LMFDB - Embedded newform 2025.2.a.t.1.4

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			1.4
Root			$1.93185$ 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)

$f(q)$	$=$	$q+1.93185 q^{2} +1.73205 q^{4} +0.896575 q^{7} -0.517638 q^{8} +O(q^{10})$	$q+1.93185 q^{2} +1.73205 q^{4} +0.896575 q^{7} -0.517638 q^{8} +4.73205 q^{11} +2.44949 q^{13} +1.73205 q^{14} -4.46410 q^{16} +0.378937 q^{17} +2.73205 q^{19} +9.14162 q^{22} -1.93185 q^{23} +4.73205 q^{26} +1.55291 q^{28} +6.46410 q^{29} -2.73205 q^{31} -7.58871 q^{32} +0.732051 q^{34} -4.24264 q^{37} +5.27792 q^{38} +4.26795 q^{41} +9.14162 q^{43} +8.19615 q^{44} -3.73205 q^{46} +4.38134 q^{47} -6.19615 q^{49} +4.24264 q^{52} +3.86370 q^{53} -0.464102 q^{56} +12.4877 q^{58} +2.53590 q^{59} +10.6603 q^{61} -5.27792 q^{62} -5.73205 q^{64} -5.13922 q^{67} +0.656339 q^{68} +3.80385 q^{71} +8.48528 q^{73} -8.19615 q^{74} +4.73205 q^{76} +4.24264 q^{77} +0.535898 q^{79} +8.24504 q^{82} -10.4171 q^{83} +17.6603 q^{86} -2.44949 q^{88} -7.39230 q^{89} +2.19615 q^{91} -3.34607 q^{92} +8.46410 q^{94} -10.2784 q^{97} -11.9700 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	1.93185	1.36603	0.683013	−	0.730406i	$-0.260669\pi$
$2$	1.93185	1.36603	0.683013	+	0.730406i	$0.260669\pi$
$3$	0	0
$3$	0	0
$4$	1.73205	0.866025
$5$	0	0
$5$	0	0
$6$	0	0
$7$	0.896575	0.338874	0.169437	−	0.985541i	$-0.445805\pi$
$7$	0.896575	0.338874	0.169437	+	0.985541i	$0.445805\pi$
$8$	−0.517638	−0.183013
$9$	0	0
$10$	0	0
$11$	4.73205	1.42677	0.713384	−	0.700774i	$-0.247162\pi$
$11$	4.73205	1.42677	0.713384	+	0.700774i	$0.247162\pi$
$12$	0	0
$13$	2.44949	0.679366	0.339683	−	0.940540i	$-0.389680\pi$
$13$	2.44949	0.679366	0.339683	+	0.940540i	$0.389680\pi$
$14$	1.73205	0.462910
$15$	0	0
$16$	−4.46410	−1.11603
$17$	0.378937	0.0919058	0.0459529	−	0.998944i	$-0.485368\pi$
$17$	0.378937	0.0919058	0.0459529	+	0.998944i	$0.485368\pi$
$18$	0	0
$19$	2.73205	0.626775	0.313388	−	0.949625i	$-0.398536\pi$
$19$	2.73205	0.626775	0.313388	+	0.949625i	$0.398536\pi$
$20$	0	0
$21$	0	0
$22$	9.14162	1.94900
$23$	−1.93185	−0.402819	−0.201409	−	0.979507i	$-0.564552\pi$
$23$	−1.93185	−0.402819	−0.201409	+	0.979507i	$0.564552\pi$
$24$	0	0
$25$	0	0
$26$	4.73205	0.928032
$27$	0	0
$28$	1.55291	0.293473
$29$	6.46410	1.20035	0.600177	−	0.799867i	$-0.295097\pi$
$29$	6.46410	1.20035	0.600177	+	0.799867i	$0.295097\pi$
$30$	0	0
$31$	−2.73205	−0.490691	−0.245345	−	0.969436i	$-0.578901\pi$
$31$	−2.73205	−0.490691	−0.245345	+	0.969436i	$0.578901\pi$
$32$	−7.58871	−1.34151
$33$	0	0
$34$	0.732051	0.125546
$35$	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	$0.613391\pi$
$38$	5.27792	0.856191
$39$	0	0
$40$	0	0
$41$	4.26795	0.666542	0.333271	−	0.942831i	$-0.391848\pi$
$41$	4.26795	0.666542	0.333271	+	0.942831i	$0.391848\pi$
$42$	0	0
$43$	9.14162	1.39408	0.697042	−	0.717030i	$-0.254499\pi$
$43$	9.14162	1.39408	0.697042	+	0.717030i	$0.254499\pi$
$44$	8.19615	1.23562
$45$	0	0
$46$	−3.73205	−0.550261
$47$	4.38134	0.639084	0.319542	−	0.947572i	$-0.396471\pi$
$47$	4.38134	0.639084	0.319542	+	0.947572i	$0.396471\pi$
$48$	0	0
$49$	−6.19615	−0.885165
$50$	0	0
$51$	0	0
$52$	4.24264	0.588348
$53$	3.86370	0.530720	0.265360	−	0.964149i	$-0.414509\pi$
$53$	3.86370	0.530720	0.265360	+	0.964149i	$0.414509\pi$
$54$	0	0
$55$	0	0
$56$	−0.464102	−0.0620182
$57$	0	0
$58$	12.4877	1.63971
$59$	2.53590	0.330146	0.165073	−	0.986281i	$-0.447214\pi$
$59$	2.53590	0.330146	0.165073	+	0.986281i	$0.447214\pi$
$60$	0	0
$61$	10.6603	1.36491	0.682453	−	0.730930i	$-0.260913\pi$
$61$	10.6603	1.36491	0.682453	+	0.730930i	$0.260913\pi$
$62$	−5.27792	−0.670296
$63$	0	0
$64$	−5.73205	−0.716506
$65$	0	0
$66$	0	0
$67$	−5.13922	−0.627855	−0.313928	−	0.949447i	$-0.601645\pi$
$67$	−5.13922	−0.627855	−0.313928	+	0.949447i	$0.601645\pi$
$68$	0.656339	0.0795928
$69$	0	0
$70$	0	0
$71$	3.80385	0.451434	0.225717	−	0.974193i	$-0.427528\pi$
$71$	3.80385	0.451434	0.225717	+	0.974193i	$0.427528\pi$
$72$	0	0
$73$	8.48528	0.993127	0.496564	−	0.868000i	$-0.334595\pi$
$73$	8.48528	0.993127	0.496564	+	0.868000i	$0.334595\pi$
$74$	−8.19615	−0.952783
$75$	0	0
$76$	4.73205	0.542803
$77$	4.24264	0.483494
$78$	0	0
$79$	0.535898	0.0602933	0.0301466	−	0.999545i	$-0.490403\pi$
$79$	0.535898	0.0602933	0.0301466	+	0.999545i	$0.490403\pi$
$80$	0	0
$81$	0	0
$82$	8.24504	0.910513
$83$	−10.4171	−1.14343	−0.571714	−	0.820453i	$-0.693721\pi$
$83$	−10.4171	−1.14343	−0.571714	+	0.820453i	$0.693721\pi$
$84$	0	0
$85$	0	0
$86$	17.6603	1.90435
$87$	0	0
$88$	−2.44949	−0.261116
$89$	−7.39230	−0.783583	−0.391791	−	0.920054i	$-0.628144\pi$
$89$	−7.39230	−0.783583	−0.391791	+	0.920054i	$0.628144\pi$
$90$	0	0
$91$	2.19615	0.230219
