LMFDB - Embedded newform 2025.2.a.g.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:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 405)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.73205$ 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)

$f(q)$	$=$	$q+0.732051 q^{2} -1.46410 q^{4} +4.73205 q^{7} -2.53590 q^{8} +5.73205 q^{11} -1.46410 q^{13} +3.46410 q^{14} +1.07180 q^{16} -2.73205 q^{17} +4.46410 q^{19} +4.19615 q^{22} -3.46410 q^{23} -1.07180 q^{26} -6.92820 q^{28} -3.19615 q^{29} -3.00000 q^{31} +5.85641 q^{32} -2.00000 q^{34} +2.73205 q^{37} +3.26795 q^{38} +7.19615 q^{41} -0.196152 q^{43} -8.39230 q^{44} -2.53590 q^{46} -8.73205 q^{47} +15.3923 q^{49} +2.14359 q^{52} +6.73205 q^{53} -12.0000 q^{56} -2.33975 q^{58} +8.26795 q^{59} +4.00000 q^{61} -2.19615 q^{62} +2.14359 q^{64} -3.46410 q^{67} +4.00000 q^{68} +3.73205 q^{71} +7.66025 q^{73} +2.00000 q^{74} -6.53590 q^{76} +27.1244 q^{77} +15.4641 q^{79} +5.26795 q^{82} +2.19615 q^{83} -0.143594 q^{86} -14.5359 q^{88} -5.19615 q^{89} -6.92820 q^{91} +5.07180 q^{92} -6.39230 q^{94} +9.66025 q^{97} +11.2679 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 4 q^{4} + 6 q^{7} - 12 q^{8} + 8 q^{11} + 4 q^{13} + 16 q^{16} - 2 q^{17} + 2 q^{19} - 2 q^{22} - 16 q^{26} + 4 q^{29} - 6 q^{31} - 16 q^{32} - 4 q^{34} + 2 q^{37} + 10 q^{38} + 4 q^{41}+ \cdots + 26 q^{98}+O(q^{100}) $ 2 * q - 2 * q^2 + 4 * q^4 + 6 * q^7 - 12 * q^8 + 8 * q^11 + 4 * q^13 + 16 * q^16 - 2 * q^17 + 2 * q^19 - 2 * q^22 - 16 * q^26 + 4 * q^29 - 6 * q^31 - 16 * q^32 - 4 * q^34 + 2 * q^37 + 10 * q^38 + 4 * q^41 + 10 * q^43 + 4 * q^44 - 12 * q^46 - 14 * q^47 + 10 * q^49 + 32 * q^52 + 10 * q^53 - 24 * q^56 - 22 * q^58 + 20 * q^59 + 8 * q^61 + 6 * q^62 + 32 * q^64 + 8 * q^68 + 4 * q^71 - 2 * q^73 + 4 * q^74 - 20 * q^76 + 30 * q^77 + 24 * q^79 + 14 * q^82 - 6 * q^83 - 28 * q^86 - 36 * q^88 + 24 * q^92 + 8 * q^94 + 2 * q^97 + 26 * q^98

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 $

$2$	0.732051	0.517638	0.258819	−	0.965926i	$-0.416667\pi$
$2$	0.732051	0.517638	0.258819	+	0.965926i	$0.416667\pi$
$3$	0	0
$3$	0	0
$4$	−1.46410	−0.732051
$5$	0	0
$5$	0	0
$6$	0	0
$7$	4.73205	1.78855	0.894274	−	0.447521i	$-0.147693\pi$
$7$	4.73205	1.78855	0.894274	+	0.447521i	$0.147693\pi$
$8$	−2.53590	−0.896575
$9$	0	0
$10$	0	0
$11$	5.73205	1.72828	0.864139	−	0.503253i	$-0.167864\pi$
$11$	5.73205	1.72828	0.864139	+	0.503253i	$0.167864\pi$
$12$	0	0
$13$	−1.46410	−0.406069	−0.203034	−	0.979172i	$-0.565080\pi$
$13$	−1.46410	−0.406069	−0.203034	+	0.979172i	$0.565080\pi$
$14$	3.46410	0.925820
$15$	0	0
$16$	1.07180	0.267949
$17$	−2.73205	−0.662620	−0.331310	−	0.943522i	$-0.607491\pi$
$17$	−2.73205	−0.662620	−0.331310	+	0.943522i	$0.607491\pi$
$18$	0	0
$19$	4.46410	1.02414	0.512068	−	0.858945i	$-0.328880\pi$
$19$	4.46410	1.02414	0.512068	+	0.858945i	$0.328880\pi$
$20$	0	0
$21$	0	0
$22$	4.19615	0.894623
$23$	−3.46410	−0.722315	−0.361158	−	0.932505i	$-0.617618\pi$
$23$	−3.46410	−0.722315	−0.361158	+	0.932505i	$0.617618\pi$
$24$	0	0
$25$	0	0
$26$	−1.07180	−0.210197
$27$	0	0
$28$	−6.92820	−1.30931
$29$	−3.19615	−0.593511	−0.296755	−	0.954954i	$-0.595905\pi$
$29$	−3.19615	−0.593511	−0.296755	+	0.954954i	$0.595905\pi$
$30$	0	0
$31$	−3.00000	−0.538816	−0.269408	−	0.963026i	$-0.586828\pi$
$31$	−3.00000	−0.538816	−0.269408	+	0.963026i	$0.586828\pi$
$32$	5.85641	1.03528
$33$	0	0
$34$	−2.00000	−0.342997
$35$	0	0
$36$	0	0
$37$	2.73205	0.449146	0.224573	−	0.974457i	$-0.427901\pi$
$37$	2.73205	0.449146	0.224573	+	0.974457i	$0.427901\pi$
$38$	3.26795	0.530131
$39$	0	0
$40$	0	0
$41$	7.19615	1.12385	0.561925	−	0.827188i	$-0.310061\pi$
$41$	7.19615	1.12385	0.561925	+	0.827188i	$0.310061\pi$
$42$	0	0
$43$	−0.196152	−0.0299130	−0.0149565	−	0.999888i	$-0.504761\pi$
$43$	−0.196152	−0.0299130	−0.0149565	+	0.999888i	$0.504761\pi$
$44$	−8.39230	−1.26519
$45$	0	0
$46$	−2.53590	−0.373898
$47$	−8.73205	−1.27370	−0.636850	−	0.770988i	$-0.719763\pi$
$47$	−8.73205	−1.27370	−0.636850	+	0.770988i	$0.719763\pi$
$48$	0	0
$49$	15.3923	2.19890
$50$	0	0
$51$	0	0
$52$	2.14359	0.297263
$53$	6.73205	0.924718	0.462359	−	0.886693i	$-0.347003\pi$
$53$	6.73205	0.924718	0.462359	+	0.886693i	$0.347003\pi$
$54$	0	0
$55$	0	0
$56$	−12.0000	−1.60357
$57$	0	0
$58$	−2.33975	−0.307224
$59$	8.26795	1.07640	0.538198	−	0.842819i	$-0.319105\pi$
$59$	8.26795	1.07640	0.538198	+	0.842819i	$0.319105\pi$
$60$	0	0
$61$	4.00000	0.512148	0.256074	−	0.966657i	$-0.417571\pi$
$61$	4.00000	0.512148	0.256074	+	0.966657i	$0.417571\pi$
$62$	−2.19615	−0.278912
$63$	0	0
$64$	2.14359	0.267949
$65$	0	0
$66$	0	0
$67$	−3.46410	−0.423207	−0.211604	−	0.977356i	$-0.567869\pi$
$67$	−3.46410	−0.423207	−0.211604	+	0.977356i	$0.567869\pi$
$68$	4.00000	0.485071
$69$	0	0
$70$	0	0
$71$	3.73205	0.442913	0.221456	−	0.975170i	$-0.428919\pi$
