LMFDB - Embedded newform 2025.4.a.bl.1.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2025,4,Mod(1,2025)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2025, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("2025.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2025 = 3^{4} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2025.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:	$119.478867762$
Analytic rank:	$0$
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} - 91 x^{14} + 3268 x^{12} - 59128 x^{10} + 571975 x^{8} - 2881141 x^{6} + 6555196 x^{4} + \cdots + 614656 $ x^16 - 91x^14 + 3268x^12 - 59128x^10 + 571975x^8 - 2881141x^6 + 6555196x^4 - 4069504*x^2 + 614656
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{5}\cdot 3^{12}\cdot 7^{2} $
Twist minimal:	no (minimal twist has level 45)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.12
Root			$2.67506$ of defining polynomial
Character	$\chi$	$=$	2025.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.67506 q^{2} -0.844033 q^{4} +15.4153 q^{7} -23.6584 q^{8} +O(q^{10})$	$q+2.67506 q^{2} -0.844033 q^{4} +15.4153 q^{7} -23.6584 q^{8} -44.5598 q^{11} +28.0233 q^{13} +41.2370 q^{14} -56.5353 q^{16} +92.6615 q^{17} +49.5811 q^{19} -119.200 q^{22} -0.922809 q^{23} +74.9642 q^{26} -13.0110 q^{28} -189.849 q^{29} -299.180 q^{31} +38.0312 q^{32} +247.875 q^{34} +57.8330 q^{37} +132.633 q^{38} +287.880 q^{41} -0.591953 q^{43} +37.6099 q^{44} -2.46857 q^{46} +598.302 q^{47} -105.368 q^{49} -23.6526 q^{52} +146.339 q^{53} -364.701 q^{56} -507.859 q^{58} +193.066 q^{59} -566.153 q^{61} -800.326 q^{62} +554.019 q^{64} -355.163 q^{67} -78.2093 q^{68} +320.703 q^{71} +636.782 q^{73} +154.707 q^{74} -41.8481 q^{76} -686.903 q^{77} -287.765 q^{79} +770.097 q^{82} +285.058 q^{83} -1.58351 q^{86} +1054.21 q^{88} +331.615 q^{89} +431.988 q^{91} +0.778881 q^{92} +1600.50 q^{94} +1821.63 q^{97} -281.866 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 54 q^{4}+O(q^{10}) $ 16 * q + 54 * q^4	$ 16 q + 54 q^{4} + 90 q^{11} + 102 q^{14} + 146 q^{16} + 4 q^{19} + 468 q^{26} + 516 q^{29} + 38 q^{31} + 212 q^{34} + 576 q^{41} + 1644 q^{44} - 290 q^{46} - 4 q^{49} + 2430 q^{56} + 2202 q^{59} + 20 q^{61} - 322 q^{64} + 2952 q^{71} + 4080 q^{74} - 396 q^{76} - 218 q^{79} + 6108 q^{86} + 4074 q^{89} - 942 q^{91} - 1078 q^{94}+O(q^{100}) $ 16 * q + 54 * q^4 + 90 * q^11 + 102 * q^14 + 146 * q^16 + 4 * q^19 + 468 * q^26 + 516 * q^29 + 38 * q^31 + 212 * q^34 + 576 * q^41 + 1644 * q^44 - 290 * q^46 - 4 * q^49 + 2430 * q^56 + 2202 * q^59 + 20 * q^61 - 322 * q^64 + 2952 * q^71 + 4080 * q^74 - 396 * q^76 - 218 * q^79 + 6108 * q^86 + 4074 * q^89 - 942 * q^91 - 1078 * q^94

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 $

$2$	2.67506	0.945778	0.472889	−	0.881122i	$-0.343211\pi$
$2$	2.67506	0.945778	0.472889	+	0.881122i	$0.343211\pi$
$3$	0	0
$3$	0	0
$4$	−0.844033	−0.105504
$5$	0	0
$5$	0	0
$6$	0	0
$7$	15.4153	0.832349	0.416175	−	0.909285i	$-0.363371\pi$
$7$	15.4153	0.832349	0.416175	+	0.909285i	$0.363371\pi$
$8$	−23.6584	−1.04556
$9$	0	0
$10$	0	0
$11$	−44.5598	−1.22139	−0.610694	−	0.791866i	$-0.709110\pi$
$11$	−44.5598	−1.22139	−0.610694	+	0.791866i	$0.709110\pi$
$12$	0	0
$13$	28.0233	0.597867	0.298933	−	0.954274i	$-0.403369\pi$
$13$	28.0233	0.597867	0.298933	+	0.954274i	$0.403369\pi$
$14$	41.2370	0.787217
$15$	0	0
$16$	−56.5353	−0.883365
$17$	92.6615	1.32198	0.660991	−	0.750394i	$-0.270136\pi$
$17$	92.6615	1.32198	0.660991	+	0.750394i	$0.270136\pi$
$18$	0	0
$19$	49.5811	0.598667	0.299334	−	0.954149i	$-0.403236\pi$
$19$	49.5811	0.598667	0.299334	+	0.954149i	$0.403236\pi$
$20$	0	0
$21$	0	0
$22$	−119.200	−1.15516
$23$	−0.922809	−0.00836604	−0.00418302	−	0.999991i	$-0.501332\pi$
$23$	−0.922809	−0.00836604	−0.00418302	+	0.999991i	$0.501332\pi$
$24$	0	0
$25$	0	0
$26$	74.9642	0.565449
$27$	0	0
$28$	−13.0110	−0.0878163
$29$	−189.849	−1.21566	−0.607830	−	0.794067i	$-0.707960\pi$
$29$	−189.849	−1.21566	−0.607830	+	0.794067i	$0.707960\pi$
$30$	0	0
$31$	−299.180	−1.73337	−0.866683	−	0.498859i	$-0.833752\pi$
$31$	−299.180	−1.73337	−0.866683	+	0.498859i	$0.833752\pi$
$32$	38.0312	0.210095
$33$	0	0
$34$	247.875	1.25030
$35$	0	0
$36$	0	0
$37$	57.8330	0.256965	0.128482	−	0.991712i	$-0.458989\pi$
$37$	57.8330	0.256965	0.128482	+	0.991712i	$0.458989\pi$
$38$	132.633	0.566206
$39$	0	0
$40$	0	0
$41$	287.880	1.09657	0.548284	−	0.836292i	$-0.315281\pi$
$41$	287.880	1.09657	0.548284	+	0.836292i	$0.315281\pi$
$42$	0	0
$43$	−0.591953	−0.00209935	−0.00104967	−	0.999999i	$-0.500334\pi$
$43$	−0.591953	−0.00209935	−0.00104967	+	0.999999i	$0.500334\pi$
$44$	37.6099	0.128862
$45$	0	0
$46$	−2.46857	−0.00791242
$47$	598.302	1.85684	0.928418	−	0.371536i	$-0.121169\pi$
$47$	598.302	1.85684	0.928418	+	0.371536i	$0.121169\pi$
$48$	0	0
$49$	−105.368	−0.307195
$50$	0	0
$51$	0	0
$52$	−23.6526	−0.0630774
$53$	146.339	0.379267	0.189634	−	0.981855i	$-0.439270\pi$
$53$	146.339	0.379267	0.189634	+	0.981855i	$0.439270\pi$