$92$	−3.34607	−0.348851
$93$	0	0
$94$	8.46410	0.873005
$95$	0	0
$96$	0	0
$97$	−10.2784	−1.04362	−0.521808	−	0.853063i	$-0.674742\pi$
$97$	−10.2784	−1.04362	−0.521808	+	0.853063i	$0.674742\pi$
$98$	−11.9700	−1.20916
$99$	0	0
$100$	0	0
$101$	2.53590	0.252331	0.126166	−	0.992009i	$-0.459733\pi$
$101$	2.53590	0.252331	0.126166	+	0.992009i	$0.459733\pi$
$102$	0	0
$103$	−9.14162	−0.900751	−0.450375	−	0.892839i	$-0.648710\pi$
$103$	−9.14162	−0.900751	−0.450375	+	0.892839i	$0.648710\pi$
$104$	−1.26795	−0.124333
$105$	0	0
$106$	7.46410	0.724978
$107$	11.7298	1.13396	0.566982	−	0.823730i	$-0.308111\pi$
$107$	11.7298	1.13396	0.566982	+	0.823730i	$0.308111\pi$
$108$	0	0
$109$	−4.66025	−0.446371	−0.223186	−	0.974776i	$-0.571646\pi$
$109$	−4.66025	−0.446371	−0.223186	+	0.974776i	$0.571646\pi$
$110$	0	0
$111$	0	0
$112$	−4.00240	−0.378192
$113$	2.55103	0.239980	0.119990	−	0.992775i	$-0.461714\pi$
$113$	2.55103	0.239980	0.119990	+	0.992775i	$0.461714\pi$
$114$	0	0
$115$	0	0
$116$	11.1962	1.03954
$117$	0	0
$118$	4.89898	0.450988
$119$	0.339746	0.0311445
$120$	0	0
$121$	11.3923	1.03566
$122$	20.5940	1.86450
$123$	0	0
$124$	−4.73205	−0.424951
$125$	0	0
$126$	0	0
$127$	−4.65874	−0.413397	−0.206698	−	0.978405i	$-0.566272\pi$
$127$	−4.65874	−0.413397	−0.206698	+	0.978405i	$0.566272\pi$
$128$	4.10394	0.362740
$129$	0	0
$130$	0	0
$131$	−0.928203	−0.0810975	−0.0405487	−	0.999178i	$-0.512911\pi$
$131$	−0.928203	−0.0810975	−0.0405487	+	0.999178i	$0.512911\pi$
$132$	0	0
$133$	2.44949	0.212398
$134$	−9.92820	−0.857666
$135$	0	0
$136$	−0.196152	−0.0168199
$137$	−18.5606	−1.58574	−0.792871	−	0.609389i	$-0.791415\pi$
$137$	−18.5606	−1.58574	−0.792871	+	0.609389i	$0.791415\pi$
$138$	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
$143$	11.5911	0.969297
$144$	0	0
$145$	0	0
$146$	16.3923	1.35664
$147$	0	0
$148$	−7.34847	−0.604040
$149$	7.73205	0.633434	0.316717	−	0.948520i	$-0.397419\pi$
$149$	7.73205	0.633434	0.316717	+	0.948520i	$0.397419\pi$
$150$	0	0
$151$	−22.5885	−1.83822	−0.919111	−	0.393998i	$-0.871092\pi$
$151$	−22.5885	−1.83822	−0.919111	+	0.393998i	$0.871092\pi$
$152$	−1.41421	−0.114708
$153$	0	0
$154$	8.19615	0.660465
$155$	0	0
$156$	0	0
$157$	−21.3891	−1.70703	−0.853517	−	0.521065i	$-0.825535\pi$
$157$	−21.3891	−1.70703	−0.853517	+	0.521065i	$0.825535\pi$
$158$	1.03528	0.0823622
$159$	0	0
$160$	0	0
$161$	−1.73205	−0.136505
$162$	0	0
$163$	−18.9396	−1.48346	−0.741731	−	0.670697i	$-0.765995\pi$
$163$	−18.9396	−1.48346	−0.741731	+	0.670697i	$0.765995\pi$
$164$	7.39230	0.577242
$165$	0	0
$166$	−20.1244	−1.56195
$167$	−17.0077	−1.31610	−0.658049	−	0.752975i	$-0.728618\pi$
$167$	−17.0077	−1.31610	−0.658049	+	0.752975i	$0.728618\pi$
$168$	0	0
$169$	−7.00000	−0.538462
$170$	0	0
$171$	0	0
$172$	15.8338	1.20731
$173$	0.757875	0.0576202	0.0288101	−	0.999585i	$-0.490828\pi$
$173$	0.757875	0.0576202	0.0288101	+	0.999585i	$0.490828\pi$
$174$	0	0
$175$	0	0
$176$	−21.1244	−1.59231
$177$	0	0
$178$	−14.2808	−1.07039
$179$	24.5885	1.83783	0.918914	−	0.394458i	$-0.129068\pi$
$179$	24.5885	1.83783	0.918914	+	0.394458i	$0.129068\pi$
$180$	0	0
$181$	8.46410	0.629132	0.314566	−	0.949236i	$-0.398141\pi$
$181$	8.46410	0.629132	0.314566	+	0.949236i	$0.398141\pi$
$182$	4.24264	0.314485
$183$	0	0
$184$	1.00000	0.0737210
$185$	0	0
$186$	0	0
$187$	1.79315	0.131128
$188$	7.58871	0.553463
$189$	0	0
$190$	0	0
$191$	0.339746	0.0245832	0.0122916	−	0.999924i	$-0.496087\pi$
$191$	0.339746	0.0245832	0.0122916	+	0.999924i	$0.496087\pi$
$192$	0	0
$193$	−20.7327	−1.49237	−0.746187	−	0.665736i	$-0.768118\pi$
$193$	−20.7327	−1.49237	−0.746187	+	0.665736i	$0.768118\pi$
$194$	−19.8564	−1.42561
$195$	0	0
$196$	−10.7321	−0.766575
$197$	−20.8343	−1.48438	−0.742190	−	0.670190i	$-0.766213\pi$
$197$	−20.8343	−1.48438	−0.742190	+	0.670190i	$0.766213\pi$
$198$	0	0
$199$	4.58846	0.325267	0.162634	−	0.986687i	$-0.448001\pi$
$199$	4.58846	0.325267	0.162634	+	0.986687i	$0.448001\pi$
$200$	0	0
$201$	0	0
$202$	4.89898	0.344691
$203$	5.79555	0.406768
$204$	0	0
$205$	0	0
$206$	−17.6603	−1.23045
$207$	0	0
$208$	−10.9348	−0.758190
$209$	12.9282	0.894263
$210$	0	0
$211$	−13.1244	−0.903518	−0.451759	−	0.892140i	$-0.649203\pi$
$211$	−13.1244	−0.903518	−0.451759	+	0.892140i	$0.649203\pi$
$212$	6.69213	0.459617
$213$	0	0
$214$	22.6603	1.54902
$215$	0	0
$216$	0	0
$217$	−2.44949	−0.166282
$218$	−9.00292	−0.609755
$219$	0	0
$220$	0	0
$221$	0.928203	0.0624377
$222$	0	0
$223$	−16.7303	−1.12035	−0.560173	−	0.828376i	$-0.689265\pi$
$223$	−16.7303	−1.12035	−0.560173	+	0.828376i	$0.689265\pi$
$224$	−6.80385	−0.454601
$225$	0	0
$226$	4.92820	0.327819
$227$	−1.89469	−0.125755	−0.0628774	−	0.998021i	$-0.520028\pi$