$71$	3.73205	0.442913	0.221456	+	0.975170i	$0.428919\pi$
$72$	0	0
$73$	7.66025	0.896565	0.448282	−	0.893892i	$-0.352036\pi$
$73$	7.66025	0.896565	0.448282	+	0.893892i	$0.352036\pi$
$74$	2.00000	0.232495
$75$	0	0
$76$	−6.53590	−0.749719
$77$	27.1244	3.09111
$78$	0	0
$79$	15.4641	1.73985	0.869924	−	0.493186i	$-0.164168\pi$
$79$	15.4641	1.73985	0.869924	+	0.493186i	$0.164168\pi$
$80$	0	0
$81$	0	0
$82$	5.26795	0.581748
$83$	2.19615	0.241059	0.120530	−	0.992710i	$-0.461541\pi$
$83$	2.19615	0.241059	0.120530	+	0.992710i	$0.461541\pi$
$84$	0	0
$85$	0	0
$86$	−0.143594	−0.0154841
$87$	0	0
$88$	−14.5359	−1.54953
$89$	−5.19615	−0.550791	−0.275396	−	0.961331i	$-0.588809\pi$
$89$	−5.19615	−0.550791	−0.275396	+	0.961331i	$0.588809\pi$
$90$	0	0
$91$	−6.92820	−0.726273
$92$	5.07180	0.528771
$93$	0	0
$94$	−6.39230	−0.659316
$95$	0	0
$96$	0	0
$97$	9.66025	0.980850	0.490425	−	0.871483i	$-0.336842\pi$
$97$	9.66025	0.980850	0.490425	+	0.871483i	$0.336842\pi$
$98$	11.2679	1.13823
$99$	0	0
$100$	0	0
$101$	2.66025	0.264705	0.132353	−	0.991203i	$-0.457747\pi$
$101$	2.66025	0.264705	0.132353	+	0.991203i	$0.457747\pi$
$102$	0	0
$103$	0.535898	0.0528036	0.0264018	−	0.999651i	$-0.491595\pi$
$103$	0.535898	0.0528036	0.0264018	+	0.999651i	$0.491595\pi$
$104$	3.71281	0.364071
$105$	0	0
$106$	4.92820	0.478669
$107$	−8.53590	−0.825196	−0.412598	−	0.910913i	$-0.635379\pi$
$107$	−8.53590	−0.825196	−0.412598	+	0.910913i	$0.635379\pi$
$108$	0	0
$109$	−6.07180	−0.581573	−0.290786	−	0.956788i	$-0.593917\pi$
$109$	−6.07180	−0.581573	−0.290786	+	0.956788i	$0.593917\pi$
$110$	0	0
$111$	0	0
$112$	5.07180	0.479240
$113$	−19.1244	−1.79907	−0.899534	−	0.436851i	$-0.856094\pi$
$113$	−19.1244	−1.79907	−0.899534	+	0.436851i	$0.856094\pi$
$114$	0	0
$115$	0	0
$116$	4.67949	0.434480
$117$	0	0
$118$	6.05256	0.557183
$119$	−12.9282	−1.18513
$120$	0	0
$121$	21.8564	1.98695
$122$	2.92820	0.265107
$123$	0	0
$124$	4.39230	0.394441
$125$	0	0
$126$	0	0
$127$	14.5885	1.29452	0.647258	−	0.762271i	$-0.275916\pi$
$127$	14.5885	1.29452	0.647258	+	0.762271i	$0.275916\pi$
$128$	−10.1436	−0.896575
$129$	0	0
$130$	0	0
$131$	−15.5885	−1.36197	−0.680985	−	0.732297i	$-0.738448\pi$
$131$	−15.5885	−1.36197	−0.680985	+	0.732297i	$0.738448\pi$
$132$	0	0
$133$	21.1244	1.83171
$134$	−2.53590	−0.219068
$135$	0	0
$136$	6.92820	0.594089
$137$	2.53590	0.216656	0.108328	−	0.994115i	$-0.465450\pi$
$137$	2.53590	0.216656	0.108328	+	0.994115i	$0.465450\pi$
$138$	0	0
$139$	0.607695	0.0515440	0.0257720	−	0.999668i	$-0.491796\pi$
$139$	0.607695	0.0515440	0.0257720	+	0.999668i	$0.491796\pi$
$140$	0	0
$141$	0	0
$142$	2.73205	0.229269
$143$	−8.39230	−0.701800
$144$	0	0
$145$	0	0
$146$	5.60770	0.464096
$147$	0	0
$148$	−4.00000	−0.328798
$149$	8.00000	0.655386	0.327693	−	0.944784i	$-0.393729\pi$
$149$	8.00000	0.655386	0.327693	+	0.944784i	$0.393729\pi$
$150$	0	0
$151$	5.39230	0.438820	0.219410	−	0.975633i	$-0.429587\pi$
$151$	5.39230	0.438820	0.219410	+	0.975633i	$0.429587\pi$
$152$	−11.3205	−0.918214
$153$	0	0
$154$	19.8564	1.60007
$155$	0	0
$156$	0	0
$157$	19.1244	1.52629	0.763145	−	0.646227i	$-0.223654\pi$
$157$	19.1244	1.52629	0.763145	+	0.646227i	$0.223654\pi$
$158$	11.3205	0.900611
$159$	0	0
$160$	0	0
$161$	−16.3923	−1.29189
$162$	0	0
$163$	−12.7321	−0.997251	−0.498626	−	0.866817i	$-0.666162\pi$
$163$	−12.7321	−0.997251	−0.498626	+	0.866817i	$0.666162\pi$
$164$	−10.5359	−0.822715
$165$	0	0
$166$	1.60770	0.124781
$167$	−17.6603	−1.36659	−0.683296	−	0.730142i	$-0.739454\pi$
$167$	−17.6603	−1.36659	−0.683296	+	0.730142i	$0.739454\pi$
$168$	0	0
$169$	−10.8564	−0.835108
$170$	0	0
$171$	0	0
$172$	0.287187	0.0218978
$173$	8.53590	0.648972	0.324486	−	0.945890i	$-0.394809\pi$
$173$	8.53590	0.648972	0.324486	+	0.945890i	$0.394809\pi$
$174$	0	0
$175$	0	0
$176$	6.14359	0.463091
$177$	0	0
$178$	−3.80385	−0.285110
$179$	−8.12436	−0.607243	−0.303621	−	0.952793i	$-0.598196\pi$
$179$	−8.12436	−0.607243	−0.303621	+	0.952793i	$0.598196\pi$
$180$	0	0
$181$	26.4641	1.96706	0.983531	−	0.180742i	$-0.0578498\pi$
$181$	26.4641	1.96706	0.983531	+	0.180742i	$0.0578498\pi$
$182$	−5.07180	−0.375947
$183$	0	0
$184$	8.78461	0.647610
$185$	0	0
$186$	0	0
$187$	−15.6603	−1.14519
$188$	12.7846	0.932413
$189$	0	0
$190$	0	0
$191$	8.12436	0.587858	0.293929	−	0.955827i	$-0.405037\pi$
$191$	8.12436	0.587858	0.293929	+	0.955827i	$0.405037\pi$
$192$	0	0
$193$	5.26795	0.379195	0.189598	−	0.981862i	$-0.439282\pi$
$193$	5.26795	0.379195	0.189598	+	0.981862i	$0.439282\pi$
$194$	7.07180	0.507725
$195$	0	0
$196$	−22.5359	−1.60971
$197$	−13.8564	−0.987228	−0.493614	−	0.869681i	$-0.664324\pi$
$197$	−13.8564	−0.987228	−0.493614	+	0.869681i	$0.664324\pi$
$198$	0	0
$199$	−2.00000	−0.141776	−0.0708881	−	0.997484i	$-0.522583\pi$
$199$	−2.00000	−0.141776	−0.0708881	+	0.997484i	$0.522583\pi$
