LMFDB - Embedded newform 4225.2.a.cb.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4225,2,Mod(1,4225)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("4225.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4225, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 4225 = 5^{2} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4225.a (trivial)

Newform invariants

comment:select newform

sage:traces = [18,0,0,16,0,16,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$33.7367948540$
Analytic rank:	$0$
Dimension:	$18$
Coefficient field:	$\mathbb{Q}[x]/(x^{18} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{18} - 26x^{16} + 281x^{14} - 1632x^{12} + 5482x^{10} - 10620x^{8} + 11052x^{6} - 5165x^{4} + 760x^{2} - 29 $ x^18 - 26x^16 + 281x^14 - 1632x^12 + 5482x^10 - 10620x^8 + 11052x^6 - 5165x^4 + 760x^2 - 29
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 845)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-2.14790$ of defining polynomial
Character	$\chi$	$=$	4225.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-2.14790 q^{2} -2.37435 q^{3} +2.61347 q^{4} +5.09986 q^{6} -3.54879 q^{7} -1.31766 q^{8} +2.63754 q^{9} +2.08852 q^{11} -6.20528 q^{12} +7.62244 q^{14} -2.39672 q^{16} -6.00072 q^{17} -5.66516 q^{18} -5.04725 q^{19} +8.42607 q^{21} -4.48593 q^{22} -2.12673 q^{23} +3.12860 q^{24} +0.860615 q^{27} -9.27464 q^{28} -10.1431 q^{29} +3.10862 q^{31} +7.78325 q^{32} -4.95888 q^{33} +12.8889 q^{34} +6.89311 q^{36} +0.0998988 q^{37} +10.8410 q^{38} +3.79617 q^{41} -18.0983 q^{42} +2.11661 q^{43} +5.45829 q^{44} +4.56800 q^{46} -5.00374 q^{47} +5.69066 q^{48} +5.59390 q^{49} +14.2478 q^{51} -5.29557 q^{53} -1.84851 q^{54} +4.67611 q^{56} +11.9839 q^{57} +21.7863 q^{58} -3.00358 q^{59} -4.76673 q^{61} -6.67699 q^{62} -9.36006 q^{63} -11.9242 q^{64} +10.6512 q^{66} -6.77655 q^{67} -15.6827 q^{68} +5.04961 q^{69} -4.89593 q^{71} -3.47539 q^{72} -4.02118 q^{73} -0.214573 q^{74} -13.1908 q^{76} -7.41173 q^{77} -4.71456 q^{79} -9.95601 q^{81} -8.15379 q^{82} +11.2703 q^{83} +22.0212 q^{84} -4.54625 q^{86} +24.0832 q^{87} -2.75197 q^{88} +1.90467 q^{89} -5.55814 q^{92} -7.38094 q^{93} +10.7475 q^{94} -18.4802 q^{96} +5.36940 q^{97} -12.0151 q^{98} +5.50856 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q + 16 q^{4} + 16 q^{6} + 18 q^{9} + 22 q^{11} + 4 q^{14} - 12 q^{16} + 28 q^{19} + 26 q^{21} + 34 q^{24} - 20 q^{29} + 32 q^{31} + 18 q^{34} + 32 q^{36} + 52 q^{41} + 50 q^{44} + 30 q^{46} + 44 q^{49}+ \cdots + 76 q^{99}+O(q^{100}) $ 18 * q + 16 * q^4 + 16 * q^6 + 18 * q^9 + 22 * q^11 + 4 * q^14 - 12 * q^16 + 28 * q^19 + 26 * q^21 + 34 * q^24 - 20 * q^29 + 32 * q^31 + 18 * q^34 + 32 * q^36 + 52 * q^41 + 50 * q^44 + 30 * q^46 + 44 * q^49 - 40 * q^51 + 90 * q^54 - 20 * q^56 + 76 * q^59 + 8 * q^61 - 68 * q^64 + 8 * q^66 + 30 * q^69 + 72 * q^71 + 30 * q^74 + 4 * q^76 + 16 * q^79 - 30 * q^81 + 78 * q^84 - 30 * q^86 + 94 * q^89 - 128 * q^94 - 18 * q^96 + 76 * 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 $

$2$	−2.14790	−1.51879	−0.759397	−	0.650628i	$-0.774506\pi$
$2$	−2.14790	−1.51879	−0.759397	+	0.650628i	$0.774506\pi$
$3$	−2.37435	−1.37083	−0.685416	−	0.728152i	$-0.740379\pi$
$3$	−2.37435	−1.37083	−0.685416	+	0.728152i	$0.740379\pi$
$4$	2.61347	1.30673
$5$	0	0
$5$	0	0
$6$	5.09986	2.08201
$7$	−3.54879	−1.34132	−0.670658	−	0.741767i	$-0.733988\pi$
$7$	−3.54879	−1.34132	−0.670658	+	0.741767i	$0.733988\pi$
$8$	−1.31766	−0.465865
$9$	2.63754	0.879179
$10$	0	0
$11$	2.08852	0.629713	0.314857	−	0.949139i	$-0.398044\pi$
$11$	2.08852	0.629713	0.314857	+	0.949139i	$0.398044\pi$
$12$	−6.20528	−1.79131
$13$	0	0
$13$	0	0
$14$	7.62244	2.03718
$15$	0	0
$16$	−2.39672	−0.599181
$17$	−6.00072	−1.45539	−0.727695	−	0.685901i	$-0.759408\pi$
$17$	−6.00072	−1.45539	−0.727695	+	0.685901i	$0.759408\pi$
$18$	−5.66516	−1.33529
$19$	−5.04725	−1.15792	−0.578959	−	0.815357i	$-0.696541\pi$
$19$	−5.04725	−1.15792	−0.578959	+	0.815357i	$0.696541\pi$
$20$	0	0
$21$	8.42607	1.83872
$22$	−4.48593	−0.956405
$23$	−2.12673	−0.443454	−0.221727	−	0.975109i	$-0.571169\pi$
$23$	−2.12673	−0.443454	−0.221727	+	0.975109i	$0.571169\pi$
$24$	3.12860	0.638622
$25$	0	0
$26$	0	0
$27$	0.860615	0.165625
$28$	−9.27464	−1.75274
$29$	−10.1431	−1.88352	−0.941762	−	0.336281i	$-0.890831\pi$
$29$	−10.1431	−1.88352	−0.941762	+	0.336281i	$0.890831\pi$
$30$	0	0
$31$	3.10862	0.558324	0.279162	−	0.960244i	$-0.409943\pi$
$31$	3.10862	0.558324	0.279162	+	0.960244i	$0.409943\pi$
$32$	7.78325	1.37590
$33$	−4.95888	−0.863231
$34$	12.8889	2.21044
$35$	0	0
$36$	6.89311	1.14885
$37$	0.0998988	0.0164233	0.00821163	−	0.999966i	$-0.497386\pi$
$37$	0.0998988	0.0164233	0.00821163	+	0.999966i	$0.497386\pi$
$38$	10.8410	1.75864
$39$	0	0
$40$	0	0
$41$	3.79617	0.592862	0.296431	−	0.955054i	$-0.404203\pi$
$41$	3.79617	0.592862	0.296431	+	0.955054i	$0.404203\pi$
$42$	−18.0983	−2.79263
$43$	2.11661	0.322779	0.161390	−	0.986891i	$-0.448402\pi$
$43$	2.11661	0.322779	0.161390	+	0.986891i	$0.448402\pi$
$44$	5.45829	0.822868
$45$	0	0
$46$	4.56800	0.673515
$47$	−5.00374	−0.729870	−0.364935	−	0.931033i	$-0.618909\pi$
$47$	−5.00374	−0.729870	−0.364935	+	0.931033i	$0.618909\pi$
$48$	5.69066	0.821376
$49$	5.59390	0.799129
$50$	0	0
$51$	14.2478	1.99509
$52$	0	0
$53$	−5.29557	−0.727402	−0.363701	−	0.931516i	$-0.618487\pi$
$53$	−5.29557	−0.727402	−0.363701	+	0.931516i	$0.618487\pi$
$54$	−1.84851	−0.251551
$55$	0	0
$56$	4.67611	0.624872
$57$	11.9839	1.58731
