LMFDB - Embedded newform 325.2.a.e.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [325,2,Mod(1,325)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(325, 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("325.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 325 = 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	325.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:	$2.59513806569$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	325.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+2.00000 q^{2} +1.00000 q^{3} +2.00000 q^{4} +2.00000 q^{6} +2.00000 q^{7} -2.00000 q^{9} +O(q^{10})$

\(q+2.00000 q^{2} +1.00000 q^{3} +2.00000 q^{4} +2.00000 q^{6} +2.00000 q^{7} -2.00000 q^{9} +2.00000 q^{11} +2.00000 q^{12} -1.00000 q^{13} +4.00000 q^{14} -4.00000 q^{16} +2.00000 q^{17} -4.00000 q^{18} +2.00000 q^{21} +4.00000 q^{22} -9.00000 q^{23} -2.00000 q^{26} -5.00000 q^{27} +4.00000 q^{28} +5.00000 q^{29} +2.00000 q^{31} -8.00000 q^{32} +2.00000 q^{33} +4.00000 q^{34} -4.00000 q^{36} -8.00000 q^{37} -1.00000 q^{39} +12.0000 q^{41} +4.00000 q^{42} +1.00000 q^{43} +4.00000 q^{44} -18.0000 q^{46} -8.00000 q^{47} -4.00000 q^{48} -3.00000 q^{49} +2.00000 q^{51} -2.00000 q^{52} +11.0000 q^{53} -10.0000 q^{54} +10.0000 q^{58} -13.0000 q^{61} +4.00000 q^{62} -4.00000 q^{63} -8.00000 q^{64} +4.00000 q^{66} +2.00000 q^{67} +4.00000 q^{68} -9.00000 q^{69} +12.0000 q^{71} +6.00000 q^{73} -16.0000 q^{74} +4.00000 q^{77} -2.00000 q^{78} +15.0000 q^{79} +1.00000 q^{81} +24.0000 q^{82} -4.00000 q^{83} +4.00000 q^{84} +2.00000 q^{86} +5.00000 q^{87} -10.0000 q^{89} -2.00000 q^{91} -18.0000 q^{92} +2.00000 q^{93} -16.0000 q^{94} -8.00000 q^{96} -8.00000 q^{97} -6.00000 q^{98} -4.00000 q^{99} +O(q^{100})\)

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	$0.250000\pi$
$3$	1.00000	0.577350	0.288675	−	0.957427i	$-0.406785\pi$
$3$	1.00000	0.577350	0.288675	+	0.957427i	$0.406785\pi$
$4$	2.00000	1.00000
$5$	0	0
$5$	0	0
$6$	2.00000	0.816497
$7$	2.00000	0.755929	0.377964	−	0.925820i	$-0.376624\pi$
$7$	2.00000	0.755929	0.377964	+	0.925820i	$0.376624\pi$
$8$	0	0
$9$	−2.00000	−0.666667
$10$	0	0
$11$	2.00000	0.603023	0.301511	−	0.953463i	$-0.402509\pi$
$11$	2.00000	0.603023	0.301511	+	0.953463i	$0.402509\pi$
$12$	2.00000	0.577350
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	4.00000	1.06904
$15$	0	0
$16$	−4.00000	−1.00000
$17$	2.00000	0.485071	0.242536	−	0.970143i	$-0.422021\pi$
$17$	2.00000	0.485071	0.242536	+	0.970143i	$0.422021\pi$
$18$	−4.00000	−0.942809
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	0	0
$21$	2.00000	0.436436
$22$	4.00000	0.852803
$23$	−9.00000	−1.87663	−0.938315	−	0.345782i	$-0.887614\pi$
$23$	−9.00000	−1.87663	−0.938315	+	0.345782i	$0.887614\pi$
$24$	0	0
$25$	0	0
$26$	−2.00000	−0.392232
$27$	−5.00000	−0.962250
$28$	4.00000	0.755929
$29$	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
$31$	2.00000	0.359211	0.179605	−	0.983739i	$-0.442518\pi$
$31$	2.00000	0.359211	0.179605	+	0.983739i	$0.442518\pi$
$32$	−8.00000	−1.41421
$33$	2.00000	0.348155
$34$	4.00000	0.685994
$35$	0	0
$36$	−4.00000	−0.666667
$37$	−8.00000	−1.31519	−0.657596	−	0.753371i	$-0.728427\pi$
$37$	−8.00000	−1.31519	−0.657596	+	0.753371i	$0.728427\pi$