$200$	0	0
$201$	0	0
$202$	1.94744	0.137021
$203$	−15.1244	−1.06152
$204$	0	0
$205$	0	0
$206$	0.392305	0.0273332
$207$	0	0
$208$	−1.56922	−0.108806
$209$	25.5885	1.76999
$210$	0	0
$211$	−8.85641	−0.609700	−0.304850	−	0.952400i	$-0.598606\pi$
$211$	−8.85641	−0.609700	−0.304850	+	0.952400i	$0.598606\pi$
$212$	−9.85641	−0.676941
$213$	0	0
$214$	−6.24871	−0.427153
$215$	0	0
$216$	0	0
$217$	−14.1962	−0.963698
$218$	−4.44486	−0.301044
$219$	0	0
$220$	0	0
$221$	4.00000	0.269069
$222$	0	0
$223$	−16.7846	−1.12398	−0.561990	−	0.827144i	$-0.689964\pi$
$223$	−16.7846	−1.12398	−0.561990	+	0.827144i	$0.689964\pi$
$224$	27.7128	1.85164
$225$	0	0
$226$	−14.0000	−0.931266
$227$	−18.0526	−1.19819	−0.599095	−	0.800678i	$-0.704473\pi$
$227$	−18.0526	−1.19819	−0.599095	+	0.800678i	$0.704473\pi$
$228$	0	0
$229$	−12.0000	−0.792982	−0.396491	−	0.918039i	$-0.629772\pi$
$229$	−12.0000	−0.792982	−0.396491	+	0.918039i	$0.629772\pi$
$230$	0	0
$231$	0	0
$232$	8.10512	0.532127
$233$	−28.0526	−1.83778	−0.918892	−	0.394509i	$-0.870915\pi$
$233$	−28.0526	−1.83778	−0.918892	+	0.394509i	$0.870915\pi$
$234$	0	0
$235$	0	0
$236$	−12.1051	−0.787976
$237$	0	0
$238$	−9.46410	−0.613467
$239$	0.535898	0.0346644	0.0173322	−	0.999850i	$-0.494483\pi$
$239$	0.535898	0.0346644	0.0173322	+	0.999850i	$0.494483\pi$
$240$	0	0
$241$	−16.3205	−1.05130	−0.525648	−	0.850702i	$-0.676177\pi$
$241$	−16.3205	−1.05130	−0.525648	+	0.850702i	$0.676177\pi$
$242$	16.0000	1.02852
$243$	0	0
$244$	−5.85641	−0.374918
$245$	0	0
$246$	0	0
$247$	−6.53590	−0.415869
$248$	7.60770	0.483089
$249$	0	0
$250$	0	0
$251$	10.3923	0.655956	0.327978	−	0.944685i	$-0.393633\pi$
$251$	10.3923	0.655956	0.327978	+	0.944685i	$0.393633\pi$
$252$	0	0
$253$	−19.8564	−1.24836
$254$	10.6795	0.670091
$255$	0	0
$256$	−11.7128	−0.732051
$257$	10.3923	0.648254	0.324127	−	0.946014i	$-0.394929\pi$
$257$	10.3923	0.648254	0.324127	+	0.946014i	$0.394929\pi$
$258$	0	0
$259$	12.9282	0.803319
$260$	0	0
$261$	0	0
$262$	−11.4115	−0.705007
$263$	21.3205	1.31468	0.657339	−	0.753595i	$-0.271682\pi$
$263$	21.3205	1.31468	0.657339	+	0.753595i	$0.271682\pi$
$264$	0	0
$265$	0	0
$266$	15.4641	0.948165
$267$	0	0
$268$	5.07180	0.309809
$269$	−10.6603	−0.649967	−0.324984	−	0.945720i	$-0.605359\pi$
$269$	−10.6603	−0.649967	−0.324984	+	0.945720i	$0.605359\pi$
$270$	0	0
$271$	−2.92820	−0.177876	−0.0889378	−	0.996037i	$-0.528347\pi$
$271$	−2.92820	−0.177876	−0.0889378	+	0.996037i	$0.528347\pi$
$272$	−2.92820	−0.177548
$273$	0	0
$274$	1.85641	0.112150
$275$	0	0
$276$	0	0
$277$	−3.80385	−0.228551	−0.114276	−	0.993449i	$-0.536455\pi$
$277$	−3.80385	−0.228551	−0.114276	+	0.993449i	$0.536455\pi$
$278$	0.444864	0.0266812
$279$	0	0
$280$	0	0
$281$	−15.4641	−0.922511	−0.461255	−	0.887267i	$-0.652601\pi$
$281$	−15.4641	−0.922511	−0.461255	+	0.887267i	$0.652601\pi$
$282$	0	0
$283$	29.3205	1.74292	0.871462	−	0.490464i	$-0.163173\pi$
$283$	29.3205	1.74292	0.871462	+	0.490464i	$0.163173\pi$
$284$	−5.46410	−0.324235
$285$	0	0
$286$	−6.14359	−0.363278
$287$	34.0526	2.01006
$288$	0	0
$289$	−9.53590	−0.560935
$290$	0	0
$291$	0	0
$292$	−11.2154	−0.656331
$293$	28.7321	1.67854	0.839272	−	0.543712i	$-0.182981\pi$
$293$	28.7321	1.67854	0.839272	+	0.543712i	$0.182981\pi$
$294$	0	0
$295$	0	0
$296$	−6.92820	−0.402694
$297$	0	0
$298$	5.85641	0.339253
$299$	5.07180	0.293310
$300$	0	0
$301$	−0.928203	−0.0535007
$302$	3.94744	0.227150
$303$	0	0
$304$	4.78461	0.274416
$305$	0	0
$306$	0	0
$307$	−14.0526	−0.802022	−0.401011	−	0.916073i	$-0.631341\pi$
$307$	−14.0526	−0.802022	−0.401011	+	0.916073i	$0.631341\pi$
$308$	−39.7128	−2.26285
$309$	0	0
$310$	0	0
$311$	−19.7321	−1.11890	−0.559451	−	0.828863i	$-0.688988\pi$
$311$	−19.7321	−1.11890	−0.559451	+	0.828863i	$0.688988\pi$
$312$	0	0
$313$	9.07180	0.512768	0.256384	−	0.966575i	$-0.417469\pi$
$313$	9.07180	0.512768	0.256384	+	0.966575i	$0.417469\pi$
$314$	14.0000	0.790066
$315$	0	0
$316$	−22.6410	−1.27366
$317$	6.19615	0.348011	0.174005	−	0.984745i	$-0.444329\pi$
$317$	6.19615	0.348011	0.174005	+	0.984745i	$0.444329\pi$
$318$	0	0
$319$	−18.3205	−1.02575
$320$	0	0
$321$	0	0
$322$	−12.0000	−0.668734
$323$	−12.1962	−0.678612
$324$	0	0
$325$	0	0
$326$	−9.32051	−0.516215
$327$	0	0
$328$	−18.2487	−1.00762
$329$	−41.3205	−2.27807
$330$	0	0
$331$	−0.464102	−0.0255093	−0.0127547	−	0.999919i	$-0.504060\pi$
$331$	−0.464102	−0.0255093	−0.0127547	+	0.999919i	$0.504060\pi$
$332$	−3.21539	−0.176467
$333$	0	0
$334$	−12.9282	−0.707400
$335$	0	0
$336$	0	0
$337$	−27.3205	−1.48824	−0.744121	−	0.668044i	$-0.767132\pi$
$337$	−27.3205	−1.48824	−0.744121	+	0.668044i	$0.767132\pi$
$338$	−7.94744	−0.432284
$339$	0	0
$340$	0	0
$341$	−17.1962	−0.931224
$342$	0	0
$343$	39.7128	2.14429
$344$	0.497423	0.0268192
$345$	0	0
$346$	6.24871	0.335933
$347$	−28.5885	−1.53471	−0.767354	−	0.641223i	$-0.778427\pi$
