LMFDB - Embedded newform 2025.2.a.v.1.1

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:	$\Q(\sqrt{3}, \sqrt{7})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 5x^{2} + 1 $ x^4 - 5*x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$0.456850$ of defining polynomial
Character	$\chi$	$=$	2025.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.18890 q^{2} +2.79129 q^{4} +3.79129 q^{7} -1.73205 q^{8} +O(q^{10})$	$q-2.18890 q^{2} +2.79129 q^{4} +3.79129 q^{7} -1.73205 q^{8} +0.456850 q^{11} +5.79129 q^{13} -8.29875 q^{14} -1.79129 q^{16} +7.48040 q^{17} +5.58258 q^{19} -1.00000 q^{22} +0.361500 q^{23} -12.6766 q^{26} +10.5826 q^{28} -5.29150 q^{29} +3.00000 q^{31} +7.38505 q^{32} -16.3739 q^{34} +4.79129 q^{37} -12.2197 q^{38} +6.47135 q^{41} -3.37386 q^{43} +1.27520 q^{44} -0.791288 q^{46} +6.10985 q^{47} +7.37386 q^{49} +16.1652 q^{52} -12.5812 q^{53} -6.56670 q^{56} +11.5826 q^{58} -8.20340 q^{59} -7.37386 q^{61} -6.56670 q^{62} -12.5826 q^{64} +1.41742 q^{67} +20.8800 q^{68} +12.3151 q^{71} -4.16515 q^{73} -10.4877 q^{74} +15.5826 q^{76} +1.73205 q^{77} -15.1652 q^{79} -14.1652 q^{82} -5.19615 q^{83} +7.38505 q^{86} -0.791288 q^{88} -2.74110 q^{89} +21.9564 q^{91} +1.00905 q^{92} -13.3739 q^{94} -5.16515 q^{97} -16.1407 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{4} + 6 q^{7}+O(q^{10}) $ 4 * q + 2 * q^4 + 6 * q^7	$ 4 q + 2 q^{4} + 6 q^{7} + 14 q^{13} + 2 q^{16} + 4 q^{19} - 4 q^{22} + 24 q^{28} + 12 q^{31} - 38 q^{34} + 10 q^{37} + 14 q^{43} + 6 q^{46} + 2 q^{49} + 28 q^{52} + 28 q^{58} - 2 q^{61} - 32 q^{64} + 24 q^{67} + 20 q^{73} + 44 q^{76} - 24 q^{79} - 20 q^{82} + 6 q^{88} + 42 q^{91} - 26 q^{94} + 16 q^{97}+O(q^{100}) $ 4 * q + 2 * q^4 + 6 * q^7 + 14 * q^13 + 2 * q^16 + 4 * q^19 - 4 * q^22 + 24 * q^28 + 12 * q^31 - 38 * q^34 + 10 * q^37 + 14 * q^43 + 6 * q^46 + 2 * q^49 + 28 * q^52 + 28 * q^58 - 2 * q^61 - 32 * q^64 + 24 * q^67 + 20 * q^73 + 44 * q^76 - 24 * q^79 - 20 * q^82 + 6 * q^88 + 42 * q^91 - 26 * q^94 + 16 * q^97

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 $

$2$	−2.18890	−1.54779	−0.773893	−	0.633316i	$-0.781693\pi$
$2$	−2.18890	−1.54779	−0.773893	+	0.633316i	$0.781693\pi$
$3$	0	0
$3$	0	0
$4$	2.79129	1.39564
$5$	0	0
$5$	0	0
$6$	0	0
$7$	3.79129	1.43297	0.716486	−	0.697601i	$-0.245749\pi$
$7$	3.79129	1.43297	0.716486	+	0.697601i	$0.245749\pi$
$8$	−1.73205	−0.612372
$9$	0	0
$10$	0	0
$11$	0.456850	0.137746	0.0688728	−	0.997625i	$-0.478060\pi$
$11$	0.456850	0.137746	0.0688728	+	0.997625i	$0.478060\pi$
$12$	0	0
$13$	5.79129	1.60621	0.803107	−	0.595835i	$-0.203179\pi$
$13$	5.79129	1.60621	0.803107	+	0.595835i	$0.203179\pi$
$14$	−8.29875	−2.21794
$15$	0	0
$16$	−1.79129	−0.447822
$17$	7.48040	1.81426	0.907132	−	0.420846i	$-0.138267\pi$
$17$	7.48040	1.81426	0.907132	+	0.420846i	$0.138267\pi$
$18$	0	0
$19$	5.58258	1.28073	0.640365	−	0.768070i	$-0.278783\pi$
$19$	5.58258	1.28073	0.640365	+	0.768070i	$0.278783\pi$
$20$	0	0
$21$	0	0
$22$	−1.00000	−0.213201
$23$	0.361500	0.0753780	0.0376890	−	0.999290i	$-0.488000\pi$
$23$	0.361500	0.0753780	0.0376890	+	0.999290i	$0.488000\pi$
$24$	0	0
$25$	0	0
$26$	−12.6766	−2.48608
$27$	0	0
$28$	10.5826	1.99992
$29$	−5.29150	−0.982607	−0.491304	−	0.870988i	$-0.663479\pi$
$29$	−5.29150	−0.982607	−0.491304	+	0.870988i	$0.663479\pi$
$30$	0	0
$31$	3.00000	0.538816	0.269408	−	0.963026i	$-0.413172\pi$
$31$	3.00000	0.538816	0.269408	+	0.963026i	$0.413172\pi$
$32$	7.38505	1.30551
$33$	0	0
$34$	−16.3739	−2.80809
$35$	0	0
$36$	0	0
$37$	4.79129	0.787683	0.393841	−	0.919178i	$-0.371146\pi$
$37$	4.79129	0.787683	0.393841	+	0.919178i	$0.371146\pi$
$38$	−12.2197	−1.98230
$39$	0	0
$40$	0	0
$41$	6.47135	1.01066	0.505328	−	0.862927i	$-0.331372\pi$
$41$	6.47135	1.01066	0.505328	+	0.862927i	$0.331372\pi$
$42$	0	0
$43$	−3.37386	−0.514509	−0.257255	−	0.966344i	$-0.582818\pi$
$43$	−3.37386	−0.514509	−0.257255	+	0.966344i	$0.582818\pi$
$44$	1.27520	0.192244
$45$	0	0
$46$	−0.791288	−0.116669
$47$	6.10985	0.891214	0.445607	−	0.895229i	$-0.352988\pi$
$47$	6.10985	0.891214	0.445607	+	0.895229i	$0.352988\pi$
$48$	0	0
$49$	7.37386	1.05341
$50$	0	0
$51$	0	0
$52$	16.1652	2.24170
$53$	−12.5812	−1.72816	−0.864081	−	0.503353i	$-0.832099\pi$
$53$	−12.5812	−1.72816	−0.864081	+	0.503353i	$0.832099\pi$
$54$	0	0
$55$	0	0
$56$	−6.56670	−0.877513
$57$	0	0
$58$	11.5826	1.52087
$59$	−8.20340	−1.06799	−0.533996	−	0.845487i	$-0.679310\pi$
$59$	−8.20340	−1.06799	−0.533996	+	0.845487i	$0.679310\pi$
$60$	0	0
$61$	−7.37386	−0.944126	−0.472063	−	0.881565i	$-0.656491\pi$
$61$	−7.37386	−0.944126	−0.472063	+	0.881565i	$0.656491\pi$
$62$	−6.56670	−0.833972
$63$	0	0
$64$	−12.5826	−1.57282
$65$	0	0
$66$	0	0
$67$	1.41742	0.173166	0.0865830	−	0.996245i	$-0.472405\pi$
