LMFDB - Embedded newform 3060.2.z.g.829.17

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3060,2,Mod(829,3060)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3060, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 0, 2, 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("3060.829"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 3060 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3060.z (of order $4$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [40,0,0,0,-4,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	no
Analytic conductor:	$24.4342230185$
Analytic rank:	$0$
Dimension:	$40$
Relative dimension:	$20$ over $\Q(i)$
Twist minimal:	no (minimal twist has level 1020)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			829.17
Character	$\chi$	$=$	3060.829
Dual form			3060.2.z.g.2809.17

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.78098 + 1.35208i) q^{5} +(2.38946 + 2.38946i) q^{7} +(4.26466 - 4.26466i) q^{11} +2.80533i q^{13} +(3.60237 - 2.00572i) q^{17} +4.71202i q^{19} +(6.40321 + 6.40321i) q^{23} +(1.34376 + 4.81605i) q^{25} +(-6.84047 - 6.84047i) q^{29} +(-1.54284 - 1.54284i) q^{31} +(1.02484 + 7.48633i) q^{35} +(4.27355 - 4.27355i) q^{37} +(-2.72196 + 2.72196i) q^{41} -2.24911 q^{43} -10.4895i q^{47} +4.41908i q^{49} +1.54211 q^{53} +(13.3614 - 1.82911i) q^{55} -3.65396i q^{59} +(-1.08330 + 1.08330i) q^{61} +(-3.79303 + 4.99623i) q^{65} -1.38282i q^{67} +(-6.44031 - 6.44031i) q^{71} +(-0.994674 + 0.994674i) q^{73} +20.3805 q^{77} +(-8.31627 + 8.31627i) q^{79} -3.99974 q^{83} +(9.12764 + 1.29855i) q^{85} +13.9632 q^{89} +(-6.70323 + 6.70323i) q^{91} +(-6.37102 + 8.39200i) q^{95} +(10.7777 - 10.7777i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 40 q - 4 q^{5} + 8 q^{11} - 24 q^{29} - 16 q^{31} - 8 q^{35} + 8 q^{41} + 28 q^{55} + 16 q^{61} - 56 q^{71} + 16 q^{79} - 40 q^{85} - 32 q^{89} + 64 q^{91} - 64 q^{95}+O(q^{100}) $ 40 * q - 4 * q^5 + 8 * q^11 - 24 * q^29 - 16 * q^31 - 8 * q^35 + 8 * q^41 + 28 * q^55 + 16 * q^61 - 56 * q^71 + 16 * q^79 - 40 * q^85 - 32 * q^89 + 64 * q^91 - 64 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3060\mathbb{Z}\right)^\times$.

$n$	$1261$	$1361$	$1531$	$1837$
$\chi(n)$	$e\left(\frac{1}{4}\right)$	$1$	$1$	$-1$

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
$5$	1.78098	+	1.35208i	0.796477	+	0.604668i
$5$	1.78098	+	1.35208i	0.796477	+	0.604668i
$6$		0			0
$7$	2.38946	+	2.38946i	0.903133	+	0.903133i	0.995706	−	0.0925731i	$-0.0295092\pi$
$7$	2.38946	+	2.38946i	0.903133	+	0.903133i	−0.0925731	+	0.995706i	$0.529509\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	4.26466	−	4.26466i	1.28584	−	1.28584i	0.348556	−	0.937288i	$-0.386672\pi$
$11$	4.26466	−	4.26466i	1.28584	−	1.28584i	0.937288	−	0.348556i	$-0.113328\pi$
$12$		0			0
$13$			2.80533i			0.778058i	0.921225	+	0.389029i	$0.127189\pi$
$13$			2.80533i			0.778058i	−0.921225	+	0.389029i	$0.872811\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	3.60237	−	2.00572i	0.873704	−	0.486458i
$17$	3.60237	−	2.00572i	0.873704	−	0.486458i
$18$		0			0
$19$			4.71202i			1.08101i	0.841340	+	0.540505i	$0.181767\pi$
$19$			4.71202i			1.08101i	−0.841340	+	0.540505i	$0.818233\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	6.40321	+	6.40321i	1.33516	+	1.33516i	0.900681	+	0.434481i	$0.143068\pi$
$23$	6.40321	+	6.40321i	1.33516	+	1.33516i	0.434481	+	0.900681i	$0.356932\pi$
$24$		0			0
$25$	1.34376	+	4.81605i	0.268753	+	0.963209i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−6.84047	−	6.84047i	−1.27024	−	1.27024i	−0.945960	−	0.324283i	$-0.894877\pi$
$29$	−6.84047	−	6.84047i	−1.27024	−	1.27024i	−0.324283	−	0.945960i	$-0.605123\pi$
$30$		0			0
$31$	−1.54284	−	1.54284i	−0.277102	−	0.277102i	0.554849	−	0.831951i	$-0.312776\pi$
$31$	−1.54284	−	1.54284i	−0.277102	−	0.277102i	−0.831951	+	0.554849i	$0.812776\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	1.02484	+	7.48633i	0.173229	+	1.26542i
$36$		0			0
$37$	4.27355	−	4.27355i	0.702568	−	0.702568i	−0.262393	−	0.964961i	$-0.584512\pi$
$37$	4.27355	−	4.27355i	0.702568	−	0.702568i	0.964961	+	0.262393i	$0.0845118\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−2.72196	+	2.72196i	−0.425099	+	0.425099i	−0.886955	−	0.461856i	$-0.847184\pi$
$41$	−2.72196	+	2.72196i	−0.425099	+	0.425099i	0.461856	+	0.886955i	$0.347184\pi$
$42$		0			0
$43$	−2.24911			−0.342986			−0.171493	−	0.985185i	$-0.554859\pi$
$43$	−2.24911			−0.342986			−0.171493	+	0.985185i	$0.554859\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		−	10.4895i		−	1.53005i	−0.644001	−	0.765025i	$-0.722727\pi$
$47$		−	10.4895i		−	1.53005i	0.644001	−	0.765025i	$-0.277273\pi$
$48$		0			0
$49$			4.41908i			0.631298i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	1.54211			0.211824			0.105912	−	0.994375i	$-0.466224\pi$
$53$	1.54211			0.211824			0.105912	+	0.994375i	$0.466224\pi$
$54$		0			0
$55$	13.3614	−	1.82911i	1.80165	−	0.246637i
$56$		0			0
$57$		0			0
$58$		0			0
$59$		−	3.65396i		−	0.475705i	−0.971301	−	0.237852i	$-0.923557\pi$
$59$		−	3.65396i		−	0.475705i	0.971301	−	0.237852i	$-0.0764435\pi$
$60$		0			0
$61$	−1.08330	+	1.08330i	−0.138702	+	0.138702i	−0.773049	−	0.634347i	$-0.781269\pi$
$61$	−1.08330	+	1.08330i	−0.138702	+	0.138702i	0.634347	+	0.773049i	$0.281269\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−3.79303	+	4.99623i	−0.470467	+	0.619706i
$66$		0			0
$67$		−	1.38282i		−	0.168938i	−0.996426	−	0.0844691i	$-0.973081\pi$
$67$		−	1.38282i		−	0.168938i	0.996426	−	0.0844691i	$-0.0269194\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−6.44031	−	6.44031i	−0.764325	−	0.764325i	0.212776	−	0.977101i	$-0.431749\pi$
$71$	−6.44031	−	6.44031i	−0.764325	−	0.764325i	−0.977101	+	0.212776i	$0.931749\pi$
