LMFDB - Embedded newform 2025.4.a.bm.1.14

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2025,4,Mod(1,2025)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage: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")

magma://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:traces = [16,0,0,60,0,0,0,0,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(14)] == traces)

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} - 118 x^{14} + 5011 x^{12} - 99412 x^{10} + 979675 x^{8} - 4710982 x^{6} + 10937485 x^{4} + \cdots + 2788900 $ x^16 - 118x^14 + 5011x^12 - 99412x^10 + 979675x^8 - 4710982x^6 + 10937485x^4 - 10898128*x^2 + 2788900
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{8}\cdot 3^{16} $
Twist minimal:	no (minimal twist has level 405)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.14
Root			$5.08832$ 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)

$f(q)$	$=$	$q+3.35627 q^{2} +3.26456 q^{4} +26.8077 q^{7} -15.8934 q^{8} +45.7770 q^{11} +8.87631 q^{13} +89.9741 q^{14} -79.4591 q^{16} +44.4832 q^{17} -150.217 q^{19} +153.640 q^{22} +202.896 q^{23} +29.7913 q^{26} +87.5154 q^{28} +184.288 q^{29} +88.9458 q^{31} -139.539 q^{32} +149.298 q^{34} +33.3201 q^{37} -504.169 q^{38} -236.995 q^{41} -273.681 q^{43} +149.441 q^{44} +680.973 q^{46} -115.339 q^{47} +375.655 q^{49} +28.9772 q^{52} +140.877 q^{53} -426.067 q^{56} +618.521 q^{58} +69.2493 q^{59} -56.4287 q^{61} +298.526 q^{62} +167.343 q^{64} +480.075 q^{67} +145.218 q^{68} -623.864 q^{71} +888.246 q^{73} +111.831 q^{74} -490.392 q^{76} +1227.18 q^{77} +776.512 q^{79} -795.421 q^{82} +1482.08 q^{83} -918.548 q^{86} -727.553 q^{88} -514.893 q^{89} +237.954 q^{91} +662.365 q^{92} -387.109 q^{94} -725.733 q^{97} +1260.80 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 60 q^{4} + 788 q^{16} - 356 q^{19} + 1520 q^{31} + 392 q^{34} + 724 q^{46} - 4 q^{49} + 1268 q^{61} + 9476 q^{64} + 228 q^{76} + 472 q^{79} + 1524 q^{91} + 10700 q^{94}+O(q^{100}) $ 16 * q + 60 * q^4 + 788 * q^16 - 356 * q^19 + 1520 * q^31 + 392 * q^34 + 724 * q^46 - 4 * q^49 + 1268 * q^61 + 9476 * q^64 + 228 * q^76 + 472 * q^79 + 1524 * q^91 + 10700 * q^94

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 $

$2$	3.35627	1.18662	0.593311	−	0.804974i	$-0.297821\pi$
$2$	3.35627	1.18662	0.593311	+	0.804974i	$0.297821\pi$
$3$	0	0
$3$	0	0
$4$	3.26456	0.408070
$5$	0	0
$5$	0	0
$6$	0	0
$7$	26.8077	1.44748	0.723741	−	0.690072i	$-0.242421\pi$
$7$	26.8077	1.44748	0.723741	+	0.690072i	$0.242421\pi$
$8$	−15.8934	−0.702397
$9$	0	0
$10$	0	0
$11$	45.7770	1.25475	0.627376	−	0.778716i	$-0.284129\pi$
$11$	45.7770	1.25475	0.627376	+	0.778716i	$0.284129\pi$
$12$	0	0
$13$	8.87631	0.189373	0.0946863	−	0.995507i	$-0.469815\pi$
$13$	8.87631	0.189373	0.0946863	+	0.995507i	$0.469815\pi$
$14$	89.9741	1.71761
$15$	0	0
$16$	−79.4591	−1.24155
$17$	44.4832	0.634634	0.317317	−	0.948320i	$-0.397218\pi$
$17$	44.4832	0.634634	0.317317	+	0.948320i	$0.397218\pi$
$18$	0	0
$19$	−150.217	−1.81380	−0.906899	−	0.421348i	$-0.861557\pi$
$19$	−150.217	−1.81380	−0.906899	+	0.421348i	$0.861557\pi$
$20$	0	0
$21$	0	0
$22$	153.640	1.48892
$23$	202.896	1.83942	0.919710	−	0.392597i	$-0.128423\pi$
$23$	202.896	1.83942	0.919710	+	0.392597i	$0.128423\pi$
$24$	0	0
$25$	0	0
$26$	29.7913	0.224714
$27$	0	0
$28$	87.5154	0.590673
$29$	184.288	1.18005	0.590025	−	0.807385i	$-0.299118\pi$
$29$	184.288	1.18005	0.590025	+	0.807385i	$0.299118\pi$
$30$	0	0
$31$	88.9458	0.515327	0.257664	−	0.966235i	$-0.417047\pi$
$31$	88.9458	0.515327	0.257664	+	0.966235i	$0.417047\pi$
$32$	−139.539	−0.770851
$33$	0	0
$34$	149.298	0.753070
$35$	0	0
$36$	0	0
$37$	33.3201	0.148048	0.0740242	−	0.997256i	$-0.476416\pi$
$37$	33.3201	0.148048	0.0740242	+	0.997256i	$0.476416\pi$
$38$	−504.169	−2.15229
$39$	0	0
$40$	0	0
$41$	−236.995	−0.902743	−0.451371	−	0.892336i	$-0.649065\pi$
$41$	−236.995	−0.902743	−0.451371	+	0.892336i	$0.649065\pi$
$42$	0	0
$43$	−273.681	−0.970604	−0.485302	−	0.874347i	$-0.661290\pi$
$43$	−273.681	−0.970604	−0.485302	+	0.874347i	$0.661290\pi$
$44$	149.441	0.512026
$45$	0	0
$46$	680.973	2.18270
$47$	−115.339	−0.357956	−0.178978	−	0.983853i	$-0.557279\pi$
$47$	−115.339	−0.357956	−0.178978	+	0.983853i	$0.557279\pi$
$48$	0	0
$49$	375.655	1.09521
$50$	0	0
$51$	0	0
$52$	28.9772	0.0772772
$53$	140.877	0.365111	0.182556	−	0.983196i	$-0.441563\pi$
$53$	140.877	0.365111	0.182556	+	0.983196i	$0.441563\pi$
$54$	0	0
$55$	0	0
$56$	−426.067	−1.01671
$57$	0	0
$58$	618.521	1.40027
$59$	69.2493	0.152805	0.0764025	−	0.997077i	$-0.475657\pi$
$59$	69.2493	0.152805	0.0764025	+	0.997077i	$0.475657\pi$
$60$	0	0
$61$	−56.4287	−0.118442	−0.0592210	−	0.998245i	$-0.518862\pi$
$61$	−56.4287	−0.118442	−0.0592210	+	0.998245i	$0.518862\pi$
