LMFDB - Embedded newform 2025.4.a.q.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:	$1$
Dimension:	$3$
Coefficient field:	3.3.2292.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 13x + 1 $ x^3 - x^2 - 13*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 3 $
Twist minimal:	no (minimal twist has level 45)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$4.10645$ 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)

$f(q)$	$=$	$q-0.174985 q^{2} -7.96938 q^{4} -8.46371 q^{7} +2.79440 q^{8} +O(q^{10})$	$q-0.174985 q^{2} -7.96938 q^{4} -8.46371 q^{7} +2.79440 q^{8} +31.5083 q^{11} -26.8696 q^{13} +1.48102 q^{14} +63.2661 q^{16} -44.3307 q^{17} -90.2082 q^{19} -5.51346 q^{22} +194.257 q^{23} +4.70176 q^{26} +67.4506 q^{28} -3.74371 q^{29} -251.664 q^{31} -33.4258 q^{32} +7.75719 q^{34} +62.2293 q^{37} +15.7850 q^{38} +204.346 q^{41} +527.661 q^{43} -251.101 q^{44} -33.9920 q^{46} +155.727 q^{47} -271.366 q^{49} +214.134 q^{52} -141.694 q^{53} -23.6510 q^{56} +0.655092 q^{58} +493.845 q^{59} -759.483 q^{61} +44.0373 q^{62} -500.280 q^{64} +543.590 q^{67} +353.288 q^{68} +928.207 q^{71} -608.739 q^{73} -10.8892 q^{74} +718.903 q^{76} -266.677 q^{77} +614.840 q^{79} -35.7573 q^{82} +1075.31 q^{83} -92.3326 q^{86} +88.0466 q^{88} -1505.15 q^{89} +227.416 q^{91} -1548.11 q^{92} -27.2499 q^{94} +332.738 q^{97} +47.4848 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - q^{2} + 11 q^{4} + 43 q^{7} - 27 q^{8}+O(q^{10}) $ 3 * q - q^2 + 11 * q^4 + 43 * q^7 - 27 * q^8	$ 3 q - q^{2} + 11 q^{4} + 43 q^{7} - 27 q^{8} - 14 q^{11} - 40 q^{13} + 27 q^{14} - 13 q^{16} - 166 q^{17} - 164 q^{19} + 376 q^{22} + 171 q^{23} - 434 q^{26} + 517 q^{28} + 335 q^{29} - 352 q^{31} - 77 q^{32} - 52 q^{34} - 402 q^{37} - 178 q^{38} - 187 q^{41} + 602 q^{43} - 982 q^{44} - 201 q^{46} + 665 q^{47} + 430 q^{49} + 456 q^{52} - 730 q^{53} - 705 q^{56} - 217 q^{58} + 298 q^{59} - 1439 q^{61} - 1614 q^{62} - 1569 q^{64} + 1849 q^{67} - 710 q^{68} - 70 q^{71} + 368 q^{73} + 320 q^{74} + 204 q^{76} - 948 q^{77} - 382 q^{79} + 575 q^{82} - 831 q^{83} - 1580 q^{86} + 1428 q^{88} - 1719 q^{89} - 710 q^{91} - 1623 q^{92} - 2077 q^{94} + 282 q^{97} + 2164 q^{98}+O(q^{100}) $ 3 * q - q^2 + 11 * q^4 + 43 * q^7 - 27 * q^8 - 14 * q^11 - 40 * q^13 + 27 * q^14 - 13 * q^16 - 166 * q^17 - 164 * q^19 + 376 * q^22 + 171 * q^23 - 434 * q^26 + 517 * q^28 + 335 * q^29 - 352 * q^31 - 77 * q^32 - 52 * q^34 - 402 * q^37 - 178 * q^38 - 187 * q^41 + 602 * q^43 - 982 * q^44 - 201 * q^46 + 665 * q^47 + 430 * q^49 + 456 * q^52 - 730 * q^53 - 705 * q^56 - 217 * q^58 + 298 * q^59 - 1439 * q^61 - 1614 * q^62 - 1569 * q^64 + 1849 * q^67 - 710 * q^68 - 70 * q^71 + 368 * q^73 + 320 * q^74 + 204 * q^76 - 948 * q^77 - 382 * q^79 + 575 * q^82 - 831 * q^83 - 1580 * q^86 + 1428 * q^88 - 1719 * q^89 - 710 * q^91 - 1623 * q^92 - 2077 * q^94 + 282 * q^97 + 2164 * q^98

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 $

$2$	−0.174985	−0.0618664	−0.0309332	−	0.999521i	$-0.509848\pi$
$2$	−0.174985	−0.0618664	−0.0309332	+	0.999521i	$0.509848\pi$
$3$	0	0
$3$	0	0
$4$	−7.96938	−0.996173
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−8.46371	−0.456998	−0.228499	−	0.973544i	$-0.573382\pi$
$7$	−8.46371	−0.456998	−0.228499	+	0.973544i	$0.573382\pi$
$8$	2.79440	0.123496
$9$	0	0
$10$	0	0
$11$	31.5083	0.863645	0.431823	−	0.901959i	$-0.357871\pi$
$11$	31.5083	0.863645	0.431823	+	0.901959i	$0.357871\pi$
$12$	0	0
$13$	−26.8696	−0.573252	−0.286626	−	0.958043i	$-0.592534\pi$
$13$	−26.8696	−0.573252	−0.286626	+	0.958043i	$0.592534\pi$
$14$	1.48102	0.0282728
$15$	0	0
$16$	63.2661	0.988532
$17$	−44.3307	−0.632457	−0.316229	−	0.948683i	$-0.602417\pi$
$17$	−44.3307	−0.632457	−0.316229	+	0.948683i	$0.602417\pi$
$18$	0	0
$19$	−90.2082	−1.08922	−0.544610	−	0.838689i	$-0.683322\pi$
$19$	−90.2082	−1.08922	−0.544610	+	0.838689i	$0.683322\pi$
$20$	0	0
$21$	0	0
$22$	−5.51346	−0.0534306
$23$	194.257	1.76111	0.880553	−	0.473947i	$-0.157171\pi$
$23$	194.257	1.76111	0.880553	+	0.473947i	$0.157171\pi$
$24$	0	0
$25$	0	0
$26$	4.70176	0.0354650
$27$	0	0
$28$	67.4506	0.455248
$29$	−3.74371	−0.0239721	−0.0119860	−	0.999928i	$-0.503815\pi$
$29$	−3.74371	−0.0239721	−0.0119860	+	0.999928i	$0.503815\pi$
$30$	0	0
$31$	−251.664	−1.45807	−0.729035	−	0.684477i	$-0.760031\pi$
$31$	−251.664	−1.45807	−0.729035	+	0.684477i	$0.760031\pi$
$32$	−33.4258	−0.184653
$33$	0	0
$34$	7.75719	0.0391278
$35$	0	0
$36$	0	0
$37$	62.2293	0.276498	0.138249	−	0.990397i	$-0.455853\pi$
$37$	62.2293	0.276498	0.138249	+	0.990397i	$0.455853\pi$
$38$	15.7850	0.0673861
$39$	0	0
$40$	0	0
$41$	204.346	0.778376	0.389188	−	0.921158i	$-0.372756\pi$
$41$	204.346	0.778376	0.389188	+	0.921158i	$0.372756\pi$
$42$	0	0
$43$	527.661	1.87134	0.935669	−	0.352878i	$-0.114797\pi$
$43$	527.661	1.87134	0.935669	+	0.352878i	$0.114797\pi$
$44$	−251.101	−0.860340
$45$	0	0
$46$	−33.9920	−0.108953
$47$	155.727	0.483301	0.241651	−	0.970363i	$-0.422311\pi$
$47$	155.727	0.483301	0.241651	+	0.970363i	$0.422311\pi$
$48$	0	0
$49$	−271.366	−0.791153
