LMFDB - Embedded newform 2025.2.a.b.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:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
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			1.1
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)

$f(q)$

$=$

$q-1.00000 q^{2} -1.00000 q^{4} +3.00000 q^{7} +3.00000 q^{8} +O(q^{10})$

$q-1.00000 q^{2} -1.00000 q^{4} +3.00000 q^{7} +3.00000 q^{8} -2.00000 q^{11} +2.00000 q^{13} -3.00000 q^{14} -1.00000 q^{16} -4.00000 q^{17} -8.00000 q^{19} +2.00000 q^{22} -3.00000 q^{23} -2.00000 q^{26} -3.00000 q^{28} -1.00000 q^{29} -5.00000 q^{32} +4.00000 q^{34} +4.00000 q^{37} +8.00000 q^{38} +5.00000 q^{41} +8.00000 q^{43} +2.00000 q^{44} +3.00000 q^{46} -7.00000 q^{47} +2.00000 q^{49} -2.00000 q^{52} +2.00000 q^{53} +9.00000 q^{56} +1.00000 q^{58} -14.0000 q^{59} +7.00000 q^{61} +7.00000 q^{64} +3.00000 q^{67} +4.00000 q^{68} +2.00000 q^{71} -4.00000 q^{73} -4.00000 q^{74} +8.00000 q^{76} -6.00000 q^{77} -6.00000 q^{79} -5.00000 q^{82} -9.00000 q^{83} -8.00000 q^{86} -6.00000 q^{88} -15.0000 q^{89} +6.00000 q^{91} +3.00000 q^{92} +7.00000 q^{94} -2.00000 q^{97} -2.00000 q^{98} +O(q^{100})$

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 $

$2$	−1.00000	−0.707107	−0.353553	−	0.935414i	$-0.615027\pi$
$2$	−1.00000	−0.707107	−0.353553	+	0.935414i	$0.615027\pi$
$3$	0	0
$3$	0	0
$4$	−1.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	3.00000	1.13389	0.566947	−	0.823754i	$-0.308125\pi$
$7$	3.00000	1.13389	0.566947	+	0.823754i	$0.308125\pi$
$8$	3.00000	1.06066
$9$	0	0
$10$	0	0
$11$	−2.00000	−0.603023	−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−2.00000	−0.603023	−0.301511	+	0.953463i	$0.597491\pi$
$12$	0	0
$13$	2.00000	0.554700	0.277350	−	0.960769i	$-0.410544\pi$
$13$	2.00000	0.554700	0.277350	+	0.960769i	$0.410544\pi$
$14$	−3.00000	−0.801784
$15$	0	0
$16$	−1.00000	−0.250000
$17$	−4.00000	−0.970143	−0.485071	−	0.874475i	$-0.661206\pi$
$17$	−4.00000	−0.970143	−0.485071	+	0.874475i	$0.661206\pi$
$18$	0	0
$19$	−8.00000	−1.83533	−0.917663	−	0.397360i	$-0.869927\pi$
$19$	−8.00000	−1.83533	−0.917663	+	0.397360i	$0.869927\pi$
$20$	0	0
$21$	0	0
$22$	2.00000	0.426401
$23$	−3.00000	−0.625543	−0.312772	−	0.949828i	$-0.601257\pi$
$23$	−3.00000	−0.625543	−0.312772	+	0.949828i	$0.601257\pi$
$24$	0	0
$25$	0	0
$26$	−2.00000	−0.392232
$27$	0	0
$28$	−3.00000	−0.566947
$29$	−1.00000	−0.185695	−0.0928477	−	0.995680i	$-0.529597\pi$
$29$	−1.00000	−0.185695	−0.0928477	+	0.995680i	$0.529597\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	−5.00000	−0.883883
$33$	0	0
$34$	4.00000	0.685994
$35$	0	0
$36$	0	0
$37$	4.00000	0.657596	0.328798	−	0.944400i	$-0.393356\pi$
$37$	4.00000	0.657596	0.328798	+	0.944400i	$0.393356\pi$
$38$	8.00000	1.29777
$39$	0	0
$40$	0	0
$41$	5.00000	0.780869	0.390434	−	0.920631i	$-0.372325\pi$
$41$	5.00000	0.780869	0.390434	+	0.920631i	$0.372325\pi$
$42$	0	0
$43$	8.00000	1.21999	0.609994	−	0.792406i	$-0.291172\pi$
$43$	8.00000	1.21999	0.609994	+	0.792406i	$0.291172\pi$
$44$	2.00000	0.301511
$45$	0	0
$46$	3.00000	0.442326
$47$	−7.00000	−1.02105	−0.510527	−	0.859861i	$-0.670550\pi$
$47$	−7.00000	−1.02105	−0.510527	+	0.859861i	$0.670550\pi$
$48$	0	0
$49$	2.00000	0.285714
$50$	0	0
$51$	0	0
$52$	−2.00000	−0.277350
$53$	2.00000	0.274721	0.137361	−	0.990521i	$-0.456138\pi$
$53$	2.00000	0.274721	0.137361	+	0.990521i	$0.456138\pi$
$54$	0	0
$55$	0	0
$56$	9.00000	1.20268
$57$	0	0
$58$	1.00000	0.131306
$59$	−14.0000	−1.82264	−0.911322	−	0.411693i	$-0.864937\pi$
$59$	−14.0000	−1.82264	−0.911322	+	0.411693i	$0.864937\pi$
$60$	0	0
$61$	7.00000	0.896258	0.448129	−	0.893969i	$-0.352090\pi$
$61$	7.00000	0.896258	0.448129	+	0.893969i	$0.352090\pi$
$62$	0	0
$63$	0	0
$64$	7.00000	0.875000
$65$	0	0
$66$	0	0
$67$	3.00000	0.366508	0.183254	−	0.983066i	$-0.441337\pi$
$67$	3.00000	0.366508	0.183254	+	0.983066i	$0.441337\pi$
$68$	4.00000	0.485071
$69$	0	0
$70$	0	0
$71$	2.00000	0.237356	0.118678	−	0.992933i	$-0.462134\pi$
$71$	2.00000	0.237356	0.118678	+	0.992933i	$0.462134\pi$
$72$	0	0
$73$	−4.00000	−0.468165	−0.234082	−	0.972217i	$-0.575209\pi$
$73$	−4.00000	−0.468165	−0.234082	+	0.972217i	$0.575209\pi$
$74$	−4.00000	−0.464991
$75$	0	0
$76$	8.00000	0.917663
$77$	−6.00000	−0.683763
$78$	0	0
$79$	−6.00000	−0.675053	−0.337526	−	0.941316i	$-0.609590\pi$
$79$	−6.00000	−0.675053	−0.337526	+	0.941316i	$0.609590\pi$
$80$	0	0
$81$	0	0
$82$	−5.00000	−0.552158
$83$	−9.00000	−0.987878	−0.493939	−	0.869496i	$-0.664443\pi$
$83$	−9.00000	−0.987878	−0.493939	+	0.869496i	$0.664443\pi$