$227$	−1.89469	−0.125755	−0.0628774	+	0.998021i	$0.520028\pi$
$228$	0	0
$229$	19.9282	1.31689	0.658446	−	0.752628i	$-0.271214\pi$
$229$	19.9282	1.31689	0.658446	+	0.752628i	$0.271214\pi$
$230$	0	0
$231$	0	0
$232$	−3.34607	−0.219680
$233$	−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
$236$	4.39230	0.285915
$237$	0	0
$238$	0.656339	0.0425441
$239$	12.9282	0.836256	0.418128	−	0.908388i	$-0.362686\pi$
$239$	12.9282	0.836256	0.418128	+	0.908388i	$0.362686\pi$
$240$	0	0
$241$	6.26795	0.403754	0.201877	−	0.979411i	$-0.435296\pi$
$241$	6.26795	0.403754	0.201877	+	0.979411i	$0.435296\pi$
$242$	22.0082	1.41474
$243$	0	0
$244$	18.4641	1.18204
$245$	0	0
$246$	0	0
$247$	6.69213	0.425810
$248$	1.41421	0.0898027
$249$	0	0
$250$	0	0
$251$	18.5885	1.17329	0.586647	−	0.809843i	$-0.300448\pi$
$251$	18.5885	1.17329	0.586647	+	0.809843i	$0.300448\pi$
$252$	0	0
$253$	−9.14162	−0.574729
$254$	−9.00000	−0.564710
$255$	0	0
$256$	19.3923	1.21202
$257$	22.4243	1.39879	0.699396	−	0.714734i	$-0.253452\pi$
$257$	22.4243	1.39879	0.699396	+	0.714734i	$0.253452\pi$
$258$	0	0
$259$	−3.80385	−0.236360
$260$	0	0
$261$	0	0
$262$	−1.79315	−0.110781
$263$	7.62587	0.470231	0.235116	−	0.971967i	$-0.424453\pi$
$263$	7.62587	0.470231	0.235116	+	0.971967i	$0.424453\pi$
$264$	0	0
$265$	0	0
$266$	4.73205	0.290141
$267$	0	0
$268$	−8.90138	−0.543739
$269$	17.1962	1.04847	0.524234	−	0.851574i	$-0.324352\pi$
$269$	17.1962	1.04847	0.524234	+	0.851574i	$0.324352\pi$
$270$	0	0
$271$	−23.5167	−1.42854	−0.714268	−	0.699873i	$-0.753240\pi$
$271$	−23.5167	−1.42854	−0.714268	+	0.699873i	$0.753240\pi$
$272$	−1.69161	−0.102569
$273$	0	0
$274$	−35.8564	−2.16616
$275$	0	0
$276$	0	0
$277$	−12.2474	−0.735878	−0.367939	−	0.929850i	$-0.619936\pi$
$277$	−12.2474	−0.735878	−0.367939	+	0.929850i	$0.619936\pi$
$278$	−15.4548	−0.926918
$279$	0	0
$280$	0	0
$281$	−16.2679	−0.970464	−0.485232	−	0.874385i	$-0.661265\pi$
$281$	−16.2679	−0.970464	−0.485232	+	0.874385i	$0.661265\pi$
$282$	0	0
$283$	−4.48288	−0.266479	−0.133240	−	0.991084i	$-0.542538\pi$
$283$	−4.48288	−0.266479	−0.133240	+	0.991084i	$0.542538\pi$
$284$	6.58846	0.390953
$285$	0	0
$286$	22.3923	1.32408
$287$	3.82654	0.225873
$288$	0	0
$289$	−16.8564	−0.991553
$290$	0	0
$291$	0	0
$292$	14.6969	0.860073
$293$	−8.86422	−0.517853	−0.258927	−	0.965897i	$-0.583369\pi$
$293$	−8.86422	−0.517853	−0.258927	+	0.965897i	$0.583369\pi$
$294$	0	0
$295$	0	0
$296$	2.19615	0.127649
$297$	0	0
$298$	14.9372	0.865287
$299$	−4.73205	−0.273662
$300$	0	0
$301$	8.19615	0.472418
$302$	−43.6375	−2.51106
$303$	0	0
$304$	−12.1962	−0.699497
$305$	0	0
$306$	0	0
$307$	−25.8719	−1.47659	−0.738295	−	0.674478i	$-0.764369\pi$
$307$	−25.8719	−1.47659	−0.738295	+	0.674478i	$0.764369\pi$
$308$	7.34847	0.418718
$309$	0	0
$310$	0	0
$311$	28.0526	1.59071	0.795357	−	0.606141i	$-0.207283\pi$
$311$	28.0526	1.59071	0.795357	+	0.606141i	$0.207283\pi$
$312$	0	0
$313$	−5.55532	−0.314005	−0.157003	−	0.987598i	$-0.550183\pi$
$313$	−5.55532	−0.314005	−0.157003	+	0.987598i	$0.550183\pi$
$314$	−41.3205	−2.33185
$315$	0	0
$316$	0.928203	0.0522155
$317$	35.1523	1.97435	0.987174	−	0.159648i	$-0.0510359\pi$
$317$	35.1523	1.97435	0.987174	+	0.159648i	$0.0510359\pi$
$318$	0	0
$319$	30.5885	1.71262
$320$	0	0
$321$	0	0
$322$	−3.34607	−0.186469
$323$	1.03528	0.0576043
$324$	0	0
$325$	0	0
$326$	−36.5885	−2.02645
$327$	0	0
$328$	−2.20925	−0.121986
$329$	3.92820	0.216569
$330$	0	0
$331$	12.3923	0.681143	0.340571	−	0.940219i	$-0.389380\pi$
$331$	12.3923	0.681143	0.340571	+	0.940219i	$0.389380\pi$
$332$	−18.0430	−0.990238
$333$	0	0
$334$	−32.8564	−1.79782
$335$	0	0
$336$	0	0
$337$	1.79315	0.0976792	0.0488396	−	0.998807i	$-0.484448\pi$
$337$	1.79315	0.0976792	0.0488396	+	0.998807i	$0.484448\pi$
$338$	−13.5230	−0.735552
$339$	0	0
$340$	0	0
$341$	−12.9282	−0.700101
$342$	0	0
$343$	−11.8313	−0.638833
$344$	−4.73205	−0.255135
$345$	0	0
$346$	1.46410	0.0787106
$347$	−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
$349$	21.5359	1.15279	0.576395	−	0.817171i	$-0.304459\pi$
$349$	21.5359	1.15279	0.576395	+	0.817171i	$0.304459\pi$
$350$	0	0
$351$	0	0
$352$	−35.9101	−1.91402
$353$	−9.04008	−0.481155	−0.240578	−	0.970630i	$-0.577337\pi$
$353$	−9.04008	−0.481155	−0.240578	+	0.970630i	$0.577337\pi$
$354$	0	0
$355$	0	0
$356$	−12.8038	−0.678603
$357$	0	0
$358$	47.5013	2.51052
$359$	9.80385	0.517427	0.258714	−	0.965954i	$-0.416701\pi$
$359$	9.80385	0.517427	0.258714	+	0.965954i	$0.416701\pi$
$360$	0	0
$361$	−11.5359	−0.607153
$362$	16.3514	0.859410
$363$	0	0
$364$	3.80385	0.199376
$365$	0	0
$366$	0	0
$367$	24.7995	1.29452	0.647262	−	0.762268i	$-0.275914\pi$
$367$	24.7995	1.29452	0.647262	+	0.762268i	$0.275914\pi$
