LMFDB - Embedded newform 2025.2.a.t.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2025,2,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, 2, names="a")

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

chi := DirichletCharacter("2025.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2025 = 3^{4} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
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:	$16.1697064093$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{24})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 4x^{2} + 1 $ x^4 - 4*x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 45)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.517638$ 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)

$f(q)$	$=$	$q-0.517638 q^{2} -1.73205 q^{4} +3.34607 q^{7} +1.93185 q^{8} +O(q^{10})$	$q-0.517638 q^{2} -1.73205 q^{4} +3.34607 q^{7} +1.93185 q^{8} +1.26795 q^{11} -2.44949 q^{13} -1.73205 q^{14} +2.46410 q^{16} +5.27792 q^{17} -0.732051 q^{19} -0.656339 q^{22} +0.517638 q^{23} +1.26795 q^{26} -5.79555 q^{28} -0.464102 q^{29} +0.732051 q^{31} -5.13922 q^{32} -2.73205 q^{34} -4.24264 q^{37} +0.378937 q^{38} +7.73205 q^{41} -0.656339 q^{43} -2.19615 q^{44} -0.267949 q^{46} -2.96713 q^{47} +4.19615 q^{49} +4.24264 q^{52} -1.03528 q^{53} +6.46410 q^{56} +0.240237 q^{58} +9.46410 q^{59} -6.66025 q^{61} -0.378937 q^{62} -2.26795 q^{64} -7.58871 q^{67} -9.14162 q^{68} +14.1962 q^{71} +8.48528 q^{73} +2.19615 q^{74} +1.26795 q^{76} +4.24264 q^{77} +7.46410 q^{79} -4.00240 q^{82} -7.96764 q^{83} +0.339746 q^{86} +2.44949 q^{88} +13.3923 q^{89} -8.19615 q^{91} -0.896575 q^{92} +1.53590 q^{94} -15.1774 q^{97} -2.17209 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q+O(q^{10}) $ 4 * q	$ 4 q + 12 q^{11} - 4 q^{16} + 4 q^{19} + 12 q^{26} + 12 q^{29} - 4 q^{31} - 4 q^{34} + 24 q^{41} + 12 q^{44} - 8 q^{46} - 4 q^{49} + 12 q^{56} + 24 q^{59} + 8 q^{61} - 16 q^{64} + 36 q^{71} - 12 q^{74} + 12 q^{76} + 16 q^{79} + 36 q^{86} + 12 q^{89} - 12 q^{91} + 20 q^{94}+O(q^{100}) $ 4 * q + 12 * q^11 - 4 * q^16 + 4 * q^19 + 12 * q^26 + 12 * q^29 - 4 * q^31 - 4 * q^34 + 24 * q^41 + 12 * q^44 - 8 * q^46 - 4 * q^49 + 12 * q^56 + 24 * q^59 + 8 * q^61 - 16 * q^64 + 36 * q^71 - 12 * q^74 + 12 * q^76 + 16 * q^79 + 36 * q^86 + 12 * q^89 - 12 * q^91 + 20 * 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 $

$2$	−0.517638	−0.366025	−0.183013	−	0.983111i	$-0.558585\pi$
$2$	−0.517638	−0.366025	−0.183013	+	0.983111i	$0.558585\pi$
$3$	0	0
$3$	0	0
$4$	−1.73205	−0.866025
$5$	0	0
$5$	0	0
$6$	0	0
$7$	3.34607	1.26469	0.632347	−	0.774685i	$-0.282092\pi$
$7$	3.34607	1.26469	0.632347	+	0.774685i	$0.282092\pi$
$8$	1.93185	0.683013
$9$	0	0
$10$	0	0
$11$	1.26795	0.382301	0.191151	−	0.981561i	$-0.438778\pi$
$11$	1.26795	0.382301	0.191151	+	0.981561i	$0.438778\pi$
$12$	0	0
$13$	−2.44949	−0.679366	−0.339683	−	0.940540i	$-0.610320\pi$
$13$	−2.44949	−0.679366	−0.339683	+	0.940540i	$0.610320\pi$
$14$	−1.73205	−0.462910
$15$	0	0
$16$	2.46410	0.616025
$17$	5.27792	1.28008	0.640041	−	0.768340i	$-0.278917\pi$
$17$	5.27792	1.28008	0.640041	+	0.768340i	$0.278917\pi$
$18$	0	0
$19$	−0.732051	−0.167944	−0.0839720	−	0.996468i	$-0.526761\pi$
$19$	−0.732051	−0.167944	−0.0839720	+	0.996468i	$0.526761\pi$
$20$	0	0
$21$	0	0
$22$	−0.656339	−0.139932
$23$	0.517638	0.107935	0.0539675	−	0.998543i	$-0.482813\pi$
$23$	0.517638	0.107935	0.0539675	+	0.998543i	$0.482813\pi$
$24$	0	0
$25$	0	0
$26$	1.26795	0.248665
$27$	0	0
$28$	−5.79555	−1.09526
$29$	−0.464102	−0.0861815	−0.0430908	−	0.999071i	$-0.513720\pi$
$29$	−0.464102	−0.0861815	−0.0430908	+	0.999071i	$0.513720\pi$
$30$	0	0
$31$	0.732051	0.131480	0.0657401	−	0.997837i	$-0.479059\pi$
$31$	0.732051	0.131480	0.0657401	+	0.997837i	$0.479059\pi$
$32$	−5.13922	−0.908494
$33$	0	0
$34$	−2.73205	−0.468543
$35$	0	0
$36$	0	0
$37$	−4.24264	−0.697486	−0.348743	−	0.937218i	$-0.613391\pi$
$37$	−4.24264	−0.697486	−0.348743	+	0.937218i	$0.613391\pi$
$38$	0.378937	0.0614718
$39$	0	0
$40$	0	0
$41$	7.73205	1.20754	0.603772	−	0.797157i	$-0.293664\pi$
$41$	7.73205	1.20754	0.603772	+	0.797157i	$0.293664\pi$
$42$	0	0
$43$	−0.656339	−0.100091	−0.0500454	−	0.998747i	$-0.515937\pi$
$43$	−0.656339	−0.100091	−0.0500454	+	0.998747i	$0.515937\pi$
$44$	−2.19615	−0.331082
$45$	0	0
$46$	−0.267949	−0.0395070
$47$	−2.96713	−0.432800	−0.216400	−	0.976305i	$-0.569432\pi$
$47$	−2.96713	−0.432800	−0.216400	+	0.976305i	$0.569432\pi$
$48$	0	0
$49$	4.19615	0.599450
$50$	0	0
$51$	0	0
$52$	4.24264	0.588348
$53$	−1.03528	−0.142206	−0.0711031	−	0.997469i	$-0.522652\pi$
$53$	−1.03528	−0.142206	−0.0711031	+	0.997469i	$0.522652\pi$
$54$	0	0
$55$	0	0
$56$	6.46410	0.863802
$57$	0	0
$58$	0.240237	0.0315446
$59$	9.46410	1.23212	0.616061	−	0.787699i	$-0.288728\pi$
$59$	9.46410	1.23212	0.616061	+	0.787699i	$0.288728\pi$
$60$	0	0
$61$	−6.66025	−0.852758	−0.426379	−	0.904545i	$-0.640211\pi$
$61$	−6.66025	−0.852758	−0.426379	+	0.904545i	$0.640211\pi$
$62$	−0.378937	−0.0481251
$63$	0	0
