LMFDB - Embedded newform 2025.4.a.y.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 = [6,-4,0,34,0,0,-40,-66,0,0,88,0,-20,180] 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:	$1$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 38x^{4} + 42x^{3} + 393x^{2} - 72x - 432 $ x^6 - 2x^5 - 38x^4 + 42x^3 + 393x^2 - 72*x - 432
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2\cdot 3^{3} $
Twist minimal:	no (minimal twist has level 405)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.07326$ of defining polynomial
Character	$\chi$	$=$	2025.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-2.07326 q^{2} -3.70159 q^{4} +4.66112 q^{7} +24.2604 q^{8} +8.89410 q^{11} -34.6172 q^{13} -9.66372 q^{14} -20.6855 q^{16} -2.66659 q^{17} +125.599 q^{19} -18.4398 q^{22} -131.975 q^{23} +71.7704 q^{26} -17.2535 q^{28} -71.2106 q^{29} +13.4250 q^{31} -151.197 q^{32} +5.52853 q^{34} -283.849 q^{37} -260.400 q^{38} +383.452 q^{41} +339.175 q^{43} -32.9223 q^{44} +273.618 q^{46} -78.2867 q^{47} -321.274 q^{49} +128.138 q^{52} -254.626 q^{53} +113.081 q^{56} +147.638 q^{58} -32.8196 q^{59} +186.983 q^{61} -27.8336 q^{62} +478.955 q^{64} -406.614 q^{67} +9.87060 q^{68} +966.124 q^{71} -276.177 q^{73} +588.492 q^{74} -464.916 q^{76} +41.4565 q^{77} -1146.85 q^{79} -794.996 q^{82} -178.045 q^{83} -703.198 q^{86} +215.775 q^{88} +806.486 q^{89} -161.355 q^{91} +488.516 q^{92} +162.309 q^{94} +1238.05 q^{97} +666.085 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 4 q^{2} + 34 q^{4} - 40 q^{7} - 66 q^{8} + 88 q^{11} - 20 q^{13} + 180 q^{14} + 58 q^{16} - 124 q^{17} - 46 q^{19} + 74 q^{22} - 210 q^{23} + 4 q^{26} - 352 q^{28} + 296 q^{29} - 104 q^{31} - 722 q^{32}+ \cdots - 1982 q^{98}+O(q^{100}) $ 6 * q - 4 * q^2 + 34 * q^4 - 40 * q^7 - 66 * q^8 + 88 * q^11 - 20 * q^13 + 180 * q^14 + 58 * q^16 - 124 * q^17 - 46 * q^19 + 74 * q^22 - 210 * q^23 + 4 * q^26 - 352 * q^28 + 296 * q^29 - 104 * q^31 - 722 * q^32 - 428 * q^34 + 204 * q^37 + 20 * q^38 + 344 * q^41 - 512 * q^43 + 716 * q^44 - 186 * q^46 - 238 * q^47 + 68 * q^49 + 468 * q^52 - 850 * q^53 + 2316 * q^56 - 890 * q^58 + 1840 * q^59 - 364 * q^61 - 1038 * q^62 - 990 * q^64 - 88 * q^67 - 236 * q^68 + 1364 * q^71 - 836 * q^73 + 1316 * q^74 - 2106 * q^76 - 840 * q^77 - 680 * q^79 - 1742 * q^82 - 2148 * q^83 + 2872 * q^86 - 1296 * q^88 + 3000 * q^89 - 3058 * q^91 - 1002 * q^92 - 3662 * q^94 + 612 * q^97 - 1982 * 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 $

$2$	−2.07326	−0.733009	−0.366504	−	0.930416i	$-0.619445\pi$
$2$	−2.07326	−0.733009	−0.366504	+	0.930416i	$0.619445\pi$
$3$	0	0
$3$	0	0
$4$	−3.70159	−0.462698
$5$	0	0
$5$	0	0
$6$	0	0
$7$	4.66112	0.251677	0.125838	−	0.992051i	$-0.459838\pi$
$7$	4.66112	0.251677	0.125838	+	0.992051i	$0.459838\pi$
$8$	24.2604	1.07217
$9$	0	0
$10$	0	0
$11$	8.89410	0.243788	0.121894	−	0.992543i	$-0.461103\pi$
$11$	8.89410	0.243788	0.121894	+	0.992543i	$0.461103\pi$
$12$	0	0
$13$	−34.6172	−0.738544	−0.369272	−	0.929321i	$-0.620393\pi$
$13$	−34.6172	−0.738544	−0.369272	+	0.929321i	$0.620393\pi$
$14$	−9.66372	−0.184481
$15$	0	0
$16$	−20.6855	−0.323212
$17$	−2.66659	−0.0380437	−0.0190218	−	0.999819i	$-0.506055\pi$
$17$	−2.66659	−0.0380437	−0.0190218	+	0.999819i	$0.506055\pi$
$18$	0	0
$19$	125.599	1.51655	0.758273	−	0.651937i	$-0.226043\pi$
$19$	125.599	1.51655	0.758273	+	0.651937i	$0.226043\pi$
$20$	0	0
$21$	0	0
$22$	−18.4398	−0.178699
$23$	−131.975	−1.19646	−0.598231	−	0.801323i	$-0.704130\pi$
$23$	−131.975	−1.19646	−0.598231	+	0.801323i	$0.704130\pi$
$24$	0	0
$25$	0	0
$26$	71.7704	0.541359
$27$	0	0
$28$	−17.2535	−0.116451
$29$	−71.2106	−0.455982	−0.227991	−	0.973663i	$-0.573216\pi$
$29$	−71.2106	−0.455982	−0.227991	+	0.973663i	$0.573216\pi$
$30$	0	0
$31$	13.4250	0.0777810	0.0388905	−	0.999243i	$-0.487618\pi$
$31$	13.4250	0.0777810	0.0388905	+	0.999243i	$0.487618\pi$
$32$	−151.197	−0.835254
$33$	0	0
$34$	5.52853	0.0278863
$35$	0	0
$36$	0	0
$37$	−283.849	−1.26120	−0.630600	−	0.776108i	$-0.717191\pi$
$37$	−283.849	−1.26120	−0.630600	+	0.776108i	$0.717191\pi$
$38$	−260.400	−1.11164
$39$	0	0
$40$	0	0
$41$	383.452	1.46061	0.730307	−	0.683119i	$-0.239377\pi$
$41$	383.452	1.46061	0.730307	+	0.683119i	$0.239377\pi$
$42$	0	0
$43$	339.175	1.20288	0.601438	−	0.798919i	$-0.294595\pi$
$43$	339.175	1.20288	0.601438	+	0.798919i	$0.294595\pi$
$44$	−32.9223	−0.112801
$45$	0	0
$46$	273.618	0.877018
$47$	−78.2867	−0.242964	−0.121482	−	0.992594i	$-0.538765\pi$
$47$	−78.2867	−0.242964	−0.121482	+	0.992594i	$0.538765\pi$
$48$	0	0
$49$	−321.274	−0.936659
$50$	0	0
$51$	0	0
$52$	128.138	0.341723
$53$	−254.626	−0.659915	−0.329958	−	0.943996i	$-0.607034\pi$
$53$	−254.626	−0.659915	−0.329958	+	0.943996i	$0.607034\pi$
$54$	0	0
$55$	0	0
$56$	113.081	0.269841
$57$	0	0
$58$	147.638	0.334239
$59$	−32.8196	−0.0724194	−0.0362097	−	0.999344i	$-0.511528\pi$
