LMFDB - Embedded newform 7225.2.a.bs.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [7225,2,Mod(1,7225)] 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("7225.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(7225, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 7225 = 5^{2} \cdot 17^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7225.a (trivial)

Newform invariants

comment:select newform

sage:traces = [12,4,8,12,0,-8,16] 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:	$57.6919154604$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 4 x^{11} - 10 x^{10} + 52 x^{9} + 21 x^{8} - 232 x^{7} + 44 x^{6} + 424 x^{5} - 137 x^{4} + \cdots + 17 $ x^12 - 4x^11 - 10x^10 + 52x^9 + 21x^8 - 232x^7 + 44x^6 + 424x^5 - 137x^4 - 316x^3 + 46x^2 + 92*x + 17
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 85)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-0.747914$ of defining polynomial
Character	$\chi$	$=$	7225.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-0.747914 q^{2} +3.07503 q^{3} -1.44062 q^{4} -2.29986 q^{6} +3.23262 q^{7} +2.57329 q^{8} +6.45581 q^{9} +2.73509 q^{11} -4.42996 q^{12} -4.31833 q^{13} -2.41772 q^{14} +0.956646 q^{16} -4.82839 q^{18} +1.26892 q^{19} +9.94040 q^{21} -2.04561 q^{22} -0.492367 q^{23} +7.91295 q^{24} +3.22974 q^{26} +10.6267 q^{27} -4.65699 q^{28} -0.444360 q^{29} -5.52836 q^{31} -5.86207 q^{32} +8.41047 q^{33} -9.30039 q^{36} +10.6999 q^{37} -0.949042 q^{38} -13.2790 q^{39} +2.17048 q^{41} -7.43457 q^{42} +2.16182 q^{43} -3.94023 q^{44} +0.368248 q^{46} +8.39597 q^{47} +2.94172 q^{48} +3.44983 q^{49} +6.22109 q^{52} +1.81698 q^{53} -7.94786 q^{54} +8.31847 q^{56} +3.90196 q^{57} +0.332343 q^{58} -3.01987 q^{59} +12.2233 q^{61} +4.13474 q^{62} +20.8692 q^{63} +2.47104 q^{64} -6.29031 q^{66} -4.21389 q^{67} -1.51404 q^{69} +3.89165 q^{71} +16.6127 q^{72} -6.47062 q^{73} -8.00263 q^{74} -1.82803 q^{76} +8.84149 q^{77} +9.93155 q^{78} -7.22150 q^{79} +13.3100 q^{81} -1.62333 q^{82} -0.227499 q^{83} -14.3204 q^{84} -1.61686 q^{86} -1.36642 q^{87} +7.03818 q^{88} -13.3408 q^{89} -13.9595 q^{91} +0.709315 q^{92} -16.9999 q^{93} -6.27946 q^{94} -18.0260 q^{96} +14.8075 q^{97} -2.58018 q^{98} +17.6572 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 4 q^{2} + 8 q^{3} + 12 q^{4} - 8 q^{6} + 16 q^{7} + 12 q^{8} + 12 q^{9} - 16 q^{11} + 16 q^{12} + 8 q^{13} + 16 q^{14} + 12 q^{16} - 4 q^{18} + 16 q^{21} + 16 q^{22} + 16 q^{23} + 16 q^{26} + 32 q^{27}+ \cdots - 56 q^{99}+O(q^{100}) $ 12 * q + 4 * q^2 + 8 * q^3 + 12 * q^4 - 8 * q^6 + 16 * q^7 + 12 * q^8 + 12 * q^9 - 16 * q^11 + 16 * q^12 + 8 * q^13 + 16 * q^14 + 12 * q^16 - 4 * q^18 + 16 * q^21 + 16 * q^22 + 16 * q^23 + 16 * q^26 + 32 * q^27 + 40 * q^28 - 16 * q^29 - 24 * q^31 + 28 * q^32 + 12 * q^36 + 24 * q^37 + 24 * q^38 - 8 * q^39 - 8 * q^41 + 16 * q^43 - 8 * q^44 - 40 * q^46 + 32 * q^47 - 24 * q^48 + 20 * q^49 + 24 * q^52 - 8 * q^54 + 24 * q^56 + 32 * q^57 - 16 * q^58 + 8 * q^59 - 24 * q^61 + 8 * q^62 + 48 * q^63 + 36 * q^64 + 40 * q^66 + 8 * q^67 + 48 * q^69 + 16 * q^71 - 12 * q^72 + 16 * q^73 + 16 * q^76 - 24 * q^77 - 24 * q^78 - 40 * q^79 + 36 * q^81 - 16 * q^82 + 40 * q^83 + 32 * q^84 - 8 * q^86 + 32 * q^87 + 48 * q^88 + 8 * q^89 - 72 * q^91 - 8 * q^92 - 24 * q^93 - 16 * q^94 - 8 * q^96 + 32 * q^97 + 60 * q^98 - 56 * 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 $

$2$	−0.747914	−0.528855	−0.264428	−	0.964406i	$-0.585183\pi$
$2$	−0.747914	−0.528855	−0.264428	+	0.964406i	$0.585183\pi$
$3$	3.07503	1.77537	0.887684	−	0.460452i	$-0.152313\pi$
$3$	3.07503	1.77537	0.887684	+	0.460452i	$0.152313\pi$
$4$	−1.44062	−0.720312
$5$	0	0
$5$	0	0
$6$	−2.29986	−0.938913
$7$	3.23262	1.22182	0.610908	−	0.791702i	$-0.290805\pi$
$7$	3.23262	1.22182	0.610908	+	0.791702i	$0.290805\pi$
$8$	2.57329	0.909796
$9$	6.45581	2.15194
$10$	0	0
$11$	2.73509	0.824660	0.412330	−	0.911035i	$-0.364715\pi$
$11$	2.73509	0.824660	0.412330	+	0.911035i	$0.364715\pi$
$12$	−4.42996	−1.27882
$13$	−4.31833	−1.19769	−0.598845	−	0.800865i	$-0.704373\pi$
$13$	−4.31833	−1.19769	−0.598845	+	0.800865i	$0.704373\pi$
$14$	−2.41772	−0.646164
$15$	0	0
$16$	0.956646	0.239162
$17$	0	0
$17$	0	0
$18$	−4.82839	−1.13806
$19$	1.26892	0.291110	0.145555	−	0.989350i	$-0.453503\pi$
$19$	1.26892	0.291110	0.145555	+	0.989350i	$0.453503\pi$
$20$	0	0
$21$	9.94040	2.16917
$22$	−2.04561	−0.436126
$23$	−0.492367	−0.102666	−0.0513328	−	0.998682i	$-0.516347\pi$
$23$	−0.492367	−0.102666	−0.0513328	+	0.998682i	$0.516347\pi$
$24$	7.91295	1.61522
$25$	0	0
$26$	3.22974	0.633404
$27$	10.6267	2.04511
$28$	−4.65699	−0.880088
$29$	−0.444360	−0.0825156	−0.0412578	−	0.999149i	$-0.513137\pi$
$29$	−0.444360	−0.0825156	−0.0412578	+	0.999149i	$0.513137\pi$
$30$	0	0
$31$	−5.52836	−0.992923	−0.496462	−	0.868059i	$-0.665368\pi$
$31$	−5.52836	−0.992923	−0.496462	+	0.868059i	$0.665368\pi$
$32$	−5.86207	−1.03628
$33$	8.41047	1.46408
$34$	0	0
$35$	0	0
$36$	−9.30039	−1.55006
$37$	10.6999	1.75906	0.879529	−	0.475845i	$-0.157858\pi$
$37$	10.6999	1.75906	0.879529	+	0.475845i	$0.157858\pi$
$38$	−0.949042	−0.153955
$39$	−13.2790	−2.12634
$40$	0	0
$41$	2.17048	0.338972	0.169486	−	0.985533i	$-0.445789\pi$
$41$	2.17048	0.338972	0.169486	+	0.985533i	$0.445789\pi$
$42$	−7.43457	−1.14718
$43$	2.16182	0.329675	0.164837	−	0.986321i	$-0.447290\pi$
$43$	2.16182	0.329675	0.164837	+	0.986321i	$0.447290\pi$
$44$	−3.94023	−0.594012
$45$	0	0
$46$	0.368248	0.0542952