$64$	−2.26795	−0.283494
$65$	0	0
$66$	0	0
$67$	−7.58871	−0.927108	−0.463554	−	0.886069i	$-0.653426\pi$
$67$	−7.58871	−0.927108	−0.463554	+	0.886069i	$0.653426\pi$
$68$	−9.14162	−1.10858
$69$	0	0
$70$	0	0
$71$	14.1962	1.68477	0.842387	−	0.538874i	$-0.181150\pi$
$71$	14.1962	1.68477	0.842387	+	0.538874i	$0.181150\pi$
$72$	0	0
$73$	8.48528	0.993127	0.496564	−	0.868000i	$-0.334595\pi$
$73$	8.48528	0.993127	0.496564	+	0.868000i	$0.334595\pi$
$74$	2.19615	0.255298
$75$	0	0
$76$	1.26795	0.145444
$77$	4.24264	0.483494
$78$	0	0
$79$	7.46410	0.839777	0.419889	−	0.907576i	$-0.362069\pi$
$79$	7.46410	0.839777	0.419889	+	0.907576i	$0.362069\pi$
$80$	0	0
$81$	0	0
$82$	−4.00240	−0.441992
$83$	−7.96764	−0.874562	−0.437281	−	0.899325i	$-0.644059\pi$
$83$	−7.96764	−0.874562	−0.437281	+	0.899325i	$0.644059\pi$
$84$	0	0
$85$	0	0
$86$	0.339746	0.0366357
$87$	0	0
$88$	2.44949	0.261116
$89$	13.3923	1.41958	0.709791	−	0.704413i	$-0.248789\pi$
$89$	13.3923	1.41958	0.709791	+	0.704413i	$0.248789\pi$
$90$	0	0
$91$	−8.19615	−0.859190
$92$	−0.896575	−0.0934745
$93$	0	0
$94$	1.53590	0.158416
$95$	0	0
$96$	0	0
$97$	−15.1774	−1.54103	−0.770516	−	0.637420i	$-0.780002\pi$
$97$	−15.1774	−1.54103	−0.770516	+	0.637420i	$0.780002\pi$
$98$	−2.17209	−0.219414
$99$	0	0
$100$	0	0
$101$	9.46410	0.941713	0.470857	−	0.882210i	$-0.343945\pi$
$101$	9.46410	0.941713	0.470857	+	0.882210i	$0.343945\pi$
$102$	0	0
$103$	0.656339	0.0646710	0.0323355	−	0.999477i	$-0.489705\pi$
$103$	0.656339	0.0646710	0.0323355	+	0.999477i	$0.489705\pi$
$104$	−4.73205	−0.464016
$105$	0	0
$106$	0.535898	0.0520511
$107$	−10.3156	−0.997246	−0.498623	−	0.866819i	$-0.666161\pi$
$107$	−10.3156	−0.997246	−0.498623	+	0.866819i	$0.666161\pi$
$108$	0	0
$109$	12.6603	1.21263	0.606316	−	0.795224i	$-0.292647\pi$
$109$	12.6603	1.21263	0.606316	+	0.795224i	$0.292647\pi$
$110$	0	0
$111$	0	0
$112$	8.24504	0.779083
$113$	17.2480	1.62255	0.811276	−	0.584663i	$-0.198773\pi$
$113$	17.2480	1.62255	0.811276	+	0.584663i	$0.198773\pi$
$114$	0	0
$115$	0	0
$116$	0.803848	0.0746354
$117$	0	0
$118$	−4.89898	−0.450988
$119$	17.6603	1.61891
$120$	0	0
$121$	−9.39230	−0.853846
$122$	3.44760	0.312131
$123$	0	0
$124$	−1.26795	−0.113865
$125$	0	0
$126$	0	0
$127$	17.3867	1.54282	0.771409	−	0.636340i	$-0.219553\pi$
$127$	17.3867	1.54282	0.771409	+	0.636340i	$0.219553\pi$
$128$	11.4524	1.01226
$129$	0	0
$130$	0	0
$131$	12.9282	1.12954	0.564771	−	0.825248i	$-0.308964\pi$
$131$	12.9282	1.12954	0.564771	+	0.825248i	$0.308964\pi$
$132$	0	0
$133$	−2.44949	−0.212398
$134$	3.92820	0.339345
$135$	0	0
$136$	10.1962	0.874313
$137$	15.7322	1.34409	0.672047	−	0.740509i	$-0.265415\pi$
$137$	15.7322	1.34409	0.672047	+	0.740509i	$0.265415\pi$
$138$	0	0
$139$	−8.00000	−0.678551	−0.339276	−	0.940687i	$-0.610182\pi$
$139$	−8.00000	−0.678551	−0.339276	+	0.940687i	$0.610182\pi$
$140$	0	0
$141$	0	0
$142$	−7.34847	−0.616670
$143$	−3.10583	−0.259722
$144$	0	0
$145$	0	0
$146$	−4.39230	−0.363510
$147$	0	0
$148$	7.34847	0.604040
$149$	4.26795	0.349644	0.174822	−	0.984600i	$-0.444065\pi$
$149$	4.26795	0.349644	0.174822	+	0.984600i	$0.444065\pi$
$150$	0	0
$151$	8.58846	0.698919	0.349459	−	0.936952i	$-0.386365\pi$
$151$	8.58846	0.698919	0.349459	+	0.936952i	$0.386365\pi$
$152$	−1.41421	−0.114708
$153$	0	0
$154$	−2.19615	−0.176971
$155$	0	0
$156$	0	0
$157$	12.9038	1.02983	0.514917	−	0.857240i	$-0.327823\pi$
$157$	12.9038	1.02983	0.514917	+	0.857240i	$0.327823\pi$
$158$	−3.86370	−0.307380
$159$	0	0
$160$	0	0
$161$	1.73205	0.136505
$162$	0	0
$163$	10.4543	0.818844	0.409422	−	0.912345i	$-0.365730\pi$
$163$	10.4543	0.818844	0.409422	+	0.912345i	$0.365730\pi$
$164$	−13.3923	−1.04576
$165$	0	0
$166$	4.12436	0.320112
$167$	9.93666	0.768922	0.384461	−	0.923141i	$-0.374387\pi$
$167$	9.93666	0.768922	0.384461	+	0.923141i	$0.374387\pi$
$168$	0	0
$169$	−7.00000	−0.538462
$170$	0	0
$171$	0	0
$172$	1.13681	0.0866811
$173$	10.5558	0.802545	0.401273	−	0.915959i	$-0.368568\pi$
$173$	10.5558	0.802545	0.401273	+	0.915959i	$0.368568\pi$
$174$	0	0
$175$	0	0
$176$	3.12436	0.235507
$177$	0	0
$178$	−6.93237	−0.519603
$179$	−6.58846	−0.492444	−0.246222	−	0.969213i	$-0.579189\pi$
$179$	−6.58846	−0.492444	−0.246222	+	0.969213i	$0.579189\pi$
$180$	0	0
$181$	1.53590	0.114162	0.0570812	−	0.998370i	$-0.481821\pi$
$181$	1.53590	0.114162	0.0570812	+	0.998370i	$0.481821\pi$
$182$	4.24264	0.314485
$183$	0	0
$184$	1.00000	0.0737210
$185$	0	0
$186$	0	0
$187$	6.69213	0.489377
$188$	5.13922	0.374816
$189$	0	0
$190$	0	0
$191$	17.6603	1.27785	0.638926	−	0.769269i	$-0.279379\pi$
$191$	17.6603	1.27785	0.638926	+	0.769269i	$0.279379\pi$
$192$	0	0
$193$	3.76217	0.270807	0.135403	−	0.990791i	$-0.456767\pi$
$193$	3.76217	0.270807	0.135403	+	0.990791i	$0.456767\pi$
$194$	7.85641	0.564057
$195$	0	0
$196$	−7.26795	−0.519139
$197$	−15.9353	−1.13534	−0.567671	−	0.823255i	$-0.692155\pi$
$197$	−15.9353	−1.13534	−0.567671	+	0.823255i	$0.692155\pi$
$198$	0	0
$199$	−26.5885	−1.88481	−0.942403	−	0.334480i	$-0.891439\pi$
