LMFDB - Embedded newform 2025.4.a.bd.1.6

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:	$1$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 44x^{6} + 567x^{4} - 2024x^{2} + 1900 $ x^8 - 44x^6 + 567x^4 - 2024*x^2 + 1900
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{5}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 405)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$1.85698$ 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)

$f(q)$	$=$	$q+1.85698 q^{2} -4.55164 q^{4} +1.02402 q^{7} -23.3081 q^{8} +O(q^{10})$	$q+1.85698 q^{2} -4.55164 q^{4} +1.02402 q^{7} -23.3081 q^{8} -16.1402 q^{11} +3.05437 q^{13} +1.90159 q^{14} -6.86951 q^{16} +69.2824 q^{17} -12.6207 q^{19} -29.9720 q^{22} -56.6563 q^{23} +5.67190 q^{26} -4.66098 q^{28} +196.187 q^{29} +213.360 q^{31} +173.708 q^{32} +128.656 q^{34} +299.951 q^{37} -23.4363 q^{38} -139.067 q^{41} -475.161 q^{43} +73.4644 q^{44} -105.209 q^{46} +193.371 q^{47} -341.951 q^{49} -13.9024 q^{52} -214.125 q^{53} -23.8680 q^{56} +364.315 q^{58} -149.848 q^{59} -495.145 q^{61} +396.204 q^{62} +377.528 q^{64} -761.598 q^{67} -315.348 q^{68} -736.184 q^{71} +701.058 q^{73} +557.001 q^{74} +57.4446 q^{76} -16.5279 q^{77} +780.649 q^{79} -258.245 q^{82} -961.505 q^{83} -882.362 q^{86} +376.198 q^{88} +520.561 q^{89} +3.12775 q^{91} +257.879 q^{92} +359.086 q^{94} +1155.19 q^{97} -634.996 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 24 q^{4}+O(q^{10}) $ 8 * q + 24 * q^4	$ 8 q + 24 q^{4} + 42 q^{14} + 4 q^{16} - 118 q^{19} - 36 q^{26} - 318 q^{29} - 416 q^{31} - 638 q^{34} + 486 q^{41} - 852 q^{44} - 598 q^{46} - 350 q^{49} + 1530 q^{56} + 1146 q^{59} - 398 q^{61} - 1640 q^{64} - 1728 q^{71} - 1218 q^{74} - 3498 q^{76} - 2596 q^{79} + 480 q^{86} - 1086 q^{89} - 2574 q^{91} - 1238 q^{94}+O(q^{100}) $ 8 * q + 24 * q^4 + 42 * q^14 + 4 * q^16 - 118 * q^19 - 36 * q^26 - 318 * q^29 - 416 * q^31 - 638 * q^34 + 486 * q^41 - 852 * q^44 - 598 * q^46 - 350 * q^49 + 1530 * q^56 + 1146 * q^59 - 398 * q^61 - 1640 * q^64 - 1728 * q^71 - 1218 * q^74 - 3498 * q^76 - 2596 * q^79 + 480 * q^86 - 1086 * q^89 - 2574 * q^91 - 1238 * 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 $

$2$	1.85698	0.656540	0.328270	−	0.944584i	$-0.393534\pi$
$2$	1.85698	0.656540	0.328270	+	0.944584i	$0.393534\pi$
$3$	0	0
$3$	0	0
$4$	−4.55164	−0.568955
$5$	0	0
$5$	0	0
$6$	0	0
$7$	1.02402	0.0552920	0.0276460	−	0.999618i	$-0.491199\pi$
$7$	1.02402	0.0552920	0.0276460	+	0.999618i	$0.491199\pi$
$8$	−23.3081	−1.03008
$9$	0	0
$10$	0	0
$11$	−16.1402	−0.442405	−0.221203	−	0.975228i	$-0.570998\pi$
$11$	−16.1402	−0.442405	−0.221203	+	0.975228i	$0.570998\pi$
$12$	0	0
$13$	3.05437	0.0651639	0.0325819	−	0.999469i	$-0.489627\pi$
$13$	3.05437	0.0651639	0.0325819	+	0.999469i	$0.489627\pi$
$14$	1.90159	0.0363015
$15$	0	0
$16$	−6.86951	−0.107336
$17$	69.2824	0.988438	0.494219	−	0.869337i	$-0.335454\pi$
$17$	69.2824	0.988438	0.494219	+	0.869337i	$0.335454\pi$
$18$	0	0
$19$	−12.6207	−0.152388	−0.0761941	−	0.997093i	$-0.524277\pi$
$19$	−12.6207	−0.152388	−0.0761941	+	0.997093i	$0.524277\pi$
$20$	0	0
$21$	0	0
$22$	−29.9720	−0.290457
$23$	−56.6563	−0.513637	−0.256819	−	0.966460i	$-0.582674\pi$
$23$	−56.6563	−0.513637	−0.256819	+	0.966460i	$0.582674\pi$
$24$	0	0
$25$	0	0
$26$	5.67190	0.0427827
$27$	0	0
$28$	−4.66098	−0.0314587
$29$	196.187	1.25624	0.628121	−	0.778116i	$-0.283824\pi$
$29$	196.187	1.25624	0.628121	+	0.778116i	$0.283824\pi$
$30$	0	0
$31$	213.360	1.23615	0.618073	−	0.786121i	$-0.287914\pi$
$31$	213.360	1.23615	0.618073	+	0.786121i	$0.287914\pi$
$32$	173.708	0.959612
$33$	0	0
$34$	128.656	0.648950
$35$	0	0
$36$	0	0
$37$	299.951	1.33275	0.666373	−	0.745619i	$-0.267846\pi$
$37$	299.951	1.33275	0.666373	+	0.745619i	$0.267846\pi$
$38$	−23.4363	−0.100049
$39$	0	0
$40$	0	0
$41$	−139.067	−0.529723	−0.264862	−	0.964286i	$-0.585326\pi$
$41$	−139.067	−0.529723	−0.264862	+	0.964286i	$0.585326\pi$
$42$	0	0
$43$	−475.161	−1.68515	−0.842573	−	0.538582i	$-0.818960\pi$
$43$	−475.161	−1.68515	−0.842573	+	0.538582i	$0.818960\pi$
$44$	73.4644	0.251708
$45$	0	0
$46$	−105.209	−0.337224
$47$	193.371	0.600130	0.300065	−	0.953919i	$-0.402992\pi$
$47$	193.371	0.600130	0.300065	+	0.953919i	$0.402992\pi$
$48$	0	0
$49$	−341.951	−0.996943
$50$	0	0
$51$	0	0
$52$	−13.9024	−0.0370753
$53$	−214.125	−0.554950	−0.277475	−	0.960733i	$-0.589498\pi$
$53$	−214.125	−0.554950	−0.277475	+	0.960733i	$0.589498\pi$
$54$	0	0
$55$	0	0
$56$	−23.8680	−0.0569553
$57$	0	0
$58$	364.315	0.824774
$59$	−149.848	−0.330653	−0.165326	−	0.986239i	$-0.552868\pi$
$59$	−149.848	−0.330653	−0.165326	+	0.986239i	$0.552868\pi$
$60$	0	0
$61$	−495.145	−1.03929	−0.519646	−	0.854382i	$-0.673936\pi$
$61$	−495.145	−1.03929	−0.519646	+	0.854382i	$0.673936\pi$
$62$	396.204	0.811580
$63$	0	0