$54$	0	0
$55$	0	0
$56$	−364.701	−0.870272
$57$	0	0
$58$	−507.859	−1.14975
$59$	193.066	0.426018	0.213009	−	0.977050i	$-0.431674\pi$
$59$	193.066	0.426018	0.213009	+	0.977050i	$0.431674\pi$
$60$	0	0
$61$	−566.153	−1.18833	−0.594167	−	0.804341i	$-0.702518\pi$
$61$	−566.153	−1.18833	−0.594167	+	0.804341i	$0.702518\pi$
$62$	−800.326	−1.63938
$63$	0	0
$64$	554.019	1.08207
$65$	0	0
$66$	0	0
$67$	−355.163	−0.647613	−0.323806	−	0.946123i	$-0.604963\pi$
$67$	−355.163	−0.647613	−0.323806	+	0.946123i	$0.604963\pi$
$68$	−78.2093	−0.139475
$69$	0	0
$70$	0	0
$71$	320.703	0.536062	0.268031	−	0.963410i	$-0.413627\pi$
$71$	320.703	0.536062	0.268031	+	0.963410i	$0.413627\pi$
$72$	0	0
$73$	636.782	1.02095	0.510477	−	0.859891i	$-0.329469\pi$
$73$	636.782	1.02095	0.510477	+	0.859891i	$0.329469\pi$
$74$	154.707	0.243031
$75$	0	0
$76$	−41.8481	−0.0631619
$77$	−686.903	−1.01662
$78$	0	0
$79$	−287.765	−0.409824	−0.204912	−	0.978780i	$-0.565691\pi$
$79$	−287.765	−0.409824	−0.204912	+	0.978780i	$0.565691\pi$
$80$	0	0
$81$	0	0
$82$	770.097	1.03711
$83$	285.058	0.376978	0.188489	−	0.982075i	$-0.439641\pi$
$83$	285.058	0.376978	0.188489	+	0.982075i	$0.439641\pi$
$84$	0	0
$85$	0	0
$86$	−1.58351	−0.00198552
$87$	0	0
$88$	1054.21	1.27704
$89$	331.615	0.394957	0.197478	−	0.980307i	$-0.436725\pi$
$89$	331.615	0.394957	0.197478	+	0.980307i	$0.436725\pi$
$90$	0	0
$91$	431.988	0.497634
$92$	0.778881	0.000882652 0
$93$	0	0
$94$	1600.50	1.75616
$95$	0	0
$96$	0	0
$97$	1821.63	1.90679	0.953393	−	0.301730i	$-0.0975642\pi$
$97$	1821.63	1.90679	0.953393	+	0.301730i	$0.0975642\pi$
$98$	−281.866	−0.290538
$99$	0	0
$100$	0	0
$101$	1147.37	1.13037	0.565184	−	0.824965i	$-0.308805\pi$
$101$	1147.37	1.13037	0.565184	+	0.824965i	$0.308805\pi$
$102$	0	0
$103$	1582.54	1.51391	0.756955	−	0.653467i	$-0.226686\pi$
$103$	1582.54	1.51391	0.756955	+	0.653467i	$0.226686\pi$
$104$	−662.986	−0.625107
$105$	0	0
$106$	391.466	0.358703
$107$	385.387	0.348194	0.174097	−	0.984728i	$-0.444299\pi$
$107$	385.387	0.348194	0.174097	+	0.984728i	$0.444299\pi$
$108$	0	0
$109$	−203.502	−0.178825	−0.0894126	−	0.995995i	$-0.528499\pi$
$109$	−203.502	−0.178825	−0.0894126	+	0.995995i	$0.528499\pi$
$110$	0	0
$111$	0	0
$112$	−871.510	−0.735268
$113$	−66.0150	−0.0549573	−0.0274786	−	0.999622i	$-0.508748\pi$
$113$	−66.0150	−0.0549573	−0.0274786	+	0.999622i	$0.508748\pi$
$114$	0	0
$115$	0	0
$116$	160.239	0.128257
$117$	0	0
$118$	516.464	0.402918
$119$	1428.41	1.10035
$120$	0	0
$121$	654.573	0.491790
$122$	−1514.49	−1.12390
$123$	0	0
$124$	252.518	0.182877
$125$	0	0
$126$	0	0
$127$	1821.52	1.27270	0.636352	−	0.771399i	$-0.280443\pi$
$127$	1821.52	1.27270	0.636352	+	0.771399i	$0.280443\pi$
$128$	1177.79	0.813301
$129$	0	0
$130$	0	0
$131$	−238.258	−0.158906	−0.0794530	−	0.996839i	$-0.525317\pi$
$131$	−238.258	−0.158906	−0.0794530	+	0.996839i	$0.525317\pi$
$132$	0	0
$133$	764.308	0.498300
$134$	−950.083	−0.612498
$135$	0	0
$136$	−2192.22	−1.38221
$137$	2047.48	1.27685	0.638424	−	0.769685i	$-0.279587\pi$
$137$	2047.48	1.27685	0.638424	+	0.769685i	$0.279587\pi$
$138$	0	0
$139$	−2179.37	−1.32987	−0.664934	−	0.746902i	$-0.731540\pi$
$139$	−2179.37	−1.32987	−0.664934	+	0.746902i	$0.731540\pi$
$140$	0	0
$141$	0	0
$142$	857.901	0.506996
$143$	−1248.71	−0.730228
$144$	0	0
$145$	0	0
$146$	1703.43	0.965596
$147$	0	0
$148$	−48.8130	−0.0271108
$149$	−155.948	−0.0857435	−0.0428717	−	0.999081i	$-0.513651\pi$
$149$	−155.948	−0.0857435	−0.0428717	+	0.999081i	$0.513651\pi$
$150$	0	0
$151$	1614.14	0.869914	0.434957	−	0.900451i	$-0.356764\pi$
$151$	1614.14	0.869914	0.434957	+	0.900451i	$0.356764\pi$
$152$	−1173.01	−0.625943
$153$	0	0
$154$	−1837.51	−0.961498
$155$	0	0
$156$	0	0
$157$	−988.508	−0.502494	−0.251247	−	0.967923i	$-0.580841\pi$
$157$	−988.508	−0.502494	−0.251247	+	0.967923i	$0.580841\pi$
$158$	−769.791	−0.387603
$159$	0	0
$160$	0	0
$161$	−14.2254	−0.00696347
$162$	0	0
$163$	2974.11	1.42914	0.714570	−	0.699564i	$-0.246622\pi$
$163$	2974.11	1.42914	0.714570	+	0.699564i	$0.246622\pi$
$164$	−242.980	−0.115692
$165$	0	0
$166$	762.548	0.356537
$167$	2438.96	1.13013	0.565067	−	0.825045i	$-0.308850\pi$
$167$	2438.96	1.13013	0.565067	+	0.825045i	$0.308850\pi$
$168$	0	0
$169$	−1411.69	−0.642555
$170$	0	0
$171$	0	0
$172$	0.499628	0.000221490 0
$173$	−2653.59	−1.16618	−0.583089	−	0.812409i	$-0.698156\pi$
$173$	−2653.59	−1.16618	−0.583089	+	0.812409i	$0.698156\pi$
$174$	0	0
$175$	0	0
$176$	2519.20	1.07893
$177$	0	0
$178$	887.092	0.373541
$179$	2102.30	0.877841	0.438921	−	0.898526i	$-0.355361\pi$
$179$	2102.30	0.877841	0.438921	+	0.898526i	$0.355361\pi$
$180$	0	0
$181$	1597.36	0.655973	0.327987	−	0.944682i	$-0.393630\pi$
$181$	1597.36	0.655973	0.327987	+	0.944682i	$0.393630\pi$
$182$	1155.60	0.470651
$183$	0	0
$184$	21.8321	0.00874721
$185$	0	0
$186$	0	0
$187$	−4128.97	−1.61465
$188$	−504.987	−0.195904
$189$	0	0
$190$	0	0
$191$	3407.43	1.29085	0.645426	−	0.763823i	$-0.276680\pi$
$191$	3407.43	1.29085	0.645426	+	0.763823i	$0.276680\pi$
$192$	0	0