$50$	0	0
$51$	0	0
$52$	214.134	0.571058
$53$	−141.694	−0.367230	−0.183615	−	0.982998i	$-0.558780\pi$
$53$	−141.694	−0.367230	−0.183615	+	0.982998i	$0.558780\pi$
$54$	0	0
$55$	0	0
$56$	−23.6510	−0.0564374
$57$	0	0
$58$	0.655092	0.00148307
$59$	493.845	1.08971	0.544857	−	0.838529i	$-0.316584\pi$
$59$	493.845	1.08971	0.544857	+	0.838529i	$0.316584\pi$
$60$	0	0
$61$	−759.483	−1.59413	−0.797065	−	0.603894i	$-0.793615\pi$
$61$	−759.483	−1.59413	−0.797065	+	0.603894i	$0.793615\pi$
$62$	44.0373	0.0902055
$63$	0	0
$64$	−500.280	−0.977108
$65$	0	0
$66$	0	0
$67$	543.590	0.991196	0.495598	−	0.868552i	$-0.334949\pi$
$67$	543.590	0.991196	0.495598	+	0.868552i	$0.334949\pi$
$68$	353.288	0.630036
$69$	0	0
$70$	0	0
$71$	928.207	1.55152	0.775760	−	0.631028i	$-0.217367\pi$
$71$	928.207	1.55152	0.775760	+	0.631028i	$0.217367\pi$
$72$	0	0
$73$	−608.739	−0.975993	−0.487997	−	0.872845i	$-0.662272\pi$
$73$	−608.739	−0.975993	−0.487997	+	0.872845i	$0.662272\pi$
$74$	−10.8892	−0.0171060
$75$	0	0
$76$	718.903	1.08505
$77$	−266.677	−0.394684
$78$	0	0
$79$	614.840	0.875632	0.437816	−	0.899065i	$-0.355752\pi$
$79$	614.840	0.875632	0.437816	+	0.899065i	$0.355752\pi$
$80$	0	0
$81$	0	0
$82$	−35.7573	−0.0481553
$83$	1075.31	1.42206	0.711028	−	0.703164i	$-0.248230\pi$
$83$	1075.31	1.42206	0.711028	+	0.703164i	$0.248230\pi$
$84$	0	0
$85$	0	0
$86$	−92.3326	−0.115773
$87$	0	0
$88$	88.0466	0.106657
$89$	−1505.15	−1.79265	−0.896324	−	0.443400i	$-0.853772\pi$
$89$	−1505.15	−1.79265	−0.896324	+	0.443400i	$0.853772\pi$
$90$	0	0
$91$	227.416	0.261975
$92$	−1548.11	−1.75437
$93$	0	0
$94$	−27.2499	−0.0299001
$95$	0	0
$96$	0	0
$97$	332.738	0.348293	0.174147	−	0.984720i	$-0.444283\pi$
$97$	332.738	0.348293	0.174147	+	0.984720i	$0.444283\pi$
$98$	47.4848	0.0489458
$99$	0	0
$100$	0	0
$101$	494.986	0.487653	0.243826	−	0.969819i	$-0.421597\pi$
$101$	494.986	0.487653	0.243826	+	0.969819i	$0.421597\pi$
$102$	0	0
$103$	630.030	0.602706	0.301353	−	0.953513i	$-0.402562\pi$
$103$	630.030	0.602706	0.301353	+	0.953513i	$0.402562\pi$
$104$	−75.0842	−0.0707943
$105$	0	0
$106$	24.7943	0.0227192
$107$	−1561.00	−1.41035	−0.705175	−	0.709034i	$-0.749131\pi$
$107$	−1561.00	−1.41035	−0.705175	+	0.709034i	$0.749131\pi$
$108$	0	0
$109$	−936.140	−0.822623	−0.411311	−	0.911495i	$-0.634929\pi$
$109$	−936.140	−0.822623	−0.411311	+	0.911495i	$0.634929\pi$
$110$	0	0
$111$	0	0
$112$	−535.466	−0.451757
$113$	1354.98	1.12802	0.564008	−	0.825770i	$-0.309259\pi$
$113$	1354.98	1.12802	0.564008	+	0.825770i	$0.309259\pi$
$114$	0	0
$115$	0	0
$116$	29.8351	0.0238803
$117$	0	0
$118$	−86.4153	−0.0674167
$119$	375.202	0.289031
$120$	0	0
$121$	−338.229	−0.254117
$122$	132.898	0.0986230
$123$	0	0
$124$	2005.60	1.45249
$125$	0	0
$126$	0	0
$127$	1182.37	0.826126	0.413063	−	0.910702i	$-0.364459\pi$
$127$	1182.37	0.826126	0.413063	+	0.910702i	$0.364459\pi$
$128$	354.947	0.245103
$129$	0	0
$130$	0	0
$131$	−2253.75	−1.50314	−0.751569	−	0.659655i	$-0.770702\pi$
$131$	−2253.75	−1.50314	−0.751569	+	0.659655i	$0.770702\pi$
$132$	0	0
$133$	763.496	0.497771
$134$	−95.1199	−0.0613217
$135$	0	0
$136$	−123.877	−0.0781059
$137$	473.711	0.295415	0.147708	−	0.989031i	$-0.452811\pi$
$137$	473.711	0.295415	0.147708	+	0.989031i	$0.452811\pi$
$138$	0	0
$139$	−137.039	−0.0836220	−0.0418110	−	0.999126i	$-0.513313\pi$
$139$	−137.039	−0.0836220	−0.0418110	+	0.999126i	$0.513313\pi$
$140$	0	0
$141$	0	0
$142$	−162.422	−0.0959870
$143$	−846.613	−0.495086
$144$	0	0
$145$	0	0
$146$	106.520	0.0603812
$147$	0	0
$148$	−495.929	−0.275440
$149$	143.111	0.0786851	0.0393426	−	0.999226i	$-0.487474\pi$
$149$	143.111	0.0786851	0.0393426	+	0.999226i	$0.487474\pi$
$150$	0	0
$151$	216.841	0.116863	0.0584314	−	0.998291i	$-0.481390\pi$
$151$	216.841	0.116863	0.0584314	+	0.998291i	$0.481390\pi$
$152$	−252.077	−0.134514
$153$	0	0
$154$	46.6644	0.0244177
$155$	0	0
$156$	0	0
$157$	−1348.43	−0.685455	−0.342728	−	0.939435i	$-0.611351\pi$
$157$	−1348.43	−0.685455	−0.342728	+	0.939435i	$0.611351\pi$
$158$	−107.588	−0.0541722
$159$	0	0
$160$	0	0
$161$	−1644.14	−0.804822
$162$	0	0
$163$	−1039.85	−0.499676	−0.249838	−	0.968288i	$-0.580377\pi$
$163$	−1039.85	−0.499676	−0.249838	+	0.968288i	$0.580377\pi$
$164$	−1628.51	−0.775397
$165$	0	0
$166$	−188.163	−0.0879774
$167$	−3327.05	−1.54164	−0.770822	−	0.637051i	$-0.780154\pi$
$167$	−3327.05	−1.54164	−0.770822	+	0.637051i	$0.780154\pi$
$168$	0	0
$169$	−1475.03	−0.671382
$170$	0	0
$171$	0	0
$172$	−4205.13	−1.86418
$173$	−1194.25	−0.524841	−0.262420	−	0.964954i	$-0.584521\pi$
$173$	−1194.25	−0.524841	−0.262420	+	0.964954i	$0.584521\pi$
$174$	0	0
$175$	0	0
$176$	1993.40	0.853741
$177$	0	0
$178$	263.378	0.110905
$179$	−2323.70	−0.970288	−0.485144	−	0.874434i	$-0.661233\pi$
$179$	−2323.70	−0.970288	−0.485144	+	0.874434i	$0.661233\pi$
$180$	0	0
$181$	2527.12	1.03779	0.518893	−	0.854839i	$-0.326344\pi$
$181$	2527.12	1.03779	0.518893	+	0.854839i	$0.326344\pi$
$182$	−39.7944	−0.0162074
$183$	0	0
$184$	542.832	0.217490
$185$	0	0
$186$	0	0
$187$	−1396.78	−0.546219
$188$	−1241.05	−0.481452
