LMFDB - Embedded newform 9025.2.a.ct.1.11

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9025,2,Mod(1,9025)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9025, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("9025.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9025 = 5^{2} \cdot 19^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9025.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$72.0649878242$
Analytic rank:	$1$
Dimension:	$24$
Twist minimal:	no (minimal twist has level 95)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.11
Character	$\chi$	$=$	9025.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.249751 q^{2} +2.30156 q^{3} -1.93762 q^{4} -0.574816 q^{6} -3.96043 q^{7} +0.983424 q^{8} +2.29718 q^{9} +O(q^{10})$	\(q-0.249751 q^{2} +2.30156 q^{3} -1.93762 q^{4} -0.574816 q^{6} -3.96043 q^{7} +0.983424 q^{8} +2.29718 q^{9} +3.13464 q^{11} -4.45956 q^{12} -2.65927 q^{13} +0.989119 q^{14} +3.62964 q^{16} -2.25718 q^{17} -0.573722 q^{18} -9.11516 q^{21} -0.782879 q^{22} +7.58941 q^{23} +2.26341 q^{24} +0.664153 q^{26} -1.61758 q^{27} +7.67382 q^{28} -1.36269 q^{29} -0.894474 q^{31} -2.87335 q^{32} +7.21457 q^{33} +0.563731 q^{34} -4.45107 q^{36} +6.62537 q^{37} -6.12046 q^{39} -6.23645 q^{41} +2.27652 q^{42} +1.77237 q^{43} -6.07376 q^{44} -1.89546 q^{46} -0.176361 q^{47} +8.35383 q^{48} +8.68497 q^{49} -5.19503 q^{51} +5.15266 q^{52} +6.77526 q^{53} +0.403992 q^{54} -3.89478 q^{56} +0.340331 q^{58} +7.81595 q^{59} -1.03941 q^{61} +0.223395 q^{62} -9.09781 q^{63} -6.54166 q^{64} -1.80184 q^{66} +15.2111 q^{67} +4.37356 q^{68} +17.4675 q^{69} -10.4376 q^{71} +2.25910 q^{72} -4.18114 q^{73} -1.65469 q^{74} -12.4145 q^{77} +1.52859 q^{78} -11.2084 q^{79} -10.6145 q^{81} +1.55756 q^{82} -13.2542 q^{83} +17.6618 q^{84} -0.442650 q^{86} -3.13630 q^{87} +3.08269 q^{88} -1.48477 q^{89} +10.5318 q^{91} -14.7054 q^{92} -2.05869 q^{93} +0.0440463 q^{94} -6.61320 q^{96} -15.3864 q^{97} -2.16908 q^{98} +7.20085 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + 18 q^{4} - 12 q^{6} + 12 q^{9}+O(q^{10}) $ 24 * q + 18 * q^4 - 12 * q^6 + 12 * q^9	$ 24 q + 18 q^{4} - 12 q^{6} + 12 q^{9} + 12 q^{11} - 24 q^{14} + 6 q^{16} - 6 q^{21} - 42 q^{24} - 12 q^{26} - 36 q^{29} - 42 q^{31} - 6 q^{34} - 6 q^{36} + 24 q^{39} - 60 q^{41} - 30 q^{44} - 6 q^{46} + 12 q^{49} - 30 q^{51} - 24 q^{54} - 18 q^{56} - 60 q^{59} + 30 q^{61} + 36 q^{66} - 66 q^{69} - 96 q^{71} + 24 q^{74} - 72 q^{79} - 96 q^{81} + 54 q^{84} - 108 q^{86} - 84 q^{89} - 96 q^{91} - 36 q^{94} - 120 q^{96}+O(q^{100}) $ 24 * q + 18 * q^4 - 12 * q^6 + 12 * q^9 + 12 * q^11 - 24 * q^14 + 6 * q^16 - 6 * q^21 - 42 * q^24 - 12 * q^26 - 36 * q^29 - 42 * q^31 - 6 * q^34 - 6 * q^36 + 24 * q^39 - 60 * q^41 - 30 * q^44 - 6 * q^46 + 12 * q^49 - 30 * q^51 - 24 * q^54 - 18 * q^56 - 60 * q^59 + 30 * q^61 + 36 * q^66 - 66 * q^69 - 96 * q^71 + 24 * q^74 - 72 * q^79 - 96 * q^81 + 54 * q^84 - 108 * q^86 - 84 * q^89 - 96 * q^91 - 36 * q^94 - 120 * q^96

Display 100 coefficients 10 coefficients

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 $

$2$	−0.249751	−0.176600	−0.0883002	−	0.996094i	$-0.528143\pi$
$2$	−0.249751	−0.176600	−0.0883002	+	0.996094i	$0.528143\pi$
$3$	2.30156	1.32881	0.664403	−	0.747374i	$-0.268686\pi$
$3$	2.30156	1.32881	0.664403	+	0.747374i	$0.268686\pi$
$4$	−1.93762	−0.968812
$5$	0	0
$5$	0	0
$6$	−0.574816	−0.234668
$7$	−3.96043	−1.49690	−0.748450	−	0.663191i	$-0.769202\pi$
$7$	−3.96043	−1.49690	−0.748450	+	0.663191i	$0.769202\pi$
$8$	0.983424	0.347693
$9$	2.29718	0.765727
$10$	0	0
$11$	3.13464	0.945131	0.472565	−	0.881296i	$-0.343328\pi$
$11$	3.13464	0.945131	0.472565	+	0.881296i	$0.343328\pi$
$12$	−4.45956	−1.28736
$13$	−2.65927	−0.737548	−0.368774	−	0.929519i	$-0.620222\pi$
$13$	−2.65927	−0.737548	−0.368774	+	0.929519i	$0.620222\pi$
$14$	0.989119	0.264353
$15$	0	0
$16$	3.62964	0.907410
$17$	−2.25718	−0.547446	−0.273723	−	0.961809i	$-0.588255\pi$
$17$	−2.25718	−0.547446	−0.273723	+	0.961809i	$0.588255\pi$
$18$	−0.573722	−0.135228
$19$	0	0
$19$	0	0
$20$	0	0
$21$	−9.11516	−1.98909
$22$	−0.782879	−0.166910
$23$	7.58941	1.58250	0.791251	−	0.611492i	$-0.209430\pi$
$23$	7.58941	1.58250	0.791251	+	0.611492i	$0.209430\pi$
$24$	2.26341	0.462017
$25$	0	0
$26$	0.664153	0.130251
$27$	−1.61758	−0.311303
$28$	7.67382	1.45022
$29$	−1.36269	−0.253044	−0.126522	−	0.991964i	$-0.540381\pi$
$29$	−1.36269	−0.253044	−0.126522	+	0.991964i	$0.540381\pi$
$30$	0	0
$31$	−0.894474	−0.160652	−0.0803261	−	0.996769i	$-0.525596\pi$
$31$	−0.894474	−0.160652	−0.0803261	+	0.996769i	$0.525596\pi$
$32$	−2.87335	−0.507942
$33$	7.21457	1.25590
$34$	0.563731	0.0966791
$35$	0	0
$36$	−4.45107	−0.741846
$37$	6.62537	1.08920	0.544602	−	0.838695i	$-0.316681\pi$
$37$	6.62537	1.08920	0.544602	+	0.838695i	$0.316681\pi$
$38$	0	0
$39$	−6.12046	−0.980058
$40$	0	0
$41$	−6.23645	−0.973970	−0.486985	−	0.873410i	$-0.661903\pi$
$41$	−6.23645	−0.973970	−0.486985	+	0.873410i	$0.661903\pi$
$42$	2.27652	0.351274
$43$	1.77237	0.270283	0.135142	−	0.990826i	$-0.456851\pi$
$43$	1.77237	0.270283	0.135142	+	0.990826i	$0.456851\pi$
$44$	−6.07376	−0.915654
$45$	0	0
$46$	−1.89546	−0.279470
$47$	−0.176361	−0.0257249	−0.0128625	−	0.999917i	$-0.504094\pi$
$47$	−0.176361	−0.0257249	−0.0128625	+	0.999917i	$0.504094\pi$
$48$	8.35383	1.20577
$49$	8.68497	1.24071
$50$	0	0
$51$	−5.19503	−0.727450
$52$	5.15266	0.714545
$53$	6.77526	0.930653	0.465326	−	0.885139i	$-0.345937\pi$
$53$	6.77526	0.930653	0.465326	+	0.885139i	$0.345937\pi$