$38$	0	0
$39$	−1.00000	−0.160128
$40$	0	0
$41$	12.0000	1.87409	0.937043	−	0.349215i	$-0.113552\pi$
$41$	12.0000	1.87409	0.937043	+	0.349215i	$0.113552\pi$
$42$	4.00000	0.617213
$43$	1.00000	0.152499	0.0762493	−	0.997089i	$-0.475706\pi$
$43$	1.00000	0.152499	0.0762493	+	0.997089i	$0.475706\pi$
$44$	4.00000	0.603023
$45$	0	0
$46$	−18.0000	−2.65396
$47$	−8.00000	−1.16692	−0.583460	−	0.812142i	$-0.698301\pi$
$47$	−8.00000	−1.16692	−0.583460	+	0.812142i	$0.698301\pi$
$48$	−4.00000	−0.577350
$49$	−3.00000	−0.428571
$50$	0	0
$51$	2.00000	0.280056
$52$	−2.00000	−0.277350
$53$	11.0000	1.51097	0.755483	−	0.655168i	$-0.227402\pi$
$53$	11.0000	1.51097	0.755483	+	0.655168i	$0.227402\pi$
$54$	−10.0000	−1.36083
$55$	0	0
$56$	0	0
$57$	0	0
$58$	10.0000	1.31306
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	−13.0000	−1.66448	−0.832240	−	0.554416i	$-0.812942\pi$
$61$	−13.0000	−1.66448	−0.832240	+	0.554416i	$0.812942\pi$
$62$	4.00000	0.508001
$63$	−4.00000	−0.503953
$64$	−8.00000	−1.00000
$65$	0	0
$66$	4.00000	0.492366
$67$	2.00000	0.244339	0.122169	−	0.992509i	$-0.461015\pi$
$67$	2.00000	0.244339	0.122169	+	0.992509i	$0.461015\pi$
$68$	4.00000	0.485071
$69$	−9.00000	−1.08347
$70$	0	0
$71$	12.0000	1.42414	0.712069	−	0.702109i	$-0.247758\pi$
$71$	12.0000	1.42414	0.712069	+	0.702109i	$0.247758\pi$
$72$	0	0
$73$	6.00000	0.702247	0.351123	−	0.936329i	$-0.385800\pi$
$73$	6.00000	0.702247	0.351123	+	0.936329i	$0.385800\pi$
$74$	−16.0000	−1.85996
$75$	0	0
$76$	0	0
$77$	4.00000	0.455842
$78$	−2.00000	−0.226455
$79$	15.0000	1.68763	0.843816	−	0.536633i	$-0.180304\pi$
$79$	15.0000	1.68763	0.843816	+	0.536633i	$0.180304\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	24.0000	2.65036
$83$	−4.00000	−0.439057	−0.219529	−	0.975606i	$-0.570452\pi$
$83$	−4.00000	−0.439057	−0.219529	+	0.975606i	$0.570452\pi$
$84$	4.00000	0.436436
$85$	0	0
$86$	2.00000	0.215666
$87$	5.00000	0.536056
$88$	0	0
$89$	−10.0000	−1.06000	−0.529999	−	0.847998i	$-0.677808\pi$
$89$	−10.0000	−1.06000	−0.529999	+	0.847998i	$0.677808\pi$
$90$	0	0
$91$	−2.00000	−0.209657
$92$	−18.0000	−1.87663
$93$	2.00000	0.207390
$94$	−16.0000	−1.65027
$95$	0	0
$96$	−8.00000	−0.816497
$97$	−8.00000	−0.812277	−0.406138	−	0.913812i	$-0.633125\pi$
$97$	−8.00000	−0.812277	−0.406138	+	0.913812i	$0.633125\pi$
$98$	−6.00000	−0.606092
$99$	−4.00000	−0.402015
$100$	0	0
$101$	−3.00000	−0.298511	−0.149256	−	0.988799i	$-0.547688\pi$
$101$	−3.00000	−0.298511	−0.149256	+	0.988799i	$0.547688\pi$
$102$	4.00000	0.396059
$103$	11.0000	1.08386	0.541931	−	0.840423i	$-0.317693\pi$
$103$	11.0000	1.08386	0.541931	+	0.840423i	$0.317693\pi$
$104$	0	0
$105$	0	0
$106$	22.0000	2.13683
$107$	7.00000	0.676716	0.338358	−	0.941018i	$-0.390129\pi$
$107$	7.00000	0.676716	0.338358	+	0.941018i	$0.390129\pi$
$108$	−10.0000	−0.962250
$109$	10.0000	0.957826	0.478913	−	0.877862i	$-0.341031\pi$
$109$	10.0000	0.957826	0.478913	+	0.877862i	$0.341031\pi$
$110$	0	0
$111$	−8.00000	−0.759326
$112$	−8.00000	−0.755929
$113$	1.00000	0.0940721	0.0470360	−	0.998893i	$-0.485022\pi$
$113$	1.00000	0.0940721	0.0470360	+	0.998893i	$0.485022\pi$
$114$	0	0
$115$	0	0
$116$	10.0000	0.928477
$117$	2.00000	0.184900
$118$	0	0
$119$	4.00000	0.366679
$120$	0	0
$121$	−7.00000	−0.636364