$47$	8.39597	1.22468	0.612339	−	0.790595i	$-0.290229\pi$
$47$	8.39597	1.22468	0.612339	+	0.790595i	$0.290229\pi$
$48$	2.94172	0.424600
$49$	3.44983	0.492833
$50$	0	0
$51$	0	0
$52$	6.22109	0.862710
$53$	1.81698	0.249581	0.124791	−	0.992183i	$-0.460174\pi$
$53$	1.81698	0.249581	0.124791	+	0.992183i	$0.460174\pi$
$54$	−7.94786	−1.08157
$55$	0	0
$56$	8.31847	1.11160
$57$	3.90196	0.516827
$58$	0.332343	0.0436388
$59$	−3.01987	−0.393153	−0.196577	−	0.980488i	$-0.562982\pi$
$59$	−3.01987	−0.393153	−0.196577	+	0.980488i	$0.562982\pi$
$60$	0	0
$61$	12.2233	1.56503	0.782514	−	0.622633i	$-0.213937\pi$
$61$	12.2233	1.56503	0.782514	+	0.622633i	$0.213937\pi$
$62$	4.13474	0.525113
$63$	20.8692	2.62927
$64$	2.47104	0.308880
$65$	0	0
$66$	−6.29031	−0.774284
$67$	−4.21389	−0.514808	−0.257404	−	0.966304i	$-0.582867\pi$
$67$	−4.21389	−0.514808	−0.257404	+	0.966304i	$0.582867\pi$
$68$	0	0
$69$	−1.51404	−0.182269
$70$	0	0
$71$	3.89165	0.461854	0.230927	−	0.972971i	$-0.425824\pi$
$71$	3.89165	0.461854	0.230927	+	0.972971i	$0.425824\pi$
$72$	16.6127	1.95782
$73$	−6.47062	−0.757329	−0.378664	−	0.925534i	$-0.623617\pi$
$73$	−6.47062	−0.757329	−0.378664	+	0.925534i	$0.623617\pi$
$74$	−8.00263	−0.930287
$75$	0	0
$76$	−1.82803	−0.209690
$77$	8.84149	1.00758
$78$	9.93155	1.12453
$79$	−7.22150	−0.812482	−0.406241	−	0.913766i	$-0.633161\pi$
$79$	−7.22150	−0.812482	−0.406241	+	0.913766i	$0.633161\pi$
$80$	0	0
$81$	13.3100	1.47889
$82$	−1.62333	−0.179267
$83$	−0.227499	−0.0249713	−0.0124856	−	0.999922i	$-0.503974\pi$
$83$	−0.227499	−0.0249713	−0.0124856	+	0.999922i	$0.503974\pi$
$84$	−14.3204	−1.56248
$85$	0	0
$86$	−1.61686	−0.174350
$87$	−1.36642	−0.146496
$88$	7.03818	0.750272
$89$	−13.3408	−1.41413	−0.707064	−	0.707150i	$-0.749981\pi$
$89$	−13.3408	−1.41413	−0.707064	+	0.707150i	$0.749981\pi$
$90$	0	0
$91$	−13.9595	−1.46336
$92$	0.709315	0.0739512
$93$	−16.9999	−1.76281
$94$	−6.27946	−0.647677
$95$	0	0
$96$	−18.0260	−1.83978
$97$	14.8075	1.50347	0.751734	−	0.659466i	$-0.229218\pi$
$97$	14.8075	1.50347	0.751734	+	0.659466i	$0.229218\pi$
$98$	−2.58018	−0.260637
$99$	17.6572	1.77461
$100$	0	0
$101$	0.284213	0.0282803	0.0141401	−	0.999900i	$-0.495499\pi$
$101$	0.284213	0.0282803	0.0141401	+	0.999900i	$0.495499\pi$
$102$	0	0
$103$	14.0842	1.38775	0.693877	−	0.720093i	$-0.255901\pi$
$103$	14.0842	1.38775	0.693877	+	0.720093i	$0.255901\pi$
$104$	−11.1123	−1.08965
$105$	0	0
$106$	−1.35894	−0.131992
$107$	18.7099	1.80875	0.904375	−	0.426738i	$-0.140337\pi$
$107$	18.7099	1.80875	0.904375	+	0.426738i	$0.140337\pi$
$108$	−15.3091	−1.47312
$109$	3.53078	0.338187	0.169094	−	0.985600i	$-0.445916\pi$
$109$	3.53078	0.338187	0.169094	+	0.985600i	$0.445916\pi$
$110$	0	0
$111$	32.9026	3.12298
$112$	3.09247	0.292211
$113$	−2.84155	−0.267310	−0.133655	−	0.991028i	$-0.542671\pi$
$113$	−2.84155	−0.267310	−0.133655	+	0.991028i	$0.542671\pi$
$114$	−2.91833	−0.273327
$115$	0	0
$116$	0.640156	0.0594370
$117$	−27.8783	−2.57735
$118$	2.25860	0.207921
$119$	0	0
$120$	0	0
$121$	−3.51930	−0.319937
$122$	−9.14195	−0.827673
$123$	6.67430	0.601801
$124$	7.96429	0.715215
$125$	0	0
$126$	−15.6083	−1.39050
$127$	−5.46515	−0.484954	−0.242477	−	0.970157i	$-0.577960\pi$
$127$	−5.46515	−0.484954	−0.242477	+	0.970157i	$0.577960\pi$
$128$	9.87602	0.872925
$129$	6.64767	0.585295
$130$	0	0
$131$	9.97951	0.871914	0.435957	−	0.899968i	$-0.356410\pi$
$131$	9.97951	0.871914	0.435957	+	0.899968i	$0.356410\pi$
$132$	−12.1163	−1.05459
$133$	4.10193	0.355682
$134$	3.15163	0.272259
$135$	0	0
$136$	0	0
$137$	3.07772	0.262947	0.131474	−	0.991320i	$-0.458029\pi$
$137$	3.07772	0.262947	0.131474	+	0.991320i	$0.458029\pi$
$138$	1.13237	0.0963941
$139$	6.70991	0.569127	0.284564	−	0.958657i	$-0.408151\pi$
$139$	6.70991	0.569127	0.284564	+	0.958657i	$0.408151\pi$
$140$	0	0
$141$	25.8178	2.17425
$142$	−2.91062	−0.244254
$143$	−11.8110	−0.987686
$144$	6.17592	0.514660
$145$	0	0
$146$	4.83947	0.400517
$147$	10.6083	0.874960
$148$	−15.4146	−1.26707
$149$	−22.9914	−1.88353	−0.941764	−	0.336276i	$-0.890833\pi$
$149$	−22.9914	−1.88353	−0.941764	+	0.336276i	$0.890833\pi$
$150$	0	0
$151$	0.195743	0.0159294	0.00796468	−	0.999968i	$-0.497465\pi$
$151$	0.195743	0.0159294	0.00796468	+	0.999968i	$0.497465\pi$
$152$	3.26530	0.264850
$153$	0	0
$154$	−6.61268	−0.532865
$155$	0	0
$156$	19.1300	1.53163
$157$	−11.8582	−0.946391	−0.473196	−	0.880957i	$-0.656900\pi$
$157$	−11.8582	−0.946391	−0.473196	+	0.880957i	$0.656900\pi$
$158$	5.40106	0.429685
$159$	5.58726	0.443099
$160$	0	0
$161$	−1.59163	−0.125438
$162$	−9.95474	−0.782118
$163$	5.34472	0.418631	0.209315	−	0.977848i	$-0.432877\pi$
$163$	5.34472	0.418631	0.209315	+	0.977848i	$0.432877\pi$
$164$	−3.12685	−0.244166
$165$	0	0
$166$	0.170150	0.0132062
$167$	11.5900	0.896859	0.448429	−	0.893818i	$-0.351984\pi$
$167$	11.5900	0.896859	0.448429	+	0.893818i	$0.351984\pi$
$168$	25.5796	1.97351
$169$	5.64797	0.434459
$170$	0	0
$171$	8.19188	0.626449
$172$	−3.11437	−0.237469
$173$	11.0271	0.838373	0.419187	−	0.907900i	$-0.362315\pi$
$173$	11.0271	0.838373	0.419187	+	0.907900i	$0.362315\pi$
$174$	1.02197	0.0774750
$175$	0	0
$176$	2.61651	0.197227
$177$	−9.28618	−0.697992
$178$	9.97781	0.747869
$179$	−21.0731	−1.57508	−0.787539	−	0.616265i	$-0.788645\pi$
$179$	−21.0731	−1.57508	−0.787539	+	0.616265i	$0.788645\pi$
$180$	0	0
$181$	22.6592	1.68425	0.842123	−	0.539285i	$-0.181305\pi$
$181$	22.6592	1.68425	0.842123	+	0.539285i	$0.181305\pi$
$182$	10.4405	0.773903
$183$	37.5869	2.77850
$184$	−1.26700	−0.0934047
