LMFDB - Embedded newform 1404.2.cj.a.125.9

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1404,2,Mod(125,1404)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1404, base_ring=CyclotomicField(12)) chi = DirichletCharacter(H, H._module([0, 10, 3])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1404.125"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 1404 = 2^{2} \cdot 3^{3} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1404.cj (of order $12$, degree $4$, 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:	$11.2109964438$
Analytic rank:	$0$
Dimension:	$56$
Relative dimension:	$14$ over $\Q(\zeta_{12})$
Twist minimal:	no (minimal twist has level 468)
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			125.9
Character	$\chi$	$=$	1404.125
Dual form			1404.2.cj.a.629.9

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.154104 + 0.575123i) q^{5} +(0.0907757 + 0.0243233i) q^{7} +(0.116136 - 0.433424i) q^{11} +(-3.43468 - 1.09680i) q^{13} -5.51273 q^{17} +(-4.88963 - 4.88963i) q^{19} +(-4.18646 + 7.25117i) q^{23} +(4.02311 - 2.32274i) q^{25} +(-5.25947 + 3.03656i) q^{29} +(1.67875 - 0.449819i) q^{31} +0.0559555i q^{35} +(6.26928 - 6.26928i) q^{37} +(1.52195 + 5.68000i) q^{41} +(-3.22894 + 1.86423i) q^{43} +(3.27565 - 12.2249i) q^{47} +(-6.05453 - 3.49558i) q^{49} -0.788507i q^{53} +0.267169 q^{55} +(-7.07279 + 1.89515i) q^{59} +(-0.455275 - 0.788559i) q^{61} +(0.101500 - 2.14439i) q^{65} +(-9.59938 + 2.57215i) q^{67} +(-0.624421 + 0.624421i) q^{71} +(-3.61085 + 3.61085i) q^{73} +(0.0210846 - 0.0365196i) q^{77} +(-4.09126 - 7.08627i) q^{79} +(-9.78070 - 2.62073i) q^{83} +(-0.849533 - 3.17050i) q^{85} +(4.46678 + 4.46678i) q^{89} +(-0.285108 - 0.183106i) q^{91} +(2.05863 - 3.56565i) q^{95} +(-4.07242 + 15.1985i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 56 q - 2 q^{7} + 12 q^{11} + 4 q^{19} - 4 q^{31} - 4 q^{37} - 24 q^{41} - 66 q^{47} + 78 q^{65} - 14 q^{67} + 28 q^{73} - 24 q^{79} + 78 q^{83} + 36 q^{85} - 8 q^{91} + 26 q^{97}+O(q^{100}) $ 56 * q - 2 * q^7 + 12 * q^11 + 4 * q^19 - 4 * q^31 - 4 * q^37 - 24 * q^41 - 66 * q^47 + 78 * q^65 - 14 * q^67 + 28 * q^73 - 24 * q^79 + 78 * q^83 + 36 * q^85 - 8 * q^91 + 26 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1404\mathbb{Z}\right)^\times$.

$n$	$677$	$703$	$1081$
$\chi(n)$	$e\left(\frac{5}{6}\right)$	$1$	$e\left(\frac{1}{4}\right)$

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
$5$	0.154104	+	0.575123i	0.0689173	+	0.257203i	0.991785	−	0.127913i	$-0.0408280\pi$
$5$	0.154104	+	0.575123i	0.0689173	+	0.257203i	−0.922868	+	0.385116i	$0.874161\pi$
$6$		0			0
$7$	0.0907757	+	0.0243233i	0.0343100	+	0.00919334i	0.275933	−	0.961177i	$-0.411013\pi$
$7$	0.0907757	+	0.0243233i	0.0343100	+	0.00919334i	−0.241623	+	0.970370i	$0.577680\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	0.116136	−	0.433424i	0.0350162	−	0.130682i	−0.946205	−	0.323567i	$-0.895118\pi$
$11$	0.116136	−	0.433424i	0.0350162	−	0.130682i	0.981221	+	0.192885i	$0.0617845\pi$
$12$		0			0
$13$	−3.43468	−	1.09680i	−0.952609	−	0.304199i
$13$	−3.43468	−	1.09680i	−0.952609	−	0.304199i
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−5.51273			−1.33703			−0.668517	−	0.743697i	$-0.733071\pi$
$17$	−5.51273			−1.33703			−0.668517	+	0.743697i	$0.733071\pi$
$18$		0			0
$19$	−4.88963	−	4.88963i	−1.12176	−	1.12176i	−0.991477	−	0.130282i	$-0.958412\pi$
$19$	−4.88963	−	4.88963i	−1.12176	−	1.12176i	−0.130282	−	0.991477i	$-0.541588\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−4.18646	+	7.25117i	−0.872938	+	1.51197i	−0.0139952	+	0.999902i	$0.504455\pi$
$23$	−4.18646	+	7.25117i	−0.872938	+	1.51197i	−0.858943	+	0.512071i	$0.828878\pi$
$24$		0			0
$25$	4.02311	−	2.32274i	0.804622	−	0.464549i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−5.25947	+	3.03656i	−0.976660	+	0.563875i	−0.901260	−	0.433279i	$-0.857357\pi$
$29$	−5.25947	+	3.03656i	−0.976660	+	0.563875i	−0.0753996	+	0.997153i	$0.524023\pi$
$30$		0			0
$31$	1.67875	−	0.449819i	0.301512	−	0.0807899i	−0.104891	−	0.994484i	$-0.533449\pi$
$31$	1.67875	−	0.449819i	0.301512	−	0.0807899i	0.406403	+	0.913694i	$0.366783\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$			0.0559555i			0.00945821i
$36$		0			0
$37$	6.26928	−	6.26928i	1.03066	−	1.03066i	0.0311493	−	0.999515i	$-0.490083\pi$
$37$	6.26928	−	6.26928i	1.03066	−	1.03066i	0.999515	−	0.0311493i	$-0.00991673\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	1.52195	+	5.68000i	0.237689	+	0.887067i	0.976918	+	0.213613i	$0.0685233\pi$
$41$	1.52195	+	5.68000i	0.237689	+	0.887067i	−0.739229	+	0.673454i	$0.764810\pi$
$42$		0			0
$43$	−3.22894	+	1.86423i	−0.492409	+	0.284292i	−0.725573	−	0.688145i	$-0.758425\pi$
$43$	−3.22894	+	1.86423i	−0.492409	+	0.284292i	0.233164	+	0.972437i	$0.425092\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	3.27565	−	12.2249i	0.477803	−	1.78318i	−0.132684	−	0.991158i	$-0.542359\pi$
$47$	3.27565	−	12.2249i	0.477803	−	1.78318i	0.610487	−	0.792027i	$-0.290974\pi$
$48$		0			0
$49$	−6.05453	−	3.49558i	−0.864933	−	0.499369i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	0.788507i		−	0.108310i	−0.998533	−	0.0541549i	$-0.982754\pi$
$53$		−	0.788507i		−	0.108310i	0.998533	−	0.0541549i	$-0.0172465\pi$
$54$		0			0
$55$	0.267169			0.0360251
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−7.07279	+	1.89515i	−0.920799	+	0.246727i	−0.687927	−	0.725780i	$-0.741479\pi$
$59$	−7.07279	+	1.89515i	−0.920799	+	0.246727i	−0.232872	+	0.972507i	$0.574812\pi$
$60$		0			0
$61$	−0.455275	−	0.788559i	−0.0582919	−	0.100965i	0.835407	−	0.549632i	$-0.185232\pi$
$61$	−0.455275	−	0.788559i	−0.0582919	−	0.100965i	−0.893699	+	0.448667i	$0.851899\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	0.101500	−	2.14439i	0.0125895	−	0.265978i
$66$		0			0
$67$	−9.59938	+	2.57215i	−1.17275	+	0.314238i	−0.792047	−	0.610460i	$-0.790985\pi$
$67$	−9.59938	+	2.57215i	−1.17275	+	0.314238i	−0.380703	+	0.924697i	$0.624318\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−0.624421	+	0.624421i	−0.0741051	+	0.0741051i	−0.743188	−	0.669083i	$-0.766687\pi$
$71$	−0.624421	+	0.624421i	−0.0741051	+	0.0741051i	0.669083	+	0.743188i	$0.266687\pi$