$54$	0.403992	0.0549763
$55$	0	0
$56$	−3.89478	−0.520462
$57$	0	0
$58$	0.340331	0.0446877
$59$	7.81595	1.01755	0.508775	−	0.860899i	$-0.330098\pi$
$59$	7.81595	1.01755	0.508775	+	0.860899i	$0.330098\pi$
$60$	0	0
$61$	−1.03941	−0.133082	−0.0665412	−	0.997784i	$-0.521196\pi$
$61$	−1.03941	−0.133082	−0.0665412	+	0.997784i	$0.521196\pi$
$62$	0.223395	0.0283712
$63$	−9.09781	−1.14622
$64$	−6.54166	−0.817707
$65$	0	0
$66$	−1.80184	−0.221792
$67$	15.2111	1.85834	0.929169	−	0.369656i	$-0.120524\pi$
$67$	15.2111	1.85834	0.929169	+	0.369656i	$0.120524\pi$
$68$	4.37356	0.530372
$69$	17.4675	2.10284
$70$	0	0
$71$	−10.4376	−1.23872	−0.619360	−	0.785107i	$-0.712608\pi$
$71$	−10.4376	−1.23872	−0.619360	+	0.785107i	$0.712608\pi$
$72$	2.25910	0.266238
$73$	−4.18114	−0.489366	−0.244683	−	0.969603i	$-0.578684\pi$
$73$	−4.18114	−0.489366	−0.244683	+	0.969603i	$0.578684\pi$
$74$	−1.65469	−0.192354
$75$	0	0
$76$	0	0
$77$	−12.4145	−1.41477
$78$	1.52859	0.173079
$79$	−11.2084	−1.26105	−0.630524	−	0.776170i	$-0.717160\pi$
$79$	−11.2084	−1.26105	−0.630524	+	0.776170i	$0.717160\pi$
$80$	0	0
$81$	−10.6145	−1.17939
$82$	1.55756	0.172004
$83$	−13.2542	−1.45484	−0.727421	−	0.686192i	$-0.759281\pi$
$83$	−13.2542	−1.45484	−0.727421	+	0.686192i	$0.759281\pi$
$84$	17.6618	1.92706
$85$	0	0
$86$	−0.442650	−0.0477321
$87$	−3.13630	−0.336247
$88$	3.08269	0.328615
$89$	−1.48477	−0.157385	−0.0786925	−	0.996899i	$-0.525075\pi$
$89$	−1.48477	−0.157385	−0.0786925	+	0.996899i	$0.525075\pi$
$90$	0	0
$91$	10.5318	1.10404
$92$	−14.7054	−1.53315
$93$	−2.05869	−0.213476
$94$	0.0440463	0.00454303
$95$	0	0
$96$	−6.61320	−0.674956
$97$	−15.3864	−1.56225	−0.781127	−	0.624372i	$-0.785355\pi$
$97$	−15.3864	−1.56225	−0.781127	+	0.624372i	$0.785355\pi$
$98$	−2.16908	−0.219110
$99$	7.20085	0.723712
$100$	0	0
$101$	5.92336	0.589396	0.294698	−	0.955590i	$-0.404781\pi$
$101$	5.92336	0.589396	0.294698	+	0.955590i	$0.404781\pi$
$102$	1.29746	0.128468
$103$	−5.00736	−0.493390	−0.246695	−	0.969093i	$-0.579345\pi$
$103$	−5.00736	−0.493390	−0.246695	+	0.969093i	$0.579345\pi$
$104$	−2.61519	−0.256440
$105$	0	0
$106$	−1.69212	−0.164354
$107$	2.81845	0.272470	0.136235	−	0.990677i	$-0.456500\pi$
$107$	2.81845	0.272470	0.136235	+	0.990677i	$0.456500\pi$
$108$	3.13426	0.301595
$109$	−6.74762	−0.646305	−0.323153	−	0.946347i	$-0.604743\pi$
$109$	−6.74762	−0.646305	−0.323153	+	0.946347i	$0.604743\pi$
$110$	0	0
$111$	15.2487	1.44734
$112$	−14.3749	−1.35830
$113$	7.76636	0.730598	0.365299	−	0.930890i	$-0.380967\pi$
$113$	7.76636	0.730598	0.365299	+	0.930890i	$0.380967\pi$
$114$	0	0
$115$	0	0
$116$	2.64037	0.245152
$117$	−6.10882	−0.564760
$118$	−1.95204	−0.179700
$119$	8.93938	0.819472
$120$	0	0
$121$	−1.17400	−0.106728
$122$	0.259593	0.0235024
$123$	−14.3536	−1.29422
$124$	1.73315	0.155642
$125$	0	0
$126$	2.27218	0.202422
$127$	−1.68249	−0.149297	−0.0746485	−	0.997210i	$-0.523783\pi$
$127$	−1.68249	−0.149297	−0.0746485	+	0.997210i	$0.523783\pi$
$128$	7.38049	0.652349
$129$	4.07921	0.359154
$130$	0	0
$131$	−7.67994	−0.670999	−0.335500	−	0.942040i	$-0.608905\pi$
$131$	−7.67994	−0.670999	−0.335500	+	0.942040i	$0.608905\pi$
$132$	−13.9791	−1.21673
$133$	0	0
$134$	−3.79899	−0.328183
$135$	0	0
$136$	−2.21976	−0.190343
$137$	9.69477	0.828280	0.414140	−	0.910213i	$-0.364082\pi$
$137$	9.69477	0.828280	0.414140	+	0.910213i	$0.364082\pi$
$138$	−4.36252	−0.371362
$139$	−6.40361	−0.543147	−0.271573	−	0.962418i	$-0.587544\pi$
$139$	−6.40361	−0.543147	−0.271573	+	0.962418i	$0.587544\pi$
$140$	0	0
$141$	−0.405906	−0.0341835
$142$	2.60681	0.218758
$143$	−8.33585	−0.697079
$144$	8.33794	0.694828
$145$	0	0
$146$	1.04424	0.0864222
$147$	19.9890	1.64866
$148$	−12.8375	−1.05523
$149$	−1.94517	−0.159355	−0.0796773	−	0.996821i	$-0.525389\pi$
$149$	−1.94517	−0.159355	−0.0796773	+	0.996821i	$0.525389\pi$
$150$	0	0
$151$	16.7168	1.36040	0.680199	−	0.733028i	$-0.261893\pi$
$151$	16.7168	1.36040	0.680199	+	0.733028i	$0.261893\pi$
$152$	0	0
$153$	−5.18514	−0.419194
$154$	3.10054	0.249848
$155$	0	0
$156$	11.8592	0.949493
$157$	−9.81019	−0.782938	−0.391469	−	0.920191i	$-0.628033\pi$
$157$	−9.81019	−0.782938	−0.391469	+	0.920191i	$0.628033\pi$
$158$	2.79932	0.222702
$159$	15.5937	1.23666
$160$	0	0
$161$	−30.0573	−2.36885
$162$	2.65098	0.208281
$163$	−5.06581	−0.396785	−0.198393	−	0.980123i	$-0.563572\pi$
$163$	−5.06581	−0.396785	−0.198393	+	0.980123i	$0.563572\pi$
$164$	12.0839	0.943595
$165$	0	0
$166$	3.31025	0.256926
$167$	−20.1611	−1.56011	−0.780056	−	0.625710i	$-0.784809\pi$
$167$	−20.1611	−1.56011	−0.780056	+	0.625710i	$0.784809\pi$
$168$	−8.96407	−0.691593
$169$	−5.92830	−0.456023
$170$	0	0
$171$	0	0
$172$	−3.43418	−0.261854
$173$	1.94875	0.148161	0.0740805	−	0.997252i	$-0.476398\pi$
$173$	1.94875	0.148161	0.0740805	+	0.997252i	$0.476398\pi$
$174$	0.783293	0.0593813
$175$	0	0
$176$	11.3776	0.857621
$177$	17.9889	1.35213
$178$	0.370821	0.0277942
$179$	−17.2710	−1.29089	−0.645447	−	0.763805i	$-0.723329\pi$
$179$	−17.2710	−1.29089	−0.645447	+	0.763805i	$0.723329\pi$
$180$	0	0
$181$	−14.4378	−1.07315	−0.536575	−	0.843852i	$-0.680282\pi$
$181$	−14.4378	−1.07315	−0.536575	+	0.843852i	$0.680282\pi$
$182$	−2.63033	−0.194973
$183$	−2.39226	−0.176841
$184$	7.46361	0.550225
$185$	0	0
$186$	0.514158	0.0376999
$187$	−7.07545	−0.517408
$188$	0.341722	0.0249226
$189$	6.40631	0.465990
$190$	0	0
$191$	−8.05176	−0.582605	−0.291302	−	0.956631i	$-0.594089\pi$
$191$	−8.05176	−0.582605	−0.291302	+	0.956631i	$0.594089\pi$
$192$	−15.0560	−1.08657