$59$	−32.8196	−0.0724194	−0.0362097	+	0.999344i	$0.511528\pi$
$60$	0	0
$61$	186.983	0.392470	0.196235	−	0.980557i	$-0.437128\pi$
$61$	186.983	0.392470	0.196235	+	0.980557i	$0.437128\pi$
$62$	−27.8336	−0.0570141
$63$	0	0
$64$	478.955	0.935460
$65$	0	0
$66$	0	0
$67$	−406.614	−0.741429	−0.370715	−	0.928747i	$-0.620887\pi$
$67$	−406.614	−0.741429	−0.370715	+	0.928747i	$0.620887\pi$
$68$	9.87060	0.0176027
$69$	0	0
$70$	0	0
$71$	966.124	1.61490	0.807450	−	0.589936i	$-0.200847\pi$
$71$	966.124	1.61490	0.807450	+	0.589936i	$0.200847\pi$
$72$	0	0
$73$	−276.177	−0.442796	−0.221398	−	0.975184i	$-0.571062\pi$
$73$	−276.177	−0.442796	−0.221398	+	0.975184i	$0.571062\pi$
$74$	588.492	0.924471
$75$	0	0
$76$	−464.916	−0.701704
$77$	41.4565	0.0613559
$78$	0	0
$79$	−1146.85	−1.63330	−0.816652	−	0.577130i	$-0.804172\pi$
$79$	−1146.85	−1.63330	−0.816652	+	0.577130i	$0.804172\pi$
$80$	0	0
$81$	0	0
$82$	−794.996	−1.07064
$83$	−178.045	−0.235457	−0.117729	−	0.993046i	$-0.537561\pi$
$83$	−178.045	−0.235457	−0.117729	+	0.993046i	$0.537561\pi$
$84$	0	0
$85$	0	0
$86$	−703.198	−0.881719
$87$	0	0
$88$	215.775	0.261383
$89$	806.486	0.960532	0.480266	−	0.877123i	$-0.340540\pi$
$89$	806.486	0.960532	0.480266	+	0.877123i	$0.340540\pi$
$90$	0	0
$91$	−161.355	−0.185874
$92$	488.516	0.553602
$93$	0	0
$94$	162.309	0.178094
$95$	0	0
$96$	0	0
$97$	1238.05	1.29592	0.647961	−	0.761673i	$-0.275622\pi$
$97$	1238.05	1.29592	0.647961	+	0.761673i	$0.275622\pi$
$98$	666.085	0.686579
$99$	0	0
$100$	0	0
$101$	566.856	0.558458	0.279229	−	0.960225i	$-0.409921\pi$
$101$	566.856	0.558458	0.279229	+	0.960225i	$0.409921\pi$
$102$	0	0
$103$	819.017	0.783497	0.391748	−	0.920072i	$-0.371870\pi$
$103$	819.017	0.783497	0.391748	+	0.920072i	$0.371870\pi$
$104$	−839.828	−0.791845
$105$	0	0
$106$	527.905	0.483724
$107$	543.772	0.491293	0.245647	−	0.969359i	$-0.421000\pi$
$107$	543.772	0.491293	0.245647	+	0.969359i	$0.421000\pi$
$108$	0	0
$109$	−1636.54	−1.43809	−0.719046	−	0.694963i	$-0.755421\pi$
$109$	−1636.54	−1.43809	−0.719046	+	0.694963i	$0.755421\pi$
$110$	0	0
$111$	0	0
$112$	−96.4178	−0.0813449
$113$	545.798	0.454375	0.227188	−	0.973851i	$-0.427047\pi$
$113$	545.798	0.454375	0.227188	+	0.973851i	$0.427047\pi$
$114$	0	0
$115$	0	0
$116$	263.592	0.210982
$117$	0	0
$118$	68.0436	0.0530841
$119$	−12.4293	−0.00957471
$120$	0	0
$121$	−1251.89	−0.940567
$122$	−387.664	−0.287684
$123$	0	0
$124$	−49.6940	−0.0359891
$125$	0	0
$126$	0	0
$127$	−36.0480	−0.0251869	−0.0125935	−	0.999921i	$-0.504009\pi$
$127$	−36.0480	−0.0251869	−0.0125935	+	0.999921i	$0.504009\pi$
$128$	216.577	0.149554
$129$	0	0
$130$	0	0
$131$	1310.15	0.873805	0.436902	−	0.899509i	$-0.356076\pi$
$131$	1310.15	0.873805	0.436902	+	0.899509i	$0.356076\pi$
$132$	0	0
$133$	585.432	0.381680
$134$	843.016	0.543474
$135$	0	0
$136$	−64.6926	−0.0407893
$137$	2197.20	1.37022	0.685109	−	0.728441i	$-0.259755\pi$
$137$	2197.20	1.37022	0.685109	+	0.728441i	$0.259755\pi$
$138$	0	0
$139$	1726.01	1.05322	0.526612	−	0.850106i	$-0.323462\pi$
$139$	1726.01	1.05322	0.526612	+	0.850106i	$0.323462\pi$
$140$	0	0
$141$	0	0
$142$	−2003.03	−1.18374
$143$	−307.889	−0.180049
$144$	0	0
$145$	0	0
$146$	572.587	0.324573
$147$	0	0
$148$	1050.69	0.583556
$149$	−1801.40	−0.990445	−0.495222	−	0.868766i	$-0.664913\pi$
$149$	−1801.40	−0.990445	−0.495222	+	0.868766i	$0.664913\pi$
$150$	0	0
$151$	3328.19	1.79367	0.896835	−	0.442365i	$-0.145860\pi$
$151$	3328.19	1.79367	0.896835	+	0.442365i	$0.145860\pi$
$152$	3047.09	1.62600
$153$	0	0
$154$	−85.9501	−0.0449744
$155$	0	0
$156$	0	0
$157$	−3830.19	−1.94702	−0.973510	−	0.228643i	$-0.926571\pi$
$157$	−3830.19	−1.94702	−0.973510	+	0.228643i	$0.926571\pi$
$158$	2377.73	1.19723
$159$	0	0
$160$	0	0
$161$	−615.151	−0.301122
$162$	0	0
$163$	2404.51	1.15543	0.577717	−	0.816237i	$-0.303944\pi$
$163$	2404.51	1.15543	0.577717	+	0.816237i	$0.303944\pi$
$164$	−1419.38	−0.675823
$165$	0	0
$166$	369.134	0.172592
$167$	−3570.95	−1.65466	−0.827331	−	0.561715i	$-0.810142\pi$
$167$	−3570.95	−1.65466	−0.827331	+	0.561715i	$0.810142\pi$
$168$	0	0
$169$	−998.653	−0.454553
$170$	0	0
$171$	0	0
$172$	−1255.49	−0.556569
$173$	159.936	0.0702874	0.0351437	−	0.999382i	$-0.488811\pi$
$173$	159.936	0.0702874	0.0351437	+	0.999382i	$0.488811\pi$
$174$	0	0
$175$	0	0
$176$	−183.979	−0.0787953
$177$	0	0
$178$	−1672.06	−0.704078
$179$	−1120.87	−0.468030	−0.234015	−	0.972233i	$-0.575187\pi$
$179$	−1120.87	−0.468030	−0.234015	+	0.972233i	$0.575187\pi$
$180$	0	0
$181$	−3856.65	−1.58377	−0.791886	−	0.610669i	$-0.790901\pi$
$181$	−3856.65	−1.58377	−0.791886	+	0.610669i	$0.790901\pi$
$182$	334.531	0.136248
$183$	0	0
$184$	−3201.77	−1.28281
$185$	0	0
$186$	0	0
$187$	−23.7169	−0.00927461
$188$	289.785	0.112419
$189$	0	0
$190$	0	0
$191$	−79.0099	−0.0299317	−0.0149659	−	0.999888i	$-0.504764\pi$
