LMFDB - Embedded newform 9409.2.a.m.1.23

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [9409,2,Mod(1,9409)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9409, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("9409.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 9409 = 97^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9409.a (trivial)

Newform invariants

comment:select newform

sage:traces = [56,0,-8,48,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	yes
Analytic conductor:	$75.1312432618$
Analytic rank:	$1$
Dimension:	$56$
Twist minimal:	no (minimal twist has level 97)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.23
Character	$\chi$	$=$	9409.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-0.650656 q^{2} -2.04155 q^{3} -1.57665 q^{4} -1.84237 q^{5} +1.32835 q^{6} -2.56780 q^{7} +2.32717 q^{8} +1.16793 q^{9} +1.19875 q^{10} -4.66112 q^{11} +3.21880 q^{12} -6.86367 q^{13} +1.67075 q^{14} +3.76130 q^{15} +1.63911 q^{16} +4.68768 q^{17} -0.759920 q^{18} -4.52836 q^{19} +2.90477 q^{20} +5.24229 q^{21} +3.03279 q^{22} -5.73987 q^{23} -4.75103 q^{24} -1.60566 q^{25} +4.46589 q^{26} +3.74027 q^{27} +4.04851 q^{28} +2.48714 q^{29} -2.44731 q^{30} -5.96236 q^{31} -5.72083 q^{32} +9.51591 q^{33} -3.05007 q^{34} +4.73084 q^{35} -1.84141 q^{36} -5.36028 q^{37} +2.94641 q^{38} +14.0125 q^{39} -4.28751 q^{40} +0.304536 q^{41} -3.41093 q^{42} -4.65947 q^{43} +7.34894 q^{44} -2.15176 q^{45} +3.73468 q^{46} -1.31590 q^{47} -3.34632 q^{48} -0.406420 q^{49} +1.04473 q^{50} -9.57013 q^{51} +10.8216 q^{52} +3.83913 q^{53} -2.43363 q^{54} +8.58752 q^{55} -5.97569 q^{56} +9.24488 q^{57} -1.61827 q^{58} +10.4680 q^{59} -5.93024 q^{60} -6.16483 q^{61} +3.87945 q^{62} -2.99900 q^{63} +0.444079 q^{64} +12.6455 q^{65} -6.19158 q^{66} +2.81947 q^{67} -7.39081 q^{68} +11.7182 q^{69} -3.07815 q^{70} -1.49606 q^{71} +2.71796 q^{72} +0.517325 q^{73} +3.48770 q^{74} +3.27804 q^{75} +7.13963 q^{76} +11.9688 q^{77} -9.11734 q^{78} +5.15153 q^{79} -3.01985 q^{80} -11.1397 q^{81} -0.198148 q^{82} +3.49084 q^{83} -8.26523 q^{84} -8.63645 q^{85} +3.03171 q^{86} -5.07762 q^{87} -10.8472 q^{88} -4.80065 q^{89} +1.40006 q^{90} +17.6245 q^{91} +9.04975 q^{92} +12.1725 q^{93} +0.856199 q^{94} +8.34293 q^{95} +11.6794 q^{96} +0.264440 q^{98} -5.44385 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 56 q - 8 q^{3} + 48 q^{4} - 24 q^{6} + 24 q^{9} - 48 q^{11} - 56 q^{12} + 24 q^{16} + 16 q^{18} - 40 q^{24} + 8 q^{25} - 32 q^{27} - 56 q^{31} - 8 q^{33} - 56 q^{35} + 40 q^{36} - 32 q^{43} - 144 q^{44}+ \cdots + 16 q^{99}+O(q^{100}) $ 56 * q - 8 * q^3 + 48 * q^4 - 24 * q^6 + 24 * q^9 - 48 * q^11 - 56 * q^12 + 24 * q^16 + 16 * q^18 - 40 * q^24 + 8 * q^25 - 32 * q^27 - 56 * q^31 - 8 * q^33 - 56 * q^35 + 40 * q^36 - 32 * q^43 - 144 * q^44 - 48 * q^47 - 112 * q^48 + 56 * q^49 + 16 * q^53 - 72 * q^54 + 24 * q^61 - 56 * q^62 + 32 * q^64 - 16 * q^65 - 64 * q^66 + 8 * q^70 + 128 * q^72 - 40 * q^73 - 72 * q^75 - 144 * q^79 - 88 * q^81 - 104 * q^85 - 64 * q^86 - 16 * q^88 - 80 * q^89 - 120 * q^91 - 48 * q^93 - 160 * q^94 - 136 * q^95 + 16 * q^96 + 96 * q^98 + 16 * q^99

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 $

$2$	−0.650656	−0.460083	−0.230042	−	0.973181i	$-0.573886\pi$
$2$	−0.650656	−0.460083	−0.230042	+	0.973181i	$0.573886\pi$
$3$	−2.04155	−1.17869	−0.589345	−	0.807882i	$-0.700614\pi$
$3$	−2.04155	−1.17869	−0.589345	+	0.807882i	$0.700614\pi$
$4$	−1.57665	−0.788323
$5$	−1.84237	−0.823934	−0.411967	−	0.911199i	$-0.635158\pi$
$5$	−1.84237	−0.823934	−0.411967	+	0.911199i	$0.635158\pi$
$6$	1.32835	0.542296
$7$	−2.56780	−0.970536	−0.485268	−	0.874365i	$-0.661278\pi$
$7$	−2.56780	−0.970536	−0.485268	+	0.874365i	$0.661278\pi$
$8$	2.32717	0.822778
$9$	1.16793	0.389309
$10$	1.19875	0.379079
$11$	−4.66112	−1.40538	−0.702690	−	0.711496i	$-0.748018\pi$
$11$	−4.66112	−1.40538	−0.702690	+	0.711496i	$0.748018\pi$
$12$	3.21880	0.929188
$13$	−6.86367	−1.90364	−0.951820	−	0.306656i	$-0.900790\pi$
$13$	−6.86367	−1.90364	−0.951820	+	0.306656i	$0.900790\pi$
$14$	1.67075	0.446527
$15$	3.76130	0.971163
$16$	1.63911	0.409777
$17$	4.68768	1.13693	0.568464	−	0.822708i	$-0.307538\pi$
$17$	4.68768	1.13693	0.568464	+	0.822708i	$0.307538\pi$
$18$	−0.759920	−0.179115
$19$	−4.52836	−1.03888	−0.519439	−	0.854508i	$-0.673859\pi$
$19$	−4.52836	−1.03888	−0.519439	+	0.854508i	$0.673859\pi$
$20$	2.90477	0.649527
$21$	5.24229	1.14396
$22$	3.03279	0.646592
$23$	−5.73987	−1.19685	−0.598423	−	0.801180i	$-0.704206\pi$
$23$	−5.73987	−1.19685	−0.598423	+	0.801180i	$0.704206\pi$
$24$	−4.75103	−0.969800
$25$	−1.60566	−0.321132
$26$	4.46589	0.875834
$27$	3.74027	0.719815
$28$	4.04851	0.765096
$29$	2.48714	0.461850	0.230925	−	0.972972i	$-0.425825\pi$
$29$	2.48714	0.461850	0.230925	+	0.972972i	$0.425825\pi$
$30$	−2.44731	−0.446816
$31$	−5.96236	−1.07087	−0.535436	−	0.844576i	$-0.679853\pi$
$31$	−5.96236	−1.07087	−0.535436	+	0.844576i	$0.679853\pi$
$32$	−5.72083	−1.01131
$33$	9.51591	1.65651
$34$	−3.05007	−0.523082
$35$	4.73084	0.799658
$36$	−1.84141	−0.306902
$37$	−5.36028	−0.881225	−0.440612	−	0.897697i	$-0.645239\pi$
$37$	−5.36028	−0.881225	−0.440612	+	0.897697i	$0.645239\pi$
$38$	2.94641	0.477970
$39$	14.0125	2.24380
$40$	−4.28751	−0.677915
$41$	0.304536	0.0475605	0.0237803	−	0.999717i	$-0.492430\pi$
$41$	0.304536	0.0475605	0.0237803	+	0.999717i	$0.492430\pi$
$42$	−3.41093	−0.526317
$43$	−4.65947	−0.710563	−0.355281	−	0.934759i	$-0.615615\pi$
$43$	−4.65947	−0.710563	−0.355281	+	0.934759i	$0.615615\pi$
$44$	7.34894	1.10789
$45$	−2.15176	−0.320765
$46$	3.73468	0.550649
$47$	−1.31590	−0.191944	−0.0959719	−	0.995384i	$-0.530596\pi$
$47$	−1.31590	−0.191944	−0.0959719	+	0.995384i	$0.530596\pi$
$48$	−3.34632	−0.483000
$49$	−0.406420	−0.0580600
$50$	1.04473	0.147748
$51$	−9.57013	−1.34009