$62$	298.526	0.611498
$63$	0	0
$64$	167.343	0.326841
$65$	0	0
$66$	0	0
$67$	480.075	0.875381	0.437691	−	0.899126i	$-0.355797\pi$
$67$	480.075	0.875381	0.437691	+	0.899126i	$0.355797\pi$
$68$	145.218	0.258975
$69$	0	0
$70$	0	0
$71$	−623.864	−1.04280	−0.521402	−	0.853311i	$-0.674591\pi$
$71$	−623.864	−1.04280	−0.521402	+	0.853311i	$0.674591\pi$
$72$	0	0
$73$	888.246	1.42413	0.712064	−	0.702114i	$-0.247761\pi$
$73$	888.246	1.42413	0.712064	+	0.702114i	$0.247761\pi$
$74$	111.831	0.175677
$75$	0	0
$76$	−490.392	−0.740156
$77$	1227.18	1.81623
$78$	0	0
$79$	776.512	1.10588	0.552939	−	0.833222i	$-0.313506\pi$
$79$	776.512	1.10588	0.552939	+	0.833222i	$0.313506\pi$
$80$	0	0
$81$	0	0
$82$	−795.421	−1.07121
$83$	1482.08	1.95999	0.979997	−	0.199014i	$-0.0637739\pi$
$83$	1482.08	1.95999	0.979997	+	0.199014i	$0.0637739\pi$
$84$	0	0
$85$	0	0
$86$	−918.548	−1.15174
$87$	0	0
$88$	−727.553	−0.881334
$89$	−514.893	−0.613242	−0.306621	−	0.951832i	$-0.599198\pi$
$89$	−514.893	−0.613242	−0.306621	+	0.951832i	$0.599198\pi$
$90$	0	0
$91$	237.954	0.274114
$92$	662.365	0.750612
$93$	0	0
$94$	−387.109	−0.424758
$95$	0	0
$96$	0	0
$97$	−725.733	−0.759660	−0.379830	−	0.925056i	$-0.624017\pi$
$97$	−725.733	−0.759660	−0.379830	+	0.925056i	$0.624017\pi$
$98$	1260.80	1.29959
$99$	0	0
$100$	0	0
$101$	238.041	0.234515	0.117257	−	0.993102i	$-0.462590\pi$
$101$	238.041	0.234515	0.117257	+	0.993102i	$0.462590\pi$
$102$	0	0
$103$	475.841	0.455203	0.227602	−	0.973754i	$-0.426912\pi$
$103$	475.841	0.455203	0.227602	+	0.973754i	$0.426912\pi$
$104$	−141.075	−0.133015
$105$	0	0
$106$	472.820	0.433249
$107$	−631.856	−0.570877	−0.285438	−	0.958397i	$-0.592139\pi$
$107$	−631.856	−0.570877	−0.285438	+	0.958397i	$0.592139\pi$
$108$	0	0
$109$	−1345.08	−1.18197	−0.590986	−	0.806682i	$-0.701261\pi$
$109$	−1345.08	−1.18197	−0.590986	+	0.806682i	$0.701261\pi$
$110$	0	0
$111$	0	0
$112$	−2130.12	−1.79712
$113$	−556.695	−0.463447	−0.231723	−	0.972782i	$-0.574436\pi$
$113$	−556.695	−0.463447	−0.231723	+	0.972782i	$0.574436\pi$
$114$	0	0
$115$	0	0
$116$	601.619	0.481543
$117$	0	0
$118$	232.420	0.181322
$119$	1192.50	0.918621
$120$	0	0
$121$	764.530	0.574403
$122$	−189.390	−0.140546
$123$	0	0
$124$	290.369	0.210289
$125$	0	0
$126$	0	0
$127$	2312.68	1.61588	0.807942	−	0.589263i	$-0.200582\pi$
$127$	2312.68	1.61588	0.807942	+	0.589263i	$0.200582\pi$
$128$	1677.96	1.15869
$129$	0	0
$130$	0	0
$131$	−264.340	−0.176301	−0.0881507	−	0.996107i	$-0.528096\pi$
$131$	−264.340	−0.176301	−0.0881507	+	0.996107i	$0.528096\pi$
$132$	0	0
$133$	−4026.98	−2.62544
$134$	1611.26	1.03875
$135$	0	0
$136$	−706.992	−0.445765
$137$	945.240	0.589469	0.294735	−	0.955579i	$-0.404769\pi$
$137$	945.240	0.589469	0.294735	+	0.955579i	$0.404769\pi$
$138$	0	0
$139$	2287.05	1.39558	0.697789	−	0.716304i	$-0.254167\pi$
$139$	2287.05	1.39558	0.697789	+	0.716304i	$0.254167\pi$
$140$	0	0
$141$	0	0
$142$	−2093.86	−1.23741
$143$	406.330	0.237616
$144$	0	0
$145$	0	0
$146$	2981.20	1.68990
$147$	0	0
$148$	108.775	0.0604141
$149$	−1352.60	−0.743687	−0.371844	−	0.928295i	$-0.621274\pi$
$149$	−1352.60	−0.743687	−0.371844	+	0.928295i	$0.621274\pi$
$150$	0	0
$151$	2078.70	1.12028	0.560140	−	0.828398i	$-0.310747\pi$
$151$	2078.70	1.12028	0.560140	+	0.828398i	$0.310747\pi$
$152$	2387.47	1.27401
$153$	0	0
$154$	4118.74	2.15518
$155$	0	0
$156$	0	0
$157$	3774.45	1.91869	0.959343	−	0.282244i	$-0.0910789\pi$
$157$	3774.45	1.91869	0.959343	+	0.282244i	$0.0910789\pi$
$158$	2606.18	1.31226
$159$	0	0
$160$	0	0
$161$	5439.18	2.66253
$162$	0	0
$163$	−1798.92	−0.864431	−0.432216	−	0.901770i	$-0.642268\pi$
$163$	−1798.92	−0.864431	−0.432216	+	0.901770i	$0.642268\pi$
$164$	−773.685	−0.368382
$165$	0	0
$166$	4974.26	2.32577
$167$	−1932.32	−0.895372	−0.447686	−	0.894191i	$-0.647752\pi$
$167$	−1932.32	−0.895372	−0.447686	+	0.894191i	$0.647752\pi$
$168$	0	0
$169$	−2118.21	−0.964138
$170$	0	0
$171$	0	0
$172$	−893.447	−0.396074
$173$	3620.91	1.59129	0.795643	−	0.605765i	$-0.207133\pi$
$173$	3620.91	1.59129	0.795643	+	0.605765i	$0.207133\pi$
$174$	0	0
$175$	0	0
$176$	−3637.40	−1.55784
$177$	0	0
$178$	−1728.12	−0.727686
$179$	−287.724	−0.120143	−0.0600713	−	0.998194i	$-0.519133\pi$
$179$	−287.724	−0.120143	−0.0600713	+	0.998194i	$0.519133\pi$
$180$	0	0
$181$	2382.17	0.978261	0.489130	−	0.872211i	$-0.337314\pi$
$181$	2382.17	0.978261	0.489130	+	0.872211i	$0.337314\pi$
$182$	798.638	0.325269
$183$	0	0
$184$	−3224.71	−1.29200
$185$	0	0
$186$	0	0
$187$	2036.31	0.796308
$188$	−376.531	−0.146071
$189$	0	0
$190$	0	0
$191$	177.777	0.0673479	0.0336740	−	0.999433i	$-0.489279\pi$
$191$	177.777	0.0673479	0.0336740	+	0.999433i	$0.489279\pi$
$192$	0	0
$193$	3526.92	1.31541	0.657703	−	0.753277i	$-0.271528\pi$