$189$	0	0
$190$	0	0
$191$	−2388.87	−0.904986	−0.452493	−	0.891768i	$-0.649465\pi$
$191$	−2388.87	−0.904986	−0.452493	+	0.891768i	$0.649465\pi$
$192$	0	0
$193$	−3547.97	−1.32326	−0.661629	−	0.749831i	$-0.730135\pi$
$193$	−3547.97	−1.32326	−0.661629	+	0.749831i	$0.730135\pi$
$194$	−58.2241	−0.0215476
$195$	0	0
$196$	2162.62	0.788125
$197$	−1239.26	−0.448192	−0.224096	−	0.974567i	$-0.571943\pi$
$197$	−1239.26	−0.448192	−0.224096	+	0.974567i	$0.571943\pi$
$198$	0	0
$199$	516.657	0.184044	0.0920222	−	0.995757i	$-0.470667\pi$
$199$	516.657	0.184044	0.0920222	+	0.995757i	$0.470667\pi$
$200$	0	0
$201$	0	0
$202$	−86.6149	−0.0301693
$203$	31.6857	0.0109552
$204$	0	0
$205$	0	0
$206$	−110.246	−0.0372873
$207$	0	0
$208$	−1699.93	−0.566678
$209$	−2842.30	−0.940700
$210$	0	0
$211$	−17.8432	−0.00582169	−0.00291084	−	0.999996i	$-0.500927\pi$
$211$	−17.8432	−0.00582169	−0.00291084	+	0.999996i	$0.500927\pi$
$212$	1129.21	0.365824
$213$	0	0
$214$	273.151	0.0872532
$215$	0	0
$216$	0	0
$217$	2130.01	0.666334
$218$	163.810	0.0508927
$219$	0	0
$220$	0	0
$221$	1191.15	0.362557
$222$	0	0
$223$	−989.553	−0.297154	−0.148577	−	0.988901i	$-0.547469\pi$
$223$	−989.553	−0.297154	−0.148577	+	0.988901i	$0.547469\pi$
$224$	282.906	0.0843860
$225$	0	0
$226$	−237.101	−0.0697863
$227$	−3427.13	−1.00206	−0.501028	−	0.865431i	$-0.667045\pi$
$227$	−3427.13	−1.00206	−0.501028	+	0.865431i	$0.667045\pi$
$228$	0	0
$229$	1099.61	0.317311	0.158656	−	0.987334i	$-0.449284\pi$
$229$	1099.61	0.317311	0.158656	+	0.987334i	$0.449284\pi$
$230$	0	0
$231$	0	0
$232$	−10.4614	−0.00296045
$233$	4459.91	1.25399	0.626993	−	0.779025i	$-0.284285\pi$
$233$	4459.91	1.25399	0.626993	+	0.779025i	$0.284285\pi$
$234$	0	0
$235$	0	0
$236$	−3935.64	−1.08554
$237$	0	0
$238$	−65.6546	−0.0178813
$239$	−6544.45	−1.77124	−0.885618	−	0.464414i	$-0.846265\pi$
$239$	−6544.45	−1.77124	−0.885618	+	0.464414i	$0.846265\pi$
$240$	0	0
$241$	−210.325	−0.0562167	−0.0281083	−	0.999605i	$-0.508948\pi$
$241$	−210.325	−0.0562167	−0.0281083	+	0.999605i	$0.508948\pi$
$242$	59.1849	0.0157213
$243$	0	0
$244$	6052.61	1.58803
$245$	0	0
$246$	0	0
$247$	2423.86	0.624398
$248$	−703.248	−0.180066
$249$	0	0
$250$	0	0
$251$	−816.143	−0.205237	−0.102618	−	0.994721i	$-0.532722\pi$
$251$	−816.143	−0.205237	−0.102618	+	0.994721i	$0.532722\pi$
$252$	0	0
$253$	6120.71	1.52097
$254$	−206.896	−0.0511094
$255$	0	0
$256$	3940.13	0.961945
$257$	−4172.35	−1.01270	−0.506350	−	0.862328i	$-0.669005\pi$
$257$	−4172.35	−1.01270	−0.506350	+	0.862328i	$0.669005\pi$
$258$	0	0
$259$	−526.691	−0.126359
$260$	0	0
$261$	0	0
$262$	394.371	0.0929937
$263$	−2817.05	−0.660482	−0.330241	−	0.943897i	$-0.607130\pi$
$263$	−2817.05	−0.660482	−0.330241	+	0.943897i	$0.607130\pi$
$264$	0	0
$265$	0	0
$266$	−133.600	−0.0307953
$267$	0	0
$268$	−4332.08	−0.987402
$269$	102.610	0.0232573	0.0116287	−	0.999932i	$-0.496298\pi$
$269$	102.610	0.0232573	0.0116287	+	0.999932i	$0.496298\pi$
$270$	0	0
$271$	−3337.27	−0.748061	−0.374031	−	0.927416i	$-0.622025\pi$
$271$	−3337.27	−0.748061	−0.374031	+	0.927416i	$0.622025\pi$
$272$	−2804.63	−0.625204
$273$	0	0
$274$	−82.8922	−0.0182763
$275$	0	0
$276$	0	0
$277$	−633.883	−0.137496	−0.0687479	−	0.997634i	$-0.521900\pi$
$277$	−633.883	−0.137496	−0.0687479	+	0.997634i	$0.521900\pi$
$278$	23.9796	0.00517339
$279$	0	0
$280$	0	0
$281$	6202.40	1.31674	0.658370	−	0.752694i	$-0.271246\pi$
$281$	6202.40	1.31674	0.658370	+	0.752694i	$0.271246\pi$
$282$	0	0
$283$	7480.12	1.57119	0.785596	−	0.618740i	$-0.212357\pi$
$283$	7480.12	1.57119	0.785596	+	0.618740i	$0.212357\pi$
$284$	−7397.24	−1.54558
$285$	0	0
$286$	148.144	0.0306292
$287$	−1729.52	−0.355716
$288$	0	0
$289$	−2947.79	−0.599998
$290$	0	0
$291$	0	0
$292$	4851.27	0.972258
$293$	−4836.51	−0.964341	−0.482171	−	0.876077i	$-0.660151\pi$
$293$	−4836.51	−0.964341	−0.482171	+	0.876077i	$0.660151\pi$
$294$	0	0
$295$	0	0
$296$	173.893	0.0341464
$297$	0	0
$298$	−25.0422	−0.00486796
$299$	−5219.61	−1.00956
$300$	0	0
$301$	−4465.97	−0.855197
$302$	−37.9439	−0.00722989
$303$	0	0
$304$	−5707.12	−1.07673
$305$	0	0
$306$	0	0
$307$	5611.86	1.04328	0.521638	−	0.853167i	$-0.325321\pi$
$307$	5611.86	1.04328	0.521638	+	0.853167i	$0.325321\pi$
$308$	2125.25	0.393173
$309$	0	0
$310$	0	0
$311$	−10921.4	−1.99131	−0.995653	−	0.0931416i	$-0.970309\pi$
$311$	−10921.4	−1.99131	−0.995653	+	0.0931416i	$0.970309\pi$
$312$	0	0
$313$	−4980.73	−0.899449	−0.449724	−	0.893167i	$-0.648478\pi$
$313$	−4980.73	−0.899449	−0.449724	+	0.893167i	$0.648478\pi$
$314$	235.955	0.0424067
$315$	0	0
$316$	−4899.89	−0.872280
$317$	−3756.12	−0.665504	−0.332752	−	0.943014i	$-0.607977\pi$
$317$	−3756.12	−0.665504	−0.332752	+	0.943014i	$0.607977\pi$
$318$	0	0
$319$	−117.958	−0.0207034
$320$	0	0
$321$	0	0
$322$	287.699	0.0497914
$323$	3998.99	0.688885
$324$	0	0
$325$	0	0
$326$	181.958	0.0309132
$327$	0	0
$328$	571.023	0.0961264
$329$	−1318.03	−0.220868
$330$	0	0
$331$	−906.954	−0.150606	−0.0753031	−	0.997161i	$-0.523992\pi$
$331$	−906.954	−0.150606	−0.0753031	+	0.997161i	$0.523992\pi$
$332$	−8569.55	−1.41661
