LMFDB - Embedded newform 9025.2.a.ck.1.1

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:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 22x^{14} + 190x^{12} - 820x^{10} + 1862x^{8} - 2154x^{6} + 1163x^{4} - 256x^{2} + 16 $ x^16 - 22x^14 + 190x^12 - 820x^10 + 1862x^8 - 2154x^6 + 1163x^4 - 256*x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 1805)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.61137$ of defining polynomial
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)

$f(q)$	$=$	$q-2.61137 q^{2} -0.146779 q^{3} +4.81924 q^{4} +0.383293 q^{6} -1.14057 q^{7} -7.36208 q^{8} -2.97846 q^{9} +O(q^{10})$	\(q-2.61137 q^{2} -0.146779 q^{3} +4.81924 q^{4} +0.383293 q^{6} -1.14057 q^{7} -7.36208 q^{8} -2.97846 q^{9} -5.36995 q^{11} -0.707361 q^{12} +2.41565 q^{13} +2.97846 q^{14} +9.58662 q^{16} +5.74451 q^{17} +7.77785 q^{18} +0.167412 q^{21} +14.0229 q^{22} +1.23526 q^{23} +1.08060 q^{24} -6.30815 q^{26} +0.877509 q^{27} -5.49670 q^{28} -3.91460 q^{29} +6.48369 q^{31} -10.3100 q^{32} +0.788193 q^{33} -15.0010 q^{34} -14.3539 q^{36} +7.37637 q^{37} -0.354565 q^{39} +4.86252 q^{41} -0.437173 q^{42} -7.59744 q^{43} -25.8791 q^{44} -3.22572 q^{46} -7.85368 q^{47} -1.40711 q^{48} -5.69909 q^{49} -0.843170 q^{51} +11.6416 q^{52} -0.0179026 q^{53} -2.29150 q^{54} +8.39699 q^{56} +10.2225 q^{58} -13.0250 q^{59} -0.0189653 q^{61} -16.9313 q^{62} +3.39715 q^{63} +7.75004 q^{64} -2.05826 q^{66} +7.07711 q^{67} +27.6842 q^{68} -0.181310 q^{69} -2.73751 q^{71} +21.9276 q^{72} -5.58836 q^{73} -19.2624 q^{74} +6.12482 q^{77} +0.925901 q^{78} -9.56774 q^{79} +8.80657 q^{81} -12.6978 q^{82} +0.390810 q^{83} +0.806797 q^{84} +19.8397 q^{86} +0.574579 q^{87} +39.5340 q^{88} -3.57717 q^{89} -2.75522 q^{91} +5.95303 q^{92} -0.951666 q^{93} +20.5088 q^{94} +1.51329 q^{96} +14.7430 q^{97} +14.8824 q^{98} +15.9942 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 12 q^{4} - 10 q^{6} + 6 q^{9}+O(q^{10}) $ 16 * q + 12 * q^4 - 10 * q^6 + 6 * q^9	$ 16 q + 12 q^{4} - 10 q^{6} + 6 q^{9} - 22 q^{11} - 6 q^{14} + 8 q^{16} + 20 q^{21} - 14 q^{24} - 16 q^{26} - 2 q^{29} - 16 q^{31} + 8 q^{34} + 18 q^{36} - 36 q^{39} - 26 q^{41} - 64 q^{44} - 2 q^{46} - 20 q^{49} + 38 q^{51} - 12 q^{54} - 6 q^{56} - 10 q^{59} - 30 q^{61} - 16 q^{64} + 4 q^{66} - 68 q^{69} + 20 q^{71} - 40 q^{74} - 12 q^{79} - 48 q^{81} + 2 q^{84} + 20 q^{86} + 86 q^{91} + 38 q^{94} + 22 q^{96} - 32 q^{99}+O(q^{100}) $ 16 * q + 12 * q^4 - 10 * q^6 + 6 * q^9 - 22 * q^11 - 6 * q^14 + 8 * q^16 + 20 * q^21 - 14 * q^24 - 16 * q^26 - 2 * q^29 - 16 * q^31 + 8 * q^34 + 18 * q^36 - 36 * q^39 - 26 * q^41 - 64 * q^44 - 2 * q^46 - 20 * q^49 + 38 * q^51 - 12 * q^54 - 6 * q^56 - 10 * q^59 - 30 * q^61 - 16 * q^64 + 4 * q^66 - 68 * q^69 + 20 * q^71 - 40 * q^74 - 12 * q^79 - 48 * q^81 + 2 * q^84 + 20 * q^86 + 86 * q^91 + 38 * q^94 + 22 * q^96 - 32 * q^99

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 $

$2$	−2.61137	−1.84652	−0.923258	−	0.384180i	$-0.874484\pi$
$2$	−2.61137	−1.84652	−0.923258	+	0.384180i	$0.874484\pi$
$3$	−0.146779	−0.0847426	−0.0423713	−	0.999102i	$-0.513491\pi$
$3$	−0.146779	−0.0847426	−0.0423713	+	0.999102i	$0.513491\pi$
$4$	4.81924	2.40962
$5$	0	0
$5$	0	0
$6$	0.383293	0.156479
$7$	−1.14057	−0.431096	−0.215548	−	0.976493i	$-0.569154\pi$
$7$	−1.14057	−0.431096	−0.215548	+	0.976493i	$0.569154\pi$
$8$	−7.36208	−2.60289
$9$	−2.97846	−0.992819
$10$	0	0
$11$	−5.36995	−1.61910	−0.809550	−	0.587051i	$-0.800289\pi$
$11$	−5.36995	−1.61910	−0.809550	+	0.587051i	$0.800289\pi$
$12$	−0.707361	−0.204198
$13$	2.41565	0.669980	0.334990	−	0.942222i	$-0.391267\pi$
$13$	2.41565	0.669980	0.334990	+	0.942222i	$0.391267\pi$
$14$	2.97846	0.796026
$15$	0	0
$16$	9.58662	2.39665
$17$	5.74451	1.39325	0.696624	−	0.717436i	$-0.254685\pi$
$17$	5.74451	1.39325	0.696624	+	0.717436i	$0.254685\pi$
$18$	7.77785	1.83326
$19$	0	0
$19$	0	0
$20$	0	0
$21$	0.167412	0.0365322
$22$	14.0229	2.98969
$23$	1.23526	0.257570	0.128785	−	0.991673i	$-0.458892\pi$
$23$	1.23526	0.257570	0.128785	+	0.991673i	$0.458892\pi$
$24$	1.08060	0.220576
$25$	0	0
$26$	−6.30815	−1.23713
$27$	0.877509	0.168877
$28$	−5.49670	−1.03878
$29$	−3.91460	−0.726923	−0.363461	−	0.931609i	$-0.618405\pi$
$29$	−3.91460	−0.726923	−0.363461	+	0.931609i	$0.618405\pi$
$30$	0	0
$31$	6.48369	1.16450	0.582252	−	0.813008i	$-0.302172\pi$
$31$	6.48369	1.16450	0.582252	+	0.813008i	$0.302172\pi$
$32$	−10.3100	−1.82257
$33$	0.788193	0.137207
$34$	−15.0010	−2.57265
$35$	0	0
$36$	−14.3539	−2.39232
$37$	7.37637	1.21267	0.606334	−	0.795210i	$-0.292639\pi$
$37$	7.37637	1.21267	0.606334	+	0.795210i	$0.292639\pi$
$38$	0	0
$39$	−0.354565	−0.0567759
$40$	0	0
$41$	4.86252	0.759398	0.379699	−	0.925110i	$-0.376028\pi$
$41$	4.86252	0.759398	0.379699	+	0.925110i	$0.376028\pi$
$42$	−0.437173	−0.0674573
$43$	−7.59744	−1.15860	−0.579299	−	0.815115i	$-0.696674\pi$
$43$	−7.59744	−1.15860	−0.579299	+	0.815115i	$0.696674\pi$
$44$	−25.8791	−3.90142
$45$	0	0
$46$	−3.22572	−0.475607
$47$	−7.85368	−1.14558	−0.572788	−	0.819703i	$-0.694138\pi$
$47$	−7.85368	−1.14558	−0.572788	+	0.819703i	$0.694138\pi$
$48$	−1.40711	−0.203099
$49$	−5.69909	−0.814156
$50$	0	0
$51$	−0.843170	−0.118067
$52$	11.6416	1.61440
$53$	−0.0179026	−0.00245911	−0.00122955	−	0.999999i	$-0.500391\pi$
$53$	−0.0179026	−0.00245911	−0.00122955	+	0.999999i	$0.500391\pi$
$54$	−2.29150	−0.311833
$55$	0	0
$56$	8.39699	1.12210
