LMFDB - Embedded newform 2025.2.a.f.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 405)
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)

$f(q)$

$=$

$q+2.00000 q^{2} +2.00000 q^{4} +O(q^{10})$

$q+2.00000 q^{2} +2.00000 q^{4} -5.00000 q^{11} -4.00000 q^{13} -4.00000 q^{16} -4.00000 q^{17} -5.00000 q^{19} -10.0000 q^{22} +6.00000 q^{23} -8.00000 q^{26} +5.00000 q^{29} -9.00000 q^{31} -8.00000 q^{32} -8.00000 q^{34} +10.0000 q^{37} -10.0000 q^{38} -7.00000 q^{41} +2.00000 q^{43} -10.0000 q^{44} +12.0000 q^{46} +2.00000 q^{47} -7.00000 q^{49} -8.00000 q^{52} +8.00000 q^{53} +10.0000 q^{58} +1.00000 q^{59} -2.00000 q^{61} -18.0000 q^{62} -8.00000 q^{64} -6.00000 q^{67} -8.00000 q^{68} -1.00000 q^{71} +8.00000 q^{73} +20.0000 q^{74} -10.0000 q^{76} +12.0000 q^{79} -14.0000 q^{82} +6.00000 q^{83} +4.00000 q^{86} +9.00000 q^{89} +12.0000 q^{92} +4.00000 q^{94} -14.0000 q^{97} -14.0000 q^{98} +O(q^{100})$

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 $

$2$	2.00000	1.41421	0.707107	−	0.707107i	$-0.250000\pi$
$2$	2.00000	1.41421	0.707107	+	0.707107i	$0.250000\pi$
$3$	0	0
$3$	0	0
$4$	2.00000	1.00000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	0	0		−	1.00000i	$-0.5\pi$
$7$	0	0			1.00000i	$0.5\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−5.00000	−1.50756	−0.753778	−	0.657129i	$-0.771771\pi$
$11$	−5.00000	−1.50756	−0.753778	+	0.657129i	$0.771771\pi$
$12$	0	0
$13$	−4.00000	−1.10940	−0.554700	−	0.832050i	$-0.687167\pi$
$13$	−4.00000	−1.10940	−0.554700	+	0.832050i	$0.687167\pi$
$14$	0	0
$15$	0	0
$16$	−4.00000	−1.00000
$17$	−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
$19$	−5.00000	−1.14708	−0.573539	−	0.819178i	$-0.694430\pi$
$19$	−5.00000	−1.14708	−0.573539	+	0.819178i	$0.694430\pi$
$20$	0	0
$21$	0	0
$22$	−10.0000	−2.13201
$23$	6.00000	1.25109	0.625543	−	0.780189i	$-0.284877\pi$
$23$	6.00000	1.25109	0.625543	+	0.780189i	$0.284877\pi$
$24$	0	0
$25$	0	0
$26$	−8.00000	−1.56893
$27$	0	0
$28$	0	0
$29$	5.00000	0.928477	0.464238	−	0.885710i	$-0.346328\pi$
$29$	5.00000	0.928477	0.464238	+	0.885710i	$0.346328\pi$
$30$	0	0
$31$	−9.00000	−1.61645	−0.808224	−	0.588875i	$-0.799571\pi$
$31$	−9.00000	−1.61645	−0.808224	+	0.588875i	$0.799571\pi$
$32$	−8.00000	−1.41421
$33$	0	0
$34$	−8.00000	−1.37199
$35$	0	0
$36$	0	0
$37$	10.0000	1.64399	0.821995	−	0.569495i	$-0.192861\pi$
$37$	10.0000	1.64399	0.821995	+	0.569495i	$0.192861\pi$
$38$	−10.0000	−1.62221
$39$	0	0
$40$	0	0
$41$	−7.00000	−1.09322	−0.546608	−	0.837389i	$-0.684081\pi$
$41$	−7.00000	−1.09322	−0.546608	+	0.837389i	$0.684081\pi$
$42$	0	0
$43$	2.00000	0.304997	0.152499	−	0.988304i	$-0.451268\pi$
$43$	2.00000	0.304997	0.152499	+	0.988304i	$0.451268\pi$
$44$	−10.0000	−1.50756
$45$	0	0
$46$	12.0000	1.76930
$47$	2.00000	0.291730	0.145865	−	0.989305i	$-0.453403\pi$
$47$	2.00000	0.291730	0.145865	+	0.989305i	$0.453403\pi$
$48$	0	0
$49$	−7.00000	−1.00000
$50$	0	0
$51$	0	0
$52$	−8.00000	−1.10940
$53$	8.00000	1.09888	0.549442	−	0.835532i	$-0.314840\pi$
$53$	8.00000	1.09888	0.549442	+	0.835532i	$0.314840\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	10.0000	1.31306
$59$	1.00000	0.130189	0.0650945	−	0.997879i	$-0.479265\pi$
$59$	1.00000	0.130189	0.0650945	+	0.997879i	$0.479265\pi$
$60$	0	0
$61$	−2.00000	−0.256074	−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000	−0.256074	−0.128037	+	0.991769i	$0.540868\pi$
$62$	−18.0000	−2.28600
$63$	0	0
$64$	−8.00000	−1.00000
$65$	0	0
$66$	0	0
$67$	−6.00000	−0.733017	−0.366508	−	0.930415i	$-0.619447\pi$
$67$	−6.00000	−0.733017	−0.366508	+	0.930415i	$0.619447\pi$
$68$	−8.00000	−0.970143
$69$	0	0
$70$	0	0
$71$	−1.00000	−0.118678	−0.0593391	−	0.998238i	$-0.518899\pi$
$71$	−1.00000	−0.118678	−0.0593391	+	0.998238i	$0.518899\pi$
$72$	0	0
$73$	8.00000	0.936329	0.468165	−	0.883641i	$-0.344915\pi$
$73$	8.00000	0.936329	0.468165	+	0.883641i	$0.344915\pi$
$74$	20.0000	2.32495
$75$	0	0
$76$	−10.0000	−1.14708
$77$	0	0
$78$	0	0
$79$	12.0000	1.35011	0.675053	−	0.737769i	$-0.264121\pi$
$79$	12.0000	1.35011	0.675053	+	0.737769i	$0.264121\pi$
$80$	0	0
$81$	0	0
$82$	−14.0000	−1.54604
$83$	6.00000	0.658586	0.329293	−	0.944228i	$-0.393190\pi$
$83$	6.00000	0.658586	0.329293	+	0.944228i	$0.393190\pi$
$84$	0	0
$85$	0	0
$86$	4.00000	0.431331