$333$	0	0
$334$	582.182	0.0953759
$335$	0	0
$336$	0	0
$337$	−9878.70	−1.59682	−0.798408	−	0.602117i	$-0.794324\pi$
$337$	−9878.70	−1.59682	−0.798408	+	0.602117i	$0.794324\pi$
$338$	258.107	0.0415360
$339$	0	0
$340$	0	0
$341$	−7929.49	−1.25925
$342$	0	0
$343$	5199.81	0.818553
$344$	1474.49	0.231103
$345$	0	0
$346$	208.976	0.0324700
$347$	−3509.71	−0.542972	−0.271486	−	0.962442i	$-0.587515\pi$
$347$	−3509.71	−0.542972	−0.271486	+	0.962442i	$0.587515\pi$
$348$	0	0
$349$	−10393.0	−1.59405	−0.797024	−	0.603947i	$-0.793594\pi$
$349$	−10393.0	−1.59405	−0.797024	+	0.603947i	$0.793594\pi$
$350$	0	0
$351$	0	0
$352$	−1053.19	−0.159475
$353$	8085.39	1.21910	0.609550	−	0.792748i	$-0.291350\pi$
$353$	8085.39	1.21910	0.609550	+	0.792748i	$0.291350\pi$
$354$	0	0
$355$	0	0
$356$	11995.1	1.78579
$357$	0	0
$358$	406.612	0.0600282
$359$	8189.49	1.20397	0.601984	−	0.798508i	$-0.294377\pi$
$359$	8189.49	1.20397	0.601984	+	0.798508i	$0.294377\pi$
$360$	0	0
$361$	1278.52	0.186400
$362$	−442.207	−0.0642041
$363$	0	0
$364$	−1812.37	−0.260972
$365$	0	0
$366$	0	0
$367$	8994.66	1.27934	0.639669	−	0.768650i	$-0.279071\pi$
$367$	8994.66	1.27934	0.639669	+	0.768650i	$0.279071\pi$
$368$	12289.9	1.74091
$369$	0	0
$370$	0	0
$371$	1199.26	0.167823
$372$	0	0
$373$	929.867	0.129080	0.0645398	−	0.997915i	$-0.479442\pi$
$373$	929.867	0.129080	0.0645398	+	0.997915i	$0.479442\pi$
$374$	244.415	0.0337926
$375$	0	0
$376$	435.164	0.0596858
$377$	100.592	0.0137420
$378$	0	0
$379$	3449.71	0.467546	0.233773	−	0.972291i	$-0.424893\pi$
$379$	3449.71	0.467546	0.233773	+	0.972291i	$0.424893\pi$
$380$	0	0
$381$	0	0
$382$	418.015	0.0559882
$383$	−5346.13	−0.713249	−0.356625	−	0.934248i	$-0.616073\pi$
$383$	−5346.13	−0.713249	−0.356625	+	0.934248i	$0.616073\pi$
$384$	0	0
$385$	0	0
$386$	620.841	0.0818652
$387$	0	0
$388$	−2651.72	−0.346960
$389$	−7723.44	−1.00667	−0.503334	−	0.864092i	$-0.667894\pi$
$389$	−7723.44	−1.00667	−0.503334	+	0.864092i	$0.667894\pi$
$390$	0	0
$391$	−8611.56	−1.11382
$392$	−758.303	−0.0977043
$393$	0	0
$394$	216.852	0.0277280
$395$	0	0
$396$	0	0
$397$	−7125.03	−0.900744	−0.450372	−	0.892841i	$-0.648709\pi$
$397$	−7125.03	−0.900744	−0.450372	+	0.892841i	$0.648709\pi$
$398$	−90.4070	−0.0113862
$399$	0	0
$400$	0	0
$401$	1793.09	0.223299	0.111649	−	0.993748i	$-0.464387\pi$
$401$	1793.09	0.223299	0.111649	+	0.993748i	$0.464387\pi$
$402$	0	0
$403$	6762.10	0.835841
$404$	−3944.73	−0.485786
$405$	0	0
$406$	−5.54451	−0.000677757 0
$407$	1960.74	0.238796
$408$	0	0
$409$	3139.73	0.379584	0.189792	−	0.981824i	$-0.439219\pi$
$409$	3139.73	0.379584	0.189792	+	0.981824i	$0.439219\pi$
$410$	0	0
$411$	0	0
$412$	−5020.95	−0.600399
$413$	−4179.77	−0.497997
$414$	0	0
$415$	0	0
$416$	898.135	0.105853
$417$	0	0
$418$	497.359	0.0581977
$419$	−2457.40	−0.286520	−0.143260	−	0.989685i	$-0.545758\pi$
$419$	−2457.40	−0.286520	−0.143260	+	0.989685i	$0.545758\pi$
$420$	0	0
$421$	−2678.67	−0.310096	−0.155048	−	0.987907i	$-0.549553\pi$
$421$	−2678.67	−0.310096	−0.155048	+	0.987907i	$0.549553\pi$
$422$	3.12228	0.000360167 0
$423$	0	0
$424$	−395.949	−0.0453514
$425$	0	0
$426$	0	0
$427$	6428.05	0.728513
$428$	12440.2	1.40495
$429$	0	0
$430$	0	0
$431$	9472.42	1.05863	0.529316	−	0.848425i	$-0.322449\pi$
$431$	9472.42	1.05863	0.529316	+	0.848425i	$0.322449\pi$
$432$	0	0
$433$	4238.21	0.470382	0.235191	−	0.971949i	$-0.424428\pi$
$433$	4238.21	0.470382	0.235191	+	0.971949i	$0.424428\pi$
$434$	−372.719	−0.0412237
$435$	0	0
$436$	7460.45	0.819474
$437$	−17523.6	−1.91823
$438$	0	0
$439$	4858.93	0.528255	0.264128	−	0.964488i	$-0.414916\pi$
$439$	4858.93	0.528255	0.264128	+	0.964488i	$0.414916\pi$
$440$	0	0
$441$	0	0
$442$	−208.432	−0.0224301
$443$	2335.75	0.250508	0.125254	−	0.992125i	$-0.460025\pi$
$443$	2335.75	0.250508	0.125254	+	0.992125i	$0.460025\pi$
$444$	0	0
$445$	0	0
$446$	173.157	0.0183839
$447$	0	0
$448$	4234.22	0.446536
$449$	−13290.5	−1.39692	−0.698460	−	0.715649i	$-0.746131\pi$
$449$	−13290.5	−1.39692	−0.698460	+	0.715649i	$0.746131\pi$
$450$	0	0
$451$	6438.58	0.672241
$452$	−10798.3	−1.12370
$453$	0	0
$454$	599.695	0.0619936
$455$	0	0
$456$	0	0
$457$	−5174.61	−0.529668	−0.264834	−	0.964294i	$-0.585317\pi$
$457$	−5174.61	−0.529668	−0.264834	+	0.964294i	$0.585317\pi$
$458$	−192.415	−0.0196309
$459$	0	0
$460$	0	0
$461$	−6340.88	−0.640617	−0.320309	−	0.947313i	$-0.603787\pi$
$461$	−6340.88	−0.640617	−0.320309	+	0.947313i	$0.603787\pi$
$462$	0	0
$463$	−3182.79	−0.319475	−0.159737	−	0.987160i	$-0.551065\pi$
$463$	−3182.79	−0.319475	−0.159737	+	0.987160i	$0.551065\pi$
$464$	−236.850	−0.0236972
$465$	0	0
$466$	−780.416	−0.0775796
$467$	−8576.23	−0.849808	−0.424904	−	0.905238i	$-0.639692\pi$
$467$	−8576.23	−0.849808	−0.424904	+	0.905238i	$0.639692\pi$
$468$	0	0
$469$	−4600.79	−0.452974
$470$	0	0
$471$	0	0
$472$	1380.00	0.134575
$473$	16625.7	1.61617
$474$	0	0
$475$	0	0
$476$	−2990.13	−0.287925
$477$	0	0
$478$	1145.18	0.109580
$479$	−2917.65	−0.278311	−0.139155	−	0.990271i	$-0.544439\pi$
$479$	−2917.65	−0.278311	−0.139155	+	0.990271i	$0.544439\pi$
