LMFDB - Embedded newform 7803.2.a.bv.1.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [7803,2,Mod(1,7803)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("7803.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(7803, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 7803 = 3^{3} \cdot 17^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7803.a (trivial)

Newform invariants

comment:select newform

sage:traces = [12,0,0,12,0,0,-8,0,0,-24,0,0,-4,0,0,20,0,0,-4,0,0,28,0,0,0,0, 0,-48,0,0,-8,0,0,0,0,0,-12,0,0,-44,0,0,-20,0,0,-36,0,0,28,0,0,8,0,0,-4, 0,0,-28,0,0,-48,0,0,40,0,0,-8,0,0,44,0,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(73)] == traces)

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

Self dual:	yes
Analytic conductor:	$62.3072686972$
Analytic rank:	$1$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 18x^{10} + 115x^{8} - 318x^{6} + 395x^{4} - 208x^{2} + 34 $ x^12 - 18x^10 + 115x^8 - 318x^6 + 395x^4 - 208*x^2 + 34
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 459)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-0.934753$ of defining polynomial
Character	$\chi$	$=$	7803.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-0.934753 q^{2} -1.12624 q^{4} -2.13241 q^{5} -2.51416 q^{7} +2.92226 q^{8} +1.99328 q^{10} -0.0514801 q^{11} +1.23986 q^{13} +2.35012 q^{14} -0.479119 q^{16} +0.633442 q^{19} +2.40160 q^{20} +0.0481212 q^{22} -5.12747 q^{23} -0.452827 q^{25} -1.15896 q^{26} +2.83154 q^{28} -1.27046 q^{29} -8.63367 q^{31} -5.39666 q^{32} +5.36122 q^{35} +8.14157 q^{37} -0.592112 q^{38} -6.23146 q^{40} +8.79368 q^{41} +5.62741 q^{43} +0.0579788 q^{44} +4.79292 q^{46} +2.31601 q^{47} -0.679008 q^{49} +0.423282 q^{50} -1.39637 q^{52} +12.2326 q^{53} +0.109777 q^{55} -7.34702 q^{56} +1.18757 q^{58} +1.61825 q^{59} -3.90422 q^{61} +8.07035 q^{62} +6.00278 q^{64} -2.64388 q^{65} +0.733387 q^{67} -5.01141 q^{70} -12.8286 q^{71} +2.40471 q^{73} -7.61036 q^{74} -0.713406 q^{76} +0.129429 q^{77} -1.77525 q^{79} +1.02168 q^{80} -8.21992 q^{82} +11.7205 q^{83} -5.26024 q^{86} -0.150438 q^{88} -8.39229 q^{89} -3.11719 q^{91} +5.77475 q^{92} -2.16490 q^{94} -1.35076 q^{95} +4.92731 q^{97} +0.634705 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 12 q^{4} - 8 q^{7} - 24 q^{10} - 4 q^{13} + 20 q^{16} - 4 q^{19} + 28 q^{22} - 48 q^{28} - 8 q^{31} - 12 q^{37} - 44 q^{40} - 20 q^{43} - 36 q^{46} + 28 q^{49} + 8 q^{52} - 4 q^{55} - 28 q^{58} - 48 q^{61}+ \cdots - 24 q^{97}+O(q^{100}) $ 12 * q + 12 * q^4 - 8 * q^7 - 24 * q^10 - 4 * q^13 + 20 * q^16 - 4 * q^19 + 28 * q^22 - 48 * q^28 - 8 * q^31 - 12 * q^37 - 44 * q^40 - 20 * q^43 - 36 * q^46 + 28 * q^49 + 8 * q^52 - 4 * q^55 - 28 * q^58 - 48 * q^61 + 40 * q^64 - 8 * q^67 + 44 * q^70 - 4 * q^73 - 28 * q^76 - 40 * q^79 - 76 * q^82 + 40 * q^88 - 52 * q^91 + 28 * q^94 - 24 * q^97