$87$	0	0
$88$	0	0
$89$	9.00000	0.953998	0.476999	−	0.878904i	$-0.341725\pi$
$89$	9.00000	0.953998	0.476999	+	0.878904i	$0.341725\pi$
$90$	0	0
$91$	0	0
$92$	12.0000	1.25109
$93$	0	0
$94$	4.00000	0.412568
$95$	0	0
$96$	0	0
$97$	−14.0000	−1.42148	−0.710742	−	0.703452i	$-0.751641\pi$
$97$	−14.0000	−1.42148	−0.710742	+	0.703452i	$0.751641\pi$
$98$	−14.0000	−1.41421
$99$	0	0
$100$	0	0
$101$	3.00000	0.298511	0.149256	−	0.988799i	$-0.452312\pi$
$101$	3.00000	0.298511	0.149256	+	0.988799i	$0.452312\pi$
$102$	0	0
$103$	−2.00000	−0.197066	−0.0985329	−	0.995134i	$-0.531415\pi$
$103$	−2.00000	−0.197066	−0.0985329	+	0.995134i	$0.531415\pi$
$104$	0	0
$105$	0	0
$106$	16.0000	1.55406
$107$	−6.00000	−0.580042	−0.290021	−	0.957020i	$-0.593662\pi$
$107$	−6.00000	−0.580042	−0.290021	+	0.957020i	$0.593662\pi$
$108$	0	0
$109$	−1.00000	−0.0957826	−0.0478913	−	0.998853i	$-0.515250\pi$
$109$	−1.00000	−0.0957826	−0.0478913	+	0.998853i	$0.515250\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−16.0000	−1.50515	−0.752577	−	0.658505i	$-0.771189\pi$
$113$	−16.0000	−1.50515	−0.752577	+	0.658505i	$0.771189\pi$
$114$	0	0
$115$	0	0
$116$	10.0000	0.928477
$117$	0	0
$118$	2.00000	0.184115
$119$	0	0
$120$	0	0
$121$	14.0000	1.27273
$122$	−4.00000	−0.362143
$123$	0	0
$124$	−18.0000	−1.61645
$125$	0	0
$126$	0	0
$127$	−16.0000	−1.41977	−0.709885	−	0.704317i	$-0.751253\pi$
$127$	−16.0000	−1.41977	−0.709885	+	0.704317i	$0.751253\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−15.0000	−1.31056	−0.655278	−	0.755388i	$-0.727449\pi$
$131$	−15.0000	−1.31056	−0.655278	+	0.755388i	$0.727449\pi$
$132$	0	0
$133$	0	0
$134$	−12.0000	−1.03664
$135$	0	0
$136$	0	0
$137$	12.0000	1.02523	0.512615	−	0.858619i	$-0.328677\pi$
$137$	12.0000	1.02523	0.512615	+	0.858619i	$0.328677\pi$
$138$	0	0
$139$	−19.0000	−1.61156	−0.805779	−	0.592216i	$-0.798253\pi$
$139$	−19.0000	−1.61156	−0.805779	+	0.592216i	$0.798253\pi$
$140$	0	0
$141$	0	0
$142$	−2.00000	−0.167836
$143$	20.0000	1.67248
$144$	0	0
$145$	0	0
$146$	16.0000	1.32417
$147$	0	0
$148$	20.0000	1.64399
$149$	2.00000	0.163846	0.0819232	−	0.996639i	$-0.473894\pi$
$149$	2.00000	0.163846	0.0819232	+	0.996639i	$0.473894\pi$
$150$	0	0
$151$	−5.00000	−0.406894	−0.203447	−	0.979086i	$-0.565214\pi$
$151$	−5.00000	−0.406894	−0.203447	+	0.979086i	$0.565214\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−2.00000	−0.159617	−0.0798087	−	0.996810i	$-0.525431\pi$
$157$	−2.00000	−0.159617	−0.0798087	+	0.996810i	$0.525431\pi$
$158$	24.0000	1.90934
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−8.00000	−0.626608	−0.313304	−	0.949653i	$-0.601436\pi$
$163$	−8.00000	−0.626608	−0.313304	+	0.949653i	$0.601436\pi$
$164$	−14.0000	−1.09322
$165$	0	0
$166$	12.0000	0.931381
$167$	12.0000	0.928588	0.464294	−	0.885681i	$-0.346308\pi$
$167$	12.0000	0.928588	0.464294	+	0.885681i	$0.346308\pi$
$168$	0	0
$169$	3.00000	0.230769
$170$	0	0
$171$	0	0
$172$	4.00000	0.304997
$173$	−6.00000	−0.456172	−0.228086	−	0.973641i	$-0.573247\pi$
$173$	−6.00000	−0.456172	−0.228086	+	0.973641i	$0.573247\pi$
$174$	0	0
$175$	0	0
$176$	20.0000	1.50756
$177$	0	0
$178$	18.0000	1.34916
$179$	−23.0000	−1.71910	−0.859550	−	0.511051i	$-0.829256\pi$
$179$	−23.0000	−1.71910	−0.859550	+	0.511051i	$0.829256\pi$
$180$	0	0
$181$	−25.0000	−1.85824	−0.929118	−	0.369784i	$-0.879432\pi$
$181$	−25.0000	−1.85824	−0.929118	+	0.369784i	$0.879432\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	20.0000	1.46254
$188$	4.00000	0.291730
$189$	0	0
$190$	0	0
$191$	−1.00000	−0.0723575	−0.0361787	−	0.999345i	$-0.511519\pi$
$191$	−1.00000	−0.0723575	−0.0361787	+	0.999345i	$0.511519\pi$
$192$	0	0
$193$	−26.0000	−1.87152	−0.935760	−	0.352636i	$-0.885285\pi$
$193$	−26.0000	−1.87152	−0.935760	+	0.352636i	$0.885285\pi$
$194$	−28.0000	−2.01028
$195$	0	0
$196$	−14.0000	−1.00000
$197$	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
$199$	16.0000	1.13421	0.567105	−	0.823646i	$-0.308063\pi$
$199$	16.0000	1.13421	0.567105	+	0.823646i	$0.308063\pi$
$200$	0	0
$201$	0	0
$202$	6.00000	0.422159
$203$	0	0
$204$	0	0
$205$	0	0
$206$	−4.00000	−0.278693
$207$	0	0
$208$	16.0000	1.10940
$209$	25.0000	1.72929
$210$	0	0
$211$	11.0000	0.757271	0.378636	−	0.925546i	$-0.376393\pi$
$211$	11.0000	0.757271	0.378636	+	0.925546i	$0.376393\pi$
