LMFDB - Embedded newform 2025.4.a.bl.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("2025.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$119.478867762$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 91 x^{14} + 3268 x^{12} - 59128 x^{10} + 571975 x^{8} - 2881141 x^{6} + 6555196 x^{4} + \cdots + 614656 $ x^16 - 91x^14 + 3268x^12 - 59128x^10 + 571975x^8 - 2881141x^6 + 6555196x^4 - 4069504*x^2 + 614656
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{5}\cdot 3^{12}\cdot 7^{2} $
Twist minimal:	no (minimal twist has level 45)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-5.02371$ 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)

$f(q)$	$=$	$q-5.02371 q^{2} +17.2377 q^{4} +5.38197 q^{7} -46.4074 q^{8} +O(q^{10})$	$q-5.02371 q^{2} +17.2377 q^{4} +5.38197 q^{7} -46.4074 q^{8} +39.1018 q^{11} -86.6271 q^{13} -27.0375 q^{14} +95.2360 q^{16} +15.4194 q^{17} -26.8412 q^{19} -196.436 q^{22} -111.138 q^{23} +435.189 q^{26} +92.7727 q^{28} -49.2347 q^{29} -179.877 q^{31} -107.179 q^{32} -77.4624 q^{34} -293.496 q^{37} +134.842 q^{38} -27.6032 q^{41} -60.5030 q^{43} +674.023 q^{44} +558.327 q^{46} +96.2351 q^{47} -314.034 q^{49} -1493.25 q^{52} +251.203 q^{53} -249.763 q^{56} +247.341 q^{58} +76.8462 q^{59} +490.091 q^{61} +903.648 q^{62} -223.452 q^{64} +238.721 q^{67} +265.794 q^{68} +640.447 q^{71} -769.257 q^{73} +1474.44 q^{74} -462.679 q^{76} +210.445 q^{77} +662.352 q^{79} +138.670 q^{82} -1302.14 q^{83} +303.950 q^{86} -1814.61 q^{88} +995.544 q^{89} -466.224 q^{91} -1915.77 q^{92} -483.457 q^{94} +814.846 q^{97} +1577.62 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 54 q^{4}+O(q^{10}) $ 16 * q + 54 * q^4	$ 16 q + 54 q^{4} + 90 q^{11} + 102 q^{14} + 146 q^{16} + 4 q^{19} + 468 q^{26} + 516 q^{29} + 38 q^{31} + 212 q^{34} + 576 q^{41} + 1644 q^{44} - 290 q^{46} - 4 q^{49} + 2430 q^{56} + 2202 q^{59} + 20 q^{61} - 322 q^{64} + 2952 q^{71} + 4080 q^{74} - 396 q^{76} - 218 q^{79} + 6108 q^{86} + 4074 q^{89} - 942 q^{91} - 1078 q^{94}+O(q^{100}) $ 16 * q + 54 * q^4 + 90 * q^11 + 102 * q^14 + 146 * q^16 + 4 * q^19 + 468 * q^26 + 516 * q^29 + 38 * q^31 + 212 * q^34 + 576 * q^41 + 1644 * q^44 - 290 * q^46 - 4 * q^49 + 2430 * q^56 + 2202 * q^59 + 20 * q^61 - 322 * q^64 + 2952 * q^71 + 4080 * q^74 - 396 * q^76 - 218 * q^79 + 6108 * q^86 + 4074 * q^89 - 942 * q^91 - 1078 * q^94

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−5.02371	−1.77615	−0.888075	−	0.459699i	$-0.847957\pi$
$2$	−5.02371	−1.77615	−0.888075	+	0.459699i	$0.847957\pi$
$3$	0	0
$3$	0	0
$4$	17.2377	2.15471
$5$	0	0
$5$	0	0
$6$	0	0
$7$	5.38197	0.290599	0.145300	−	0.989388i	$-0.453585\pi$
$7$	5.38197	0.290599	0.145300	+	0.989388i	$0.453585\pi$
$8$	−46.4074	−2.05094
$9$	0	0
$10$	0	0
$11$	39.1018	1.07178	0.535892	−	0.844286i	$-0.319975\pi$
$11$	39.1018	1.07178	0.535892	+	0.844286i	$0.319975\pi$
$12$	0	0
$13$	−86.6271	−1.84816	−0.924078	−	0.382204i	$-0.875165\pi$
$13$	−86.6271	−1.84816	−0.924078	+	0.382204i	$0.875165\pi$
$14$	−27.0375	−0.516148
$15$	0	0
$16$	95.2360	1.48806
$17$	15.4194	0.219985	0.109993	−	0.993932i	$-0.464917\pi$
$17$	15.4194	0.219985	0.109993	+	0.993932i	$0.464917\pi$
$18$	0	0
$19$	−26.8412	−0.324094	−0.162047	−	0.986783i	$-0.551810\pi$
$19$	−26.8412	−0.324094	−0.162047	+	0.986783i	$0.551810\pi$
$20$	0	0
$21$	0	0
$22$	−196.436	−1.90365
$23$	−111.138	−1.00756	−0.503781	−	0.863831i	$-0.668058\pi$
$23$	−111.138	−1.00756	−0.503781	+	0.863831i	$0.668058\pi$
$24$	0	0
$25$	0	0
$26$	435.189	3.28260
$27$	0	0
$28$	92.7727	0.626157
$29$	−49.2347	−0.315264	−0.157632	−	0.987498i	$-0.550386\pi$
$29$	−49.2347	−0.315264	−0.157632	+	0.987498i	$0.550386\pi$
$30$	0	0
$31$	−179.877	−1.04215	−0.521077	−	0.853510i	$-0.674470\pi$
$31$	−179.877	−1.04215	−0.521077	+	0.853510i	$0.674470\pi$
$32$	−107.179	−0.592085
$33$	0	0
$34$	−77.4624	−0.390726
$35$	0	0
$36$	0	0
$37$	−293.496	−1.30407	−0.652033	−	0.758191i	$-0.726084\pi$
$37$	−293.496	−1.30407	−0.652033	+	0.758191i	$0.726084\pi$
$38$	134.842	0.575640
$39$	0	0
$40$	0	0
$41$	−27.6032	−0.105144	−0.0525719	−	0.998617i	$-0.516742\pi$
$41$	−27.6032	−0.105144	−0.0525719	+	0.998617i	$0.516742\pi$
$42$	0	0
$43$	−60.5030	−0.214573	−0.107286	−	0.994228i	$-0.534216\pi$
$43$	−60.5030	−0.214573	−0.107286	+	0.994228i	$0.534216\pi$
$44$	674.023	2.30938
$45$	0	0
$46$	558.327	1.78958
$47$	96.2351	0.298667	0.149333	−	0.988787i	$-0.452287\pi$
$47$	96.2351	0.298667	0.149333	+	0.988787i	$0.452287\pi$
$48$	0	0
$49$	−314.034	−0.915552
$50$	0	0
$51$	0	0
$52$	−1493.25	−3.98224
$53$	251.203	0.651046	0.325523	−	0.945534i	$-0.394460\pi$
$53$	251.203	0.651046	0.325523	+	0.945534i	$0.394460\pi$
$54$	0	0
$55$	0	0
$56$	−249.763	−0.596001
$57$	0	0
$58$	247.341	0.559957
$59$	76.8462	0.169568	0.0847841	−	0.996399i	$-0.472980\pi$
$59$	76.8462	0.169568	0.0847841	+	0.996399i	$0.472980\pi$
$60$	0	0
$61$	490.091	1.02868	0.514342	−	0.857585i	$-0.328036\pi$