$480$	0	0
$481$	−1672.07	−0.158503
$482$	36.8036	0.00347792
$483$	0	0
$484$	2695.48	0.253144
$485$	0	0
$486$	0	0
$487$	14061.0	1.30834	0.654172	−	0.756346i	$-0.273017\pi$
$487$	14061.0	1.30834	0.654172	+	0.756346i	$0.273017\pi$
$488$	−2122.30	−0.196869
$489$	0	0
$490$	0	0
$491$	932.662	0.0857239	0.0428620	−	0.999081i	$-0.486352\pi$
$491$	932.662	0.0857239	0.0428620	+	0.999081i	$0.486352\pi$
$492$	0	0
$493$	165.961	0.0151613
$494$	−424.137	−0.0386292
$495$	0	0
$496$	−15921.8	−1.44135
$497$	−7856.08	−0.709041
$498$	0	0
$499$	−14430.8	−1.29461	−0.647305	−	0.762231i	$-0.724104\pi$
$499$	−14430.8	−1.29461	−0.647305	+	0.762231i	$0.724104\pi$
$500$	0	0
$501$	0	0
$502$	142.812	0.0126973
$503$	−3230.55	−0.286368	−0.143184	−	0.989696i	$-0.545734\pi$
$503$	−3230.55	−0.286368	−0.143184	+	0.989696i	$0.545734\pi$
$504$	0	0
$505$	0	0
$506$	−1071.03	−0.0940970
$507$	0	0
$508$	−9422.72	−0.822964
$509$	360.756	0.0314150	0.0157075	−	0.999877i	$-0.495000\pi$
$509$	360.756	0.0314150	0.0157075	+	0.999877i	$0.495000\pi$
$510$	0	0
$511$	5152.19	0.446027
$512$	−3529.04	−0.304615
$513$	0	0
$514$	730.096	0.0626521
$515$	0	0
$516$	0	0
$517$	4906.70	0.417401
$518$	92.1628	0.00781738
$519$	0	0
$520$	0	0
$521$	11698.8	0.983747	0.491873	−	0.870667i	$-0.336312\pi$
$521$	11698.8	0.983747	0.491873	+	0.870667i	$0.336312\pi$
$522$	0	0
$523$	−18557.3	−1.55154	−0.775770	−	0.631015i	$-0.782638\pi$
$523$	−18557.3	−1.55154	−0.775770	+	0.631015i	$0.782638\pi$
$524$	17961.0	1.49738
$525$	0	0
$526$	492.940	0.0408616
$527$	11156.4	0.922166
$528$	0	0
$529$	25568.9	2.10150
$530$	0	0
$531$	0	0
$532$	−6084.59	−0.495866
$533$	−5490.68	−0.446206
$534$	0	0
$535$	0	0
$536$	1519.01	0.122409
$537$	0	0
$538$	−17.9551	−0.00143885
$539$	−8550.26	−0.683276
$540$	0	0
$541$	22755.3	1.80837	0.904185	−	0.427140i	$-0.140479\pi$
$541$	22755.3	1.80837	0.904185	+	0.427140i	$0.140479\pi$
$542$	583.971	0.0462799
$543$	0	0
$544$	1481.79	0.116785
$545$	0	0
$546$	0	0
$547$	−5798.09	−0.453215	−0.226608	−	0.973986i	$-0.572763\pi$
$547$	−5798.09	−0.453215	−0.226608	+	0.973986i	$0.572763\pi$
$548$	−3775.18	−0.294284
$549$	0	0
$550$	0	0
$551$	337.713	0.0261108
$552$	0	0
$553$	−5203.83	−0.400162
$554$	110.920	0.00850637
$555$	0	0
$556$	1092.11	0.0833019
$557$	15740.3	1.19738	0.598688	−	0.800982i	$-0.295689\pi$
$557$	15740.3	1.19738	0.598688	+	0.800982i	$0.295689\pi$
$558$	0	0
$559$	−14178.0	−1.07275
$560$	0	0
$561$	0	0
$562$	−1085.32	−0.0814620
$563$	3819.12	0.285891	0.142946	−	0.989731i	$-0.454343\pi$
$563$	3819.12	0.285891	0.142946	+	0.989731i	$0.454343\pi$
$564$	0	0
$565$	0	0
$566$	−1308.91	−0.0972040
$567$	0	0
$568$	2593.78	0.191607
$569$	−8891.12	−0.655070	−0.327535	−	0.944839i	$-0.606218\pi$
$569$	−8891.12	−0.655070	−0.327535	+	0.944839i	$0.606218\pi$
$570$	0	0
$571$	8386.63	0.614657	0.307329	−	0.951603i	$-0.400565\pi$
$571$	8386.63	0.614657	0.307329	+	0.951603i	$0.400565\pi$
$572$	6746.98	0.493192
$573$	0	0
$574$	302.640	0.0220069
$575$	0	0
$576$	0	0
$577$	−16922.3	−1.22095	−0.610473	−	0.792037i	$-0.709021\pi$
$577$	−16922.3	−1.22095	−0.610473	+	0.792037i	$0.709021\pi$
$578$	515.818	0.0371197
$579$	0	0
$580$	0	0
$581$	−9101.12	−0.649876
$582$	0	0
$583$	−4464.54	−0.317156
$584$	−1701.06	−0.120531
$585$	0	0
$586$	846.315	0.0596603
$587$	15238.6	1.07149	0.535744	−	0.844380i	$-0.320031\pi$
$587$	15238.6	1.07149	0.535744	+	0.844380i	$0.320031\pi$
$588$	0	0
$589$	22702.1	1.58816
$590$	0	0
$591$	0	0
$592$	3937.00	0.273327
$593$	−16960.2	−1.17449	−0.587245	−	0.809409i	$-0.699787\pi$
$593$	−16960.2	−1.17449	−0.587245	+	0.809409i	$0.699787\pi$
$594$	0	0
$595$	0	0
$596$	−1140.50	−0.0783839
$597$	0	0
$598$	913.351	0.0624577
$599$	24912.1	1.69930	0.849651	−	0.527345i	$-0.176813\pi$
$599$	24912.1	1.69930	0.849651	+	0.527345i	$0.176813\pi$
$600$	0	0
$601$	−4351.27	−0.295328	−0.147664	−	0.989038i	$-0.547175\pi$
$601$	−4351.27	−0.295328	−0.147664	+	0.989038i	$0.547175\pi$
$602$	781.476	0.0529080
$603$	0	0
$604$	−1728.09	−0.116416
$605$	0	0
$606$	0	0
$607$	−27823.7	−1.86051	−0.930254	−	0.366915i	$-0.880414\pi$
$607$	−27823.7	−1.86051	−0.930254	+	0.366915i	$0.880414\pi$
$608$	3015.28	0.201128
$609$	0	0
$610$	0	0
$611$	−4184.33	−0.277054
$612$	0	0
$613$	−20034.6	−1.32005	−0.660024	−	0.751244i	$-0.729454\pi$
$613$	−20034.6	−1.32005	−0.660024	+	0.751244i	$0.729454\pi$
$614$	−981.989	−0.0645437
$615$	0	0
$616$	−745.201	−0.0487419
$617$	7553.89	0.492882	0.246441	−	0.969158i	$-0.420739\pi$
$617$	7553.89	0.492882	0.246441	+	0.969158i	$0.420739\pi$
$618$	0	0
$619$	12385.1	0.804198	0.402099	−	0.915596i	$-0.368281\pi$
$619$	12385.1	0.804198	0.402099	+	0.915596i	$0.368281\pi$
$620$	0	0
$621$	0	0
$622$	1911.08	0.123195
$623$	12739.2	0.819236
$624$	0	0
$625$	0	0
$626$	871.551	0.0556457
$627$	0	0
$628$	10746.2	0.682832
$629$	−2758.67	−0.174873
$630$	0	0
$631$	−1325.17	−0.0836043	−0.0418022	−	0.999126i	$-0.513310\pi$
$631$	−1325.17	−0.0836043	−0.0418022	+	0.999126i	$0.513310\pi$
$632$	1718.11	0.108137
$633$	0	0
$634$	657.264	0.0411724
$635$	0	0
$636$	0	0
$637$	7291.47	0.453530