$67$	1.41742	0.173166	0.0865830	+	0.996245i	$0.472405\pi$
$68$	20.8800	2.53207
$69$	0	0
$70$	0	0
$71$	12.3151	1.46153	0.730764	−	0.682630i	$-0.239164\pi$
$71$	12.3151	1.46153	0.730764	+	0.682630i	$0.239164\pi$
$72$	0	0
$73$	−4.16515	−0.487494	−0.243747	−	0.969839i	$-0.578377\pi$
$73$	−4.16515	−0.487494	−0.243747	+	0.969839i	$0.578377\pi$
$74$	−10.4877	−1.21917
$75$	0	0
$76$	15.5826	1.78744
$77$	1.73205	0.197386
$78$	0	0
$79$	−15.1652	−1.70621	−0.853106	−	0.521737i	$-0.825284\pi$
$79$	−15.1652	−1.70621	−0.853106	+	0.521737i	$0.825284\pi$
$80$	0	0
$81$	0	0
$82$	−14.1652	−1.56428
$83$	−5.19615	−0.570352	−0.285176	−	0.958475i	$-0.592052\pi$
$83$	−5.19615	−0.570352	−0.285176	+	0.958475i	$0.592052\pi$
$84$	0	0
$85$	0	0
$86$	7.38505	0.796351
$87$	0	0
$88$	−0.791288	−0.0843516
$89$	−2.74110	−0.290556	−0.145278	−	0.989391i	$-0.546408\pi$
$89$	−2.74110	−0.290556	−0.145278	+	0.989391i	$0.546408\pi$
$90$	0	0
$91$	21.9564	2.30166
$92$	1.00905	0.105201
$93$	0	0
$94$	−13.3739	−1.37941
$95$	0	0
$96$	0	0
$97$	−5.16515	−0.524442	−0.262221	−	0.965008i	$-0.584455\pi$
$97$	−5.16515	−0.524442	−0.262221	+	0.965008i	$0.584455\pi$
$98$	−16.1407	−1.63045
$99$	0	0
$100$	0	0
$101$	−13.4949	−1.34279	−0.671397	−	0.741098i	$-0.734305\pi$
$101$	−13.4949	−1.34279	−0.671397	+	0.741098i	$0.734305\pi$
$102$	0	0
$103$	−5.95644	−0.586905	−0.293453	−	0.955974i	$-0.594804\pi$
$103$	−5.95644	−0.586905	−0.293453	+	0.955974i	$0.594804\pi$
$104$	−10.0308	−0.983601
$105$	0	0
$106$	27.5390	2.67483
$107$	−14.8655	−1.43710	−0.718549	−	0.695476i	$-0.755193\pi$
$107$	−14.8655	−1.43710	−0.718549	+	0.695476i	$0.755193\pi$
$108$	0	0
$109$	2.79129	0.267357	0.133678	−	0.991025i	$-0.457321\pi$
$109$	2.79129	0.267357	0.133678	+	0.991025i	$0.457321\pi$
$110$	0	0
$111$	0	0
$112$	−6.79129	−0.641716
$113$	−3.19795	−0.300838	−0.150419	−	0.988622i	$-0.548062\pi$
$113$	−3.19795	−0.300838	−0.150419	+	0.988622i	$0.548062\pi$
$114$	0	0
$115$	0	0
$116$	−14.7701	−1.37137
$117$	0	0
$118$	17.9564	1.65302
$119$	28.3604	2.59979
$120$	0	0
$121$	−10.7913	−0.981026
$122$	16.1407	1.46131
$123$	0	0
$124$	8.37386	0.751995
$125$	0	0
$126$	0	0
$127$	17.0000	1.50851	0.754253	−	0.656584i	$-0.227999\pi$
$127$	17.0000	1.50851	0.754253	+	0.656584i	$0.227999\pi$
$128$	12.7719	1.12889
$129$	0	0
$130$	0	0
$131$	−2.74110	−0.239491	−0.119746	−	0.992805i	$-0.538208\pi$
$131$	−2.74110	−0.239491	−0.119746	+	0.992805i	$0.538208\pi$
$132$	0	0
$133$	21.1652	1.83525
$134$	−3.10260	−0.268024
$135$	0	0
$136$	−12.9564	−1.11101
$137$	13.4949	1.15295	0.576474	−	0.817116i	$-0.304428\pi$
$137$	13.4949	1.15295	0.576474	+	0.817116i	$0.304428\pi$
$138$	0	0
$139$	−12.3739	−1.04954	−0.524769	−	0.851245i	$-0.675848\pi$
$139$	−12.3739	−1.04954	−0.524769	+	0.851245i	$0.675848\pi$
$140$	0	0
$141$	0	0
$142$	−26.9564	−2.26213
$143$	2.64575	0.221249
$144$	0	0
$145$	0	0
$146$	9.11710	0.754537
$147$	0	0
$148$	13.3739	1.09932
$149$	5.38685	0.441308	0.220654	−	0.975352i	$-0.429181\pi$
$149$	5.38685	0.441308	0.220654	+	0.975352i	$0.429181\pi$
$150$	0	0
$151$	−10.3739	−0.844213	−0.422107	−	0.906546i	$-0.638709\pi$
$151$	−10.3739	−0.844213	−0.422107	+	0.906546i	$0.638709\pi$
$152$	−9.66930	−0.784284
$153$	0	0
$154$	−3.79129	−0.305511
$155$	0	0
$156$	0	0
$157$	−6.58258	−0.525347	−0.262673	−	0.964885i	$-0.584604\pi$
$157$	−6.58258	−0.525347	−0.262673	+	0.964885i	$0.584604\pi$
$158$	33.1950	2.64085
$159$	0	0
$160$	0	0
$161$	1.37055	0.108015
$162$	0	0
$163$	−8.16515	−0.639544	−0.319772	−	0.947495i	$-0.603606\pi$
$163$	−8.16515	−0.639544	−0.319772	+	0.947495i	$0.603606\pi$
$164$	18.0634	1.41052
$165$	0	0
$166$	11.3739	0.882783
$167$	3.10260	0.240087	0.120043	−	0.992769i	$-0.461697\pi$
$167$	3.10260	0.240087	0.120043	+	0.992769i	$0.461697\pi$
$168$	0	0
$169$	20.5390	1.57992
$170$	0	0
$171$	0	0
$172$	−9.41742	−0.718072
$173$	6.92820	0.526742	0.263371	−	0.964695i	$-0.415166\pi$
$173$	6.92820	0.526742	0.263371	+	0.964695i	$0.415166\pi$
$174$	0	0
$175$	0	0
$176$	−0.818350	−0.0616855
$177$	0	0
$178$	6.00000	0.449719
$179$	16.6929	1.24768	0.623841	−	0.781551i	$-0.285571\pi$
$179$	16.6929	1.24768	0.623841	+	0.781551i	$0.285571\pi$
$180$	0	0
$181$	0.582576	0.0433025	0.0216513	−	0.999766i	$-0.493108\pi$
$181$	0.582576	0.0433025	0.0216513	+	0.999766i	$0.493108\pi$
$182$	−48.0605	−3.56248
$183$	0	0
$184$	−0.626136	−0.0461594
$185$	0	0
$186$	0	0
$187$	3.41742	0.249907
$188$	17.0544	1.24382
$189$	0	0
$190$	0	0
$191$	20.2523	1.46541	0.732703	−	0.680549i	$-0.238259\pi$
$191$	20.2523	1.46541	0.732703	+	0.680549i	$0.238259\pi$
$192$	0	0
$193$	2.58258	0.185898	0.0929489	−	0.995671i	$-0.470371\pi$
$193$	2.58258	0.185898	0.0929489	+	0.995671i	$0.470371\pi$
$194$	11.3060	0.811724
$195$	0	0
$196$	20.5826	1.47018
$197$	−2.09355	−0.149159	−0.0745797	−	0.997215i	$-0.523762\pi$