$58$	21.7863	2.86068
$59$	−3.00358	−0.391033	−0.195517	−	0.980700i	$-0.562638\pi$
$59$	−3.00358	−0.391033	−0.195517	+	0.980700i	$0.562638\pi$
$60$	0	0
$61$	−4.76673	−0.610317	−0.305158	−	0.952302i	$-0.598709\pi$
$61$	−4.76673	−0.610317	−0.305158	+	0.952302i	$0.598709\pi$
$62$	−6.67699	−0.847979
$63$	−9.36006	−1.17926
$64$	−11.9242	−1.49052
$65$	0	0
$66$	10.6512	1.31107
$67$	−6.77655	−0.827887	−0.413944	−	0.910303i	$-0.635849\pi$
$67$	−6.77655	−0.827887	−0.413944	+	0.910303i	$0.635849\pi$
$68$	−15.6827	−1.90181
$69$	5.04961	0.607901
$70$	0	0
$71$	−4.89593	−0.581040	−0.290520	−	0.956869i	$-0.593828\pi$
$71$	−4.89593	−0.581040	−0.290520	+	0.956869i	$0.593828\pi$
$72$	−3.47539	−0.409578
$73$	−4.02118	−0.470643	−0.235321	−	0.971918i	$-0.575614\pi$
$73$	−4.02118	−0.470643	−0.235321	+	0.971918i	$0.575614\pi$
$74$	−0.214573	−0.0249436
$75$	0	0
$76$	−13.1908	−1.51309
$77$	−7.41173	−0.844645
$78$	0	0
$79$	−4.71456	−0.530429	−0.265215	−	0.964189i	$-0.585443\pi$
$79$	−4.71456	−0.530429	−0.265215	+	0.964189i	$0.585443\pi$
$80$	0	0
$81$	−9.95601	−1.10622
$82$	−8.15379	−0.900436
$83$	11.2703	1.23707	0.618536	−	0.785757i	$-0.287726\pi$
$83$	11.2703	1.23707	0.618536	+	0.785757i	$0.287726\pi$
$84$	22.0212	2.40271
$85$	0	0
$86$	−4.54625	−0.490235
$87$	24.0832	2.58199
$88$	−2.75197	−0.293361
$89$	1.90467	0.201895	0.100948	−	0.994892i	$-0.467813\pi$
$89$	1.90467	0.201895	0.100948	+	0.994892i	$0.467813\pi$
$90$	0	0
$91$	0	0
$92$	−5.55814	−0.579477
$93$	−7.38094	−0.765368
$94$	10.7475	1.10852
$95$	0	0
$96$	−18.4802	−1.88612
$97$	5.36940	0.545180	0.272590	−	0.962130i	$-0.412120\pi$
$97$	5.36940	0.545180	0.272590	+	0.962130i	$0.412120\pi$
$98$	−12.0151	−1.21371
$99$	5.50856	0.553631
$100$	0	0
$101$	−11.2109	−1.11553	−0.557765	−	0.829999i	$-0.688341\pi$
$101$	−11.2109	−1.11553	−0.557765	+	0.829999i	$0.688341\pi$
$102$	−30.6029	−3.03013
$103$	−11.8123	−1.16390	−0.581951	−	0.813224i	$-0.697710\pi$
$103$	−11.8123	−1.16390	−0.581951	+	0.813224i	$0.697710\pi$
$104$	0	0
$105$	0	0
$106$	11.3743	1.10477
$107$	5.51439	0.533097	0.266548	−	0.963822i	$-0.414117\pi$
$107$	5.51439	0.533097	0.266548	+	0.963822i	$0.414117\pi$
$108$	2.24919	0.216428
$109$	−12.0982	−1.15880	−0.579401	−	0.815043i	$-0.696713\pi$
$109$	−12.0982	−1.15880	−0.579401	+	0.815043i	$0.696713\pi$
$110$	0	0
$111$	−0.237195	−0.0225135
$112$	8.50547	0.803691
$113$	−5.96413	−0.561058	−0.280529	−	0.959846i	$-0.590510\pi$
$113$	−5.96413	−0.561058	−0.280529	+	0.959846i	$0.590510\pi$
$114$	−25.7403	−2.41080
$115$	0	0
$116$	−26.5086	−2.46126
$117$	0	0
$118$	6.45139	0.593898
$119$	21.2953	1.95214
$120$	0	0
$121$	−6.63807	−0.603461
$122$	10.2384	0.926945
$123$	−9.01344	−0.812714
$124$	8.12427	0.729581
$125$	0	0
$126$	20.1045	1.79105
$127$	−11.9901	−1.06395	−0.531973	−	0.846761i	$-0.678549\pi$
$127$	−11.9901	−1.06395	−0.531973	+	0.846761i	$0.678549\pi$
$128$	10.0454	0.887899
$129$	−5.02556	−0.442476
$130$	0	0
$131$	−19.5464	−1.70777	−0.853886	−	0.520459i	$-0.825761\pi$
$131$	−19.5464	−1.70777	−0.853886	+	0.520459i	$0.825761\pi$
$132$	−12.9599	−1.12801
$133$	17.9116	1.55313
$134$	14.5553	1.25739
$135$	0	0
$136$	7.90694	0.678015
$137$	−12.8391	−1.09692	−0.548461	−	0.836176i	$-0.684786\pi$
$137$	−12.8391	−1.09692	−0.548461	+	0.836176i	$0.684786\pi$
$138$	−10.8460	−0.923276
$139$	5.38253	0.456540	0.228270	−	0.973598i	$-0.426693\pi$
$139$	5.38253	0.456540	0.228270	+	0.973598i	$0.426693\pi$
$140$	0	0
$141$	11.8806	1.00053
$142$	10.5160	0.882479
$143$	0	0
$144$	−6.32145	−0.526787
$145$	0	0
$146$	8.63708	0.714809
$147$	−13.2819	−1.09547
$148$	0.261082	0.0214608
$149$	15.5515	1.27403	0.637013	−	0.770853i	$-0.280170\pi$
$149$	15.5515	1.27403	0.637013	+	0.770853i	$0.280170\pi$
$150$	0	0
$151$	11.6494	0.948012	0.474006	−	0.880522i	$-0.342808\pi$
$151$	11.6494	0.948012	0.474006	+	0.880522i	$0.342808\pi$
$152$	6.65058	0.539433
$153$	−15.8271	−1.27955
$154$	15.9196	1.28284
$155$	0	0
$156$	0	0
$157$	−2.02732	−0.161798	−0.0808989	−	0.996722i	$-0.525779\pi$
$157$	−2.02732	−0.161798	−0.0808989	+	0.996722i	$0.525779\pi$
$158$	10.1264	0.805613
$159$	12.5735	0.997146
$160$	0	0
$161$	7.54732	0.594812
$162$	21.3845	1.68012
$163$	−8.48371	−0.664495	−0.332248	−	0.943192i	$-0.607807\pi$
$163$	−8.48371	−0.664495	−0.332248	+	0.943192i	$0.607807\pi$
$164$	9.92117	0.774713
$165$	0	0
$166$	−24.2074	−1.87886
$167$	4.52950	0.350503	0.175252	−	0.984524i	$-0.443926\pi$
$167$	4.52950	0.350503	0.175252	+	0.984524i	$0.443926\pi$
$168$	−11.1027	−0.856594
$169$	0	0
$170$	0	0
$171$	−13.3123	−1.01802
$172$	5.53168	0.421787
$173$	−1.03121	−0.0784013	−0.0392006	−	0.999231i	$-0.512481\pi$
$173$	−1.03121	−0.0784013	−0.0392006	+	0.999231i	$0.512481\pi$
$174$	−51.7283	−3.92151
$175$	0	0
$176$	−5.00561	−0.377312
$177$	7.13155	0.536040
$178$	−4.09105	−0.306637
$179$	−17.4031	−1.30077	−0.650386	−	0.759604i	$-0.725393\pi$
$179$	−17.4031	−1.30077	−0.650386	+	0.759604i	$0.725393\pi$
$180$	0	0
$181$	−22.2540	−1.65412	−0.827062	−	0.562111i	$-0.809989\pi$
$181$	−22.2540	−1.65412	−0.827062	+	0.562111i	$0.809989\pi$
$182$	0	0
$183$	11.3179	0.836642
$184$	2.80232	0.206590
$185$	0	0
$186$	15.8535	1.16244
$187$	−12.5326	−0.916478
$188$	−13.0771	−0.953746
$189$	−3.05414	−0.222156
$190$	0	0
$191$	−7.05937	−0.510798	−0.255399	−	0.966836i	$-0.582207\pi$
$191$	−7.05937	−0.510798	−0.255399	+	0.966836i	$0.582207\pi$
$192$	28.3122	2.04325
$193$	−21.7870	−1.56826	−0.784130	−	0.620596i	$-0.786891\pi$
$193$	−21.7870	−1.56826	−0.784130	+	0.620596i	$0.786891\pi$