$193$	−819.130	−0.305504	−0.152752	−	0.988265i	$-0.548814\pi$
$193$	−819.130	−0.305504	−0.152752	+	0.988265i	$0.548814\pi$
$194$	4872.97	1.80340
$195$	0	0
$196$	88.9340	0.0324103
$197$	359.979	0.130190	0.0650950	−	0.997879i	$-0.479265\pi$
$197$	359.979	0.130190	0.0650950	+	0.997879i	$0.479265\pi$
$198$	0	0
$199$	−1338.34	−0.476745	−0.238372	−	0.971174i	$-0.576614\pi$
$199$	−1338.34	−0.476745	−0.238372	+	0.971174i	$0.576614\pi$
$200$	0	0
$201$	0	0
$202$	3069.28	1.06908
$203$	−2926.59	−1.01185
$204$	0	0
$205$	0	0
$206$	4233.41	1.43182
$207$	0	0
$208$	−1584.31	−0.528135
$209$	−2209.32	−0.731205
$210$	0	0
$211$	1068.23	0.348530	0.174265	−	0.984699i	$-0.444245\pi$
$211$	1068.23	0.348530	0.174265	+	0.984699i	$0.444245\pi$
$212$	−123.515	−0.0400143
$213$	0	0
$214$	1030.94	0.329315
$215$	0	0
$216$	0	0
$217$	−4611.96	−1.44277
$218$	−544.381	−0.169129
$219$	0	0
$220$	0	0
$221$	2596.68	0.790370
$222$	0	0
$223$	1518.83	0.456091	0.228046	−	0.973650i	$-0.426766\pi$
$223$	1518.83	0.456091	0.228046	+	0.973650i	$0.426766\pi$
$224$	586.263	0.174872
$225$	0	0
$226$	−176.594	−0.0519774
$227$	−4235.12	−1.23830	−0.619152	−	0.785271i	$-0.712524\pi$
$227$	−4235.12	−1.23830	−0.619152	+	0.785271i	$0.712524\pi$
$228$	0	0
$229$	−5536.52	−1.59766	−0.798829	−	0.601558i	$-0.794547\pi$
$229$	−5536.52	−1.59766	−0.798829	+	0.601558i	$0.794547\pi$
$230$	0	0
$231$	0	0
$232$	4491.53	1.27105
$233$	6337.30	1.78185	0.890923	−	0.454154i	$-0.150059\pi$
$233$	6337.30	1.78185	0.890923	+	0.454154i	$0.150059\pi$
$234$	0	0
$235$	0	0
$236$	−162.954	−0.0449466
$237$	0	0
$238$	3821.08	1.04069
$239$	−2542.87	−0.688220	−0.344110	−	0.938929i	$-0.611819\pi$
$239$	−2542.87	−0.688220	−0.344110	+	0.938929i	$0.611819\pi$
$240$	0	0
$241$	2051.42	0.548314	0.274157	−	0.961685i	$-0.411601\pi$
$241$	2051.42	0.548314	0.274157	+	0.961685i	$0.411601\pi$
$242$	1751.02	0.465124
$243$	0	0
$244$	477.852	0.125374
$245$	0	0
$246$	0	0
$247$	1389.43	0.357923
$248$	7078.11	1.81234
$249$	0	0
$250$	0	0
$251$	2770.89	0.696801	0.348401	−	0.937346i	$-0.386725\pi$
$251$	2770.89	0.696801	0.348401	+	0.937346i	$0.386725\pi$
$252$	0	0
$253$	41.1201	0.0102182
$254$	4872.67	1.20370
$255$	0	0
$256$	−1281.50	−0.312865
$257$	−426.755	−0.103581	−0.0517904	−	0.998658i	$-0.516493\pi$
$257$	−426.755	−0.103581	−0.0517904	+	0.998658i	$0.516493\pi$
$258$	0	0
$259$	891.514	0.213884
$260$	0	0
$261$	0	0
$262$	−637.355	−0.150290
$263$	3762.39	0.882125	0.441063	−	0.897476i	$-0.354602\pi$
$263$	3762.39	0.882125	0.441063	+	0.897476i	$0.354602\pi$
$264$	0	0
$265$	0	0
$266$	2044.57	0.471281
$267$	0	0
$268$	299.769	0.0683258
$269$	1980.25	0.448839	0.224420	−	0.974493i	$-0.427951\pi$
$269$	1980.25	0.448839	0.224420	+	0.974493i	$0.427951\pi$
$270$	0	0
$271$	5659.70	1.26864	0.634321	−	0.773070i	$-0.281280\pi$
$271$	5659.70	1.26864	0.634321	+	0.773070i	$0.281280\pi$
$272$	−5238.65	−1.16779
$273$	0	0
$274$	5477.14	1.20761
$275$	0	0
$276$	0	0
$277$	−6169.57	−1.33824	−0.669121	−	0.743153i	$-0.733329\pi$
$277$	−6169.57	−1.33824	−0.669121	+	0.743153i	$0.733329\pi$
$278$	−5829.95	−1.25776
$279$	0	0
$280$	0	0
$281$	−3343.45	−0.709800	−0.354900	−	0.934904i	$-0.615485\pi$
$281$	−3343.45	−0.709800	−0.354900	+	0.934904i	$0.615485\pi$
$282$	0	0
$283$	5103.32	1.07195	0.535973	−	0.844235i	$-0.319945\pi$
$283$	5103.32	1.07195	0.535973	+	0.844235i	$0.319945\pi$
$284$	−270.684	−0.0565568
$285$	0	0
$286$	−3340.39	−0.690633
$287$	4437.76	0.912727
$288$	0	0
$289$	3673.14	0.747638
$290$	0	0
$291$	0	0
$292$	−537.465	−0.107715
$293$	4130.28	0.823527	0.411764	−	0.911291i	$-0.364913\pi$
$293$	4130.28	0.823527	0.411764	+	0.911291i	$0.364913\pi$
$294$	0	0
$295$	0	0
$296$	−1368.23	−0.268672
$297$	0	0
$298$	−417.172	−0.0810943
$299$	−25.8602	−0.00500178
$300$	0	0
$301$	−9.12515	−0.00174739
$302$	4317.93	0.822746
$303$	0	0
$304$	−2803.08	−0.528841
$305$	0	0
$306$	0	0
$307$	3382.52	0.628830	0.314415	−	0.949286i	$-0.398192\pi$
$307$	3382.52	0.628830	0.314415	+	0.949286i	$0.398192\pi$
$308$	579.769	0.107258
$309$	0	0
$310$	0	0
$311$	6021.22	1.09785	0.548926	−	0.835871i	$-0.315037\pi$
$311$	6021.22	1.09785	0.548926	+	0.835871i	$0.315037\pi$
$312$	0	0
$313$	−10192.9	−1.84070	−0.920350	−	0.391096i	$-0.872096\pi$
$313$	−10192.9	−1.84070	−0.920350	+	0.391096i	$0.872096\pi$
$314$	−2644.32	−0.475247
$315$	0	0
$316$	242.884	0.0432382
$317$	−5400.59	−0.956869	−0.478435	−	0.878123i	$-0.658796\pi$
$317$	−5400.59	−0.956869	−0.478435	+	0.878123i	$0.658796\pi$
$318$	0	0
$319$	8459.65	1.48479
$320$	0	0
$321$	0	0
$322$	−38.0538	−0.00658589
$323$	4594.25	0.791428
$324$	0	0
$325$	0	0
$326$	7955.92	1.35165
$327$	0	0
$328$	−6810.76	−1.14653
$329$	9223.02	1.54554
$330$	0	0
$331$	−2302.90	−0.382413	−0.191206	−	0.981550i	$-0.561240\pi$
$331$	−2302.90	−0.382413	−0.191206	+	0.981550i	$0.561240\pi$
$332$	−240.598	−0.0397727
$333$	0	0
$334$	6524.37	1.06886
$335$	0	0
$336$	0	0
$337$	9566.33	1.54632	0.773162	−	0.634209i	$-0.218674\pi$
$337$	9566.33	1.54632	0.773162	+	0.634209i	$0.218674\pi$
