LMFDB - Embedded newform 9409.2.a.n.1.3

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 = [56,8,16,40,0] 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:	$0$
Dimension:	$56$
Twist minimal:	no (minimal twist has level 97)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
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)

$f(q)$	$=$	\(q-2.36394 q^{2} +0.429115 q^{3} +3.58823 q^{4} -1.05721 q^{5} -1.01440 q^{6} +1.83323 q^{7} -3.75448 q^{8} -2.81586 q^{9} +2.49919 q^{10} -2.02550 q^{11} +1.53976 q^{12} -1.26591 q^{13} -4.33365 q^{14} -0.453666 q^{15} +1.69892 q^{16} +5.49328 q^{17} +6.65653 q^{18} -2.74211 q^{19} -3.79352 q^{20} +0.786666 q^{21} +4.78816 q^{22} -7.47844 q^{23} -1.61110 q^{24} -3.88230 q^{25} +2.99254 q^{26} -2.49568 q^{27} +6.57804 q^{28} +5.15694 q^{29} +1.07244 q^{30} +1.37907 q^{31} +3.49281 q^{32} -0.869172 q^{33} -12.9858 q^{34} -1.93811 q^{35} -10.1039 q^{36} +3.18081 q^{37} +6.48219 q^{38} -0.543222 q^{39} +3.96928 q^{40} -10.4437 q^{41} -1.85963 q^{42} +1.04450 q^{43} -7.26794 q^{44} +2.97696 q^{45} +17.6786 q^{46} -9.22056 q^{47} +0.729032 q^{48} -3.63928 q^{49} +9.17754 q^{50} +2.35725 q^{51} -4.54238 q^{52} +13.7521 q^{53} +5.89963 q^{54} +2.14138 q^{55} -6.88281 q^{56} -1.17668 q^{57} -12.1907 q^{58} +5.42706 q^{59} -1.62786 q^{60} +4.56051 q^{61} -3.26005 q^{62} -5.16211 q^{63} -11.6546 q^{64} +1.33834 q^{65} +2.05467 q^{66} -15.3080 q^{67} +19.7111 q^{68} -3.20911 q^{69} +4.58159 q^{70} +2.17481 q^{71} +10.5721 q^{72} -8.19777 q^{73} -7.51925 q^{74} -1.66595 q^{75} -9.83931 q^{76} -3.71320 q^{77} +1.28415 q^{78} -3.60300 q^{79} -1.79612 q^{80} +7.37665 q^{81} +24.6884 q^{82} +14.6991 q^{83} +2.82274 q^{84} -5.80757 q^{85} -2.46914 q^{86} +2.21292 q^{87} +7.60468 q^{88} -5.69432 q^{89} -7.03737 q^{90} -2.32071 q^{91} -26.8343 q^{92} +0.591782 q^{93} +21.7969 q^{94} +2.89899 q^{95} +1.49882 q^{96} +8.60304 q^{98} +5.70352 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 56 q + 8 q^{2} + 16 q^{3} + 40 q^{4} + 16 q^{6} + 24 q^{8} + 40 q^{9} + 32 q^{11} + 48 q^{12} + 8 q^{16} + 24 q^{18} + 16 q^{22} + 8 q^{25} + 64 q^{27} + 48 q^{31} + 56 q^{32} + 80 q^{33} + 48 q^{35} + 8 q^{36}+ \cdots - 16 q^{99}+O(q^{100}) $ 56 * q + 8 * q^2 + 16 * q^3 + 40 * q^4 + 16 * q^6 + 24 * q^8 + 40 * q^9 + 32 * q^11 + 48 * q^12 + 8 * q^16 + 24 * q^18 + 16 * q^22 + 8 * q^25 + 64 * q^27 + 48 * q^31 + 56 * q^32 + 80 * q^33 + 48 * q^35 + 8 * q^36 + 32 * q^43 + 80 * q^44 + 16 * q^47 + 112 * q^48 + 8 * q^49 + 24 * q^50 + 48 * q^53 + 96 * q^54 + 32 * q^61 + 80 * q^62 - 56 * q^64 + 128 * q^65 + 192 * q^66 + 80 * q^70 - 88 * q^72 + 32 * q^73 + 80 * q^75 + 48 * q^79 + 72 * q^81 + 144 * q^86 + 176 * q^88 + 96 * q^89 + 64 * q^91 + 64 * q^94 + 176 * q^95 + 32 * q^96 - 136 * 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 $

$2$	−2.36394	−1.67156	−0.835780	−	0.549064i	$-0.814984\pi$
$2$	−2.36394	−1.67156	−0.835780	+	0.549064i	$0.814984\pi$
$3$	0.429115	0.247750	0.123875	−	0.992298i	$-0.460468\pi$
$3$	0.429115	0.247750	0.123875	+	0.992298i	$0.460468\pi$
$4$	3.58823	1.79411
$5$	−1.05721	−0.472800	−0.236400	−	0.971656i	$-0.575968\pi$
$5$	−1.05721	−0.472800	−0.236400	+	0.971656i	$0.575968\pi$
$6$	−1.01440	−0.414129
$7$	1.83323	0.692895	0.346447	−	0.938069i	$-0.387388\pi$
$7$	1.83323	0.692895	0.346447	+	0.938069i	$0.387388\pi$
$8$	−3.75448	−1.32741
$9$	−2.81586	−0.938620
$10$	2.49919	0.790314
$11$	−2.02550	−0.610710	−0.305355	−	0.952239i	$-0.598775\pi$
$11$	−2.02550	−0.610710	−0.305355	+	0.952239i	$0.598775\pi$
$12$	1.53976	0.444491
$13$	−1.26591	−0.351101	−0.175550	−	0.984470i	$-0.556171\pi$
$13$	−1.26591	−0.351101	−0.175550	+	0.984470i	$0.556171\pi$
$14$	−4.33365	−1.15822
$15$	−0.453666	−0.117136
$16$	1.69892	0.424729
$17$	5.49328	1.33232	0.666158	−	0.745811i	$-0.267938\pi$
$17$	5.49328	1.33232	0.666158	+	0.745811i	$0.267938\pi$
$18$	6.65653	1.56896
$19$	−2.74211	−0.629083	−0.314541	−	0.949244i	$-0.601851\pi$
$19$	−2.74211	−0.629083	−0.314541	+	0.949244i	$0.601851\pi$
$20$	−3.79352	−0.848257
$21$	0.786666	0.171665
$22$	4.78816	1.02084
$23$	−7.47844	−1.55936	−0.779681	−	0.626177i	$-0.784619\pi$
$23$	−7.47844	−1.55936	−0.779681	+	0.626177i	$0.784619\pi$
$24$	−1.61110	−0.328865
$25$	−3.88230	−0.776460
$26$	2.99254	0.586886
$27$	−2.49568	−0.480293
$28$	6.57804	1.24313
$29$	5.15694	0.957619	0.478809	−	0.877919i	$-0.341068\pi$
$29$	5.15694	0.957619	0.478809	+	0.877919i	$0.341068\pi$
$30$	1.07244	0.195800
$31$	1.37907	0.247689	0.123845	−	0.992302i	$-0.460478\pi$
$31$	1.37907	0.247689	0.123845	+	0.992302i	$0.460478\pi$
$32$	3.49281	0.617448
$33$	−0.869172	−0.151303
$34$	−12.9858	−2.22705
$35$	−1.93811	−0.327601
$36$	−10.1039	−1.68399
$37$	3.18081	0.522922	0.261461	−	0.965214i	$-0.415796\pi$
$37$	3.18081	0.522922	0.261461	+	0.965214i	$0.415796\pi$
$38$	6.48219	1.05155
$39$	−0.543222	−0.0869852
$40$	3.96928	0.627599
$41$	−10.4437	−1.63104	−0.815518	−	0.578731i	$-0.803548\pi$
$41$	−10.4437	−1.63104	−0.815518	+	0.578731i	$0.803548\pi$
$42$	−1.85963	−0.286948
$43$	1.04450	0.159285	0.0796425	−	0.996823i	$-0.474622\pi$
$43$	1.04450	0.159285	0.0796425	+	0.996823i	$0.474622\pi$
$44$	−7.26794	−1.09568
$45$	2.97696	0.443780
$46$	17.6786	2.60657
$47$	−9.22056	−1.34496	−0.672478	−	0.740117i	$-0.734770\pi$
$47$	−9.22056	−1.34496	−0.672478	+	0.740117i	$0.734770\pi$
$48$	0.729032	0.105227
$49$	−3.63928	−0.519897
$50$	9.17754	1.29790
$51$	2.35725	0.330081
$52$	−4.54238	−0.629915
$53$	13.7521	1.88900	0.944501	−	0.328507i	$-0.106546\pi$
$53$	13.7521	1.88900	0.944501	+	0.328507i	$0.106546\pi$
$54$	5.89963	0.802838
$55$	2.14138	0.288744
$56$	−6.88281	−0.919754
$57$	−1.17668	−0.155855
$58$	−12.1907	−1.60072
$59$	5.42706	0.706543	0.353272	−	0.935521i	$-0.385069\pi$