$212$	16.0000	1.09888
$213$	0	0
$214$	−12.0000	−0.820303
$215$	0	0
$216$	0	0
$217$	0	0
$218$	−2.00000	−0.135457
$219$	0	0
$220$	0	0
$221$	16.0000	1.07628
$222$	0	0
$223$	−8.00000	−0.535720	−0.267860	−	0.963458i	$-0.586316\pi$
$223$	−8.00000	−0.535720	−0.267860	+	0.963458i	$0.586316\pi$
$224$	0	0
$225$	0	0
$226$	−32.0000	−2.12861
$227$	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
$229$	6.00000	0.396491	0.198246	−	0.980152i	$-0.436476\pi$
$229$	6.00000	0.396491	0.198246	+	0.980152i	$0.436476\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	6.00000	0.393073	0.196537	−	0.980497i	$-0.437031\pi$
$233$	6.00000	0.393073	0.196537	+	0.980497i	$0.437031\pi$
$234$	0	0
$235$	0	0
$236$	2.00000	0.130189
$237$	0	0
$238$	0	0
$239$	16.0000	1.03495	0.517477	−	0.855697i	$-0.326871\pi$
$239$	16.0000	1.03495	0.517477	+	0.855697i	$0.326871\pi$
$240$	0	0
$241$	−11.0000	−0.708572	−0.354286	−	0.935137i	$-0.615276\pi$
$241$	−11.0000	−0.708572	−0.354286	+	0.935137i	$0.615276\pi$
$242$	28.0000	1.79991
$243$	0	0
$244$	−4.00000	−0.256074
$245$	0	0
$246$	0	0
$247$	20.0000	1.27257
$248$	0	0
$249$	0	0
$250$	0	0
$251$	12.0000	0.757433	0.378717	−	0.925513i	$-0.376365\pi$
$251$	12.0000	0.757433	0.378717	+	0.925513i	$0.376365\pi$
$252$	0	0
$253$	−30.0000	−1.88608
$254$	−32.0000	−2.00786
$255$	0	0
$256$	16.0000	1.00000
$257$	−6.00000	−0.374270	−0.187135	−	0.982334i	$-0.559920\pi$
$257$	−6.00000	−0.374270	−0.187135	+	0.982334i	$0.559920\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	−30.0000	−1.85341
$263$	10.0000	0.616626	0.308313	−	0.951285i	$-0.400236\pi$
$263$	10.0000	0.616626	0.308313	+	0.951285i	$0.400236\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	−12.0000	−0.733017
$269$	31.0000	1.89010	0.945052	−	0.326921i	$-0.106011\pi$
$269$	31.0000	1.89010	0.945052	+	0.326921i	$0.106011\pi$
$270$	0	0
$271$	−8.00000	−0.485965	−0.242983	−	0.970031i	$-0.578126\pi$
$271$	−8.00000	−0.485965	−0.242983	+	0.970031i	$0.578126\pi$
$272$	16.0000	0.970143
$273$	0	0
$274$	24.0000	1.44989
$275$	0	0
$276$	0	0
$277$	18.0000	1.08152	0.540758	−	0.841178i	$-0.318138\pi$
$277$	18.0000	1.08152	0.540758	+	0.841178i	$0.318138\pi$
$278$	−38.0000	−2.27909
$279$	0	0
$280$	0	0
$281$	−6.00000	−0.357930	−0.178965	−	0.983855i	$-0.557275\pi$
$281$	−6.00000	−0.357930	−0.178965	+	0.983855i	$0.557275\pi$
$282$	0	0
$283$	−6.00000	−0.356663	−0.178331	−	0.983970i	$-0.557070\pi$
$283$	−6.00000	−0.356663	−0.178331	+	0.983970i	$0.557070\pi$
$284$	−2.00000	−0.118678
$285$	0	0
$286$	40.0000	2.36525
$287$	0	0
$288$	0	0
$289$	−1.00000	−0.0588235
$290$	0	0
$291$	0	0
$292$	16.0000	0.936329
$293$	18.0000	1.05157	0.525786	−	0.850617i	$-0.323771\pi$
$293$	18.0000	1.05157	0.525786	+	0.850617i	$0.323771\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	4.00000	0.231714
$299$	−24.0000	−1.38796
$300$	0	0
$301$	0	0
$302$	−10.0000	−0.575435
$303$	0	0
$304$	20.0000	1.14708
$305$	0	0
$306$	0	0
$307$	−10.0000	−0.570730	−0.285365	−	0.958419i	$-0.592115\pi$
$307$	−10.0000	−0.570730	−0.285365	+	0.958419i	$0.592115\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	9.00000	0.510343	0.255172	−	0.966896i	$-0.417868\pi$
$311$	9.00000	0.510343	0.255172	+	0.966896i	$0.417868\pi$
$312$	0	0
$313$	4.00000	0.226093	0.113047	−	0.993590i	$-0.463939\pi$
$313$	4.00000	0.226093	0.113047	+	0.993590i	$0.463939\pi$
$314$	−4.00000	−0.225733
$315$	0	0
$316$	24.0000	1.35011
$317$	−2.00000	−0.112331	−0.0561656	−	0.998421i	$-0.517887\pi$
$317$	−2.00000	−0.112331	−0.0561656	+	0.998421i	$0.517887\pi$
$318$	0	0
$319$	−25.0000	−1.39973
$320$	0	0
$321$	0	0
$322$	0	0
$323$	20.0000	1.11283
$324$	0	0
$325$	0	0
$326$	−16.0000	−0.886158
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−21.0000	−1.15426	−0.577132	−	0.816651i	$-0.695828\pi$
$331$	−21.0000	−1.15426	−0.577132	+	0.816651i	$0.695828\pi$
$332$	12.0000	0.658586
$333$	0	0
$334$	24.0000	1.31322
$335$	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	$0.430081\pi$
$338$	6.00000	0.326357
$339$	0	0
$340$	0	0
$341$	45.0000	2.43689
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	−12.0000	−0.645124
$347$	20.0000	1.07366	0.536828	−	0.843692i	$-0.319622\pi$
$347$	20.0000	1.07366	0.536828	+	0.843692i	$0.319622\pi$
$348$	0	0