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 $

$2$	−0.934753	−0.660970	−0.330485	−	0.943811i	$-0.607212\pi$
$2$	−0.934753	−0.660970	−0.330485	+	0.943811i	$0.607212\pi$
$3$	0	0
$3$	0	0
$4$	−1.12624	−0.563118
$5$	−2.13241	−0.953643	−0.476821	−	0.879000i	$-0.658211\pi$
$5$	−2.13241	−0.953643	−0.476821	+	0.879000i	$0.658211\pi$
$6$	0	0
$7$	−2.51416	−0.950262	−0.475131	−	0.879915i	$-0.657599\pi$
$7$	−2.51416	−0.950262	−0.475131	+	0.879915i	$0.657599\pi$
$8$	2.92226	1.03317
$9$	0	0
$10$	1.99328	0.630330
$11$	−0.0514801	−0.0155218	−0.00776092	−	0.999970i	$-0.502470\pi$
$11$	−0.0514801	−0.0155218	−0.00776092	+	0.999970i	$0.502470\pi$
$12$	0	0
$13$	1.23986	0.343874	0.171937	−	0.985108i	$-0.444997\pi$
$13$	1.23986	0.343874	0.171937	+	0.985108i	$0.444997\pi$
$14$	2.35012	0.628095
$15$	0	0
$16$	−0.479119	−0.119780
$17$	0	0
$17$	0	0
$18$	0	0
$19$	0.633442	0.145322	0.0726608	−	0.997357i	$-0.476851\pi$
$19$	0.633442	0.145322	0.0726608	+	0.997357i	$0.476851\pi$
$20$	2.40160	0.537014
$21$	0	0
$22$	0.0481212	0.0102595
$23$	−5.12747	−1.06915	−0.534576	−	0.845120i	$-0.679529\pi$
$23$	−5.12747	−1.06915	−0.534576	+	0.845120i	$0.679529\pi$
$24$	0	0
$25$	−0.452827	−0.0905655
$26$	−1.15896	−0.227291
$27$	0	0
$28$	2.83154	0.535110
$29$	−1.27046	−0.235919	−0.117959	−	0.993018i	$-0.537635\pi$
$29$	−1.27046	−0.235919	−0.117959	+	0.993018i	$0.537635\pi$
$30$	0	0
$31$	−8.63367	−1.55065	−0.775327	−	0.631560i	$-0.782415\pi$
$31$	−8.63367	−1.55065	−0.775327	+	0.631560i	$0.782415\pi$
$32$	−5.39666	−0.954004
$33$	0	0
$34$	0	0
$35$	5.36122	0.906211
$36$	0	0
$37$	8.14157	1.33847	0.669233	−	0.743052i	$-0.266623\pi$
$37$	8.14157	1.33847	0.669233	+	0.743052i	$0.266623\pi$
$38$	−0.592112	−0.0960533
$39$	0	0
$40$	−6.23146	−0.985280
$41$	8.79368	1.37334	0.686671	−	0.726968i	$-0.259071\pi$
$41$	8.79368	1.37334	0.686671	+	0.726968i	$0.259071\pi$
$42$	0	0
$43$	5.62741	0.858173	0.429086	−	0.903263i	$-0.358836\pi$
$43$	5.62741	0.858173	0.429086	+	0.903263i	$0.358836\pi$
$44$	0.0579788	0.00874063
$45$	0	0
$46$	4.79292	0.706678
$47$	2.31601	0.337825	0.168912	−	0.985631i	$-0.445974\pi$
$47$	2.31601	0.337825	0.168912	+	0.985631i	$0.445974\pi$
$48$	0	0
$49$	−0.679008	−0.0970012
$50$	0.423282	0.0598611
$51$	0	0
$52$	−1.39637	−0.193642
$53$	12.2326	1.68027	0.840136	−	0.542376i	$-0.182475\pi$
$53$	12.2326	1.68027	0.840136	+	0.542376i	$0.182475\pi$
$54$	0	0
$55$	0.109777	0.0148023
$56$	−7.34702	−0.981787
$57$	0	0
$58$	1.18757	0.155935
$59$	1.61825	0.210678	0.105339	−	0.994436i	$-0.466407\pi$
$59$	1.61825	0.210678	0.105339	+	0.994436i	$0.466407\pi$
$60$	0	0
$61$	−3.90422	−0.499885	−0.249942	−	0.968261i	$-0.580412\pi$
$61$	−3.90422	−0.499885	−0.249942	+	0.968261i	$0.580412\pi$
$62$	8.07035	1.02494
$63$	0	0
$64$	6.00278	0.750348
$65$	−2.64388	−0.327933
$66$	0	0
$67$	0.733387	0.0895975	0.0447987	−	0.998996i	$-0.485735\pi$
$67$	0.733387	0.0895975	0.0447987	+	0.998996i	$0.485735\pi$
$68$	0	0
$69$	0	0
$70$	−5.01141	−0.598979
$71$	−12.8286	−1.52247	−0.761235	−	0.648476i	$-0.775407\pi$
$71$	−12.8286	−1.52247	−0.761235	+	0.648476i	$0.775407\pi$
$72$	0	0
$73$	2.40471	0.281450	0.140725	−	0.990049i	$-0.455057\pi$
$73$	2.40471	0.281450	0.140725	+	0.990049i	$0.455057\pi$
$74$	−7.61036	−0.884687
$75$	0	0
$76$	−0.713406	−0.0818333
$77$	0.129429	0.0147498
$78$	0	0
$79$	−1.77525	−0.199731	−0.0998657	−	0.995001i	$-0.531841\pi$
$79$	−1.77525	−0.199731	−0.0998657	+	0.995001i	$0.531841\pi$
$80$	1.02168	0.114227
$81$	0	0
$82$	−8.21992	−0.907739
$83$	11.7205	1.28649	0.643247	−	0.765658i	$-0.277587\pi$
$83$	11.7205	1.28649	0.643247	+	0.765658i	$0.277587\pi$
$84$	0	0
$85$	0	0
$86$	−5.26024	−0.567227
$87$	0	0
$88$	−0.150438	−0.0160368
$89$	−8.39229	−0.889580	−0.444790	−	0.895635i	$-0.646722\pi$
$89$	−8.39229	−0.889580	−0.444790	+	0.895635i	$0.646722\pi$
$90$	0	0
$91$	−3.11719	−0.326771
$92$	5.77475	0.602059
$93$	0	0
$94$	−2.16490	−0.223292
$95$	−1.35076	−0.138585
$96$	0	0
$97$	4.92731	0.500293	0.250146	−	0.968208i	$-0.419521\pi$
$97$	4.92731	0.500293	0.250146	+	0.968208i	$0.419521\pi$
$98$	0.634705	0.0641149
$99$	0	0
$100$	0.509991	0.0509991
$101$	−10.4205	−1.03688	−0.518438	−	0.855115i	$-0.673486\pi$
$101$	−10.4205	−1.03688	−0.518438	+	0.855115i	$0.673486\pi$
$102$	0	0
$103$	9.39654	0.925868	0.462934	−	0.886393i	$-0.346797\pi$
$103$	9.39654	0.925868	0.462934	+	0.886393i	$0.346797\pi$
$104$	3.62318	0.355282
$105$	0	0
$106$	−11.4344	−1.11061
$107$	19.6443	1.89908	0.949542	−	0.313640i	$-0.101548\pi$
$107$	19.6443	1.89908	0.949542	+	0.313640i	$0.101548\pi$
$108$	0	0
$109$	−16.1580	−1.54766	−0.773829	−	0.633394i	$-0.781661\pi$
$109$	−16.1580	−1.54766	−0.773829	+	0.633394i	$0.781661\pi$
$110$	−0.102614	−0.00978387
$111$	0	0
$112$	1.20458	0.113822
$113$	0.195949	0.0184333	0.00921666	−	0.999958i	$-0.497066\pi$
$113$	0.195949	0.0184333	0.00921666	+	0.999958i	$0.497066\pi$
$114$	0	0
$115$	10.9339	1.01959
$116$	1.43084	0.132850
$117$	0	0
$118$	−1.51266	−0.139252
$119$	0	0
$120$	0	0
$121$	−10.9973	−0.999759
$122$	3.64949	0.330409
$123$	0	0
$124$	9.72355	0.873201
$125$	11.6277	1.04001
$126$	0	0
$127$	19.4041	1.72184	0.860920	−	0.508741i	$-0.169889\pi$
$127$	19.4041	1.72184	0.860920	+	0.508741i	$0.169889\pi$
$128$	5.18220	0.458046
$129$	0	0
$130$	2.47138	0.216754
$131$	13.3116	1.16304	0.581519	−	0.813533i	$-0.302459\pi$
$131$	13.3116	1.16304	0.581519	+	0.813533i	$0.302459\pi$
$132$	0	0
$133$	−1.59257	−0.138094
$134$	−0.685536	−0.0592213
$135$	0	0
$136$	0	0
$137$	10.5293	0.899577	0.449789	−	0.893135i	$-0.351499\pi$
$137$	10.5293	0.899577	0.449789	+	0.893135i	$0.351499\pi$
$138$	0	0
$139$	−7.66894	−0.650471	−0.325236	−	0.945633i	$-0.605444\pi$
$139$	−7.66894	−0.650471	−0.325236	+	0.945633i	$0.605444\pi$
$140$	−6.03800	−0.510304
$141$	0	0
$142$	11.9915	1.00631
$143$	−0.0638279	−0.00533756
$144$	0	0
$145$	2.70914	0.224982
$146$	−2.24781	−0.186030
$147$	0	0
$148$	−9.16933	−0.753715
$149$	9.47788	0.776458	0.388229	−	0.921563i	$-0.373087\pi$
$149$	9.47788	0.776458	0.388229	+	0.921563i	$0.373087\pi$