$185$	0	0
$186$	12.7145	0.932269
$187$	0	0
$188$	−12.0954	−0.882150
$189$	34.3521	2.49875
$190$	0	0
$191$	−3.50162	−0.253368	−0.126684	−	0.991943i	$-0.540433\pi$
$191$	−3.50162	−0.253368	−0.126684	+	0.991943i	$0.540433\pi$
$192$	7.59851	0.548375
$193$	−12.2942	−0.884959	−0.442479	−	0.896779i	$-0.645901\pi$
$193$	−12.2942	−0.884959	−0.442479	+	0.896779i	$0.645901\pi$
$194$	−11.0747	−0.795117
$195$	0	0
$196$	−4.96991	−0.354993
$197$	−16.9757	−1.20947	−0.604734	−	0.796427i	$-0.706721\pi$
$197$	−16.9757	−1.20947	−0.604734	+	0.796427i	$0.706721\pi$
$198$	−13.2061	−0.938514
$199$	11.5779	0.820735	0.410367	−	0.911920i	$-0.365400\pi$
$199$	11.5779	0.820735	0.410367	+	0.911920i	$0.365400\pi$
$200$	0	0
$201$	−12.9578	−0.913975
$202$	−0.212567	−0.0149562
$203$	−1.43645	−0.100819
$204$	0	0
$205$	0	0
$206$	−10.5338	−0.733921
$207$	−3.17862	−0.220930
$208$	−4.13111	−0.286441
$209$	3.47060	0.240066
$210$	0	0
$211$	−6.66380	−0.458755	−0.229378	−	0.973338i	$-0.573669\pi$
$211$	−6.66380	−0.458755	−0.229378	+	0.973338i	$0.573669\pi$
$212$	−2.61758	−0.179776
$213$	11.9669	0.819962
$214$	−13.9934	−0.956567
$215$	0	0
$216$	27.3456	1.86063
$217$	−17.8711	−1.21317
$218$	−2.64072	−0.178852
$219$	−19.8973	−1.34454
$220$	0	0
$221$	0	0
$222$	−24.6083	−1.65160
$223$	12.5928	0.843273	0.421637	−	0.906765i	$-0.361456\pi$
$223$	12.5928	0.843273	0.421637	+	0.906765i	$0.361456\pi$
$224$	−18.9499	−1.26614
$225$	0	0
$226$	2.12523	0.141368
$227$	8.04265	0.533809	0.266905	−	0.963723i	$-0.413999\pi$
$227$	8.04265	0.533809	0.266905	+	0.963723i	$0.413999\pi$
$228$	−5.62126	−0.372277
$229$	−21.6295	−1.42932	−0.714658	−	0.699474i	$-0.753418\pi$
$229$	−21.6295	−1.42932	−0.714658	+	0.699474i	$0.753418\pi$
$230$	0	0
$231$	27.1879	1.78883
$232$	−1.14347	−0.0750724
$233$	21.9506	1.43803	0.719014	−	0.694995i	$-0.244594\pi$
$233$	21.9506	1.43803	0.719014	+	0.694995i	$0.244594\pi$
$234$	20.8506	1.36304
$235$	0	0
$236$	4.35050	0.283193
$237$	−22.2063	−1.44246
$238$	0	0
$239$	5.90132	0.381725	0.190862	−	0.981617i	$-0.438872\pi$
$239$	5.90132	0.381725	0.190862	+	0.981617i	$0.438872\pi$
$240$	0	0
$241$	−13.3115	−0.857466	−0.428733	−	0.903431i	$-0.641040\pi$
$241$	−13.3115	−0.857466	−0.428733	+	0.903431i	$0.641040\pi$
$242$	2.63214	0.169200
$243$	9.04855	0.580464
$244$	−17.6091	−1.12731
$245$	0	0
$246$	−4.99180	−0.318266
$247$	−5.47960	−0.348659
$248$	−14.2261	−0.903358
$249$	−0.699566	−0.0443332
$250$	0	0
$251$	3.59367	0.226831	0.113415	−	0.993548i	$-0.463821\pi$
$251$	3.59367	0.226831	0.113415	+	0.993548i	$0.463821\pi$
$252$	−30.0646	−1.89389
$253$	−1.34667	−0.0846641
$254$	4.08746	0.256470
$255$	0	0
$256$	−12.3285	−0.770531
$257$	−24.5491	−1.53133	−0.765665	−	0.643239i	$-0.777590\pi$
$257$	−24.5491	−1.53133	−0.765665	+	0.643239i	$0.777590\pi$
$258$	−4.97189	−0.309536
$259$	34.5888	2.14924
$260$	0	0
$261$	−2.86870	−0.177568
$262$	−7.46382	−0.461116
$263$	−14.6401	−0.902749	−0.451375	−	0.892335i	$-0.649066\pi$
$263$	−14.6401	−0.902749	−0.451375	+	0.892335i	$0.649066\pi$
$264$	21.6426	1.33201
$265$	0	0
$266$	−3.06789	−0.188104
$267$	−41.0235	−2.51060
$268$	6.07063	0.370823
$269$	−18.7597	−1.14380	−0.571900	−	0.820323i	$-0.693794\pi$
$269$	−18.7597	−1.14380	−0.571900	+	0.820323i	$0.693794\pi$
$270$	0	0
$271$	−5.76388	−0.350131	−0.175065	−	0.984557i	$-0.556014\pi$
$271$	−5.76388	−0.350131	−0.175065	+	0.984557i	$0.556014\pi$
$272$	0	0
$273$	−42.9259	−2.59800
$274$	−2.30187	−0.139061
$275$	0	0
$276$	2.18117	0.131291
$277$	0.123252	0.00740547	0.00370273	−	0.999993i	$-0.498821\pi$
$277$	0.123252	0.00740547	0.00370273	+	0.999993i	$0.498821\pi$
$278$	−5.01844	−0.300986
$279$	−35.6900	−2.13671
$280$	0	0
$281$	1.82363	0.108789	0.0543944	−	0.998520i	$-0.482677\pi$
$281$	1.82363	0.108789	0.0543944	+	0.998520i	$0.482677\pi$
$282$	−19.3095	−1.14987
$283$	−28.6278	−1.70174	−0.850872	−	0.525373i	$-0.823926\pi$
$283$	−28.6278	−1.70174	−0.850872	+	0.525373i	$0.823926\pi$
$284$	−5.60641	−0.332679
$285$	0	0
$286$	8.83362	0.522343
$287$	7.01634	0.414162
$288$	−37.8444	−2.23000
$289$	0	0
$290$	0	0
$291$	45.5333	2.66921
$292$	9.32173	0.545513
$293$	1.41607	0.0827278	0.0413639	−	0.999144i	$-0.486830\pi$
$293$	1.41607	0.0827278	0.0413639	+	0.999144i	$0.486830\pi$
$294$	−7.93412	−0.462727
$295$	0	0
$296$	27.5340	1.60038
$297$	29.0649	1.68652
$298$	17.1956	0.996113
$299$	2.12620	0.122961
$300$	0	0
$301$	6.98835	0.402802
$302$	−0.146399	−0.00842433
$303$	0.873965	0.0502080
$304$	1.21391	0.0696222
$305$	0	0
$306$	0	0
$307$	21.7364	1.24056	0.620281	−	0.784379i	$-0.287018\pi$
$307$	21.7364	1.24056	0.620281	+	0.784379i	$0.287018\pi$
$308$	−12.7373	−0.725773
$309$	43.3092	2.46378
$310$	0	0
$311$	−8.65102	−0.490554	−0.245277	−	0.969453i	$-0.578879\pi$
$311$	−8.65102	−0.490554	−0.245277	+	0.969453i	$0.578879\pi$
$312$	−34.1707	−1.93454
$313$	10.2820	0.581175	0.290587	−	0.956848i	$-0.406149\pi$
$313$	10.2820	0.581175	0.290587	+	0.956848i	$0.406149\pi$
$314$	8.86896	0.500504
$315$	0	0
$316$	10.4035	0.585241
$317$	1.12766	0.0633359	0.0316680	−	0.999498i	$-0.489918\pi$
$317$	1.12766	0.0633359	0.0316680	+	0.999498i	$0.489918\pi$
$318$	−4.17879	−0.234335
$319$	−1.21536	−0.0680473
$320$	0	0
$321$	57.5334	3.21120
$322$	1.19041	0.0663387
$323$	0	0
$324$	−19.1747	−1.06526
$325$	0	0
$326$	−3.99739	−0.221395
$327$	10.8573	0.600407
$328$	5.58528	0.308396
$329$	27.1410	1.49633
$330$	0	0
$331$	−34.2599	−1.88310	−0.941548	−	0.336878i	$-0.890629\pi$
$331$	−34.2599	−1.88310	−0.941548	+	0.336878i	$0.890629\pi$