$61$	490.091	1.02868	0.514342	+	0.857585i	$0.328036\pi$
$62$	903.648	1.85102
$63$	0	0
$64$	−223.452	−0.436430
$65$	0	0
$66$	0	0
$67$	238.721	0.435290	0.217645	−	0.976028i	$-0.430163\pi$
$67$	238.721	0.435290	0.217645	+	0.976028i	$0.430163\pi$
$68$	265.794	0.474004
$69$	0	0
$70$	0	0
$71$	640.447	1.07052	0.535261	−	0.844687i	$-0.320213\pi$
$71$	640.447	1.07052	0.535261	+	0.844687i	$0.320213\pi$
$72$	0	0
$73$	−769.257	−1.23335	−0.616676	−	0.787217i	$-0.711521\pi$
$73$	−769.257	−1.23335	−0.616676	+	0.787217i	$0.711521\pi$
$74$	1474.44	2.31622
$75$	0	0
$76$	−462.679	−0.698328
$77$	210.445	0.311460
$78$	0	0
$79$	662.352	0.943296	0.471648	−	0.881787i	$-0.343659\pi$
$79$	662.352	0.943296	0.471648	+	0.881787i	$0.343659\pi$
$80$	0	0
$81$	0	0
$82$	138.670	0.186751
$83$	−1302.14	−1.72203	−0.861016	−	0.508578i	$-0.830171\pi$
$83$	−1302.14	−1.72203	−0.861016	+	0.508578i	$0.830171\pi$
$84$	0	0
$85$	0	0
$86$	303.950	0.381113
$87$	0	0
$88$	−1814.61	−2.19816
$89$	995.544	1.18570	0.592851	−	0.805312i	$-0.298002\pi$
$89$	995.544	1.18570	0.592851	+	0.805312i	$0.298002\pi$
$90$	0	0
$91$	−466.224	−0.537073
$92$	−1915.77	−2.17100
$93$	0	0
$94$	−483.457	−0.530477
$95$	0	0
$96$	0	0
$97$	814.846	0.852939	0.426470	−	0.904502i	$-0.359757\pi$
$97$	814.846	0.852939	0.426470	+	0.904502i	$0.359757\pi$
$98$	1577.62	1.62616
$99$	0	0
$100$	0	0
$101$	652.604	0.642936	0.321468	−	0.946920i	$-0.395824\pi$
$101$	652.604	0.642936	0.321468	+	0.946920i	$0.395824\pi$
$102$	0	0
$103$	−153.955	−0.147278	−0.0736391	−	0.997285i	$-0.523461\pi$
$103$	−153.955	−0.147278	−0.0736391	+	0.997285i	$0.523461\pi$
$104$	4020.14	3.79045
$105$	0	0
$106$	−1261.97	−1.15636
$107$	348.878	0.315209	0.157604	−	0.987502i	$-0.449623\pi$
$107$	348.878	0.315209	0.157604	+	0.987502i	$0.449623\pi$
$108$	0	0
$109$	1125.26	0.988815	0.494407	−	0.869230i	$-0.335385\pi$
$109$	1125.26	0.988815	0.494407	+	0.869230i	$0.335385\pi$
$110$	0	0
$111$	0	0
$112$	512.557	0.432430
$113$	352.076	0.293102	0.146551	−	0.989203i	$-0.453183\pi$
$113$	352.076	0.293102	0.146551	+	0.989203i	$0.453183\pi$
$114$	0	0
$115$	0	0
$116$	−848.692	−0.679303
$117$	0	0
$118$	−386.053	−0.301179
$119$	82.9866	0.0639275
$120$	0	0
$121$	197.948	0.148721
$122$	−2462.08	−1.82710
$123$	0	0
$124$	−3100.65	−2.24554
$125$	0	0
$126$	0	0
$127$	1502.08	1.04951	0.524757	−	0.851252i	$-0.324156\pi$
$127$	1502.08	1.04951	0.524757	+	0.851252i	$0.324156\pi$
$128$	1979.99	1.36725
$129$	0	0
$130$	0	0
$131$	1574.71	1.05025	0.525127	−	0.851024i	$-0.324018\pi$
$131$	1574.71	1.05025	0.525127	+	0.851024i	$0.324018\pi$
$132$	0	0
$133$	−144.458	−0.0941814
$134$	−1199.26	−0.773140
$135$	0	0
$136$	−715.573	−0.451175
$137$	892.298	0.556454	0.278227	−	0.960515i	$-0.410253\pi$
$137$	892.298	0.556454	0.278227	+	0.960515i	$0.410253\pi$
$138$	0	0
$139$	652.706	0.398287	0.199143	−	0.979970i	$-0.436184\pi$
$139$	652.706	0.398287	0.199143	+	0.979970i	$0.436184\pi$
$140$	0	0
$141$	0	0
$142$	−3217.42	−1.90141
$143$	−3387.27	−1.98082
$144$	0	0
$145$	0	0
$146$	3864.52	2.19062
$147$	0	0
$148$	−5059.19	−2.80988
$149$	−3437.85	−1.89020	−0.945101	−	0.326779i	$-0.894037\pi$
$149$	−3437.85	−1.89020	−0.945101	+	0.326779i	$0.894037\pi$
$150$	0	0
$151$	1670.92	0.900513	0.450256	−	0.892899i	$-0.351333\pi$
$151$	1670.92	0.900513	0.450256	+	0.892899i	$0.351333\pi$
$152$	1245.63	0.664696
$153$	0	0
$154$	−1057.21	−0.553199
$155$	0	0
$156$	0	0
$157$	666.105	0.338605	0.169302	−	0.985564i	$-0.445849\pi$
$157$	666.105	0.338605	0.169302	+	0.985564i	$0.445849\pi$
$158$	−3327.46	−1.67544
$159$	0	0
$160$	0	0
$161$	−598.143	−0.292797
$162$	0	0
$163$	−889.255	−0.427312	−0.213656	−	0.976909i	$-0.568537\pi$
$163$	−889.255	−0.427312	−0.213656	+	0.976909i	$0.568537\pi$
$164$	−475.815	−0.226554
$165$	0	0
$166$	6541.59	3.05859
$167$	403.862	0.187137	0.0935683	−	0.995613i	$-0.470173\pi$
$167$	403.862	0.187137	0.0935683	+	0.995613i	$0.470173\pi$
$168$	0	0
$169$	5307.25	2.41568
$170$	0	0
$171$	0	0
$172$	−1042.93	−0.462342
$173$	236.061	0.103742	0.0518712	−	0.998654i	$-0.483481\pi$
$173$	236.061	0.103742	0.0518712	+	0.998654i	$0.483481\pi$
$174$	0	0
$175$	0	0
$176$	3723.90	1.59488
$177$	0	0
$178$	−5001.33	−2.10599
$179$	−2404.31	−1.00395	−0.501973	−	0.864883i	$-0.667392\pi$
$179$	−2404.31	−1.00395	−0.501973	+	0.864883i	$0.667392\pi$
$180$	0	0
$181$	−1218.41	−0.500354	−0.250177	−	0.968200i	$-0.580489\pi$
$181$	−1218.41	−0.500354	−0.250177	+	0.968200i	$0.580489\pi$
$182$	2342.18	0.953921
$183$	0	0
$184$	5157.64	2.06645
$185$	0	0
$186$	0	0
$187$	602.925	0.235777
$188$	1658.87	0.643539
$189$	0	0
$190$	0	0
$191$	−175.943	−0.0666533	−0.0333267	−	0.999445i	$-0.510610\pi$
$191$	−175.943	−0.0666533	−0.0333267	+	0.999445i	$0.510610\pi$
$192$	0	0
$193$	−2215.26	−0.826207	−0.413103	−	0.910684i	$-0.635555\pi$
$193$	−2215.26	−0.826207	−0.413103	+	0.910684i	$0.635555\pi$
$194$	−4093.55	−1.51495