$64$	377.528	0.737360
$65$	0	0
$66$	0	0
$67$	−761.598	−1.38872	−0.694358	−	0.719630i	$-0.744311\pi$
$67$	−761.598	−1.38872	−0.694358	+	0.719630i	$0.744311\pi$
$68$	−315.348	−0.562377
$69$	0	0
$70$	0	0
$71$	−736.184	−1.23055	−0.615274	−	0.788313i	$-0.710955\pi$
$71$	−736.184	−1.23055	−0.615274	+	0.788313i	$0.710955\pi$
$72$	0	0
$73$	701.058	1.12401	0.562004	−	0.827135i	$-0.310031\pi$
$73$	701.058	1.12401	0.562004	+	0.827135i	$0.310031\pi$
$74$	557.001	0.875001
$75$	0	0
$76$	57.4446	0.0867020
$77$	−16.5279	−0.0244615
$78$	0	0
$79$	780.649	1.11177	0.555886	−	0.831259i	$-0.312379\pi$
$79$	780.649	1.11177	0.555886	+	0.831259i	$0.312379\pi$
$80$	0	0
$81$	0	0
$82$	−258.245	−0.347785
$83$	−961.505	−1.27155	−0.635776	−	0.771873i	$-0.719320\pi$
$83$	−961.505	−1.27155	−0.635776	+	0.771873i	$0.719320\pi$
$84$	0	0
$85$	0	0
$86$	−882.362	−1.10637
$87$	0	0
$88$	376.198	0.455714
$89$	520.561	0.619993	0.309996	−	0.950738i	$-0.399672\pi$
$89$	520.561	0.619993	0.309996	+	0.950738i	$0.399672\pi$
$90$	0	0
$91$	3.12775	0.00360304
$92$	257.879	0.292236
$93$	0	0
$94$	359.086	0.394010
$95$	0	0
$96$	0	0
$97$	1155.19	1.20919	0.604596	−	0.796532i	$-0.293334\pi$
$97$	1155.19	1.20919	0.604596	+	0.796532i	$0.293334\pi$
$98$	−634.996	−0.654533
$99$	0	0
$100$	0	0
$101$	1388.75	1.36817	0.684086	−	0.729401i	$-0.260201\pi$
$101$	1388.75	1.36817	0.684086	+	0.729401i	$0.260201\pi$
$102$	0	0
$103$	−1003.13	−0.959622	−0.479811	−	0.877372i	$-0.659295\pi$
$103$	−1003.13	−0.959622	−0.479811	+	0.877372i	$0.659295\pi$
$104$	−71.1916	−0.0671242
$105$	0	0
$106$	−397.625	−0.364347
$107$	−1626.48	−1.46952	−0.734758	−	0.678329i	$-0.762704\pi$
$107$	−1626.48	−1.46952	−0.734758	+	0.678329i	$0.762704\pi$
$108$	0	0
$109$	−268.408	−0.235861	−0.117930	−	0.993022i	$-0.537626\pi$
$109$	−268.408	−0.235861	−0.117930	+	0.993022i	$0.537626\pi$
$110$	0	0
$111$	0	0
$112$	−7.03453	−0.00593483
$113$	−1190.42	−0.991022	−0.495511	−	0.868602i	$-0.665019\pi$
$113$	−1190.42	−0.991022	−0.495511	+	0.868602i	$0.665019\pi$
$114$	0	0
$115$	0	0
$116$	−892.972	−0.714745
$117$	0	0
$118$	−278.264	−0.217087
$119$	70.9468	0.0546528
$120$	0	0
$121$	−1070.49	−0.804278
$122$	−919.472	−0.682337
$123$	0	0
$124$	−971.135	−0.703311
$125$	0	0
$126$	0	0
$127$	−396.224	−0.276844	−0.138422	−	0.990373i	$-0.544203\pi$
$127$	−396.224	−0.276844	−0.138422	+	0.990373i	$0.544203\pi$
$128$	−688.605	−0.475505
$129$	0	0
$130$	0	0
$131$	995.717	0.664093	0.332046	−	0.943263i	$-0.392261\pi$
$131$	995.717	0.664093	0.332046	+	0.943263i	$0.392261\pi$
$132$	0	0
$133$	−12.9238	−0.00842585
$134$	−1414.27	−0.911748
$135$	0	0
$136$	−1614.84	−1.01817
$137$	−2198.88	−1.37126	−0.685631	−	0.727949i	$-0.740474\pi$
$137$	−2198.88	−1.37126	−0.685631	+	0.727949i	$0.740474\pi$
$138$	0	0
$139$	−475.975	−0.290444	−0.145222	−	0.989399i	$-0.546390\pi$
$139$	−475.975	−0.290444	−0.145222	+	0.989399i	$0.546390\pi$
$140$	0	0
$141$	0	0
$142$	−1367.08	−0.807905
$143$	−49.2982	−0.0288288
$144$	0	0
$145$	0	0
$146$	1301.85	0.737957
$147$	0	0
$148$	−1365.27	−0.758271
$149$	−1440.38	−0.791948	−0.395974	−	0.918262i	$-0.629593\pi$
$149$	−1440.38	−0.791948	−0.395974	+	0.918262i	$0.629593\pi$
$150$	0	0
$151$	−347.351	−0.187199	−0.0935994	−	0.995610i	$-0.529837\pi$
$151$	−347.351	−0.187199	−0.0935994	+	0.995610i	$0.529837\pi$
$152$	294.163	0.156972
$153$	0	0
$154$	−30.6920	−0.0160599
$155$	0	0
$156$	0	0
$157$	520.092	0.264381	0.132191	−	0.991224i	$-0.457799\pi$
$157$	520.092	0.264381	0.132191	+	0.991224i	$0.457799\pi$
$158$	1449.65	0.729923
$159$	0	0
$160$	0	0
$161$	−58.0174	−0.0284001
$162$	0	0
$163$	3756.90	1.80530	0.902648	−	0.430380i	$-0.141620\pi$
$163$	3756.90	1.80530	0.902648	+	0.430380i	$0.141620\pi$
$164$	632.984	0.301389
$165$	0	0
$166$	−1785.49	−0.834826
$167$	−936.744	−0.434056	−0.217028	−	0.976165i	$-0.569636\pi$
$167$	−936.744	−0.434056	−0.217028	+	0.976165i	$0.569636\pi$
$168$	0	0
$169$	−2187.67	−0.995754
$170$	0	0
$171$	0	0
$172$	2162.76	0.958772
$173$	2493.98	1.09603	0.548017	−	0.836467i	$-0.315383\pi$
$173$	2493.98	1.09603	0.548017	+	0.836467i	$0.315383\pi$
$174$	0	0
$175$	0	0
$176$	110.875	0.0474860
$177$	0	0
$178$	966.670	0.407050
$179$	−2769.95	−1.15662	−0.578312	−	0.815816i	$-0.696288\pi$
$179$	−2769.95	−1.15662	−0.578312	+	0.815816i	$0.696288\pi$
$180$	0	0
$181$	−477.139	−0.195942	−0.0979708	−	0.995189i	$-0.531235\pi$
$181$	−477.139	−0.195942	−0.0979708	+	0.995189i	$0.531235\pi$
$182$	5.80815	0.00236554
$183$	0	0
$184$	1320.55	0.529089
$185$	0	0
$186$	0	0
$187$	−1118.23	−0.437290
$188$	−880.156	−0.341447
$189$	0	0
$190$	0	0
$191$	5099.95	1.93204	0.966019	−	0.258469i	$-0.0832181\pi$
$191$	5099.95	1.93204	0.966019	+	0.258469i	$0.0832181\pi$
$192$	0	0
$193$	−4538.00	−1.69250	−0.846250	−	0.532786i	$-0.821145\pi$
$193$	−4538.00	−1.69250	−0.846250	+	0.532786i	$0.821145\pi$