$332$	0.327741	0.0179871
$333$	69.0767	3.78538
$334$	−8.66830	−0.474309
$335$	0	0
$336$	9.50945	0.518783
$337$	13.7463	0.748806	0.374403	−	0.927266i	$-0.377848\pi$
$337$	13.7463	0.748806	0.374403	+	0.927266i	$0.377848\pi$
$338$	−4.22420	−0.229766
$339$	−8.73784	−0.474574
$340$	0	0
$341$	−15.1206	−0.818824
$342$	−6.12683	−0.331301
$343$	−11.4763	−0.619665
$344$	5.56300	0.299937
$345$	0	0
$346$	−8.24731	−0.443378
$347$	18.8786	1.01346	0.506728	−	0.862106i	$-0.330855\pi$
$347$	18.8786	1.01346	0.506728	+	0.862106i	$0.330855\pi$
$348$	1.96850	0.105523
$349$	−7.04988	−0.377371	−0.188686	−	0.982038i	$-0.560423\pi$
$349$	−7.04988	−0.377371	−0.188686	+	0.982038i	$0.560423\pi$
$350$	0	0
$351$	−45.8896	−2.44941
$352$	−16.0333	−0.854577
$353$	−8.60779	−0.458146	−0.229073	−	0.973409i	$-0.573569\pi$
$353$	−8.60779	−0.458146	−0.229073	+	0.973409i	$0.573569\pi$
$354$	6.94527	0.369137
$355$	0	0
$356$	19.2191	1.01861
$357$	0	0
$358$	15.7609	0.832988
$359$	9.18100	0.484555	0.242277	−	0.970207i	$-0.422106\pi$
$359$	9.18100	0.484555	0.242277	+	0.970207i	$0.422106\pi$
$360$	0	0
$361$	−17.3898	−0.915255
$362$	−16.9472	−0.890723
$363$	−10.8220	−0.568005
$364$	20.1104	1.05407
$365$	0	0
$366$	−28.1118	−1.46943
$367$	6.81607	0.355796	0.177898	−	0.984049i	$-0.443070\pi$
$367$	6.81607	0.355796	0.177898	+	0.984049i	$0.443070\pi$
$368$	−0.471021	−0.0245537
$369$	14.0122	0.729446
$370$	0	0
$371$	5.87360	0.304942
$372$	24.4904	1.26977
$373$	10.4647	0.541841	0.270920	−	0.962602i	$-0.412672\pi$
$373$	10.4647	0.541841	0.270920	+	0.962602i	$0.412672\pi$
$374$	0	0
$375$	0	0
$376$	21.6053	1.11421
$377$	1.91889	0.0988281
$378$	−25.6924	−1.32148
$379$	−25.3419	−1.30173	−0.650864	−	0.759195i	$-0.725593\pi$
$379$	−25.3419	−1.30173	−0.650864	+	0.759195i	$0.725593\pi$
$380$	0	0
$381$	−16.8055	−0.860972
$382$	2.61891	0.133995
$383$	−7.69048	−0.392965	−0.196483	−	0.980507i	$-0.562952\pi$
$383$	−7.69048	−0.392965	−0.196483	+	0.980507i	$0.562952\pi$
$384$	30.3691	1.54976
$385$	0	0
$386$	9.19504	0.468015
$387$	13.9563	0.709439
$388$	−21.3320	−1.08297
$389$	15.8379	0.803013	0.401506	−	0.915856i	$-0.368487\pi$
$389$	15.8379	0.803013	0.401506	+	0.915856i	$0.368487\pi$
$390$	0	0
$391$	0	0
$392$	8.87742	0.448377
$393$	30.6873	1.54797
$394$	12.6964	0.639634
$395$	0	0
$396$	−25.4374	−1.27828
$397$	29.1640	1.46370	0.731850	−	0.681466i	$-0.238657\pi$
$397$	29.1640	1.46370	0.731850	+	0.681466i	$0.238657\pi$
$398$	−8.65927	−0.434050
$399$	12.6135	0.631467
$400$	0	0
$401$	−16.3859	−0.818271	−0.409136	−	0.912474i	$-0.634170\pi$
$401$	−16.3859	−0.818271	−0.409136	+	0.912474i	$0.634170\pi$
$402$	9.69135	0.483360
$403$	23.8733	1.18921
$404$	−0.409445	−0.0203706
$405$	0	0
$406$	1.07434	0.0533186
$407$	29.2652	1.45062
$408$	0	0
$409$	33.9971	1.68105	0.840525	−	0.541772i	$-0.182247\pi$
$409$	33.9971	1.68105	0.840525	+	0.541772i	$0.182247\pi$
$410$	0	0
$411$	9.46408	0.466829
$412$	−20.2900	−0.999616
$413$	−9.76209	−0.480361
$414$	2.37734	0.116840
$415$	0	0
$416$	25.3144	1.24114
$417$	20.6332	1.01041
$418$	−2.59571	−0.126960
$419$	0.892935	0.0436227	0.0218114	−	0.999762i	$-0.493057\pi$
$419$	0.892935	0.0436227	0.0218114	+	0.999762i	$0.493057\pi$
$420$	0	0
$421$	33.6725	1.64110	0.820550	−	0.571575i	$-0.193668\pi$
$421$	33.6725	1.64110	0.820550	+	0.571575i	$0.193668\pi$
$422$	4.98395	0.242615
$423$	54.2027	2.63543
$424$	4.67562	0.227068
$425$	0	0
$426$	−8.95025	−0.433641
$427$	39.5131	1.91218
$428$	−26.9539	−1.30286
$429$	−36.3192	−1.75351
$430$	0	0
$431$	16.2970	0.784997	0.392498	−	0.919753i	$-0.371611\pi$
$431$	16.2970	0.784997	0.392498	+	0.919753i	$0.371611\pi$
$432$	10.1660	0.489112
$433$	34.9947	1.68174	0.840869	−	0.541239i	$-0.182044\pi$
$433$	34.9947	1.68174	0.840869	+	0.541239i	$0.182044\pi$
$434$	13.3660	0.641591
$435$	0	0
$436$	−5.08653	−0.243600
$437$	−0.624773	−0.0298869
$438$	14.8815	0.711066
$439$	22.2777	1.06326	0.531628	−	0.846978i	$-0.321580\pi$
$439$	22.2777	1.06326	0.531628	+	0.846978i	$0.321580\pi$
$440$	0	0
$441$	22.2714	1.06054
$442$	0	0
$443$	8.79907	0.418056	0.209028	−	0.977910i	$-0.432970\pi$
$443$	8.79907	0.418056	0.209028	+	0.977910i	$0.432970\pi$
$444$	−47.4003	−2.24952
$445$	0	0
$446$	−9.41830	−0.445970
$447$	−70.6992	−3.34396
$448$	7.98792	0.377394
$449$	8.86683	0.418452	0.209226	−	0.977867i	$-0.432906\pi$
$449$	8.86683	0.418452	0.209226	+	0.977867i	$0.432906\pi$
$450$	0	0
$451$	5.93646	0.279537
$452$	4.09360	0.192547
$453$	0.601916	0.0282805
$454$	−6.01521	−0.282308
$455$	0	0
$456$	10.0409	0.470207
$457$	18.4756	0.864254	0.432127	−	0.901813i	$-0.357763\pi$
$457$	18.4756	0.864254	0.432127	+	0.901813i	$0.357763\pi$
$458$	16.1770	0.755902
$459$	0	0
$460$	0	0
$461$	−24.5700	−1.14434	−0.572169	−	0.820136i	$-0.693898\pi$
$461$	−24.5700	−1.14434	−0.572169	+	0.820136i	$0.693898\pi$
$462$	−20.3342	−0.946032
$463$	9.90931	0.460525	0.230262	−	0.973129i	$-0.426042\pi$
$463$	9.90931	0.460525	0.230262	+	0.973129i	$0.426042\pi$
$464$	−0.425096	−0.0197346
$465$	0	0
$466$	−16.4171	−0.760509
$467$	29.4041	1.36066	0.680329	−	0.732907i	$-0.261837\pi$
$467$	29.4041	1.36066	0.680329	+	0.732907i	$0.261837\pi$
$468$	40.1621	1.85650
$469$	−13.6219	−0.629001
$470$	0	0
$471$	−36.4645	−1.68019
$472$	−7.77100	−0.357689
$473$	5.91277	0.271870
$474$	16.6084	0.762850
$475$	0	0
$476$	0	0
$477$	11.7301	0.537083
$478$	−4.41369	−0.201877
$479$	28.6003	1.30678	0.653391	−	0.757021i	$-0.273346\pi$
$479$	28.6003	1.30678	0.653391	+	0.757021i	$0.273346\pi$
$480$	0	0
$481$	−46.2058	−2.10680