$52$	10.8216	1.50068
$53$	3.83913	0.527346	0.263673	−	0.964612i	$-0.415066\pi$
$53$	3.83913	0.527346	0.263673	+	0.964612i	$0.415066\pi$
$54$	−2.43363	−0.331175
$55$	8.58752	1.15794
$56$	−5.97569	−0.798535
$57$	9.24488	1.22451
$58$	−1.61827	−0.212490
$59$	10.4680	1.36282	0.681409	−	0.731903i	$-0.261368\pi$
$59$	10.4680	1.36282	0.681409	+	0.731903i	$0.261368\pi$
$60$	−5.93024	−0.765590
$61$	−6.16483	−0.789325	−0.394663	−	0.918826i	$-0.629139\pi$
$61$	−6.16483	−0.789325	−0.394663	+	0.918826i	$0.629139\pi$
$62$	3.87945	0.492690
$63$	−2.99900	−0.377839
$64$	0.444079	0.0555098
$65$	12.6455	1.56848
$66$	−6.19158	−0.762131
$67$	2.81947	0.344453	0.172226	−	0.985057i	$-0.444904\pi$
$67$	2.81947	0.344453	0.172226	+	0.985057i	$0.444904\pi$
$68$	−7.39081	−0.896267
$69$	11.7182	1.41071
$70$	−3.07815	−0.367909
$71$	−1.49606	−0.177549	−0.0887747	−	0.996052i	$-0.528295\pi$
$71$	−1.49606	−0.177549	−0.0887747	+	0.996052i	$0.528295\pi$
$72$	2.71796	0.320315
$73$	0.517325	0.0605483	0.0302741	−	0.999542i	$-0.490362\pi$
$73$	0.517325	0.0605483	0.0302741	+	0.999542i	$0.490362\pi$
$74$	3.48770	0.405437
$75$	3.27804	0.378515
$76$	7.13963	0.818971
$77$	11.9688	1.36397
$78$	−9.11734	−1.03234
$79$	5.15153	0.579593	0.289796	−	0.957088i	$-0.406412\pi$
$79$	5.15153	0.579593	0.289796	+	0.957088i	$0.406412\pi$
$80$	−3.01985	−0.337629
$81$	−11.1397	−1.23775
$82$	−0.198148	−0.0218818
$83$	3.49084	0.383169	0.191585	−	0.981476i	$-0.438637\pi$
$83$	3.49084	0.383169	0.191585	+	0.981476i	$0.438637\pi$
$84$	−8.26523	−0.901811
$85$	−8.63645	−0.936755
$86$	3.03171	0.326918
$87$	−5.07762	−0.544378
$88$	−10.8472	−1.15632
$89$	−4.80065	−0.508868	−0.254434	−	0.967090i	$-0.581889\pi$
$89$	−4.80065	−0.508868	−0.254434	+	0.967090i	$0.581889\pi$
$90$	1.40006	0.147579
$91$	17.6245	1.84755
$92$	9.04975	0.943502
$93$	12.1725	1.26223
$94$	0.856199	0.0883102
$95$	8.34293	0.855967
$96$	11.6794	1.19202
$97$	0	0
$97$	0	0
$98$	0.264440	0.0267124
$99$	−5.44385	−0.547128
$100$	2.53156	0.253156
$101$	14.5160	1.44439	0.722197	−	0.691688i	$-0.243133\pi$
$101$	14.5160	1.44439	0.722197	+	0.691688i	$0.243133\pi$
$102$	6.22686	0.616551
$103$	−17.1720	−1.69201	−0.846004	−	0.533176i	$-0.820998\pi$
$103$	−17.1720	−1.69201	−0.846004	+	0.533176i	$0.820998\pi$
$104$	−15.9729	−1.56627
$105$	−9.65825	−0.942549
$106$	−2.49796	−0.242623
$107$	17.6808	1.70927	0.854635	−	0.519230i	$-0.173781\pi$
$107$	17.6808	1.70927	0.854635	+	0.519230i	$0.173781\pi$
$108$	−5.89708	−0.567447
$109$	−7.65959	−0.733655	−0.366828	−	0.930289i	$-0.619556\pi$
$109$	−7.65959	−0.733655	−0.366828	+	0.930289i	$0.619556\pi$
$110$	−5.58752	−0.532749
$111$	10.9433	1.03869
$112$	−4.20889	−0.397703
$113$	2.99989	0.282206	0.141103	−	0.989995i	$-0.454935\pi$
$113$	2.99989	0.282206	0.141103	+	0.989995i	$0.454935\pi$
$114$	−6.01524	−0.563379
$115$	10.5750	0.986123
$116$	−3.92134	−0.364087
$117$	−8.01628	−0.741105
$118$	−6.81107	−0.627010
$119$	−12.0370	−1.10343
$120$	8.75317	0.799051
$121$	10.7260	0.975093
$122$	4.01118	0.363155
$123$	−0.621726	−0.0560591
$124$	9.40054	0.844193
$125$	12.1701	1.08853
$126$	1.95132	0.173837
$127$	−16.8734	−1.49727	−0.748636	−	0.662982i	$-0.769291\pi$
$127$	−16.8734	−1.49727	−0.748636	+	0.662982i	$0.769291\pi$
$128$	11.1527	0.985770
$129$	9.51254	0.837533
$130$	−8.22784	−0.721629
$131$	0.299972	0.0262086	0.0131043	−	0.999914i	$-0.495829\pi$
$131$	0.299972	0.0262086	0.0131043	+	0.999914i	$0.495829\pi$
$132$	−15.0032	−1.30586
$133$	11.6279	1.00827
$134$	−1.83450	−0.158477
$135$	−6.89097	−0.593080
$136$	10.9090	0.935440
$137$	11.8331	1.01097	0.505485	−	0.862836i	$-0.331314\pi$
$137$	11.8331	1.01097	0.505485	+	0.862836i	$0.331314\pi$
$138$	−7.62455	−0.649044
$139$	15.1982	1.28910	0.644548	−	0.764564i	$-0.277046\pi$
$139$	15.1982	1.28910	0.644548	+	0.764564i	$0.277046\pi$
$140$	−7.45886	−0.630389
$141$	2.68648	0.226242
$142$	0.973419	0.0816875
$143$	31.9924	2.67534
$144$	1.91436	0.159530
$145$	−4.58224	−0.380534
$146$	−0.336601	−0.0278573
$147$	0.829727	0.0684347
$148$	8.45127	0.694690
$149$	−1.92460	−0.157669	−0.0788347	−	0.996888i	$-0.525120\pi$
$149$	−1.92460	−0.157669	−0.0788347	+	0.996888i	$0.525120\pi$
$150$	−2.13288	−0.174149
$151$	2.83138	0.230414	0.115207	−	0.993342i	$-0.463247\pi$
$151$	2.83138	0.230414	0.115207	+	0.993342i	$0.463247\pi$
$152$	−10.5383	−0.854765
$153$	5.47487	0.442617
$154$	−7.78758	−0.627541
$155$	10.9849	0.882328
$156$	−22.0928	−1.76884
$157$	16.3421	1.30424	0.652122	−	0.758114i	$-0.273879\pi$
$157$	16.3421	1.30424	0.652122	+	0.758114i	$0.273879\pi$
$158$	−3.35188	−0.266661
$159$	−7.83779	−0.621577
$160$	10.5399	0.833253
$161$	14.7388	1.16158
$162$	7.24813	0.569467
$163$	−4.77132	−0.373719	−0.186859	−	0.982387i	$-0.559831\pi$
$163$	−4.77132	−0.373719	−0.186859	+	0.982387i	$0.559831\pi$
$164$	−0.480146	−0.0374931
$165$	−17.5319	−1.36485
$166$	−2.27134	−0.176290
$167$	−23.9473	−1.85309	−0.926547	−	0.376179i	$-0.877238\pi$
$167$	−23.9473	−1.85309	−0.926547	+	0.376179i	$0.877238\pi$
$168$	12.1997	0.941226
$169$	34.1100	2.62385
$170$	5.61936	0.430985
$171$	−5.28880	−0.404445
$172$	7.34634	0.560153
$173$	10.1092	0.768585	0.384292	−	0.923211i	$-0.374445\pi$
$173$	10.1092	0.768585	0.384292	+	0.923211i	$0.374445\pi$
$174$	3.30379	0.250459
$175$	4.12301	0.311670
$176$	−7.64007	−0.575892
$177$	−21.3709	−1.60634
$178$	3.12357	0.234122
$179$	−8.81018	−0.658503	−0.329252	−	0.944242i	$-0.606796\pi$
$179$	−8.81018	−0.658503	−0.329252	+	0.944242i	$0.606796\pi$
$180$	3.39256	0.252867
$181$	−22.0762	−1.64091	−0.820454	−	0.571713i	$-0.806279\pi$
$181$	−22.0762	−1.64091	−0.820454	+	0.571713i	$0.806279\pi$
$182$	−11.4675	−0.850028
$183$	12.5858	0.930370
$184$	−13.3576	−0.984739
$185$	9.87564	0.726071
$186$	−7.92009	−0.580729
$187$	−21.8498	−1.59782
$188$	2.07471	0.151314
$189$	−9.60425	−0.698606
$190$	−5.42838	−0.393816