$57$	0	0
$58$	10.2225	1.34227
$59$	−13.0250	−1.69571	−0.847857	−	0.530225i	$-0.822108\pi$
$59$	−13.0250	−1.69571	−0.847857	+	0.530225i	$0.822108\pi$
$60$	0	0
$61$	−0.0189653	−0.00242826	−0.00121413	−	0.999999i	$-0.500386\pi$
$61$	−0.0189653	−0.00242826	−0.00121413	+	0.999999i	$0.500386\pi$
$62$	−16.9313	−2.15028
$63$	3.39715	0.428000
$64$	7.75004	0.968754
$65$	0	0
$66$	−2.05826	−0.253355
$67$	7.07711	0.864606	0.432303	−	0.901728i	$-0.357701\pi$
$67$	7.07711	0.864606	0.432303	+	0.901728i	$0.357701\pi$
$68$	27.6842	3.35720
$69$	−0.181310	−0.0218271
$70$	0	0
$71$	−2.73751	−0.324883	−0.162442	−	0.986718i	$-0.551937\pi$
$71$	−2.73751	−0.324883	−0.162442	+	0.986718i	$0.551937\pi$
$72$	21.9276	2.58420
$73$	−5.58836	−0.654068	−0.327034	−	0.945013i	$-0.606049\pi$
$73$	−5.58836	−0.654068	−0.327034	+	0.945013i	$0.606049\pi$
$74$	−19.2624	−2.23921
$75$	0	0
$76$	0	0
$77$	6.12482	0.697988
$78$	0.925901	0.104838
$79$	−9.56774	−1.07645	−0.538227	−	0.842800i	$-0.680906\pi$
$79$	−9.56774	−1.07645	−0.538227	+	0.842800i	$0.680906\pi$
$80$	0	0
$81$	8.80657	0.978508
$82$	−12.6978	−1.40224
$83$	0.390810	0.0428970	0.0214485	−	0.999770i	$-0.493172\pi$
$83$	0.390810	0.0428970	0.0214485	+	0.999770i	$0.493172\pi$
$84$	0.806797	0.0880288
$85$	0	0
$86$	19.8397	2.13937
$87$	0.574579	0.0616014
$88$	39.5340	4.21434
$89$	−3.57717	−0.379179	−0.189590	−	0.981863i	$-0.560716\pi$
$89$	−3.57717	−0.379179	−0.189590	+	0.981863i	$0.560716\pi$
$90$	0	0
$91$	−2.75522	−0.288826
$92$	5.95303	0.620646
$93$	−0.951666	−0.0986832
$94$	20.5088	2.11532
$95$	0	0
$96$	1.51329	0.154450
$97$	14.7430	1.49693	0.748463	−	0.663176i	$-0.230792\pi$
$97$	14.7430	1.49693	0.748463	+	0.663176i	$0.230792\pi$
$98$	14.8824	1.50335
$99$	15.9942	1.60747
$100$	0	0
$101$	−10.5437	−1.04914	−0.524569	−	0.851368i	$-0.675774\pi$
$101$	−10.5437	−1.04914	−0.524569	+	0.851368i	$0.675774\pi$
$102$	2.20183	0.218013
$103$	9.12063	0.898682	0.449341	−	0.893360i	$-0.351659\pi$
$103$	9.12063	0.898682	0.449341	+	0.893360i	$0.351659\pi$
$104$	−17.7842	−1.74388
$105$	0	0
$106$	0.0467502	0.00454078
$107$	−3.56368	−0.344514	−0.172257	−	0.985052i	$-0.555106\pi$
$107$	−3.56368	−0.344514	−0.172257	+	0.985052i	$0.555106\pi$
$108$	4.22893	0.406929
$109$	10.6645	1.02147	0.510736	−	0.859737i	$-0.329373\pi$
$109$	10.6645	1.02147	0.510736	+	0.859737i	$0.329373\pi$
$110$	0	0
$111$	−1.08269	−0.102765
$112$	−10.9342	−1.03319
$113$	12.8769	1.21135	0.605676	−	0.795711i	$-0.292903\pi$
$113$	12.8769	1.21135	0.605676	+	0.795711i	$0.292903\pi$
$114$	0	0
$115$	0	0
$116$	−18.8654	−1.75161
$117$	−7.19490	−0.665169
$118$	34.0131	3.13116
$119$	−6.55203	−0.600624
$120$	0	0
$121$	17.8363	1.62149
$122$	0.0495254	0.00448382
$123$	−0.713713	−0.0643534
$124$	31.2465	2.80602
$125$	0	0
$126$	−8.87120	−0.790309
$127$	7.57260	0.671960	0.335980	−	0.941869i	$-0.390933\pi$
$127$	7.57260	0.671960	0.335980	+	0.941869i	$0.390933\pi$
$128$	0.381857	0.0337517
$129$	1.11514	0.0981827
$130$	0	0
$131$	11.2785	0.985406	0.492703	−	0.870197i	$-0.336009\pi$
$131$	11.2785	0.985406	0.492703	+	0.870197i	$0.336009\pi$
$132$	3.79849	0.330616
$133$	0	0
$134$	−18.4809	−1.59651
$135$	0	0
$136$	−42.2915	−3.62647
$137$	13.8445	1.18282	0.591408	−	0.806373i	$-0.298572\pi$
$137$	13.8445	1.18282	0.591408	+	0.806373i	$0.298572\pi$
$138$	0.473467	0.0403042
$139$	−1.97238	−0.167295	−0.0836475	−	0.996495i	$-0.526657\pi$
$139$	−1.97238	−0.167295	−0.0836475	+	0.996495i	$0.526657\pi$
$140$	0	0
$141$	1.15275	0.0970791
$142$	7.14866	0.599902
$143$	−12.9719	−1.08477
$144$	−28.5533	−2.37944
$145$	0	0
$146$	14.5933	1.20775
$147$	0.836504	0.0689937
$148$	35.5485	2.92207
$149$	11.1782	0.915753	0.457876	−	0.889016i	$-0.348610\pi$
$149$	11.1782	0.915753	0.457876	+	0.889016i	$0.348610\pi$
$150$	0	0
$151$	10.4524	0.850603	0.425301	−	0.905052i	$-0.360168\pi$
$151$	10.4524	0.850603	0.425301	+	0.905052i	$0.360168\pi$
$152$	0	0
$153$	−17.1098	−1.38324
$154$	−15.9942	−1.28885
$155$	0	0
$156$	−1.70874	−0.136808
$157$	5.62072	0.448583	0.224291	−	0.974522i	$-0.427993\pi$
$157$	5.62072	0.448583	0.224291	+	0.974522i	$0.427993\pi$
$158$	24.9849	1.98769
$159$	0.00262771	0.000208391 0
$160$	0	0
$161$	−1.40891	−0.111037
$162$	−22.9972	−1.80683
$163$	5.97258	0.467808	0.233904	−	0.972260i	$-0.424850\pi$
$163$	5.97258	0.467808	0.233904	+	0.972260i	$0.424850\pi$
$164$	23.4337	1.82986
$165$	0	0
$166$	−1.02055	−0.0792100
$167$	−10.2447	−0.792756	−0.396378	−	0.918087i	$-0.629733\pi$
$167$	−10.2447	−0.792756	−0.396378	+	0.918087i	$0.629733\pi$
$168$	−1.23250	−0.0950893
$169$	−7.16464	−0.551126
$170$	0	0
$171$	0	0
$172$	−36.6139	−2.79178
$173$	−2.49506	−0.189696	−0.0948479	−	0.995492i	$-0.530236\pi$
$173$	−2.49506	−0.189696	−0.0948479	+	0.995492i	$0.530236\pi$
$174$	−1.50044	−0.113748
$175$	0	0
$176$	−51.4796	−3.88042
$177$	1.91179	0.143699
$178$	9.34131	0.700161
$179$	22.5409	1.68478	0.842392	−	0.538865i	$-0.181147\pi$
$179$	22.5409	1.68478	0.842392	+	0.538865i	$0.181147\pi$
$180$	0	0
$181$	−6.81802	−0.506779	−0.253390	−	0.967364i	$-0.581545\pi$
$181$	−6.81802	−0.506779	−0.253390	+	0.967364i	$0.581545\pi$
$182$	7.19490	0.533322
$183$	0.00278370	0.000205777 0
$184$	−9.09410	−0.670426
$185$	0	0
$186$	2.48515	0.182220
$187$	−30.8477	−2.25581
$188$	−37.8488	−2.76041
$189$	−1.00086	−0.0728021
$190$	0	0
$191$	−4.97452	−0.359944	−0.179972	−	0.983672i	$-0.557601\pi$
$191$	−4.97452	−0.359944	−0.179972	+	0.983672i	$0.557601\pi$