$193$	−6.13649	−0.441714	−0.220857	−	0.975306i	$-0.570885\pi$
$193$	−6.13649	−0.441714	−0.220857	+	0.975306i	$0.570885\pi$
$194$	3.84277	0.275895
$195$	0	0
$196$	−16.8282	−1.20201
$197$	−17.4160	−1.24084	−0.620418	−	0.784271i	$-0.713037\pi$
$197$	−17.4160	−1.24084	−0.620418	+	0.784271i	$0.713037\pi$
$198$	−1.79842	−0.127808
$199$	0.0105972	0.000751213 0	0.000375607	−	1.00000i	$-0.499880\pi$
$199$	0.0105972	0.000751213 0	0.000375607		1.00000i	$0.499880\pi$
$200$	0	0
$201$	35.0094	2.46937
$202$	−1.47936	−0.104088
$203$	5.39681	0.378782
$204$	10.0660	0.704762
$205$	0	0
$206$	1.25059	0.0871329
$207$	17.4343	1.21176
$208$	−9.65218	−0.669258
$209$	0	0
$210$	0	0
$211$	−27.1262	−1.86745	−0.933723	−	0.357997i	$-0.883460\pi$
$211$	−27.1262	−1.86745	−0.933723	+	0.357997i	$0.883460\pi$
$212$	−13.1279	−0.901628
$213$	−24.0228	−1.64602
$214$	−0.703909	−0.0481183
$215$	0	0
$216$	−1.59077	−0.108238
$217$	3.54250	0.240480
$218$	1.68522	0.114138
$219$	−9.62316	−0.650273
$220$	0	0
$221$	6.00243	0.403767
$222$	−3.80837	−0.255601
$223$	−7.65043	−0.512311	−0.256155	−	0.966636i	$-0.582456\pi$
$223$	−7.65043	−0.512311	−0.256155	+	0.966636i	$0.582456\pi$
$224$	11.3797	0.760338
$225$	0	0
$226$	−1.93965	−0.129024
$227$	25.9617	1.72314	0.861571	−	0.507638i	$-0.169481\pi$
$227$	25.9617	1.72314	0.861571	+	0.507638i	$0.169481\pi$
$228$	0	0
$229$	−6.42130	−0.424332	−0.212166	−	0.977234i	$-0.568052\pi$
$229$	−6.42130	−0.424332	−0.212166	+	0.977234i	$0.568052\pi$
$230$	0	0
$231$	−28.5728	−1.87995
$232$	−1.34010	−0.0879817
$233$	19.0954	1.25098	0.625492	−	0.780231i	$-0.284898\pi$
$233$	19.0954	1.25098	0.625492	+	0.780231i	$0.284898\pi$
$234$	1.52568	0.0997369
$235$	0	0
$236$	−15.1444	−0.985815
$237$	−25.7969	−1.67569
$238$	−2.23262	−0.144719
$239$	−8.39178	−0.542819	−0.271409	−	0.962464i	$-0.587490\pi$
$239$	−8.39178	−0.542819	−0.271409	+	0.962464i	$0.587490\pi$
$240$	0	0
$241$	8.61689	0.555063	0.277531	−	0.960717i	$-0.410484\pi$
$241$	8.61689	0.555063	0.277531	+	0.960717i	$0.410484\pi$
$242$	0.293208	0.0188481
$243$	−19.5772	−1.25588
$244$	2.01398	0.128932
$245$	0	0
$246$	3.58481	0.228559
$247$	0	0
$248$	−0.879647	−0.0558576
$249$	−30.5054	−1.93320
$250$	0	0
$251$	29.2619	1.84700	0.923498	−	0.383604i	$-0.125317\pi$
$251$	29.2619	1.84700	0.923498	+	0.383604i	$0.125317\pi$
$252$	17.6281	1.11047
$253$	23.7901	1.49567
$254$	0.420203	0.0263659
$255$	0	0
$256$	11.2400	0.702502
$257$	−5.48068	−0.341876	−0.170938	−	0.985282i	$-0.554680\pi$
$257$	−5.48068	−0.341876	−0.170938	+	0.985282i	$0.554680\pi$
$258$	−1.01878	−0.0634268
$259$	−26.2393	−1.63043
$260$	0	0
$261$	−3.13033	−0.193763
$262$	1.91807	0.118499
$263$	0.568281	0.0350417	0.0175209	−	0.999846i	$-0.494423\pi$
$263$	0.568281	0.0350417	0.0175209	+	0.999846i	$0.494423\pi$
$264$	7.09499	0.436666
$265$	0	0
$266$	0	0
$267$	−3.41728	−0.209134
$268$	−29.4735	−1.80038
$269$	4.44312	0.270902	0.135451	−	0.990784i	$-0.456752\pi$
$269$	4.44312	0.270902	0.135451	+	0.990784i	$0.456752\pi$
$270$	0	0
$271$	17.7660	1.07921	0.539605	−	0.841919i	$-0.318574\pi$
$271$	17.7660	1.07921	0.539605	+	0.841919i	$0.318574\pi$
$272$	−8.19274	−0.496758
$273$	24.2396	1.46705
$274$	−2.42127	−0.146275
$275$	0	0
$276$	−33.8454	−2.03726
$277$	−8.10450	−0.486952	−0.243476	−	0.969907i	$-0.578288\pi$
$277$	−8.10450	−0.486952	−0.243476	+	0.969907i	$0.578288\pi$
$278$	1.59930	0.0959199
$279$	−2.05477	−0.123016
$280$	0	0
$281$	−7.05224	−0.420701	−0.210351	−	0.977626i	$-0.567461\pi$
$281$	−7.05224	−0.420701	−0.210351	+	0.977626i	$0.567461\pi$
$282$	0.101375	0.00603681
$283$	−1.29862	−0.0771949	−0.0385975	−	0.999255i	$-0.512289\pi$
$283$	−1.29862	−0.0771949	−0.0385975	+	0.999255i	$0.512289\pi$
$284$	20.2242	1.20009
$285$	0	0
$286$	2.08188	0.123104
$287$	24.6990	1.45794
$288$	−6.60061	−0.388945
$289$	−11.9052	−0.700303
$290$	0	0
$291$	−35.4128	−2.07593
$292$	8.10149	0.474104
$293$	14.9498	0.873376	0.436688	−	0.899613i	$-0.356151\pi$
$293$	14.9498	0.873376	0.436688	+	0.899613i	$0.356151\pi$
$294$	−4.99226	−0.291155
$295$	0	0
$296$	6.51555	0.378708
$297$	−5.07054	−0.294223
$298$	0.485808	0.0281421
$299$	−20.1823	−1.16717
$300$	0	0
$301$	−7.01932	−0.404587
$302$	−4.17504	−0.240247
$303$	13.6330	0.783194
$304$	0	0
$305$	0	0
$306$	1.29499	0.0740298
$307$	5.01371	0.286147	0.143074	−	0.989712i	$-0.454301\pi$
$307$	5.01371	0.286147	0.143074	+	0.989712i	$0.454301\pi$
$308$	24.0547	1.37064
$309$	−11.5248	−0.655620
$310$	0	0
$311$	−16.7443	−0.949481	−0.474741	−	0.880126i	$-0.657458\pi$
$311$	−16.7443	−0.949481	−0.474741	+	0.880126i	$0.657458\pi$
$312$	−6.01901	−0.340759
$313$	−5.73858	−0.324363	−0.162182	−	0.986761i	$-0.551853\pi$
$313$	−5.73858	−0.324363	−0.162182	+	0.986761i	$0.551853\pi$
$314$	2.45010	0.138267
$315$	0	0
$316$	21.7178	1.22172
$317$	7.94635	0.446312	0.223156	−	0.974783i	$-0.428364\pi$
$317$	7.94635	0.446312	0.223156	+	0.974783i	$0.428364\pi$
$318$	−3.89453	−0.218394
$319$	−4.27153	−0.239160
$320$	0	0
$321$	6.48683	0.362060
$322$	7.50683	0.418339
$323$	0	0
$324$	20.5669	1.14261
$325$	0	0
$326$	1.26519	0.0700724
$327$	−15.5301	−0.858814
$328$	−6.13308	−0.338643
$329$	0.698465	0.0385076
$330$	0	0
$331$	20.8177	1.14424	0.572121	−	0.820169i	$-0.306121\pi$
$331$	20.8177	1.14424	0.572121	+	0.820169i	$0.306121\pi$
$332$	25.6817	1.40947
$333$	15.2197	0.834033
$334$	5.03525	0.275516
$335$	0	0
$336$	−33.0847	−1.80492
$337$	−27.3618	−1.49049	−0.745245	−	0.666790i	$-0.767668\pi$
$337$	−27.3618	−1.49049	−0.745245	+	0.666790i	$0.767668\pi$
$338$	1.48060	0.0805339
$339$	17.8748	0.970824