$84$	0	0
$85$	0	0
$86$	−8.00000	−0.862662
$87$	0	0
$88$	−6.00000	−0.639602
$89$	−15.0000	−1.59000	−0.794998	−	0.606612i	$-0.792528\pi$
$89$	−15.0000	−1.59000	−0.794998	+	0.606612i	$0.792528\pi$
$90$	0	0
$91$	6.00000	0.628971
$92$	3.00000	0.312772
$93$	0	0
$94$	7.00000	0.721995
$95$	0	0
$96$	0	0
$97$	−2.00000	−0.203069	−0.101535	−	0.994832i	$-0.532375\pi$
$97$	−2.00000	−0.203069	−0.101535	+	0.994832i	$0.532375\pi$
$98$	−2.00000	−0.202031
$99$	0	0
$100$	0	0
$101$	−18.0000	−1.79107	−0.895533	−	0.444994i	$-0.853206\pi$
$101$	−18.0000	−1.79107	−0.895533	+	0.444994i	$0.853206\pi$
$102$	0	0
$103$	−8.00000	−0.788263	−0.394132	−	0.919054i	$-0.628955\pi$
$103$	−8.00000	−0.788263	−0.394132	+	0.919054i	$0.628955\pi$
$104$	6.00000	0.588348
$105$	0	0
$106$	−2.00000	−0.194257
$107$	−3.00000	−0.290021	−0.145010	−	0.989430i	$-0.546322\pi$
$107$	−3.00000	−0.290021	−0.145010	+	0.989430i	$0.546322\pi$
$108$	0	0
$109$	5.00000	0.478913	0.239457	−	0.970907i	$-0.423031\pi$
$109$	5.00000	0.478913	0.239457	+	0.970907i	$0.423031\pi$
$110$	0	0
$111$	0	0
$112$	−3.00000	−0.283473
$113$	8.00000	0.752577	0.376288	−	0.926503i	$-0.377200\pi$
$113$	8.00000	0.752577	0.376288	+	0.926503i	$0.377200\pi$
$114$	0	0
$115$	0	0
$116$	1.00000	0.0928477
$117$	0	0
$118$	14.0000	1.28880
$119$	−12.0000	−1.10004
$120$	0	0
$121$	−7.00000	−0.636364
$122$	−7.00000	−0.633750
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	5.00000	0.443678	0.221839	−	0.975083i	$-0.428794\pi$
$127$	5.00000	0.443678	0.221839	+	0.975083i	$0.428794\pi$
$128$	3.00000	0.265165
$129$	0	0
$130$	0	0
$131$	−6.00000	−0.524222	−0.262111	−	0.965038i	$-0.584419\pi$
$131$	−6.00000	−0.524222	−0.262111	+	0.965038i	$0.584419\pi$
$132$	0	0
$133$	−24.0000	−2.08106
$134$	−3.00000	−0.259161
$135$	0	0
$136$	−12.0000	−1.02899
$137$	−12.0000	−1.02523	−0.512615	−	0.858619i	$-0.671323\pi$
$137$	−12.0000	−1.02523	−0.512615	+	0.858619i	$0.671323\pi$
$138$	0	0
$139$	−16.0000	−1.35710	−0.678551	−	0.734553i	$-0.737392\pi$
$139$	−16.0000	−1.35710	−0.678551	+	0.734553i	$0.737392\pi$
$140$	0	0
$141$	0	0
$142$	−2.00000	−0.167836
$143$	−4.00000	−0.334497
$144$	0	0
$145$	0	0
$146$	4.00000	0.331042
$147$	0	0
$148$	−4.00000	−0.328798
$149$	17.0000	1.39269	0.696347	−	0.717705i	$-0.254807\pi$
$149$	17.0000	1.39269	0.696347	+	0.717705i	$0.254807\pi$
$150$	0	0
$151$	−2.00000	−0.162758	−0.0813788	−	0.996683i	$-0.525932\pi$
$151$	−2.00000	−0.162758	−0.0813788	+	0.996683i	$0.525932\pi$
$152$	−24.0000	−1.94666
$153$	0	0
$154$	6.00000	0.483494
$155$	0	0
$156$	0	0
$157$	−14.0000	−1.11732	−0.558661	−	0.829396i	$-0.688685\pi$
$157$	−14.0000	−1.11732	−0.558661	+	0.829396i	$0.688685\pi$
$158$	6.00000	0.477334
$159$	0	0
$160$	0	0
$161$	−9.00000	−0.709299
$162$	0	0
$163$	4.00000	0.313304	0.156652	−	0.987654i	$-0.449930\pi$
$163$	4.00000	0.313304	0.156652	+	0.987654i	$0.449930\pi$
$164$	−5.00000	−0.390434
$165$	0	0
$166$	9.00000	0.698535
$167$	9.00000	0.696441	0.348220	−	0.937413i	$-0.386786\pi$
$167$	9.00000	0.696441	0.348220	+	0.937413i	$0.386786\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	0	0
$171$	0	0
$172$	−8.00000	−0.609994
$173$	0	0		−	1.00000i	$-0.5\pi$
$173$	0	0			1.00000i	$0.5\pi$
$174$	0	0
$175$	0	0
$176$	2.00000	0.150756
$177$	0	0
$178$	15.0000	1.12430
$179$	−2.00000	−0.149487	−0.0747435	−	0.997203i	$-0.523814\pi$
$179$	−2.00000	−0.149487	−0.0747435	+	0.997203i	$0.523814\pi$
$180$	0	0
$181$	−7.00000	−0.520306	−0.260153	−	0.965567i	$-0.583773\pi$
$181$	−7.00000	−0.520306	−0.260153	+	0.965567i	$0.583773\pi$
$182$	−6.00000	−0.444750
$183$	0	0
$184$	−9.00000	−0.663489
$185$	0	0
$186$	0	0
$187$	8.00000	0.585018
$188$	7.00000	0.510527
$189$	0	0
$190$	0	0
$191$	8.00000	0.578860	0.289430	−	0.957199i	$-0.406534\pi$
$191$	8.00000	0.578860	0.289430	+	0.957199i	$0.406534\pi$
$192$	0	0
$193$	10.0000	0.719816	0.359908	−	0.932988i	$-0.382808\pi$
$193$	10.0000	0.719816	0.359908	+	0.932988i	$0.382808\pi$
$194$	2.00000	0.143592
$195$	0	0
$196$	−2.00000	−0.142857
$197$	12.0000	0.854965	0.427482	−	0.904024i	$-0.359401\pi$
$197$	12.0000	0.854965	0.427482	+	0.904024i	$0.359401\pi$
$198$	0	0
$199$	4.00000	0.283552	0.141776	−	0.989899i	$-0.454719\pi$
$199$	4.00000	0.283552	0.141776	+	0.989899i	$0.454719\pi$
$200$	0	0
$201$	0	0
$202$	18.0000	1.26648
$203$	−3.00000	−0.210559
$204$	0	0
$205$	0	0
$206$	8.00000	0.557386
$207$	0	0
$208$	−2.00000	−0.138675
$209$	16.0000	1.10674
$210$	0	0
$211$	−22.0000	−1.51454	−0.757271	−	0.653101i	$-0.773468\pi$
$211$	−22.0000	−1.51454	−0.757271	+	0.653101i	$0.773468\pi$
$212$	−2.00000	−0.137361