$194$	−11.5329	−0.828016
$195$	0	0
$196$	14.6195	1.04425
$197$	−22.4438	−1.59905	−0.799526	−	0.600631i	$-0.794916\pi$
$197$	−22.4438	−1.59905	−0.799526	+	0.600631i	$0.794916\pi$
$198$	−11.8318	−0.840851
$199$	−12.2097	−0.865520	−0.432760	−	0.901509i	$-0.642460\pi$
$199$	−12.2097	−0.865520	−0.432760	+	0.901509i	$0.642460\pi$
$200$	0	0
$201$	16.0899	1.13489
$202$	24.0800	1.69426
$203$	35.9957	2.52640
$204$	37.2362	2.60705
$205$	0	0
$206$	25.3716	1.76773
$207$	−5.60933	−0.389876
$208$	0	0
$209$	−10.5413	−0.729156
$210$	0	0
$211$	17.3180	1.19222	0.596111	−	0.802902i	$-0.296712\pi$
$211$	17.3180	1.19222	0.596111	+	0.802902i	$0.296712\pi$
$212$	−13.8398	−0.950521
$213$	11.6246	0.796507
$214$	−11.8444	−0.809664
$215$	0	0
$216$	−1.13400	−0.0771591
$217$	−11.0318	−0.748889
$218$	25.9858	1.75998
$219$	9.54768	0.645172
$220$	0	0
$221$	0	0
$222$	0.509470	0.0341934
$223$	5.28335	0.353800	0.176900	−	0.984229i	$-0.443393\pi$
$223$	5.28335	0.353800	0.176900	+	0.984229i	$0.443393\pi$
$224$	−27.6211	−1.84551
$225$	0	0
$226$	12.8103	0.852131
$227$	−2.94943	−0.195760	−0.0978802	−	0.995198i	$-0.531206\pi$
$227$	−2.94943	−0.195760	−0.0978802	+	0.995198i	$0.531206\pi$
$228$	31.3196	2.07419
$229$	−3.09371	−0.204438	−0.102219	−	0.994762i	$-0.532594\pi$
$229$	−3.09371	−0.204438	−0.102219	+	0.994762i	$0.532594\pi$
$230$	0	0
$231$	17.5980	1.15787
$232$	13.3652	0.877467
$233$	6.38449	0.418262	0.209131	−	0.977888i	$-0.432937\pi$
$233$	6.38449	0.418262	0.209131	+	0.977888i	$0.432937\pi$
$234$	0	0
$235$	0	0
$236$	−7.84976	−0.510976
$237$	11.1940	0.727129
$238$	−45.7401	−2.96489
$239$	23.8468	1.54252	0.771260	−	0.636521i	$-0.219627\pi$
$239$	23.8468	1.54252	0.771260	+	0.636521i	$0.219627\pi$
$240$	0	0
$241$	−27.5319	−1.77349	−0.886744	−	0.462261i	$-0.847038\pi$
$241$	−27.5319	−1.77349	−0.886744	+	0.462261i	$0.847038\pi$
$242$	14.2579	0.916533
$243$	21.0572	1.35082
$244$	−12.4577	−0.797522
$245$	0	0
$246$	19.3600	1.23435
$247$	0	0
$248$	−4.09611	−0.260104
$249$	−26.7595	−1.69582
$250$	0	0
$251$	0.230820	0.0145692	0.00728462	−	0.999973i	$-0.497681\pi$
$251$	0.230820	0.0145692	0.00728462	+	0.999973i	$0.497681\pi$
$252$	−24.4622	−1.54097
$253$	−4.44173	−0.279249
$254$	25.7534	1.61591
$255$	0	0
$256$	2.27181	0.141988
$257$	12.9414	0.807260	0.403630	−	0.914922i	$-0.367748\pi$
$257$	12.9414	0.807260	0.403630	+	0.914922i	$0.367748\pi$
$258$	10.7944	0.672030
$259$	−0.354520	−0.0220288
$260$	0	0
$261$	−26.7528	−1.65595
$262$	41.9836	2.59375
$263$	9.89693	0.610271	0.305135	−	0.952309i	$-0.401298\pi$
$263$	9.89693	0.610271	0.305135	+	0.952309i	$0.401298\pi$
$264$	6.53415	0.402149
$265$	0	0
$266$	−38.4723	−2.35889
$267$	−4.52236	−0.276764
$268$	−17.7103	−1.08183
$269$	12.6400	0.770674	0.385337	−	0.922776i	$-0.374085\pi$
$269$	12.6400	0.770674	0.385337	+	0.922776i	$0.374085\pi$
$270$	0	0
$271$	19.4775	1.18318	0.591588	−	0.806240i	$-0.298501\pi$
$271$	19.4775	1.18318	0.591588	+	0.806240i	$0.298501\pi$
$272$	14.3821	0.872042
$273$	0	0
$274$	27.5772	1.66600
$275$	0	0
$276$	13.1970	0.794365
$277$	27.8610	1.67401	0.837004	−	0.547197i	$-0.184305\pi$
$277$	27.8610	1.67401	0.837004	+	0.547197i	$0.184305\pi$
$278$	−11.5611	−0.693390
$279$	8.19909	0.490867
$280$	0	0
$281$	−22.0533	−1.31559	−0.657796	−	0.753197i	$-0.728511\pi$
$281$	−22.0533	−1.31559	−0.657796	+	0.753197i	$0.728511\pi$
$282$	−25.5184	−1.51960
$283$	−25.1358	−1.49417	−0.747083	−	0.664730i	$-0.768546\pi$
$283$	−25.1358	−1.49417	−0.747083	+	0.664730i	$0.768546\pi$
$284$	−12.7953	−0.759264
$285$	0	0
$286$	0	0
$287$	−13.4718	−0.795216
$288$	20.5286	1.20966
$289$	19.0087	1.11816
$290$	0	0
$291$	−12.7488	−0.747350
$292$	−10.5092	−0.615005
$293$	7.24385	0.423190	0.211595	−	0.977357i	$-0.432134\pi$
$293$	7.24385	0.423190	0.211595	+	0.977357i	$0.432134\pi$
$294$	28.5281	1.66379
$295$	0	0
$296$	−0.131633	−0.00765102
$297$	1.79741	0.104297
$298$	−33.4030	−1.93498
$299$	0	0
$300$	0	0
$301$	−7.51139	−0.432949
$302$	−25.0216	−1.43983
$303$	26.6187	1.52920
$304$	12.0969	0.693802
$305$	0	0
$306$	33.9951	1.94337
$307$	−29.4770	−1.68234	−0.841170	−	0.540770i	$-0.818133\pi$
$307$	−29.4770	−1.68234	−0.841170	+	0.540770i	$0.818133\pi$
$308$	−19.3703	−1.10373
$309$	28.0466	1.59551
$310$	0	0
$311$	−16.3474	−0.926978	−0.463489	−	0.886103i	$-0.653403\pi$
$311$	−16.3474	−0.926978	−0.463489	+	0.886103i	$0.653403\pi$
$312$	0	0
$313$	−29.4517	−1.66471	−0.832355	−	0.554244i	$-0.813008\pi$
$313$	−29.4517	−1.66471	−0.832355	+	0.554244i	$0.813008\pi$
$314$	4.35448	0.245737
$315$	0	0
$316$	−12.3213	−0.693130
$317$	16.8218	0.944804	0.472402	−	0.881383i	$-0.343387\pi$
$317$	16.8218	0.944804	0.472402	+	0.881383i	$0.343387\pi$
$318$	−27.0067	−1.51446
$319$	−21.1841	−1.18608
$320$	0	0
$321$	−13.0931	−0.730786
$322$	−16.2109	−0.903397
$323$	30.2871	1.68522
$324$	−26.0197	−1.44554
$325$	0	0
$326$	18.2221	1.00923
$327$	28.7255	1.58852
$328$	−5.00208	−0.276194
$329$	17.7572	0.978987
$330$	0	0
$331$	2.15049	0.118202	0.0591009	−	0.998252i	$-0.481177\pi$
$331$	2.15049	0.118202	0.0591009	+	0.998252i	$0.481177\pi$
$332$	29.4544	1.61652
$333$	0.263487	0.0144390
$334$	−9.72890	−0.532342
$335$	0	0
$336$	−20.1950	−1.10173
$337$	−2.37962	−0.129626	−0.0648131	−	0.997897i	$-0.520645\pi$
$337$	−2.37962	−0.129626	−0.0648131	+	0.997897i	$0.520645\pi$
$338$	0	0
$339$	14.1609	0.769116
$340$	0	0
$341$	6.49242	0.351584
$342$	28.5935	1.54616
$343$	4.98994	0.269431
$344$	−2.78898	−0.150372
$345$	0	0
$346$	2.21493	0.119075
