 20 * q^43 - 22 * q^49 - 22 * q^53 - 2 * q^55 + 14 * q^61 - 12 * q^65 - 38 * q^77 - 26 * q^79 + 18 * q^91 - 4 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4680\mathbb{Z}\right)^\times$.

$n$	$937$	$1081$	$2081$	$2341$	$3511$
$\chi(n)$	$1$	$-1$	$1$	$1$	$1$

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			0
$3$		0			0
$4$		0			0
$5$			1.00000i			0.447214i
$5$			1.00000i			0.447214i
$6$		0			0
$7$			1.70156i			0.643130i	0.946888	+	0.321565i	$0.104209\pi$
$7$			1.70156i			0.643130i	−0.946888	+	0.321565i	$0.895791\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$			3.70156i			1.11606i	0.829820	+	0.558031i	$0.188443\pi$
$11$			3.70156i			1.11606i	−0.829820	+	0.558031i	$0.811557\pi$
$12$		0			0
$13$	2.00000	+	3.00000i	0.554700	+	0.832050i
$13$	2.00000	+	3.00000i	0.554700	+	0.832050i
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−0.298438			−0.0723818			−0.0361909	−	0.999345i	$-0.511522\pi$
$17$	−0.298438			−0.0723818			−0.0361909	+	0.999345i	$0.511522\pi$
$18$		0			0
$19$		−	5.40312i		−	1.23956i	−0.784775	−	0.619781i	$-0.787221\pi$
$19$		−	5.40312i		−	1.23956i	0.784775	−	0.619781i	$-0.212779\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	3.70156			0.771829			0.385915	−	0.922535i	$-0.373886\pi$
$23$	3.70156			0.771829			0.385915	+	0.922535i	$0.373886\pi$
$24$		0			0
$25$	−1.00000			−0.200000
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−2.00000			−0.371391			−0.185695	−	0.982607i	$-0.559454\pi$
$29$	−2.00000			−0.371391			−0.185695	+	0.982607i	$0.559454\pi$
$30$		0			0
$31$			3.40312i			0.611219i	0.952157	+	0.305610i	$0.0988602\pi$
$31$			3.40312i			0.611219i	−0.952157	+	0.305610i	$0.901140\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−1.70156			−0.287616
$36$		0			0
$37$			3.70156i			0.608533i	0.952587	+	0.304267i	$0.0984113\pi$
$37$			3.70156i			0.608533i	−0.952587	+	0.304267i	$0.901589\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			2.29844i			0.358956i	0.983762	+	0.179478i	$0.0574408\pi$
$41$			2.29844i			0.358956i	−0.983762	+	0.179478i	$0.942559\pi$
$42$		0			0
$43$	11.4031			1.73896			0.869480	−	0.493968i	$-0.164454\pi$
$43$	11.4031			1.73896			0.869480	+	0.493968i	$0.164454\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			8.00000i			1.16692i	0.812142	+	0.583460i	$0.198301\pi$
$47$			8.00000i			1.16692i	−0.812142	+	0.583460i	$0.801699\pi$
$48$		0			0
$49$	4.10469			0.586384
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−2.29844			−0.315715			−0.157857	−	0.987462i	$-0.550459\pi$
$53$	−2.29844			−0.315715			−0.157857	+	0.987462i	$0.550459\pi$
$54$		0			0
$55$	−3.70156			−0.499119
$56$		0			0
$57$		0			0
$58$		0			0
$59$			13.4031i			1.74494i	0.488669	+	0.872469i	$0.337482\pi$
$59$			13.4031i			1.74494i	−0.488669	+	0.872469i	$0.662518\pi$
$60$		0			0
$61$	0.298438			0.0382111			0.0191055	−	0.999817i	$-0.493918\pi$
$61$	0.298438			0.0382111			0.0191055	+	0.999817i	$0.493918\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−3.00000	+	2.00000i	−0.372104	+	0.248069i
$66$		0			0
$67$		−	8.00000i		−	0.977356i	−0.872464	−	0.488678i	$-0.837479\pi$
$67$		−	8.00000i		−	0.977356i	0.872464	−	0.488678i	$-0.162521\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			5.70156i			0.676651i	0.941029	+	0.338325i	$0.109860\pi$
$71$			5.70156i			0.676651i	−0.941029	+	0.338325i	$0.890140\pi$
$72$		0			0
$73$		−	12.8062i		−	1.49886i	−0.662085	−	0.749429i	$-0.730328\pi$
$73$		−	12.8062i		−	1.49886i	0.662085	−	0.749429i	$-0.269672\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−6.29844			−0.717774
$78$		0			0
$79$	−9.70156			−1.09151			−0.545756	−	0.837944i	$-0.683757\pi$
$79$	−9.70156			−1.09151			−0.545756	+	0.837944i	$0.683757\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		−	3.40312i		−	0.373541i	−0.982404	−	0.186771i	$-0.940198\pi$
$83$		−	3.40312i		−	0.373541i	0.982404	−	0.186771i	$-0.0598021\pi$
$84$		0			0
$85$		−	0.298438i		−	0.0323701i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	1.70156i		−	0.180365i	−0.995925	−	0.0901826i	$-0.971255\pi$
$89$		−	1.70156i		−	0.180365i	0.995925	−	0.0901826i	$-0.0287451\pi$
$90$		0			0
$91$	−5.10469	+	3.40312i	−0.535117	+	0.356744i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	5.40312			0.554349
$96$		0			0
$97$		−	3.10469i		−	0.315233i	−0.987500	−	0.157617i	$-0.949619\pi$
$97$		−	3.10469i		−	0.315233i	0.987500	−	0.157617i	$-0.0503810\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−16.8062			−1.67228			−0.836142	−	0.548513i	$-0.815194\pi$
$101$	−16.8062			−1.67228			−0.836142	+	0.548513i	$0.815194\pi$
$102$		0			0
$103$	9.40312			0.926517			0.463259	−	0.886223i	$-0.346680\pi$
$103$	9.40312			0.926517			0.463259	+	0.886223i	$0.346680\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−13.1047			−1.26688			−0.633439	−	0.773793i	$-0.718357\pi$