$213$	0	0
$214$	3.00000	0.205076
$215$	0	0
$216$	0	0
$217$	0	0
$218$	−5.00000	−0.338643
$219$	0	0
$220$	0	0
$221$	−8.00000	−0.538138
$222$	0	0
$223$	19.0000	1.27233	0.636167	−	0.771551i	$-0.280519\pi$
$223$	19.0000	1.27233	0.636167	+	0.771551i	$0.280519\pi$
$224$	−15.0000	−1.00223
$225$	0	0
$226$	−8.00000	−0.532152
$227$	4.00000	0.265489	0.132745	−	0.991150i	$-0.457621\pi$
$227$	4.00000	0.265489	0.132745	+	0.991150i	$0.457621\pi$
$228$	0	0
$229$	15.0000	0.991228	0.495614	−	0.868543i	$-0.334943\pi$
$229$	15.0000	0.991228	0.495614	+	0.868543i	$0.334943\pi$
$230$	0	0
$231$	0	0
$232$	−3.00000	−0.196960
$233$	−24.0000	−1.57229	−0.786146	−	0.618041i	$-0.787927\pi$
$233$	−24.0000	−1.57229	−0.786146	+	0.618041i	$0.787927\pi$
$234$	0	0
$235$	0	0
$236$	14.0000	0.911322
$237$	0	0
$238$	12.0000	0.777844
$239$	−8.00000	−0.517477	−0.258738	−	0.965947i	$-0.583307\pi$
$239$	−8.00000	−0.517477	−0.258738	+	0.965947i	$0.583307\pi$
$240$	0	0
$241$	−11.0000	−0.708572	−0.354286	−	0.935137i	$-0.615276\pi$
$241$	−11.0000	−0.708572	−0.354286	+	0.935137i	$0.615276\pi$
$242$	7.00000	0.449977
$243$	0	0
$244$	−7.00000	−0.448129
$245$	0	0
$246$	0	0
$247$	−16.0000	−1.01806
$248$	0	0
$249$	0	0
$250$	0	0
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	0	0
$253$	6.00000	0.377217
$254$	−5.00000	−0.313728
$255$	0	0
$256$	−17.0000	−1.06250
$257$	6.00000	0.374270	0.187135	−	0.982334i	$-0.440080\pi$
$257$	6.00000	0.374270	0.187135	+	0.982334i	$0.440080\pi$
$258$	0	0
$259$	12.0000	0.745644
$260$	0	0
$261$	0	0
$262$	6.00000	0.370681
$263$	16.0000	0.986602	0.493301	−	0.869859i	$-0.335790\pi$
$263$	16.0000	0.986602	0.493301	+	0.869859i	$0.335790\pi$
$264$	0	0
$265$	0	0
$266$	24.0000	1.47153
$267$	0	0
$268$	−3.00000	−0.183254
$269$	25.0000	1.52428	0.762138	−	0.647414i	$-0.224150\pi$
$269$	25.0000	1.52428	0.762138	+	0.647414i	$0.224150\pi$
$270$	0	0
$271$	−8.00000	−0.485965	−0.242983	−	0.970031i	$-0.578126\pi$
$271$	−8.00000	−0.485965	−0.242983	+	0.970031i	$0.578126\pi$
$272$	4.00000	0.242536
$273$	0	0
$274$	12.0000	0.724947
$275$	0	0
$276$	0	0
$277$	−12.0000	−0.721010	−0.360505	−	0.932757i	$-0.617396\pi$
$277$	−12.0000	−0.721010	−0.360505	+	0.932757i	$0.617396\pi$
$278$	16.0000	0.959616
$279$	0	0
$280$	0	0
$281$	−15.0000	−0.894825	−0.447412	−	0.894328i	$-0.647654\pi$
$281$	−15.0000	−0.894825	−0.447412	+	0.894328i	$0.647654\pi$
$282$	0	0
$283$	−21.0000	−1.24832	−0.624160	−	0.781296i	$-0.714559\pi$
$283$	−21.0000	−1.24832	−0.624160	+	0.781296i	$0.714559\pi$
$284$	−2.00000	−0.118678
$285$	0	0
$286$	4.00000	0.236525
$287$	15.0000	0.885422
$288$	0	0
$289$	−1.00000	−0.0588235
$290$	0	0
$291$	0	0
$292$	4.00000	0.234082
$293$	−12.0000	−0.701047	−0.350524	−	0.936554i	$-0.613996\pi$
$293$	−12.0000	−0.701047	−0.350524	+	0.936554i	$0.613996\pi$
$294$	0	0
$295$	0	0
$296$	12.0000	0.697486
$297$	0	0
$298$	−17.0000	−0.984784
$299$	−6.00000	−0.346989
$300$	0	0
$301$	24.0000	1.38334
$302$	2.00000	0.115087
$303$	0	0
$304$	8.00000	0.458831
$305$	0	0
$306$	0	0
$307$	−7.00000	−0.399511	−0.199756	−	0.979846i	$-0.564015\pi$
$307$	−7.00000	−0.399511	−0.199756	+	0.979846i	$0.564015\pi$
$308$	6.00000	0.341882
$309$	0	0
$310$	0	0
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	0	0
$313$	−14.0000	−0.791327	−0.395663	−	0.918396i	$-0.629485\pi$
$313$	−14.0000	−0.791327	−0.395663	+	0.918396i	$0.629485\pi$
$314$	14.0000	0.790066
$315$	0	0
$316$	6.00000	0.337526
$317$	34.0000	1.90963	0.954815	−	0.297200i	$-0.0960529\pi$
$317$	34.0000	1.90963	0.954815	+	0.297200i	$0.0960529\pi$
$318$	0	0
$319$	2.00000	0.111979
$320$	0	0
$321$	0	0
$322$	9.00000	0.501550
$323$	32.0000	1.78053
$324$	0	0
$325$	0	0
$326$	−4.00000	−0.221540
$327$	0	0
$328$	15.0000	0.828236
$329$	−21.0000	−1.15777
$330$	0	0
$331$	−6.00000	−0.329790	−0.164895	−	0.986311i	$-0.552728\pi$
$331$	−6.00000	−0.329790	−0.164895	+	0.986311i	$0.552728\pi$
$332$	9.00000	0.493939
$333$	0	0
$334$	−9.00000	−0.492458
$335$	0	0
$336$	0	0
$337$	8.00000	0.435788	0.217894	−	0.975972i	$-0.430081\pi$
$337$	8.00000	0.435788	0.217894	+	0.975972i	$0.430081\pi$
$338$	9.00000	0.489535
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−15.0000	−0.809924
$344$	24.0000	1.29399
$345$	0	0
$346$	0	0
$347$	−4.00000	−0.214731	−0.107366	−	0.994220i	$-0.534242\pi$
$347$	−4.00000	−0.214731	−0.107366	+	0.994220i	$0.534242\pi$
$348$	0	0
$349$	−5.00000	−0.267644	−0.133822	−	0.991005i	$-0.542725\pi$
$349$	−5.00000	−0.267644	−0.133822	+	0.991005i	$0.542725\pi$
$350$	0	0
$351$	0	0
$352$	10.0000	0.533002
$353$	−24.0000	−1.27739	−0.638696	−	0.769460i	$-0.720526\pi$