$347$	26.0916	1.40067	0.700336	−	0.713813i	$-0.253033\pi$
$347$	26.0916	1.40067	0.700336	+	0.713813i	$0.253033\pi$
$348$	62.9407	3.37398
$349$	16.3589	0.875674	0.437837	−	0.899054i	$-0.355745\pi$
$349$	16.3589	0.875674	0.437837	+	0.899054i	$0.355745\pi$
$350$	0	0
$351$	0	0
$352$	16.2555	0.866421
$353$	−2.76979	−0.147421	−0.0737105	−	0.997280i	$-0.523484\pi$
$353$	−2.76979	−0.147421	−0.0737105	+	0.997280i	$0.523484\pi$
$354$	−15.3179	−0.814135
$355$	0	0
$356$	4.97780	0.263823
$357$	−50.5625	−2.67605
$358$	37.3802	1.97560
$359$	17.9581	0.947791	0.473896	−	0.880581i	$-0.342847\pi$
$359$	17.9581	0.947791	0.473896	+	0.880581i	$0.342847\pi$
$360$	0	0
$361$	6.47470	0.340774
$362$	47.7992	2.51227
$363$	15.7611	0.827243
$364$	0	0
$365$	0	0
$366$	−24.3097	−1.27069
$367$	27.0501	1.41200	0.706001	−	0.708211i	$-0.250497\pi$
$367$	27.0501	1.41200	0.706001	+	0.708211i	$0.250497\pi$
$368$	5.09719	0.265709
$369$	10.0125	0.521232
$370$	0	0
$371$	18.7928	0.975676
$372$	−19.2899	−1.00013
$373$	0.658276	0.0340842	0.0170421	−	0.999855i	$-0.494575\pi$
$373$	0.658276	0.0340842	0.0170421	+	0.999855i	$0.494575\pi$
$374$	26.9189	1.39194
$375$	0	0
$376$	6.59325	0.340021
$377$	0	0
$378$	6.55999	0.337409
$379$	−17.9715	−0.923135	−0.461568	−	0.887105i	$-0.652713\pi$
$379$	−17.9715	−0.923135	−0.461568	+	0.887105i	$0.652713\pi$
$380$	0	0
$381$	28.4686	1.45849
$382$	15.1628	0.775797
$383$	−1.37967	−0.0704978	−0.0352489	−	0.999379i	$-0.511222\pi$
$383$	−1.37967	−0.0704978	−0.0352489	+	0.999379i	$0.511222\pi$
$384$	−23.8514	−1.21716
$385$	0	0
$386$	46.7962	2.38186
$387$	5.58262	0.283781
$388$	14.0327	0.712405
$389$	6.64124	0.336724	0.168362	−	0.985725i	$-0.446152\pi$
$389$	6.64124	0.336724	0.168362	+	0.985725i	$0.446152\pi$
$390$	0	0
$391$	12.7619	0.645399
$392$	−7.37089	−0.372286
$393$	46.4099	2.34107
$394$	48.2070	2.42863
$395$	0	0
$396$	14.3964	0.723448
$397$	17.4163	0.874097	0.437048	−	0.899438i	$-0.356024\pi$
$397$	17.4163	0.874097	0.437048	+	0.899438i	$0.356024\pi$
$398$	26.2251	1.31455
$399$	−42.5284	−2.12908
$400$	0	0
$401$	−27.1562	−1.35612	−0.678058	−	0.735009i	$-0.737178\pi$
$401$	−27.1562	−1.35612	−0.678058	+	0.735009i	$0.737178\pi$
$402$	−34.5595	−1.72367
$403$	0	0
$404$	−29.2994	−1.45770
$405$	0	0
$406$	−77.3150	−3.83708
$407$	0.208641	0.0103420
$408$	−18.7738	−0.929444
$409$	−3.12154	−0.154350	−0.0771751	−	0.997018i	$-0.524590\pi$
$409$	−3.12154	−0.154350	−0.0771751	+	0.997018i	$0.524590\pi$
$410$	0	0
$411$	30.4846	1.50370
$412$	−30.8711	−1.52091
$413$	10.6591	0.524499
$414$	12.0483	0.592141
$415$	0	0
$416$	0	0
$417$	−12.7800	−0.625840
$418$	22.6416	1.10744
$419$	−3.35864	−0.164080	−0.0820402	−	0.996629i	$-0.526144\pi$
$419$	−3.35864	−0.164080	−0.0820402	+	0.996629i	$0.526144\pi$
$420$	0	0
$421$	−9.50793	−0.463388	−0.231694	−	0.972789i	$-0.574427\pi$
$421$	−9.50793	−0.463388	−0.231694	+	0.972789i	$0.574427\pi$
$422$	−37.1974	−1.81074
$423$	−13.1975	−0.641686
$424$	6.97778	0.338871
$425$	0	0
$426$	−24.9685	−1.20973
$427$	16.9161	0.818628
$428$	14.4117	0.696615
$429$	0	0
$430$	0	0
$431$	37.7029	1.81609	0.908043	−	0.418878i	$-0.137576\pi$
$431$	37.7029	1.81609	0.908043	+	0.418878i	$0.137576\pi$
$432$	−2.06266	−0.0992397
$433$	−20.9632	−1.00742	−0.503712	−	0.863871i	$-0.668033\pi$
$433$	−20.9632	−1.00742	−0.503712	+	0.863871i	$0.668033\pi$
$434$	23.6952	1.13741
$435$	0	0
$436$	−31.6184	−1.51425
$437$	10.7341	0.513484
$438$	−20.5074	−0.979883
$439$	−10.4601	−0.499231	−0.249616	−	0.968345i	$-0.580304\pi$
$439$	−10.4601	−0.499231	−0.249616	+	0.968345i	$0.580304\pi$
$440$	0	0
$441$	14.7541	0.702577
$442$	0	0
$443$	19.6909	0.935543	0.467772	−	0.883849i	$-0.345057\pi$
$443$	19.6909	0.935543	0.467772	+	0.883849i	$0.345057\pi$
$444$	−0.619901	−0.0294192
$445$	0	0
$446$	−11.3481	−0.537349
$447$	−36.9246	−1.74648
$448$	42.3164	1.99926
$449$	7.93865	0.374648	0.187324	−	0.982298i	$-0.440019\pi$
$449$	7.93865	0.374648	0.187324	+	0.982298i	$0.440019\pi$
$450$	0	0
$451$	7.92839	0.373333
$452$	−15.5871	−0.733154
$453$	−27.6597	−1.29956
$454$	6.33507	0.297319
$455$	0	0
$456$	−15.7908	−0.739472
$457$	−7.10597	−0.332403	−0.166202	−	0.986092i	$-0.553150\pi$
$457$	−7.10597	−0.332403	−0.166202	+	0.986092i	$0.553150\pi$
$458$	6.64498	0.310500
$459$	−5.16431	−0.241050
$460$	0	0
$461$	15.3424	0.714566	0.357283	−	0.933996i	$-0.383703\pi$
$461$	15.3424	0.714566	0.357283	+	0.933996i	$0.383703\pi$
$462$	−37.7988	−1.75856
$463$	−10.1047	−0.469606	−0.234803	−	0.972043i	$-0.575445\pi$
$463$	−10.1047	−0.469606	−0.234803	+	0.972043i	$0.575445\pi$
$464$	24.3102	1.12857
$465$	0	0
$466$	−13.7132	−0.635253
$467$	14.2884	0.661190	0.330595	−	0.943773i	$-0.392751\pi$
$467$	14.2884	0.661190	0.330595	+	0.943773i	$0.392751\pi$
$468$	0	0
$469$	24.0485	1.11046
$470$	0	0
$471$	4.81357	0.221797
$472$	3.95771	0.182169
$473$	4.42058	0.203258
$474$	−24.0436	−1.10436
$475$	0	0
$476$	55.6546	2.55092
$477$	−13.9673	−0.639516
$478$	−51.2204	−2.34277
$479$	33.8677	1.54745	0.773727	−	0.633519i	$-0.218390\pi$
$479$	33.8677	1.54745	0.773727	+	0.633519i	$0.218390\pi$
$480$	0	0
$481$	0	0
$482$	59.1358	2.69356
$483$	−17.9200	−0.815388
$484$	−17.3484	−0.788563
$485$	0	0
$486$	−45.2287	−2.05162
$487$	13.0574	0.591689	0.295844	−	0.955236i	$-0.404399\pi$
$487$	13.0574	0.591689	0.295844	+	0.955236i	$0.404399\pi$
$488$	6.28095	0.284325
$489$	20.1433	0.910911
$490$	0	0
$491$	40.8205	1.84220	0.921101	−	0.389325i	$-0.127292\pi$
$491$	40.8205	1.84220	0.921101	+	0.389325i	$0.127292\pi$