$72$		0			0
$73$	−3.61085	+	3.61085i	−0.422618	+	0.422618i	−0.886104	−	0.463486i	$-0.846598\pi$
$73$	−3.61085	+	3.61085i	−0.422618	+	0.422618i	0.463486	+	0.886104i	$0.346598\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	0.0210846	−	0.0365196i	0.00240281	−	0.00416179i
$78$		0			0
$79$	−4.09126	−	7.08627i	−0.460303	−	0.797268i	0.538673	−	0.842515i	$-0.318926\pi$
$79$	−4.09126	−	7.08627i	−0.460303	−	0.797268i	−0.998976	+	0.0452472i	$0.985592\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−9.78070	−	2.62073i	−1.07357	−	0.287662i	−0.321611	−	0.946872i	$-0.604224\pi$
$83$	−9.78070	−	2.62073i	−1.07357	−	0.287662i	−0.751960	+	0.659209i	$0.770891\pi$
$84$		0			0
$85$	−0.849533	−	3.17050i	−0.0921448	−	0.343889i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	4.46678	+	4.46678i	0.473478	+	0.473478i	0.903038	−	0.429560i	$-0.141331\pi$
$89$	4.46678	+	4.46678i	0.473478	+	0.473478i	−0.429560	+	0.903038i	$0.641331\pi$
$90$		0			0
$91$	−0.285108	−	0.183106i	−0.0298874	−	0.0191947i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	2.05863	−	3.56565i	0.211211	−	0.365828i
$96$		0			0
$97$	−4.07242	+	15.1985i	−0.413491	+	1.54317i	0.374347	+	0.927289i	$0.377867\pi$
$97$	−4.07242	+	15.1985i	−0.413491	+	1.54317i	−0.787838	+	0.615882i	$0.788800\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−2.40675	−	4.16862i	−0.239481	−	0.414793i	0.721084	−	0.692847i	$-0.243644\pi$
$101$	−2.40675	−	4.16862i	−0.239481	−	0.414793i	−0.960565	+	0.278054i	$0.910311\pi$
$102$		0			0
$103$	−5.26666	−	3.04071i	−0.518940	−	0.299610i	0.217561	−	0.976047i	$-0.430190\pi$
$103$	−5.26666	−	3.04071i	−0.518940	−	0.299610i	−0.736501	+	0.676437i	$0.763523\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	6.16867i		−	0.596348i	−0.954512	−	0.298174i	$-0.903622\pi$
$107$		−	6.16867i		−	0.596348i	0.954512	−	0.298174i	$-0.0963775\pi$
$108$		0			0
$109$	3.28996	+	3.28996i	0.315121	+	0.315121i	0.846890	−	0.531769i	$-0.178472\pi$
$109$	3.28996	+	3.28996i	0.315121	+	0.315121i	−0.531769	+	0.846890i	$0.678472\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−0.843186	−	0.486814i	−0.0793203	−	0.0457956i	0.459815	−	0.888015i	$-0.347916\pi$
$113$	−0.843186	−	0.486814i	−0.0793203	−	0.0457956i	−0.539136	+	0.842219i	$0.681249\pi$
$114$		0			0
$115$	−4.81547	−	1.29030i	−0.449045	−	0.120321i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−0.500422	−	0.134088i	−0.0458736	−	0.0122918i
$120$		0			0
$121$	9.35191	+	5.39933i	0.850174	+	0.490848i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	4.06094	+	4.06094i	0.363221	+	0.363221i
$126$		0			0
$127$		−	5.76365i		−	0.511441i	−0.966751	−	0.255721i	$-0.917687\pi$
$127$		−	5.76365i		−	0.511441i	0.966751	−	0.255721i	$-0.0823127\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−0.554624	−	0.320212i	−0.0484577	−	0.0279771i	0.475575	−	0.879675i	$-0.342240\pi$
$131$	−0.554624	−	0.320212i	−0.0484577	−	0.0279771i	−0.524033	+	0.851698i	$0.675573\pi$
$132$		0			0
$133$	−0.324928	−	0.562792i	−0.0281748	−	0.0488003i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	1.83310	−	6.84124i	0.156613	−	0.584486i	−0.842349	−	0.538932i	$-0.818828\pi$
$137$	1.83310	−	6.84124i	0.156613	−	0.584486i	0.998962	−	0.0455542i	$-0.0145054\pi$
$138$		0			0
$139$	7.31643	−	12.6724i	0.620571	−	1.07486i	−0.368808	−	0.929506i	$-0.620234\pi$
$139$	7.31643	−	12.6724i	0.620571	−	1.07486i	0.989379	−	0.145355i	$-0.0464326\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−0.874269	+	1.36129i	−0.0731101	+	0.113837i
$144$		0			0
$145$	−2.55690	−	2.55690i	−0.212339	−	0.212339i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	4.89656	+	18.2742i	0.401142	+	1.49708i	0.811061	+	0.584961i	$0.198890\pi$
$149$	4.89656	+	18.2742i	0.401142	+	1.49708i	−0.409919	+	0.912122i	$0.634443\pi$
$150$		0			0
$151$	6.61159	+	1.77157i	0.538043	+	0.144168i	0.517600	−	0.855623i	$-0.326826\pi$
$151$	6.61159	+	1.77157i	0.538043	+	0.144168i	0.0204434	+	0.999791i	$0.493492\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	0.517403	+	0.896168i	0.0415588	+	0.0719820i
$156$		0			0
$157$	−5.25770	+	9.10660i	−0.419610	+	0.726786i	−0.995900	−	0.0904592i	$-0.971167\pi$
$157$	−5.25770	+	9.10660i	−0.419610	+	0.726786i	0.576290	+	0.817245i	$0.304500\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−0.556402	+	0.556402i	−0.0438506	+	0.0438506i
$162$		0			0
$163$	−7.35881	+	7.35881i	−0.576387	+	0.576387i	−0.933906	−	0.357519i	$-0.883623\pi$
$163$	−7.35881	+	7.35881i	−0.576387	+	0.576387i	0.357519	+	0.933906i	$0.383623\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−21.6849	+	5.81045i	−1.67803	+	0.449626i	−0.967257	−	0.253799i	$-0.918320\pi$
$167$	−21.6849	+	5.81045i	−1.67803	+	0.449626i	−0.710770	+	0.703425i	$0.751653\pi$
$168$		0			0
$169$	10.5940	+	7.53434i	0.814927	+	0.579564i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−8.21662	−	14.2316i	−0.624698	−	1.08201i	−0.988599	−	0.150571i	$-0.951889\pi$
$173$	−8.21662	−	14.2316i	−0.624698	−	1.08201i	0.363901	−	0.931438i	$-0.381445\pi$
$174$		0			0
$175$	0.421697	−	0.112993i	0.0318773	−	0.00854150i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	17.4503			1.30430			0.652149	−	0.758090i	$-0.273867\pi$
$179$	17.4503			1.30430			0.652149	+	0.758090i	$0.273867\pi$
$180$		0			0
$181$		−	13.2331i		−	0.983608i	−0.870706	−	0.491804i	$-0.836338\pi$
$181$		−	13.2331i		−	0.983608i	0.870706	−	0.491804i	$-0.163662\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	4.57173	+	2.63949i	0.336120	+	0.194059i
$186$		0			0
$187$	−0.640224	+	2.38935i	−0.0468178	+	0.174727i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−11.2787	+	6.51178i	−0.816100	+	0.471176i	−0.849070	−	0.528281i	$-0.822837\pi$
$191$	−11.2787	+	6.51178i	−0.816100	+	0.471176i	0.0329696	+	0.999456i	$0.489504\pi$
$192$		0			0
$193$	0.663942	+	2.47786i	0.0477916	+	0.178361i	0.985696	−	0.168533i	$-0.0539031\pi$
$193$	0.663942	+	2.47786i	0.0477916	+	0.178361i	−0.937904	+	0.346894i	$0.887236\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	14.0351	−	14.0351i	0.999960	−	0.999960i	−4.02191e−5	−	1.00000i	$-0.500013\pi$