$191$	−79.0099	−0.0299317	−0.0149659	+	0.999888i	$0.504764\pi$
$192$	0	0
$193$	−4571.49	−1.70499	−0.852496	−	0.522734i	$-0.824912\pi$
$193$	−4571.49	−1.70499	−0.852496	+	0.522734i	$0.824912\pi$
$194$	−2566.79	−0.949922
$195$	0	0
$196$	1189.22	0.433391
$197$	4081.61	1.47615	0.738077	−	0.674717i	$-0.235734\pi$
$197$	4081.61	1.47615	0.738077	+	0.674717i	$0.235734\pi$
$198$	0	0
$199$	−2518.98	−0.897316	−0.448658	−	0.893703i	$-0.648098\pi$
$199$	−2518.98	−0.897316	−0.448658	+	0.893703i	$0.648098\pi$
$200$	0	0
$201$	0	0
$202$	−1175.24	−0.409355
$203$	−331.921	−0.114760
$204$	0	0
$205$	0	0
$206$	−1698.04	−0.574310
$207$	0	0
$208$	716.075	0.238706
$209$	1117.09	0.369717
$210$	0	0
$211$	227.244	0.0741428	0.0370714	−	0.999313i	$-0.488197\pi$
$211$	227.244	0.0741428	0.0370714	+	0.999313i	$0.488197\pi$
$212$	942.519	0.305342
$213$	0	0
$214$	−1127.38	−0.360122
$215$	0	0
$216$	0	0
$217$	62.5758	0.0195757
$218$	3392.97	1.05413
$219$	0	0
$220$	0	0
$221$	92.3096	0.0280969
$222$	0	0
$223$	515.223	0.154717	0.0773585	−	0.997003i	$-0.475351\pi$
$223$	515.223	0.154717	0.0773585	+	0.997003i	$0.475351\pi$
$224$	−704.748	−0.210214
$225$	0	0
$226$	−1131.58	−0.333061
$227$	3956.04	1.15670	0.578352	−	0.815787i	$-0.303696\pi$
$227$	3956.04	1.15670	0.578352	+	0.815787i	$0.303696\pi$
$228$	0	0
$229$	−1676.57	−0.483803	−0.241901	−	0.970301i	$-0.577771\pi$
$229$	−1676.57	−0.483803	−0.241901	+	0.970301i	$0.577771\pi$
$230$	0	0
$231$	0	0
$232$	−1727.60	−0.488890
$233$	−4610.01	−1.29619	−0.648095	−	0.761560i	$-0.724434\pi$
$233$	−4610.01	−1.29619	−0.648095	+	0.761560i	$0.724434\pi$
$234$	0	0
$235$	0	0
$236$	121.485	0.0335084
$237$	0	0
$238$	25.7692	0.00701834
$239$	−6908.24	−1.86969	−0.934846	−	0.355052i	$-0.884463\pi$
$239$	−6908.24	−1.86969	−0.934846	+	0.355052i	$0.884463\pi$
$240$	0	0
$241$	−2980.96	−0.796765	−0.398383	−	0.917219i	$-0.630428\pi$
$241$	−2980.96	−0.796765	−0.398383	+	0.917219i	$0.630428\pi$
$242$	2595.51	0.689444
$243$	0	0
$244$	−692.133	−0.181595
$245$	0	0
$246$	0	0
$247$	−4347.88	−1.12004
$248$	325.698	0.0833945
$249$	0	0
$250$	0	0
$251$	7132.90	1.79372	0.896862	−	0.442310i	$-0.145841\pi$
$251$	7132.90	1.79372	0.896862	+	0.442310i	$0.145841\pi$
$252$	0	0
$253$	−1173.80	−0.291684
$254$	74.7369	0.0184622
$255$	0	0
$256$	−4280.66	−1.04508
$257$	−6562.56	−1.59285	−0.796423	−	0.604740i	$-0.793277\pi$
$257$	−6562.56	−1.59285	−0.796423	+	0.604740i	$0.793277\pi$
$258$	0	0
$259$	−1323.05	−0.317415
$260$	0	0
$261$	0	0
$262$	−2716.28	−0.640506
$263$	−5663.50	−1.32786	−0.663928	−	0.747797i	$-0.731112\pi$
$263$	−5663.50	−1.32786	−0.663928	+	0.747797i	$0.731112\pi$
$264$	0	0
$265$	0	0
$266$	−1213.75	−0.279775
$267$	0	0
$268$	1505.12	0.343058
$269$	4018.43	0.910811	0.455406	−	0.890284i	$-0.349494\pi$
$269$	4018.43	0.910811	0.455406	+	0.890284i	$0.349494\pi$
$270$	0	0
$271$	−1518.33	−0.340340	−0.170170	−	0.985415i	$-0.554432\pi$
$271$	−1518.33	−0.340340	−0.170170	+	0.985415i	$0.554432\pi$
$272$	55.1598	0.0122962
$273$	0	0
$274$	−4555.38	−1.00438
$275$	0	0
$276$	0	0
$277$	−2182.44	−0.473393	−0.236697	−	0.971584i	$-0.576065\pi$
$277$	−2182.44	−0.473393	−0.236697	+	0.971584i	$0.576065\pi$
$278$	−3578.47	−0.772022
$279$	0	0
$280$	0	0
$281$	5102.57	1.08325	0.541626	−	0.840620i	$-0.317809\pi$
$281$	5102.57	1.08325	0.541626	+	0.840620i	$0.317809\pi$
$282$	0	0
$283$	2577.94	0.541493	0.270747	−	0.962651i	$-0.412729\pi$
$283$	2577.94	0.541493	0.270747	+	0.962651i	$0.412729\pi$
$284$	−3576.19	−0.747212
$285$	0	0
$286$	638.333	0.131977
$287$	1787.32	0.367603
$288$	0	0
$289$	−4905.89	−0.998553
$290$	0	0
$291$	0	0
$292$	1022.29	0.204881
$293$	8339.85	1.66286	0.831432	−	0.555626i	$-0.187521\pi$
$293$	8339.85	1.66286	0.831432	+	0.555626i	$0.187521\pi$
$294$	0	0
$295$	0	0
$296$	−6886.29	−1.35222
$297$	0	0
$298$	3734.77	0.726004
$299$	4568.59	0.883640
$300$	0	0
$301$	1580.94	0.302736
$302$	−6900.21	−1.31478
$303$	0	0
$304$	−2598.08	−0.490166
$305$	0	0
$306$	0	0
$307$	−7042.31	−1.30921	−0.654603	−	0.755973i	$-0.727164\pi$
$307$	−7042.31	−1.30921	−0.654603	+	0.755973i	$0.727164\pi$
$308$	−153.455	−0.0283893
$309$	0	0
$310$	0	0
$311$	2342.92	0.427186	0.213593	−	0.976923i	$-0.431483\pi$
$311$	2342.92	0.427186	0.213593	+	0.976923i	$0.431483\pi$
$312$	0	0
$313$	3833.47	0.692269	0.346135	−	0.938185i	$-0.387494\pi$
$313$	3833.47	0.692269	0.346135	+	0.938185i	$0.387494\pi$
$314$	7940.98	1.42718
$315$	0	0
$316$	4245.18	0.755727
$317$	−5889.98	−1.04358	−0.521789	−	0.853075i	$-0.674735\pi$
$317$	−5889.98	−1.04358	−0.521789	+	0.853075i	$0.674735\pi$
$318$	0	0
$319$	−633.355	−0.111163
$320$	0	0
$321$	0	0
$322$	1275.37	0.220725
$323$	−334.921	−0.0576950
$324$	0	0
$325$	0	0
$326$	−4985.18	−0.846943
$327$	0	0
$328$	9302.72	1.56603
$329$	−364.904	−0.0611484
$330$	0	0
$331$	−6338.95	−1.05263	−0.526315	−	0.850290i	$-0.676427\pi$
$331$	−6338.95	−1.05263	−0.526315	+	0.850290i	$0.676427\pi$