$195$	0	0
$196$	−5413.22	−1.97275
$197$	−1054.08	−0.381218	−0.190609	−	0.981666i	$-0.561046\pi$
$197$	−1054.08	−0.381218	−0.190609	+	0.981666i	$0.561046\pi$
$198$	0	0
$199$	−3484.04	−1.24109	−0.620546	−	0.784170i	$-0.713089\pi$
$199$	−3484.04	−1.24109	−0.620546	+	0.784170i	$0.713089\pi$
$200$	0	0
$201$	0	0
$202$	−3278.49	−1.14195
$203$	−264.980	−0.0916155
$204$	0	0
$205$	0	0
$206$	773.426	0.261588
$207$	0	0
$208$	−8250.01	−2.75017
$209$	−1049.54	−0.347359
$210$	0	0
$211$	−4196.90	−1.36932	−0.684660	−	0.728863i	$-0.740049\pi$
$211$	−4196.90	−1.36932	−0.684660	+	0.728863i	$0.740049\pi$
$212$	4330.16	1.40282
$213$	0	0
$214$	−1752.66	−0.559858
$215$	0	0
$216$	0	0
$217$	−968.091	−0.302849
$218$	−5653.01	−1.75628
$219$	0	0
$220$	0	0
$221$	−1335.73	−0.406567
$222$	0	0
$223$	−3034.56	−0.911251	−0.455625	−	0.890172i	$-0.650584\pi$
$223$	−3034.56	−0.911251	−0.455625	+	0.890172i	$0.650584\pi$
$224$	−576.834	−0.172059
$225$	0	0
$226$	−1768.73	−0.520593
$227$	−3236.16	−0.946219	−0.473109	−	0.881004i	$-0.656868\pi$
$227$	−3236.16	−0.946219	−0.473109	+	0.881004i	$0.656868\pi$
$228$	0	0
$229$	−2974.56	−0.858360	−0.429180	−	0.903219i	$-0.641197\pi$
$229$	−2974.56	−0.858360	−0.429180	+	0.903219i	$0.641197\pi$
$230$	0	0
$231$	0	0
$232$	2284.86	0.646587
$233$	2927.42	0.823097	0.411549	−	0.911388i	$-0.364988\pi$
$233$	2927.42	0.823097	0.411549	+	0.911388i	$0.364988\pi$
$234$	0	0
$235$	0	0
$236$	1324.65	0.365370
$237$	0	0
$238$	−416.901	−0.113545
$239$	−2878.69	−0.779110	−0.389555	−	0.921003i	$-0.627371\pi$
$239$	−2878.69	−0.779110	−0.389555	+	0.921003i	$0.627371\pi$
$240$	0	0
$241$	−1117.58	−0.298713	−0.149357	−	0.988783i	$-0.547720\pi$
$241$	−1117.58	−0.298713	−0.149357	+	0.988783i	$0.547720\pi$
$242$	−994.435	−0.264152
$243$	0	0
$244$	8448.03	2.21651
$245$	0	0
$246$	0	0
$247$	2325.17	0.598976
$248$	8347.60	2.13739
$249$	0	0
$250$	0	0
$251$	4612.00	1.15979	0.579894	−	0.814692i	$-0.303094\pi$
$251$	4612.00	1.15979	0.579894	+	0.814692i	$0.303094\pi$
$252$	0	0
$253$	−4345.71	−1.07989
$254$	−7546.04	−1.86410
$255$	0	0
$256$	−8159.28	−1.99201
$257$	3787.72	0.919345	0.459672	−	0.888089i	$-0.347967\pi$
$257$	3787.72	0.919345	0.459672	+	0.888089i	$0.347967\pi$
$258$	0	0
$259$	−1579.59	−0.378960
$260$	0	0
$261$	0	0
$262$	−7910.89	−1.86541
$263$	4989.56	1.16985	0.584923	−	0.811089i	$-0.301125\pi$
$263$	4989.56	1.16985	0.584923	+	0.811089i	$0.301125\pi$
$264$	0	0
$265$	0	0
$266$	725.717	0.167280
$267$	0	0
$268$	4114.99	0.937922
$269$	5845.57	1.32495	0.662473	−	0.749086i	$-0.269507\pi$
$269$	5845.57	1.32495	0.662473	+	0.749086i	$0.269507\pi$
$270$	0	0
$271$	2766.09	0.620030	0.310015	−	0.950732i	$-0.399666\pi$
$271$	2766.09	0.620030	0.310015	+	0.950732i	$0.399666\pi$
$272$	1468.48	0.327351
$273$	0	0
$274$	−4482.65	−0.988346
$275$	0	0
$276$	0	0
$277$	1893.80	0.410785	0.205393	−	0.978680i	$-0.434153\pi$
$277$	1893.80	0.410785	0.205393	+	0.978680i	$0.434153\pi$
$278$	−3279.01	−0.707417
$279$	0	0
$280$	0	0
$281$	−4484.38	−0.952012	−0.476006	−	0.879442i	$-0.657916\pi$
$281$	−4484.38	−0.952012	−0.476006	+	0.879442i	$0.657916\pi$
$282$	0	0
$283$	5797.64	1.21779	0.608894	−	0.793251i	$-0.291613\pi$
$283$	5797.64	1.21779	0.608894	+	0.793251i	$0.291613\pi$
$284$	11039.8	2.30666
$285$	0	0
$286$	17016.7	3.51824
$287$	−148.560	−0.0305547
$288$	0	0
$289$	−4675.24	−0.951607
$290$	0	0
$291$	0	0
$292$	−13260.2	−2.65752
$293$	7910.26	1.57721	0.788605	−	0.614900i	$-0.210804\pi$
$293$	7910.26	1.57721	0.788605	+	0.614900i	$0.210804\pi$
$294$	0	0
$295$	0	0
$296$	13620.4	2.67456
$297$	0	0
$298$	17270.8	3.35728
$299$	9627.59	1.86213
$300$	0	0
$301$	−325.626	−0.0623546
$302$	−8394.21	−1.59945
$303$	0	0
$304$	−2556.24	−0.482272
$305$	0	0
$306$	0	0
$307$	4426.37	0.822887	0.411443	−	0.911435i	$-0.365025\pi$
$307$	4426.37	0.822887	0.411443	+	0.911435i	$0.365025\pi$
$308$	3627.58	0.671105
$309$	0	0
$310$	0	0
$311$	−9697.24	−1.76810	−0.884052	−	0.467389i	$-0.845195\pi$
$311$	−9697.24	−1.76810	−0.884052	+	0.467389i	$0.845195\pi$
$312$	0	0
$313$	4523.09	0.816806	0.408403	−	0.912802i	$-0.366086\pi$
$313$	4523.09	0.816806	0.408403	+	0.912802i	$0.366086\pi$
$314$	−3346.32	−0.601413
$315$	0	0
$316$	11417.4	2.03253
$317$	5322.21	0.942981	0.471491	−	0.881871i	$-0.343716\pi$
$317$	5322.21	0.942981	0.471491	+	0.881871i	$0.343716\pi$
$318$	0	0
$319$	−1925.17	−0.337895
$320$	0	0
$321$	0	0
$322$	3004.90	0.520051
$323$	−413.874	−0.0712958
$324$	0	0
$325$	0	0
$326$	4467.36	0.758970
$327$	0	0
$328$	1280.99	0.215643
$329$	517.934	0.0867922
$330$	0	0
$331$	−7833.11	−1.30075	−0.650373	−	0.759615i	$-0.725387\pi$
$331$	−7833.11	−1.30075	−0.650373	+	0.759615i	$0.725387\pi$
$332$	−22445.9	−3.71048
$333$	0	0
$334$	−2028.89	−0.332383
$335$	0	0
$336$	0	0
$337$	9928.08	1.60480	0.802399	−	0.596787i	$-0.203556\pi$
$337$	9928.08	1.60480	0.802399	+	0.596787i	$0.203556\pi$
$338$	−26662.1	−4.29061
$339$	0	0
$340$	0	0