$107$	−13.1047			−1.26688			−0.633439	+	0.773793i	$0.718357\pi$
$108$		0			0
$109$		−	6.00000i		−	0.574696i	−0.957826	−	0.287348i	$-0.907226\pi$
$109$		−	6.00000i		−	0.574696i	0.957826	−	0.287348i	$-0.0927736\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−14.0000			−1.31701			−0.658505	−	0.752577i	$-0.728811\pi$
$113$	−14.0000			−1.31701			−0.658505	+	0.752577i	$0.728811\pi$
$114$		0			0
$115$			3.70156i			0.345172i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		−	0.507811i		−	0.0465509i
$120$		0			0
$121$	−2.70156			−0.245597
$122$		0			0
$123$		0			0
$124$		0			0
$125$		−	1.00000i		−	0.0894427i
$126$		0			0
$127$	−16.8062			−1.49131			−0.745657	−	0.666330i	$-0.767864\pi$
$127$	−16.8062			−1.49131			−0.745657	+	0.666330i	$0.767864\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	12.0000			1.04844			0.524222	−	0.851581i	$-0.324356\pi$
$131$	12.0000			1.04844			0.524222	+	0.851581i	$0.324356\pi$
$132$		0			0
$133$	9.19375			0.797199
$134$		0			0
$135$		0			0
$136$		0			0
$137$			16.8062i			1.43585i	0.696118	+	0.717927i	$0.254909\pi$
$137$			16.8062i			1.43585i	−0.696118	+	0.717927i	$0.745091\pi$
$138$		0			0
$139$	−9.10469			−0.772249			−0.386125	−	0.922447i	$-0.626187\pi$
$139$	−9.10469			−0.772249			−0.386125	+	0.922447i	$0.626187\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−11.1047	+	7.40312i	−0.928621	+	0.619080i
$144$		0			0
$145$		−	2.00000i		−	0.166091i
$146$		0			0
$147$		0			0
$148$		0			0
$149$			8.29844i			0.679835i	0.940455	+	0.339917i	$0.110399\pi$
$149$			8.29844i			0.679835i	−0.940455	+	0.339917i	$0.889601\pi$
$150$		0			0
$151$		−	7.40312i		−	0.602458i	−0.953552	−	0.301229i	$-0.902603\pi$
$151$		−	7.40312i		−	0.602458i	0.953552	−	0.301229i	$-0.0973968\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−3.40312			−0.273346
$156$		0			0
$157$	2.80625			0.223963			0.111982	−	0.993710i	$-0.464280\pi$
$157$	2.80625			0.223963			0.111982	+	0.993710i	$0.464280\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			6.29844i			0.496386i
$162$		0			0
$163$			13.7016i			1.07319i	0.843840	+	0.536595i	$0.180290\pi$
$163$			13.7016i			1.07319i	−0.843840	+	0.536595i	$0.819710\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			8.00000i			0.619059i	0.950890	+	0.309529i	$0.100171\pi$
$167$			8.00000i			0.619059i	−0.950890	+	0.309529i	$0.899829\pi$
$168$		0			0
$169$	−5.00000	+	12.0000i	−0.384615	+	0.923077i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−23.4031			−1.77931			−0.889653	−	0.456637i	$-0.849054\pi$
$173$	−23.4031			−1.77931			−0.889653	+	0.456637i	$0.849054\pi$
$174$		0			0
$175$		−	1.70156i		−	0.128626i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	18.8062			1.40565			0.702823	−	0.711365i	$-0.251923\pi$
$179$	18.8062			1.40565			0.702823	+	0.711365i	$0.251923\pi$
$180$		0			0
$181$	−11.1047			−0.825405			−0.412702	−	0.910866i	$-0.635415\pi$
$181$	−11.1047			−0.825405			−0.412702	+	0.910866i	$0.635415\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−3.70156			−0.272144
$186$		0			0
$187$		−	1.10469i		−	0.0807827i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−4.00000			−0.289430			−0.144715	−	0.989473i	$-0.546227\pi$
$191$	−4.00000			−0.289430			−0.144715	+	0.989473i	$0.546227\pi$
$192$		0			0
$193$			6.50781i			0.468442i	0.972183	+	0.234221i	$0.0752540\pi$
$193$			6.50781i			0.468442i	−0.972183	+	0.234221i	$0.924746\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	12.8062i		−	0.912407i	−0.889875	−	0.456204i	$-0.849209\pi$
$197$		−	12.8062i		−	0.912407i	0.889875	−	0.456204i	$-0.150791\pi$
$198$		0			0
$199$	10.8062			0.766035			0.383017	−	0.923741i	$-0.374885\pi$
$199$	10.8062			0.766035			0.383017	+	0.923741i	$0.374885\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		−	3.40312i		−	0.238852i
$204$		0			0
$205$	−2.29844			−0.160530
$206$		0			0
$207$		0			0
$208$		0			0
$209$	20.0000			1.38343
$210$		0			0
$211$	25.6125			1.76324			0.881619	−	0.471963i	$-0.156454\pi$
$211$	25.6125			1.76324			0.881619	+	0.471963i	$0.156454\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$			11.4031i			0.777687i
$216$		0			0
$217$	−5.79063			−0.393093
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−0.596876	−	0.895314i	−0.0401502	−	0.0602253i
$222$		0			0
$223$			24.0000i			1.60716i	0.595198	+	0.803579i	$0.297074\pi$
$223$			24.0000i			1.60716i	−0.595198	+	0.803579i	$0.702926\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		−	14.8062i		−	0.982725i	−0.870955	−	0.491363i	$-0.836499\pi$
$227$		−	14.8062i		−	0.982725i	0.870955	−	0.491363i	$-0.163501\pi$
$228$		0			0
$229$			8.80625i			0.581933i	0.956733	+	0.290967i	$0.0939769\pi$
$229$			8.80625i			0.581933i	−0.956733	+	0.290967i	$0.906023\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−3.70156			−0.242497			−0.121249	−	0.992622i	$-0.538690\pi$