$347$	−28.5885	−1.53471	−0.767354	+	0.641223i	$0.778427\pi$
$348$	0	0
$349$	−18.8564	−1.00936	−0.504680	−	0.863306i	$-0.668390\pi$
$349$	−18.8564	−1.00936	−0.504680	+	0.863306i	$0.668390\pi$
$350$	0	0
$351$	0	0
$352$	33.5692	1.78925
$353$	25.5167	1.35811	0.679057	−	0.734085i	$-0.262389\pi$
$353$	25.5167	1.35811	0.679057	+	0.734085i	$0.262389\pi$
$354$	0	0
$355$	0	0
$356$	7.60770	0.403207
$357$	0	0
$358$	−5.94744	−0.314332
$359$	6.12436	0.323231	0.161616	−	0.986854i	$-0.448330\pi$
$359$	6.12436	0.323231	0.161616	+	0.986854i	$0.448330\pi$
$360$	0	0
$361$	0.928203	0.0488528
$362$	19.3731	1.01823
$363$	0	0
$364$	10.1436	0.531669
$365$	0	0
$366$	0	0
$367$	−31.1769	−1.62742	−0.813711	−	0.581270i	$-0.802556\pi$
$367$	−31.1769	−1.62742	−0.813711	+	0.581270i	$0.802556\pi$
$368$	−3.71281	−0.193544
$369$	0	0
$370$	0	0
$371$	31.8564	1.65390
$372$	0	0
$373$	−20.0526	−1.03828	−0.519141	−	0.854689i	$-0.673748\pi$
$373$	−20.0526	−1.03828	−0.519141	+	0.854689i	$0.673748\pi$
$374$	−11.4641	−0.592795
$375$	0	0
$376$	22.1436	1.14197
$377$	4.67949	0.241006
$378$	0	0
$379$	2.39230	0.122884	0.0614422	−	0.998111i	$-0.480430\pi$
$379$	2.39230	0.122884	0.0614422	+	0.998111i	$0.480430\pi$
$380$	0	0
$381$	0	0
$382$	5.94744	0.304298
$383$	2.53590	0.129578	0.0647892	−	0.997899i	$-0.479363\pi$
$383$	2.53590	0.129578	0.0647892	+	0.997899i	$0.479363\pi$
$384$	0	0
$385$	0	0
$386$	3.85641	0.196286
$387$	0	0
$388$	−14.1436	−0.718032
$389$	−27.4641	−1.39249	−0.696243	−	0.717807i	$-0.745146\pi$
$389$	−27.4641	−1.39249	−0.696243	+	0.717807i	$0.745146\pi$
$390$	0	0
$391$	9.46410	0.478620
$392$	−39.0333	−1.97148
$393$	0	0
$394$	−10.1436	−0.511027
$395$	0	0
$396$	0	0
$397$	−14.3923	−0.722329	−0.361165	−	0.932502i	$-0.617621\pi$
$397$	−14.3923	−0.722329	−0.361165	+	0.932502i	$0.617621\pi$
$398$	−1.46410	−0.0733888
$399$	0	0
$400$	0	0
$401$	−24.9282	−1.24486	−0.622428	−	0.782677i	$-0.713853\pi$
$401$	−24.9282	−1.24486	−0.622428	+	0.782677i	$0.713853\pi$
$402$	0	0
$403$	4.39230	0.218796
$404$	−3.89488	−0.193778
$405$	0	0
$406$	−11.0718	−0.549484
$407$	15.6603	0.776250
$408$	0	0
$409$	−17.8564	−0.882942	−0.441471	−	0.897275i	$-0.645543\pi$
$409$	−17.8564	−0.882942	−0.441471	+	0.897275i	$0.645543\pi$
$410$	0	0
$411$	0	0
$412$	−0.784610	−0.0386549
$413$	39.1244	1.92518
$414$	0	0
$415$	0	0
$416$	−8.57437	−0.420393
$417$	0	0
$418$	18.7321	0.916215
$419$	−20.3923	−0.996229	−0.498115	−	0.867111i	$-0.665974\pi$
$419$	−20.3923	−0.996229	−0.498115	+	0.867111i	$0.665974\pi$
$420$	0	0
$421$	33.7846	1.64656	0.823281	−	0.567635i	$-0.192141\pi$
$421$	33.7846	1.64656	0.823281	+	0.567635i	$0.192141\pi$
$422$	−6.48334	−0.315604
$423$	0	0
$424$	−17.0718	−0.829080
$425$	0	0
$426$	0	0
$427$	18.9282	0.916000
$428$	12.4974	0.604086
$429$	0	0
$430$	0	0
$431$	21.3397	1.02790	0.513950	−	0.857820i	$-0.328182\pi$
$431$	21.3397	1.02790	0.513950	+	0.857820i	$0.328182\pi$
$432$	0	0
$433$	35.4641	1.70430	0.852148	−	0.523301i	$-0.175300\pi$
$433$	35.4641	1.70430	0.852148	+	0.523301i	$0.175300\pi$
$434$	−10.3923	−0.498847
$435$	0	0
$436$	8.88973	0.425741
$437$	−15.4641	−0.739748
$438$	0	0
$439$	5.39230	0.257361	0.128680	−	0.991686i	$-0.458926\pi$
$439$	5.39230	0.257361	0.128680	+	0.991686i	$0.458926\pi$
$440$	0	0
$441$	0	0
$442$	2.92820	0.139280
$443$	−0.339746	−0.0161418	−0.00807091	−	0.999967i	$-0.502569\pi$
$443$	−0.339746	−0.0161418	−0.00807091	+	0.999967i	$0.502569\pi$
$444$	0	0
$445$	0	0
$446$	−12.2872	−0.581815
$447$	0	0
$448$	10.1436	0.479240
$449$	−8.12436	−0.383412	−0.191706	−	0.981452i	$-0.561402\pi$
$449$	−8.12436	−0.383412	−0.191706	+	0.981452i	$0.561402\pi$
$450$	0	0
$451$	41.2487	1.94233
$452$	28.0000	1.31701
$453$	0	0
$454$	−13.2154	−0.620229
$455$	0	0
$456$	0	0
$457$	−2.73205	−0.127800	−0.0639000	−	0.997956i	$-0.520354\pi$
$457$	−2.73205	−0.127800	−0.0639000	+	0.997956i	$0.520354\pi$
$458$	−8.78461	−0.410478
$459$	0	0
$460$	0	0
$461$	37.0526	1.72571	0.862855	−	0.505452i	$-0.168674\pi$
$461$	37.0526	1.72571	0.862855	+	0.505452i	$0.168674\pi$
$462$	0	0
$463$	10.3923	0.482971	0.241486	−	0.970404i	$-0.422365\pi$
$463$	10.3923	0.482971	0.241486	+	0.970404i	$0.422365\pi$
$464$	−3.42563	−0.159031
$465$	0	0
$466$	−20.5359	−0.951307
$467$	37.3731	1.72942	0.864710	−	0.502272i	$-0.167502\pi$
$467$	37.3731	1.72942	0.864710	+	0.502272i	$0.167502\pi$
$468$	0	0
$469$	−16.3923	−0.756926
$470$	0	0
$471$	0	0
$472$	−20.9667	−0.965070
$473$	−1.12436	−0.0516979
$474$	0	0
$475$	0	0
$476$	18.9282	0.867573
$477$	0	0
$478$	0.392305	0.0179436
$479$	11.8756	0.542612	0.271306	−	0.962493i	$-0.412544\pi$
$479$	11.8756	0.542612	0.271306	+	0.962493i	$0.412544\pi$
$480$	0	0
$481$	−4.00000	−0.182384
$482$	−11.9474	−0.544191
$483$	0	0
$484$	−32.0000	−1.45455
$485$	0	0
$486$	0	0
$487$	−24.3923	−1.10532	−0.552660	−	0.833407i	$-0.686387\pi$