$353$	−24.0000	−1.27739	−0.638696	+	0.769460i	$0.720526\pi$
$354$	0	0
$355$	0	0
$356$	15.0000	0.794998
$357$	0	0
$358$	2.00000	0.105703
$359$	24.0000	1.26667	0.633336	−	0.773877i	$-0.281685\pi$
$359$	24.0000	1.26667	0.633336	+	0.773877i	$0.281685\pi$
$360$	0	0
$361$	45.0000	2.36842
$362$	7.00000	0.367912
$363$	0	0
$364$	−6.00000	−0.314485
$365$	0	0
$366$	0	0
$367$	−24.0000	−1.25279	−0.626395	−	0.779506i	$-0.715470\pi$
$367$	−24.0000	−1.25279	−0.626395	+	0.779506i	$0.715470\pi$
$368$	3.00000	0.156386
$369$	0	0
$370$	0	0
$371$	6.00000	0.311504
$372$	0	0
$373$	−10.0000	−0.517780	−0.258890	−	0.965907i	$-0.583357\pi$
$373$	−10.0000	−0.517780	−0.258890	+	0.965907i	$0.583357\pi$
$374$	−8.00000	−0.413670
$375$	0	0
$376$	−21.0000	−1.08299
$377$	−2.00000	−0.103005
$378$	0	0
$379$	−26.0000	−1.33553	−0.667765	−	0.744372i	$-0.732749\pi$
$379$	−26.0000	−1.33553	−0.667765	+	0.744372i	$0.732749\pi$
$380$	0	0
$381$	0	0
$382$	−8.00000	−0.409316
$383$	−36.0000	−1.83951	−0.919757	−	0.392488i	$-0.871614\pi$
$383$	−36.0000	−1.83951	−0.919757	+	0.392488i	$0.871614\pi$
$384$	0	0
$385$	0	0
$386$	−10.0000	−0.508987
$387$	0	0
$388$	2.00000	0.101535
$389$	33.0000	1.67317	0.836583	−	0.547840i	$-0.184550\pi$
$389$	33.0000	1.67317	0.836583	+	0.547840i	$0.184550\pi$
$390$	0	0
$391$	12.0000	0.606866
$392$	6.00000	0.303046
$393$	0	0
$394$	−12.0000	−0.604551
$395$	0	0
$396$	0	0
$397$	−34.0000	−1.70641	−0.853206	−	0.521575i	$-0.825345\pi$
$397$	−34.0000	−1.70641	−0.853206	+	0.521575i	$0.825345\pi$
$398$	−4.00000	−0.200502
$399$	0	0
$400$	0	0
$401$	−18.0000	−0.898877	−0.449439	−	0.893311i	$-0.648376\pi$
$401$	−18.0000	−0.898877	−0.449439	+	0.893311i	$0.648376\pi$
$402$	0	0
$403$	0	0
$404$	18.0000	0.895533
$405$	0	0
$406$	3.00000	0.148888
$407$	−8.00000	−0.396545
$408$	0	0
$409$	14.0000	0.692255	0.346128	−	0.938187i	$-0.387496\pi$
$409$	14.0000	0.692255	0.346128	+	0.938187i	$0.387496\pi$
$410$	0	0
$411$	0	0
$412$	8.00000	0.394132
$413$	−42.0000	−2.06668
$414$	0	0
$415$	0	0
$416$	−10.0000	−0.490290
$417$	0	0
$418$	−16.0000	−0.782586
$419$	26.0000	1.27018	0.635092	−	0.772437i	$-0.280962\pi$
$419$	26.0000	1.27018	0.635092	+	0.772437i	$0.280962\pi$
$420$	0	0
$421$	34.0000	1.65706	0.828529	−	0.559946i	$-0.189178\pi$
$421$	34.0000	1.65706	0.828529	+	0.559946i	$0.189178\pi$
$422$	22.0000	1.07094
$423$	0	0
$424$	6.00000	0.291386
$425$	0	0
$426$	0	0
$427$	21.0000	1.01626
$428$	3.00000	0.145010
$429$	0	0
$430$	0	0
$431$	−30.0000	−1.44505	−0.722525	−	0.691345i	$-0.757018\pi$
$431$	−30.0000	−1.44505	−0.722525	+	0.691345i	$0.757018\pi$
$432$	0	0
$433$	−28.0000	−1.34559	−0.672797	−	0.739827i	$-0.734907\pi$
$433$	−28.0000	−1.34559	−0.672797	+	0.739827i	$0.734907\pi$
$434$	0	0
$435$	0	0
$436$	−5.00000	−0.239457
$437$	24.0000	1.14808
$438$	0	0
$439$	28.0000	1.33637	0.668184	−	0.743996i	$-0.267072\pi$
$439$	28.0000	1.33637	0.668184	+	0.743996i	$0.267072\pi$
$440$	0	0
$441$	0	0
$442$	8.00000	0.380521
$443$	−15.0000	−0.712672	−0.356336	−	0.934358i	$-0.615974\pi$
$443$	−15.0000	−0.712672	−0.356336	+	0.934358i	$0.615974\pi$
$444$	0	0
$445$	0	0
$446$	−19.0000	−0.899676
$447$	0	0
$448$	21.0000	0.992157
$449$	−26.0000	−1.22702	−0.613508	−	0.789689i	$-0.710242\pi$
$449$	−26.0000	−1.22702	−0.613508	+	0.789689i	$0.710242\pi$
$450$	0	0
$451$	−10.0000	−0.470882
$452$	−8.00000	−0.376288
$453$	0	0
$454$	−4.00000	−0.187729
$455$	0	0
$456$	0	0
$457$	20.0000	0.935561	0.467780	−	0.883845i	$-0.345054\pi$
$457$	20.0000	0.935561	0.467780	+	0.883845i	$0.345054\pi$
$458$	−15.0000	−0.700904
$459$	0	0
$460$	0	0
$461$	9.00000	0.419172	0.209586	−	0.977790i	$-0.432788\pi$
$461$	9.00000	0.419172	0.209586	+	0.977790i	$0.432788\pi$
$462$	0	0
$463$	36.0000	1.67306	0.836531	−	0.547920i	$-0.184580\pi$
$463$	36.0000	1.67306	0.836531	+	0.547920i	$0.184580\pi$
$464$	1.00000	0.0464238
$465$	0	0
$466$	24.0000	1.11178
$467$	−20.0000	−0.925490	−0.462745	−	0.886492i	$-0.653135\pi$
$467$	−20.0000	−0.925490	−0.462745	+	0.886492i	$0.653135\pi$
$468$	0	0
$469$	9.00000	0.415581
$470$	0	0
$471$	0	0
$472$	−42.0000	−1.93321
$473$	−16.0000	−0.735681
$474$	0	0
$475$	0	0
$476$	12.0000	0.550019
$477$	0	0
$478$	8.00000	0.365911
$479$	−18.0000	−0.822441	−0.411220	−	0.911536i	$-0.634897\pi$
$479$	−18.0000	−0.822441	−0.411220	+	0.911536i	$0.634897\pi$
$480$	0	0
$481$	8.00000	0.364769
$482$	11.0000	0.501036
$483$	0	0
$484$	7.00000	0.318182
$485$	0	0
$486$	0	0
$487$	16.0000	0.725029	0.362515	−	0.931978i	$-0.381918\pi$
$487$	16.0000	0.725029	0.362515	+	0.931978i	$0.381918\pi$
$488$	21.0000	0.950625
$489$	0	0
$490$	0	0