$191$	12.4964	0.904207	0.452104	−	0.891965i	$-0.350674\pi$
$191$	12.4964	0.904207	0.452104	+	0.891965i	$0.350674\pi$
$192$	−0.906609	−0.0654289
$193$	19.1903	1.38135	0.690676	−	0.723165i	$-0.257313\pi$
$193$	19.1903	1.38135	0.690676	+	0.723165i	$0.257313\pi$
$194$	0	0
$195$	−25.8163	−1.84875
$196$	0.640781	0.0457700
$197$	8.38989	0.597755	0.298878	−	0.954291i	$-0.403388\pi$
$197$	8.38989	0.597755	0.298878	+	0.954291i	$0.403388\pi$
$198$	3.54208	0.251724
$199$	2.67707	0.189772	0.0948862	−	0.995488i	$-0.469751\pi$
$199$	2.67707	0.189772	0.0948862	+	0.995488i	$0.469751\pi$
$200$	−3.73664	−0.264220
$201$	−5.75609	−0.406003
$202$	−9.44491	−0.664542
$203$	−6.38647	−0.448242
$204$	15.0887	1.05642
$205$	−0.561069	−0.0391868
$206$	11.1731	0.778465
$207$	−6.70376	−0.465943
$208$	−11.2503	−0.780068
$209$	21.1072	1.46002
$210$	6.28420	0.433651
$211$	−16.2742	−1.12036	−0.560180	−	0.828371i	$-0.689268\pi$
$211$	−16.2742	−1.12036	−0.560180	+	0.828371i	$0.689268\pi$
$212$	−6.05296	−0.415719
$213$	3.05428	0.209276
$214$	−11.5041	−0.786406
$215$	8.58448	0.585457
$216$	8.70423	0.592248
$217$	15.3101	1.03932
$218$	4.98376	0.337543
$219$	−1.05614	−0.0713677
$220$	−13.5395	−0.912832
$221$	−32.1747	−2.16430
$222$	−7.12031	−0.477884
$223$	2.67702	0.179266	0.0896331	−	0.995975i	$-0.471431\pi$
$223$	2.67702	0.179266	0.0896331	+	0.995975i	$0.471431\pi$
$224$	14.6899	0.981512
$225$	−1.87530	−0.125020
$226$	−1.95190	−0.129838
$227$	19.1448	1.27069	0.635343	−	0.772230i	$-0.280859\pi$
$227$	19.1448	1.27069	0.635343	+	0.772230i	$0.280859\pi$
$228$	−14.5759	−0.965313
$229$	1.28997	0.0852439	0.0426220	−	0.999091i	$-0.486429\pi$
$229$	1.28997	0.0852439	0.0426220	+	0.999091i	$0.486429\pi$
$230$	−6.88068	−0.453699
$231$	−24.4349	−1.60770
$232$	5.78799	0.380000
$233$	−15.2617	−0.999827	−0.499914	−	0.866075i	$-0.666635\pi$
$233$	−15.2617	−0.999827	−0.499914	+	0.866075i	$0.666635\pi$
$234$	5.21584	0.340970
$235$	2.42438	0.158149
$236$	−16.5043	−1.07434
$237$	−10.5171	−0.683160
$238$	7.83195	0.507670
$239$	2.51891	0.162934	0.0814672	−	0.996676i	$-0.474039\pi$
$239$	2.51891	0.162934	0.0814672	+	0.996676i	$0.474039\pi$
$240$	6.16517	0.397960
$241$	17.9879	1.15870	0.579351	−	0.815078i	$-0.303306\pi$
$241$	17.9879	1.15870	0.579351	+	0.815078i	$0.303306\pi$
$242$	−6.97896	−0.448624
$243$	11.5215	0.739106
$244$	9.71975	0.622244
$245$	0.748777	0.0478376
$246$	0.404530	0.0257919
$247$	31.0812	1.97765
$248$	−13.8754	−0.881090
$249$	−7.12673	−0.451638
$250$	−7.91855	−0.500813
$251$	9.24296	0.583411	0.291705	−	0.956508i	$-0.405777\pi$
$251$	9.24296	0.583411	0.291705	+	0.956508i	$0.405777\pi$
$252$	4.72837	0.297859
$253$	26.7542	1.68202
$254$	10.9788	0.688870
$255$	17.6318	1.10414
$256$	−8.14474	−0.509046
$257$	22.2436	1.38752	0.693760	−	0.720206i	$-0.255953\pi$
$257$	22.2436	1.38752	0.693760	+	0.720206i	$0.255953\pi$
$258$	−6.18940	−0.385335
$259$	13.7641	0.855260
$260$	−19.9374	−1.23647
$261$	2.90480	0.179803
$262$	−0.195178	−0.0120582
$263$	1.33704	0.0824454	0.0412227	−	0.999150i	$-0.486875\pi$
$263$	1.33704	0.0824454	0.0412227	+	0.999150i	$0.486875\pi$
$264$	22.1451	1.36294
$265$	−7.07312	−0.434498
$266$	−7.56577	−0.463887
$267$	9.80077	0.599797
$268$	−4.44530	−0.271540
$269$	−22.6137	−1.37878	−0.689392	−	0.724389i	$-0.742122\pi$
$269$	−22.6137	−1.37878	−0.689392	+	0.724389i	$0.742122\pi$
$270$	4.48365	0.272866
$271$	−19.5304	−1.18638	−0.593192	−	0.805061i	$-0.702133\pi$
$271$	−19.5304	−1.18638	−0.593192	+	0.805061i	$0.702133\pi$
$272$	7.68361	0.465887
$273$	−35.9813	−2.17769
$274$	−7.69928	−0.465130
$275$	7.48417	0.451313
$276$	−18.4755	−1.11210
$277$	0.0691749	0.00415632	0.00207816	−	0.999998i	$-0.499339\pi$
$277$	0.0691749	0.00415632	0.00207816	+	0.999998i	$0.499339\pi$
$278$	−9.88881	−0.593091
$279$	−6.96361	−0.416901
$280$	11.0095	0.657941
$281$	−4.77423	−0.284807	−0.142403	−	0.989809i	$-0.545483\pi$
$281$	−4.77423	−0.284807	−0.142403	+	0.989809i	$0.545483\pi$
$282$	−1.74797	−0.104090
$283$	1.10209	0.0655125	0.0327563	−	0.999463i	$-0.489571\pi$
$283$	1.10209	0.0655125	0.0327563	+	0.999463i	$0.489571\pi$
$284$	2.35875	0.139966
$285$	−17.0325	−1.00892
$286$	−20.8161	−1.23088
$287$	−0.781987	−0.0461592
$288$	−6.68152	−0.393712
$289$	4.97432	0.292607
$290$	2.98146	0.175078
$291$	0	0
$292$	−0.815638	−0.0477316
$293$	−12.0929	−0.706472	−0.353236	−	0.935534i	$-0.614919\pi$
$293$	−12.0929	−0.706472	−0.353236	+	0.935534i	$0.614919\pi$
$294$	−0.539867	−0.0314857
$295$	−19.2860	−1.12287
$296$	−12.4743	−0.725052
$297$	−17.4338	−1.01161
$298$	1.25225	0.0725410
$299$	39.3966	2.27837
$300$	−5.16831	−0.298392
$301$	11.9646	0.689627
$302$	−1.84225	−0.106010
$303$	−29.6351	−1.70249
$304$	−7.42247	−0.425708
$305$	11.3579	0.650352
$306$	−3.56226	−0.203641
$307$	−11.7411	−0.670098	−0.335049	−	0.942201i	$-0.608753\pi$
$307$	−11.7411	−0.670098	−0.335049	+	0.942201i	$0.608753\pi$
$308$	−18.8706	−1.07525
$309$	35.0575	1.99435
$310$	−7.14739	−0.405945
$311$	−10.3352	−0.586055	−0.293028	−	0.956104i	$-0.594663\pi$
$311$	−10.3352	−0.586055	−0.293028	+	0.956104i	$0.594663\pi$
$312$	32.6095	1.84615
$313$	−1.91530	−0.108259	−0.0541296	−	0.998534i	$-0.517238\pi$
$313$	−1.91530	−0.108259	−0.0541296	+	0.998534i	$0.517238\pi$
$314$	−10.6331	−0.600061
$315$	5.52528	0.311314
$316$	−8.12215	−0.456906
$317$	15.4087	0.865442	0.432721	−	0.901528i	$-0.357554\pi$
$317$	15.4087	0.865442	0.432721	+	0.901528i	$0.357554\pi$
$318$	5.09970	0.285977
$319$	−11.5929	−0.649075
$320$	−0.818159	−0.0457365
$321$	−36.0963	−2.01470
$322$	−9.58991	−0.534425
$323$	−21.2275	−1.18113
$324$	17.5634	0.975745
$325$	11.0207	0.611320
$326$	3.10449	0.171942
$327$	15.6374	0.864752
$328$	0.708706	0.0391318
$329$	3.37897	0.186288
$330$	11.4072	0.627946
$331$	−0.564788	−0.0310436	−0.0155218	−	0.999880i	$-0.504941\pi$
$331$	−0.564788	−0.0310436	−0.0155218	+	0.999880i	$0.504941\pi$