$59$	5.42706	0.706543	0.353272	+	0.935521i	$0.385069\pi$
$60$	−1.62786	−0.210156
$61$	4.56051	0.583914	0.291957	−	0.956431i	$-0.405694\pi$
$61$	4.56051	0.583914	0.291957	+	0.956431i	$0.405694\pi$
$62$	−3.26005	−0.414027
$63$	−5.16211	−0.650365
$64$	−11.6546	−1.45683
$65$	1.33834	0.166001
$66$	2.05467	0.252913
$67$	−15.3080	−1.87017	−0.935087	−	0.354417i	$-0.884679\pi$
$67$	−15.3080	−1.87017	−0.935087	+	0.354417i	$0.884679\pi$
$68$	19.7111	2.39033
$69$	−3.20911	−0.386332
$70$	4.58159	0.547604
$71$	2.17481	0.258102	0.129051	−	0.991638i	$-0.458807\pi$
$71$	2.17481	0.258102	0.129051	+	0.991638i	$0.458807\pi$
$72$	10.5721	1.24593
$73$	−8.19777	−0.959477	−0.479738	−	0.877412i	$-0.659268\pi$
$73$	−8.19777	−0.959477	−0.479738	+	0.877412i	$0.659268\pi$
$74$	−7.51925	−0.874095
$75$	−1.66595	−0.192368
$76$	−9.83931	−1.12865
$77$	−3.71320	−0.423158
$78$	1.28415	0.145401
$79$	−3.60300	−0.405369	−0.202684	−	0.979244i	$-0.564967\pi$
$79$	−3.60300	−0.405369	−0.202684	+	0.979244i	$0.564967\pi$
$80$	−1.79612	−0.200812
$81$	7.37665	0.819627
$82$	24.6884	2.72638
$83$	14.6991	1.61344	0.806718	−	0.590937i	$-0.201242\pi$
$83$	14.6991	1.61344	0.806718	+	0.590937i	$0.201242\pi$
$84$	2.82274	0.307986
$85$	−5.80757	−0.629919
$86$	−2.46914	−0.266255
$87$	2.21292	0.237250
$88$	7.60468	0.810662
$89$	−5.69432	−0.603597	−0.301798	−	0.953372i	$-0.597587\pi$
$89$	−5.69432	−0.603597	−0.301798	+	0.953372i	$0.597587\pi$
$90$	−7.03737	−0.741804
$91$	−2.32071	−0.243276
$92$	−26.8343	−2.79767
$93$	0.591782	0.0613649
$94$	21.7969	2.24818
$95$	2.89899	0.297430
$96$	1.49882	0.152973
$97$	0	0
$97$	0	0
$98$	8.60304	0.869038
$99$	5.70352	0.573225
$100$	−13.9306	−1.39306
$101$	−5.65301	−0.562496	−0.281248	−	0.959635i	$-0.590748\pi$
$101$	−5.65301	−0.562496	−0.281248	+	0.959635i	$0.590748\pi$
$102$	−5.57241	−0.551751
$103$	−15.7188	−1.54882	−0.774412	−	0.632682i	$-0.781954\pi$
$103$	−15.7188	−1.54882	−0.774412	+	0.632682i	$0.781954\pi$
$104$	4.75284	0.466054
$105$	−0.831674	−0.0811631
$106$	−32.5093	−3.15758
$107$	13.7606	1.33029	0.665145	−	0.746714i	$-0.268370\pi$
$107$	13.7606	1.33029	0.665145	+	0.746714i	$0.268370\pi$
$108$	−8.95505	−0.861700
$109$	−4.08834	−0.391592	−0.195796	−	0.980645i	$-0.562729\pi$
$109$	−4.08834	−0.391592	−0.195796	+	0.980645i	$0.562729\pi$
$110$	−5.06210	−0.482653
$111$	1.36493	0.129554
$112$	3.11450	0.294293
$113$	7.54899	0.710149	0.355075	−	0.934838i	$-0.384455\pi$
$113$	7.54899	0.710149	0.355075	+	0.934838i	$0.384455\pi$
$114$	2.78161	0.260521
$115$	7.90630	0.737267
$116$	18.5043	1.71808
$117$	3.56463	0.329550
$118$	−12.8293	−1.18103
$119$	10.0704	0.923155
$120$	1.70328	0.155487
$121$	−6.89736	−0.627033
$122$	−10.7808	−0.976047
$123$	−4.48157	−0.404089
$124$	4.94843	0.444382
$125$	9.39048	0.839910
$126$	12.2029	1.08712
$127$	10.9204	0.969031	0.484515	−	0.874783i	$-0.338996\pi$
$127$	10.9204	0.969031	0.484515	+	0.874783i	$0.338996\pi$
$128$	20.5653	1.81773
$129$	0.448212	0.0394628
$130$	−3.16376	−0.277480
$131$	−10.1783	−0.889281	−0.444640	−	0.895709i	$-0.646669\pi$
$131$	−10.1783	−0.889281	−0.444640	+	0.895709i	$0.646669\pi$
$132$	−3.11879	−0.271455
$133$	−5.02691	−0.435888
$134$	36.1873	3.12611
$135$	2.63846	0.227083
$136$	−20.6244	−1.76853
$137$	10.8298	0.925252	0.462626	−	0.886554i	$-0.346907\pi$
$137$	10.8298	0.925252	0.462626	+	0.886554i	$0.346907\pi$
$138$	7.58616	0.645777
$139$	−0.712051	−0.0603954	−0.0301977	−	0.999544i	$-0.509614\pi$
$139$	−0.712051	−0.0603954	−0.0301977	+	0.999544i	$0.509614\pi$
$140$	−6.95439	−0.587753
$141$	−3.95668	−0.333213
$142$	−5.14112	−0.431433
$143$	2.56410	0.214421
$144$	−4.78391	−0.398659
$145$	−5.45198	−0.452762
$146$	19.3791	1.60382
$147$	−1.56167	−0.128804
$148$	11.4135	0.938181
$149$	−14.3096	−1.17229	−0.586143	−	0.810207i	$-0.699354\pi$
$149$	−14.3096	−1.17229	−0.586143	+	0.810207i	$0.699354\pi$
$150$	3.93822	0.321555
$151$	−10.1068	−0.822479	−0.411240	−	0.911527i	$-0.634904\pi$
$151$	−10.1068	−0.822479	−0.411240	+	0.911527i	$0.634904\pi$
$152$	10.2952	0.835050
$153$	−15.4683	−1.25054
$154$	8.77779	0.707334
$155$	−1.45798	−0.117107
$156$	−1.94921	−0.156061
$157$	0.423904	0.0338312	0.0169156	−	0.999857i	$-0.494615\pi$
$157$	0.423904	0.0338312	0.0169156	+	0.999857i	$0.494615\pi$
$158$	8.51728	0.677598
$159$	5.90126	0.468000
$160$	−3.69265	−0.291929
$161$	−13.7097	−1.08047
$162$	−17.4380	−1.37006
$163$	9.82774	0.769768	0.384884	−	0.922965i	$-0.374241\pi$
$163$	9.82774	0.769768	0.384884	+	0.922965i	$0.374241\pi$
$164$	−37.4745	−2.92626
$165$	0.918900	0.0715363
$166$	−34.7478	−2.69696
$167$	12.3857	0.958433	0.479217	−	0.877697i	$-0.340921\pi$
$167$	12.3857	0.958433	0.479217	+	0.877697i	$0.340921\pi$
$168$	−2.95352	−0.227869
$169$	−11.3975	−0.876728
$170$	13.7288	1.05295
$171$	7.72139	0.590470
$172$	3.74791	0.285775
$173$	−18.3232	−1.39309	−0.696544	−	0.717514i	$-0.745280\pi$
$173$	−18.3232	−1.39309	−0.696544	+	0.717514i	$0.745280\pi$
$174$	−5.23122	−0.396578
$175$	−7.11714	−0.538005
$176$	−3.44115	−0.259387
$177$	2.32884	0.175046
$178$	13.4610	1.00895
$179$	1.75590	0.131242	0.0656210	−	0.997845i	$-0.479097\pi$
$179$	1.75590	0.131242	0.0656210	+	0.997845i	$0.479097\pi$
$180$	10.6820	0.796191
$181$	−3.22171	−0.239468	−0.119734	−	0.992806i	$-0.538204\pi$
$181$	−3.22171	−0.239468	−0.119734	+	0.992806i	$0.538204\pi$
$182$	5.48602	0.406651
$183$	1.95699	0.144665
$184$	28.0776	2.06991
$185$	−3.36279	−0.247237
$186$	−1.39894	−0.102575
$187$	−11.1266	−0.813659
$188$	−33.0855	−2.41301
$189$	−4.57514	−0.332793
$190$	−6.85305	−0.497173
$191$	22.7565	1.64660	0.823302	−	0.567604i	$-0.192130\pi$
$191$	22.7565	1.64660	0.823302	+	0.567604i	$0.192130\pi$
$192$	−5.00119	−0.360929
$193$	−0.260716	−0.0187668	−0.00938338	−	0.999956i	$-0.502987\pi$
$193$	−0.260716	−0.0187668	−0.00938338	+	0.999956i	$0.502987\pi$