$338$	−3776.37	−0.607714
$339$	0	0
$340$	0	0
$341$	13331.4	2.11711
$342$	0	0
$343$	−6911.73	−1.08804
$344$	14.0046	0.00219500
$345$	0	0
$346$	−7098.52	−1.10294
$347$	81.5218	0.0126119	0.00630593	−	0.999980i	$-0.497993\pi$
$347$	81.5218	0.0126119	0.00630593	+	0.999980i	$0.497993\pi$
$348$	0	0
$349$	93.5719	0.0143518	0.00717591	−	0.999974i	$-0.497716\pi$
$349$	93.5719	0.0143518	0.00717591	+	0.999974i	$0.497716\pi$
$350$	0	0
$351$	0	0
$352$	−1694.66	−0.256607
$353$	−6675.41	−1.00651	−0.503253	−	0.864139i	$-0.667864\pi$
$353$	−6675.41	−1.00651	−0.503253	+	0.864139i	$0.667864\pi$
$354$	0	0
$355$	0	0
$356$	−279.894	−0.0416696
$357$	0	0
$358$	5623.80	0.830243
$359$	8334.50	1.22529	0.612644	−	0.790359i	$-0.290106\pi$
$359$	8334.50	1.22529	0.612644	+	0.790359i	$0.290106\pi$
$360$	0	0
$361$	−4400.72	−0.641598
$362$	4273.05	0.620405
$363$	0	0
$364$	−364.613	−0.0525025
$365$	0	0
$366$	0	0
$367$	125.761	0.0178874	0.00894369	−	0.999960i	$-0.497153\pi$
$367$	125.761	0.0178874	0.00894369	+	0.999960i	$0.497153\pi$
$368$	52.1713	0.00739026
$369$	0	0
$370$	0	0
$371$	2255.86	0.315683
$372$	0	0
$373$	−1435.76	−0.199305	−0.0996525	−	0.995022i	$-0.531773\pi$
$373$	−1435.76	−0.199305	−0.0996525	+	0.995022i	$0.531773\pi$
$374$	−11045.3	−1.52710
$375$	0	0
$376$	−14154.8	−1.94144
$377$	−5320.21	−0.726803
$378$	0	0
$379$	−2125.04	−0.288010	−0.144005	−	0.989577i	$-0.545998\pi$
$379$	−2125.04	−0.288010	−0.144005	+	0.989577i	$0.545998\pi$
$380$	0	0
$381$	0	0
$382$	9115.09	1.22086
$383$	−7060.76	−0.942005	−0.471002	−	0.882132i	$-0.656108\pi$
$383$	−7060.76	−0.942005	−0.471002	+	0.882132i	$0.656108\pi$
$384$	0	0
$385$	0	0
$386$	−2191.22	−0.288939
$387$	0	0
$388$	−1537.51	−0.201174
$389$	9819.12	1.27982	0.639909	−	0.768451i	$-0.278972\pi$
$389$	9819.12	1.27982	0.639909	+	0.768451i	$0.278972\pi$
$390$	0	0
$391$	−85.5088	−0.0110598
$392$	2492.83	0.321191
$393$	0	0
$394$	962.967	0.123131
$395$	0	0
$396$	0	0
$397$	−10995.5	−1.39005	−0.695023	−	0.718987i	$-0.744606\pi$
$397$	−10995.5	−1.39005	−0.695023	+	0.718987i	$0.744606\pi$
$398$	−3580.14	−0.450895
$399$	0	0
$400$	0	0
$401$	6531.81	0.813424	0.406712	−	0.913556i	$-0.366675\pi$
$401$	6531.81	0.813424	0.406712	+	0.913556i	$0.366675\pi$
$402$	0	0
$403$	−8384.02	−1.03632
$404$	−968.415	−0.119259
$405$	0	0
$406$	−7828.82	−0.956989
$407$	−2577.02	−0.313854
$408$	0	0
$409$	−5005.34	−0.605130	−0.302565	−	0.953129i	$-0.597843\pi$
$409$	−5005.34	−0.605130	−0.302565	+	0.953129i	$0.597843\pi$
$410$	0	0
$411$	0	0
$412$	−1335.72	−0.159724
$413$	2976.17	0.354596
$414$	0	0
$415$	0	0
$416$	1065.76	0.125609
$417$	0	0
$418$	−5910.07	−0.691558
$419$	73.4944	0.00856907	0.00428453	−	0.999991i	$-0.498636\pi$
$419$	73.4944	0.00856907	0.00428453	+	0.999991i	$0.498636\pi$
$420$	0	0
$421$	5030.81	0.582391	0.291195	−	0.956664i	$-0.405947\pi$
$421$	5030.81	0.582391	0.291195	+	0.956664i	$0.405947\pi$
$422$	2857.58	0.329632
$423$	0	0
$424$	−3462.13	−0.396547
$425$	0	0
$426$	0	0
$427$	−8727.43	−0.989110
$428$	−325.280	−0.0367360
$429$	0	0
$430$	0	0
$431$	1951.29	0.218075	0.109037	−	0.994038i	$-0.465223\pi$
$431$	1951.29	0.218075	0.109037	+	0.994038i	$0.465223\pi$
$432$	0	0
$433$	3805.50	0.422357	0.211178	−	0.977448i	$-0.432270\pi$
$433$	3805.50	0.422357	0.211178	+	0.977448i	$0.432270\pi$
$434$	−12337.3	−1.36454
$435$	0	0
$436$	171.762	0.0188668
$437$	−45.7538	−0.00500847
$438$	0	0
$439$	11904.4	1.29422	0.647111	−	0.762395i	$-0.275977\pi$
$439$	11904.4	1.29422	0.647111	+	0.762395i	$0.275977\pi$
$440$	0	0
$441$	0	0
$442$	6946.29	0.747514
$443$	−9068.45	−0.972585	−0.486292	−	0.873796i	$-0.661651\pi$
$443$	−9068.45	−0.972585	−0.486292	+	0.873796i	$0.661651\pi$
$444$	0	0
$445$	0	0
$446$	4062.97	0.431361
$447$	0	0
$448$	8540.37	0.900658
$449$	−7332.48	−0.770693	−0.385346	−	0.922772i	$-0.625918\pi$
$449$	−7332.48	−0.770693	−0.385346	+	0.922772i	$0.625918\pi$
$450$	0	0
$451$	−12827.9	−1.33934
$452$	55.7189	0.00579822
$453$	0	0
$454$	−11329.2	−1.17116
$455$	0	0
$456$	0	0
$457$	−1170.43	−0.119804	−0.0599019	−	0.998204i	$-0.519079\pi$
$457$	−1170.43	−0.119804	−0.0599019	+	0.998204i	$0.519079\pi$
$458$	−14810.5	−1.51103
$459$	0	0
$460$	0	0
$461$	14747.8	1.48996	0.744981	−	0.667086i	$-0.232459\pi$
$461$	14747.8	1.48996	0.744981	+	0.667086i	$0.232459\pi$
$462$	0	0
$463$	−2085.51	−0.209334	−0.104667	−	0.994507i	$-0.533378\pi$
$463$	−2085.51	−0.209334	−0.104667	+	0.994507i	$0.533378\pi$
$464$	10733.2	1.07387
$465$	0	0
$466$	16952.7	1.68523
$467$	9943.76	0.985315	0.492658	−	0.870223i	$-0.336025\pi$
$467$	9943.76	0.985315	0.492658	+	0.870223i	$0.336025\pi$
$468$	0	0
$469$	−5474.95	−0.539040
$470$	0	0
$471$	0	0
$472$	−4567.62	−0.445428
$473$	26.3773	0.00256412
$474$	0	0
$475$	0	0
$476$	−1205.62	−0.116092
$477$	0	0
$478$	−6802.34	−0.650903
$479$	2664.20	0.254134	0.127067	−	0.991894i	$-0.459444\pi$
$479$	2664.20	0.254134	0.127067	+	0.991894i	$0.459444\pi$
$480$	0	0
$481$	1620.67	0.153631
$482$	5487.68	0.518583
$483$	0	0
$484$	−552.481	−0.0518859
$485$	0	0
$486$	0	0