$332$	−5.50382	−0.302061
$333$	−6.26042	−0.343069
$334$	15.5814	0.852578
$335$	−5.19451	−0.283807
$336$	8.59267	0.468769
$337$	−28.7187	−1.56441	−0.782204	−	0.623022i	$-0.785905\pi$
$337$	−28.7187	−1.56441	−0.782204	+	0.623022i	$0.785905\pi$
$338$	−22.1939	−1.20719
$339$	−6.12443	−0.332633
$340$	13.6166	0.738466
$341$	27.7913	1.50498
$342$	3.44119	0.186078
$343$	19.0182	1.02689
$344$	−10.8434	−0.584635
$345$	−21.5894	−1.16233
$346$	−6.57759	−0.353613
$347$	18.1465	0.974157	0.487078	−	0.873358i	$-0.338063\pi$
$347$	18.1465	0.974157	0.487078	+	0.873358i	$0.338063\pi$
$348$	8.00562	0.429146
$349$	3.71361	0.198785	0.0993925	−	0.995048i	$-0.468310\pi$
$349$	3.71361	0.198785	0.0993925	+	0.995048i	$0.468310\pi$
$350$	−2.68266	−0.143394
$351$	−25.6720	−1.37027
$352$	26.6655	1.42127
$353$	24.5441	1.30635	0.653174	−	0.757208i	$-0.273437\pi$
$353$	24.5441	1.30635	0.653174	+	0.757208i	$0.273437\pi$
$354$	13.9051	0.739050
$355$	2.75630	0.146289
$356$	7.56893	0.401152
$357$	24.5742	1.30060
$358$	5.73240	0.302967
$359$	10.2352	0.540193	0.270096	−	0.962833i	$-0.412944\pi$
$359$	10.2352	0.540193	0.270096	+	0.962833i	$0.412944\pi$
$360$	−5.00750	−0.263919
$361$	1.50606	0.0792662
$362$	14.3640	0.754954
$363$	−21.8977	−1.14933
$364$	−27.7876	−1.45647
$365$	−0.953106	−0.0498878
$366$	−8.18903	−0.428048
$367$	−10.2657	−0.535868	−0.267934	−	0.963437i	$-0.586341\pi$
$367$	−10.2657	−0.535868	−0.267934	+	0.963437i	$0.586341\pi$
$368$	−9.40827	−0.490440
$369$	0.355676	0.0185158
$370$	−6.42564	−0.334053
$371$	−9.85812	−0.511808
$372$	−19.1917	−0.995042
$373$	−24.9947	−1.29418	−0.647089	−	0.762415i	$-0.724014\pi$
$373$	−24.9947	−1.29418	−0.647089	+	0.762415i	$0.724014\pi$
$374$	14.2167	0.735129
$375$	−24.8459	−1.28303
$376$	−3.06232	−0.157927
$377$	−17.0709	−0.879197
$378$	6.24906	0.321417
$379$	−15.0677	−0.773974	−0.386987	−	0.922085i	$-0.626484\pi$
$379$	−15.0677	−0.773974	−0.386987	+	0.922085i	$0.626484\pi$
$380$	−13.1539	−0.674779
$381$	34.4479	1.76482
$382$	−8.13085	−0.416011
$383$	−4.98783	−0.254866	−0.127433	−	0.991847i	$-0.540674\pi$
$383$	−4.98783	−0.254866	−0.127433	+	0.991847i	$0.540674\pi$
$384$	−22.7688	−1.16192
$385$	−22.0510	−1.12382
$386$	−12.4863	−0.635537
$387$	−5.44193	−0.276629
$388$	0	0
$389$	22.6518	1.14849	0.574247	−	0.818682i	$-0.305295\pi$
$389$	22.6518	1.14849	0.574247	+	0.818682i	$0.305295\pi$
$390$	16.7976	0.850577
$391$	−26.9067	−1.36073
$392$	−0.945807	−0.0477705
$393$	−0.612407	−0.0308918
$394$	−5.45894	−0.275017
$395$	−9.49105	−0.477546
$396$	8.58303	0.431313
$397$	17.3358	0.870059	0.435030	−	0.900416i	$-0.356738\pi$
$397$	17.3358	0.870059	0.435030	+	0.900416i	$0.356738\pi$
$398$	−1.74185	−0.0873111
$399$	−23.7390	−1.18843
$400$	−2.63185	−0.131592
$401$	33.0496	1.65042	0.825210	−	0.564827i	$-0.191057\pi$
$401$	33.0496	1.65042	0.825210	+	0.564827i	$0.191057\pi$
$402$	3.74523	0.186795
$403$	40.9237	2.03856
$404$	−22.8866	−1.13865
$405$	20.5235	1.01982
$406$	4.15540	0.206229
$407$	24.9849	1.23846
$408$	−22.2713	−1.10259
$409$	27.6495	1.36718	0.683589	−	0.729867i	$-0.260418\pi$
$409$	27.6495	1.36718	0.683589	+	0.729867i	$0.260418\pi$
$410$	0.365063	0.0180292
$411$	−24.1579	−1.19162
$412$	27.0742	1.33385
$413$	−26.8797	−1.32266
$414$	4.36184	0.214373
$415$	−6.43143	−0.315707
$416$	39.2659	1.92517
$417$	−31.0279	−1.51944
$418$	−13.7335	−0.671730
$419$	7.39493	0.361266	0.180633	−	0.983551i	$-0.442185\pi$
$419$	7.39493	0.361266	0.180633	+	0.983551i	$0.442185\pi$
$420$	15.2276	0.743033
$421$	−23.0464	−1.12321	−0.561606	−	0.827405i	$-0.689816\pi$
$421$	−23.0464	−1.12321	−0.561606	+	0.827405i	$0.689816\pi$
$422$	10.5889	0.515459
$423$	−1.53688	−0.0747255
$424$	8.93431	0.433888
$425$	−7.52682	−0.365104
$426$	−1.98728	−0.0962842
$427$	15.8300	0.766069
$428$	−27.8764	−1.34746
$429$	−65.3141	−3.15339
$430$	−5.58555	−0.269359
$431$	−23.5543	−1.13457	−0.567286	−	0.823521i	$-0.692006\pi$
$431$	−23.5543	−1.13457	−0.567286	+	0.823521i	$0.692006\pi$
$432$	6.13070	0.294963
$433$	−23.0763	−1.10898	−0.554488	−	0.832192i	$-0.687086\pi$
$433$	−23.0763	−1.10898	−0.554488	+	0.832192i	$0.687086\pi$
$434$	−9.96164	−0.478174
$435$	9.35488	0.448532
$436$	12.0765	0.578357
$437$	25.9922	1.24338
$438$	0.687187	0.0328351
$439$	4.75841	0.227107	0.113553	−	0.993532i	$-0.463777\pi$
$439$	4.75841	0.227107	0.113553	+	0.993532i	$0.463777\pi$
$440$	19.9846	0.952728
$441$	−0.474669	−0.0226033
$442$	20.9347	0.995760
$443$	13.9758	0.664011	0.332006	−	0.943277i	$-0.392275\pi$
$443$	13.9758	0.664011	0.332006	+	0.943277i	$0.392275\pi$
$444$	−17.2537	−0.818824
$445$	8.84459	0.419274
$446$	−1.74182	−0.0824774
$447$	3.92917	0.185843
$448$	−1.14030	−0.0538743
$449$	28.3851	1.33958	0.669788	−	0.742552i	$-0.266385\pi$
$449$	28.3851	1.33958	0.669788	+	0.742552i	$0.266385\pi$
$450$	1.22017	0.0575195
$451$	−1.41948	−0.0668406
$452$	−4.72977	−0.222470
$453$	−5.78040	−0.271587
$454$	−12.4567	−0.584621
$455$	−32.4709	−1.52226
$456$	21.5144	1.00750
$457$	2.08253	0.0974164	0.0487082	−	0.998813i	$-0.484490\pi$
$457$	2.08253	0.0974164	0.0487082	+	0.998813i	$0.484490\pi$
$458$	−0.839330	−0.0392193
$459$	17.5332	0.818378
$460$	−16.6730	−0.777384
$461$	−16.0119	−0.745747	−0.372874	−	0.927882i	$-0.621627\pi$
$461$	−16.0119	−0.745747	−0.372874	+	0.927882i	$0.621627\pi$
$462$	15.8987	0.739676
$463$	−33.6885	−1.56564	−0.782818	−	0.622250i	$-0.786219\pi$
$463$	−33.6885	−1.56564	−0.782818	+	0.622250i	$0.786219\pi$
$464$	4.07669	0.189256
$465$	−22.4262	−1.03999
$466$	9.93012	0.460004
$467$	−34.8296	−1.61172	−0.805860	−	0.592105i	$-0.798297\pi$
$467$	−34.8296	−1.61172	−0.805860	+	0.592105i	$0.798297\pi$
$468$	12.6388	0.584230
$469$	−7.23982	−0.334304
$470$	−1.57744	−0.0727618
$471$	−33.3633	−1.53730
$472$	24.3608	1.12130
$473$	21.7183	0.998611
$474$	6.84303	0.314311
$475$	7.27101	0.333617