$340$	0	0
$341$	−2.80386	−0.151837
$342$	0	0
$343$	−6.67319	−0.360318
$344$	1.74299	0.0939756
$345$	0	0
$346$	−0.486703	−0.0261653
$347$	20.1999	1.08439	0.542194	−	0.840254i	$-0.317594\pi$
$347$	20.1999	1.08439	0.542194	+	0.840254i	$0.317594\pi$
$348$	6.07698	0.325760
$349$	−15.6240	−0.836332	−0.418166	−	0.908371i	$-0.637327\pi$
$349$	−15.6240	−0.836332	−0.418166	+	0.908371i	$0.637327\pi$
$350$	0	0
$351$	4.30158	0.229601
$352$	−9.00694	−0.480072
$353$	−4.59743	−0.244696	−0.122348	−	0.992487i	$-0.539042\pi$
$353$	−4.59743	−0.244696	−0.122348	+	0.992487i	$0.539042\pi$
$354$	−4.49274	−0.238786
$355$	0	0
$356$	2.87692	0.152476
$357$	20.5745	1.08892
$358$	4.31344	0.227972
$359$	−10.8302	−0.571598	−0.285799	−	0.958290i	$-0.592259\pi$
$359$	−10.8302	−0.571598	−0.285799	+	0.958290i	$0.592259\pi$
$360$	0	0
$361$	0	0
$362$	3.60584	0.189519
$363$	−2.70204	−0.141820
$364$	−20.4067	−1.06960
$365$	0	0
$366$	0.597468	0.0312302
$367$	−6.66237	−0.347773	−0.173887	−	0.984766i	$-0.555633\pi$
$367$	−6.66237	−0.347773	−0.173887	+	0.984766i	$0.555633\pi$
$368$	27.5468	1.43598
$369$	−14.3263	−0.745795
$370$	0	0
$371$	−26.8329	−1.39309
$372$	3.98896	0.206818
$373$	−18.1212	−0.938279	−0.469139	−	0.883124i	$-0.655436\pi$
$373$	−18.1212	−0.938279	−0.469139	+	0.883124i	$0.655436\pi$
$374$	1.76710	0.0913744
$375$	0	0
$376$	−0.173438	−0.00894438
$377$	3.62374	0.186632
$378$	−1.59998	−0.0822940
$379$	−31.9277	−1.64002	−0.820008	−	0.572352i	$-0.806031\pi$
$379$	−31.9277	−1.64002	−0.820008	+	0.572352i	$0.806031\pi$
$380$	0	0
$381$	−3.87236	−0.198387
$382$	2.01093	0.102888
$383$	13.7847	0.704367	0.352183	−	0.935931i	$-0.385439\pi$
$383$	13.7847	0.704367	0.352183	+	0.935931i	$0.385439\pi$
$384$	16.9866	0.866846
$385$	0	0
$386$	1.53259	0.0780069
$387$	4.07145	0.206963
$388$	29.8131	1.51353
$389$	−24.6819	−1.25142	−0.625712	−	0.780054i	$-0.715191\pi$
$389$	−24.6819	−1.25142	−0.625712	+	0.780054i	$0.715191\pi$
$390$	0	0
$391$	−17.1306	−0.866334
$392$	8.54101	0.431386
$393$	−17.6758	−0.891628
$394$	4.34965	0.219132
$395$	0	0
$396$	−13.9525	−0.701141
$397$	−17.8679	−0.896762	−0.448381	−	0.893842i	$-0.647999\pi$
$397$	−17.8679	−0.896762	−0.448381	+	0.893842i	$0.647999\pi$
$398$	−0.00264665	−0.000132665 0
$399$	0	0
$400$	0	0
$401$	22.9321	1.14517	0.572587	−	0.819844i	$-0.305940\pi$
$401$	22.9321	1.14517	0.572587	+	0.819844i	$0.305940\pi$
$402$	−8.74361	−0.436092
$403$	2.37864	0.118489
$404$	−11.4772	−0.571014
$405$	0	0
$406$	−1.34786	−0.0668930
$407$	20.7682	1.02944
$408$	−5.10892	−0.252929
$409$	−25.1805	−1.24509	−0.622547	−	0.782583i	$-0.713902\pi$
$409$	−25.1805	−1.24509	−0.622547	+	0.782583i	$0.713902\pi$
$410$	0	0
$411$	22.3131	1.10062
$412$	9.70239	0.478003
$413$	−30.9545	−1.52317
$414$	−4.35422	−0.213998
$415$	0	0
$416$	7.64101	0.374631
$417$	−14.7383	−0.721737
$418$	0	0
$419$	0.249578	0.0121927	0.00609635	−	0.999981i	$-0.498059\pi$
$419$	0.249578	0.0121927	0.00609635	+	0.999981i	$0.498059\pi$
$420$	0	0
$421$	−30.1682	−1.47031	−0.735154	−	0.677900i	$-0.762890\pi$
$421$	−30.1682	−1.47031	−0.735154	+	0.677900i	$0.762890\pi$
$422$	6.77479	0.329792
$423$	−0.405134	−0.0196983
$424$	6.66295	0.323581
$425$	0	0
$426$	5.99972	0.290687
$427$	4.11649	0.199211
$428$	−5.46110	−0.263972
$429$	−19.1855	−0.926283
$430$	0	0
$431$	−15.4165	−0.742588	−0.371294	−	0.928515i	$-0.621086\pi$
$431$	−15.4165	−0.742588	−0.371294	+	0.928515i	$0.621086\pi$
$432$	−5.87123	−0.282480
$433$	12.5378	0.602529	0.301264	−	0.953541i	$-0.402591\pi$
$433$	12.5378	0.602529	0.301264	+	0.953541i	$0.402591\pi$
$434$	−0.884741	−0.0424689
$435$	0	0
$436$	13.0744	0.626148
$437$	0	0
$438$	2.40339	0.114838
$439$	−25.3236	−1.20863	−0.604315	−	0.796746i	$-0.706553\pi$
$439$	−25.3236	−1.20863	−0.604315	+	0.796746i	$0.706553\pi$
$440$	0	0
$441$	19.9509	0.950045
$442$	−1.49911	−0.0713055
$443$	−7.41736	−0.352410	−0.176205	−	0.984354i	$-0.556382\pi$
$443$	−7.41736	−0.352410	−0.176205	+	0.984354i	$0.556382\pi$
$444$	−29.5462	−1.40220
$445$	0	0
$446$	1.91070	0.0904742
$447$	−4.47693	−0.211752
$448$	25.9077	1.22403
$449$	−24.0580	−1.13537	−0.567684	−	0.823247i	$-0.692160\pi$
$449$	−24.0580	−1.13537	−0.567684	+	0.823247i	$0.692160\pi$
$450$	0	0
$451$	−19.5491	−0.920529
$452$	−15.0483	−0.707812
$453$	38.4748	1.80771
$454$	−6.48396	−0.304307
$455$	0	0
$456$	0	0
$457$	6.39908	0.299337	0.149668	−	0.988736i	$-0.452179\pi$
$457$	6.39908	0.299337	0.149668	+	0.988736i	$0.452179\pi$
$458$	1.60372	0.0749371
$459$	3.65116	0.170422
$460$	0	0
$461$	−15.2501	−0.710268	−0.355134	−	0.934815i	$-0.615565\pi$
$461$	−15.2501	−0.710268	−0.355134	+	0.934815i	$0.615565\pi$
$462$	7.13607	0.332000
$463$	−36.4298	−1.69304	−0.846518	−	0.532359i	$-0.821306\pi$
$463$	−36.4298	−1.69304	−0.846518	+	0.532359i	$0.821306\pi$
$464$	−4.94605	−0.229615
$465$	0	0
$466$	−4.76910	−0.220924
$467$	28.1967	1.30479	0.652395	−	0.757880i	$-0.273765\pi$
$467$	28.1967	1.30479	0.652395	+	0.757880i	$0.273765\pi$
$468$	11.8366	0.547147
$469$	−60.2426	−2.78175
$470$	0	0
$471$	−22.5787	−1.04037
$472$	7.68640	0.353795
$473$	5.55574	0.255453
$474$	6.44279	0.295927
$475$	0	0
$476$	−17.3212	−0.793914
$477$	15.5640	0.712626
$478$	2.09585	0.0958620
$479$	29.4311	1.34474	0.672371	−	0.740215i	$-0.265276\pi$
$479$	29.4311	1.34474	0.672371	+	0.740215i	$0.265276\pi$
$480$	0	0
$481$	−17.6186	−0.803340
$482$	−2.15207	−0.0980243
$483$	−69.1787	−3.14774
$484$	2.27478	0.103399
$485$	0	0
$486$	4.88941	0.221788
$487$	−2.72017	−0.123263	−0.0616313	−	0.998099i	$-0.519630\pi$