$341$	−7033.49	−1.11696
$342$	0	0
$343$	−3536.14	−0.556658
$344$	2807.79	0.440075
$345$	0	0
$346$	−1185.90	−0.184262
$347$	−4040.33	−0.625062	−0.312531	−	0.949908i	$-0.601177\pi$
$347$	−4040.33	−0.625062	−0.312531	+	0.949908i	$0.601177\pi$
$348$	0	0
$349$	4799.72	0.736169	0.368085	−	0.929792i	$-0.380014\pi$
$349$	4799.72	0.736169	0.368085	+	0.929792i	$0.380014\pi$
$350$	0	0
$351$	0	0
$352$	−4190.88	−0.634588
$353$	−3189.48	−0.480903	−0.240451	−	0.970661i	$-0.577295\pi$
$353$	−3189.48	−0.480903	−0.240451	+	0.970661i	$0.577295\pi$
$354$	0	0
$355$	0	0
$356$	17160.9	2.55484
$357$	0	0
$358$	12078.5	1.78316
$359$	1159.83	0.170511	0.0852554	−	0.996359i	$-0.472829\pi$
$359$	1159.83	0.170511	0.0852554	+	0.996359i	$0.472829\pi$
$360$	0	0
$361$	−6138.55	−0.894963
$362$	6120.96	0.888703
$363$	0	0
$364$	−8036.62	−1.15724
$365$	0	0
$366$	0	0
$367$	11875.2	1.68904	0.844520	−	0.535524i	$-0.179886\pi$
$367$	11875.2	1.68904	0.844520	+	0.535524i	$0.179886\pi$
$368$	−10584.4	−1.49932
$369$	0	0
$370$	0	0
$371$	1351.97	0.189194
$372$	0	0
$373$	−2643.26	−0.366924	−0.183462	−	0.983027i	$-0.558730\pi$
$373$	−2643.26	−0.366924	−0.183462	+	0.983027i	$0.558730\pi$
$374$	−3028.92	−0.418774
$375$	0	0
$376$	−4466.02	−0.612546
$377$	4265.06	0.582657
$378$	0	0
$379$	10452.7	1.41667	0.708335	−	0.705876i	$-0.249447\pi$
$379$	10452.7	1.41667	0.708335	+	0.705876i	$0.249447\pi$
$380$	0	0
$381$	0	0
$382$	883.887	0.118386
$383$	10879.8	1.45153	0.725763	−	0.687945i	$-0.241487\pi$
$383$	10879.8	1.45153	0.725763	+	0.687945i	$0.241487\pi$
$384$	0	0
$385$	0	0
$386$	11128.8	1.46747
$387$	0	0
$388$	14046.1	1.83784
$389$	−1677.66	−0.218665	−0.109333	−	0.994005i	$-0.534871\pi$
$389$	−1677.66	−0.218665	−0.109333	+	0.994005i	$0.534871\pi$
$390$	0	0
$391$	−1713.68	−0.221649
$392$	14573.5	1.87774
$393$	0	0
$394$	5295.38	0.677100
$395$	0	0
$396$	0	0
$397$	13147.5	1.66210	0.831049	−	0.556199i	$-0.187741\pi$
$397$	13147.5	1.66210	0.831049	+	0.556199i	$0.187741\pi$
$398$	17502.8	2.20437
$399$	0	0
$400$	0	0
$401$	6521.91	0.812191	0.406095	−	0.913831i	$-0.366890\pi$
$401$	6521.91	0.812191	0.406095	+	0.913831i	$0.366890\pi$
$402$	0	0
$403$	15582.2	1.92606
$404$	11249.4	1.38534
$405$	0	0
$406$	1331.18	0.162723
$407$	−11476.2	−1.39768
$408$	0	0
$409$	5642.67	0.682181	0.341091	−	0.940030i	$-0.389204\pi$
$409$	5642.67	0.682181	0.341091	+	0.940030i	$0.389204\pi$
$410$	0	0
$411$	0	0
$412$	−2653.83	−0.317342
$413$	413.584	0.0492764
$414$	0	0
$415$	0	0
$416$	9284.59	1.09427
$417$	0	0
$418$	5272.57	0.616961
$419$	9621.73	1.12184	0.560922	−	0.827869i	$-0.310447\pi$
$419$	9621.73	1.12184	0.560922	+	0.827869i	$0.310447\pi$
$420$	0	0
$421$	10030.3	1.16115	0.580577	−	0.814205i	$-0.302827\pi$
$421$	10030.3	1.16115	0.580577	+	0.814205i	$0.302827\pi$
$422$	21084.0	2.43212
$423$	0	0
$424$	−11657.7	−1.33525
$425$	0	0
$426$	0	0
$427$	2637.66	0.298935
$428$	6013.85	0.679183
$429$	0	0
$430$	0	0
$431$	−3867.23	−0.432200	−0.216100	−	0.976371i	$-0.569334\pi$
$431$	−3867.23	−0.432200	−0.216100	+	0.976371i	$0.569334\pi$
$432$	0	0
$433$	−2345.27	−0.260292	−0.130146	−	0.991495i	$-0.541545\pi$
$433$	−2345.27	−0.260292	−0.130146	+	0.991495i	$0.541545\pi$
$434$	4863.41	0.537906
$435$	0	0
$436$	19396.9	2.13061
$437$	2983.08	0.326545
$438$	0	0
$439$	−14117.9	−1.53487	−0.767437	−	0.641125i	$-0.778468\pi$
$439$	−14117.9	−1.53487	−0.767437	+	0.641125i	$0.778468\pi$
$440$	0	0
$441$	0	0
$442$	6710.34	0.722123
$443$	−17796.6	−1.90867	−0.954337	−	0.298731i	$-0.903437\pi$
$443$	−17796.6	−1.90867	−0.954337	+	0.298731i	$0.903437\pi$
$444$	0	0
$445$	0	0
$446$	15244.7	1.61852
$447$	0	0
$448$	−1202.61	−0.126826
$449$	11518.0	1.21062	0.605310	−	0.795990i	$-0.293049\pi$
$449$	11518.0	1.21062	0.605310	+	0.795990i	$0.293049\pi$
$450$	0	0
$451$	−1079.33	−0.112691
$452$	6068.97	0.631550
$453$	0	0
$454$	16257.5	1.68063
$455$	0	0
$456$	0	0
$457$	9857.71	1.00902	0.504512	−	0.863405i	$-0.331672\pi$
$457$	9857.71	1.00902	0.504512	+	0.863405i	$0.331672\pi$
$458$	14943.3	1.52458
$459$	0	0
$460$	0	0
$461$	16081.6	1.62472	0.812358	−	0.583159i	$-0.198184\pi$
$461$	16081.6	1.62472	0.812358	+	0.583159i	$0.198184\pi$
$462$	0	0
$463$	18895.8	1.89668	0.948338	−	0.317263i	$-0.102764\pi$
$463$	18895.8	1.89668	0.948338	+	0.317263i	$0.102764\pi$
$464$	−4688.92	−0.469133
$465$	0	0
$466$	−14706.5	−1.46194
$467$	−10368.7	−1.02742	−0.513712	−	0.857963i	$-0.671730\pi$
$467$	−10368.7	−1.02742	−0.513712	+	0.857963i	$0.671730\pi$
$468$	0	0
$469$	1284.79	0.126495
$470$	0	0
$471$	0	0
$472$	−3566.23	−0.347774
$473$	−2365.77	−0.229976
$474$	0	0
$475$	0	0
$476$	1430.50	0.137745
$477$	0	0
$478$	14461.7	1.38382
$479$	9554.02	0.911345	0.455672	−	0.890148i	$-0.349399\pi$
$479$	9554.02	0.911345	0.455672	+	0.890148i	$0.349399\pi$
$480$	0	0
$481$	25424.7	2.41012
$482$	5614.42	0.530559
$483$	0	0
$484$	3412.17	0.320451
$485$	0	0
$486$	0	0
$487$	−17514.2	−1.62966	−0.814831	−	0.579699i	$-0.803170\pi$