$193$	3526.92	1.31541	0.657703	+	0.753277i	$0.271528\pi$
$194$	−2435.76	−0.901428
$195$	0	0
$196$	1226.35	0.446920
$197$	−1835.81	−0.663941	−0.331970	−	0.943290i	$-0.607713\pi$
$197$	−1835.81	−0.663941	−0.331970	+	0.943290i	$0.607713\pi$
$198$	0	0
$199$	2296.18	0.817948	0.408974	−	0.912546i	$-0.365887\pi$
$199$	2296.18	0.817948	0.408974	+	0.912546i	$0.365887\pi$
$200$	0	0
$201$	0	0
$202$	798.930	0.278280
$203$	4940.35	1.70810
$204$	0	0
$205$	0	0
$206$	1597.05	0.540154
$207$	0	0
$208$	−705.304	−0.235115
$209$	−6876.48	−2.27587
$210$	0	0
$211$	−1212.29	−0.395533	−0.197767	−	0.980249i	$-0.563369\pi$
$211$	−1212.29	−0.395533	−0.197767	+	0.980249i	$0.563369\pi$
$212$	459.900	0.148991
$213$	0	0
$214$	−2120.68	−0.677414
$215$	0	0
$216$	0	0
$217$	2384.44	0.745927
$218$	−4514.44	−1.40255
$219$	0	0
$220$	0	0
$221$	394.847	0.120182
$222$	0	0
$223$	−319.218	−0.0958584	−0.0479292	−	0.998851i	$-0.515262\pi$
$223$	−319.218	−0.0958584	−0.0479292	+	0.998851i	$0.515262\pi$
$224$	−3740.72	−1.11579
$225$	0	0
$226$	−1868.42	−0.549936
$227$	−2506.98	−0.733013	−0.366507	−	0.930415i	$-0.619446\pi$
$227$	−2506.98	−0.733013	−0.366507	+	0.930415i	$0.619446\pi$
$228$	0	0
$229$	−1348.94	−0.389260	−0.194630	−	0.980877i	$-0.562351\pi$
$229$	−1348.94	−0.389260	−0.194630	+	0.980877i	$0.562351\pi$
$230$	0	0
$231$	0	0
$232$	−2928.97	−0.828864
$233$	2591.55	0.728662	0.364331	−	0.931270i	$-0.381298\pi$
$233$	2591.55	0.728662	0.364331	+	0.931270i	$0.381298\pi$
$234$	0	0
$235$	0	0
$236$	226.068	0.0623551
$237$	0	0
$238$	4002.34	1.09006
$239$	−2870.08	−0.776780	−0.388390	−	0.921495i	$-0.626969\pi$
$239$	−2870.08	−0.776780	−0.388390	+	0.921495i	$0.626969\pi$
$240$	0	0
$241$	5672.20	1.51609	0.758047	−	0.652200i	$-0.226154\pi$
$241$	5672.20	1.51609	0.758047	+	0.652200i	$0.226154\pi$
$242$	2565.97	0.681598
$243$	0	0
$244$	−184.215	−0.0483326
$245$	0	0
$246$	0	0
$247$	−1333.37	−0.343484
$248$	−1413.65	−0.361964
$249$	0	0
$250$	0	0
$251$	2728.73	0.686199	0.343099	−	0.939299i	$-0.388523\pi$
$251$	2728.73	0.686199	0.343099	+	0.939299i	$0.388523\pi$
$252$	0	0
$253$	9287.95	2.30802
$254$	7761.98	1.91744
$255$	0	0
$256$	4292.94	1.04808
$257$	571.884	0.138806	0.0694030	−	0.997589i	$-0.477891\pi$
$257$	571.884	0.138806	0.0694030	+	0.997589i	$0.477891\pi$
$258$	0	0
$259$	893.237	0.214298
$260$	0	0
$261$	0	0
$262$	−887.196	−0.209203
$263$	−3712.10	−0.870335	−0.435168	−	0.900349i	$-0.643311\pi$
$263$	−3712.10	−0.870335	−0.435168	+	0.900349i	$0.643311\pi$
$264$	0	0
$265$	0	0
$266$	−13515.6	−3.11540
$267$	0	0
$268$	1567.23	0.357216
$269$	7883.48	1.78686	0.893428	−	0.449205i	$-0.148293\pi$
$269$	7883.48	1.78686	0.893428	+	0.449205i	$0.148293\pi$
$270$	0	0
$271$	3689.94	0.827115	0.413558	−	0.910478i	$-0.364286\pi$
$271$	3689.94	0.827115	0.413558	+	0.910478i	$0.364286\pi$
$272$	−3534.60	−0.787929
$273$	0	0
$274$	3172.48	0.699477
$275$	0	0
$276$	0	0
$277$	−6846.77	−1.48514	−0.742568	−	0.669771i	$-0.766392\pi$
$277$	−6846.77	−1.48514	−0.742568	+	0.669771i	$0.766392\pi$
$278$	7675.97	1.65602
$279$	0	0
$280$	0	0
$281$	5767.95	1.22451	0.612255	−	0.790660i	$-0.290263\pi$
$281$	5767.95	1.22451	0.612255	+	0.790660i	$0.290263\pi$
$282$	0	0
$283$	−4376.30	−0.919238	−0.459619	−	0.888116i	$-0.652014\pi$
$283$	−4376.30	−0.919238	−0.459619	+	0.888116i	$0.652014\pi$
$284$	−2036.64	−0.425536
$285$	0	0
$286$	1363.75	0.281960
$287$	−6353.31	−1.30670
$288$	0	0
$289$	−2934.24	−0.597240
$290$	0	0
$291$	0	0
$292$	2899.73	0.581143
$293$	−2949.60	−0.588114	−0.294057	−	0.955788i	$-0.595006\pi$
$293$	−2949.60	−0.588114	−0.294057	+	0.955788i	$0.595006\pi$
$294$	0	0
$295$	0	0
$296$	−529.571	−0.103989
$297$	0	0
$298$	−4539.70	−0.882475
$299$	1800.96	0.348336
$300$	0	0
$301$	−7336.77	−1.40493
$302$	6976.68	1.32935
$303$	0	0
$304$	11936.1	2.25192
$305$	0	0
$306$	0	0
$307$	−2812.48	−0.522855	−0.261427	−	0.965223i	$-0.584193\pi$
$307$	−2812.48	−0.522855	−0.261427	+	0.965223i	$0.584193\pi$
$308$	4006.19	0.741149
$309$	0	0
$310$	0	0
$311$	−5345.68	−0.974680	−0.487340	−	0.873212i	$-0.662033\pi$
$311$	−5345.68	−0.974680	−0.487340	+	0.873212i	$0.662033\pi$
$312$	0	0
$313$	7674.90	1.38598	0.692989	−	0.720948i	$-0.256293\pi$
$313$	7674.90	1.38598	0.692989	+	0.720948i	$0.256293\pi$
$314$	12668.1	2.27675
$315$	0	0
$316$	2534.97	0.451275
$317$	1214.13	0.215118	0.107559	−	0.994199i	$-0.465696\pi$
$317$	1214.13	0.215118	0.107559	+	0.994199i	$0.465696\pi$
$318$	0	0
$319$	8436.16	1.48067
$320$	0	0
$321$	0	0
$322$	18255.4	3.15941
$323$	−6682.14	−1.15110
$324$	0	0
$325$	0	0
$326$	−6037.66	−1.02575
$327$	0	0
$328$	3766.67	0.634084
$329$	−3091.98	−0.518135
$330$	0	0
$331$	−830.157	−0.137854	−0.0689268	−	0.997622i	$-0.521957\pi$
$331$	−830.157	−0.137854	−0.0689268	+	0.997622i	$0.521957\pi$
$332$	4838.33	0.799814
