LMFDB - Embedded newform 2025.4.a.w.1.3

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 = [5,0,0,36,0,0,49,-45,0,0,12,0,108,-78] 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:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 38x^{3} - 15x^{2} + 304x + 60 $ x^5 - 38x^3 - 15x^2 + 304*x + 60
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-0.196411$ 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)

$f(q)$	$=$	$q+0.196411 q^{2} -7.96142 q^{4} +16.8177 q^{7} -3.13500 q^{8} +62.1569 q^{11} +60.5269 q^{13} +3.30317 q^{14} +63.0756 q^{16} +43.7901 q^{17} +85.8457 q^{19} +12.2083 q^{22} +215.038 q^{23} +11.8881 q^{26} -133.892 q^{28} -48.3855 q^{29} +257.340 q^{31} +37.4688 q^{32} +8.60087 q^{34} -236.321 q^{37} +16.8610 q^{38} -134.307 q^{41} +91.4691 q^{43} -494.857 q^{44} +42.2359 q^{46} -274.947 q^{47} -60.1666 q^{49} -481.880 q^{52} +44.3393 q^{53} -52.7233 q^{56} -9.50345 q^{58} +769.187 q^{59} +232.793 q^{61} +50.5444 q^{62} -497.246 q^{64} -844.628 q^{67} -348.632 q^{68} +120.076 q^{71} +767.243 q^{73} -46.4160 q^{74} -683.454 q^{76} +1045.33 q^{77} -174.405 q^{79} -26.3794 q^{82} -1336.51 q^{83} +17.9656 q^{86} -194.862 q^{88} +690.349 q^{89} +1017.92 q^{91} -1712.01 q^{92} -54.0026 q^{94} +416.030 q^{97} -11.8174 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 36 q^{4} + 49 q^{7} - 45 q^{8} + 12 q^{11} + 108 q^{13} - 78 q^{14} + 168 q^{16} + 57 q^{17} + 126 q^{19} + 200 q^{22} + 228 q^{23} - 225 q^{26} + 373 q^{28} + 336 q^{29} - 22 q^{31} - 1110 q^{32}+ \cdots + 3705 q^{98}+O(q^{100}) $ 5 * q + 36 * q^4 + 49 * q^7 - 45 * q^8 + 12 * q^11 + 108 * q^13 - 78 * q^14 + 168 * q^16 + 57 * q^17 + 126 * q^19 + 200 * q^22 + 228 * q^23 - 225 * q^26 + 373 * q^28 + 336 * q^29 - 22 * q^31 - 1110 * q^32 + 526 * q^34 + 176 * q^37 + 180 * q^38 + 603 * q^41 + 318 * q^43 - 1143 * q^44 - 687 * q^46 + 1071 * q^47 + 848 * q^49 + 1166 * q^52 + 582 * q^53 - 2580 * q^56 + 1120 * q^58 - 570 * q^59 - 384 * q^61 + 1695 * q^62 + 851 * q^64 + 778 * q^67 - 1731 * q^68 + 306 * q^71 + 1641 * q^73 + 3282 * q^74 - 1160 * q^76 - 2598 * q^77 + 1067 * q^79 + 2545 * q^82 - 954 * q^83 + 4017 * q^86 + 3525 * q^88 + 1167 * q^89 + 424 * q^91 + 906 * q^92 - 1490 * q^94 + 2087 * q^97 + 3705 * q^98

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0.196411	0.0694418	0.0347209	−	0.999397i	$-0.488946\pi$
$2$	0.196411	0.0694418	0.0347209	+	0.999397i	$0.488946\pi$
$3$	0	0
$3$	0	0
$4$	−7.96142	−0.995178
$5$	0	0
$5$	0	0
$6$	0	0
$7$	16.8177	0.908068	0.454034	−	0.890984i	$-0.349984\pi$
$7$	16.8177	0.908068	0.454034	+	0.890984i	$0.349984\pi$
$8$	−3.13500	−0.138549
$9$	0	0
$10$	0	0
$11$	62.1569	1.70373	0.851864	−	0.523764i	$-0.175472\pi$
$11$	62.1569	1.70373	0.851864	+	0.523764i	$0.175472\pi$
$12$	0	0
$13$	60.5269	1.29132	0.645659	−	0.763626i	$-0.276583\pi$
$13$	60.5269	1.29132	0.645659	+	0.763626i	$0.276583\pi$
$14$	3.30317	0.0630579
$15$	0	0
$16$	63.0756	0.985557
$17$	43.7901	0.624745	0.312373	−	0.949960i	$-0.398876\pi$
$17$	43.7901	0.624745	0.312373	+	0.949960i	$0.398876\pi$
$18$	0	0
$19$	85.8457	1.03654	0.518272	−	0.855216i	$-0.326575\pi$
$19$	85.8457	1.03654	0.518272	+	0.855216i	$0.326575\pi$
$20$	0	0
$21$	0	0
$22$	12.2083	0.118310
$23$	215.038	1.94950	0.974751	−	0.223296i	$-0.0716818\pi$
$23$	215.038	1.94950	0.974751	+	0.223296i	$0.0716818\pi$
$24$	0	0
$25$	0	0
$26$	11.8881	0.0896714
$27$	0	0
$28$	−133.892	−0.903689
$29$	−48.3855	−0.309826	−0.154913	−	0.987928i	$-0.549510\pi$
$29$	−48.3855	−0.309826	−0.154913	+	0.987928i	$0.549510\pi$
$30$	0	0
$31$	257.340	1.49095	0.745477	−	0.666531i	$-0.232222\pi$
$31$	257.340	1.49095	0.745477	+	0.666531i	$0.232222\pi$
$32$	37.4688	0.206988
$33$	0	0
$34$	8.60087	0.0433834
$35$	0	0
$36$	0	0
$37$	−236.321	−1.05002	−0.525012	−	0.851095i	$-0.675939\pi$
$37$	−236.321	−1.05002	−0.525012	+	0.851095i	$0.675939\pi$
$38$	16.8610	0.0719796
$39$	0	0
$40$	0	0
$41$	−134.307	−0.511591	−0.255796	−	0.966731i	$-0.582337\pi$
$41$	−134.307	−0.511591	−0.255796	+	0.966731i	$0.582337\pi$
$42$	0	0
$43$	91.4691	0.324393	0.162197	−	0.986758i	$-0.448142\pi$
$43$	91.4691	0.324393	0.162197	+	0.986758i	$0.448142\pi$
$44$	−494.857	−1.69551
$45$	0	0
$46$	42.2359	0.135377
$47$	−274.947	−0.853300	−0.426650	−	0.904417i	$-0.640306\pi$
$47$	−274.947	−0.853300	−0.426650	+	0.904417i	$0.640306\pi$
$48$	0	0
$49$	−60.1666	−0.175413
$50$	0	0
$51$	0	0
$52$	−481.880	−1.28509
$53$	44.3393	0.114915	0.0574573	−	0.998348i	$-0.481701\pi$
$53$	44.3393	0.114915	0.0574573	+	0.998348i	$0.481701\pi$
$54$	0	0
$55$	0	0
$56$	−52.7233	−0.125812
$57$	0	0
$58$	−9.50345	−0.0215149
$59$	769.187	1.69728	0.848641	−	0.528969i	$-0.177421\pi$
$59$	769.187	1.69728	0.848641	+	0.528969i	$0.177421\pi$
$60$	0	0
$61$	232.793	0.488624	0.244312	−	0.969697i	$-0.421438\pi$