$638$	20.6408	0.00128084
$639$	0	0
$640$	0	0
$641$	22511.6	1.38714	0.693568	−	0.720392i	$-0.256038\pi$
$641$	22511.6	1.38714	0.693568	+	0.720392i	$0.256038\pi$
$642$	0	0
$643$	−5614.61	−0.344353	−0.172176	−	0.985066i	$-0.555080\pi$
$643$	−5614.61	−0.344353	−0.172176	+	0.985066i	$0.555080\pi$
$644$	13102.8	0.801741
$645$	0	0
$646$	−699.762	−0.0426188
$647$	−11753.6	−0.714188	−0.357094	−	0.934068i	$-0.616232\pi$
$647$	−11753.6	−0.714188	−0.357094	+	0.934068i	$0.616232\pi$
$648$	0	0
$649$	15560.2	0.941127
$650$	0	0
$651$	0	0
$652$	8286.95	0.497764
$653$	−25858.2	−1.54963	−0.774816	−	0.632187i	$-0.782157\pi$
$653$	−25858.2	−1.54963	−0.774816	+	0.632187i	$0.782157\pi$
$654$	0	0
$655$	0	0
$656$	12928.1	0.769450
$657$	0	0
$658$	230.635	0.0136643
$659$	17695.5	1.04601	0.523004	−	0.852330i	$-0.324811\pi$
$659$	17695.5	1.04601	0.523004	+	0.852330i	$0.324811\pi$
$660$	0	0
$661$	6230.09	0.366600	0.183300	−	0.983057i	$-0.441322\pi$
$661$	6230.09	0.366600	0.183300	+	0.983057i	$0.441322\pi$
$662$	158.703	0.00931747
$663$	0	0
$664$	3004.84	0.175618
$665$	0	0
$666$	0	0
$667$	−727.243	−0.0422174
$668$	26514.5	1.53574
$669$	0	0
$670$	0	0
$671$	−23930.0	−1.37676
$672$	0	0
$673$	6051.68	0.346620	0.173310	−	0.984867i	$-0.444554\pi$
$673$	6051.68	0.346620	0.173310	+	0.984867i	$0.444554\pi$
$674$	1728.62	0.0987892
$675$	0	0
$676$	11755.0	0.668812
$677$	28098.6	1.59515	0.797576	−	0.603218i	$-0.206115\pi$
$677$	28098.6	1.59515	0.797576	+	0.603218i	$0.206115\pi$
$678$	0	0
$679$	−2816.20	−0.159169
$680$	0	0
$681$	0	0
$682$	1387.54	0.0779056
$683$	−8335.71	−0.466994	−0.233497	−	0.972358i	$-0.575017\pi$
$683$	−8335.71	−0.466994	−0.233497	+	0.972358i	$0.575017\pi$
$684$	0	0
$685$	0	0
$686$	−909.888	−0.0506409
$687$	0	0
$688$	33383.0	1.84988
$689$	3807.26	0.210515
$690$	0	0
$691$	−16885.1	−0.929579	−0.464790	−	0.885421i	$-0.653870\pi$
$691$	−16885.1	−0.929579	−0.464790	+	0.885421i	$0.653870\pi$
$692$	9517.46	0.522832
$693$	0	0
$694$	614.146	0.0335917
$695$	0	0
$696$	0	0
$697$	−9058.78	−0.492290
$698$	1818.61	0.0986180
$699$	0	0
$700$	0	0
$701$	−16875.2	−0.909226	−0.454613	−	0.890689i	$-0.650222\pi$
$701$	−16875.2	−0.909226	−0.454613	+	0.890689i	$0.650222\pi$
$702$	0	0
$703$	−5613.59	−0.301167
$704$	−15762.9	−0.843875
$705$	0	0
$706$	−1414.82	−0.0754213
$707$	−4189.42	−0.222856
$708$	0	0
$709$	−19007.4	−1.00682	−0.503412	−	0.864046i	$-0.667922\pi$
$709$	−19007.4	−1.00682	−0.503412	+	0.864046i	$0.667922\pi$
$710$	0	0
$711$	0	0
$712$	−4205.99	−0.221385
$713$	−48887.5	−2.56782
$714$	0	0
$715$	0	0
$716$	18518.5	0.966574
$717$	0	0
$718$	−1433.03	−0.0744852
$719$	−17588.1	−0.912275	−0.456138	−	0.889909i	$-0.650767\pi$
$719$	−17588.1	−0.912275	−0.456138	+	0.889909i	$0.650767\pi$
$720$	0	0
$721$	−5332.40	−0.275435
$722$	−223.721	−0.0115319
$723$	0	0
$724$	−20139.6	−1.03381
$725$	0	0
$726$	0	0
$727$	−5605.38	−0.285959	−0.142979	−	0.989726i	$-0.545668\pi$
$727$	−5605.38	−0.285959	−0.142979	+	0.989726i	$0.545668\pi$
$728$	635.491	0.0323528
$729$	0	0
$730$	0	0
$731$	−23391.6	−1.18354
$732$	0	0
$733$	−14920.1	−0.751823	−0.375911	−	0.926656i	$-0.622670\pi$
$733$	−14920.1	−0.751823	−0.375911	+	0.926656i	$0.622670\pi$
$734$	−1573.93	−0.0791481
$735$	0	0
$736$	−6493.20	−0.325194
$737$	17127.6	0.856042
$738$	0	0
$739$	−27418.8	−1.36484	−0.682421	−	0.730959i	$-0.739073\pi$
$739$	−27418.8	−1.36484	−0.682421	+	0.730959i	$0.739073\pi$
$740$	0	0
$741$	0	0
$742$	−209.852	−0.0103826
$743$	−24544.2	−1.21190	−0.605948	−	0.795504i	$-0.707206\pi$
$743$	−24544.2	−1.21190	−0.605948	+	0.795504i	$0.707206\pi$
$744$	0	0
$745$	0	0
$746$	−162.712	−0.00798569
$747$	0	0
$748$	11131.5	0.544128
$749$	13211.8	0.644526
$750$	0	0
$751$	1534.57	0.0745634	0.0372817	−	0.999305i	$-0.488130\pi$
$751$	1534.57	0.0745634	0.0372817	+	0.999305i	$0.488130\pi$
$752$	9852.26	0.477759
$753$	0	0
$754$	−17.6020	−0.000850170 0
$755$	0	0
$756$	0	0
$757$	18051.1	0.866681	0.433341	−	0.901230i	$-0.357335\pi$
$757$	18051.1	0.866681	0.433341	+	0.901230i	$0.357335\pi$
$758$	−603.647	−0.0289254
$759$	0	0
$760$	0	0
$761$	−12924.6	−0.615658	−0.307829	−	0.951442i	$-0.599603\pi$
$761$	−12924.6	−0.615658	−0.307829	+	0.951442i	$0.599603\pi$
$762$	0	0
$763$	7923.22	0.375937
$764$	19037.8	0.901522
$765$	0	0
$766$	935.491	0.0441262
$767$	−13269.4	−0.624681
$768$	0	0
$769$	−3372.32	−0.158139	−0.0790695	−	0.996869i	$-0.525195\pi$
$769$	−3372.32	−0.158139	−0.0790695	+	0.996869i	$0.525195\pi$
$770$	0	0
$771$	0	0
$772$	28275.2	1.31819
$773$	−27152.6	−1.26341	−0.631703	−	0.775211i	$-0.717644\pi$
$773$	−27152.6	−1.26341	−0.631703	+	0.775211i	$0.717644\pi$
$774$	0	0
$775$	0	0
$776$	929.802	0.0430128
$777$	0	0
$778$	1351.48	0.0622790
$779$	−18433.7	−0.847823
$780$	0	0
$781$	29246.2	1.33996
$782$	1506.89	0.0689083
$783$	0	0
$784$	−17168.2	−0.782080
$785$	0	0
$786$	0	0
$787$	25212.1	1.14195	0.570974	−	0.820968i	$-0.306566\pi$
$787$	25212.1	1.14195	0.570974	+	0.820968i	$0.306566\pi$
$788$	9876.16	0.446477
$789$	0	0
$790$	0	0
$791$	−11468.2	−0.515500
$792$	0	0
$793$	20407.0	0.913838
$794$	1246.77	0.0557258
$795$	0	0