$487$	3071.50	0.285797	0.142898	−	0.989737i	$-0.454358\pi$
$487$	3071.50	0.285797	0.142898	+	0.989737i	$0.454358\pi$
$488$	13394.2	1.24248
$489$	0	0
$490$	0	0
$491$	−16089.6	−1.47885	−0.739423	−	0.673241i	$-0.764902\pi$
$491$	−16089.6	−1.47885	−0.739423	+	0.673241i	$0.764902\pi$
$492$	0	0
$493$	−17591.7	−1.60708
$494$	3716.80	0.338516
$495$	0	0
$496$	16914.3	1.53119
$497$	4943.74	0.446191
$498$	0	0
$499$	12060.6	1.08197	0.540987	−	0.841031i	$-0.318051\pi$
$499$	12060.6	1.08197	0.540987	+	0.841031i	$0.318051\pi$
$500$	0	0
$501$	0	0
$502$	7412.31	0.659019
$503$	1324.90	0.117444	0.0587222	−	0.998274i	$-0.481297\pi$
$503$	1324.90	0.117444	0.0587222	+	0.998274i	$0.481297\pi$
$504$	0	0
$505$	0	0
$506$	109.999	0.00966413
$507$	0	0
$508$	−1537.42	−0.134276
$509$	12121.1	1.05551	0.527757	−	0.849395i	$-0.323033\pi$
$509$	12121.1	1.05551	0.527757	+	0.849395i	$0.323033\pi$
$510$	0	0
$511$	9816.20	0.849791
$512$	−12850.4	−1.10920
$513$	0	0
$514$	−1141.60	−0.0979644
$515$	0	0
$516$	0	0
$517$	−26660.2	−2.26792
$518$	2384.86	0.202287
$519$	0	0
$520$	0	0
$521$	−17997.6	−1.51342	−0.756708	−	0.653753i	$-0.773193\pi$
$521$	−17997.6	−1.51342	−0.756708	+	0.653753i	$0.773193\pi$
$522$	0	0
$523$	−6451.03	−0.539357	−0.269679	−	0.962950i	$-0.586917\pi$
$523$	−6451.03	−0.539357	−0.269679	+	0.962950i	$0.586917\pi$
$524$	201.098	0.0167653
$525$	0	0
$526$	10064.6	0.834294
$527$	−27722.5	−2.29148
$528$	0	0
$529$	−12166.1	−0.999930
$530$	0	0
$531$	0	0
$532$	−645.101	−0.0525727
$533$	8067.35	0.655602
$534$	0	0
$535$	0	0
$536$	8402.57	0.677119
$537$	0	0
$538$	5297.29	0.424502
$539$	4695.17	0.375204
$540$	0	0
$541$	−520.899	−0.0413959	−0.0206980	−	0.999786i	$-0.506589\pi$
$541$	−520.899	−0.0413959	−0.0206980	+	0.999786i	$0.506589\pi$
$542$	15140.1	1.19985
$543$	0	0
$544$	3524.02	0.277741
$545$	0	0
$546$	0	0
$547$	−8098.33	−0.633015	−0.316508	−	0.948590i	$-0.602510\pi$
$547$	−8098.33	−0.633015	−0.316508	+	0.948590i	$0.602510\pi$
$548$	−1728.14	−0.134713
$549$	0	0
$550$	0	0
$551$	−9412.94	−0.727776
$552$	0	0
$553$	−4436.00	−0.341117
$554$	−16504.0	−1.26568
$555$	0	0
$556$	1839.46	0.140307
$557$	−6687.22	−0.508701	−0.254351	−	0.967112i	$-0.581862\pi$
$557$	−6687.22	−0.508701	−0.254351	+	0.967112i	$0.581862\pi$
$558$	0	0
$559$	−16.5885	−0.00125513
$560$	0	0
$561$	0	0
$562$	−8943.95	−0.671313
$563$	2427.54	0.181720	0.0908602	−	0.995864i	$-0.471038\pi$
$563$	2427.54	0.181720	0.0908602	+	0.995864i	$0.471038\pi$
$564$	0	0
$565$	0	0
$566$	13651.7	1.01382
$567$	0	0
$568$	−7587.30	−0.560486
$569$	−2553.65	−0.188145	−0.0940727	−	0.995565i	$-0.529989\pi$
$569$	−2553.65	−0.188145	−0.0940727	+	0.995565i	$0.529989\pi$
$570$	0	0
$571$	9036.15	0.662261	0.331130	−	0.943585i	$-0.392570\pi$
$571$	9036.15	0.662261	0.331130	+	0.943585i	$0.392570\pi$
$572$	1053.95	0.0770421
$573$	0	0
$574$	11871.3	0.863237
$575$	0	0
$576$	0	0
$577$	−5427.02	−0.391560	−0.195780	−	0.980648i	$-0.562724\pi$
$577$	−5427.02	−0.391560	−0.195780	+	0.980648i	$0.562724\pi$
$578$	9825.90	0.707099
$579$	0	0
$580$	0	0
$581$	4394.26	0.313777
$582$	0	0
$583$	−6520.82	−0.463233
$584$	−15065.2	−1.06747
$585$	0	0
$586$	11048.8	0.778874
$587$	5339.38	0.375434	0.187717	−	0.982223i	$-0.439891\pi$
$587$	5339.38	0.375434	0.187717	+	0.982223i	$0.439891\pi$
$588$	0	0
$589$	−14833.7	−1.03771
$590$	0	0
$591$	0	0
$592$	−3269.61	−0.226993
$593$	4617.67	0.319772	0.159886	−	0.987135i	$-0.448887\pi$
$593$	4617.67	0.319772	0.159886	+	0.987135i	$0.448887\pi$
$594$	0	0
$595$	0	0
$596$	131.626	0.00904629
$597$	0	0
$598$	−69.1776	−0.00473057
$599$	−10888.4	−0.742716	−0.371358	−	0.928490i	$-0.621108\pi$
$599$	−10888.4	−0.742716	−0.371358	+	0.928490i	$0.621108\pi$
$600$	0	0
$601$	−11273.2	−0.765131	−0.382566	−	0.923928i	$-0.624959\pi$
$601$	−11273.2	−0.765131	−0.382566	+	0.923928i	$0.624959\pi$
$602$	−24.4104	−0.00165264
$603$	0	0
$604$	−1362.39	−0.0917795
$605$	0	0
$606$	0	0
$607$	−10224.7	−0.683702	−0.341851	−	0.939754i	$-0.611054\pi$
$607$	−10224.7	−0.683702	−0.341851	+	0.939754i	$0.611054\pi$
$608$	1885.63	0.125777
$609$	0	0
$610$	0	0
$611$	16766.4	1.11014
$612$	0	0
$613$	7119.74	0.469109	0.234554	−	0.972103i	$-0.424637\pi$
$613$	7119.74	0.469109	0.234554	+	0.972103i	$0.424637\pi$
$614$	9048.46	0.594733
$615$	0	0
$616$	16251.0	1.06294
$617$	5358.95	0.349665	0.174832	−	0.984598i	$-0.444062\pi$
$617$	5358.95	0.349665	0.174832	+	0.984598i	$0.444062\pi$
$618$	0	0
$619$	−21597.2	−1.40237	−0.701183	−	0.712981i	$-0.747344\pi$
$619$	−21597.2	−1.40237	−0.701183	+	0.712981i	$0.747344\pi$
$620$	0	0
$621$	0	0
$622$	16107.1	1.03832
$623$	5111.96	0.328742
$624$	0	0
$625$	0	0
$626$	−27266.8	−1.74089
$627$	0	0
$628$	834.333	0.0530152
$629$	5358.89	0.339703
$630$	0	0
$631$	−1384.23	−0.0873305	−0.0436652	−	0.999046i	$-0.513903\pi$
$631$	−1384.23	−0.0873305	−0.0436652	+	0.999046i	$0.513903\pi$
$632$	6808.06	0.428497
$633$	0	0
$634$	−14446.9	−0.904986
$635$	0	0
$636$	0	0
$637$	−2952.76	−0.183662
$638$	22630.1	1.40429
$639$	0	0
$640$	0	0
$641$	1623.14	0.100016	0.0500080	−	0.998749i	$-0.484075\pi$