$368$	8.62398	0.449556
$369$	0	0
$370$	0	0
$371$	3.46410	0.179847
$372$	0	0
$373$	32.1480	1.66456	0.832280	−	0.554356i	$-0.187035\pi$
$373$	32.1480	1.66456	0.832280	+	0.554356i	$0.187035\pi$
$374$	3.46410	0.179124
$375$	0	0
$376$	−2.26795	−0.116961
$377$	15.8338	0.815480
$378$	0	0
$379$	12.5359	0.643926	0.321963	−	0.946752i	$-0.395657\pi$
$379$	12.5359	0.643926	0.321963	+	0.946752i	$0.395657\pi$
$380$	0	0
$381$	0	0
$382$	0.656339	0.0335812
$383$	−20.4553	−1.04522	−0.522609	−	0.852572i	$-0.675041\pi$
$383$	−20.4553	−1.04522	−0.522609	+	0.852572i	$0.675041\pi$
$384$	0	0
$385$	0	0
$386$	−40.0526	−2.03862
$387$	0	0
$388$	−17.8028	−0.903799
$389$	−24.1244	−1.22315	−0.611577	−	0.791185i	$-0.709464\pi$
$389$	−24.1244	−1.22315	−0.611577	+	0.791185i	$0.709464\pi$
$390$	0	0
$391$	−0.732051	−0.0370214
$392$	3.20736	0.161996
$393$	0	0
$394$	−40.2487	−2.02770
$395$	0	0
$396$	0	0
$397$	29.3939	1.47524	0.737618	−	0.675218i	$-0.235950\pi$
$397$	29.3939	1.47524	0.737618	+	0.675218i	$0.235950\pi$
$398$	8.86422	0.444323
$399$	0	0
$400$	0	0
$401$	−30.9282	−1.54448	−0.772240	−	0.635330i	$-0.780864\pi$
$401$	−30.9282	−1.54448	−0.772240	+	0.635330i	$0.780864\pi$
$402$	0	0
$403$	−6.69213	−0.333359
$404$	4.39230	0.218525
$405$	0	0
$406$	11.1962	0.555656
$407$	−20.0764	−0.995150
$408$	0	0
$409$	−23.4641	−1.16023	−0.580113	−	0.814536i	$-0.696991\pi$
$409$	−23.4641	−1.16023	−0.580113	+	0.814536i	$0.696991\pi$
$410$	0	0
$411$	0	0
$412$	−15.8338	−0.780073
$413$	2.27362	0.111878
$414$	0	0
$415$	0	0
$416$	−18.5885	−0.911374
$417$	0	0
$418$	24.9754	1.22159
$419$	−16.0526	−0.784219	−0.392109	−	0.919919i	$-0.628255\pi$
$419$	−16.0526	−0.784219	−0.392109	+	0.919919i	$0.628255\pi$
$420$	0	0
$421$	−12.5359	−0.610962	−0.305481	−	0.952198i	$-0.598817\pi$
$421$	−12.5359	−0.610962	−0.305481	+	0.952198i	$0.598817\pi$
$422$	−25.3543	−1.23423
$423$	0	0
$424$	−2.00000	−0.0971286
$425$	0	0
$426$	0	0
$427$	9.55772	0.462531
$428$	20.3166	0.982041
$429$	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
$433$	30.5307	1.46721	0.733606	−	0.679575i	$-0.237836\pi$
$433$	30.5307	1.46721	0.733606	+	0.679575i	$0.237836\pi$
$434$	−4.73205	−0.227146
$435$	0	0
$436$	−8.07180	−0.386569
$437$	−5.27792	−0.252477
$438$	0	0
$439$	9.32051	0.444844	0.222422	−	0.974951i	$-0.428604\pi$
$439$	9.32051	0.444844	0.222422	+	0.974951i	$0.428604\pi$
$440$	0	0
$441$	0	0
$442$	1.79315	0.0852915
$443$	13.9019	0.660499	0.330250	−	0.943894i	$-0.392867\pi$
$443$	13.9019	0.660499	0.330250	+	0.943894i	$0.392867\pi$
$444$	0	0
$445$	0	0
$446$	−32.3205	−1.53042
$447$	0	0
$448$	−5.13922	−0.242805
$449$	−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
$451$	20.1962	0.951000
$452$	4.41851	0.207829
$453$	0	0
$454$	−3.66025	−0.171784
$455$	0	0
$456$	0	0
$457$	−34.5975	−1.61840	−0.809201	−	0.587533i	$-0.800099\pi$
$457$	−34.5975	−1.61840	−0.809201	+	0.587533i	$0.800099\pi$
$458$	38.4983	1.79891
$459$	0	0
$460$	0	0
$461$	18.7128	0.871543	0.435771	−	0.900057i	$-0.356476\pi$
$461$	18.7128	0.871543	0.435771	+	0.900057i	$0.356476\pi$
$462$	0	0
$463$	0.175865	0.00817316	0.00408658	−	0.999992i	$-0.498699\pi$
$463$	0.175865	0.00817316	0.00408658	+	0.999992i	$0.498699\pi$
$464$	−28.8564	−1.33963
$465$	0	0
$466$	−13.6603	−0.632799
$467$	5.75839	0.266467	0.133233	−	0.991085i	$-0.457464\pi$
$467$	5.75839	0.266467	0.133233	+	0.991085i	$0.457464\pi$
$468$	0	0
$469$	−4.60770	−0.212764
$470$	0	0
$471$	0	0
$472$	−1.31268	−0.0604209
$473$	43.2586	1.98903
$474$	0	0
$475$	0	0
$476$	0.588457	0.0269719
$477$	0	0
$478$	24.9754	1.14235
$479$	31.8564	1.45556	0.727778	−	0.685813i	$-0.240553\pi$
$479$	31.8564	1.45556	0.727778	+	0.685813i	$0.240553\pi$
$480$	0	0
$481$	−10.3923	−0.473848
$482$	12.1087	0.551538
$483$	0	0
$484$	19.7321	0.896911
$485$	0	0
$486$	0	0
$487$	1.13681	0.0515139	0.0257569	−	0.999668i	$-0.491800\pi$
$487$	1.13681	0.0515139	0.0257569	+	0.999668i	$0.491800\pi$
$488$	−5.51815	−0.249795
$489$	0	0
$490$	0	0
$491$	−21.7128	−0.979886	−0.489943	−	0.871755i	$-0.662982\pi$
$491$	−21.7128	−0.979886	−0.489943	+	0.871755i	$0.662982\pi$
$492$	0	0
$493$	2.44949	0.110319
$494$	12.9282	0.581667
$495$	0	0
$496$	12.1962	0.547623
$497$	3.41044	0.152979
$498$	0	0
$499$	−3.85641	−0.172636	−0.0863182	−	0.996268i	$-0.527510\pi$
$499$	−3.85641	−0.172636	−0.0863182	+	0.996268i	$0.527510\pi$
$500$	0	0
$501$	0	0
$502$	35.9101	1.60275
$503$	−29.3567	−1.30895	−0.654476	−	0.756083i	$-0.727111\pi$
$503$	−29.3567	−1.30895	−0.654476	+	0.756083i	$0.727111\pi$
$504$	0	0
$505$	0	0
$506$	−17.6603	−0.785094
$507$	0	0
$508$	−8.06918	−0.358012
$509$	−10.8564	−0.481202	−0.240601	−	0.970624i	$-0.577344\pi$
$509$	−10.8564	−0.481202	−0.240601	+	0.970624i	$0.577344\pi$