$796$	−4117.44	−0.183340
$797$	39176.9	1.74118	0.870588	−	0.492013i	$-0.163739\pi$
$797$	39176.9	1.74118	0.870588	+	0.492013i	$0.163739\pi$
$798$	0	0
$799$	−6903.50	−0.305667
$800$	0	0
$801$	0	0
$802$	−313.764	−0.0138147
$803$	−19180.3	−0.842912
$804$	0	0
$805$	0	0
$806$	−1183.26	−0.0517105
$807$	0	0
$808$	1383.19	0.0602231
$809$	36739.0	1.59663	0.798316	−	0.602239i	$-0.205725\pi$
$809$	36739.0	1.59663	0.798316	+	0.602239i	$0.205725\pi$
$810$	0	0
$811$	−29660.0	−1.28422	−0.642111	−	0.766611i	$-0.721941\pi$
$811$	−29660.0	−1.28422	−0.642111	+	0.766611i	$0.721941\pi$
$812$	−252.515	−0.0109132
$813$	0	0
$814$	−343.099	−0.0147735
$815$	0	0
$816$	0	0
$817$	−47599.4	−2.03830
$818$	−549.405	−0.0234835
$819$	0	0
$820$	0	0
$821$	−20113.2	−0.855002	−0.427501	−	0.904015i	$-0.640606\pi$
$821$	−20113.2	−0.855002	−0.427501	+	0.904015i	$0.640606\pi$
$822$	0	0
$823$	−5374.05	−0.227615	−0.113808	−	0.993503i	$-0.536305\pi$
$823$	−5374.05	−0.227615	−0.113808	+	0.993503i	$0.536305\pi$
$824$	1760.55	0.0744318
$825$	0	0
$826$	731.395	0.0308093
$827$	27865.2	1.17167	0.585833	−	0.810432i	$-0.300767\pi$
$827$	27865.2	1.17167	0.585833	+	0.810432i	$0.300767\pi$
$828$	0	0
$829$	24363.1	1.02070	0.510352	−	0.859965i	$-0.329515\pi$
$829$	24363.1	1.02070	0.510352	+	0.859965i	$0.329515\pi$
$830$	0	0
$831$	0	0
$832$	13442.3	0.560129
$833$	12029.8	0.500370
$834$	0	0
$835$	0	0
$836$	22651.4	0.937099
$837$	0	0
$838$	430.007	0.0177259
$839$	35530.6	1.46204	0.731020	−	0.682356i	$-0.239045\pi$
$839$	35530.6	1.46204	0.731020	+	0.682356i	$0.239045\pi$
$840$	0	0
$841$	−24375.0	−0.999425
$842$	468.726	0.0191845
$843$	0	0
$844$	142.199	0.00579940
$845$	0	0
$846$	0	0
$847$	2862.68	0.116131
$848$	−8964.43	−0.363019
$849$	0	0
$850$	0	0
$851$	12088.5	0.486943
$852$	0	0
$853$	−34447.5	−1.38272	−0.691360	−	0.722511i	$-0.742988\pi$
$853$	−34447.5	−1.38272	−0.691360	+	0.722511i	$0.742988\pi$
$854$	−1124.81	−0.0450705
$855$	0	0
$856$	−4362.05	−0.174173
$857$	20006.8	0.797457	0.398729	−	0.917069i	$-0.369452\pi$
$857$	20006.8	0.797457	0.398729	+	0.917069i	$0.369452\pi$
$858$	0	0
$859$	1786.38	0.0709552	0.0354776	−	0.999370i	$-0.488705\pi$
$859$	1786.38	0.0709552	0.0354776	+	0.999370i	$0.488705\pi$
$860$	0	0
$861$	0	0
$862$	−1657.53	−0.0654937
$863$	12151.9	0.479322	0.239661	−	0.970857i	$-0.422964\pi$
$863$	12151.9	0.479322	0.239661	+	0.970857i	$0.422964\pi$
$864$	0	0
$865$	0	0
$866$	−741.621	−0.0291008
$867$	0	0
$868$	−16974.9	−0.663784
$869$	19372.5	0.756235
$870$	0	0
$871$	−14606.0	−0.568205
$872$	−2615.94	−0.101591
$873$	0	0
$874$	3066.36	0.118674
$875$	0	0
$876$	0	0
$877$	18873.3	0.726688	0.363344	−	0.931655i	$-0.381635\pi$
$877$	18873.3	0.726688	0.363344	+	0.931655i	$0.381635\pi$
$878$	−850.238	−0.0326813
$879$	0	0
$880$	0	0
$881$	−23587.2	−0.902014	−0.451007	−	0.892520i	$-0.648935\pi$
$881$	−23587.2	−0.902014	−0.451007	+	0.892520i	$0.648935\pi$
$882$	0	0
$883$	29504.5	1.12447	0.562234	−	0.826978i	$-0.309942\pi$
$883$	29504.5	1.12447	0.562234	+	0.826978i	$0.309942\pi$
$884$	−9492.70	−0.361170
$885$	0	0
$886$	−408.721	−0.0154980
$887$	2476.04	0.0937286	0.0468643	−	0.998901i	$-0.485077\pi$
$887$	2476.04	0.0937286	0.0468643	+	0.998901i	$0.485077\pi$
$888$	0	0
$889$	−10007.2	−0.377538
$890$	0	0
$891$	0	0
$892$	7886.13	0.296017
$893$	−14047.9	−0.526422
$894$	0	0
$895$	0	0
$896$	−3004.17	−0.112012
$897$	0	0
$898$	2325.63	0.0864224
$899$	942.157	0.0349529
$900$	0	0
$901$	6281.40	0.232257
$902$	−1126.65	−0.0415891
$903$	0	0
$904$	3786.35	0.139305
$905$	0	0
$906$	0	0
$907$	26751.0	0.979331	0.489666	−	0.871910i	$-0.337119\pi$
$907$	26751.0	0.979331	0.489666	+	0.871910i	$0.337119\pi$
$908$	27312.1	0.998221
$909$	0	0
$910$	0	0
$911$	−12668.2	−0.460721	−0.230360	−	0.973105i	$-0.573990\pi$
$911$	−12668.2	−0.460721	−0.230360	+	0.973105i	$0.573990\pi$
$912$	0	0
$913$	33881.1	1.22815
$914$	905.478	0.0327686
$915$	0	0
$916$	−8763.21	−0.316097
$917$	19075.1	0.686930
$918$	0	0
$919$	46565.5	1.67144	0.835721	−	0.549155i	$-0.185050\pi$
$919$	46565.5	1.67144	0.835721	+	0.549155i	$0.185050\pi$
$920$	0	0
$921$	0	0
$922$	1109.56	0.0396327
$923$	−24940.5	−0.889412
$924$	0	0
$925$	0	0
$926$	556.940	0.0197648
$927$	0	0
$928$	125.136	0.00442651
$929$	−4610.92	−0.162841	−0.0814205	−	0.996680i	$-0.525946\pi$
$929$	−4610.92	−0.162841	−0.0814205	+	0.996680i	$0.525946\pi$
$930$	0	0
$931$	24479.4	0.861740
$932$	−35542.8	−1.24919
$933$	0	0
$934$	1500.71	0.0525746
$935$	0	0
$936$	0	0
$937$	6625.43	0.230996	0.115498	−	0.993308i	$-0.463154\pi$
$937$	6625.43	0.230996	0.115498	+	0.993308i	$0.463154\pi$
$938$	805.068	0.0280239
$939$	0	0
$940$	0	0
$941$	43442.6	1.50498	0.752491	−	0.658602i	$-0.228852\pi$
$941$	43442.6	1.50498	0.752491	+	0.658602i	$0.228852\pi$
$942$	0	0
$943$	39695.6	1.37080
$944$	31243.7	1.07722
$945$	0	0
$946$	−2909.24	−0.0999868
$947$	32922.4	1.12971	0.564855	−	0.825190i	$-0.308932\pi$
$947$	32922.4	1.12971	0.564855	+	0.825190i	$0.308932\pi$
$948$	0	0
$949$	16356.6	0.559490
$950$	0	0
$951$	0	0
$952$	1048.46	0.0356942
$953$	20253.5	0.688430	0.344215	−	0.938891i	$-0.388145\pi$