$61$	232.793	0.488624	0.244312	+	0.969697i	$0.421438\pi$
$62$	50.5444	0.103535
$63$	0	0
$64$	−497.246	−0.971183
$65$	0	0
$66$	0	0
$67$	−844.628	−1.54012	−0.770058	−	0.637974i	$-0.779773\pi$
$67$	−844.628	−1.54012	−0.770058	+	0.637974i	$0.779773\pi$
$68$	−348.632	−0.621732
$69$	0	0
$70$	0	0
$71$	120.076	0.200710	0.100355	−	0.994952i	$-0.468002\pi$
$71$	120.076	0.200710	0.100355	+	0.994952i	$0.468002\pi$
$72$	0	0
$73$	767.243	1.23012	0.615061	−	0.788479i	$-0.289131\pi$
$73$	767.243	1.23012	0.615061	+	0.788479i	$0.289131\pi$
$74$	−46.4160	−0.0729156
$75$	0	0
$76$	−683.454	−1.03155
$77$	1045.33	1.54710
$78$	0	0
$79$	−174.405	−0.248381	−0.124190	−	0.992258i	$-0.539633\pi$
$79$	−174.405	−0.248381	−0.124190	+	0.992258i	$0.539633\pi$
$80$	0	0
$81$	0	0
$82$	−26.3794	−0.0355258
$83$	−1336.51	−1.76748	−0.883740	−	0.467978i	$-0.844983\pi$
$83$	−1336.51	−1.76748	−0.883740	+	0.467978i	$0.844983\pi$
$84$	0	0
$85$	0	0
$86$	17.9656	0.0225265
$87$	0	0
$88$	−194.862	−0.236049
$89$	690.349	0.822212	0.411106	−	0.911588i	$-0.365143\pi$
$89$	690.349	0.822212	0.411106	+	0.911588i	$0.365143\pi$
$90$	0	0
$91$	1017.92	1.17260
$92$	−1712.01	−1.94010
$93$	0	0
$94$	−54.0026	−0.0592547
$95$	0	0
$96$	0	0
$97$	416.030	0.435479	0.217739	−	0.976007i	$-0.430132\pi$
$97$	416.030	0.435479	0.217739	+	0.976007i	$0.430132\pi$
$98$	−11.8174	−0.0121810
$99$	0	0
$100$	0	0
$101$	−283.692	−0.279489	−0.139745	−	0.990188i	$-0.544628\pi$
$101$	−283.692	−0.279489	−0.139745	+	0.990188i	$0.544628\pi$
$102$	0	0
$103$	−333.701	−0.319229	−0.159614	−	0.987179i	$-0.551025\pi$
$103$	−333.701	−0.319229	−0.159614	+	0.987179i	$0.551025\pi$
$104$	−189.752	−0.178910
$105$	0	0
$106$	8.70873	0.00797987
$107$	−307.906	−0.278191	−0.139095	−	0.990279i	$-0.544419\pi$
$107$	−307.906	−0.278191	−0.139095	+	0.990279i	$0.544419\pi$
$108$	0	0
$109$	−1042.84	−0.916383	−0.458192	−	0.888853i	$-0.651503\pi$
$109$	−1042.84	−0.916383	−0.458192	+	0.888853i	$0.651503\pi$
$110$	0	0
$111$	0	0
$112$	1060.78	0.894952
$113$	1173.89	0.977257	0.488629	−	0.872492i	$-0.337497\pi$
$113$	1173.89	0.977257	0.488629	+	0.872492i	$0.337497\pi$
$114$	0	0
$115$	0	0
$116$	385.218	0.308332
$117$	0	0
$118$	151.077	0.117862
$119$	736.447	0.567311
$120$	0	0
$121$	2532.48	1.90269
$122$	45.7231	0.0339309
$123$	0	0
$124$	−2048.79	−1.48377
$125$	0	0
$126$	0	0
$127$	1040.77	0.727189	0.363594	−	0.931557i	$-0.381549\pi$
$127$	1040.77	0.727189	0.363594	+	0.931557i	$0.381549\pi$
$128$	−397.415	−0.274428
$129$	0	0
$130$	0	0
$131$	−1919.76	−1.28038	−0.640192	−	0.768215i	$-0.721145\pi$
$131$	−1919.76	−1.28038	−0.640192	+	0.768215i	$0.721145\pi$
$132$	0	0
$133$	1443.72	0.941253
$134$	−165.894	−0.106948
$135$	0	0
$136$	−137.282	−0.0865577
$137$	1726.42	1.07663	0.538314	−	0.842744i	$-0.319061\pi$
$137$	1726.42	1.07663	0.538314	+	0.842744i	$0.319061\pi$
$138$	0	0
$139$	−1891.22	−1.15404	−0.577018	−	0.816731i	$-0.695784\pi$
$139$	−1891.22	−1.15404	−0.577018	+	0.816731i	$0.695784\pi$
$140$	0	0
$141$	0	0
$142$	23.5842	0.0139376
$143$	3762.16	2.20005
$144$	0	0
$145$	0	0
$146$	150.695	0.0854219
$147$	0	0
$148$	1881.45	1.04496
$149$	−2550.80	−1.40248	−0.701239	−	0.712926i	$-0.747370\pi$
$149$	−2550.80	−1.40248	−0.701239	+	0.712926i	$0.747370\pi$
$150$	0	0
$151$	1390.39	0.749325	0.374663	−	0.927161i	$-0.377759\pi$
$151$	1390.39	0.749325	0.374663	+	0.927161i	$0.377759\pi$
$152$	−269.126	−0.143612
$153$	0	0
$154$	205.315	0.107433
$155$	0	0
$156$	0	0
$157$	980.571	0.498459	0.249229	−	0.968444i	$-0.419823\pi$
$157$	980.571	0.498459	0.249229	+	0.968444i	$0.419823\pi$
$158$	−34.2550	−0.0172480
$159$	0	0
$160$	0	0
$161$	3616.43	1.77028
$162$	0	0
$163$	−2201.31	−1.05779	−0.528896	−	0.848686i	$-0.677394\pi$
$163$	−2201.31	−1.05779	−0.528896	+	0.848686i	$0.677394\pi$
$164$	1069.28	0.509124
$165$	0	0
$166$	−262.505	−0.122737
$167$	−818.796	−0.379403	−0.189702	−	0.981842i	$-0.560752\pi$
$167$	−818.796	−0.379403	−0.189702	+	0.981842i	$0.560752\pi$
$168$	0	0
$169$	1466.50	0.667501
$170$	0	0
$171$	0	0
$172$	−728.225	−0.322829
$173$	−3385.35	−1.48776	−0.743882	−	0.668311i	$-0.767017\pi$
$173$	−3385.35	−1.48776	−0.743882	+	0.668311i	$0.767017\pi$
$174$	0	0
$175$	0	0
$176$	3920.58	1.67912
$177$	0	0
$178$	135.592	0.0570959
$179$	−3635.90	−1.51821	−0.759107	−	0.650966i	$-0.774364\pi$
$179$	−3635.90	−1.51821	−0.759107	+	0.650966i	$0.774364\pi$
$180$	0	0
$181$	−1056.53	−0.433876	−0.216938	−	0.976185i	$-0.569607\pi$
$181$	−1056.53	−0.433876	−0.216938	+	0.976185i	$0.569607\pi$
$182$	199.931	0.0814278
$183$	0	0
$184$	−674.144	−0.270101
$185$	0	0
$186$	0	0
$187$	2721.86	1.06440
$188$	2188.97	0.849186
$189$	0	0
$190$	0	0
$191$	−4169.70	−1.57963	−0.789814	−	0.613346i	$-0.789823\pi$
$191$	−4169.70	−1.57963	−0.789814	+	0.613346i	$0.789823\pi$
$192$	0	0
$193$	4146.74	1.54658	0.773288	−	0.634055i	$-0.218611\pi$