$332$	659.049	0.108946
$333$	0	0
$334$	7403.51	1.21288
$335$	0	0
$336$	0	0
$337$	−968.342	−0.156525	−0.0782625	−	0.996933i	$-0.524937\pi$
$337$	−968.342	−0.156525	−0.0782625	+	0.996933i	$0.524937\pi$
$338$	2070.47	0.333191
$339$	0	0
$340$	0	0
$341$	119.404	0.0189621
$342$	0	0
$343$	−3096.26	−0.487412
$344$	8228.54	1.28969
$345$	0	0
$346$	−331.590	−0.0515213
$347$	−6457.71	−0.999043	−0.499521	−	0.866302i	$-0.666491\pi$
$347$	−6457.71	−0.999043	−0.499521	+	0.866302i	$0.666491\pi$
$348$	0	0
$349$	−6155.18	−0.944066	−0.472033	−	0.881581i	$-0.656480\pi$
$349$	−6155.18	−0.944066	−0.472033	+	0.881581i	$0.656480\pi$
$350$	0	0
$351$	0	0
$352$	−1344.76	−0.203625
$353$	−3663.53	−0.552380	−0.276190	−	0.961103i	$-0.589072\pi$
$353$	−3663.53	−0.552380	−0.276190	+	0.961103i	$0.589072\pi$
$354$	0	0
$355$	0	0
$356$	−2985.28	−0.444437
$357$	0	0
$358$	2323.85	0.343070
$359$	12112.6	1.78073	0.890364	−	0.455250i	$-0.150450\pi$
$359$	12112.6	1.78073	0.890364	+	0.455250i	$0.150450\pi$
$360$	0	0
$361$	8916.11	1.29991
$362$	7995.85	1.16092
$363$	0	0
$364$	597.269	0.0860038
$365$	0	0
$366$	0	0
$367$	−11616.2	−1.65220	−0.826101	−	0.563522i	$-0.809446\pi$
$367$	−11616.2	−1.65220	−0.826101	+	0.563522i	$0.809446\pi$
$368$	2729.97	0.386711
$369$	0	0
$370$	0	0
$371$	−1186.84	−0.166085
$372$	0	0
$373$	2215.72	0.307576	0.153788	−	0.988104i	$-0.450853\pi$
$373$	2215.72	0.307576	0.153788	+	0.988104i	$0.450853\pi$
$374$	49.1713	0.00679837
$375$	0	0
$376$	−1899.27	−0.260499
$377$	2465.11	0.336763
$378$	0	0
$379$	−6539.83	−0.886354	−0.443177	−	0.896434i	$-0.646149\pi$
$379$	−6539.83	−0.886354	−0.443177	+	0.896434i	$0.646149\pi$
$380$	0	0
$381$	0	0
$382$	163.808	0.0219402
$383$	−8495.26	−1.13339	−0.566694	−	0.823928i	$-0.691778\pi$
$383$	−8495.26	−1.13339	−0.566694	+	0.823928i	$0.691778\pi$
$384$	0	0
$385$	0	0
$386$	9477.90	1.24977
$387$	0	0
$388$	−4582.74	−0.599621
$389$	−3536.84	−0.460989	−0.230494	−	0.973074i	$-0.574034\pi$
$389$	−3536.84	−0.460989	−0.230494	+	0.973074i	$0.574034\pi$
$390$	0	0
$391$	351.922	0.0455178
$392$	−7794.25	−1.00426
$393$	0	0
$394$	−8462.23	−1.08203
$395$	0	0
$396$	0	0
$397$	−8586.22	−1.08547	−0.542733	−	0.839905i	$-0.682610\pi$
$397$	−8586.22	−1.08547	−0.542733	+	0.839905i	$0.682610\pi$
$398$	5222.51	0.657740
$399$	0	0
$400$	0	0
$401$	−7232.79	−0.900719	−0.450359	−	0.892847i	$-0.648704\pi$
$401$	−7232.79	−0.900719	−0.450359	+	0.892847i	$0.648704\pi$
$402$	0	0
$403$	−464.737	−0.0574447
$404$	−2098.27	−0.258398
$405$	0	0
$406$	688.160	0.0841202
$407$	−2524.58	−0.307466
$408$	0	0
$409$	1230.96	0.148819	0.0744096	−	0.997228i	$-0.476293\pi$
$409$	1230.96	0.148819	0.0744096	+	0.997228i	$0.476293\pi$
$410$	0	0
$411$	0	0
$412$	−3031.66	−0.362523
$413$	−152.976	−0.0182263
$414$	0	0
$415$	0	0
$416$	5234.01	0.616871
$417$	0	0
$418$	−2316.02	−0.271005
$419$	−582.439	−0.0679093	−0.0339547	−	0.999423i	$-0.510810\pi$
$419$	−582.439	−0.0679093	−0.0339547	+	0.999423i	$0.510810\pi$
$420$	0	0
$421$	−12562.3	−1.45427	−0.727135	−	0.686494i	$-0.759149\pi$
$421$	−12562.3	−1.45427	−0.727135	+	0.686494i	$0.759149\pi$
$422$	−471.136	−0.0543473
$423$	0	0
$424$	−6177.33	−0.707542
$425$	0	0
$426$	0	0
$427$	871.549	0.0987757
$428$	−2012.82	−0.227321
$429$	0	0
$430$	0	0
$431$	−9612.85	−1.07433	−0.537163	−	0.843478i	$-0.680504\pi$
$431$	−9612.85	−1.07433	−0.537163	+	0.843478i	$0.680504\pi$
$432$	0	0
$433$	−8285.20	−0.919541	−0.459771	−	0.888038i	$-0.652068\pi$
$433$	−8285.20	−0.919541	−0.459771	+	0.888038i	$0.652068\pi$
$434$	−129.736	−0.0143491
$435$	0	0
$436$	6057.79	0.665403
$437$	−16575.9	−1.81449
$438$	0	0
$439$	10769.0	1.17079	0.585393	−	0.810750i	$-0.300940\pi$
$439$	10769.0	1.17079	0.585393	+	0.810750i	$0.300940\pi$
$440$	0	0
$441$	0	0
$442$	−191.382	−0.0205953
$443$	−8809.93	−0.944858	−0.472429	−	0.881369i	$-0.656623\pi$
$443$	−8809.93	−0.944858	−0.472429	+	0.881369i	$0.656623\pi$
$444$	0	0
$445$	0	0
$446$	−1068.19	−0.113409
$447$	0	0
$448$	2232.47	0.235434
$449$	−9060.10	−0.952277	−0.476139	−	0.879370i	$-0.657964\pi$
$449$	−9060.10	−0.952277	−0.476139	+	0.879370i	$0.657964\pi$
$450$	0	0
$451$	3410.46	0.356081
$452$	−2020.32	−0.210239
$453$	0	0
$454$	−8201.91	−0.847874
$455$	0	0
$456$	0	0
$457$	−2477.55	−0.253600	−0.126800	−	0.991928i	$-0.540471\pi$
$457$	−2477.55	−0.253600	−0.126800	+	0.991928i	$0.540471\pi$
$458$	3475.97	0.354631
$459$	0	0
$460$	0	0
$461$	−15301.5	−1.54591	−0.772954	−	0.634462i	$-0.781222\pi$
$461$	−15301.5	−1.54591	−0.772954	+	0.634462i	$0.781222\pi$
$462$	0	0
$463$	683.012	0.0685578	0.0342789	−	0.999412i	$-0.489087\pi$
$463$	683.012	0.0685578	0.0342789	+	0.999412i	$0.489087\pi$
$464$	1473.03	0.147379
$465$	0	0
$466$	9557.76	0.950118
$467$	−6569.32	−0.650946	−0.325473	−	0.945551i	$-0.605523\pi$
$467$	−6569.32	−0.650946	−0.325473	+	0.945551i	$0.605523\pi$
$468$	0	0
$469$	−1895.28	−0.186601
$470$	0	0