$333$	0	0
$334$	−6485.38	−1.06247
$335$	0	0
$336$	0	0
$337$	−8322.52	−1.34527	−0.672636	−	0.739973i	$-0.734838\pi$
$337$	−8322.52	−1.34527	−0.672636	+	0.739973i	$0.734838\pi$
$338$	−7109.29	−1.14407
$339$	0	0
$340$	0	0
$341$	4071.67	0.646608
$342$	0	0
$343$	875.416	0.137808
$344$	4349.73	0.681750
$345$	0	0
$346$	12152.8	1.88825
$347$	2390.92	0.369888	0.184944	−	0.982749i	$-0.440790\pi$
$347$	2390.92	0.369888	0.184944	+	0.982749i	$0.440790\pi$
$348$	0	0
$349$	4959.07	0.760610	0.380305	−	0.924861i	$-0.375819\pi$
$349$	4959.07	0.760610	0.380305	+	0.924861i	$0.375819\pi$
$350$	0	0
$351$	0	0
$352$	−6387.67	−0.967227
$353$	−5718.50	−0.862224	−0.431112	−	0.902299i	$-0.641879\pi$
$353$	−5718.50	−0.862224	−0.431112	+	0.902299i	$0.641879\pi$
$354$	0	0
$355$	0	0
$356$	−1680.90	−0.250245
$357$	0	0
$358$	−965.680	−0.142564
$359$	−9065.08	−1.33269	−0.666346	−	0.745643i	$-0.732143\pi$
$359$	−9065.08	−1.33269	−0.666346	+	0.745643i	$0.732143\pi$
$360$	0	0
$361$	15706.2	2.28986
$362$	7995.20	1.16082
$363$	0	0
$364$	776.814	0.111857
$365$	0	0
$366$	0	0
$367$	−5809.24	−0.826267	−0.413133	−	0.910670i	$-0.635566\pi$
$367$	−5809.24	−0.826267	−0.413133	+	0.910670i	$0.635566\pi$
$368$	−16121.9	−2.28373
$369$	0	0
$370$	0	0
$371$	3776.59	0.528492
$372$	0	0
$373$	−7788.72	−1.08119	−0.540596	−	0.841282i	$-0.681801\pi$
$373$	−7788.72	−1.08119	−0.540596	+	0.841282i	$0.681801\pi$
$374$	6834.40	0.944916
$375$	0	0
$376$	1833.13	0.251427
$377$	1635.80	0.223469
$378$	0	0
$379$	4678.18	0.634042	0.317021	−	0.948419i	$-0.397317\pi$
$379$	4678.18	0.634042	0.317021	+	0.948419i	$0.397317\pi$
$380$	0	0
$381$	0	0
$382$	596.666	0.0799165
$383$	132.836	0.0177222	0.00886108	−	0.999961i	$-0.497179\pi$
$383$	132.836	0.0177222	0.00886108	+	0.999961i	$0.497179\pi$
$384$	0	0
$385$	0	0
$386$	11837.3	1.56089
$387$	0	0
$388$	−2369.20	−0.309994
$389$	−2705.00	−0.352568	−0.176284	−	0.984339i	$-0.556408\pi$
$389$	−2705.00	−0.352568	−0.176284	+	0.984339i	$0.556408\pi$
$390$	0	0
$391$	9025.46	1.16736
$392$	−5970.45	−0.769269
$393$	0	0
$394$	−6161.49	−0.787846
$395$	0	0
$396$	0	0
$397$	7194.44	0.909518	0.454759	−	0.890614i	$-0.349725\pi$
$397$	7194.44	0.909518	0.454759	+	0.890614i	$0.349725\pi$
$398$	7706.59	0.970594
$399$	0	0
$400$	0	0
$401$	−9972.50	−1.24190	−0.620951	−	0.783849i	$-0.713254\pi$
$401$	−9972.50	−1.24190	−0.620951	+	0.783849i	$0.713254\pi$
$402$	0	0
$403$	789.510	0.0975889
$404$	777.099	0.0956983
$405$	0	0
$406$	16581.2	2.02687
$407$	1525.29	0.185764
$408$	0	0
$409$	−1606.15	−0.194179	−0.0970893	−	0.995276i	$-0.530953\pi$
$409$	−1606.15	−0.194179	−0.0970893	+	0.995276i	$0.530953\pi$
$410$	0	0
$411$	0	0
$412$	1553.41	0.185755
$413$	1856.42	0.221183
$414$	0	0
$415$	0	0
$416$	−1238.59	−0.145978
$417$	0	0
$418$	−23079.3	−2.70059
$419$	6817.60	0.794896	0.397448	−	0.917625i	$-0.369896\pi$
$419$	6817.60	0.794896	0.397448	+	0.917625i	$0.369896\pi$
$420$	0	0
$421$	−12131.3	−1.40437	−0.702187	−	0.711993i	$-0.747793\pi$
$421$	−12131.3	−1.40437	−0.702187	+	0.711993i	$0.747793\pi$
$422$	−4068.78	−0.469348
$423$	0	0
$424$	−2239.01	−0.256453
$425$	0	0
$426$	0	0
$427$	−1512.73	−0.171443
$428$	−2062.73	−0.232957
$429$	0	0
$430$	0	0
$431$	−11446.1	−1.27921	−0.639603	−	0.768705i	$-0.720901\pi$
$431$	−11446.1	−1.27921	−0.639603	+	0.768705i	$0.720901\pi$
$432$	0	0
$433$	−10673.5	−1.18460	−0.592302	−	0.805716i	$-0.701781\pi$
$433$	−10673.5	−1.18460	−0.592302	+	0.805716i	$0.701781\pi$
$434$	8002.82	0.885133
$435$	0	0
$436$	−4391.08	−0.482327
$437$	−30478.4	−3.33634
$438$	0	0
$439$	−8406.03	−0.913890	−0.456945	−	0.889495i	$-0.651056\pi$
$439$	−8406.03	−0.913890	−0.456945	+	0.889495i	$0.651056\pi$
$440$	0	0
$441$	0	0
$442$	1325.21	0.142611
$443$	−11772.7	−1.26261	−0.631307	−	0.775533i	$-0.717481\pi$
$443$	−11772.7	−1.26261	−0.631307	+	0.775533i	$0.717481\pi$
$444$	0	0
$445$	0	0
$446$	−1071.38	−0.113748
$447$	0	0
$448$	4486.08	0.473097
$449$	−7167.46	−0.753348	−0.376674	−	0.926346i	$-0.622932\pi$
$449$	−7167.46	−0.753348	−0.376674	+	0.926346i	$0.622932\pi$
$450$	0	0
$451$	−10848.9	−1.13272
$452$	−1817.36	−0.189119
$453$	0	0
$454$	−8414.10	−0.869809
$455$	0	0
$456$	0	0
$457$	8218.86	0.841274	0.420637	−	0.907229i	$-0.361807\pi$
$457$	8218.86	0.841274	0.420637	+	0.907229i	$0.361807\pi$
$458$	−4527.42	−0.461905
$459$	0	0
$460$	0	0
$461$	1005.57	0.101592	0.0507962	−	0.998709i	$-0.483824\pi$
$461$	1005.57	0.101592	0.0507962	+	0.998709i	$0.483824\pi$
$462$	0	0
$463$	−11967.8	−1.20128	−0.600640	−	0.799520i	$-0.705087\pi$
$463$	−11967.8	−1.20128	−0.600640	+	0.799520i	$0.705087\pi$
$464$	−14643.4	−1.46509
$465$	0	0
$466$	8697.95	0.864646
$467$	3461.85	0.343031	0.171516	−	0.985181i	$-0.445134\pi$
$467$	3461.85	0.343031	0.171516	+	0.985181i	$0.445134\pi$
$468$	0	0