$641$	1623.14	0.100016	0.0500080	+	0.998749i	$0.484075\pi$
$642$	0	0
$643$	−28170.8	−1.72776	−0.863879	−	0.503699i	$-0.831972\pi$
$643$	−28170.8	−1.72776	−0.863879	+	0.503699i	$0.831972\pi$
$644$	12.0067	0.000734675 0
$645$	0	0
$646$	12289.9	0.748515
$647$	−16695.3	−1.01447	−0.507235	−	0.861808i	$-0.669332\pi$
$647$	−16695.3	−1.01447	−0.507235	+	0.861808i	$0.669332\pi$
$648$	0	0
$649$	−8602.97	−0.520333
$650$	0	0
$651$	0	0
$652$	−2510.24	−0.150780
$653$	4489.96	0.269075	0.134537	−	0.990909i	$-0.457045\pi$
$653$	4489.96	0.269075	0.134537	+	0.990909i	$0.457045\pi$
$654$	0	0
$655$	0	0
$656$	−16275.4	−0.968669
$657$	0	0
$658$	24672.2	1.46173
$659$	15300.9	0.904459	0.452229	−	0.891902i	$-0.350629\pi$
$659$	15300.9	0.904459	0.452229	+	0.891902i	$0.350629\pi$
$660$	0	0
$661$	−20634.9	−1.21423	−0.607113	−	0.794615i	$-0.707673\pi$
$661$	−20634.9	−1.21423	−0.607113	+	0.794615i	$0.707673\pi$
$662$	−6160.39	−0.361678
$663$	0	0
$664$	−6744.00	−0.394154
$665$	0	0
$666$	0	0
$667$	175.195	0.0101703
$668$	−2058.56	−0.119234
$669$	0	0
$670$	0	0
$671$	25227.6	1.45142
$672$	0	0
$673$	17553.8	1.00542	0.502712	−	0.864454i	$-0.332336\pi$
$673$	17553.8	1.00542	0.502712	+	0.864454i	$0.332336\pi$
$674$	25590.5	1.46248
$675$	0	0
$676$	1191.52	0.0677922
$677$	15182.6	0.861915	0.430958	−	0.902372i	$-0.358176\pi$
$677$	15182.6	0.861915	0.430958	+	0.902372i	$0.358176\pi$
$678$	0	0
$679$	28081.0	1.58711
$680$	0	0
$681$	0	0
$682$	35662.3	2.00232
$683$	1735.90	0.0972510	0.0486255	−	0.998817i	$-0.484516\pi$
$683$	1735.90	0.0972510	0.0486255	+	0.998817i	$0.484516\pi$
$684$	0	0
$685$	0	0
$686$	−18489.3	−1.02905
$687$	0	0
$688$	33.4663	0.00185449
$689$	4100.90	0.226751
$690$	0	0
$691$	−27648.0	−1.52211	−0.761055	−	0.648687i	$-0.775318\pi$
$691$	−27648.0	−1.52211	−0.761055	+	0.648687i	$0.775318\pi$
$692$	2239.72	0.123037
$693$	0	0
$694$	218.076	0.0119280
$695$	0	0
$696$	0	0
$697$	26675.4	1.44964
$698$	250.311	0.0135736
$699$	0	0
$700$	0	0
$701$	−19116.9	−1.03001	−0.515005	−	0.857187i	$-0.672210\pi$
$701$	−19116.9	−1.03001	−0.515005	+	0.857187i	$0.672210\pi$
$702$	0	0
$703$	2867.42	0.153836
$704$	−24686.9	−1.32162
$705$	0	0
$706$	−17857.2	−0.951931
$707$	17687.0	0.940861
$708$	0	0
$709$	−10045.3	−0.532099	−0.266050	−	0.963959i	$-0.585719\pi$
$709$	−10045.3	−0.532099	−0.266050	+	0.963959i	$0.585719\pi$
$710$	0	0
$711$	0	0
$712$	−7845.47	−0.412952
$713$	276.086	0.0145014
$714$	0	0
$715$	0	0
$716$	−1774.41	−0.0926159
$717$	0	0
$718$	22295.3	1.15885
$719$	−22559.7	−1.17014	−0.585072	−	0.810981i	$-0.698934\pi$
$719$	−22559.7	−1.17014	−0.585072	+	0.810981i	$0.698934\pi$
$720$	0	0
$721$	24395.4	1.26010
$722$	−11772.2	−0.606809
$723$	0	0
$724$	−1348.23	−0.0692079
$725$	0	0
$726$	0	0
$727$	−3668.07	−0.187127	−0.0935635	−	0.995613i	$-0.529826\pi$
$727$	−3668.07	−0.187127	−0.0935635	+	0.995613i	$0.529826\pi$
$728$	−10220.1	−0.520307
$729$	0	0
$730$	0	0
$731$	−54.8513	−0.00277530
$732$	0	0
$733$	31428.4	1.58368	0.791838	−	0.610731i	$-0.209124\pi$
$733$	31428.4	1.58368	0.791838	+	0.610731i	$0.209124\pi$
$734$	336.419	0.0169175
$735$	0	0
$736$	−35.0955	−0.00175766
$737$	15826.0	0.790987
$738$	0	0
$739$	17467.0	0.869465	0.434732	−	0.900560i	$-0.356843\pi$
$739$	17467.0	0.869465	0.434732	+	0.900560i	$0.356843\pi$
$740$	0	0
$741$	0	0
$742$	6034.57	0.298566
$743$	−24993.4	−1.23408	−0.617038	−	0.786933i	$-0.711668\pi$
$743$	−24993.4	−1.23408	−0.617038	+	0.786933i	$0.711668\pi$
$744$	0	0
$745$	0	0
$746$	−3840.74	−0.188498
$747$	0	0
$748$	3484.99	0.170353
$749$	5940.87	0.289819
$750$	0	0
$751$	−9348.04	−0.454214	−0.227107	−	0.973870i	$-0.572927\pi$
$751$	−9348.04	−0.454214	−0.227107	+	0.973870i	$0.572927\pi$
$752$	−33825.2	−1.64026
$753$	0	0
$754$	−14231.9	−0.687395
$755$	0	0
$756$	0	0
$757$	11708.4	0.562153	0.281076	−	0.959685i	$-0.409309\pi$
$757$	11708.4	0.562153	0.281076	+	0.959685i	$0.409309\pi$
$758$	−5684.61	−0.272394
$759$	0	0
$760$	0	0
$761$	−19410.7	−0.924621	−0.462310	−	0.886718i	$-0.652979\pi$
$761$	−19410.7	−0.924621	−0.462310	+	0.886718i	$0.652979\pi$
$762$	0	0
$763$	−3137.05	−0.148845
$764$	−2875.98	−0.136190
$765$	0	0
$766$	−18888.0	−0.890927
$767$	5410.35	0.254702
$768$	0	0
$769$	−9857.66	−0.462258	−0.231129	−	0.972923i	$-0.574242\pi$
$769$	−9857.66	−0.462258	−0.231129	+	0.972923i	$0.574242\pi$
$770$	0	0
$771$	0	0
$772$	691.373	0.0322319
$773$	−27574.5	−1.28303	−0.641517	−	0.767108i	$-0.721695\pi$
$773$	−27574.5	−1.28303	−0.641517	+	0.767108i	$0.721695\pi$
$774$	0	0
$775$	0	0
$776$	−43096.7	−1.99366
$777$	0	0
$778$	26266.8	1.21042
$779$	14273.4	0.656479
$780$	0	0
$781$	−14290.4	−0.654741
$782$	−228.741	−0.0104601
$783$	0	0
$784$	5957.01	0.271365
$785$	0	0
$786$	0	0
$787$	−11777.0	−0.533425	−0.266712	−	0.963776i	$-0.585937\pi$
$787$	−11777.0	−0.533425	−0.266712	+	0.963776i	$0.585937\pi$
$788$	−303.834	−0.0137356
$789$	0	0
$790$	0	0
$791$	−1017.64	−0.0457436
$792$	0	0
$793$	−15865.5	−0.710466
$794$	−29413.7	−1.31468
$795$	0	0
$796$	1129.60	0.0502985