$72$		0			0
$73$	−0.994674	+	0.994674i	−0.116418	+	0.116418i	−0.762916	−	0.646498i	$-0.776233\pi$
$73$	−0.994674	+	0.994674i	−0.116418	+	0.116418i	0.646498	+	0.762916i	$0.276233\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	20.3805			2.32258
$78$		0			0
$79$	−8.31627	+	8.31627i	−0.935654	+	0.935654i	−0.998051	−	0.0623974i	$-0.980125\pi$
$79$	−8.31627	+	8.31627i	−0.935654	+	0.935654i	0.0623974	+	0.998051i	$0.480125\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−3.99974			−0.439029			−0.219514	−	0.975609i	$-0.570447\pi$
$83$	−3.99974			−0.439029			−0.219514	+	0.975609i	$0.570447\pi$
$84$		0			0
$85$	9.12764	+	1.29855i	0.990031	+	0.140848i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	13.9632			1.48010			0.740049	−	0.672553i	$-0.234802\pi$
$89$	13.9632			1.48010			0.740049	+	0.672553i	$0.234802\pi$
$90$		0			0
$91$	−6.70323	+	6.70323i	−0.702690	+	0.702690i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−6.37102	+	8.39200i	−0.653653	+	0.861001i
$96$		0			0
$97$	10.7777	−	10.7777i	1.09431	−	1.09431i	0.0992474	−	0.995063i	$-0.468356\pi$
$97$	10.7777	−	10.7777i	1.09431	−	1.09431i	0.995063	−	0.0992474i	$-0.0316435\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−13.8478			−1.37791			−0.688954	−	0.724805i	$-0.741930\pi$
$101$	−13.8478			−1.37791			−0.688954	+	0.724805i	$0.741930\pi$
$102$		0			0
$103$		−	3.52129i		−	0.346963i	−0.984837	−	0.173481i	$-0.944498\pi$
$103$		−	3.52129i		−	0.346963i	0.984837	−	0.173481i	$-0.0555016\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−8.77511	+	8.77511i	−0.848322	+	0.848322i	−0.989924	−	0.141602i	$-0.954775\pi$
$107$	−8.77511	+	8.77511i	−0.848322	+	0.848322i	0.141602	+	0.989924i	$0.454775\pi$
$108$		0			0
$109$	−8.32449	+	8.32449i	−0.797341	+	0.797341i	−0.982676	−	0.185334i	$-0.940663\pi$
$109$	−8.32449	+	8.32449i	−0.797341	+	0.797341i	0.185334	+	0.982676i	$0.440663\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	7.33189	+	7.33189i	0.689726	+	0.689726i	0.962171	−	0.272445i	$-0.0878324\pi$
$113$	7.33189	+	7.33189i	0.689726	+	0.689726i	−0.272445	+	0.962171i	$0.587832\pi$
$114$		0			0
$115$	2.74633	+	20.0616i	0.256096	+	1.87076i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	13.4003	+	3.81515i	1.22841	+	0.349734i
$120$		0			0
$121$		−	25.3747i		−	2.30679i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−4.11846	+	10.3941i	−0.368367	+	0.929681i
$126$		0			0
$127$	−15.4196			−1.36827			−0.684133	−	0.729357i	$-0.739819\pi$
$127$	−15.4196			−1.36827			−0.684133	+	0.729357i	$0.739819\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−7.12212	−	7.12212i	−0.622262	−	0.622262i	0.323847	−	0.946109i	$-0.395024\pi$
$131$	−7.12212	−	7.12212i	−0.622262	−	0.622262i	−0.946109	+	0.323847i	$0.895024\pi$
$132$		0			0
$133$	−11.2592	+	11.2592i	−0.976297	+	0.976297i
$134$		0			0
$135$		0			0
$136$		0			0
$137$			5.39737i			0.461128i	0.973057	+	0.230564i	$0.0740572\pi$
$137$			5.39737i			0.461128i	−0.973057	+	0.230564i	$0.925943\pi$
$138$		0			0
$139$	−5.39376	−	5.39376i	−0.457493	−	0.457493i	0.440339	−	0.897832i	$-0.354858\pi$
$139$	−5.39376	−	5.39376i	−0.457493	−	0.457493i	−0.897832	+	0.440339i	$0.854858\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	11.9638	+	11.9638i	1.00046	+	1.00046i
$144$		0			0
$145$	−2.93387	−	21.4316i	−0.243644	−	1.77980i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−0.288811			−0.0236603			−0.0118301	−	0.999930i	$-0.503766\pi$
$149$	−0.288811			−0.0236603			−0.0118301	+	0.999930i	$0.503766\pi$
$150$		0			0
$151$			0.348985i			0.0284000i	0.999899	+	0.0142000i	$0.00452015\pi$
$151$			0.348985i			0.0284000i	−0.999899	+	0.0142000i	$0.995480\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−0.661720	−	4.83379i	−0.0531506	−	0.388260i
$156$		0			0
$157$			12.6572i			1.01015i	0.863075	+	0.505076i	$0.168535\pi$
$157$			12.6572i			1.01015i	−0.863075	+	0.505076i	$0.831465\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			30.6005i			2.41166i
$162$		0			0
$163$	10.1621	+	10.1621i	0.795958	+	0.795958i	0.982455	−	0.186497i	$-0.0597136\pi$
$163$	10.1621	+	10.1621i	0.795958	+	0.795958i	−0.186497	+	0.982455i	$0.559714\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−4.22338	+	4.22338i	−0.326815	+	0.326815i	−0.851374	−	0.524559i	$-0.824230\pi$
$167$	−4.22338	+	4.22338i	−0.326815	+	0.326815i	0.524559	+	0.851374i	$0.324230\pi$
$168$		0			0
$169$	5.13013			0.394626
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−0.809262	+	0.809262i	−0.0615271	+	0.0615271i	−0.737201	−	0.675674i	$-0.763853\pi$
$173$	−0.809262	+	0.809262i	−0.0615271	+	0.0615271i	0.675674	+	0.737201i	$0.263853\pi$
$174$		0			0
$175$	−8.29690	+	14.7186i	−0.627187	+	1.11263i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	22.2296i		−	1.66152i	−0.556633	−	0.830759i	$-0.687907\pi$
$179$		−	22.2296i		−	1.66152i	0.556633	−	0.830759i	$-0.312093\pi$
$180$		0			0
$181$	−3.76373	+	3.76373i	−0.279756	+	0.279756i	−0.833012	−	0.553256i	$-0.813385\pi$
$181$	−3.76373	+	3.76373i	−0.279756	+	0.279756i	0.553256	+	0.833012i	$0.313385\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	13.3893	−	1.83292i	0.984399	−	0.134759i
$186$		0			0
$187$	6.80919	−	23.9166i	0.497937	−	1.74896i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	10.4029			0.752728			0.376364	−	0.926472i	$-0.377174\pi$
$191$	10.4029			0.752728			0.376364	+	0.926472i	$0.377174\pi$
$192$		0			0
$193$	11.8621	+	11.8621i	0.853851	+	0.853851i	0.990605	−	0.136754i	$-0.0436669\pi$
$193$	11.8621	+	11.8621i	0.853851	+	0.853851i	−0.136754	+	0.990605i	$0.543667\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	14.5178	+	14.5178i	1.03435	+	1.03435i	0.999389	+	0.0349653i	$0.0111321\pi$
$197$	14.5178	+	14.5178i	1.03435	+	1.03435i	0.0349653	+	0.999389i	$0.488868\pi$