$193$	4146.74	1.54658	0.773288	+	0.634055i	$0.218611\pi$
$194$	81.7129	0.0302404
$195$	0	0
$196$	479.012	0.174567
$197$	4708.33	1.70282	0.851409	−	0.524503i	$-0.175749\pi$
$197$	4708.33	1.70282	0.851409	+	0.524503i	$0.175749\pi$
$198$	0	0
$199$	167.199	0.0595598	0.0297799	−	0.999556i	$-0.490519\pi$
$199$	167.199	0.0595598	0.0297799	+	0.999556i	$0.490519\pi$
$200$	0	0
$201$	0	0
$202$	−55.7203	−0.0194082
$203$	−813.731	−0.281343
$204$	0	0
$205$	0	0
$206$	−65.5427	−0.0221678
$207$	0	0
$208$	3817.77	1.27267
$209$	5335.90	1.76599
$210$	0	0
$211$	−2077.65	−0.677874	−0.338937	−	0.940809i	$-0.610067\pi$
$211$	−2077.65	−0.677874	−0.338937	+	0.940809i	$0.610067\pi$
$212$	−353.004	−0.114360
$213$	0	0
$214$	−60.4762	−0.0193181
$215$	0	0
$216$	0	0
$217$	4327.85	1.35389
$218$	−204.825	−0.0636353
$219$	0	0
$220$	0	0
$221$	2650.48	0.806744
$222$	0	0
$223$	4494.10	1.34954	0.674770	−	0.738028i	$-0.264243\pi$
$223$	4494.10	1.34954	0.674770	+	0.738028i	$0.264243\pi$
$224$	630.137	0.187959
$225$	0	0
$226$	230.565	0.0678625
$227$	−4874.68	−1.42530	−0.712652	−	0.701518i	$-0.752506\pi$
$227$	−4874.68	−1.42530	−0.712652	+	0.701518i	$0.752506\pi$
$228$	0	0
$229$	2677.36	0.772598	0.386299	−	0.922374i	$-0.373753\pi$
$229$	2677.36	0.772598	0.386299	+	0.922374i	$0.373753\pi$
$230$	0	0
$231$	0	0
$232$	151.689	0.0429261
$233$	2621.78	0.737162	0.368581	−	0.929596i	$-0.379844\pi$
$233$	2621.78	0.737162	0.368581	+	0.929596i	$0.379844\pi$
$234$	0	0
$235$	0	0
$236$	−6123.82	−1.68910
$237$	0	0
$238$	144.646	0.0393951
$239$	1636.80	0.442994	0.221497	−	0.975161i	$-0.428906\pi$
$239$	1636.80	0.442994	0.221497	+	0.975161i	$0.428906\pi$
$240$	0	0
$241$	−1330.75	−0.355691	−0.177845	−	0.984058i	$-0.556913\pi$
$241$	−1330.75	−0.355691	−0.177845	+	0.984058i	$0.556913\pi$
$242$	497.407	0.132126
$243$	0	0
$244$	−1853.36	−0.486268
$245$	0	0
$246$	0	0
$247$	5195.97	1.33851
$248$	−806.761	−0.206570
$249$	0	0
$250$	0	0
$251$	−2130.07	−0.535654	−0.267827	−	0.963467i	$-0.586305\pi$
$251$	−2130.07	−0.535654	−0.267827	+	0.963467i	$0.586305\pi$
$252$	0	0
$253$	13366.1	3.32142
$254$	204.418	0.0504973
$255$	0	0
$256$	3899.91	0.952126
$257$	−7413.59	−1.79941	−0.899703	−	0.436503i	$-0.856217\pi$
$257$	−7413.59	−1.79941	−0.899703	+	0.436503i	$0.856217\pi$
$258$	0	0
$259$	−3974.36	−0.953493
$260$	0	0
$261$	0	0
$262$	−377.062	−0.0889121
$263$	−6954.31	−1.63050	−0.815249	−	0.579110i	$-0.803400\pi$
$263$	−6954.31	−1.63050	−0.815249	+	0.579110i	$0.803400\pi$
$264$	0	0
$265$	0	0
$266$	283.563	0.0653623
$267$	0	0
$268$	6724.44	1.53269
$269$	−6456.20	−1.46335	−0.731675	−	0.681653i	$-0.761261\pi$
$269$	−6456.20	−1.46335	−0.731675	+	0.681653i	$0.761261\pi$
$270$	0	0
$271$	6907.22	1.54828	0.774140	−	0.633015i	$-0.218183\pi$
$271$	6907.22	1.54828	0.774140	+	0.633015i	$0.218183\pi$
$272$	2762.09	0.615722
$273$	0	0
$274$	339.088	0.0747630
$275$	0	0
$276$	0	0
$277$	2178.56	0.472553	0.236276	−	0.971686i	$-0.424073\pi$
$277$	2178.56	0.472553	0.236276	+	0.971686i	$0.424073\pi$
$278$	−371.456	−0.0801384
$279$	0	0
$280$	0	0
$281$	−3471.48	−0.736980	−0.368490	−	0.929632i	$-0.620125\pi$
$281$	−3471.48	−0.736980	−0.368490	+	0.929632i	$0.620125\pi$
$282$	0	0
$283$	3181.51	0.668272	0.334136	−	0.942525i	$-0.391555\pi$
$283$	3181.51	0.668272	0.334136	+	0.942525i	$0.391555\pi$
$284$	−955.975	−0.199742
$285$	0	0
$286$	738.930	0.152776
$287$	−2258.73	−0.464559
$288$	0	0
$289$	−2995.42	−0.609694
$290$	0	0
$291$	0	0
$292$	−6108.34	−1.22419
$293$	4955.18	0.988002	0.494001	−	0.869461i	$-0.335534\pi$
$293$	4955.18	0.988002	0.494001	+	0.869461i	$0.335534\pi$
$294$	0	0
$295$	0	0
$296$	740.866	0.145480
$297$	0	0
$298$	−501.005	−0.0973907
$299$	13015.6	2.51743
$300$	0	0
$301$	1538.30	0.294571
$302$	273.088	0.0520345
$303$	0	0
$304$	5414.77	1.02157
$305$	0	0
$306$	0	0
$307$	3151.77	0.585931	0.292966	−	0.956123i	$-0.405358\pi$
$307$	3151.77	0.585931	0.292966	+	0.956123i	$0.405358\pi$
$308$	−8322.33	−1.53964
$309$	0	0
$310$	0	0
$311$	931.002	0.169750	0.0848750	−	0.996392i	$-0.472951\pi$
$311$	931.002	0.169750	0.0848750	+	0.996392i	$0.472951\pi$
$312$	0	0
$313$	−3460.09	−0.624843	−0.312422	−	0.949944i	$-0.601140\pi$
$313$	−3460.09	−0.624843	−0.312422	+	0.949944i	$0.601140\pi$
$314$	192.595	0.0346139
$315$	0	0
$316$	1388.51	0.247183
$317$	−706.536	−0.125183	−0.0625915	−	0.998039i	$-0.519937\pi$
$317$	−706.536	−0.125183	−0.0625915	+	0.998039i	$0.519937\pi$
$318$	0	0
$319$	−3007.49	−0.527860
$320$	0	0
$321$	0	0
$322$	710.308	0.122931
$323$	3759.19	0.647576
$324$	0	0
$325$	0	0
$326$	−432.363	−0.0734551
$327$	0	0
$328$	421.053	0.0708803
$329$	−4623.96	−0.774855
$330$	0	0
$331$	11165.4	1.85410	0.927051	−	0.374936i	$-0.122335\pi$
$331$	11165.4	1.85410	0.927051	+	0.374936i	$0.122335\pi$
$332$	10640.5	1.75896
$333$	0	0
$334$	−160.821	−0.0263465