$487$	−2.72017	−0.123263	−0.0616313	+	0.998099i	$0.519630\pi$
$488$	−1.02218	−0.0462718
$489$	−11.6593	−0.527251
$490$	0	0
$491$	18.9629	0.855783	0.427891	−	0.903830i	$-0.359257\pi$
$491$	18.9629	0.855783	0.427891	+	0.903830i	$0.359257\pi$
$492$	27.8118	1.25385
$493$	3.07582	0.138528
$494$	0	0
$495$	0	0
$496$	−3.24662	−0.145777
$497$	41.3375	1.85424
$498$	7.61875	0.341404
$499$	−6.02343	−0.269646	−0.134823	−	0.990870i	$-0.543047\pi$
$499$	−6.02343	−0.269646	−0.134823	+	0.990870i	$0.543047\pi$
$500$	0	0
$501$	−46.4020	−2.07309
$502$	−7.30818	−0.326180
$503$	18.6186	0.830164	0.415082	−	0.909784i	$-0.363753\pi$
$503$	18.6186	0.830164	0.415082	+	0.909784i	$0.363753\pi$
$504$	−8.94701	−0.398532
$505$	0	0
$506$	−5.94159	−0.264136
$507$	−13.6443	−0.605967
$508$	3.26004	0.144641
$509$	−26.1610	−1.15957	−0.579784	−	0.814770i	$-0.696863\pi$
$509$	−26.1610	−1.15957	−0.579784	+	0.814770i	$0.696863\pi$
$510$	0	0
$511$	16.5591	0.732532
$512$	−17.5682	−0.776411
$513$	0	0
$514$	1.36880	0.0603754
$515$	0	0
$516$	−7.90397	−0.347953
$517$	−0.552830	−0.0243134
$518$	6.55327	0.287934
$519$	4.48518	0.196877
$520$	0	0
$521$	3.06917	0.134463	0.0672315	−	0.997737i	$-0.478583\pi$
$521$	3.06917	0.134463	0.0672315	+	0.997737i	$0.478583\pi$
$522$	0.781803	0.0342186
$523$	−11.5164	−0.503576	−0.251788	−	0.967782i	$-0.581019\pi$
$523$	−11.5164	−0.503576	−0.251788	+	0.967782i	$0.581019\pi$
$524$	14.8808	0.650072
$525$	0	0
$526$	−0.141929	−0.00618838
$527$	2.01899	0.0879484
$528$	26.1863	1.13961
$529$	34.5992	1.50431
$530$	0	0
$531$	17.9547	0.779166
$532$	0	0
$533$	16.5844	0.718350
$534$	0.853468	0.0369332
$535$	0	0
$536$	14.9590	0.646131
$537$	−39.7502	−1.71535
$538$	−1.10967	−0.0478413
$539$	27.2243	1.17263
$540$	0	0
$541$	−29.8468	−1.28322	−0.641608	−	0.767033i	$-0.721732\pi$
$541$	−29.8468	−1.28322	−0.641608	+	0.767033i	$0.721732\pi$
$542$	−4.43708	−0.190589
$543$	−33.2294	−1.42601
$544$	6.48567	0.278071
$545$	0	0
$546$	−6.05386	−0.259081
$547$	1.17678	0.0503155	0.0251578	−	0.999683i	$-0.491991\pi$
$547$	1.17678	0.0503155	0.0251578	+	0.999683i	$0.491991\pi$
$548$	−18.7848	−0.802448
$549$	−2.38771	−0.101905
$550$	0	0
$551$	0	0
$552$	17.1780	0.731142
$553$	44.3902	1.88766
$554$	2.02410	0.0859959
$555$	0	0
$556$	12.4078	0.526207
$557$	−45.9826	−1.94834	−0.974172	−	0.225807i	$-0.927498\pi$
$557$	−45.9826	−1.94834	−0.974172	+	0.225807i	$0.927498\pi$
$558$	0.513180	0.0217246
$559$	−4.71319	−0.199347
$560$	0	0
$561$	−16.2846	−0.687535
$562$	1.76130	0.0742960
$563$	−23.7250	−0.999890	−0.499945	−	0.866057i	$-0.666647\pi$
$563$	−23.7250	−0.999890	−0.499945	+	0.866057i	$0.666647\pi$
$564$	0.786493	0.0331173
$565$	0	0
$566$	0.324331	0.0136327
$567$	42.0379	1.76543
$568$	−10.2646	−0.430694
$569$	−7.42753	−0.311378	−0.155689	−	0.987806i	$-0.549760\pi$
$569$	−7.42753	−0.311378	−0.155689	+	0.987806i	$0.549760\pi$
$570$	0	0
$571$	35.2355	1.47456	0.737279	−	0.675588i	$-0.236110\pi$
$571$	35.2355	1.47456	0.737279	+	0.675588i	$0.236110\pi$
$572$	16.1518	0.675339
$573$	−18.5316	−0.774169
$574$	−6.16859	−0.257472
$575$	0	0
$576$	−15.0274	−0.626140
$577$	31.5680	1.31419	0.657096	−	0.753807i	$-0.271784\pi$
$577$	31.5680	1.31419	0.657096	+	0.753807i	$0.271784\pi$
$578$	2.97332	0.123674
$579$	−14.1235	−0.586953
$580$	0	0
$581$	52.4924	2.17775
$582$	8.84436	0.366610
$583$	21.2380	0.879589
$584$	−4.11184	−0.170149
$585$	0	0
$586$	−3.73372	−0.154238
$587$	37.2187	1.53618	0.768090	−	0.640342i	$-0.221207\pi$
$587$	37.2187	1.53618	0.768090	+	0.640342i	$0.221207\pi$
$588$	−38.7311	−1.59725
$589$	0	0
$590$	0	0
$591$	−40.0839	−1.64883
$592$	24.0477	0.988354
$593$	−37.5290	−1.54113	−0.770566	−	0.637360i	$-0.780026\pi$
$593$	−37.5290	−1.54113	−0.770566	+	0.637360i	$0.780026\pi$
$594$	1.26637	0.0519598
$595$	0	0
$596$	3.76901	0.154385
$597$	0.0243900	0.000998217 0
$598$	5.04053	0.206123
$599$	−2.13127	−0.0870813	−0.0435406	−	0.999052i	$-0.513864\pi$
$599$	−2.13127	−0.0870813	−0.0435406	+	0.999052i	$0.513864\pi$
$600$	0	0
$601$	0.0853415	0.00348115	0.00174058	−	0.999998i	$-0.499446\pi$
$601$	0.0853415	0.00348115	0.00174058	+	0.999998i	$0.499446\pi$
$602$	1.75308	0.0714502
$603$	34.9428	1.42298
$604$	−32.3910	−1.31797
$605$	0	0
$606$	−3.40484	−0.138312
$607$	−21.0800	−0.855610	−0.427805	−	0.903871i	$-0.640713\pi$
$607$	−21.0800	−0.855610	−0.427805	+	0.903871i	$0.640713\pi$
$608$	0	0
$609$	12.4211	0.503328
$610$	0	0
$611$	0.468991	0.0189734
$612$	10.0469	0.406120
$613$	−17.8897	−0.722559	−0.361280	−	0.932458i	$-0.617660\pi$
$613$	−17.8897	−0.722559	−0.361280	+	0.932458i	$0.617660\pi$
$614$	−1.25218	−0.0505337
$615$	0	0
$616$	−12.2087	−0.491904
$617$	−1.52526	−0.0614045	−0.0307022	−	0.999529i	$-0.509774\pi$
$617$	−1.52526	−0.0614045	−0.0307022	+	0.999529i	$0.509774\pi$
$618$	2.87831	0.115783
$619$	11.8121	0.474769	0.237384	−	0.971416i	$-0.423710\pi$
$619$	11.8121	0.474769	0.237384	+	0.971416i	$0.423710\pi$
$620$	0	0
$621$	−12.2765	−0.492638
$622$	4.18190	0.167679
$623$	5.88031	0.235589
$624$	−22.2151	−0.889314
$625$	0	0
$626$	1.43321	0.0572827
$627$	0	0
$628$	19.0085	0.758520
$629$	−14.9546	−0.596280
$630$	0	0
$631$	8.02047	0.319290	0.159645	−	0.987174i	$-0.448965\pi$
$631$	8.02047	0.319290	0.159645	+	0.987174i	$0.448965\pi$
$632$	−11.0227	−0.438458
$633$	−62.4326	−2.48147
$634$	−1.98461	−0.0788188
$635$	0	0
$636$	−30.2147	−1.19809
$637$	−23.0956	−0.915083
$638$	1.06682	0.0422357
$639$	−23.9771	−0.948521
$640$	0	0
$641$	42.0011	1.65894	0.829472	−	0.558548i	$-0.188642\pi$
$641$	42.0011	1.65894	0.829472	+	0.558548i	$0.188642\pi$