$492$	−23.5563	−1.06200
$493$	60.8658	2.74126
$494$	0	0
$495$	0	0
$496$	−7.45050	−0.334537
$497$	17.3746	0.779358
$498$	57.4767	2.57559
$499$	−34.5723	−1.54767	−0.773833	−	0.633389i	$-0.781663\pi$
$499$	−34.5723	−1.54767	−0.773833	+	0.633389i	$0.781663\pi$
$500$	0	0
$501$	−10.7546	−0.480481
$502$	−0.495779	−0.0221277
$503$	44.0662	1.96481	0.982407	−	0.186754i	$-0.0597967\pi$
$503$	44.0662	1.96481	0.982407	+	0.186754i	$0.0597967\pi$
$504$	12.3334	0.549374
$505$	0	0
$506$	9.54038	0.424122
$507$	0	0
$508$	−31.3356	−1.39029
$509$	42.1772	1.86947	0.934735	−	0.355345i	$-0.115637\pi$
$509$	42.1772	1.86947	0.934735	+	0.355345i	$0.115637\pi$
$510$	0	0
$511$	14.2703	0.631281
$512$	−24.9705	−1.10355
$513$	−4.34374	−0.191781
$514$	−27.7967	−1.22606
$515$	0	0
$516$	−13.1341	−0.578198
$517$	−10.4504	−0.459609
$518$	0.761473	0.0334572
$519$	2.44845	0.107475
$520$	0	0
$521$	28.6488	1.25513	0.627564	−	0.778565i	$-0.284052\pi$
$521$	28.6488	1.25513	0.627564	+	0.778565i	$0.284052\pi$
$522$	57.4622	2.51505
$523$	20.6845	0.904471	0.452235	−	0.891899i	$-0.350627\pi$
$523$	20.6845	0.904471	0.452235	+	0.891899i	$0.350627\pi$
$524$	−51.0838	−2.23160
$525$	0	0
$526$	−21.2576	−0.926875
$527$	−18.6540	−0.812579
$528$	11.8851	0.517232
$529$	−18.4770	−0.803348
$530$	0	0
$531$	−7.92206	−0.343788
$532$	46.8114	2.02953
$533$	0	0
$534$	9.71357	0.420347
$535$	0	0
$536$	8.92922	0.385684
$537$	41.3211	1.78314
$538$	−27.1494	−1.17049
$539$	11.6830	0.503222
$540$	0	0
$541$	−25.8647	−1.11201	−0.556005	−	0.831179i	$-0.687666\pi$
$541$	−25.8647	−1.11201	−0.556005	+	0.831179i	$0.687666\pi$
$542$	−41.8358	−1.79700
$543$	52.8387	2.26752
$544$	−46.7051	−2.00247
$545$	0	0
$546$	0	0
$547$	29.5937	1.26533	0.632667	−	0.774424i	$-0.281960\pi$
$547$	29.5937	1.26533	0.632667	+	0.774424i	$0.281960\pi$
$548$	−33.5547	−1.43338
$549$	−12.5724	−0.536578
$550$	0	0
$551$	51.1947	2.18097
$552$	−6.65369	−0.283200
$553$	16.7310	0.711473
$554$	−59.8427	−2.54247
$555$	0	0
$556$	14.0671	0.596576
$557$	−28.3520	−1.20131	−0.600657	−	0.799507i	$-0.705094\pi$
$557$	−28.3520	−1.20131	−0.600657	+	0.799507i	$0.705094\pi$
$558$	−17.6108	−0.745525
$559$	0	0
$560$	0	0
$561$	29.7569	1.25634
$562$	47.3683	1.99811
$563$	13.2071	0.556615	0.278307	−	0.960492i	$-0.410227\pi$
$563$	13.2071	0.556615	0.278307	+	0.960492i	$0.410227\pi$
$564$	31.0496	1.30742
$565$	0	0
$566$	53.9891	2.26933
$567$	35.3318	1.48380
$568$	6.45119	0.270686
$569$	−10.8594	−0.455248	−0.227624	−	0.973749i	$-0.573096\pi$
$569$	−10.8594	−0.455248	−0.227624	+	0.973749i	$0.573096\pi$
$570$	0	0
$571$	3.37601	0.141282	0.0706409	−	0.997502i	$-0.477496\pi$
$571$	3.37601	0.141282	0.0706409	+	0.997502i	$0.477496\pi$
$572$	0	0
$573$	16.7614	0.700218
$574$	28.9361	1.20777
$575$	0	0
$576$	−31.4505	−1.31044
$577$	−0.958534	−0.0399043	−0.0199521	−	0.999801i	$-0.506351\pi$
$577$	−0.958534	−0.0399043	−0.0199521	+	0.999801i	$0.506351\pi$
$578$	−40.8287	−1.69825
$579$	51.7299	2.14982
$580$	0	0
$581$	−39.9958	−1.65930
$582$	27.3832	1.13507
$583$	−11.0599	−0.458055
$584$	5.29856	0.219256
$585$	0	0
$586$	−15.5590	−0.642738
$587$	21.6825	0.894934	0.447467	−	0.894300i	$-0.352326\pi$
$587$	21.6825	0.894934	0.447467	+	0.894300i	$0.352326\pi$
$588$	−34.7118	−1.43149
$589$	−15.6900	−0.646493
$590$	0	0
$591$	53.2894	2.19203
$592$	−0.239430	−0.00984051
$593$	−37.9527	−1.55853	−0.779264	−	0.626695i	$-0.784407\pi$
$593$	−37.9527	−1.55853	−0.779264	+	0.626695i	$0.784407\pi$
$594$	−3.86066	−0.158405
$595$	0	0
$596$	40.6433	1.66481
$597$	28.9900	1.18648
$598$	0	0
$599$	−4.17043	−0.170399	−0.0851995	−	0.996364i	$-0.527153\pi$
$599$	−4.17043	−0.170399	−0.0851995	+	0.996364i	$0.527153\pi$
$600$	0	0
$601$	42.2234	1.72233	0.861165	−	0.508326i	$-0.169736\pi$
$601$	42.2234	1.72233	0.861165	+	0.508326i	$0.169736\pi$
$602$	16.1337	0.657560
$603$	−17.8734	−0.727861
$604$	30.4452	1.23880
$605$	0	0
$606$	−57.1742	−2.32254
$607$	−8.63499	−0.350483	−0.175242	−	0.984525i	$-0.556071\pi$
$607$	−8.63499	−0.350483	−0.175242	+	0.984525i	$0.556071\pi$
$608$	−39.2840	−1.59318
$609$	−85.4663	−3.46327
$610$	0	0
$611$	0	0
$612$	−41.3637	−1.67203
$613$	35.7619	1.44441	0.722205	−	0.691679i	$-0.243129\pi$
$613$	35.7619	1.44441	0.722205	+	0.691679i	$0.243129\pi$
$614$	63.3136	2.55513
$615$	0	0
$616$	9.76617	0.393490
$617$	−1.13565	−0.0457197	−0.0228599	−	0.999739i	$-0.507277\pi$
$617$	−1.13565	−0.0457197	−0.0228599	+	0.999739i	$0.507277\pi$
$618$	−60.2412	−2.42326
$619$	−47.6316	−1.91447	−0.957237	−	0.289305i	$-0.906576\pi$
$619$	−47.6316	−1.91447	−0.957237	+	0.289305i	$0.906576\pi$
$620$	0	0
$621$	−1.83030	−0.0734473
$622$	35.1126	1.40789
$623$	−6.75929	−0.270805
$624$	0	0
$625$	0	0
$626$	63.2593	2.52835
$627$	25.0287	0.999550
$628$	−5.29833	−0.211427
$629$	−0.599465	−0.0239022
$630$	0	0
$631$	−7.67241	−0.305434	−0.152717	−	0.988270i	$-0.548802\pi$
$631$	−7.67241	−0.305434	−0.152717	+	0.988270i	$0.548802\pi$
$632$	6.21221	0.247108
$633$	−41.1191	−1.63434
$634$	−36.1314	−1.43496
$635$	0	0
$636$	32.8605	1.30300
$637$	0	0
$638$	45.5012	1.80141
$639$	−12.9132	−0.510838
$640$	0	0
$641$	−15.8544	−0.626213	−0.313106	−	0.949718i	$-0.601370\pi$
$641$	−15.8544	−0.626213	−0.313106	+	0.949718i	$0.601370\pi$
$642$	28.1226	1.10991
$643$	20.4282	0.805611	0.402806	−	0.915285i	$-0.368035\pi$
$643$	20.4282	0.805611	0.402806	+	0.915285i	$0.368035\pi$
$644$	19.7247	0.777261
$645$	0	0
$646$	−65.0537	−2.55950
$647$	−23.8101	−0.936073	−0.468036	−	0.883709i	$-0.655038\pi$