$122$	−26.0000	−2.35393
$123$	12.0000	1.08200
$124$	4.00000	0.359211
$125$	0	0
$126$	−8.00000	−0.712697
$127$	7.00000	0.621150	0.310575	−	0.950549i	$-0.399478\pi$
$127$	7.00000	0.621150	0.310575	+	0.950549i	$0.399478\pi$
$128$	0	0
$129$	1.00000	0.0880451
$130$	0	0
$131$	17.0000	1.48530	0.742648	−	0.669681i	$-0.233569\pi$
$131$	17.0000	1.48530	0.742648	+	0.669681i	$0.233569\pi$
$132$	4.00000	0.348155
$133$	0	0
$134$	4.00000	0.345547
$135$	0	0
$136$	0	0
$137$	−8.00000	−0.683486	−0.341743	−	0.939793i	$-0.611017\pi$
$137$	−8.00000	−0.683486	−0.341743	+	0.939793i	$0.611017\pi$
$138$	−18.0000	−1.53226
$139$	15.0000	1.27228	0.636142	−	0.771572i	$-0.280529\pi$
$139$	15.0000	1.27228	0.636142	+	0.771572i	$0.280529\pi$
$140$	0	0
$141$	−8.00000	−0.673722
$142$	24.0000	2.01404
$143$	−2.00000	−0.167248
$144$	8.00000	0.666667
$145$	0	0
$146$	12.0000	0.993127
$147$	−3.00000	−0.247436
$148$	−16.0000	−1.31519
$149$	0	0		−	1.00000i	$-0.5\pi$
$149$	0	0			1.00000i	$0.5\pi$
$150$	0	0
$151$	−8.00000	−0.651031	−0.325515	−	0.945537i	$-0.605538\pi$
$151$	−8.00000	−0.651031	−0.325515	+	0.945537i	$0.605538\pi$
$152$	0	0
$153$	−4.00000	−0.323381
$154$	8.00000	0.644658
$155$	0	0
$156$	−2.00000	−0.160128
$157$	−18.0000	−1.43656	−0.718278	−	0.695756i	$-0.755069\pi$
$157$	−18.0000	−1.43656	−0.718278	+	0.695756i	$0.755069\pi$
$158$	30.0000	2.38667
$159$	11.0000	0.872357
$160$	0	0
$161$	−18.0000	−1.41860
$162$	2.00000	0.157135
$163$	−24.0000	−1.87983	−0.939913	−	0.341415i	$-0.889094\pi$
$163$	−24.0000	−1.87983	−0.939913	+	0.341415i	$0.889094\pi$
$164$	24.0000	1.87409
$165$	0	0
$166$	−8.00000	−0.620920
$167$	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
$169$	1.00000	0.0769231
$170$	0	0
$171$	0	0
$172$	2.00000	0.152499
$173$	6.00000	0.456172	0.228086	−	0.973641i	$-0.426753\pi$
$173$	6.00000	0.456172	0.228086	+	0.973641i	$0.426753\pi$
$174$	10.0000	0.758098
$175$	0	0
$176$	−8.00000	−0.603023
$177$	0	0
$178$	−20.0000	−1.49906
$179$	15.0000	1.12115	0.560576	−	0.828103i	$-0.310580\pi$
$179$	15.0000	1.12115	0.560576	+	0.828103i	$0.310580\pi$
$180$	0	0
$181$	2.00000	0.148659	0.0743294	−	0.997234i	$-0.476318\pi$
$181$	2.00000	0.148659	0.0743294	+	0.997234i	$0.476318\pi$
$182$	−4.00000	−0.296500
$183$	−13.0000	−0.960988
$184$	0	0
$185$	0	0
$186$	4.00000	0.293294
$187$	4.00000	0.292509
$188$	−16.0000	−1.16692
$189$	−10.0000	−0.727393
$190$	0	0
$191$	−3.00000	−0.217072	−0.108536	−	0.994092i	$-0.534616\pi$
$191$	−3.00000	−0.217072	−0.108536	+	0.994092i	$0.534616\pi$
$192$	−8.00000	−0.577350
$193$	16.0000	1.15171	0.575853	−	0.817554i	$-0.304670\pi$
$193$	16.0000	1.15171	0.575853	+	0.817554i	$0.304670\pi$
$194$	−16.0000	−1.14873
$195$	0	0
$196$	−6.00000	−0.428571
$197$	12.0000	0.854965	0.427482	−	0.904024i	$-0.359401\pi$
$197$	12.0000	0.854965	0.427482	+	0.904024i	$0.359401\pi$
$198$	−8.00000	−0.568535
$199$	−25.0000	−1.77220	−0.886102	−	0.463491i	$-0.846597\pi$
$199$	−25.0000	−1.77220	−0.886102	+	0.463491i	$0.846597\pi$
$200$	0	0
$201$	2.00000	0.141069
$202$	−6.00000	−0.422159
$203$	10.0000	0.701862
$204$	4.00000	0.280056
$205$	0	0
$206$	22.0000	1.53281
$207$	18.0000	1.25109
$208$	4.00000	0.277350
$209$	0	0
$210$	0	0
$211$	12.0000	0.826114	0.413057	−	0.910705i	$-0.364461\pi$