$349$	13.0000	0.695874	0.347937	−	0.937518i	$-0.386882\pi$
$349$	13.0000	0.695874	0.347937	+	0.937518i	$0.386882\pi$
$350$	0	0
$351$	0	0
$352$	40.0000	2.13201
$353$	−18.0000	−0.958043	−0.479022	−	0.877803i	$-0.659008\pi$
$353$	−18.0000	−0.958043	−0.479022	+	0.877803i	$0.659008\pi$
$354$	0	0
$355$	0	0
$356$	18.0000	0.953998
$357$	0	0
$358$	−46.0000	−2.43118
$359$	27.0000	1.42501	0.712503	−	0.701669i	$-0.247562\pi$
$359$	27.0000	1.42501	0.712503	+	0.701669i	$0.247562\pi$
$360$	0	0
$361$	6.00000	0.315789
$362$	−50.0000	−2.62794
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−18.0000	−0.939592	−0.469796	−	0.882775i	$-0.655673\pi$
$367$	−18.0000	−0.939592	−0.469796	+	0.882775i	$0.655673\pi$
$368$	−24.0000	−1.25109
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−16.0000	−0.828449	−0.414224	−	0.910175i	$-0.635947\pi$
$373$	−16.0000	−0.828449	−0.414224	+	0.910175i	$0.635947\pi$
$374$	40.0000	2.06835
$375$	0	0
$376$	0	0
$377$	−20.0000	−1.03005
$378$	0	0
$379$	4.00000	0.205466	0.102733	−	0.994709i	$-0.467241\pi$
$379$	4.00000	0.205466	0.102733	+	0.994709i	$0.467241\pi$
$380$	0	0
$381$	0	0
$382$	−2.00000	−0.102329
$383$	−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
$386$	−52.0000	−2.64673
$387$	0	0
$388$	−28.0000	−1.42148
$389$	−6.00000	−0.304212	−0.152106	−	0.988364i	$-0.548606\pi$
$389$	−6.00000	−0.304212	−0.152106	+	0.988364i	$0.548606\pi$
$390$	0	0
$391$	−24.0000	−1.21373
$392$	0	0
$393$	0	0
$394$	24.0000	1.20910
$395$	0	0
$396$	0	0
$397$	38.0000	1.90717	0.953583	−	0.301131i	$-0.0973643\pi$
$397$	38.0000	1.90717	0.953583	+	0.301131i	$0.0973643\pi$
$398$	32.0000	1.60402
$399$	0	0
$400$	0	0
$401$	−30.0000	−1.49813	−0.749064	−	0.662497i	$-0.769497\pi$
$401$	−30.0000	−1.49813	−0.749064	+	0.662497i	$0.769497\pi$
$402$	0	0
$403$	36.0000	1.79329
$404$	6.00000	0.298511
$405$	0	0
$406$	0	0
$407$	−50.0000	−2.47841
$408$	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
$412$	−4.00000	−0.197066
$413$	0	0
$414$	0	0
$415$	0	0
$416$	32.0000	1.56893
$417$	0	0
$418$	50.0000	2.44558
$419$	−16.0000	−0.781651	−0.390826	−	0.920465i	$-0.627810\pi$
$419$	−16.0000	−0.781651	−0.390826	+	0.920465i	$0.627810\pi$
$420$	0	0
$421$	13.0000	0.633581	0.316791	−	0.948495i	$-0.397395\pi$
$421$	13.0000	0.633581	0.316791	+	0.948495i	$0.397395\pi$
$422$	22.0000	1.07094
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	0	0
$428$	−12.0000	−0.580042
$429$	0	0
$430$	0	0
$431$	−3.00000	−0.144505	−0.0722525	−	0.997386i	$-0.523019\pi$
$431$	−3.00000	−0.144505	−0.0722525	+	0.997386i	$0.523019\pi$
$432$	0	0
$433$	26.0000	1.24948	0.624740	−	0.780833i	$-0.285205\pi$
$433$	26.0000	1.24948	0.624740	+	0.780833i	$0.285205\pi$
$434$	0	0
$435$	0	0
$436$	−2.00000	−0.0957826
$437$	−30.0000	−1.43509
$438$	0	0
$439$	−29.0000	−1.38409	−0.692047	−	0.721852i	$-0.743291\pi$
$439$	−29.0000	−1.38409	−0.692047	+	0.721852i	$0.743291\pi$
$440$	0	0
$441$	0	0
$442$	32.0000	1.52208
$443$	6.00000	0.285069	0.142534	−	0.989790i	$-0.454475\pi$
$443$	6.00000	0.285069	0.142534	+	0.989790i	$0.454475\pi$
$444$	0	0
$445$	0	0
$446$	−16.0000	−0.757622
$447$	0	0
$448$	0	0
$449$	−17.0000	−0.802280	−0.401140	−	0.916017i	$-0.631386\pi$
$449$	−17.0000	−0.802280	−0.401140	+	0.916017i	$0.631386\pi$
$450$	0	0
$451$	35.0000	1.64809
$452$	−32.0000	−1.50515
$453$	0	0
$454$	8.00000	0.375459
$455$	0	0
$456$	0	0
$457$	38.0000	1.77757	0.888783	−	0.458329i	$-0.151552\pi$
$457$	38.0000	1.77757	0.888783	+	0.458329i	$0.151552\pi$
$458$	12.0000	0.560723
$459$	0	0
$460$	0	0
$461$	15.0000	0.698620	0.349310	−	0.937007i	$-0.386416\pi$
$461$	15.0000	0.698620	0.349310	+	0.937007i	$0.386416\pi$
$462$	0	0
$463$	6.00000	0.278844	0.139422	−	0.990233i	$-0.455476\pi$
$463$	6.00000	0.278844	0.139422	+	0.990233i	$0.455476\pi$
$464$	−20.0000	−0.928477
$465$	0	0
$466$	12.0000	0.555889
$467$	4.00000	0.185098	0.0925490	−	0.995708i	$-0.470499\pi$
$467$	4.00000	0.185098	0.0925490	+	0.995708i	$0.470499\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−10.0000	−0.459800
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	32.0000	1.46365
$479$	15.0000	0.685367	0.342684	−	0.939451i	$-0.388664\pi$
$479$	15.0000	0.685367	0.342684	+	0.939451i	$0.388664\pi$
$480$	0	0
$481$	−40.0000	−1.82384
$482$	−22.0000	−1.00207
$483$	0	0
$484$	28.0000	1.27273