$197$	14.0351	−	14.0351i	0.999960	−	0.999960i	1.00000		4.02191e-5i	$1.28021e-5\pi$
$198$		0			0
$199$			17.2254i			1.22108i	0.791986	+	0.610539i	$0.209047\pi$
$199$			17.2254i			1.22108i	−0.791986	+	0.610539i	$0.790953\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−0.551291	+	0.147718i	−0.0386931	+	0.0103678i
$204$		0			0
$205$	−3.03216	+	1.75062i	−0.211775	+	0.122269i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−2.68715	+	1.55142i	−0.185874	+	0.107314i
$210$		0			0
$211$	10.2869	−	17.8175i	0.708181	−	1.22660i	−0.257351	−	0.966318i	$-0.582850\pi$
$211$	10.2869	−	17.8175i	0.708181	−	1.22660i	0.965531	−	0.260287i	$-0.0838171\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−1.56975	−	1.56975i	−0.107056	−	0.107056i
$216$		0			0
$217$	0.163331			0.0110876
$218$		0			0
$219$		0			0
$220$		0			0
$221$	18.9345	+	6.04638i	1.27367	+	0.406724i
$222$		0			0
$223$	7.37990	−	27.5422i	0.494195	−	1.84436i	−0.0402997	−	0.999188i	$-0.512831\pi$
$223$	7.37990	−	27.5422i	0.494195	−	1.84436i	0.534494	−	0.845172i	$-0.320502\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	15.6985	+	4.20639i	1.04194	+	0.279188i	0.738919	−	0.673794i	$-0.235337\pi$
$227$	15.6985	+	4.20639i	1.04194	+	0.279188i	0.303026	+	0.952982i	$0.402003\pi$
$228$		0			0
$229$	4.60954	+	17.2030i	0.304607	+	1.13681i	0.933283	+	0.359141i	$0.116930\pi$
$229$	4.60954	+	17.2030i	0.304607	+	1.13681i	−0.628677	+	0.777667i	$0.716403\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	17.9589			1.17653			0.588263	−	0.808669i	$-0.299812\pi$
$233$	17.9589			1.17653			0.588263	+	0.808669i	$0.299812\pi$
$234$		0			0
$235$	7.53562			0.491569
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−2.04851	−	7.64515i	−0.132507	−	0.494523i	0.867489	−	0.497457i	$-0.165733\pi$
$239$	−2.04851	−	7.64515i	−0.132507	−	0.494523i	−0.999996	+	0.00293378i	$0.999066\pi$
$240$		0			0
$241$	−11.3575	−	3.04324i	−0.731603	−	0.196032i	−0.126260	−	0.991997i	$-0.540297\pi$
$241$	−11.3575	−	3.04324i	−0.731603	−	0.196032i	−0.605343	+	0.795965i	$0.706964\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	1.07737	−	4.02078i	0.0688304	−	0.256878i
$246$		0			0
$247$	11.4314	+	22.1573i	0.727360	+	1.40983i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	6.78689			0.428385			0.214192	−	0.976792i	$-0.431288\pi$
$251$	6.78689			0.428385			0.214192	+	0.976792i	$0.431288\pi$
$252$		0			0
$253$	2.65663	+	2.65663i	0.167021	+	0.167021i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−7.74724	+	13.4186i	−0.483259	+	0.837030i	−0.999815	−	0.0192238i	$-0.993881\pi$
$257$	−7.74724	+	13.4186i	−0.483259	+	0.837030i	0.516556	+	0.856253i	$0.327214\pi$
$258$		0			0
$259$	0.721588	−	0.416609i	0.0448373	−	0.0258868i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−18.8929	+	10.9078i	−1.16499	+	0.672606i	−0.952494	−	0.304556i	$-0.901492\pi$
$263$	−18.8929	+	10.9078i	−1.16499	+	0.672606i	−0.212494	+	0.977162i	$0.568159\pi$
$264$		0			0
$265$	0.453489	−	0.121512i	0.0278576	−	0.00746442i
$266$		0			0
$267$		0			0
$268$		0			0
$269$		−	1.62378i		−	0.0990035i	−0.998774	−	0.0495017i	$-0.984237\pi$
$269$		−	1.62378i		−	0.0990035i	0.998774	−	0.0495017i	$-0.0157633\pi$
$270$		0			0
$271$	3.57292	−	3.57292i	0.217039	−	0.217039i	−0.590210	−	0.807250i	$-0.700955\pi$
$271$	3.57292	−	3.57292i	0.217039	−	0.217039i	0.807250	+	0.590210i	$0.200955\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−0.539506	−	2.01346i	−0.0325334	−	0.121416i
$276$		0			0
$277$	9.52727	−	5.50057i	0.572438	−	0.330497i	−0.185684	−	0.982609i	$-0.559450\pi$
$277$	9.52727	−	5.50057i	0.572438	−	0.330497i	0.758123	+	0.652112i	$0.226117\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−2.52855	+	9.43668i	−0.150841	+	0.562945i	0.848585	+	0.529059i	$0.177455\pi$
$281$	−2.52855	+	9.43668i	−0.150841	+	0.562945i	−0.999426	+	0.0338861i	$0.989212\pi$
$282$		0			0
$283$	−3.28632	−	1.89736i	−0.195351	−	0.112786i	0.399134	−	0.916893i	$-0.369311\pi$
$283$	−3.28632	−	1.89736i	−0.195351	−	0.112786i	−0.594485	+	0.804106i	$0.702644\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			0.552625i			0.0326204i
$288$		0			0
$289$	13.3902			0.787659
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−12.1275	+	3.24954i	−0.708494	+	0.189840i	−0.595032	−	0.803702i	$-0.702861\pi$
$293$	−12.1275	+	3.24954i	−0.708494	+	0.189840i	−0.113462	+	0.993542i	$0.536194\pi$
$294$		0			0
$295$	−2.17989	−	3.77568i	−0.126918	−	0.219829i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	22.3323	−	20.3137i	1.29151	−	1.17477i
$300$		0			0
$301$	−0.338454	+	0.0906884i	−0.0195081	+	0.00522719i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	0.383359	−	0.383359i	0.0219511	−	0.0219511i
$306$		0			0
$307$	−3.99494	+	3.99494i	−0.228003	+	0.228003i	−0.811858	−	0.583855i	$-0.801544\pi$
$307$	−3.99494	+	3.99494i	−0.228003	+	0.228003i	0.583855	+	0.811858i	$0.301544\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	9.11900	−	15.7946i	0.517091	−	0.895628i	−0.482712	−	0.875779i	$-0.660348\pi$
$311$	9.11900	−	15.7946i	0.517091	−	0.895628i	0.999803	−	0.0198485i	$-0.00631838\pi$
$312$		0			0
$313$	2.02810	+	3.51277i	0.114635	+	0.198553i	0.917634	−	0.397427i	$-0.130097\pi$
$313$	2.02810	+	3.51277i	0.114635	+	0.198553i	−0.802999	+	0.595980i	$0.796764\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	8.81239	+	2.36127i	0.494953	+	0.132622i	0.497657	−	0.867374i	$-0.334194\pi$
$317$	8.81239	+	2.36127i	0.494953	+	0.132622i	−0.00270356	+	0.999996i	$0.500861\pi$
$318$		0			0
$319$	0.705305	+	2.63223i	0.0394895	+	0.147377i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	26.9552	+	26.9552i	1.49983	+	1.49983i
$324$		0			0
$325$	−16.3657	+	3.56532i	−0.907804	+	0.197768i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	0.594699	−	1.03005i	0.0327868	−	0.0567885i
$330$		0			0
$331$	−2.75963	+	10.2991i	−0.151683	+	0.566088i	0.847684	+	0.530502i	$0.177997\pi$
$331$	−2.75963	+	10.2991i	−0.151683	+	0.566088i	−0.999367	+	0.0355863i	$0.988670\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−2.95860	−	5.12445i	−0.161646	−	0.279979i
$336$		0			0
$337$	28.8295	+	16.6447i	1.57044	+	0.906694i	0.996115	+	0.0880675i	$0.0280691\pi$