$194$	2145.16	0.793884
$195$	0	0
$196$	1556.44	0.567215
$197$	835.536	0.302180	0.151090	−	0.988520i	$-0.451722\pi$
$197$	835.536	0.302180	0.151090	+	0.988520i	$0.451722\pi$
$198$	0	0
$199$	−5045.21	−1.79721	−0.898607	−	0.438753i	$-0.855420\pi$
$199$	−5045.21	−1.79721	−0.898607	+	0.438753i	$0.855420\pi$
$200$	0	0
$201$	0	0
$202$	2578.87	0.898261
$203$	200.900	0.0694602
$204$	0	0
$205$	0	0
$206$	−1862.78	−0.630031
$207$	0	0
$208$	−20.9820	−0.00699444
$209$	203.700	0.0674173
$210$	0	0
$211$	−4048.83	−1.32101	−0.660504	−	0.750822i	$-0.729658\pi$
$211$	−4048.83	−1.32101	−0.660504	+	0.750822i	$0.729658\pi$
$212$	974.620	0.315741
$213$	0	0
$214$	−3020.34	−0.964797
$215$	0	0
$216$	0	0
$217$	218.485	0.0683490
$218$	−498.427	−0.154852
$219$	0	0
$220$	0	0
$221$	211.614	0.0644105
$222$	0	0
$223$	−1976.14	−0.593419	−0.296709	−	0.954968i	$-0.595889\pi$
$223$	−1976.14	−0.593419	−0.296709	+	0.954968i	$0.595889\pi$
$224$	177.881	0.0530589
$225$	0	0
$226$	−2210.59	−0.650646
$227$	742.725	0.217165	0.108582	−	0.994087i	$-0.465369\pi$
$227$	742.725	0.217165	0.108582	+	0.994087i	$0.465369\pi$
$228$	0	0
$229$	−1151.56	−0.332301	−0.166151	−	0.986100i	$-0.553134\pi$
$229$	−1151.56	−0.332301	−0.166151	+	0.986100i	$0.553134\pi$
$230$	0	0
$231$	0	0
$232$	−4572.75	−1.29403
$233$	1954.50	0.549544	0.274772	−	0.961509i	$-0.411398\pi$
$233$	1954.50	0.549544	0.274772	+	0.961509i	$0.411398\pi$
$234$	0	0
$235$	0	0
$236$	682.052	0.188126
$237$	0	0
$238$	131.746	0.0358817
$239$	−1567.50	−0.424239	−0.212120	−	0.977244i	$-0.568037\pi$
$239$	−1567.50	−0.424239	−0.212120	+	0.977244i	$0.568037\pi$
$240$	0	0
$241$	−7402.25	−1.97851	−0.989255	−	0.146197i	$-0.953297\pi$
$241$	−7402.25	−1.97851	−0.989255	+	0.146197i	$0.953297\pi$
$242$	−1987.88	−0.528041
$243$	0	0
$244$	2253.72	0.591310
$245$	0	0
$246$	0	0
$247$	−38.5482	−0.00993021
$248$	−4973.01	−1.27333
$249$	0	0
$250$	0	0
$251$	−6720.19	−1.68994	−0.844970	−	0.534814i	$-0.820382\pi$
$251$	−6720.19	−1.68994	−0.844970	+	0.534814i	$0.820382\pi$
$252$	0	0
$253$	914.445	0.227236
$254$	−735.779	−0.181759
$255$	0	0
$256$	−4298.95	−1.04955
$257$	224.241	0.0544270	0.0272135	−	0.999630i	$-0.491337\pi$
$257$	224.241	0.0544270	0.0272135	+	0.999630i	$0.491337\pi$
$258$	0	0
$259$	307.156	0.0736902
$260$	0	0
$261$	0	0
$262$	1849.02	0.436004
$263$	−3935.69	−0.922757	−0.461379	−	0.887203i	$-0.652645\pi$
$263$	−3935.69	−0.922757	−0.461379	+	0.887203i	$0.652645\pi$
$264$	0	0
$265$	0	0
$266$	−23.9993	−0.00553191
$267$	0	0
$268$	3466.52	0.790116
$269$	−3509.03	−0.795352	−0.397676	−	0.917526i	$-0.630183\pi$
$269$	−3509.03	−0.795352	−0.397676	+	0.917526i	$0.630183\pi$
$270$	0	0
$271$	−3557.73	−0.797480	−0.398740	−	0.917064i	$-0.630552\pi$
$271$	−3557.73	−0.797480	−0.398740	+	0.917064i	$0.630552\pi$
$272$	−475.936	−0.106095
$273$	0	0
$274$	−4083.27	−0.900290
$275$	0	0
$276$	0	0
$277$	4502.64	0.976668	0.488334	−	0.872657i	$-0.337605\pi$
$277$	4502.64	0.976668	0.488334	+	0.872657i	$0.337605\pi$
$278$	−883.875	−0.190688
$279$	0	0
$280$	0	0
$281$	8534.68	1.81187	0.905936	−	0.423414i	$-0.139168\pi$
$281$	8534.68	1.81187	0.905936	+	0.423414i	$0.139168\pi$
$282$	0	0
$283$	4611.25	0.968589	0.484294	−	0.874905i	$-0.339076\pi$
$283$	4611.25	0.968589	0.484294	+	0.874905i	$0.339076\pi$
$284$	3350.84	0.700126
$285$	0	0
$286$	−91.5457	−0.0189273
$287$	−142.408	−0.0292895
$288$	0	0
$289$	−112.947	−0.0229895
$290$	0	0
$291$	0	0
$292$	−3190.96	−0.639510
$293$	7923.38	1.57983	0.789913	−	0.613219i	$-0.210126\pi$
$293$	7923.38	1.57983	0.789913	+	0.613219i	$0.210126\pi$
$294$	0	0
$295$	0	0
$296$	−6991.28	−1.37284
$297$	0	0
$298$	−2674.75	−0.519946
$299$	−173.050	−0.0334706
$300$	0	0
$301$	−486.575	−0.0931752
$302$	−645.022	−0.122904
$303$	0	0
$304$	86.6977	0.0163568
$305$	0	0
$306$	0	0
$307$	−3908.89	−0.726684	−0.363342	−	0.931656i	$-0.618364\pi$
$307$	−3908.89	−0.726684	−0.363342	+	0.931656i	$0.618364\pi$
$308$	75.2292	0.0139175
$309$	0	0
$310$	0	0
$311$	−5864.44	−1.06927	−0.534633	−	0.845084i	$-0.679550\pi$
$311$	−5864.44	−1.06927	−0.534633	+	0.845084i	$0.679550\pi$
$312$	0	0
$313$	−2580.04	−0.465918	−0.232959	−	0.972487i	$-0.574841\pi$
$313$	−2580.04	−0.465918	−0.232959	+	0.972487i	$0.574841\pi$
$314$	965.799	0.173577
$315$	0	0
$316$	−3553.23	−0.632547
$317$	5113.14	0.905939	0.452969	−	0.891526i	$-0.350365\pi$
$317$	5113.14	0.905939	0.452969	+	0.891526i	$0.350365\pi$
$318$	0	0
$319$	−3166.50	−0.555768
$320$	0	0
$321$	0	0
$322$	−107.737	−0.0186458
$323$	−874.389	−0.150626
$324$	0	0
$325$	0	0
$326$	6976.48	1.18525
$327$	0	0
$328$	3241.39	0.545659
$329$	198.017	0.0331824
$330$	0	0
$331$	−9659.43	−1.60402	−0.802009	−	0.597311i	$-0.796236\pi$
$331$	−9659.43	−1.60402	−0.802009	+	0.597311i	$0.796236\pi$
$332$	4376.42	0.723456
$333$	0	0
$334$	−1739.51	−0.284976
$335$	0	0
$336$	0	0