$335$	0	0
$336$	0	0
$337$	7333.37	1.18538	0.592692	−	0.805430i	$-0.298065\pi$
$337$	7333.37	1.18538	0.592692	+	0.805430i	$0.298065\pi$
$338$	288.037	0.0463525
$339$	0	0
$340$	0	0
$341$	15995.4	2.54018
$342$	0	0
$343$	−6780.32	−1.06735
$344$	−286.756	−0.0449443
$345$	0	0
$346$	−664.920	−0.103313
$347$	8646.97	1.33773	0.668867	−	0.743382i	$-0.266780\pi$
$347$	8646.97	1.33773	0.668867	+	0.743382i	$0.266780\pi$
$348$	0	0
$349$	−9020.31	−1.38351	−0.691757	−	0.722130i	$-0.743163\pi$
$349$	−9020.31	−1.38351	−0.691757	+	0.722130i	$0.743163\pi$
$350$	0	0
$351$	0	0
$352$	2328.94	0.352650
$353$	−7085.03	−1.06827	−0.534133	−	0.845400i	$-0.679362\pi$
$353$	−7085.03	−1.06827	−0.534133	+	0.845400i	$0.679362\pi$
$354$	0	0
$355$	0	0
$356$	−5496.16	−0.818247
$357$	0	0
$358$	−714.132	−0.105427
$359$	−7857.15	−1.15511	−0.577555	−	0.816352i	$-0.695993\pi$
$359$	−7857.15	−1.15511	−0.577555	+	0.816352i	$0.695993\pi$
$360$	0	0
$361$	510.483	0.0744252
$362$	−207.515	−0.0301292
$363$	0	0
$364$	−8104.09	−1.16695
$365$	0	0
$366$	0	0
$367$	491.735	0.0699411	0.0349705	−	0.999388i	$-0.488866\pi$
$367$	491.735	0.0699411	0.0349705	+	0.999388i	$0.488866\pi$
$368$	13563.7	1.92134
$369$	0	0
$370$	0	0
$371$	745.683	0.104350
$372$	0	0
$373$	8619.88	1.19657	0.598284	−	0.801284i	$-0.295849\pi$
$373$	8619.88	1.19657	0.598284	+	0.801284i	$0.295849\pi$
$374$	534.603	0.0739135
$375$	0	0
$376$	861.958	0.118224
$377$	−2928.62	−0.400084
$378$	0	0
$379$	−513.349	−0.0695751	−0.0347875	−	0.999395i	$-0.511075\pi$
$379$	−513.349	−0.0695751	−0.0347875	+	0.999395i	$0.511075\pi$
$380$	0	0
$381$	0	0
$382$	−818.976	−0.109692
$383$	−430.848	−0.0574811	−0.0287406	−	0.999587i	$-0.509150\pi$
$383$	−430.848	−0.0574811	−0.0287406	+	0.999587i	$0.509150\pi$
$384$	0	0
$385$	0	0
$386$	814.467	0.107397
$387$	0	0
$388$	−3312.19	−0.433379
$389$	−10014.6	−1.30530	−0.652651	−	0.757658i	$-0.726343\pi$
$389$	−10014.6	−1.30530	−0.652651	+	0.757658i	$0.726343\pi$
$390$	0	0
$391$	9416.54	1.21794
$392$	188.622	0.0243032
$393$	0	0
$394$	924.769	0.118247
$395$	0	0
$396$	0	0
$397$	−8645.55	−1.09297	−0.546483	−	0.837470i	$-0.684034\pi$
$397$	−8645.55	−1.09297	−0.546483	+	0.837470i	$0.684034\pi$
$398$	32.8397	0.00413594
$399$	0	0
$400$	0	0
$401$	−860.487	−0.107159	−0.0535794	−	0.998564i	$-0.517063\pi$
$401$	−860.487	−0.107159	−0.0535794	+	0.998564i	$0.517063\pi$
$402$	0	0
$403$	15576.0	1.92530
$404$	2258.59	0.278141
$405$	0	0
$406$	−159.826	−0.0195370
$407$	−14689.0	−1.78895
$408$	0	0
$409$	−7540.70	−0.911647	−0.455823	−	0.890070i	$-0.650655\pi$
$409$	−7540.70	−0.911647	−0.455823	+	0.890070i	$0.650655\pi$
$410$	0	0
$411$	0	0
$412$	2656.74	0.317690
$413$	12935.9	1.54125
$414$	0	0
$415$	0	0
$416$	2267.87	0.267287
$417$	0	0
$418$	1048.03	0.122634
$419$	1833.54	0.213781	0.106891	−	0.994271i	$-0.465911\pi$
$419$	1833.54	0.213781	0.106891	+	0.994271i	$0.465911\pi$
$420$	0	0
$421$	−9319.22	−1.07884	−0.539419	−	0.842037i	$-0.681356\pi$
$421$	−9319.22	−1.07884	−0.539419	+	0.842037i	$0.681356\pi$
$422$	−408.074	−0.0470728
$423$	0	0
$424$	−139.004	−0.0159213
$425$	0	0
$426$	0	0
$427$	3915.03	0.443704
$428$	2451.37	0.276849
$429$	0	0
$430$	0	0
$431$	315.173	0.0352235	0.0176118	−	0.999845i	$-0.494394\pi$
$431$	315.173	0.0352235	0.0176118	+	0.999845i	$0.494394\pi$
$432$	0	0
$433$	8285.74	0.919602	0.459801	−	0.888022i	$-0.347921\pi$
$433$	8285.74	0.919602	0.459801	+	0.888022i	$0.347921\pi$
$434$	850.038	0.0940164
$435$	0	0
$436$	8302.48	0.911964
$437$	18460.1	2.02075
$438$	0	0
$439$	−11980.5	−1.30250	−0.651251	−	0.758863i	$-0.725755\pi$
$439$	−11980.5	−1.30250	−0.651251	+	0.758863i	$0.725755\pi$
$440$	0	0
$441$	0	0
$442$	520.584	0.0560218
$443$	−8130.09	−0.871946	−0.435973	−	0.899960i	$-0.643596\pi$
$443$	−8130.09	−0.871946	−0.435973	+	0.899960i	$0.643596\pi$
$444$	0	0
$445$	0	0
$446$	882.692	0.0937145
$447$	0	0
$448$	−8362.51	−0.881900
$449$	1686.35	0.177247	0.0886233	−	0.996065i	$-0.471753\pi$
$449$	1686.35	0.177247	0.0886233	+	0.996065i	$0.471753\pi$
$450$	0	0
$451$	−8348.11	−0.871612
$452$	−9345.82	−0.972545
$453$	0	0
$454$	−957.441	−0.0989757
$455$	0	0
$456$	0	0
$457$	−10804.5	−1.10594	−0.552971	−	0.833201i	$-0.686506\pi$
$457$	−10804.5	−1.10594	−0.552971	+	0.833201i	$0.686506\pi$
$458$	525.864	0.0536506
$459$	0	0
$460$	0	0
$461$	−12593.1	−1.27227	−0.636136	−	0.771577i	$-0.719468\pi$
$461$	−12593.1	−1.27227	−0.636136	+	0.771577i	$0.719468\pi$
$462$	0	0
$463$	490.650	0.0492493	0.0246247	−	0.999697i	$-0.492161\pi$
$463$	490.650	0.0492493	0.0246247	+	0.999697i	$0.492161\pi$
$464$	−3051.95	−0.305351
$465$	0	0
$466$	514.947	0.0511898
$467$	−10112.0	−1.00198	−0.500992	−	0.865452i	$-0.667031\pi$
$467$	−10112.0	−1.00198	−0.500992	+	0.865452i	$0.667031\pi$
$468$	0	0
$469$	−14204.7	−1.39853
$470$	0	0
$471$	0	0