$491$	20.0000	0.902587	0.451294	−	0.892375i	$-0.350963\pi$
$491$	20.0000	0.902587	0.451294	+	0.892375i	$0.350963\pi$
$492$	0	0
$493$	4.00000	0.180151
$494$	16.0000	0.719874
$495$	0	0
$496$	0	0
$497$	6.00000	0.269137
$498$	0	0
$499$	−32.0000	−1.43252	−0.716258	−	0.697835i	$-0.754147\pi$
$499$	−32.0000	−1.43252	−0.716258	+	0.697835i	$0.754147\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	7.00000	0.312115	0.156057	−	0.987748i	$-0.450122\pi$
$503$	7.00000	0.312115	0.156057	+	0.987748i	$0.450122\pi$
$504$	0	0
$505$	0	0
$506$	−6.00000	−0.266733
$507$	0	0
$508$	−5.00000	−0.221839
$509$	43.0000	1.90594	0.952971	−	0.303062i	$-0.0980090\pi$
$509$	43.0000	1.90594	0.952971	+	0.303062i	$0.0980090\pi$
$510$	0	0
$511$	−12.0000	−0.530849
$512$	11.0000	0.486136
$513$	0	0
$514$	−6.00000	−0.264649
$515$	0	0
$516$	0	0
$517$	14.0000	0.615719
$518$	−12.0000	−0.527250
$519$	0	0
$520$	0	0
$521$	11.0000	0.481919	0.240959	−	0.970535i	$-0.422538\pi$
$521$	11.0000	0.481919	0.240959	+	0.970535i	$0.422538\pi$
$522$	0	0
$523$	29.0000	1.26808	0.634041	−	0.773300i	$-0.281395\pi$
$523$	29.0000	1.26808	0.634041	+	0.773300i	$0.281395\pi$
$524$	6.00000	0.262111
$525$	0	0
$526$	−16.0000	−0.697633
$527$	0	0
$528$	0	0
$529$	−14.0000	−0.608696
$530$	0	0
$531$	0	0
$532$	24.0000	1.04053
$533$	10.0000	0.433148
$534$	0	0
$535$	0	0
$536$	9.00000	0.388741
$537$	0	0
$538$	−25.0000	−1.07783
$539$	−4.00000	−0.172292
$540$	0	0
$541$	−39.0000	−1.67674	−0.838370	−	0.545101i	$-0.816491\pi$
$541$	−39.0000	−1.67674	−0.838370	+	0.545101i	$0.816491\pi$
$542$	8.00000	0.343629
$543$	0	0
$544$	20.0000	0.857493
$545$	0	0
$546$	0	0
$547$	29.0000	1.23995	0.619975	−	0.784621i	$-0.287143\pi$
$547$	29.0000	1.23995	0.619975	+	0.784621i	$0.287143\pi$
$548$	12.0000	0.512615
$549$	0	0
$550$	0	0
$551$	8.00000	0.340811
$552$	0	0
$553$	−18.0000	−0.765438
$554$	12.0000	0.509831
$555$	0	0
$556$	16.0000	0.678551
$557$	30.0000	1.27114	0.635570	−	0.772043i	$-0.280765\pi$
$557$	30.0000	1.27114	0.635570	+	0.772043i	$0.280765\pi$
$558$	0	0
$559$	16.0000	0.676728
$560$	0	0
$561$	0	0
$562$	15.0000	0.632737
$563$	21.0000	0.885044	0.442522	−	0.896758i	$-0.354084\pi$
$563$	21.0000	0.885044	0.442522	+	0.896758i	$0.354084\pi$
$564$	0	0
$565$	0	0
$566$	21.0000	0.882696
$567$	0	0
$568$	6.00000	0.251754
$569$	−6.00000	−0.251533	−0.125767	−	0.992060i	$-0.540139\pi$
$569$	−6.00000	−0.251533	−0.125767	+	0.992060i	$0.540139\pi$
$570$	0	0
$571$	32.0000	1.33916	0.669579	−	0.742741i	$-0.266474\pi$
$571$	32.0000	1.33916	0.669579	+	0.742741i	$0.266474\pi$
$572$	4.00000	0.167248
$573$	0	0
$574$	−15.0000	−0.626088
$575$	0	0
$576$	0	0
$577$	10.0000	0.416305	0.208153	−	0.978096i	$-0.433255\pi$
$577$	10.0000	0.416305	0.208153	+	0.978096i	$0.433255\pi$
$578$	1.00000	0.0415945
$579$	0	0
$580$	0	0
$581$	−27.0000	−1.12015
$582$	0	0
$583$	−4.00000	−0.165663
$584$	−12.0000	−0.496564
$585$	0	0
$586$	12.0000	0.495715
$587$	33.0000	1.36206	0.681028	−	0.732257i	$-0.261533\pi$
$587$	33.0000	1.36206	0.681028	+	0.732257i	$0.261533\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	0	0
$592$	−4.00000	−0.164399
$593$	−20.0000	−0.821302	−0.410651	−	0.911793i	$-0.634698\pi$
$593$	−20.0000	−0.821302	−0.410651	+	0.911793i	$0.634698\pi$
$594$	0	0
$595$	0	0
$596$	−17.0000	−0.696347
$597$	0	0
$598$	6.00000	0.245358
$599$	−10.0000	−0.408589	−0.204294	−	0.978909i	$-0.565490\pi$
$599$	−10.0000	−0.408589	−0.204294	+	0.978909i	$0.565490\pi$
$600$	0	0
$601$	2.00000	0.0815817	0.0407909	−	0.999168i	$-0.487012\pi$
$601$	2.00000	0.0815817	0.0407909	+	0.999168i	$0.487012\pi$
$602$	−24.0000	−0.978167
$603$	0	0
$604$	2.00000	0.0813788
$605$	0	0
$606$	0	0
$607$	41.0000	1.66414	0.832069	−	0.554672i	$-0.187156\pi$
$607$	41.0000	1.66414	0.832069	+	0.554672i	$0.187156\pi$
$608$	40.0000	1.62221
$609$	0	0
$610$	0	0
$611$	−14.0000	−0.566379
$612$	0	0
$613$	44.0000	1.77714	0.888572	−	0.458738i	$-0.151698\pi$
$613$	44.0000	1.77714	0.888572	+	0.458738i	$0.151698\pi$
$614$	7.00000	0.282497
$615$	0	0
$616$	−18.0000	−0.725241
$617$	36.0000	1.44931	0.724653	−	0.689114i	$-0.242000\pi$
$617$	36.0000	1.44931	0.724653	+	0.689114i	$0.242000\pi$
$618$	0	0
$619$	−4.00000	−0.160774	−0.0803868	−	0.996764i	$-0.525616\pi$
$619$	−4.00000	−0.160774	−0.0803868	+	0.996764i	$0.525616\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−45.0000	−1.80289
$624$	0	0
$625$	0	0
$626$	14.0000	0.559553
$627$	0	0
$628$	14.0000	0.558661
$629$	−16.0000	−0.637962
$630$	0	0
$631$	16.0000	0.636950	0.318475	−	0.947931i	$-0.396829\pi$
$631$	16.0000	0.636950	0.318475	+	0.947931i	$0.396829\pi$