$487$	−24.3923	−1.10532	−0.552660	+	0.833407i	$0.686387\pi$
$488$	−10.1436	−0.459179
$489$	0	0
$490$	0	0
$491$	13.8756	0.626199	0.313100	−	0.949720i	$-0.398633\pi$
$491$	13.8756	0.626199	0.313100	+	0.949720i	$0.398633\pi$
$492$	0	0
$493$	8.73205	0.393272
$494$	−4.78461	−0.215270
$495$	0	0
$496$	−3.21539	−0.144375
$497$	17.6603	0.792171
$498$	0	0
$499$	24.3205	1.08874	0.544368	−	0.838847i	$-0.316770\pi$
$499$	24.3205	1.08874	0.544368	+	0.838847i	$0.316770\pi$
$500$	0	0
$501$	0	0
$502$	7.60770	0.339548
$503$	−7.32051	−0.326405	−0.163203	−	0.986593i	$-0.552182\pi$
$503$	−7.32051	−0.326405	−0.163203	+	0.986593i	$0.552182\pi$
$504$	0	0
$505$	0	0
$506$	−14.5359	−0.646200
$507$	0	0
$508$	−21.3590	−0.947652
$509$	6.78461	0.300723	0.150361	−	0.988631i	$-0.451956\pi$
$509$	6.78461	0.300723	0.150361	+	0.988631i	$0.451956\pi$
$510$	0	0
$511$	36.2487	1.60355
$512$	11.7128	0.517638
$513$	0	0
$514$	7.60770	0.335561
$515$	0	0
$516$	0	0
$517$	−50.0526	−2.20131
$518$	9.46410	0.415829
$519$	0	0
$520$	0	0
$521$	−19.4641	−0.852738	−0.426369	−	0.904549i	$-0.640207\pi$
$521$	−19.4641	−0.852738	−0.426369	+	0.904549i	$0.640207\pi$
$522$	0	0
$523$	−22.2487	−0.972868	−0.486434	−	0.873717i	$-0.661703\pi$
$523$	−22.2487	−0.972868	−0.486434	+	0.873717i	$0.661703\pi$
$524$	22.8231	0.997031
$525$	0	0
$526$	15.6077	0.680528
$527$	8.19615	0.357030
$528$	0	0
$529$	−11.0000	−0.478261
$530$	0	0
$531$	0	0
$532$	−30.9282	−1.34091
$533$	−10.5359	−0.456360
$534$	0	0
$535$	0	0
$536$	8.78461	0.379437
$537$	0	0
$538$	−7.80385	−0.336448
$539$	88.2295	3.80031
$540$	0	0
$541$	−24.4641	−1.05179	−0.525897	−	0.850548i	$-0.676270\pi$
$541$	−24.4641	−1.05179	−0.525897	+	0.850548i	$0.676270\pi$
$542$	−2.14359	−0.0920752
$543$	0	0
$544$	−16.0000	−0.685994
$545$	0	0
$546$	0	0
$547$	33.8564	1.44760	0.723798	−	0.690012i	$-0.242395\pi$
$547$	33.8564	1.44760	0.723798	+	0.690012i	$0.242395\pi$
$548$	−3.71281	−0.158604
$549$	0	0
$550$	0	0
$551$	−14.2679	−0.607835
$552$	0	0
$553$	73.1769	3.11180
$554$	−2.78461	−0.118307
$555$	0	0
$556$	−0.889727	−0.0377328
$557$	−2.53590	−0.107449	−0.0537247	−	0.998556i	$-0.517109\pi$
$557$	−2.53590	−0.107449	−0.0537247	+	0.998556i	$0.517109\pi$
$558$	0	0
$559$	0.287187	0.0121467
$560$	0	0
$561$	0	0
$562$	−11.3205	−0.477527
$563$	−16.7321	−0.705172	−0.352586	−	0.935779i	$-0.614698\pi$
$563$	−16.7321	−0.705172	−0.352586	+	0.935779i	$0.614698\pi$
$564$	0	0
$565$	0	0
$566$	21.4641	0.902203
$567$	0	0
$568$	−9.46410	−0.397105
$569$	32.9090	1.37962	0.689808	−	0.723993i	$-0.257695\pi$
$569$	32.9090	1.37962	0.689808	+	0.723993i	$0.257695\pi$
$570$	0	0
$571$	−39.7846	−1.66493	−0.832467	−	0.554075i	$-0.813072\pi$
$571$	−39.7846	−1.66493	−0.832467	+	0.554075i	$0.813072\pi$
$572$	12.2872	0.513753
$573$	0	0
$574$	24.9282	1.04048
$575$	0	0
$576$	0	0
$577$	−15.2679	−0.635613	−0.317807	−	0.948156i	$-0.602946\pi$
$577$	−15.2679	−0.635613	−0.317807	+	0.948156i	$0.602946\pi$
$578$	−6.98076	−0.290361
$579$	0	0
$580$	0	0
$581$	10.3923	0.431145
$582$	0	0
$583$	38.5885	1.59817
$584$	−19.4256	−0.803838
$585$	0	0
$586$	21.0333	0.868878
$587$	−11.6603	−0.481270	−0.240635	−	0.970616i	$-0.577356\pi$
$587$	−11.6603	−0.481270	−0.240635	+	0.970616i	$0.577356\pi$
$588$	0	0
$589$	−13.3923	−0.551820
$590$	0	0
$591$	0	0
$592$	2.92820	0.120348
$593$	−0.143594	−0.00589668	−0.00294834	−	0.999996i	$-0.500938\pi$
$593$	−0.143594	−0.00589668	−0.00294834	+	0.999996i	$0.500938\pi$
$594$	0	0
$595$	0	0
$596$	−11.7128	−0.479776
$597$	0	0
$598$	3.71281	0.151828
$599$	−27.1962	−1.11120	−0.555602	−	0.831448i	$-0.687512\pi$
$599$	−27.1962	−1.11120	−0.555602	+	0.831448i	$0.687512\pi$
$600$	0	0
$601$	−31.2487	−1.27466	−0.637331	−	0.770590i	$-0.719961\pi$
$601$	−31.2487	−1.27466	−0.637331	+	0.770590i	$0.719961\pi$
$602$	−0.679492	−0.0276940
$603$	0	0
$604$	−7.89488	−0.321238
$605$	0	0
$606$	0	0
$607$	16.1962	0.657382	0.328691	−	0.944438i	$-0.393393\pi$
$607$	16.1962	0.657382	0.328691	+	0.944438i	$0.393393\pi$
$608$	26.1436	1.06026
$609$	0	0
$610$	0	0
$611$	12.7846	0.517210
$612$	0	0
$613$	−1.46410	−0.0591345	−0.0295673	−	0.999563i	$-0.509413\pi$
$613$	−1.46410	−0.0591345	−0.0295673	+	0.999563i	$0.509413\pi$
$614$	−10.2872	−0.415157
$615$	0	0
$616$	−68.7846	−2.77141
$617$	−6.92820	−0.278919	−0.139459	−	0.990228i	$-0.544536\pi$
$617$	−6.92820	−0.278919	−0.139459	+	0.990228i	$0.544536\pi$
$618$	0	0
$619$	−11.8564	−0.476549	−0.238275	−	0.971198i	$-0.576582\pi$
$619$	−11.8564	−0.476549	−0.238275	+	0.971198i	$0.576582\pi$
$620$	0	0
$621$	0	0
$622$	−14.4449	−0.579186
$623$	−24.5885	−0.985116
$624$	0	0
$625$	0	0
$626$	6.64102	0.265428
$627$	0	0
$628$	−28.0000	−1.11732
$629$	−7.46410	−0.297613
$630$	0	0
$631$	−32.7128	−1.30228	−0.651138	−	0.758959i	$-0.725708\pi$
$631$	−32.7128	−1.30228	−0.651138	+	0.758959i	$0.725708\pi$