$953$	20253.5	0.688430	0.344215	+	0.938891i	$0.388145\pi$
$954$	0	0
$955$	0	0
$956$	52155.2	1.76446
$957$	0	0
$958$	510.544	0.0172181
$959$	−4009.36	−0.135004
$960$	0	0
$961$	33543.7	1.12597
$962$	292.587	0.00980602
$963$	0	0
$964$	1676.16	0.0560015
$965$	0	0
$966$	0	0
$967$	−3237.13	−0.107651	−0.0538257	−	0.998550i	$-0.517142\pi$
$967$	−3237.13	−0.107651	−0.0538257	+	0.998550i	$0.517142\pi$
$968$	−945.146	−0.0313824
$969$	0	0
$970$	0	0
$971$	1731.06	0.0572114	0.0286057	−	0.999591i	$-0.490893\pi$
$971$	1731.06	0.0572114	0.0286057	+	0.999591i	$0.490893\pi$
$972$	0	0
$973$	1159.86	0.0382151
$974$	−2460.45	−0.0809425
$975$	0	0
$976$	−48049.5	−1.57585
$977$	34546.9	1.13127	0.565636	−	0.824655i	$-0.308631\pi$
$977$	34546.9	1.13127	0.565636	+	0.824655i	$0.308631\pi$
$978$	0	0
$979$	−47424.7	−1.54821
$980$	0	0
$981$	0	0
$982$	−163.201	−0.00530343
$983$	20453.2	0.663637	0.331818	−	0.943343i	$-0.392338\pi$
$983$	20453.2	0.663637	0.331818	+	0.943343i	$0.392338\pi$
$984$	0	0
$985$	0	0
$986$	−29.0407	−0.000937975 0
$987$	0	0
$988$	−19316.6	−0.622008
$989$	102502.	3.29563
$990$	0	0
$991$	−5387.77	−0.172703	−0.0863513	−	0.996265i	$-0.527521\pi$
$991$	−5387.77	−0.172703	−0.0863513	+	0.996265i	$0.527521\pi$
$992$	8412.05	0.269237
$993$	0	0
$994$	1374.69	0.0438658
$995$	0	0
$996$	0	0
$997$	21844.9	0.693916	0.346958	−	0.937881i	$-0.387215\pi$
$997$	21844.9	0.693916	0.346958	+	0.937881i	$0.387215\pi$
$998$	2525.16	0.0800929
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2025.4.a.q.1.2		3
3.2	odd	2		2025.4.a.s.1.2		3
5.4	even	2		405.4.a.j.1.2		3
9.2	odd	6		225.4.e.c.76.2		6
9.5	odd	6		225.4.e.c.151.2		6
15.14	odd	2		405.4.a.h.1.2		3
45.2	even	12		225.4.k.c.49.4		12
45.4	even	6		135.4.e.b.46.2		6
45.14	odd	6		45.4.e.b.16.2	✓	6
45.23	even	12		225.4.k.c.124.4		12
45.29	odd	6		45.4.e.b.31.2	yes	6
45.32	even	12		225.4.k.c.124.3		12
45.34	even	6		135.4.e.b.91.2		6
45.38	even	12		225.4.k.c.49.3		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
45.4.e.b.16.2	✓	6	45.14	odd	6
45.4.e.b.31.2	yes	6	45.29	odd	6
135.4.e.b.46.2		6	45.4	even	6
135.4.e.b.91.2		6	45.34	even	6
225.4.e.c.76.2		6	9.2	odd	6
225.4.e.c.151.2		6	9.5	odd	6
225.4.k.c.49.3		12	45.38	even	12
225.4.k.c.49.4		12	45.2	even	12
225.4.k.c.124.3		12	45.32	even	12
225.4.k.c.124.4		12	45.23	even	12
405.4.a.h.1.2		3	15.14	odd	2
405.4.a.j.1.2		3	5.4	even	2
2025.4.a.q.1.2		3	1.1	even	1	trivial
2025.4.a.s.1.2		3	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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q-0.174985 q^{2} -7.96938 q^{4} -8.46371 q^{7} +2.79440 q^{8} +O(q^{10})\)	\(q-0.174985 q^{2} -7.96938 q^{4} -8.46371 q^{7} +2.79440 q^{8} +31.5083 q^{11} -26.8696 q^{13} +1.48102 q^{14} +63.2661 q^{16} -44.3307 q^{17} -90.2082 q^{19} -5.51346 q^{22} +194.257 q^{23} +4.70176 q^{26} +67.4506 q^{28} -3.74371 q^{29} -251.664 q^{31} -33.4258 q^{32} +7.75719 q^{34} +62.2293 q^{37} +15.7850 q^{38} +204.346 q^{41} +527.661 q^{43} -251.101 q^{44} -33.9920 q^{46} +155.727 q^{47} -271.366 q^{49} +214.134 q^{52} -141.694 q^{53} -23.6510 q^{56} +0.655092 q^{58} +493.845 q^{59} -759.483 q^{61} +44.0373 q^{62} -500.280 q^{64} +543.590 q^{67} +353.288 q^{68} +928.207 q^{71} -608.739 q^{73} -10.8892 q^{74} +718.903 q^{76} -266.677 q^{77} +614.840 q^{79} -35.7573 q^{82} +1075.31 q^{83} -92.3326 q^{86} +88.0466 q^{88} -1505.15 q^{89} +227.416 q^{91} -1548.11 q^{92} -27.2499 q^{94} +332.738 q^{97} +47.4848 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - q^{2} + 11 q^{4} + 43 q^{7} - 27 q^{8}+O(q^{10}) \) 3 * q - q^2 + 11 * q^4 + 43 * q^7 - 27 * q^8	\( 3 q - q^{2} + 11 q^{4} + 43 q^{7} - 27 q^{8} - 14 q^{11} - 40 q^{13} + 27 q^{14} - 13 q^{16} - 166 q^{17} - 164 q^{19} + 376 q^{22} + 171 q^{23} - 434 q^{26} + 517 q^{28} + 335 q^{29} - 352 q^{31} - 77 q^{32} - 52 q^{34} - 402 q^{37} - 178 q^{38} - 187 q^{41} + 602 q^{43} - 982 q^{44} - 201 q^{46} + 665 q^{47} + 430 q^{49} + 456 q^{52} - 730 q^{53} - 705 q^{56} - 217 q^{58} + 298 q^{59} - 1439 q^{61} - 1614 q^{62} - 1569 q^{64} + 1849 q^{67} - 710 q^{68} - 70 q^{71} + 368 q^{73} + 320 q^{74} + 204 q^{76} - 948 q^{77} - 382 q^{79} + 575 q^{82} - 831 q^{83} - 1580 q^{86} + 1428 q^{88} - 1719 q^{89} - 710 q^{91} - 1623 q^{92} - 2077 q^{94} + 282 q^{97} + 2164 q^{98}+O(q^{100}) \) 3 * q - q^2 + 11 * q^4 + 43 * q^7 - 27 * q^8 - 14 * q^11 - 40 * q^13 + 27 * q^14 - 13 * q^16 - 166 * q^17 - 164 * q^19 + 376 * q^22 + 171 * q^23 - 434 * q^26 + 517 * q^28 + 335 * q^29 - 352 * q^31 - 77 * q^32 - 52 * q^34 - 402 * q^37 - 178 * q^38 - 187 * q^41 + 602 * q^43 - 982 * q^44 - 201 * q^46 + 665 * q^47 + 430 * q^49 + 456 * q^52 - 730 * q^53 - 705 * q^56 - 217 * q^58 + 298 * q^59 - 1439 * q^61 - 1614 * q^62 - 1569 * q^64 + 1849 * q^67 - 710 * q^68 - 70 * q^71 + 368 * q^73 + 320 * q^74 + 204 * q^76 - 948 * q^77 - 382 * q^79 + 575 * q^82 - 831 * q^83 - 1580 * q^86 + 1428 * q^88 - 1719 * q^89 - 710 * q^91 - 1623 * q^92 - 2077 * q^94 + 282 * q^97 + 2164 * q^98

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