$197$	−2.09355	−0.149159	−0.0745797	+	0.997215i	$0.523762\pi$
$198$	0	0
$199$	−10.3739	−0.735384	−0.367692	−	0.929948i	$-0.619852\pi$
$199$	−10.3739	−0.735384	−0.367692	+	0.929948i	$0.619852\pi$
$200$	0	0
$201$	0	0
$202$	29.5390	2.07836
$203$	−20.0616	−1.40805
$204$	0	0
$205$	0	0
$206$	13.0381	0.908404
$207$	0	0
$208$	−10.3739	−0.719298
$209$	2.55040	0.176415
$210$	0	0
$211$	−19.3303	−1.33075	−0.665376	−	0.746508i	$-0.731729\pi$
$211$	−19.3303	−1.33075	−0.665376	+	0.746508i	$0.731729\pi$
$212$	−35.1178	−2.41190
$213$	0	0
$214$	32.5390	2.22432
$215$	0	0
$216$	0	0
$217$	11.3739	0.772108
$218$	−6.10985	−0.413811
$219$	0	0
$220$	0	0
$221$	43.3212	2.91410
$222$	0	0
$223$	17.7477	1.18848	0.594238	−	0.804289i	$-0.297454\pi$
$223$	17.7477	1.18848	0.594238	+	0.804289i	$0.297454\pi$
$224$	27.9989	1.87075
$225$	0	0
$226$	7.00000	0.465633
$227$	14.6748	0.973998	0.486999	−	0.873403i	$-0.338092\pi$
$227$	14.6748	0.973998	0.486999	+	0.873403i	$0.338092\pi$
$228$	0	0
$229$	−1.74773	−0.115493	−0.0577465	−	0.998331i	$-0.518392\pi$
$229$	−1.74773	−0.115493	−0.0577465	+	0.998331i	$0.518392\pi$
$230$	0	0
$231$	0	0
$232$	9.16515	0.601722
$233$	−8.66025	−0.567352	−0.283676	−	0.958920i	$-0.591554\pi$
$233$	−8.66025	−0.567352	−0.283676	+	0.958920i	$0.591554\pi$
$234$	0	0
$235$	0	0
$236$	−22.8981	−1.49054
$237$	0	0
$238$	−62.0780	−4.02392
$239$	−0.913701	−0.0591024	−0.0295512	−	0.999563i	$-0.509408\pi$
$239$	−0.913701	−0.0591024	−0.0295512	+	0.999563i	$0.509408\pi$
$240$	0	0
$241$	−12.4174	−0.799877	−0.399938	−	0.916542i	$-0.630968\pi$
$241$	−12.4174	−0.799877	−0.399938	+	0.916542i	$0.630968\pi$
$242$	23.6211	1.51842
$243$	0	0
$244$	−20.5826	−1.31766
$245$	0	0
$246$	0	0
$247$	32.3303	2.05713
$248$	−5.19615	−0.329956
$249$	0	0
$250$	0	0
$251$	1.08450	0.0684530	0.0342265	−	0.999414i	$-0.489103\pi$
$251$	1.08450	0.0684530	0.0342265	+	0.999414i	$0.489103\pi$
$252$	0	0
$253$	0.165151	0.0103830
$254$	−37.2113	−2.33485
$255$	0	0
$256$	−2.79129	−0.174455
$257$	−23.5257	−1.46749	−0.733747	−	0.679423i	$-0.762230\pi$
$257$	−23.5257	−1.46749	−0.733747	+	0.679423i	$0.762230\pi$
$258$	0	0
$259$	18.1652	1.12873
$260$	0	0
$261$	0	0
$262$	6.00000	0.370681
$263$	−0.913701	−0.0563412	−0.0281706	−	0.999603i	$-0.508968\pi$
$263$	−0.913701	−0.0563412	−0.0281706	+	0.999603i	$0.508968\pi$
$264$	0	0
$265$	0	0
$266$	−46.3284	−2.84058
$267$	0	0
$268$	3.95644	0.241678
$269$	3.19795	0.194983	0.0974913	−	0.995236i	$-0.468918\pi$
$269$	3.19795	0.194983	0.0974913	+	0.995236i	$0.468918\pi$
$270$	0	0
$271$	−20.1652	−1.22495	−0.612473	−	0.790492i	$-0.709825\pi$
$271$	−20.1652	−1.22495	−0.612473	+	0.790492i	$0.709825\pi$
$272$	−13.3996	−0.812467
$273$	0	0
$274$	−29.5390	−1.78452
$275$	0	0
$276$	0	0
$277$	−18.0000	−1.08152	−0.540758	−	0.841178i	$-0.681862\pi$
$277$	−18.0000	−1.08152	−0.540758	+	0.841178i	$0.681862\pi$
$278$	27.0852	1.62446
$279$	0	0
$280$	0	0
$281$	30.7400	1.83379	0.916896	−	0.399126i	$-0.130686\pi$
$281$	30.7400	1.83379	0.916896	+	0.399126i	$0.130686\pi$
$282$	0	0
$283$	29.5390	1.75591	0.877956	−	0.478741i	$-0.158907\pi$
$283$	29.5390	1.75591	0.877956	+	0.478741i	$0.158907\pi$
$284$	34.3749	2.03977
$285$	0	0
$286$	−5.79129	−0.342446
$287$	24.5348	1.44824
$288$	0	0
$289$	38.9564	2.29156
$290$	0	0
$291$	0	0
$292$	−11.6261	−0.680368
$293$	20.3477	1.18872	0.594362	−	0.804198i	$-0.297405\pi$
$293$	20.3477	1.18872	0.594362	+	0.804198i	$0.297405\pi$
$294$	0	0
$295$	0	0
$296$	−8.29875	−0.482355
$297$	0	0
$298$	−11.7913	−0.683051
$299$	2.09355	0.121073
$300$	0	0
$301$	−12.7913	−0.737278
$302$	22.7074	1.30666
$303$	0	0
$304$	−10.0000	−0.573539
$305$	0	0
$306$	0	0
$307$	21.9129	1.25063	0.625317	−	0.780371i	$-0.284970\pi$
$307$	21.9129	1.25063	0.625317	+	0.780371i	$0.284970\pi$
$308$	4.83465	0.275480
$309$	0	0
$310$	0	0
$311$	14.1425	0.801945	0.400973	−	0.916090i	$-0.368672\pi$
$311$	14.1425	0.801945	0.400973	+	0.916090i	$0.368672\pi$
$312$	0	0
$313$	−11.7913	−0.666483	−0.333241	−	0.942842i	$-0.608142\pi$
$313$	−11.7913	−0.666483	−0.333241	+	0.942842i	$0.608142\pi$
$314$	14.4086	0.813125
$315$	0	0
$316$	−42.3303	−2.38127
$317$	18.1389	1.01878	0.509390	−	0.860536i	$-0.329871\pi$
$317$	18.1389	1.01878	0.509390	+	0.860536i	$0.329871\pi$
$318$	0	0
$319$	−2.41742	−0.135350
$320$	0	0
$321$	0	0
$322$	−3.00000	−0.167183
$323$	41.7599	2.32358
$324$	0	0
$325$	0	0
$326$	17.8727	0.989878
$327$	0	0
$328$	−11.2087	−0.618898
$329$	23.1642	1.27708
$330$	0	0
$331$	−1.58258	−0.0869862	−0.0434931	−	0.999054i	$-0.513849\pi$
$331$	−1.58258	−0.0869862	−0.0434931	+	0.999054i	$0.513849\pi$
$332$	−14.5040	−0.796008
$333$	0	0
$334$	−6.79129	−0.371603
$335$	0	0
$336$	0	0
$337$	−1.16515	−0.0634698	−0.0317349	−	0.999496i	$-0.510103\pi$
$337$	−1.16515	−0.0634698	−0.0317349	+	0.999496i	$0.510103\pi$