$642$	−1.62009	−0.0639399
$643$	12.7588	0.503158	0.251579	−	0.967837i	$-0.419050\pi$
$643$	12.7588	0.503158	0.251579	+	0.967837i	$0.419050\pi$
$644$	58.2398	2.29497
$645$	0	0
$646$	0	0
$647$	44.1646	1.73629	0.868145	−	0.496311i	$-0.165312\pi$
$647$	44.1646	1.73629	0.868145	+	0.496311i	$0.165312\pi$
$648$	−10.4386	−0.410065
$649$	24.5002	0.961718
$650$	0	0
$651$	8.15327	0.319552
$652$	9.81565	0.384410
$653$	−21.7731	−0.852048	−0.426024	−	0.904712i	$-0.640086\pi$
$653$	−21.7731	−0.852048	−0.426024	+	0.904712i	$0.640086\pi$
$654$	3.87864	0.151667
$655$	0	0
$656$	−22.6361	−0.883790
$657$	−9.60485	−0.374721
$658$	−0.174442	−0.00680046
$659$	41.1821	1.60423	0.802114	−	0.597171i	$-0.203709\pi$
$659$	41.1821	1.60423	0.802114	+	0.597171i	$0.203709\pi$
$660$	0	0
$661$	13.6157	0.529588	0.264794	−	0.964305i	$-0.414696\pi$
$661$	13.6157	0.529588	0.264794	+	0.964305i	$0.414696\pi$
$662$	−5.19923	−0.202074
$663$	13.8150	0.536529
$664$	−13.0345	−0.505838
$665$	0	0
$666$	−3.80112	−0.147290
$667$	−10.3420	−0.400443
$668$	39.0646	1.51146
$669$	−17.6079	−0.680762
$670$	0	0
$671$	−3.25817	−0.125780
$672$	26.1911	1.01034
$673$	−16.0356	−0.618129	−0.309064	−	0.951041i	$-0.600016\pi$
$673$	−16.0356	−0.618129	−0.309064	+	0.951041i	$0.600016\pi$
$674$	6.83362	0.263221
$675$	0	0
$676$	11.4868	0.441801
$677$	21.8215	0.838668	0.419334	−	0.907832i	$-0.362264\pi$
$677$	21.8215	0.838668	0.419334	+	0.907832i	$0.362264\pi$
$678$	−4.46423	−0.171448
$679$	60.9367	2.33854
$680$	0	0
$681$	59.7525	2.28972
$682$	0.700265	0.0268145
$683$	13.5921	0.520088	0.260044	−	0.965597i	$-0.416263\pi$
$683$	13.5921	0.520088	0.260044	+	0.965597i	$0.416263\pi$
$684$	0	0
$685$	0	0
$686$	1.66663	0.0636323
$687$	−14.7790	−0.563855
$688$	6.43305	0.245258
$689$	−18.0172	−0.686401
$690$	0	0
$691$	18.1458	0.690300	0.345150	−	0.938547i	$-0.387828\pi$
$691$	18.1458	0.690300	0.345150	+	0.938547i	$0.387828\pi$
$692$	−3.77596	−0.143540
$693$	−28.5184	−1.08332
$694$	−5.04494	−0.191503
$695$	0	0
$696$	−3.08432	−0.116911
$697$	14.0768	0.533196
$698$	3.90210	0.147697
$699$	43.9493	1.66232
$700$	0	0
$701$	−26.8748	−1.01505	−0.507523	−	0.861638i	$-0.669439\pi$
$701$	−26.8748	−1.01505	−0.507523	+	0.861638i	$0.669439\pi$
$702$	−1.07432	−0.0405477
$703$	0	0
$704$	−20.5058	−0.772840
$705$	0	0
$706$	1.14821	0.0432134
$707$	−23.4590	−0.882267
$708$	−34.8557	−1.30996
$709$	20.6855	0.776861	0.388431	−	0.921478i	$-0.373017\pi$
$709$	20.6855	0.776861	0.388431	+	0.921478i	$0.373017\pi$
$710$	0	0
$711$	−25.7478	−0.965619
$712$	−1.46016	−0.0547216
$713$	−6.78853	−0.254232
$714$	−5.13850	−0.192304
$715$	0	0
$716$	33.4647	1.25063
$717$	−19.3142	−0.721301
$718$	2.70486	0.100944
$719$	−29.1011	−1.08529	−0.542644	−	0.839963i	$-0.682577\pi$
$719$	−29.1011	−1.08529	−0.542644	+	0.839963i	$0.682577\pi$
$720$	0	0
$721$	19.8313	0.738556
$722$	0	0
$723$	19.8323	0.737571
$724$	27.9750	1.03968
$725$	0	0
$726$	0.674837	0.0250455
$727$	10.8609	0.402808	0.201404	−	0.979508i	$-0.435450\pi$
$727$	10.8609	0.402808	0.201404	+	0.979508i	$0.435450\pi$
$728$	10.3573	0.383865
$729$	−13.2146	−0.489428
$730$	0	0
$731$	−4.00054	−0.147965
$732$	4.63530	0.171326
$733$	49.4817	1.82765	0.913825	−	0.406108i	$-0.133114\pi$
$733$	49.4817	1.82765	0.913825	+	0.406108i	$0.133114\pi$
$734$	1.66393	0.0614169
$735$	0	0
$736$	−21.8071	−0.803819
$737$	47.6815	1.75637
$738$	3.57799	0.131708
$739$	−31.5052	−1.15894	−0.579468	−	0.814995i	$-0.696740\pi$
$739$	−31.5052	−1.15894	−0.579468	+	0.814995i	$0.696740\pi$
$740$	0	0
$741$	0	0
$742$	6.70153	0.246021
$743$	35.6669	1.30849	0.654246	−	0.756282i	$-0.272986\pi$
$743$	35.6669	1.30849	0.654246	+	0.756282i	$0.272986\pi$
$744$	−2.02456	−0.0742240
$745$	0	0
$746$	4.52577	0.165700
$747$	−30.4474	−1.11401
$748$	13.7096	0.501271
$749$	−11.1623	−0.407860
$750$	0	0
$751$	2.10024	0.0766389	0.0383194	−	0.999266i	$-0.487800\pi$
$751$	2.10024	0.0766389	0.0383194	+	0.999266i	$0.487800\pi$
$752$	−0.640127	−0.0233430
$753$	67.3481	2.45430
$754$	−0.905032	−0.0329593
$755$	0	0
$756$	−12.4130	−0.451457
$757$	1.15817	0.0420945	0.0210473	−	0.999778i	$-0.493300\pi$
$757$	1.15817	0.0420945	0.0210473	+	0.999778i	$0.493300\pi$
$758$	7.97396	0.289627
$759$	54.7544	1.98746
$760$	0	0
$761$	−44.6406	−1.61822	−0.809111	−	0.587656i	$-0.800051\pi$
$761$	−44.6406	−1.61822	−0.809111	+	0.587656i	$0.800051\pi$
$762$	0.967123	0.0350352
$763$	26.7235	0.967454
$764$	15.6013	0.564435
$765$	0	0
$766$	−3.44275	−0.124391
$767$	−20.7847	−0.750492
$768$	25.8696	0.933489
$769$	29.7532	1.07293	0.536465	−	0.843923i	$-0.319759\pi$
$769$	29.7532	1.07293	0.536465	+	0.843923i	$0.319759\pi$
$770$	0	0
$771$	−12.6141	−0.454287
$772$	11.8902	0.427938
$773$	13.6612	0.491360	0.245680	−	0.969351i	$-0.420989\pi$
$773$	13.6612	0.491360	0.245680	+	0.969351i	$0.420989\pi$
$774$	−1.01685	−0.0365498
$775$	0	0
$776$	−15.1314	−0.543185
$777$	−60.3913	−2.16652
$778$	6.16433	0.221002
$779$	0	0
$780$	0	0
$781$	−32.7183	−1.17075
$782$	4.27839	0.152995
$783$	2.20425	0.0787736
$784$	31.5233	1.12583
$785$	0	0
$786$	4.41455	0.157462
$787$	−49.9989	−1.78227	−0.891134	−	0.453740i	$-0.850089\pi$
$787$	−49.9989	−1.78227	−0.891134	+	0.453740i	$0.850089\pi$
$788$	33.7456	1.20214
$789$	1.30793	0.0465637
$790$	0	0
$791$	−30.7581	−1.09363
$792$	7.08149	0.251630
$793$	2.76406	0.0981547
$794$	4.46251	0.158369
$795$	0	0
$796$	−0.0205333	−0.000727785 0
$797$	−11.0649	−0.391939	−0.195969	−	0.980610i	$-0.562785\pi$
$797$	−11.0649	−0.391939	−0.195969	+	0.980610i	$0.562785\pi$
$798$	0	0
$799$	0.398078	0.0140830
$800$	0	0