$194$	0	0
$195$	0.574302	0.0411266
$196$	−13.0585	−0.932753
$197$	12.9183	0.920392	0.460196	−	0.887817i	$-0.347779\pi$
$197$	12.9183	0.920392	0.460196	+	0.887817i	$0.347779\pi$
$198$	−13.4828	−0.958180
$199$	15.0611	1.06765	0.533827	−	0.845593i	$-0.320753\pi$
$199$	15.0611	1.06765	0.533827	+	0.845593i	$0.320753\pi$
$200$	14.5760	1.03068
$201$	−6.56892	−0.463336
$202$	13.3634	0.940246
$203$	9.45384	0.663529
$204$	8.45835	0.592203
$205$	11.0412	0.771154
$206$	37.1585	2.58895
$207$	21.0582	1.46365
$208$	−2.15068	−0.149123
$209$	5.55413	0.384187
$210$	1.96603	0.135669
$211$	11.0194	0.758605	0.379303	−	0.925273i	$-0.376164\pi$
$211$	11.0194	0.758605	0.379303	+	0.925273i	$0.376164\pi$
$212$	49.3458	3.38909
$213$	0.933243	0.0639448
$214$	−32.5293	−2.22366
$215$	−1.10426	−0.0753100
$216$	9.36996	0.637545
$217$	2.52816	0.171623
$218$	9.66460	0.654569
$219$	−3.51779	−0.237710
$220$	7.68376	0.518039
$221$	−6.95401	−0.467777
$222$	−3.22663	−0.216557
$223$	3.24145	0.217064	0.108532	−	0.994093i	$-0.465385\pi$
$223$	3.24145	0.217064	0.108532	+	0.994093i	$0.465385\pi$
$224$	6.40312	0.427826
$225$	10.9320	0.728801
$226$	−17.8454	−1.18706
$227$	3.06614	0.203507	0.101753	−	0.994810i	$-0.467555\pi$
$227$	3.06614	0.203507	0.101753	+	0.994810i	$0.467555\pi$
$228$	−4.22220	−0.279622
$229$	7.66542	0.506545	0.253272	−	0.967395i	$-0.418493\pi$
$229$	7.66542	0.506545	0.253272	+	0.967395i	$0.418493\pi$
$230$	−18.6900	−1.23239
$231$	−1.59339	−0.104837
$232$	−19.3616	−1.27115
$233$	−20.4374	−1.33890	−0.669450	−	0.742857i	$-0.733470\pi$
$233$	−20.4374	−1.33890	−0.669450	+	0.742857i	$0.733470\pi$
$234$	−8.42659	−0.550863
$235$	9.74810	0.635896
$236$	19.4735	1.26762
$237$	−1.54610	−0.100430
$238$	−23.8059	−1.54311
$239$	−1.62314	−0.104992	−0.0524961	−	0.998621i	$-0.516718\pi$
$239$	−1.62314	−0.104992	−0.0524961	+	0.998621i	$0.516718\pi$
$240$	−0.770742	−0.0497512
$241$	24.7120	1.59184	0.795918	−	0.605404i	$-0.206988\pi$
$241$	24.7120	1.59184	0.795918	+	0.605404i	$0.206988\pi$
$242$	16.3050	1.04812
$243$	10.6525	0.683355
$244$	16.3642	1.04761
$245$	3.84749	0.245807
$246$	10.5942	0.675459
$247$	3.47127	0.220872
$248$	−5.17770	−0.328785
$249$	6.30761	0.399729
$250$	−22.1986	−1.40396
$251$	8.46810	0.534501	0.267251	−	0.963627i	$-0.413885\pi$
$251$	8.46810	0.534501	0.267251	+	0.963627i	$0.413885\pi$
$252$	−18.5228	−1.16683
$253$	15.1476	0.952318
$254$	−25.8153	−1.61979
$255$	−2.49212	−0.156062
$256$	−25.3059	−1.58162
$257$	29.1991	1.82139	0.910695	−	0.413080i	$-0.135547\pi$
$257$	29.1991	1.82139	0.910695	+	0.413080i	$0.135547\pi$
$258$	−1.05955	−0.0659645
$259$	5.83115	0.362330
$260$	4.80226	0.297824
$261$	−14.5212	−0.898840
$262$	24.0609	1.48649
$263$	10.6463	0.656481	0.328241	−	0.944594i	$-0.393544\pi$
$263$	10.6463	0.656481	0.328241	+	0.944594i	$0.393544\pi$
$264$	3.26329	0.200841
$265$	−14.5390	−0.893121
$266$	11.8833	0.728613
$267$	−2.44352	−0.149541
$268$	−54.9287	−3.35531
$269$	14.4291	0.879760	0.439880	−	0.898057i	$-0.355021\pi$
$269$	14.4291	0.879760	0.439880	+	0.898057i	$0.355021\pi$
$270$	−6.23717	−0.379582
$271$	−11.0284	−0.669929	−0.334964	−	0.942231i	$-0.608724\pi$
$271$	−11.0284	−0.669929	−0.334964	+	0.942231i	$0.608724\pi$
$272$	9.33263	0.565874
$273$	−0.995851	−0.0602716
$274$	−25.6010	−1.54661
$275$	7.86359	0.474192
$276$	−11.5150	−0.693123
$277$	−15.6035	−0.937524	−0.468762	−	0.883324i	$-0.655300\pi$
$277$	−15.6035	−0.937524	−0.468762	+	0.883324i	$0.655300\pi$
$278$	1.68325	0.100955
$279$	−3.88328	−0.232486
$280$	7.27660	0.434860
$281$	−14.2676	−0.851134	−0.425567	−	0.904927i	$-0.639925\pi$
$281$	−14.2676	−0.851134	−0.425567	+	0.904927i	$0.639925\pi$
$282$	9.35338	0.556985
$283$	13.4202	0.797747	0.398873	−	0.917006i	$-0.369401\pi$
$283$	13.4202	0.797747	0.398873	+	0.917006i	$0.369401\pi$
$284$	7.80370	0.463064
$285$	1.24400	0.0736883
$286$	−6.06139	−0.358417
$287$	−19.1457	−1.13014
$288$	−9.83527	−0.579549
$289$	13.1761	0.775066
$290$	12.8882	0.756819
$291$	0	0
$292$	−29.4155	−1.72141
$293$	−26.3939	−1.54195	−0.770973	−	0.636868i	$-0.780230\pi$
$293$	−26.3939	−1.54195	−0.770973	+	0.636868i	$0.780230\pi$
$294$	3.69170	0.215304
$295$	−5.73756	−0.334054
$296$	−11.9423	−0.694130
$297$	5.05498	0.293320
$298$	33.8270	1.95955
$299$	9.46705	0.547493
$300$	−5.97782	−0.345130
$301$	1.91481	0.110368
$302$	23.8919	1.37482
$303$	−2.42580	−0.139358
$304$	−4.65862	−0.267190
$305$	−4.82143	−0.276074
$306$	36.5662	2.09035
$307$	30.0254	1.71364	0.856820	−	0.515616i	$-0.172437\pi$
$307$	30.0254	1.71364	0.856820	+	0.515616i	$0.172437\pi$
$308$	−13.3238	−0.759194
$309$	−6.74520	−0.383721
$310$	3.44657	0.195752
$311$	−8.97430	−0.508886	−0.254443	−	0.967088i	$-0.581892\pi$
$311$	−8.97430	−0.508886	−0.254443	+	0.967088i	$0.581892\pi$
$312$	2.03952	0.115465
$313$	21.4497	1.21241	0.606203	−	0.795310i	$-0.292692\pi$
$313$	21.4497	1.21241	0.606203	+	0.795310i	$0.292692\pi$
$314$	−1.00209	−0.0565509
$315$	5.45745	0.307493
$316$	−12.9284	−0.727278
$317$	−5.74021	−0.322402	−0.161201	−	0.986922i	$-0.551537\pi$
$317$	−5.74021	−0.322402	−0.161201	+	0.986922i	$0.551537\pi$
$318$	−13.9502	−0.782291
$319$	−10.4454	−0.584828
$320$	12.3214	0.688789
$321$	5.90490	0.329579
$322$	32.4089	1.80608
$323$	−15.0632	−0.838137
$324$	26.4691	1.47050
$325$	4.91465	0.272616
$326$	−23.2322	−1.28671
$327$	−1.75437	−0.0970168
$328$	39.2107	2.16505
$329$	−16.9034	−0.931914
$330$	−2.17223	−0.119577
$331$	−27.9801	−1.53793	−0.768963	−	0.639293i	$-0.779227\pi$
$331$	−27.9801	−1.53793	−0.768963	+	0.639293i	$0.779227\pi$
$332$	52.7437	2.89469
$333$	−8.95671	−0.490825
$334$	−29.2791	−1.60208
$335$	16.1839	0.884219
$336$	1.33648	0.0729110
$337$	−1.22028	−0.0664727	−0.0332363	−	0.999448i	$-0.510581\pi$