$198$		0			0
$199$	11.1693	+	11.1693i	0.791771	+	0.791771i	0.981782	−	0.190011i	$-0.0608524\pi$
$199$	11.1693	+	11.1693i	0.791771	+	0.791771i	−0.190011	+	0.981782i	$0.560852\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		−	32.6901i		−	2.29440i
$204$		0			0
$205$	−8.52806	+	1.16745i	−0.595626	+	0.0815379i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	20.0952	+	20.0952i	1.39001	+	1.39001i
$210$		0			0
$211$	12.5528	−	12.5528i	0.864174	−	0.864174i	−0.127646	−	0.991820i	$-0.540742\pi$
$211$	12.5528	−	12.5528i	0.864174	−	0.864174i	0.991820	+	0.127646i	$0.0407422\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−4.00561	−	3.04097i	−0.273180	−	0.207392i
$216$		0			0
$217$		−	7.37311i		−	0.500519i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	5.62670	+	10.1058i	0.378493	+	0.679792i
$222$		0			0
$223$	−4.73119			−0.316824			−0.158412	−	0.987373i	$-0.550637\pi$
$223$	−4.73119			−0.316824			−0.158412	+	0.987373i	$0.550637\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−6.01353	−	6.01353i	−0.399132	−	0.399132i	0.478795	−	0.877927i	$-0.341074\pi$
$227$	−6.01353	−	6.01353i	−0.399132	−	0.399132i	−0.877927	+	0.478795i	$0.841074\pi$
$228$		0			0
$229$		−	4.61922i		−	0.305247i	−0.988284	−	0.152623i	$-0.951228\pi$
$229$		−	4.61922i		−	0.305247i	0.988284	−	0.152623i	$-0.0487721\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−6.47363	+	6.47363i	−0.424101	+	0.424101i	−0.886613	−	0.462512i	$-0.846948\pi$
$233$	−6.47363	+	6.47363i	−0.424101	+	0.424101i	0.462512	+	0.886613i	$0.346948\pi$
$234$		0			0
$235$	14.1826	−	18.6815i	0.925172	−	1.21865i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−7.81975			−0.505818			−0.252909	−	0.967490i	$-0.581387\pi$
$239$	−7.81975			−0.505818			−0.252909	+	0.967490i	$0.581387\pi$
$240$		0			0
$241$	−6.68972	−	6.68972i	−0.430923	−	0.430923i	0.458019	−	0.888942i	$-0.348559\pi$
$241$	−6.68972	−	6.68972i	−0.430923	−	0.430923i	−0.888942	+	0.458019i	$0.848559\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−5.97495	+	7.87029i	−0.381726	+	0.502814i
$246$		0			0
$247$	−13.2188			−0.841089
$248$		0			0
$249$		0			0
$250$		0			0
$251$	4.73604			0.298936			0.149468	−	0.988767i	$-0.452244\pi$
$251$	4.73604			0.298936			0.149468	+	0.988767i	$0.452244\pi$
$252$		0			0
$253$	54.6151			3.43362
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−2.35995			−0.147209			−0.0736047	−	0.997287i	$-0.523450\pi$
$257$	−2.35995			−0.147209			−0.0736047	+	0.997287i	$0.523450\pi$
$258$		0			0
$259$	20.4230			1.26902
$260$		0			0
$261$		0			0
$262$		0			0
$263$	18.7175			1.15417			0.577084	−	0.816685i	$-0.304191\pi$
$263$	18.7175			1.15417			0.577084	+	0.816685i	$0.304191\pi$
$264$		0			0
$265$	2.74646	+	2.08505i	0.168713	+	0.128084i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−4.44388	−	4.44388i	−0.270948	−	0.270948i	0.558534	−	0.829482i	$-0.311364\pi$
$269$	−4.44388	−	4.44388i	−0.270948	−	0.270948i	−0.829482	+	0.558534i	$0.811364\pi$
$270$		0			0
$271$	−3.55255			−0.215802			−0.107901	−	0.994162i	$-0.534413\pi$
$271$	−3.55255			−0.215802			−0.107901	+	0.994162i	$0.534413\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	26.2695	+	14.8081i	1.58411	+	0.892963i
$276$		0			0
$277$	−8.28598	+	8.28598i	−0.497856	+	0.497856i	−0.910770	−	0.412914i	$-0.864511\pi$
$277$	−8.28598	+	8.28598i	−0.497856	+	0.497856i	0.412914	+	0.910770i	$0.364511\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			10.7011i			0.638373i	0.947692	+	0.319187i	$0.103410\pi$
$281$			10.7011i			0.638373i	−0.947692	+	0.319187i	$0.896590\pi$
$282$		0			0
$283$	−10.6946	−	10.6946i	−0.635729	−	0.635729i	0.313770	−	0.949499i	$-0.398408\pi$
$283$	−10.6946	−	10.6946i	−0.635729	−	0.635729i	−0.949499	+	0.313770i	$0.898408\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−13.0081			−0.767842
$288$		0			0
$289$	8.95418	−	14.4507i	0.526717	−	0.850041i
$290$		0			0
$291$		0			0
$292$		0			0
$293$			24.7337i			1.44496i	0.691394	+	0.722478i	$0.256997\pi$
$293$			24.7337i			1.44496i	−0.691394	+	0.722478i	$0.743003\pi$
$294$		0			0
$295$	4.94044	−	6.50762i	0.287644	−	0.378888i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−17.9631	+	17.9631i	−1.03883	+	1.03883i
$300$		0			0
$301$	−5.37416	−	5.37416i	−0.309761	−	0.309761i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−3.39404	+	0.464625i	−0.194342	+	0.0266044i
$306$		0			0
$307$		−	16.7478i		−	0.955846i	−0.878402	−	0.477923i	$-0.841390\pi$
$307$		−	16.7478i		−	0.955846i	0.878402	−	0.477923i	$-0.158610\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	2.87130	+	2.87130i	0.162816	+	0.162816i	0.783813	−	0.620997i	$-0.213272\pi$
$311$	2.87130	+	2.87130i	0.162816	+	0.162816i	−0.620997	+	0.783813i	$0.713272\pi$
$312$		0			0
$313$	−22.8159	−	22.8159i	−1.28963	−	1.28963i	−0.935009	−	0.354624i	$-0.884609\pi$
$313$	−22.8159	−	22.8159i	−1.28963	−	1.28963i	−0.354624	−	0.935009i	$-0.615391\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−12.8512	−	12.8512i	−0.721793	−	0.721793i	0.247177	−	0.968970i	$-0.420497\pi$
$317$	−12.8512	−	12.8512i	−0.721793	−	0.721793i	−0.968970	+	0.247177i	$0.920497\pi$
$318$		0			0
$319$	−58.3446			−3.26667
$320$		0			0
$321$		0			0
$322$		0			0
$323$	9.45098	+	16.9744i	0.525867	+	0.944483i
$324$		0			0
$325$	−13.5106	+	3.76970i	−0.749433	+	0.209105i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	25.0643	−	25.0643i	1.38184	−	1.38184i
$330$		0			0
$331$			14.5405i			0.799221i	0.916685	+	0.399610i	$0.130855\pi$
$331$			14.5405i			0.799221i	−0.916685	+	0.399610i	$0.869145\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	1.86968	−	2.46277i	0.102152	−	0.134556i
$336$		0			0
$337$	12.9706	−	12.9706i	0.706551	−	0.706551i	−0.259257	−	0.965808i	$-0.583478\pi$