$338$	−44.9579	−2.44539
$339$	0	0
$340$	0	0
$341$	1.37055	0.0742195
$342$	0	0
$343$	1.41742	0.0765337
$344$	5.84370	0.315071
$345$	0	0
$346$	−15.1652	−0.815284
$347$	−11.5722	−0.621226	−0.310613	−	0.950536i	$-0.600534\pi$
$347$	−11.5722	−0.621226	−0.310613	+	0.950536i	$0.600534\pi$
$348$	0	0
$349$	−2.33030	−0.124738	−0.0623691	−	0.998053i	$-0.519866\pi$
$349$	−2.33030	−0.124738	−0.0623691	+	0.998053i	$0.519866\pi$
$350$	0	0
$351$	0	0
$352$	3.37386	0.179828
$353$	2.81655	0.149910	0.0749549	−	0.997187i	$-0.476119\pi$
$353$	2.81655	0.149910	0.0749549	+	0.997187i	$0.476119\pi$
$354$	0	0
$355$	0	0
$356$	−7.65120	−0.405513
$357$	0	0
$358$	−36.5390	−1.93115
$359$	−36.0116	−1.90062	−0.950309	−	0.311309i	$-0.899233\pi$
$359$	−36.0116	−1.90062	−0.950309	+	0.311309i	$0.899233\pi$
$360$	0	0
$361$	12.1652	0.640271
$362$	−1.27520	−0.0670231
$363$	0	0
$364$	61.2867	3.21230
$365$	0	0
$366$	0	0
$367$	9.62614	0.502480	0.251240	−	0.967925i	$-0.419162\pi$
$367$	9.62614	0.502480	0.251240	+	0.967925i	$0.419162\pi$
$368$	−0.647551	−0.0337559
$369$	0	0
$370$	0	0
$371$	−47.6990	−2.47641
$372$	0	0
$373$	2.16515	0.112107	0.0560536	−	0.998428i	$-0.482148\pi$
$373$	2.16515	0.112107	0.0560536	+	0.998428i	$0.482148\pi$
$374$	−7.48040	−0.386802
$375$	0	0
$376$	−10.5826	−0.545755
$377$	−30.6446	−1.57828
$378$	0	0
$379$	−1.37386	−0.0705706	−0.0352853	−	0.999377i	$-0.511234\pi$
$379$	−1.37386	−0.0705706	−0.0352853	+	0.999377i	$0.511234\pi$
$380$	0	0
$381$	0	0
$382$	−44.3303	−2.26814
$383$	−22.8782	−1.16902	−0.584510	−	0.811387i	$-0.698713\pi$
$383$	−22.8782	−1.16902	−0.584510	+	0.811387i	$0.698713\pi$
$384$	0	0
$385$	0	0
$386$	−5.65300	−0.287730
$387$	0	0
$388$	−14.4174	−0.731934
$389$	20.3477	1.03167	0.515834	−	0.856689i	$-0.327482\pi$
$389$	20.3477	1.03167	0.515834	+	0.856689i	$0.327482\pi$
$390$	0	0
$391$	2.70417	0.136756
$392$	−12.7719	−0.645079
$393$	0	0
$394$	4.58258	0.230867
$395$	0	0
$396$	0	0
$397$	−22.4955	−1.12901	−0.564507	−	0.825428i	$-0.690934\pi$
$397$	−22.4955	−1.12901	−0.564507	+	0.825428i	$0.690934\pi$
$398$	22.7074	1.13822
$399$	0	0
$400$	0	0
$401$	−33.1950	−1.65768	−0.828840	−	0.559486i	$-0.810999\pi$
$401$	−33.1950	−1.65768	−0.828840	+	0.559486i	$0.810999\pi$
$402$	0	0
$403$	17.3739	0.865454
$404$	−37.6682	−1.87406
$405$	0	0
$406$	43.9129	2.17936
$407$	2.18890	0.108500
$408$	0	0
$409$	17.7913	0.879723	0.439861	−	0.898066i	$-0.355028\pi$
$409$	17.7913	0.879723	0.439861	+	0.898066i	$0.355028\pi$
$410$	0	0
$411$	0	0
$412$	−16.6261	−0.819111
$413$	−31.1015	−1.53040
$414$	0	0
$415$	0	0
$416$	42.7690	2.09692
$417$	0	0
$418$	−5.58258	−0.273053
$419$	−35.6700	−1.74259	−0.871296	−	0.490758i	$-0.836720\pi$
$419$	−35.6700	−1.74259	−0.871296	+	0.490758i	$0.836720\pi$
$420$	0	0
$421$	−20.0000	−0.974740	−0.487370	−	0.873195i	$-0.662044\pi$
$421$	−20.0000	−0.974740	−0.487370	+	0.873195i	$0.662044\pi$
$422$	42.3121	2.05972
$423$	0	0
$424$	21.7913	1.05828
$425$	0	0
$426$	0	0
$427$	−27.9564	−1.35291
$428$	−41.4938	−2.00568
$429$	0	0
$430$	0	0
$431$	19.7756	0.952555	0.476278	−	0.879295i	$-0.341986\pi$
$431$	19.7756	0.952555	0.476278	+	0.879295i	$0.341986\pi$
$432$	0	0
$433$	−20.4174	−0.981199	−0.490599	−	0.871385i	$-0.663222\pi$
$433$	−20.4174	−0.981199	−0.490599	+	0.871385i	$0.663222\pi$
$434$	−24.8963	−1.19506
$435$	0	0
$436$	7.79129	0.373135
$437$	2.01810	0.0965389
$438$	0	0
$439$	−10.3739	−0.495117	−0.247559	−	0.968873i	$-0.579628\pi$
$439$	−10.3739	−0.495117	−0.247559	+	0.968873i	$0.579628\pi$
$440$	0	0
$441$	0	0
$442$	−94.8258	−4.51040
$443$	−3.10260	−0.147409	−0.0737045	−	0.997280i	$-0.523482\pi$
$443$	−3.10260	−0.147409	−0.0737045	+	0.997280i	$0.523482\pi$
$444$	0	0
$445$	0	0
$446$	−38.8480	−1.83951
$447$	0	0
$448$	−47.7042	−2.25381
$449$	−13.6856	−0.645864	−0.322932	−	0.946422i	$-0.604668\pi$
$449$	−13.6856	−0.645864	−0.322932	+	0.946422i	$0.604668\pi$
$450$	0	0
$451$	2.95644	0.139213
$452$	−8.92640	−0.419863
$453$	0	0
$454$	−32.1216	−1.50754
$455$	0	0
$456$	0	0
$457$	27.5826	1.29026	0.645129	−	0.764073i	$-0.276804\pi$
$457$	27.5826	1.29026	0.645129	+	0.764073i	$0.276804\pi$
$458$	3.82560	0.178759
$459$	0	0
$460$	0	0
$461$	−26.9144	−1.25353	−0.626763	−	0.779210i	$-0.715621\pi$
$461$	−26.9144	−1.25353	−0.626763	+	0.779210i	$0.715621\pi$
$462$	0	0
$463$	−17.3739	−0.807432	−0.403716	−	0.914884i	$-0.632282\pi$
$463$	−17.3739	−0.807432	−0.403716	+	0.914884i	$0.632282\pi$
$464$	9.47860	0.440033
$465$	0	0
$466$	18.9564	0.878140
$467$	−0.190700	−0.00882456	−0.00441228	−	0.999990i	$-0.501404\pi$
$467$	−0.190700	−0.00882456	−0.00441228	+	0.999990i	$0.501404\pi$
$468$	0	0
$469$	5.37386	0.248142
$470$	0	0
$471$	0	0
$472$	14.2087	0.654009
$473$	−1.54135	−0.0708714
$474$	0	0
$475$	0	0
$476$	79.1619	3.62838
$477$	0	0
$478$	2.00000	0.0914779