$472$	−2411.40	−0.235156
$473$	5685.44	0.552678
$474$	0	0
$475$	0	0
$476$	−5863.17	−0.564575
$477$	0	0
$478$	321.485	0.0307623
$479$	675.778	0.0644615	0.0322308	−	0.999480i	$-0.489739\pi$
$479$	675.778	0.0644615	0.0322308	+	0.999480i	$0.489739\pi$
$480$	0	0
$481$	−14303.8	−1.35591
$482$	−261.375	−0.0246998
$483$	0	0
$484$	−20162.1	−1.89351
$485$	0	0
$486$	0	0
$487$	1430.49	0.133104	0.0665518	−	0.997783i	$-0.478800\pi$
$487$	1430.49	0.133104	0.0665518	+	0.997783i	$0.478800\pi$
$488$	−729.806	−0.0676983
$489$	0	0
$490$	0	0
$491$	1535.95	0.141174	0.0705870	−	0.997506i	$-0.477513\pi$
$491$	1535.95	0.141174	0.0705870	+	0.997506i	$0.477513\pi$
$492$	0	0
$493$	−2118.81	−0.193563
$494$	1020.55	0.0929485
$495$	0	0
$496$	16231.9	1.46942
$497$	2019.39	0.182258
$498$	0	0
$499$	−3618.46	−0.324618	−0.162309	−	0.986740i	$-0.551894\pi$
$499$	−3618.46	−0.324618	−0.162309	+	0.986740i	$0.551894\pi$
$500$	0	0
$501$	0	0
$502$	−418.370	−0.0371968
$503$	−16435.1	−1.45687	−0.728436	−	0.685114i	$-0.759752\pi$
$503$	−16435.1	−1.45687	−0.728436	+	0.685114i	$0.759752\pi$
$504$	0	0
$505$	0	0
$506$	2625.25	0.230645
$507$	0	0
$508$	−8285.97	−0.723682
$509$	9670.87	0.842149	0.421075	−	0.907026i	$-0.361653\pi$
$509$	9670.87	0.842149	0.421075	+	0.907026i	$0.361653\pi$
$510$	0	0
$511$	12903.2	1.11703
$512$	3945.30	0.340546
$513$	0	0
$514$	−1456.11	−0.124954
$515$	0	0
$516$	0	0
$517$	−17089.8	−1.45379
$518$	−780.608	−0.0662123
$519$	0	0
$520$	0	0
$521$	7895.74	0.663951	0.331976	−	0.943288i	$-0.392285\pi$
$521$	7895.74	0.663951	0.331976	+	0.943288i	$0.392285\pi$
$522$	0	0
$523$	17372.5	1.45248	0.726240	−	0.687442i	$-0.241266\pi$
$523$	17372.5	1.45248	0.726240	+	0.687442i	$0.241266\pi$
$524$	15284.0	1.27421
$525$	0	0
$526$	−1365.90	−0.113225
$527$	11268.9	0.931467
$528$	0	0
$529$	34074.4	2.80055
$530$	0	0
$531$	0	0
$532$	−11494.1	−0.936714
$533$	−8129.18	−0.660627
$534$	0	0
$535$	0	0
$536$	2647.91	0.213381
$537$	0	0
$538$	−1268.07	−0.101618
$539$	−3739.77	−0.298856
$540$	0	0
$541$	14758.6	1.17287	0.586433	−	0.809997i	$-0.300532\pi$
$541$	14758.6	1.17287	0.586433	+	0.809997i	$0.300532\pi$
$542$	1356.65	0.107515
$543$	0	0
$544$	1640.76	0.129314
$545$	0	0
$546$	0	0
$547$	14610.7	1.14206	0.571031	−	0.820929i	$-0.306544\pi$
$547$	14610.7	1.14206	0.571031	+	0.820929i	$0.306544\pi$
$548$	−13744.8	−1.07144
$549$	0	0
$550$	0	0
$551$	−4153.69	−0.321149
$552$	0	0
$553$	−2933.08	−0.225546
$554$	427.894	0.0328149
$555$	0	0
$556$	15056.8	1.14847
$557$	5282.05	0.401809	0.200905	−	0.979611i	$-0.435612\pi$
$557$	5282.05	0.401809	0.200905	+	0.979611i	$0.435612\pi$
$558$	0	0
$559$	5536.34	0.418895
$560$	0	0
$561$	0	0
$562$	−681.838	−0.0511772
$563$	−14862.8	−1.11259	−0.556297	−	0.830983i	$-0.687778\pi$
$563$	−14862.8	−1.11259	−0.556297	+	0.830983i	$0.687778\pi$
$564$	0	0
$565$	0	0
$566$	624.884	0.0464060
$567$	0	0
$568$	−376.438	−0.0278081
$569$	−14831.0	−1.09270	−0.546351	−	0.837556i	$-0.683984\pi$
$569$	−14831.0	−1.09270	−0.546351	+	0.837556i	$0.683984\pi$
$570$	0	0
$571$	5005.19	0.366832	0.183416	−	0.983035i	$-0.441285\pi$
$571$	5005.19	0.366832	0.183416	+	0.983035i	$0.441285\pi$
$572$	−29952.1	−2.18944
$573$	0	0
$574$	−443.639	−0.0322599
$575$	0	0
$576$	0	0
$577$	18453.4	1.33141	0.665705	−	0.746215i	$-0.268131\pi$
$577$	18453.4	1.33141	0.665705	+	0.746215i	$0.268131\pi$
$578$	−588.335	−0.0423382
$579$	0	0
$580$	0	0
$581$	−22476.9	−1.60499
$582$	0	0
$583$	2755.99	0.195783
$584$	−2405.31	−0.170432
$585$	0	0
$586$	973.252	0.0686087
$587$	11462.9	0.806006	0.403003	−	0.915199i	$-0.367966\pi$
$587$	11462.9	0.806006	0.403003	+	0.915199i	$0.367966\pi$
$588$	0	0
$589$	22091.5	1.54544
$590$	0	0
$591$	0	0
$592$	−14906.1	−1.03486
$593$	9937.56	0.688173	0.344086	−	0.938938i	$-0.388189\pi$
$593$	9937.56	0.688173	0.344086	+	0.938938i	$0.388189\pi$
$594$	0	0
$595$	0	0
$596$	20308.0	1.39572
$597$	0	0
$598$	2556.40	0.174815
$599$	17608.1	1.20108	0.600542	−	0.799593i	$-0.294952\pi$
$599$	17608.1	1.20108	0.600542	+	0.799593i	$0.294952\pi$
$600$	0	0
$601$	−20493.9	−1.39096	−0.695478	−	0.718547i	$-0.744807\pi$
$601$	−20493.9	−1.39096	−0.695478	+	0.718547i	$0.744807\pi$
$602$	302.138	0.0204556
$603$	0	0
$604$	−11069.5	−0.745712
$605$	0	0
$606$	0	0
$607$	8917.27	0.596278	0.298139	−	0.954522i	$-0.403634\pi$
$607$	8917.27	0.596278	0.298139	+	0.954522i	$0.403634\pi$
$608$	3216.53	0.214552
$609$	0	0
$610$	0	0
$611$	−16641.7	−1.10188
$612$	0	0
$613$	−14907.7	−0.982247	−0.491124	−	0.871090i	$-0.663414\pi$
$613$	−14907.7	−0.982247	−0.491124	+	0.871090i	$0.663414\pi$
$614$	619.042	0.0406881
$615$	0	0
$616$	−3277.12	−0.214349
$617$	−13575.6	−0.885794	−0.442897	−	0.896573i	$-0.646049\pi$
$617$	−13575.6	−0.885794	−0.442897	+	0.896573i	$0.646049\pi$
$618$	0	0
$619$	5132.01	0.333236	0.166618	−	0.986022i	$-0.446715\pi$