$199$	−26.5885	−1.88481	−0.942403	+	0.334480i	$0.891439\pi$
$200$	0	0
$201$	0	0
$202$	−4.89898	−0.344691
$203$	−1.55291	−0.108993
$204$	0	0
$205$	0	0
$206$	−0.339746	−0.0236712
$207$	0	0
$208$	−6.03579	−0.418507
$209$	−0.928203	−0.0642052
$210$	0	0
$211$	11.1244	0.765832	0.382916	−	0.923783i	$-0.374920\pi$
$211$	11.1244	0.765832	0.382916	+	0.923783i	$0.374920\pi$
$212$	1.79315	0.123154
$213$	0	0
$214$	5.33975	0.365018
$215$	0	0
$216$	0	0
$217$	2.44949	0.166282
$218$	−6.55343	−0.443854
$219$	0	0
$220$	0	0
$221$	−12.9282	−0.869645
$222$	0	0
$223$	−4.48288	−0.300196	−0.150098	−	0.988671i	$-0.547959\pi$
$223$	−4.48288	−0.300196	−0.150098	+	0.988671i	$0.547959\pi$
$224$	−17.1962	−1.14897
$225$	0	0
$226$	−8.92820	−0.593895
$227$	−26.3896	−1.75154	−0.875769	−	0.482730i	$-0.839645\pi$
$227$	−26.3896	−1.75154	−0.875769	+	0.482730i	$0.839645\pi$
$228$	0	0
$229$	6.07180	0.401236	0.200618	−	0.979670i	$-0.435705\pi$
$229$	6.07180	0.401236	0.200618	+	0.979670i	$0.435705\pi$
$230$	0	0
$231$	0	0
$232$	−0.896575	−0.0588631
$233$	−7.07107	−0.463241	−0.231621	−	0.972806i	$-0.574403\pi$
$233$	−7.07107	−0.463241	−0.231621	+	0.972806i	$0.574403\pi$
$234$	0	0
$235$	0	0
$236$	−16.3923	−1.06705
$237$	0	0
$238$	−9.14162	−0.592563
$239$	−0.928203	−0.0600405	−0.0300202	−	0.999549i	$-0.509557\pi$
$239$	−0.928203	−0.0600405	−0.0300202	+	0.999549i	$0.509557\pi$
$240$	0	0
$241$	9.73205	0.626897	0.313448	−	0.949605i	$-0.398516\pi$
$241$	9.73205	0.626897	0.313448	+	0.949605i	$0.398516\pi$
$242$	4.86181	0.312529
$243$	0	0
$244$	11.5359	0.738510
$245$	0	0
$246$	0	0
$247$	1.79315	0.114095
$248$	1.41421	0.0898027
$249$	0	0
$250$	0	0
$251$	−12.5885	−0.794576	−0.397288	−	0.917694i	$-0.630049\pi$
$251$	−12.5885	−0.794576	−0.397288	+	0.917694i	$0.630049\pi$
$252$	0	0
$253$	0.656339	0.0412637
$254$	−9.00000	−0.564710
$255$	0	0
$256$	−1.39230	−0.0870191
$257$	−16.7675	−1.04593	−0.522964	−	0.852355i	$-0.675174\pi$
$257$	−16.7675	−1.04593	−0.522964	+	0.852355i	$0.675174\pi$
$258$	0	0
$259$	−14.1962	−0.882106
$260$	0	0
$261$	0	0
$262$	−6.69213	−0.413441
$263$	−21.7680	−1.34227	−0.671136	−	0.741334i	$-0.734194\pi$
$263$	−21.7680	−1.34227	−0.671136	+	0.741334i	$0.734194\pi$
$264$	0	0
$265$	0	0
$266$	1.26795	0.0777430
$267$	0	0
$268$	13.1440	0.802899
$269$	6.80385	0.414838	0.207419	−	0.978252i	$-0.433494\pi$
$269$	6.80385	0.414838	0.207419	+	0.978252i	$0.433494\pi$
$270$	0	0
$271$	21.5167	1.30704	0.653522	−	0.756908i	$-0.273291\pi$
$271$	21.5167	1.30704	0.653522	+	0.756908i	$0.273291\pi$
$272$	13.0053	0.788564
$273$	0	0
$274$	−8.14359	−0.491972
$275$	0	0
$276$	0	0
$277$	12.2474	0.735878	0.367939	−	0.929850i	$-0.380064\pi$
$277$	12.2474	0.735878	0.367939	+	0.929850i	$0.380064\pi$
$278$	4.14110	0.248367
$279$	0	0
$280$	0	0
$281$	−19.7321	−1.17712	−0.588558	−	0.808455i	$-0.700304\pi$
$281$	−19.7321	−1.17712	−0.588558	+	0.808455i	$0.700304\pi$
$282$	0	0
$283$	−16.7303	−0.994515	−0.497257	−	0.867603i	$-0.665660\pi$
$283$	−16.7303	−0.994515	−0.497257	+	0.867603i	$0.665660\pi$
$284$	−24.5885	−1.45906
$285$	0	0
$286$	1.60770	0.0950650
$287$	25.8719	1.52717
$288$	0	0
$289$	10.8564	0.638612
$290$	0	0
$291$	0	0
$292$	−14.6969	−0.860073
$293$	−13.7632	−0.804055	−0.402027	−	0.915628i	$-0.631694\pi$
$293$	−13.7632	−0.804055	−0.402027	+	0.915628i	$0.631694\pi$
$294$	0	0
$295$	0	0
$296$	−8.19615	−0.476392
$297$	0	0
$298$	−2.20925	−0.127979
$299$	−1.26795	−0.0733274
$300$	0	0
$301$	−2.19615	−0.126584
$302$	−4.44571	−0.255822
$303$	0	0
$304$	−1.80385	−0.103458
$305$	0	0
$306$	0	0
$307$	−3.82654	−0.218392	−0.109196	−	0.994020i	$-0.534828\pi$
$307$	−3.82654	−0.218392	−0.109196	+	0.994020i	$0.534828\pi$
$308$	−7.34847	−0.418718
$309$	0	0
$310$	0	0
$311$	−10.0526	−0.570028	−0.285014	−	0.958523i	$-0.591998\pi$
$311$	−10.0526	−0.570028	−0.285014	+	0.958523i	$0.591998\pi$
$312$	0	0
$313$	14.0406	0.793622	0.396811	−	0.917900i	$-0.370117\pi$
$313$	14.0406	0.793622	0.396811	+	0.917900i	$0.370117\pi$
$314$	−6.67949	−0.376946
$315$	0	0
$316$	−12.9282	−0.727268
$317$	−4.03957	−0.226885	−0.113442	−	0.993545i	$-0.536188\pi$
$317$	−4.03957	−0.226885	−0.113442	+	0.993545i	$0.536188\pi$
$318$	0	0
$319$	−0.588457	−0.0329473
$320$	0	0
$321$	0	0
$322$	−0.896575	−0.0499642
$323$	−3.86370	−0.214982
$324$	0	0
$325$	0	0
$326$	−5.41154	−0.299718
$327$	0	0
$328$	14.9372	0.824768
$329$	−9.92820	−0.547360
$330$	0	0
$331$	−8.39230	−0.461283	−0.230641	−	0.973039i	$-0.574082\pi$
$331$	−8.39230	−0.461283	−0.230641	+	0.973039i	$0.574082\pi$
$332$	13.8004	0.757393
$333$	0	0
$334$	−5.14359	−0.281445
$335$	0	0
$336$	0	0
$337$	6.69213	0.364544	0.182272	−	0.983248i	$-0.441655\pi$
$337$	6.69213	0.364544	0.182272	+	0.983248i	$0.441655\pi$
$338$	3.62347	0.197091
$339$	0	0
$340$	0	0
$341$	0.928203	0.0502650
$342$	0	0
$343$	−9.38186	−0.506573
$344$	−1.26795	−0.0683632
$345$	0	0
$346$	−5.46410	−0.293752
$347$	−9.89949	−0.531433	−0.265716	−	0.964051i	$-0.585608\pi$
$347$	−9.89949	−0.531433	−0.265716	+	0.964051i	$0.585608\pi$
$348$	0	0
$349$	28.4641	1.52365	0.761824	−	0.647784i	$-0.224304\pi$