$211$	12.0000	0.826114	0.413057	+	0.910705i	$0.364461\pi$
$212$	22.0000	1.51097
$213$	12.0000	0.822226
$214$	14.0000	0.957020
$215$	0	0
$216$	0	0
$217$	4.00000	0.271538
$218$	20.0000	1.35457
$219$	6.00000	0.405442
$220$	0	0
$221$	−2.00000	−0.134535
$222$	−16.0000	−1.07385
$223$	−14.0000	−0.937509	−0.468755	−	0.883328i	$-0.655297\pi$
$223$	−14.0000	−0.937509	−0.468755	+	0.883328i	$0.655297\pi$
$224$	−16.0000	−1.06904
$225$	0	0
$226$	2.00000	0.133038
$227$	−8.00000	−0.530979	−0.265489	−	0.964114i	$-0.585534\pi$
$227$	−8.00000	−0.530979	−0.265489	+	0.964114i	$0.585534\pi$
$228$	0	0
$229$	−10.0000	−0.660819	−0.330409	−	0.943838i	$-0.607187\pi$
$229$	−10.0000	−0.660819	−0.330409	+	0.943838i	$0.607187\pi$
$230$	0	0
$231$	4.00000	0.263181
$232$	0	0
$233$	1.00000	0.0655122	0.0327561	−	0.999463i	$-0.489572\pi$
$233$	1.00000	0.0655122	0.0327561	+	0.999463i	$0.489572\pi$
$234$	4.00000	0.261488
$235$	0	0
$236$	0	0
$237$	15.0000	0.974355
$238$	8.00000	0.518563
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	0	0
$241$	22.0000	1.41714	0.708572	−	0.705638i	$-0.249340\pi$
$241$	22.0000	1.41714	0.708572	+	0.705638i	$0.249340\pi$
$242$	−14.0000	−0.899954
$243$	16.0000	1.02640
$244$	−26.0000	−1.66448
$245$	0	0
$246$	24.0000	1.53018
$247$	0	0
$248$	0	0
$249$	−4.00000	−0.253490
$250$	0	0
$251$	12.0000	0.757433	0.378717	−	0.925513i	$-0.376365\pi$
$251$	12.0000	0.757433	0.378717	+	0.925513i	$0.376365\pi$
$252$	−8.00000	−0.503953
$253$	−18.0000	−1.13165
$254$	14.0000	0.878438
$255$	0	0
$256$	16.0000	1.00000
$257$	17.0000	1.06043	0.530215	−	0.847863i	$-0.322111\pi$
$257$	17.0000	1.06043	0.530215	+	0.847863i	$0.322111\pi$
$258$	2.00000	0.124515
$259$	−16.0000	−0.994192
$260$	0	0
$261$	−10.0000	−0.618984
$262$	34.0000	2.10053
$263$	−4.00000	−0.246651	−0.123325	−	0.992366i	$-0.539356\pi$
$263$	−4.00000	−0.246651	−0.123325	+	0.992366i	$0.539356\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−10.0000	−0.611990
$268$	4.00000	0.244339
$269$	−15.0000	−0.914566	−0.457283	−	0.889321i	$-0.651177\pi$
$269$	−15.0000	−0.914566	−0.457283	+	0.889321i	$0.651177\pi$
$270$	0	0
$271$	−28.0000	−1.70088	−0.850439	−	0.526073i	$-0.823664\pi$
$271$	−28.0000	−1.70088	−0.850439	+	0.526073i	$0.823664\pi$
$272$	−8.00000	−0.485071
$273$	−2.00000	−0.121046
$274$	−16.0000	−0.966595
$275$	0	0
$276$	−18.0000	−1.08347
$277$	−23.0000	−1.38194	−0.690968	−	0.722885i	$-0.742815\pi$
$277$	−23.0000	−1.38194	−0.690968	+	0.722885i	$0.742815\pi$
$278$	30.0000	1.79928
$279$	−4.00000	−0.239474
$280$	0	0
$281$	−18.0000	−1.07379	−0.536895	−	0.843649i	$-0.680403\pi$
$281$	−18.0000	−1.07379	−0.536895	+	0.843649i	$0.680403\pi$
$282$	−16.0000	−0.952786
$283$	−4.00000	−0.237775	−0.118888	−	0.992908i	$-0.537933\pi$
$283$	−4.00000	−0.237775	−0.118888	+	0.992908i	$0.537933\pi$
$284$	24.0000	1.42414
$285$	0	0
$286$	−4.00000	−0.236525
$287$	24.0000	1.41668
$288$	16.0000	0.942809
$289$	−13.0000	−0.764706
$290$	0	0
$291$	−8.00000	−0.468968
$292$	12.0000	0.702247
$293$	−14.0000	−0.817889	−0.408944	−	0.912559i	$-0.634103\pi$
$293$	−14.0000	−0.817889	−0.408944	+	0.912559i	$0.634103\pi$
$294$	−6.00000	−0.349927
$295$	0	0
$296$	0	0
$297$	−10.0000	−0.580259
$298$	0	0
$299$	9.00000	0.520483
$300$	0	0
$301$	2.00000	0.115278