$482$	9.95583	0.453476
$483$	−4.89432	−0.222699
$484$	5.06999	0.230454
$485$	0	0
$486$	−6.76754	−0.306982
$487$	0.617883	0.0279990	0.0139995	−	0.999902i	$-0.495544\pi$
$487$	0.617883	0.0279990	0.0139995	+	0.999902i	$0.495544\pi$
$488$	31.4540	1.42386
$489$	16.4352	0.743224
$490$	0	0
$491$	25.4667	1.14930	0.574648	−	0.818401i	$-0.305139\pi$
$491$	25.4667	1.14930	0.574648	+	0.818401i	$0.305139\pi$
$492$	−9.61515	−0.433485
$493$	0	0
$494$	4.09827	0.184390
$495$	0	0
$496$	−5.28869	−0.237469
$497$	12.5802	0.564301
$498$	0.523216	0.0234459
$499$	−36.4603	−1.63219	−0.816093	−	0.577921i	$-0.803864\pi$
$499$	−36.4603	−1.63219	−0.816093	+	0.577921i	$0.803864\pi$
$500$	0	0
$501$	35.6395	1.59226
$502$	−2.68776	−0.119961
$503$	−30.1881	−1.34602	−0.673011	−	0.739632i	$-0.734999\pi$
$503$	−30.1881	−1.34602	−0.673011	+	0.739632i	$0.734999\pi$
$504$	53.7025	2.39210
$505$	0	0
$506$	1.00719	0.0447751
$507$	17.3677	0.771326
$508$	7.87323	0.349318
$509$	40.1857	1.78120	0.890600	−	0.454787i	$-0.150285\pi$
$509$	40.1857	1.78120	0.890600	+	0.454787i	$0.150285\pi$
$510$	0	0
$511$	−20.9171	−0.925316
$512$	−10.5314	−0.465426
$513$	13.4844	0.595351
$514$	18.3606	0.809852
$515$	0	0
$516$	−9.57679	−0.421595
$517$	22.9637	1.00994
$518$	−25.8695	−1.13664
$519$	33.9086	1.48842
$520$	0	0
$521$	−35.3560	−1.54897	−0.774487	−	0.632590i	$-0.781992\pi$
$521$	−35.3560	−1.54897	−0.774487	+	0.632590i	$0.781992\pi$
$522$	2.14554	0.0939079
$523$	−24.5035	−1.07146	−0.535732	−	0.844388i	$-0.679964\pi$
$523$	−24.5035	−1.07146	−0.535732	+	0.844388i	$0.679964\pi$
$524$	−14.3767	−0.628050
$525$	0	0
$526$	10.9496	0.477424
$527$	0	0
$528$	8.04585	0.350150
$529$	−22.7576	−0.989460
$530$	0	0
$531$	−19.4957	−0.846041
$532$	−5.90934	−0.256202
$533$	−9.37286	−0.405984
$534$	30.6821	1.32774
$535$	0	0
$536$	−10.8436	−0.468371
$537$	−64.8004	−2.79634
$538$	14.0307	0.604905
$539$	9.43558	0.406419
$540$	0	0
$541$	25.5508	1.09851	0.549257	−	0.835653i	$-0.314911\pi$
$541$	25.5508	1.09851	0.549257	+	0.835653i	$0.314911\pi$
$542$	4.31089	0.185169
$543$	69.6778	2.99016
$544$	0	0
$545$	0	0
$546$	32.1049	1.37396
$547$	−11.7618	−0.502896	−0.251448	−	0.967871i	$-0.580907\pi$
$547$	−11.7618	−0.502896	−0.251448	+	0.967871i	$0.580907\pi$
$548$	−4.43384	−0.189404
$549$	78.9110	3.36784
$550$	0	0
$551$	−0.563857	−0.0240211
$552$	−3.89607	−0.165828
$553$	−23.3444	−0.992703
$554$	−0.0921816	−0.00391642
$555$	0	0
$556$	−9.66646	−0.409949
$557$	35.6812	1.51186	0.755931	−	0.654651i	$-0.227184\pi$
$557$	35.6812	1.51186	0.755931	+	0.654651i	$0.227184\pi$
$558$	26.6931	1.13001
$559$	−9.33546	−0.394848
$560$	0	0
$561$	0	0
$562$	−1.36392	−0.0575335
$563$	−30.1232	−1.26954	−0.634771	−	0.772700i	$-0.718906\pi$
$563$	−30.1232	−1.26954	−0.634771	+	0.772700i	$0.718906\pi$
$564$	−37.1938	−1.56614
$565$	0	0
$566$	21.4111	0.899976
$567$	43.0262	1.80693
$568$	10.0144	0.420193
$569$	−24.3985	−1.02284	−0.511420	−	0.859331i	$-0.670880\pi$
$569$	−24.3985	−1.02284	−0.511420	+	0.859331i	$0.670880\pi$
$570$	0	0
$571$	−33.2818	−1.39280	−0.696399	−	0.717654i	$-0.745216\pi$
$571$	−33.2818	−1.39280	−0.696399	+	0.717654i	$0.745216\pi$
$572$	17.0152	0.711442
$573$	−10.7676	−0.449822
$574$	−5.24762	−0.219032
$575$	0	0
$576$	15.9525	0.664689
$577$	−18.0611	−0.751895	−0.375947	−	0.926641i	$-0.622683\pi$
$577$	−18.0611	−0.751895	−0.375947	+	0.926641i	$0.622683\pi$
$578$	0	0
$579$	−37.8051	−1.57113
$580$	0	0
$581$	−0.735418	−0.0305103
$582$	−34.0550	−1.41163
$583$	4.96959	0.205820
$584$	−16.6508	−0.689015
$585$	0	0
$586$	−1.05910	−0.0437510
$587$	−41.8303	−1.72652	−0.863261	−	0.504758i	$-0.831582\pi$
$587$	−41.8303	−1.72652	−0.863261	+	0.504758i	$0.831582\pi$
$588$	−15.2826	−0.630244
$589$	−7.01504	−0.289050
$590$	0	0
$591$	−52.2008	−2.14725
$592$	10.2360	0.420699
$593$	−20.6659	−0.848648	−0.424324	−	0.905510i	$-0.639488\pi$
$593$	−20.6659	−0.848648	−0.424324	+	0.905510i	$0.639488\pi$
$594$	−21.7381	−0.891925
$595$	0	0
$596$	33.1219	1.35673
$597$	35.6023	1.45711
$598$	−1.59022	−0.0650288
$599$	9.21817	0.376644	0.188322	−	0.982107i	$-0.439695\pi$
$599$	9.21817	0.376644	0.188322	+	0.982107i	$0.439695\pi$
$600$	0	0
$601$	1.62070	0.0661098	0.0330549	−	0.999454i	$-0.489476\pi$
$601$	1.62070	0.0661098	0.0330549	+	0.999454i	$0.489476\pi$
$602$	−5.22669	−0.213024
$603$	−27.2040	−1.10783
$604$	−0.281993	−0.0114741
$605$	0	0
$606$	−0.653651	−0.0265527
$607$	−25.0157	−1.01536	−0.507679	−	0.861547i	$-0.669496\pi$
$607$	−25.0157	−1.01536	−0.507679	+	0.861547i	$0.669496\pi$
$608$	−7.43849	−0.301671
$609$	−4.41712	−0.178991
$610$	0	0
$611$	−36.2566	−1.46678
$612$	0	0
$613$	−4.83538	−0.195299	−0.0976495	−	0.995221i	$-0.531132\pi$
$613$	−4.83538	−0.195299	−0.0976495	+	0.995221i	$0.531132\pi$
$614$	−16.2570	−0.656078
$615$	0	0
$616$	22.7517	0.916694
$617$	16.8134	0.676883	0.338441	−	0.940987i	$-0.390100\pi$
$617$	16.8134	0.676883	0.338441	+	0.940987i	$0.390100\pi$
$618$	−32.3916	−1.30298
$619$	−12.7224	−0.511356	−0.255678	−	0.966762i	$-0.582299\pi$
$619$	−12.7224	−0.511356	−0.255678	+	0.966762i	$0.582299\pi$
$620$	0	0
$621$	−5.23223	−0.209962
$622$	6.47022	0.259432
$623$	−43.1259	−1.72780
$624$	−12.7033	−0.508539
$625$	0	0
$626$	−7.69008	−0.307357
$627$	10.6722	0.426206
$628$	17.0833	0.681697
$629$	0	0
$630$	0	0
$631$	4.23998	0.168791	0.0843955	−	0.996432i	$-0.473104\pi$
$631$	4.23998	0.168791	0.0843955	+	0.996432i	$0.473104\pi$
$632$	−18.5830	−0.739193
$633$	−20.4914	−0.814459
$634$	−0.843396	−0.0334955
$635$	0	0
$636$	−8.04915	−0.319169