$801$	−3.41078	−0.120514
$802$	−5.72730	−0.202238
$803$	−13.1064	−0.462515
$804$	−67.8350	−2.39236
$805$	0	0
$806$	−0.594068	−0.0209251
$807$	10.2261	0.359976
$808$	5.82518	0.204929
$809$	6.05426	0.212857	0.106428	−	0.994320i	$-0.466059\pi$
$809$	6.05426	0.212857	0.106428	+	0.994320i	$0.466059\pi$
$810$	0	0
$811$	−6.97028	−0.244760	−0.122380	−	0.992483i	$-0.539053\pi$
$811$	−6.97028	−0.244760	−0.122380	+	0.992483i	$0.539053\pi$
$812$	−10.4570	−0.366969
$813$	40.8896	1.43406
$814$	−5.18686	−0.181799
$815$	0	0
$816$	−18.8561	−0.660095
$817$	0	0
$818$	6.28883	0.219884
$819$	24.1935	0.845390
$820$	0	0
$821$	9.65335	0.336904	0.168452	−	0.985710i	$-0.446123\pi$
$821$	9.65335	0.336904	0.168452	+	0.985710i	$0.446123\pi$
$822$	−5.57271	−0.194371
$823$	14.4949	0.505261	0.252631	−	0.967563i	$-0.418704\pi$
$823$	14.4949	0.505261	0.252631	+	0.967563i	$0.418704\pi$
$824$	−4.92436	−0.171548
$825$	0	0
$826$	7.73090	0.268993
$827$	−27.7016	−0.963279	−0.481639	−	0.876369i	$-0.659959\pi$
$827$	−27.7016	−0.963279	−0.481639	+	0.876369i	$0.659959\pi$
$828$	−33.7810	−1.17397
$829$	32.2071	1.11860	0.559299	−	0.828966i	$-0.311070\pi$
$829$	32.2071	1.11860	0.559299	+	0.828966i	$0.311070\pi$
$830$	0	0
$831$	−18.6530	−0.647065
$832$	17.3960	0.603098
$833$	−19.6035	−0.679221
$834$	3.68090	0.127459
$835$	0	0
$836$	0	0
$837$	1.44688	0.0500116
$838$	−0.0623324	−0.00215324
$839$	−45.1856	−1.55998	−0.779990	−	0.625791i	$-0.784776\pi$
$839$	−45.1856	−1.55998	−0.779990	+	0.625791i	$0.784776\pi$
$840$	0	0
$841$	−27.1431	−0.935969
$842$	7.53453	0.259657
$843$	−16.2312	−0.559031
$844$	52.5604	1.80920
$845$	0	0
$846$	0.101182	0.00347872
$847$	4.64956	0.159761
$848$	24.5917	0.844483
$849$	−2.98885	−0.102577
$850$	0	0
$851$	50.2826	1.72367
$852$	46.5473	1.59468
$853$	−41.8087	−1.43150	−0.715751	−	0.698355i	$-0.753916\pi$
$853$	−41.8087	−1.43150	−0.715751	+	0.698355i	$0.753916\pi$
$854$	−1.02810	−0.0351808
$855$	0	0
$856$	2.77173	0.0947358
$857$	42.4772	1.45099	0.725497	−	0.688226i	$-0.241610\pi$
$857$	42.4772	1.45099	0.725497	+	0.688226i	$0.241610\pi$
$858$	4.79158	0.163582
$859$	2.27865	0.0777467	0.0388733	−	0.999244i	$-0.487623\pi$
$859$	2.27865	0.0777467	0.0388733	+	0.999244i	$0.487623\pi$
$860$	0	0
$861$	56.8463	1.93732
$862$	3.85029	0.131141
$863$	41.6831	1.41891	0.709455	−	0.704751i	$-0.248941\pi$
$863$	41.6831	1.41891	0.709455	+	0.704751i	$0.248941\pi$
$864$	4.64788	0.158124
$865$	0	0
$866$	−3.13133	−0.106407
$867$	−27.4004	−0.930567
$868$	−6.86403	−0.232980
$869$	−35.1345	−1.19186
$870$	0	0
$871$	−40.4505	−1.37061
$872$	−6.63578	−0.224716
$873$	−35.3454	−1.19626
$874$	0	0
$875$	0	0
$876$	18.6461	0.629992
$877$	−5.34370	−0.180444	−0.0902220	−	0.995922i	$-0.528758\pi$
$877$	−5.34370	−0.180444	−0.0902220	+	0.995922i	$0.528758\pi$
$878$	6.32458	0.213444
$879$	34.4078	1.16055
$880$	0	0
$881$	19.1989	0.646827	0.323414	−	0.946258i	$-0.395169\pi$
$881$	19.1989	0.646827	0.323414	+	0.946258i	$0.395169\pi$
$882$	−4.98276	−0.167778
$883$	28.7091	0.966138	0.483069	−	0.875582i	$-0.339522\pi$
$883$	28.7091	0.966138	0.483069	+	0.875582i	$0.339522\pi$
$884$	−11.6305	−0.391175
$885$	0	0
$886$	1.85249	0.0622357
$887$	10.1948	0.342307	0.171153	−	0.985244i	$-0.445251\pi$
$887$	10.1948	0.342307	0.171153	+	0.985244i	$0.445251\pi$
$888$	14.9959	0.503230
$889$	6.66338	0.223483
$890$	0	0
$891$	−33.2727	−1.11468
$892$	14.8237	0.496333
$893$	0	0
$894$	1.11812	0.0373954
$895$	0	0
$896$	−29.2299	−0.976501
$897$	−46.4507	−1.55094
$898$	6.00850	0.200506
$899$	1.21889	0.0406521
$900$	0	0
$901$	−15.2930	−0.509482
$902$	4.88239	0.162566
$903$	−16.1554	−0.537618
$904$	7.63763	0.254024
$905$	0	0
$906$	−9.60911	−0.319241
$907$	−39.0849	−1.29779	−0.648896	−	0.760877i	$-0.724769\pi$
$907$	−39.0849	−1.29779	−0.648896	+	0.760877i	$0.724769\pi$
$908$	−50.3041	−1.66940
$909$	13.6070	0.451317
$910$	0	0
$911$	−36.6728	−1.21502	−0.607512	−	0.794310i	$-0.707832\pi$
$911$	−36.6728	−1.21502	−0.607512	+	0.794310i	$0.707832\pi$
$912$	0	0
$913$	−41.5473	−1.37502
$914$	−1.59818	−0.0528629
$915$	0	0
$916$	12.4421	0.411098
$917$	30.4158	1.00442
$918$	−0.911881	−0.0300965
$919$	−20.4855	−0.675753	−0.337877	−	0.941190i	$-0.609709\pi$
$919$	−20.4855	−0.675753	−0.337877	+	0.941190i	$0.609709\pi$
$920$	0	0
$921$	11.5394	0.380235
$922$	3.80872	0.125434
$923$	27.7564	0.913615
$924$	55.3633	1.82132
$925$	0	0
$926$	9.09837	0.298991
$927$	−11.5028	−0.377802
$928$	3.91547	0.128532
$929$	17.3877	0.570472	0.285236	−	0.958457i	$-0.407928\pi$
$929$	17.3877	0.570472	0.285236	+	0.958457i	$0.407928\pi$
$930$	0	0
$931$	0	0
$932$	−36.9998	−1.21197
$933$	−38.5380	−1.26168
$934$	−7.04215	−0.230426
$935$	0	0
$936$	−6.00756	−0.196363
$937$	33.2075	1.08484	0.542421	−	0.840107i	$-0.317508\pi$
$937$	33.2075	1.08484	0.542421	+	0.840107i	$0.317508\pi$
$938$	15.0456	0.491257
$939$	−13.2077	−0.431016
$940$	0	0
$941$	36.7788	1.19895	0.599477	−	0.800392i	$-0.295375\pi$
$941$	36.7788	1.19895	0.599477	+	0.800392i	$0.295375\pi$
$942$	5.63906	0.183730
$943$	−47.3310	−1.54131
$944$	28.3691	0.923335
$945$	0	0
$946$	−1.38755	−0.0451131
$947$	34.9882	1.13696	0.568482	−	0.822696i	$-0.307531\pi$
$947$	34.9882	1.13696	0.568482	+	0.822696i	$0.307531\pi$
$948$	49.9847	1.62343
$949$	11.1188	0.360931
$950$	0	0
$951$	18.2890	0.593062
$952$	8.79120	0.284925
$953$	−53.8985	−1.74594	−0.872972	−	0.487771i	$-0.837810\pi$
$953$	−53.8985	−1.74594	−0.872972	+	0.487771i	$0.837810\pi$
$954$	−3.88712	−0.125850
$955$	0	0
$956$	16.2601	0.525890
$957$	−9.83119	−0.317797
$958$	−7.35043	−0.237482