$510$	0	0
$511$	7.60770	0.336545
$512$	29.2552	1.29291
$513$	0	0
$514$	43.3205	1.91079
$515$	0	0
$516$	0	0
$517$	20.7327	0.911824
$518$	−7.34847	−0.322873
$519$	0	0
$520$	0	0
$521$	1.39230	0.0609980	0.0304990	−	0.999535i	$-0.490290\pi$
$521$	1.39230	0.0609980	0.0304990	+	0.999535i	$0.490290\pi$
$522$	0	0
$523$	30.9468	1.35321	0.676604	−	0.736347i	$-0.263451\pi$
$523$	30.9468	1.35321	0.676604	+	0.736347i	$0.263451\pi$
$524$	−1.60770	−0.0702325
$525$	0	0
$526$	14.7321	0.642348
$527$	−1.03528	−0.0450973
$528$	0	0
$529$	−19.2679	−0.837737
$530$	0	0
$531$	0	0
$532$	4.24264	0.183942
$533$	10.4543	0.452826
$534$	0	0
$535$	0	0
$536$	2.66025	0.114905
$537$	0	0
$538$	33.2204	1.43223
$539$	−29.3205	−1.26292
$540$	0	0
$541$	21.3923	0.919727	0.459864	−	0.887990i	$-0.347898\pi$
$541$	21.3923	0.919727	0.459864	+	0.887990i	$0.347898\pi$
$542$	−45.4307	−1.95142
$543$	0	0
$544$	−2.87564	−0.123292
$545$	0	0
$546$	0	0
$547$	30.2905	1.29513	0.647563	−	0.762012i	$-0.275788\pi$
$547$	30.2905	1.29513	0.647563	+	0.762012i	$0.275788\pi$
$548$	−32.1480	−1.37329
$549$	0	0
$550$	0	0
$551$	17.6603	0.752352
$552$	0	0
$553$	0.480473	0.0204318
$554$	−23.6603	−1.00523
$555$	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
$559$	22.3923	0.947094
$560$	0	0
$561$	0	0
$562$	−31.4273	−1.32568
$563$	−27.5636	−1.16167	−0.580833	−	0.814023i	$-0.697273\pi$
$563$	−27.5636	−1.16167	−0.580833	+	0.814023i	$0.697273\pi$
$564$	0	0
$565$	0	0
$566$	−8.66025	−0.364018
$567$	0	0
$568$	−1.96902	−0.0826181
$569$	−5.32051	−0.223047	−0.111524	−	0.993762i	$-0.535573\pi$
$569$	−5.32051	−0.223047	−0.111524	+	0.993762i	$0.535573\pi$
$570$	0	0
$571$	5.46410	0.228666	0.114333	−	0.993443i	$-0.463527\pi$
$571$	5.46410	0.228666	0.114333	+	0.993443i	$0.463527\pi$
$572$	20.0764	0.839436
$573$	0	0
$574$	7.39230	0.308549
$575$	0	0
$576$	0	0
$577$	28.5617	1.18904	0.594519	−	0.804082i	$-0.297343\pi$
$577$	28.5617	1.18904	0.594519	+	0.804082i	$0.297343\pi$
$578$	−32.5641	−1.35449
$579$	0	0
$580$	0	0
$581$	−9.33975	−0.387478
$582$	0	0
$583$	18.2832	0.757214
$584$	−4.39230	−0.181755
$585$	0	0
$586$	−17.1244	−0.707401
$587$	22.0082	0.908377	0.454189	−	0.890906i	$-0.349929\pi$
$587$	22.0082	0.908377	0.454189	+	0.890906i	$0.349929\pi$
$588$	0	0
$589$	−7.46410	−0.307553
$590$	0	0
$591$	0	0
$592$	18.9396	0.778412
$593$	−28.9406	−1.18845	−0.594224	−	0.804299i	$-0.702541\pi$
$593$	−28.9406	−1.18845	−0.594224	+	0.804299i	$0.702541\pi$
$594$	0	0
$595$	0	0
$596$	13.3923	0.548570
$597$	0	0
$598$	−9.14162	−0.373829
$599$	33.7128	1.37747	0.688734	−	0.725014i	$-0.258167\pi$
$599$	33.7128	1.37747	0.688734	+	0.725014i	$0.258167\pi$
$600$	0	0
$601$	−16.9282	−0.690516	−0.345258	−	0.938508i	$-0.612209\pi$
$601$	−16.9282	−0.690516	−0.345258	+	0.938508i	$0.612209\pi$
$602$	15.8338	0.645335
$603$	0	0
$604$	−39.1244	−1.59195
$605$	0	0
$606$	0	0
$607$	8.72552	0.354158	0.177079	−	0.984197i	$-0.443335\pi$
$607$	8.72552	0.354158	0.177079	+	0.984197i	$0.443335\pi$
$608$	−20.7327	−0.840823
$609$	0	0
$610$	0	0
$611$	10.7321	0.434172
$612$	0	0
$613$	−24.3190	−0.982236	−0.491118	−	0.871093i	$-0.663412\pi$
$613$	−24.3190	−0.982236	−0.491118	+	0.871093i	$0.663412\pi$
$614$	−49.9808	−2.01706
$615$	0	0
$616$	−2.19615	−0.0884855
$617$	13.1069	0.527662	0.263831	−	0.964569i	$-0.415014\pi$
$617$	13.1069	0.527662	0.263831	+	0.964569i	$0.415014\pi$
$618$	0	0
$619$	19.8038	0.795984	0.397992	−	0.917389i	$-0.369707\pi$
$619$	19.8038	0.795984	0.397992	+	0.917389i	$0.369707\pi$
$620$	0	0
$621$	0	0
$622$	54.1934	2.17296
$623$	−6.62776	−0.265536
$624$	0	0
$625$	0	0
$626$	−10.7321	−0.428939
$627$	0	0
$628$	−37.0470	−1.47833
$629$	−1.60770	−0.0641030
$630$	0	0
$631$	−33.3205	−1.32647	−0.663234	−	0.748412i	$-0.730817\pi$
$631$	−33.3205	−1.32647	−0.663234	+	0.748412i	$0.730817\pi$
$632$	−0.277401	−0.0110344
$633$	0	0
$634$	67.9090	2.69701
$635$	0	0
$636$	0	0
$637$	−15.1774	−0.601351
$638$	59.0924	2.33949
$639$	0	0
$640$	0	0
$641$	−25.0526	−0.989517	−0.494758	−	0.869031i	$-0.664743\pi$
$641$	−25.0526	−0.989517	−0.494758	+	0.869031i	$0.664743\pi$
$642$	0	0
$643$	25.2156	0.994406	0.497203	−	0.867634i	$-0.334360\pi$
$643$	25.2156	0.994406	0.497203	+	0.867634i	$0.334360\pi$
$644$	−3.00000	−0.118217
$645$	0	0
$646$	2.00000	0.0786889
$647$	12.3861	0.486950	0.243475	−	0.969907i	$-0.421713\pi$
$647$	12.3861	0.486950	0.243475	+	0.969907i	$0.421713\pi$
$648$	0	0
$649$	12.0000	0.471041
$650$	0	0
$651$	0	0
$652$	−32.8043	−1.28472
$653$	0.453267	0.0177377	0.00886885	−	0.999961i	$-0.497177\pi$
$653$	0.453267	0.0177377	0.00886885	+	0.999961i	$0.497177\pi$
$654$	0	0
$655$	0	0
$656$	−19.0526	−0.743877
$657$	0	0
$658$	7.58871	0.295839