$637$	−14.8975	−0.590261
$638$	0.908988	0.0359872
$639$	25.1238	0.993880
$640$	0	0
$641$	21.2203	0.838151	0.419075	−	0.907951i	$-0.362354\pi$
$641$	21.2203	0.838151	0.419075	+	0.907951i	$0.362354\pi$
$642$	−43.0300	−1.69826
$643$	8.16667	0.322062	0.161031	−	0.986949i	$-0.448518\pi$
$643$	8.16667	0.322062	0.161031	+	0.986949i	$0.448518\pi$
$644$	2.29295	0.0903548
$645$	0	0
$646$	0	0
$647$	−23.4331	−0.921249	−0.460624	−	0.887595i	$-0.652374\pi$
$647$	−23.4331	−0.921249	−0.460624	+	0.887595i	$0.652374\pi$
$648$	34.2505	1.34549
$649$	−8.25960	−0.324218
$650$	0	0
$651$	−54.9541	−2.15382
$652$	−7.69973	−0.301545
$653$	16.8073	0.657721	0.328860	−	0.944379i	$-0.393335\pi$
$653$	16.8073	0.657721	0.328860	+	0.944379i	$0.393335\pi$
$654$	−8.12029	−0.317529
$655$	0	0
$656$	2.07638	0.0810692
$657$	−41.7731	−1.62972
$658$	−20.2991	−0.791342
$659$	14.0972	0.549150	0.274575	−	0.961566i	$-0.411463\pi$
$659$	14.0972	0.549150	0.274575	+	0.961566i	$0.411463\pi$
$660$	0	0
$661$	−10.3907	−0.404152	−0.202076	−	0.979370i	$-0.564769\pi$
$661$	−10.3907	−0.404152	−0.202076	+	0.979370i	$0.564769\pi$
$662$	25.6235	0.995886
$663$	0	0
$664$	−0.585422	−0.0227188
$665$	0	0
$666$	−51.6634	−2.00192
$667$	0.218788	0.00847151
$668$	−16.6968	−0.646018
$669$	38.7231	1.49712
$670$	0	0
$671$	33.4317	1.29062
$672$	−58.2714	−2.24787
$673$	19.8244	0.764174	0.382087	−	0.924126i	$-0.375205\pi$
$673$	19.8244	0.764174	0.382087	+	0.924126i	$0.375205\pi$
$674$	−10.2810	−0.396010
$675$	0	0
$676$	−8.13660	−0.312946
$677$	−3.34594	−0.128595	−0.0642974	−	0.997931i	$-0.520481\pi$
$677$	−3.34594	−0.128595	−0.0642974	+	0.997931i	$0.520481\pi$
$678$	6.53516	0.250981
$679$	47.8669	1.83696
$680$	0	0
$681$	24.7314	0.947709
$682$	11.3089	0.433039
$683$	50.2828	1.92402	0.962009	−	0.273017i	$-0.0880215\pi$
$683$	50.2828	1.92402	0.962009	+	0.273017i	$0.0880215\pi$
$684$	−11.8014	−0.451239
$685$	0	0
$686$	8.58333	0.327713
$687$	−66.5113	−2.53756
$688$	2.06810	0.0788456
$689$	−7.84631	−0.298921
$690$	0	0
$691$	−38.9162	−1.48044	−0.740221	−	0.672364i	$-0.765279\pi$
$691$	−38.9162	−1.48044	−0.740221	+	0.672364i	$0.765279\pi$
$692$	−15.8859	−0.603890
$693$	57.0790	2.16825
$694$	−14.1196	−0.535972
$695$	0	0
$696$	−3.51620	−0.133281
$697$	0	0
$698$	5.27270	0.199575
$699$	67.4986	2.55303
$700$	0	0
$701$	−37.5419	−1.41794	−0.708969	−	0.705239i	$-0.750840\pi$
$701$	−37.5419	−1.41794	−0.708969	+	0.705239i	$0.750840\pi$
$702$	34.3215	1.29538
$703$	13.5773	0.512079
$704$	6.75850	0.254721
$705$	0	0
$706$	6.43789	0.242293
$707$	0.918754	0.0345533
$708$	13.3779	0.502772
$709$	13.7546	0.516566	0.258283	−	0.966069i	$-0.416843\pi$
$709$	13.7546	0.516566	0.258283	+	0.966069i	$0.416843\pi$
$710$	0	0
$711$	−46.6206	−1.74841
$712$	−34.3299	−1.28657
$713$	2.72198	0.101939
$714$	0	0
$715$	0	0
$716$	30.3584	1.13455
$717$	18.1467	0.677703
$718$	−6.86660	−0.256259
$719$	0.528983	0.0197278	0.00986388	−	0.999951i	$-0.496860\pi$
$719$	0.528983	0.0197278	0.00986388	+	0.999951i	$0.496860\pi$
$720$	0	0
$721$	45.5288	1.69558
$722$	13.0061	0.484038
$723$	−40.9331	−1.52232
$724$	−32.6434	−1.21318
$725$	0	0
$726$	8.09390	0.300393
$727$	−26.7632	−0.992591	−0.496296	−	0.868154i	$-0.665307\pi$
$727$	−26.7632	−0.992591	−0.496296	+	0.868154i	$0.665307\pi$
$728$	−35.9219	−1.33135
$729$	−12.1055	−0.448351
$730$	0	0
$731$	0	0
$732$	−54.1486	−2.00139
$733$	−17.3486	−0.640787	−0.320393	−	0.947285i	$-0.603815\pi$
$733$	−17.3486	−0.640787	−0.320393	+	0.947285i	$0.603815\pi$
$734$	−5.09783	−0.188164
$735$	0	0
$736$	2.88629	0.106390
$737$	−11.5253	−0.424542
$738$	−10.4799	−0.385772
$739$	30.2467	1.11264	0.556321	−	0.830967i	$-0.312212\pi$
$739$	30.2467	1.11264	0.556321	+	0.830967i	$0.312212\pi$
$740$	0	0
$741$	−16.8499	−0.618998
$742$	−4.39295	−0.161270
$743$	−7.18076	−0.263437	−0.131718	−	0.991287i	$-0.542049\pi$
$743$	−7.18076	−0.263437	−0.131718	+	0.991287i	$0.542049\pi$
$744$	−43.7457	−1.60379
$745$	0	0
$746$	−7.82669	−0.286555
$747$	−1.46869	−0.0537366
$748$	0	0
$749$	60.4819	2.20996
$750$	0	0
$751$	−31.2967	−1.14203	−0.571016	−	0.820939i	$-0.693451\pi$
$751$	−31.2967	−1.14203	−0.571016	+	0.820939i	$0.693451\pi$
$752$	8.03197	0.292896
$753$	11.0506	0.402708
$754$	−1.43517	−0.0522658
$755$	0	0
$756$	−49.4884	−1.79988
$757$	−10.7757	−0.391649	−0.195824	−	0.980639i	$-0.562738\pi$
$757$	−10.7757	−0.391649	−0.195824	+	0.980639i	$0.562738\pi$
$758$	18.9536	0.688425
$759$	−4.14104	−0.150310
$760$	0	0
$761$	33.1409	1.20136	0.600678	−	0.799491i	$-0.294898\pi$
$761$	33.1409	1.20136	0.600678	+	0.799491i	$0.294898\pi$
$762$	12.5691	0.455329
$763$	11.4137	0.413203
$764$	5.04452	0.182504
$765$	0	0
$766$	5.75182	0.207822
$767$	13.0408	0.470876
$768$	−37.9105	−1.36798
$769$	−11.4864	−0.414210	−0.207105	−	0.978319i	$-0.566404\pi$
$769$	−11.4864	−0.414210	−0.207105	+	0.978319i	$0.566404\pi$
$770$	0	0
$771$	−75.4892	−2.71868
$772$	17.7114	0.637446
$773$	3.20903	0.115421	0.0577104	−	0.998333i	$-0.481620\pi$
$773$	3.20903	0.115421	0.0577104	+	0.998333i	$0.481620\pi$
$774$	−10.4381	−0.375191
$775$	0	0
$776$	38.1039	1.36785
$777$	106.362	3.81570
$778$	−11.8454	−0.424678
$779$	2.75416	0.0986781
$780$	0	0
$781$	10.6440	0.380873
$782$	0	0
$783$	−4.72208	−0.168754
$784$	3.30027	0.117867
$785$	0	0
$786$	−22.9515	−0.818651
$787$	3.37029	0.120138	0.0600690	−	0.998194i	$-0.480868\pi$
$787$	3.37029	0.120138	0.0600690	+	0.998194i	$0.480868\pi$
$788$	24.4556	0.871195
$789$	−45.0188	−1.60271
$790$	0	0
$791$	−9.18564	−0.326604
$792$	45.4371	1.61454