$479$	−13.7810	−0.629668	−0.314834	−	0.949147i	$-0.601949\pi$
$479$	−13.7810	−0.629668	−0.314834	+	0.949147i	$0.601949\pi$
$480$	0	0
$481$	27.7477	1.26519
$482$	27.1805	1.23804
$483$	0	0
$484$	−30.1216	−1.36916
$485$	0	0
$486$	0	0
$487$	4.00000	0.181257	0.0906287	−	0.995885i	$-0.471112\pi$
$487$	4.00000	0.181257	0.0906287	+	0.995885i	$0.471112\pi$
$488$	12.7719	0.578157
$489$	0	0
$490$	0	0
$491$	−7.02355	−0.316969	−0.158484	−	0.987362i	$-0.550661\pi$
$491$	−7.02355	−0.316969	−0.158484	+	0.987362i	$0.550661\pi$
$492$	0	0
$493$	−39.5826	−1.78271
$494$	−70.7678	−3.18400
$495$	0	0
$496$	−5.37386	−0.241294
$497$	46.6899	2.09433
$498$	0	0
$499$	31.7913	1.42317	0.711587	−	0.702598i	$-0.247977\pi$
$499$	31.7913	1.42317	0.711587	+	0.702598i	$0.247977\pi$
$500$	0	0
$501$	0	0
$502$	−2.37386	−0.105951
$503$	−32.3012	−1.44024	−0.720120	−	0.693850i	$-0.755913\pi$
$503$	−32.3012	−1.44024	−0.720120	+	0.693850i	$0.755913\pi$
$504$	0	0
$505$	0	0
$506$	−0.361500	−0.0160706
$507$	0	0
$508$	47.4519	2.10534
$509$	−15.7792	−0.699399	−0.349699	−	0.936862i	$-0.613716\pi$
$509$	−15.7792	−0.699399	−0.349699	+	0.936862i	$0.613716\pi$
$510$	0	0
$511$	−15.7913	−0.698565
$512$	−19.4340	−0.858868
$513$	0	0
$514$	51.4955	2.27137
$515$	0	0
$516$	0	0
$517$	2.79129	0.122761
$518$	−39.7617	−1.74703
$519$	0	0
$520$	0	0
$521$	−19.1479	−0.838885	−0.419443	−	0.907782i	$-0.637774\pi$
$521$	−19.1479	−0.838885	−0.419443	+	0.907782i	$0.637774\pi$
$522$	0	0
$523$	20.1652	0.881761	0.440880	−	0.897566i	$-0.354666\pi$
$523$	20.1652	0.881761	0.440880	+	0.897566i	$0.354666\pi$
$524$	−7.65120	−0.334244
$525$	0	0
$526$	2.00000	0.0872041
$527$	22.4412	0.977555
$528$	0	0
$529$	−22.8693	−0.994318
$530$	0	0
$531$	0	0
$532$	59.0780	2.56136
$533$	37.4775	1.62333
$534$	0	0
$535$	0	0
$536$	−2.45505	−0.106042
$537$	0	0
$538$	−7.00000	−0.301791
$539$	3.36875	0.145102
$540$	0	0
$541$	−6.46099	−0.277779	−0.138890	−	0.990308i	$-0.544353\pi$
$541$	−6.46099	−0.277779	−0.138890	+	0.990308i	$0.544353\pi$
$542$	44.1395	1.89595
$543$	0	0
$544$	55.2432	2.36853
$545$	0	0
$546$	0	0
$547$	−0.669697	−0.0286342	−0.0143171	−	0.999898i	$-0.504557\pi$
$547$	−0.669697	−0.0286342	−0.0143171	+	0.999898i	$0.504557\pi$
$548$	37.6682	1.60910
$549$	0	0
$550$	0	0
$551$	−29.5402	−1.25846
$552$	0	0
$553$	−57.4955	−2.44496
$554$	39.4002	1.67395
$555$	0	0
$556$	−34.5390	−1.46478
$557$	−6.92820	−0.293557	−0.146779	−	0.989169i	$-0.546891\pi$
$557$	−6.92820	−0.293557	−0.146779	+	0.989169i	$0.546891\pi$
$558$	0	0
$559$	−19.5390	−0.826412
$560$	0	0
$561$	0	0
$562$	−67.2867	−2.83832
$563$	−19.7756	−0.833440	−0.416720	−	0.909035i	$-0.636821\pi$
$563$	−19.7756	−0.833440	−0.416720	+	0.909035i	$0.636821\pi$
$564$	0	0
$565$	0	0
$566$	−64.6580	−2.71778
$567$	0	0
$568$	−21.3303	−0.895000
$569$	6.92820	0.290445	0.145223	−	0.989399i	$-0.453610\pi$
$569$	6.92820	0.290445	0.145223	+	0.989399i	$0.453610\pi$
$570$	0	0
$571$	8.00000	0.334790	0.167395	−	0.985890i	$-0.446465\pi$
$571$	8.00000	0.334790	0.167395	+	0.985890i	$0.446465\pi$
$572$	7.38505	0.308785
$573$	0	0
$574$	−53.7042	−2.24157
$575$	0	0
$576$	0	0
$577$	−1.83485	−0.0763857	−0.0381929	−	0.999270i	$-0.512160\pi$
$577$	−1.83485	−0.0763857	−0.0381929	+	0.999270i	$0.512160\pi$
$578$	−85.2718	−3.54684
$579$	0	0
$580$	0	0
$581$	−19.7001	−0.817298
$582$	0	0
$583$	−5.74773	−0.238047
$584$	7.21425	0.298528
$585$	0	0
$586$	−44.5390	−1.83989
$587$	−11.4014	−0.470584	−0.235292	−	0.971925i	$-0.575605\pi$
$587$	−11.4014	−0.470584	−0.235292	+	0.971925i	$0.575605\pi$
$588$	0	0
$589$	16.7477	0.690078
$590$	0	0
$591$	0	0
$592$	−8.58258	−0.352742
$593$	−0.266150	−0.0109295	−0.00546473	−	0.999985i	$-0.501739\pi$
$593$	−0.266150	−0.0109295	−0.00546473	+	0.999985i	$0.501739\pi$
$594$	0	0
$595$	0	0
$596$	15.0363	0.615909
$597$	0	0
$598$	−4.58258	−0.187395
$599$	24.8009	1.01334	0.506669	−	0.862141i	$-0.330877\pi$
$599$	24.8009	1.01334	0.506669	+	0.862141i	$0.330877\pi$
$600$	0	0
$601$	37.5390	1.53125	0.765624	−	0.643288i	$-0.222430\pi$
$601$	37.5390	1.53125	0.765624	+	0.643288i	$0.222430\pi$
$602$	27.9989	1.14115
$603$	0	0
$604$	−28.9564	−1.17822
$605$	0	0
$606$	0	0
$607$	15.2523	0.619071	0.309535	−	0.950888i	$-0.399827\pi$
$607$	15.2523	0.619071	0.309535	+	0.950888i	$0.399827\pi$
$608$	41.2276	1.67200
$609$	0	0
$610$	0	0
$611$	35.3839	1.43148
$612$	0	0
$613$	33.9129	1.36973	0.684864	−	0.728671i	$-0.259862\pi$
$613$	33.9129	1.36973	0.684864	+	0.728671i	$0.259862\pi$
$614$	−47.9651	−1.93571
$615$	0	0
$616$	−3.00000	−0.120873
$617$	−30.7400	−1.23754	−0.618772	−	0.785570i	$-0.712370\pi$
$617$	−30.7400	−1.23754	−0.618772	+	0.785570i	$0.712370\pi$
$618$	0	0
$619$	−23.5826	−0.947864	−0.473932	−	0.880562i	$-0.657166\pi$
$619$	−23.5826	−0.947864	−0.473932	+	0.880562i	$0.657166\pi$