$469$	12869.7	1.26710
$470$	0	0
$471$	0	0
$472$	−1100.61	−0.107330
$473$	−12528.3	−1.21787
$474$	0	0
$475$	0	0
$476$	3892.97	0.374861
$477$	0	0
$478$	−9632.78	−0.921743
$479$	−7222.36	−0.688931	−0.344466	−	0.938799i	$-0.611940\pi$
$479$	−7222.36	−0.688931	−0.344466	+	0.938799i	$0.611940\pi$
$480$	0	0
$481$	295.760	0.0280363
$482$	19037.4	1.79903
$483$	0	0
$484$	2495.85	0.234396
$485$	0	0
$486$	0	0
$487$	−3426.70	−0.318847	−0.159424	−	0.987210i	$-0.550964\pi$
$487$	−3426.70	−0.318847	−0.159424	+	0.987210i	$0.550964\pi$
$488$	896.846	0.0831933
$489$	0	0
$490$	0	0
$491$	−12674.5	−1.16496	−0.582478	−	0.812846i	$-0.697917\pi$
$491$	−12674.5	−1.16496	−0.582478	+	0.812846i	$0.697917\pi$
$492$	0	0
$493$	8197.74	0.748900
$494$	−4475.16	−0.407585
$495$	0	0
$496$	−7067.56	−0.639804
$497$	−16724.4	−1.50944
$498$	0	0
$499$	19261.3	1.72796	0.863981	−	0.503525i	$-0.167964\pi$
$499$	19261.3	1.72796	0.863981	+	0.503525i	$0.167964\pi$
$500$	0	0
$501$	0	0
$502$	9158.35	0.814258
$503$	−10348.4	−0.917318	−0.458659	−	0.888612i	$-0.651670\pi$
$503$	−10348.4	−0.917318	−0.458659	+	0.888612i	$0.651670\pi$
$504$	0	0
$505$	0	0
$506$	31172.9	2.73874
$507$	0	0
$508$	7549.87	0.659393
$509$	4368.10	0.380379	0.190189	−	0.981747i	$-0.439090\pi$
$509$	4368.10	0.380379	0.190189	+	0.981747i	$0.439090\pi$
$510$	0	0
$511$	23811.9	2.06140
$512$	984.613	0.0849886
$513$	0	0
$514$	1919.40	0.164710
$515$	0	0
$516$	0	0
$517$	−5279.87	−0.449146
$518$	2997.95	0.254290
$519$	0	0
$520$	0	0
$521$	14033.2	1.18005	0.590026	−	0.807384i	$-0.299117\pi$
$521$	14033.2	1.18005	0.590026	+	0.807384i	$0.299117\pi$
$522$	0	0
$523$	−4491.29	−0.375507	−0.187754	−	0.982216i	$-0.560121\pi$
$523$	−4491.29	−0.375507	−0.187754	+	0.982216i	$0.560121\pi$
$524$	−862.952	−0.0719432
$525$	0	0
$526$	−12458.8	−1.03276
$527$	3956.60	0.327044
$528$	0	0
$529$	28999.7	2.38347
$530$	0	0
$531$	0	0
$532$	−13146.3	−1.07136
$533$	−2103.64	−0.170955
$534$	0	0
$535$	0	0
$536$	−7630.05	−0.614865
$537$	0	0
$538$	26459.1	2.12032
$539$	17196.4	1.37421
$540$	0	0
$541$	−15900.4	−1.26361	−0.631804	−	0.775128i	$-0.717685\pi$
$541$	−15900.4	−1.26361	−0.631804	+	0.775128i	$0.717685\pi$
$542$	12384.5	0.981472
$543$	0	0
$544$	−6207.14	−0.489208
$545$	0	0
$546$	0	0
$547$	−17433.9	−1.36274	−0.681371	−	0.731938i	$-0.738616\pi$
$547$	−17433.9	−1.36274	−0.681371	+	0.731938i	$0.738616\pi$
$548$	3085.79	0.240544
$549$	0	0
$550$	0	0
$551$	−27683.3	−2.14037
$552$	0	0
$553$	20816.5	1.60074
$554$	−22979.6	−1.76229
$555$	0	0
$556$	7466.21	0.569493
$557$	−16296.2	−1.23966	−0.619830	−	0.784736i	$-0.712799\pi$
$557$	−16296.2	−1.23966	−0.619830	+	0.784736i	$0.712799\pi$
$558$	0	0
$559$	−2429.28	−0.183806
$560$	0	0
$561$	0	0
$562$	19358.8	1.45303
$563$	−8428.30	−0.630924	−0.315462	−	0.948938i	$-0.602160\pi$
$563$	−8428.30	−0.630924	−0.315462	+	0.948938i	$0.602160\pi$
$564$	0	0
$565$	0	0
$566$	−14688.1	−1.09079
$567$	0	0
$568$	9915.34	0.732462
$569$	10218.0	0.752833	0.376417	−	0.926450i	$-0.377156\pi$
$569$	10218.0	0.752833	0.376417	+	0.926450i	$0.377156\pi$
$570$	0	0
$571$	25235.0	1.84948	0.924739	−	0.380602i	$-0.124283\pi$
$571$	25235.0	1.84948	0.924739	+	0.380602i	$0.124283\pi$
$572$	1326.49	0.0969638
$573$	0	0
$574$	−21323.4	−1.55056
$575$	0	0
$576$	0	0
$577$	12949.0	0.934272	0.467136	−	0.884186i	$-0.345286\pi$
$577$	12949.0	0.934272	0.467136	+	0.884186i	$0.345286\pi$
$578$	−9848.11	−0.708698
$579$	0	0
$580$	0	0
$581$	39731.2	2.83706
$582$	0	0
$583$	6448.90	0.458124
$584$	−14117.3	−1.00030
$585$	0	0
$586$	−9899.66	−0.697869
$587$	−16247.2	−1.14241	−0.571203	−	0.820809i	$-0.693523\pi$
$587$	−16247.2	−1.14241	−0.571203	+	0.820809i	$0.693523\pi$
$588$	0	0
$589$	−13361.2	−0.934699
$590$	0	0
$591$	0	0
$592$	−2647.59	−0.183809
$593$	6054.60	0.419279	0.209640	−	0.977779i	$-0.432771\pi$
$593$	6054.60	0.419279	0.209640	+	0.977779i	$0.432771\pi$
$594$	0	0
$595$	0	0
$596$	−4415.64	−0.303476
$597$	0	0
$598$	6044.53	0.413343
$599$	7657.43	0.522327	0.261164	−	0.965295i	$-0.415894\pi$
$599$	7657.43	0.522327	0.261164	+	0.965295i	$0.415894\pi$
$600$	0	0
$601$	−859.110	−0.0583092	−0.0291546	−	0.999575i	$-0.509282\pi$
$601$	−859.110	−0.0583092	−0.0291546	+	0.999575i	$0.509282\pi$
$602$	−24624.2	−1.66712
$603$	0	0
$604$	6786.04	0.457152
$605$	0	0
$606$	0	0
$607$	−15408.7	−1.03035	−0.515173	−	0.857086i	$-0.672272\pi$
$607$	−15408.7	−1.03035	−0.515173	+	0.857086i	$0.672272\pi$
$608$	20961.1	1.39817
$609$	0	0
$610$	0	0
$611$	−1023.79	−0.0677871
$612$	0	0
$613$	18224.4	1.20078	0.600389	−	0.799708i	$-0.295012\pi$
$613$	18224.4	1.20078	0.600389	+	0.799708i	$0.295012\pi$
$614$	−9439.43	−0.620431
$615$	0	0
$616$	−19504.1	−1.27572
$617$	1623.92	0.105959	0.0529793	−	0.998596i	$-0.483128\pi$
$617$	1623.92	0.105959	0.0529793	+	0.998596i	$0.483128\pi$