$632$	−39.2154	−1.55990
$633$	0	0
$634$	4.53590	0.180144
$635$	0	0
$636$	0	0
$637$	−22.5359	−0.892905
$638$	−13.4115	−0.530968
$639$	0	0
$640$	0	0
$641$	−9.33975	−0.368898	−0.184449	−	0.982842i	$-0.559050\pi$
$641$	−9.33975	−0.368898	−0.184449	+	0.982842i	$0.559050\pi$
$642$	0	0
$643$	−41.6603	−1.64292	−0.821460	−	0.570266i	$-0.806840\pi$
$643$	−41.6603	−1.64292	−0.821460	+	0.570266i	$0.806840\pi$
$644$	24.0000	0.945732
$645$	0	0
$646$	−8.92820	−0.351275
$647$	11.4641	0.450700	0.225350	−	0.974278i	$-0.427647\pi$
$647$	11.4641	0.450700	0.225350	+	0.974278i	$0.427647\pi$
$648$	0	0
$649$	47.3923	1.86031
$650$	0	0
$651$	0	0
$652$	18.6410	0.730039
$653$	−17.4641	−0.683423	−0.341712	−	0.939805i	$-0.611007\pi$
$653$	−17.4641	−0.683423	−0.341712	+	0.939805i	$0.611007\pi$
$654$	0	0
$655$	0	0
$656$	7.71281	0.301135
$657$	0	0
$658$	−30.2487	−1.17922
$659$	1.46410	0.0570333	0.0285167	−	0.999593i	$-0.490922\pi$
$659$	1.46410	0.0570333	0.0285167	+	0.999593i	$0.490922\pi$
$660$	0	0
$661$	18.3205	0.712585	0.356293	−	0.934374i	$-0.384041\pi$
$661$	18.3205	0.712585	0.356293	+	0.934374i	$0.384041\pi$
$662$	−0.339746	−0.0132046
$663$	0	0
$664$	−5.56922	−0.216128
$665$	0	0
$666$	0	0
$667$	11.0718	0.428702
$668$	25.8564	1.00041
$669$	0	0
$670$	0	0
$671$	22.9282	0.885133
$672$	0	0
$673$	10.3923	0.400594	0.200297	−	0.979735i	$-0.435809\pi$
$673$	10.3923	0.400594	0.200297	+	0.979735i	$0.435809\pi$
$674$	−20.0000	−0.770371
$675$	0	0
$676$	15.8949	0.611342
$677$	−18.0000	−0.691796	−0.345898	−	0.938272i	$-0.612426\pi$
$677$	−18.0000	−0.691796	−0.345898	+	0.938272i	$0.612426\pi$
$678$	0	0
$679$	45.7128	1.75430
$680$	0	0
$681$	0	0
$682$	−12.5885	−0.482037
$683$	−19.6077	−0.750268	−0.375134	−	0.926971i	$-0.622403\pi$
$683$	−19.6077	−0.750268	−0.375134	+	0.926971i	$0.622403\pi$
$684$	0	0
$685$	0	0
$686$	29.0718	1.10997
$687$	0	0
$688$	−0.210236	−0.00801515
$689$	−9.85641	−0.375499
$690$	0	0
$691$	−17.7128	−0.673827	−0.336914	−	0.941536i	$-0.609383\pi$
$691$	−17.7128	−0.673827	−0.336914	+	0.941536i	$0.609383\pi$
$692$	−12.4974	−0.475081
$693$	0	0
$694$	−20.9282	−0.794424
$695$	0	0
$696$	0	0
$697$	−19.6603	−0.744685
$698$	−13.8038	−0.522483
$699$	0	0
$700$	0	0
$701$	20.8038	0.785750	0.392875	−	0.919592i	$-0.371480\pi$
$701$	20.8038	0.785750	0.392875	+	0.919592i	$0.371480\pi$
$702$	0	0
$703$	12.1962	0.459987
$704$	12.2872	0.463091
$705$	0	0
$706$	18.6795	0.703012
$707$	12.5885	0.473438
$708$	0	0
$709$	−22.5359	−0.846353	−0.423177	−	0.906047i	$-0.639085\pi$
$709$	−22.5359	−0.846353	−0.423177	+	0.906047i	$0.639085\pi$
$710$	0	0
$711$	0	0
$712$	13.1769	0.493826
$713$	10.3923	0.389195
$714$	0	0
$715$	0	0
$716$	11.8949	0.444533
$717$	0	0
$718$	4.48334	0.167317
$719$	−8.41154	−0.313698	−0.156849	−	0.987623i	$-0.550134\pi$
$719$	−8.41154	−0.313698	−0.156849	+	0.987623i	$0.550134\pi$
$720$	0	0
$721$	2.53590	0.0944418
$722$	0.679492	0.0252881
$723$	0	0
$724$	−38.7461	−1.43999
$725$	0	0
$726$	0	0
$727$	8.39230	0.311253	0.155627	−	0.987816i	$-0.450260\pi$
$727$	8.39230	0.311253	0.155627	+	0.987816i	$0.450260\pi$
$728$	17.5692	0.651159
$729$	0	0
$730$	0	0
$731$	0.535898	0.0198209
$732$	0	0
$733$	−34.7846	−1.28480	−0.642399	−	0.766370i	$-0.722061\pi$
$733$	−34.7846	−1.28480	−0.642399	+	0.766370i	$0.722061\pi$
$734$	−22.8231	−0.842415
$735$	0	0
$736$	−20.2872	−0.747796
$737$	−19.8564	−0.731420
$738$	0	0
$739$	22.4641	0.826355	0.413178	−	0.910650i	$-0.364419\pi$
$739$	22.4641	0.826355	0.413178	+	0.910650i	$0.364419\pi$
$740$	0	0
$741$	0	0
$742$	23.3205	0.856123
$743$	19.9090	0.730389	0.365195	−	0.930931i	$-0.381002\pi$
$743$	19.9090	0.730389	0.365195	+	0.930931i	$0.381002\pi$
$744$	0	0
$745$	0	0
$746$	−14.6795	−0.537454
$747$	0	0
$748$	22.9282	0.838338
$749$	−40.3923	−1.47590
$750$	0	0
$751$	48.7846	1.78018	0.890088	−	0.455789i	$-0.150643\pi$
$751$	48.7846	1.78018	0.890088	+	0.455789i	$0.150643\pi$
$752$	−9.35898	−0.341287
$753$	0	0
$754$	3.42563	0.124754
$755$	0	0
$756$	0	0
$757$	9.17691	0.333541	0.166770	−	0.985996i	$-0.446666\pi$
$757$	9.17691	0.333541	0.166770	+	0.985996i	$0.446666\pi$
$758$	1.75129	0.0636097
$759$	0	0
$760$	0	0
$761$	−23.4449	−0.849876	−0.424938	−	0.905223i	$-0.639704\pi$
$761$	−23.4449	−0.849876	−0.424938	+	0.905223i	$0.639704\pi$
$762$	0	0
$763$	−28.7321	−1.04017
$764$	−11.8949	−0.430342
$765$	0	0
$766$	1.85641	0.0670747
$767$	−12.1051	−0.437090
$768$	0	0
$769$	−9.53590	−0.343873	−0.171937	−	0.985108i	$-0.555002\pi$
$769$	−9.53590	−0.343873	−0.171937	+	0.985108i	$0.555002\pi$
$770$	0	0
$771$	0	0
$772$	−7.71281	−0.277590
$773$	1.51666	0.0545505	0.0272752	−	0.999628i	$-0.491317\pi$
$773$	1.51666	0.0545505	0.0272752	+	0.999628i	$0.491317\pi$
$774$	0	0
$775$	0	0
$776$	−24.4974	−0.879406
$777$	0	0
$778$	−20.1051	−0.720803
$779$	32.1244	1.15097
$780$	0	0
$781$	21.3923	0.765477