$620$	0	0
$621$	0	0
$622$	−30.9564	−1.24124
$623$	−10.3923	−0.416359
$624$	0	0
$625$	0	0
$626$	25.8100	1.03157
$627$	0	0
$628$	−18.3739	−0.733197
$629$	35.8408	1.42906
$630$	0	0
$631$	−18.2867	−0.727984	−0.363992	−	0.931402i	$-0.618586\pi$
$631$	−18.2867	−0.727984	−0.363992	+	0.931402i	$0.618586\pi$
$632$	26.2668	1.04484
$633$	0	0
$634$	−39.7042	−1.57685
$635$	0	0
$636$	0	0
$637$	42.7042	1.69200
$638$	5.29150	0.209493
$639$	0	0
$640$	0	0
$641$	25.6193	1.01190	0.505950	−	0.862563i	$-0.331142\pi$
$641$	25.6193	1.01190	0.505950	+	0.862563i	$0.331142\pi$
$642$	0	0
$643$	46.2867	1.82537	0.912685	−	0.408663i	$-0.134005\pi$
$643$	46.2867	1.82537	0.912685	+	0.408663i	$0.134005\pi$
$644$	3.82560	0.150750
$645$	0	0
$646$	−91.4083	−3.59641
$647$	8.85095	0.347967	0.173983	−	0.984749i	$-0.444336\pi$
$647$	8.85095	0.347967	0.173983	+	0.984749i	$0.444336\pi$
$648$	0	0
$649$	−3.74773	−0.147111
$650$	0	0
$651$	0	0
$652$	−22.7913	−0.892576
$653$	−18.1588	−0.710607	−0.355304	−	0.934751i	$-0.615623\pi$
$653$	−18.1588	−0.710607	−0.355304	+	0.934751i	$0.615623\pi$
$654$	0	0
$655$	0	0
$656$	−11.5921	−0.452594
$657$	0	0
$658$	−50.7042	−1.97665
$659$	−3.29330	−0.128289	−0.0641444	−	0.997941i	$-0.520432\pi$
$659$	−3.29330	−0.128289	−0.0641444	+	0.997941i	$0.520432\pi$
$660$	0	0
$661$	−36.5826	−1.42290	−0.711449	−	0.702738i	$-0.751961\pi$
$661$	−36.5826	−1.42290	−0.711449	+	0.702738i	$0.751961\pi$
$662$	3.46410	0.134636
$663$	0	0
$664$	9.00000	0.349268
$665$	0	0
$666$	0	0
$667$	−1.91288	−0.0740670
$668$	8.66025	0.335075
$669$	0	0
$670$	0	0
$671$	−3.36875	−0.130049
$672$	0	0
$673$	−36.4955	−1.40680	−0.703398	−	0.710796i	$-0.748335\pi$
$673$	−36.4955	−1.40680	−0.703398	+	0.710796i	$0.748335\pi$
$674$	2.55040	0.0982378
$675$	0	0
$676$	57.3303	2.20501
$677$	−23.8118	−0.915160	−0.457580	−	0.889168i	$-0.651284\pi$
$677$	−23.8118	−0.915160	−0.457580	+	0.889168i	$0.651284\pi$
$678$	0	0
$679$	−19.5826	−0.751510
$680$	0	0
$681$	0	0
$682$	−3.00000	−0.114876
$683$	40.4847	1.54910	0.774552	−	0.632510i	$-0.217975\pi$
$683$	40.4847	1.54910	0.774552	+	0.632510i	$0.217975\pi$
$684$	0	0
$685$	0	0
$686$	−3.10260	−0.118458
$687$	0	0
$688$	6.04356	0.230409
$689$	−72.8614	−2.77580
$690$	0	0
$691$	42.0780	1.60072	0.800362	−	0.599517i	$-0.204641\pi$
$691$	42.0780	1.60072	0.800362	+	0.599517i	$0.204641\pi$
$692$	19.3386	0.735144
$693$	0	0
$694$	25.3303	0.961525
$695$	0	0
$696$	0	0
$697$	48.4083	1.83360
$698$	5.10080	0.193068
$699$	0	0
$700$	0	0
$701$	−18.4249	−0.695899	−0.347950	−	0.937513i	$-0.613122\pi$
$701$	−18.4249	−0.695899	−0.347950	+	0.937513i	$0.613122\pi$
$702$	0	0
$703$	26.7477	1.00881
$704$	−5.74835	−0.216649
$705$	0	0
$706$	−6.16515	−0.232029
$707$	−51.1631	−1.92419
$708$	0	0
$709$	30.0780	1.12960	0.564802	−	0.825226i	$-0.308953\pi$
$709$	30.0780	1.12960	0.564802	+	0.825226i	$0.308953\pi$
$710$	0	0
$711$	0	0
$712$	4.74773	0.177929
$713$	1.08450	0.0406149
$714$	0	0
$715$	0	0
$716$	46.5946	1.74132
$717$	0	0
$718$	78.8258	2.94175
$719$	−48.1359	−1.79517	−0.897583	−	0.440844i	$-0.854679\pi$
$719$	−48.1359	−1.79517	−0.897583	+	0.440844i	$0.854679\pi$
$720$	0	0
$721$	−22.5826	−0.841019
$722$	−26.6283	−0.991003
$723$	0	0
$724$	1.62614	0.0604349
$725$	0	0
$726$	0	0
$727$	36.3739	1.34903	0.674516	−	0.738260i	$-0.264352\pi$
$727$	36.3739	1.34903	0.674516	+	0.738260i	$0.264352\pi$
$728$	−38.0297	−1.40947
$729$	0	0
$730$	0	0
$731$	−25.2379	−0.933456
$732$	0	0
$733$	9.37386	0.346232	0.173116	−	0.984901i	$-0.444617\pi$
$733$	9.37386	0.346232	0.173116	+	0.984901i	$0.444617\pi$
$734$	−21.0707	−0.777732
$735$	0	0
$736$	2.66970	0.0984063
$737$	0.647551	0.0238528
$738$	0	0
$739$	9.70417	0.356974	0.178487	−	0.983942i	$-0.442880\pi$
$739$	9.70417	0.356974	0.178487	+	0.983942i	$0.442880\pi$
$740$	0	0
$741$	0	0
$742$	104.408	3.83295
$743$	12.0290	0.441301	0.220651	−	0.975353i	$-0.429182\pi$
$743$	12.0290	0.441301	0.220651	+	0.975353i	$0.429182\pi$
$744$	0	0
$745$	0	0
$746$	−4.73930	−0.173518
$747$	0	0
$748$	9.53901	0.348781
$749$	−56.3592	−2.05932
$750$	0	0
$751$	19.6261	0.716168	0.358084	−	0.933689i	$-0.383430\pi$
$751$	19.6261	0.716168	0.358084	+	0.933689i	$0.383430\pi$
$752$	−10.9445	−0.399105
$753$	0	0
$754$	67.0780	2.44284
$755$	0	0
$756$	0	0
$757$	18.1216	0.658640	0.329320	−	0.944218i	$-0.393180\pi$
$757$	18.1216	0.658640	0.329320	+	0.944218i	$0.393180\pi$
$758$	3.00725	0.109228
$759$	0	0
$760$	0	0
$761$	0.361500	0.0131044	0.00655218	−	0.999979i	$-0.497914\pi$
$761$	0.361500	0.0131044	0.00655218	+	0.999979i	$0.497914\pi$
$762$	0	0
$763$	10.5826	0.383115
$764$	56.5300	2.04518
$765$	0	0
$766$	50.0780	1.80939
$767$	−47.5083	−1.71542
$768$	0	0
$769$	17.9564	0.647526	0.323763	−	0.946138i	$-0.395052\pi$
$769$	17.9564	0.647526	0.323763	+	0.946138i	$0.395052\pi$
$770$	0	0