$337$	12.9706	−	12.9706i	0.706551	−	0.706551i	0.965808	+	0.259257i	$0.0834778\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−13.1593			−0.712619
$342$		0			0
$343$	6.16701	−	6.16701i	0.332987	−	0.332987i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	7.42014	+	7.42014i	0.398334	+	0.398334i	0.877645	−	0.479311i	$-0.159113\pi$
$347$	7.42014	+	7.42014i	0.398334	+	0.398334i	−0.479311	+	0.877645i	$0.659113\pi$
$348$		0			0
$349$			28.8613i			1.54491i	0.635069	+	0.772455i	$0.280971\pi$
$349$			28.8613i			1.54491i	−0.635069	+	0.772455i	$0.719029\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			12.6440i			0.672973i	0.941688	+	0.336486i	$0.109239\pi$
$353$			12.6440i			0.672973i	−0.941688	+	0.336486i	$0.890761\pi$
$354$		0			0
$355$	−2.76224	−	20.1779i	−0.146604	−	1.07093i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	34.1499i		−	1.80236i	−0.433441	−	0.901182i	$-0.642701\pi$
$359$		−	34.1499i		−	1.80236i	0.433441	−	0.901182i	$-0.357299\pi$
$360$		0			0
$361$	−3.20311			−0.168585
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−3.11637	+	0.426614i	−0.163118	+	0.0223300i
$366$		0			0
$367$	10.7838	+	10.7838i	0.562907	+	0.562907i	0.930132	−	0.367225i	$-0.119692\pi$
$367$	10.7838	+	10.7838i	0.562907	+	0.562907i	−0.367225	+	0.930132i	$0.619692\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	3.68481	+	3.68481i	0.191306	+	0.191306i
$372$		0			0
$373$			26.1065i			1.35174i	0.737020	+	0.675870i	$0.236232\pi$
$373$			26.1065i			1.35174i	−0.737020	+	0.675870i	$0.763768\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	19.1898	−	19.1898i	0.988323	−	0.988323i
$378$		0			0
$379$	2.87742	+	2.87742i	0.147803	+	0.147803i	0.777136	−	0.629333i	$-0.216672\pi$
$379$	2.87742	+	2.87742i	0.147803	+	0.147803i	−0.629333	+	0.777136i	$0.716672\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−35.3658			−1.80711			−0.903554	−	0.428475i	$-0.859051\pi$
$383$	−35.3658			−1.80711			−0.903554	+	0.428475i	$0.859051\pi$
$384$		0			0
$385$	36.2972	+	27.5561i	1.84988	+	1.40439i
$386$		0			0
$387$		0			0
$388$		0			0
$389$		−	19.7757i		−	1.00267i	−0.865254	−	0.501334i	$-0.832843\pi$
$389$		−	19.7757i		−	1.00267i	0.865254	−	0.501334i	$-0.167157\pi$
$390$		0			0
$391$	35.9098	+	10.2237i	1.81604	+	0.517035i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−26.0554	+	3.56684i	−1.31099	+	0.179467i
$396$		0			0
$397$	8.17926	+	8.17926i	0.410505	+	0.410505i	0.881914	−	0.471409i	$-0.156255\pi$
$397$	8.17926	+	8.17926i	0.410505	+	0.410505i	−0.471409	+	0.881914i	$0.656255\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	7.82988	−	7.82988i	0.391006	−	0.391006i	−0.484040	−	0.875046i	$-0.660831\pi$
$401$	7.82988	−	7.82988i	0.391006	−	0.391006i	0.875046	+	0.484040i	$0.160831\pi$
$402$		0			0
$403$	4.32816	−	4.32816i	0.215601	−	0.215601i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	36.4505i		−	1.80678i
$408$		0			0
$409$	−7.46124			−0.368935			−0.184467	−	0.982839i	$-0.559056\pi$
$409$	−7.46124			−0.368935			−0.184467	+	0.982839i	$0.559056\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	8.73101	−	8.73101i	0.429625	−	0.429625i
$414$		0			0
$415$	−7.12345	−	5.40797i	−0.349676	−	0.265467i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−4.12125	+	4.12125i	−0.201336	+	0.201336i	−0.800572	−	0.599236i	$-0.795471\pi$
$419$	−4.12125	+	4.12125i	−0.201336	+	0.201336i	0.599236	+	0.800572i	$0.295471\pi$
$420$		0			0
$421$	23.7970			1.15980			0.579899	−	0.814689i	$-0.303092\pi$
$421$	23.7970			1.15980			0.579899	+	0.814689i	$0.303092\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	14.5004	+	14.6540i	0.703371	+	0.710823i
$426$		0			0
$427$	−5.17701			−0.250533
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−22.9770	+	22.9770i	−1.10676	+	1.10676i	−0.113192	+	0.993573i	$0.536107\pi$
$431$	−22.9770	+	22.9770i	−1.10676	+	1.10676i	−0.993573	+	0.113192i	$0.963893\pi$
$432$		0			0
$433$	3.63235			0.174560			0.0872798	−	0.996184i	$-0.472183\pi$
$433$	3.63235			0.174560			0.0872798	+	0.996184i	$0.472183\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−30.1720	+	30.1720i	−1.44332	+	1.44332i
$438$		0			0
$439$	−22.3229	−	22.3229i	−1.06541	−	1.06541i	−0.997705	−	0.0677083i	$-0.978431\pi$
$439$	−22.3229	−	22.3229i	−1.06541	−	1.06541i	−0.0677083	−	0.997705i	$-0.521569\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		−	36.7744i		−	1.74720i	−0.486642	−	0.873601i	$-0.661779\pi$
$443$		−	36.7744i		−	1.74720i	0.486642	−	0.873601i	$-0.338221\pi$
$444$		0			0
$445$	24.8682	+	18.8794i	1.17886	+	0.894968i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−8.75527	+	8.75527i	−0.413187	+	0.413187i	−0.882847	−	0.469660i	$-0.844376\pi$
$449$	−8.75527	+	8.75527i	−0.413187	+	0.413187i	0.469660	+	0.882847i	$0.344376\pi$
$450$		0			0
$451$			23.2165i			1.09322i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−21.0016	+	2.87501i	−0.984571	+	0.134782i
$456$		0			0
$457$	−19.8700			−0.929479			−0.464739	−	0.885447i	$-0.653852\pi$
$457$	−19.8700			−0.929479			−0.464739	+	0.885447i	$0.653852\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	18.4951i		−	0.861402i	−0.902495	−	0.430701i	$-0.858266\pi$
$461$		−	18.4951i		−	0.861402i	0.902495	−	0.430701i	$-0.141734\pi$
$462$		0			0
$463$			39.0259i			1.81369i	0.421465	+	0.906845i	$0.361516\pi$
$463$			39.0259i			1.81369i	−0.421465	+	0.906845i	$0.638484\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−16.7619			−0.775650			−0.387825	−	0.921733i	$-0.626773\pi$
$467$	−16.7619			−0.775650			−0.387825	+	0.921733i	$0.626773\pi$
$468$		0			0
$469$	3.30420	−	3.30420i	0.152574	−	0.152574i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−9.59168	+	9.59168i	−0.441026	+	0.441026i
$474$		0			0