$192$	−1.13754	−0.0820948
$193$	−7.01693	−0.505090	−0.252545	−	0.967585i	$-0.581267\pi$
$193$	−7.01693	−0.505090	−0.252545	+	0.967585i	$0.581267\pi$
$194$	−38.4995	−2.76410
$195$	0	0
$196$	−27.4653	−1.96181
$197$	22.0852	1.57351	0.786754	−	0.617266i	$-0.211760\pi$
$197$	22.0852	1.57351	0.786754	+	0.617266i	$0.211760\pi$
$198$	−41.7666	−2.96822
$199$	2.40961	0.170812	0.0854062	−	0.996346i	$-0.472781\pi$
$199$	2.40961	0.170812	0.0854062	+	0.996346i	$0.472781\pi$
$200$	0	0
$201$	−1.03877	−0.0732690
$202$	27.5335	1.93725
$203$	4.46489	0.313374
$204$	−4.06344	−0.284498
$205$	0	0
$206$	−23.8173	−1.65943
$207$	−3.67917	−0.255720
$208$	23.1579	1.60571
$209$	0	0
$210$	0	0
$211$	19.0046	1.30833	0.654166	−	0.756351i	$-0.273020\pi$
$211$	19.0046	1.30833	0.654166	+	0.756351i	$0.273020\pi$
$212$	−0.0862768	−0.00592552
$213$	0.401808	0.0275315
$214$	9.30609	0.636151
$215$	0	0
$216$	−6.46029	−0.439567
$217$	−7.39512	−0.502014
$218$	−27.8489	−1.88617
$219$	0.820251	0.0554274
$220$	0	0
$221$	13.8767	0.933449
$222$	2.82731	0.189757
$223$	−22.0446	−1.47622	−0.738108	−	0.674683i	$-0.764280\pi$
$223$	−22.0446	−1.47622	−0.738108	+	0.674683i	$0.764280\pi$
$224$	11.7593	0.785704
$225$	0	0
$226$	−33.6262	−2.23678
$227$	−11.7969	−0.782990	−0.391495	−	0.920180i	$-0.628042\pi$
$227$	−11.7969	−0.782990	−0.391495	+	0.920180i	$0.628042\pi$
$228$	0	0
$229$	−1.19467	−0.0789463	−0.0394731	−	0.999221i	$-0.512568\pi$
$229$	−1.19467	−0.0789463	−0.0394731	+	0.999221i	$0.512568\pi$
$230$	0	0
$231$	−0.898992	−0.0591493
$232$	28.8196	1.89210
$233$	−15.6081	−1.02252	−0.511260	−	0.859426i	$-0.670821\pi$
$233$	−15.6081	−1.02252	−0.511260	+	0.859426i	$0.670821\pi$
$234$	18.7885	1.22825
$235$	0	0
$236$	−62.7708	−4.08603
$237$	1.40434	0.0912216
$238$	17.1098	1.10906
$239$	1.05474	0.0682255	0.0341128	−	0.999418i	$-0.489139\pi$
$239$	1.05474	0.0682255	0.0341128	+	0.999418i	$0.489139\pi$
$240$	0	0
$241$	9.59410	0.618010	0.309005	−	0.951060i	$-0.400004\pi$
$241$	9.59410	0.618010	0.309005	+	0.951060i	$0.400004\pi$
$242$	−46.5772	−2.99410
$243$	−3.92514	−0.251798
$244$	−0.0913985	−0.00585119
$245$	0	0
$246$	1.86377	0.118830
$247$	0	0
$248$	−47.7334	−3.03108
$249$	−0.0573625	−0.00363520
$250$	0	0
$251$	−15.1640	−0.957142	−0.478571	−	0.878049i	$-0.658845\pi$
$251$	−15.1640	−0.957142	−0.478571	+	0.878049i	$0.658845\pi$
$252$	16.3717	1.03132
$253$	−6.63329	−0.417031
$254$	−19.7748	−1.24078
$255$	0	0
$256$	−16.4972	−1.03108
$257$	−23.0689	−1.43900	−0.719499	−	0.694493i	$-0.755629\pi$
$257$	−23.0689	−1.43900	−0.719499	+	0.694493i	$0.755629\pi$
$258$	−2.91204	−0.181296
$259$	−8.41329	−0.522777
$260$	0	0
$261$	11.6595	0.721703
$262$	−29.4523	−1.81957
$263$	3.05610	0.188447	0.0942235	−	0.995551i	$-0.469963\pi$
$263$	3.05610	0.188447	0.0942235	+	0.995551i	$0.469963\pi$
$264$	−5.80274	−0.357134
$265$	0	0
$266$	0	0
$267$	0.525052	0.0321327
$268$	34.1063	2.08337
$269$	−1.05451	−0.0642948	−0.0321474	−	0.999483i	$-0.510235\pi$
$269$	−1.05451	−0.0642948	−0.0321474	+	0.999483i	$0.510235\pi$
$270$	0	0
$271$	−1.67598	−0.101809	−0.0509043	−	0.998704i	$-0.516210\pi$
$271$	−1.67598	−0.101809	−0.0509043	+	0.998704i	$0.516210\pi$
$272$	55.0704	3.33913
$273$	0.404408	0.0244759
$274$	−36.1531	−2.18409
$275$	0	0
$276$	−0.873776	−0.0525952
$277$	13.8944	0.834834	0.417417	−	0.908715i	$-0.362935\pi$
$277$	13.8944	0.834834	0.417417	+	0.908715i	$0.362935\pi$
$278$	5.15061	0.308913
$279$	−19.3114	−1.15614
$280$	0	0
$281$	1.25004	0.0745713	0.0372856	−	0.999305i	$-0.488129\pi$
$281$	1.25004	0.0745713	0.0372856	+	0.999305i	$0.488129\pi$
$282$	−3.01026	−0.179258
$283$	−15.7261	−0.934819	−0.467409	−	0.884041i	$-0.654813\pi$
$283$	−15.7261	−0.934819	−0.467409	+	0.884041i	$0.654813\pi$
$284$	−13.1927	−0.782846
$285$	0	0
$286$	33.8744	2.00304
$287$	−5.54606	−0.327374
$288$	30.7080	1.80948
$289$	15.9994	0.941140
$290$	0	0
$291$	−2.16396	−0.126854
$292$	−26.9316	−1.57606
$293$	−11.9316	−0.697051	−0.348526	−	0.937299i	$-0.613318\pi$
$293$	−11.9316	−0.697051	−0.348526	+	0.937299i	$0.613318\pi$
$294$	−2.18442	−0.127398
$295$	0	0
$296$	−54.3055	−3.15644
$297$	−4.71218	−0.273428
$298$	−29.1904	−1.69095
$299$	2.98396	0.172567
$300$	0	0
$301$	8.66544	0.499467
$302$	−27.2950	−1.57065
$303$	1.54759	0.0889067
$304$	0	0
$305$	0	0
$306$	44.6799	2.55418
$307$	−24.3857	−1.39176	−0.695882	−	0.718156i	$-0.744986\pi$
$307$	−24.3857	−1.39176	−0.695882	+	0.718156i	$0.744986\pi$
$308$	29.5170	1.68189
$309$	−1.33871	−0.0761567
$310$	0	0
$311$	−5.06773	−0.287364	−0.143682	−	0.989624i	$-0.545894\pi$
$311$	−5.06773	−0.287364	−0.143682	+	0.989624i	$0.545894\pi$
$312$	2.61034	0.147781
$313$	18.9537	1.07133	0.535664	−	0.844431i	$-0.320061\pi$
$313$	18.9537	1.07133	0.535664	+	0.844431i	$0.320061\pi$
$314$	−14.6778	−0.828315
$315$	0	0
$316$	−46.1093	−2.59385
$317$	−12.1377	−0.681720	−0.340860	−	0.940114i	$-0.610718\pi$
$317$	−12.1377	−0.681720	−0.340860	+	0.940114i	$0.610718\pi$
$318$	−0.00686192	−0.000384798 0
$319$	21.0212	1.17696
$320$	0	0
$321$	0.523072	0.0291950
$322$	3.67917	0.205032
$323$	0	0
$324$	42.4410	2.35783
$325$	0	0
$326$	−15.5966	−0.863815
$327$	−1.56532	−0.0865623
$328$	−35.7983	−1.97663
$329$	8.95769	0.493853
$330$	0	0
$331$	19.3649	1.06439	0.532197	−	0.846621i	$-0.321367\pi$
$331$	19.3649	1.06439	0.532197	+	0.846621i	$0.321367\pi$
$332$	1.88341	0.103366
$333$	−21.9702	−1.20396
$334$	26.7526	1.46384
$335$	0	0
$336$	1.60491	0.0875551