$619$	5132.01	0.333236	0.166618	+	0.986022i	$0.446715\pi$
$620$	0	0
$621$	0	0
$622$	182.859	0.0117878
$623$	11610.0	0.746624
$624$	0	0
$625$	0	0
$626$	−679.600	−0.0433902
$627$	0	0
$628$	−7806.74	−0.496055
$629$	−10348.5	−0.655997
$630$	0	0
$631$	−28063.9	−1.77054	−0.885268	−	0.465081i	$-0.846025\pi$
$631$	−28063.9	−1.77054	−0.885268	+	0.465081i	$0.846025\pi$
$632$	546.759	0.0344128
$633$	0	0
$634$	−138.771	−0.00869293
$635$	0	0
$636$	0	0
$637$	−3641.69	−0.226514
$638$	−590.705	−0.0366555
$639$	0	0
$640$	0	0
$641$	−694.011	−0.0427641	−0.0213820	−	0.999771i	$-0.506807\pi$
$641$	−694.011	−0.0427641	−0.0213820	+	0.999771i	$0.506807\pi$
$642$	0	0
$643$	−14359.2	−0.880674	−0.440337	−	0.897833i	$-0.645141\pi$
$643$	−14359.2	−0.880674	−0.440337	+	0.897833i	$0.645141\pi$
$644$	−28792.0	−1.76174
$645$	0	0
$646$	738.347	0.0449689
$647$	5808.50	0.352945	0.176473	−	0.984306i	$-0.443531\pi$
$647$	5808.50	0.352945	0.176473	+	0.984306i	$0.443531\pi$
$648$	0	0
$649$	47810.3	2.89171
$650$	0	0
$651$	0	0
$652$	17525.6	1.05269
$653$	20870.5	1.25073	0.625365	−	0.780332i	$-0.284950\pi$
$653$	20870.5	1.25073	0.625365	+	0.780332i	$0.284950\pi$
$654$	0	0
$655$	0	0
$656$	−8471.50	−0.504202
$657$	0	0
$658$	−908.197	−0.0538073
$659$	16435.7	0.971541	0.485771	−	0.874086i	$-0.338539\pi$
$659$	16435.7	0.971541	0.485771	+	0.874086i	$0.338539\pi$
$660$	0	0
$661$	7547.42	0.444116	0.222058	−	0.975033i	$-0.428723\pi$
$661$	7547.42	0.444116	0.222058	+	0.975033i	$0.428723\pi$
$662$	2193.01	0.128752
$663$	0	0
$664$	4189.95	0.244882
$665$	0	0
$666$	0	0
$667$	−10404.7	−0.604007
$668$	6518.78	0.377574
$669$	0	0
$670$	0	0
$671$	14469.7	0.832482
$672$	0	0
$673$	7812.92	0.447498	0.223749	−	0.974647i	$-0.428170\pi$
$673$	7812.92	0.447498	0.223749	+	0.974647i	$0.428170\pi$
$674$	1440.36	0.0823151
$675$	0	0
$676$	−11675.4	−0.664283
$677$	10773.0	0.611579	0.305789	−	0.952099i	$-0.401080\pi$
$677$	10773.0	0.611579	0.305789	+	0.952099i	$0.401080\pi$
$678$	0	0
$679$	6996.64	0.395444
$680$	0	0
$681$	0	0
$682$	3141.68	0.176395
$683$	−7346.43	−0.411572	−0.205786	−	0.978597i	$-0.565975\pi$
$683$	−7346.43	−0.411572	−0.205786	+	0.978597i	$0.565975\pi$
$684$	0	0
$685$	0	0
$686$	−1331.73	−0.0741190
$687$	0	0
$688$	5769.47	0.319708
$689$	2683.72	0.148391
$690$	0	0
$691$	−33029.1	−1.81836	−0.909179	−	0.416405i	$-0.863290\pi$
$691$	−33029.1	−1.81836	−0.909179	+	0.416405i	$0.863290\pi$
$692$	26952.2	1.48059
$693$	0	0
$694$	1698.36	0.0928947
$695$	0	0
$696$	0	0
$697$	−5881.32	−0.319614
$698$	−1771.69	−0.0960737
$699$	0	0
$700$	0	0
$701$	−2881.41	−0.155249	−0.0776244	−	0.996983i	$-0.524733\pi$
$701$	−2881.41	−0.155249	−0.0776244	+	0.996983i	$0.524733\pi$
$702$	0	0
$703$	−20287.1	−1.08840
$704$	−30907.2	−1.65463
$705$	0	0
$706$	−1391.58	−0.0741823
$707$	−4771.03	−0.253795
$708$	0	0
$709$	6858.12	0.363275	0.181638	−	0.983366i	$-0.441860\pi$
$709$	6858.12	0.363275	0.181638	+	0.983366i	$0.441860\pi$
$710$	0	0
$711$	0	0
$712$	−2164.24	−0.113916
$713$	55337.8	2.90662
$714$	0	0
$715$	0	0
$716$	28947.0	1.51089
$717$	0	0
$718$	−1543.23	−0.0802129
$719$	9827.84	0.509759	0.254879	−	0.966973i	$-0.417964\pi$
$719$	9827.84	0.509759	0.254879	+	0.966973i	$0.417964\pi$
$720$	0	0
$721$	−5612.08	−0.289882
$722$	100.264	0.00516822
$723$	0	0
$724$	8411.52	0.431784
$725$	0	0
$726$	0	0
$727$	35550.2	1.81360	0.906798	−	0.421566i	$-0.138519\pi$
$727$	35550.2	1.81360	0.906798	+	0.421566i	$0.138519\pi$
$728$	−3191.18	−0.162463
$729$	0	0
$730$	0	0
$731$	4005.45	0.202663
$732$	0	0
$733$	28654.1	1.44388	0.721940	−	0.691956i	$-0.243251\pi$
$733$	28654.1	1.44388	0.721940	+	0.691956i	$0.243251\pi$
$734$	96.5823	0.00485683
$735$	0	0
$736$	8057.21	0.403523
$737$	−52499.4	−2.62394
$738$	0	0
$739$	5183.64	0.258029	0.129014	−	0.991643i	$-0.458819\pi$
$739$	5183.64	0.258029	0.129014	+	0.991643i	$0.458819\pi$
$740$	0	0
$741$	0	0
$742$	146.460	0.00724627
$743$	−14138.1	−0.698085	−0.349043	−	0.937107i	$-0.613493\pi$
$743$	−14138.1	−0.698085	−0.349043	+	0.937107i	$0.613493\pi$
$744$	0	0
$745$	0	0
$746$	1693.04	0.0830919
$747$	0	0
$748$	−21669.9	−1.05926
$749$	−5178.26	−0.252616
$750$	0	0
$751$	−31181.5	−1.51509	−0.757544	−	0.652784i	$-0.773601\pi$
$751$	−31181.5	−1.51509	−0.757544	+	0.652784i	$0.773601\pi$
$752$	−17342.4	−0.840976
$753$	0	0
$754$	−575.214	−0.0277826
$755$	0	0
$756$	0	0
$757$	−3247.70	−0.155931	−0.0779655	−	0.996956i	$-0.524842\pi$
$757$	−3247.70	−0.155931	−0.0779655	+	0.996956i	$0.524842\pi$
$758$	−100.827	−0.00483142
$759$	0	0
$760$	0	0
$761$	24531.8	1.16856	0.584282	−	0.811551i	$-0.301376\pi$
$761$	24531.8	1.16856	0.584282	+	0.811551i	$0.301376\pi$
$762$	0	0
$763$	−17538.1	−0.832138
$764$	33196.8	1.57201
$765$	0	0
$766$	−84.6232	−0.00399159
$767$	46556.5	2.19173
$768$	0	0
$769$	−25554.8	−1.19835	−0.599174	−	0.800619i	$-0.704504\pi$