$487$	−17514.2	−1.62966	−0.814831	+	0.579699i	$0.803170\pi$
$488$	−22743.8	−2.10977
$489$	0	0
$490$	0	0
$491$	3866.54	0.355386	0.177693	−	0.984086i	$-0.443137\pi$
$491$	3866.54	0.355386	0.177693	+	0.984086i	$0.443137\pi$
$492$	0	0
$493$	−759.169	−0.0693534
$494$	−11681.0	−1.06387
$495$	0	0
$496$	−17130.7	−1.55079
$497$	3446.87	0.311093
$498$	0	0
$499$	4855.96	0.435637	0.217818	−	0.975989i	$-0.430106\pi$
$499$	4855.96	0.435637	0.217818	+	0.975989i	$0.430106\pi$
$500$	0	0
$501$	0	0
$502$	−23169.3	−2.05996
$503$	−5481.79	−0.485927	−0.242963	−	0.970035i	$-0.578119\pi$
$503$	−5481.79	−0.485927	−0.242963	+	0.970035i	$0.578119\pi$
$504$	0	0
$505$	0	0
$506$	21831.6	1.91805
$507$	0	0
$508$	25892.4	2.26140
$509$	20427.1	1.77881	0.889405	−	0.457120i	$-0.151119\pi$
$509$	20427.1	1.77881	0.889405	+	0.457120i	$0.151119\pi$
$510$	0	0
$511$	−4140.12	−0.358411
$512$	25150.0	2.17086
$513$	0	0
$514$	−19028.4	−1.63289
$515$	0	0
$516$	0	0
$517$	3762.96	0.320106
$518$	7935.39	0.673091
$519$	0	0
$520$	0	0
$521$	5768.55	0.485076	0.242538	−	0.970142i	$-0.422020\pi$
$521$	5768.55	0.485076	0.242538	+	0.970142i	$0.422020\pi$
$522$	0	0
$523$	1753.72	0.146625	0.0733126	−	0.997309i	$-0.476643\pi$
$523$	1753.72	0.146625	0.0733126	+	0.997309i	$0.476643\pi$
$524$	27144.4	2.26299
$525$	0	0
$526$	−25066.1	−2.07782
$527$	−2773.58	−0.229258
$528$	0	0
$529$	184.728	0.0151827
$530$	0	0
$531$	0	0
$532$	−2490.13	−0.202934
$533$	2391.18	0.194322
$534$	0	0
$535$	0	0
$536$	−11078.4	−0.892751
$537$	0	0
$538$	−29366.4	−2.35330
$539$	−12279.3	−0.981274
$540$	0	0
$541$	7146.18	0.567908	0.283954	−	0.958838i	$-0.408354\pi$
$541$	7146.18	0.567908	0.283954	+	0.958838i	$0.408354\pi$
$542$	−13896.1	−1.10127
$543$	0	0
$544$	−1652.63	−0.130250
$545$	0	0
$546$	0	0
$547$	−7407.89	−0.579047	−0.289523	−	0.957171i	$-0.593497\pi$
$547$	−7407.89	−0.579047	−0.289523	+	0.957171i	$0.593497\pi$
$548$	15381.1	1.19900
$549$	0	0
$550$	0	0
$551$	1321.52	0.102175
$552$	0	0
$553$	3564.76	0.274121
$554$	−9513.91	−0.729616
$555$	0	0
$556$	11251.1	0.858192
$557$	−11116.3	−0.845626	−0.422813	−	0.906217i	$-0.638957\pi$
$557$	−11116.3	−0.845626	−0.422813	+	0.906217i	$0.638957\pi$
$558$	0	0
$559$	5241.20	0.396564
$560$	0	0
$561$	0	0
$562$	22528.2	1.69092
$563$	−11206.6	−0.838901	−0.419450	−	0.907778i	$-0.637777\pi$
$563$	−11206.6	−0.838901	−0.419450	+	0.907778i	$0.637777\pi$
$564$	0	0
$565$	0	0
$566$	−29125.7	−2.16298
$567$	0	0
$568$	−29721.5	−2.19557
$569$	−15873.8	−1.16953	−0.584765	−	0.811203i	$-0.698813\pi$
$569$	−15873.8	−1.16953	−0.584765	+	0.811203i	$0.698813\pi$
$570$	0	0
$571$	−2388.45	−0.175050	−0.0875250	−	0.996162i	$-0.527896\pi$
$571$	−2388.45	−0.175050	−0.0875250	+	0.996162i	$0.527896\pi$
$572$	−58388.7	−4.26810
$573$	0	0
$574$	746.320	0.0542697
$575$	0	0
$576$	0	0
$577$	10429.0	0.752455	0.376227	−	0.926527i	$-0.377221\pi$
$577$	10429.0	0.752455	0.376227	+	0.926527i	$0.377221\pi$
$578$	23487.1	1.69020
$579$	0	0
$580$	0	0
$581$	−7008.09	−0.500421
$582$	0	0
$583$	9822.50	0.697781
$584$	35699.2	2.52953
$585$	0	0
$586$	−39738.9	−2.80136
$587$	−3148.05	−0.221352	−0.110676	−	0.993857i	$-0.535302\pi$
$587$	−3148.05	−0.221352	−0.110676	+	0.993857i	$0.535302\pi$
$588$	0	0
$589$	4828.10	0.337756
$590$	0	0
$591$	0	0
$592$	−27951.4	−1.94053
$593$	−12123.4	−0.839542	−0.419771	−	0.907630i	$-0.637890\pi$
$593$	−12123.4	−0.839542	−0.419771	+	0.907630i	$0.637890\pi$
$594$	0	0
$595$	0	0
$596$	−59260.6	−4.07283
$597$	0	0
$598$	−48366.2	−3.30743
$599$	15842.8	1.08067	0.540333	−	0.841451i	$-0.318298\pi$
$599$	15842.8	1.08067	0.540333	+	0.841451i	$0.318298\pi$
$600$	0	0
$601$	−25690.1	−1.74363	−0.871815	−	0.489836i	$-0.837057\pi$
$601$	−25690.1	−1.74363	−0.871815	+	0.489836i	$0.837057\pi$
$602$	1635.85	0.110751
$603$	0	0
$604$	28802.8	1.94034
$605$	0	0
$606$	0	0
$607$	14523.3	0.971139	0.485570	−	0.874198i	$-0.338612\pi$
$607$	14523.3	0.971139	0.485570	+	0.874198i	$0.338612\pi$
$608$	2876.81	0.191891
$609$	0	0
$610$	0	0
$611$	−8336.56	−0.551982
$612$	0	0
$613$	26426.7	1.74121	0.870606	−	0.491981i	$-0.163727\pi$
$613$	26426.7	1.74121	0.870606	+	0.491981i	$0.163727\pi$
$614$	−22236.8	−1.46157
$615$	0	0
$616$	−9766.19	−0.638784
$617$	3757.77	0.245190	0.122595	−	0.992457i	$-0.460878\pi$
$617$	3757.77	0.245190	0.122595	+	0.992457i	$0.460878\pi$
$618$	0	0
$619$	−23480.5	−1.52465	−0.762327	−	0.647192i	$-0.775943\pi$
$619$	−23480.5	−1.52465	−0.762327	+	0.647192i	$0.775943\pi$
$620$	0	0
$621$	0	0
$622$	48716.1	3.14042
$623$	5357.99	0.344564
$624$	0	0
$625$	0	0
$626$	−22722.7	−1.45077
$627$	0	0
$628$	11482.1	0.729595
$629$	−4525.52	−0.286875
$630$	0	0
$631$	8654.12	0.545983	0.272991	−	0.962016i	$-0.411987\pi$
$631$	8654.12	0.545983	0.272991	+	0.962016i	$0.411987\pi$
$632$	−30738.0	−1.93464
$633$	0	0
$634$	−26737.2	−1.67488
$635$	0	0
$636$	0	0
$637$	27203.9	1.69208
$638$	9671.48	0.600153
$639$	0	0
$640$	0	0