$349$	28.4641	1.52365	0.761824	+	0.647784i	$0.224304\pi$
$350$	0	0
$351$	0	0
$352$	−6.51626	−0.347318
$353$	20.3538	1.08332	0.541662	−	0.840597i	$-0.317795\pi$
$353$	20.3538	1.08332	0.541662	+	0.840597i	$0.317795\pi$
$354$	0	0
$355$	0	0
$356$	−23.1962	−1.22939
$357$	0	0
$358$	3.41044	0.180247
$359$	20.1962	1.06591	0.532956	−	0.846143i	$-0.321081\pi$
$359$	20.1962	1.06591	0.532956	+	0.846143i	$0.321081\pi$
$360$	0	0
$361$	−18.4641	−0.971795
$362$	−0.795040	−0.0417863
$363$	0	0
$364$	14.1962	0.744081
$365$	0	0
$366$	0	0
$367$	34.5975	1.80597	0.902986	−	0.429669i	$-0.141370\pi$
$367$	34.5975	1.80597	0.902986	+	0.429669i	$0.141370\pi$
$368$	1.27551	0.0664907
$369$	0	0
$370$	0	0
$371$	−3.46410	−0.179847
$372$	0	0
$373$	27.2490	1.41090	0.705450	−	0.708760i	$-0.250745\pi$
$373$	27.2490	1.41090	0.705450	+	0.708760i	$0.250745\pi$
$374$	−3.46410	−0.179124
$375$	0	0
$376$	−5.73205	−0.295608
$377$	1.13681	0.0585488
$378$	0	0
$379$	19.4641	0.999804	0.499902	−	0.866082i	$-0.333369\pi$
$379$	19.4641	0.999804	0.499902	+	0.866082i	$0.333369\pi$
$380$	0	0
$381$	0	0
$382$	−9.14162	−0.467726
$383$	−10.6574	−0.544566	−0.272283	−	0.962217i	$-0.587779\pi$
$383$	−10.6574	−0.544566	−0.272283	+	0.962217i	$0.587779\pi$
$384$	0	0
$385$	0	0
$386$	−1.94744	−0.0991221
$387$	0	0
$388$	26.2880	1.33457
$389$	0.124356	0.00630508	0.00315254	−	0.999995i	$-0.498997\pi$
$389$	0.124356	0.00630508	0.00315254	+	0.999995i	$0.498997\pi$
$390$	0	0
$391$	2.73205	0.138166
$392$	8.10634	0.409432
$393$	0	0
$394$	8.24871	0.415564
$395$	0	0
$396$	0	0
$397$	−29.3939	−1.47524	−0.737618	−	0.675218i	$-0.764050\pi$
$397$	−29.3939	−1.47524	−0.737618	+	0.675218i	$0.764050\pi$
$398$	13.7632	0.689887
$399$	0	0
$400$	0	0
$401$	−17.0718	−0.852525	−0.426262	−	0.904600i	$-0.640170\pi$
$401$	−17.0718	−0.852525	−0.426262	+	0.904600i	$0.640170\pi$
$402$	0	0
$403$	−1.79315	−0.0893232
$404$	−16.3923	−0.815548
$405$	0	0
$406$	0.803848	0.0398943
$407$	−5.37945	−0.266650
$408$	0	0
$409$	−16.5359	−0.817648	−0.408824	−	0.912613i	$-0.634061\pi$
$409$	−16.5359	−0.817648	−0.408824	+	0.912613i	$0.634061\pi$
$410$	0	0
$411$	0	0
$412$	−1.13681	−0.0560067
$413$	31.6675	1.55826
$414$	0	0
$415$	0	0
$416$	12.5885	0.617200
$417$	0	0
$418$	0.480473	0.0235007
$419$	22.0526	1.07734	0.538669	−	0.842517i	$-0.318927\pi$
$419$	22.0526	1.07734	0.538669	+	0.842517i	$0.318927\pi$
$420$	0	0
$421$	−19.4641	−0.948622	−0.474311	−	0.880357i	$-0.657303\pi$
$421$	−19.4641	−0.948622	−0.474311	+	0.880357i	$0.657303\pi$
$422$	−5.75839	−0.280314
$423$	0	0
$424$	−2.00000	−0.0971286
$425$	0	0
$426$	0	0
$427$	−22.2856	−1.07848
$428$	17.8671	0.863641
$429$	0	0
$430$	0	0
$431$	−6.00000	−0.289010	−0.144505	−	0.989504i	$-0.546159\pi$
$431$	−6.00000	−0.289010	−0.144505	+	0.989504i	$0.546159\pi$
$432$	0	0
$433$	−13.5601	−0.651658	−0.325829	−	0.945429i	$-0.605643\pi$
$433$	−13.5601	−0.651658	−0.325829	+	0.945429i	$0.605643\pi$
$434$	−1.26795	−0.0608635
$435$	0	0
$436$	−21.9282	−1.05017
$437$	−0.378937	−0.0181270
$438$	0	0
$439$	−25.3205	−1.20848	−0.604241	−	0.796802i	$-0.706524\pi$
$439$	−25.3205	−1.20848	−0.604241	+	0.796802i	$0.706524\pi$
$440$	0	0
$441$	0	0
$442$	6.69213	0.318312
$443$	1.65445	0.0786053	0.0393027	−	0.999227i	$-0.487486\pi$
$443$	1.65445	0.0786053	0.0393027	+	0.999227i	$0.487486\pi$
$444$	0	0
$445$	0	0
$446$	2.32051	0.109879
$447$	0	0
$448$	−7.58871	−0.358533
$449$	−12.0000	−0.566315	−0.283158	−	0.959073i	$-0.591382\pi$
$449$	−12.0000	−0.566315	−0.283158	+	0.959073i	$0.591382\pi$
$450$	0	0
$451$	9.80385	0.461645
$452$	−29.8744	−1.40517
$453$	0	0
$454$	13.6603	0.641107
$455$	0	0
$456$	0	0
$457$	−24.7995	−1.16007	−0.580036	−	0.814591i	$-0.696962\pi$
$457$	−24.7995	−1.16007	−0.580036	+	0.814591i	$0.696962\pi$
$458$	−3.14299	−0.146862
$459$	0	0
$460$	0	0
$461$	−36.7128	−1.70989	−0.854943	−	0.518722i	$-0.826408\pi$
$461$	−36.7128	−1.70989	−0.854943	+	0.518722i	$0.826408\pi$
$462$	0	0
$463$	−34.1170	−1.58555	−0.792776	−	0.609514i	$-0.791365\pi$
$463$	−34.1170	−1.58555	−0.792776	+	0.609514i	$0.791365\pi$
$464$	−1.14359	−0.0530900
$465$	0	0
$466$	3.66025	0.169558
$467$	25.3543	1.17326	0.586629	−	0.809856i	$-0.300455\pi$
$467$	25.3543	1.17326	0.586629	+	0.809856i	$0.300455\pi$
$468$	0	0
$469$	−25.3923	−1.17251
$470$	0	0
$471$	0	0
$472$	18.2832	0.841554
$473$	−0.832204	−0.0382648
$474$	0	0
$475$	0	0
$476$	−30.5885	−1.40202
$477$	0	0
$478$	0.480473	0.0219763
$479$	4.14359	0.189326	0.0946628	−	0.995509i	$-0.469823\pi$
$479$	4.14359	0.189326	0.0946628	+	0.995509i	$0.469823\pi$
$480$	0	0
$481$	10.3923	0.473848
$482$	−5.03768	−0.229460
$483$	0	0
$484$	16.2679	0.739452
$485$	0	0
$486$	0	0
$487$	15.8338	0.717496	0.358748	−	0.933435i	$-0.383204\pi$
$487$	15.8338	0.717496	0.358748	+	0.933435i	$0.383204\pi$
$488$	−12.8666	−0.582445
$489$	0	0
$490$	0	0
$491$	33.7128	1.52144	0.760719	−	0.649081i	$-0.224847\pi$
$491$	33.7128	1.52144	0.760719	+	0.649081i	$0.224847\pi$
$492$	0	0
$493$	−2.44949	−0.110319
$494$	−0.928203	−0.0417618
$495$	0	0
$496$	1.80385	0.0809951