$471$	0	0
$472$	−796.218	−0.0776460
$473$	3016.66	0.293248
$474$	0	0
$475$	0	0
$476$	46.0081	0.00443020
$477$	0	0
$478$	14322.6	1.37050
$479$	11288.7	1.07681	0.538405	−	0.842686i	$-0.319027\pi$
$479$	11288.7	1.07681	0.538405	+	0.842686i	$0.319027\pi$
$480$	0	0
$481$	9826.03	0.931452
$482$	6180.31	0.584036
$483$	0	0
$484$	4634.00	0.435199
$485$	0	0
$486$	0	0
$487$	8937.53	0.831618	0.415809	−	0.909452i	$-0.363498\pi$
$487$	8937.53	0.831618	0.415809	+	0.909452i	$0.363498\pi$
$488$	4536.28	0.420795
$489$	0	0
$490$	0	0
$491$	3693.14	0.339448	0.169724	−	0.985492i	$-0.445712\pi$
$491$	3693.14	0.339448	0.169724	+	0.985492i	$0.445712\pi$
$492$	0	0
$493$	189.889	0.0173472
$494$	9014.29	0.820996
$495$	0	0
$496$	−277.705	−0.0251397
$497$	4503.22	0.406433
$498$	0	0
$499$	−11186.4	−1.00355	−0.501777	−	0.864997i	$-0.667320\pi$
$499$	−11186.4	−1.00355	−0.501777	+	0.864997i	$0.667320\pi$
$500$	0	0
$501$	0	0
$502$	−14788.4	−1.31482
$503$	−6553.81	−0.580954	−0.290477	−	0.956882i	$-0.593814\pi$
$503$	−6553.81	−0.580954	−0.290477	+	0.956882i	$0.593814\pi$
$504$	0	0
$505$	0	0
$506$	2433.59	0.213807
$507$	0	0
$508$	133.435	0.0116540
$509$	−15027.7	−1.30863	−0.654314	−	0.756223i	$-0.727043\pi$
$509$	−15027.7	−1.30863	−0.654314	+	0.756223i	$0.727043\pi$
$510$	0	0
$511$	−1287.29	−0.111441
$512$	7142.32	0.616502
$513$	0	0
$514$	13605.9	1.16757
$515$	0	0
$516$	0	0
$517$	−696.290	−0.0592318
$518$	2743.03	0.232668
$519$	0	0
$520$	0	0
$521$	−835.969	−0.0702965	−0.0351483	−	0.999382i	$-0.511190\pi$
$521$	−835.969	−0.0702965	−0.0351483	+	0.999382i	$0.511190\pi$
$522$	0	0
$523$	13032.4	1.08961	0.544805	−	0.838563i	$-0.316604\pi$
$523$	13032.4	1.08961	0.544805	+	0.838563i	$0.316604\pi$
$524$	−4849.64	−0.404308
$525$	0	0
$526$	11741.9	0.973330
$527$	−35.7991	−0.00295907
$528$	0	0
$529$	5250.35	0.431524
$530$	0	0
$531$	0	0
$532$	−2167.03	−0.176603
$533$	−13274.0	−1.07873
$534$	0	0
$535$	0	0
$536$	−9864.63	−0.794939
$537$	0	0
$538$	−8331.26	−0.667633
$539$	−2857.44	−0.228347
$540$	0	0
$541$	−4182.33	−0.332370	−0.166185	−	0.986095i	$-0.553145\pi$
$541$	−4182.33	−0.332370	−0.166185	+	0.986095i	$0.553145\pi$
$542$	3147.90	0.249472
$543$	0	0
$544$	403.180	0.0317761
$545$	0	0
$546$	0	0
$547$	−5013.90	−0.391918	−0.195959	−	0.980612i	$-0.562782\pi$
$547$	−5013.90	−0.391918	−0.195959	+	0.980612i	$0.562782\pi$
$548$	−8133.14	−0.633997
$549$	0	0
$550$	0	0
$551$	−8943.98	−0.691518
$552$	0	0
$553$	−5345.62	−0.411065
$554$	4524.76	0.347001
$555$	0	0
$556$	−6388.97	−0.487325
$557$	2611.86	0.198686	0.0993430	−	0.995053i	$-0.468326\pi$
$557$	2611.86	0.198686	0.0993430	+	0.995053i	$0.468326\pi$
$558$	0	0
$559$	−11741.3	−0.888377
$560$	0	0
$561$	0	0
$562$	−10579.0	−0.794033
$563$	18674.0	1.39790	0.698948	−	0.715173i	$-0.253652\pi$
$563$	18674.0	1.39790	0.698948	+	0.715173i	$0.253652\pi$
$564$	0	0
$565$	0	0
$566$	−5344.74	−0.396919
$567$	0	0
$568$	23438.6	1.73145
$569$	−23671.0	−1.74401	−0.872004	−	0.489498i	$-0.837180\pi$
$569$	−23671.0	−1.74401	−0.872004	+	0.489498i	$0.837180\pi$
$570$	0	0
$571$	−9355.54	−0.685669	−0.342835	−	0.939396i	$-0.611387\pi$
$571$	−9355.54	−0.685669	−0.342835	+	0.939396i	$0.611387\pi$
$572$	1139.68	0.0833082
$573$	0	0
$574$	−3705.57	−0.269456
$575$	0	0
$576$	0	0
$577$	21695.2	1.56531	0.782653	−	0.622459i	$-0.213866\pi$
$577$	21695.2	1.56531	0.782653	+	0.622459i	$0.213866\pi$
$578$	10171.2	0.731948
$579$	0	0
$580$	0	0
$581$	−829.889	−0.0592592
$582$	0	0
$583$	−2264.67	−0.160880
$584$	−6700.18	−0.474752
$585$	0	0
$586$	−17290.7	−1.21889
$587$	−737.149	−0.0518320	−0.0259160	−	0.999664i	$-0.508250\pi$
$587$	−737.149	−0.0518320	−0.0259160	+	0.999664i	$0.508250\pi$
$588$	0	0
$589$	1686.17	0.117958
$590$	0	0
$591$	0	0
$592$	5871.56	0.407635
$593$	21908.1	1.51713	0.758565	−	0.651597i	$-0.225901\pi$
$593$	21908.1	1.51713	0.758565	+	0.651597i	$0.225901\pi$
$594$	0	0
$595$	0	0
$596$	6668.03	0.458277
$597$	0	0
$598$	−9471.89	−0.647716
$599$	−7595.84	−0.518126	−0.259063	−	0.965860i	$-0.583414\pi$
$599$	−7595.84	−0.518126	−0.259063	+	0.965860i	$0.583414\pi$
$600$	0	0
$601$	−6502.52	−0.441337	−0.220668	−	0.975349i	$-0.570824\pi$
$601$	−6502.52	−0.441337	−0.220668	+	0.975349i	$0.570824\pi$
$602$	−3277.69	−0.221908
$603$	0	0
$604$	−12319.6	−0.829929
$605$	0	0
$606$	0	0
$607$	26156.8	1.74905	0.874524	−	0.484981i	$-0.161174\pi$
$607$	26156.8	1.74905	0.874524	+	0.484981i	$0.161174\pi$
$608$	−18990.2	−1.26670
$609$	0	0
$610$	0	0
$611$	2710.06	0.179439
$612$	0	0
$613$	−18172.8	−1.19738	−0.598690	−	0.800981i	$-0.704312\pi$
$613$	−18172.8	−1.19738	−0.598690	+	0.800981i	$0.704312\pi$
$614$	14600.6	0.959659
$615$	0	0
$616$	1005.75	0.0657840
$617$	26584.8	1.73462	0.867312	−	0.497766i	$-0.165846\pi$
$617$	26584.8	1.73462	0.867312	+	0.497766i	$0.165846\pi$
$618$	0	0
$619$	21687.8	1.40825	0.704124	−	0.710077i	$-0.251340\pi$
$619$	21687.8	1.40825	0.704124	+	0.710077i	$0.251340\pi$