$337$	−10014.6	−1.61878	−0.809392	−	0.587269i	$-0.800203\pi$
$337$	−10014.6	−1.61878	−0.809392	+	0.587269i	$0.800203\pi$
$338$	−4062.45	−0.653753
$339$	0	0
$340$	0	0
$341$	−3443.67	−0.546877
$342$	0	0
$343$	−701.406	−0.110415
$344$	11075.1	1.73584
$345$	0	0
$346$	4631.27	0.719591
$347$	−302.277	−0.0467639	−0.0233819	−	0.999727i	$-0.507443\pi$
$347$	−302.277	−0.0467639	−0.0233819	+	0.999727i	$0.507443\pi$
$348$	0	0
$349$	4268.50	0.654692	0.327346	−	0.944904i	$-0.393846\pi$
$349$	4268.50	0.654692	0.327346	+	0.944904i	$0.393846\pi$
$350$	0	0
$351$	0	0
$352$	−2803.69	−0.424537
$353$	5331.06	0.803807	0.401904	−	0.915682i	$-0.368349\pi$
$353$	5331.06	0.803807	0.401904	+	0.915682i	$0.368349\pi$
$354$	0	0
$355$	0	0
$356$	−2369.40	−0.352748
$357$	0	0
$358$	−5143.73	−0.759370
$359$	5000.05	0.735077	0.367538	−	0.930008i	$-0.380201\pi$
$359$	5000.05	0.735077	0.367538	+	0.930008i	$0.380201\pi$
$360$	0	0
$361$	−6699.72	−0.976778
$362$	−886.036	−0.128644
$363$	0	0
$364$	−14.2364	−0.00204997
$365$	0	0
$366$	0	0
$367$	11958.3	1.70087	0.850433	−	0.526084i	$-0.176340\pi$
$367$	11958.3	1.70087	0.850433	+	0.526084i	$0.176340\pi$
$368$	389.201	0.0551318
$369$	0	0
$370$	0	0
$371$	−219.269	−0.0306843
$372$	0	0
$373$	8400.04	1.16605	0.583026	−	0.812453i	$-0.301868\pi$
$373$	8400.04	1.16605	0.583026	+	0.812453i	$0.301868\pi$
$374$	−2076.53	−0.287099
$375$	0	0
$376$	−4507.12	−0.618183
$377$	599.228	0.0818616
$378$	0	0
$379$	−3416.45	−0.463038	−0.231519	−	0.972830i	$-0.574370\pi$
$379$	−3416.45	−0.463038	−0.231519	+	0.972830i	$0.574370\pi$
$380$	0	0
$381$	0	0
$382$	9470.49	1.26846
$383$	−6883.21	−0.918317	−0.459159	−	0.888354i	$-0.651849\pi$
$383$	−6883.21	−0.918317	−0.459159	+	0.888354i	$0.651849\pi$
$384$	0	0
$385$	0	0
$386$	−8426.97	−1.11120
$387$	0	0
$388$	−5258.00	−0.687976
$389$	6131.20	0.799137	0.399568	−	0.916703i	$-0.369160\pi$
$389$	6131.20	0.799137	0.399568	+	0.916703i	$0.369160\pi$
$390$	0	0
$391$	−3925.29	−0.507699
$392$	7970.24	1.02693
$393$	0	0
$394$	1551.57	0.198393
$395$	0	0
$396$	0	0
$397$	−1078.96	−0.136401	−0.0682006	−	0.997672i	$-0.521726\pi$
$397$	−1078.96	−0.136401	−0.0682006	+	0.997672i	$0.521726\pi$
$398$	−9368.85	−1.17994
$399$	0	0
$400$	0	0
$401$	1481.85	0.184539	0.0922694	−	0.995734i	$-0.470588\pi$
$401$	1481.85	0.184539	0.0922694	+	0.995734i	$0.470588\pi$
$402$	0	0
$403$	651.680	0.0805521
$404$	−6321.07	−0.778428
$405$	0	0
$406$	373.067	0.0456034
$407$	−4841.27	−0.589613
$408$	0	0
$409$	−6855.84	−0.828850	−0.414425	−	0.910084i	$-0.636017\pi$
$409$	−6855.84	−0.828850	−0.414425	+	0.910084i	$0.636017\pi$
$410$	0	0
$411$	0	0
$412$	4565.87	0.545981
$413$	−153.447	−0.0182824
$414$	0	0
$415$	0	0
$416$	530.570	0.0625320
$417$	0	0
$418$	378.266	0.0442622
$419$	−1750.38	−0.204085	−0.102043	−	0.994780i	$-0.532538\pi$
$419$	−1750.38	−0.204085	−0.102043	+	0.994780i	$0.532538\pi$
$420$	0	0
$421$	185.281	0.0214490	0.0107245	−	0.999942i	$-0.496586\pi$
$421$	185.281	0.0214490	0.0107245	+	0.999942i	$0.496586\pi$
$422$	−7518.58	−0.867296
$423$	0	0
$424$	4990.85	0.571644
$425$	0	0
$426$	0	0
$427$	−507.039	−0.0574645
$428$	7403.17	0.836088
$429$	0	0
$430$	0	0
$431$	2340.26	0.261546	0.130773	−	0.991412i	$-0.458254\pi$
$431$	2340.26	0.261546	0.130773	+	0.991412i	$0.458254\pi$
$432$	0	0
$433$	−12295.7	−1.36465	−0.682326	−	0.731048i	$-0.739032\pi$
$433$	−12295.7	−1.36465	−0.682326	+	0.731048i	$0.739032\pi$
$434$	405.722	0.0448739
$435$	0	0
$436$	1221.70	0.134194
$437$	715.040	0.0782723
$438$	0	0
$439$	−4488.49	−0.487981	−0.243991	−	0.969778i	$-0.578457\pi$
$439$	−4488.49	−0.487981	−0.243991	+	0.969778i	$0.578457\pi$
$440$	0	0
$441$	0	0
$442$	392.963	0.0422881
$443$	9811.11	1.05223	0.526117	−	0.850412i	$-0.323647\pi$
$443$	9811.11	1.05223	0.526117	+	0.850412i	$0.323647\pi$
$444$	0	0
$445$	0	0
$446$	−3669.65	−0.389603
$447$	0	0
$448$	386.598	0.0407701
$449$	−4230.71	−0.444676	−0.222338	−	0.974970i	$-0.571369\pi$
$449$	−4230.71	−0.444676	−0.222338	+	0.974970i	$0.571369\pi$
$450$	0	0
$451$	2244.58	0.234352
$452$	5418.37	0.563846
$453$	0	0
$454$	1379.22	0.142577
$455$	0	0
$456$	0	0
$457$	−4686.07	−0.479662	−0.239831	−	0.970815i	$-0.577092\pi$
$457$	−4686.07	−0.479662	−0.239831	+	0.970815i	$0.577092\pi$
$458$	−2138.42	−0.218169
$459$	0	0
$460$	0	0
$461$	12995.4	1.31292	0.656460	−	0.754361i	$-0.272053\pi$
$461$	12995.4	1.31292	0.656460	+	0.754361i	$0.272053\pi$
$462$	0	0
$463$	950.920	0.0954492	0.0477246	−	0.998861i	$-0.484803\pi$
$463$	950.920	0.0954492	0.0477246	+	0.998861i	$0.484803\pi$
$464$	−1347.71	−0.134840
$465$	0	0
$466$	3629.47	0.360798
$467$	18320.3	1.81534	0.907670	−	0.419685i	$-0.137860\pi$
$467$	18320.3	1.81534	0.907670	+	0.419685i	$0.137860\pi$
$468$	0	0
$469$	−779.893	−0.0767849
$470$	0	0
$471$	0	0
$472$	3492.66	0.340599
$473$	7669.19	0.745518
$474$	0	0
$475$	0	0