$497$	47.5013	2.13072
$498$	0	0
$499$	23.8564	1.06796	0.533980	−	0.845497i	$-0.320696\pi$
$499$	23.8564	1.06796	0.533980	+	0.845497i	$0.320696\pi$
$500$	0	0
$501$	0	0
$502$	6.51626	0.290835
$503$	2.48665	0.110874	0.0554372	−	0.998462i	$-0.482345\pi$
$503$	2.48665	0.110874	0.0554372	+	0.998462i	$0.482345\pi$
$504$	0	0
$505$	0	0
$506$	−0.339746	−0.0151036
$507$	0	0
$508$	−30.1146	−1.33612
$509$	16.8564	0.747147	0.373574	−	0.927601i	$-0.378132\pi$
$509$	16.8564	0.747147	0.373574	+	0.927601i	$0.378132\pi$
$510$	0	0
$511$	28.3923	1.25600
$512$	−22.1841	−0.980408
$513$	0	0
$514$	8.67949	0.382836
$515$	0	0
$516$	0	0
$517$	−3.76217	−0.165460
$518$	7.34847	0.322873
$519$	0	0
$520$	0	0
$521$	−19.3923	−0.849592	−0.424796	−	0.905289i	$-0.639654\pi$
$521$	−19.3923	−0.849592	−0.424796	+	0.905289i	$0.639654\pi$
$522$	0	0
$523$	−35.1894	−1.53873	−0.769363	−	0.638812i	$-0.779426\pi$
$523$	−35.1894	−1.53873	−0.769363	+	0.638812i	$0.779426\pi$
$524$	−22.3923	−0.978212
$525$	0	0
$526$	11.2679	0.491306
$527$	3.86370	0.168306
$528$	0	0
$529$	−22.7321	−0.988350
$530$	0	0
$531$	0	0
$532$	4.24264	0.183942
$533$	−18.9396	−0.820364
$534$	0	0
$535$	0	0
$536$	−14.6603	−0.633227
$537$	0	0
$538$	−3.52193	−0.151841
$539$	5.32051	0.229171
$540$	0	0
$541$	0.607695	0.0261269	0.0130634	−	0.999915i	$-0.495842\pi$
$541$	0.607695	0.0261269	0.0130634	+	0.999915i	$0.495842\pi$
$542$	−11.1378	−0.478411
$543$	0	0
$544$	−27.1244	−1.16295
$545$	0	0
$546$	0	0
$547$	−26.0478	−1.11372	−0.556862	−	0.830605i	$-0.687995\pi$
$547$	−26.0478	−1.11372	−0.556862	+	0.830605i	$0.687995\pi$
$548$	−27.2490	−1.16402
$549$	0	0
$550$	0	0
$551$	0.339746	0.0144737
$552$	0	0
$553$	24.9754	1.06206
$554$	−6.33975	−0.269350
$555$	0	0
$556$	13.8564	0.587643
$557$	−31.1127	−1.31829	−0.659144	−	0.752017i	$-0.729081\pi$
$557$	−31.1127	−1.31829	−0.659144	+	0.752017i	$0.729081\pi$
$558$	0	0
$559$	1.60770	0.0679983
$560$	0	0
$561$	0	0
$562$	10.2141	0.430854
$563$	9.17878	0.386840	0.193420	−	0.981116i	$-0.438042\pi$
$563$	9.17878	0.386840	0.193420	+	0.981116i	$0.438042\pi$
$564$	0	0
$565$	0	0
$566$	8.66025	0.364018
$567$	0	0
$568$	27.4249	1.15072
$569$	29.3205	1.22918	0.614590	−	0.788847i	$-0.289322\pi$
$569$	29.3205	1.22918	0.614590	+	0.788847i	$0.289322\pi$
$570$	0	0
$571$	−1.46410	−0.0612707	−0.0306354	−	0.999531i	$-0.509753\pi$
$571$	−1.46410	−0.0612707	−0.0306354	+	0.999531i	$0.509753\pi$
$572$	5.37945	0.224926
$573$	0	0
$574$	−13.3923	−0.558984
$575$	0	0
$576$	0	0
$577$	13.8647	0.577196	0.288598	−	0.957450i	$-0.406811\pi$
$577$	13.8647	0.577196	0.288598	+	0.957450i	$0.406811\pi$
$578$	−5.61969	−0.233748
$579$	0	0
$580$	0	0
$581$	−26.6603	−1.10605
$582$	0	0
$583$	−1.31268	−0.0543656
$584$	16.3923	0.678318
$585$	0	0
$586$	7.12436	0.294304
$587$	4.86181	0.200669	0.100334	−	0.994954i	$-0.468009\pi$
$587$	4.86181	0.200669	0.100334	+	0.994954i	$0.468009\pi$
$588$	0	0
$589$	−0.535898	−0.0220813
$590$	0	0
$591$	0	0
$592$	−10.4543	−0.429669
$593$	−19.1427	−0.786094	−0.393047	−	0.919518i	$-0.628579\pi$
$593$	−19.1427	−0.786094	−0.393047	+	0.919518i	$0.628579\pi$
$594$	0	0
$595$	0	0
$596$	−7.39230	−0.302801
$597$	0	0
$598$	0.656339	0.0268397
$599$	−21.7128	−0.887161	−0.443581	−	0.896234i	$-0.646292\pi$
$599$	−21.7128	−0.887161	−0.443581	+	0.896234i	$0.646292\pi$
$600$	0	0
$601$	−3.07180	−0.125301	−0.0626506	−	0.998036i	$-0.519955\pi$
$601$	−3.07180	−0.125301	−0.0626506	+	0.998036i	$0.519955\pi$
$602$	1.13681	0.0463330
$603$	0	0
$604$	−14.8756	−0.605281
$605$	0	0
$606$	0	0
$607$	20.9730	0.851266	0.425633	−	0.904896i	$-0.360051\pi$
$607$	20.9730	0.851266	0.425633	+	0.904896i	$0.360051\pi$
$608$	3.76217	0.152576
$609$	0	0
$610$	0	0
$611$	7.26795	0.294030
$612$	0	0
$613$	−9.62209	−0.388633	−0.194316	−	0.980939i	$-0.562249\pi$
$613$	−9.62209	−0.388633	−0.194316	+	0.980939i	$0.562249\pi$
$614$	1.98076	0.0799371
$615$	0	0
$616$	8.19615	0.330232
$617$	18.0058	0.724888	0.362444	−	0.932006i	$-0.381942\pi$
$617$	18.0058	0.724888	0.362444	+	0.932006i	$0.381942\pi$
$618$	0	0
$619$	30.1962	1.21369	0.606843	−	0.794822i	$-0.292436\pi$
$619$	30.1962	1.21369	0.606843	+	0.794822i	$0.292436\pi$
$620$	0	0
$621$	0	0
$622$	5.20359	0.208645
$623$	44.8115	1.79534
$624$	0	0
$625$	0	0
$626$	−7.26795	−0.290486
$627$	0	0
$628$	−22.3500	−0.891863
$629$	−22.3923	−0.892840
$630$	0	0
$631$	1.32051	0.0525686	0.0262843	−	0.999655i	$-0.491632\pi$
$631$	1.32051	0.0525686	0.0262843	+	0.999655i	$0.491632\pi$
$632$	14.4195	0.573578
$633$	0	0
$634$	2.09103	0.0830456
$635$	0	0
$636$	0	0
$637$	−10.2784	−0.407246
$638$	0.304608	0.0120595
$639$	0	0
$640$	0	0
$641$	13.0526	0.515545	0.257773	−	0.966206i	$-0.417011\pi$
$641$	13.0526	0.515545	0.257773	+	0.966206i	$0.417011\pi$
$642$	0	0
$643$	12.9682	0.511414	0.255707	−	0.966754i	$-0.417692\pi$
$643$	12.9682	0.511414	0.255707	+	0.966754i	$0.417692\pi$
$644$	−3.00000	−0.118217
$645$	0	0
$646$	2.00000	0.0786889
$647$	−19.4572	−0.764942	−0.382471	−	0.923967i	$-0.624927\pi$
$647$	−19.4572	−0.764942	−0.382471	+	0.923967i	$0.624927\pi$