$793$	−52.7841	−1.87442
$794$	−21.8122	−0.774085
$795$	0	0
$796$	−16.6794	−0.591185
$797$	21.3498	0.756247	0.378124	−	0.925755i	$-0.376569\pi$
$797$	21.3498	0.756247	0.378124	+	0.925755i	$0.376569\pi$
$798$	−9.43385	−0.333955
$799$	0	0
$800$	0	0
$801$	−86.1259	−3.04311
$802$	12.2552	0.432747
$803$	−17.6977	−0.624538
$804$	18.6674	0.658347
$805$	0	0
$806$	−17.8552	−0.628922
$807$	−57.6867	−2.03067
$808$	0.731364	0.0257293
$809$	−2.72968	−0.0959705	−0.0479852	−	0.998848i	$-0.515280\pi$
$809$	−2.72968	−0.0959705	−0.0479852	+	0.998848i	$0.515280\pi$
$810$	0	0
$811$	−37.3353	−1.31102	−0.655510	−	0.755187i	$-0.727546\pi$
$811$	−37.3353	−1.31102	−0.655510	+	0.755187i	$0.727546\pi$
$812$	2.06938	0.0726211
$813$	−17.7241	−0.621611
$814$	−21.8879	−0.767170
$815$	0	0
$816$	0	0
$817$	2.74317	0.0959715
$818$	−25.4269	−0.889032
$819$	−90.1199	−3.14905
$820$	0	0
$821$	42.2293	1.47381	0.736906	−	0.675995i	$-0.236286\pi$
$821$	42.2293	1.47381	0.736906	+	0.675995i	$0.236286\pi$
$822$	−7.07832	−0.246885
$823$	20.7606	0.723668	0.361834	−	0.932243i	$-0.382151\pi$
$823$	20.7606	0.723668	0.361834	+	0.932243i	$0.382151\pi$
$824$	36.2427	1.26257
$825$	0	0
$826$	7.30120	0.254041
$827$	11.5073	0.400149	0.200075	−	0.979781i	$-0.435881\pi$
$827$	11.5073	0.400149	0.200075	+	0.979781i	$0.435881\pi$
$828$	4.57920	0.159138
$829$	−11.7508	−0.408121	−0.204060	−	0.978958i	$-0.565414\pi$
$829$	−11.7508	−0.408121	−0.204060	+	0.978958i	$0.565414\pi$
$830$	0	0
$831$	0.379002	0.0131474
$832$	−10.6708	−0.369942
$833$	0	0
$834$	−15.4318	−0.534361
$835$	0	0
$836$	−4.99983	−0.172923
$837$	−58.7483	−2.03064
$838$	−0.667839	−0.0230701
$839$	36.5993	1.26355	0.631775	−	0.775152i	$-0.282327\pi$
$839$	36.5993	1.26355	0.631775	+	0.775152i	$0.282327\pi$
$840$	0	0
$841$	−28.8025	−0.993191
$842$	−25.1842	−0.867904
$843$	5.60772	0.193140
$844$	9.60003	0.330447
$845$	0	0
$846$	−40.5390	−1.39376
$847$	−11.3766	−0.390903
$848$	1.73821	0.0596902
$849$	−88.0312	−3.02122
$850$	0	0
$851$	−5.26829	−0.180595
$852$	−17.2399	−0.590628
$853$	38.0771	1.30374	0.651868	−	0.758332i	$-0.273986\pi$
$853$	38.0771	1.30374	0.651868	+	0.758332i	$0.273986\pi$
$854$	−29.5524	−1.01126
$855$	0	0
$856$	48.1459	1.64559
$857$	−20.2277	−0.690965	−0.345482	−	0.938425i	$-0.612285\pi$
$857$	−20.2277	−0.690965	−0.345482	+	0.938425i	$0.612285\pi$
$858$	27.1636	0.927351
$859$	−2.60725	−0.0889581	−0.0444790	−	0.999010i	$-0.514163\pi$
$859$	−2.60725	−0.0889581	−0.0444790	+	0.999010i	$0.514163\pi$
$860$	0	0
$861$	21.5755	0.735290
$862$	−12.1887	−0.415150
$863$	−11.2563	−0.383169	−0.191584	−	0.981476i	$-0.561363\pi$
$863$	−11.2563	−0.383169	−0.191584	+	0.981476i	$0.561363\pi$
$864$	−62.2945	−2.11930
$865$	0	0
$866$	−26.1730	−0.889396
$867$	0	0
$868$	25.7455	0.873860
$869$	−19.7514	−0.670021
$870$	0	0
$871$	18.1970	0.616580
$872$	9.08573	0.307682
$873$	95.5940	3.23537
$874$	0.467276	0.0158059
$875$	0	0
$876$	28.6646	0.968487
$877$	6.00107	0.202642	0.101321	−	0.994854i	$-0.467693\pi$
$877$	6.00107	0.202642	0.101321	+	0.994854i	$0.467693\pi$
$878$	−16.6618	−0.562309
$879$	4.35446	0.146872
$880$	0	0
$881$	34.3314	1.15665	0.578327	−	0.815805i	$-0.303706\pi$
$881$	34.3314	1.15665	0.578327	+	0.815805i	$0.303706\pi$
$882$	−16.6571	−0.560874
$883$	−11.8244	−0.397921	−0.198961	−	0.980007i	$-0.563757\pi$
$883$	−11.8244	−0.397921	−0.198961	+	0.980007i	$0.563757\pi$
$884$	0	0
$885$	0	0
$886$	−6.58095	−0.221091
$887$	−24.2352	−0.813738	−0.406869	−	0.913487i	$-0.633379\pi$
$887$	−24.2352	−0.813738	−0.406869	+	0.913487i	$0.633379\pi$
$888$	84.6680	2.84127
$889$	−17.6668	−0.592524
$890$	0	0
$891$	36.4040	1.21958
$892$	−18.1414	−0.607420
$893$	10.6538	0.356515
$894$	52.8769	1.76847
$895$	0	0
$896$	31.9254	1.06655
$897$	6.53813	0.218302
$898$	−6.63163	−0.221300
$899$	2.45659	0.0819317
$900$	0	0
$901$	0	0
$902$	−4.43996	−0.147835
$903$	21.4894	0.715122
$904$	−7.31213	−0.243198
$905$	0	0
$906$	−0.450182	−0.0149563
$907$	−44.7796	−1.48688	−0.743441	−	0.668802i	$-0.766808\pi$
$907$	−44.7796	−1.48688	−0.743441	+	0.668802i	$0.766808\pi$
$908$	−11.5864	−0.384509
$909$	1.83483	0.0608574
$910$	0	0
$911$	9.94858	0.329611	0.164806	−	0.986326i	$-0.447300\pi$
$911$	9.94858	0.329611	0.164806	+	0.986326i	$0.447300\pi$
$912$	3.73279	0.123605
$913$	−0.622230	−0.0205928
$914$	−13.8182	−0.457066
$915$	0	0
$916$	31.1600	1.02955
$917$	32.2600	1.06532
$918$	0	0
$919$	−33.3601	−1.10045	−0.550224	−	0.835017i	$-0.685458\pi$
$919$	−33.3601	−1.10045	−0.550224	+	0.835017i	$0.685458\pi$
$920$	0	0
$921$	66.8401	2.20246
$922$	18.3762	0.605189
$923$	−16.8054	−0.553158
$924$	−39.1675	−1.28852
$925$	0	0
$926$	−7.41132	−0.243551
$927$	90.9247	2.98636
$928$	2.60487	0.0855091
$929$	−24.8961	−0.816813	−0.408407	−	0.912800i	$-0.633915\pi$
$929$	−24.8961	−0.816813	−0.408407	+	0.912800i	$0.633915\pi$
$930$	0	0
$931$	4.37755	0.143468
$932$	−31.6225	−1.03583
$933$	−26.6021	−0.870915
$934$	−21.9917	−0.719592
$935$	0	0
$936$	−71.7390	−2.34486
$937$	−31.2460	−1.02076	−0.510381	−	0.859948i	$-0.670496\pi$
$937$	−31.2460	−1.02076	−0.510381	+	0.859948i	$0.670496\pi$
$938$	10.1880	0.332650
$939$	31.6176	1.03180
$940$	0	0
$941$	1.73372	0.0565177	0.0282589	−	0.999601i	$-0.491004\pi$
$941$	1.73372	0.0565177	0.0282589	+	0.999601i	$0.491004\pi$
$942$	27.2723	0.888579
$943$	−1.06867	−0.0348008
$944$	−2.88895	−0.0940272
$945$	0	0
$946$	−4.42225	−0.143780
$947$	−10.8513	−0.352619	−0.176309	−	0.984335i	$-0.556416\pi$
$947$	−10.8513	−0.352619	−0.176309	+	0.984335i	$0.556416\pi$