$476$	18.9781	0.869860
$477$	4.48383	0.205301
$478$	−1.63894	−0.0749634
$479$	−14.4795	−0.661587	−0.330794	−	0.943703i	$-0.607316\pi$
$479$	−14.4795	−0.661587	−0.330794	+	0.943703i	$0.607316\pi$
$480$	−21.5177	−0.982146
$481$	36.7912	1.67754
$482$	−11.7039	−0.533100
$483$	−30.0901	−1.36915
$484$	−16.9112	−0.768689
$485$	0	0
$486$	−7.49655	−0.340050
$487$	−0.262184	−0.0118807	−0.00594035	−	0.999982i	$-0.501891\pi$
$487$	−0.262184	−0.0118807	−0.00594035	+	0.999982i	$0.501891\pi$
$488$	−14.3466	−0.649439
$489$	9.74090	0.440499
$490$	−0.487197	−0.0220093
$491$	14.7374	0.665091	0.332545	−	0.943087i	$-0.392093\pi$
$491$	14.7374	0.665091	0.332545	+	0.943087i	$0.392093\pi$
$492$	0.980242	0.0441927
$493$	11.6589	0.525091
$494$	−20.2232	−0.909884
$495$	10.0296	0.450797
$496$	−9.77295	−0.438819
$497$	3.84157	0.172318
$498$	4.63705	0.207791
$499$	−11.3367	−0.507503	−0.253751	−	0.967270i	$-0.581664\pi$
$499$	−11.3367	−0.507503	−0.253751	+	0.967270i	$0.581664\pi$
$500$	−19.1879	−0.858111
$501$	48.8895	2.18422
$502$	−6.01399	−0.268418
$503$	−26.3827	−1.17635	−0.588174	−	0.808734i	$-0.700153\pi$
$503$	−26.3827	−1.17635	−0.588174	+	0.808734i	$0.700153\pi$
$504$	−6.97918	−0.310877
$505$	−26.7438	−1.19009
$506$	−17.4078	−0.773871
$507$	−69.6373	−3.09270
$508$	26.6034	1.18033
$509$	−1.35821	−0.0602015	−0.0301008	−	0.999547i	$-0.509583\pi$
$509$	−1.35821	−0.0602015	−0.0301008	+	0.999547i	$0.509583\pi$
$510$	−11.4722	−0.507998
$511$	−1.32839	−0.0587643
$512$	−17.0060	−0.751566
$513$	−16.9373	−0.747799
$514$	−14.4730	−0.638375
$515$	31.6373	1.39410
$516$	−14.9979	−0.660247
$517$	6.13357	0.269754
$518$	−8.95570	−0.393491
$519$	−20.6384	−0.905923
$520$	29.4281	1.29051
$521$	−35.9794	−1.57629	−0.788143	−	0.615492i	$-0.788957\pi$
$521$	−35.9794	−1.57629	−0.788143	+	0.615492i	$0.788957\pi$
$522$	−1.89003	−0.0827242
$523$	9.81356	0.429117	0.214559	−	0.976711i	$-0.431169\pi$
$523$	9.81356	0.429117	0.214559	+	0.976711i	$0.431169\pi$
$524$	−0.472949	−0.0206609
$525$	−8.41733	−0.367363
$526$	−0.869953	−0.0379318
$527$	−27.9496	−1.21751
$528$	15.5976	0.678798
$529$	9.94614	0.432441
$530$	4.60217	0.199905
$531$	12.2259	0.530558
$532$	−18.3331	−0.794841
$533$	−2.09024	−0.0905382
$534$	−6.37693	−0.275957
$535$	−32.5747	−1.40833
$536$	6.56137	0.283408
$537$	17.9864	0.776171
$538$	14.7138	0.634355
$539$	1.89437	0.0815964
$540$	10.8646	0.467539
$541$	29.9403	1.28724	0.643618	−	0.765347i	$-0.277433\pi$
$541$	29.9403	1.28724	0.643618	+	0.765347i	$0.277433\pi$
$542$	12.7075	0.545836
$543$	45.0696	1.93412
$544$	−26.8174	−1.14979
$545$	14.1118	0.604484
$546$	23.4115	1.00192
$547$	−37.4063	−1.59938	−0.799689	−	0.600414i	$-0.795002\pi$
$547$	−37.4063	−1.59938	−0.799689	+	0.600414i	$0.795002\pi$
$548$	−18.6566	−0.796971
$549$	−7.20008	−0.307292
$550$	−4.86962	−0.207641
$551$	−11.2627	−0.479806
$552$	27.2703	1.16070
$553$	−13.2281	−0.562516
$554$	−0.0450091	−0.00191225
$555$	−20.1616	−0.855813
$556$	−23.9622	−1.01622
$557$	20.8187	0.882116	0.441058	−	0.897479i	$-0.354603\pi$
$557$	20.8187	0.882116	0.441058	+	0.897479i	$0.354603\pi$
$558$	4.53092	0.191809
$559$	31.9811	1.35266
$560$	7.75435	0.327681
$561$	44.6075	1.88333
$562$	3.10638	0.131035
$563$	−26.8099	−1.12990	−0.564951	−	0.825125i	$-0.691105\pi$
$563$	−26.8099	−1.12990	−0.564951	+	0.825125i	$0.691105\pi$
$564$	−4.23563	−0.178352
$565$	−5.52692	−0.232519
$566$	−0.717083	−0.0301412
$567$	28.6046	1.20128
$568$	−3.48158	−0.146084
$569$	−12.2909	−0.515261	−0.257630	−	0.966244i	$-0.582942\pi$
$569$	−12.2909	−0.515261	−0.257630	+	0.966244i	$0.582942\pi$
$570$	11.0823	0.464187
$571$	34.3236	1.43640	0.718200	−	0.695837i	$-0.244966\pi$
$571$	34.3236	1.43640	0.718200	+	0.695837i	$0.244966\pi$
$572$	−50.4407	−2.10903
$573$	−25.5120	−1.06578
$574$	0.508805	0.0212371
$575$	9.21629	0.384346
$576$	0.518652	0.0216105
$577$	4.29547	0.178823	0.0894114	−	0.995995i	$-0.471501\pi$
$577$	4.29547	0.178823	0.0894114	+	0.995995i	$0.471501\pi$
$578$	−3.23657	−0.134624
$579$	−39.1781	−1.62818
$580$	7.22457	0.299984
$581$	−8.96377	−0.371880
$582$	0	0
$583$	−17.8947	−0.741121
$584$	1.20390	0.0498178
$585$	14.7690	0.610622
$586$	7.86830	0.325036
$587$	39.8276	1.64386	0.821930	−	0.569588i	$-0.192897\pi$
$587$	39.8276	1.64386	0.821930	+	0.569588i	$0.192897\pi$
$588$	−1.30819	−0.0539487
$589$	26.9997	1.11250
$590$	12.5485	0.516615
$591$	−17.1284	−0.704568
$592$	−8.78607	−0.361105
$593$	−5.90990	−0.242690	−0.121345	−	0.992610i	$-0.538721\pi$
$593$	−5.90990	−0.242690	−0.121345	+	0.992610i	$0.538721\pi$
$594$	11.3434	0.465427
$595$	22.1767	0.909154
$596$	3.03441	0.124294
$597$	−5.46537	−0.223683
$598$	−25.6337	−1.04824
$599$	−4.83047	−0.197368	−0.0986838	−	0.995119i	$-0.531463\pi$
$599$	−4.83047	−0.197368	−0.0986838	+	0.995119i	$0.531463\pi$
$600$	7.62854	0.311434
$601$	40.3140	1.64444	0.822221	−	0.569169i	$-0.192735\pi$
$601$	40.3140	1.64444	0.822221	+	0.569169i	$0.192735\pi$
$602$	−7.78482	−0.317286
$603$	3.29294	0.134099
$604$	−4.46408	−0.181641
$605$	−19.7613	−0.803413
$606$	19.2823	0.783288
$607$	15.6280	0.634321	0.317161	−	0.948372i	$-0.397271\pi$
$607$	15.6280	0.634321	0.317161	+	0.948372i	$0.397271\pi$
$608$	25.9060	1.05063
$609$	13.0383	0.528339
$610$	−7.39010	−0.299216
$611$	9.03191	0.365392
$612$	−8.63194	−0.348925
$613$	−28.9912	−1.17094	−0.585472	−	0.810693i	$-0.699091\pi$
$613$	−28.9912	−1.17094	−0.585472	+	0.810693i	$0.699091\pi$
$614$	7.63939	0.308301
$615$	1.14545	0.0461890
$616$	27.8534	1.12225
$617$	−8.33897	−0.335714	−0.167857	−	0.985811i	$-0.553685\pi$
$617$	−8.33897	−0.335714	−0.167857	+	0.985811i	$0.553685\pi$
$618$	−22.8104	−0.917569
$619$	−35.0398	−1.40837	−0.704185	−	0.710016i	$-0.748688\pi$
$619$	−35.0398	−1.40837	−0.704185	+	0.710016i	$0.748688\pi$
$620$	−17.3193	−0.695560
$621$	−21.4687	−0.861508
$622$	6.72466	0.269634
$623$	12.3271	0.493874