$337$	4.13312	0.225146	0.112573	−	0.993643i	$-0.464091\pi$
$337$	4.13312	0.225146	0.112573	+	0.993643i	$0.464091\pi$
$338$	18.7095	1.01766
$339$	−1.89005	−0.102653
$340$	0	0
$341$	−34.8171	−1.88545
$342$	0	0
$343$	14.4842	0.782076
$344$	55.9330	3.01570
$345$	0	0
$346$	6.51552	0.350276
$347$	4.77587	0.256382	0.128191	−	0.991749i	$-0.459083\pi$
$347$	4.77587	0.256382	0.128191	+	0.991749i	$0.459083\pi$
$348$	2.76904	0.148436
$349$	0.557017	0.0298165	0.0149082	−	0.999889i	$-0.495254\pi$
$349$	0.557017	0.0298165	0.0149082	+	0.999889i	$0.495254\pi$
$350$	0	0
$351$	2.11975	0.113144
$352$	55.3643	2.95093
$353$	24.4841	1.30316	0.651580	−	0.758580i	$-0.274107\pi$
$353$	24.4841	1.30316	0.651580	+	0.758580i	$0.274107\pi$
$354$	−4.99240	−0.265343
$355$	0	0
$356$	−17.2393	−0.913679
$357$	0.961697	0.0508984
$358$	−58.8625	−3.11098
$359$	20.9706	1.10679	0.553393	−	0.832921i	$-0.313333\pi$
$359$	20.9706	1.10679	0.553393	+	0.832921i	$0.313333\pi$
$360$	0	0
$361$	0	0
$362$	17.8044	0.935776
$363$	−2.61799	−0.137409
$364$	−13.2781	−0.695961
$365$	0	0
$366$	−0.00726927	−0.000379971 0
$367$	−21.4343	−1.11886	−0.559431	−	0.828877i	$-0.688980\pi$
$367$	−21.4343	−1.11886	−0.559431	+	0.828877i	$0.688980\pi$
$368$	11.8420	0.617306
$369$	−14.4828	−0.753945
$370$	0	0
$371$	0.0204192	0.00106011
$372$	−4.58631	−0.237789
$373$	27.7981	1.43933	0.719666	−	0.694321i	$-0.244295\pi$
$373$	27.7981	1.43933	0.719666	+	0.694321i	$0.244295\pi$
$374$	80.5547	4.16539
$375$	0	0
$376$	57.8194	2.98181
$377$	−9.45630	−0.487024
$378$	2.61362	0.134430
$379$	−1.97507	−0.101452	−0.0507261	−	0.998713i	$-0.516154\pi$
$379$	−1.97507	−0.101452	−0.0507261	+	0.998713i	$0.516154\pi$
$380$	0	0
$381$	−1.11150	−0.0569436
$382$	12.9903	0.664642
$383$	10.0103	0.511504	0.255752	−	0.966742i	$-0.417677\pi$
$383$	10.0103	0.511504	0.255752	+	0.966742i	$0.417677\pi$
$384$	−0.0560484	−0.00286021
$385$	0	0
$386$	18.3238	0.932656
$387$	22.6286	1.15028
$388$	71.0502	3.60703
$389$	12.0247	0.609678	0.304839	−	0.952404i	$-0.401397\pi$
$389$	12.0247	0.609678	0.304839	+	0.952404i	$0.401397\pi$
$390$	0	0
$391$	7.09597	0.358859
$392$	41.9572	2.11916
$393$	−1.65544	−0.0835059
$394$	−57.6727	−2.90551
$395$	0	0
$396$	77.0797	3.87340
$397$	−1.87227	−0.0939666	−0.0469833	−	0.998896i	$-0.514961\pi$
$397$	−1.87227	−0.0939666	−0.0469833	+	0.998896i	$0.514961\pi$
$398$	−6.29237	−0.315408
$399$	0	0
$400$	0	0
$401$	−23.5289	−1.17498	−0.587489	−	0.809232i	$-0.699883\pi$
$401$	−23.5289	−1.17498	−0.587489	+	0.809232i	$0.699883\pi$
$402$	2.71260	0.135292
$403$	15.6623	0.780196
$404$	−50.8127	−2.52803
$405$	0	0
$406$	−11.6595	−0.578649
$407$	−39.6107	−1.96343
$408$	6.20749	0.307316
$409$	−16.0211	−0.792193	−0.396097	−	0.918209i	$-0.629636\pi$
$409$	−16.0211	−0.792193	−0.396097	+	0.918209i	$0.629636\pi$
$410$	0	0
$411$	−2.03207	−0.100235
$412$	43.9545	2.16548
$413$	14.8560	0.731016
$414$	9.60767	0.472191
$415$	0	0
$416$	−24.9054	−1.22109
$417$	0.289503	0.0141770
$418$	0	0
$419$	−10.2305	−0.499794	−0.249897	−	0.968272i	$-0.580397\pi$
$419$	−10.2305	−0.499794	−0.249897	+	0.968272i	$0.580397\pi$
$420$	0	0
$421$	25.6300	1.24913	0.624565	−	0.780973i	$-0.285276\pi$
$421$	25.6300	1.24913	0.624565	+	0.780973i	$0.285276\pi$
$422$	−49.6281	−2.41586
$423$	23.3918	1.13735
$424$	0.131800	0.00640078
$425$	0	0
$426$	−1.04927	−0.0508373
$427$	0.0216313	0.00104681
$428$	−17.1743	−0.830149
$429$	1.90400	0.0919259
$430$	0	0
$431$	−26.3929	−1.27130	−0.635650	−	0.771978i	$-0.719268\pi$
$431$	−26.3929	−1.27130	−0.635650	+	0.771978i	$0.719268\pi$
$432$	8.41234	0.404739
$433$	−11.1339	−0.535061	−0.267531	−	0.963549i	$-0.586208\pi$
$433$	−11.1339	−0.535061	−0.267531	+	0.963549i	$0.586208\pi$
$434$	19.3114	0.926976
$435$	0	0
$436$	51.3948	2.46136
$437$	0	0
$438$	−2.14198	−0.102348
$439$	32.5878	1.55533	0.777666	−	0.628677i	$-0.216403\pi$
$439$	32.5878	1.55533	0.777666	+	0.628677i	$0.216403\pi$
$440$	0	0
$441$	16.9745	0.808309
$442$	−36.2372	−1.72363
$443$	−41.7702	−1.98456	−0.992281	−	0.124013i	$-0.960424\pi$
$443$	−41.7702	−1.98456	−0.992281	+	0.124013i	$0.960424\pi$
$444$	−5.21776	−0.247624
$445$	0	0
$446$	57.5666	2.72586
$447$	−1.64072	−0.0776033
$448$	−8.83948	−0.417626
$449$	−40.0139	−1.88837	−0.944187	−	0.329409i	$-0.893151\pi$
$449$	−40.0139	−1.88837	−0.944187	+	0.329409i	$0.893151\pi$
$450$	0	0
$451$	−26.1115	−1.22954
$452$	62.0567	2.91890
$453$	−1.53419	−0.0720823
$454$	30.8061	1.44580
$455$	0	0
$456$	0	0
$457$	30.1103	1.40850	0.704249	−	0.709953i	$-0.251284\pi$
$457$	30.1103	1.40850	0.704249	+	0.709953i	$0.251284\pi$
$458$	3.11973	0.145776
$459$	5.04086	0.235287
$460$	0	0
$461$	−19.8065	−0.922479	−0.461239	−	0.887276i	$-0.652595\pi$
$461$	−19.8065	−0.922479	−0.461239	+	0.887276i	$0.652595\pi$
$462$	2.34760	0.109220
$463$	−17.0673	−0.793186	−0.396593	−	0.917994i	$-0.629808\pi$
$463$	−17.0673	−0.793186	−0.396593	+	0.917994i	$0.629808\pi$
$464$	−37.5278	−1.74218
$465$	0	0
$466$	40.7585	1.88810
$467$	24.1262	1.11643	0.558214	−	0.829697i	$-0.311487\pi$
$467$	24.1262	1.11643	0.558214	+	0.829697i	$0.311487\pi$
$468$	−34.6740	−1.60281
$469$	−8.07196	−0.372728
$470$	0	0
$471$	−0.825001	−0.0380141
$472$	95.8913	4.41376
$473$	40.7979	1.87589
$474$	−3.66724	−0.168442
$475$	0	0
$476$	−31.5758	−1.44728
$477$	0.0533220	0.00244145
$478$	−2.75432	−0.125979
$479$	−32.5068	−1.48527	−0.742637	−	0.669695i	$-0.766425\pi$
$479$	−32.5068	−1.48527	−0.742637	+	0.669695i	$0.766425\pi$