$948$	31.9910	1.03902
$949$	27.9423	0.907044
$950$	0	0
$951$	3.46760	0.112445
$952$	0	0
$953$	16.9007	0.547466	0.273733	−	0.961806i	$-0.411742\pi$
$953$	16.9007	0.547466	0.273733	+	0.961806i	$0.411742\pi$
$954$	−8.77308	−0.284039
$955$	0	0
$956$	−8.50159	−0.274961
$957$	−3.73728	−0.120809
$958$	−21.3906	−0.691098
$959$	9.94910	0.321273
$960$	0	0
$961$	−0.437196	−0.0141031
$962$	34.5580	1.11419
$963$	120.787	3.89231
$964$	19.1768	0.617643
$965$	0	0
$966$	3.66053	0.117776
$967$	−31.2356	−1.00447	−0.502234	−	0.864732i	$-0.667489\pi$
$967$	−31.2356	−1.00447	−0.502234	+	0.864732i	$0.667489\pi$
$968$	−9.05619	−0.291077
$969$	0	0
$970$	0	0
$971$	17.3888	0.558033	0.279016	−	0.960286i	$-0.409992\pi$
$971$	17.3888	0.558033	0.279016	+	0.960286i	$0.409992\pi$
$972$	−13.0356	−0.418116
$973$	21.6906	0.695368
$974$	−0.462124	−0.0148074
$975$	0	0
$976$	11.6933	0.374294
$977$	9.17338	0.293482	0.146741	−	0.989175i	$-0.453122\pi$
$977$	9.17338	0.293482	0.146741	+	0.989175i	$0.453122\pi$
$978$	−12.2921	−0.393058
$979$	−36.4884	−1.16617
$980$	0	0
$981$	22.7940	0.727757
$982$	−19.0469	−0.607811
$983$	−37.2969	−1.18959	−0.594793	−	0.803879i	$-0.702766\pi$
$983$	−37.2969	−1.18959	−0.594793	+	0.803879i	$0.702766\pi$
$984$	17.1749	0.547516
$985$	0	0
$986$	0	0
$987$	83.4593	2.65654
$988$	7.89405	0.251143
$989$	−1.06441	−0.0338463
$990$	0	0
$991$	35.0682	1.11398	0.556990	−	0.830519i	$-0.311956\pi$
$991$	35.0682	1.11398	0.556990	+	0.830519i	$0.311956\pi$
$992$	32.4077	1.02894
$993$	−105.350	−3.34319
$994$	−9.40894	−0.298433
$995$	0	0
$996$	1.00781	0.0319338
$997$	−37.3576	−1.18313	−0.591563	−	0.806259i	$-0.701489\pi$
$997$	−37.3576	−1.18313	−0.591563	+	0.806259i	$0.701489\pi$
$998$	27.2692	0.863190
$999$	113.705	3.59747

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7225.2.a.bs.1.4		12
5.4	even	2		1445.2.a.p.1.9		12
17.5	odd	16		425.2.m.b.76.5		24
17.7	odd	16		425.2.m.b.151.5		24
17.16	even	2		7225.2.a.bq.1.4		12
85.4	even	4		1445.2.d.j.866.7		24
85.7	even	16		425.2.n.f.49.2		24
85.22	even	16		425.2.n.c.399.5		24
85.24	odd	16		85.2.l.a.66.2	✓	24
85.39	odd	16		85.2.l.a.76.2	yes	24
85.58	even	16		425.2.n.c.49.5		24
85.64	even	4		1445.2.d.j.866.8		24
85.73	even	16		425.2.n.f.399.2		24
85.84	even	2		1445.2.a.q.1.9		12
255.194	even	16		765.2.be.b.406.5		24
255.209	even	16		765.2.be.b.586.5		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
85.2.l.a.66.2	✓	24	85.24	odd	16
85.2.l.a.76.2	yes	24	85.39	odd	16
425.2.m.b.76.5		24	17.5	odd	16
425.2.m.b.151.5		24	17.7	odd	16
425.2.n.c.49.5		24	85.58	even	16
425.2.n.c.399.5		24	85.22	even	16
425.2.n.f.49.2		24	85.7	even	16
425.2.n.f.399.2		24	85.73	even	16
765.2.be.b.406.5		24	255.194	even	16
765.2.be.b.586.5		24	255.209	even	16
1445.2.a.p.1.9		12	5.4	even	2
1445.2.a.q.1.9		12	85.84	even	2
1445.2.d.j.866.7		24	85.4	even	4
1445.2.d.j.866.8		24	85.64	even	4
7225.2.a.bq.1.4		12	17.16	even	2
7225.2.a.bs.1.4		12	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 7225 = 5^{2} \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7225.a (trivial)

\(f(q)\)	\(=\)	\(q-0.747914 q^{2} +3.07503 q^{3} -1.44062 q^{4} -2.29986 q^{6} +3.23262 q^{7} +2.57329 q^{8} +6.45581 q^{9} +2.73509 q^{11} -4.42996 q^{12} -4.31833 q^{13} -2.41772 q^{14} +0.956646 q^{16} -4.82839 q^{18} +1.26892 q^{19} +9.94040 q^{21} -2.04561 q^{22} -0.492367 q^{23} +7.91295 q^{24} +3.22974 q^{26} +10.6267 q^{27} -4.65699 q^{28} -0.444360 q^{29} -5.52836 q^{31} -5.86207 q^{32} +8.41047 q^{33} -9.30039 q^{36} +10.6999 q^{37} -0.949042 q^{38} -13.2790 q^{39} +2.17048 q^{41} -7.43457 q^{42} +2.16182 q^{43} -3.94023 q^{44} +0.368248 q^{46} +8.39597 q^{47} +2.94172 q^{48} +3.44983 q^{49} +6.22109 q^{52} +1.81698 q^{53} -7.94786 q^{54} +8.31847 q^{56} +3.90196 q^{57} +0.332343 q^{58} -3.01987 q^{59} +12.2233 q^{61} +4.13474 q^{62} +20.8692 q^{63} +2.47104 q^{64} -6.29031 q^{66} -4.21389 q^{67} -1.51404 q^{69} +3.89165 q^{71} +16.6127 q^{72} -6.47062 q^{73} -8.00263 q^{74} -1.82803 q^{76} +8.84149 q^{77} +9.93155 q^{78} -7.22150 q^{79} +13.3100 q^{81} -1.62333 q^{82} -0.227499 q^{83} -14.3204 q^{84} -1.61686 q^{86} -1.36642 q^{87} +7.03818 q^{88} -13.3408 q^{89} -13.9595 q^{91} +0.709315 q^{92} -16.9999 q^{93} -6.27946 q^{94} -18.0260 q^{96} +14.8075 q^{97} -2.58018 q^{98} +17.6572 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 4 q^{2} + 8 q^{3} + 12 q^{4} - 8 q^{6} + 16 q^{7} + 12 q^{8} + 12 q^{9} - 16 q^{11} + 16 q^{12} + 8 q^{13} + 16 q^{14} + 12 q^{16} - 4 q^{18} + 16 q^{21} + 16 q^{22} + 16 q^{23} + 16 q^{26} + 32 q^{27}+ \cdots - 56 q^{99}+O(q^{100}) \) 12 * q + 4 * q^2 + 8 * q^3 + 12 * q^4 - 8 * q^6 + 16 * q^7 + 12 * q^8 + 12 * q^9 - 16 * q^11 + 16 * q^12 + 8 * q^13 + 16 * q^14 + 12 * q^16 - 4 * q^18 + 16 * q^21 + 16 * q^22 + 16 * q^23 + 16 * q^26 + 32 * q^27 + 40 * q^28 - 16 * q^29 - 24 * q^31 + 28 * q^32 + 12 * q^36 + 24 * q^37 + 24 * q^38 - 8 * q^39 - 8 * q^41 + 16 * q^43 - 8 * q^44 - 40 * q^46 + 32 * q^47 - 24 * q^48 + 20 * q^49 + 24 * q^52 - 8 * q^54 + 24 * q^56 + 32 * q^57 - 16 * q^58 + 8 * q^59 - 24 * q^61 + 8 * q^62 + 48 * q^63 + 36 * q^64 + 40 * q^66 + 8 * q^67 + 48 * q^69 + 16 * q^71 - 12 * q^72 + 16 * q^73 + 16 * q^76 - 24 * q^77 - 24 * q^78 - 40 * q^79 + 36 * q^81 - 16 * q^82 + 40 * q^83 + 32 * q^84 - 8 * q^86 + 32 * q^87 + 48 * q^88 + 8 * q^89 - 72 * q^91 - 8 * q^92 - 24 * q^93 - 16 * q^94 - 8 * q^96 + 32 * q^97 + 60 * q^98 - 56 * q^99

Introduction

L-functions

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