$337$	−1.22028	−0.0664727	−0.0332363	+	0.999448i	$0.510581\pi$
$338$	26.9430	1.46550
$339$	3.23939	0.175939
$340$	−20.8389	−1.13015
$341$	−2.79331	−0.151266
$342$	−18.2529	−0.987006
$343$	−19.5042	−1.05313
$344$	−3.92156	−0.211436
$345$	3.39272	0.182658
$346$	43.3150	2.32863
$347$	−2.06189	−0.110688	−0.0553440	−	0.998467i	$-0.517626\pi$
$347$	−2.06189	−0.110688	−0.0553440	+	0.998467i	$0.517626\pi$
$348$	7.94046	0.425653
$349$	−11.9556	−0.639967	−0.319983	−	0.947423i	$-0.603677\pi$
$349$	−11.9556	−0.639967	−0.319983	+	0.947423i	$0.603677\pi$
$350$	16.8245	0.899308
$351$	3.15931	0.168631
$352$	−7.07468	−0.377082
$353$	16.9597	0.902671	0.451336	−	0.892354i	$-0.350948\pi$
$353$	16.9597	0.902671	0.451336	+	0.892354i	$0.350948\pi$
$354$	−5.50524	−0.292600
$355$	−2.29923	−0.122031
$356$	−20.4325	−1.08292
$357$	4.32138	0.228712
$358$	−4.15085	−0.219379
$359$	−18.2612	−0.963788	−0.481894	−	0.876229i	$-0.660051\pi$
$359$	−18.2612	−0.963788	−0.481894	+	0.876229i	$0.660051\pi$
$360$	−11.1769	−0.589077
$361$	−11.4808	−0.604255
$362$	7.61593	0.400284
$363$	−2.95976	−0.155347
$364$	−8.32722	−0.436465
$365$	8.66679	0.453641
$366$	−4.62620	−0.241816
$367$	36.1690	1.88800	0.944002	−	0.329939i	$-0.107028\pi$
$367$	36.1690	1.88800	0.944002	+	0.329939i	$0.107028\pi$
$368$	−12.7052	−0.662307
$369$	29.4081	1.53092
$370$	7.94945	0.413272
$371$	25.2108	1.30888
$372$	2.12345	0.110096
$373$	−10.5388	−0.545677	−0.272838	−	0.962060i	$-0.587962\pi$
$373$	−10.5388	−0.545677	−0.272838	+	0.962060i	$0.587962\pi$
$374$	26.3027	1.36008
$375$	4.02960	0.208088
$376$	34.6184	1.78531
$377$	−6.52823	−0.336221
$378$	10.8154	0.556283
$379$	22.7256	1.16734	0.583669	−	0.811992i	$-0.301617\pi$
$379$	22.7256	1.16734	0.583669	+	0.811992i	$0.301617\pi$
$380$	10.4022	0.533624
$381$	4.68612	0.240077
$382$	−53.7951	−2.75240
$383$	−10.9452	−0.559275	−0.279638	−	0.960106i	$-0.590214\pi$
$383$	−10.9452	−0.559275	−0.279638	+	0.960106i	$0.590214\pi$
$384$	8.82488	0.450343
$385$	3.92564	0.200069
$386$	0.616318	0.0313698
$387$	−2.94117	−0.149508
$388$	0	0
$389$	12.0964	0.613310	0.306655	−	0.951821i	$-0.400790\pi$
$389$	12.0964	0.613310	0.306655	+	0.951821i	$0.400790\pi$
$390$	−1.35762	−0.0687456
$391$	−41.0812	−2.07756
$392$	13.6636	0.690115
$393$	−4.36766	−0.220319
$394$	−30.5382	−1.53849
$395$	3.80914	0.191658
$396$	20.4655	1.02843
$397$	−8.50508	−0.426858	−0.213429	−	0.976959i	$-0.568463\pi$
$397$	−8.50508	−0.426858	−0.213429	+	0.976959i	$0.568463\pi$
$398$	−35.6036	−1.78465
$399$	−2.15712	−0.107991
$400$	−6.59571	−0.329785
$401$	−6.19223	−0.309225	−0.154613	−	0.987975i	$-0.549413\pi$
$401$	−6.19223	−0.309225	−0.154613	+	0.987975i	$0.549413\pi$
$402$	15.5285	0.774493
$403$	−1.74579	−0.0869639
$404$	−20.2843	−1.00918
$405$	−7.79869	−0.387520
$406$	−22.3483	−1.10913
$407$	−6.44272	−0.319354
$408$	−8.85025	−0.438152
$409$	11.7675	0.581864	0.290932	−	0.956744i	$-0.406035\pi$
$409$	11.7675	0.581864	0.290932	+	0.956744i	$0.406035\pi$
$410$	−26.1009	−1.28903
$411$	4.64723	0.229231
$412$	−56.4028	−2.77877
$413$	9.94904	0.489560
$414$	−49.7805	−2.44658
$415$	−15.5401	−0.762833
$416$	−4.42159	−0.216786
$417$	−0.305552	−0.0149629
$418$	−13.1297	−0.642192
$419$	−24.7721	−1.21019	−0.605097	−	0.796152i	$-0.706866\pi$
$419$	−24.7721	−1.21019	−0.605097	+	0.796152i	$0.706866\pi$
$420$	−2.98423	−0.145616
$421$	−19.5161	−0.951157	−0.475578	−	0.879673i	$-0.657761\pi$
$421$	−19.5161	−0.951157	−0.475578	+	0.879673i	$0.657761\pi$
$422$	−26.0492	−1.26805
$423$	25.9638	1.26240
$424$	−51.6321	−2.50748
$425$	−21.3266	−1.03449
$426$	−2.20613	−0.106887
$427$	8.36046	0.404591
$428$	49.3763	2.38669
$429$	1.10030	0.0531228
$430$	2.61041	0.125885
$431$	18.1750	0.875458	0.437729	−	0.899107i	$-0.355783\pi$
$431$	18.1750	0.875458	0.437729	+	0.899107i	$0.355783\pi$
$432$	−4.23995	−0.203994
$433$	−22.9270	−1.10180	−0.550900	−	0.834571i	$-0.685716\pi$
$433$	−22.9270	−1.10180	−0.550900	+	0.834571i	$0.685716\pi$
$434$	−5.97642	−0.286877
$435$	−2.33953	−0.112172
$436$	−14.6699	−0.702560
$437$	20.5067	0.980968
$438$	8.31586	0.397347
$439$	−17.8020	−0.849641	−0.424821	−	0.905278i	$-0.639663\pi$
$439$	−17.8020	−0.849641	−0.424821	+	0.905278i	$0.639663\pi$
$440$	−8.03977	−0.383281
$441$	10.2477	0.487985
$442$	16.4389	0.781918
$443$	1.77491	0.0843286	0.0421643	−	0.999111i	$-0.486575\pi$
$443$	1.77491	0.0843286	0.0421643	+	0.999111i	$0.486575\pi$
$444$	4.89769	0.232434
$445$	6.02011	0.285380
$446$	−7.66261	−0.362835
$447$	−6.14046	−0.290434
$448$	−21.3656	−1.00943
$449$	−9.65284	−0.455546	−0.227773	−	0.973714i	$-0.573144\pi$
$449$	−9.65284	−0.455546	−0.227773	+	0.973714i	$0.573144\pi$
$450$	−25.8427	−1.21823
$451$	21.1537	0.996091
$452$	27.0875	1.27409
$453$	−4.33698	−0.203769
$454$	−7.24818	−0.340174
$455$	2.45348	0.115021
$456$	4.41782	0.206883
$457$	20.4646	0.957294	0.478647	−	0.878007i	$-0.341127\pi$
$457$	20.4646	0.957294	0.478647	+	0.878007i	$0.341127\pi$
$458$	−18.1206	−0.846720
$459$	−13.7094	−0.639902
$460$	28.3696	1.32274
$461$	37.3985	1.74182	0.870912	−	0.491439i	$-0.163529\pi$
$461$	37.3985	1.74182	0.870912	+	0.491439i	$0.163529\pi$
$462$	3.76668	0.175242
$463$	28.8345	1.34005	0.670026	−	0.742338i	$-0.266283\pi$
$463$	28.8345	1.34005	0.670026	+	0.742338i	$0.266283\pi$
$464$	8.76121	0.406729
$465$	−0.625640	−0.0290134
$466$	48.3129	2.23805
$467$	41.8497	1.93657	0.968286	−	0.249844i	$-0.0803793\pi$
$467$	41.8497	1.93657	0.968286	+	0.249844i	$0.0803793\pi$
$468$	12.7907	0.591251
$469$	−28.0631	−1.29583
$470$	−23.0439	−1.06294
$471$	0.181904	0.00838168
$472$	−20.3758	−0.937871
$473$	−2.11564	−0.0972770
$474$	3.65490	0.167875
$475$	10.6457	0.488458
$476$	36.1350	1.65624
$477$	−38.7241	−1.77306
$478$	3.83701	0.175501
$479$	−19.1195	−0.873593	−0.436797	−	0.899560i	$-0.643887\pi$