$476$	−322.924	−0.0310949
$477$	0	0
$478$	−2910.81	−0.278530
$479$	10559.3	1.00724	0.503619	−	0.863926i	$-0.332002\pi$
$479$	10559.3	1.00724	0.503619	+	0.863926i	$0.332002\pi$
$480$	0	0
$481$	916.161	0.0868469
$482$	−13745.8	−1.29897
$483$	0	0
$484$	4872.50	0.457597
$485$	0	0
$486$	0	0
$487$	−677.906	−0.0630778	−0.0315389	−	0.999503i	$-0.510041\pi$
$487$	−677.906	−0.0630778	−0.0315389	+	0.999503i	$0.510041\pi$
$488$	11540.9	1.07056
$489$	0	0
$490$	0	0
$491$	−8179.39	−0.751794	−0.375897	−	0.926662i	$-0.622665\pi$
$491$	−8179.39	−0.751794	−0.375897	+	0.926662i	$0.622665\pi$
$492$	0	0
$493$	13592.3	1.24172
$494$	−71.5831	−0.00651958
$495$	0	0
$496$	−1465.68	−0.132683
$497$	−753.869	−0.0680395
$498$	0	0
$499$	7533.28	0.675823	0.337912	−	0.941178i	$-0.390280\pi$
$499$	7533.28	0.675823	0.337912	+	0.941178i	$0.390280\pi$
$500$	0	0
$501$	0	0
$502$	−12479.2	−1.10951
$503$	5729.79	0.507910	0.253955	−	0.967216i	$-0.418268\pi$
$503$	5729.79	0.507910	0.253955	+	0.967216i	$0.418268\pi$
$504$	0	0
$505$	0	0
$506$	1698.10	0.149190
$507$	0	0
$508$	1803.47	0.157512
$509$	−17391.8	−1.51450	−0.757250	−	0.653126i	$-0.773457\pi$
$509$	−17391.8	−1.51450	−0.757250	+	0.653126i	$0.773457\pi$
$510$	0	0
$511$	717.899	0.0621487
$512$	−2474.21	−0.213566
$513$	0	0
$514$	416.409	0.0357335
$515$	0	0
$516$	0	0
$517$	−3121.05	−0.265501
$518$	570.382	0.0483806
$519$	0	0
$520$	0	0
$521$	−3803.92	−0.319871	−0.159935	−	0.987127i	$-0.551129\pi$
$521$	−3803.92	−0.319871	−0.159935	+	0.987127i	$0.551129\pi$
$522$	0	0
$523$	−997.925	−0.0834345	−0.0417172	−	0.999129i	$-0.513283\pi$
$523$	−997.925	−0.0834345	−0.0417172	+	0.999129i	$0.513283\pi$
$524$	−4532.14	−0.377839
$525$	0	0
$526$	−7308.49	−0.605828
$527$	14782.1	1.22185
$528$	0	0
$529$	−8957.06	−0.736177
$530$	0	0
$531$	0	0
$532$	58.8246	0.00479393
$533$	−424.763	−0.0345188
$534$	0	0
$535$	0	0
$536$	17751.4	1.43049
$537$	0	0
$538$	−6516.20	−0.522181
$539$	5519.17	0.441053
$540$	0	0
$541$	−4211.74	−0.334707	−0.167354	−	0.985897i	$-0.553522\pi$
$541$	−4211.74	−0.334707	−0.167354	+	0.985897i	$0.553522\pi$
$542$	−6606.63	−0.523578
$543$	0	0
$544$	12034.9	0.948517
$545$	0	0
$546$	0	0
$547$	−13471.6	−1.05303	−0.526514	−	0.850167i	$-0.676501\pi$
$547$	−13471.6	−1.05303	−0.526514	+	0.850167i	$0.676501\pi$
$548$	10008.5	0.780186
$549$	0	0
$550$	0	0
$551$	−2476.01	−0.191436
$552$	0	0
$553$	799.403	0.0614721
$554$	8361.29	0.641222
$555$	0	0
$556$	2166.47	0.165249
$557$	−13291.0	−1.01105	−0.505526	−	0.862811i	$-0.668702\pi$
$557$	−13291.0	−1.01105	−0.505526	+	0.862811i	$0.668702\pi$
$558$	0	0
$559$	−1451.32	−0.109811
$560$	0	0
$561$	0	0
$562$	15848.7	1.18957
$563$	13629.1	1.02024	0.510121	−	0.860103i	$-0.329601\pi$
$563$	13629.1	1.02024	0.510121	+	0.860103i	$0.329601\pi$
$564$	0	0
$565$	0	0
$566$	8562.99	0.635918
$567$	0	0
$568$	17159.0	1.26757
$569$	−19977.4	−1.47187	−0.735935	−	0.677052i	$-0.763257\pi$
$569$	−19977.4	−1.47187	−0.735935	+	0.677052i	$0.763257\pi$
$570$	0	0
$571$	−21864.1	−1.60242	−0.801211	−	0.598383i	$-0.795810\pi$
$571$	−21864.1	−1.60242	−0.801211	+	0.598383i	$0.795810\pi$
$572$	224.388	0.0164023
$573$	0	0
$574$	−264.448	−0.0192297
$575$	0	0
$576$	0	0
$577$	10696.0	0.771714	0.385857	−	0.922559i	$-0.373906\pi$
$577$	10696.0	0.771714	0.385857	+	0.922559i	$0.373906\pi$
$578$	−209.741	−0.0150935
$579$	0	0
$580$	0	0
$581$	−984.603	−0.0703067
$582$	0	0
$583$	3456.02	0.245513
$584$	−16340.3	−1.15782
$585$	0	0
$586$	14713.5	1.03722
$587$	26421.1	1.85778	0.928889	−	0.370358i	$-0.120765\pi$
$587$	26421.1	1.85778	0.928889	+	0.370358i	$0.120765\pi$
$588$	0	0
$589$	−2692.74	−0.188374
$590$	0	0
$591$	0	0
$592$	−2060.51	−0.143052
$593$	25567.0	1.77051	0.885253	−	0.465110i	$-0.153985\pi$
$593$	25567.0	1.77051	0.885253	+	0.465110i	$0.153985\pi$
$594$	0	0
$595$	0	0
$596$	6556.07	0.450582
$597$	0	0
$598$	−321.349	−0.0219748
$599$	3565.31	0.243196	0.121598	−	0.992579i	$-0.461198\pi$
$599$	3565.31	0.243196	0.121598	+	0.992579i	$0.461198\pi$
$600$	0	0
$601$	8979.40	0.609446	0.304723	−	0.952441i	$-0.401436\pi$
$601$	8979.40	0.609446	0.304723	+	0.952441i	$0.401436\pi$
$602$	−903.559	−0.0611733
$603$	0	0
$604$	1581.01	0.106508
$605$	0	0
$606$	0	0
$607$	20174.5	1.34903	0.674513	−	0.738263i	$-0.264353\pi$
$607$	20174.5	1.34903	0.674513	+	0.738263i	$0.264353\pi$
$608$	−2192.31	−0.146234
$609$	0	0
$610$	0	0
$611$	590.628	0.0391068
$612$	0	0
$613$	−7366.17	−0.485346	−0.242673	−	0.970108i	$-0.578024\pi$
$613$	−7366.17	−0.485346	−0.242673	+	0.970108i	$0.578024\pi$
$614$	−7258.72	−0.477098
$615$	0	0
$616$	385.235	0.0251973
$617$	−1688.85	−0.110195	−0.0550975	−	0.998481i	$-0.517547\pi$
$617$	−1688.85	−0.110195	−0.0550975	+	0.998481i	$0.517547\pi$
$618$	0	0
$619$	−15073.5	−0.978765	−0.489383	−	0.872069i	$-0.662778\pi$
$619$	−15073.5	−0.978765	−0.489383	+	0.872069i	$0.662778\pi$