$659$	36.2487	1.41205	0.706025	−	0.708187i	$-0.250487\pi$
$659$	36.2487	1.41205	0.706025	+	0.708187i	$0.250487\pi$
$660$	0	0
$661$	−30.7846	−1.19738	−0.598691	−	0.800980i	$-0.704312\pi$
$661$	−30.7846	−1.19738	−0.598691	+	0.800980i	$0.704312\pi$
$662$	23.9401	0.930458
$663$	0	0
$664$	5.39230	0.209262
$665$	0	0
$666$	0	0
$667$	−12.4877	−0.483525
$668$	−29.4582	−1.13977
$669$	0	0
$670$	0	0
$671$	50.4449	1.94740
$672$	0	0
$673$	−1.61729	−0.0623418	−0.0311709	−	0.999514i	$-0.509924\pi$
$673$	−1.61729	−0.0623418	−0.0311709	+	0.999514i	$0.509924\pi$
$674$	3.46410	0.133432
$675$	0	0
$676$	−12.1244	−0.466321
$677$	−33.0817	−1.27143	−0.635717	−	0.771922i	$-0.719295\pi$
$677$	−33.0817	−1.27143	−0.635717	+	0.771922i	$0.719295\pi$
$678$	0	0
$679$	−9.21539	−0.353654
$680$	0	0
$681$	0	0
$682$	−24.9754	−0.956356
$683$	19.6975	0.753702	0.376851	−	0.926274i	$-0.377007\pi$
$683$	19.6975	0.753702	0.376851	+	0.926274i	$0.377007\pi$
$684$	0	0
$685$	0	0
$686$	−22.8564	−0.872662
$687$	0	0
$688$	−40.8091	−1.55583
$689$	9.46410	0.360554
$690$	0	0
$691$	20.2487	0.770297	0.385149	−	0.922855i	$-0.374150\pi$
$691$	20.2487	0.770297	0.385149	+	0.922855i	$0.374150\pi$
$692$	1.31268	0.0499005
$693$	0	0
$694$	−19.1244	−0.725951
$695$	0	0
$696$	0	0
$697$	1.61729	0.0612591
$698$	41.6042	1.57474
$699$	0	0
$700$	0	0
$701$	−23.1962	−0.876107	−0.438053	−	0.898949i	$-0.644332\pi$
$701$	−23.1962	−0.876107	−0.438053	+	0.898949i	$0.644332\pi$
$702$	0	0
$703$	−11.5911	−0.437167
$704$	−27.1244	−1.02229
$705$	0	0
$706$	−17.4641	−0.657270
$707$	2.27362	0.0855084
$708$	0	0
$709$	−31.7846	−1.19370	−0.596848	−	0.802354i	$-0.703580\pi$
$709$	−31.7846	−1.19370	−0.596848	+	0.802354i	$0.703580\pi$
$710$	0	0
$711$	0	0
$712$	3.82654	0.143406
$713$	5.27792	0.197660
$714$	0	0
$715$	0	0
$716$	42.5885	1.59161
$717$	0	0
$718$	18.9396	0.706819
$719$	19.6077	0.731244	0.365622	−	0.930763i	$-0.380856\pi$
$719$	19.6077	0.731244	0.365622	+	0.930763i	$0.380856\pi$
$720$	0	0
$721$	−8.19615	−0.305241
$722$	−22.2856	−0.829386
$723$	0	0
$724$	14.6603	0.544844
$725$	0	0
$726$	0	0
$727$	28.3214	1.05038	0.525192	−	0.850984i	$-0.323994\pi$
$727$	28.3214	1.05038	0.525192	+	0.850984i	$0.323994\pi$
$728$	−1.13681	−0.0421331
$729$	0	0
$730$	0	0
$731$	3.46410	0.128124
$732$	0	0
$733$	41.4655	1.53156	0.765781	−	0.643102i	$-0.222353\pi$
$733$	41.4655	1.53156	0.765781	+	0.643102i	$0.222353\pi$
$734$	47.9090	1.76835
$735$	0	0
$736$	14.6603	0.540384
$737$	−24.3190	−0.895803
$738$	0	0
$739$	−45.8564	−1.68686	−0.843428	−	0.537243i	$-0.819466\pi$
$739$	−45.8564	−1.68686	−0.843428	+	0.537243i	$0.819466\pi$
$740$	0	0
$741$	0	0
$742$	6.69213	0.245676
$743$	25.3171	0.928796	0.464398	−	0.885627i	$-0.346271\pi$
$743$	25.3171	0.928796	0.464398	+	0.885627i	$0.346271\pi$
$744$	0	0
$745$	0	0
$746$	62.1051	2.27383
$747$	0	0
$748$	3.10583	0.113560
$749$	10.5167	0.384270
$750$	0	0
$751$	34.4449	1.25691	0.628455	−	0.777846i	$-0.283687\pi$
$751$	34.4449	1.25691	0.628455	+	0.777846i	$0.283687\pi$
$752$	−19.5588	−0.713234
$753$	0	0
$754$	30.5885	1.11397
$755$	0	0
$756$	0	0
$757$	7.34847	0.267085	0.133542	−	0.991043i	$-0.457365\pi$
$757$	7.34847	0.267085	0.133542	+	0.991043i	$0.457365\pi$
$758$	24.2175	0.879619
$759$	0	0
$760$	0	0
$761$	2.07180	0.0751026	0.0375513	−	0.999295i	$-0.488044\pi$
$761$	2.07180	0.0751026	0.0375513	+	0.999295i	$0.488044\pi$
$762$	0	0
$763$	−4.17827	−0.151264
$764$	0.588457	0.0212896
$765$	0	0
$766$	−39.5167	−1.42779
$767$	6.21166	0.224290
$768$	0	0
$769$	−27.6410	−0.996761	−0.498380	−	0.866959i	$-0.666072\pi$
$769$	−27.6410	−0.996761	−0.498380	+	0.866959i	$0.666072\pi$
$770$	0	0
$771$	0	0
$772$	−35.9101	−1.29243
$773$	−7.72741	−0.277935	−0.138968	−	0.990297i	$-0.544378\pi$
$773$	−7.72741	−0.277935	−0.138968	+	0.990297i	$0.544378\pi$
$774$	0	0
$775$	0	0
$776$	5.32051	0.190995
$777$	0	0
$778$	−46.6047	−1.67086
$779$	11.6603	0.417772
$780$	0	0
$781$	18.0000	0.644091
$782$	−1.41421	−0.0505722
$783$	0	0
$784$	27.6603	0.987866
$785$	0	0
$786$	0	0
$787$	−16.3142	−0.581539	−0.290770	−	0.956793i	$-0.593911\pi$
$787$	−16.3142	−0.581539	−0.290770	+	0.956793i	$0.593911\pi$
$788$	−36.0860	−1.28551
$789$	0	0
$790$	0	0
$791$	2.28719	0.0813230
$792$	0	0
$793$	26.1122	0.927271
$794$	56.7846	2.01521
$795$	0	0
$796$	7.94744	0.281690
$797$	15.9081	0.563493	0.281747	−	0.959489i	$-0.409086\pi$
$797$	15.9081	0.563493	0.281747	+	0.959489i	$0.409086\pi$
$798$	0	0
$799$	1.66025	0.0587356
$800$	0	0
$801$	0	0
$802$	−59.7487	−2.10980
$803$	40.1528	1.41696
$804$	0	0
$805$	0	0
$806$	−12.9282	−0.455377
$807$	0	0
$808$	−1.31268	−0.0461798
$809$	37.1769	1.30707	0.653535	−	0.756896i	$-0.273285\pi$
$809$	37.1769	1.30707	0.653535	+	0.756896i	$0.273285\pi$