$782$	6.92820	0.247752
$783$	0	0
$784$	16.4974	0.589194
$785$	0	0
$786$	0	0
$787$	48.0526	1.71289	0.856444	−	0.516239i	$-0.172668\pi$
$787$	48.0526	1.71289	0.856444	+	0.516239i	$0.172668\pi$
$788$	20.2872	0.722701
$789$	0	0
$790$	0	0
$791$	−90.4974	−3.21772
$792$	0	0
$793$	−5.85641	−0.207967
$794$	−10.5359	−0.373905
$795$	0	0
$796$	2.92820	0.103787
$797$	−15.6077	−0.552853	−0.276426	−	0.961035i	$-0.589150\pi$
$797$	−15.6077	−0.552853	−0.276426	+	0.961035i	$0.589150\pi$
$798$	0	0
$799$	23.8564	0.843979
$800$	0	0
$801$	0	0
$802$	−18.2487	−0.644384
$803$	43.9090	1.54951
$804$	0	0
$805$	0	0
$806$	3.21539	0.113257
$807$	0	0
$808$	−6.74613	−0.237328
$809$	−45.4449	−1.59776	−0.798878	−	0.601493i	$-0.794573\pi$
$809$	−45.4449	−1.59776	−0.798878	+	0.601493i	$0.794573\pi$
$810$	0	0
$811$	−18.4641	−0.648362	−0.324181	−	0.945995i	$-0.605089\pi$
$811$	−18.4641	−0.648362	−0.324181	+	0.945995i	$0.605089\pi$
$812$	22.1436	0.777088
$813$	0	0
$814$	11.4641	0.401817
$815$	0	0
$816$	0	0
$817$	−0.875644	−0.0306349
$818$	−13.0718	−0.457045
$819$	0	0
$820$	0	0
$821$	37.7321	1.31686	0.658429	−	0.752643i	$-0.271221\pi$
$821$	37.7321	1.31686	0.658429	+	0.752643i	$0.271221\pi$
$822$	0	0
$823$	−3.85641	−0.134426	−0.0672129	−	0.997739i	$-0.521411\pi$
$823$	−3.85641	−0.134426	−0.0672129	+	0.997739i	$0.521411\pi$
$824$	−1.35898	−0.0473424
$825$	0	0
$826$	28.6410	0.996548
$827$	−11.6077	−0.403639	−0.201820	−	0.979423i	$-0.564685\pi$
$827$	−11.6077	−0.403639	−0.201820	+	0.979423i	$0.564685\pi$
$828$	0	0
$829$	−23.7846	−0.826074	−0.413037	−	0.910714i	$-0.635532\pi$
$829$	−23.7846	−0.826074	−0.413037	+	0.910714i	$0.635532\pi$
$830$	0	0
$831$	0	0
$832$	−3.13844	−0.108806
$833$	−42.0526	−1.45703
$834$	0	0
$835$	0	0
$836$	−37.4641	−1.29572
$837$	0	0
$838$	−14.9282	−0.515686
$839$	−33.1962	−1.14606	−0.573029	−	0.819535i	$-0.694232\pi$
$839$	−33.1962	−1.14606	−0.573029	+	0.819535i	$0.694232\pi$
$840$	0	0
$841$	−18.7846	−0.647745
$842$	24.7321	0.852323
$843$	0	0
$844$	12.9667	0.446331
$845$	0	0
$846$	0	0
$847$	103.426	3.55375
$848$	7.21539	0.247778
$849$	0	0
$850$	0	0
$851$	−9.46410	−0.324425
$852$	0	0
$853$	−10.4833	−0.358943	−0.179471	−	0.983763i	$-0.557439\pi$
$853$	−10.4833	−0.358943	−0.179471	+	0.983763i	$0.557439\pi$
$854$	13.8564	0.474156
$855$	0	0
$856$	21.6462	0.739851
$857$	44.4974	1.52000	0.760002	−	0.649921i	$-0.225198\pi$
$857$	44.4974	1.52000	0.760002	+	0.649921i	$0.225198\pi$
$858$	0	0
$859$	−11.0000	−0.375315	−0.187658	−	0.982235i	$-0.560090\pi$
$859$	−11.0000	−0.375315	−0.187658	+	0.982235i	$0.560090\pi$
$860$	0	0
$861$	0	0
$862$	15.6218	0.532080
$863$	−23.1244	−0.787162	−0.393581	−	0.919290i	$-0.628764\pi$
$863$	−23.1244	−0.787162	−0.393581	+	0.919290i	$0.628764\pi$
$864$	0	0
$865$	0	0
$866$	25.9615	0.882209
$867$	0	0
$868$	20.7846	0.705476
$869$	88.6410	3.00694
$870$	0	0
$871$	5.07180	0.171851
$872$	15.3975	0.521424
$873$	0	0
$874$	−11.3205	−0.382922
$875$	0	0
$876$	0	0
$877$	−33.4641	−1.13000	−0.565001	−	0.825090i	$-0.691124\pi$
$877$	−33.4641	−1.13000	−0.565001	+	0.825090i	$0.691124\pi$
$878$	3.94744	0.133220
$879$	0	0
$880$	0	0
$881$	47.0526	1.58524	0.792620	−	0.609715i	$-0.208716\pi$
$881$	47.0526	1.58524	0.792620	+	0.609715i	$0.208716\pi$
$882$	0	0
$883$	30.1962	1.01618	0.508091	−	0.861304i	$-0.330351\pi$
$883$	30.1962	1.01618	0.508091	+	0.861304i	$0.330351\pi$
$884$	−5.85641	−0.196972
$885$	0	0
$886$	−0.248711	−0.00835562
$887$	33.8038	1.13502	0.567511	−	0.823366i	$-0.307906\pi$
$887$	33.8038	1.13502	0.567511	+	0.823366i	$0.307906\pi$
$888$	0	0
$889$	69.0333	2.31530
$890$	0	0
$891$	0	0
$892$	24.5744	0.822811
$893$	−38.9808	−1.30444
$894$	0	0
$895$	0	0
$896$	−48.0000	−1.60357
$897$	0	0
$898$	−5.94744	−0.198469
$899$	9.58846	0.319793
$900$	0	0
$901$	−18.3923	−0.612737
$902$	30.1962	1.00542
$903$	0	0
$904$	48.4974	1.61300
$905$	0	0
$906$	0	0
$907$	30.0000	0.996134	0.498067	−	0.867139i	$-0.334043\pi$
$907$	30.0000	0.996134	0.498067	+	0.867139i	$0.334043\pi$
$908$	26.4308	0.877136
$909$	0	0
$910$	0	0
$911$	−23.5885	−0.781520	−0.390760	−	0.920493i	$-0.627788\pi$
$911$	−23.5885	−0.781520	−0.390760	+	0.920493i	$0.627788\pi$
$912$	0	0
$913$	12.5885	0.416617
$914$	−2.00000	−0.0661541
$915$	0	0
$916$	17.5692	0.580503
$917$	−73.7654	−2.43595
$918$	0	0
$919$	−56.9615	−1.87899	−0.939494	−	0.342566i	$-0.888704\pi$
$919$	−56.9615	−1.87899	−0.939494	+	0.342566i	$0.888704\pi$
$920$	0	0
$921$	0	0
$922$	27.1244	0.893293
$923$	−5.46410	−0.179853
$924$	0	0
$925$	0	0
$926$	7.60770	0.250004
$927$	0	0
$928$	−18.7180	−0.614447
$929$	44.3731	1.45583	0.727917	−	0.685666i	$-0.240489\pi$
$929$	44.3731	1.45583	0.727917	+	0.685666i	$0.240489\pi$
$930$	0	0
$931$	68.7128	2.25197
$932$	41.0718	1.34535
$933$	0	0
$934$	27.3590	0.895213
$935$	0	0
$936$	0	0
$937$	4.14359	0.135365	0.0676827	−	0.997707i	$-0.478439\pi$