$769$	−25554.8	−1.19835	−0.599174	+	0.800619i	$0.704504\pi$
$770$	0	0
$771$	0	0
$772$	−33014.0	−1.53912
$773$	−28912.2	−1.34528	−0.672639	−	0.739971i	$-0.734839\pi$
$773$	−28912.2	−1.34528	−0.672639	+	0.739971i	$0.734839\pi$
$774$	0	0
$775$	0	0
$776$	−1304.25	−0.0603350
$777$	0	0
$778$	−1966.99	−0.0906426
$779$	−11529.7	−0.530287
$780$	0	0
$781$	7463.54	0.341955
$782$	1849.51	0.0845761
$783$	0	0
$784$	−3795.05	−0.172879
$785$	0	0
$786$	0	0
$787$	5252.83	0.237920	0.118960	−	0.992899i	$-0.462044\pi$
$787$	5252.83	0.237920	0.118960	+	0.992899i	$0.462044\pi$
$788$	−37485.0	−1.69461
$789$	0	0
$790$	0	0
$791$	19742.0	0.887416
$792$	0	0
$793$	14090.2	0.630969
$794$	−1698.08	−0.0758976
$795$	0	0
$796$	−1331.14	−0.0592726
$797$	27734.8	1.23264	0.616321	−	0.787495i	$-0.288622\pi$
$797$	27734.8	1.23264	0.616321	+	0.787495i	$0.288622\pi$
$798$	0	0
$799$	−12040.0	−0.533095
$800$	0	0
$801$	0	0
$802$	−169.009	−0.00744130
$803$	47689.4	2.09579
$804$	0	0
$805$	0	0
$806$	3059.29	0.133696
$807$	0	0
$808$	889.374	0.0387229
$809$	31620.9	1.37420	0.687102	−	0.726561i	$-0.258882\pi$
$809$	31620.9	1.37420	0.687102	+	0.726561i	$0.258882\pi$
$810$	0	0
$811$	−21878.4	−0.947291	−0.473646	−	0.880716i	$-0.657062\pi$
$811$	−21878.4	−0.947291	−0.473646	+	0.880716i	$0.657062\pi$
$812$	6478.45	0.279987
$813$	0	0
$814$	−2885.07	−0.124228
$815$	0	0
$816$	0	0
$817$	7852.23	0.336248
$818$	−1481.08	−0.0633064
$819$	0	0
$820$	0	0
$821$	11725.2	0.498432	0.249216	−	0.968448i	$-0.419827\pi$
$821$	11725.2	0.498432	0.249216	+	0.968448i	$0.419827\pi$
$822$	0	0
$823$	−24827.0	−1.05154	−0.525769	−	0.850627i	$-0.676222\pi$
$823$	−24827.0	−1.05154	−0.525769	+	0.850627i	$0.676222\pi$
$824$	1046.15	0.0442288
$825$	0	0
$826$	2540.76	0.107027
$827$	−12490.3	−0.525187	−0.262594	−	0.964906i	$-0.584578\pi$
$827$	−12490.3	−0.525187	−0.262594	+	0.964906i	$0.584578\pi$
$828$	0	0
$829$	−11539.8	−0.483466	−0.241733	−	0.970343i	$-0.577716\pi$
$829$	−11539.8	−0.483466	−0.241733	+	0.970343i	$0.577716\pi$
$830$	0	0
$831$	0	0
$832$	−30096.7	−1.25411
$833$	−2634.70	−0.109588
$834$	0	0
$835$	0	0
$836$	−42481.4	−1.75747
$837$	0	0
$838$	360.128	0.0148454
$839$	1808.81	0.0744303	0.0372152	−	0.999307i	$-0.488151\pi$
$839$	1808.81	0.0744303	0.0372152	+	0.999307i	$0.488151\pi$
$840$	0	0
$841$	−22047.8	−0.904008
$842$	−1830.40	−0.0749165
$843$	0	0
$844$	16541.1	0.674605
$845$	0	0
$846$	0	0
$847$	42590.3	1.72777
$848$	2796.73	0.113255
$849$	0	0
$850$	0	0
$851$	−50817.9	−2.04702
$852$	0	0
$853$	−28693.1	−1.15174	−0.575869	−	0.817542i	$-0.695336\pi$
$853$	−28693.1	−1.15174	−0.575869	+	0.817542i	$0.695336\pi$
$854$	768.955	0.0308116
$855$	0	0
$856$	965.286	0.0385430
$857$	−3588.57	−0.143038	−0.0715188	−	0.997439i	$-0.522785\pi$
$857$	−3588.57	−0.143038	−0.0715188	+	0.997439i	$0.522785\pi$
$858$	0	0
$859$	44877.4	1.78253	0.891267	−	0.453478i	$-0.149817\pi$
$859$	44877.4	1.78253	0.891267	+	0.453478i	$0.149817\pi$
$860$	0	0
$861$	0	0
$862$	61.9034	0.00244598
$863$	−48455.9	−1.91131	−0.955654	−	0.294492i	$-0.904850\pi$
$863$	−48455.9	−1.91131	−0.955654	+	0.294492i	$0.904850\pi$
$864$	0	0
$865$	0	0
$866$	1627.41	0.0638588
$867$	0	0
$868$	−34455.9	−1.34736
$869$	−10840.5	−0.423173
$870$	0	0
$871$	−51122.7	−1.98878
$872$	3269.30	0.126964
$873$	0	0
$874$	3625.77	0.140324
$875$	0	0
$876$	0	0
$877$	9925.44	0.382164	0.191082	−	0.981574i	$-0.438800\pi$
$877$	9925.44	0.382164	0.191082	+	0.981574i	$0.438800\pi$
$878$	−2353.10	−0.0904481
$879$	0	0
$880$	0	0
$881$	−30842.9	−1.17948	−0.589741	−	0.807592i	$-0.700770\pi$
$881$	−30842.9	−1.17948	−0.589741	+	0.807592i	$0.700770\pi$
$882$	0	0
$883$	−19132.7	−0.729180	−0.364590	−	0.931168i	$-0.618791\pi$
$883$	−19132.7	−0.729180	−0.364590	+	0.931168i	$0.618791\pi$
$884$	−21101.6	−0.802854
$885$	0	0
$886$	−1596.84	−0.0605495
$887$	−20374.8	−0.771273	−0.385636	−	0.922651i	$-0.626018\pi$
$887$	−20374.8	−0.771273	−0.385636	+	0.922651i	$0.626018\pi$
$888$	0	0
$889$	17503.2	0.660337
$890$	0	0
$891$	0	0
$892$	−35779.5	−1.34303
$893$	−23603.0	−0.884484
$894$	0	0
$895$	0	0
$896$	−6683.58	−0.249200
$897$	0	0
$898$	331.218	0.0123083
$899$	−12451.5	−0.461937
$900$	0	0
$901$	1941.62	0.0717923
$902$	−1639.66	−0.0605263
$903$	0	0
$904$	−3680.14	−0.135398
$905$	0	0
$906$	0	0
$907$	−18954.4	−0.693903	−0.346951	−	0.937883i	$-0.612783\pi$
$907$	−18954.4	−0.693903	−0.346951	+	0.937883i	$0.612783\pi$
$908$	38809.4	1.41843
$909$	0	0
$910$	0	0
$911$	41799.7	1.52018	0.760092	−	0.649816i	$-0.225154\pi$
$911$	41799.7	1.52018	0.760092	+	0.649816i	$0.225154\pi$
$912$	0	0
$913$	−83073.2	−3.01130
$914$	−2122.13	−0.0767986
$915$	0	0
$916$	−21315.6	−0.768873
$917$	−32285.9	−1.16268
$918$	0	0
$919$	−26828.5	−0.962993	−0.481496	−	0.876448i	$-0.659907\pi$
$919$	−26828.5	−0.962993	−0.481496	+	0.876448i	$0.659907\pi$