$620$	0	0
$621$	0	0
$622$	−4857.48	−0.313131
$623$	3759.13	0.241744
$624$	0	0
$625$	0	0
$626$	−7947.78	−0.507439
$627$	0	0
$628$	14177.8	0.900883
$629$	756.907	0.0479807
$630$	0	0
$631$	−7685.15	−0.484851	−0.242426	−	0.970170i	$-0.577943\pi$
$631$	−7685.15	−0.484851	−0.242426	+	0.970170i	$0.577943\pi$
$632$	−27823.2	−1.75118
$633$	0	0
$634$	12211.5	0.764951
$635$	0	0
$636$	0	0
$637$	11121.6	0.691764
$638$	1313.11	0.0814835
$639$	0	0
$640$	0	0
$641$	−8055.69	−0.496381	−0.248191	−	0.968711i	$-0.579836\pi$
$641$	−8055.69	−0.496381	−0.248191	+	0.968711i	$0.579836\pi$
$642$	0	0
$643$	7122.02	0.436804	0.218402	−	0.975859i	$-0.429916\pi$
$643$	7122.02	0.436804	0.218402	+	0.975859i	$0.429916\pi$
$644$	2277.03	0.139329
$645$	0	0
$646$	694.378	0.0422909
$647$	−10280.7	−0.624692	−0.312346	−	0.949968i	$-0.601115\pi$
$647$	−10280.7	−0.624692	−0.312346	+	0.949968i	$0.601115\pi$
$648$	0	0
$649$	−291.901	−0.0176550
$650$	0	0
$651$	0	0
$652$	−8900.51	−0.534618
$653$	−8985.03	−0.538456	−0.269228	−	0.963077i	$-0.586768\pi$
$653$	−8985.03	−0.538456	−0.269228	+	0.963077i	$0.586768\pi$
$654$	0	0
$655$	0	0
$656$	−7931.92	−0.472087
$657$	0	0
$658$	756.541	0.0448223
$659$	−12538.7	−0.741183	−0.370591	−	0.928796i	$-0.620845\pi$
$659$	−12538.7	−0.741183	−0.370591	+	0.928796i	$0.620845\pi$
$660$	0	0
$661$	−12479.9	−0.734360	−0.367180	−	0.930150i	$-0.619677\pi$
$661$	−12479.9	−0.734360	−0.367180	+	0.930150i	$0.619677\pi$
$662$	13142.3	0.771586
$663$	0	0
$664$	−4319.45	−0.252451
$665$	0	0
$666$	0	0
$667$	9398.01	0.545566
$668$	13218.2	0.765609
$669$	0	0
$670$	0	0
$671$	1663.04	0.0956797
$672$	0	0
$673$	−13185.6	−0.755228	−0.377614	−	0.925963i	$-0.623255\pi$
$673$	−13185.6	−0.755228	−0.377614	+	0.925963i	$0.623255\pi$
$674$	2007.63	0.114734
$675$	0	0
$676$	3696.60	0.210321
$677$	26568.3	1.50827	0.754136	−	0.656718i	$-0.228056\pi$
$677$	26568.3	1.50827	0.754136	+	0.656718i	$0.228056\pi$
$678$	0	0
$679$	5770.68	0.326154
$680$	0	0
$681$	0	0
$682$	−247.555	−0.0138994
$683$	−7602.40	−0.425912	−0.212956	−	0.977062i	$-0.568309\pi$
$683$	−7602.40	−0.425912	−0.212956	+	0.977062i	$0.568309\pi$
$684$	0	0
$685$	0	0
$686$	6419.36	0.357277
$687$	0	0
$688$	−7016.02	−0.388784
$689$	8814.41	0.487376
$690$	0	0
$691$	6257.31	0.344485	0.172243	−	0.985055i	$-0.444899\pi$
$691$	6257.31	0.344485	0.172243	+	0.985055i	$0.444899\pi$
$692$	−592.018	−0.0325219
$693$	0	0
$694$	13388.5	0.732307
$695$	0	0
$696$	0	0
$697$	−1022.51	−0.0555671
$698$	12761.3	0.692009
$699$	0	0
$700$	0	0
$701$	−769.271	−0.0414479	−0.0207239	−	0.999785i	$-0.506597\pi$
$701$	−769.271	−0.0414479	−0.0207239	+	0.999785i	$0.506597\pi$
$702$	0	0
$703$	−35651.1	−1.91267
$704$	4259.88	0.228054
$705$	0	0
$706$	7595.46	0.404900
$707$	2642.18	0.140551
$708$	0	0
$709$	−8450.32	−0.447614	−0.223807	−	0.974633i	$-0.571849\pi$
$709$	−8450.32	−0.447614	−0.223807	+	0.974633i	$0.571849\pi$
$710$	0	0
$711$	0	0
$712$	19565.7	1.02985
$713$	−1771.77	−0.0930620
$714$	0	0
$715$	0	0
$716$	4148.98	0.216557
$717$	0	0
$718$	−25112.7	−1.30529
$719$	−25350.0	−1.31488	−0.657438	−	0.753508i	$-0.728360\pi$
$719$	−25350.0	−1.31488	−0.657438	+	0.753508i	$0.728360\pi$
$720$	0	0
$721$	3817.54	0.197188
$722$	−18485.4	−0.952848
$723$	0	0
$724$	14275.7	0.732809
$725$	0	0
$726$	0	0
$727$	3825.15	0.195140	0.0975701	−	0.995229i	$-0.468893\pi$
$727$	3825.15	0.195140	0.0975701	+	0.995229i	$0.468893\pi$
$728$	−3914.54	−0.199289
$729$	0	0
$730$	0	0
$731$	−904.440	−0.0457618
$732$	0	0
$733$	3692.83	0.186081	0.0930407	−	0.995662i	$-0.470341\pi$
$733$	3692.83	0.186081	0.0930407	+	0.995662i	$0.470341\pi$
$734$	24083.3	1.21108
$735$	0	0
$736$	19954.2	0.999350
$737$	−3616.46	−0.180752
$738$	0	0
$739$	4181.32	0.208136	0.104068	−	0.994570i	$-0.466814\pi$
$739$	4181.32	0.208136	0.104068	+	0.994570i	$0.466814\pi$
$740$	0	0
$741$	0	0
$742$	2460.63	0.121742
$743$	927.982	0.0458201	0.0229101	−	0.999738i	$-0.492707\pi$
$743$	927.982	0.0458201	0.0229101	+	0.999738i	$0.492707\pi$
$744$	0	0
$745$	0	0
$746$	−4593.78	−0.225456
$747$	0	0
$748$	87.7902	0.00429135
$749$	2534.59	0.123647
$750$	0	0
$751$	−22369.5	−1.08692	−0.543458	−	0.839436i	$-0.682885\pi$
$751$	−22369.5	−1.08692	−0.543458	+	0.839436i	$0.682885\pi$
$752$	1619.40	0.0785287
$753$	0	0
$754$	−5110.81	−0.246850
$755$	0	0
$756$	0	0
$757$	−35390.0	−1.69917	−0.849586	−	0.527451i	$-0.823148\pi$
$757$	−35390.0	−1.69917	−0.849586	+	0.527451i	$0.823148\pi$
$758$	13558.8	0.649705
$759$	0	0
$760$	0	0
$761$	33434.6	1.59265	0.796323	−	0.604872i	$-0.206776\pi$
$761$	33434.6	1.59265	0.796323	+	0.604872i	$0.206776\pi$
$762$	0	0
$763$	−7628.10	−0.361934
$764$	292.462	0.0138494
$765$	0	0
$766$	17612.9	0.830783
$767$	1136.12	0.0534849
$768$	0	0
$769$	25021.4	1.17334	0.586668	−	0.809828i	$-0.300440\pi$
$769$	25021.4	1.17334	0.586668	+	0.809828i	$0.300440\pi$
$770$	0	0
$771$	0	0