$475$	−22.6933	+	6.33184i	−1.04124	+	0.290525i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−15.7464	−	15.7464i	−0.719470	−	0.719470i	0.249027	−	0.968497i	$-0.419889\pi$
$479$	−15.7464	−	15.7464i	−0.719470	−	0.719470i	−0.968497	+	0.249027i	$0.919889\pi$
$480$		0			0
$481$	11.9887	+	11.9887i	0.546638	+	0.546638i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	33.7672	−	4.62254i	1.53329	−	0.209899i
$486$		0			0
$487$	−8.81863	−	8.81863i	−0.399610	−	0.399610i	0.478485	−	0.878096i	$-0.341186\pi$
$487$	−8.81863	−	8.81863i	−0.399610	−	0.399610i	−0.878096	+	0.478485i	$0.841186\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	28.9417i		−	1.30612i	−0.757307	−	0.653060i	$-0.773485\pi$
$491$		−	28.9417i		−	1.30612i	0.757307	−	0.653060i	$-0.226515\pi$
$492$		0			0
$493$	−38.3620	−	10.9219i	−1.72774	−	0.491896i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	30.7778i		−	1.38057i
$498$		0			0
$499$	−16.9409	+	16.9409i	−0.758380	+	0.758380i	−0.976027	−	0.217647i	$-0.930162\pi$
$499$	−16.9409	+	16.9409i	−0.758380	+	0.758380i	0.217647	+	0.976027i	$0.430162\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−30.4137	−	30.4137i	−1.35608	−	1.35608i	−0.878693	−	0.477387i	$-0.841584\pi$
$503$	−30.4137	−	30.4137i	−1.35608	−	1.35608i	−0.477387	−	0.878693i	$-0.658416\pi$
$504$		0			0
$505$	−24.6626	−	18.7233i	−1.09747	−	0.833178i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−7.44875			−0.330160			−0.165080	−	0.986280i	$-0.552788\pi$
$509$	−7.44875			−0.330160			−0.165080	+	0.986280i	$0.552788\pi$
$510$		0			0
$511$	−4.75348			−0.210281
$512$		0			0
$513$		0			0
$514$		0			0
$515$	4.76106	−	6.27133i	0.209797	−	0.276348i
$516$		0			0
$517$	−44.7341	−	44.7341i	−1.96740	−	1.96740i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	6.53353	−	6.53353i	0.286239	−	0.286239i	−0.549352	−	0.835591i	$-0.685125\pi$
$521$	6.53353	−	6.53353i	0.286239	−	0.286239i	0.835591	+	0.549352i	$0.185125\pi$
$522$		0			0
$523$			9.89276i			0.432580i	0.976329	+	0.216290i	$0.0693957\pi$
$523$			9.89276i			0.432580i	−0.976329	+	0.216290i	$0.930604\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−8.65237	−	2.46338i	−0.376903	−	0.107306i
$528$		0			0
$529$			59.0023i			2.56532i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−7.63600	−	7.63600i	−0.330752	−	0.330752i
$534$		0			0
$535$	−27.4929	+	3.76363i	−1.18862	+	0.162716i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	18.8459	+	18.8459i	0.811750	+	0.811750i
$540$		0			0
$541$	3.93386	+	3.93386i	0.169130	+	0.169130i	0.786597	−	0.617467i	$-0.211841\pi$
$541$	3.93386	+	3.93386i	0.169130	+	0.169130i	−0.617467	+	0.786597i	$0.711841\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−26.0811	+	3.57036i	−1.11719	+	0.152937i
$546$		0			0
$547$	15.4239	−	15.4239i	0.659477	−	0.659477i	−0.295779	−	0.955256i	$-0.595579\pi$
$547$	15.4239	−	15.4239i	0.659477	−	0.659477i	0.955256	+	0.295779i	$0.0955792\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	32.2324	−	32.2324i	1.37315	−	1.37315i
$552$		0			0
$553$	−39.7429			−1.69004
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	16.8327i		−	0.713224i	−0.934252	−	0.356612i	$-0.883932\pi$
$557$		−	16.8327i		−	0.713224i	0.934252	−	0.356612i	$-0.116068\pi$
$558$		0			0
$559$		−	6.30948i		−	0.266863i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−37.2966			−1.57187			−0.785933	−	0.618311i	$-0.787817\pi$
$563$	−37.2966			−1.57187			−0.785933	+	0.618311i	$0.787817\pi$
$564$		0			0
$565$	3.14463	+	22.9712i	0.132296	+	0.966407i
$566$		0			0
$567$		0			0
$568$		0			0
$569$			28.6070i			1.19927i	0.800275	+	0.599633i	$0.204687\pi$
$569$			28.6070i			1.19927i	−0.800275	+	0.599633i	$0.795313\pi$
$570$		0			0
$571$	−14.0413	+	14.0413i	−0.587611	+	0.587611i	−0.936984	−	0.349373i	$-0.886395\pi$
$571$	−14.0413	+	14.0413i	−0.587611	+	0.587611i	0.349373	+	0.936984i	$0.386395\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−22.2338	+	39.4426i	−0.927212	+	1.64487i
$576$		0			0
$577$		−	36.5537i		−	1.52175i	−0.648899	−	0.760874i	$-0.724770\pi$
$577$		−	36.5537i		−	1.52175i	0.648899	−	0.760874i	$-0.275230\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−9.55724	−	9.55724i	−0.396501	−	0.396501i
$582$		0			0
$583$	6.57656	−	6.57656i	0.272373	−	0.272373i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	20.8032			0.858638			0.429319	−	0.903153i	$-0.358754\pi$
$587$	20.8032			0.858638			0.429319	+	0.903153i	$0.358754\pi$
$588$		0			0
$589$	7.26987	−	7.26987i	0.299550	−	0.299550i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−0.498805			−0.0204835			−0.0102417	−	0.999948i	$-0.503260\pi$
$593$	−0.498805			−0.0204835			−0.0102417	+	0.999948i	$0.503260\pi$
$594$		0			0
$595$	18.7073	+	24.9130i	0.766925	+	1.02133i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	26.1051			1.06662			0.533312	−	0.845919i	$-0.320947\pi$
$599$	26.1051			1.06662			0.533312	+	0.845919i	$0.320947\pi$
$600$		0			0
$601$	28.2569	−	28.2569i	1.15262	−	1.15262i	0.166598	−	0.986025i	$-0.446722\pi$
$601$	28.2569	−	28.2569i	1.15262	−	1.15262i	0.986025	−	0.166598i	$-0.0532781\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	34.3086	−	45.1917i	1.39484	−	1.83730i
$606$		0			0
$607$	−1.13156	+	1.13156i	−0.0459285	+	0.0459285i	−0.729698	−	0.683770i	$-0.760339\pi$
$607$	−1.13156	+	1.13156i	−0.0459285	+	0.0459285i	0.683770	+	0.729698i	$0.260339\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	29.4265			1.19047
$612$		0			0
$613$		−	47.5168i		−	1.91918i	−0.281395	−	0.959592i	$-0.590797\pi$
$613$		−	47.5168i		−	1.91918i	0.281395	−	0.959592i	$-0.409203\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	24.9113	−	24.9113i	1.00289	−	1.00289i	0.00289372	−	0.999996i	$-0.499079\pi$
$617$	24.9113	−	24.9113i	1.00289	−	1.00289i	0.999996	−	0.00289372i	$-0.000921101\pi$