$632$	−18.0000	−0.716002
$633$	0	0
$634$	−34.0000	−1.35031
$635$	0	0
$636$	0	0
$637$	4.00000	0.158486
$638$	−2.00000	−0.0791808
$639$	0	0
$640$	0	0
$641$	33.0000	1.30342	0.651711	−	0.758468i	$-0.274052\pi$
$641$	33.0000	1.30342	0.651711	+	0.758468i	$0.274052\pi$
$642$	0	0
$643$	9.00000	0.354925	0.177463	−	0.984128i	$-0.443211\pi$
$643$	9.00000	0.354925	0.177463	+	0.984128i	$0.443211\pi$
$644$	9.00000	0.354650
$645$	0	0
$646$	−32.0000	−1.25902
$647$	17.0000	0.668339	0.334169	−	0.942513i	$-0.391544\pi$
$647$	17.0000	0.668339	0.334169	+	0.942513i	$0.391544\pi$
$648$	0	0
$649$	28.0000	1.09910
$650$	0	0
$651$	0	0
$652$	−4.00000	−0.156652
$653$	4.00000	0.156532	0.0782660	−	0.996933i	$-0.475062\pi$
$653$	4.00000	0.156532	0.0782660	+	0.996933i	$0.475062\pi$
$654$	0	0
$655$	0	0
$656$	−5.00000	−0.195217
$657$	0	0
$658$	21.0000	0.818665
$659$	−8.00000	−0.311636	−0.155818	−	0.987786i	$-0.549801\pi$
$659$	−8.00000	−0.311636	−0.155818	+	0.987786i	$0.549801\pi$
$660$	0	0
$661$	−14.0000	−0.544537	−0.272268	−	0.962221i	$-0.587774\pi$
$661$	−14.0000	−0.544537	−0.272268	+	0.962221i	$0.587774\pi$
$662$	6.00000	0.233197
$663$	0	0
$664$	−27.0000	−1.04780
$665$	0	0
$666$	0	0
$667$	3.00000	0.116160
$668$	−9.00000	−0.348220
$669$	0	0
$670$	0	0
$671$	−14.0000	−0.540464
$672$	0	0
$673$	−6.00000	−0.231283	−0.115642	−	0.993291i	$-0.536892\pi$
$673$	−6.00000	−0.231283	−0.115642	+	0.993291i	$0.536892\pi$
$674$	−8.00000	−0.308148
$675$	0	0
$676$	9.00000	0.346154
$677$	42.0000	1.61419	0.807096	−	0.590421i	$-0.201038\pi$
$677$	42.0000	1.61419	0.807096	+	0.590421i	$0.201038\pi$
$678$	0	0
$679$	−6.00000	−0.230259
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−12.0000	−0.459167	−0.229584	−	0.973289i	$-0.573736\pi$
$683$	−12.0000	−0.459167	−0.229584	+	0.973289i	$0.573736\pi$
$684$	0	0
$685$	0	0
$686$	15.0000	0.572703
$687$	0	0
$688$	−8.00000	−0.304997
$689$	4.00000	0.152388
$690$	0	0
$691$	−14.0000	−0.532585	−0.266293	−	0.963892i	$-0.585799\pi$
$691$	−14.0000	−0.532585	−0.266293	+	0.963892i	$0.585799\pi$
$692$	0	0
$693$	0	0
$694$	4.00000	0.151838
$695$	0	0
$696$	0	0
$697$	−20.0000	−0.757554
$698$	5.00000	0.189253
$699$	0	0
$700$	0	0
$701$	23.0000	0.868698	0.434349	−	0.900745i	$-0.356978\pi$
$701$	23.0000	0.868698	0.434349	+	0.900745i	$0.356978\pi$
$702$	0	0
$703$	−32.0000	−1.20690
$704$	−14.0000	−0.527645
$705$	0	0
$706$	24.0000	0.903252
$707$	−54.0000	−2.03088
$708$	0	0
$709$	−41.0000	−1.53979	−0.769894	−	0.638172i	$-0.779691\pi$
$709$	−41.0000	−1.53979	−0.769894	+	0.638172i	$0.779691\pi$
$710$	0	0
$711$	0	0
$712$	−45.0000	−1.68645
$713$	0	0
$714$	0	0
$715$	0	0
$716$	2.00000	0.0747435
$717$	0	0
$718$	−24.0000	−0.895672
$719$	6.00000	0.223762	0.111881	−	0.993722i	$-0.464312\pi$
$719$	6.00000	0.223762	0.111881	+	0.993722i	$0.464312\pi$
$720$	0	0
$721$	−24.0000	−0.893807
$722$	−45.0000	−1.67473
$723$	0	0
$724$	7.00000	0.260153
$725$	0	0
$726$	0	0
$727$	−23.0000	−0.853023	−0.426511	−	0.904482i	$-0.640258\pi$
$727$	−23.0000	−0.853023	−0.426511	+	0.904482i	$0.640258\pi$
$728$	18.0000	0.667124
$729$	0	0
$730$	0	0
$731$	−32.0000	−1.18356
$732$	0	0
$733$	34.0000	1.25582	0.627909	−	0.778287i	$-0.283911\pi$
$733$	34.0000	1.25582	0.627909	+	0.778287i	$0.283911\pi$
$734$	24.0000	0.885856
$735$	0	0
$736$	15.0000	0.552907
$737$	−6.00000	−0.221013
$738$	0	0
$739$	−2.00000	−0.0735712	−0.0367856	−	0.999323i	$-0.511712\pi$
$739$	−2.00000	−0.0735712	−0.0367856	+	0.999323i	$0.511712\pi$
$740$	0	0
$741$	0	0
$742$	−6.00000	−0.220267
$743$	29.0000	1.06391	0.531953	−	0.846774i	$-0.321458\pi$
$743$	29.0000	1.06391	0.531953	+	0.846774i	$0.321458\pi$
$744$	0	0
$745$	0	0
$746$	10.0000	0.366126
$747$	0	0
$748$	−8.00000	−0.292509
$749$	−9.00000	−0.328853
$750$	0	0
$751$	10.0000	0.364905	0.182453	−	0.983215i	$-0.441596\pi$
$751$	10.0000	0.364905	0.182453	+	0.983215i	$0.441596\pi$
$752$	7.00000	0.255264
$753$	0	0
$754$	2.00000	0.0728357
$755$	0	0
$756$	0	0
$757$	26.0000	0.944986	0.472493	−	0.881334i	$-0.343354\pi$
$757$	26.0000	0.944986	0.472493	+	0.881334i	$0.343354\pi$
$758$	26.0000	0.944363
$759$	0	0
$760$	0	0
$761$	15.0000	0.543750	0.271875	−	0.962333i	$-0.412356\pi$
$761$	15.0000	0.543750	0.271875	+	0.962333i	$0.412356\pi$
$762$	0	0
$763$	15.0000	0.543036
$764$	−8.00000	−0.289430
$765$	0	0
$766$	36.0000	1.30073
$767$	−28.0000	−1.01102
$768$	0	0
$769$	5.00000	0.180305	0.0901523	−	0.995928i	$-0.471265\pi$
$769$	5.00000	0.180305	0.0901523	+	0.995928i	$0.471265\pi$
$770$	0	0
$771$	0	0
$772$	−10.0000	−0.359908
$773$	24.0000	0.863220	0.431610	−	0.902060i	$-0.357946\pi$