$618$	0	0
$619$	−10054.6	−0.652875	−0.326438	−	0.945219i	$-0.605848\pi$
$619$	−10054.6	−0.652875	−0.326438	+	0.945219i	$0.605848\pi$
$620$	0	0
$621$	0	0
$622$	−17941.5	−1.15658
$623$	−13803.1	−0.887657
$624$	0	0
$625$	0	0
$626$	25759.1	1.64463
$627$	0	0
$628$	12321.9	0.782957
$629$	1482.19	0.0939565
$630$	0	0
$631$	−4893.59	−0.308733	−0.154367	−	0.988014i	$-0.549334\pi$
$631$	−4893.59	−0.308733	−0.154367	+	0.988014i	$0.549334\pi$
$632$	−12341.4	−0.776766
$633$	0	0
$634$	4074.96	0.255264
$635$	0	0
$636$	0	0
$637$	3334.43	0.207402
$638$	28314.0	1.75700
$639$	0	0
$640$	0	0
$641$	5255.04	0.323809	0.161905	−	0.986806i	$-0.448236\pi$
$641$	5255.04	0.323809	0.161905	+	0.986806i	$0.448236\pi$
$642$	0	0
$643$	−9278.37	−0.569056	−0.284528	−	0.958668i	$-0.591837\pi$
$643$	−9278.37	−0.569056	−0.284528	+	0.958668i	$0.591837\pi$
$644$	17756.5	1.08650
$645$	0	0
$646$	−22427.1	−1.36592
$647$	17660.2	1.07310	0.536549	−	0.843869i	$-0.319728\pi$
$647$	17660.2	1.07310	0.536549	+	0.843869i	$0.319728\pi$
$648$	0	0
$649$	3170.02	0.191732
$650$	0	0
$651$	0	0
$652$	−5872.68	−0.352748
$653$	−23852.1	−1.42941	−0.714704	−	0.699427i	$-0.753439\pi$
$653$	−23852.1	−1.42941	−0.714704	+	0.699427i	$0.753439\pi$
$654$	0	0
$655$	0	0
$656$	18831.4	1.12080
$657$	0	0
$658$	−10377.5	−0.614830
$659$	−29672.3	−1.75398	−0.876988	−	0.480513i	$-0.840451\pi$
$659$	−29672.3	−1.75398	−0.876988	+	0.480513i	$0.840451\pi$
$660$	0	0
$661$	18219.2	1.07208	0.536039	−	0.844193i	$-0.319920\pi$
$661$	18219.2	1.07208	0.536039	+	0.844193i	$0.319920\pi$
$662$	−2786.23	−0.163580
$663$	0	0
$664$	−23555.3	−1.37669
$665$	0	0
$666$	0	0
$667$	37391.3	2.17061
$668$	−6308.15	−0.365374
$669$	0	0
$670$	0	0
$671$	−2583.14	−0.148615
$672$	0	0
$673$	−8414.83	−0.481973	−0.240987	−	0.970528i	$-0.577471\pi$
$673$	−8414.83	−0.481973	−0.240987	+	0.970528i	$0.577471\pi$
$674$	−27932.6	−1.59633
$675$	0	0
$676$	−6915.02	−0.393435
$677$	−16544.4	−0.939223	−0.469612	−	0.882873i	$-0.655606\pi$
$677$	−16544.4	−0.939223	−0.469612	+	0.882873i	$0.655606\pi$
$678$	0	0
$679$	−19455.3	−1.09959
$680$	0	0
$681$	0	0
$682$	13665.6	0.767278
$683$	−7515.53	−0.421045	−0.210522	−	0.977589i	$-0.567517\pi$
$683$	−7515.53	−0.421045	−0.210522	+	0.977589i	$0.567517\pi$
$684$	0	0
$685$	0	0
$686$	2938.13	0.163525
$687$	0	0
$688$	21746.5	1.20505
$689$	1250.46	0.0691421
$690$	0	0
$691$	−3854.43	−0.212199	−0.106099	−	0.994356i	$-0.533836\pi$
$691$	−3854.43	−0.212199	−0.106099	+	0.994356i	$0.533836\pi$
$692$	11820.7	0.649356
$693$	0	0
$694$	8024.57	0.438917
$695$	0	0
$696$	0	0
$697$	−10542.3	−0.572911
$698$	16644.0	0.902555
$699$	0	0
$700$	0	0
$701$	−11894.5	−0.640871	−0.320436	−	0.947270i	$-0.603829\pi$
$701$	−11894.5	−0.640871	−0.320436	+	0.947270i	$0.603829\pi$
$702$	0	0
$703$	−5005.25	−0.268530
$704$	7660.44	0.410104
$705$	0	0
$706$	−19192.8	−1.02313
$707$	6381.35	0.339456
$708$	0	0
$709$	−488.396	−0.0258704	−0.0129352	−	0.999916i	$-0.504118\pi$
$709$	−488.396	−0.0258704	−0.0129352	+	0.999916i	$0.504118\pi$
$710$	0	0
$711$	0	0
$712$	8183.42	0.430740
$713$	18046.7	0.947903
$714$	0	0
$715$	0	0
$716$	−939.291	−0.0490265
$717$	0	0
$718$	−30424.9	−1.58140
$719$	29408.3	1.52537	0.762687	−	0.646767i	$-0.223880\pi$
$719$	29408.3	1.52537	0.762687	+	0.646767i	$0.223880\pi$
$720$	0	0
$721$	12756.2	0.658899
$722$	52714.2	2.71720
$723$	0	0
$724$	7776.72	0.399198
$725$	0	0
$726$	0	0
$727$	−29640.7	−1.51212	−0.756060	−	0.654502i	$-0.772878\pi$
$727$	−29640.7	−1.51212	−0.756060	+	0.654502i	$0.772878\pi$
$728$	−3781.90	−0.192537
$729$	0	0
$730$	0	0
$731$	−12174.2	−0.615978
$732$	0	0
$733$	−27400.3	−1.38070	−0.690350	−	0.723475i	$-0.742544\pi$
$733$	−27400.3	−1.38070	−0.690350	+	0.723475i	$0.742544\pi$
$734$	−19497.4	−0.980466
$735$	0	0
$736$	−28311.8	−1.41792
$737$	21976.4	1.09839
$738$	0	0
$739$	−2753.90	−0.137082	−0.0685412	−	0.997648i	$-0.521834\pi$
$739$	−2753.90	−0.137082	−0.0685412	+	0.997648i	$0.521834\pi$
$740$	0	0
$741$	0	0
$742$	12675.2	0.627120
$743$	36997.6	1.82680	0.913400	−	0.407064i	$-0.133447\pi$
$743$	36997.6	1.82680	0.913400	+	0.407064i	$0.133447\pi$
$744$	0	0
$745$	0	0
$746$	−26141.0	−1.28296
$747$	0	0
$748$	6647.64	0.324949
$749$	−16938.6	−0.826334
$750$	0	0
$751$	−33238.3	−1.61503	−0.807513	−	0.589850i	$-0.799187\pi$
$751$	−33238.3	−1.61503	−0.807513	+	0.589850i	$0.799187\pi$
$752$	9164.74	0.444420
$753$	0	0
$754$	5490.19	0.265173
$755$	0	0
$756$	0	0
$757$	−9552.62	−0.458647	−0.229324	−	0.973350i	$-0.573651\pi$
$757$	−9552.62	−0.458647	−0.229324	+	0.973350i	$0.573651\pi$
$758$	15701.2	0.752367
$759$	0	0
$760$	0	0
$761$	4931.73	0.234921	0.117461	−	0.993078i	$-0.462525\pi$
$761$	4931.73	0.234921	0.117461	+	0.993078i	$0.462525\pi$
$762$	0	0
$763$	−36058.5	−1.71088