$772$	16921.8	0.788897
$773$	3065.27	0.142626	0.0713132	−	0.997454i	$-0.477281\pi$
$773$	3065.27	0.142626	0.0713132	+	0.997454i	$0.477281\pi$
$774$	0	0
$775$	0	0
$776$	30035.5	1.38945
$777$	0	0
$778$	7332.79	0.337909
$779$	48161.2	2.21509
$780$	0	0
$781$	8592.81	0.393694
$782$	−729.627	−0.0333650
$783$	0	0
$784$	6645.73	0.302739
$785$	0	0
$786$	0	0
$787$	35683.1	1.61622	0.808111	−	0.589031i	$-0.200490\pi$
$787$	35683.1	1.61622	0.808111	+	0.589031i	$0.200490\pi$
$788$	−15108.4	−0.683014
$789$	0	0
$790$	0	0
$791$	2544.03	0.114356
$792$	0	0
$793$	−6472.81	−0.289856
$794$	17801.5	0.795656
$795$	0	0
$796$	9324.23	0.415187
$797$	4525.07	0.201112	0.100556	−	0.994931i	$-0.467938\pi$
$797$	4525.07	0.201112	0.100556	+	0.994931i	$0.467938\pi$
$798$	0	0
$799$	208.758	0.00924323
$800$	0	0
$801$	0	0
$802$	14995.5	0.660235
$803$	−2456.35	−0.107948
$804$	0	0
$805$	0	0
$806$	963.521	0.0421074
$807$	0	0
$808$	13752.2	0.598762
$809$	−16234.6	−0.705535	−0.352767	−	0.935711i	$-0.614759\pi$
$809$	−16234.6	−0.705535	−0.352767	+	0.935711i	$0.614759\pi$
$810$	0	0
$811$	2197.06	0.0951286	0.0475643	−	0.998868i	$-0.484854\pi$
$811$	2197.06	0.0951286	0.0475643	+	0.998868i	$0.484854\pi$
$812$	1228.64	0.0530993
$813$	0	0
$814$	5234.11	0.225375
$815$	0	0
$816$	0	0
$817$	42600.0	1.82422
$818$	−2552.10	−0.109086
$819$	0	0
$820$	0	0
$821$	−20141.6	−0.856208	−0.428104	−	0.903729i	$-0.640818\pi$
$821$	−20141.6	−0.856208	−0.428104	+	0.903729i	$0.640818\pi$
$822$	0	0
$823$	−15517.0	−0.657217	−0.328608	−	0.944466i	$-0.606580\pi$
$823$	−15517.0	−0.657217	−0.328608	+	0.944466i	$0.606580\pi$
$824$	19869.7	0.840042
$825$	0	0
$826$	317.159	0.0133600
$827$	−5582.51	−0.234732	−0.117366	−	0.993089i	$-0.537445\pi$
$827$	−5582.51	−0.234732	−0.117366	+	0.993089i	$0.537445\pi$
$828$	0	0
$829$	26037.6	1.09086	0.545430	−	0.838157i	$-0.316367\pi$
$829$	26037.6	1.09086	0.545430	+	0.838157i	$0.316367\pi$
$830$	0	0
$831$	0	0
$832$	−16580.1	−0.690878
$833$	856.705	0.0356339
$834$	0	0
$835$	0	0
$836$	−4135.01	−0.171067
$837$	0	0
$838$	1207.55	0.0497781
$839$	−18088.0	−0.744298	−0.372149	−	0.928173i	$-0.621379\pi$
$839$	−18088.0	−0.744298	−0.372149	+	0.928173i	$0.621379\pi$
$840$	0	0
$841$	−19318.0	−0.792080
$842$	26044.9	1.06599
$843$	0	0
$844$	−841.164	−0.0343057
$845$	0	0
$846$	0	0
$847$	−5835.23	−0.236719
$848$	5267.07	0.213292
$849$	0	0
$850$	0	0
$851$	37460.9	1.50898
$852$	0	0
$853$	25349.0	1.01751	0.508753	−	0.860912i	$-0.330107\pi$
$853$	25349.0	1.01751	0.508753	+	0.860912i	$0.330107\pi$
$854$	−1806.95	−0.0724034
$855$	0	0
$856$	13192.1	0.526750
$857$	563.797	0.0224725	0.0112363	−	0.999937i	$-0.496423\pi$
$857$	563.797	0.0224725	0.0112363	+	0.999937i	$0.496423\pi$
$858$	0	0
$859$	−20558.9	−0.816600	−0.408300	−	0.912848i	$-0.633878\pi$
$859$	−20558.9	−0.816600	−0.408300	+	0.912848i	$0.633878\pi$
$860$	0	0
$861$	0	0
$862$	19929.9	0.787490
$863$	7056.31	0.278331	0.139166	−	0.990269i	$-0.455558\pi$
$863$	7056.31	0.278331	0.139166	+	0.990269i	$0.455558\pi$
$864$	0	0
$865$	0	0
$866$	17177.4	0.674032
$867$	0	0
$868$	−231.630	−0.00905763
$869$	−10200.2	−0.398181
$870$	0	0
$871$	14075.8	0.547578
$872$	−39703.2	−1.54188
$873$	0	0
$874$	34366.2	1.33004
$875$	0	0
$876$	0	0
$877$	−2516.30	−0.0968864	−0.0484432	−	0.998826i	$-0.515426\pi$
$877$	−2516.30	−0.0968864	−0.0484432	+	0.998826i	$0.515426\pi$
$878$	−22326.9	−0.858196
$879$	0	0
$880$	0	0
$881$	31571.2	1.20733	0.603666	−	0.797237i	$-0.293706\pi$
$881$	31571.2	1.20733	0.603666	+	0.797237i	$0.293706\pi$
$882$	0	0
$883$	−47743.5	−1.81959	−0.909794	−	0.415059i	$-0.863761\pi$
$883$	−47743.5	−1.81959	−0.909794	+	0.415059i	$0.863761\pi$
$884$	−341.692	−0.0130004
$885$	0	0
$886$	18265.3	0.692589
$887$	33028.9	1.25028	0.625141	−	0.780512i	$-0.285041\pi$
$887$	33028.9	1.25028	0.625141	+	0.780512i	$0.285041\pi$
$888$	0	0
$889$	−168.024	−0.00633897
$890$	0	0
$891$	0	0
$892$	−1907.14	−0.0715873
$893$	−9832.74	−0.368466
$894$	0	0
$895$	0	0
$896$	1009.49	0.0376392
$897$	0	0
$898$	18784.0	0.698028
$899$	−956.006	−0.0354667
$900$	0	0
$901$	678.981	0.0251056
$902$	−7070.78	−0.261010
$903$	0	0
$904$	13241.3	0.487168
$905$	0	0
$906$	0	0
$907$	43330.5	1.58629	0.793146	−	0.609032i	$-0.208442\pi$
$907$	43330.5	1.58629	0.793146	+	0.609032i	$0.208442\pi$
$908$	−14643.6	−0.535205
$909$	0	0
$910$	0	0
$911$	22354.4	0.812992	0.406496	−	0.913653i	$-0.366751\pi$
$911$	22354.4	0.812992	0.406496	+	0.913653i	$0.366751\pi$
$912$	0	0
$913$	−1583.55	−0.0574018
$914$	5136.62	0.185891
$915$	0	0
$916$	6205.97	0.223855
$917$	6106.77	0.219916
$918$	0	0
$919$	1306.08	0.0468811	0.0234406	−	0.999725i	$-0.492538\pi$
$919$	1306.08	0.0468811	0.0234406	+	0.999725i	$0.492538\pi$
$920$	0	0
$921$	0	0
$922$	31724.1	1.13316
$923$	−33444.5	−1.19267
$924$	0	0
$925$	0	0
$926$	−1416.06	−0.0502535
$927$	0	0
$928$	10766.8	0.380861