$920$	0	0
$921$	0	0
$922$	−2473.42	−0.0883489
$923$	7267.82	0.259180
$924$	0	0
$925$	0	0
$926$	96.3691	0.00341996
$927$	0	0
$928$	−1812.95	−0.0641302
$929$	−24737.6	−0.873643	−0.436821	−	0.899548i	$-0.643896\pi$
$929$	−24737.6	−0.873643	−0.436821	+	0.899548i	$0.643896\pi$
$930$	0	0
$931$	−5165.04	−0.181823
$932$	−20873.1	−0.733607
$933$	0	0
$934$	−1986.10	−0.0695796
$935$	0	0
$936$	0	0
$937$	5084.38	0.177267	0.0886337	−	0.996064i	$-0.471750\pi$
$937$	5084.38	0.177267	0.0886337	+	0.996064i	$0.471750\pi$
$938$	−2789.95	−0.0971164
$939$	0	0
$940$	0	0
$941$	−47884.9	−1.65888	−0.829438	−	0.558599i	$-0.811339\pi$
$941$	−47884.9	−1.65888	−0.829438	+	0.558599i	$0.811339\pi$
$942$	0	0
$943$	−28881.1	−0.997348
$944$	48517.0	1.67277
$945$	0	0
$946$	1116.68	0.0383789
$947$	−25966.0	−0.891005	−0.445503	−	0.895281i	$-0.646975\pi$
$947$	−25966.0	−0.891005	−0.445503	+	0.895281i	$0.646975\pi$
$948$	0	0
$949$	46438.8	1.58848
$950$	0	0
$951$	0	0
$952$	−2308.76	−0.0786002
$953$	−45778.3	−1.55604	−0.778020	−	0.628240i	$-0.783776\pi$
$953$	−45778.3	−1.55604	−0.778020	+	0.628240i	$0.783776\pi$
$954$	0	0
$955$	0	0
$956$	−13031.2	−0.440857
$957$	0	0
$958$	132.730	0.00447633
$959$	29034.3	0.977652
$960$	0	0
$961$	36432.8	1.22295
$962$	−2809.42	−0.0941572
$963$	0	0
$964$	10594.7	0.353975
$965$	0	0
$966$	0	0
$967$	40166.5	1.33575	0.667873	−	0.744275i	$-0.267205\pi$
$967$	40166.5	1.33575	0.667873	+	0.744275i	$0.267205\pi$
$968$	−7939.32	−0.263615
$969$	0	0
$970$	0	0
$971$	8509.27	0.281231	0.140616	−	0.990064i	$-0.455092\pi$
$971$	8509.27	0.281231	0.140616	+	0.990064i	$0.455092\pi$
$972$	0	0
$973$	−31805.8	−1.04794
$974$	280.963	0.00924296
$975$	0	0
$976$	14683.6	0.481567
$977$	36887.7	1.20792	0.603962	−	0.797013i	$-0.293588\pi$
$977$	36887.7	1.20792	0.603962	+	0.797013i	$0.293588\pi$
$978$	0	0
$979$	42909.9	1.40082
$980$	0	0
$981$	0	0
$982$	301.677	0.00980337
$983$	30685.4	0.995637	0.497818	−	0.867281i	$-0.334135\pi$
$983$	30685.4	0.995637	0.497818	+	0.867281i	$0.334135\pi$
$984$	0	0
$985$	0	0
$986$	−416.157	−0.0134413
$987$	0	0
$988$	−41367.3	−1.33205
$989$	19669.3	0.632405
$990$	0	0
$991$	4019.80	0.128853	0.0644265	−	0.997922i	$-0.479478\pi$
$991$	4019.80	0.128853	0.0644265	+	0.997922i	$0.479478\pi$
$992$	9642.20	0.308609
$993$	0	0
$994$	396.632	0.0126563
$995$	0	0
$996$	0	0
$997$	−6729.02	−0.213751	−0.106876	−	0.994272i	$-0.534085\pi$
$997$	−6729.02	−0.213751	−0.106876	+	0.994272i	$0.534085\pi$
$998$	−710.706	−0.0225421
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2025.4.a.w.1.3	yes	5
3.2	odd	2		2025.4.a.x.1.3	yes	5
5.4	even	2		2025.4.a.v.1.3	yes	5
15.14	odd	2		2025.4.a.u.1.3	✓	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2025.4.a.u.1.3	✓	5	15.14	odd	2
2025.4.a.v.1.3	yes	5	5.4	even	2
2025.4.a.w.1.3	yes	5	1.1	even	1	trivial
2025.4.a.x.1.3	yes	5	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+0.196411 q^{2} -7.96142 q^{4} +16.8177 q^{7} -3.13500 q^{8} +62.1569 q^{11} +60.5269 q^{13} +3.30317 q^{14} +63.0756 q^{16} +43.7901 q^{17} +85.8457 q^{19} +12.2083 q^{22} +215.038 q^{23} +11.8881 q^{26} -133.892 q^{28} -48.3855 q^{29} +257.340 q^{31} +37.4688 q^{32} +8.60087 q^{34} -236.321 q^{37} +16.8610 q^{38} -134.307 q^{41} +91.4691 q^{43} -494.857 q^{44} +42.2359 q^{46} -274.947 q^{47} -60.1666 q^{49} -481.880 q^{52} +44.3393 q^{53} -52.7233 q^{56} -9.50345 q^{58} +769.187 q^{59} +232.793 q^{61} +50.5444 q^{62} -497.246 q^{64} -844.628 q^{67} -348.632 q^{68} +120.076 q^{71} +767.243 q^{73} -46.4160 q^{74} -683.454 q^{76} +1045.33 q^{77} -174.405 q^{79} -26.3794 q^{82} -1336.51 q^{83} +17.9656 q^{86} -194.862 q^{88} +690.349 q^{89} +1017.92 q^{91} -1712.01 q^{92} -54.0026 q^{94} +416.030 q^{97} -11.8174 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 36 q^{4} + 49 q^{7} - 45 q^{8} + 12 q^{11} + 108 q^{13} - 78 q^{14} + 168 q^{16} + 57 q^{17} + 126 q^{19} + 200 q^{22} + 228 q^{23} - 225 q^{26} + 373 q^{28} + 336 q^{29} - 22 q^{31} - 1110 q^{32}+ \cdots + 3705 q^{98}+O(q^{100}) \) 5 * q + 36 * q^4 + 49 * q^7 - 45 * q^8 + 12 * q^11 + 108 * q^13 - 78 * q^14 + 168 * q^16 + 57 * q^17 + 126 * q^19 + 200 * q^22 + 228 * q^23 - 225 * q^26 + 373 * q^28 + 336 * q^29 - 22 * q^31 - 1110 * q^32 + 526 * q^34 + 176 * q^37 + 180 * q^38 + 603 * q^41 + 318 * q^43 - 1143 * q^44 - 687 * q^46 + 1071 * q^47 + 848 * q^49 + 1166 * q^52 + 582 * q^53 - 2580 * q^56 + 1120 * q^58 - 570 * q^59 - 384 * q^61 + 1695 * q^62 + 851 * q^64 + 778 * q^67 - 1731 * q^68 + 306 * q^71 + 1641 * q^73 + 3282 * q^74 - 1160 * q^76 - 2598 * q^77 + 1067 * q^79 + 2545 * q^82 - 954 * q^83 + 4017 * q^86 + 3525 * q^88 + 1167 * q^89 + 424 * q^91 + 906 * q^92 - 1490 * q^94 + 2087 * q^97 + 3705 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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