$618$		0			0
$619$	25.4359	−	25.4359i	1.02235	−	1.02235i	0.0226092	−	0.999744i	$-0.492803\pi$
$619$	25.4359	−	25.4359i	1.02235	−	1.02235i	0.999744	−	0.0226092i	$-0.00719734\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	33.3646	+	33.3646i	1.33673	+	1.33673i
$624$		0			0
$625$	−21.3886	+	12.9433i	−0.855544	+	0.517730i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	6.82338	−	23.9665i	0.272066	−	0.955606i
$630$		0			0
$631$			17.7721i			0.707497i	0.935340	+	0.353749i	$0.115093\pi$
$631$			17.7721i			0.707497i	−0.935340	+	0.353749i	$0.884907\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−27.4619	−	20.8485i	−1.08979	−	0.827347i
$636$		0			0
$637$	−12.3970			−0.491186
$638$		0			0
$639$		0			0
$640$		0			0
$641$	4.52009	+	4.52009i	0.178533	+	0.178533i	0.790716	−	0.612183i	$-0.209708\pi$
$641$	4.52009	+	4.52009i	0.178533	+	0.178533i	−0.612183	+	0.790716i	$0.709708\pi$
$642$		0			0
$643$	−13.7299	+	13.7299i	−0.541453	+	0.541453i	−0.923955	−	0.382502i	$-0.875063\pi$
$643$	−13.7299	+	13.7299i	−0.541453	+	0.541453i	0.382502	+	0.923955i	$0.375063\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	18.1216i		−	0.712432i	−0.934404	−	0.356216i	$-0.884067\pi$
$647$		−	18.1216i		−	0.712432i	0.934404	−	0.356216i	$-0.115933\pi$
$648$		0			0
$649$	−15.5829	−	15.5829i	−0.611682	−	0.611682i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−11.8153	−	11.8153i	−0.462370	−	0.462370i	0.437062	−	0.899432i	$-0.356019\pi$
$653$	−11.8153	−	11.8153i	−0.462370	−	0.462370i	−0.899432	+	0.437062i	$0.856019\pi$
$654$		0			0
$655$	−3.05466	−	22.3140i	−0.119356	−	0.871880i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−15.5043			−0.603960			−0.301980	−	0.953314i	$-0.597648\pi$
$659$	−15.5043			−0.603960			−0.301980	+	0.953314i	$0.597648\pi$
$660$		0			0
$661$		−	24.3150i		−	0.945743i	−0.881131	−	0.472872i	$-0.843217\pi$
$661$		−	24.3150i		−	0.945743i	0.881131	−	0.472872i	$-0.156783\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−35.2757	+	4.82905i	−1.36793	+	0.187263i
$666$		0			0
$667$		−	87.6019i		−	3.39196i
$668$		0			0
$669$		0			0
$670$		0			0
$671$			9.23981i			0.356699i
$672$		0			0
$673$	27.9718	+	27.9718i	1.07823	+	1.07823i	0.996668	+	0.0815665i	$0.0259923\pi$
$673$	27.9718	+	27.9718i	1.07823	+	1.07823i	0.0815665	+	0.996668i	$0.474008\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−16.7842	+	16.7842i	−0.645069	+	0.645069i	−0.951797	−	0.306728i	$-0.900766\pi$
$677$	−16.7842	+	16.7842i	−0.645069	+	0.645069i	0.306728	+	0.951797i	$0.400766\pi$
$678$		0			0
$679$	51.5059			1.97661
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−26.0516	+	26.0516i	−0.996838	+	0.996838i	−0.999995	−	0.00315743i	$-0.998995\pi$
$683$	−26.0516	+	26.0516i	−0.996838	+	0.996838i	0.00315743	+	0.999995i	$0.498995\pi$
$684$		0			0
$685$	−7.29767	+	9.61259i	−0.278830	+	0.367278i
$686$		0			0
$687$		0			0
$688$		0			0
$689$			4.32611i			0.164812i
$690$		0			0
$691$	−4.56148	+	4.56148i	−0.173527	+	0.173527i	−0.788527	−	0.615000i	$-0.789156\pi$
$691$	−4.56148	+	4.56148i	−0.173527	+	0.173527i	0.615000	+	0.788527i	$0.289156\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−2.31338	−	16.8990i	−0.0877514	−	0.641015i
$696$		0			0
$697$	−4.34603	+	15.2650i	−0.164618	+	0.578204i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	18.3026			0.691280			0.345640	−	0.938367i	$-0.387662\pi$
$701$	18.3026			0.691280			0.345640	+	0.938367i	$0.387662\pi$
$702$		0			0
$703$	20.1370	+	20.1370i	0.759483	+	0.759483i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−33.0889	−	33.0889i	−1.24443	−	1.24443i
$708$		0			0
$709$	17.8191	+	17.8191i	0.669209	+	0.669209i	0.957533	−	0.288324i	$-0.0930981\pi$
$709$	17.8191	+	17.8191i	0.669209	+	0.669209i	−0.288324	+	0.957533i	$0.593098\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	19.7582i		−	0.739951i
$714$		0			0
$715$	5.13124	+	37.4832i	0.191898	+	1.40179i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	9.05095	+	9.05095i	0.337544	+	0.337544i	0.855442	−	0.517899i	$-0.173286\pi$
$719$	9.05095	+	9.05095i	0.337544	+	0.337544i	−0.517899	+	0.855442i	$0.673286\pi$
$720$		0			0
$721$	8.41399	−	8.41399i	0.313353	−	0.313353i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	23.7520	−	42.1360i	0.882129	−	1.56489i
$726$		0			0
$727$		−	12.9739i		−	0.481177i	−0.970627	−	0.240588i	$-0.922660\pi$
$727$		−	12.9739i		−	0.481177i	0.970627	−	0.240588i	$-0.0773404\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−8.10212	+	4.51108i	−0.299668	+	0.166848i
$732$		0			0
$733$	30.8931			1.14106			0.570531	−	0.821276i	$-0.306737\pi$
$733$	30.8931			1.14106			0.570531	+	0.821276i	$0.306737\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−5.89726	−	5.89726i	−0.217228	−	0.217228i
$738$		0			0
$739$		−	26.3986i		−	0.971087i	−0.874213	−	0.485543i	$-0.838622\pi$
$739$		−	26.3986i		−	0.971087i	0.874213	−	0.485543i	$-0.161378\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−7.13107	+	7.13107i	−0.261614	+	0.261614i	−0.825709	−	0.564096i	$-0.809225\pi$
$743$	−7.13107	+	7.13107i	−0.261614	+	0.261614i	0.564096	+	0.825709i	$0.309225\pi$
$744$		0			0
$745$	−0.514365	−	0.390495i	−0.0188449	−	0.0143066i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−41.9356			−1.53230
$750$		0			0
$751$	−4.06370	−	4.06370i	−0.148287	−	0.148287i	0.629066	−	0.777352i	$-0.283438\pi$
$751$	−4.06370	−	4.06370i	−0.148287	−	0.148287i	−0.777352	+	0.629066i	$0.783438\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−0.471855	+	0.621534i	−0.0171726	+	0.0226199i
$756$		0			0
$757$	−8.51336			−0.309423			−0.154712	−	0.987960i	$-0.549445\pi$
$757$	−8.51336			−0.309423			−0.154712	+	0.987960i	$0.549445\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−54.1864			−1.96426			−0.982128	−	0.188216i	$-0.939729\pi$