$624$	22.9681	0.919458
$625$	−14.3936	−0.575742
$626$	1.24620	0.0498083
$627$	−43.0915	−1.72091
$628$	−25.7658	−1.02817
$629$	−25.1273	−1.00189
$630$	−3.59506	−0.143231
$631$	−32.9488	−1.31167	−0.655835	−	0.754904i	$-0.727683\pi$
$631$	−32.9488	−1.31167	−0.655835	+	0.754904i	$0.727683\pi$
$632$	11.9885	0.476876
$633$	33.2246	1.32056
$634$	−10.0258	−0.398175
$635$	31.0871	1.23365
$636$	12.3574	0.490003
$637$	2.78953	0.110525
$638$	7.54296	0.298629
$639$	−1.74729	−0.0691216
$640$	−20.5475	−0.812210
$641$	−46.4379	−1.83419	−0.917093	−	0.398673i	$-0.869471\pi$
$641$	−46.4379	−1.83419	−0.917093	+	0.398673i	$0.869471\pi$
$642$	23.4863	0.926929
$643$	−14.1962	−0.559842	−0.279921	−	0.960023i	$-0.590308\pi$
$643$	−14.1962	−0.559842	−0.279921	+	0.960023i	$0.590308\pi$
$644$	−23.2379	−0.915702
$645$	−17.5257	−0.690072
$646$	13.8118	0.543418
$647$	−24.3060	−0.955567	−0.477784	−	0.878478i	$-0.658560\pi$
$647$	−24.3060	−0.955567	−0.477784	+	0.878478i	$0.658560\pi$
$648$	−25.9240	−1.01839
$649$	−48.7926	−1.91528
$650$	−7.17071	−0.281258
$651$	−31.2564	−1.22504
$652$	7.52269	0.294611
$653$	46.1296	1.80519	0.902596	−	0.430489i	$-0.141659\pi$
$653$	46.1296	1.80519	0.902596	+	0.430489i	$0.141659\pi$
$654$	−10.1746	−0.397858
$655$	−0.552660	−0.0215942
$656$	0.499167	0.0194892
$657$	0.604198	0.0235720
$658$	−2.19855	−0.0857082
$659$	0.618447	0.0240913	0.0120456	−	0.999927i	$-0.496166\pi$
$659$	0.618447	0.0240913	0.0120456	+	0.999927i	$0.496166\pi$
$660$	27.6415	1.07595
$661$	18.5990	0.723416	0.361708	−	0.932291i	$-0.382194\pi$
$661$	18.5990	0.723416	0.361708	+	0.932291i	$0.382194\pi$
$662$	0.367483	0.0142826
$663$	65.6863	2.55104
$664$	8.12377	0.315263
$665$	−21.4230	−0.830747
$666$	4.07338	0.157840
$667$	−14.2759	−0.552764
$668$	37.7564	1.46084
$669$	−5.46526	−0.211299
$670$	3.37984	0.130575
$671$	28.7350	1.10930
$672$	−29.9902	−1.15690
$673$	−15.5779	−0.600485	−0.300243	−	0.953863i	$-0.597068\pi$
$673$	−15.5779	−0.600485	−0.300243	+	0.953863i	$0.597068\pi$
$674$	18.6860	0.719758
$675$	−6.00560	−0.231156
$676$	−53.7795	−2.06844
$677$	20.6285	0.792816	0.396408	−	0.918074i	$-0.370257\pi$
$677$	20.6285	0.792816	0.396408	+	0.918074i	$0.370257\pi$
$678$	3.98490	0.153039
$679$	0	0
$680$	−20.0985	−0.770741
$681$	−39.0851	−1.49774
$682$	−18.0826	−0.692417
$683$	−15.8823	−0.607718	−0.303859	−	0.952717i	$-0.598275\pi$
$683$	−15.8823	−0.607718	−0.303859	+	0.952717i	$0.598275\pi$
$684$	8.33857	0.318833
$685$	−21.8010	−0.832972
$686$	−12.3743	−0.472453
$687$	−2.63355	−0.100476
$688$	−7.63737	−0.291172
$689$	−26.3506	−1.00388
$690$	14.0473	0.534770
$691$	11.2426	0.427690	0.213845	−	0.976868i	$-0.431401\pi$
$691$	11.2426	0.427690	0.213845	+	0.976868i	$0.431401\pi$
$692$	−15.9386	−0.605893
$693$	13.9787	0.531007
$694$	−11.8072	−0.448193
$695$	−28.0008	−1.06213
$696$	−11.8165	−0.447902
$697$	1.42757	0.0540730
$698$	−2.41628	−0.0914576
$699$	31.1575	1.17849
$700$	−6.50053	−0.245697
$701$	−47.2986	−1.78645	−0.893223	−	0.449615i	$-0.851561\pi$
$701$	−47.2986	−1.78645	−0.893223	+	0.449615i	$0.851561\pi$
$702$	16.7036	0.630438
$703$	24.2733	0.915484
$704$	−2.06990	−0.0780124
$705$	−4.94950	−0.186409
$706$	−15.9697	−0.601029
$707$	−37.2741	−1.40184
$708$	33.6944	1.26631
$709$	3.41346	0.128195	0.0640975	−	0.997944i	$-0.479583\pi$
$709$	3.41346	0.128195	0.0640975	+	0.997944i	$0.479583\pi$
$710$	−1.79340	−0.0673051
$711$	6.01662	0.225641
$712$	−11.1719	−0.418685
$713$	34.2232	1.28167
$714$	−15.9893	−0.598385
$715$	−58.9419	−2.20430
$716$	13.8905	0.519114
$717$	−5.14247	−0.192049
$718$	−6.65959	−0.248534
$719$	34.4387	1.28435	0.642173	−	0.766560i	$-0.278033\pi$
$719$	34.4387	1.28435	0.642173	+	0.766560i	$0.278033\pi$
$720$	−3.52696	−0.131442
$721$	44.0942	1.64216
$722$	−0.979926	−0.0364691
$723$	−36.7232	−1.36575
$724$	34.8063	1.29357
$725$	−3.99350	−0.148315
$726$	14.2479	0.528789
$727$	−5.68610	−0.210886	−0.105443	−	0.994425i	$-0.533626\pi$
$727$	−5.68610	−0.210886	−0.105443	+	0.994425i	$0.533626\pi$
$728$	41.0152	1.52012
$729$	9.89743	0.366571
$730$	0.620144	0.0229526
$731$	−21.8421	−0.807859
$732$	−19.8434	−0.733432
$733$	−0.445092	−0.0164399	−0.00821993	−	0.999966i	$-0.502617\pi$
$733$	−0.445092	−0.0164399	−0.00821993	+	0.999966i	$0.502617\pi$
$734$	6.67947	0.246544
$735$	−1.52867	−0.0563857
$736$	32.8368	1.21038
$737$	−13.1419	−0.484087
$738$	−0.231423	−0.00851880
$739$	7.64189	0.281111	0.140556	−	0.990073i	$-0.455111\pi$
$739$	7.64189	0.281111	0.140556	+	0.990073i	$0.455111\pi$
$740$	−15.5704	−0.572379
$741$	−63.4538	−2.33103
$742$	6.41424	0.235474
$743$	2.37475	0.0871210	0.0435605	−	0.999051i	$-0.486130\pi$
$743$	2.37475	0.0871210	0.0435605	+	0.999051i	$0.486130\pi$
$744$	28.3274	1.03853
$745$	3.54583	0.129909
$746$	16.2630	0.595430
$747$	4.07705	0.149171
$748$	34.4494	1.25960
$749$	−45.4007	−1.65891
$750$	16.1661	0.590303
$751$	−32.7998	−1.19688	−0.598441	−	0.801167i	$-0.704213\pi$
$751$	−32.7998	−1.19688	−0.598441	+	0.801167i	$0.704213\pi$
$752$	−2.15690	−0.0786541
$753$	−18.8700	−0.687660
$754$	11.1073	0.404504
$755$	−5.21645	−0.189846
$756$	15.1425	0.550727
$757$	−20.5491	−0.746870	−0.373435	−	0.927656i	$-0.621820\pi$
$757$	−20.5491	−0.746870	−0.373435	+	0.927656i	$0.621820\pi$
$758$	9.80387	0.356093
$759$	−54.6201	−1.98258
$760$	19.4154	0.704271
$761$	−24.8727	−0.901635	−0.450818	−	0.892616i	$-0.648868\pi$
$761$	−24.8727	−0.901635	−0.450818	+	0.892616i	$0.648868\pi$
$762$	−22.4137	−0.811963
$763$	19.6683	0.712039
$764$	−19.7024	−0.712808
$765$	−10.0868	−0.364687
$766$	3.24536	0.117260
$767$	−71.8489	−2.59431
$768$	16.6279	0.600008
$769$	3.69186	0.133132	0.0665660	−	0.997782i	$-0.478796\pi$
$769$	3.69186	0.133132	0.0665660	+	0.997782i	$0.478796\pi$
$770$	14.3476	0.517052
$771$	−45.4115	−1.63546
$772$	−30.2564	−1.08895
$773$	29.0765	1.04581	0.522904	−	0.852391i	$-0.324848\pi$