$937$	4.14359	0.135365	0.0676827	+	0.997707i	$0.478439\pi$
$938$	−12.0000	−0.391814
$939$	0	0
$940$	0	0
$941$	−35.1769	−1.14673	−0.573367	−	0.819298i	$-0.694363\pi$
$941$	−35.1769	−1.14673	−0.573367	+	0.819298i	$0.694363\pi$
$942$	0	0
$943$	−24.9282	−0.811774
$944$	8.86156	0.288419
$945$	0	0
$946$	−0.823085	−0.0267608
$947$	−57.7128	−1.87541	−0.937707	−	0.347427i	$-0.887056\pi$
$947$	−57.7128	−1.87541	−0.937707	+	0.347427i	$0.887056\pi$
$948$	0	0
$949$	−11.2154	−0.364067
$950$	0	0
$951$	0	0
$952$	32.7846	1.06256
$953$	36.3923	1.17886	0.589431	−	0.807819i	$-0.299352\pi$
$953$	36.3923	1.17886	0.589431	+	0.807819i	$0.299352\pi$
$954$	0	0
$955$	0	0
$956$	−0.784610	−0.0253761
$957$	0	0
$958$	8.69358	0.280877
$959$	12.0000	0.387500
$960$	0	0
$961$	−22.0000	−0.709677
$962$	−2.92820	−0.0944091
$963$	0	0
$964$	23.8949	0.769602
$965$	0	0
$966$	0	0
$967$	25.8038	0.829796	0.414898	−	0.909868i	$-0.363817\pi$
$967$	25.8038	0.829796	0.414898	+	0.909868i	$0.363817\pi$
$968$	−55.4256	−1.78145
$969$	0	0
$970$	0	0
$971$	−17.4449	−0.559832	−0.279916	−	0.960024i	$-0.590307\pi$
$971$	−17.4449	−0.559832	−0.279916	+	0.960024i	$0.590307\pi$
$972$	0	0
$973$	2.87564	0.0921889
$974$	−17.8564	−0.572156
$975$	0	0
$976$	4.28719	0.137230
$977$	5.46410	0.174812	0.0874060	−	0.996173i	$-0.472142\pi$
$977$	5.46410	0.174812	0.0874060	+	0.996173i	$0.472142\pi$
$978$	0	0
$979$	−29.7846	−0.951920
$980$	0	0
$981$	0	0
$982$	10.1577	0.324144
$983$	48.5885	1.54973	0.774866	−	0.632126i	$-0.217818\pi$
$983$	48.5885	1.54973	0.774866	+	0.632126i	$0.217818\pi$
$984$	0	0
$985$	0	0
$986$	6.39230	0.203572
$987$	0	0
$988$	9.56922	0.304437
$989$	0.679492	0.0216066
$990$	0	0
$991$	30.8564	0.980186	0.490093	−	0.871670i	$-0.336963\pi$
$991$	30.8564	0.980186	0.490093	+	0.871670i	$0.336963\pi$
$992$	−17.5692	−0.557823
$993$	0	0
$994$	12.9282	0.410058
$995$	0	0
$996$	0	0
$997$	−25.5167	−0.808121	−0.404060	−	0.914732i	$-0.632401\pi$
$997$	−25.5167	−0.808121	−0.404060	+	0.914732i	$0.632401\pi$
$998$	17.8038	0.563571
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2025.2.a.g.1.2		2
3.2	odd	2		2025.2.a.m.1.1		2
5.2	odd	4		2025.2.b.h.649.3		4
5.3	odd	4		2025.2.b.h.649.2		4
5.4	even	2		405.2.a.h.1.1	yes	2
15.2	even	4		2025.2.b.g.649.2		4
15.8	even	4		2025.2.b.g.649.3		4
15.14	odd	2		405.2.a.g.1.2	✓	2
20.19	odd	2		6480.2.a.bi.1.2		2
45.4	even	6		405.2.e.i.136.2		4
45.14	odd	6		405.2.e.l.136.1		4
45.29	odd	6		405.2.e.l.271.1		4
45.34	even	6		405.2.e.i.271.2		4
60.59	even	2		6480.2.a.br.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
405.2.a.g.1.2	✓	2	15.14	odd	2
405.2.a.h.1.1	yes	2	5.4	even	2
405.2.e.i.136.2		4	45.4	even	6
405.2.e.i.271.2		4	45.34	even	6
405.2.e.l.136.1		4	45.14	odd	6
405.2.e.l.271.1		4	45.29	odd	6
2025.2.a.g.1.2		2	1.1	even	1	trivial
2025.2.a.m.1.1		2	3.2	odd	2
2025.2.b.g.649.2		4	15.2	even	4
2025.2.b.g.649.3		4	15.8	even	4
2025.2.b.h.649.2		4	5.3	odd	4
2025.2.b.h.649.3		4	5.2	odd	4
6480.2.a.bi.1.2		2	20.19	odd	2
6480.2.a.br.1.2		2	60.59	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+0.732051 q^{2} -1.46410 q^{4} +4.73205 q^{7} -2.53590 q^{8} +5.73205 q^{11} -1.46410 q^{13} +3.46410 q^{14} +1.07180 q^{16} -2.73205 q^{17} +4.46410 q^{19} +4.19615 q^{22} -3.46410 q^{23} -1.07180 q^{26} -6.92820 q^{28} -3.19615 q^{29} -3.00000 q^{31} +5.85641 q^{32} -2.00000 q^{34} +2.73205 q^{37} +3.26795 q^{38} +7.19615 q^{41} -0.196152 q^{43} -8.39230 q^{44} -2.53590 q^{46} -8.73205 q^{47} +15.3923 q^{49} +2.14359 q^{52} +6.73205 q^{53} -12.0000 q^{56} -2.33975 q^{58} +8.26795 q^{59} +4.00000 q^{61} -2.19615 q^{62} +2.14359 q^{64} -3.46410 q^{67} +4.00000 q^{68} +3.73205 q^{71} +7.66025 q^{73} +2.00000 q^{74} -6.53590 q^{76} +27.1244 q^{77} +15.4641 q^{79} +5.26795 q^{82} +2.19615 q^{83} -0.143594 q^{86} -14.5359 q^{88} -5.19615 q^{89} -6.92820 q^{91} +5.07180 q^{92} -6.39230 q^{94} +9.66025 q^{97} +11.2679 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 4 q^{4} + 6 q^{7} - 12 q^{8} + 8 q^{11} + 4 q^{13} + 16 q^{16} - 2 q^{17} + 2 q^{19} - 2 q^{22} - 16 q^{26} + 4 q^{29} - 6 q^{31} - 16 q^{32} - 4 q^{34} + 2 q^{37} + 10 q^{38} + 4 q^{41}+ \cdots + 26 q^{98}+O(q^{100}) \) 2 * q - 2 * q^2 + 4 * q^4 + 6 * q^7 - 12 * q^8 + 8 * q^11 + 4 * q^13 + 16 * q^16 - 2 * q^17 + 2 * q^19 - 2 * q^22 - 16 * q^26 + 4 * q^29 - 6 * q^31 - 16 * q^32 - 4 * q^34 + 2 * q^37 + 10 * q^38 + 4 * q^41 + 10 * q^43 + 4 * q^44 - 12 * q^46 - 14 * q^47 + 10 * q^49 + 32 * q^52 + 10 * q^53 - 24 * q^56 - 22 * q^58 + 20 * q^59 + 8 * q^61 + 6 * q^62 + 32 * q^64 + 8 * q^68 + 4 * q^71 - 2 * q^73 + 4 * q^74 - 20 * q^76 + 30 * q^77 + 24 * q^79 + 14 * q^82 - 6 * q^83 - 28 * q^86 - 36 * q^88 + 24 * q^92 + 8 * q^94 + 2 * q^97 + 26 * q^98

Introduction

L-functions

Modular forms

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