$302$	−16.0000	−0.920697
$303$	−3.00000	−0.172345
$304$	0	0
$305$	0	0
$306$	−8.00000	−0.457330
$307$	2.00000	0.114146	0.0570730	−	0.998370i	$-0.481823\pi$
$307$	2.00000	0.114146	0.0570730	+	0.998370i	$0.481823\pi$
$308$	8.00000	0.455842
$309$	11.0000	0.625768
$310$	0	0
$311$	−13.0000	−0.737162	−0.368581	−	0.929596i	$-0.620156\pi$
$311$	−13.0000	−0.737162	−0.368581	+	0.929596i	$0.620156\pi$
$312$	0	0
$313$	−19.0000	−1.07394	−0.536972	−	0.843600i	$-0.680432\pi$
$313$	−19.0000	−1.07394	−0.536972	+	0.843600i	$0.680432\pi$
$314$	−36.0000	−2.03160
$315$	0	0
$316$	30.0000	1.68763
$317$	−18.0000	−1.01098	−0.505490	−	0.862832i	$-0.668688\pi$
$317$	−18.0000	−1.01098	−0.505490	+	0.862832i	$0.668688\pi$
$318$	22.0000	1.23370
$319$	10.0000	0.559893
$320$	0	0
$321$	7.00000	0.390702
$322$	−36.0000	−2.00620
$323$	0	0
$324$	2.00000	0.111111
$325$	0	0
$326$	−48.0000	−2.65847
$327$	10.0000	0.553001
$328$	0	0
$329$	−16.0000	−0.882109
$330$	0	0
$331$	−18.0000	−0.989369	−0.494685	−	0.869072i	$-0.664716\pi$
$331$	−18.0000	−0.989369	−0.494685	+	0.869072i	$0.664716\pi$
$332$	−8.00000	−0.439057
$333$	16.0000	0.876795
$334$	24.0000	1.31322
$335$	0	0
$336$	−8.00000	−0.436436
$337$	−13.0000	−0.708155	−0.354078	−	0.935216i	$-0.615205\pi$
$337$	−13.0000	−0.708155	−0.354078	+	0.935216i	$0.615205\pi$
$338$	2.00000	0.108786
$339$	1.00000	0.0543125
$340$	0	0
$341$	4.00000	0.216612
$342$	0	0
$343$	−20.0000	−1.07990
$344$	0	0
$345$	0	0
$346$	12.0000	0.645124
$347$	17.0000	0.912608	0.456304	−	0.889824i	$-0.349173\pi$
$347$	17.0000	0.912608	0.456304	+	0.889824i	$0.349173\pi$
$348$	10.0000	0.536056
$349$	−30.0000	−1.60586	−0.802932	−	0.596071i	$-0.796728\pi$
$349$	−30.0000	−1.60586	−0.802932	+	0.596071i	$0.796728\pi$
$350$	0	0
$351$	5.00000	0.266880
$352$	−16.0000	−0.852803
$353$	−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
$356$	−20.0000	−1.06000
$357$	4.00000	0.211702
$358$	30.0000	1.58555
$359$	20.0000	1.05556	0.527780	−	0.849381i	$-0.323025\pi$
$359$	20.0000	1.05556	0.527780	+	0.849381i	$0.323025\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	4.00000	0.210235
$363$	−7.00000	−0.367405
$364$	−4.00000	−0.209657
$365$	0	0
$366$	−26.0000	−1.35904
$367$	−23.0000	−1.20059	−0.600295	−	0.799779i	$-0.704950\pi$
$367$	−23.0000	−1.20059	−0.600295	+	0.799779i	$0.704950\pi$
$368$	36.0000	1.87663
$369$	−24.0000	−1.24939
$370$	0	0
$371$	22.0000	1.14218
$372$	4.00000	0.207390
$373$	1.00000	0.0517780	0.0258890	−	0.999665i	$-0.491758\pi$
$373$	1.00000	0.0517780	0.0258890	+	0.999665i	$0.491758\pi$
$374$	8.00000	0.413670
$375$	0	0
$376$	0	0
$377$	−5.00000	−0.257513
$378$	−20.0000	−1.02869
$379$	10.0000	0.513665	0.256833	−	0.966456i	$-0.417321\pi$
$379$	10.0000	0.513665	0.256833	+	0.966456i	$0.417321\pi$
$380$	0	0
$381$	7.00000	0.358621
$382$	−6.00000	−0.306987
$383$	−24.0000	−1.22634	−0.613171	−	0.789950i	$-0.710106\pi$
$383$	−24.0000	−1.22634	−0.613171	+	0.789950i	$0.710106\pi$
$384$	0	0
$385$	0	0
$386$	32.0000	1.62876
$387$	−2.00000	−0.101666
$388$	−16.0000	−0.812277
$389$	35.0000	1.77457	0.887285	−	0.461221i	$-0.152589\pi$
$389$	35.0000	1.77457	0.887285	+	0.461221i	$0.152589\pi$
$390$	0	0
$391$	−18.0000	−0.910299
$392$	0	0
$393$	17.0000	0.857537
$394$	24.0000	1.20910
$395$	0	0
$396$	−8.00000	−0.402015