$337$	28.8295	+	16.6447i	1.57044	+	0.906694i	0.574326	+	0.818627i	$0.305264\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		−	0.779850i		−	0.0422312i
$342$		0			0
$343$	−0.929747	−	0.929747i	−0.0502016	−	0.0502016i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−26.5898	−	15.3516i	−1.42742	−	0.824119i	−0.430500	−	0.902591i	$-0.641663\pi$
$347$	−26.5898	−	15.3516i	−1.42742	−	0.824119i	−0.996916	+	0.0784717i	$0.974996\pi$
$348$		0			0
$349$	21.1835	+	5.67610i	1.13393	+	0.303835i	0.776506	−	0.630109i	$-0.216990\pi$
$349$	21.1835	+	5.67610i	1.13393	+	0.303835i	0.357420	+	0.933944i	$0.383657\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	28.2274	+	7.56350i	1.50239	+	0.402565i	0.913900	−	0.405939i	$-0.133056\pi$
$353$	28.2274	+	7.56350i	1.50239	+	0.402565i	0.588491	+	0.808504i	$0.299722\pi$
$354$		0			0
$355$	−0.455345	−	0.262893i	−0.0241672	−	0.0139529i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−9.20779	−	9.20779i	−0.485969	−	0.485969i	0.421063	−	0.907031i	$-0.361657\pi$
$359$	−9.20779	−	9.20779i	−0.485969	−	0.485969i	−0.907031	+	0.421063i	$0.861657\pi$
$360$		0			0
$361$			28.8171i			1.51669i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−2.63313	−	1.52024i	−0.137824	−	0.0795729i
$366$		0			0
$367$	−1.31371	−	2.27542i	−0.0685752	−	0.118776i	0.829699	−	0.558211i	$-0.188512\pi$
$367$	−1.31371	−	2.27542i	−0.0685752	−	0.118776i	−0.898274	+	0.439435i	$0.855179\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	0.0191791	−	0.0715773i	0.000995728	−	0.00371611i
$372$		0			0
$373$	7.30630	−	12.6549i	0.378306	−	0.655245i	−0.612510	−	0.790463i	$-0.709840\pi$
$373$	7.30630	−	12.6549i	0.378306	−	0.655245i	0.990816	+	0.135218i	$0.0431734\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	21.3951	−	4.66100i	1.10190	−	0.240054i
$378$		0			0
$379$	−0.202557	−	0.202557i	−0.0104047	−	0.0104047i	0.701885	−	0.712290i	$-0.252342\pi$
$379$	−0.202557	−	0.202557i	−0.0104047	−	0.0104047i	−0.712290	+	0.701885i	$0.752342\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	3.19302	+	11.9165i	0.163155	+	0.608904i	0.998268	+	0.0588257i	$0.0187356\pi$
$383$	3.19302	+	11.9165i	0.163155	+	0.608904i	−0.835113	+	0.550079i	$0.814598\pi$
$384$		0			0
$385$	0.0242525	+	0.00649843i	0.00123602	+	0.000331191i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−10.0814	−	17.4615i	−0.511148	−	0.885335i	−0.999917	−	0.0129213i	$-0.995887\pi$
$389$	−10.0814	−	17.4615i	−0.511148	−	0.885335i	0.488768	−	0.872414i	$-0.337446\pi$
$390$		0			0
$391$	23.0789	−	39.9737i	1.16715	−	2.02156i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	3.44500	−	3.44500i	0.173337	−	0.173337i
$396$		0			0
$397$	−10.1506	+	10.1506i	−0.509444	+	0.509444i	−0.914356	−	0.404912i	$-0.867302\pi$
$397$	−10.1506	+	10.1506i	−0.509444	+	0.509444i	0.404912	+	0.914356i	$0.367302\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	34.1423	−	9.14839i	1.70498	−	0.456849i	0.730797	−	0.682595i	$-0.239149\pi$
$401$	34.1423	−	9.14839i	1.70498	−	0.456849i	0.974186	+	0.225746i	$0.0724819\pi$
$402$		0			0
$403$	−6.25932	−	0.296272i	−0.311799	−	0.0147583i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−1.98917	−	3.44534i	−0.0985995	−	0.170779i
$408$		0			0
$409$	−31.2053	+	8.36144i	−1.54300	+	0.413447i	−0.927235	−	0.374480i	$-0.877821\pi$
$409$	−31.2053	+	8.36144i	−1.54300	+	0.413447i	−0.615769	+	0.787927i	$0.711155\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−0.688134			−0.0338609
$414$		0			0
$415$		−	6.02897i		−	0.295951i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−24.2744	−	14.0148i	−1.18588	−	0.684668i	−0.228512	−	0.973541i	$-0.573386\pi$
$419$	−24.2744	−	14.0148i	−1.18588	−	0.684668i	−0.957367	+	0.288873i	$0.906719\pi$
$420$		0			0
$421$	−1.32618	+	4.94935i	−0.0646338	+	0.241217i	−0.990683	−	0.136185i	$-0.956516\pi$
$421$	−1.32618	+	4.94935i	−0.0646338	+	0.241217i	0.926050	+	0.377402i	$0.123182\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−22.1783	+	12.8047i	−1.07581	+	0.621117i
$426$		0			0
$427$	−0.0221475	−	0.0826558i	−0.00107179	−	0.00399999i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−7.88442	+	7.88442i	−0.379779	+	0.379779i	−0.871022	−	0.491243i	$-0.836543\pi$
$431$	−7.88442	+	7.88442i	−0.379779	+	0.379779i	0.491243	+	0.871022i	$0.336543\pi$
$432$		0			0
$433$			21.4637i			1.03148i	0.856746	+	0.515739i	$0.172483\pi$
$433$			21.4637i			1.03148i	−0.856746	+	0.515739i	$0.827517\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	55.9259	−	14.9853i	2.67530	−	0.716843i
$438$		0			0
$439$	−8.91004	+	5.14422i	−0.425253	+	0.245520i	−0.697322	−	0.716758i	$-0.745625\pi$
$439$	−8.91004	+	5.14422i	−0.425253	+	0.245520i	0.272069	+	0.962278i	$0.412292\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−33.9414	+	19.5961i	−1.61260	+	0.931037i	−0.623840	+	0.781552i	$0.714428\pi$
$443$	−33.9414	+	19.5961i	−1.61260	+	0.931037i	−0.988764	+	0.149486i	$0.952238\pi$
$444$		0			0
$445$	−1.88060	+	3.25730i	−0.0891491	+	0.154411i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	1.87941	+	1.87941i	0.0886947	+	0.0886947i	0.750062	−	0.661367i	$-0.230024\pi$
$449$	1.87941	+	1.87941i	0.0886947	+	0.0886947i	−0.661367	+	0.750062i	$0.730024\pi$
$450$		0			0
$451$	2.63860			0.124247
$452$		0			0
$453$		0			0
$454$		0			0
$455$	0.0613722	−	0.192189i	0.00287717	−	0.00900998i
$456$		0			0
$457$	−10.1980	+	38.0595i	−0.477042	+	1.78035i	0.136451	+	0.990647i	$0.456430\pi$
$457$	−10.1980	+	38.0595i	−0.477042	+	1.78035i	−0.613494	+	0.789700i	$0.710236\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−4.06644	−	1.08960i	−0.189393	−	0.0507477i	0.162876	−	0.986647i	$-0.447923\pi$
$461$	−4.06644	−	1.08960i	−0.189393	−	0.0507477i	−0.352269	+	0.935899i	$0.614590\pi$
$462$		0			0
$463$	6.24235	+	23.2968i	0.290107	+	1.08269i	0.945026	+	0.326994i	$0.106036\pi$
$463$	6.24235	+	23.2968i	0.290107	+	1.08269i	−0.654920	+	0.755698i	$0.727298\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	24.1608			1.11803			0.559013	−	0.829159i	$-0.311180\pi$
$467$	24.1608			1.11803			0.559013	+	0.829159i	$0.311180\pi$
$468$		0			0
$469$	−0.933953			−0.0431260
$470$		0			0
$471$		0			0
$472$		0			0