$959$	−38.3954	−1.23985
$960$	0	0
$961$	−30.1999	−0.974191
$962$	4.40026	0.141870
$963$	6.47449	0.208637
$964$	−16.6963	−0.537752
$965$	0	0
$966$	17.2774	0.555892
$967$	49.8831	1.60413	0.802066	−	0.597236i	$-0.203734\pi$
$967$	49.8831	1.60413	0.802066	+	0.597236i	$0.203734\pi$
$968$	−1.15454	−0.0371084
$969$	0	0
$970$	0	0
$971$	−13.8066	−0.443074	−0.221537	−	0.975152i	$-0.571107\pi$
$971$	−13.8066	−0.443074	−0.221537	+	0.975152i	$0.571107\pi$
$972$	37.9332	1.21671
$973$	25.3610	0.813037
$974$	0.679364	0.0217682
$975$	0	0
$976$	−3.77267	−0.120760
$977$	31.6433	1.01236	0.506180	−	0.862428i	$-0.331057\pi$
$977$	31.6433	1.01236	0.506180	+	0.862428i	$0.331057\pi$
$978$	2.91191	0.0931127
$979$	−4.65421	−0.148749
$980$	0	0
$981$	−15.5005	−0.494893
$982$	−4.73599	−0.151132
$983$	−40.1898	−1.28186	−0.640928	−	0.767601i	$-0.721450\pi$
$983$	−40.1898	−1.28186	−0.640928	+	0.767601i	$0.721450\pi$
$984$	−14.1157	−0.449991
$985$	0	0
$986$	−0.768188	−0.0244641
$987$	1.60756	0.0511692
$988$	0	0
$989$	13.4512	0.427724
$990$	0	0
$991$	19.4757	0.618667	0.309333	−	0.950954i	$-0.399894\pi$
$991$	19.4757	0.618667	0.309333	+	0.950954i	$0.399894\pi$
$992$	2.57014	0.0816020
$993$	47.9131	1.52048
$994$	−10.3241	−0.327459
$995$	0	0
$996$	59.1081	1.87291
$997$	−14.1426	−0.447901	−0.223950	−	0.974601i	$-0.571895\pi$
$997$	−14.1426	−0.447901	−0.223950	+	0.974601i	$0.571895\pi$
$998$	1.50436	0.0476195
$999$	−10.7171	−0.339073

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9025.2.a.ct.1.11		24
5.2	odd	4		1805.2.b.l.1084.11		24
5.3	odd	4		1805.2.b.l.1084.14		24
5.4	even	2	inner	9025.2.a.ct.1.14		24
19.2	odd	18		475.2.l.f.251.4		48
19.10	odd	18		475.2.l.f.176.4		48
19.18	odd	2		9025.2.a.cu.1.14		24
95.2	even	36		95.2.p.a.4.5	yes	48
95.18	even	4		1805.2.b.k.1084.11		24
95.29	odd	18		475.2.l.f.176.5		48
95.37	even	4		1805.2.b.k.1084.14		24
95.48	even	36		95.2.p.a.24.5	yes	48
95.59	odd	18		475.2.l.f.251.5		48
95.67	even	36		95.2.p.a.24.4	yes	48
95.78	even	36		95.2.p.a.4.4	✓	48
95.94	odd	2		9025.2.a.cu.1.11		24
285.2	odd	36		855.2.da.b.289.4		48
285.143	odd	36		855.2.da.b.784.4		48
285.173	odd	36		855.2.da.b.289.5		48
285.257	odd	36		855.2.da.b.784.5		48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
95.2.p.a.4.4	✓	48	95.78	even	36
95.2.p.a.4.5	yes	48	95.2	even	36
95.2.p.a.24.4	yes	48	95.67	even	36
95.2.p.a.24.5	yes	48	95.48	even	36
475.2.l.f.176.4		48	19.10	odd	18
475.2.l.f.176.5		48	95.29	odd	18
475.2.l.f.251.4		48	19.2	odd	18
475.2.l.f.251.5		48	95.59	odd	18
855.2.da.b.289.4		48	285.2	odd	36
855.2.da.b.289.5		48	285.173	odd	36
855.2.da.b.784.4		48	285.143	odd	36
855.2.da.b.784.5		48	285.257	odd	36
1805.2.b.k.1084.11		24	95.18	even	4
1805.2.b.k.1084.14		24	95.37	even	4
1805.2.b.l.1084.11		24	5.2	odd	4
1805.2.b.l.1084.14		24	5.3	odd	4
9025.2.a.ct.1.11		24	1.1	even	1	trivial
9025.2.a.ct.1.14		24	5.4	even	2	inner
9025.2.a.cu.1.11		24	95.94	odd	2
9025.2.a.cu.1.14		24	19.18	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 9025 = 5^{2} \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9025.a (trivial)

\(f(q)\)	\(=\)	\(q-0.249751 q^{2} +2.30156 q^{3} -1.93762 q^{4} -0.574816 q^{6} -3.96043 q^{7} +0.983424 q^{8} +2.29718 q^{9} +O(q^{10})\)	\(q-0.249751 q^{2} +2.30156 q^{3} -1.93762 q^{4} -0.574816 q^{6} -3.96043 q^{7} +0.983424 q^{8} +2.29718 q^{9} +3.13464 q^{11} -4.45956 q^{12} -2.65927 q^{13} +0.989119 q^{14} +3.62964 q^{16} -2.25718 q^{17} -0.573722 q^{18} -9.11516 q^{21} -0.782879 q^{22} +7.58941 q^{23} +2.26341 q^{24} +0.664153 q^{26} -1.61758 q^{27} +7.67382 q^{28} -1.36269 q^{29} -0.894474 q^{31} -2.87335 q^{32} +7.21457 q^{33} +0.563731 q^{34} -4.45107 q^{36} +6.62537 q^{37} -6.12046 q^{39} -6.23645 q^{41} +2.27652 q^{42} +1.77237 q^{43} -6.07376 q^{44} -1.89546 q^{46} -0.176361 q^{47} +8.35383 q^{48} +8.68497 q^{49} -5.19503 q^{51} +5.15266 q^{52} +6.77526 q^{53} +0.403992 q^{54} -3.89478 q^{56} +0.340331 q^{58} +7.81595 q^{59} -1.03941 q^{61} +0.223395 q^{62} -9.09781 q^{63} -6.54166 q^{64} -1.80184 q^{66} +15.2111 q^{67} +4.37356 q^{68} +17.4675 q^{69} -10.4376 q^{71} +2.25910 q^{72} -4.18114 q^{73} -1.65469 q^{74} -12.4145 q^{77} +1.52859 q^{78} -11.2084 q^{79} -10.6145 q^{81} +1.55756 q^{82} -13.2542 q^{83} +17.6618 q^{84} -0.442650 q^{86} -3.13630 q^{87} +3.08269 q^{88} -1.48477 q^{89} +10.5318 q^{91} -14.7054 q^{92} -2.05869 q^{93} +0.0440463 q^{94} -6.61320 q^{96} -15.3864 q^{97} -2.16908 q^{98} +7.20085 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 18 q^{4} - 12 q^{6} + 12 q^{9}+O(q^{10}) \) 24 * q + 18 * q^4 - 12 * q^6 + 12 * q^9	\( 24 q + 18 q^{4} - 12 q^{6} + 12 q^{9} + 12 q^{11} - 24 q^{14} + 6 q^{16} - 6 q^{21} - 42 q^{24} - 12 q^{26} - 36 q^{29} - 42 q^{31} - 6 q^{34} - 6 q^{36} + 24 q^{39} - 60 q^{41} - 30 q^{44} - 6 q^{46} + 12 q^{49} - 30 q^{51} - 24 q^{54} - 18 q^{56} - 60 q^{59} + 30 q^{61} + 36 q^{66} - 66 q^{69} - 96 q^{71} + 24 q^{74} - 72 q^{79} - 96 q^{81} + 54 q^{84} - 108 q^{86} - 84 q^{89} - 96 q^{91} - 36 q^{94} - 120 q^{96}+O(q^{100}) \) 24 * q + 18 * q^4 - 12 * q^6 + 12 * q^9 + 12 * q^11 - 24 * q^14 + 6 * q^16 - 6 * q^21 - 42 * q^24 - 12 * q^26 - 36 * q^29 - 42 * q^31 - 6 * q^34 - 6 * q^36 + 24 * q^39 - 60 * q^41 - 30 * q^44 - 6 * q^46 + 12 * q^49 - 30 * q^51 - 24 * q^54 - 18 * q^56 - 60 * q^59 + 30 * q^61 + 36 * q^66 - 66 * q^69 - 96 * q^71 + 24 * q^74 - 72 * q^79 - 96 * q^81 + 54 * q^84 - 108 * q^86 - 84 * q^89 - 96 * q^91 - 36 * q^94 - 120 * q^96

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more