$771$	0	0
$772$	7.20871	0.259447
$773$	−13.5704	−0.488092	−0.244046	−	0.969764i	$-0.578475\pi$
$773$	−13.5704	−0.488092	−0.244046	+	0.969764i	$0.578475\pi$
$774$	0	0
$775$	0	0
$776$	8.94630	0.321154
$777$	0	0
$778$	−44.5390	−1.59680
$779$	36.1268	1.29438
$780$	0	0
$781$	5.62614	0.201319
$782$	−5.91915	−0.211668
$783$	0	0
$784$	−13.2087	−0.471740
$785$	0	0
$786$	0	0
$787$	−10.7913	−0.384668	−0.192334	−	0.981330i	$-0.561606\pi$
$787$	−10.7913	−0.384668	−0.192334	+	0.981330i	$0.561606\pi$
$788$	−5.84370	−0.208173
$789$	0	0
$790$	0	0
$791$	−12.1244	−0.431092
$792$	0	0
$793$	−42.7042	−1.51647
$794$	49.2403	1.74747
$795$	0	0
$796$	−28.9564	−1.02633
$797$	40.5801	1.43742	0.718710	−	0.695310i	$-0.244733\pi$
$797$	40.5801	1.43742	0.718710	+	0.695310i	$0.244733\pi$
$798$	0	0
$799$	45.7042	1.61690
$800$	0	0
$801$	0	0
$802$	72.6606	2.56574
$803$	−1.90285	−0.0671501
$804$	0	0
$805$	0	0
$806$	−38.0297	−1.33954
$807$	0	0
$808$	23.3739	0.822290
$809$	48.9742	1.72184	0.860920	−	0.508740i	$-0.169889\pi$
$809$	48.9742	1.72184	0.860920	+	0.508740i	$0.169889\pi$
$810$	0	0
$811$	−20.7042	−0.727022	−0.363511	−	0.931590i	$-0.618422\pi$
$811$	−20.7042	−0.727022	−0.363511	+	0.931590i	$0.618422\pi$
$812$	−55.9977	−1.96513
$813$	0	0
$814$	−4.79129	−0.167935
$815$	0	0
$816$	0	0
$817$	−18.8348	−0.658948
$818$	−38.9434	−1.36162
$819$	0	0
$820$	0	0
$821$	12.1244	0.423143	0.211571	−	0.977363i	$-0.432142\pi$
$821$	12.1244	0.423143	0.211571	+	0.977363i	$0.432142\pi$
$822$	0	0
$823$	23.4174	0.816280	0.408140	−	0.912919i	$-0.366178\pi$
$823$	23.4174	0.816280	0.408140	+	0.912919i	$0.366178\pi$
$824$	10.3169	0.359405
$825$	0	0
$826$	68.0780	2.36874
$827$	32.7382	1.13842	0.569209	−	0.822193i	$-0.307250\pi$
$827$	32.7382	1.13842	0.569209	+	0.822193i	$0.307250\pi$
$828$	0	0
$829$	−9.49545	−0.329791	−0.164895	−	0.986311i	$-0.552729\pi$
$829$	−9.49545	−0.329791	−0.164895	+	0.986311i	$0.552729\pi$
$830$	0	0
$831$	0	0
$832$	−72.8693	−2.52629
$833$	55.1595	1.91116
$834$	0	0
$835$	0	0
$836$	7.11890	0.246212
$837$	0	0
$838$	78.0780	2.69716
$839$	39.5909	1.36683	0.683415	−	0.730030i	$-0.260494\pi$
$839$	39.5909	1.36683	0.683415	+	0.730030i	$0.260494\pi$
$840$	0	0
$841$	−1.00000	−0.0344828
$842$	43.7780	1.50869
$843$	0	0
$844$	−53.9564	−1.85726
$845$	0	0
$846$	0	0
$847$	−40.9129	−1.40578
$848$	22.5366	0.773909
$849$	0	0
$850$	0	0
$851$	1.73205	0.0593739
$852$	0	0
$853$	39.6606	1.35795	0.678977	−	0.734160i	$-0.262424\pi$
$853$	39.6606	1.35795	0.678977	+	0.734160i	$0.262424\pi$
$854$	61.1939	2.09401
$855$	0	0
$856$	25.7477	0.880039
$857$	−2.47495	−0.0845427	−0.0422714	−	0.999106i	$-0.513459\pi$
$857$	−2.47495	−0.0845427	−0.0422714	+	0.999106i	$0.513459\pi$
$858$	0	0
$859$	−41.0000	−1.39890	−0.699451	−	0.714681i	$-0.746572\pi$
$859$	−41.0000	−1.39890	−0.699451	+	0.714681i	$0.746572\pi$
$860$	0	0
$861$	0	0
$862$	−43.2867	−1.47435
$863$	36.3930	1.23883	0.619416	−	0.785063i	$-0.287370\pi$
$863$	36.3930	1.23883	0.619416	+	0.785063i	$0.287370\pi$
$864$	0	0
$865$	0	0
$866$	44.6917	1.51869
$867$	0	0
$868$	31.7477	1.07759
$869$	−6.92820	−0.235023
$870$	0	0
$871$	8.20871	0.278142
$872$	−4.83465	−0.163722
$873$	0	0
$874$	−4.41742	−0.149422
$875$	0	0
$876$	0	0
$877$	−11.7042	−0.395222	−0.197611	−	0.980281i	$-0.563318\pi$
$877$	−11.7042	−0.395222	−0.197611	+	0.980281i	$0.563318\pi$
$878$	22.7074	0.766336
$879$	0	0
$880$	0	0
$881$	−48.9742	−1.64998	−0.824991	−	0.565146i	$-0.808820\pi$
$881$	−48.9742	−1.64998	−0.824991	+	0.565146i	$0.808820\pi$
$882$	0	0
$883$	46.0000	1.54802	0.774012	−	0.633171i	$-0.218247\pi$
$883$	46.0000	1.54802	0.774012	+	0.633171i	$0.218247\pi$
$884$	120.922	4.06704
$885$	0	0
$886$	6.79129	0.228158
$887$	−33.1196	−1.11205	−0.556023	−	0.831167i	$-0.687673\pi$
$887$	−33.1196	−1.11205	−0.556023	+	0.831167i	$0.687673\pi$
$888$	0	0
$889$	64.4519	2.16165
$890$	0	0
$891$	0	0
$892$	49.5390	1.65869
$893$	34.1087	1.14140
$894$	0	0
$895$	0	0
$896$	48.4220	1.61766
$897$	0	0
$898$	29.9564	0.999659
$899$	−15.8745	−0.529444
$900$	0	0
$901$	−94.1125	−3.13534
$902$	−6.47135	−0.215472
$903$	0	0
$904$	5.53901	0.184225
$905$	0	0
$906$	0	0
$907$	−28.1216	−0.933762	−0.466881	−	0.884320i	$-0.654622\pi$
$907$	−28.1216	−0.933762	−0.466881	+	0.884320i	$0.654622\pi$
$908$	40.9615	1.35935
$909$	0	0
$910$	0	0
$911$	−18.5203	−0.613604	−0.306802	−	0.951773i	$-0.599259\pi$
$911$	−18.5203	−0.613604	−0.306802	+	0.951773i	$0.599259\pi$
$912$	0	0
$913$	−2.37386	−0.0785634
$914$	−60.3755	−1.99705
$915$	0	0
$916$	−4.87841	−0.161187
$917$	−10.3923	−0.343184
$918$	0	0
$919$	8.12159	0.267907	0.133953	−	0.990988i	$-0.457233\pi$
$919$	8.12159	0.267907	0.133953	+	0.990988i	$0.457233\pi$
$920$	0	0
$921$	0	0
$922$	58.9129	1.94019
$923$	71.3200	2.34753
$924$	0	0
$925$	0	0
$926$	38.0297	1.24973