$479$	−19.1195	−0.873593	−0.436797	+	0.899560i	$0.643887\pi$
$480$	−1.58457	−0.0723254
$481$	−4.02662	−0.183598
$482$	−58.4177	−2.66085
$483$	−5.88303	−0.267687
$484$	−24.7493	−1.12497
$485$	0	0
$486$	−25.1818	−1.14227
$487$	−27.2384	−1.23429	−0.617144	−	0.786850i	$-0.711710\pi$
$487$	−27.2384	−1.23429	−0.617144	+	0.786850i	$0.711710\pi$
$488$	−17.1223	−0.775092
$489$	4.21724	0.190710
$490$	−9.09525	−0.410881
$491$	7.57555	0.341880	0.170940	−	0.985281i	$-0.445320\pi$
$491$	7.57555	0.341880	0.170940	+	0.985281i	$0.445320\pi$
$492$	−16.0809	−0.724982
$493$	28.3285	1.27585
$494$	−8.20588	−0.369200
$495$	−6.02983	−0.271021
$496$	2.34293	0.105201
$497$	3.98692	0.178838
$498$	−14.9108	−0.668170
$499$	29.5822	1.32428	0.662141	−	0.749380i	$-0.269648\pi$
$499$	29.5822	1.32428	0.662141	+	0.749380i	$0.269648\pi$
$500$	33.6952	1.50689
$501$	5.31489	0.237452
$502$	−20.0181	−0.893451
$503$	−16.3111	−0.727275	−0.363637	−	0.931541i	$-0.618465\pi$
$503$	−16.3111	−0.727275	−0.363637	+	0.931541i	$0.618465\pi$
$504$	19.3810	0.863300
$505$	5.97644	0.265948
$506$	−35.8080	−1.59186
$507$	−4.89083	−0.217209
$508$	39.1850	1.73855
$509$	41.7974	1.85264	0.926318	−	0.376743i	$-0.122956\pi$
$509$	41.7974	1.85264	0.926318	+	0.376743i	$0.122956\pi$
$510$	5.89122	0.260868
$511$	−15.0284	−0.664817
$512$	18.6911	0.826037
$513$	6.84341	0.302144
$514$	−69.0250	−3.04456
$515$	16.6182	0.732284
$516$	1.60829	0.0708008
$517$	18.6762	0.821379
$518$	−13.7845	−0.605656
$519$	−7.86277	−0.345137
$520$	−5.02476	−0.220350
$521$	−1.72704	−0.0756631	−0.0378316	−	0.999284i	$-0.512045\pi$
$521$	−1.72704	−0.0756631	−0.0378316	+	0.999284i	$0.512045\pi$
$522$	34.3273	1.50247
$523$	24.7516	1.08231	0.541156	−	0.840922i	$-0.317987\pi$
$523$	24.7516	1.08231	0.541156	+	0.840922i	$0.317987\pi$
$524$	−36.5220	−1.59547
$525$	−3.05407	−0.133291
$526$	−25.1673	−1.09735
$527$	7.57564	0.330000
$528$	−1.47665	−0.0642630
$529$	32.9270	1.43161
$530$	34.3693	1.49290
$531$	−15.2818	−0.663176
$532$	−18.0377	−0.782033
$533$	13.2208	0.572658
$534$	5.77634	0.249967
$535$	−14.5479	−0.628961
$536$	57.4737	2.48249
$537$	0.753483	0.0325152
$538$	−34.1096	−1.47057
$539$	7.37134	0.317506
$540$	9.46739	0.407412
$541$	42.7620	1.83848	0.919242	−	0.393693i	$-0.128803\pi$
$541$	42.7620	1.83848	0.919242	+	0.393693i	$0.128803\pi$
$542$	26.0706	1.11983
$543$	−1.38248	−0.0593281
$544$	19.1870	0.822635
$545$	4.32224	0.185145
$546$	2.35413	0.100748
$547$	−34.6477	−1.48143	−0.740713	−	0.671821i	$-0.765512\pi$
$547$	−34.6477	−1.48143	−0.740713	+	0.671821i	$0.765512\pi$
$548$	38.8598	1.66001
$549$	−12.8418	−0.548073
$550$	−18.5891	−0.792641
$551$	−14.1409	−0.602422
$552$	12.0485	0.512820
$553$	−6.60511	−0.280878
$554$	36.8858	1.56713
$555$	−1.44303	−0.0612530
$556$	−2.55500	−0.108356
$557$	11.2288	0.475777	0.237889	−	0.971292i	$-0.423545\pi$
$557$	11.2288	0.475777	0.237889	+	0.971292i	$0.423545\pi$
$558$	9.17985	0.388614
$559$	−1.32225	−0.0559251
$560$	−3.29269	−0.139142
$561$	−4.77460	−0.201584
$562$	33.7278	1.42272
$563$	−24.8759	−1.04839	−0.524197	−	0.851597i	$-0.675635\pi$
$563$	−24.8759	−1.04839	−0.524197	+	0.851597i	$0.675635\pi$
$564$	−14.1975	−0.597822
$565$	−7.98089	−0.335759
$566$	−31.7245	−1.33348
$567$	13.5231	0.567916
$568$	−8.16526	−0.342607
$569$	−19.5331	−0.818868	−0.409434	−	0.912340i	$-0.634274\pi$
$569$	−19.5331	−0.818868	−0.409434	+	0.912340i	$0.634274\pi$
$570$	−2.94075	−0.123174
$571$	37.8845	1.58542	0.792709	−	0.609601i	$-0.208670\pi$
$571$	37.8845	1.58542	0.792709	+	0.609601i	$0.208670\pi$
$572$	9.20058	0.384695
$573$	9.76517	0.407946
$574$	45.2594	1.88909
$575$	29.0335	1.21078
$576$	32.8178	1.36741
$577$	4.48078	0.186537	0.0932687	−	0.995641i	$-0.470268\pi$
$577$	4.48078	0.186537	0.0932687	+	0.995641i	$0.470268\pi$
$578$	−31.1476	−1.29557
$579$	−0.111877	−0.00464946
$580$	−19.5629	−0.812307
$581$	26.9468	1.11794
$582$	0	0
$583$	−27.8549	−1.15363
$584$	30.7783	1.27362
$585$	−3.76858	−0.155811
$586$	62.3936	2.57745
$587$	−16.6439	−0.686966	−0.343483	−	0.939159i	$-0.611607\pi$
$587$	−16.6439	−0.686966	−0.343483	+	0.939159i	$0.611607\pi$
$588$	−5.60362	−0.231090
$589$	−3.78157	−0.155817
$590$	13.5633	0.558391
$591$	5.54345	0.228027
$592$	5.40393	0.222100
$593$	15.4466	0.634315	0.317157	−	0.948373i	$-0.397272\pi$
$593$	15.4466	0.634315	0.317157	+	0.948373i	$0.397272\pi$
$594$	−11.9497	−0.490302
$595$	−10.6466	−0.436468
$596$	−51.3460	−2.10321
$597$	6.46296	0.264511
$598$	−22.3796	−0.915168
$599$	−14.4344	−0.589772	−0.294886	−	0.955532i	$-0.595282\pi$
$599$	−14.4344	−0.589772	−0.294886	+	0.955532i	$0.595282\pi$
$600$	6.25479	0.255351
$601$	−26.1849	−1.06810	−0.534052	−	0.845451i	$-0.679331\pi$
$601$	−26.1849	−1.06810	−0.534052	+	0.845451i	$0.679331\pi$
$602$	−4.52650	−0.184486
$603$	43.1053	1.75538
$604$	−36.2655	−1.47562
$605$	7.29198	0.296461
$606$	5.73444	0.232946
$607$	−37.5991	−1.52610	−0.763050	−	0.646339i	$-0.776299\pi$
$607$	−37.5991	−1.52610	−0.763050	+	0.646339i	$0.776299\pi$
$608$	−9.57767	−0.388426
$609$	4.05679	0.164389
$610$	11.3976	0.461475
$611$	11.6724	0.472216
$612$	−55.5038	−2.24361
$613$	−31.9240	−1.28940	−0.644699	−	0.764437i	$-0.723017\pi$
$613$	−31.9240	−1.28940	−0.644699	+	0.764437i	$0.723017\pi$
$614$	−70.9783	−2.86445
$615$	4.73797	0.191053
$616$	13.9411	0.561703
$617$	−8.30489	−0.334342	−0.167171	−	0.985928i	$-0.553463\pi$
$617$	−8.30489	−0.334342	−0.167171	+	0.985928i	$0.553463\pi$
$618$	15.9453	0.641413
$619$	−36.7863	−1.47857	−0.739283	−	0.673395i	$-0.764836\pi$
$619$	−36.7863	−1.47857	−0.739283	+	0.673395i	$0.764836\pi$
$620$	−5.23155	−0.210104
$621$	18.6638	0.748951
$622$	21.2147	0.850633
$623$	−10.4390	−0.418229
$624$	−0.922890	−0.0369452
$625$	9.48376	0.379350
$626$	−50.7058	−2.02661
$627$	2.38336	0.0951824
$628$	1.52106	0.0606971