$473$	0.433007	+	1.61600i	0.0199097	+	0.0743039i
$474$		0			0
$475$	−31.0289	−	8.31417i	−1.42370	−	0.381480i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	2.49787	−	9.32218i	0.114131	−	0.425941i	−0.885090	−	0.465420i	$-0.845903\pi$
$479$	2.49787	−	9.32218i	0.114131	−	0.425941i	0.999220	+	0.0394790i	$0.0125698\pi$
$480$		0			0
$481$	−28.4092	+	14.6568i	−1.29535	+	0.668293i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−9.36857			−0.425405
$486$		0			0
$487$	15.4742	+	15.4742i	0.701205	+	0.701205i	0.964669	−	0.263464i	$-0.0848651\pi$
$487$	15.4742	+	15.4742i	0.701205	+	0.701205i	−0.263464	+	0.964669i	$0.584865\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−6.89886	+	11.9492i	−0.311341	+	0.539258i	−0.978653	−	0.205520i	$-0.934111\pi$
$491$	−6.89886	+	11.9492i	−0.311341	+	0.539258i	0.667312	+	0.744778i	$0.267445\pi$
$492$		0			0
$493$	28.9941	−	16.7397i	1.30583	−	0.753919i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−0.0718702	+	0.0414943i	−0.00322382	+	0.00186127i
$498$		0			0
$499$	−0.634735	+	0.170077i	−0.0284146	+	0.00761368i	−0.272998	−	0.962014i	$-0.588015\pi$
$499$	−0.634735	+	0.170077i	−0.0284146	+	0.00761368i	0.244584	+	0.969628i	$0.421349\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	20.4283i		−	0.910852i	−0.890274	−	0.455426i	$-0.849487\pi$
$503$		−	20.4283i		−	0.910852i	0.890274	−	0.455426i	$-0.150513\pi$
$504$		0			0
$505$	2.02658	−	2.02658i	0.0901817	−	0.0901817i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	0.00431483	+	0.0161032i	0.000191252	+	0.000713760i	0.966021	−	0.258462i	$-0.0832157\pi$
$509$	0.00431483	+	0.0161032i	0.000191252	+	0.000713760i	−0.965830	+	0.259176i	$0.916549\pi$
$510$		0			0
$511$	−0.415605	+	0.239950i	−0.0183853	+	0.0106147i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	0.937170	−	3.49756i	0.0412966	−	0.154121i
$516$		0			0
$517$	−4.91814	−	2.83949i	−0.216300	−	0.124881i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	31.5428i		−	1.38191i	−0.722896	−	0.690957i	$-0.757189\pi$
$521$		−	31.5428i		−	1.38191i	0.722896	−	0.690957i	$-0.242811\pi$
$522$		0			0
$523$	2.86422			0.125243			0.0626217	−	0.998037i	$-0.480054\pi$
$523$	2.86422			0.125243			0.0626217	+	0.998037i	$0.480054\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−9.25449	+	2.47973i	−0.403132	+	0.108019i
$528$		0			0
$529$	−23.5530	−	40.7949i	−1.02404	−	1.77369i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	1.00243	−	21.1783i	0.0434200	−	0.917333i
$534$		0			0
$535$	3.54774	−	0.950615i	0.153382	−	0.0410987i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−2.21822	+	2.21822i	−0.0955453	+	0.0955453i
$540$		0			0
$541$	24.0790	−	24.0790i	1.03524	−	1.03524i	0.0358828	−	0.999356i	$-0.488576\pi$
$541$	24.0790	−	24.0790i	1.03524	−	1.03524i	0.999356	−	0.0358828i	$-0.0114243\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−1.38514	+	2.39913i	−0.0593327	+	0.102767i
$546$		0			0
$547$	3.28072	+	5.68237i	0.140273	+	0.242960i	0.927599	−	0.373576i	$-0.121869\pi$
$547$	3.28072	+	5.68237i	0.140273	+	0.242960i	−0.787326	+	0.616537i	$0.788535\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	40.5646	+	10.8692i	1.72811	+	0.463045i
$552$		0			0
$553$	−0.199026	−	0.742774i	−0.00846343	−	0.0315860i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−31.2392	−	31.2392i	−1.32365	−	1.32365i	−0.910801	−	0.412846i	$-0.864535\pi$
$557$	−31.2392	−	31.2392i	−1.32365	−	1.32365i	−0.412846	−	0.910801i	$-0.635465\pi$
$558$		0			0
$559$	13.1351	−	2.86152i	0.555554	−	0.121029i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−2.67778	+	4.63804i	−0.112855	+	0.195470i	−0.916920	−	0.399071i	$-0.869333\pi$
$563$	−2.67778	+	4.63804i	−0.112855	+	0.195470i	0.804065	+	0.594541i	$0.202666\pi$
$564$		0			0
$565$	0.150040	−	0.559956i	0.00631222	−	0.0235575i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	6.91658	+	11.9799i	0.289958	+	0.502222i	0.973799	−	0.227408i	$-0.0730252\pi$
$569$	6.91658	+	11.9799i	0.289958	+	0.502222i	−0.683841	+	0.729631i	$0.739692\pi$
$570$		0			0
$571$	−0.279180	−	0.161184i	−0.0116833	−	0.00674536i	0.494147	−	0.869378i	$-0.335480\pi$
$571$	−0.279180	−	0.161184i	−0.0116833	−	0.00674536i	−0.505830	+	0.862633i	$0.668814\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			38.8963i			1.62209i
$576$		0			0
$577$	−14.4319	−	14.4319i	−0.600807	−	0.600807i	0.339720	−	0.940527i	$-0.389668\pi$
$577$	−14.4319	−	14.4319i	−0.600807	−	0.600807i	−0.940527	+	0.339720i	$0.889668\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−0.824105	−	0.475797i	−0.0341896	−	0.0197394i
$582$		0			0
$583$	−0.341758	−	0.0915738i	−0.0141542	−	0.00379260i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−22.2892	−	5.97237i	−0.919973	−	0.246506i	−0.232399	−	0.972621i	$-0.574657\pi$
$587$	−22.2892	−	5.97237i	−0.919973	−	0.246506i	−0.687574	+	0.726115i	$0.741324\pi$
$588$		0			0
$589$	−10.4079	−	6.00901i	−0.428851	−	0.247597i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−1.94470	−	1.94470i	−0.0798591	−	0.0798591i	0.666049	−	0.745908i	$-0.267984\pi$
$593$	−1.94470	−	1.94470i	−0.0798591	−	0.0798591i	−0.745908	+	0.666049i	$0.767984\pi$
$594$		0			0
$595$		−	0.308468i		−	0.0126460i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−11.1324	−	6.42730i	−0.454858	−	0.262613i	0.255022	−	0.966935i	$-0.417917\pi$
$599$	−11.1324	−	6.42730i	−0.454858	−	0.262613i	−0.709880	+	0.704323i	$0.751251\pi$
$600$		0			0
$601$	−7.22889	−	12.5208i	−0.294873	−	0.510734i	0.680083	−	0.733135i	$-0.261944\pi$
$601$	−7.22889	−	12.5208i	−0.294873	−	0.510734i	−0.974955	+	0.222401i	$0.928611\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−1.66411	+	6.21056i	−0.0676559	+	0.252495i
$606$		0			0
$607$	2.33525	−	4.04478i	0.0947850	−	0.164172i	−0.814734	−	0.579835i	$-0.803117\pi$
$607$	2.33525	−	4.04478i	0.0947850	−	0.164172i	0.909519	+	0.415663i	$0.136450\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−24.6591	+	38.3959i	−0.997601	+	1.55333i
$612$		0			0
$613$	−32.2764	−	32.2764i	−1.30363	−	1.30363i	−0.925924	−	0.377709i	$-0.876712\pi$
$613$	−32.2764	−	32.2764i	−1.30363	−	1.30363i	−0.377709	−	0.925924i	$-0.623288\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	7.81372	+	29.1612i	0.314569	+	1.17399i	0.924390	+	0.381448i	$0.124574\pi$