$929$	−50663.9	−1.78927	−0.894633	−	0.446802i	$-0.852563\pi$
$929$	−50663.9	−1.78927	−0.894633	+	0.446802i	$0.852563\pi$
$930$	0	0
$931$	−40351.7	−1.42049
$932$	17064.4	0.599745
$933$	0	0
$934$	13619.9	0.477149
$935$	0	0
$936$	0	0
$937$	12652.4	0.441126	0.220563	−	0.975373i	$-0.429211\pi$
$937$	12652.4	0.441126	0.220563	+	0.975373i	$0.429211\pi$
$938$	3929.40	0.136780
$939$	0	0
$940$	0	0
$941$	−40677.4	−1.40919	−0.704594	−	0.709610i	$-0.748871\pi$
$941$	−40677.4	−1.40919	−0.704594	+	0.709610i	$0.748871\pi$
$942$	0	0
$943$	−50606.0	−1.74757
$944$	678.891	0.0234068
$945$	0	0
$946$	−6254.32	−0.214953
$947$	−26916.8	−0.923631	−0.461816	−	0.886976i	$-0.652802\pi$
$947$	−26916.8	−0.923631	−0.461816	+	0.886976i	$0.652802\pi$
$948$	0	0
$949$	9560.46	0.327024
$950$	0	0
$951$	0	0
$952$	−301.540	−0.0102657
$953$	29644.9	1.00765	0.503827	−	0.863805i	$-0.331925\pi$
$953$	29644.9	1.00765	0.503827	+	0.863805i	$0.331925\pi$
$954$	0	0
$955$	0	0
$956$	25571.4	0.865104
$957$	0	0
$958$	−23404.3	−0.789310
$959$	10241.4	0.344852
$960$	0	0
$961$	−29610.8	−0.993950
$962$	−20371.9	−0.682762
$963$	0	0
$964$	11034.3	0.368662
$965$	0	0
$966$	0	0
$967$	−21452.1	−0.713395	−0.356697	−	0.934220i	$-0.616097\pi$
$967$	−21452.1	−0.713395	−0.356697	+	0.934220i	$0.616097\pi$
$968$	−30371.5	−1.00845
$969$	0	0
$970$	0	0
$971$	42684.6	1.41073	0.705363	−	0.708846i	$-0.250784\pi$
$971$	42684.6	1.41073	0.705363	+	0.708846i	$0.250784\pi$
$972$	0	0
$973$	8045.14	0.265072
$974$	−18529.8	−0.609583
$975$	0	0
$976$	−3867.84	−0.126851
$977$	60386.2	1.97741	0.988704	−	0.149884i	$-0.0478901\pi$
$977$	60386.2	1.97741	0.988704	+	0.149884i	$0.0478901\pi$
$978$	0	0
$979$	7172.97	0.234167
$980$	0	0
$981$	0	0
$982$	−7656.85	−0.248818
$983$	−59009.2	−1.91465	−0.957325	−	0.289014i	$-0.906672\pi$
$983$	−59009.2	−1.91465	−0.957325	+	0.289014i	$0.906672\pi$
$984$	0	0
$985$	0	0
$986$	−393.690	−0.0127157
$987$	0	0
$988$	16094.1	0.518239
$989$	−44762.6	−1.43920
$990$	0	0
$991$	−30899.0	−0.990454	−0.495227	−	0.868764i	$-0.664915\pi$
$991$	−30899.0	−0.990454	−0.495227	+	0.868764i	$0.664915\pi$
$992$	−2029.83	−0.0649668
$993$	0	0
$994$	−9336.36	−0.297919
$995$	0	0
$996$	0	0
$997$	33866.8	1.07580	0.537901	−	0.843008i	$-0.319218\pi$
$997$	33866.8	1.07580	0.537901	+	0.843008i	$0.319218\pi$
$998$	23192.4	0.735614
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2025.4.a.y.1.3		6
3.2	odd	2		2025.4.a.z.1.4		6
5.4	even	2		405.4.a.l.1.4	yes	6
15.14	odd	2		405.4.a.k.1.3	✓	6
45.4	even	6		405.4.e.w.136.3		12
45.14	odd	6		405.4.e.x.136.4		12
45.29	odd	6		405.4.e.x.271.4		12
45.34	even	6		405.4.e.w.271.3		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
405.4.a.k.1.3	✓	6	15.14	odd	2
405.4.a.l.1.4	yes	6	5.4	even	2
405.4.e.w.136.3		12	45.4	even	6
405.4.e.w.271.3		12	45.34	even	6
405.4.e.x.136.4		12	45.14	odd	6
405.4.e.x.271.4		12	45.29	odd	6
2025.4.a.y.1.3		6	1.1	even	1	trivial
2025.4.a.z.1.4		6	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)

\(f(q)\)	\(=\)	\(q-2.07326 q^{2} -3.70159 q^{4} +4.66112 q^{7} +24.2604 q^{8} +8.89410 q^{11} -34.6172 q^{13} -9.66372 q^{14} -20.6855 q^{16} -2.66659 q^{17} +125.599 q^{19} -18.4398 q^{22} -131.975 q^{23} +71.7704 q^{26} -17.2535 q^{28} -71.2106 q^{29} +13.4250 q^{31} -151.197 q^{32} +5.52853 q^{34} -283.849 q^{37} -260.400 q^{38} +383.452 q^{41} +339.175 q^{43} -32.9223 q^{44} +273.618 q^{46} -78.2867 q^{47} -321.274 q^{49} +128.138 q^{52} -254.626 q^{53} +113.081 q^{56} +147.638 q^{58} -32.8196 q^{59} +186.983 q^{61} -27.8336 q^{62} +478.955 q^{64} -406.614 q^{67} +9.87060 q^{68} +966.124 q^{71} -276.177 q^{73} +588.492 q^{74} -464.916 q^{76} +41.4565 q^{77} -1146.85 q^{79} -794.996 q^{82} -178.045 q^{83} -703.198 q^{86} +215.775 q^{88} +806.486 q^{89} -161.355 q^{91} +488.516 q^{92} +162.309 q^{94} +1238.05 q^{97} +666.085 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 4 q^{2} + 34 q^{4} - 40 q^{7} - 66 q^{8} + 88 q^{11} - 20 q^{13} + 180 q^{14} + 58 q^{16} - 124 q^{17} - 46 q^{19} + 74 q^{22} - 210 q^{23} + 4 q^{26} - 352 q^{28} + 296 q^{29} - 104 q^{31} - 722 q^{32}+ \cdots - 1982 q^{98}+O(q^{100}) \) 6 * q - 4 * q^2 + 34 * q^4 - 40 * q^7 - 66 * q^8 + 88 * q^11 - 20 * q^13 + 180 * q^14 + 58 * q^16 - 124 * q^17 - 46 * q^19 + 74 * q^22 - 210 * q^23 + 4 * q^26 - 352 * q^28 + 296 * q^29 - 104 * q^31 - 722 * q^32 - 428 * q^34 + 204 * q^37 + 20 * q^38 + 344 * q^41 - 512 * q^43 + 716 * q^44 - 186 * q^46 - 238 * q^47 + 68 * q^49 + 468 * q^52 - 850 * q^53 + 2316 * q^56 - 890 * q^58 + 1840 * q^59 - 364 * q^61 - 1038 * q^62 - 990 * q^64 - 88 * q^67 - 236 * q^68 + 1364 * q^71 - 836 * q^73 + 1316 * q^74 - 2106 * q^76 - 840 * q^77 - 680 * q^79 - 1742 * q^82 - 2148 * q^83 + 2872 * q^86 - 1296 * q^88 + 3000 * q^89 - 3058 * q^91 - 1002 * q^92 - 3662 * q^94 + 612 * q^97 - 1982 * q^98

Introduction

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