$647$	−23.8101	−0.936073	−0.468036	+	0.883709i	$0.655038\pi$
$648$	13.1187	0.515351
$649$	−6.27305	−0.246239
$650$	0	0
$651$	26.1934	1.02660
$652$	−22.1719	−0.868318
$653$	−31.1678	−1.21969	−0.609844	−	0.792521i	$-0.708768\pi$
$653$	−31.1678	−1.21969	−0.609844	+	0.792521i	$0.708768\pi$
$654$	−61.6994	−2.41264
$655$	0	0
$656$	−9.09838	−0.355232
$657$	−10.6060	−0.413779
$658$	−38.1407	−1.48688
$659$	35.0918	1.36698	0.683492	−	0.729958i	$-0.260460\pi$
$659$	35.0918	1.36698	0.683492	+	0.729958i	$0.260460\pi$
$660$	0	0
$661$	−16.3641	−0.636489	−0.318244	−	0.948009i	$-0.603093\pi$
$661$	−16.3641	−0.636489	−0.318244	+	0.948009i	$0.603093\pi$
$662$	−4.61904	−0.179524
$663$	0	0
$664$	−14.8504	−0.576308
$665$	0	0
$666$	−0.565943	−0.0219298
$667$	21.5716	0.835257
$668$	11.8377	0.458014
$669$	−12.5445	−0.485000
$670$	0	0
$671$	−9.95542	−0.384325
$672$	65.5822	2.52989
$673$	−47.1689	−1.81823	−0.909114	−	0.416548i	$-0.863240\pi$
$673$	−47.1689	−1.81823	−0.909114	+	0.416548i	$0.863240\pi$
$674$	5.11118	0.196875
$675$	0	0
$676$	0	0
$677$	−6.68421	−0.256895	−0.128448	−	0.991716i	$-0.540999\pi$
$677$	−6.68421	−0.256895	−0.128448	+	0.991716i	$0.540999\pi$
$678$	−30.4162	−1.16813
$679$	−19.0549	−0.731259
$680$	0	0
$681$	7.00297	0.268354
$682$	−13.9451	−0.533984
$683$	−21.4574	−0.821043	−0.410522	−	0.911851i	$-0.634653\pi$
$683$	−21.4574	−0.821043	−0.410522	+	0.911851i	$0.634653\pi$
$684$	−34.7912	−1.33028
$685$	0	0
$686$	−10.7179	−0.409211
$687$	7.34556	0.280251
$688$	−5.07292	−0.193403
$689$	0	0
$690$	0	0
$691$	−36.7873	−1.39946	−0.699728	−	0.714409i	$-0.746696\pi$
$691$	−36.7873	−1.39946	−0.699728	+	0.714409i	$0.746696\pi$
$692$	−2.69503	−0.102450
$693$	−19.5487	−0.742594
$694$	−56.0422	−2.12733
$695$	0	0
$696$	−31.7336	−1.20286
$697$	−22.7798	−0.862846
$698$	−35.1373	−1.32997
$699$	−15.1590	−0.573367
$700$	0	0
$701$	26.7626	1.01081	0.505404	−	0.862883i	$-0.331344\pi$
$701$	26.7626	1.01081	0.505404	+	0.862883i	$0.331344\pi$
$702$	0	0
$703$	−0.504214	−0.0190168
$704$	−24.9039	−0.938602
$705$	0	0
$706$	5.94923	0.223902
$707$	39.7853	1.49628
$708$	18.6381	0.700462
$709$	−23.4847	−0.881988	−0.440994	−	0.897510i	$-0.645374\pi$
$709$	−23.4847	−0.881988	−0.440994	+	0.897510i	$0.645374\pi$
$710$	0	0
$711$	−12.4348	−0.466342
$712$	−2.50972	−0.0940558
$713$	−6.61120	−0.247591
$714$	108.603	4.06437
$715$	0	0
$716$	−45.4825	−1.69976
$717$	−56.6206	−2.11453
$718$	−38.5721	−1.43950
$719$	−17.3284	−0.646241	−0.323121	−	0.946358i	$-0.604732\pi$
$719$	−17.3284	−0.646241	−0.323121	+	0.946358i	$0.604732\pi$
$720$	0	0
$721$	41.9194	1.56116
$722$	−13.9070	−0.517565
$723$	65.3704	2.43115
$724$	−58.1600	−2.16150
$725$	0	0
$726$	−33.8532	−1.25641
$727$	2.13255	0.0790919	0.0395460	−	0.999218i	$-0.487409\pi$
$727$	2.13255	0.0790919	0.0395460	+	0.999218i	$0.487409\pi$
$728$	0	0
$729$	−20.1291	−0.745524
$730$	0	0
$731$	−12.7012	−0.469769
$732$	29.5789	1.09327
$733$	4.01857	0.148429	0.0742147	−	0.997242i	$-0.476355\pi$
$733$	4.01857	0.148429	0.0742147	+	0.997242i	$0.476355\pi$
$734$	−58.1008	−2.14454
$735$	0	0
$736$	−16.5529	−0.610147
$737$	−14.1530	−0.521332
$738$	−21.5059	−0.791644
$739$	2.00491	0.0737519	0.0368759	−	0.999320i	$-0.488259\pi$
$739$	2.00491	0.0737519	0.0368759	+	0.999320i	$0.488259\pi$
$740$	0	0
$741$	0	0
$742$	−40.3651	−1.48185
$743$	−6.48999	−0.238094	−0.119047	−	0.992889i	$-0.537984\pi$
$743$	−6.48999	−0.238094	−0.119047	+	0.992889i	$0.537984\pi$
$744$	9.72561	0.356558
$745$	0	0
$746$	−1.41391	−0.0517669
$747$	29.7257	1.08761
$748$	−32.7537	−1.19759
$749$	−19.5694	−0.715051
$750$	0	0
$751$	−28.7393	−1.04871	−0.524355	−	0.851500i	$-0.675694\pi$
$751$	−28.7393	−1.04871	−0.524355	+	0.851500i	$0.675694\pi$
$752$	11.9926	0.437324
$753$	−0.548048	−0.0199720
$754$	0	0
$755$	0	0
$756$	−7.98190	−0.290299
$757$	−43.1705	−1.56906	−0.784530	−	0.620091i	$-0.787095\pi$
$757$	−43.1705	−1.56906	−0.784530	+	0.620091i	$0.787095\pi$
$758$	38.6010	1.40205
$759$	10.5462	0.382803
$760$	0	0
$761$	5.01133	0.181661	0.0908303	−	0.995866i	$-0.471048\pi$
$761$	5.01133	0.181661	0.0908303	+	0.995866i	$0.471048\pi$
$762$	−61.1477	−2.21515
$763$	42.9341	1.55432
$764$	−18.4494	−0.667477
$765$	0	0
$766$	2.96339	0.107072
$767$	0	0
$768$	−5.39406	−0.194642
$769$	−0.117089	−0.00422232	−0.00211116	−	0.999998i	$-0.500672\pi$
$769$	−0.117089	−0.00422232	−0.00211116	+	0.999998i	$0.500672\pi$
$770$	0	0
$771$	−30.7273	−1.10662
$772$	−56.9395	−2.04930
$773$	−21.8395	−0.785512	−0.392756	−	0.919643i	$-0.628478\pi$
$773$	−21.8395	−0.785512	−0.392756	+	0.919643i	$0.628478\pi$
$774$	−11.9909	−0.431004
$775$	0	0
$776$	−7.07507	−0.253980
$777$	0.841754	0.0301978
$778$	−14.2647	−0.511415
$779$	−19.1602	−0.686486
$780$	0	0
$781$	−10.2253	−0.365888
$782$	−27.4113	−0.980227
$783$	−8.72929	−0.311960
$784$	−13.4070	−0.478823
$785$	0	0
$786$	−99.6837	−3.55560
$787$	27.7621	0.989613	0.494807	−	0.869003i	$-0.335239\pi$
$787$	27.7621	0.989613	0.494807	+	0.869003i	$0.335239\pi$
$788$	−58.6561	−2.08954
$789$	−23.4988	−0.836579
$790$	0	0
$791$	21.1654	0.752556
$792$	−7.25843	−0.257917
$793$	0	0
$794$	−37.4084	−1.32757
$795$	0	0
$796$	−31.9096	−1.13100
$797$	−41.9565	−1.48618	−0.743088	−	0.669194i	$-0.766640\pi$
$797$	−41.9565	−1.48618	−0.743088	+	0.669194i	$0.766640\pi$
$798$	91.3468	3.23364
$799$	30.0260	1.06224
$800$	0	0
$801$	5.02365	0.177502
$802$	58.3287	2.05966
$803$	−8.39832	−0.296370
$804$	42.0504	1.48300
$805$	0	0
$806$	0	0
$807$	−30.0117	−1.05646
$808$	14.7723	0.519686