$620$	0	0
$621$	0	0
$622$	−10890.1	−0.702017
$623$	533.066	0.0342807
$624$	0	0
$625$	0	0
$626$	−4791.07	−0.305894
$627$	0	0
$628$	−2367.27	−0.150421
$629$	20781.3	1.31734
$630$	0	0
$631$	−5188.82	−0.327359	−0.163680	−	0.986514i	$-0.552336\pi$
$631$	−5188.82	−0.327359	−0.163680	+	0.986514i	$0.552336\pi$
$632$	−18195.5	−1.14522
$633$	0	0
$634$	9494.98	0.594785
$635$	0	0
$636$	0	0
$637$	−1044.45	−0.0649647
$638$	−5880.12	−0.364884
$639$	0	0
$640$	0	0
$641$	−13861.3	−0.854117	−0.427058	−	0.904224i	$-0.640450\pi$
$641$	−13861.3	−0.854117	−0.427058	+	0.904224i	$0.640450\pi$
$642$	0	0
$643$	−27120.4	−1.66333	−0.831667	−	0.555274i	$-0.812613\pi$
$643$	−27120.4	−1.66333	−0.831667	+	0.555274i	$0.812613\pi$
$644$	264.074	0.0161583
$645$	0	0
$646$	−1623.72	−0.0988923
$647$	941.957	0.0572367	0.0286184	−	0.999590i	$-0.490889\pi$
$647$	941.957	0.0572367	0.0286184	+	0.999590i	$0.490889\pi$
$648$	0	0
$649$	2418.57	0.146282
$650$	0	0
$651$	0	0
$652$	−17100.0	−1.02713
$653$	−6306.43	−0.377932	−0.188966	−	0.981984i	$-0.560514\pi$
$653$	−6306.43	−0.377932	−0.188966	+	0.981984i	$0.560514\pi$
$654$	0	0
$655$	0	0
$656$	955.324	0.0568584
$657$	0	0
$658$	367.712	0.0217856
$659$	−3003.37	−0.177533	−0.0887667	−	0.996052i	$-0.528293\pi$
$659$	−3003.37	−0.177533	−0.0887667	+	0.996052i	$0.528293\pi$
$660$	0	0
$661$	14926.2	0.878310	0.439155	−	0.898411i	$-0.355278\pi$
$661$	14926.2	0.878310	0.439155	+	0.898411i	$0.355278\pi$
$662$	−17937.3	−1.05310
$663$	0	0
$664$	22410.9	1.30980
$665$	0	0
$666$	0	0
$667$	−11115.2	−0.645253
$668$	4263.72	0.246958
$669$	0	0
$670$	0	0
$671$	7991.74	0.459788
$672$	0	0
$673$	12069.9	0.691321	0.345660	−	0.938360i	$-0.387655\pi$
$673$	12069.9	0.691321	0.345660	+	0.938360i	$0.387655\pi$
$674$	−18596.9	−1.06280
$675$	0	0
$676$	9957.48	0.566539
$677$	−8131.34	−0.461614	−0.230807	−	0.973000i	$-0.574137\pi$
$677$	−8131.34	−0.461614	−0.230807	+	0.973000i	$0.574137\pi$
$678$	0	0
$679$	1182.94	0.0668587
$680$	0	0
$681$	0	0
$682$	−6394.81	−0.359047
$683$	−28805.3	−1.61377	−0.806884	−	0.590710i	$-0.798848\pi$
$683$	−28805.3	−1.61377	−0.806884	+	0.590710i	$0.798848\pi$
$684$	0	0
$685$	0	0
$686$	−1302.49	−0.0724919
$687$	0	0
$688$	3264.12	0.180877
$689$	−654.018	−0.0361627
$690$	0	0
$691$	21170.5	1.16551	0.582754	−	0.812649i	$-0.301975\pi$
$691$	21170.5	1.16551	0.582754	+	0.812649i	$0.301975\pi$
$692$	−11351.7	−0.623594
$693$	0	0
$694$	−561.321	−0.0307024
$695$	0	0
$696$	0	0
$697$	−9634.92	−0.523599
$698$	7926.51	0.429832
$699$	0	0
$700$	0	0
$701$	31508.9	1.69768	0.848841	−	0.528649i	$-0.177301\pi$
$701$	31508.9	1.69768	0.848841	+	0.528649i	$0.177301\pi$
$702$	0	0
$703$	−3785.57	−0.203095
$704$	−6093.39	−0.326212
$705$	0	0
$706$	9899.66	0.527732
$707$	1422.11	0.0756490
$708$	0	0
$709$	−16548.3	−0.876567	−0.438283	−	0.898837i	$-0.644413\pi$
$709$	−16548.3	−0.876567	−0.438283	+	0.898837i	$0.644413\pi$
$710$	0	0
$711$	0	0
$712$	−12133.3	−0.638644
$713$	−12088.2	−0.634931
$714$	0	0
$715$	0	0
$716$	12607.8	0.658066
$717$	0	0
$718$	9284.97	0.482607
$719$	32635.3	1.69275	0.846377	−	0.532584i	$-0.178779\pi$
$719$	32635.3	1.69275	0.846377	+	0.532584i	$0.178779\pi$
$720$	0	0
$721$	−1027.22	−0.0530594
$722$	−12441.2	−0.641294
$723$	0	0
$724$	2171.76	0.111482
$725$	0	0
$726$	0	0
$727$	32435.2	1.65468	0.827341	−	0.561701i	$-0.189853\pi$
$727$	32435.2	1.65468	0.827341	+	0.561701i	$0.189853\pi$
$728$	−72.9018	−0.00371143
$729$	0	0
$730$	0	0
$731$	−32920.3	−1.66566
$732$	0	0
$733$	15278.0	0.769859	0.384930	−	0.922946i	$-0.374226\pi$
$733$	15278.0	0.769859	0.384930	+	0.922946i	$0.374226\pi$
$734$	22206.3	1.11669
$735$	0	0
$736$	−9841.67	−0.492893
$737$	12292.3	0.614375
$738$	0	0
$739$	−20611.4	−1.02599	−0.512993	−	0.858393i	$-0.671463\pi$
$739$	−20611.4	−1.02599	−0.512993	+	0.858393i	$0.671463\pi$
$740$	0	0
$741$	0	0
$742$	−407.177	−0.0201455
$743$	−21784.4	−1.07563	−0.537815	−	0.843063i	$-0.680750\pi$
$743$	−21784.4	−1.07563	−0.537815	+	0.843063i	$0.680750\pi$
$744$	0	0
$745$	0	0
$746$	15598.7	0.765561
$747$	0	0
$748$	5089.79	0.248798
$749$	−1665.56	−0.0812525
$750$	0	0
$751$	18299.2	0.889146	0.444573	−	0.895743i	$-0.353355\pi$
$751$	18299.2	0.889146	0.444573	+	0.895743i	$0.353355\pi$
$752$	−1328.37	−0.0644156
$753$	0	0
$754$	1112.75	0.0537455
$755$	0	0
$756$	0	0
$757$	27136.5	1.30290	0.651448	−	0.758693i	$-0.274162\pi$
$757$	27136.5	1.30290	0.651448	+	0.758693i	$0.274162\pi$
$758$	−6344.27	−0.304003
$759$	0	0
$760$	0	0
$761$	5534.03	0.263611	0.131806	−	0.991276i	$-0.457922\pi$
$761$	5534.03	0.263611	0.131806	+	0.991276i	$0.457922\pi$
$762$	0	0
$763$	−274.856	−0.0130412
$764$	−23213.1	−1.09924
$765$	0	0
$766$	−12782.0	−0.602912
$767$	−457.691	−0.0215466
$768$	0	0
$769$	−30272.2	−1.41956	−0.709782	−	0.704422i	$-0.751206\pi$
$769$	−30272.2	−1.41956	−0.709782	+	0.704422i	$0.751206\pi$