$773$	24.0000	0.863220	0.431610	+	0.902060i	$0.357946\pi$
$774$	0	0
$775$	0	0
$776$	−6.00000	−0.215387
$777$	0	0
$778$	−33.0000	−1.18311
$779$	−40.0000	−1.43315
$780$	0	0
$781$	−4.00000	−0.143131
$782$	−12.0000	−0.429119
$783$	0	0
$784$	−2.00000	−0.0714286
$785$	0	0
$786$	0	0
$787$	−28.0000	−0.998092	−0.499046	−	0.866575i	$-0.666316\pi$
$787$	−28.0000	−0.998092	−0.499046	+	0.866575i	$0.666316\pi$
$788$	−12.0000	−0.427482
$789$	0	0
$790$	0	0
$791$	24.0000	0.853342
$792$	0	0
$793$	14.0000	0.497155
$794$	34.0000	1.20661
$795$	0	0
$796$	−4.00000	−0.141776
$797$	−26.0000	−0.920967	−0.460484	−	0.887668i	$-0.652324\pi$
$797$	−26.0000	−0.920967	−0.460484	+	0.887668i	$0.652324\pi$
$798$	0	0
$799$	28.0000	0.990569
$800$	0	0
$801$	0	0
$802$	18.0000	0.635602
$803$	8.00000	0.282314
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	−54.0000	−1.89971
$809$	26.0000	0.914111	0.457056	−	0.889438i	$-0.348904\pi$
$809$	26.0000	0.914111	0.457056	+	0.889438i	$0.348904\pi$
$810$	0	0
$811$	−42.0000	−1.47482	−0.737410	−	0.675446i	$-0.763951\pi$
$811$	−42.0000	−1.47482	−0.737410	+	0.675446i	$0.763951\pi$
$812$	3.00000	0.105279
$813$	0	0
$814$	8.00000	0.280400
$815$	0	0
$816$	0	0
$817$	−64.0000	−2.23908
$818$	−14.0000	−0.489499
$819$	0	0
$820$	0	0
$821$	15.0000	0.523504	0.261752	−	0.965135i	$-0.415700\pi$
$821$	15.0000	0.523504	0.261752	+	0.965135i	$0.415700\pi$
$822$	0	0
$823$	−53.0000	−1.84746	−0.923732	−	0.383040i	$-0.874877\pi$
$823$	−53.0000	−1.84746	−0.923732	+	0.383040i	$0.874877\pi$
$824$	−24.0000	−0.836080
$825$	0	0
$826$	42.0000	1.46137
$827$	−37.0000	−1.28662	−0.643308	−	0.765607i	$-0.722439\pi$
$827$	−37.0000	−1.28662	−0.643308	+	0.765607i	$0.722439\pi$
$828$	0	0
$829$	−3.00000	−0.104194	−0.0520972	−	0.998642i	$-0.516591\pi$
$829$	−3.00000	−0.104194	−0.0520972	+	0.998642i	$0.516591\pi$
$830$	0	0
$831$	0	0
$832$	14.0000	0.485363
$833$	−8.00000	−0.277184
$834$	0	0
$835$	0	0
$836$	−16.0000	−0.553372
$837$	0	0
$838$	−26.0000	−0.898155
$839$	−40.0000	−1.38095	−0.690477	−	0.723355i	$-0.742599\pi$
$839$	−40.0000	−1.38095	−0.690477	+	0.723355i	$0.742599\pi$
$840$	0	0
$841$	−28.0000	−0.965517
$842$	−34.0000	−1.17172
$843$	0	0
$844$	22.0000	0.757271
$845$	0	0
$846$	0	0
$847$	−21.0000	−0.721569
$848$	−2.00000	−0.0686803
$849$	0	0
$850$	0	0
$851$	−12.0000	−0.411355
$852$	0	0
$853$	54.0000	1.84892	0.924462	−	0.381273i	$-0.124514\pi$
$853$	54.0000	1.84892	0.924462	+	0.381273i	$0.124514\pi$
$854$	−21.0000	−0.718605
$855$	0	0
$856$	−9.00000	−0.307614
$857$	−10.0000	−0.341593	−0.170797	−	0.985306i	$-0.554634\pi$
$857$	−10.0000	−0.341593	−0.170797	+	0.985306i	$0.554634\pi$
$858$	0	0
$859$	22.0000	0.750630	0.375315	−	0.926897i	$-0.377534\pi$
$859$	22.0000	0.750630	0.375315	+	0.926897i	$0.377534\pi$
$860$	0	0
$861$	0	0
$862$	30.0000	1.02180
$863$	−17.0000	−0.578687	−0.289343	−	0.957225i	$-0.593437\pi$
$863$	−17.0000	−0.578687	−0.289343	+	0.957225i	$0.593437\pi$
$864$	0	0
$865$	0	0
$866$	28.0000	0.951479
$867$	0	0
$868$	0	0
$869$	12.0000	0.407072
$870$	0	0
$871$	6.00000	0.203302
$872$	15.0000	0.507964
$873$	0	0
$874$	−24.0000	−0.811812
$875$	0	0
$876$	0	0
$877$	18.0000	0.607817	0.303908	−	0.952701i	$-0.401708\pi$
$877$	18.0000	0.607817	0.303908	+	0.952701i	$0.401708\pi$
$878$	−28.0000	−0.944954
$879$	0	0
$880$	0	0
$881$	−35.0000	−1.17918	−0.589590	−	0.807703i	$-0.700711\pi$
$881$	−35.0000	−1.17918	−0.589590	+	0.807703i	$0.700711\pi$
$882$	0	0
$883$	−23.0000	−0.774012	−0.387006	−	0.922077i	$-0.626491\pi$
$883$	−23.0000	−0.774012	−0.387006	+	0.922077i	$0.626491\pi$
$884$	8.00000	0.269069
$885$	0	0
$886$	15.0000	0.503935
$887$	−36.0000	−1.20876	−0.604381	−	0.796696i	$-0.706579\pi$
$887$	−36.0000	−1.20876	−0.604381	+	0.796696i	$0.706579\pi$
$888$	0	0
$889$	15.0000	0.503084
$890$	0	0
$891$	0	0
$892$	−19.0000	−0.636167
$893$	56.0000	1.87397
$894$	0	0
$895$	0	0
$896$	9.00000	0.300669
$897$	0	0
$898$	26.0000	0.867631
$899$	0	0
$900$	0	0
$901$	−8.00000	−0.266519
$902$	10.0000	0.332964
$903$	0	0
$904$	24.0000	0.798228
$905$	0	0
$906$	0	0
$907$	51.0000	1.69343	0.846714	−	0.532049i	$-0.178578\pi$
$907$	51.0000	1.69343	0.846714	+	0.532049i	$0.178578\pi$
$908$	−4.00000	−0.132745
$909$	0	0
$910$	0	0
$911$	−50.0000	−1.65657	−0.828287	−	0.560304i	$-0.810684\pi$
$911$	−50.0000	−1.65657	−0.828287	+	0.560304i	$0.810684\pi$
$912$	0	0
$913$	18.0000	0.595713
$914$	−20.0000	−0.661541
$915$	0	0
$916$	−15.0000	−0.495614
$917$	−18.0000	−0.594412
$918$	0	0
$919$	10.0000	0.329870	0.164935	−	0.986304i	$-0.447259\pi$
$919$	10.0000	0.329870	0.164935	+	0.986304i	$0.447259\pi$