$648$	0	0
$649$	12.0000	0.471041
$650$	0	0
$651$	0	0
$652$	−18.1074	−0.709140
$653$	−48.5365	−1.89938	−0.949691	−	0.313190i	$-0.898602\pi$
$653$	−48.5365	−1.89938	−0.949691	+	0.313190i	$0.898602\pi$
$654$	0	0
$655$	0	0
$656$	19.0526	0.743877
$657$	0	0
$658$	5.13922	0.200348
$659$	−12.2487	−0.477142	−0.238571	−	0.971125i	$-0.576679\pi$
$659$	−12.2487	−0.477142	−0.238571	+	0.971125i	$0.576679\pi$
$660$	0	0
$661$	10.7846	0.419473	0.209736	−	0.977758i	$-0.432739\pi$
$661$	10.7846	0.419473	0.209736	+	0.977758i	$0.432739\pi$
$662$	4.34418	0.168841
$663$	0	0
$664$	−15.3923	−0.597337
$665$	0	0
$666$	0	0
$667$	−0.240237	−0.00930200
$668$	−17.2108	−0.665906
$669$	0	0
$670$	0	0
$671$	−8.44486	−0.326010
$672$	0	0
$673$	−40.8091	−1.57308	−0.786538	−	0.617542i	$-0.788129\pi$
$673$	−40.8091	−1.57308	−0.786538	+	0.617542i	$0.788129\pi$
$674$	−3.46410	−0.133432
$675$	0	0
$676$	12.1244	0.466321
$677$	−3.68784	−0.141735	−0.0708676	−	0.997486i	$-0.522577\pi$
$677$	−3.68784	−0.141735	−0.0708676	+	0.997486i	$0.522577\pi$
$678$	0	0
$679$	−50.7846	−1.94893
$680$	0	0
$681$	0	0
$682$	−0.480473	−0.0183983
$683$	0.101536	0.00388517	0.00194258	−	0.999998i	$-0.499382\pi$
$683$	0.101536	0.00388517	0.00194258	+	0.999998i	$0.499382\pi$
$684$	0	0
$685$	0	0
$686$	4.85641	0.185418
$687$	0	0
$688$	−1.61729	−0.0616584
$689$	2.53590	0.0966100
$690$	0	0
$691$	−28.2487	−1.07463	−0.537316	−	0.843381i	$-0.680562\pi$
$691$	−28.2487	−1.07463	−0.537316	+	0.843381i	$0.680562\pi$
$692$	−18.2832	−0.695025
$693$	0	0
$694$	5.12436	0.194518
$695$	0	0
$696$	0	0
$697$	40.8091	1.54576
$698$	−14.7341	−0.557694
$699$	0	0
$700$	0	0
$701$	−12.8038	−0.483595	−0.241797	−	0.970327i	$-0.577737\pi$
$701$	−12.8038	−0.483595	−0.241797	+	0.970327i	$0.577737\pi$
$702$	0	0
$703$	3.10583	0.117139
$704$	−2.87564	−0.108380
$705$	0	0
$706$	−10.5359	−0.396524
$707$	31.6675	1.19098
$708$	0	0
$709$	9.78461	0.367469	0.183734	−	0.982976i	$-0.441181\pi$
$709$	9.78461	0.367469	0.183734	+	0.982976i	$0.441181\pi$
$710$	0	0
$711$	0	0
$712$	25.8719	0.969592
$713$	0.378937	0.0141913
$714$	0	0
$715$	0	0
$716$	11.4115	0.426469
$717$	0	0
$718$	−10.4543	−0.390151
$719$	40.3923	1.50638	0.753189	−	0.657804i	$-0.228514\pi$
$719$	40.3923	1.50638	0.753189	+	0.657804i	$0.228514\pi$
$720$	0	0
$721$	2.19615	0.0817890
$722$	9.55772	0.355702
$723$	0	0
$724$	−2.66025	−0.0988676
$725$	0	0
$726$	0	0
$727$	1.37705	0.0510719	0.0255360	−	0.999674i	$-0.491871\pi$
$727$	1.37705	0.0510719	0.0255360	+	0.999674i	$0.491871\pi$
$728$	−15.8338	−0.586838
$729$	0	0
$730$	0	0
$731$	−3.46410	−0.128124
$732$	0	0
$733$	−7.52433	−0.277918	−0.138959	−	0.990298i	$-0.544376\pi$
$733$	−7.52433	−0.277918	−0.138959	+	0.990298i	$0.544376\pi$
$734$	−17.9090	−0.661032
$735$	0	0
$736$	−2.66025	−0.0980583
$737$	−9.62209	−0.354434
$738$	0	0
$739$	−18.1436	−0.667423	−0.333711	−	0.942675i	$-0.608301\pi$
$739$	−18.1436	−0.667423	−0.333711	+	0.942675i	$0.608301\pi$
$740$	0	0
$741$	0	0
$742$	1.79315	0.0658286
$743$	32.6656	1.19839	0.599193	−	0.800605i	$-0.295488\pi$
$743$	32.6656	1.19839	0.599193	+	0.800605i	$0.295488\pi$
$744$	0	0
$745$	0	0
$746$	−14.1051	−0.516425
$747$	0	0
$748$	−11.5911	−0.423813
$749$	−34.5167	−1.26121
$750$	0	0
$751$	−24.4449	−0.892006	−0.446003	−	0.895032i	$-0.647153\pi$
$751$	−24.4449	−0.892006	−0.446003	+	0.895032i	$0.647153\pi$
$752$	−7.31130	−0.266616
$753$	0	0
$754$	−0.588457	−0.0214303
$755$	0	0
$756$	0	0
$757$	−7.34847	−0.267085	−0.133542	−	0.991043i	$-0.542635\pi$
$757$	−7.34847	−0.267085	−0.133542	+	0.991043i	$0.542635\pi$
$758$	−10.0754	−0.365954
$759$	0	0
$760$	0	0
$761$	15.9282	0.577397	0.288698	−	0.957420i	$-0.406777\pi$
$761$	15.9282	0.577397	0.288698	+	0.957420i	$0.406777\pi$
$762$	0	0
$763$	42.3620	1.53361
$764$	−30.5885	−1.10665
$765$	0	0
$766$	5.51666	0.199325
$767$	−23.1822	−0.837061
$768$	0	0
$769$	41.6410	1.50161	0.750807	−	0.660522i	$-0.229665\pi$
$769$	41.6410	1.50161	0.750807	+	0.660522i	$0.229665\pi$
$770$	0	0
$771$	0	0
$772$	−6.51626	−0.234526
$773$	2.07055	0.0744726	0.0372363	−	0.999306i	$-0.488145\pi$
$773$	2.07055	0.0744726	0.0372363	+	0.999306i	$0.488145\pi$
$774$	0	0
$775$	0	0
$776$	−29.3205	−1.05254
$777$	0	0
$778$	−0.0643712	−0.00230782
$779$	−5.66025	−0.202800
$780$	0	0
$781$	18.0000	0.644091
$782$	−1.41421	−0.0505722
$783$	0	0
$784$	10.3397	0.369277
$785$	0	0
$786$	0	0
$787$	−26.1122	−0.930799	−0.465399	−	0.885101i	$-0.654089\pi$
$787$	−26.1122	−0.930799	−0.465399	+	0.885101i	$0.654089\pi$
$788$	27.6007	0.983235
$789$	0	0
$790$	0	0
$791$	57.7128	2.05203
$792$	0	0
$793$	16.3142	0.579335
$794$	15.2154	0.539974
$795$	0	0
$796$	46.0526	1.63229
$797$	−52.6776	−1.86594	−0.932969	−	0.359957i	$-0.882791\pi$
$797$	−52.6776	−1.86594	−0.932969	+	0.359957i	$0.882791\pi$
$798$	0	0
$799$	−15.6603	−0.554020
$800$	0	0
$801$	0	0
$802$	8.83701	0.312046
$803$	10.7589	0.379674
$804$	0	0
$805$	0	0
$806$	0.928203	0.0326946
$807$	0	0
$808$	18.2832	0.643202
$809$	−25.1769	−0.885173	−0.442587	−	0.896726i	$-0.645939\pi$
$809$	−25.1769	−0.885173	−0.442587	+	0.896726i	$0.645939\pi$