$485$	0	0
$486$	0	0
$487$	−8.00000	−0.362515	−0.181257	−	0.983436i	$-0.558017\pi$
$487$	−8.00000	−0.362515	−0.181257	+	0.983436i	$0.558017\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−43.0000	−1.94056	−0.970281	−	0.241979i	$-0.922203\pi$
$491$	−43.0000	−1.94056	−0.970281	+	0.241979i	$0.922203\pi$
$492$	0	0
$493$	−20.0000	−0.900755
$494$	40.0000	1.79969
$495$	0	0
$496$	36.0000	1.61645
$497$	0	0
$498$	0	0
$499$	7.00000	0.313363	0.156682	−	0.987649i	$-0.449920\pi$
$499$	7.00000	0.313363	0.156682	+	0.987649i	$0.449920\pi$
$500$	0	0
$501$	0	0
$502$	24.0000	1.07117
$503$	−20.0000	−0.891756	−0.445878	−	0.895094i	$-0.647108\pi$
$503$	−20.0000	−0.891756	−0.445878	+	0.895094i	$0.647108\pi$
$504$	0	0
$505$	0	0
$506$	−60.0000	−2.66733
$507$	0	0
$508$	−32.0000	−1.41977
$509$	−2.00000	−0.0886484	−0.0443242	−	0.999017i	$-0.514113\pi$
$509$	−2.00000	−0.0886484	−0.0443242	+	0.999017i	$0.514113\pi$
$510$	0	0
$511$	0	0
$512$	32.0000	1.41421
$513$	0	0
$514$	−12.0000	−0.529297
$515$	0	0
$516$	0	0
$517$	−10.0000	−0.439799
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−10.0000	−0.438108	−0.219054	−	0.975713i	$-0.570297\pi$
$521$	−10.0000	−0.438108	−0.219054	+	0.975713i	$0.570297\pi$
$522$	0	0
$523$	−16.0000	−0.699631	−0.349816	−	0.936819i	$-0.613756\pi$
$523$	−16.0000	−0.699631	−0.349816	+	0.936819i	$0.613756\pi$
$524$	−30.0000	−1.31056
$525$	0	0
$526$	20.0000	0.872041
$527$	36.0000	1.56818
$528$	0	0
$529$	13.0000	0.565217
$530$	0	0
$531$	0	0
$532$	0	0
$533$	28.0000	1.21281
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	62.0000	2.67301
$539$	35.0000	1.50756
$540$	0	0
$541$	3.00000	0.128980	0.0644900	−	0.997918i	$-0.479458\pi$
$541$	3.00000	0.128980	0.0644900	+	0.997918i	$0.479458\pi$
$542$	−16.0000	−0.687259
$543$	0	0
$544$	32.0000	1.37199
$545$	0	0
$546$	0	0
$547$	−28.0000	−1.19719	−0.598597	−	0.801050i	$-0.704275\pi$
$547$	−28.0000	−1.19719	−0.598597	+	0.801050i	$0.704275\pi$
$548$	24.0000	1.02523
$549$	0	0
$550$	0	0
$551$	−25.0000	−1.06504
$552$	0	0
$553$	0	0
$554$	36.0000	1.52949
$555$	0	0
$556$	−38.0000	−1.61156
$557$	−24.0000	−1.01691	−0.508456	−	0.861088i	$-0.669784\pi$
$557$	−24.0000	−1.01691	−0.508456	+	0.861088i	$0.669784\pi$
$558$	0	0
$559$	−8.00000	−0.338364
$560$	0	0
$561$	0	0
$562$	−12.0000	−0.506189
$563$	−18.0000	−0.758610	−0.379305	−	0.925272i	$-0.623837\pi$
$563$	−18.0000	−0.758610	−0.379305	+	0.925272i	$0.623837\pi$
$564$	0	0
$565$	0	0
$566$	−12.0000	−0.504398
$567$	0	0
$568$	0	0
$569$	3.00000	0.125767	0.0628833	−	0.998021i	$-0.479970\pi$
$569$	3.00000	0.125767	0.0628833	+	0.998021i	$0.479970\pi$
$570$	0	0
$571$	5.00000	0.209243	0.104622	−	0.994512i	$-0.466637\pi$
$571$	5.00000	0.209243	0.104622	+	0.994512i	$0.466637\pi$
$572$	40.0000	1.67248
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	16.0000	0.666089	0.333044	−	0.942911i	$-0.391924\pi$
$577$	16.0000	0.666089	0.333044	+	0.942911i	$0.391924\pi$
$578$	−2.00000	−0.0831890
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−40.0000	−1.65663
$584$	0	0
$585$	0	0
$586$	36.0000	1.48715
$587$	18.0000	0.742940	0.371470	−	0.928445i	$-0.378854\pi$
$587$	18.0000	0.742940	0.371470	+	0.928445i	$0.378854\pi$
$588$	0	0
$589$	45.0000	1.85419
$590$	0	0
$591$	0	0
$592$	−40.0000	−1.64399
$593$	−14.0000	−0.574911	−0.287456	−	0.957794i	$-0.592809\pi$
$593$	−14.0000	−0.574911	−0.287456	+	0.957794i	$0.592809\pi$
$594$	0	0
$595$	0	0
$596$	4.00000	0.163846
$597$	0	0
$598$	−48.0000	−1.96287
$599$	17.0000	0.694601	0.347301	−	0.937754i	$-0.387098\pi$
$599$	17.0000	0.694601	0.347301	+	0.937754i	$0.387098\pi$
$600$	0	0
$601$	−19.0000	−0.775026	−0.387513	−	0.921864i	$-0.626666\pi$
$601$	−19.0000	−0.775026	−0.387513	+	0.921864i	$0.626666\pi$
$602$	0	0
$603$	0	0
$604$	−10.0000	−0.406894
$605$	0	0
$606$	0	0
$607$	−34.0000	−1.38002	−0.690009	−	0.723801i	$-0.742393\pi$
$607$	−34.0000	−1.38002	−0.690009	+	0.723801i	$0.742393\pi$
$608$	40.0000	1.62221
$609$	0	0
$610$	0	0
$611$	−8.00000	−0.323645
$612$	0	0
$613$	−4.00000	−0.161558	−0.0807792	−	0.996732i	$-0.525741\pi$
$613$	−4.00000	−0.161558	−0.0807792	+	0.996732i	$0.525741\pi$
$614$	−20.0000	−0.807134
$615$	0	0
$616$	0	0
$617$	−24.0000	−0.966204	−0.483102	−	0.875564i	$-0.660490\pi$
$617$	−24.0000	−0.966204	−0.483102	+	0.875564i	$0.660490\pi$