$773$	29.0765	1.04581	0.522904	+	0.852391i	$0.324848\pi$
$774$	3.54082	0.127272
$775$	9.57353	0.343891
$776$	0	0
$777$	−28.1001	−1.00809
$778$	−14.7386	−0.528403
$779$	−1.37905	−0.0494096
$780$	40.7032	1.45741
$781$	6.97330	0.249524
$782$	17.5070	0.626049
$783$	9.30257	0.332447
$784$	−0.666166	−0.0237916
$785$	−30.1083	−1.07461
$786$	0.398466	0.0142128
$787$	−52.1926	−1.86047	−0.930233	−	0.366969i	$-0.880396\pi$
$787$	−52.1926	−1.86047	−0.930233	+	0.366969i	$0.880396\pi$
$788$	−13.2279	−0.471224
$789$	−2.72963	−0.0971775
$790$	6.17541	0.219711
$791$	−7.70311	−0.273891
$792$	−12.6688	−0.450165
$793$	42.3134	1.50259
$794$	−11.2797	−0.400300
$795$	14.4401	0.512139
$796$	−4.22079	−0.149602
$797$	−25.7730	−0.912926	−0.456463	−	0.889742i	$-0.650884\pi$
$797$	−25.7730	−0.912926	−0.456463	+	0.889742i	$0.650884\pi$
$798$	15.4459	0.546779
$799$	−6.16852	−0.218227
$800$	9.18571	0.324764
$801$	−5.60681	−0.198107
$802$	−21.5039	−0.759330
$803$	−2.41131	−0.0850934
$804$	9.07531	0.320062
$805$	−27.1544	−0.957068
$806$	−26.6273	−0.937906
$807$	46.1671	1.62516
$808$	33.7811	1.18842
$809$	−9.38194	−0.329852	−0.164926	−	0.986306i	$-0.552738\pi$
$809$	−9.38194	−0.329852	−0.164926	+	0.986306i	$0.552738\pi$
$810$	−13.3538	−0.469204
$811$	7.97690	0.280107	0.140053	−	0.990144i	$-0.455273\pi$
$811$	7.97690	0.280107	0.140053	+	0.990144i	$0.455273\pi$
$812$	10.0692	0.353360
$813$	39.8722	1.39838
$814$	−16.2566	−0.569793
$815$	8.79056	0.307920
$816$	−15.6865	−0.549136
$817$	21.0998	0.738187
$818$	−17.9903	−0.629016
$819$	20.5842	0.719269
$820$	0.884608	0.0308918
$821$	48.8084	1.70343	0.851713	−	0.524008i	$-0.175564\pi$
$821$	48.8084	1.70343	0.851713	+	0.524008i	$0.175564\pi$
$822$	15.7185	0.548244
$823$	26.2841	0.916205	0.458102	−	0.888899i	$-0.348529\pi$
$823$	26.2841	0.916205	0.458102	+	0.888899i	$0.348529\pi$
$824$	−39.9621	−1.39215
$825$	−15.2793	−0.531958
$826$	17.4894	0.608535
$827$	12.0210	0.418012	0.209006	−	0.977914i	$-0.432977\pi$
$827$	12.0210	0.418012	0.209006	+	0.977914i	$0.432977\pi$
$828$	10.5695	0.367314
$829$	−31.0007	−1.07670	−0.538350	−	0.842721i	$-0.680952\pi$
$829$	−31.0007	−1.07670	−0.538350	+	0.842721i	$0.680952\pi$
$830$	4.18465	0.145251
$831$	−0.141224	−0.00489901
$832$	−3.04801	−0.105671
$833$	−1.90517	−0.0660101
$834$	20.1885	0.699071
$835$	44.1198	1.52683
$836$	−33.2786	−1.15097
$837$	−22.3008	−0.770830
$838$	−4.81156	−0.166212
$839$	−46.5457	−1.60694	−0.803469	−	0.595347i	$-0.797015\pi$
$839$	−46.5457	−1.60694	−0.803469	+	0.595347i	$0.797015\pi$
$840$	−22.4764	−0.775508
$841$	−22.8141	−0.786694
$842$	14.9953	0.516771
$843$	9.74684	0.335699
$844$	25.6586	0.883206
$845$	−62.8434	−2.16188
$846$	0.999979	0.0343800
$847$	−27.5423	−0.946363
$848$	6.29275	0.216094
$849$	−2.24998	−0.0772190
$850$	4.89737	0.167978
$851$	30.7673	1.05469
$852$	−4.81551	−0.164977
$853$	−9.67911	−0.331406	−0.165703	−	0.986176i	$-0.552989\pi$
$853$	−9.67911	−0.331406	−0.165703	+	0.986176i	$0.552989\pi$
$854$	−10.2999	−0.352455
$855$	9.74395	0.333236
$856$	41.1462	1.40635
$857$	−23.7610	−0.811660	−0.405830	−	0.913949i	$-0.633017\pi$
$857$	−23.7610	−0.811660	−0.405830	+	0.913949i	$0.633017\pi$
$858$	42.4970	1.45082
$859$	45.2604	1.54426	0.772132	−	0.635462i	$-0.219190\pi$
$859$	45.2604	1.54426	0.772132	+	0.635462i	$0.219190\pi$
$860$	−13.5347	−0.461529
$861$	1.59647	0.0544074
$862$	15.3258	0.521997
$863$	−5.34189	−0.181840	−0.0909200	−	0.995858i	$-0.528981\pi$
$863$	−5.34189	−0.181840	−0.0909200	+	0.995858i	$0.528981\pi$
$864$	−21.3974	−0.727955
$865$	−18.6248	−0.633264
$866$	15.0147	0.510222
$867$	−10.1553	−0.344893
$868$	−24.1387	−0.819320
$869$	−24.0119	−0.814548
$870$	−6.08681	−0.206362
$871$	−19.3519	−0.655714
$872$	−17.8251	−0.603635
$873$	0	0
$874$	−16.9120	−0.572057
$875$	−31.2503	−1.05645
$876$	1.66517	0.0562608
$877$	33.8470	1.14293	0.571467	−	0.820625i	$-0.306375\pi$
$877$	33.8470	1.14293	0.571467	+	0.820625i	$0.306375\pi$
$878$	−3.09609	−0.104488
$879$	24.6882	0.832712
$880$	14.0759	0.474497
$881$	−2.07525	−0.0699170	−0.0349585	−	0.999389i	$-0.511130\pi$
$881$	−2.07525	−0.0699170	−0.0349585	+	0.999389i	$0.511130\pi$
$882$	0.308847	0.0103994
$883$	8.90806	0.299780	0.149890	−	0.988703i	$-0.452108\pi$
$883$	8.90806	0.299780	0.149890	+	0.988703i	$0.452108\pi$
$884$	50.7281	1.70617
$885$	39.3733	1.32352
$886$	−9.09346	−0.305501
$887$	−5.99572	−0.201317	−0.100658	−	0.994921i	$-0.532095\pi$
$887$	−5.99572	−0.201317	−0.100658	+	0.994921i	$0.532095\pi$
$888$	25.4668	0.854611
$889$	43.3274	1.45316
$890$	−5.75479	−0.192901
$891$	51.9236	1.73951
$892$	−4.22071	−0.141320
$893$	5.95887	0.199406
$894$	−2.55654	−0.0855034
$895$	16.2316	0.542564
$896$	−28.6379	−0.956725
$897$	−80.4302	−2.68549
$898$	−18.4690	−0.616317
$899$	−14.8292	−0.494583
$900$	2.95668	0.0985560
$901$	17.9966	0.599554
$902$	0.923593	0.0307523
$903$	−24.4263	−0.812856
$904$	6.98125	0.232193
$905$	40.6725	1.35200
$906$	3.76105	0.124953
$907$	−26.7442	−0.888027	−0.444014	−	0.896020i	$-0.646446\pi$
$907$	−26.7442	−0.888027	−0.444014	+	0.896020i	$0.646446\pi$
$908$	−30.1846	−1.00171
$909$	16.9536	0.562316
$910$	21.1274	0.700367
$911$	5.85914	0.194122	0.0970611	−	0.995278i	$-0.469056\pi$
$911$	5.85914	0.194122	0.0970611	+	0.995278i	$0.469056\pi$
$912$	15.1533	0.501777
$913$	−16.2712	−0.538499
$914$	−1.35501	−0.0448197
$915$	−23.1878	−0.766564
$916$	−2.03383	−0.0671998
$917$	−0.770266	−0.0254364
$918$	−11.4081	−0.376522
$919$	−29.8167	−0.983563	−0.491781	−	0.870719i	$-0.663654\pi$
$919$	−29.8167	−0.983563	−0.491781	+	0.870719i	$0.663654\pi$
$920$	24.6098	0.811360
$921$	23.9700	0.789837
$922$	10.4182	0.343106
$923$	10.2684	0.337990
$924$	38.5252	1.26739
$925$	8.60679	0.282990
$926$	21.9196	0.720323
$927$	−20.0557	−0.658715
$928$	−14.2285	−0.467074
$929$	−9.81538	−0.322032	−0.161016	−	0.986952i	$-0.551477\pi$