$810$	0	0
$811$	−39.5692	−1.38946	−0.694732	−	0.719269i	$-0.744477\pi$
$811$	−39.5692	−1.38946	−0.694732	+	0.719269i	$0.744477\pi$
$812$	2.68973	0.0943909
$813$	0	0
$814$	2.78461	0.0976005
$815$	0	0
$816$	0	0
$817$	0.480473	0.0168096
$818$	8.55961	0.299280
$819$	0	0
$820$	0	0
$821$	29.4449	1.02763	0.513816	−	0.857900i	$-0.328231\pi$
$821$	29.4449	1.02763	0.513816	+	0.857900i	$0.328231\pi$
$822$	0	0
$823$	19.1798	0.668566	0.334283	−	0.942473i	$-0.391506\pi$
$823$	19.1798	0.668566	0.334283	+	0.942473i	$0.391506\pi$
$824$	1.26795	0.0441711
$825$	0	0
$826$	−16.3923	−0.570361
$827$	23.8014	0.827656	0.413828	−	0.910355i	$-0.364192\pi$
$827$	23.8014	0.827656	0.413828	+	0.910355i	$0.364192\pi$
$828$	0	0
$829$	0.411543	0.0142935	0.00714673	−	0.999974i	$-0.497725\pi$
$829$	0.411543	0.0142935	0.00714673	+	0.999974i	$0.497725\pi$
$830$	0	0
$831$	0	0
$832$	5.55532	0.192596
$833$	22.1469	0.767346
$834$	0	0
$835$	0	0
$836$	1.60770	0.0556033
$837$	0	0
$838$	−11.4152	−0.394333
$839$	4.73205	0.163369	0.0816843	−	0.996658i	$-0.473970\pi$
$839$	4.73205	0.163369	0.0816843	+	0.996658i	$0.473970\pi$
$840$	0	0
$841$	−28.7846	−0.992573
$842$	10.0754	0.347220
$843$	0	0
$844$	−19.2679	−0.663230
$845$	0	0
$846$	0	0
$847$	−31.4273	−1.07985
$848$	−2.55103	−0.0876026
$849$	0	0
$850$	0	0
$851$	−2.19615	−0.0752831
$852$	0	0
$853$	43.4345	1.48717	0.743584	−	0.668642i	$-0.233124\pi$
$853$	43.4345	1.48717	0.743584	+	0.668642i	$0.233124\pi$
$854$	11.5359	0.394750
$855$	0	0
$856$	−19.9282	−0.681132
$857$	13.7632	0.470142	0.235071	−	0.971978i	$-0.424468\pi$
$857$	13.7632	0.470142	0.235071	+	0.971978i	$0.424468\pi$
$858$	0	0
$859$	36.4449	1.24348	0.621741	−	0.783223i	$-0.286426\pi$
$859$	36.4449	1.24348	0.621741	+	0.783223i	$0.286426\pi$
$860$	0	0
$861$	0	0
$862$	3.10583	0.105785
$863$	44.9131	1.52886	0.764429	−	0.644708i	$-0.223021\pi$
$863$	44.9131	1.52886	0.764429	+	0.644708i	$0.223021\pi$
$864$	0	0
$865$	0	0
$866$	7.01924	0.238523
$867$	0	0
$868$	−4.24264	−0.144005
$869$	9.46410	0.321048
$870$	0	0
$871$	18.5885	0.629846
$872$	24.4577	0.828243
$873$	0	0
$874$	0.196152	0.00663495
$875$	0	0
$876$	0	0
$877$	−29.2180	−0.986622	−0.493311	−	0.869853i	$-0.664214\pi$
$877$	−29.2180	−0.986622	−0.493311	+	0.869853i	$0.664214\pi$
$878$	13.1069	0.442335
$879$	0	0
$880$	0	0
$881$	−39.5885	−1.33377	−0.666885	−	0.745161i	$-0.732373\pi$
$881$	−39.5885	−1.33377	−0.666885	+	0.745161i	$0.732373\pi$
$882$	0	0
$883$	54.4336	1.83184	0.915919	−	0.401364i	$-0.131464\pi$
$883$	54.4336	1.83184	0.915919	+	0.401364i	$0.131464\pi$
$884$	22.3923	0.753135
$885$	0	0
$886$	−0.856406	−0.0287715
$887$	−8.58682	−0.288317	−0.144159	−	0.989555i	$-0.546048\pi$
$887$	−8.58682	−0.288317	−0.144159	+	0.989555i	$0.546048\pi$
$888$	0	0
$889$	58.1769	1.95119
$890$	0	0
$891$	0	0
$892$	7.76457	0.259977
$893$	2.17209	0.0726862
$894$	0	0
$895$	0	0
$896$	38.3205	1.28020
$897$	0	0
$898$	6.21166	0.207286
$899$	−0.339746	−0.0113312
$900$	0	0
$901$	−5.46410	−0.182036
$902$	−5.07484	−0.168974
$903$	0	0
$904$	33.3205	1.10822
$905$	0	0
$906$	0	0
$907$	11.5267	0.382739	0.191370	−	0.981518i	$-0.438707\pi$
$907$	11.5267	0.382739	0.191370	+	0.981518i	$0.438707\pi$
$908$	45.7081	1.51688
$909$	0	0
$910$	0	0
$911$	49.8564	1.65182	0.825908	−	0.563805i	$-0.190663\pi$
$911$	49.8564	1.65182	0.825908	+	0.563805i	$0.190663\pi$
$912$	0	0
$913$	−10.1026	−0.334346
$914$	12.8372	0.424616
$915$	0	0
$916$	−10.5167	−0.347480
$917$	43.2586	1.42853
$918$	0	0
$919$	47.1769	1.55622	0.778111	−	0.628126i	$-0.216178\pi$
$919$	47.1769	1.55622	0.778111	+	0.628126i	$0.216178\pi$
$920$	0	0
$921$	0	0
$922$	19.0040	0.625862
$923$	−34.7733	−1.14458
$924$	0	0
$925$	0	0
$926$	17.6603	0.580352
$927$	0	0
$928$	2.38512	0.0782954
$929$	−3.46410	−0.113653	−0.0568267	−	0.998384i	$-0.518098\pi$
$929$	−3.46410	−0.113653	−0.0568267	+	0.998384i	$0.518098\pi$
$930$	0	0
$931$	−3.07180	−0.100674
$932$	12.2474	0.401179
$933$	0	0
$934$	−13.1244	−0.429442
$935$	0	0
$936$	0	0
$937$	−28.5617	−0.933069	−0.466535	−	0.884503i	$-0.654498\pi$
$937$	−28.5617	−0.933069	−0.466535	+	0.884503i	$0.654498\pi$
$938$	13.1440	0.429168
$939$	0	0
$940$	0	0
$941$	26.3205	0.858024	0.429012	−	0.903299i	$-0.358862\pi$
$941$	26.3205	0.858024	0.429012	+	0.903299i	$0.358862\pi$
$942$	0	0
$943$	4.00240	0.130336
$944$	23.3205	0.759018
$945$	0	0
$946$	0.430781	0.0140059
$947$	−4.76028	−0.154688	−0.0773441	−	0.997004i	$-0.524644\pi$
$947$	−4.76028	−0.154688	−0.0773441	+	0.997004i	$0.524644\pi$
$948$	0	0
$949$	−20.7846	−0.674697
$950$	0	0
$951$	0	0
$952$	34.1170	1.10574
$953$	−51.1619	−1.65730	−0.828648	−	0.559770i	$-0.810889\pi$
$953$	−51.1619	−1.65730	−0.828648	+	0.559770i	$0.810889\pi$
$954$	0	0
$955$	0	0
$956$	1.60770	0.0519966
$957$	0	0
$958$	−2.14488	−0.0692980
$959$	52.6410	1.69987
$960$	0	0
$961$	−30.4641	−0.982713
$962$	−5.37945	−0.173441
$963$	0	0
$964$	−16.8564	−0.542908
$965$	0	0
$966$	0	0
$967$	−55.9222	−1.79834	−0.899168	−	0.437604i	$-0.855827\pi$
$967$	−55.9222	−1.79834	−0.899168	+	0.437604i	$0.855827\pi$