$810$	0	0
$811$	43.5692	1.52992	0.764961	−	0.644076i	$-0.222758\pi$
$811$	43.5692	1.52992	0.764961	+	0.644076i	$0.222758\pi$
$812$	10.0382	0.352272
$813$	0	0
$814$	−38.7846	−1.35940
$815$	0	0
$816$	0	0
$817$	24.9754	0.873778
$818$	−45.3292	−1.58490
$819$	0	0
$820$	0	0
$821$	−29.4449	−1.02763	−0.513816	−	0.857900i	$-0.671769\pi$
$821$	−29.4449	−1.02763	−0.513816	+	0.857900i	$0.671769\pi$
$822$	0	0
$823$	2.03339	0.0708794	0.0354397	−	0.999372i	$-0.488717\pi$
$823$	2.03339	0.0708794	0.0354397	+	0.999372i	$0.488717\pi$
$824$	4.73205	0.164849
$825$	0	0
$826$	4.39230	0.152828
$827$	11.5539	0.401770	0.200885	−	0.979615i	$-0.435618\pi$
$827$	11.5539	0.401770	0.200885	+	0.979615i	$0.435618\pi$
$828$	0	0
$829$	31.5885	1.09711	0.548556	−	0.836114i	$-0.315178\pi$
$829$	31.5885	1.09711	0.548556	+	0.836114i	$0.315178\pi$
$830$	0	0
$831$	0	0
$832$	−14.0406	−0.486770
$833$	−2.34795	−0.0813518
$834$	0	0
$835$	0	0
$836$	22.3923	0.774454
$837$	0	0
$838$	−31.0112	−1.07126
$839$	1.26795	0.0437745	0.0218872	−	0.999760i	$-0.493033\pi$
$839$	1.26795	0.0437745	0.0218872	+	0.999760i	$0.493033\pi$
$840$	0	0
$841$	12.7846	0.440849
$842$	−24.2175	−0.834590
$843$	0	0
$844$	−22.7321	−0.782469
$845$	0	0
$846$	0	0
$847$	10.2141	0.350959
$848$	−17.2480	−0.592298
$849$	0	0
$850$	0	0
$851$	8.19615	0.280960
$852$	0	0
$853$	−34.9492	−1.19664	−0.598319	−	0.801258i	$-0.704164\pi$
$853$	−34.9492	−1.19664	−0.598319	+	0.801258i	$0.704164\pi$
$854$	18.4641	0.631829
$855$	0	0
$856$	−6.07180	−0.207530
$857$	8.86422	0.302796	0.151398	−	0.988473i	$-0.451623\pi$
$857$	8.86422	0.302796	0.151398	+	0.988473i	$0.451623\pi$
$858$	0	0
$859$	−22.4449	−0.765809	−0.382904	−	0.923788i	$-0.625076\pi$
$859$	−22.4449	−0.765809	−0.382904	+	0.923788i	$0.625076\pi$
$860$	0	0
$861$	0	0
$862$	−11.5911	−0.394795
$863$	13.0697	0.444898	0.222449	−	0.974944i	$-0.428595\pi$
$863$	13.0697	0.444898	0.222449	+	0.974944i	$0.428595\pi$
$864$	0	0
$865$	0	0
$866$	58.9808	2.00425
$867$	0	0
$868$	−4.24264	−0.144005
$869$	2.53590	0.0860245
$870$	0	0
$871$	−12.5885	−0.426544
$872$	2.41233	0.0816916
$873$	0	0
$874$	−10.1962	−0.344890
$875$	0	0
$876$	0	0
$877$	−4.72311	−0.159488	−0.0797441	−	0.996815i	$-0.525410\pi$
$877$	−4.72311	−0.159488	−0.0797441	+	0.996815i	$0.525410\pi$
$878$	18.0058	0.607668
$879$	0	0
$880$	0	0
$881$	−8.41154	−0.283392	−0.141696	−	0.989910i	$-0.545256\pi$
$881$	−8.41154	−0.283392	−0.141696	+	0.989910i	$0.545256\pi$
$882$	0	0
$883$	17.6913	0.595359	0.297679	−	0.954666i	$-0.403787\pi$
$883$	17.6913	0.595359	0.297679	+	0.954666i	$0.403787\pi$
$884$	1.60770	0.0540726
$885$	0	0
$886$	26.8564	0.902259
$887$	−28.1827	−0.946284	−0.473142	−	0.880986i	$-0.656880\pi$
$887$	−28.1827	−0.946284	−0.473142	+	0.880986i	$0.656880\pi$
$888$	0	0
$889$	−4.17691	−0.140089
$890$	0	0
$891$	0	0
$892$	−28.9778	−0.970248
$893$	11.9700	0.400562
$894$	0	0
$895$	0	0
$896$	3.67949	0.122923
$897$	0	0
$898$	−23.1822	−0.773601
$899$	−17.6603	−0.589002
$900$	0	0
$901$	1.46410	0.0487763
$902$	39.0160	1.29909
$903$	0	0
$904$	−1.32051	−0.0439194
$905$	0	0
$906$	0	0
$907$	−49.7105	−1.65061	−0.825305	−	0.564687i	$-0.808997\pi$
$907$	−49.7105	−1.65061	−0.825305	+	0.564687i	$0.808997\pi$
$908$	−3.28169	−0.108907
$909$	0	0
$910$	0	0
$911$	22.1436	0.733650	0.366825	−	0.930290i	$-0.380445\pi$
$911$	22.1436	0.733650	0.366825	+	0.930290i	$0.380445\pi$
$912$	0	0
$913$	−49.2944	−1.63141
$914$	−66.8372	−2.21078
$915$	0	0
$916$	34.5167	1.14046
$917$	−0.832204	−0.0274818
$918$	0	0
$919$	−15.1769	−0.500640	−0.250320	−	0.968163i	$-0.580536\pi$
$919$	−15.1769	−0.500640	−0.250320	+	0.968163i	$0.580536\pi$
$920$	0	0
$921$	0	0
$922$	36.1504	1.19055
$923$	9.31749	0.306689
$924$	0	0
$925$	0	0
$926$	0.339746	0.0111647
$927$	0	0
$928$	−49.0542	−1.61028
$929$	3.46410	0.113653	0.0568267	−	0.998384i	$-0.481902\pi$
$929$	3.46410	0.113653	0.0568267	+	0.998384i	$0.481902\pi$
$930$	0	0
$931$	−16.9282	−0.554799
$932$	−12.2474	−0.401179
$933$	0	0
$934$	11.1244	0.364000
$935$	0	0
$936$	0	0
$937$	−13.8647	−0.452941	−0.226471	−	0.974018i	$-0.572719\pi$
$937$	−13.8647	−0.452941	−0.226471	+	0.974018i	$0.572719\pi$
$938$	−8.90138	−0.290640
$939$	0	0
$940$	0	0
$941$	−8.32051	−0.271241	−0.135620	−	0.990761i	$-0.543303\pi$
$941$	−8.32051	−0.271241	−0.135620	+	0.990761i	$0.543303\pi$
$942$	0	0
$943$	−8.24504	−0.268496
$944$	−11.3205	−0.368451
$945$	0	0
$946$	83.5692	2.71707
$947$	−2.31079	−0.0750906	−0.0375453	−	0.999295i	$-0.511954\pi$
$947$	−2.31079	−0.0750906	−0.0375453	+	0.999295i	$0.511954\pi$
$948$	0	0
$949$	20.7846	0.674697
$950$	0	0
$951$	0	0
$952$	−0.175865	−0.00569983
$953$	37.0197	1.19919	0.599594	−	0.800305i	$-0.295329\pi$