$150$	0	0
$151$	17.3658	1.41321	0.706603	−	0.707610i	$-0.250227\pi$
$151$	17.3658	1.41321	0.706603	+	0.707610i	$0.250227\pi$
$152$	1.85108	0.150143
$153$	0	0
$154$	−0.120984	−0.00974919
$155$	18.4105	1.47877
$156$	0	0
$157$	−10.1833	−0.812712	−0.406356	−	0.913715i	$-0.633201\pi$
$157$	−10.1833	−0.812712	−0.406356	+	0.913715i	$0.633201\pi$
$158$	1.65942	0.132016
$159$	0	0
$160$	11.5079	0.909779
$161$	12.8913	1.01598
$162$	0	0
$163$	−18.4347	−1.44392	−0.721960	−	0.691934i	$-0.756759\pi$
$163$	−18.4347	−1.44392	−0.721960	+	0.691934i	$0.756759\pi$
$164$	−9.90377	−0.773354
$165$	0	0
$166$	−10.9558	−0.850335
$167$	11.4107	0.882983	0.441492	−	0.897265i	$-0.354449\pi$
$167$	11.4107	0.882983	0.441492	+	0.897265i	$0.354449\pi$
$168$	0	0
$169$	−11.4628	−0.881751
$170$	0	0
$171$	0	0
$172$	−6.33780	−0.483253
$173$	15.4099	1.17159	0.585796	−	0.810459i	$-0.300782\pi$
$173$	15.4099	1.17159	0.585796	+	0.810459i	$0.300782\pi$
$174$	0	0
$175$	1.13848	0.0860610
$176$	0.0246651	0.00185920
$177$	0	0
$178$	7.84472	0.587986
$179$	12.0348	0.899526	0.449763	−	0.893148i	$-0.351508\pi$
$179$	12.0348	0.899526	0.449763	+	0.893148i	$0.351508\pi$
$180$	0	0
$181$	−11.7291	−0.871815	−0.435908	−	0.899991i	$-0.643573\pi$
$181$	−11.7291	−0.871815	−0.435908	+	0.899991i	$0.643573\pi$
$182$	2.91381	0.215986
$183$	0	0
$184$	−14.9838	−1.10462
$185$	−17.3612	−1.27642
$186$	0	0
$187$	0	0
$188$	−2.60838	−0.190235
$189$	0	0
$190$	1.26263	0.0916005
$191$	−9.72587	−0.703739	−0.351870	−	0.936049i	$-0.614454\pi$
$191$	−9.72587	−0.703739	−0.351870	+	0.936049i	$0.614454\pi$
$192$	0	0
$193$	−17.8920	−1.28790	−0.643948	−	0.765069i	$-0.722705\pi$
$193$	−17.8920	−1.28790	−0.643948	+	0.765069i	$0.722705\pi$
$194$	−4.60582	−0.330679
$195$	0	0
$196$	0.764724	0.0546231
$197$	17.0649	1.21583	0.607913	−	0.794004i	$-0.292007\pi$
$197$	17.0649	1.21583	0.607913	+	0.794004i	$0.292007\pi$
$198$	0	0
$199$	17.0421	1.20808	0.604042	−	0.796952i	$-0.293556\pi$
$199$	17.0421	1.20808	0.604042	+	0.796952i	$0.293556\pi$
$200$	−1.32328	−0.0935700
$201$	0	0
$202$	9.74057	0.685344
$203$	3.19414	0.224185
$204$	0	0
$205$	−18.7517	−1.30968
$206$	−8.78344	−0.611971
$207$	0	0
$208$	−0.594039	−0.0411892
$209$	−0.0326097	−0.00225566
$210$	0	0
$211$	8.90221	0.612854	0.306427	−	0.951894i	$-0.400867\pi$
$211$	8.90221	0.612854	0.306427	+	0.951894i	$0.400867\pi$
$212$	−13.7768	−0.946191
$213$	0	0
$214$	−18.3626	−1.25524
$215$	−12.0000	−0.818390
$216$	0	0
$217$	21.7064	1.47353
$218$	15.1038	1.02296
$219$	0	0
$220$	−0.123634	−0.00833543
$221$	0	0
$222$	0	0
$223$	−28.1597	−1.88571	−0.942855	−	0.333204i	$-0.891870\pi$
$223$	−28.1597	−1.88571	−0.942855	+	0.333204i	$0.891870\pi$
$224$	13.5681	0.906554
$225$	0	0
$226$	−0.183164	−0.0121839
$227$	11.6806	0.775265	0.387633	−	0.921814i	$-0.373293\pi$
$227$	11.6806	0.775265	0.387633	+	0.921814i	$0.373293\pi$
$228$	0	0
$229$	9.51632	0.628856	0.314428	−	0.949281i	$-0.398187\pi$
$229$	9.51632	0.628856	0.314428	+	0.949281i	$0.398187\pi$
$230$	−10.2205	−0.673918
$231$	0	0
$232$	−3.71262	−0.243745
$233$	−17.2298	−1.12876	−0.564381	−	0.825514i	$-0.690885\pi$
$233$	−17.2298	−1.12876	−0.564381	+	0.825514i	$0.690885\pi$
$234$	0	0
$235$	−4.93869	−0.322164
$236$	−1.82253	−0.118636
$237$	0	0
$238$	0	0
$239$	3.28677	0.212603	0.106302	−	0.994334i	$-0.466099\pi$
$239$	3.28677	0.212603	0.106302	+	0.994334i	$0.466099\pi$
$240$	0	0
$241$	−1.80700	−0.116399	−0.0581995	−	0.998305i	$-0.518536\pi$
$241$	−1.80700	−0.116399	−0.0581995	+	0.998305i	$0.518536\pi$
$242$	10.2798	0.660811
$243$	0	0
$244$	4.39708	0.281494
$245$	1.44792	0.0925045
$246$	0	0
$247$	0.785377	0.0499724
$248$	−25.2298	−1.60210
$249$	0	0
$250$	−10.8690	−0.687416
$251$	−1.65010	−0.104153	−0.0520767	−	0.998643i	$-0.516584\pi$
$251$	−1.65010	−0.104153	−0.0520767	+	0.998643i	$0.516584\pi$
$252$	0	0
$253$	0.263963	0.0165952
$254$	−18.1381	−1.13808
$255$	0	0
$256$	−16.8496	−1.05310
$257$	−15.2226	−0.949557	−0.474778	−	0.880105i	$-0.657472\pi$
$257$	−15.2226	−0.949557	−0.474778	+	0.880105i	$0.657472\pi$
$258$	0	0
$259$	−20.4692	−1.27189
$260$	2.97764	0.184665
$261$	0	0
$262$	−12.4430	−0.768734
$263$	−15.2960	−0.943194	−0.471597	−	0.881814i	$-0.656322\pi$
$263$	−15.2960	−0.943194	−0.471597	+	0.881814i	$0.656322\pi$
$264$	0	0
$265$	−26.0848	−1.60238
$266$	1.48866	0.0912758
$267$	0	0
$268$	−0.825967	−0.0504540
$269$	1.02224	0.0623270	0.0311635	−	0.999514i	$-0.490079\pi$
$269$	1.02224	0.0623270	0.0311635	+	0.999514i	$0.490079\pi$
$270$	0	0
$271$	0.264346	0.0160579	0.00802893	−	0.999968i	$-0.497444\pi$
$271$	0.264346	0.0160579	0.00802893	+	0.999968i	$0.497444\pi$
$272$	0	0
$273$	0	0
$274$	−9.84228	−0.594594
$275$	0.0233116	0.00140574
$276$	0	0
$277$	−4.70468	−0.282677	−0.141338	−	0.989961i	$-0.545141\pi$
$277$	−4.70468	−0.282677	−0.141338	+	0.989961i	$0.545141\pi$
$278$	7.16857	0.429942
$279$	0	0
$280$	15.6669	0.936274
$281$	27.3390	1.63091	0.815455	−	0.578821i	$-0.196487\pi$
$281$	27.3390	1.63091	0.815455	+	0.578821i	$0.196487\pi$
$282$	0	0
$283$	−5.19513	−0.308818	−0.154409	−	0.988007i	$-0.549347\pi$
$283$	−5.19513	−0.308818	−0.154409	+	0.988007i	$0.549347\pi$
$284$	14.4480	0.857331
$285$	0	0
$286$	0.0596633	0.00352797
$287$	−22.1087	−1.30504
$288$	0	0
$289$	0	0
$290$	−2.53238	−0.148706
$291$	0	0
$292$	−2.70827	−0.158489
$293$	−34.0196	−1.98745	−0.993724	−	0.111859i	$-0.964319\pi$
$293$	−34.0196	−1.98745	−0.993724	+	0.111859i	$0.964319\pi$
$294$	0	0
$295$	−3.45077	−0.200911
$296$	23.7918	1.38287
$297$	0	0
$298$	−8.85948	−0.513216
$299$	−6.35733	−0.367654
$300$	0	0
$301$	−14.1482	−0.815489
$302$	−16.2327	−0.934087
$303$	0	0
$304$	−0.303494	−0.0174066
$305$	8.32540	0.476711
$306$	0	0
$307$	13.8619	0.791143	0.395571	−	0.918435i	$-0.370547\pi$
$307$	13.8619	0.791143	0.395571	+	0.918435i	$0.370547\pi$
$308$	−0.145768	−0.00830589
$309$	0	0
$310$	−17.2093	−0.977423
$311$	24.5658	1.39300	0.696500	−	0.717557i	$-0.254739\pi$
$311$	24.5658	1.39300	0.696500	+	0.717557i	$0.254739\pi$
$312$	0	0