$770$	0	0
$771$	0	0
$772$	20655.3	0.962956
$773$	−16238.3	−0.755562	−0.377781	−	0.925895i	$-0.623313\pi$
$773$	−16238.3	−0.755562	−0.377781	+	0.925895i	$0.623313\pi$
$774$	0	0
$775$	0	0
$776$	−26925.3	−1.24557
$777$	0	0
$778$	11385.5	0.524666
$779$	1755.12	0.0807236
$780$	0	0
$781$	11882.2	0.544401
$782$	−7289.17	−0.333325
$783$	0	0
$784$	2349.04	0.107008
$785$	0	0
$786$	0	0
$787$	−17287.5	−0.783017	−0.391508	−	0.920174i	$-0.628047\pi$
$787$	−17287.5	−0.783017	−0.391508	+	0.920174i	$0.628047\pi$
$788$	−3803.06	−0.171927
$789$	0	0
$790$	0	0
$791$	−1219.02	−0.0547956
$792$	0	0
$793$	−1512.36	−0.0677243
$794$	−2003.60	−0.0895529
$795$	0	0
$796$	22964.0	1.02253
$797$	−1836.82	−0.0816354	−0.0408177	−	0.999167i	$-0.512996\pi$
$797$	−1836.82	−0.0816354	−0.0408177	+	0.999167i	$0.512996\pi$
$798$	0	0
$799$	13397.2	0.593192
$800$	0	0
$801$	0	0
$802$	2751.76	0.121157
$803$	−11315.2	−0.497267
$804$	0	0
$805$	0	0
$806$	1210.15	0.0528857
$807$	0	0
$808$	−32369.0	−1.40933
$809$	14459.9	0.628410	0.314205	−	0.949355i	$-0.398262\pi$
$809$	14459.9	0.628410	0.314205	+	0.949355i	$0.398262\pi$
$810$	0	0
$811$	30701.8	1.32933	0.664664	−	0.747142i	$-0.268575\pi$
$811$	30701.8	1.32933	0.664664	+	0.747142i	$0.268575\pi$
$812$	−914.424	−0.0395197
$813$	0	0
$814$	−8990.12	−0.387105
$815$	0	0
$816$	0	0
$817$	5996.84	0.256797
$818$	−12731.1	−0.544173
$819$	0	0
$820$	0	0
$821$	−29132.0	−1.23838	−0.619192	−	0.785239i	$-0.712540\pi$
$821$	−29132.0	−1.23838	−0.619192	+	0.785239i	$0.712540\pi$
$822$	0	0
$823$	−35618.9	−1.50862	−0.754312	−	0.656516i	$-0.772029\pi$
$823$	−35618.9	−1.50862	−0.754312	+	0.656516i	$0.772029\pi$
$824$	23381.0	0.988490
$825$	0	0
$826$	−284.948	−0.0120032
$827$	9616.94	0.404370	0.202185	−	0.979347i	$-0.435196\pi$
$827$	9616.94	0.404370	0.202185	+	0.979347i	$0.435196\pi$
$828$	0	0
$829$	38846.8	1.62751	0.813755	−	0.581208i	$-0.197420\pi$
$829$	38846.8	1.62751	0.813755	+	0.581208i	$0.197420\pi$
$830$	0	0
$831$	0	0
$832$	1153.11	0.0480492
$833$	−23691.2	−0.985417
$834$	0	0
$835$	0	0
$836$	−927.168	−0.0383574
$837$	0	0
$838$	−3250.41	−0.133990
$839$	−3938.86	−0.162079	−0.0810396	−	0.996711i	$-0.525824\pi$
$839$	−3938.86	−0.162079	−0.0810396	+	0.996711i	$0.525824\pi$
$840$	0	0
$841$	14100.3	0.578144
$842$	344.062	0.0140822
$843$	0	0
$844$	18428.8	0.751594
$845$	0	0
$846$	0	0
$847$	−1096.21	−0.0444701
$848$	1470.93	0.0595661
$849$	0	0
$850$	0	0
$851$	−16994.1	−0.684548
$852$	0	0
$853$	−27660.3	−1.11028	−0.555142	−	0.831756i	$-0.687336\pi$
$853$	−27660.3	−1.11028	−0.555142	+	0.831756i	$0.687336\pi$
$854$	−941.561	−0.0377278
$855$	0	0
$856$	37910.3	1.51372
$857$	21674.2	0.863918	0.431959	−	0.901893i	$-0.357823\pi$
$857$	21674.2	0.863918	0.431959	+	0.901893i	$0.357823\pi$
$858$	0	0
$859$	−9236.03	−0.366856	−0.183428	−	0.983033i	$-0.558719\pi$
$859$	−9236.03	−0.366856	−0.183428	+	0.983033i	$0.558719\pi$
$860$	0	0
$861$	0	0
$862$	4345.81	0.171715
$863$	8055.86	0.317758	0.158879	−	0.987298i	$-0.449212\pi$
$863$	8055.86	0.317758	0.158879	+	0.987298i	$0.449212\pi$
$864$	0	0
$865$	0	0
$866$	−22832.9	−0.895950
$867$	0	0
$868$	−994.464	−0.0388875
$869$	−12599.8	−0.491853
$870$	0	0
$871$	−2326.20	−0.0904941
$872$	6256.08	0.242956
$873$	0	0
$874$	1327.81	0.0513889
$875$	0	0
$876$	0	0
$877$	−34062.1	−1.31151	−0.655756	−	0.754973i	$-0.727650\pi$
$877$	−34062.1	−1.31151	−0.655756	+	0.754973i	$0.727650\pi$
$878$	−8335.01	−0.320379
$879$	0	0
$880$	0	0
$881$	29946.6	1.14521	0.572603	−	0.819833i	$-0.305934\pi$
$881$	29946.6	1.14521	0.572603	+	0.819833i	$0.305934\pi$
$882$	0	0
$883$	45422.6	1.73114	0.865568	−	0.500792i	$-0.166958\pi$
$883$	45422.6	1.73114	0.865568	+	0.500792i	$0.166958\pi$
$884$	−963.191	−0.0366466
$885$	0	0
$886$	18219.0	0.690835
$887$	46885.3	1.77481	0.887405	−	0.460991i	$-0.152506\pi$
$887$	46885.3	1.77481	0.887405	+	0.460991i	$0.152506\pi$
$888$	0	0
$889$	−405.743	−0.0153073
$890$	0	0
$891$	0	0
$892$	8994.69	0.337628
$893$	−2440.47	−0.0914528
$894$	0	0
$895$	0	0
$896$	−705.147	−0.0262916
$897$	0	0
$898$	−7856.33	−0.291948
$899$	41858.4	1.55290
$900$	0	0
$901$	−14835.1	−0.548534
$902$	4168.12	0.153862
$903$	0	0
$904$	27746.5	1.02083
$905$	0	0
$906$	0	0
$907$	−31490.5	−1.15284	−0.576420	−	0.817153i	$-0.695551\pi$
$907$	−31490.5	−1.15284	−0.576420	+	0.817153i	$0.695551\pi$
$908$	−3380.61	−0.123557
$909$	0	0
$910$	0	0
$911$	−40798.1	−1.48376	−0.741878	−	0.670535i	$-0.766065\pi$
$911$	−40798.1	−1.48376	−0.741878	+	0.670535i	$0.766065\pi$
$912$	0	0
$913$	15518.9	0.562542
$914$	−8701.93	−0.314917
$915$	0	0
$916$	5241.47	0.189064
$917$	1019.64	0.0367190
$918$	0	0
$919$	−25901.7	−0.929725	−0.464862	−	0.885383i	$-0.653896\pi$
$919$	−25901.7	−0.929725	−0.464862	+	0.885383i	$0.653896\pi$
$920$	0	0
$921$	0	0
$922$	24132.2	0.861985
$923$	−2248.58	−0.0801873
$924$	0	0
$925$	0	0