$480$	0	0
$481$	17.8187	0.812464
$482$	−25.0537	−1.14117
$483$	0.206797	0.00940960
$484$	85.9576	3.90717
$485$	0	0
$486$	10.2500	0.464949
$487$	−30.4975	−1.38197	−0.690986	−	0.722868i	$-0.742823\pi$
$487$	−30.4975	−1.38197	−0.690986	+	0.722868i	$0.742823\pi$
$488$	0.139624	0.00632049
$489$	−0.876646	−0.0396433
$490$	0	0
$491$	−19.4673	−0.878548	−0.439274	−	0.898353i	$-0.644764\pi$
$491$	−19.4673	−0.878548	−0.439274	+	0.898353i	$0.644764\pi$
$492$	−3.43956	−0.155067
$493$	−22.4875	−1.01278
$494$	0	0
$495$	0	0
$496$	62.1567	2.79092
$497$	3.12234	0.140056
$498$	0.149795	0.00671246
$499$	1.81090	0.0810670	0.0405335	−	0.999178i	$-0.487094\pi$
$499$	1.81090	0.0810670	0.0405335	+	0.999178i	$0.487094\pi$
$500$	0	0
$501$	1.50370	0.0671802
$502$	39.5987	1.76738
$503$	−26.5154	−1.18226	−0.591132	−	0.806575i	$-0.701319\pi$
$503$	−26.5154	−1.18226	−0.591132	+	0.806575i	$0.701319\pi$
$504$	−25.0101	−1.11404
$505$	0	0
$506$	17.3220	0.770055
$507$	1.05162	0.0467039
$508$	36.4942	1.61917
$509$	−11.3299	−0.502191	−0.251095	−	0.967962i	$-0.580791\pi$
$509$	−11.3299	−0.502191	−0.251095	+	0.967962i	$0.580791\pi$
$510$	0	0
$511$	6.37393	0.281966
$512$	42.3167	1.87015
$513$	0	0
$514$	60.2414	2.65713
$515$	0	0
$516$	5.37414	0.236583
$517$	42.1738	1.85480
$518$	21.9702	0.965315
$519$	0.366221	0.0160753
$520$	0	0
$521$	−25.1821	−1.10325	−0.551624	−	0.834093i	$-0.685992\pi$
$521$	−25.1821	−1.10325	−0.551624	+	0.834093i	$0.685992\pi$
$522$	−30.4472	−1.33264
$523$	2.56239	0.112045	0.0560227	−	0.998429i	$-0.482158\pi$
$523$	2.56239	0.112045	0.0560227	+	0.998429i	$0.482158\pi$
$524$	54.3538	2.37446
$525$	0	0
$526$	−7.98060	−0.347971
$527$	37.2456	1.62244
$528$	7.55610	0.328837
$529$	−21.4741	−0.933658
$530$	0	0
$531$	38.7945	1.68354
$532$	0	0
$533$	11.7461	0.508782
$534$	−1.37110	−0.0593335
$535$	0	0
$536$	−52.1022	−2.25047
$537$	−3.30852	−0.142773
$538$	2.75372	0.118721
$539$	30.6038	1.31820
$540$	0	0
$541$	27.1770	1.16843	0.584215	−	0.811599i	$-0.301402\pi$
$541$	27.1770	1.16843	0.584215	+	0.811599i	$0.301402\pi$
$542$	4.37660	0.187991
$543$	1.00074	0.0429458
$544$	−59.2260	−2.53929
$545$	0	0
$546$	−1.05606	−0.0451951
$547$	−15.6526	−0.669259	−0.334629	−	0.942350i	$-0.608611\pi$
$547$	−15.6526	−0.669259	−0.334629	+	0.942350i	$0.608611\pi$
$548$	66.7200	2.85014
$549$	0.0564874	0.00241082
$550$	0	0
$551$	0	0
$552$	1.33482	0.0568136
$553$	10.9127	0.464055
$554$	−36.2834	−1.54153
$555$	0	0
$556$	−9.50537	−0.403118
$557$	−38.7469	−1.64176	−0.820880	−	0.571101i	$-0.806516\pi$
$557$	−38.7469	−1.64176	−0.820880	+	0.571101i	$0.806516\pi$
$558$	50.4291	2.13484
$559$	−18.3528	−0.776239
$560$	0	0
$561$	4.52778	0.191163
$562$	−3.26432	−0.137697
$563$	−38.8502	−1.63734	−0.818670	−	0.574265i	$-0.805288\pi$
$563$	−38.8502	−1.63734	−0.818670	+	0.574265i	$0.805288\pi$
$564$	5.55539	0.233924
$565$	0	0
$566$	41.0666	1.72616
$567$	−10.0445	−0.421831
$568$	20.1538	0.845635
$569$	−19.2093	−0.805294	−0.402647	−	0.915355i	$-0.631910\pi$
$569$	−19.2093	−0.805294	−0.402647	+	0.915355i	$0.631910\pi$
$570$	0	0
$571$	−8.84542	−0.370169	−0.185085	−	0.982723i	$-0.559256\pi$
$571$	−8.84542	−0.370169	−0.185085	+	0.982723i	$0.559256\pi$
$572$	−62.5148	−2.61387
$573$	0.730153	0.0305026
$574$	14.4828	0.604500
$575$	0	0
$576$	−23.0831	−0.961797
$577$	−15.0333	−0.625845	−0.312922	−	0.949779i	$-0.601308\pi$
$577$	−15.0333	−0.625845	−0.312922	+	0.949779i	$0.601308\pi$
$578$	−41.7802	−1.73783
$579$	1.02993	0.0428026
$580$	0	0
$581$	−0.445748	−0.0184927
$582$	5.65089	0.234237
$583$	0.0961358	0.00398154
$584$	41.1419	1.70247
$585$	0	0
$586$	31.1578	1.28712
$587$	42.7945	1.76632	0.883159	−	0.469073i	$-0.155412\pi$
$587$	42.7945	1.76632	0.883159	+	0.469073i	$0.155412\pi$
$588$	4.03132	0.166249
$589$	0	0
$590$	0	0
$591$	−3.24164	−0.133343
$592$	70.7145	2.90635
$593$	−30.9216	−1.26980	−0.634900	−	0.772595i	$-0.718959\pi$
$593$	−30.9216	−1.26980	−0.634900	+	0.772595i	$0.718959\pi$
$594$	12.3052	0.504890
$595$	0	0
$596$	53.8704	2.20662
$597$	−0.353678	−0.0144751
$598$	−7.79221	−0.318647
$599$	18.2839	0.747058	0.373529	−	0.927618i	$-0.378148\pi$
$599$	18.2839	0.747058	0.373529	+	0.927618i	$0.378148\pi$
$600$	0	0
$601$	−40.5660	−1.65472	−0.827361	−	0.561670i	$-0.810159\pi$
$601$	−40.5660	−1.65472	−0.827361	+	0.561670i	$0.810159\pi$
$602$	−22.6286	−0.922275
$603$	−21.0788	−0.858397
$604$	50.3726	2.04963
$605$	0	0
$606$	−4.04133	−0.164168
$607$	−9.33324	−0.378825	−0.189412	−	0.981898i	$-0.560658\pi$
$607$	−9.33324	−0.378825	−0.189412	+	0.981898i	$0.560658\pi$
$608$	0	0
$609$	−0.655350	−0.0265561
$610$	0	0
$611$	−18.9717	−0.767514
$612$	−82.4561	−3.33309
$613$	−40.6646	−1.64243	−0.821213	−	0.570621i	$-0.806702\pi$
$613$	−40.6646	−1.64243	−0.821213	+	0.570621i	$0.806702\pi$
$614$	63.6800	2.56991
$615$	0	0
$616$	−45.0914	−1.81678
$617$	−18.4246	−0.741745	−0.370872	−	0.928684i	$-0.620941\pi$
$617$	−18.4246	−0.741745	−0.370872	+	0.928684i	$0.620941\pi$
$618$	3.49587	0.140625
$619$	−2.68372	−0.107868	−0.0539340	−	0.998545i	$-0.517176\pi$
$619$	−2.68372	−0.107868	−0.0539340	+	0.998545i	$0.517176\pi$
$620$	0	0
$621$	1.08395	0.0434975
$622$	13.2337	0.530623
$623$	4.08003	0.163463
$624$	−3.39908	−0.136072
$625$	0	0
$626$	−49.4952	−1.97823
$627$	0	0
$628$	27.0876	1.08091
$629$	42.3736	1.68955
$630$	0	0
$631$	−18.6512	−0.742492	−0.371246	−	0.928535i	$-0.621069\pi$