$618$	0	0
$619$	−28.0000	−1.12542	−0.562708	−	0.826656i	$-0.690240\pi$
$619$	−28.0000	−1.12542	−0.562708	+	0.826656i	$0.690240\pi$
$620$	0	0
$621$	0	0
$622$	18.0000	0.721734
$623$	0	0
$624$	0	0
$625$	0	0
$626$	8.00000	0.319744
$627$	0	0
$628$	−4.00000	−0.159617
$629$	−40.0000	−1.59490
$630$	0	0
$631$	−17.0000	−0.676759	−0.338380	−	0.941010i	$-0.609879\pi$
$631$	−17.0000	−0.676759	−0.338380	+	0.941010i	$0.609879\pi$
$632$	0	0
$633$	0	0
$634$	−4.00000	−0.158860
$635$	0	0
$636$	0	0
$637$	28.0000	1.10940
$638$	−50.0000	−1.97952
$639$	0	0
$640$	0	0
$641$	−3.00000	−0.118493	−0.0592464	−	0.998243i	$-0.518870\pi$
$641$	−3.00000	−0.118493	−0.0592464	+	0.998243i	$0.518870\pi$
$642$	0	0
$643$	−6.00000	−0.236617	−0.118308	−	0.992977i	$-0.537747\pi$
$643$	−6.00000	−0.236617	−0.118308	+	0.992977i	$0.537747\pi$
$644$	0	0
$645$	0	0
$646$	40.0000	1.57378
$647$	−10.0000	−0.393141	−0.196570	−	0.980490i	$-0.562980\pi$
$647$	−10.0000	−0.393141	−0.196570	+	0.980490i	$0.562980\pi$
$648$	0	0
$649$	−5.00000	−0.196267
$650$	0	0
$651$	0	0
$652$	−16.0000	−0.626608
$653$	−8.00000	−0.313064	−0.156532	−	0.987673i	$-0.550031\pi$
$653$	−8.00000	−0.313064	−0.156532	+	0.987673i	$0.550031\pi$
$654$	0	0
$655$	0	0
$656$	28.0000	1.09322
$657$	0	0
$658$	0	0
$659$	−44.0000	−1.71400	−0.856998	−	0.515319i	$-0.827673\pi$
$659$	−44.0000	−1.71400	−0.856998	+	0.515319i	$0.827673\pi$
$660$	0	0
$661$	25.0000	0.972387	0.486194	−	0.873851i	$-0.338385\pi$
$661$	25.0000	0.972387	0.486194	+	0.873851i	$0.338385\pi$
$662$	−42.0000	−1.63238
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	30.0000	1.16160
$668$	24.0000	0.928588
$669$	0	0
$670$	0	0
$671$	10.0000	0.386046
$672$	0	0
$673$	−42.0000	−1.61898	−0.809491	−	0.587133i	$-0.800257\pi$
$673$	−42.0000	−1.61898	−0.809491	+	0.587133i	$0.800257\pi$
$674$	16.0000	0.616297
$675$	0	0
$676$	6.00000	0.230769
$677$	6.00000	0.230599	0.115299	−	0.993331i	$-0.463217\pi$
$677$	6.00000	0.230599	0.115299	+	0.993331i	$0.463217\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	90.0000	3.44628
$683$	48.0000	1.83667	0.918334	−	0.395805i	$-0.129534\pi$
$683$	48.0000	1.83667	0.918334	+	0.395805i	$0.129534\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	−8.00000	−0.304997
$689$	−32.0000	−1.21910
$690$	0	0
$691$	4.00000	0.152167	0.0760836	−	0.997101i	$-0.475758\pi$
$691$	4.00000	0.152167	0.0760836	+	0.997101i	$0.475758\pi$
$692$	−12.0000	−0.456172
$693$	0	0
$694$	40.0000	1.51838
$695$	0	0
$696$	0	0
$697$	28.0000	1.06058
$698$	26.0000	0.984115
$699$	0	0
$700$	0	0
$701$	−19.0000	−0.717620	−0.358810	−	0.933411i	$-0.616817\pi$
$701$	−19.0000	−0.717620	−0.358810	+	0.933411i	$0.616817\pi$
$702$	0	0
$703$	−50.0000	−1.88579
$704$	40.0000	1.50756
$705$	0	0
$706$	−36.0000	−1.35488
$707$	0	0
$708$	0	0
$709$	22.0000	0.826227	0.413114	−	0.910679i	$-0.364441\pi$
$709$	22.0000	0.826227	0.413114	+	0.910679i	$0.364441\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−54.0000	−2.02232
$714$	0	0
$715$	0	0
$716$	−46.0000	−1.71910
$717$	0	0
$718$	54.0000	2.01526
$719$	−27.0000	−1.00693	−0.503465	−	0.864016i	$-0.667942\pi$
$719$	−27.0000	−1.00693	−0.503465	+	0.864016i	$0.667942\pi$
$720$	0	0
$721$	0	0
$722$	12.0000	0.446594
$723$	0	0
$724$	−50.0000	−1.85824
$725$	0	0
$726$	0	0
$727$	−44.0000	−1.63187	−0.815935	−	0.578144i	$-0.803777\pi$
$727$	−44.0000	−1.63187	−0.815935	+	0.578144i	$0.803777\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−8.00000	−0.295891
$732$	0	0
$733$	46.0000	1.69905	0.849524	−	0.527549i	$-0.176889\pi$
$733$	46.0000	1.69905	0.849524	+	0.527549i	$0.176889\pi$
$734$	−36.0000	−1.32878
$735$	0	0
$736$	−48.0000	−1.76930
$737$	30.0000	1.10506
$738$	0	0
$739$	−35.0000	−1.28750	−0.643748	−	0.765238i	$-0.722621\pi$
$739$	−35.0000	−1.28750	−0.643748	+	0.765238i	$0.722621\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−4.00000	−0.146746	−0.0733729	−	0.997305i	$-0.523376\pi$
$743$	−4.00000	−0.146746	−0.0733729	+	0.997305i	$0.523376\pi$
$744$	0	0
$745$	0	0
$746$	−32.0000	−1.17160
$747$	0	0
$748$	40.0000	1.46254
$749$	0	0
$750$	0	0
$751$	−32.0000	−1.16770	−0.583848	−	0.811863i	$-0.698454\pi$
$751$	−32.0000	−1.16770	−0.583848	+	0.811863i	$0.698454\pi$
$752$	−8.00000	−0.291730
$753$	0	0
$754$	−40.0000	−1.45671
$755$	0	0