$761$	−54.1864			−1.96426			−0.982128	+	0.188216i	$0.939729\pi$
$762$		0			0
$763$	−39.7821			−1.44021
$764$		0			0
$765$		0			0
$766$		0			0
$767$	10.2506			0.370126
$768$		0			0
$769$	21.0180			0.757927			0.378964	−	0.925412i	$-0.376281\pi$
$769$	21.0180			0.757927			0.378964	+	0.925412i	$0.376281\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	44.8315			1.61248			0.806239	−	0.591590i	$-0.201499\pi$
$773$	44.8315			1.61248			0.806239	+	0.591590i	$0.201499\pi$
$774$		0			0
$775$	5.35716	−	9.50358i	0.192435	−	0.341379i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−12.8259	−	12.8259i	−0.459537	−	0.459537i
$780$		0			0
$781$	−54.9315			−1.96560
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−17.1135	+	22.5421i	−0.610806	+	0.804563i
$786$		0			0
$787$	−16.9154	+	16.9154i	−0.602970	+	0.602970i	−0.941100	−	0.338130i	$-0.890206\pi$
$787$	−16.9154	+	16.9154i	−0.602970	+	0.602970i	0.338130	+	0.941100i	$0.390206\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			35.0386i			1.24583i
$792$		0			0
$793$	−3.03901	−	3.03901i	−0.107918	−	0.107918i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−7.89001			−0.279478			−0.139739	−	0.990188i	$-0.544626\pi$
$797$	−7.89001			−0.279478			−0.139739	+	0.990188i	$0.544626\pi$
$798$		0			0
$799$	−21.0390	−	37.7871i	−0.744305	−	1.33681i
$800$		0			0
$801$		0			0
$802$		0			0
$803$			8.48389i			0.299390i
$804$		0			0
$805$	−41.3743	+	54.4988i	−1.45825	+	1.92083i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	34.2754	−	34.2754i	1.20506	−	1.20506i	0.232452	−	0.972608i	$-0.425325\pi$
$809$	34.2754	−	34.2754i	1.20506	−	1.20506i	0.972608	−	0.232452i	$-0.0746747\pi$
$810$		0			0
$811$	6.17565	+	6.17565i	0.216856	+	0.216856i	0.807172	−	0.590316i	$-0.200997\pi$
$811$	6.17565	+	6.17565i	0.216856	+	0.216856i	−0.590316	+	0.807172i	$0.700997\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	4.35851	+	31.8385i	0.152672	+	1.11525i
$816$		0			0
$817$		−	10.5978i		−	0.370771i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−16.7192	−	16.7192i	−0.583505	−	0.583505i	0.352360	−	0.935865i	$-0.385379\pi$
$821$	−16.7192	−	16.7192i	−0.583505	−	0.583505i	−0.935865	+	0.352360i	$0.885379\pi$
$822$		0			0
$823$	18.3320	+	18.3320i	0.639013	+	0.639013i	0.950312	−	0.311299i	$-0.100764\pi$
$823$	18.3320	+	18.3320i	0.639013	+	0.639013i	−0.311299	+	0.950312i	$0.600764\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−25.2223	−	25.2223i	−0.877065	−	0.877065i	0.116165	−	0.993230i	$-0.462940\pi$
$827$	−25.2223	−	25.2223i	−0.877065	−	0.877065i	−0.993230	+	0.116165i	$0.962940\pi$
$828$		0			0
$829$	−6.97218			−0.242154			−0.121077	−	0.992643i	$-0.538635\pi$
$829$	−6.97218			−0.242154			−0.121077	+	0.992643i	$0.538635\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	8.86344	+	15.9192i	0.307100	+	0.551567i
$834$		0			0
$835$	−13.2321	+	1.81140i	−0.457915	+	0.0626861i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−34.0874	+	34.0874i	−1.17683	+	1.17683i	−0.196278	+	0.980548i	$0.562885\pi$
$839$	−34.0874	+	34.0874i	−1.17683	+	1.17683i	−0.980548	+	0.196278i	$0.937115\pi$
$840$		0			0
$841$			64.5840i			2.22704i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	9.13665	+	6.93634i	0.314310	+	0.238618i
$846$		0			0
$847$	60.6319	−	60.6319i	2.08334	−	2.08334i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	54.7289			1.87608
$852$		0			0
$853$	14.3998	−	14.3998i	0.493041	−	0.493041i	−0.416222	−	0.909263i	$-0.636646\pi$
$853$	14.3998	−	14.3998i	0.493041	−	0.493041i	0.909263	+	0.416222i	$0.136646\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	1.90013	+	1.90013i	0.0649072	+	0.0649072i	0.738815	−	0.673908i	$-0.235386\pi$
$857$	1.90013	+	1.90013i	0.0649072	+	0.0649072i	−0.673908	+	0.738815i	$0.735386\pi$
$858$		0			0
$859$			13.2517i			0.452143i	0.974111	+	0.226072i	$0.0725883\pi$
$859$			13.2517i			0.452143i	−0.974111	+	0.226072i	$0.927412\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	37.8899i		−	1.28979i	−0.764273	−	0.644893i	$-0.776902\pi$
$863$		−	37.8899i		−	1.28979i	0.764273	−	0.644893i	$-0.223098\pi$
$864$		0			0
$865$	−2.53546	+	0.347091i	−0.0862084	+	0.0118015i
$866$		0			0
$867$		0			0
$868$		0			0
$869$			70.9322i			2.40621i
$870$		0			0
$871$	3.87926			0.131444
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−34.6774	+	14.9955i	−1.17231	+	0.506941i
$876$		0			0
$877$	6.90096	+	6.90096i	0.233029	+	0.233029i	0.813956	−	0.580927i	$-0.197310\pi$
$877$	6.90096	+	6.90096i	0.233029	+	0.233029i	−0.580927	+	0.813956i	$0.697310\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−4.85040	−	4.85040i	−0.163414	−	0.163414i	0.620663	−	0.784077i	$-0.286863\pi$
$881$	−4.85040	−	4.85040i	−0.163414	−	0.163414i	−0.784077	+	0.620663i	$0.786863\pi$
$882$		0			0
$883$			37.4138i			1.25908i	0.776970	+	0.629538i	$0.216756\pi$
$883$			37.4138i			1.25908i	−0.776970	+	0.629538i	$0.783244\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	13.1504	−	13.1504i	0.441546	−	0.441546i	−0.450985	−	0.892531i	$-0.648927\pi$
$887$	13.1504	−	13.1504i	0.441546	−	0.441546i	0.892531	+	0.450985i	$0.148927\pi$
$888$		0			0
$889$	−36.8445	−	36.8445i	−1.23573	−	1.23573i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	49.4267			1.65400
$894$		0			0
$895$	30.0562	−	39.5904i	1.00467	−	1.32336i
$896$		0			0
$897$		0			0
$898$		0			0
$899$			21.1074i			0.703973i
$900$		0			0
$901$	5.55524	−	3.09303i	0.185072	−	0.103044i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−11.7920	+	1.61426i	−0.391979	+	0.0536597i
$906$		0			0
$907$	−12.6188	−	12.6188i	−0.419001	−	0.419001i	0.465859	−	0.884859i	$-0.345746\pi$