$631$	−18.6512	−0.742492	−0.371246	+	0.928535i	$0.621069\pi$
$632$	70.4385	2.80189
$633$	−2.78947	−0.110872
$634$	31.6959	1.25881
$635$	0	0
$636$	0.0126636	0.000502144 0
$637$	−13.7670	−0.545469
$638$	−54.8941	−2.17328
$639$	8.15357	0.322550
$640$	0	0
$641$	4.88420	0.192914	0.0964572	−	0.995337i	$-0.469249\pi$
$641$	4.88420	0.192914	0.0964572	+	0.995337i	$0.469249\pi$
$642$	−1.36593	−0.0539091
$643$	−3.68757	−0.145424	−0.0727118	−	0.997353i	$-0.523165\pi$
$643$	−3.68757	−0.145424	−0.0727118	+	0.997353i	$0.523165\pi$
$644$	−6.78986	−0.267558
$645$	0	0
$646$	0	0
$647$	26.3559	1.03616	0.518079	−	0.855333i	$-0.326647\pi$
$647$	26.3559	1.03616	0.518079	+	0.855333i	$0.326647\pi$
$648$	−64.8347	−2.54695
$649$	69.9437	2.74553
$650$	0	0
$651$	1.08544	0.0425419
$652$	28.7833	1.12724
$653$	23.9281	0.936379	0.468189	−	0.883628i	$-0.344907\pi$
$653$	23.9281	0.936379	0.468189	+	0.883628i	$0.344907\pi$
$654$	4.08762	0.159839
$655$	0	0
$656$	46.6151	1.82001
$657$	16.6447	0.649371
$658$	−23.3918	−0.911908
$659$	−5.97176	−0.232627	−0.116313	−	0.993213i	$-0.537108\pi$
$659$	−5.97176	−0.232627	−0.116313	+	0.993213i	$0.537108\pi$
$660$	0	0
$661$	−11.3007	−0.439545	−0.219773	−	0.975551i	$-0.570532\pi$
$661$	−11.3007	−0.439545	−0.219773	+	0.975551i	$0.570532\pi$
$662$	−50.5690	−1.96542
$663$	−2.03680	−0.0791029
$664$	−2.87718	−0.111656
$665$	0	0
$666$	57.3723	2.22313
$667$	−4.83555	−0.187233
$668$	−49.3715	−1.91024
$669$	3.23567	0.125098
$670$	0	0
$671$	0.101843	0.00393160
$672$	−1.72602	−0.0665826
$673$	−22.9796	−0.885798	−0.442899	−	0.896571i	$-0.646050\pi$
$673$	−22.9796	−0.885798	−0.442899	+	0.896571i	$0.646050\pi$
$674$	−10.7931	−0.415735
$675$	0	0
$676$	−34.5281	−1.32801
$677$	−30.2320	−1.16191	−0.580955	−	0.813936i	$-0.697321\pi$
$677$	−30.2320	−1.16191	−0.580955	+	0.813936i	$0.697321\pi$
$678$	4.93561	0.189551
$679$	−16.8155	−0.645319
$680$	0	0
$681$	1.73154	0.0663526
$682$	90.9202	3.48151
$683$	38.7762	1.48373	0.741866	−	0.670549i	$-0.233941\pi$
$683$	38.7762	1.48373	0.741866	+	0.670549i	$0.233941\pi$
$684$	0	0
$685$	0	0
$686$	−37.8237	−1.44412
$687$	0.175352	0.00669011
$688$	−72.8338	−2.77676
$689$	−0.0432463	−0.00164755
$690$	0	0
$691$	−34.4907	−1.31209	−0.656044	−	0.754723i	$-0.727771\pi$
$691$	−34.4907	−1.31209	−0.656044	+	0.754723i	$0.727771\pi$
$692$	−12.0243	−0.457095
$693$	−18.2425	−0.692975
$694$	−12.4716	−0.473414
$695$	0	0
$696$	−4.23010	−0.160341
$697$	27.9328	1.05803
$698$	−1.45458	−0.0550566
$699$	2.29093	0.0866511
$700$	0	0
$701$	−2.43041	−0.0917952	−0.0458976	−	0.998946i	$-0.514615\pi$
$701$	−2.43041	−0.0917952	−0.0458976	+	0.998946i	$0.514615\pi$
$702$	−5.53546	−0.208922
$703$	0	0
$704$	−41.6173	−1.56851
$705$	0	0
$706$	−63.9371	−2.40630
$707$	12.0259	0.452279
$708$	9.21340	0.346261
$709$	−14.9095	−0.559939	−0.279969	−	0.960009i	$-0.590324\pi$
$709$	−14.9095	−0.559939	−0.279969	+	0.960009i	$0.590324\pi$
$710$	0	0
$711$	28.4971	1.06872
$712$	26.3354	0.986962
$713$	8.00905	0.299941
$714$	−2.51135	−0.0939848
$715$	0	0
$716$	108.630	4.05969
$717$	−0.154813	−0.00578161
$718$	−54.7619	−2.04370
$719$	−47.2596	−1.76248	−0.881242	−	0.472665i	$-0.843292\pi$
$719$	−47.2596	−1.76248	−0.881242	+	0.472665i	$0.843292\pi$
$720$	0	0
$721$	−10.4027	−0.387419
$722$	0	0
$723$	−1.40821	−0.0523718
$724$	−32.8577	−1.22115
$725$	0	0
$726$	6.83654	0.253728
$727$	−41.1123	−1.52477	−0.762386	−	0.647123i	$-0.775972\pi$
$727$	−41.1123	−1.52477	−0.762386	+	0.647123i	$0.775972\pi$
$728$	20.2842	0.751782
$729$	−25.8436	−0.957170
$730$	0	0
$731$	−43.6436	−1.61422
$732$	0.0134153	0.000495845 0
$733$	−25.6070	−0.945816	−0.472908	−	0.881112i	$-0.656796\pi$
$733$	−25.6070	−0.945816	−0.472908	+	0.881112i	$0.656796\pi$
$734$	55.9729	2.06600
$735$	0	0
$736$	−12.7356	−0.469440
$737$	−38.0037	−1.39988
$738$	37.8199	1.39217
$739$	10.4997	0.386237	0.193119	−	0.981175i	$-0.438140\pi$
$739$	10.4997	0.386237	0.193119	+	0.981175i	$0.438140\pi$
$740$	0	0
$741$	0	0
$742$	−0.0533220	−0.00195751
$743$	−4.64951	−0.170574	−0.0852871	−	0.996356i	$-0.527181\pi$
$743$	−4.64951	−0.170574	−0.0852871	+	0.996356i	$0.527181\pi$
$744$	7.00624	0.256861
$745$	0	0
$746$	−72.5911	−2.65775
$747$	−1.16401	−0.0425889
$748$	−148.663	−5.43564
$749$	4.06464	0.148519
$750$	0	0
$751$	−52.7364	−1.92438	−0.962190	−	0.272381i	$-0.912189\pi$
$751$	−52.7364	−1.92438	−0.962190	+	0.272381i	$0.912189\pi$
$752$	−75.2902	−2.74555
$753$	2.22575	0.0811107
$754$	24.6939	0.899298
$755$	0	0
$756$	−4.82340	−0.175425
$757$	−13.3049	−0.483575	−0.241788	−	0.970329i	$-0.577734\pi$
$757$	−13.3049	−0.483575	−0.241788	+	0.970329i	$0.577734\pi$
$758$	5.15762	0.187333
$759$	0.973624	0.0353403
$760$	0	0
$761$	−20.3115	−0.736292	−0.368146	−	0.929768i	$-0.620007\pi$
$761$	−20.3115	−0.736292	−0.368146	+	0.929768i	$0.620007\pi$
$762$	2.90252	0.105147
$763$	−12.1636	−0.440353
$764$	−23.9734	−0.867328
$765$	0	0
$766$	−26.1407	−0.944501
$767$	−31.4639	−1.13610
$768$	2.42144	0.0873762
$769$	−21.2618	−0.766721	−0.383360	−	0.923599i	$-0.625233\pi$
$769$	−21.2618	−0.766721	−0.383360	+	0.923599i	$0.625233\pi$
$770$	0	0
$771$	3.38602	0.121945
$772$	−33.8163	−1.21707
$773$	36.0232	1.29566	0.647832	−	0.761783i	$-0.275676\pi$
$773$	36.0232	1.29566	0.647832	+	0.761783i	$0.275676\pi$
$774$	−59.0917	−2.12401
$775$	0	0
$776$	−108.539	−3.89633
$777$	1.23489	0.0443014
$778$	−31.4010	−1.12578
$779$	0	0
$780$	0	0
$781$	14.7003	0.526018
$782$	−18.5302	−0.662638
$783$	−3.43510	−0.122760