$641$	−19177.6	−1.18170	−0.590850	−	0.806781i	$-0.701208\pi$
$641$	−19177.6	−1.18170	−0.590850	+	0.806781i	$0.701208\pi$
$642$	0	0
$643$	−9835.57	−0.603230	−0.301615	−	0.953430i	$-0.597526\pi$
$643$	−9835.57	−0.603230	−0.301615	+	0.953430i	$0.597526\pi$
$644$	−10310.6	−0.630892
$645$	0	0
$646$	2079.18	0.126632
$647$	7621.18	0.463090	0.231545	−	0.972824i	$-0.425622\pi$
$647$	7621.18	0.463090	0.231545	+	0.972824i	$0.425622\pi$
$648$	0	0
$649$	3004.82	0.181741
$650$	0	0
$651$	0	0
$652$	−15328.7	−0.920733
$653$	−4396.32	−0.263463	−0.131731	−	0.991285i	$-0.542054\pi$
$653$	−4396.32	−0.263463	−0.131731	+	0.991285i	$0.542054\pi$
$654$	0	0
$655$	0	0
$656$	−2628.82	−0.156460
$657$	0	0
$658$	−2601.95	−0.154156
$659$	−10820.3	−0.639601	−0.319801	−	0.947485i	$-0.603616\pi$
$659$	−10820.3	−0.639601	−0.319801	+	0.947485i	$0.603616\pi$
$660$	0	0
$661$	−28915.5	−1.70149	−0.850744	−	0.525581i	$-0.823848\pi$
$661$	−28915.5	−1.70149	−0.850744	+	0.525581i	$0.823848\pi$
$662$	39351.3	2.31032
$663$	0	0
$664$	60429.0	3.53178
$665$	0	0
$666$	0	0
$667$	5471.87	0.317649
$668$	6961.65	0.403225
$669$	0	0
$670$	0	0
$671$	19163.4	1.10253
$672$	0	0
$673$	−5061.16	−0.289886	−0.144943	−	0.989440i	$-0.546300\pi$
$673$	−5061.16	−0.289886	−0.144943	+	0.989440i	$0.546300\pi$
$674$	−49875.8	−2.85036
$675$	0	0
$676$	91484.6	5.20509
$677$	12059.3	0.684601	0.342300	−	0.939591i	$-0.388794\pi$
$677$	12059.3	0.684601	0.342300	+	0.939591i	$0.388794\pi$
$678$	0	0
$679$	4385.48	0.247863
$680$	0	0
$681$	0	0
$682$	35334.2	1.98390
$683$	−6110.33	−0.342321	−0.171160	−	0.985243i	$-0.554752\pi$
$683$	−6110.33	−0.342321	−0.171160	+	0.985243i	$0.554752\pi$
$684$	0	0
$685$	0	0
$686$	17764.5	0.988708
$687$	0	0
$688$	−5762.06	−0.319297
$689$	−21761.0	−1.20323
$690$	0	0
$691$	27209.7	1.49798	0.748991	−	0.662580i	$-0.230538\pi$
$691$	27209.7	1.49798	0.748991	+	0.662580i	$0.230538\pi$
$692$	4069.15	0.223535
$693$	0	0
$694$	20297.5	1.11020
$695$	0	0
$696$	0	0
$697$	−425.624	−0.0231300
$698$	−24112.4	−1.30755
$699$	0	0
$700$	0	0
$701$	30424.7	1.63927	0.819633	−	0.572888i	$-0.194177\pi$
$701$	30424.7	1.63927	0.819633	+	0.572888i	$0.194177\pi$
$702$	0	0
$703$	7877.77	0.422640
$704$	−8737.37	−0.467759
$705$	0	0
$706$	16023.0	0.854156
$707$	3512.30	0.186837
$708$	0	0
$709$	1846.41	0.0978045	0.0489022	−	0.998804i	$-0.484428\pi$
$709$	1846.41	0.0978045	0.0489022	+	0.998804i	$0.484428\pi$
$710$	0	0
$711$	0	0
$712$	−46200.6	−2.43180
$713$	19991.2	1.05004
$714$	0	0
$715$	0	0
$716$	−41444.6	−2.16321
$717$	0	0
$718$	−5826.64	−0.302853
$719$	−13301.3	−0.689923	−0.344961	−	0.938617i	$-0.612108\pi$
$719$	−13301.3	−0.689923	−0.344961	+	0.938617i	$0.612108\pi$
$720$	0	0
$721$	−828.583	−0.0427989
$722$	30838.3	1.58959
$723$	0	0
$724$	−21002.6	−1.07812
$725$	0	0
$726$	0	0
$727$	−25915.1	−1.32206	−0.661031	−	0.750358i	$-0.729881\pi$
$727$	−25915.1	−1.32206	−0.661031	+	0.750358i	$0.729881\pi$
$728$	21636.3	1.10150
$729$	0	0
$730$	0	0
$731$	−932.918	−0.0472028
$732$	0	0
$733$	22247.8	1.12106	0.560532	−	0.828132i	$-0.310597\pi$
$733$	22247.8	1.12106	0.560532	+	0.828132i	$0.310597\pi$
$734$	−59657.3	−2.99999
$735$	0	0
$736$	11911.7	0.596563
$737$	9334.41	0.466536
$738$	0	0
$739$	−23675.6	−1.17851	−0.589256	−	0.807947i	$-0.700579\pi$
$739$	−23675.6	−1.17851	−0.589256	+	0.807947i	$0.700579\pi$
$740$	0	0
$741$	0	0
$742$	−6791.91	−0.336036
$743$	33555.1	1.65682	0.828411	−	0.560120i	$-0.189245\pi$
$743$	33555.1	1.65682	0.828411	+	0.560120i	$0.189245\pi$
$744$	0	0
$745$	0	0
$746$	13279.0	0.651713
$747$	0	0
$748$	10393.0	0.508030
$749$	1877.65	0.0915993
$750$	0	0
$751$	21889.8	1.06361	0.531804	−	0.846868i	$-0.321514\pi$
$751$	21889.8	1.06361	0.531804	+	0.846868i	$0.321514\pi$
$752$	9165.04	0.444434
$753$	0	0
$754$	−21426.4	−1.03489
$755$	0	0
$756$	0	0
$757$	24362.3	1.16970	0.584851	−	0.811141i	$-0.301153\pi$
$757$	24362.3	1.16970	0.584851	+	0.811141i	$0.301153\pi$
$758$	−52511.2	−2.51622
$759$	0	0
$760$	0	0
$761$	27685.3	1.31878	0.659391	−	0.751801i	$-0.270814\pi$
$761$	27685.3	1.31878	0.659391	+	0.751801i	$0.270814\pi$
$762$	0	0
$763$	6056.14	0.287349
$764$	−3032.85	−0.143619
$765$	0	0
$766$	−54657.2	−2.57813
$767$	−6656.96	−0.313388
$768$	0	0
$769$	−6996.11	−0.328070	−0.164035	−	0.986454i	$-0.552451\pi$
$769$	−6996.11	−0.328070	−0.164035	+	0.986454i	$0.552451\pi$
$770$	0	0
$771$	0	0
$772$	−38185.9	−1.78023
$773$	−19650.7	−0.914342	−0.457171	−	0.889379i	$-0.651137\pi$
$773$	−19650.7	−0.914342	−0.457171	+	0.889379i	$0.651137\pi$
$774$	0	0
$775$	0	0
$776$	−37814.9	−1.74932
$777$	0	0
$778$	8428.08	0.388382
$779$	740.901	0.0340764
$780$	0	0
$781$	25042.6	1.14737
$782$	8609.05	0.393681
$783$	0	0
$784$	−29907.4	−1.36240
$785$	0	0
$786$	0	0
$787$	1312.96	0.0594689	0.0297344	−	0.999558i	$-0.490534\pi$
$787$	1312.96	0.0594689	0.0297344	+	0.999558i	$0.490534\pi$
$788$	−18169.8	−0.821413
$789$	0	0
$790$	0	0
$791$	1894.86	0.0851752