$397$	−18.0000	−0.903394	−0.451697	−	0.892171i	$-0.649181\pi$
$397$	−18.0000	−0.903394	−0.451697	+	0.892171i	$0.649181\pi$
$398$	−50.0000	−2.50627
$399$	0	0
$400$	0	0
$401$	22.0000	1.09863	0.549314	−	0.835616i	$-0.314889\pi$
$401$	22.0000	1.09863	0.549314	+	0.835616i	$0.314889\pi$
$402$	4.00000	0.199502
$403$	−2.00000	−0.0996271
$404$	−6.00000	−0.298511
$405$	0	0
$406$	20.0000	0.992583
$407$	−16.0000	−0.793091
$408$	0	0
$409$	−30.0000	−1.48340	−0.741702	−	0.670729i	$-0.765981\pi$
$409$	−30.0000	−1.48340	−0.741702	+	0.670729i	$0.765981\pi$
$410$	0	0
$411$	−8.00000	−0.394611
$412$	22.0000	1.08386
$413$	0	0
$414$	36.0000	1.76930
$415$	0	0
$416$	8.00000	0.392232
$417$	15.0000	0.734553
$418$	0	0
$419$	35.0000	1.70986	0.854931	−	0.518742i	$-0.173599\pi$
$419$	35.0000	1.70986	0.854931	+	0.518742i	$0.173599\pi$
$420$	0	0
$421$	22.0000	1.07221	0.536107	−	0.844150i	$-0.319894\pi$
$421$	22.0000	1.07221	0.536107	+	0.844150i	$0.319894\pi$
$422$	24.0000	1.16830
$423$	16.0000	0.777947
$424$	0	0
$425$	0	0
$426$	24.0000	1.16280
$427$	−26.0000	−1.25823
$428$	14.0000	0.676716
$429$	−2.00000	−0.0965609
$430$	0	0
$431$	2.00000	0.0963366	0.0481683	−	0.998839i	$-0.484662\pi$
$431$	2.00000	0.0963366	0.0481683	+	0.998839i	$0.484662\pi$
$432$	20.0000	0.962250
$433$	1.00000	0.0480569	0.0240285	−	0.999711i	$-0.492351\pi$
$433$	1.00000	0.0480569	0.0240285	+	0.999711i	$0.492351\pi$
$434$	8.00000	0.384012
$435$	0	0
$436$	20.0000	0.957826
$437$	0	0
$438$	12.0000	0.573382
$439$	0	0		−	1.00000i	$-0.5\pi$
$439$	0	0			1.00000i	$0.5\pi$
$440$	0	0
$441$	6.00000	0.285714
$442$	−4.00000	−0.190261
$443$	−4.00000	−0.190046	−0.0950229	−	0.995475i	$-0.530292\pi$
$443$	−4.00000	−0.190046	−0.0950229	+	0.995475i	$0.530292\pi$
$444$	−16.0000	−0.759326
$445$	0	0
$446$	−28.0000	−1.32584
$447$	0	0
$448$	−16.0000	−0.755929
$449$	0	0		−	1.00000i	$-0.5\pi$
$449$	0	0			1.00000i	$0.5\pi$
$450$	0	0
$451$	24.0000	1.13012
$452$	2.00000	0.0940721
$453$	−8.00000	−0.375873
$454$	−16.0000	−0.750917
$455$	0	0
$456$	0	0
$457$	32.0000	1.49690	0.748448	−	0.663193i	$-0.230799\pi$
$457$	32.0000	1.49690	0.748448	+	0.663193i	$0.230799\pi$
$458$	−20.0000	−0.934539
$459$	−10.0000	−0.466760
$460$	0	0
$461$	−38.0000	−1.76984	−0.884918	−	0.465746i	$-0.845786\pi$
$461$	−38.0000	−1.76984	−0.884918	+	0.465746i	$0.845786\pi$
$462$	8.00000	0.372194
$463$	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
$466$	2.00000	0.0926482
$467$	17.0000	0.786666	0.393333	−	0.919396i	$-0.371322\pi$
$467$	17.0000	0.786666	0.393333	+	0.919396i	$0.371322\pi$
$468$	4.00000	0.184900
$469$	4.00000	0.184703
$470$	0	0
$471$	−18.0000	−0.829396
$472$	0	0
$473$	2.00000	0.0919601
$474$	30.0000	1.37795
$475$	0	0
$476$	8.00000	0.366679
$477$	−22.0000	−1.00731
$478$	0	0
$479$	−30.0000	−1.37073	−0.685367	−	0.728197i	$-0.740358\pi$
$479$	−30.0000	−1.37073	−0.685367	+	0.728197i	$0.740358\pi$
$480$	0	0
$481$	8.00000	0.364769
$482$	44.0000	2.00415
$483$	−18.0000	−0.819028
$484$	−14.0000	−0.636364
$485$	0	0
$486$	32.0000	1.45155
$487$	32.0000	1.45006	0.725029	−	0.688718i	$-0.241826\pi$
$487$	32.0000	1.45006	0.725029	+	0.688718i	$0.241826\pi$
$488$	0	0
$489$	−24.0000	−1.08532
$490$	0	0
$491$	27.0000	1.21849	0.609246	−	0.792981i	$-0.291472\pi$