$784$	−54.6350	−1.95125
$785$	0	0
$786$	4.32296	0.154195
$787$	41.0609	1.46366	0.731831	−	0.681486i	$-0.238666\pi$
$787$	41.0609	1.46366	0.731831	+	0.681486i	$0.238666\pi$
$788$	106.434	3.79156
$789$	−0.448570	−0.0159695
$790$	0	0
$791$	−14.6870	−0.522210
$792$	−117.750	−4.18407
$793$	−0.0458135	−0.00162689
$794$	4.88919	0.173511
$795$	0	0
$796$	11.6125	0.411593
$797$	33.9392	1.20219	0.601093	−	0.799179i	$-0.294732\pi$
$797$	33.9392	1.20219	0.601093	+	0.799179i	$0.294732\pi$
$798$	0	0
$799$	−45.1155	−1.59607
$800$	0	0
$801$	10.6545	0.376456
$802$	61.4426	2.16961
$803$	30.0092	1.05900
$804$	−5.00607	−0.176551
$805$	0	0
$806$	−40.9001	−1.44064
$807$	0.154780	0.00544851
$808$	77.6237	2.73079
$809$	−51.3707	−1.80610	−0.903048	−	0.429539i	$-0.858676\pi$
$809$	−51.3707	−1.80610	−0.903048	+	0.429539i	$0.858676\pi$
$810$	0	0
$811$	−20.6118	−0.723779	−0.361890	−	0.932221i	$-0.617868\pi$
$811$	−20.6118	−0.723779	−0.361890	+	0.932221i	$0.617868\pi$
$812$	21.5174	0.755112
$813$	0.245998	0.00862752
$814$	103.438	3.62551
$815$	0	0
$816$	−8.08315	−0.282967
$817$	0	0
$818$	41.8370	1.46280
$819$	8.20631	0.286752
$820$	0	0
$821$	38.7002	1.35065	0.675323	−	0.737522i	$-0.264004\pi$
$821$	38.7002	1.35065	0.675323	+	0.737522i	$0.264004\pi$
$822$	5.30650	0.185085
$823$	−18.7887	−0.654932	−0.327466	−	0.944863i	$-0.606195\pi$
$823$	−18.7887	−0.654932	−0.327466	+	0.944863i	$0.606195\pi$
$824$	−67.1468	−2.33917
$825$	0	0
$826$	−38.7945	−1.34983
$827$	−36.8031	−1.27977	−0.639885	−	0.768471i	$-0.721018\pi$
$827$	−36.8031	−1.27977	−0.639885	+	0.768471i	$0.721018\pi$
$828$	−17.7308	−0.616189
$829$	22.6730	0.787466	0.393733	−	0.919225i	$-0.371184\pi$
$829$	22.6730	0.787466	0.393733	+	0.919225i	$0.371184\pi$
$830$	0	0
$831$	−2.03940	−0.0707460
$832$	18.7214	0.649047
$833$	−32.7385	−1.13432
$834$	−0.755998	−0.0261781
$835$	0	0
$836$	0	0
$837$	5.68950	0.196658
$838$	26.7157	0.922878
$839$	2.70578	0.0934139	0.0467070	−	0.998909i	$-0.485127\pi$
$839$	2.70578	0.0934139	0.0467070	+	0.998909i	$0.485127\pi$
$840$	0	0
$841$	−13.6759	−0.471583
$842$	−66.9293	−2.30654
$843$	−0.183479	−0.00631937
$844$	91.5879	3.15259
$845$	0	0
$846$	−61.0847	−2.10013
$847$	−20.3436	−0.699016
$848$	−0.171625	−0.00589363
$849$	2.30825	0.0792190
$850$	0	0
$851$	9.11175	0.312347
$852$	1.93641	0.0663404
$853$	28.0139	0.959178	0.479589	−	0.877493i	$-0.340786\pi$
$853$	28.0139	0.959178	0.479589	+	0.877493i	$0.340786\pi$
$854$	−0.0564874	−0.00193296
$855$	0	0
$856$	26.2361	0.896732
$857$	37.9069	1.29487	0.647437	−	0.762119i	$-0.275841\pi$
$857$	37.9069	1.29487	0.647437	+	0.762119i	$0.275841\pi$
$858$	−4.97204	−0.169743
$859$	−25.5377	−0.871336	−0.435668	−	0.900107i	$-0.643488\pi$
$859$	−25.5377	−0.871336	−0.435668	+	0.900107i	$0.643488\pi$
$860$	0	0
$861$	0.814042	0.0277425
$862$	68.9215	2.34747
$863$	46.3212	1.57679	0.788395	−	0.615169i	$-0.210912\pi$
$863$	46.3212	1.57679	0.788395	+	0.615169i	$0.210912\pi$
$864$	−9.04714	−0.307790
$865$	0	0
$866$	29.0747	0.987999
$867$	−2.34836	−0.0797546
$868$	−35.6389	−1.20966
$869$	51.3783	1.74289
$870$	0	0
$871$	17.0958	0.579269
$872$	−78.5128	−2.65878
$873$	−43.9114	−1.48618
$874$	0	0
$875$	0	0
$876$	3.95299	0.133559
$877$	−43.7693	−1.47798	−0.738992	−	0.673715i	$-0.764698\pi$
$877$	−43.7693	−1.47798	−0.738992	+	0.673715i	$0.764698\pi$
$878$	−85.0988	−2.87195
$879$	1.75130	0.0590700
$880$	0	0
$881$	−34.6665	−1.16795	−0.583973	−	0.811773i	$-0.698503\pi$
$881$	−34.6665	−1.16795	−0.583973	+	0.811773i	$0.698503\pi$
$882$	−44.3267	−1.49256
$883$	43.5573	1.46582	0.732910	−	0.680325i	$-0.238161\pi$
$883$	43.5573	1.46582	0.732910	+	0.680325i	$0.238161\pi$
$884$	66.8753	2.24926
$885$	0	0
$886$	109.077	3.66452
$887$	48.2351	1.61957	0.809787	−	0.586723i	$-0.199582\pi$
$887$	48.2351	1.61957	0.809787	+	0.586723i	$0.199582\pi$
$888$	7.97087	0.267485
$889$	−8.63710	−0.289679
$890$	0	0
$891$	−47.2908	−1.58430
$892$	−106.238	−3.55712
$893$	0	0
$894$	4.28452	0.143296
$895$	0	0
$896$	−0.435536	−0.0145502
$897$	−0.437981	−0.0146238
$898$	104.491	3.48691
$899$	−25.3810	−0.846505
$900$	0	0
$901$	−0.102841	−0.00342615
$902$	68.1867	2.27037
$903$	−1.27190	−0.0423262
$904$	−94.8005	−3.15302
$905$	0	0
$906$	4.00632	0.133101
$907$	−44.1877	−1.46723	−0.733613	−	0.679567i	$-0.762168\pi$
$907$	−44.1877	−1.46723	−0.733613	+	0.679567i	$0.762168\pi$
$908$	−56.8523	−1.88671
$909$	31.4040	1.04160
$910$	0	0
$911$	−33.6536	−1.11499	−0.557496	−	0.830180i	$-0.688238\pi$
$911$	−33.6536	−1.11499	−0.557496	+	0.830180i	$0.688238\pi$
$912$	0	0
$913$	−2.09863	−0.0694545
$914$	−78.6290	−2.60082
$915$	0	0
$916$	−5.75742	−0.190231
$917$	−12.8639	−0.424805
$918$	−13.1635	−0.434461
$919$	4.04484	0.133427	0.0667134	−	0.997772i	$-0.478749\pi$
$919$	4.04484	0.133427	0.0667134	+	0.997772i	$0.478749\pi$
$920$	0	0
$921$	3.57929	0.117942
$922$	51.7220	1.70337
$923$	−6.61287	−0.217665
$924$	−4.33246	−0.142527
$925$	0	0
$926$	44.5691	1.46463
$927$	−27.1654	−0.892229
$928$	40.3596	1.32487
$929$	−40.2101	−1.31925	−0.659625	−	0.751594i	$-0.729285\pi$
$929$	−40.2101	−1.31925	−0.659625	+	0.751594i	$0.729285\pi$
$930$	0	0
$931$	0	0
$932$	−75.2192	−2.46389
$933$	0.743833	0.0243520
$934$	−63.0024	−2.06150
$935$	0	0
$936$	52.9695	1.73136
$937$	13.1716	0.430297	0.215149	−	0.976581i	$-0.430976\pi$
$937$	13.1716	0.430297	0.215149	+	0.976581i	$0.430976\pi$
$938$	21.0788	0.688249
$939$	−2.78200	−0.0907872
$940$	0	0