$953$	37.0197	1.19919	0.599594	+	0.800305i	$0.295329\pi$
$954$	0	0
$955$	0	0
$956$	22.3923	0.724219
$957$	0	0
$958$	61.5419	1.98833
$959$	−16.6410	−0.537366
$960$	0	0
$961$	−23.5359	−0.759223
$962$	−20.0764	−0.647289
$963$	0	0
$964$	10.8564	0.349661
$965$	0	0
$966$	0	0
$967$	34.7090	1.11616	0.558082	−	0.829786i	$-0.311537\pi$
$967$	34.7090	1.11616	0.558082	+	0.829786i	$0.311537\pi$
$968$	−5.89709	−0.189540
$969$	0	0
$970$	0	0
$971$	27.8038	0.892268	0.446134	−	0.894966i	$-0.352800\pi$
$971$	27.8038	0.892268	0.446134	+	0.894966i	$0.352800\pi$
$972$	0	0
$973$	−7.17260	−0.229943
$974$	2.19615	0.0703693
$975$	0	0
$976$	−47.5885	−1.52327
$977$	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
$979$	−34.9808	−1.11799
$980$	0	0
$981$	0	0
$982$	−41.9459	−1.33855
$983$	−35.8730	−1.14417	−0.572085	−	0.820194i	$-0.693865\pi$
$983$	−35.8730	−1.14417	−0.572085	+	0.820194i	$0.693865\pi$
$984$	0	0
$985$	0	0
$986$	4.73205	0.150699
$987$	0	0
$988$	11.5911	0.368762
$989$	−17.6603	−0.561563
$990$	0	0
$991$	25.0718	0.796432	0.398216	−	0.917292i	$-0.369629\pi$
$991$	25.0718	0.796432	0.398216	+	0.917292i	$0.369629\pi$
$992$	20.7327	0.658265
$993$	0	0
$994$	6.58846	0.208973
$995$	0	0
$996$	0	0
$997$	15.1774	0.480673	0.240337	−	0.970690i	$-0.422742\pi$
$997$	15.1774	0.480673	0.240337	+	0.970690i	$0.422742\pi$
$998$	−7.45001	−0.235826
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2025.2.a.t.1.4		4
3.2	odd	2		2025.2.a.r.1.1		4
5.2	odd	4		405.2.b.d.244.4		4
5.3	odd	4		405.2.b.d.244.1		4
5.4	even	2	inner	2025.2.a.t.1.1		4
9.2	odd	6		675.2.e.d.226.4		8
9.4	even	3		225.2.e.d.151.1		8
9.5	odd	6		675.2.e.d.451.4		8
9.7	even	3		225.2.e.d.76.1		8
15.2	even	4		405.2.b.c.244.1		4
15.8	even	4		405.2.b.c.244.4		4
15.14	odd	2		2025.2.a.r.1.4		4
45.2	even	12		135.2.j.a.64.1		8
45.4	even	6		225.2.e.d.151.4		8
45.7	odd	12		45.2.j.a.4.4	yes	8
45.13	odd	12		45.2.j.a.34.4	yes	8
45.14	odd	6		675.2.e.d.451.1		8
45.22	odd	12		45.2.j.a.34.1	yes	8
45.23	even	12		135.2.j.a.19.1		8
45.29	odd	6		675.2.e.d.226.1		8
45.32	even	12		135.2.j.a.19.4		8
45.34	even	6		225.2.e.d.76.4		8
45.38	even	12		135.2.j.a.64.4		8
45.43	odd	12		45.2.j.a.4.1	✓	8
180.7	even	12		720.2.by.d.49.4		8
180.23	odd	12		2160.2.by.c.289.4		8
180.43	even	12		720.2.by.d.49.1		8
180.47	odd	12		2160.2.by.c.1009.4		8
180.67	even	12		720.2.by.d.529.1		8
180.83	odd	12		2160.2.by.c.1009.2		8
180.103	even	12		720.2.by.d.529.4		8
180.167	odd	12		2160.2.by.c.289.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
45.2.j.a.4.1	✓	8	45.43	odd	12
45.2.j.a.4.4	yes	8	45.7	odd	12
45.2.j.a.34.1	yes	8	45.22	odd	12
45.2.j.a.34.4	yes	8	45.13	odd	12
135.2.j.a.19.1		8	45.23	even	12
135.2.j.a.19.4		8	45.32	even	12
135.2.j.a.64.1		8	45.2	even	12
135.2.j.a.64.4		8	45.38	even	12
225.2.e.d.76.1		8	9.7	even	3
225.2.e.d.76.4		8	45.34	even	6
225.2.e.d.151.1		8	9.4	even	3
225.2.e.d.151.4		8	45.4	even	6
405.2.b.c.244.1		4	15.2	even	4
405.2.b.c.244.4		4	15.8	even	4
405.2.b.d.244.1		4	5.3	odd	4
405.2.b.d.244.4		4	5.2	odd	4
675.2.e.d.226.1		8	45.29	odd	6
675.2.e.d.226.4		8	9.2	odd	6
675.2.e.d.451.1		8	45.14	odd	6
675.2.e.d.451.4		8	9.5	odd	6
720.2.by.d.49.1		8	180.43	even	12
720.2.by.d.49.4		8	180.7	even	12
720.2.by.d.529.1		8	180.67	even	12
720.2.by.d.529.4		8	180.103	even	12
2025.2.a.r.1.1		4	3.2	odd	2
2025.2.a.r.1.4		4	15.14	odd	2
2025.2.a.t.1.1		4	5.4	even	2	inner
2025.2.a.t.1.4		4	1.1	even	1	trivial
2160.2.by.c.289.2		8	180.167	odd	12
2160.2.by.c.289.4		8	180.23	odd	12
2160.2.by.c.1009.2		8	180.83	odd	12
2160.2.by.c.1009.4		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)

\(f(q)\)	\(=\)	\(q+1.93185 q^{2} +1.73205 q^{4} +0.896575 q^{7} -0.517638 q^{8} +O(q^{10})\)	\(q+1.93185 q^{2} +1.73205 q^{4} +0.896575 q^{7} -0.517638 q^{8} +4.73205 q^{11} +2.44949 q^{13} +1.73205 q^{14} -4.46410 q^{16} +0.378937 q^{17} +2.73205 q^{19} +9.14162 q^{22} -1.93185 q^{23} +4.73205 q^{26} +1.55291 q^{28} +6.46410 q^{29} -2.73205 q^{31} -7.58871 q^{32} +0.732051 q^{34} -4.24264 q^{37} +5.27792 q^{38} +4.26795 q^{41} +9.14162 q^{43} +8.19615 q^{44} -3.73205 q^{46} +4.38134 q^{47} -6.19615 q^{49} +4.24264 q^{52} +3.86370 q^{53} -0.464102 q^{56} +12.4877 q^{58} +2.53590 q^{59} +10.6603 q^{61} -5.27792 q^{62} -5.73205 q^{64} -5.13922 q^{67} +0.656339 q^{68} +3.80385 q^{71} +8.48528 q^{73} -8.19615 q^{74} +4.73205 q^{76} +4.24264 q^{77} +0.535898 q^{79} +8.24504 q^{82} -10.4171 q^{83} +17.6603 q^{86} -2.44949 q^{88} -7.39230 q^{89} +2.19615 q^{91} -3.34607 q^{92} +8.46410 q^{94} -10.2784 q^{97} -11.9700 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 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

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