$907$	−12.6188	−	12.6188i	−0.419001	−	0.419001i	−0.884859	+	0.465859i	$0.845746\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−4.01480	+	4.01480i	−0.133016	+	0.133016i	−0.770480	−	0.637464i	$-0.779983\pi$
$911$	−4.01480	+	4.01480i	−0.133016	+	0.133016i	0.637464	+	0.770480i	$0.279983\pi$
$912$		0			0
$913$	−17.0575	+	17.0575i	−0.564522	+	0.564522i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		−	34.0361i		−	1.12397i
$918$		0			0
$919$	13.4225			0.442768			0.221384	−	0.975187i	$-0.428943\pi$
$919$	13.4225			0.442768			0.221384	+	0.975187i	$0.428943\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	18.0672	−	18.0672i	0.594689	−	0.594689i
$924$		0			0
$925$	26.3243	+	14.8390i	0.865536	+	0.487903i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−26.6853	+	26.6853i	−0.875517	+	0.875517i	−0.993067	−	0.117550i	$-0.962496\pi$
$929$	−26.6853	+	26.6853i	−0.875517	+	0.875517i	0.117550	+	0.993067i	$0.462496\pi$
$930$		0			0
$931$	−20.8228			−0.682440
$932$		0			0
$933$		0			0
$934$		0			0
$935$	44.4642	−	33.3884i	1.45413	−	1.09192i
$936$		0			0
$937$	45.0578			1.47198			0.735988	−	0.676995i	$-0.236718\pi$
$937$	45.0578			1.47198			0.735988	+	0.676995i	$0.236718\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−35.4120	+	35.4120i	−1.15440	+	1.15440i	−0.168738	+	0.985661i	$0.553969\pi$
$941$	−35.4120	+	35.4120i	−1.15440	+	1.15440i	−0.985661	+	0.168738i	$0.946031\pi$
$942$		0			0
$943$	−34.8586			−1.13515
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−15.5934	+	15.5934i	−0.506718	+	0.506718i	−0.913518	−	0.406799i	$-0.866645\pi$
$947$	−15.5934	+	15.5934i	−0.506718	+	0.506718i	0.406799	+	0.913518i	$0.366645\pi$
$948$		0			0
$949$	−2.79039	−	2.79039i	−0.0905798	−	0.0905798i
$950$		0			0
$951$		0			0
$952$		0			0
$953$			36.0170i			1.16671i	0.812219	+	0.583353i	$0.198259\pi$
$953$			36.0170i			1.16671i	−0.812219	+	0.583353i	$0.801741\pi$
$954$		0			0
$955$	18.5273	+	14.0656i	0.599531	+	0.455151i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−12.8968	+	12.8968i	−0.416460	+	0.416460i
$960$		0			0
$961$		−	26.2393i		−	0.846429i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	5.08763	+	37.1646i	0.163777	+	1.19637i
$966$		0			0
$967$	−19.0087			−0.611279			−0.305639	−	0.952147i	$-0.598870\pi$
$967$	−19.0087			−0.611279			−0.305639	+	0.952147i	$0.598870\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			0.994297i			0.0319085i	0.999873	+	0.0159543i	$0.00507861\pi$
$971$			0.994297i			0.0319085i	−0.999873	+	0.0159543i	$0.994921\pi$
$972$		0			0
$973$		−	25.7764i		−	0.826354i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	23.6574			0.756868			0.378434	−	0.925628i	$-0.376463\pi$
$977$	23.6574			0.756868			0.378434	+	0.925628i	$0.376463\pi$
$978$		0			0
$979$	59.5484	−	59.5484i	1.90317	−	1.90317i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−32.4392	+	32.4392i	−1.03465	+	1.03465i	−0.0352715	+	0.999378i	$0.511230\pi$
$983$	−32.4392	+	32.4392i	−1.03465	+	1.03465i	−0.999378	+	0.0352715i	$0.988770\pi$
$984$		0			0
$985$	6.22668	+	45.4852i	0.198399	+	1.44928i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−14.4015	−	14.4015i	−0.457941	−	0.457941i
$990$		0			0
$991$	−36.2049	−	36.2049i	−1.15009	−	1.15009i	−0.986535	−	0.163552i	$-0.947705\pi$
$991$	−36.2049	−	36.2049i	−1.15009	−	1.15009i	−0.163552	−	0.986535i	$-0.552295\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	4.79050	+	34.9941i	0.151869	+	1.10939i
$996$		0			0
$997$	9.20588	+	9.20588i	0.291553	+	0.291553i	0.837694	−	0.546140i	$-0.183903\pi$
$997$	9.20588	+	9.20588i	0.291553	+	0.291553i	−0.546140	+	0.837694i	$0.683903\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3060.2.z.g.829.17		40
3.2	odd	2		1020.2.y.a.829.18	yes	40
5.4	even	2	inner	3060.2.z.g.829.15		40
15.14	odd	2		1020.2.y.a.829.3	yes	40
17.4	even	4	inner	3060.2.z.g.2809.15		40
51.38	odd	4		1020.2.y.a.769.3	✓	40
85.4	even	4	inner	3060.2.z.g.2809.17		40
255.89	odd	4		1020.2.y.a.769.18	yes	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1020.2.y.a.769.3	✓	40	51.38	odd	4
1020.2.y.a.769.18	yes	40	255.89	odd	4
1020.2.y.a.829.3	yes	40	15.14	odd	2
1020.2.y.a.829.18	yes	40	3.2	odd	2
3060.2.z.g.829.15		40	5.4	even	2	inner
3060.2.z.g.829.17		40	1.1	even	1	trivial
3060.2.z.g.2809.15		40	17.4	even	4	inner
3060.2.z.g.2809.17		40	85.4	even	4	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:	\( N \)	\(=\)	\( 3060 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3060.z (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(1.78098 + 1.35208i) q^{5} +(2.38946 + 2.38946i) q^{7} +(4.26466 - 4.26466i) q^{11} +2.80533i q^{13} +(3.60237 - 2.00572i) q^{17} +4.71202i q^{19} +(6.40321 + 6.40321i) q^{23} +(1.34376 + 4.81605i) q^{25} +(-6.84047 - 6.84047i) q^{29} +(-1.54284 - 1.54284i) q^{31} +(1.02484 + 7.48633i) q^{35} +(4.27355 - 4.27355i) q^{37} +(-2.72196 + 2.72196i) q^{41} -2.24911 q^{43} -10.4895i q^{47} +4.41908i q^{49} +1.54211 q^{53} +(13.3614 - 1.82911i) q^{55} -3.65396i q^{59} +(-1.08330 + 1.08330i) q^{61} +(-3.79303 + 4.99623i) q^{65} -1.38282i q^{67} +(-6.44031 - 6.44031i) q^{71} +(-0.994674 + 0.994674i) q^{73} +20.3805 q^{77} +(-8.31627 + 8.31627i) q^{79} -3.99974 q^{83} +(9.12764 + 1.29855i) q^{85} +13.9632 q^{89} +(-6.70323 + 6.70323i) q^{91} +(-6.37102 + 8.39200i) q^{95} +(10.7777 - 10.7777i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q - 4 q^{5} + 8 q^{11} - 24 q^{29} - 16 q^{31} - 8 q^{35} + 8 q^{41} + 28 q^{55} + 16 q^{61} - 56 q^{71} + 16 q^{79} - 40 q^{85} - 32 q^{89} + 64 q^{91} - 64 q^{95}+O(q^{100}) \) 40 * q - 4 * q^5 + 8 * q^11 - 24 * q^29 - 16 * q^31 - 8 * q^35 + 8 * q^41 + 28 * q^55 + 16 * q^61 - 56 * q^71 + 16 * q^79 - 40 * q^85 - 32 * q^89 + 64 * q^91 - 64 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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