$491$	27.0000	1.21849	0.609246	+	0.792981i	$0.291472\pi$
$492$	24.0000	1.08200
$493$	10.0000	0.450377
$494$	0	0
$495$	0	0
$496$	−8.00000	−0.359211
$497$	24.0000	1.07655
$498$	−8.00000	−0.358489
$499$	10.0000	0.447661	0.223831	−	0.974628i	$-0.428144\pi$
$499$	10.0000	0.447661	0.223831	+	0.974628i	$0.428144\pi$
$500$	0	0
$501$	12.0000	0.536120
$502$	24.0000	1.07117
$503$	21.0000	0.936344	0.468172	−	0.883637i	$-0.344913\pi$
$503$	21.0000	0.936344	0.468172	+	0.883637i	$0.344913\pi$
$504$	0	0
$505$	0	0
$506$	−36.0000	−1.60040
$507$	1.00000	0.0444116
$508$	14.0000	0.621150
$509$	0	0		−	1.00000i	$-0.5\pi$
$509$	0	0			1.00000i	$0.5\pi$
$510$	0	0
$511$	12.0000	0.530849
$512$	32.0000	1.41421
$513$	0	0
$514$	34.0000	1.49968
$515$	0	0
$516$	2.00000	0.0880451
$517$	−16.0000	−0.703679
$518$	−32.0000	−1.40600
$519$	6.00000	0.263371
$520$	0	0
$521$	−3.00000	−0.131432	−0.0657162	−	0.997838i	$-0.520933\pi$
$521$	−3.00000	−0.131432	−0.0657162	+	0.997838i	$0.520933\pi$
$522$	−20.0000	−0.875376
$523$	36.0000	1.57417	0.787085	−	0.616844i	$-0.211589\pi$
$523$	36.0000	1.57417	0.787085	+	0.616844i	$0.211589\pi$
$524$	34.0000	1.48530
$525$	0	0
$526$	−8.00000	−0.348817
$527$	4.00000	0.174243
$528$	−8.00000	−0.348155
$529$	58.0000	2.52174
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−12.0000	−0.519778
$534$	−20.0000	−0.865485
$535$	0	0
$536$	0	0
$537$	15.0000	0.647298
$538$	−30.0000	−1.29339
$539$	−6.00000	−0.258438
$540$	0	0
$541$	−28.0000	−1.20381	−0.601907	−	0.798566i	$-0.705592\pi$
$541$	−28.0000	−1.20381	−0.601907	+	0.798566i	$0.705592\pi$
$542$	−56.0000	−2.40541
$543$	2.00000	0.0858282
$544$	−16.0000	−0.685994
$545$	0	0
$546$	−4.00000	−0.171184
$547$	12.0000	0.513083	0.256541	−	0.966533i	$-0.417417\pi$
$547$	12.0000	0.513083	0.256541	+	0.966533i	$0.417417\pi$
$548$	−16.0000	−0.683486
$549$	26.0000	1.10965
$550$	0	0
$551$	0	0
$552$	0	0
$553$	30.0000	1.27573
$554$	−46.0000	−1.95435
$555$	0	0
$556$	30.0000	1.27228
$557$	22.0000	0.932170	0.466085	−	0.884740i	$-0.345664\pi$
$557$	22.0000	0.932170	0.466085	+	0.884740i	$0.345664\pi$
$558$	−8.00000	−0.338667
$559$	−1.00000	−0.0422955
$560$	0	0
$561$	4.00000	0.168880
$562$	−36.0000	−1.51857
$563$	11.0000	0.463595	0.231797	−	0.972764i	$-0.425539\pi$
$563$	11.0000	0.463595	0.231797	+	0.972764i	$0.425539\pi$
$564$	−16.0000	−0.673722
$565$	0	0
$566$	−8.00000	−0.336265
$567$	2.00000	0.0839921
$568$	0	0
$569$	25.0000	1.04805	0.524027	−	0.851701i	$-0.324429\pi$
$569$	25.0000	1.04805	0.524027	+	0.851701i	$0.324429\pi$
$570$	0	0
$571$	12.0000	0.502184	0.251092	−	0.967963i	$-0.419210\pi$
$571$	12.0000	0.502184	0.251092	+	0.967963i	$0.419210\pi$
$572$	−4.00000	−0.167248
$573$	−3.00000	−0.125327
$574$	48.0000	2.00348
$575$	0	0
$576$	16.0000	0.666667
$577$	2.00000	0.0832611	0.0416305	−	0.999133i	$-0.486745\pi$
$577$	2.00000	0.0832611	0.0416305	+	0.999133i	$0.486745\pi$
$578$	−26.0000	−1.08146
$579$	16.0000	0.664937
$580$	0	0
$581$	−8.00000	−0.331896
$582$	−16.0000	−0.663221
$583$	22.0000	0.911147
$584$	0	0
$585$	0	0
$586$	−28.0000	−1.15667
$587$	−38.0000	−1.56843	−0.784214	−	0.620491i	$-0.786934\pi$
$587$	−38.0000	−1.56843	−0.784214	+	0.620491i	$0.786934\pi$
$588$	−6.00000	−0.247436
$589$	0	0
$590$	0	0
$591$	12.0000	0.493614
$592$	32.0000	1.31519