$809$	8.06707	0.283623	0.141812	−	0.989894i	$-0.454707\pi$
$809$	8.06707	0.283623	0.141812	+	0.989894i	$0.454707\pi$
$810$	0	0
$811$	16.0953	0.565181	0.282591	−	0.959241i	$-0.408806\pi$
$811$	16.0953	0.565181	0.282591	+	0.959241i	$0.408806\pi$
$812$	94.0735	3.30133
$813$	−46.2465	−1.62194
$814$	−0.448140	−0.0157073
$815$	0	0
$816$	−34.1481	−1.19542
$817$	−10.6830	−0.373752
$818$	6.70475	0.234426
$819$	0	0
$820$	0	0
$821$	−16.5777	−0.578564	−0.289282	−	0.957244i	$-0.593417\pi$
$821$	−16.5777	−0.578564	−0.289282	+	0.957244i	$0.593417\pi$
$822$	−65.4779	−2.28380
$823$	24.9895	0.871079	0.435540	−	0.900170i	$-0.356558\pi$
$823$	24.9895	0.871079	0.435540	+	0.900170i	$0.356558\pi$
$824$	15.5647	0.542221
$825$	0	0
$826$	−22.8946	−0.796606
$827$	9.28575	0.322897	0.161449	−	0.986881i	$-0.448383\pi$
$827$	9.28575	0.322897	0.161449	+	0.986881i	$0.448383\pi$
$828$	−14.6598	−0.509464
$829$	48.3689	1.67992	0.839961	−	0.542646i	$-0.182578\pi$
$829$	48.3689	1.67992	0.839961	+	0.542646i	$0.182578\pi$
$830$	0	0
$831$	−66.1519	−2.29478
$832$	0	0
$833$	−33.5675	−1.16304
$834$	27.4502	0.950521
$835$	0	0
$836$	−27.5493	−0.952813
$837$	2.67532	0.0924727
$838$	7.21402	0.249204
$839$	27.5771	0.952068	0.476034	−	0.879427i	$-0.342074\pi$
$839$	27.5771	0.952068	0.476034	+	0.879427i	$0.342074\pi$
$840$	0	0
$841$	73.8822	2.54766
$842$	20.4221	0.703791
$843$	52.3623	1.80345
$844$	45.2601	1.55792
$845$	0	0
$846$	28.3470	0.974589
$847$	23.5571	0.809432
$848$	12.6920	0.435846
$849$	59.6811	2.04825
$850$	0	0
$851$	−0.212458	−0.00728297
$852$	30.3806	1.04082
$853$	−36.3384	−1.24420	−0.622101	−	0.782937i	$-0.713721\pi$
$853$	−36.3384	−1.24420	−0.622101	+	0.782937i	$0.713721\pi$
$854$	−36.3341	−1.24333
$855$	0	0
$856$	−7.26612	−0.248351
$857$	18.4246	0.629373	0.314687	−	0.949196i	$-0.398101\pi$
$857$	18.4246	0.629373	0.314687	+	0.949196i	$0.398101\pi$
$858$	0	0
$859$	−36.8252	−1.25646	−0.628229	−	0.778028i	$-0.716220\pi$
$859$	−36.8252	−1.25646	−0.628229	+	0.778028i	$0.716220\pi$
$860$	0	0
$861$	31.9868	1.09011
$862$	−80.9820	−2.75826
$863$	−12.0557	−0.410380	−0.205190	−	0.978722i	$-0.565781\pi$
$863$	−12.0557	−0.410380	−0.205190	+	0.978722i	$0.565781\pi$
$864$	6.69838	0.227884
$865$	0	0
$866$	45.0267	1.53007
$867$	−45.1332	−1.53281
$868$	−28.8313	−0.978599
$869$	−9.84646	−0.334018
$870$	0	0
$871$	0	0
$872$	15.9414	0.539845
$873$	14.1620	0.479311
$874$	−23.0558	−0.779876
$875$	0	0
$876$	24.9525	0.843068
$877$	−18.4435	−0.622793	−0.311396	−	0.950280i	$-0.600797\pi$
$877$	−18.4435	−0.622793	−0.311396	+	0.950280i	$0.600797\pi$
$878$	22.4671	0.758230
$879$	−17.1994	−0.580122
$880$	0	0
$881$	43.4371	1.46343	0.731716	−	0.681610i	$-0.238720\pi$
$881$	43.4371	1.46343	0.731716	+	0.681610i	$0.238720\pi$
$882$	−31.6904	−1.06707
$883$	−26.7242	−0.899341	−0.449670	−	0.893195i	$-0.648459\pi$
$883$	−26.7242	−0.899341	−0.449670	+	0.893195i	$0.648459\pi$
$884$	0	0
$885$	0	0
$886$	−42.2941	−1.42090
$887$	−42.3744	−1.42279	−0.711396	−	0.702791i	$-0.751937\pi$
$887$	−42.3744	−1.42279	−0.711396	+	0.702791i	$0.751937\pi$
$888$	0.312543	0.0104883
$889$	42.5502	1.42709
$890$	0	0
$891$	−20.7934	−0.696604
$892$	13.8079	0.462322
$893$	25.2551	0.845130
$894$	79.3104	2.65254
$895$	0	0
$896$	−35.6491	−1.19095
$897$	0	0
$898$	−17.0514	−0.569013
$899$	−31.5310	−1.05162
$900$	0	0
$901$	31.7772	1.05865
$902$	−17.0294	−0.567016
$903$	17.8347	0.593500
$904$	7.85872	0.261377
$905$	0	0
$906$	59.4101	1.97377
$907$	−20.4291	−0.678337	−0.339168	−	0.940726i	$-0.610146\pi$
$907$	−20.4291	−0.678337	−0.339168	+	0.940726i	$0.610146\pi$
$908$	−7.70823	−0.255807
$909$	−29.5693	−0.980751
$910$	0	0
$911$	23.7626	0.787290	0.393645	−	0.919263i	$-0.371214\pi$
$911$	23.7626	0.787290	0.393645	+	0.919263i	$0.371214\pi$
$912$	−28.7222	−0.951086
$913$	23.5382	0.779000
$914$	15.2629	0.504852
$915$	0	0
$916$	−8.08532	−0.267146
$917$	69.3659	2.29066
$918$	11.0924	0.366104
$919$	−6.47619	−0.213630	−0.106815	−	0.994279i	$-0.534065\pi$
$919$	−6.47619	−0.213630	−0.106815	+	0.994279i	$0.534065\pi$
$920$	0	0
$921$	69.9887	2.30621
$922$	−32.9539	−1.08528
$923$	0	0
$924$	45.9919	1.51302
$925$	0	0
$926$	21.7039	0.713235
$927$	−31.1554	−1.02328
$928$	−78.9462	−2.59153
$929$	−20.8206	−0.683101	−0.341551	−	0.939863i	$-0.610952\pi$
$929$	−20.8206	−0.683101	−0.341551	+	0.939863i	$0.610952\pi$
$930$	0	0
$931$	−28.2338	−0.925326
$932$	16.6857	0.546557
$933$	38.8145	1.27073
$934$	−30.6901	−1.00421
$935$	0	0
$936$	0	0
$937$	18.1838	0.594039	0.297019	−	0.954871i	$-0.404007\pi$
$937$	18.1838	0.594039	0.297019	+	0.954871i	$0.404007\pi$
$938$	−51.6538	−1.68656
$939$	69.9287	2.28204
$940$	0	0
$941$	−4.73191	−0.154256	−0.0771279	−	0.997021i	$-0.524575\pi$
$941$	−4.73191	−0.154256	−0.0771279	+	0.997021i	$0.524575\pi$
$942$	−10.3391	−0.336864
$943$	−8.07344	−0.262907
$944$	7.19876	0.234300
$945$	0	0
$946$	−9.49495	−0.308708
$947$	−29.5954	−0.961721	−0.480861	−	0.876797i	$-0.659676\pi$
$947$	−29.5954	−0.961721	−0.480861	+	0.876797i	$0.659676\pi$
$948$	29.2552	0.950164
$949$	0	0
$950$	0	0
$951$	−39.9408	−1.29517
$952$	−28.0601	−0.909432
$953$	−1.49339	−0.0483755	−0.0241877	−	0.999707i	$-0.507700\pi$
$953$	−1.49339	−0.0483755	−0.0241877	+	0.999707i	$0.507700\pi$
$954$	30.0002	0.971293
$955$	0	0
$956$	62.3228	2.01566
$957$	50.2984	1.62592
$958$	−72.7443	−2.35026
$959$	45.5634	1.47132
$960$	0	0
$961$	−21.3365	−0.688274
$962$	0	0
$963$	14.5444	0.468687
$964$	−71.9538	−2.31748
$965$	0	0
$966$	38.4903	1.23841