$617$	7.81372	+	29.1612i	0.314569	+	1.17399i	−0.609822	+	0.792538i	$0.708759\pi$
$618$		0			0
$619$	29.3839	+	7.87339i	1.18104	+	0.316458i	0.795338	−	0.606166i	$-0.207293\pi$
$619$	29.3839	+	7.87339i	1.18104	+	0.316458i	0.385700	+	0.922624i	$0.373960\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	0.296829	+	0.514122i	0.0118922	+	0.0205979i
$624$		0			0
$625$	9.90398	−	17.1542i	0.396159	−	0.686168i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−34.5609	+	34.5609i	−1.37803	+	1.37803i
$630$		0			0
$631$	18.0342	−	18.0342i	0.717931	−	0.717931i	−0.250251	−	0.968181i	$-0.580513\pi$
$631$	18.0342	−	18.0342i	0.717931	−	0.717931i	0.968181	+	0.250251i	$0.0805130\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	3.31481	−	0.888201i	0.131544	−	0.0352472i
$636$		0			0
$637$	16.9614	+	18.6468i	0.672035	+	0.738815i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	18.9014	+	32.7382i	0.746562	+	1.29308i	0.949461	+	0.313884i	$0.101630\pi$
$641$	18.9014	+	32.7382i	0.746562	+	1.29308i	−0.202900	+	0.979200i	$0.565037\pi$
$642$		0			0
$643$	−37.4118	+	10.0245i	−1.47538	+	0.395326i	−0.904771	−	0.425898i	$-0.859958\pi$
$643$	−37.4118	+	10.0245i	−1.47538	+	0.395326i	−0.570606	+	0.821224i	$0.693292\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−37.4085			−1.47068			−0.735339	−	0.677699i	$-0.762977\pi$
$647$	−37.4085			−1.47068			−0.735339	+	0.677699i	$0.762977\pi$
$648$		0			0
$649$			3.28561i			0.128972i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−22.3962	−	12.9304i	−0.876431	−	0.506008i	−0.00695093	−	0.999976i	$-0.502213\pi$
$653$	−22.3962	−	12.9304i	−0.876431	−	0.506008i	−0.869480	+	0.493968i	$0.835546\pi$
$654$		0			0
$655$	0.0986919	−	0.368323i	0.00385621	−	0.0143916i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	21.5416	−	12.4370i	0.839140	−	0.484478i	−0.0178318	−	0.999841i	$-0.505676\pi$
$659$	21.5416	−	12.4370i	0.839140	−	0.484478i	0.856972	+	0.515363i	$0.172343\pi$
$660$		0			0
$661$	0.570777	+	2.13017i	0.0222007	+	0.0828540i	0.976137	−	0.217154i	$-0.0696773\pi$
$661$	0.570777	+	2.13017i	0.0222007	+	0.0828540i	−0.953937	+	0.300008i	$0.903011\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	0.273602	−	0.273602i	0.0106098	−	0.0106098i
$666$		0			0
$667$		−	50.8498i		−	1.96891i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−0.394654	+	0.105747i	−0.0152354	+	0.00408232i
$672$		0			0
$673$	−19.8145	+	11.4399i	−0.763795	+	0.440977i	−0.830656	−	0.556785i	$-0.812035\pi$
$673$	−19.8145	+	11.4399i	−0.763795	+	0.440977i	0.0668619	+	0.997762i	$0.478701\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−8.77596	+	5.06680i	−0.337287	+	0.194733i	−0.659072	−	0.752080i	$-0.729051\pi$
$677$	−8.77596	+	5.06680i	−0.337287	+	0.194733i	0.321784	+	0.946813i	$0.395717\pi$
$678$		0			0
$679$	−0.739353	+	1.28060i	−0.0283738	+	0.0491448i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−17.0297	−	17.0297i	−0.651622	−	0.651622i	0.301761	−	0.953384i	$-0.402425\pi$
$683$	−17.0297	−	17.0297i	−0.651622	−	0.651622i	−0.953384	+	0.301761i	$0.902425\pi$
$684$		0			0
$685$	4.21704			0.161125
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−0.864837	+	2.70827i	−0.0329477	+	0.103177i
$690$		0			0
$691$	5.92606	−	22.1164i	0.225438	−	0.841346i	−0.756791	−	0.653657i	$-0.773234\pi$
$691$	5.92606	−	22.1164i	0.225438	−	0.841346i	0.982229	−	0.187689i	$-0.0600996\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	8.41570	+	2.25498i	0.319226	+	0.0855362i
$696$		0			0
$697$	−8.39011	−	31.3123i	−0.317798	−	1.18604i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−37.0143			−1.39801			−0.699005	−	0.715116i	$-0.746374\pi$
$701$	−37.0143			−1.39801			−0.699005	+	0.715116i	$0.746374\pi$
$702$		0			0
$703$	−61.3090			−2.31231
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−0.117080	−	0.436950i	−0.00440326	−	0.0164332i
$708$		0			0
$709$	9.54653	+	2.55799i	0.358528	+	0.0960672i	0.433587	−	0.901112i	$-0.357248\pi$
$709$	9.54653	+	2.55799i	0.358528	+	0.0960672i	−0.0750590	+	0.997179i	$0.523915\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−3.76630	+	14.0560i	−0.141049	+	0.526403i
$714$		0			0
$715$	−0.917640	−	0.293032i	−0.0343178	−	0.0109588i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−15.8636			−0.591613			−0.295807	−	0.955248i	$-0.595588\pi$
$719$	−15.8636			−0.591613			−0.295807	+	0.955248i	$0.595588\pi$
$720$		0			0
$721$	−0.404125	−	0.404125i	−0.0150504	−	0.0150504i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−14.1063	+	24.4328i	−0.523894	+	0.907411i
$726$		0			0
$727$	28.3237	−	16.3527i	1.05047	−	0.606489i	0.127689	−	0.991814i	$-0.459244\pi$
$727$	28.3237	−	16.3527i	1.05047	−	0.606489i	0.922781	+	0.385325i	$0.125911\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	17.8003	−	10.2770i	0.658367	−	0.380109i
$732$		0			0
$733$	9.16892	−	2.45680i	0.338662	−	0.0907442i	−0.0854802	−	0.996340i	$-0.527242\pi$
$733$	9.16892	−	2.45680i	0.338662	−	0.0907442i	0.424142	+	0.905596i	$0.360576\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			4.45932i			0.164261i
$738$		0			0
$739$	−15.9355	+	15.9355i	−0.586198	+	0.586198i	−0.936599	−	0.350402i	$-0.886045\pi$
$739$	−15.9355	+	15.9355i	−0.586198	+	0.586198i	0.350402	+	0.936599i	$0.386045\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−7.90302	−	29.4945i	−0.289934	−	1.08205i	−0.945158	−	0.326613i	$-0.894093\pi$
$743$	−7.90302	−	29.4945i	−0.289934	−	1.08205i	0.655224	−	0.755434i	$-0.272574\pi$
$744$		0			0
$745$	−9.75535	+	5.63226i	−0.357408	+	0.206350i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	0.150042	−	0.559965i	0.00548242	−	0.0204607i
$750$		0			0
$751$	−27.5324	−	15.8958i	−1.00467	−	0.580048i	−0.0950448	−	0.995473i	$-0.530299\pi$
$751$	−27.5324	−	15.8958i	−1.00467	−	0.580048i	−0.909627	+	0.415425i	$0.863633\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$			4.07548i			0.148322i
$756$		0			0
$757$	9.24174			0.335897			0.167948	−	0.985796i	$-0.446286\pi$
$757$	9.24174			0.335897			0.167948	+	0.985796i	$0.446286\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−31.9876	+	8.57104i	−1.15955	+	0.310700i	−0.786785	−	0.617227i	$-0.788256\pi$