$756$	0	0
$757$	−28.0000	−1.01768	−0.508839	−	0.860862i	$-0.669925\pi$
$757$	−28.0000	−1.01768	−0.508839	+	0.860862i	$0.669925\pi$
$758$	8.00000	0.290573
$759$	0	0
$760$	0	0
$761$	27.0000	0.978749	0.489375	−	0.872074i	$-0.337225\pi$
$761$	27.0000	0.978749	0.489375	+	0.872074i	$0.337225\pi$
$762$	0	0
$763$	0	0
$764$	−2.00000	−0.0723575
$765$	0	0
$766$	−72.0000	−2.60147
$767$	−4.00000	−0.144432
$768$	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
$772$	−52.0000	−1.87152
$773$	18.0000	0.647415	0.323708	−	0.946157i	$-0.395071\pi$
$773$	18.0000	0.647415	0.323708	+	0.946157i	$0.395071\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	−12.0000	−0.430221
$779$	35.0000	1.25401
$780$	0	0
$781$	5.00000	0.178914
$782$	−48.0000	−1.71648
$783$	0	0
$784$	28.0000	1.00000
$785$	0	0
$786$	0	0
$787$	−4.00000	−0.142585	−0.0712923	−	0.997455i	$-0.522712\pi$
$787$	−4.00000	−0.142585	−0.0712923	+	0.997455i	$0.522712\pi$
$788$	24.0000	0.854965
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	8.00000	0.284088
$794$	76.0000	2.69714
$795$	0	0
$796$	32.0000	1.13421
$797$	4.00000	0.141687	0.0708436	−	0.997487i	$-0.477431\pi$
$797$	4.00000	0.141687	0.0708436	+	0.997487i	$0.477431\pi$
$798$	0	0
$799$	−8.00000	−0.283020
$800$	0	0
$801$	0	0
$802$	−60.0000	−2.11867
$803$	−40.0000	−1.41157
$804$	0	0
$805$	0	0
$806$	72.0000	2.53609
$807$	0	0
$808$	0	0
$809$	−7.00000	−0.246107	−0.123053	−	0.992400i	$-0.539269\pi$
$809$	−7.00000	−0.246107	−0.123053	+	0.992400i	$0.539269\pi$
$810$	0	0
$811$	−21.0000	−0.737410	−0.368705	−	0.929547i	$-0.620199\pi$
$811$	−21.0000	−0.737410	−0.368705	+	0.929547i	$0.620199\pi$
$812$	0	0
$813$	0	0
$814$	−100.000	−3.50500
$815$	0	0
$816$	0	0
$817$	−10.0000	−0.349856
$818$	28.0000	0.978997
$819$	0	0
$820$	0	0
$821$	−3.00000	−0.104701	−0.0523504	−	0.998629i	$-0.516671\pi$
$821$	−3.00000	−0.104701	−0.0523504	+	0.998629i	$0.516671\pi$
$822$	0	0
$823$	46.0000	1.60346	0.801730	−	0.597687i	$-0.203913\pi$
$823$	46.0000	1.60346	0.801730	+	0.597687i	$0.203913\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	44.0000	1.53003	0.765015	−	0.644013i	$-0.222732\pi$
$827$	44.0000	1.53003	0.765015	+	0.644013i	$0.222732\pi$
$828$	0	0
$829$	27.0000	0.937749	0.468874	−	0.883265i	$-0.344660\pi$
$829$	27.0000	0.937749	0.468874	+	0.883265i	$0.344660\pi$
$830$	0	0
$831$	0	0
$832$	32.0000	1.10940
$833$	28.0000	0.970143
$834$	0	0
$835$	0	0
$836$	50.0000	1.72929
$837$	0	0
$838$	−32.0000	−1.10542
$839$	−13.0000	−0.448810	−0.224405	−	0.974496i	$-0.572044\pi$
$839$	−13.0000	−0.448810	−0.224405	+	0.974496i	$0.572044\pi$
$840$	0	0
$841$	−4.00000	−0.137931
$842$	26.0000	0.896019
$843$	0	0
$844$	22.0000	0.757271
$845$	0	0
$846$	0	0
$847$	0	0
$848$	−32.0000	−1.09888
$849$	0	0
$850$	0	0
$851$	60.0000	2.05677
$852$	0	0
$853$	0	0		−	1.00000i	$-0.5\pi$
$853$	0	0			1.00000i	$0.5\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	32.0000	1.09310	0.546550	−	0.837427i	$-0.315941\pi$
$857$	32.0000	1.09310	0.546550	+	0.837427i	$0.315941\pi$
$858$	0	0
$859$	55.0000	1.87658	0.938288	−	0.345855i	$-0.112411\pi$
$859$	55.0000	1.87658	0.938288	+	0.345855i	$0.112411\pi$
$860$	0	0
$861$	0	0
$862$	−6.00000	−0.204361
$863$	−44.0000	−1.49778	−0.748889	−	0.662696i	$-0.769412\pi$
$863$	−44.0000	−1.49778	−0.748889	+	0.662696i	$0.769412\pi$
$864$	0	0
$865$	0	0
$866$	52.0000	1.76703
$867$	0	0
$868$	0	0
$869$	−60.0000	−2.03536
$870$	0	0
$871$	24.0000	0.813209
$872$	0	0
$873$	0	0
$874$	−60.0000	−2.02953
$875$	0	0
$876$	0	0
$877$	24.0000	0.810422	0.405211	−	0.914223i	$-0.367198\pi$
$877$	24.0000	0.810422	0.405211	+	0.914223i	$0.367198\pi$
$878$	−58.0000	−1.95741
$879$	0	0
$880$	0	0
$881$	25.0000	0.842271	0.421136	−	0.906998i	$-0.361632\pi$
$881$	25.0000	0.842271	0.421136	+	0.906998i	$0.361632\pi$
$882$	0	0
$883$	52.0000	1.74994	0.874970	−	0.484178i	$-0.160881\pi$
$883$	52.0000	1.74994	0.874970	+	0.484178i	$0.160881\pi$
$884$	32.0000	1.07628
$885$	0	0
$886$	12.0000	0.403148
$887$	6.00000	0.201460	0.100730	−	0.994914i	$-0.467882\pi$
$887$	6.00000	0.201460	0.100730	+	0.994914i	$0.467882\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	−16.0000	−0.535720
$893$	−10.0000	−0.334637
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	−34.0000	−1.13459
$899$	−45.0000	−1.50083
$900$	0	0