$929$	−9.81538	−0.322032	−0.161016	+	0.986952i	$0.551477\pi$
$930$	14.5918	0.478483
$931$	1.84042	0.0603172
$932$	24.0623	0.788187
$933$	21.0998	0.690777
$934$	22.6621	0.741526
$935$	40.2555	1.31650
$936$	−18.6552	−0.609765
$937$	−8.35265	−0.272869	−0.136435	−	0.990649i	$-0.543564\pi$
$937$	−8.35265	−0.272869	−0.136435	+	0.990649i	$0.543564\pi$
$938$	4.71063	0.153808
$939$	3.91019	0.127604
$940$	−3.82239	−0.124673
$941$	21.2826	0.693794	0.346897	−	0.937903i	$-0.387235\pi$
$941$	21.2826	0.693794	0.346897	+	0.937903i	$0.387235\pi$
$942$	21.7080	0.707286
$943$	−1.74800	−0.0569227
$944$	17.1582	0.558451
$945$	17.6946	0.575606
$946$	−14.1312	−0.459444
$947$	19.5756	0.636123	0.318061	−	0.948070i	$-0.396968\pi$
$947$	19.5756	0.636123	0.318061	+	0.948070i	$0.396968\pi$
$948$	16.5818	0.538551
$949$	−3.55075	−0.115262
$950$	−4.73093	−0.153492
$951$	−31.4577	−1.02009
$952$	−28.0121	−0.907878
$953$	−3.09413	−0.100229	−0.0501143	−	0.998743i	$-0.515959\pi$
$953$	−3.09413	−0.100229	−0.0501143	+	0.998743i	$0.515959\pi$
$954$	−2.91743	−0.0944554
$955$	−23.0230	−0.745007
$956$	−3.97142	−0.128445
$957$	23.6674	0.765058
$958$	9.42120	0.304385
$959$	−30.3850	−0.981182
$960$	1.67031	0.0539091
$961$	4.54978	0.146767
$962$	−23.9384	−0.771806
$963$	20.6499	0.665434
$964$	−28.3605	−0.913432
$965$	−35.3558	−1.13814
$966$	19.5783	0.629921
$967$	34.0569	1.09520	0.547598	−	0.836742i	$-0.315543\pi$
$967$	34.0569	1.09520	0.547598	+	0.836742i	$0.315543\pi$
$968$	24.9613	0.802285
$969$	43.3370	1.39219
$970$	0	0
$971$	−2.42672	−0.0778773	−0.0389386	−	0.999242i	$-0.512398\pi$
$971$	−2.42672	−0.0778773	−0.0389386	+	0.999242i	$0.512398\pi$
$972$	−18.1654	−0.582654
$973$	−39.0259	−1.25111
$974$	0.170592	0.00546611
$975$	−22.4994	−0.720557
$976$	−10.1048	−0.323447
$977$	5.09843	0.163113	0.0815565	−	0.996669i	$-0.474011\pi$
$977$	5.09843	0.163113	0.0815565	+	0.996669i	$0.474011\pi$
$978$	−6.33797	−0.202666
$979$	22.3764	0.715153
$980$	−1.18056	−0.0377115
$981$	−8.94584	−0.285619
$982$	−9.58900	−0.305997
$983$	54.1134	1.72595	0.862975	−	0.505247i	$-0.168599\pi$
$983$	54.1134	1.72595	0.862975	+	0.505247i	$0.168599\pi$
$984$	−1.44686	−0.0461242
$985$	−15.4573	−0.492511
$986$	−7.58594	−0.241586
$987$	−6.89833	−0.219576
$988$	−49.0041	−1.55903
$989$	26.7448	0.850434
$990$	−6.52583	−0.207404
$991$	17.5470	0.557400	0.278700	−	0.960378i	$-0.410096\pi$
$991$	17.5470	0.557400	0.278700	+	0.960378i	$0.410096\pi$
$992$	34.1097	1.08298
$993$	1.15304	0.0365907
$994$	−2.49954	−0.0792807
$995$	−4.93216	−0.156360
$996$	11.2363	0.356037
$997$	7.28357	0.230673	0.115337	−	0.993326i	$-0.463205\pi$
$997$	7.28357	0.230673	0.115337	+	0.993326i	$0.463205\pi$
$998$	7.37633	0.233493
$999$	−20.0489	−0.634318

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9409.2.a.m.1.23		56
97.3	even	48		97.2.i.a.9.4	✓	56
97.65	even	48		97.2.i.a.54.4	yes	56
97.96	even	2	inner	9409.2.a.m.1.24		56
291.65	odd	48		873.2.bo.b.442.4		56
291.197	odd	48		873.2.bo.b.397.4		56

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
97.2.i.a.9.4	✓	56	97.3	even	48
97.2.i.a.54.4	yes	56	97.65	even	48
873.2.bo.b.397.4		56	291.197	odd	48
873.2.bo.b.442.4		56	291.65	odd	48
9409.2.a.m.1.23		56	1.1	even	1	trivial
9409.2.a.m.1.24		56	97.96	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 9409 = 97^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9409.a (trivial)

\(f(q)\)	\(=\)	\(q-0.650656 q^{2} -2.04155 q^{3} -1.57665 q^{4} -1.84237 q^{5} +1.32835 q^{6} -2.56780 q^{7} +2.32717 q^{8} +1.16793 q^{9} +1.19875 q^{10} -4.66112 q^{11} +3.21880 q^{12} -6.86367 q^{13} +1.67075 q^{14} +3.76130 q^{15} +1.63911 q^{16} +4.68768 q^{17} -0.759920 q^{18} -4.52836 q^{19} +2.90477 q^{20} +5.24229 q^{21} +3.03279 q^{22} -5.73987 q^{23} -4.75103 q^{24} -1.60566 q^{25} +4.46589 q^{26} +3.74027 q^{27} +4.04851 q^{28} +2.48714 q^{29} -2.44731 q^{30} -5.96236 q^{31} -5.72083 q^{32} +9.51591 q^{33} -3.05007 q^{34} +4.73084 q^{35} -1.84141 q^{36} -5.36028 q^{37} +2.94641 q^{38} +14.0125 q^{39} -4.28751 q^{40} +0.304536 q^{41} -3.41093 q^{42} -4.65947 q^{43} +7.34894 q^{44} -2.15176 q^{45} +3.73468 q^{46} -1.31590 q^{47} -3.34632 q^{48} -0.406420 q^{49} +1.04473 q^{50} -9.57013 q^{51} +10.8216 q^{52} +3.83913 q^{53} -2.43363 q^{54} +8.58752 q^{55} -5.97569 q^{56} +9.24488 q^{57} -1.61827 q^{58} +10.4680 q^{59} -5.93024 q^{60} -6.16483 q^{61} +3.87945 q^{62} -2.99900 q^{63} +0.444079 q^{64} +12.6455 q^{65} -6.19158 q^{66} +2.81947 q^{67} -7.39081 q^{68} +11.7182 q^{69} -3.07815 q^{70} -1.49606 q^{71} +2.71796 q^{72} +0.517325 q^{73} +3.48770 q^{74} +3.27804 q^{75} +7.13963 q^{76} +11.9688 q^{77} -9.11734 q^{78} +5.15153 q^{79} -3.01985 q^{80} -11.1397 q^{81} -0.198148 q^{82} +3.49084 q^{83} -8.26523 q^{84} -8.63645 q^{85} +3.03171 q^{86} -5.07762 q^{87} -10.8472 q^{88} -4.80065 q^{89} +1.40006 q^{90} +17.6245 q^{91} +9.04975 q^{92} +12.1725 q^{93} +0.856199 q^{94} +8.34293 q^{95} +11.6794 q^{96} +0.264440 q^{98} -5.44385 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 56 q - 8 q^{3} + 48 q^{4} - 24 q^{6} + 24 q^{9} - 48 q^{11} - 56 q^{12} + 24 q^{16} + 16 q^{18} - 40 q^{24} + 8 q^{25} - 32 q^{27} - 56 q^{31} - 8 q^{33} - 56 q^{35} + 40 q^{36} - 32 q^{43} - 144 q^{44}+ \cdots + 16 q^{99}+O(q^{100}) \) 56 * q - 8 * q^3 + 48 * q^4 - 24 * q^6 + 24 * q^9 - 48 * q^11 - 56 * q^12 + 24 * q^16 + 16 * q^18 - 40 * q^24 + 8 * q^25 - 32 * q^27 - 56 * q^31 - 8 * q^33 - 56 * q^35 + 40 * q^36 - 32 * q^43 - 144 * q^44 - 48 * q^47 - 112 * q^48 + 56 * q^49 + 16 * q^53 - 72 * q^54 + 24 * q^61 - 56 * q^62 + 32 * q^64 - 16 * q^65 - 64 * q^66 + 8 * q^70 + 128 * q^72 - 40 * q^73 - 72 * q^75 - 144 * q^79 - 88 * q^81 - 104 * q^85 - 64 * q^86 - 16 * q^88 - 80 * q^89 - 120 * q^91 - 48 * q^93 - 160 * q^94 - 136 * q^95 + 16 * q^96 + 96 * q^98 + 16 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more