$920$	0	0
$921$	0	0
$922$	−9.00000	−0.296399
$923$	4.00000	0.131662
$924$	0	0
$925$	0	0
$926$	−36.0000	−1.18303
$927$	0	0
$928$	5.00000	0.164133
$929$	14.0000	0.459325	0.229663	−	0.973270i	$-0.426238\pi$
$929$	14.0000	0.459325	0.229663	+	0.973270i	$0.426238\pi$
$930$	0	0
$931$	−16.0000	−0.524379
$932$	24.0000	0.786146
$933$	0	0
$934$	20.0000	0.654420
$935$	0	0
$936$	0	0
$937$	0	0		−	1.00000i	$-0.5\pi$
$937$	0	0			1.00000i	$0.5\pi$
$938$	−9.00000	−0.293860
$939$	0	0
$940$	0	0
$941$	−7.00000	−0.228193	−0.114097	−	0.993470i	$-0.536397\pi$
$941$	−7.00000	−0.228193	−0.114097	+	0.993470i	$0.536397\pi$
$942$	0	0
$943$	−15.0000	−0.488467
$944$	14.0000	0.455661
$945$	0	0
$946$	16.0000	0.520205
$947$	−57.0000	−1.85225	−0.926126	−	0.377215i	$-0.876882\pi$
$947$	−57.0000	−1.85225	−0.926126	+	0.377215i	$0.876882\pi$
$948$	0	0
$949$	−8.00000	−0.259691
$950$	0	0
$951$	0	0
$952$	−36.0000	−1.16677
$953$	26.0000	0.842223	0.421111	−	0.907009i	$-0.361640\pi$
$953$	26.0000	0.842223	0.421111	+	0.907009i	$0.361640\pi$
$954$	0	0
$955$	0	0
$956$	8.00000	0.258738
$957$	0	0
$958$	18.0000	0.581554
$959$	−36.0000	−1.16250
$960$	0	0
$961$	−31.0000	−1.00000
$962$	−8.00000	−0.257930
$963$	0	0
$964$	11.0000	0.354286
$965$	0	0
$966$	0	0
$967$	−41.0000	−1.31847	−0.659236	−	0.751936i	$-0.729120\pi$
$967$	−41.0000	−1.31847	−0.659236	+	0.751936i	$0.729120\pi$
$968$	−21.0000	−0.674966
$969$	0	0
$970$	0	0
$971$	−36.0000	−1.15529	−0.577647	−	0.816286i	$-0.696029\pi$
$971$	−36.0000	−1.15529	−0.577647	+	0.816286i	$0.696029\pi$
$972$	0	0
$973$	−48.0000	−1.53881
$974$	−16.0000	−0.512673
$975$	0	0
$976$	−7.00000	−0.224065
$977$	38.0000	1.21573	0.607864	−	0.794041i	$-0.292027\pi$
$977$	38.0000	1.21573	0.607864	+	0.794041i	$0.292027\pi$
$978$	0	0
$979$	30.0000	0.958804
$980$	0	0
$981$	0	0
$982$	−20.0000	−0.638226
$983$	−3.00000	−0.0956851	−0.0478426	−	0.998855i	$-0.515235\pi$
$983$	−3.00000	−0.0956851	−0.0478426	+	0.998855i	$0.515235\pi$
$984$	0	0
$985$	0	0
$986$	−4.00000	−0.127386
$987$	0	0
$988$	16.0000	0.509028
$989$	−24.0000	−0.763156
$990$	0	0
$991$	26.0000	0.825917	0.412959	−	0.910750i	$-0.364495\pi$
$991$	26.0000	0.825917	0.412959	+	0.910750i	$0.364495\pi$
$992$	0	0
$993$	0	0
$994$	−6.00000	−0.190308
$995$	0	0
$996$	0	0
$997$	18.0000	0.570066	0.285033	−	0.958518i	$-0.407995\pi$
$997$	18.0000	0.570066	0.285033	+	0.958518i	$0.407995\pi$
$998$	32.0000	1.01294
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2025.2.a.b.1.1		1
3.2	odd	2		2025.2.a.e.1.1		1
5.2	odd	4		2025.2.b.c.649.1		2
5.3	odd	4		2025.2.b.c.649.2		2
5.4	even	2		405.2.a.e.1.1		1
9.2	odd	6		675.2.e.a.226.1		2
9.4	even	3		225.2.e.a.151.1		2
9.5	odd	6		675.2.e.a.451.1		2
9.7	even	3		225.2.e.a.76.1		2
15.2	even	4		2025.2.b.d.649.2		2
15.8	even	4		2025.2.b.d.649.1		2
15.14	odd	2		405.2.a.b.1.1		1
20.19	odd	2		6480.2.a.k.1.1		1
45.2	even	12		675.2.k.a.199.2		4
45.4	even	6		45.2.e.a.16.1	✓	2
45.7	odd	12		225.2.k.a.49.1		4
45.13	odd	12		225.2.k.a.124.1		4
45.14	odd	6		135.2.e.a.46.1		2
45.22	odd	12		225.2.k.a.124.2		4
45.23	even	12		675.2.k.a.424.2		4
45.29	odd	6		135.2.e.a.91.1		2
45.32	even	12		675.2.k.a.424.1		4
45.34	even	6		45.2.e.a.31.1	yes	2
45.38	even	12		675.2.k.a.199.1		4
45.43	odd	12		225.2.k.a.49.2		4
60.59	even	2		6480.2.a.x.1.1		1
180.59	even	6		2160.2.q.a.721.1		2
180.79	odd	6		720.2.q.d.481.1		2
180.119	even	6		2160.2.q.a.1441.1		2
180.139	odd	6		720.2.q.d.241.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
45.2.e.a.16.1	✓	2	45.4	even	6
45.2.e.a.31.1	yes	2	45.34	even	6
135.2.e.a.46.1		2	45.14	odd	6
135.2.e.a.91.1		2	45.29	odd	6
225.2.e.a.76.1		2	9.7	even	3
225.2.e.a.151.1		2	9.4	even	3
225.2.k.a.49.1		4	45.7	odd	12
225.2.k.a.49.2		4	45.43	odd	12
225.2.k.a.124.1		4	45.13	odd	12
225.2.k.a.124.2		4	45.22	odd	12
405.2.a.b.1.1		1	15.14	odd	2
405.2.a.e.1.1		1	5.4	even	2
675.2.e.a.226.1		2	9.2	odd	6
675.2.e.a.451.1		2	9.5	odd	6
675.2.k.a.199.1		4	45.38	even	12
675.2.k.a.199.2		4	45.2	even	12
675.2.k.a.424.1		4	45.32	even	12
675.2.k.a.424.2		4	45.23	even	12
720.2.q.d.241.1		2	180.139	odd	6
720.2.q.d.481.1		2	180.79	odd	6
2025.2.a.b.1.1		1	1.1	even	1	trivial
2025.2.a.e.1.1		1	3.2	odd	2
2025.2.b.c.649.1		2	5.2	odd	4
2025.2.b.c.649.2		2	5.3	odd	4
2025.2.b.d.649.1		2	15.8	even	4
2025.2.b.d.649.2		2	15.2	even	4
2160.2.q.a.721.1		2	180.59	even	6
2160.2.q.a.1441.1		2	180.119	even	6
6480.2.a.k.1.1		1	20.19	odd	2
6480.2.a.x.1.1		1	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)

Introduction

L-functions

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