$968$	−18.1445	−0.583188
$969$	0	0
$970$	0	0
$971$	38.1962	1.22577	0.612886	−	0.790171i	$-0.290008\pi$
$971$	38.1962	1.22577	0.612886	+	0.790171i	$0.290008\pi$
$972$	0	0
$973$	−26.7685	−0.858159
$974$	−8.19615	−0.262622
$975$	0	0
$976$	−16.4115	−0.525321
$977$	15.5563	0.497692	0.248846	−	0.968543i	$-0.419949\pi$
$977$	15.5563	0.497692	0.248846	+	0.968543i	$0.419949\pi$
$978$	0	0
$979$	16.9808	0.542708
$980$	0	0
$981$	0	0
$982$	−17.4510	−0.556885
$983$	−33.4235	−1.06604	−0.533022	−	0.846101i	$-0.678944\pi$
$983$	−33.4235	−1.06604	−0.533022	+	0.846101i	$0.678944\pi$
$984$	0	0
$985$	0	0
$986$	1.26795	0.0403797
$987$	0	0
$988$	−3.10583	−0.0988096
$989$	−0.339746	−0.0108033
$990$	0	0
$991$	38.9282	1.23660	0.618298	−	0.785944i	$-0.287823\pi$
$991$	38.9282	1.23660	0.618298	+	0.785944i	$0.287823\pi$
$992$	−3.76217	−0.119449
$993$	0	0
$994$	−24.5885	−0.779899
$995$	0	0
$996$	0	0
$997$	10.2784	0.325521	0.162761	−	0.986666i	$-0.447960\pi$
$997$	10.2784	0.325521	0.162761	+	0.986666i	$0.447960\pi$
$998$	−12.3490	−0.390900
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2025.2.a.t.1.2		4
3.2	odd	2		2025.2.a.r.1.3		4
5.2	odd	4		405.2.b.d.244.2		4
5.3	odd	4		405.2.b.d.244.3		4
5.4	even	2	inner	2025.2.a.t.1.3		4
9.2	odd	6		675.2.e.d.226.2		8
9.4	even	3		225.2.e.d.151.3		8
9.5	odd	6		675.2.e.d.451.2		8
9.7	even	3		225.2.e.d.76.3		8
15.2	even	4		405.2.b.c.244.3		4
15.8	even	4		405.2.b.c.244.2		4
15.14	odd	2		2025.2.a.r.1.2		4
45.2	even	12		135.2.j.a.64.3		8
45.4	even	6		225.2.e.d.151.2		8
45.7	odd	12		45.2.j.a.4.2	✓	8
45.13	odd	12		45.2.j.a.34.2	yes	8
45.14	odd	6		675.2.e.d.451.3		8
45.22	odd	12		45.2.j.a.34.3	yes	8
45.23	even	12		135.2.j.a.19.3		8
45.29	odd	6		675.2.e.d.226.3		8
45.32	even	12		135.2.j.a.19.2		8
45.34	even	6		225.2.e.d.76.2		8
45.38	even	12		135.2.j.a.64.2		8
45.43	odd	12		45.2.j.a.4.3	yes	8
180.7	even	12		720.2.by.d.49.2		8
180.23	odd	12		2160.2.by.c.289.3		8
180.43	even	12		720.2.by.d.49.3		8
180.47	odd	12		2160.2.by.c.1009.3		8
180.67	even	12		720.2.by.d.529.3		8
180.83	odd	12		2160.2.by.c.1009.1		8
180.103	even	12		720.2.by.d.529.2		8
180.167	odd	12		2160.2.by.c.289.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
45.2.j.a.4.2	✓	8	45.7	odd	12
45.2.j.a.4.3	yes	8	45.43	odd	12
45.2.j.a.34.2	yes	8	45.13	odd	12
45.2.j.a.34.3	yes	8	45.22	odd	12
135.2.j.a.19.2		8	45.32	even	12
135.2.j.a.19.3		8	45.23	even	12
135.2.j.a.64.2		8	45.38	even	12
135.2.j.a.64.3		8	45.2	even	12
225.2.e.d.76.2		8	45.34	even	6
225.2.e.d.76.3		8	9.7	even	3
225.2.e.d.151.2		8	45.4	even	6
225.2.e.d.151.3		8	9.4	even	3
405.2.b.c.244.2		4	15.8	even	4
405.2.b.c.244.3		4	15.2	even	4
405.2.b.d.244.2		4	5.2	odd	4
405.2.b.d.244.3		4	5.3	odd	4
675.2.e.d.226.2		8	9.2	odd	6
675.2.e.d.226.3		8	45.29	odd	6
675.2.e.d.451.2		8	9.5	odd	6
675.2.e.d.451.3		8	45.14	odd	6
720.2.by.d.49.2		8	180.7	even	12
720.2.by.d.49.3		8	180.43	even	12
720.2.by.d.529.2		8	180.103	even	12
720.2.by.d.529.3		8	180.67	even	12
2025.2.a.r.1.2		4	15.14	odd	2
2025.2.a.r.1.3		4	3.2	odd	2
2025.2.a.t.1.2		4	1.1	even	1	trivial
2025.2.a.t.1.3		4	5.4	even	2	inner
2160.2.by.c.289.1		8	180.167	odd	12
2160.2.by.c.289.3		8	180.23	odd	12
2160.2.by.c.1009.1		8	180.83	odd	12
2160.2.by.c.1009.3		8	180.47	odd	12

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:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2025.a (trivial)

\(f(q)\)	\(=\)	\(q-0.517638 q^{2} -1.73205 q^{4} +3.34607 q^{7} +1.93185 q^{8} +O(q^{10})\)	\(q-0.517638 q^{2} -1.73205 q^{4} +3.34607 q^{7} +1.93185 q^{8} +1.26795 q^{11} -2.44949 q^{13} -1.73205 q^{14} +2.46410 q^{16} +5.27792 q^{17} -0.732051 q^{19} -0.656339 q^{22} +0.517638 q^{23} +1.26795 q^{26} -5.79555 q^{28} -0.464102 q^{29} +0.732051 q^{31} -5.13922 q^{32} -2.73205 q^{34} -4.24264 q^{37} +0.378937 q^{38} +7.73205 q^{41} -0.656339 q^{43} -2.19615 q^{44} -0.267949 q^{46} -2.96713 q^{47} +4.19615 q^{49} +4.24264 q^{52} -1.03528 q^{53} +6.46410 q^{56} +0.240237 q^{58} +9.46410 q^{59} -6.66025 q^{61} -0.378937 q^{62} -2.26795 q^{64} -7.58871 q^{67} -9.14162 q^{68} +14.1962 q^{71} +8.48528 q^{73} +2.19615 q^{74} +1.26795 q^{76} +4.24264 q^{77} +7.46410 q^{79} -4.00240 q^{82} -7.96764 q^{83} +0.339746 q^{86} +2.44949 q^{88} +13.3923 q^{89} -8.19615 q^{91} -0.896575 q^{92} +1.53590 q^{94} -15.1774 q^{97} -2.17209 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 4 * q	\( 4 q + 12 q^{11} - 4 q^{16} + 4 q^{19} + 12 q^{26} + 12 q^{29} - 4 q^{31} - 4 q^{34} + 24 q^{41} + 12 q^{44} - 8 q^{46} - 4 q^{49} + 12 q^{56} + 24 q^{59} + 8 q^{61} - 16 q^{64} + 36 q^{71} - 12 q^{74} + 12 q^{76} + 16 q^{79} + 36 q^{86} + 12 q^{89} - 12 q^{91} + 20 q^{94}+O(q^{100}) \) 4 * q + 12 * q^11 - 4 * q^16 + 4 * q^19 + 12 * q^26 + 12 * q^29 - 4 * q^31 - 4 * q^34 + 24 * q^41 + 12 * q^44 - 8 * q^46 - 4 * q^49 + 12 * q^56 + 24 * q^59 + 8 * q^61 - 16 * q^64 + 36 * q^71 - 12 * q^74 + 12 * q^76 + 16 * q^79 + 36 * q^86 + 12 * q^89 - 12 * q^91 + 20 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more