$313$	−30.5328	−1.72581	−0.862907	−	0.505362i	$-0.831359\pi$
$313$	−30.5328	−1.72581	−0.862907	+	0.505362i	$0.831359\pi$
$314$	9.51883	0.537179
$315$	0	0
$316$	1.99935	0.112472
$317$	−26.6677	−1.49781	−0.748904	−	0.662678i	$-0.769420\pi$
$317$	−26.6677	−1.49781	−0.748904	+	0.662678i	$0.769420\pi$
$318$	0	0
$319$	0.0654034	0.00366189
$320$	−12.8004	−0.715564
$321$	0	0
$322$	−12.0502	−0.671529
$323$	0	0
$324$	0	0
$325$	−0.561441	−0.0311431
$326$	17.2319	0.954389
$327$	0	0
$328$	25.6974	1.41890
$329$	−5.82282	−0.321022
$330$	0	0
$331$	17.5621	0.965302	0.482651	−	0.875813i	$-0.339674\pi$
$331$	17.5621	0.965302	0.482651	+	0.875813i	$0.339674\pi$
$332$	−13.2001	−0.724449
$333$	0	0
$334$	−10.6662	−0.583626
$335$	−1.56388	−0.0854440
$336$	0	0
$337$	−22.9406	−1.24965	−0.624826	−	0.780764i	$-0.714830\pi$
$337$	−22.9406	−1.24965	−0.624826	+	0.780764i	$0.714830\pi$
$338$	10.7148	0.582811
$339$	0	0
$340$	0	0
$341$	0.444462	0.0240690
$342$	0	0
$343$	19.3062	1.04244
$344$	16.4448	0.886642
$345$	0	0
$346$	−14.4044	−0.774388
$347$	1.39673	0.0749805	0.0374903	−	0.999297i	$-0.488064\pi$
$347$	1.39673	0.0749805	0.0374903	+	0.999297i	$0.488064\pi$
$348$	0	0
$349$	6.33758	0.339243	0.169621	−	0.985509i	$-0.445746\pi$
$349$	6.33758	0.339243	0.169621	+	0.985509i	$0.445746\pi$
$350$	−1.06420	−0.0568838
$351$	0	0
$352$	0.277821	0.0148079
$353$	−31.0008	−1.65001	−0.825004	−	0.565127i	$-0.808827\pi$
$353$	−31.0008	−1.65001	−0.825004	+	0.565127i	$0.808827\pi$
$354$	0	0
$355$	27.3558	1.45189
$356$	9.45170	0.500939
$357$	0	0
$358$	−11.2496	−0.594560
$359$	−10.7623	−0.568013	−0.284007	−	0.958822i	$-0.591664\pi$
$359$	−10.7623	−0.568013	−0.284007	+	0.958822i	$0.591664\pi$
$360$	0	0
$361$	−18.5988	−0.978882
$362$	10.9638	0.576244
$363$	0	0
$364$	3.51070	0.184011
$365$	−5.12782	−0.268402
$366$	0	0
$367$	−16.3676	−0.854381	−0.427191	−	0.904162i	$-0.640497\pi$
$367$	−16.3676	−0.854381	−0.427191	+	0.904162i	$0.640497\pi$
$368$	2.45667	0.128063
$369$	0	0
$370$	16.2284	0.843675
$371$	−30.7546	−1.59670
$372$	0	0
$373$	−27.1937	−1.40803	−0.704017	−	0.710183i	$-0.748612\pi$
$373$	−27.1937	−1.40803	−0.704017	+	0.710183i	$0.748612\pi$
$374$	0	0
$375$	0	0
$376$	6.76799	0.349032
$377$	−1.57519	−0.0811263
$378$	0	0
$379$	−8.30429	−0.426563	−0.213281	−	0.976991i	$-0.568415\pi$
$379$	−8.30429	−0.426563	−0.213281	+	0.976991i	$0.568415\pi$
$380$	1.52127	0.0780397
$381$	0	0
$382$	9.09129	0.465151
$383$	−14.4740	−0.739585	−0.369792	−	0.929114i	$-0.620571\pi$
$383$	−14.4740	−0.739585	−0.369792	+	0.929114i	$0.620571\pi$
$384$	0	0
$385$	−0.275996	−0.0140661
$386$	16.7246	0.851261
$387$	0	0
$388$	−5.54932	−0.281724
$389$	19.0549	0.966120	0.483060	−	0.875587i	$-0.339525\pi$
$389$	19.0549	0.966120	0.483060	+	0.875587i	$0.339525\pi$
$390$	0	0
$391$	0	0
$392$	−1.98424	−0.100219
$393$	0	0
$394$	−15.9515	−0.803625
$395$	3.78556	0.190472
$396$	0	0
$397$	0.288705	0.0144897	0.00724484	−	0.999974i	$-0.497694\pi$
$397$	0.288705	0.0144897	0.00724484	+	0.999974i	$0.497694\pi$
$398$	−15.9302	−0.798508
$399$	0	0
$400$	0.216958	0.0108479
$401$	16.8676	0.842326	0.421163	−	0.906985i	$-0.361622\pi$
$401$	16.8676	0.842326	0.421163	+	0.906985i	$0.361622\pi$
$402$	0	0
$403$	−10.7045	−0.533230
$404$	11.7359	0.583883
$405$	0	0
$406$	−2.98573	−0.148179
$407$	−0.419129	−0.0207754
$408$	0	0
$409$	−14.3422	−0.709174	−0.354587	−	0.935023i	$-0.615379\pi$
$409$	−14.3422	−0.709174	−0.354587	+	0.935023i	$0.615379\pi$
$410$	17.5282	0.865658
$411$	0	0
$412$	−10.5827	−0.521373
$413$	−4.06853	−0.200199
$414$	0	0
$415$	−24.9930	−1.22686
$416$	−6.69108	−0.328057
$417$	0	0
$418$	0.0304820	0.00149092
$419$	17.9333	0.876099	0.438049	−	0.898951i	$-0.355670\pi$
$419$	17.9333	0.876099	0.438049	+	0.898951i	$0.355670\pi$
$420$	0	0
$421$	−24.3033	−1.18447	−0.592235	−	0.805765i	$-0.701754\pi$
$421$	−24.3033	−1.18447	−0.592235	+	0.805765i	$0.701754\pi$
$422$	−8.32137	−0.405078
$423$	0	0
$424$	35.7467	1.73601
$425$	0	0
$426$	0	0
$427$	9.81583	0.475022
$428$	−22.1241	−1.06941
$429$	0	0
$430$	11.2170	0.540932
$431$	−12.1988	−0.587596	−0.293798	−	0.955868i	$-0.594919\pi$
$431$	−12.1988	−0.587596	−0.293798	+	0.955868i	$0.594919\pi$
$432$	0	0
$433$	28.6849	1.37851	0.689253	−	0.724521i	$-0.257939\pi$
$433$	28.6849	1.37851	0.689253	+	0.724521i	$0.257939\pi$
$434$	−20.2901	−0.973958
$435$	0	0
$436$	18.1978	0.871515
$437$	−3.24796	−0.155371
$438$	0	0
$439$	−31.7217	−1.51399	−0.756996	−	0.653420i	$-0.773334\pi$
$439$	−31.7217	−1.51399	−0.756996	+	0.653420i	$0.773334\pi$
$440$	0.320796	0.0152933
$441$	0	0
$442$	0	0
$443$	20.8705	0.991586	0.495793	−	0.868441i	$-0.334878\pi$
$443$	20.8705	0.991586	0.495793	+	0.868441i	$0.334878\pi$
$444$	0	0
$445$	17.8958	0.848342
$446$	26.3223	1.24640
$447$	0	0
$448$	−15.0920	−0.713028
$449$	−8.02255	−0.378607	−0.189304	−	0.981919i	$-0.560623\pi$
$449$	−8.02255	−0.378607	−0.189304	+	0.981919i	$0.560623\pi$
$450$	0	0
$451$	−0.452700	−0.0213168
$452$	−0.220685	−0.0103801
$453$	0	0
$454$	−10.9184	−0.512427
$455$	6.64714	0.311623
$456$	0	0
$457$	0.0642173	0.00300396	0.00150198	−	0.999999i	$-0.499522\pi$
$457$	0.0642173	0.00300396	0.00150198	+	0.999999i	$0.499522\pi$
$458$	−8.89541	−0.415655
$459$	0	0
$460$	−12.3141	−0.574149
$461$	−3.44572	−0.160483	−0.0802416	−	0.996775i	$-0.525569\pi$
$461$	−3.44572	−0.160483	−0.0802416	+	0.996775i	$0.525569\pi$
$462$	0	0
$463$	−19.9151	−0.925534	−0.462767	−	0.886480i	$-0.653143\pi$
$463$	−19.9151	−0.925534	−0.462767	+	0.886480i	$0.653143\pi$
$464$	0.608702	0.0282583
$465$	0	0
$466$	16.1056	0.746078
$467$	−11.4241	−0.528646	−0.264323	−	0.964434i	$-0.585149\pi$
$467$	−11.4241	−0.528646	−0.264323	+	0.964434i	$0.585149\pi$
$468$	0	0
$469$	−1.84385	−0.0851411
$470$	4.61645	0.212941
$471$	0	0
$472$	4.72894	0.217667
$473$	−0.289700	−0.0133204
$474$	0	0
$475$	−0.286840	−0.0131611
$476$	0	0
$477$	0	0
$478$	−3.07232	−0.140525
$479$	−25.8162	−1.17957	−0.589786	−	0.807560i	$-0.700788\pi$
$479$	−25.8162	−1.17957	−0.589786	+	0.807560i	$0.700788\pi$
$480$	0	0
$481$	10.0944	0.460264