$967$	−10.0656	−0.323687	−0.161844	−	0.986816i	$-0.551744\pi$
$967$	−10.0656	−0.323687	−0.161844	+	0.986816i	$0.551744\pi$
$968$	8.74675	0.281131
$969$	−71.9122	−2.31015
$970$	0	0
$971$	−12.3186	−0.395322	−0.197661	−	0.980270i	$-0.563334\pi$
$971$	−12.3186	−0.395322	−0.197661	+	0.980270i	$0.563334\pi$
$972$	55.0323	1.76516
$973$	−19.1015	−0.612365
$974$	−28.0460	−0.898653
$975$	0	0
$976$	11.4245	0.365690
$977$	−14.8608	−0.475438	−0.237719	−	0.971334i	$-0.576400\pi$
$977$	−14.8608	−0.475438	−0.237719	+	0.971334i	$0.576400\pi$
$978$	−43.2657	−1.38349
$979$	3.97796	0.127136
$980$	0	0
$981$	−31.9096	−1.01879
$982$	−87.6782	−2.79792
$983$	51.5408	1.64390	0.821948	−	0.569562i	$-0.192887\pi$
$983$	51.5408	1.64390	0.821948	+	0.569562i	$0.192887\pi$
$984$	11.8767	0.378615
$985$	0	0
$986$	−130.734	−4.16341
$987$	−42.1618	−1.34203
$988$	0	0
$989$	−4.50145	−0.143138
$990$	0	0
$991$	−5.57520	−0.177102	−0.0885511	−	0.996072i	$-0.528224\pi$
$991$	−5.57520	−0.177102	−0.0885511	+	0.996072i	$0.528224\pi$
$992$	24.1951	0.768197
$993$	−5.10602	−0.162035
$994$	−37.3189	−1.18368
$995$	0	0
$996$	−69.9351	−2.21598
$997$	25.7010	0.813959	0.406979	−	0.913437i	$-0.366582\pi$
$997$	25.7010	0.813959	0.406979	+	0.913437i	$0.366582\pi$
$998$	74.2577	2.35059
$999$	0.0859745	0.00272011

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4225.2.a.cb.1.3		18
5.2	odd	4		845.2.b.h.339.3	yes	18
5.3	odd	4		845.2.b.h.339.16	yes	18
5.4	even	2	inner	4225.2.a.cb.1.16		18
13.12	even	2		4225.2.a.ca.1.16		18
65.2	even	12		845.2.l.g.654.31		72
65.3	odd	12		845.2.n.h.529.3		36
65.7	even	12		845.2.l.g.699.6		72
65.8	even	4		845.2.d.e.844.5		36
65.12	odd	4		845.2.b.g.339.16	yes	18
65.17	odd	12		845.2.n.i.484.16		36
65.18	even	4		845.2.d.e.844.31		36
65.22	odd	12		845.2.n.h.484.3		36
65.23	odd	12		845.2.n.i.529.16		36
65.28	even	12		845.2.l.g.654.6		72
65.32	even	12		845.2.l.g.699.32		72
65.33	even	12		845.2.l.g.699.31		72
65.37	even	12		845.2.l.g.654.5		72
65.38	odd	4		845.2.b.g.339.3	✓	18
65.42	odd	12		845.2.n.h.529.16		36
65.43	odd	12		845.2.n.i.484.3		36
65.47	even	4		845.2.d.e.844.32		36
65.48	odd	12		845.2.n.h.484.16		36
65.57	even	4		845.2.d.e.844.6		36
65.58	even	12		845.2.l.g.699.5		72
65.62	odd	12		845.2.n.i.529.3		36
65.63	even	12		845.2.l.g.654.32		72
65.64	even	2		4225.2.a.ca.1.3		18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
845.2.b.g.339.3	✓	18	65.38	odd	4
845.2.b.g.339.16	yes	18	65.12	odd	4
845.2.b.h.339.3	yes	18	5.2	odd	4
845.2.b.h.339.16	yes	18	5.3	odd	4
845.2.d.e.844.5		36	65.8	even	4
845.2.d.e.844.6		36	65.57	even	4
845.2.d.e.844.31		36	65.18	even	4
845.2.d.e.844.32		36	65.47	even	4
845.2.l.g.654.5		72	65.37	even	12
845.2.l.g.654.6		72	65.28	even	12
845.2.l.g.654.31		72	65.2	even	12
845.2.l.g.654.32		72	65.63	even	12
845.2.l.g.699.5		72	65.58	even	12
845.2.l.g.699.6		72	65.7	even	12
845.2.l.g.699.31		72	65.33	even	12
845.2.l.g.699.32		72	65.32	even	12
845.2.n.h.484.3		36	65.22	odd	12
845.2.n.h.484.16		36	65.48	odd	12
845.2.n.h.529.3		36	65.3	odd	12
845.2.n.h.529.16		36	65.42	odd	12
845.2.n.i.484.3		36	65.43	odd	12
845.2.n.i.484.16		36	65.17	odd	12
845.2.n.i.529.3		36	65.62	odd	12
845.2.n.i.529.16		36	65.23	odd	12
4225.2.a.ca.1.3		18	65.64	even	2
4225.2.a.ca.1.16		18	13.12	even	2
4225.2.a.cb.1.3		18	1.1	even	1	trivial
4225.2.a.cb.1.16		18	5.4	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:	\( N \)	\(=\)	\( 4225 = 5^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4225.a (trivial)

\(f(q)\)	\(=\)	\(q-2.14790 q^{2} -2.37435 q^{3} +2.61347 q^{4} +5.09986 q^{6} -3.54879 q^{7} -1.31766 q^{8} +2.63754 q^{9} +2.08852 q^{11} -6.20528 q^{12} +7.62244 q^{14} -2.39672 q^{16} -6.00072 q^{17} -5.66516 q^{18} -5.04725 q^{19} +8.42607 q^{21} -4.48593 q^{22} -2.12673 q^{23} +3.12860 q^{24} +0.860615 q^{27} -9.27464 q^{28} -10.1431 q^{29} +3.10862 q^{31} +7.78325 q^{32} -4.95888 q^{33} +12.8889 q^{34} +6.89311 q^{36} +0.0998988 q^{37} +10.8410 q^{38} +3.79617 q^{41} -18.0983 q^{42} +2.11661 q^{43} +5.45829 q^{44} +4.56800 q^{46} -5.00374 q^{47} +5.69066 q^{48} +5.59390 q^{49} +14.2478 q^{51} -5.29557 q^{53} -1.84851 q^{54} +4.67611 q^{56} +11.9839 q^{57} +21.7863 q^{58} -3.00358 q^{59} -4.76673 q^{61} -6.67699 q^{62} -9.36006 q^{63} -11.9242 q^{64} +10.6512 q^{66} -6.77655 q^{67} -15.6827 q^{68} +5.04961 q^{69} -4.89593 q^{71} -3.47539 q^{72} -4.02118 q^{73} -0.214573 q^{74} -13.1908 q^{76} -7.41173 q^{77} -4.71456 q^{79} -9.95601 q^{81} -8.15379 q^{82} +11.2703 q^{83} +22.0212 q^{84} -4.54625 q^{86} +24.0832 q^{87} -2.75197 q^{88} +1.90467 q^{89} -5.55814 q^{92} -7.38094 q^{93} +10.7475 q^{94} -18.4802 q^{96} +5.36940 q^{97} -12.0151 q^{98} +5.50856 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q + 16 q^{4} + 16 q^{6} + 18 q^{9} + 22 q^{11} + 4 q^{14} - 12 q^{16} + 28 q^{19} + 26 q^{21} + 34 q^{24} - 20 q^{29} + 32 q^{31} + 18 q^{34} + 32 q^{36} + 52 q^{41} + 50 q^{44} + 30 q^{46} + 44 q^{49}+ \cdots + 76 q^{99}+O(q^{100}) \) 18 * q + 16 * q^4 + 16 * q^6 + 18 * q^9 + 22 * q^11 + 4 * q^14 - 12 * q^16 + 28 * q^19 + 26 * q^21 + 34 * q^24 - 20 * q^29 + 32 * q^31 + 18 * q^34 + 32 * q^36 + 52 * q^41 + 50 * q^44 + 30 * q^46 + 44 * q^49 - 40 * q^51 + 90 * q^54 - 20 * q^56 + 76 * q^59 + 8 * q^61 - 68 * q^64 + 8 * q^66 + 30 * q^69 + 72 * q^71 + 30 * q^74 + 4 * q^76 + 16 * q^79 - 30 * q^81 + 78 * q^84 - 30 * q^86 + 94 * q^89 - 128 * q^94 - 18 * q^96 + 76 * q^99

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