$233$	−3.70156			−0.242497			−0.121249	+	0.992622i	$0.538690\pi$
$234$		0			0
$235$	−8.00000			−0.521862
$236$		0			0
$237$		0			0
$238$		0			0
$239$			9.70156i			0.627542i	0.949499	+	0.313771i	$0.101592\pi$
$239$			9.70156i			0.627542i	−0.949499	+	0.313771i	$0.898408\pi$
$240$		0			0
$241$			11.4031i			0.734540i	0.930114	+	0.367270i	$0.119707\pi$
$241$			11.4031i			0.734540i	−0.930114	+	0.367270i	$0.880293\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$			4.10469i			0.262239i
$246$		0			0
$247$	16.2094	−	10.8062i	1.03138	−	0.687585i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−16.5969			−1.04759			−0.523793	−	0.851846i	$-0.675483\pi$
$251$	−16.5969			−1.04759			−0.523793	+	0.851846i	$0.675483\pi$
$252$		0			0
$253$			13.7016i			0.861410i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	4.80625			0.299806			0.149903	−	0.988701i	$-0.452104\pi$
$257$	4.80625			0.299806			0.149903	+	0.988701i	$0.452104\pi$
$258$		0			0
$259$	−6.29844			−0.391366
$260$		0			0
$261$		0			0
$262$		0			0
$263$	20.2094			1.24616			0.623082	−	0.782157i	$-0.285880\pi$
$263$	20.2094			1.24616			0.623082	+	0.782157i	$0.285880\pi$
$264$		0			0
$265$		−	2.29844i		−	0.141192i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	13.4031			0.817203			0.408601	−	0.912713i	$-0.366017\pi$
$269$	13.4031			0.817203			0.408601	+	0.912713i	$0.366017\pi$
$270$		0			0
$271$			15.4031i			0.935673i	0.883815	+	0.467837i	$0.154966\pi$
$271$			15.4031i			0.935673i	−0.883815	+	0.467837i	$0.845034\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		−	3.70156i		−	0.223213i
$276$		0			0
$277$	−14.8062			−0.889621			−0.444811	−	0.895625i	$-0.646729\pi$
$277$	−14.8062			−0.889621			−0.444811	+	0.895625i	$0.646729\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	4.59688i		−	0.274227i	−0.990555	−	0.137113i	$-0.956218\pi$
$281$		−	4.59688i		−	0.274227i	0.990555	−	0.137113i	$-0.0437824\pi$
$282$		0			0
$283$	0.596876			0.0354806			0.0177403	−	0.999843i	$-0.494353\pi$
$283$	0.596876			0.0354806			0.0177403	+	0.999843i	$0.494353\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−3.91093			−0.230855
$288$		0			0
$289$	−16.9109			−0.994761
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	6.00000i		−	0.350524i	−0.984522	−	0.175262i	$-0.943923\pi$
$293$		−	6.00000i		−	0.350524i	0.984522	−	0.175262i	$-0.0560772\pi$
$294$		0			0
$295$	−13.4031			−0.780360
$296$		0			0
$297$		0			0
$298$		0			0
$299$	7.40312	+	11.1047i	0.428134	+	0.642201i
$300$		0			0
$301$			19.4031i			1.11838i
$302$		0			0
$303$		0			0
$304$		0			0
$305$			0.298438i			0.0170885i
$306$		0			0
$307$		−	9.10469i		−	0.519632i	−0.965658	−	0.259816i	$-0.916338\pi$
$307$		−	9.10469i		−	0.519632i	0.965658	−	0.259816i	$-0.0836619\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	7.40312			0.419793			0.209896	−	0.977724i	$-0.432687\pi$
$311$	7.40312			0.419793			0.209896	+	0.977724i	$0.432687\pi$
$312$		0			0
$313$	−32.2094			−1.82058			−0.910291	−	0.413970i	$-0.864142\pi$
$313$	−32.2094			−1.82058			−0.910291	+	0.413970i	$0.864142\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	3.19375i		−	0.179379i	−0.995970	−	0.0896895i	$-0.971413\pi$
$317$		−	3.19375i		−	0.179379i	0.995970	−	0.0896895i	$-0.0285875\pi$
$318$		0			0
$319$		−	7.40312i		−	0.414495i
$320$		0			0
$321$		0			0
$322$		0			0
$323$			1.61250i			0.0897217i
$324$		0			0
$325$	−2.00000	−	3.00000i	−0.110940	−	0.166410i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−13.6125			−0.750481
$330$		0			0
$331$			2.59688i			0.142737i	0.997450	+	0.0713686i	$0.0227367\pi$
$331$			2.59688i			0.142737i	−0.997450	+	0.0713686i	$0.977263\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	8.00000			0.437087
$336$		0			0
$337$	−32.8062			−1.78707			−0.893535	−	0.448993i	$-0.851783\pi$
$337$	−32.8062			−1.78707			−0.893535	+	0.448993i	$0.851783\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−12.5969			−0.682159
$342$		0			0
$343$			18.8953i			1.02025i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	0.507811			0.0272607			0.0136304	−	0.999907i	$-0.495661\pi$
$347$	0.507811			0.0272607			0.0136304	+	0.999907i	$0.495661\pi$
$348$		0			0
$349$		−	16.8062i		−	0.899618i	−0.893125	−	0.449809i	$-0.851492\pi$
$349$		−	16.8062i		−	0.899618i	0.893125	−	0.449809i	$-0.148508\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	8.20937i		−	0.436941i	−0.975844	−	0.218470i	$-0.929893\pi$
$353$		−	8.20937i		−	0.436941i	0.975844	−	0.218470i	$-0.0701067\pi$
$354$		0			0
$355$	−5.70156			−0.302607
$356$		0			0
$357$		0			0
$358$		0			0
$359$			16.0000i			0.844448i	0.906492	+	0.422224i	$0.138750\pi$
$359$			16.0000i			0.844448i	−0.906492	+	0.422224i	$0.861250\pi$
$360$		0			0
$361$	−10.1938			−0.536513
$362$		0			0