$482$	1.68910	0.0769363
$483$	0	0
$484$	12.3856	0.562983
$485$	−10.5071	−0.477101
$486$	0	0
$487$	−38.0584	−1.72459	−0.862294	−	0.506407i	$-0.830973\pi$
$487$	−38.0584	−1.72459	−0.862294	+	0.506407i	$0.830973\pi$
$488$	−11.4092	−0.516468
$489$	0	0
$490$	−1.35345	−0.0611427
$491$	−21.5889	−0.974295	−0.487147	−	0.873320i	$-0.661963\pi$
$491$	−21.5889	−0.974295	−0.487147	+	0.873320i	$0.661963\pi$
$492$	0	0
$493$	0	0
$494$	−0.734134	−0.0330302
$495$	0	0
$496$	4.13656	0.185737
$497$	32.2530	1.44675
$498$	0	0
$499$	−24.6501	−1.10349	−0.551746	−	0.834012i	$-0.686038\pi$
$499$	−24.6501	−1.10349	−0.551746	+	0.834012i	$0.686038\pi$
$500$	−13.0955	−0.585648
$501$	0	0
$502$	1.54244	0.0688423
$503$	−18.4268	−0.821609	−0.410805	−	0.911723i	$-0.634752\pi$
$503$	−18.4268	−0.821609	−0.410805	+	0.911723i	$0.634752\pi$
$504$	0	0
$505$	22.2207	0.988809
$506$	−0.246740	−0.0109689
$507$	0	0
$508$	−21.8537	−0.969599
$509$	5.56944	0.246861	0.123431	−	0.992353i	$-0.460610\pi$
$509$	5.56944	0.246861	0.123431	+	0.992353i	$0.460610\pi$
$510$	0	0
$511$	−6.04581	−0.267451
$512$	5.38586	0.238024
$513$	0	0
$514$	14.2293	0.627629
$515$	−20.0373	−0.882947
$516$	0	0
$517$	−0.119228	−0.00524366
$518$	19.1337	0.840684
$519$	0	0
$520$	−7.72611	−0.338812
$521$	−13.4537	−0.589419	−0.294710	−	0.955587i	$-0.595223\pi$
$521$	−13.4537	−0.589419	−0.294710	+	0.955587i	$0.595223\pi$
$522$	0	0
$523$	42.3379	1.85131	0.925655	−	0.378370i	$-0.123515\pi$
$523$	42.3379	1.85131	0.925655	+	0.378370i	$0.123515\pi$
$524$	−14.9920	−0.654928
$525$	0	0
$526$	14.2980	0.623423
$527$	0	0
$528$	0	0
$529$	3.29098	0.143086
$530$	24.3829	1.05912
$531$	0	0
$532$	1.79361	0.0777631
$533$	10.9029	0.472257
$534$	0	0
$535$	−41.8897	−1.81105
$536$	2.14315	0.0925699
$537$	0	0
$538$	−0.955541	−0.0411963
$539$	0.0349554	0.00150564
$540$	0	0
$541$	−26.2172	−1.12716	−0.563582	−	0.826060i	$-0.690577\pi$
$541$	−26.2172	−1.12716	−0.563582	+	0.826060i	$0.690577\pi$
$542$	−0.247098	−0.0106138
$543$	0	0
$544$	0	0
$545$	34.4556	1.47591
$546$	0	0
$547$	23.2431	0.993803	0.496902	−	0.867807i	$-0.334471\pi$
$547$	23.2431	0.993803	0.496902	+	0.867807i	$0.334471\pi$
$548$	−11.8585	−0.506568
$549$	0	0
$550$	−0.0217906	−0.000929154 0
$551$	−0.804764	−0.0342841
$552$	0	0
$553$	4.46326	0.189797
$554$	4.39772	0.186841
$555$	0	0
$556$	8.63704	0.366292
$557$	−10.6676	−0.452003	−0.226001	−	0.974127i	$-0.572565\pi$
$557$	−10.6676	−0.452003	−0.226001	+	0.974127i	$0.572565\pi$
$558$	0	0
$559$	6.97718	0.295103
$560$	−2.56866	−0.108546
$561$	0	0
$562$	−25.5552	−1.07798
$563$	−26.8484	−1.13153	−0.565763	−	0.824568i	$-0.691418\pi$
$563$	−26.8484	−1.13153	−0.565763	+	0.824568i	$0.691418\pi$
$564$	0	0
$565$	−0.417843	−0.0175788
$566$	4.85616	0.204120
$567$	0	0
$568$	−37.4884	−1.57298
$569$	43.7194	1.83281	0.916406	−	0.400249i	$-0.131076\pi$
$569$	43.7194	1.83281	0.916406	+	0.400249i	$0.131076\pi$
$570$	0	0
$571$	−5.22623	−0.218711	−0.109355	−	0.994003i	$-0.534879\pi$
$571$	−5.22623	−0.218711	−0.109355	+	0.994003i	$0.534879\pi$
$572$	0.0718853	0.00300568
$573$	0	0
$574$	20.6662	0.862590
$575$	2.32186	0.0968283
$576$	0	0
$577$	7.37007	0.306820	0.153410	−	0.988163i	$-0.450974\pi$
$577$	7.37007	0.306820	0.153410	+	0.988163i	$0.450974\pi$
$578$	0	0
$579$	0	0
$580$	−3.05114	−0.126692
$581$	−29.4673	−1.22251
$582$	0	0
$583$	−0.629733	−0.0260809
$584$	7.02718	0.290787
$585$	0	0
$586$	31.8000	1.31364
$587$	−4.82529	−0.199161	−0.0995806	−	0.995029i	$-0.531750\pi$
$587$	−4.82529	−0.199161	−0.0995806	+	0.995029i	$0.531750\pi$
$588$	0	0
$589$	−5.46893	−0.225343
$590$	3.22561	0.132796
$591$	0	0
$592$	−3.90078	−0.160321
$593$	17.4847	0.718012	0.359006	−	0.933335i	$-0.383116\pi$
$593$	17.4847	0.718012	0.359006	+	0.933335i	$0.383116\pi$
$594$	0	0
$595$	0	0
$596$	−10.6743	−0.437238
$597$	0	0
$598$	5.94253	0.243008
$599$	−34.7504	−1.41986	−0.709931	−	0.704271i	$-0.751274\pi$
$599$	−34.7504	−1.41986	−0.709931	+	0.704271i	$0.751274\pi$
$600$	0	0
$601$	−30.5956	−1.24802	−0.624010	−	0.781416i	$-0.714498\pi$
$601$	−30.5956	−1.24802	−0.624010	+	0.781416i	$0.714498\pi$
$602$	13.2251	0.539014
$603$	0	0
$604$	−19.5580	−0.795802
$605$	23.4509	0.953413
$606$	0	0
$607$	−18.8527	−0.765209	−0.382604	−	0.923912i	$-0.624973\pi$
$607$	−18.8527	−0.765209	−0.382604	+	0.923912i	$0.624973\pi$
$608$	−3.41847	−0.138637
$609$	0	0
$610$	−7.78220	−0.315092
$611$	2.87152	0.116169
$612$	0	0
$613$	27.5903	1.11436	0.557181	−	0.830391i	$-0.311883\pi$
$613$	27.5903	1.11436	0.557181	+	0.830391i	$0.311883\pi$
$614$	−12.9575	−0.522922
$615$	0	0
$616$	0.378225	0.0152391
$617$	−28.1344	−1.13265	−0.566324	−	0.824183i	$-0.691635\pi$
$617$	−28.1344	−1.13265	−0.566324	+	0.824183i	$0.691635\pi$
$618$	0	0
$619$	6.29344	0.252955	0.126477	−	0.991969i	$-0.459633\pi$
$619$	6.29344	0.252955	0.126477	+	0.991969i	$0.459633\pi$
$620$	−20.7346	−0.832722
$621$	0	0
$622$	−22.9630	−0.920732
$623$	21.0995	0.845335
$624$	0	0
$625$	−22.5308	−0.901232
$626$	28.5406	1.14071
$627$	0	0
$628$	11.4688	0.457653
$629$	0	0
$630$	0	0
$631$	−5.75820	−0.229231	−0.114615	−	0.993410i	$-0.536564\pi$
$631$	−5.75820	−0.229231	−0.114615	+	0.993410i	$0.536564\pi$
$632$	−5.18774	−0.206357
$633$	0	0
$634$	24.9277	0.990007
$635$	−41.3776	−1.64202
$636$	0	0
$637$	−0.841873	−0.0333562
$638$	−0.0611361	−0.00242040
$639$	0	0
$640$	−11.0506	−0.436812
$641$	−43.3802	−1.71342	−0.856708	−	0.515802i	$-0.827494\pi$
$641$	−43.3802	−1.71342	−0.856708	+	0.515802i	$0.827494\pi$
$642$	0	0
$643$	34.0941	1.34454	0.672269	−	0.740307i	$-0.265320\pi$
$643$	34.0941	1.34454	0.672269	+	0.740307i	$0.265320\pi$
$644$	−14.5186	−0.572114
$645$	0	0
$646$	0	0
$647$	−25.0959	−0.986623	−0.493312	−	0.869853i	$-0.664214\pi$
$647$	−25.0959	−0.986623	−0.493312	+	0.869853i	$0.664214\pi$
$648$	0	0
$649$	−0.0833075	−0.00327011
$650$	0.524809	0.0205847
$651$	0	0
$652$	20.7619	0.813098
$653$	−10.6508	−0.416797	−0.208398	−	0.978044i	$-0.566825\pi$
$653$	−10.6508	−0.416797	−0.208398	+	0.978044i	$0.566825\pi$
$654$	0	0
$655$	−28.3858	−1.10912