$629$	17.4731	0.696697
$630$	−12.9011	−0.513992
$631$	−18.6322	−0.741735	−0.370868	−	0.928686i	$-0.620940\pi$
$631$	−18.6322	−0.741735	−0.370868	+	0.928686i	$0.620940\pi$
$632$	13.5274	0.538090
$633$	4.72858	0.187944
$634$	13.5695	0.538915
$635$	−11.5452	−0.458158
$636$	21.1751	0.839645
$637$	4.60700	0.182536
$638$	24.6922	0.977575
$639$	−6.12395	−0.242260
$640$	−21.7419	−0.859424
$641$	31.5101	1.24458	0.622288	−	0.782789i	$-0.286203\pi$
$641$	31.5101	1.24458	0.622288	+	0.782789i	$0.286203\pi$
$642$	−13.9588	−0.550911
$643$	−5.25512	−0.207242	−0.103621	−	0.994617i	$-0.533043\pi$
$643$	−5.25512	−0.207242	−0.103621	+	0.994617i	$0.533043\pi$
$644$	−49.1934	−1.93849
$645$	−0.473855	−0.0186580
$646$	35.6085	1.40100
$647$	31.0836	1.22202	0.611012	−	0.791621i	$-0.290763\pi$
$647$	31.0836	1.22202	0.611012	+	0.791621i	$0.290763\pi$
$648$	−27.6955	−1.08798
$649$	−10.9925	−0.431493
$650$	−11.6180	−0.455694
$651$	1.08487	0.0425195
$652$	35.2642	1.38105
$653$	−1.99501	−0.0780709	−0.0390354	−	0.999238i	$-0.512429\pi$
$653$	−1.99501	−0.0780709	−0.0390354	+	0.999238i	$0.512429\pi$
$654$	4.14723	0.162169
$655$	10.7606	0.420452
$656$	−17.7430	−0.692749
$657$	23.0838	0.900584
$658$	39.9586	1.55775
$659$	20.7661	0.808931	0.404466	−	0.914553i	$-0.367458\pi$
$659$	20.7661	0.808931	0.404466	+	0.914553i	$0.367458\pi$
$660$	3.29722	0.128344
$661$	−1.31914	−0.0513084	−0.0256542	−	0.999671i	$-0.508167\pi$
$661$	−1.31914	−0.0513084	−0.0256542	+	0.999671i	$0.508167\pi$
$662$	66.1434	2.57074
$663$	−2.98407	−0.115892
$664$	−55.1874	−2.14169
$665$	5.31451	0.206088
$666$	21.1732	0.820443
$667$	−38.5658	−1.49327
$668$	44.4426	1.71954
$669$	1.39096	0.0537775
$670$	−38.2577	−1.47802
$671$	−9.23730	−0.356602
$672$	2.74768	0.105994
$673$	2.95386	0.113863	0.0569314	−	0.998378i	$-0.481868\pi$
$673$	2.95386	0.113863	0.0569314	+	0.998378i	$0.481868\pi$
$674$	2.88466	0.111113
$675$	9.68896	0.372928
$676$	−40.8967	−1.57295
$677$	−36.6429	−1.40830	−0.704150	−	0.710051i	$-0.748672\pi$
$677$	−36.6429	−1.40830	−0.704150	+	0.710051i	$0.748672\pi$
$678$	−7.65773	−0.294093
$679$	0	0
$680$	21.8044	0.836160
$681$	1.31573	0.0504188
$682$	6.60323	0.252851
$683$	15.2912	0.585103	0.292552	−	0.956250i	$-0.405496\pi$
$683$	15.2912	0.585103	0.292552	+	0.956250i	$0.405496\pi$
$684$	27.7061	1.05937
$685$	−11.4494	−0.437459
$686$	46.1069	1.76037
$687$	3.28935	0.125496
$688$	1.77452	0.0676530
$689$	−17.4090	−0.663231
$690$	−8.02019	−0.305323
$691$	−10.3168	−0.392471	−0.196236	−	0.980557i	$-0.562872\pi$
$691$	−10.3168	−0.392471	−0.196236	+	0.980557i	$0.562872\pi$
$692$	−65.7478	−2.49936
$693$	10.4558	0.397185
$694$	4.87418	0.185022
$695$	0.752790	0.0285549
$696$	−8.30836	−0.314928
$697$	−57.3703	−2.17306
$698$	28.2623	1.06974
$699$	−8.77001	−0.331712
$700$	−25.5379	−0.965242
$701$	38.1415	1.44058	0.720292	−	0.693671i	$-0.244008\pi$
$701$	38.1415	1.44058	0.720292	+	0.693671i	$0.244008\pi$
$702$	−7.46842	−0.281877
$703$	−8.72212	−0.328961
$704$	23.6064	0.889701
$705$	4.18306	0.157543
$706$	−40.0916	−1.50887
$707$	−10.3633	−0.389751
$708$	8.35639	0.314052
$709$	−1.54532	−0.0580355	−0.0290178	−	0.999579i	$-0.509238\pi$
$709$	−1.54532	−0.0580355	−0.0290178	+	0.999579i	$0.509238\pi$
$710$	5.43526	0.203982
$711$	10.1455	0.380487
$712$	21.3792	0.801219
$713$	−10.3133	−0.386237
$714$	−10.2155	−0.382305
$715$	−2.71080	−0.101378
$716$	6.30056	0.235463
$717$	−0.696515	−0.0260118
$718$	43.1684	1.61103
$719$	37.7318	1.40716	0.703579	−	0.710617i	$-0.251584\pi$
$719$	37.7318	1.40716	0.703579	+	0.710617i	$0.251584\pi$
$720$	5.05762	0.188486
$721$	−28.8162	−1.07317
$722$	27.1401	1.01005
$723$	10.6043	0.394377
$724$	−11.5602	−0.429632
$725$	−20.0208	−0.743553
$726$	6.99671	0.259672
$727$	14.6956	0.545031	0.272516	−	0.962151i	$-0.412144\pi$
$727$	14.6956	0.545031	0.272516	+	0.962151i	$0.412144\pi$
$728$	8.71304	0.322927
$729$	−17.5588	−0.650326
$730$	−20.4878	−0.758288
$731$	5.73774	0.212218
$732$	7.02211	0.259545
$733$	−20.1071	−0.742671	−0.371336	−	0.928499i	$-0.621100\pi$
$733$	−20.1071	−0.742671	−0.371336	+	0.928499i	$0.621100\pi$
$734$	−85.5014	−3.15591
$735$	1.65102	0.0608987
$736$	−26.1208	−0.962824
$737$	31.0064	1.14213
$738$	−69.5190	−2.55903
$739$	16.1345	0.593516	0.296758	−	0.954953i	$-0.404095\pi$
$739$	16.1345	0.593516	0.296758	+	0.954953i	$0.404095\pi$
$740$	−12.0665	−0.443572
$741$	1.48957	0.0547209
$742$	−59.5969	−2.18787
$743$	−8.11443	−0.297689	−0.148845	−	0.988861i	$-0.547555\pi$
$743$	−8.11443	−0.297689	−0.148845	+	0.988861i	$0.547555\pi$
$744$	−2.22183	−0.0814563
$745$	15.1283	0.554257
$746$	24.9131	0.912132
$747$	−41.3906	−1.51440
$748$	−39.9248	−1.45980
$749$	25.2264	0.921751
$750$	−9.52575	−0.347831
$751$	20.1552	0.735474	0.367737	−	0.929930i	$-0.380133\pi$
$751$	20.1552	0.735474	0.367737	+	0.929930i	$0.380133\pi$
$752$	−15.6650	−0.571243
$753$	3.63379	0.132423
$754$	15.4324	0.562013
$755$	10.6850	0.388868
$756$	−16.4166	−0.597067
$757$	−2.89755	−0.105313	−0.0526567	−	0.998613i	$-0.516769\pi$
$757$	−2.89755	−0.105313	−0.0526567	+	0.998613i	$0.516769\pi$
$758$	−53.7221	−1.95128
$759$	6.50005	0.235937
$760$	−10.8842	−0.394812
$761$	−44.6936	−1.62014	−0.810071	−	0.586332i	$-0.800571\pi$
$761$	−44.6936	−1.62014	−0.810071	+	0.586332i	$0.800571\pi$
$762$	−11.0777	−0.401304
$763$	−7.49485	−0.271332
$764$	81.6555	2.95419
$765$	16.3533	0.591255
$766$	25.8739	0.934862
$767$	−6.87018	−0.248068
$768$	−10.8591	−0.391846
$769$	−18.0949	−0.652520	−0.326260	−	0.945280i	$-0.605788\pi$
$769$	−18.0949	−0.652520	−0.326260	+	0.945280i	$0.605788\pi$
$770$	−9.27999	−0.334428
$771$	12.5298	0.451249
$772$	−0.935508	−0.0336697
$773$	6.53353	0.234995	0.117497	−	0.993073i	$-0.462513\pi$
$773$	6.53353	0.234995	0.117497	+	0.993073i	$0.462513\pi$