$363$		0			0
$364$		0			0
$365$	12.8062			0.670310
$366$		0			0
$367$	18.0000			0.939592			0.469796	−	0.882775i	$-0.344327\pi$
$367$	18.0000			0.939592			0.469796	+	0.882775i	$0.344327\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		−	3.91093i		−	0.203046i
$372$		0			0
$373$	17.6125			0.911941			0.455970	−	0.889995i	$-0.349292\pi$
$373$	17.6125			0.911941			0.455970	+	0.889995i	$0.349292\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−4.00000	−	6.00000i	−0.206010	−	0.309016i
$378$		0			0
$379$			20.8062i			1.06875i	0.845249	+	0.534373i	$0.179452\pi$
$379$			20.8062i			1.06875i	−0.845249	+	0.534373i	$0.820548\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		−	19.4031i		−	0.991453i	−0.868479	−	0.495727i	$-0.834902\pi$
$383$		−	19.4031i		−	0.991453i	0.868479	−	0.495727i	$-0.165098\pi$
$384$		0			0
$385$		−	6.29844i		−	0.320998i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	10.0000			0.507020			0.253510	−	0.967333i	$-0.418415\pi$
$389$	10.0000			0.507020			0.253510	+	0.967333i	$0.418415\pi$
$390$		0			0
$391$	−1.10469			−0.0558664
$392$		0			0
$393$		0			0
$394$		0			0
$395$		−	9.70156i		−	0.488139i
$396$		0			0
$397$			23.7016i			1.18955i	0.803893	+	0.594774i	$0.202758\pi$
$397$			23.7016i			1.18955i	−0.803893	+	0.594774i	$0.797242\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			10.2094i			0.509832i	0.966963	+	0.254916i	$0.0820478\pi$
$401$			10.2094i			0.509832i	−0.966963	+	0.254916i	$0.917952\pi$
$402$		0			0
$403$	−10.2094	+	6.80625i	−0.508565	+	0.339043i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−13.7016			−0.679161
$408$		0			0
$409$		−	18.8062i		−	0.929909i	−0.885335	−	0.464955i	$-0.846071\pi$
$409$		−	18.8062i		−	0.929909i	0.885335	−	0.464955i	$-0.153929\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−22.8062			−1.12222
$414$		0			0
$415$	3.40312			0.167053
$416$		0			0
$417$		0			0
$418$		0			0
$419$	8.59688			0.419985			0.209992	−	0.977703i	$-0.432656\pi$
$419$	8.59688			0.419985			0.209992	+	0.977703i	$0.432656\pi$
$420$		0			0
$421$		−	12.8062i		−	0.624138i	−0.950059	−	0.312069i	$-0.898978\pi$
$421$		−	12.8062i		−	0.624138i	0.950059	−	0.312069i	$-0.101022\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	0.298438			0.0144764
$426$		0			0
$427$			0.507811i			0.0245747i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	5.19375i		−	0.250174i	−0.992146	−	0.125087i	$-0.960079\pi$
$431$		−	5.19375i		−	0.250174i	0.992146	−	0.125087i	$-0.0399210\pi$
$432$		0			0
$433$	27.6125			1.32697			0.663486	−	0.748189i	$-0.269076\pi$
$433$	27.6125			1.32697			0.663486	+	0.748189i	$0.269076\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		−	20.0000i		−	0.956730i
$438$		0			0
$439$	29.1047			1.38909			0.694545	−	0.719449i	$-0.255606\pi$
$439$	29.1047			1.38909			0.694545	+	0.719449i	$0.255606\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	17.7016			0.841027			0.420513	−	0.907286i	$-0.361850\pi$
$443$	17.7016			0.841027			0.420513	+	0.907286i	$0.361850\pi$
$444$		0			0
$445$	1.70156			0.0806618
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	27.9109i		−	1.31720i	−0.752494	−	0.658599i	$-0.771149\pi$
$449$		−	27.9109i		−	1.31720i	0.752494	−	0.658599i	$-0.228851\pi$
$450$		0			0
$451$	−8.50781			−0.400617
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−3.40312	−	5.10469i	−0.159541	−	0.239311i
$456$		0			0
$457$		−	35.1047i		−	1.64213i	−0.570836	−	0.821064i	$-0.693381\pi$
$457$		−	35.1047i		−	1.64213i	0.570836	−	0.821064i	$-0.306619\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	36.7172i		−	1.71009i	−0.518554	−	0.855045i	$-0.673529\pi$
$461$		−	36.7172i		−	1.71009i	0.518554	−	0.855045i	$-0.326471\pi$
$462$		0			0
$463$			24.5078i			1.13897i	0.822000	+	0.569487i	$0.192858\pi$
$463$			24.5078i			1.13897i	−0.822000	+	0.569487i	$0.807142\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−10.8953			−0.504175			−0.252087	−	0.967704i	$-0.581117\pi$
$467$	−10.8953			−0.504175			−0.252087	+	0.967704i	$0.581117\pi$
$468$		0			0
$469$	13.6125			0.628567
$470$		0			0
$471$		0			0
$472$		0			0
$473$			42.2094i			1.94079i
$474$		0			0
$475$			5.40312i			0.247912i
$476$		0			0
$477$		0			0
$478$		0			0
$479$			31.3141i			1.43078i	0.698727	+	0.715388i	$0.253750\pi$
$479$			31.3141i			1.43078i	−0.698727	+	0.715388i	$0.746250\pi$
$480$		0			0
$481$	−11.1047	+	7.40312i	−0.506330	+	0.337553i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	3.10469			0.140977
$486$		0			0
$487$			16.5078i			0.748040i	0.927421	+	0.374020i	$0.122021\pi$
$487$			16.5078i			0.748040i	−0.927421	+	0.374020i	$0.877979\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−17.6125			−0.794841			−0.397420	−	0.917637i	$-0.630095\pi$
$491$	−17.6125			−0.794841			−0.397420	+	0.917637i	$0.630095\pi$
$492$		0			0