$656$	−4.21322	−0.164499
$657$	0	0
$658$	5.44290	0.212186
$659$	38.7396	1.50908	0.754540	−	0.656255i	$-0.227860\pi$
$659$	38.7396	1.50908	0.754540	+	0.656255i	$0.227860\pi$
$660$	0	0
$661$	45.9763	1.78827	0.894135	−	0.447798i	$-0.147792\pi$
$661$	45.9763	1.78827	0.894135	+	0.447798i	$0.147792\pi$
$662$	−16.4163	−0.638036
$663$	0	0
$664$	34.2504	1.32917
$665$	3.39602	0.131692
$666$	0	0
$667$	6.51425	0.252233
$668$	−12.8511	−0.497224
$669$	0	0
$670$	1.46184	0.0564759
$671$	0.200990	0.00775912
$672$	0	0
$673$	18.3510	0.707380	0.353690	−	0.935363i	$-0.384927\pi$
$673$	18.3510	0.707380	0.353690	+	0.935363i	$0.384927\pi$
$674$	21.4438	0.825983
$675$	0	0
$676$	12.9098	0.496530
$677$	9.65888	0.371221	0.185610	−	0.982623i	$-0.440574\pi$
$677$	9.65888	0.371221	0.185610	+	0.982623i	$0.440574\pi$
$678$	0	0
$679$	−12.3880	−0.475410
$680$	0	0
$681$	0	0
$682$	−0.415463	−0.0159089
$683$	−5.72299	−0.218984	−0.109492	−	0.993988i	$-0.534922\pi$
$683$	−5.72299	−0.218984	−0.109492	+	0.993988i	$0.534922\pi$
$684$	0	0
$685$	−22.4527	−0.857875
$686$	−18.0466	−0.689021
$687$	0	0
$688$	−2.69620	−0.102792
$689$	15.1666	0.577802
$690$	0	0
$691$	11.4136	0.434193	0.217096	−	0.976150i	$-0.430341\pi$
$691$	11.4136	0.434193	0.217096	+	0.976150i	$0.430341\pi$
$692$	−17.3552	−0.659745
$693$	0	0
$694$	−1.30560	−0.0495599
$695$	16.3533	0.620317
$696$	0	0
$697$	0	0
$698$	−5.92407	−0.224229
$699$	0	0
$700$	−1.28220	−0.0484625
$701$	38.3043	1.44673	0.723367	−	0.690464i	$-0.242594\pi$
$701$	38.3043	1.44673	0.723367	+	0.690464i	$0.242594\pi$
$702$	0	0
$703$	5.15722	0.194508
$704$	−0.309024	−0.0116468
$705$	0	0
$706$	28.9781	1.09061
$707$	26.1987	0.985304
$708$	0	0
$709$	−22.0651	−0.828673	−0.414336	−	0.910124i	$-0.635986\pi$
$709$	−22.0651	−0.828673	−0.414336	+	0.910124i	$0.635986\pi$
$710$	−25.5709	−0.959658
$711$	0	0
$712$	−24.5244	−0.919092
$713$	44.2689	1.65788
$714$	0	0
$715$	0.136107	0.00509012
$716$	−13.5541	−0.506539
$717$	0	0
$718$	10.0601	0.375440
$719$	−43.3181	−1.61549	−0.807746	−	0.589531i	$-0.799313\pi$
$719$	−43.3181	−1.61549	−0.807746	+	0.589531i	$0.799313\pi$
$720$	0	0
$721$	−23.6244	−0.879818
$722$	17.3852	0.647012
$723$	0	0
$724$	13.2097	0.490935
$725$	0.575299	0.0213661
$726$	0	0
$727$	−0.285170	−0.0105764	−0.00528818	−	0.999986i	$-0.501683\pi$
$727$	−0.285170	−0.0105764	−0.00528818	+	0.999986i	$0.501683\pi$
$728$	−9.10925	−0.337611
$729$	0	0
$730$	4.79325	0.177406
$731$	0	0
$732$	0	0
$733$	17.8571	0.659568	0.329784	−	0.944056i	$-0.393024\pi$
$733$	17.8571	0.659568	0.329784	+	0.944056i	$0.393024\pi$
$734$	15.2997	0.564721
$735$	0	0
$736$	27.6712	1.01998
$737$	−0.0377548	−0.00139072
$738$	0	0
$739$	11.9813	0.440738	0.220369	−	0.975417i	$-0.429274\pi$
$739$	11.9813	0.440738	0.220369	+	0.975417i	$0.429274\pi$
$740$	19.5528	0.718775
$741$	0	0
$742$	28.7480	1.05537
$743$	6.24869	0.229242	0.114621	−	0.993409i	$-0.463435\pi$
$743$	6.24869	0.229242	0.114621	+	0.993409i	$0.463435\pi$
$744$	0	0
$745$	−20.2107	−0.740464
$746$	25.4194	0.930669
$747$	0	0
$748$	0	0
$749$	−49.3888	−1.80463
$750$	0	0
$751$	−7.34373	−0.267976	−0.133988	−	0.990983i	$-0.542778\pi$
$751$	−7.34373	−0.267976	−0.133988	+	0.990983i	$0.542778\pi$
$752$	−1.10964	−0.0404646
$753$	0	0
$754$	1.47241	0.0536221
$755$	−37.0309	−1.34769
$756$	0	0
$757$	31.1599	1.13253	0.566264	−	0.824224i	$-0.308388\pi$
$757$	31.1599	1.13253	0.566264	+	0.824224i	$0.308388\pi$
$758$	7.76246	0.281945
$759$	0	0
$760$	−3.94727	−0.143182
$761$	32.7906	1.18866	0.594330	−	0.804222i	$-0.297417\pi$
$761$	32.7906	1.18866	0.594330	+	0.804222i	$0.297417\pi$
$762$	0	0
$763$	40.6239	1.47068
$764$	10.9536	0.396288
$765$	0	0
$766$	13.5296	0.488844
$767$	2.00639	0.0724467
$768$	0	0
$769$	−42.3229	−1.52620	−0.763102	−	0.646278i	$-0.776325\pi$
$769$	−42.3229	−1.52620	−0.763102	+	0.646278i	$0.776325\pi$
$770$	0.257988	0.00929725
$771$	0	0
$772$	20.1507	0.725238
$773$	−47.1035	−1.69420	−0.847098	−	0.531437i	$-0.821652\pi$
$773$	−47.1035	−1.69420	−0.847098	+	0.531437i	$0.821652\pi$
$774$	0	0
$775$	3.90956	0.140436
$776$	14.3989	0.516890
$777$	0	0
$778$	−17.8116	−0.638577
$779$	5.57029	0.199576
$780$	0	0
$781$	0.660416	0.0236315
$782$	0	0
$783$	0	0
$784$	0.325326	0.0116188
$785$	21.7149	0.775037
$786$	0	0
$787$	8.28287	0.295252	0.147626	−	0.989043i	$-0.452837\pi$
$787$	8.28287	0.295252	0.147626	+	0.989043i	$0.452837\pi$
$788$	−19.2191	−0.684654
$789$	0	0
$790$	−3.53857	−0.125897
$791$	−0.492646	−0.0175165
$792$	0	0
$793$	−4.84067	−0.171897
$794$	−0.269868	−0.00957725
$795$	0	0
$796$	−19.1935	−0.680294
$797$	21.6186	0.765768	0.382884	−	0.923796i	$-0.374931\pi$
$797$	21.6186	0.765768	0.382884	+	0.923796i	$0.374931\pi$
$798$	0	0
$799$	0	0
$800$	2.44376	0.0863998
$801$	0	0
$802$	−15.7670	−0.556753
$803$	−0.123794	−0.00436861
$804$	0	0
$805$	−27.4895	−0.968877
$806$	10.0061	0.352449
$807$	0	0
$808$	−30.4513	−1.07127
$809$	−39.2750	−1.38084	−0.690418	−	0.723411i	$-0.742573\pi$
$809$	−39.2750	−1.38084	−0.690418	+	0.723411i	$0.742573\pi$
$810$	0	0
$811$	55.3481	1.94354	0.971768	−	0.235939i	$-0.0758167\pi$
$811$	55.3481	1.94354	0.971768	+	0.235939i	$0.0758167\pi$
$812$	−3.59736	−0.126242
$813$	0	0
$814$	0.391782	0.0137320
$815$	39.3104	1.37698
$816$	0	0
$817$	3.56464	0.124711
$818$	13.4064	0.468743
$819$	0	0
$820$	21.1189	0.737504
$821$	12.7592	0.445300	0.222650	−	0.974898i	$-0.428529\pi$
$821$	12.7592	0.445300	0.222650	+	0.974898i	$0.428529\pi$
$822$	0	0
$823$	50.1243	1.74722	0.873612	−	0.486623i	$-0.161772\pi$
$823$	50.1243	1.74722	0.873612	+	0.486623i	$0.161772\pi$
$824$	27.4591	0.956584
$825$	0	0
$826$	3.80307	0.132326
$827$	0.250279	0.00870306	0.00435153	−	0.999991i	$-0.498615\pi$
$827$	0.250279	0.00870306	0.00435153	+	0.999991i	$0.498615\pi$
$828$	0	0
$829$	3.00006	0.104196	0.0520982	−	0.998642i	$-0.483409\pi$
$829$	3.00006	0.104196	0.0520982	+	0.998642i	$0.483409\pi$
$830$	23.3623	0.810916
$831$	0	0
$832$	7.44259	0.258025
$833$	0	0
$834$	0	0
$835$	−24.3322	−0.842051
$836$	0.0367262	0.00127020