$792$	0	0
$793$	−42455.1	−1.90117
$794$	−66049.1	−2.95214
$795$	0	0
$796$	−60056.8	−2.67419
$797$	−2391.63	−0.106294	−0.0531468	−	0.998587i	$-0.516925\pi$
$797$	−2391.63	−0.106294	−0.0531468	+	0.998587i	$0.516925\pi$
$798$	0	0
$799$	1483.88	0.0657022
$800$	0	0
$801$	0	0
$802$	−32764.2	−1.44257
$803$	−30079.3	−1.32189
$804$	0	0
$805$	0	0
$806$	−78280.4	−3.42098
$807$	0	0
$808$	−30285.7	−1.31862
$809$	7059.58	0.306801	0.153400	−	0.988164i	$-0.450978\pi$
$809$	7059.58	0.306801	0.153400	+	0.988164i	$0.450978\pi$
$810$	0	0
$811$	12699.2	0.549849	0.274925	−	0.961466i	$-0.411347\pi$
$811$	12699.2	0.549849	0.274925	+	0.961466i	$0.411347\pi$
$812$	−4567.64	−0.197405
$813$	0	0
$814$	57653.2	2.48248
$815$	0	0
$816$	0	0
$817$	1623.97	0.0695417
$818$	−28347.1	−1.21166
$819$	0	0
$820$	0	0
$821$	−1624.01	−0.0690356	−0.0345178	−	0.999404i	$-0.510990\pi$
$821$	−1624.01	−0.0690356	−0.0345178	+	0.999404i	$0.510990\pi$
$822$	0	0
$823$	21151.7	0.895870	0.447935	−	0.894066i	$-0.352160\pi$
$823$	21151.7	0.895870	0.447935	+	0.894066i	$0.352160\pi$
$824$	7144.66	0.302058
$825$	0	0
$826$	−2077.73	−0.0875223
$827$	18269.2	0.768178	0.384089	−	0.923296i	$-0.374516\pi$
$827$	18269.2	0.768178	0.384089	+	0.923296i	$0.374516\pi$
$828$	0	0
$829$	19893.4	0.833446	0.416723	−	0.909034i	$-0.363179\pi$
$829$	19893.4	0.833446	0.416723	+	0.909034i	$0.363179\pi$
$830$	0	0
$831$	0	0
$832$	19357.0	0.806591
$833$	−4842.21	−0.201408
$834$	0	0
$835$	0	0
$836$	−18091.6	−0.748457
$837$	0	0
$838$	−48336.8	−1.99256
$839$	11705.8	0.481678	0.240839	−	0.970565i	$-0.422577\pi$
$839$	11705.8	0.481678	0.240839	+	0.970565i	$0.422577\pi$
$840$	0	0
$841$	−21964.9	−0.900608
$842$	−50389.2	−2.06238
$843$	0	0
$844$	−72344.8	−2.95048
$845$	0	0
$846$	0	0
$847$	1065.35	0.0432183
$848$	23923.6	0.968797
$849$	0	0
$850$	0	0
$851$	32618.6	1.31393
$852$	0	0
$853$	1058.28	0.0424792	0.0212396	−	0.999774i	$-0.493239\pi$
$853$	1058.28	0.0424792	0.0212396	+	0.999774i	$0.493239\pi$
$854$	−13250.8	−0.530953
$855$	0	0
$856$	−16190.5	−0.646473
$857$	24035.3	0.958030	0.479015	−	0.877807i	$-0.340994\pi$
$857$	24035.3	0.958030	0.479015	+	0.877807i	$0.340994\pi$
$858$	0	0
$859$	−35094.2	−1.39395	−0.696973	−	0.717098i	$-0.745470\pi$
$859$	−35094.2	−1.39395	−0.696973	+	0.717098i	$0.745470\pi$
$860$	0	0
$861$	0	0
$862$	19427.9	0.767651
$863$	41958.9	1.65504	0.827518	−	0.561439i	$-0.189752\pi$
$863$	41958.9	1.65504	0.827518	+	0.561439i	$0.189752\pi$
$864$	0	0
$865$	0	0
$866$	11782.0	0.462318
$867$	0	0
$868$	−16687.6	−0.652552
$869$	25899.1	1.01101
$870$	0	0
$871$	−20679.7	−0.804483
$872$	−52220.6	−2.02800
$873$	0	0
$874$	−14986.1	−0.579993
$875$	0	0
$876$	0	0
$877$	−19611.1	−0.755097	−0.377548	−	0.925990i	$-0.623233\pi$
$877$	−19611.1	−0.755097	−0.377548	+	0.925990i	$0.623233\pi$
$878$	70924.1	2.72617
$879$	0	0
$880$	0	0
$881$	−10465.2	−0.400204	−0.200102	−	0.979775i	$-0.564127\pi$
$881$	−10465.2	−0.400204	−0.200102	+	0.979775i	$0.564127\pi$
$882$	0	0
$883$	10812.2	0.412074	0.206037	−	0.978544i	$-0.433943\pi$
$883$	10812.2	0.412074	0.206037	+	0.978544i	$0.433943\pi$
$884$	−23025.0	−0.876033
$885$	0	0
$886$	89405.1	3.39009
$887$	36000.6	1.36277	0.681387	−	0.731923i	$-0.261377\pi$
$887$	36000.6	1.36277	0.681387	+	0.731923i	$0.261377\pi$
$888$	0	0
$889$	8084.17	0.304988
$890$	0	0
$891$	0	0
$892$	−52308.7	−1.96348
$893$	−2583.06	−0.0967960
$894$	0	0
$895$	0	0
$896$	10656.3	0.397322
$897$	0	0
$898$	−57863.1	−2.15024
$899$	8856.18	0.328554
$900$	0	0
$901$	3873.40	0.143220
$902$	5422.26	0.200157
$903$	0	0
$904$	−16338.9	−0.601134
$905$	0	0
$906$	0	0
$907$	5211.64	0.190794	0.0953968	−	0.995439i	$-0.469588\pi$
$907$	5211.64	0.190794	0.0953968	+	0.995439i	$0.469588\pi$
$908$	−55783.9	−2.03883
$909$	0	0
$910$	0	0
$911$	16733.1	0.608555	0.304277	−	0.952584i	$-0.401585\pi$
$911$	16733.1	0.608555	0.304277	+	0.952584i	$0.401585\pi$
$912$	0	0
$913$	−50916.1	−1.84565
$914$	−49522.3	−1.79218
$915$	0	0
$916$	−51274.5	−1.84952
$917$	8475.05	0.305203
$918$	0	0
$919$	16722.8	0.600255	0.300127	−	0.953899i	$-0.402971\pi$
$919$	16722.8	0.600255	0.300127	+	0.953899i	$0.402971\pi$
$920$	0	0
$921$	0	0
$922$	−80789.2	−2.88574
$923$	−55480.0	−1.97849
$924$	0	0
$925$	0	0
$926$	−94926.8	−3.36878
$927$	0	0
$928$	5276.93	0.186663
$929$	35344.5	1.24824	0.624120	−	0.781328i	$-0.285457\pi$
$929$	35344.5	1.24824	0.624120	+	0.781328i	$0.285457\pi$
$930$	0	0
$931$	8429.05	0.296725
$932$	50461.9	1.77353
$933$	0	0
$934$	52089.5	1.82486
$935$	0	0
$936$	0	0
$937$	2015.96	0.0702867	0.0351433	−	0.999382i	$-0.488811\pi$
$937$	2015.96	0.0702867	0.0351433	+	0.999382i	$0.488811\pi$
$938$	−6454.41	−0.224674
$939$	0	0
$940$	0	0
$941$	−39880.3	−1.38157	−0.690787	−	0.723058i	$-0.742736\pi$
$941$	−39880.3	−1.38157	−0.690787	+	0.723058i	$0.742736\pi$
$942$	0	0
$943$	3067.77	0.105939
$944$	7318.52	0.252328
$945$	0	0
$946$	11885.0	0.408471