$493$	0.596876			0.0268819
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−9.70156			−0.435175
$498$		0			0
$499$			14.5969i			0.653446i	0.945120	+	0.326723i	$0.105944\pi$
$499$			14.5969i			0.653446i	−0.945120	+	0.326723i	$0.894056\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	10.5969			0.472491			0.236246	−	0.971693i	$-0.424083\pi$
$503$	10.5969			0.472491			0.236246	+	0.971693i	$0.424083\pi$
$504$		0			0
$505$		−	16.8062i		−	0.747868i
$506$		0			0
$507$		0			0
$508$		0			0
$509$			36.7172i			1.62746i	0.581243	+	0.813730i	$0.302567\pi$
$509$			36.7172i			1.62746i	−0.581243	+	0.813730i	$0.697433\pi$
$510$		0			0
$511$	21.7906			0.963961
$512$		0			0
$513$		0			0
$514$		0			0
$515$			9.40312i			0.414351i
$516$		0			0
$517$	−29.6125			−1.30236
$518$		0			0
$519$		0			0
$520$		0			0
$521$	32.2094			1.41112			0.705559	−	0.708651i	$-0.250696\pi$
$521$	32.2094			1.41112			0.705559	+	0.708651i	$0.250696\pi$
$522$		0			0
$523$	19.4031			0.848439			0.424220	−	0.905559i	$-0.360548\pi$
$523$	19.4031			0.848439			0.424220	+	0.905559i	$0.360548\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	1.01562i		−	0.0442412i
$528$		0			0
$529$	−9.29844			−0.404280
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−6.89531	+	4.59688i	−0.298669	+	0.199113i
$534$		0			0
$535$		−	13.1047i		−	0.566565i
$536$		0			0
$537$		0			0
$538$		0			0
$539$			15.1938i			0.654441i
$540$		0			0
$541$			14.5969i			0.627569i	0.949494	+	0.313784i	$0.101597\pi$
$541$			14.5969i			0.627569i	−0.949494	+	0.313784i	$0.898403\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	6.00000			0.257012
$546$		0			0
$547$	−31.4031			−1.34270			−0.671350	−	0.741140i	$-0.734285\pi$
$547$	−31.4031			−1.34270			−0.671350	+	0.741140i	$0.734285\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$			10.8062i			0.460362i
$552$		0			0
$553$		−	16.5078i		−	0.701984i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	14.0000i		−	0.593199i	−0.955002	−	0.296600i	$-0.904147\pi$
$557$		−	14.0000i		−	0.593199i	0.955002	−	0.296600i	$-0.0958526\pi$
$558$		0			0
$559$	22.8062	+	34.2094i	0.964602	+	1.44690i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−18.8953			−0.796343			−0.398171	−	0.917311i	$-0.630355\pi$
$563$	−18.8953			−0.796343			−0.398171	+	0.917311i	$0.630355\pi$
$564$		0			0
$565$		−	14.0000i		−	0.588984i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	26.0000			1.08998			0.544988	−	0.838444i	$-0.316534\pi$
$569$	26.0000			1.08998			0.544988	+	0.838444i	$0.316534\pi$
$570$		0			0
$571$	19.3141			0.808268			0.404134	−	0.914700i	$-0.367573\pi$
$571$	19.3141			0.808268			0.404134	+	0.914700i	$0.367573\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−3.70156			−0.154366
$576$		0			0
$577$		−	1.49219i		−	0.0621207i	−0.999518	−	0.0310603i	$-0.990112\pi$
$577$		−	1.49219i		−	0.0621207i	0.999518	−	0.0310603i	$-0.00988840\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	5.79063			0.240236
$582$		0			0
$583$		−	8.50781i		−	0.352358i
$584$		0			0
$585$		0			0
$586$		0			0
$587$			6.20937i			0.256288i	0.991756	+	0.128144i	$0.0409020\pi$
$587$			6.20937i			0.256288i	−0.991756	+	0.128144i	$0.959098\pi$
$588$		0			0
$589$	18.3875			0.757644
$590$		0			0
$591$		0			0
$592$		0			0
$593$			22.0000i			0.903432i	0.892162	+	0.451716i	$0.149188\pi$
$593$			22.0000i			0.903432i	−0.892162	+	0.451716i	$0.850812\pi$
$594$		0			0
$595$	0.507811			0.0208182
$596$		0			0
$597$		0			0
$598$		0			0
$599$	17.1938			0.702518			0.351259	−	0.936278i	$-0.385754\pi$
$599$	17.1938			0.702518			0.351259	+	0.936278i	$0.385754\pi$
$600$		0			0
$601$	36.1203			1.47338			0.736689	−	0.676232i	$-0.236388\pi$
$601$	36.1203			1.47338			0.736689	+	0.676232i	$0.236388\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		−	2.70156i		−	0.109834i
$606$		0			0
$607$	−4.20937			−0.170853			−0.0854266	−	0.996344i	$-0.527225\pi$
$607$	−4.20937			−0.170853			−0.0854266	+	0.996344i	$0.527225\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−24.0000	+	16.0000i	−0.970936	+	0.647291i
$612$		0			0
$613$		−	3.10469i		−	0.125397i	−0.998033	−	0.0626986i	$-0.980029\pi$
$613$		−	3.10469i		−	0.125397i	0.998033	−	0.0626986i	$-0.0199707\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	27.6125i		−	1.11164i	−0.831304	−	0.555819i	$-0.812405\pi$
$617$		−	27.6125i		−	1.11164i	0.831304	−	0.555819i	$-0.187595\pi$
$618$		0			0
$619$		−	22.0000i		−	0.884255i	−0.896952	−	0.442127i	$-0.854224\pi$
$619$		−	22.0000i		−	0.884255i	0.896952	−	0.442127i	$-0.145776\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	2.89531			0.115998
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	1.10469i		−	0.0440467i
$630$		0			0
$631$		−	6.20937i		−	0.247191i	−0.992333	−	0.123596i	$-0.960557\pi$