$774$	6.95276	0.249912
$775$	−5.35398	−0.192321
$776$	0	0
$777$	2.50223	0.0897672
$778$	−28.5951	−1.02518
$779$	28.6378	1.02606
$780$	2.06073	0.0737858
$781$	−4.40506	−0.157626
$782$	97.1135	3.47277
$783$	−12.8700	−0.459938
$784$	−6.18283	−0.220815
$785$	−0.448157	−0.0159954
$786$	10.3249	0.368277
$787$	43.5362	1.55190	0.775948	−	0.630797i	$-0.217272\pi$
$787$	43.5362	1.55190	0.775948	+	0.630797i	$0.217272\pi$
$788$	46.3539	1.65129
$789$	4.56851	0.162643
$790$	−9.00458	−0.320369
$791$	13.8390	0.492059
$792$	−21.4137	−0.760903
$793$	−5.77321	−0.205013
$794$	20.1055	0.713519
$795$	−6.23889	−0.221271
$796$	54.0427	1.91549
$797$	4.76111	0.168647	0.0843236	−	0.996438i	$-0.473127\pi$
$797$	4.76111	0.168647	0.0843236	+	0.996438i	$0.473127\pi$
$798$	5.09932	0.180514
$799$	−50.6511	−1.79191
$800$	−13.5601	−0.479423
$801$	16.0344	0.566548
$802$	14.6381	0.516889
$803$	16.6046	0.585962
$804$	−23.5708	−0.831276
$805$	14.4941	0.510848
$806$	4.12694	0.145365
$807$	6.19176	0.217960
$808$	21.2241	0.746662
$809$	43.6289	1.53391	0.766955	−	0.641700i	$-0.221771\pi$
$809$	43.6289	1.53391	0.766955	+	0.641700i	$0.221771\pi$
$810$	18.4357	0.647763
$811$	46.2477	1.62398	0.811988	−	0.583674i	$-0.198386\pi$
$811$	46.2477	1.62398	0.811988	+	0.583674i	$0.198386\pi$
$812$	33.9225	1.19045
$813$	−4.73246	−0.165975
$814$	15.2302	0.533819
$815$	−10.3900	−0.363946
$816$	4.00477	0.140195
$817$	−2.86414	−0.100203
$818$	−27.8176	−0.972621
$819$	6.53478	0.228344
$820$	39.6185	1.38354
$821$	−28.3808	−0.990497	−0.495249	−	0.868751i	$-0.664923\pi$
$821$	−28.3808	−0.990497	−0.495249	+	0.868751i	$0.664923\pi$
$822$	−10.9858	−0.383173
$823$	15.5024	0.540380	0.270190	−	0.962807i	$-0.412914\pi$
$823$	15.5024	0.540380	0.270190	+	0.962807i	$0.412914\pi$
$824$	59.0161	2.05592
$825$	3.37439	0.117481
$826$	−23.5190	−0.818329
$827$	−17.1235	−0.595442	−0.297721	−	0.954653i	$-0.596226\pi$
$827$	−17.1235	−0.595442	−0.297721	+	0.954653i	$0.596226\pi$
$828$	75.5617	2.62595
$829$	18.7826	0.652346	0.326173	−	0.945310i	$-0.394241\pi$
$829$	18.7826	0.652346	0.326173	+	0.945310i	$0.394241\pi$
$830$	36.7359	1.27512
$831$	−6.69571	−0.232271
$832$	14.7538	0.511494
$833$	−19.9916	−0.692667
$834$	0.722308	0.0250115
$835$	−13.0943	−0.453147
$836$	19.9295	0.689276
$837$	−3.44172	−0.118963
$838$	58.5597	2.02291
$839$	3.01979	0.104255	0.0521274	−	0.998640i	$-0.483400\pi$
$839$	3.01979	0.104255	0.0521274	+	0.998640i	$0.483400\pi$
$840$	3.12250	0.107737
$841$	−2.40601	−0.0829659
$842$	46.1350	1.58992
$843$	−6.12245	−0.210868
$844$	39.5400	1.36102
$845$	12.0495	0.414517
$846$	−61.3770	−2.11018
$847$	−12.6444	−0.434468
$848$	23.3638	0.802315
$849$	5.75880	0.197642
$850$	50.4148	1.72921
$851$	−23.7875	−0.815424
$852$	3.34869	0.114724
$853$	19.4106	0.664607	0.332304	−	0.943172i	$-0.392174\pi$
$853$	19.4106	0.664607	0.332304	+	0.943172i	$0.392174\pi$
$854$	−19.7636	−0.676298
$855$	−8.16316	−0.279174
$856$	−51.6640	−1.76584
$857$	30.6788	1.04797	0.523984	−	0.851728i	$-0.324445\pi$
$857$	30.6788	1.04797	0.523984	+	0.851728i	$0.324445\pi$
$858$	−2.60104	−0.0887979
$859$	−21.5358	−0.734791	−0.367396	−	0.930065i	$-0.619750\pi$
$859$	−21.5358	−0.734791	−0.367396	+	0.930065i	$0.619750\pi$
$860$	−3.96234	−0.135115
$861$	−8.21573	−0.279991
$862$	−42.9646	−1.46338
$863$	−52.9673	−1.80303	−0.901514	−	0.432749i	$-0.857544\pi$
$863$	−52.9673	−1.80303	−0.901514	+	0.432749i	$0.857544\pi$
$864$	−8.71692	−0.296556
$865$	19.3715	0.658652
$866$	54.1981	1.84173
$867$	5.65408	0.192023
$868$	9.07160	0.307910
$869$	7.29786	0.247563
$870$	5.53051	0.187502
$871$	19.3786	0.656620
$872$	15.3496	0.519802
$873$	0	0
$874$	−48.4766	−1.63975
$875$	17.2149	0.581970
$876$	−12.6226	−0.426479
$877$	−43.2098	−1.45909	−0.729545	−	0.683932i	$-0.760268\pi$
$877$	−43.2098	−1.45909	−0.729545	+	0.683932i	$0.760268\pi$
$878$	42.0828	1.42023
$879$	−11.3260	−0.382017
$880$	3.63803	0.122638
$881$	−19.0913	−0.643202	−0.321601	−	0.946875i	$-0.604221\pi$
$881$	−19.0913	−0.643202	−0.321601	+	0.946875i	$0.604221\pi$
$882$	−24.2250	−0.815697
$883$	27.7369	0.933423	0.466711	−	0.884410i	$-0.345439\pi$
$883$	27.7369	0.933423	0.466711	+	0.884410i	$0.345439\pi$
$884$	−24.9526	−0.839246
$885$	−2.46208	−0.0827618
$886$	−4.19579	−0.140960
$887$	21.9226	0.736087	0.368044	−	0.929809i	$-0.380028\pi$
$887$	21.9226	0.736087	0.368044	+	0.929809i	$0.380028\pi$
$888$	−5.12461	−0.171971
$889$	20.0196	0.671437
$890$	−14.2312	−0.477031
$891$	−14.9414	−0.500555
$892$	11.6311	0.389437
$893$	25.2838	0.846089
$894$	14.5157	0.485478
$895$	−1.85636	−0.0620513
$896$	37.7009	1.25950
$897$	4.06246	0.135641
$898$	22.8188	0.761472
$899$	7.11180	0.237192
$900$	39.2265	1.30755
$901$	75.5444	2.51675
$902$	−50.0062	−1.66503
$903$	0.821674	0.0273436
$904$	−28.3425	−0.942658
$905$	3.40603	0.113220
$906$	10.2524	0.340612
$907$	−5.65826	−0.187879	−0.0939397	−	0.995578i	$-0.529946\pi$
$907$	−5.65826	−0.187879	−0.0939397	+	0.995578i	$0.529946\pi$
$908$	11.0020	0.365115
$909$	15.9181	0.527970
$910$	−5.79989	−0.192264
$911$	12.4649	0.412980	0.206490	−	0.978449i	$-0.433796\pi$
$911$	12.4649	0.412980	0.206490	+	0.978449i	$0.433796\pi$
$912$	−1.99908	−0.0661963
$913$	−29.7730	−0.985342
$914$	−48.3772	−1.60018
$915$	−2.06895	−0.0683974
$916$	27.5052	0.908799
$917$	−18.6591	−0.616178
$918$	32.4083	1.06963
$919$	29.6872	0.979291	0.489645	−	0.871922i	$-0.337126\pi$
$919$	29.6872	0.979291	0.489645	+	0.871922i	$0.337126\pi$
$920$	−29.6840	−0.978654
$921$	12.8844	0.424554
$922$	−88.4080	−2.91156
$923$	−2.75311	−0.0906199
$924$	−5.71744	−0.188090
$925$	−12.3489	−0.406028
$926$	−68.1631	−2.23998
$927$	44.2621	1.45376
$928$	18.0122	0.591280
$929$	15.1270	0.496301	0.248151	−	0.968721i	$-0.420177\pi$
$929$	15.1270	0.496301	0.248151	+	0.968721i	$0.420177\pi$