$947$	−35010.2	−1.20135	−0.600675	−	0.799493i	$-0.705102\pi$
$947$	−35010.2	−1.20135	−0.600675	+	0.799493i	$0.705102\pi$
$948$	0	0
$949$	66638.5	2.27943
$950$	0	0
$951$	0	0
$952$	−3851.19	−0.131111
$953$	49213.8	1.67281	0.836407	−	0.548109i	$-0.184652\pi$
$953$	49213.8	1.67281	0.836407	+	0.548109i	$0.184652\pi$
$954$	0	0
$955$	0	0
$956$	−49622.0	−1.67876
$957$	0	0
$958$	−47996.6	−1.61868
$959$	4802.33	0.161705
$960$	0	0
$961$	2564.57	0.0860855
$962$	−127726.	−4.28073
$963$	0	0
$964$	−19264.5	−0.643640
$965$	0	0
$966$	0	0
$967$	−1219.34	−0.0405495	−0.0202747	−	0.999794i	$-0.506454\pi$
$967$	−1219.34	−0.0405495	−0.0202747	+	0.999794i	$0.506454\pi$
$968$	−9186.26	−0.305018
$969$	0	0
$970$	0	0
$971$	1660.33	0.0548740	0.0274370	−	0.999624i	$-0.491265\pi$
$971$	1660.33	0.0548740	0.0274370	+	0.999624i	$0.491265\pi$
$972$	0	0
$973$	3512.85	0.115742
$974$	87986.4	2.89452
$975$	0	0
$976$	46674.3	1.53075
$977$	−47307.7	−1.54914	−0.774569	−	0.632489i	$-0.782033\pi$
$977$	−47307.7	−1.54914	−0.774569	+	0.632489i	$0.782033\pi$
$978$	0	0
$979$	38927.5	1.27082
$980$	0	0
$981$	0	0
$982$	−19424.4	−0.631218
$983$	41203.0	1.33690	0.668450	−	0.743757i	$-0.266958\pi$
$983$	41203.0	1.33690	0.668450	+	0.743757i	$0.266958\pi$
$984$	0	0
$985$	0	0
$986$	3813.84	0.123182
$987$	0	0
$988$	40080.5	1.29062
$989$	6724.20	0.216195
$990$	0	0
$991$	−29805.3	−0.955395	−0.477698	−	0.878524i	$-0.658529\pi$
$991$	−29805.3	−0.955395	−0.477698	+	0.878524i	$0.658529\pi$
$992$	19279.0	0.617044
$993$	0	0
$994$	−17316.1	−0.552548
$995$	0	0
$996$	0	0
$997$	39832.8	1.26531	0.632657	−	0.774433i	$-0.281964\pi$
$997$	39832.8	1.26531	0.632657	+	0.774433i	$0.281964\pi$
$998$	−24395.0	−0.773756
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2025.4.a.bl.1.2		16
3.2	odd	2		2025.4.a.bk.1.15		16
5.2	odd	4		405.4.b.f.244.2		16
5.3	odd	4		405.4.b.f.244.15		16
5.4	even	2	inner	2025.4.a.bl.1.15		16
9.2	odd	6		225.4.e.g.76.2		32
9.5	odd	6		225.4.e.g.151.2		32
15.2	even	4		405.4.b.e.244.15		16
15.8	even	4		405.4.b.e.244.2		16
15.14	odd	2		2025.4.a.bk.1.2		16
45.2	even	12		45.4.j.a.4.15	yes	32
45.7	odd	12		135.4.j.a.64.2		32
45.13	odd	12		135.4.j.a.19.2		32
45.14	odd	6		225.4.e.g.151.15		32
45.22	odd	12		135.4.j.a.19.15		32
45.23	even	12		45.4.j.a.34.15	yes	32
45.29	odd	6		225.4.e.g.76.15		32
45.32	even	12		45.4.j.a.34.2	yes	32
45.38	even	12		45.4.j.a.4.2	✓	32
45.43	odd	12		135.4.j.a.64.15		32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
45.4.j.a.4.2	✓	32	45.38	even	12
45.4.j.a.4.15	yes	32	45.2	even	12
45.4.j.a.34.2	yes	32	45.32	even	12
45.4.j.a.34.15	yes	32	45.23	even	12
135.4.j.a.19.2		32	45.13	odd	12
135.4.j.a.19.15		32	45.22	odd	12
135.4.j.a.64.2		32	45.7	odd	12
135.4.j.a.64.15		32	45.43	odd	12
225.4.e.g.76.2		32	9.2	odd	6
225.4.e.g.76.15		32	45.29	odd	6
225.4.e.g.151.2		32	9.5	odd	6
225.4.e.g.151.15		32	45.14	odd	6
405.4.b.e.244.2		16	15.8	even	4
405.4.b.e.244.15		16	15.2	even	4
405.4.b.f.244.2		16	5.2	odd	4
405.4.b.f.244.15		16	5.3	odd	4
2025.4.a.bk.1.2		16	15.14	odd	2
2025.4.a.bk.1.15		16	3.2	odd	2
2025.4.a.bl.1.2		16	1.1	even	1	trivial
2025.4.a.bl.1.15		16	5.4	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-5.02371 q^{2} +17.2377 q^{4} +5.38197 q^{7} -46.4074 q^{8} +O(q^{10})\)	\(q-5.02371 q^{2} +17.2377 q^{4} +5.38197 q^{7} -46.4074 q^{8} +39.1018 q^{11} -86.6271 q^{13} -27.0375 q^{14} +95.2360 q^{16} +15.4194 q^{17} -26.8412 q^{19} -196.436 q^{22} -111.138 q^{23} +435.189 q^{26} +92.7727 q^{28} -49.2347 q^{29} -179.877 q^{31} -107.179 q^{32} -77.4624 q^{34} -293.496 q^{37} +134.842 q^{38} -27.6032 q^{41} -60.5030 q^{43} +674.023 q^{44} +558.327 q^{46} +96.2351 q^{47} -314.034 q^{49} -1493.25 q^{52} +251.203 q^{53} -249.763 q^{56} +247.341 q^{58} +76.8462 q^{59} +490.091 q^{61} +903.648 q^{62} -223.452 q^{64} +238.721 q^{67} +265.794 q^{68} +640.447 q^{71} -769.257 q^{73} +1474.44 q^{74} -462.679 q^{76} +210.445 q^{77} +662.352 q^{79} +138.670 q^{82} -1302.14 q^{83} +303.950 q^{86} -1814.61 q^{88} +995.544 q^{89} -466.224 q^{91} -1915.77 q^{92} -483.457 q^{94} +814.846 q^{97} +1577.62 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 54 q^{4}+O(q^{10}) \) 16 * q + 54 * q^4	\( 16 q + 54 q^{4} + 90 q^{11} + 102 q^{14} + 146 q^{16} + 4 q^{19} + 468 q^{26} + 516 q^{29} + 38 q^{31} + 212 q^{34} + 576 q^{41} + 1644 q^{44} - 290 q^{46} - 4 q^{49} + 2430 q^{56} + 2202 q^{59} + 20 q^{61} - 322 q^{64} + 2952 q^{71} + 4080 q^{74} - 396 q^{76} - 218 q^{79} + 6108 q^{86} + 4074 q^{89} - 942 q^{91} - 1078 q^{94}+O(q^{100}) \) 16 * q + 54 * q^4 + 90 * q^11 + 102 * q^14 + 146 * q^16 + 4 * q^19 + 468 * q^26 + 516 * q^29 + 38 * q^31 + 212 * q^34 + 576 * q^41 + 1644 * q^44 - 290 * q^46 - 4 * q^49 + 2430 * q^56 + 2202 * q^59 + 20 * q^61 - 322 * q^64 + 2952 * q^71 + 4080 * q^74 - 396 * q^76 - 218 * q^79 + 6108 * q^86 + 4074 * q^89 - 942 * q^91 - 1078 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more