$901$	−32.0000	−1.06607
$902$	70.0000	2.33075
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	18.0000	0.597680	0.298840	−	0.954303i	$-0.403400\pi$
$907$	18.0000	0.597680	0.298840	+	0.954303i	$0.403400\pi$
$908$	8.00000	0.265489
$909$	0	0
$910$	0	0
$911$	49.0000	1.62344	0.811721	−	0.584045i	$-0.198531\pi$
$911$	49.0000	1.62344	0.811721	+	0.584045i	$0.198531\pi$
$912$	0	0
$913$	−30.0000	−0.992855
$914$	76.0000	2.51386
$915$	0	0
$916$	12.0000	0.396491
$917$	0	0
$918$	0	0
$919$	−11.0000	−0.362857	−0.181428	−	0.983404i	$-0.558072\pi$
$919$	−11.0000	−0.362857	−0.181428	+	0.983404i	$0.558072\pi$
$920$	0	0
$921$	0	0
$922$	30.0000	0.987997
$923$	4.00000	0.131662
$924$	0	0
$925$	0	0
$926$	12.0000	0.394344
$927$	0	0
$928$	−40.0000	−1.31306
$929$	−1.00000	−0.0328089	−0.0164045	−	0.999865i	$-0.505222\pi$
$929$	−1.00000	−0.0328089	−0.0164045	+	0.999865i	$0.505222\pi$
$930$	0	0
$931$	35.0000	1.14708
$932$	12.0000	0.393073
$933$	0	0
$934$	8.00000	0.261768
$935$	0	0
$936$	0	0
$937$	−42.0000	−1.37208	−0.686040	−	0.727564i	$-0.740653\pi$
$937$	−42.0000	−1.37208	−0.686040	+	0.727564i	$0.740653\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−46.0000	−1.49956	−0.749779	−	0.661689i	$-0.769840\pi$
$941$	−46.0000	−1.49956	−0.749779	+	0.661689i	$0.769840\pi$
$942$	0	0
$943$	−42.0000	−1.36771
$944$	−4.00000	−0.130189
$945$	0	0
$946$	−20.0000	−0.650256
$947$	18.0000	0.584921	0.292461	−	0.956278i	$-0.405526\pi$
$947$	18.0000	0.584921	0.292461	+	0.956278i	$0.405526\pi$
$948$	0	0
$949$	−32.0000	−1.03876
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−16.0000	−0.518291	−0.259145	−	0.965838i	$-0.583441\pi$
$953$	−16.0000	−0.518291	−0.259145	+	0.965838i	$0.583441\pi$
$954$	0	0
$955$	0	0
$956$	32.0000	1.03495
$957$	0	0
$958$	30.0000	0.969256
$959$	0	0
$960$	0	0
$961$	50.0000	1.61290
$962$	−80.0000	−2.57930
$963$	0	0
$964$	−22.0000	−0.708572
$965$	0	0
$966$	0	0
$967$	40.0000	1.28631	0.643157	−	0.765735i	$-0.277624\pi$
$967$	40.0000	1.28631	0.643157	+	0.765735i	$0.277624\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	27.0000	0.866471	0.433236	−	0.901281i	$-0.357372\pi$
$971$	27.0000	0.866471	0.433236	+	0.901281i	$0.357372\pi$
$972$	0	0
$973$	0	0
$974$	−16.0000	−0.512673
$975$	0	0
$976$	8.00000	0.256074
$977$	−28.0000	−0.895799	−0.447900	−	0.894084i	$-0.647828\pi$
$977$	−28.0000	−0.895799	−0.447900	+	0.894084i	$0.647828\pi$
$978$	0	0
$979$	−45.0000	−1.43821
$980$	0	0
$981$	0	0
$982$	−86.0000	−2.74437
$983$	48.0000	1.53096	0.765481	−	0.643458i	$-0.222501\pi$
$983$	48.0000	1.53096	0.765481	+	0.643458i	$0.222501\pi$
$984$	0	0
$985$	0	0
$986$	−40.0000	−1.27386
$987$	0	0
$988$	40.0000	1.27257
$989$	12.0000	0.381578
$990$	0	0
$991$	−1.00000	−0.0317660	−0.0158830	−	0.999874i	$-0.505056\pi$
$991$	−1.00000	−0.0317660	−0.0158830	+	0.999874i	$0.505056\pi$
$992$	72.0000	2.28600
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−48.0000	−1.52018	−0.760088	−	0.649821i	$-0.774844\pi$
$997$	−48.0000	−1.52018	−0.760088	+	0.649821i	$0.774844\pi$
$998$	14.0000	0.443162
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2025.2.a.f.1.1		1
3.2	odd	2		2025.2.a.a.1.1		1
5.2	odd	4		2025.2.b.a.649.2		2
5.3	odd	4		2025.2.b.a.649.1		2
5.4	even	2		405.2.a.a.1.1	✓	1
15.2	even	4		2025.2.b.b.649.1		2
15.8	even	4		2025.2.b.b.649.2		2
15.14	odd	2		405.2.a.f.1.1	yes	1
20.19	odd	2		6480.2.a.f.1.1		1
45.4	even	6		405.2.e.g.136.1		2
45.14	odd	6		405.2.e.a.136.1		2
45.29	odd	6		405.2.e.a.271.1		2
45.34	even	6		405.2.e.g.271.1		2
60.59	even	2		6480.2.a.r.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
405.2.a.a.1.1	✓	1	5.4	even	2
405.2.a.f.1.1	yes	1	15.14	odd	2
405.2.e.a.136.1		2	45.14	odd	6
405.2.e.a.271.1		2	45.29	odd	6
405.2.e.g.136.1		2	45.4	even	6
405.2.e.g.271.1		2	45.34	even	6
2025.2.a.a.1.1		1	3.2	odd	2
2025.2.a.f.1.1		1	1.1	even	1	trivial
2025.2.b.a.649.1		2	5.3	odd	4
2025.2.b.a.649.2		2	5.2	odd	4
2025.2.b.b.649.1		2	15.2	even	4
2025.2.b.b.649.2		2	15.8	even	4
6480.2.a.f.1.1		1	20.19	odd	2
6480.2.a.r.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)

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L-functions

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Groups

Database

Properties

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