$631$		−	6.20937i		−	0.247191i	0.992333	−	0.123596i	$-0.0394426\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		−	16.8062i		−	0.666936i
$636$		0			0
$637$	8.20937	+	12.3141i	0.325267	+	0.487901i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	19.0156			0.751072			0.375536	−	0.926808i	$-0.377459\pi$
$641$	19.0156			0.751072			0.375536	+	0.926808i	$0.377459\pi$
$642$		0			0
$643$		−	24.5078i		−	0.966494i	−0.875484	−	0.483247i	$-0.839457\pi$
$643$		−	24.5078i		−	0.966494i	0.875484	−	0.483247i	$-0.160543\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	29.9109			1.17592			0.587960	−	0.808890i	$-0.299931\pi$
$647$	29.9109			1.17592			0.587960	+	0.808890i	$0.299931\pi$
$648$		0			0
$649$	−49.6125			−1.94746
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−11.4031			−0.446239			−0.223119	−	0.974791i	$-0.571624\pi$
$653$	−11.4031			−0.446239			−0.223119	+	0.974791i	$0.571624\pi$
$654$		0			0
$655$			12.0000i			0.468879i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−21.0156			−0.818652			−0.409326	−	0.912388i	$-0.634236\pi$
$659$	−21.0156			−0.818652			−0.409326	+	0.912388i	$0.634236\pi$
$660$		0			0
$661$		−	25.4031i		−	0.988067i	−0.869443	−	0.494034i	$-0.835522\pi$
$661$		−	25.4031i		−	0.988067i	0.869443	−	0.494034i	$-0.164478\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$			9.19375i			0.356518i
$666$		0			0
$667$	−7.40312			−0.286650
$668$		0			0
$669$		0			0
$670$		0			0
$671$			1.10469i			0.0426459i
$672$		0			0
$673$	14.5969			0.562668			0.281334	−	0.959610i	$-0.409223\pi$
$673$	14.5969			0.562668			0.281334	+	0.959610i	$0.409223\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	9.70156			0.372861			0.186431	−	0.982468i	$-0.440308\pi$
$677$	9.70156			0.372861			0.186431	+	0.982468i	$0.440308\pi$
$678$		0			0
$679$	5.28282			0.202736
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	36.5969i		−	1.40034i	−0.713976	−	0.700170i	$-0.753107\pi$
$683$		−	36.5969i		−	1.40034i	0.713976	−	0.700170i	$-0.246893\pi$
$684$		0			0
$685$	−16.8062			−0.642134
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−4.59688	−	6.89531i	−0.175127	−	0.262691i
$690$		0			0
$691$		−	31.6125i		−	1.20260i	−0.799025	−	0.601298i	$-0.794650\pi$
$691$		−	31.6125i		−	1.20260i	0.799025	−	0.601298i	$-0.205350\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		−	9.10469i		−	0.345360i
$696$		0			0
$697$		−	0.685941i		−	0.0259819i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−35.0156			−1.32252			−0.661261	−	0.750156i	$-0.729978\pi$
$701$	−35.0156			−1.32252			−0.661261	+	0.750156i	$0.729978\pi$
$702$		0			0
$703$	20.0000			0.754314
$704$		0			0
$705$		0			0
$706$		0			0
$707$		−	28.5969i		−	1.07550i
$708$		0			0
$709$			28.2094i			1.05943i	0.848177	+	0.529713i	$0.177700\pi$
$709$			28.2094i			1.05943i	−0.848177	+	0.529713i	$0.822300\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			12.5969i			0.471757i
$714$		0			0
$715$	−7.40312	−	11.1047i	−0.276861	−	0.415292i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−15.4031			−0.574440			−0.287220	−	0.957865i	$-0.592731\pi$
$719$	−15.4031			−0.574440			−0.287220	+	0.957865i	$0.592731\pi$
$720$		0			0
$721$			16.0000i			0.595871i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	2.00000			0.0742781
$726$		0			0
$727$	−11.1938			−0.415153			−0.207577	−	0.978219i	$-0.566558\pi$
$727$	−11.1938			−0.415153			−0.207577	+	0.978219i	$0.566558\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−3.40312			−0.125869
$732$		0			0
$733$		−	43.7016i		−	1.61415i	−0.590446	−	0.807077i	$-0.701048\pi$
$733$		−	43.7016i		−	1.61415i	0.590446	−	0.807077i	$-0.298952\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	29.6125			1.09079
$738$		0			0
$739$			36.8062i			1.35394i	0.736011	+	0.676970i	$0.236707\pi$
$739$			36.8062i			1.35394i	−0.736011	+	0.676970i	$0.763293\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	40.0000i		−	1.46746i	−0.679442	−	0.733729i	$-0.737778\pi$
$743$		−	40.0000i		−	1.46746i	0.679442	−	0.733729i	$-0.262222\pi$
$744$		0			0
$745$	−8.29844			−0.304031
$746$		0			0
$747$		0			0
$748$		0			0
$749$		−	22.2984i		−	0.814767i
$750$		0			0
$751$	42.2984			1.54349			0.771746	−	0.635931i	$-0.219384\pi$
$751$	42.2984			1.54349			0.771746	+	0.635931i	$0.219384\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	7.40312			0.269427
$756$		0			0
$757$	6.80625			0.247377			0.123689	−	0.992321i	$-0.460528\pi$
$757$	6.80625			0.247377			0.123689	+	0.992321i	$0.460528\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	11.4031i		−	0.413363i	−0.978408	−	0.206681i	$-0.933734\pi$
$761$		−	11.4031i		−	0.413363i	0.978408	−	0.206681i	$-0.0662664\pi$
$762$		0			0
$763$	10.2094			0.369604
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−40.2094	+	26.8062i	−1.45188	+	0.967918i
$768$		0			0
$769$		−	30.8062i		−	1.11090i	−0.831549	−	0.555451i	$-0.812546\pi$