$764$	580.361	0.0274826
$765$	0	0
$766$	445.832	0.0210295
$767$	614.678	0.0289371
$768$	0	0
$769$	−4719.32	−0.221305	−0.110652	−	0.993859i	$-0.535294\pi$
$769$	−4719.32	−0.221305	−0.110652	+	0.993859i	$0.535294\pi$
$770$	0	0
$771$	0	0
$772$	11513.8	0.536777
$773$	−4926.29	−0.229219	−0.114609	−	0.993411i	$-0.536562\pi$
$773$	−4926.29	−0.229219	−0.114609	+	0.993411i	$0.536562\pi$
$774$	0	0
$775$	0	0
$776$	11534.4	0.533583
$777$	0	0
$778$	−9078.71	−0.418364
$779$	35600.7	1.63739
$780$	0	0
$781$	−28558.6	−1.30846
$782$	30291.9	1.38521
$783$	0	0
$784$	−29849.2	−1.35975
$785$	0	0
$786$	0	0
$787$	24413.7	1.10579	0.552895	−	0.833251i	$-0.313523\pi$
$787$	24413.7	1.10579	0.552895	+	0.833251i	$0.313523\pi$
$788$	−5993.12	−0.270934
$789$	0	0
$790$	0	0
$791$	−14923.8	−0.670831
$792$	0	0
$793$	−500.879	−0.0224297
$794$	24146.5	1.07925
$795$	0	0
$796$	7495.99	0.333779
$797$	−7060.69	−0.313805	−0.156902	−	0.987614i	$-0.550151\pi$
$797$	−7060.69	−0.313805	−0.156902	+	0.987614i	$0.550151\pi$
$798$	0	0
$799$	−5130.66	−0.227171
$800$	0	0
$801$	0	0
$802$	−33470.4	−1.47367
$803$	40661.2	1.78693
$804$	0	0
$805$	0	0
$806$	2649.81	0.115801
$807$	0	0
$808$	−3783.29	−0.164722
$809$	5807.08	0.252368	0.126184	−	0.992007i	$-0.459727\pi$
$809$	5807.08	0.252368	0.126184	+	0.992007i	$0.459727\pi$
$810$	0	0
$811$	−12087.6	−0.523372	−0.261686	−	0.965153i	$-0.584278\pi$
$811$	−12087.6	−0.523372	−0.261686	+	0.965153i	$0.584278\pi$
$812$	16128.1	0.697025
$813$	0	0
$814$	5119.30	0.220432
$815$	0	0
$816$	0	0
$817$	41111.6	1.76048
$818$	−5390.68	−0.230416
$819$	0	0
$820$	0	0
$821$	−16998.7	−0.722605	−0.361303	−	0.932449i	$-0.617668\pi$
$821$	−16998.7	−0.722605	−0.361303	+	0.932449i	$0.617668\pi$
$822$	0	0
$823$	14070.5	0.595948	0.297974	−	0.954574i	$-0.403689\pi$
$823$	14070.5	0.595948	0.297974	+	0.954574i	$0.403689\pi$
$824$	−7562.74	−0.319734
$825$	0	0
$826$	6230.64	0.262460
$827$	−43037.6	−1.80963	−0.904815	−	0.425805i	$-0.859991\pi$
$827$	−43037.6	−1.80963	−0.904815	+	0.425805i	$0.859991\pi$
$828$	0	0
$829$	−2897.22	−0.121381	−0.0606903	−	0.998157i	$-0.519330\pi$
$829$	−2897.22	−0.121381	−0.0606903	+	0.998157i	$0.519330\pi$
$830$	0	0
$831$	0	0
$832$	1485.38	0.0618948
$833$	16710.4	0.695054
$834$	0	0
$835$	0	0
$836$	−22448.7	−0.928712
$837$	0	0
$838$	22881.7	0.943241
$839$	38561.1	1.58674	0.793371	−	0.608739i	$-0.208324\pi$
$839$	38561.1	1.58674	0.793371	+	0.608739i	$0.208324\pi$
$840$	0	0
$841$	9573.17	0.392520
$842$	−40715.8	−1.66646
$843$	0	0
$844$	−3957.59	−0.161405
$845$	0	0
$846$	0	0
$847$	20495.3	0.831438
$848$	−11193.9	−0.453303
$849$	0	0
$850$	0	0
$851$	6760.51	0.272323
$852$	0	0
$853$	−40858.5	−1.64006	−0.820028	−	0.572323i	$-0.806042\pi$
$853$	−40858.5	−1.64006	−0.820028	+	0.572323i	$0.806042\pi$
$854$	−5077.12	−0.203437
$855$	0	0
$856$	10042.4	0.400982
$857$	32227.9	1.28458	0.642289	−	0.766462i	$-0.277985\pi$
$857$	32227.9	1.28458	0.642289	+	0.766462i	$0.277985\pi$
$858$	0	0
$859$	−4540.27	−0.180340	−0.0901699	−	0.995926i	$-0.528741\pi$
$859$	−4540.27	−0.180340	−0.0901699	+	0.995926i	$0.528741\pi$
$860$	0	0
$861$	0	0
$862$	−38416.1	−1.51793
$863$	−1552.00	−0.0612174	−0.0306087	−	0.999531i	$-0.509745\pi$
$863$	−1552.00	−0.0612174	−0.0306087	+	0.999531i	$0.509745\pi$
$864$	0	0
$865$	0	0
$866$	−35823.0	−1.40568
$867$	0	0
$868$	7784.13	0.304390
$869$	35546.3	1.38760
$870$	0	0
$871$	4261.30	0.165773
$872$	21377.9	0.830214
$873$	0	0
$874$	−102294.	−3.95897
$875$	0	0
$876$	0	0
$877$	−7704.91	−0.296666	−0.148333	−	0.988937i	$-0.547391\pi$
$877$	−7704.91	−0.296666	−0.148333	+	0.988937i	$0.547391\pi$
$878$	−28212.9	−1.08444
$879$	0	0
$880$	0	0
$881$	5469.44	0.209160	0.104580	−	0.994516i	$-0.466650\pi$
$881$	5469.44	0.209160	0.104580	+	0.994516i	$0.466650\pi$
$882$	0	0
$883$	−9303.70	−0.354580	−0.177290	−	0.984159i	$-0.556733\pi$
$883$	−9303.70	−0.354580	−0.177290	+	0.984159i	$0.556733\pi$
$884$	1289.00	0.0490427
$885$	0	0
$886$	−39512.4	−1.49824
$887$	30162.4	1.14178	0.570888	−	0.821028i	$-0.306599\pi$
$887$	30162.4	1.14178	0.570888	+	0.821028i	$0.306599\pi$
$888$	0	0
$889$	61997.7	2.33896
$890$	0	0
$891$	0	0
$892$	−1042.11	−0.0391169
$893$	17325.9	0.649260
$894$	0	0
$895$	0	0
$896$	44982.3	1.67718
$897$	0	0
$898$	−24055.9	−0.893939
$899$	16391.7	0.608112
$900$	0	0
$901$	6266.65	0.231712
$902$	−36411.9	−1.34411
$903$	0	0
$904$	8847.80	0.325524
$905$	0	0
$906$	0	0
$907$	−33409.5	−1.22309	−0.611547	−	0.791208i	$-0.709452\pi$
$907$	−33409.5	−1.22309	−0.611547	+	0.791208i	$0.709452\pi$
$908$	−8184.17	−0.299120
$909$	0	0
$910$	0	0
$911$	−36399.1	−1.32377	−0.661885	−	0.749605i	$-0.730243\pi$
$911$	−36399.1	−1.32377	−0.661885	+	0.749605i	$0.730243\pi$
$912$	0	0
$913$	67845.1	2.45931
$914$	27584.7	0.998273
$915$	0	0
$916$	−4403.70	−0.158845