$797$	24659.8	1.09598	0.547990	−	0.836485i	$-0.315393\pi$
$797$	24659.8	1.09598	0.547990	+	0.836485i	$0.315393\pi$
$798$	0	0
$799$	55439.5	2.45471
$800$	0	0
$801$	0	0
$802$	17473.0	0.769318
$803$	−28374.9	−1.24698
$804$	0	0
$805$	0	0
$806$	−22427.8	−0.980130
$807$	0	0
$808$	−27144.8	−1.18187
$809$	−19263.0	−0.837144	−0.418572	−	0.908184i	$-0.637469\pi$
$809$	−19263.0	−0.837144	−0.418572	+	0.908184i	$0.637469\pi$
$810$	0	0
$811$	−13597.4	−0.588741	−0.294370	−	0.955691i	$-0.595110\pi$
$811$	−13597.4	−0.588741	−0.294370	+	0.955691i	$0.595110\pi$
$812$	2470.14	0.106755
$813$	0	0
$814$	−6893.71	−0.296836
$815$	0	0
$816$	0	0
$817$	−29.3497	−0.00125681
$818$	−13389.6	−0.572318
$819$	0	0
$820$	0	0
$821$	−28673.6	−1.21890	−0.609449	−	0.792825i	$-0.708609\pi$
$821$	−28673.6	−1.21890	−0.609449	+	0.792825i	$0.708609\pi$
$822$	0	0
$823$	8437.62	0.357372	0.178686	−	0.983906i	$-0.442815\pi$
$823$	8437.62	0.357372	0.178686	+	0.983906i	$0.442815\pi$
$824$	−37440.4	−1.58289
$825$	0	0
$826$	7961.46	0.335369
$827$	11315.7	0.475799	0.237900	−	0.971290i	$-0.423541\pi$
$827$	11315.7	0.475799	0.237900	+	0.971290i	$0.423541\pi$
$828$	0	0
$829$	6773.57	0.283783	0.141891	−	0.989882i	$-0.454682\pi$
$829$	6773.57	0.283783	0.141891	+	0.989882i	$0.454682\pi$
$830$	0	0
$831$	0	0
$832$	15525.4	0.646932
$833$	−9763.54	−0.406106
$834$	0	0
$835$	0	0
$836$	1864.74	0.0771452
$837$	0	0
$838$	196.602	0.00810443
$839$	−8913.89	−0.366796	−0.183398	−	0.983039i	$-0.558710\pi$
$839$	−8913.89	−0.366796	−0.183398	+	0.983039i	$0.558710\pi$
$840$	0	0
$841$	11653.8	0.477831
$842$	13457.7	0.550812
$843$	0	0
$844$	−901.621	−0.0367714
$845$	0	0
$846$	0	0
$847$	10090.4	0.409341
$848$	−8273.31	−0.335031
$849$	0	0
$850$	0	0
$851$	−53.3688	−0.00214978
$852$	0	0
$853$	31669.2	1.27120	0.635600	−	0.772018i	$-0.280753\pi$
$853$	31669.2	1.27120	0.635600	+	0.772018i	$0.280753\pi$
$854$	−23346.4	−0.935478
$855$	0	0
$856$	−9117.63	−0.364059
$857$	−44054.8	−1.75599	−0.877995	−	0.478670i	$-0.841119\pi$
$857$	−44054.8	−1.75599	−0.877995	+	0.478670i	$0.841119\pi$
$858$	0	0
$859$	19069.4	0.757438	0.378719	−	0.925512i	$-0.376365\pi$
$859$	19069.4	0.757438	0.378719	+	0.925512i	$0.376365\pi$
$860$	0	0
$861$	0	0
$862$	5219.82	0.206250
$863$	−6256.14	−0.246769	−0.123384	−	0.992359i	$-0.539375\pi$
$863$	−6256.14	−0.246769	−0.123384	+	0.992359i	$0.539375\pi$
$864$	0	0
$865$	0	0
$866$	10179.9	0.399456
$867$	0	0
$868$	3892.65	0.152218
$869$	12822.8	0.500555
$870$	0	0
$871$	−9952.84	−0.387186
$872$	4814.52	0.186973
$873$	0	0
$874$	−122.394	−0.00473690
$875$	0	0
$876$	0	0
$877$	−6957.46	−0.267887	−0.133943	−	0.990989i	$-0.542764\pi$
$877$	−6957.46	−0.267887	−0.133943	+	0.990989i	$0.542764\pi$
$878$	31844.9	1.22405
$879$	0	0
$880$	0	0
$881$	−10626.4	−0.406371	−0.203186	−	0.979140i	$-0.565129\pi$
$881$	−10626.4	−0.406371	−0.203186	+	0.979140i	$0.565129\pi$
$882$	0	0
$883$	31013.5	1.18198	0.590990	−	0.806679i	$-0.298737\pi$
$883$	31013.5	1.18198	0.590990	+	0.806679i	$0.298737\pi$
$884$	−2191.69	−0.0833873
$885$	0	0
$886$	−24258.7	−0.919849
$887$	−9397.37	−0.355730	−0.177865	−	0.984055i	$-0.556919\pi$
$887$	−9397.37	−0.355730	−0.177865	+	0.984055i	$0.556919\pi$
$888$	0	0
$889$	28079.3	1.05933
$890$	0	0
$891$	0	0
$892$	−1281.94	−0.0481195
$893$	29664.5	1.11163
$894$	0	0
$895$	0	0
$896$	18155.9	0.676950
$897$	0	0
$898$	−19614.8	−0.728904
$899$	56799.2	2.10718
$900$	0	0
$901$	13560.0	0.501385
$902$	−34315.3	−1.26671
$903$	0	0
$904$	1561.81	0.0574612
$905$	0	0
$906$	0	0
$907$	−27560.5	−1.00896	−0.504482	−	0.863422i	$-0.668316\pi$
$907$	−27560.5	−1.00896	−0.504482	+	0.863422i	$0.668316\pi$
$908$	3574.58	0.130646
$909$	0	0
$910$	0	0
$911$	35870.1	1.30453	0.652266	−	0.757990i	$-0.273819\pi$
$911$	35870.1	1.30453	0.652266	+	0.757990i	$0.273819\pi$
$912$	0	0
$913$	−12702.1	−0.460436
$914$	−3130.97	−0.113308
$915$	0	0
$916$	4673.01	0.168560
$917$	−3672.82	−0.132265
$918$	0	0
$919$	3081.79	0.110619	0.0553095	−	0.998469i	$-0.482385\pi$
$919$	3081.79	0.110619	0.0553095	+	0.998469i	$0.482385\pi$
$920$	0	0
$921$	0	0
$922$	39451.2	1.40917
$923$	8987.16	0.320494
$924$	0	0
$925$	0	0
$926$	−5578.86	−0.197984
$927$	0	0
$928$	−7220.20	−0.255404
$929$	4270.03	0.150802	0.0754011	−	0.997153i	$-0.475976\pi$
$929$	4270.03	0.150802	0.0754011	+	0.997153i	$0.475976\pi$
$930$	0	0
$931$	−5224.25	−0.183908
$932$	−5348.89	−0.187992
$933$	0	0
$934$	26600.2	0.931890
$935$	0	0
$936$	0	0
$937$	−49225.3	−1.71624	−0.858122	−	0.513446i	$-0.828369\pi$
$937$	−49225.3	−1.71624	−0.858122	+	0.513446i	$0.828369\pi$
$938$	−14645.8	−0.509812
$939$	0	0
$940$	0	0
$941$	−34569.7	−1.19760	−0.598800	−	0.800899i	$-0.704356\pi$
$941$	−34569.7	−1.19760	−0.598800	+	0.800899i	$0.704356\pi$
$942$	0	0
$943$	−265.658	−0.00917393
$944$	−10915.1	−0.376329
$945$	0	0
$946$	70.5610	0.00242509
$947$	27342.4	0.938234	0.469117	−	0.883136i	$-0.344572\pi$
$947$	27342.4	0.938234	0.469117	+	0.883136i	$0.344572\pi$
$948$	0	0
$949$	17844.7	0.610395
$950$	0	0
$951$	0	0
$952$	−33793.7	−1.15048