$769$		−	30.8062i		−	1.11090i	0.831549	−	0.555451i	$-0.187454\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			44.8062i			1.61157i	0.592209	+	0.805784i	$0.298256\pi$
$773$			44.8062i			1.61157i	−0.592209	+	0.805784i	$0.701744\pi$
$774$		0			0
$775$		−	3.40312i		−	0.122244i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	12.4187			0.444948
$780$		0			0
$781$	−21.1047			−0.755185
$782$		0			0
$783$		0			0
$784$		0			0
$785$			2.80625i			0.100159i
$786$		0			0
$787$		−	52.0000i		−	1.85360i	−0.375555	−	0.926800i	$-0.622548\pi$
$787$		−	52.0000i		−	1.85360i	0.375555	−	0.926800i	$-0.377452\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	23.8219i		−	0.847008i
$792$		0			0
$793$	0.596876	+	0.895314i	0.0211957	+	0.0317935i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−53.5234			−1.89590			−0.947949	−	0.318423i	$-0.896847\pi$
$797$	−53.5234			−1.89590			−0.947949	+	0.318423i	$0.896847\pi$
$798$		0			0
$799$		−	2.38750i		−	0.0844638i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	47.4031			1.67282
$804$		0			0
$805$	−6.29844			−0.221991
$806$		0			0
$807$		0			0
$808$		0			0
$809$	29.4031			1.03376			0.516879	−	0.856058i	$-0.327094\pi$
$809$	29.4031			1.03376			0.516879	+	0.856058i	$0.327094\pi$
$810$		0			0
$811$			32.8062i			1.15198i	0.817456	+	0.575992i	$0.195384\pi$
$811$			32.8062i			1.15198i	−0.817456	+	0.575992i	$0.804616\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−13.7016			−0.479945
$816$		0			0
$817$		−	61.6125i		−	2.15555i
$818$		0			0
$819$		0			0
$820$		0			0
$821$			6.50781i			0.227124i	0.993531	+	0.113562i	$0.0362261\pi$
$821$			6.50781i			0.227124i	−0.993531	+	0.113562i	$0.963774\pi$
$822$		0			0
$823$	47.6125			1.65967			0.829834	−	0.558011i	$-0.188435\pi$
$823$	47.6125			1.65967			0.829834	+	0.558011i	$0.188435\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	5.19375i		−	0.180604i	−0.995914	−	0.0903022i	$-0.971217\pi$
$827$		−	5.19375i		−	0.180604i	0.995914	−	0.0903022i	$-0.0287833\pi$
$828$		0			0
$829$	14.0000			0.486240			0.243120	−	0.969996i	$-0.421829\pi$
$829$	14.0000			0.486240			0.243120	+	0.969996i	$0.421829\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−1.22499			−0.0424435
$834$		0			0
$835$	−8.00000			−0.276851
$836$		0			0
$837$		0			0
$838$		0			0
$839$			2.29844i			0.0793509i	0.999213	+	0.0396754i	$0.0126324\pi$
$839$			2.29844i			0.0793509i	−0.999213	+	0.0396754i	$0.987368\pi$
$840$		0			0
$841$	−25.0000			−0.862069
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−12.0000	−	5.00000i	−0.412813	−	0.172005i
$846$		0			0
$847$		−	4.59688i		−	0.157951i
$848$		0			0
$849$		0			0
$850$		0			0
$851$			13.7016i			0.469684i
$852$		0			0
$853$			7.10469i			0.243260i	0.992576	+	0.121630i	$0.0388121\pi$
$853$			7.10469i			0.243260i	−0.992576	+	0.121630i	$0.961188\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−45.3141			−1.54790			−0.773949	−	0.633247i	$-0.781722\pi$
$857$	−45.3141			−1.54790			−0.773949	+	0.633247i	$0.781722\pi$
$858$		0			0
$859$	12.5078			0.426761			0.213380	−	0.976969i	$-0.431553\pi$
$859$	12.5078			0.426761			0.213380	+	0.976969i	$0.431553\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	43.4031i		−	1.47746i	−0.674002	−	0.738730i	$-0.735426\pi$
$863$		−	43.4031i		−	1.47746i	0.674002	−	0.738730i	$-0.264574\pi$
$864$		0			0
$865$		−	23.4031i		−	0.795730i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	35.9109i		−	1.21820i
$870$		0			0
$871$	24.0000	−	16.0000i	0.813209	−	0.542139i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	1.70156			0.0575233
$876$		0			0
$877$			0.387503i			0.0130850i	0.999979	+	0.00654252i	$0.00208256\pi$
$877$			0.387503i			0.0130850i	−0.999979	+	0.00654252i	$0.997917\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−0.209373			−0.00705395			−0.00352697	−	0.999994i	$-0.501123\pi$
$881$	−0.209373			−0.00705395			−0.00352697	+	0.999994i	$0.501123\pi$
$882$		0			0
$883$	−44.4187			−1.49481			−0.747405	−	0.664369i	$-0.768700\pi$
$883$	−44.4187			−1.49481			−0.747405	+	0.664369i	$0.768700\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	5.31406			0.178429			0.0892143	−	0.996012i	$-0.471564\pi$
$887$	5.31406			0.178429			0.0892143	+	0.996012i	$0.471564\pi$
$888$		0			0
$889$		−	28.5969i		−	0.959108i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	43.2250			1.44647
$894$		0			0
$895$			18.8062i			0.628624i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		−	6.80625i		−	0.227001i
$900$		0			0
$901$	0.685941			0.0228520
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	11.1047i		−	0.369132i
$906$		0			0
$907$	24.0000			0.796907			0.398453	−	0.917189i	$-0.369547\pi$
$907$	24.0000			0.796907			0.398453	+	0.917189i	$0.369547\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	51.8219			1.71694			0.858468	−	0.512868i	$-0.171417\pi$