$930$	1.47898	0.0484976
$931$	9.97929	0.327058
$932$	−73.3341	−2.40214
$933$	−3.85101	−0.126076
$934$	−98.9303	−3.23710
$935$	11.7632	0.384698
$936$	−13.3833	−0.437448
$937$	17.1692	0.560894	0.280447	−	0.959870i	$-0.409517\pi$
$937$	17.1692	0.560894	0.280447	+	0.959870i	$0.409517\pi$
$938$	66.3396	2.16607
$939$	9.20438	0.300374
$940$	34.9784	1.14087
$941$	−1.71456	−0.0558932	−0.0279466	−	0.999609i	$-0.508897\pi$
$941$	−1.71456	−0.0558932	−0.0279466	+	0.999609i	$0.508897\pi$
$942$	−0.430010	−0.0140105
$943$	78.1028	2.54338
$944$	9.22013	0.300090
$945$	4.83690	0.157344
$946$	5.00124	0.162604
$947$	−4.23908	−0.137752	−0.0688759	−	0.997625i	$-0.521941\pi$
$947$	−4.23908	−0.137752	−0.0688759	+	0.997625i	$0.521941\pi$
$948$	−5.54776	−0.180183
$949$	10.3777	0.336873
$950$	−25.1658	−0.816486
$951$	−2.46321	−0.0798751
$952$	−37.8092	−1.22540
$953$	27.3612	0.886316	0.443158	−	0.896443i	$-0.353858\pi$
$953$	27.3612	0.886316	0.443158	+	0.896443i	$0.353858\pi$
$954$	91.5416	2.96377
$955$	−24.0585	−0.778514
$956$	−5.82420	−0.188368
$957$	−4.48226	−0.144891
$958$	45.1975	1.46026
$959$	19.8535	0.641102
$960$	5.28732	0.170647
$961$	−29.0982	−0.938650
$962$	9.51871	0.306896
$963$	−38.7480	−1.24864
$964$	88.6721	2.85594
$965$	0.275632	0.00887292
$966$	13.9072	0.447456
$967$	3.50058	0.112571	0.0562855	−	0.998415i	$-0.482074\pi$
$967$	3.50058	0.112571	0.0562855	+	0.998415i	$0.482074\pi$
$968$	25.8960	0.832329
$969$	−6.46384	−0.207648
$970$	0	0
$971$	36.9161	1.18469	0.592347	−	0.805683i	$-0.298201\pi$
$971$	36.9161	1.18469	0.592347	+	0.805683i	$0.298201\pi$
$972$	38.2234	1.22602
$973$	−1.30535	−0.0418477
$974$	64.3899	2.06319
$975$	2.10895	0.0675405
$976$	7.74793	0.248005
$977$	29.2312	0.935190	0.467595	−	0.883943i	$-0.345121\pi$
$977$	29.2312	0.935190	0.467595	+	0.883943i	$0.345121\pi$
$978$	−9.96931	−0.318783
$979$	11.5338	0.368623
$980$	13.8057	0.441006
$981$	11.5122	0.367556
$982$	−17.9082	−0.571473
$983$	49.1228	1.56678	0.783388	−	0.621533i	$-0.213490\pi$
$983$	49.1228	1.56678	0.783388	+	0.621533i	$0.213490\pi$
$984$	16.8259	0.536391
$985$	−13.6574	−0.435162
$986$	−66.9669	−2.13266
$987$	−7.25350	−0.230882
$988$	12.4557	0.396269
$989$	−7.81124	−0.248383
$990$	14.2542	0.453027
$991$	−25.5908	−0.812917	−0.406459	−	0.913669i	$-0.633236\pi$
$991$	−25.5908	−0.812917	−0.406459	+	0.913669i	$0.633236\pi$
$992$	4.81685	0.152935
$993$	−12.0067	−0.381021
$994$	−9.42484	−0.298938
$995$	−15.9228	−0.504787
$996$	22.6331	0.717158
$997$	33.0937	1.04809	0.524043	−	0.851692i	$-0.324423\pi$
$997$	33.0937	1.04809	0.524043	+	0.851692i	$0.324423\pi$
$998$	−69.9306	−2.21362
$999$	−7.93826	−0.251156

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9409.2.a.n.1.3		56
97.51	odd	32		97.2.h.a.79.7	yes	56
97.78	odd	32		97.2.h.a.70.7	✓	56
97.96	even	2	inner	9409.2.a.n.1.4		56
291.245	even	32		873.2.bh.b.370.1		56
291.272	even	32		873.2.bh.b.361.1		56

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
97.2.h.a.70.7	✓	56	97.78	odd	32
97.2.h.a.79.7	yes	56	97.51	odd	32
873.2.bh.b.361.1		56	291.272	even	32
873.2.bh.b.370.1		56	291.245	even	32
9409.2.a.n.1.3		56	1.1	even	1	trivial
9409.2.a.n.1.4		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)

\(f(q)\)	\(=\)	\(q-2.36394 q^{2} +0.429115 q^{3} +3.58823 q^{4} -1.05721 q^{5} -1.01440 q^{6} +1.83323 q^{7} -3.75448 q^{8} -2.81586 q^{9} +2.49919 q^{10} -2.02550 q^{11} +1.53976 q^{12} -1.26591 q^{13} -4.33365 q^{14} -0.453666 q^{15} +1.69892 q^{16} +5.49328 q^{17} +6.65653 q^{18} -2.74211 q^{19} -3.79352 q^{20} +0.786666 q^{21} +4.78816 q^{22} -7.47844 q^{23} -1.61110 q^{24} -3.88230 q^{25} +2.99254 q^{26} -2.49568 q^{27} +6.57804 q^{28} +5.15694 q^{29} +1.07244 q^{30} +1.37907 q^{31} +3.49281 q^{32} -0.869172 q^{33} -12.9858 q^{34} -1.93811 q^{35} -10.1039 q^{36} +3.18081 q^{37} +6.48219 q^{38} -0.543222 q^{39} +3.96928 q^{40} -10.4437 q^{41} -1.85963 q^{42} +1.04450 q^{43} -7.26794 q^{44} +2.97696 q^{45} +17.6786 q^{46} -9.22056 q^{47} +0.729032 q^{48} -3.63928 q^{49} +9.17754 q^{50} +2.35725 q^{51} -4.54238 q^{52} +13.7521 q^{53} +5.89963 q^{54} +2.14138 q^{55} -6.88281 q^{56} -1.17668 q^{57} -12.1907 q^{58} +5.42706 q^{59} -1.62786 q^{60} +4.56051 q^{61} -3.26005 q^{62} -5.16211 q^{63} -11.6546 q^{64} +1.33834 q^{65} +2.05467 q^{66} -15.3080 q^{67} +19.7111 q^{68} -3.20911 q^{69} +4.58159 q^{70} +2.17481 q^{71} +10.5721 q^{72} -8.19777 q^{73} -7.51925 q^{74} -1.66595 q^{75} -9.83931 q^{76} -3.71320 q^{77} +1.28415 q^{78} -3.60300 q^{79} -1.79612 q^{80} +7.37665 q^{81} +24.6884 q^{82} +14.6991 q^{83} +2.82274 q^{84} -5.80757 q^{85} -2.46914 q^{86} +2.21292 q^{87} +7.60468 q^{88} -5.69432 q^{89} -7.03737 q^{90} -2.32071 q^{91} -26.8343 q^{92} +0.591782 q^{93} +21.7969 q^{94} +2.89899 q^{95} +1.49882 q^{96} +8.60304 q^{98} +5.70352 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 56 q + 8 q^{2} + 16 q^{3} + 40 q^{4} + 16 q^{6} + 24 q^{8} + 40 q^{9} + 32 q^{11} + 48 q^{12} + 8 q^{16} + 24 q^{18} + 16 q^{22} + 8 q^{25} + 64 q^{27} + 48 q^{31} + 56 q^{32} + 80 q^{33} + 48 q^{35} + 8 q^{36}+ \cdots - 16 q^{99}+O(q^{100}) \) 56 * q + 8 * q^2 + 16 * q^3 + 40 * q^4 + 16 * q^6 + 24 * q^8 + 40 * q^9 + 32 * q^11 + 48 * q^12 + 8 * q^16 + 24 * q^18 + 16 * q^22 + 8 * q^25 + 64 * q^27 + 48 * q^31 + 56 * q^32 + 80 * q^33 + 48 * q^35 + 8 * q^36 + 32 * q^43 + 80 * q^44 + 16 * q^47 + 112 * q^48 + 8 * q^49 + 24 * q^50 + 48 * q^53 + 96 * q^54 + 32 * q^61 + 80 * q^62 - 56 * q^64 + 128 * q^65 + 192 * q^66 + 80 * q^70 - 88 * q^72 + 32 * q^73 + 80 * q^75 + 48 * q^79 + 72 * q^81 + 144 * q^86 + 176 * q^88 + 96 * q^89 + 64 * q^91 + 64 * q^94 + 176 * q^95 + 32 * q^96 - 136 * q^98 - 16 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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