$761$	−31.9876	+	8.57104i	−1.15955	+	0.310700i	−0.372763	+	0.927927i	$0.621589\pi$
$762$		0			0
$763$	0.218626	+	0.378671i	0.00791478	+	0.0137088i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	26.3714	+	1.24823i	0.952216	+	0.0450711i
$768$		0			0
$769$	9.84061	−	2.63678i	0.354862	−	0.0950849i	−0.0769842	−	0.997032i	$-0.524529\pi$
$769$	9.84061	−	2.63678i	0.354862	−	0.0950849i	0.431846	+	0.901947i	$0.357862\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−12.1060	+	12.1060i	−0.435421	+	0.435421i	−0.890468	−	0.455046i	$-0.849623\pi$
$773$	−12.1060	+	12.1060i	−0.435421	+	0.435421i	0.455046	+	0.890468i	$0.349623\pi$
$774$		0			0
$775$	5.70897	−	5.70897i	0.205072	−	0.205072i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	20.3313	−	35.2149i	0.728446	−	1.26171i
$780$		0			0
$781$	0.198121	+	0.343156i	0.00708934	+	0.0122791i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−6.04765	−	1.62046i	−0.215850	−	0.0578368i
$786$		0			0
$787$	−7.76240	−	28.9697i	−0.276700	−	1.03266i	−0.954694	−	0.297590i	$-0.903817\pi$
$787$	−7.76240	−	28.9697i	−0.276700	−	1.03266i	0.677994	−	0.735067i	$-0.262849\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−0.0646999	−	0.0646999i	−0.00230047	−	0.00230047i
$792$		0			0
$793$	0.698829	+	3.20779i	0.0248161	+	0.113912i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	14.1855	−	24.5700i	0.502476	−	0.870314i	−0.497520	−	0.867453i	$-0.665756\pi$
$797$	14.1855	−	24.5700i	0.502476	−	0.870314i	0.999996	−	0.00286136i	$-0.000910799\pi$
$798$		0			0
$799$	−18.0578	+	67.3926i	−0.638839	+	2.38418i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	1.14568	+	1.98438i	0.0404302	+	0.0700271i
$804$		0			0
$805$	−0.405743	−	0.234256i	−0.0143006	−	0.00825644i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	41.8676i		−	1.47199i	−0.676989	−	0.735993i	$-0.736715\pi$
$809$		−	41.8676i		−	1.47199i	0.676989	−	0.735993i	$-0.263285\pi$
$810$		0			0
$811$	24.9544	+	24.9544i	0.876267	+	0.876267i	0.993146	−	0.116879i	$-0.0372890\pi$
$811$	24.9544	+	24.9544i	0.876267	+	0.876267i	−0.116879	+	0.993146i	$0.537289\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−5.36625	−	3.09820i	−0.187971	−	0.108525i
$816$		0			0
$817$	24.9037	+	6.67294i	0.871272	+	0.233457i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−18.4066	−	4.93204i	−0.642396	−	0.172129i	−0.0771075	−	0.997023i	$-0.524568\pi$
$821$	−18.4066	−	4.93204i	−0.642396	−	0.172129i	−0.565288	+	0.824893i	$0.691235\pi$
$822$		0			0
$823$	−8.28351	−	4.78249i	−0.288745	−	0.166707i	0.348631	−	0.937260i	$-0.386647\pi$
$823$	−8.28351	−	4.78249i	−0.288745	−	0.166707i	−0.637376	+	0.770553i	$0.719980\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−13.8587	−	13.8587i	−0.481915	−	0.481915i	0.423828	−	0.905743i	$-0.360686\pi$
$827$	−13.8587	−	13.8587i	−0.481915	−	0.481915i	−0.905743	+	0.423828i	$0.860686\pi$
$828$		0			0
$829$		−	45.6637i		−	1.58597i	−0.609244	−	0.792983i	$-0.708527\pi$
$829$		−	45.6637i		−	1.58597i	0.609244	−	0.792983i	$-0.291473\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	33.3770	+	19.2702i	1.15644	+	0.667673i
$834$		0			0
$835$	−6.68345	−	11.5761i	−0.231290	−	0.400606i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−3.65147	+	13.6275i	−0.126063	+	0.470472i	−0.999875	−	0.0157905i	$-0.994974\pi$
$839$	−3.65147	+	13.6275i	−0.126063	+	0.470472i	0.873813	+	0.486263i	$0.161640\pi$
$840$		0			0
$841$	3.94137	−	6.82665i	0.135909	−	0.235402i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−2.70059	+	7.25395i	−0.0929031	+	0.249544i
$846$		0			0
$847$	0.717597	+	0.717597i	0.0246569	+	0.0246569i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	19.2135	+	71.7058i	0.658631	+	2.45804i
$852$		0			0
$853$	−36.4596	−	9.76933i	−1.24835	−	0.334495i	−0.426653	−	0.904415i	$-0.640308\pi$
$853$	−36.4596	−	9.76933i	−1.24835	−	0.334495i	−0.821700	+	0.569920i	$0.806974\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−4.38094	−	7.58800i	−0.149650	−	0.259201i	0.781448	−	0.623970i	$-0.214481\pi$
$857$	−4.38094	−	7.58800i	−0.149650	−	0.259201i	−0.931098	+	0.364769i	$0.881148\pi$
$858$		0			0
$859$	−17.2450	+	29.8692i	−0.588391	+	1.01912i	0.406052	+	0.913850i	$0.366905\pi$
$859$	−17.2450	+	29.8692i	−0.588391	+	1.01912i	−0.994443	+	0.105274i	$0.966428\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−27.1745	+	27.1745i	−0.925029	+	0.925029i	−0.997379	−	0.0723499i	$-0.976950\pi$
$863$	−27.1745	+	27.1745i	−0.925029	+	0.925029i	0.0723499	+	0.997379i	$0.476950\pi$
$864$		0			0
$865$	6.91872	−	6.91872i	0.235243	−	0.235243i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−3.54650	+	0.950282i	−0.120307	+	0.0322361i
$870$		0			0
$871$	35.7919	+	1.69413i	1.21276	+	0.0574035i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	0.269859	+	0.467410i	0.00912291	+	0.0158013i
$876$		0			0
$877$	−4.20346	+	1.12631i	−0.141941	+	0.0380329i	−0.329090	−	0.944299i	$-0.606742\pi$
$877$	−4.20346	+	1.12631i	−0.141941	+	0.0380329i	0.187149	+	0.982332i	$0.440075\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	17.7491			0.597983			0.298992	−	0.954256i	$-0.403350\pi$
$881$	17.7491			0.597983			0.298992	+	0.954256i	$0.403350\pi$
$882$		0			0
$883$		−	13.6618i		−	0.459757i	−0.973219	−	0.229879i	$-0.926167\pi$
$883$		−	13.6618i		−	0.459757i	0.973219	−	0.229879i	$-0.0738329\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	27.8424	+	16.0748i	0.934858	+	0.539740i	0.888345	−	0.459177i	$-0.151856\pi$
$887$	27.8424	+	16.0748i	0.934858	+	0.539740i	0.0465133	+	0.998918i	$0.485189\pi$
$888$		0			0
$889$	0.140191	−	0.523200i	0.00470185	−	0.0175476i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−75.7920	+	43.7586i	−2.53628	+	1.46432i
$894$		0			0
$895$	2.68916	+	10.0361i	0.0898888	+	0.335470i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−7.46343	+	7.46343i	−0.248919	+	0.248919i
$900$		0			0
$901$			4.34683i			0.144814i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	7.61066	−	2.03927i	0.252987	−	0.0677877i
$906$		0			0
$907$	−22.2907	+	12.8695i	−0.740149	+	0.427325i	−0.822123	−	0.569309i	$-0.807211\pi$