$911$	51.8219			1.71694			0.858468	+	0.512868i	$0.171417\pi$
$912$		0			0
$913$	12.5969			0.416896
$914$		0			0
$915$		0			0
$916$		0			0
$917$			20.4187i			0.674286i
$918$		0			0
$919$	−13.7016			−0.451973			−0.225986	−	0.974130i	$-0.572560\pi$
$919$	−13.7016			−0.451973			−0.225986	+	0.974130i	$0.572560\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−17.1047	+	11.4031i	−0.563008	+	0.375338i
$924$		0			0
$925$		−	3.70156i		−	0.121707i
$926$		0			0
$927$		0			0
$928$		0			0
$929$			24.5078i			0.804075i	0.915623	+	0.402038i	$0.131698\pi$
$929$			24.5078i			0.804075i	−0.915623	+	0.402038i	$0.868302\pi$
$930$		0			0
$931$		−	22.1781i		−	0.726859i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	1.10469			0.0361271
$936$		0			0
$937$	−59.6125			−1.94746			−0.973728	−	0.227716i	$-0.926874\pi$
$937$	−59.6125			−1.94746			−0.973728	+	0.227716i	$0.926874\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$			1.49219i			0.0486440i	0.999704	+	0.0243220i	$0.00774269\pi$
$941$			1.49219i			0.0486440i	−0.999704	+	0.0243220i	$0.992257\pi$
$942$		0			0
$943$			8.50781i			0.277052i
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	20.0000i		−	0.649913i	−0.945729	−	0.324956i	$-0.894650\pi$
$947$		−	20.0000i		−	0.649913i	0.945729	−	0.324956i	$-0.105350\pi$
$948$		0			0
$949$	38.4187	−	25.6125i	1.24713	−	0.831417i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	56.1203			1.81792			0.908958	−	0.416889i	$-0.136880\pi$
$953$	56.1203			1.81792			0.908958	+	0.416889i	$0.136880\pi$
$954$		0			0
$955$		−	4.00000i		−	0.129437i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−28.5969			−0.923441
$960$		0			0
$961$	19.4187			0.626411
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−6.50781			−0.209494
$966$		0			0
$967$			8.00000i			0.257263i	0.991692	+	0.128631i	$0.0410584\pi$
$967$			8.00000i			0.257263i	−0.991692	+	0.128631i	$0.958942\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−38.2094			−1.22620			−0.613099	−	0.790006i	$-0.710077\pi$
$971$	−38.2094			−1.22620			−0.613099	+	0.790006i	$0.710077\pi$
$972$		0			0
$973$		−	15.4922i		−	0.496657i
$974$		0			0
$975$		0			0
$976$		0			0
$977$			25.4031i			0.812718i	0.913714	+	0.406359i	$0.133202\pi$
$977$			25.4031i			0.812718i	−0.913714	+	0.406359i	$0.866798\pi$
$978$		0			0
$979$	6.29844			0.201299
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	28.4187i		−	0.906417i	−0.891404	−	0.453209i	$-0.850279\pi$
$983$		−	28.4187i		−	0.906417i	0.891404	−	0.453209i	$-0.149721\pi$
$984$		0			0
$985$	12.8062			0.408041
$986$		0			0
$987$		0			0
$988$		0			0
$989$	42.2094			1.34218
$990$		0			0
$991$	14.8953			0.473165			0.236583	−	0.971611i	$-0.423973\pi$
$991$	14.8953			0.473165			0.236583	+	0.971611i	$0.423973\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$			10.8062i			0.342581i
$996$		0			0
$997$	−44.0000			−1.39349			−0.696747	−	0.717317i	$-0.745370\pi$
$997$	−44.0000			−1.39349			−0.696747	+	0.717317i	$0.745370\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4680.2.g.h.2521.4		4
3.2	odd	2		1560.2.g.f.961.2	✓	4
12.11	even	2		3120.2.g.r.961.1		4
13.12	even	2	inner	4680.2.g.h.2521.1		4
39.38	odd	2		1560.2.g.f.961.3	yes	4
156.155	even	2		3120.2.g.r.961.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1560.2.g.f.961.2	✓	4	3.2	odd	2
1560.2.g.f.961.3	yes	4	39.38	odd	2
3120.2.g.r.961.1		4	12.11	even	2
3120.2.g.r.961.4		4	156.155	even	2
4680.2.g.h.2521.1		4	13.12	even	2	inner
4680.2.g.h.2521.4		4	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 4680 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4680.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{5} +1.70156i q^{7} +3.70156i q^{11} +(2.00000 + 3.00000i) q^{13} -0.298438 q^{17} -5.40312i q^{19} +3.70156 q^{23} -1.00000 q^{25} -2.00000 q^{29} +3.40312i q^{31} -1.70156 q^{35} +3.70156i q^{37} +2.29844i q^{41} +11.4031 q^{43} +8.00000i q^{47} +4.10469 q^{49} -2.29844 q^{53} -3.70156 q^{55} +13.4031i q^{59} +0.298438 q^{61} +(-3.00000 + 2.00000i) q^{65} -8.00000i q^{67} +5.70156i q^{71} -12.8062i q^{73} -6.29844 q^{77} -9.70156 q^{79} -3.40312i q^{83} -0.298438i q^{85} -1.70156i q^{89} +(-5.10469 + 3.40312i) q^{91} +5.40312 q^{95} -3.10469i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 8 q^{13} - 14 q^{17} + 2 q^{23} - 4 q^{25} - 8 q^{29} + 6 q^{35} + 20 q^{43} - 22 q^{49} - 22 q^{53} - 2 q^{55} + 14 q^{61} - 12 q^{65} - 38 q^{77} - 26 q^{79} + 18 q^{91} - 4 q^{95}+O(q^{100}) \) 4 * q + 8 * q^13 - 14 * q^17 + 2 * q^23 - 4 * q^25 - 8 * q^29 + 6 * q^35 + 20 * q^43 - 22 * q^49 - 22 * q^53 - 2 * q^55 + 14 * q^61 - 12 * q^65 - 38 * q^77 - 26 * q^79 + 18 * q^91 - 4 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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