$837$	0	0
$838$	−16.7632	−0.579075
$839$	50.1076	1.72991	0.864954	−	0.501851i	$-0.167348\pi$
$839$	50.1076	1.72991	0.864954	+	0.501851i	$0.167348\pi$
$840$	0	0
$841$	−27.3859	−0.944342
$842$	22.7176	0.782900
$843$	0	0
$844$	−10.0260	−0.345109
$845$	24.4433	0.840875
$846$	0	0
$847$	27.6491	0.950034
$848$	−5.86085	−0.201262
$849$	0	0
$850$	0	0
$851$	−41.7457	−1.43102
$852$	0	0
$853$	−25.7516	−0.881717	−0.440858	−	0.897577i	$-0.645326\pi$
$853$	−25.7516	−0.881717	−0.440858	+	0.897577i	$0.645326\pi$
$854$	−9.17538	−0.313975
$855$	0	0
$856$	57.4057	1.96209
$857$	38.0376	1.29934	0.649670	−	0.760217i	$-0.274907\pi$
$857$	38.0376	1.29934	0.649670	+	0.760217i	$0.274907\pi$
$858$	0	0
$859$	14.4106	0.491683	0.245842	−	0.969310i	$-0.420936\pi$
$859$	14.4106	0.491683	0.245842	+	0.969310i	$0.420936\pi$
$860$	13.5148	0.460850
$861$	0	0
$862$	11.4029	0.388384
$863$	12.2991	0.418666	0.209333	−	0.977844i	$-0.432871\pi$
$863$	12.2991	0.418666	0.209333	+	0.977844i	$0.432871\pi$
$864$	0	0
$865$	−32.8602	−1.11728
$866$	−26.8133	−0.911152
$867$	0	0
$868$	−24.4466	−0.829770
$869$	0.0913901	0.00310020
$870$	0	0
$871$	0.909294	0.0308103
$872$	−47.2180	−1.59900
$873$	0	0
$874$	3.03604	0.102696
$875$	−29.2338	−0.988282
$876$	0	0
$877$	−15.2249	−0.514108	−0.257054	−	0.966397i	$-0.582752\pi$
$877$	−15.2249	−0.514108	−0.257054	+	0.966397i	$0.582752\pi$
$878$	29.6519	1.00070
$879$	0	0
$880$	−0.0525961	−0.00177301
$881$	5.07852	0.171100	0.0855499	−	0.996334i	$-0.472735\pi$
$881$	5.07852	0.171100	0.0855499	+	0.996334i	$0.472735\pi$
$882$	0	0
$883$	−45.2155	−1.52162	−0.760811	−	0.648973i	$-0.775199\pi$
$883$	−45.2155	−1.52162	−0.760811	+	0.648973i	$0.775199\pi$
$884$	0	0
$885$	0	0
$886$	−19.5087	−0.655409
$887$	−39.2414	−1.31760	−0.658798	−	0.752320i	$-0.728935\pi$
$887$	−39.2414	−1.31760	−0.658798	+	0.752320i	$0.728935\pi$
$888$	0	0
$889$	−48.7851	−1.63620
$890$	−16.7282	−0.560729
$891$	0	0
$892$	31.7144	1.06188
$893$	1.46706	0.0490933
$894$	0	0
$895$	−25.6632	−0.857826
$896$	−13.0289	−0.435264
$897$	0	0
$898$	7.49910	0.250248
$899$	10.9687	0.365828
$900$	0	0
$901$	0	0
$902$	0.423162	0.0140898
$903$	0	0
$904$	0.572613	0.0190448
$905$	25.0112	0.831400
$906$	0	0
$907$	−30.4548	−1.01124	−0.505618	−	0.862757i	$-0.668736\pi$
$907$	−30.4548	−1.01124	−0.505618	+	0.862757i	$0.668736\pi$
$908$	−13.1551	−0.436566
$909$	0	0
$910$	−6.21343	−0.205973
$911$	10.4870	0.347450	0.173725	−	0.984794i	$-0.444420\pi$
$911$	10.4870	0.347450	0.173725	+	0.984794i	$0.444420\pi$
$912$	0	0
$913$	−0.603374	−0.0199688
$914$	−0.0600274	−0.00198553
$915$	0	0
$916$	−10.7176	−0.354120
$917$	−33.4674	−1.10519
$918$	0	0
$919$	21.2263	0.700190	0.350095	−	0.936714i	$-0.386149\pi$
$919$	21.2263	0.700190	0.350095	+	0.936714i	$0.386149\pi$
$920$	31.9516	1.05341
$921$	0	0
$922$	3.22090	0.106075
$923$	−15.9056	−0.523538
$924$	0	0
$925$	−3.68673	−0.121219
$926$	18.6157	0.611751
$927$	0	0
$928$	6.85625	0.225067
$929$	−12.3636	−0.405635	−0.202818	−	0.979217i	$-0.565010\pi$
$929$	−12.3636	−0.405635	−0.202818	+	0.979217i	$0.565010\pi$
$930$	0	0
$931$	−0.430113	−0.0140964
$932$	19.4048	0.635626
$933$	0	0
$934$	10.6788	0.349420
$935$	0	0
$936$	0	0
$937$	3.45931	0.113011	0.0565054	−	0.998402i	$-0.482004\pi$
$937$	3.45931	0.113011	0.0565054	+	0.998402i	$0.482004\pi$
$938$	1.72355	0.0562758
$939$	0	0
$940$	5.56213	0.181417
$941$	41.0791	1.33914	0.669571	−	0.742748i	$-0.266478\pi$
$941$	41.0791	1.33914	0.669571	+	0.742748i	$0.266478\pi$
$942$	0	0
$943$	−45.0894	−1.46831
$944$	−0.775333	−0.0252349
$945$	0	0
$946$	0.270798	0.00880440
$947$	13.6512	0.443606	0.221803	−	0.975092i	$-0.428806\pi$
$947$	13.6512	0.443606	0.221803	+	0.975092i	$0.428806\pi$
$948$	0	0
$949$	2.98149	0.0967832
$950$	0.268125	0.00869911
$951$	0	0
$952$	0	0
$953$	−21.1693	−0.685741	−0.342871	−	0.939383i	$-0.611399\pi$
$953$	−21.1693	−0.685741	−0.342871	+	0.939383i	$0.611399\pi$
$954$	0	0
$955$	20.7395	0.671116
$956$	−3.70168	−0.119721
$957$	0	0
$958$	24.1318	0.779662
$959$	−26.4723	−0.854834
$960$	0	0
$961$	43.5403	1.40453
$962$	−9.43575	−0.304221
$963$	0	0
$964$	2.03511	0.0655464
$965$	38.1531	1.22819
$966$	0	0
$967$	23.5134	0.756139	0.378069	−	0.925777i	$-0.376588\pi$
$967$	23.5134	0.756139	0.378069	+	0.925777i	$0.376588\pi$
$968$	−32.1371	−1.03293
$969$	0	0
$970$	9.82150	0.315349
$971$	−7.86563	−0.252420	−0.126210	−	0.992004i	$-0.540281\pi$
$971$	−7.86563	−0.252420	−0.126210	+	0.992004i	$0.540281\pi$
$972$	0	0
$973$	19.2809	0.618118
$974$	35.5752	1.13990
$975$	0	0
$976$	1.87059	0.0598760
$977$	−48.0339	−1.53674	−0.768371	−	0.640005i	$-0.778932\pi$
$977$	−48.0339	−1.53674	−0.768371	+	0.640005i	$0.778932\pi$
$978$	0	0
$979$	0.432036	0.0138079
$980$	−1.63070	−0.0520910
$981$	0	0
$982$	20.1803	0.643980
$983$	−11.5978	−0.369911	−0.184956	−	0.982747i	$-0.559214\pi$
$983$	−11.5978	−0.369911	−0.184956	+	0.982747i	$0.559214\pi$
$984$	0	0
$985$	−36.3894	−1.15946
$986$	0	0
$987$	0	0
$988$	−0.884520	−0.0281403
$989$	−28.8544	−0.917517
$990$	0	0
$991$	−36.3964	−1.15617	−0.578084	−	0.815977i	$-0.696200\pi$
$991$	−36.3964	−1.15617	−0.578084	+	0.815977i	$0.696200\pi$
$992$	46.5930	1.47933
$993$	0	0
$994$	−30.1486	−0.956257
$995$	−36.3408	−1.15208
$996$	0	0
$997$	25.1521	0.796574	0.398287	−	0.917261i	$-0.369605\pi$
$997$	25.1521	0.796574	0.398287	+	0.917261i	$0.369605\pi$
$998$	23.0418	0.729375
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7803.2.a.bv.1.5		12
3.2	odd	2	inner	7803.2.a.bv.1.8		12
17.2	even	8		459.2.f.c.55.5	✓	24
17.9	even	8		459.2.f.c.217.8	yes	24
17.16	even	2		7803.2.a.bw.1.5		12
51.2	odd	8		459.2.f.c.55.8	yes	24
51.26	odd	8		459.2.f.c.217.5	yes	24
51.50	odd	2		7803.2.a.bw.1.8		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
459.2.f.c.55.5	✓	24	17.2	even	8
459.2.f.c.55.8	yes	24	51.2	odd	8
459.2.f.c.217.5	yes	24	51.26	odd	8
459.2.f.c.217.8	yes	24	17.9	even	8
7803.2.a.bv.1.5		12	1.1	even	1	trivial
7803.2.a.bv.1.8		12	3.2	odd	2	inner
7803.2.a.bw.1.5		12	17.16	even	2
7803.2.a.bw.1.8		12	51.50	odd	2