$927$	0	0
$928$	−39.0780	−1.28280
$929$	15.7792	0.517697	0.258849	−	0.965918i	$-0.416657\pi$
$929$	15.7792	0.517697	0.258849	+	0.965918i	$0.416657\pi$
$930$	0	0
$931$	41.1652	1.34913
$932$	−24.1733	−0.791822
$933$	0	0
$934$	0.417424	0.0136585
$935$	0	0
$936$	0	0
$937$	1.41742	0.0463052	0.0231526	−	0.999732i	$-0.492630\pi$
$937$	1.41742	0.0463052	0.0231526	+	0.999732i	$0.492630\pi$
$938$	−11.7629	−0.384071
$939$	0	0
$940$	0	0
$941$	40.1033	1.30733	0.653665	−	0.756784i	$-0.273230\pi$
$941$	40.1033	1.30733	0.653665	+	0.756784i	$0.273230\pi$
$942$	0	0
$943$	2.33939	0.0761812
$944$	14.6947	0.478270
$945$	0	0
$946$	3.37386	0.109694
$947$	50.8016	1.65083	0.825415	−	0.564527i	$-0.190941\pi$
$947$	50.8016	1.65083	0.825415	+	0.564527i	$0.190941\pi$
$948$	0	0
$949$	−24.1216	−0.783020
$950$	0	0
$951$	0	0
$952$	−49.1216	−1.59204
$953$	−4.71940	−0.152876	−0.0764382	−	0.997074i	$-0.524355\pi$
$953$	−4.71940	−0.152876	−0.0764382	+	0.997074i	$0.524355\pi$
$954$	0	0
$955$	0	0
$956$	−2.55040	−0.0824859
$957$	0	0
$958$	30.1652	0.974592
$959$	51.1631	1.65214
$960$	0	0
$961$	−22.0000	−0.709677
$962$	−60.7370	−1.95824
$963$	0	0
$964$	−34.6606	−1.11634
$965$	0	0
$966$	0	0
$967$	40.4955	1.30225	0.651123	−	0.758972i	$-0.274298\pi$
$967$	40.4955	1.30225	0.651123	+	0.758972i	$0.274298\pi$
$968$	18.6911	0.600753
$969$	0	0
$970$	0	0
$971$	−17.6820	−0.567443	−0.283721	−	0.958907i	$-0.591569\pi$
$971$	−17.6820	−0.567443	−0.283721	+	0.958907i	$0.591569\pi$
$972$	0	0
$973$	−46.9129	−1.50396
$974$	−8.75560	−0.280548
$975$	0	0
$976$	13.2087	0.422801
$977$	25.5239	0.816582	0.408291	−	0.912852i	$-0.366125\pi$
$977$	25.5239	0.816582	0.408291	+	0.912852i	$0.366125\pi$
$978$	0	0
$979$	−1.25227	−0.0400228
$980$	0	0
$981$	0	0
$982$	15.3739	0.490600
$983$	16.1606	0.515442	0.257721	−	0.966219i	$-0.417028\pi$
$983$	16.1606	0.515442	0.257721	+	0.966219i	$0.417028\pi$
$984$	0	0
$985$	0	0
$986$	86.6423	2.75925
$987$	0	0
$988$	90.2432	2.87102
$989$	−1.21965	−0.0387827
$990$	0	0
$991$	36.4519	1.15793	0.578966	−	0.815351i	$-0.303456\pi$
$991$	36.4519	1.15793	0.578966	+	0.815351i	$0.303456\pi$
$992$	22.1552	0.703427
$993$	0	0
$994$	−102.200	−3.24158
$995$	0	0
$996$	0	0
$997$	−41.4083	−1.31142	−0.655708	−	0.755015i	$-0.727630\pi$
$997$	−41.4083	−1.31142	−0.655708	+	0.755015i	$0.727630\pi$
$998$	−69.5880	−2.20277
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2025.2.a.v.1.1	yes	4
3.2	odd	2	inner	2025.2.a.v.1.4	yes	4
5.2	odd	4		2025.2.b.p.649.2		8
5.3	odd	4		2025.2.b.p.649.7		8
5.4	even	2		2025.2.a.u.1.4	yes	4
15.2	even	4		2025.2.b.p.649.8		8
15.8	even	4		2025.2.b.p.649.1		8
15.14	odd	2		2025.2.a.u.1.1	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2025.2.a.u.1.1	✓	4	15.14	odd	2
2025.2.a.u.1.4	yes	4	5.4	even	2
2025.2.a.v.1.1	yes	4	1.1	even	1	trivial
2025.2.a.v.1.4	yes	4	3.2	odd	2	inner
2025.2.b.p.649.1		8	15.8	even	4
2025.2.b.p.649.2		8	5.2	odd	4
2025.2.b.p.649.7		8	5.3	odd	4
2025.2.b.p.649.8		8	15.2	even	4

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)

\(f(q)\)	\(=\)	\(q-2.18890 q^{2} +2.79129 q^{4} +3.79129 q^{7} -1.73205 q^{8} +O(q^{10})\)	\(q-2.18890 q^{2} +2.79129 q^{4} +3.79129 q^{7} -1.73205 q^{8} +0.456850 q^{11} +5.79129 q^{13} -8.29875 q^{14} -1.79129 q^{16} +7.48040 q^{17} +5.58258 q^{19} -1.00000 q^{22} +0.361500 q^{23} -12.6766 q^{26} +10.5826 q^{28} -5.29150 q^{29} +3.00000 q^{31} +7.38505 q^{32} -16.3739 q^{34} +4.79129 q^{37} -12.2197 q^{38} +6.47135 q^{41} -3.37386 q^{43} +1.27520 q^{44} -0.791288 q^{46} +6.10985 q^{47} +7.37386 q^{49} +16.1652 q^{52} -12.5812 q^{53} -6.56670 q^{56} +11.5826 q^{58} -8.20340 q^{59} -7.37386 q^{61} -6.56670 q^{62} -12.5826 q^{64} +1.41742 q^{67} +20.8800 q^{68} +12.3151 q^{71} -4.16515 q^{73} -10.4877 q^{74} +15.5826 q^{76} +1.73205 q^{77} -15.1652 q^{79} -14.1652 q^{82} -5.19615 q^{83} +7.38505 q^{86} -0.791288 q^{88} -2.74110 q^{89} +21.9564 q^{91} +1.00905 q^{92} -13.3739 q^{94} -5.16515 q^{97} -16.1407 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{4} + 6 q^{7}+O(q^{10}) \) 4 * q + 2 * q^4 + 6 * q^7	\( 4 q + 2 q^{4} + 6 q^{7} + 14 q^{13} + 2 q^{16} + 4 q^{19} - 4 q^{22} + 24 q^{28} + 12 q^{31} - 38 q^{34} + 10 q^{37} + 14 q^{43} + 6 q^{46} + 2 q^{49} + 28 q^{52} + 28 q^{58} - 2 q^{61} - 32 q^{64} + 24 q^{67} + 20 q^{73} + 44 q^{76} - 24 q^{79} - 20 q^{82} + 6 q^{88} + 42 q^{91} - 26 q^{94} + 16 q^{97}+O(q^{100}) \) 4 * q + 2 * q^4 + 6 * q^7 + 14 * q^13 + 2 * q^16 + 4 * q^19 - 4 * q^22 + 24 * q^28 + 12 * q^31 - 38 * q^34 + 10 * q^37 + 14 * q^43 + 6 * q^46 + 2 * q^49 + 28 * q^52 + 28 * q^58 - 2 * q^61 - 32 * q^64 + 24 * q^67 + 20 * q^73 + 44 * q^76 - 24 * q^79 - 20 * q^82 + 6 * q^88 + 42 * q^91 - 26 * q^94 + 16 * q^97

Introduction

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