$907$	−22.2907	+	12.8695i	−0.740149	+	0.427325i	0.0819746	+	0.996634i	$0.473877\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	11.7787	−	6.80041i	0.390244	−	0.225308i	−0.292022	−	0.956412i	$-0.594328\pi$
$911$	11.7787	−	6.80041i	0.390244	−	0.225308i	0.682266	+	0.731104i	$0.260995\pi$
$912$		0			0
$913$	−2.27177	+	3.93483i	−0.0751847	+	0.130224i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−0.0425578	−	0.0425578i	−0.00140538	−	0.00140538i
$918$		0			0
$919$	3.46854			0.114417			0.0572083	−	0.998362i	$-0.481780\pi$
$919$	3.46854			0.114417			0.0572083	+	0.998362i	$0.481780\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	2.82955	−	1.45982i	0.0931358	−	0.0480505i
$924$		0			0
$925$	10.6601	−	39.7839i	0.350501	−	1.30809i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	26.2691	+	7.03879i	0.861862	+	0.230935i	0.662565	−	0.749004i	$-0.269468\pi$
$929$	26.2691	+	7.03879i	0.861862	+	0.230935i	0.199296	+	0.979939i	$0.436134\pi$
$930$		0			0
$931$	12.5123	+	46.6966i	0.410074	+	1.53042i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−1.47283			−0.0481668
$936$		0			0
$937$	40.3459			1.31804			0.659022	−	0.752124i	$-0.270970\pi$
$937$	40.3459			1.31804			0.659022	+	0.752124i	$0.270970\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−13.2517	−	49.4561i	−0.431994	−	1.61222i	−0.748159	−	0.663520i	$-0.769062\pi$
$941$	−13.2517	−	49.4561i	−0.431994	−	1.61222i	0.316165	−	0.948704i	$-0.397605\pi$
$942$		0			0
$943$	−47.5583	−	12.7432i	−1.54871	−	0.414976i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	6.28521	−	23.4567i	0.204242	−	0.762241i	−0.785438	−	0.618941i	$-0.787562\pi$
$947$	6.28521	−	23.4567i	0.204242	−	0.762241i	0.989679	−	0.143300i	$-0.0457713\pi$
$948$		0			0
$949$	16.3625	−	8.44172i	0.531149	−	0.274030i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−35.8048			−1.15983			−0.579915	−	0.814677i	$-0.696914\pi$
$953$	−35.8048			−1.15983			−0.579915	+	0.814677i	$0.696914\pi$
$954$		0			0
$955$	−5.48317	−	5.48317i	−0.177431	−	0.177431i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	0.332803	−	0.576431i	0.0107468	−	0.0186139i
$960$		0			0
$961$	−24.2309	+	13.9897i	−0.781643	+	0.451282i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−1.32276	+	0.763697i	−0.0425812	+	0.0245843i
$966$		0			0
$967$	−55.3566	+	14.8327i	−1.78015	+	0.476989i	−0.990611	−	0.136714i	$-0.956346\pi$
$967$	−55.3566	+	14.8327i	−1.78015	+	0.476989i	−0.789537	+	0.613703i	$0.789679\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		−	33.6934i		−	1.08127i	−0.841257	−	0.540636i	$-0.818184\pi$
$971$		−	33.6934i		−	1.08127i	0.841257	−	0.540636i	$-0.181816\pi$
$972$		0			0
$973$	0.972389	−	0.972389i	0.0311734	−	0.0311734i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−13.0463	−	48.6894i	−0.417387	−	1.55771i	−0.780006	−	0.625773i	$-0.784784\pi$
$977$	−13.0463	−	48.6894i	−0.417387	−	1.55771i	0.362618	−	0.931938i	$-0.381883\pi$
$978$		0			0
$979$	2.45476	−	1.41726i	0.0784546	−	0.0452958i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−2.24381	+	8.37401i	−0.0715664	+	0.267090i	−0.992433	−	0.122789i	$-0.960816\pi$
$983$	−2.24381	+	8.37401i	−0.0715664	+	0.267090i	0.920866	+	0.389878i	$0.127483\pi$
$984$		0			0
$985$	10.2348	+	5.90905i	0.326107	+	0.188278i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	31.2181i		−	0.992679i
$990$		0			0
$991$	−41.3299			−1.31289			−0.656443	−	0.754376i	$-0.727940\pi$
$991$	−41.3299			−1.31289			−0.656443	+	0.754376i	$0.727940\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−9.90675	+	2.65450i	−0.314065	+	0.0841535i
$996$		0			0
$997$	−21.3446	−	36.9700i	−0.675991	−	1.17085i	−0.976178	−	0.216970i	$-0.930383\pi$
$997$	−21.3446	−	36.9700i	−0.675991	−	1.17085i	0.300187	−	0.953880i	$-0.402951\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1404.2.cj.a.125.9		56
3.2	odd	2		468.2.cg.a.437.1	yes	56
9.2	odd	6	inner	1404.2.cj.a.1061.9		56
9.7	even	3		468.2.cg.a.281.1	yes	56
13.5	odd	4	inner	1404.2.cj.a.1097.9		56
39.5	even	4		468.2.cg.a.5.1	✓	56
117.70	odd	12		468.2.cg.a.317.1	yes	56
117.83	even	12	inner	1404.2.cj.a.629.9		56

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
468.2.cg.a.5.1	✓	56	39.5	even	4
468.2.cg.a.281.1	yes	56	9.7	even	3
468.2.cg.a.317.1	yes	56	117.70	odd	12
468.2.cg.a.437.1	yes	56	3.2	odd	2
1404.2.cj.a.125.9		56	1.1	even	1	trivial
1404.2.cj.a.629.9		56	117.83	even	12	inner
1404.2.cj.a.1061.9		56	9.2	odd	6	inner
1404.2.cj.a.1097.9		56	13.5	odd	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:	\( N \)	\(=\)	\( 1404 = 2^{2} \cdot 3^{3} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1404.cj (of order \(12\), degree \(4\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.154104 + 0.575123i) q^{5} +(0.0907757 + 0.0243233i) q^{7} +(0.116136 - 0.433424i) q^{11} +(-3.43468 - 1.09680i) q^{13} -5.51273 q^{17} +(-4.88963 - 4.88963i) q^{19} +(-4.18646 + 7.25117i) q^{23} +(4.02311 - 2.32274i) q^{25} +(-5.25947 + 3.03656i) q^{29} +(1.67875 - 0.449819i) q^{31} +0.0559555i q^{35} +(6.26928 - 6.26928i) q^{37} +(1.52195 + 5.68000i) q^{41} +(-3.22894 + 1.86423i) q^{43} +(3.27565 - 12.2249i) q^{47} +(-6.05453 - 3.49558i) q^{49} -0.788507i q^{53} +0.267169 q^{55} +(-7.07279 + 1.89515i) q^{59} +(-0.455275 - 0.788559i) q^{61} +(0.101500 - 2.14439i) q^{65} +(-9.59938 + 2.57215i) q^{67} +(-0.624421 + 0.624421i) q^{71} +(-3.61085 + 3.61085i) q^{73} +(0.0210846 - 0.0365196i) q^{77} +(-4.09126 - 7.08627i) q^{79} +(-9.78070 - 2.62073i) q^{83} +(-0.849533 - 3.17050i) q^{85} +(4.46678 + 4.46678i) q^{89} +(-0.285108 - 0.183106i) q^{91} +(2.05863 - 3.56565i) q^{95} +(-4.07242 + 15.1985i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 56 q - 2 q^{7} + 12 q^{11} + 4 q^{19} - 4 q^{31} - 4 q^{37} - 24 q^{41} - 66 q^{47} + 78 q^{65} - 14 q^{67} + 28 q^{73} - 24 q^{79} + 78 q^{83} + 36 q^{85} - 8 q^{91} + 26 q^{97}+O(q^{100}) \) 56 * q - 2 * q^7 + 12 * q^11 + 4 * q^19 - 4 * q^31 - 4 * q^37 - 24 * q^41 - 66 * q^47 + 78 * q^65 - 14 * q^67 + 28 * q^73 - 24 * q^79 + 78 * q^83 + 36 * q^85 - 8 * q^91 + 26 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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