$917$	−7086.35	−0.255193
$918$	0	0
$919$	12102.6	0.434414	0.217207	−	0.976126i	$-0.430305\pi$
$919$	12102.6	0.434414	0.217207	+	0.976126i	$0.430305\pi$
$920$	0	0
$921$	0	0
$922$	3374.97	0.120552
$923$	−5537.61	−0.197478
$924$	0	0
$925$	0	0
$926$	−40167.3	−1.42546
$927$	0	0
$928$	−25715.4	−0.909643
$929$	−51713.2	−1.82633	−0.913163	−	0.407595i	$-0.866367\pi$
$929$	−51713.2	−1.82633	−0.913163	+	0.407595i	$0.866367\pi$
$930$	0	0
$931$	−56429.9	−1.98648
$932$	8460.27	0.297345
$933$	0	0
$934$	11618.9	0.407048
$935$	0	0
$936$	0	0
$937$	13806.9	0.481378	0.240689	−	0.970602i	$-0.422627\pi$
$937$	13806.9	0.481378	0.240689	+	0.970602i	$0.422627\pi$
$938$	43194.3	1.50357
$939$	0	0
$940$	0	0
$941$	33843.2	1.17243	0.586216	−	0.810155i	$-0.300617\pi$
$941$	33843.2	1.17243	0.586216	+	0.810155i	$0.300617\pi$
$942$	0	0
$943$	−48085.3	−1.66052
$944$	−5502.49	−0.189715
$945$	0	0
$946$	−42048.3	−1.44515
$947$	2536.01	0.0870215	0.0435107	−	0.999053i	$-0.486146\pi$
$947$	2536.01	0.0870215	0.0435107	+	0.999053i	$0.486146\pi$
$948$	0	0
$949$	7884.35	0.269691
$950$	0	0
$951$	0	0
$952$	−18952.9	−0.645237
$953$	47593.7	1.61774	0.808872	−	0.587985i	$-0.200079\pi$
$953$	47593.7	1.61774	0.808872	+	0.587985i	$0.200079\pi$
$954$	0	0
$955$	0	0
$956$	−9369.55	−0.316980
$957$	0	0
$958$	−24240.2	−0.817500
$959$	25339.7	0.853246
$960$	0	0
$961$	−21879.6	−0.734438
$962$	992.649	0.0332685
$963$	0	0
$964$	18517.2	0.618672
$965$	0	0
$966$	0	0
$967$	−3399.71	−0.113058	−0.0565291	−	0.998401i	$-0.518003\pi$
$967$	−3399.71	−0.113058	−0.0565291	+	0.998401i	$0.518003\pi$
$968$	−12151.0	−0.403459
$969$	0	0
$970$	0	0
$971$	−33995.3	−1.12354	−0.561771	−	0.827293i	$-0.689880\pi$
$971$	−33995.3	−1.12354	−0.561771	+	0.827293i	$0.689880\pi$
$972$	0	0
$973$	61310.7	2.02007
$974$	−11500.9	−0.378351
$975$	0	0
$976$	4483.78	0.147051
$977$	10128.3	0.331660	0.165830	−	0.986154i	$-0.446970\pi$
$977$	10128.3	0.331660	0.165830	+	0.986154i	$0.446970\pi$
$978$	0	0
$979$	−23570.2	−0.769467
$980$	0	0
$981$	0	0
$982$	−42539.2	−1.38236
$983$	−38181.1	−1.23885	−0.619424	−	0.785057i	$-0.712634\pi$
$983$	−38181.1	−1.23885	−0.619424	+	0.785057i	$0.712634\pi$
$984$	0	0
$985$	0	0
$986$	27513.8	0.888661
$987$	0	0
$988$	−4352.87	−0.140165
$989$	−55528.7	−1.78535
$990$	0	0
$991$	−4067.67	−0.130387	−0.0651936	−	0.997873i	$-0.520766\pi$
$991$	−4067.67	−0.130387	−0.0651936	+	0.997873i	$0.520766\pi$
$992$	−12411.4	−0.397240
$993$	0	0
$994$	−56131.6	−1.79113
$995$	0	0
$996$	0	0
$997$	−33469.8	−1.06319	−0.531594	−	0.846999i	$-0.678407\pi$
$997$	−33469.8	−1.06319	−0.531594	+	0.846999i	$0.678407\pi$
$998$	64646.0	2.05044
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2025.4.a.bm.1.14		16
3.2	odd	2	inner	2025.4.a.bm.1.4		16
5.2	odd	4		405.4.b.d.244.14	yes	16
5.3	odd	4		405.4.b.d.244.4	yes	16
5.4	even	2	inner	2025.4.a.bm.1.3		16
15.2	even	4		405.4.b.d.244.3	✓	16
15.8	even	4		405.4.b.d.244.13	yes	16
15.14	odd	2	inner	2025.4.a.bm.1.13		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
405.4.b.d.244.3	✓	16	15.2	even	4
405.4.b.d.244.4	yes	16	5.3	odd	4
405.4.b.d.244.13	yes	16	15.8	even	4
405.4.b.d.244.14	yes	16	5.2	odd	4
2025.4.a.bm.1.3		16	5.4	even	2	inner
2025.4.a.bm.1.4		16	3.2	odd	2	inner
2025.4.a.bm.1.13		16	15.14	odd	2	inner
2025.4.a.bm.1.14		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 $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q+3.35627 q^{2} +3.26456 q^{4} +26.8077 q^{7} -15.8934 q^{8} +45.7770 q^{11} +8.87631 q^{13} +89.9741 q^{14} -79.4591 q^{16} +44.4832 q^{17} -150.217 q^{19} +153.640 q^{22} +202.896 q^{23} +29.7913 q^{26} +87.5154 q^{28} +184.288 q^{29} +88.9458 q^{31} -139.539 q^{32} +149.298 q^{34} +33.3201 q^{37} -504.169 q^{38} -236.995 q^{41} -273.681 q^{43} +149.441 q^{44} +680.973 q^{46} -115.339 q^{47} +375.655 q^{49} +28.9772 q^{52} +140.877 q^{53} -426.067 q^{56} +618.521 q^{58} +69.2493 q^{59} -56.4287 q^{61} +298.526 q^{62} +167.343 q^{64} +480.075 q^{67} +145.218 q^{68} -623.864 q^{71} +888.246 q^{73} +111.831 q^{74} -490.392 q^{76} +1227.18 q^{77} +776.512 q^{79} -795.421 q^{82} +1482.08 q^{83} -918.548 q^{86} -727.553 q^{88} -514.893 q^{89} +237.954 q^{91} +662.365 q^{92} -387.109 q^{94} -725.733 q^{97} +1260.80 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 60 q^{4} + 788 q^{16} - 356 q^{19} + 1520 q^{31} + 392 q^{34} + 724 q^{46} - 4 q^{49} + 1268 q^{61} + 9476 q^{64} + 228 q^{76} + 472 q^{79} + 1524 q^{91} + 10700 q^{94}+O(q^{100}) \) 16 * q + 60 * q^4 + 788 * q^16 - 356 * q^19 + 1520 * q^31 + 392 * q^34 + 724 * q^46 - 4 * q^49 + 1268 * q^61 + 9476 * q^64 + 228 * q^76 + 472 * q^79 + 1524 * q^91 + 10700 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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