$941$	33.9107	1.10546	0.552729	−	0.833361i	$-0.313586\pi$
$941$	33.9107	1.10546	0.552729	+	0.833361i	$0.313586\pi$
$942$	2.15438	0.0701936
$943$	6.00648	0.195598
$944$	−124.866	−4.06404
$945$	0	0
$946$	−106.538	−3.46386
$947$	−34.4561	−1.11967	−0.559836	−	0.828603i	$-0.689136\pi$
$947$	−34.4561	−1.11967	−0.559836	+	0.828603i	$0.689136\pi$
$948$	6.76785	0.219810
$949$	−13.4995	−0.438213
$950$	0	0
$951$	1.78155	0.0577707
$952$	48.2366	1.56336
$953$	6.03770	0.195580	0.0977902	−	0.995207i	$-0.468823\pi$
$953$	6.03770	0.195580	0.0977902	+	0.995207i	$0.468823\pi$
$954$	−0.139243	−0.00450817
$955$	0	0
$956$	5.08305	0.164398
$957$	−3.08546	−0.0997388
$958$	84.8872	2.74258
$959$	−15.7907	−0.509907
$960$	0	0
$961$	11.0382	0.356072
$962$	−46.5313	−1.50023
$963$	10.6143	0.342040
$964$	46.2363	1.48917
$965$	0	0
$966$	−0.540023	−0.0173750
$967$	12.5734	0.404334	0.202167	−	0.979351i	$-0.435202\pi$
$967$	12.5734	0.404334	0.202167	+	0.979351i	$0.435202\pi$
$968$	−131.313	−4.22055
$969$	0	0
$970$	0	0
$971$	−20.5595	−0.659787	−0.329894	−	0.944018i	$-0.607013\pi$
$971$	−20.5595	−0.659787	−0.329894	+	0.944018i	$0.607013\pi$
$972$	−18.9162	−0.606738
$973$	2.24964	0.0721202
$974$	79.6401	2.55183
$975$	0	0
$976$	−0.181813	−0.00581970
$977$	−1.26370	−0.0404295	−0.0202147	−	0.999796i	$-0.506435\pi$
$977$	−1.26370	−0.0404295	−0.0202147	+	0.999796i	$0.506435\pi$
$978$	2.28924	0.0732020
$979$	19.2092	0.613930
$980$	0	0
$981$	−31.7637	−1.01414
$982$	50.8364	1.62225
$983$	35.5329	1.13332	0.566662	−	0.823950i	$-0.308235\pi$
$983$	35.5329	1.13332	0.566662	+	0.823950i	$0.308235\pi$
$984$	5.25442	0.167505
$985$	0	0
$986$	58.7230	1.87012
$987$	−1.31480	−0.0418504
$988$	0	0
$989$	−9.38483	−0.298420
$990$	0	0
$991$	4.48854	0.142583	0.0712916	−	0.997456i	$-0.477288\pi$
$991$	4.48854	0.142583	0.0712916	+	0.997456i	$0.477288\pi$
$992$	−66.8470	−2.12239
$993$	−2.84236	−0.0901995
$994$	−8.15357	−0.258615
$995$	0	0
$996$	−0.276444	−0.00875946
$997$	−8.36943	−0.265063	−0.132531	−	0.991179i	$-0.542310\pi$
$997$	−8.36943	−0.265063	−0.132531	+	0.991179i	$0.542310\pi$
$998$	−4.72892	−0.149691
$999$	6.47283	0.204791

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9025.2.a.ck.1.1		16
5.2	odd	4		1805.2.b.j.1084.1	yes	16
5.3	odd	4		1805.2.b.j.1084.16	yes	16
5.4	even	2	inner	9025.2.a.ck.1.16		16
19.18	odd	2		9025.2.a.cl.1.16		16
95.18	even	4		1805.2.b.i.1084.1	✓	16
95.37	even	4		1805.2.b.i.1084.16	yes	16
95.94	odd	2		9025.2.a.cl.1.1		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1805.2.b.i.1084.1	✓	16	95.18	even	4
1805.2.b.i.1084.16	yes	16	95.37	even	4
1805.2.b.j.1084.1	yes	16	5.2	odd	4
1805.2.b.j.1084.16	yes	16	5.3	odd	4
9025.2.a.ck.1.1		16	1.1	even	1	trivial
9025.2.a.ck.1.16		16	5.4	even	2	inner
9025.2.a.cl.1.1		16	95.94	odd	2
9025.2.a.cl.1.16		16	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)

\(f(q)\)	\(=\)	\(q-2.61137 q^{2} -0.146779 q^{3} +4.81924 q^{4} +0.383293 q^{6} -1.14057 q^{7} -7.36208 q^{8} -2.97846 q^{9} +O(q^{10})\)	\(q-2.61137 q^{2} -0.146779 q^{3} +4.81924 q^{4} +0.383293 q^{6} -1.14057 q^{7} -7.36208 q^{8} -2.97846 q^{9} -5.36995 q^{11} -0.707361 q^{12} +2.41565 q^{13} +2.97846 q^{14} +9.58662 q^{16} +5.74451 q^{17} +7.77785 q^{18} +0.167412 q^{21} +14.0229 q^{22} +1.23526 q^{23} +1.08060 q^{24} -6.30815 q^{26} +0.877509 q^{27} -5.49670 q^{28} -3.91460 q^{29} +6.48369 q^{31} -10.3100 q^{32} +0.788193 q^{33} -15.0010 q^{34} -14.3539 q^{36} +7.37637 q^{37} -0.354565 q^{39} +4.86252 q^{41} -0.437173 q^{42} -7.59744 q^{43} -25.8791 q^{44} -3.22572 q^{46} -7.85368 q^{47} -1.40711 q^{48} -5.69909 q^{49} -0.843170 q^{51} +11.6416 q^{52} -0.0179026 q^{53} -2.29150 q^{54} +8.39699 q^{56} +10.2225 q^{58} -13.0250 q^{59} -0.0189653 q^{61} -16.9313 q^{62} +3.39715 q^{63} +7.75004 q^{64} -2.05826 q^{66} +7.07711 q^{67} +27.6842 q^{68} -0.181310 q^{69} -2.73751 q^{71} +21.9276 q^{72} -5.58836 q^{73} -19.2624 q^{74} +6.12482 q^{77} +0.925901 q^{78} -9.56774 q^{79} +8.80657 q^{81} -12.6978 q^{82} +0.390810 q^{83} +0.806797 q^{84} +19.8397 q^{86} +0.574579 q^{87} +39.5340 q^{88} -3.57717 q^{89} -2.75522 q^{91} +5.95303 q^{92} -0.951666 q^{93} +20.5088 q^{94} +1.51329 q^{96} +14.7430 q^{97} +14.8824 q^{98} +15.9942 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 12 q^{4} - 10 q^{6} + 6 q^{9}+O(q^{10}) \) 16 * q + 12 * q^4 - 10 * q^6 + 6 * q^9	\( 16 q + 12 q^{4} - 10 q^{6} + 6 q^{9} - 22 q^{11} - 6 q^{14} + 8 q^{16} + 20 q^{21} - 14 q^{24} - 16 q^{26} - 2 q^{29} - 16 q^{31} + 8 q^{34} + 18 q^{36} - 36 q^{39} - 26 q^{41} - 64 q^{44} - 2 q^{46} - 20 q^{49} + 38 q^{51} - 12 q^{54} - 6 q^{56} - 10 q^{59} - 30 q^{61} - 16 q^{64} + 4 q^{66} - 68 q^{69} + 20 q^{71} - 40 q^{74} - 12 q^{79} - 48 q^{81} + 2 q^{84} + 20 q^{86} + 86 q^{91} + 38 q^{94} + 22 q^{96} - 32 q^{99}+O(q^{100}) \) 16 * q + 12 * q^4 - 10 * q^6 + 6 * q^9 - 22 * q^11 - 6 * q^14 + 8 * q^16 + 20 * q^21 - 14 * q^24 - 16 * q^26 - 2 * q^29 - 16 * q^31 + 8 * q^34 + 18 * q^36 - 36 * q^39 - 26 * q^41 - 64 * q^44 - 2 * q^46 - 20 * q^49 + 38 * q^51 - 12 * q^54 - 6 * q^56 - 10 * q^59 - 30 * q^61 - 16 * q^64 + 4 * q^66 - 68 * q^69 + 20 * q^71 - 40 * q^74 - 12 * q^79 - 48 * q^81 + 2 * q^84 + 20 * q^86 + 86 * q^91 + 38 * q^94 + 22 * q^96 - 32 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

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Database

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