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:	\( N \)	\(=\)	\( 7803 = 3^{3} \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7803.a (trivial)

\(f(q)\)	\(=\)	\(q-0.934753 q^{2} -1.12624 q^{4} -2.13241 q^{5} -2.51416 q^{7} +2.92226 q^{8} +1.99328 q^{10} -0.0514801 q^{11} +1.23986 q^{13} +2.35012 q^{14} -0.479119 q^{16} +0.633442 q^{19} +2.40160 q^{20} +0.0481212 q^{22} -5.12747 q^{23} -0.452827 q^{25} -1.15896 q^{26} +2.83154 q^{28} -1.27046 q^{29} -8.63367 q^{31} -5.39666 q^{32} +5.36122 q^{35} +8.14157 q^{37} -0.592112 q^{38} -6.23146 q^{40} +8.79368 q^{41} +5.62741 q^{43} +0.0579788 q^{44} +4.79292 q^{46} +2.31601 q^{47} -0.679008 q^{49} +0.423282 q^{50} -1.39637 q^{52} +12.2326 q^{53} +0.109777 q^{55} -7.34702 q^{56} +1.18757 q^{58} +1.61825 q^{59} -3.90422 q^{61} +8.07035 q^{62} +6.00278 q^{64} -2.64388 q^{65} +0.733387 q^{67} -5.01141 q^{70} -12.8286 q^{71} +2.40471 q^{73} -7.61036 q^{74} -0.713406 q^{76} +0.129429 q^{77} -1.77525 q^{79} +1.02168 q^{80} -8.21992 q^{82} +11.7205 q^{83} -5.26024 q^{86} -0.150438 q^{88} -8.39229 q^{89} -3.11719 q^{91} +5.77475 q^{92} -2.16490 q^{94} -1.35076 q^{95} +4.92731 q^{97} +0.634705 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 12 q^{4} - 8 q^{7} - 24 q^{10} - 4 q^{13} + 20 q^{16} - 4 q^{19} + 28 q^{22} - 48 q^{28} - 8 q^{31} - 12 q^{37} - 44 q^{40} - 20 q^{43} - 36 q^{46} + 28 q^{49} + 8 q^{52} - 4 q^{55} - 28 q^{58} - 48 q^{61}+ \cdots - 24 q^{97}+O(q^{100}) \) 12 * q + 12 * q^4 - 8 * q^7 - 24 * q^10 - 4 * q^13 + 20 * q^16 - 4 * q^19 + 28 * q^22 - 48 * q^28 - 8 * q^31 - 12 * q^37 - 44 * q^40 - 20 * q^43 - 36 * q^46 + 28 * q^49 + 8 * q^52 - 4 * q^55 - 28 * q^58 - 48 * q^61 + 40 * q^64 - 8 * q^67 + 44 * q^70 - 4 * q^73 - 28 * q^76 - 40 * q^79 - 76 * q^82 + 40 * q^88 - 52 * q^91 + 28 * q^94 - 24 * q^97

Introduction

L-functions

Modular forms

Varieties

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Representations

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Database

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