$926$	1765.84	0.0626663
$927$	0	0
$928$	34079.3	1.20550
$929$	−6125.72	−0.216338	−0.108169	−	0.994132i	$-0.534499\pi$
$929$	−6125.72	−0.216338	−0.108169	+	0.994132i	$0.534499\pi$
$930$	0	0
$931$	4315.65	0.151922
$932$	−8896.19	−0.312666
$933$	0	0
$934$	34020.4	1.19184
$935$	0	0
$936$	0	0
$937$	32640.5	1.13801	0.569007	−	0.822333i	$-0.307328\pi$
$937$	32640.5	1.13801	0.569007	+	0.822333i	$0.307328\pi$
$938$	−1448.24	−0.0504124
$939$	0	0
$940$	0	0
$941$	−23106.0	−0.800460	−0.400230	−	0.916415i	$-0.631070\pi$
$941$	−23106.0	−0.800460	−0.400230	+	0.916415i	$0.631070\pi$
$942$	0	0
$943$	7879.04	0.272086
$944$	1029.38	0.0354909
$945$	0	0
$946$	14241.5	0.489463
$947$	−2309.51	−0.0792491	−0.0396245	−	0.999215i	$-0.512616\pi$
$947$	−2309.51	−0.0792491	−0.0396245	+	0.999215i	$0.512616\pi$
$948$	0	0
$949$	2141.29	0.0732447
$950$	0	0
$951$	0	0
$952$	−1653.63	−0.0562968
$953$	−27818.8	−0.945580	−0.472790	−	0.881175i	$-0.656753\pi$
$953$	−27818.8	−0.945580	−0.472790	+	0.881175i	$0.656753\pi$
$954$	0	0
$955$	0	0
$956$	7134.69	0.241373
$957$	0	0
$958$	19608.4	0.661293
$959$	−2251.70	−0.0758199
$960$	0	0
$961$	15731.3	0.528056
$962$	1701.29	0.0570185
$963$	0	0
$964$	33692.4	1.12568
$965$	0	0
$966$	0	0
$967$	−19865.9	−0.660646	−0.330323	−	0.943868i	$-0.607158\pi$
$967$	−19865.9	−0.660646	−0.330323	+	0.943868i	$0.607158\pi$
$968$	24951.2	0.828472
$969$	0	0
$970$	0	0
$971$	10529.6	0.348004	0.174002	−	0.984745i	$-0.444330\pi$
$971$	10529.6	0.348004	0.174002	+	0.984745i	$0.444330\pi$
$972$	0	0
$973$	−487.409	−0.0160592
$974$	−1258.86	−0.0414131
$975$	0	0
$976$	3401.40	0.111553
$977$	−23428.3	−0.767184	−0.383592	−	0.923503i	$-0.625313\pi$
$977$	−23428.3	−0.767184	−0.383592	+	0.923503i	$0.625313\pi$
$978$	0	0
$979$	−8401.97	−0.274288
$980$	0	0
$981$	0	0
$982$	−15188.9	−0.493583
$983$	−60100.0	−1.95004	−0.975021	−	0.222114i	$-0.928704\pi$
$983$	−60100.0	−1.95004	−0.975021	+	0.222114i	$0.928704\pi$
$984$	0	0
$985$	0	0
$986$	25240.6	0.815238
$987$	0	0
$988$	175.457	0.00564984
$989$	26920.9	0.865555
$990$	0	0
$991$	13491.2	0.432455	0.216228	−	0.976343i	$-0.430625\pi$
$991$	13491.2	0.432455	0.216228	+	0.976343i	$0.430625\pi$
$992$	37062.3	1.18622
$993$	0	0
$994$	−1399.92	−0.0446707
$995$	0	0
$996$	0	0
$997$	−30289.6	−0.962167	−0.481084	−	0.876675i	$-0.659757\pi$
$997$	−30289.6	−0.962167	−0.481084	+	0.876675i	$0.659757\pi$
$998$	13989.1	0.443705
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2025.4.a.bd.1.6		8
3.2	odd	2		2025.4.a.bc.1.3		8
5.2	odd	4		405.4.b.c.244.6	yes	8
5.3	odd	4		405.4.b.c.244.3	yes	8
5.4	even	2	inner	2025.4.a.bd.1.3		8
15.2	even	4		405.4.b.b.244.3	✓	8
15.8	even	4		405.4.b.b.244.6	yes	8
15.14	odd	2		2025.4.a.bc.1.6		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
405.4.b.b.244.3	✓	8	15.2	even	4
405.4.b.b.244.6	yes	8	15.8	even	4
405.4.b.c.244.3	yes	8	5.3	odd	4
405.4.b.c.244.6	yes	8	5.2	odd	4
2025.4.a.bc.1.3		8	3.2	odd	2
2025.4.a.bc.1.6		8	15.14	odd	2
2025.4.a.bd.1.3		8	5.4	even	2	inner
2025.4.a.bd.1.6		8	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)

\(f(q)\)	\(=\)	\(q+1.85698 q^{2} -4.55164 q^{4} +1.02402 q^{7} -23.3081 q^{8} +O(q^{10})\)	\(q+1.85698 q^{2} -4.55164 q^{4} +1.02402 q^{7} -23.3081 q^{8} -16.1402 q^{11} +3.05437 q^{13} +1.90159 q^{14} -6.86951 q^{16} +69.2824 q^{17} -12.6207 q^{19} -29.9720 q^{22} -56.6563 q^{23} +5.67190 q^{26} -4.66098 q^{28} +196.187 q^{29} +213.360 q^{31} +173.708 q^{32} +128.656 q^{34} +299.951 q^{37} -23.4363 q^{38} -139.067 q^{41} -475.161 q^{43} +73.4644 q^{44} -105.209 q^{46} +193.371 q^{47} -341.951 q^{49} -13.9024 q^{52} -214.125 q^{53} -23.8680 q^{56} +364.315 q^{58} -149.848 q^{59} -495.145 q^{61} +396.204 q^{62} +377.528 q^{64} -761.598 q^{67} -315.348 q^{68} -736.184 q^{71} +701.058 q^{73} +557.001 q^{74} +57.4446 q^{76} -16.5279 q^{77} +780.649 q^{79} -258.245 q^{82} -961.505 q^{83} -882.362 q^{86} +376.198 q^{88} +520.561 q^{89} +3.12775 q^{91} +257.879 q^{92} +359.086 q^{94} +1155.19 q^{97} -634.996 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 24 q^{4}+O(q^{10}) \) 8 * q + 24 * q^4	\( 8 q + 24 q^{4} + 42 q^{14} + 4 q^{16} - 118 q^{19} - 36 q^{26} - 318 q^{29} - 416 q^{31} - 638 q^{34} + 486 q^{41} - 852 q^{44} - 598 q^{46} - 350 q^{49} + 1530 q^{56} + 1146 q^{59} - 398 q^{61} - 1640 q^{64} - 1728 q^{71} - 1218 q^{74} - 3498 q^{76} - 2596 q^{79} + 480 q^{86} - 1086 q^{89} - 2574 q^{91} - 1238 q^{94}+O(q^{100}) \) 8 * q + 24 * q^4 + 42 * q^14 + 4 * q^16 - 118 * q^19 - 36 * q^26 - 318 * q^29 - 416 * q^31 - 638 * q^34 + 486 * q^41 - 852 * q^44 - 598 * q^46 - 350 * q^49 + 1530 * q^56 + 1146 * q^59 - 398 * q^61 - 1640 * q^64 - 1728 * q^71 - 1218 * q^74 - 3498 * q^76 - 2596 * q^79 + 480 * q^86 - 1086 * q^89 - 2574 * q^91 - 1238 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more