$953$	45941.5	1.56158	0.780792	−	0.624791i	$-0.214816\pi$
$953$	45941.5	1.56158	0.780792	+	0.624791i	$0.214816\pi$
$954$	0	0
$955$	0	0
$956$	2146.27	0.0726101
$957$	0	0
$958$	7126.90	0.240355
$959$	31562.6	1.06278
$960$	0	0
$961$	59717.8	2.00456
$962$	4335.40	0.145300
$963$	0	0
$964$	−1731.47	−0.0578494
$965$	0	0
$966$	0	0
$967$	47338.8	1.57426	0.787132	−	0.616785i	$-0.211565\pi$
$967$	47338.8	1.57426	0.787132	+	0.616785i	$0.211565\pi$
$968$	−15486.1	−0.514197
$969$	0	0
$970$	0	0
$971$	−50168.3	−1.65806	−0.829031	−	0.559202i	$-0.811107\pi$
$971$	−50168.3	−1.65806	−0.829031	+	0.559202i	$0.811107\pi$
$972$	0	0
$973$	−33595.7	−1.10691
$974$	8216.46	0.270300
$975$	0	0
$976$	32007.6	1.04973
$977$	47642.1	1.56009	0.780044	−	0.625725i	$-0.215197\pi$
$977$	47642.1	1.56009	0.780044	+	0.625725i	$0.215197\pi$
$978$	0	0
$979$	−14776.7	−0.482396
$980$	0	0
$981$	0	0
$982$	−43040.7	−1.39866
$983$	−11158.1	−0.362044	−0.181022	−	0.983479i	$-0.557940\pi$
$983$	−11158.1	−0.362044	−0.181022	+	0.983479i	$0.557940\pi$
$984$	0	0
$985$	0	0
$986$	−47059.0	−1.51994
$987$	0	0
$988$	−1172.72	−0.0377624
$989$	0.546260	1.75632e−5 0
$990$	0	0
$991$	−57670.7	−1.84861	−0.924304	−	0.381658i	$-0.875353\pi$
$991$	−57670.7	−1.84861	−0.924304	+	0.381658i	$0.875353\pi$
$992$	−11378.2	−0.364171
$993$	0	0
$994$	13224.8	0.421998
$995$	0	0
$996$	0	0
$997$	51730.8	1.64326	0.821630	−	0.570021i	$-0.193065\pi$
$997$	51730.8	1.64326	0.821630	+	0.570021i	$0.193065\pi$
$998$	32262.8	1.02331
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2025.4.a.bl.1.12		16
3.2	odd	2		2025.4.a.bk.1.5		16
5.2	odd	4		405.4.b.f.244.12		16
5.3	odd	4		405.4.b.f.244.5		16
5.4	even	2	inner	2025.4.a.bl.1.5		16
9.2	odd	6		225.4.e.g.76.12		32
9.5	odd	6		225.4.e.g.151.12		32
15.2	even	4		405.4.b.e.244.5		16
15.8	even	4		405.4.b.e.244.12		16
15.14	odd	2		2025.4.a.bk.1.12		16
45.2	even	12		45.4.j.a.4.5	✓	32
45.7	odd	12		135.4.j.a.64.12		32
45.13	odd	12		135.4.j.a.19.12		32
45.14	odd	6		225.4.e.g.151.5		32
45.22	odd	12		135.4.j.a.19.5		32
45.23	even	12		45.4.j.a.34.5	yes	32
45.29	odd	6		225.4.e.g.76.5		32
45.32	even	12		45.4.j.a.34.12	yes	32
45.38	even	12		45.4.j.a.4.12	yes	32
45.43	odd	12		135.4.j.a.64.5		32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
45.4.j.a.4.5	✓	32	45.2	even	12
45.4.j.a.4.12	yes	32	45.38	even	12
45.4.j.a.34.5	yes	32	45.23	even	12
45.4.j.a.34.12	yes	32	45.32	even	12
135.4.j.a.19.5		32	45.22	odd	12
135.4.j.a.19.12		32	45.13	odd	12
135.4.j.a.64.5		32	45.43	odd	12
135.4.j.a.64.12		32	45.7	odd	12
225.4.e.g.76.5		32	45.29	odd	6
225.4.e.g.76.12		32	9.2	odd	6
225.4.e.g.151.5		32	45.14	odd	6
225.4.e.g.151.12		32	9.5	odd	6
405.4.b.e.244.5		16	15.2	even	4
405.4.b.e.244.12		16	15.8	even	4
405.4.b.f.244.5		16	5.3	odd	4
405.4.b.f.244.12		16	5.2	odd	4
2025.4.a.bk.1.5		16	3.2	odd	2
2025.4.a.bk.1.12		16	15.14	odd	2
2025.4.a.bl.1.5		16	5.4	even	2	inner
2025.4.a.bl.1.12		16	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 2025 = 3^{4} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2025.a (trivial)

\(f(q)\)	\(=\)	\(q+2.67506 q^{2} -0.844033 q^{4} +15.4153 q^{7} -23.6584 q^{8} +O(q^{10})\)	\(q+2.67506 q^{2} -0.844033 q^{4} +15.4153 q^{7} -23.6584 q^{8} -44.5598 q^{11} +28.0233 q^{13} +41.2370 q^{14} -56.5353 q^{16} +92.6615 q^{17} +49.5811 q^{19} -119.200 q^{22} -0.922809 q^{23} +74.9642 q^{26} -13.0110 q^{28} -189.849 q^{29} -299.180 q^{31} +38.0312 q^{32} +247.875 q^{34} +57.8330 q^{37} +132.633 q^{38} +287.880 q^{41} -0.591953 q^{43} +37.6099 q^{44} -2.46857 q^{46} +598.302 q^{47} -105.368 q^{49} -23.6526 q^{52} +146.339 q^{53} -364.701 q^{56} -507.859 q^{58} +193.066 q^{59} -566.153 q^{61} -800.326 q^{62} +554.019 q^{64} -355.163 q^{67} -78.2093 q^{68} +320.703 q^{71} +636.782 q^{73} +154.707 q^{74} -41.8481 q^{76} -686.903 q^{77} -287.765 q^{79} +770.097 q^{82} +285.058 q^{83} -1.58351 q^{86} +1054.21 q^{88} +331.615 q^{89} +431.988 q^{91} +0.778881 q^{92} +1600.50 q^{94} +1821.63 q^{97} -281.866 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 54 q^{4}+O(q^{10}) \) 16 * q + 54 * q^4	\( 16 q + 54 q^{4} + 90 q^{11} + 102 q^{14} + 146 q^{16} + 4 q^{19} + 468 q^{26} + 516 q^{29} + 38 q^{31} + 212 q^{34} + 576 q^{41} + 1644 q^{44} - 290 q^{46} - 4 q^{49} + 2430 q^{56} + 2202 q^{59} + 20 q^{61} - 322 q^{64} + 2952 q^{71} + 4080 q^{74} - 396 q^{76} - 218 q^{79} + 6108 q^{86} + 4074 q^{89} - 942 q^{91} - 1078 q^{94}+O(q^{100}) \) 16 * q + 54 * q^4 + 90 * q^11 + 102 * q^14 + 146 * q^16 + 4 * q^19 + 468 * q^26 + 516 * q^29 + 38 * q^31 + 212 * q^34 + 576 * q^41 + 1644 * q^44 - 290 * q^46 - 4 * q^49 + 2430 * q^56 + 2202 * q^59 + 20 * q^61 - 322 * q^64 + 2952 * q^71 + 4080 * q^74 - 396 * q^76 - 218 * q^79 + 6108 * q^86 + 4074 * q^89 - 942 * q^91 - 1078 * q^94

Introduction

L-functions

Modular forms

Varieties

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Representations

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Database

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