$593$	26.0000	1.06769	0.533846	−	0.845582i	$-0.320746\pi$
$593$	26.0000	1.06769	0.533846	+	0.845582i	$0.320746\pi$
$594$	−20.0000	−0.820610
$595$	0	0
$596$	0	0
$597$	−25.0000	−1.02318
$598$	18.0000	0.736075
$599$	5.00000	0.204294	0.102147	−	0.994769i	$-0.467429\pi$
$599$	5.00000	0.204294	0.102147	+	0.994769i	$0.467429\pi$
$600$	0	0
$601$	−23.0000	−0.938190	−0.469095	−	0.883148i	$-0.655420\pi$
$601$	−23.0000	−0.938190	−0.469095	+	0.883148i	$0.655420\pi$
$602$	4.00000	0.163028
$603$	−4.00000	−0.162893
$604$	−16.0000	−0.651031
$605$	0	0
$606$	−6.00000	−0.243733
$607$	−8.00000	−0.324710	−0.162355	−	0.986732i	$-0.551909\pi$
$607$	−8.00000	−0.324710	−0.162355	+	0.986732i	$0.551909\pi$
$608$	0	0
$609$	10.0000	0.405220
$610$	0	0
$611$	8.00000	0.323645
$612$	−8.00000	−0.323381
$613$	−24.0000	−0.969351	−0.484675	−	0.874694i	$-0.661062\pi$
$613$	−24.0000	−0.969351	−0.484675	+	0.874694i	$0.661062\pi$
$614$	4.00000	0.161427
$615$	0	0
$616$	0	0
$617$	−48.0000	−1.93241	−0.966204	−	0.257780i	$-0.917009\pi$
$617$	−48.0000	−1.93241	−0.966204	+	0.257780i	$0.917009\pi$
$618$	22.0000	0.884970
$619$	40.0000	1.60774	0.803868	−	0.594808i	$-0.202772\pi$
$619$	40.0000	1.60774	0.803868	+	0.594808i	$0.202772\pi$
$620$	0	0
$621$	45.0000	1.80579
$622$	−26.0000	−1.04251
$623$	−20.0000	−0.801283
$624$	4.00000	0.160128
$625$	0	0
$626$	−38.0000	−1.51879
$627$	0	0
$628$	−36.0000	−1.43656
$629$	−16.0000	−0.637962
$630$	0	0
$631$	−8.00000	−0.318475	−0.159237	−	0.987240i	$-0.550904\pi$
$631$	−8.00000	−0.318475	−0.159237	+	0.987240i	$0.550904\pi$
$632$	0	0
$633$	12.0000	0.476957
$634$	−36.0000	−1.42974
$635$	0	0
$636$	22.0000	0.872357
$637$	3.00000	0.118864
$638$	20.0000	0.791808
$639$	−24.0000	−0.949425
$640$	0	0
$641$	−13.0000	−0.513469	−0.256735	−	0.966482i	$-0.582647\pi$
$641$	−13.0000	−0.513469	−0.256735	+	0.966482i	$0.582647\pi$
$642$	14.0000	0.552536
$643$	26.0000	1.02534	0.512670	−	0.858586i	$-0.328656\pi$
$643$	26.0000	1.02534	0.512670	+	0.858586i	$0.328656\pi$
$644$	−36.0000	−1.41860
$645$	0	0
$646$	0	0
$647$	32.0000	1.25805	0.629025	−	0.777385i	$-0.283454\pi$
$647$	32.0000	1.25805	0.629025	+	0.777385i	$0.283454\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	4.00000	0.156772
$652$	−48.0000	−1.87983
$653$	6.00000	0.234798	0.117399	−	0.993085i	$-0.462544\pi$
$653$	6.00000	0.234798	0.117399	+	0.993085i	$0.462544\pi$
$654$	20.0000	0.782062
$655$	0	0
$656$	−48.0000	−1.87409
$657$	−12.0000	−0.468165
$658$	−32.0000	−1.24749
$659$	−15.0000	−0.584317	−0.292159	−	0.956370i	$-0.594373\pi$
$659$	−15.0000	−0.584317	−0.292159	+	0.956370i	$0.594373\pi$
$660$	0	0
$661$	2.00000	0.0777910	0.0388955	−	0.999243i	$-0.487616\pi$
$661$	2.00000	0.0777910	0.0388955	+	0.999243i	$0.487616\pi$
$662$	−36.0000	−1.39918
$663$	−2.00000	−0.0776736
$664$	0	0
$665$	0	0
$666$	32.0000	1.23997
$667$	−45.0000	−1.74241
$668$	24.0000	0.928588
$669$	−14.0000	−0.541271
$670$	0	0
$671$	−26.0000	−1.00372
$672$	−16.0000	−0.617213
$673$	6.00000	0.231283	0.115642	−	0.993291i	$-0.463108\pi$
$673$	6.00000	0.231283	0.115642	+	0.993291i	$0.463108\pi$
$674$	−26.0000	−1.00148
$675$	0	0
$676$	2.00000	0.0769231
$677$	27.0000	1.03769	0.518847	−	0.854867i	$-0.326361\pi$
$677$	27.0000	1.03769	0.518847	+	0.854867i	$0.326361\pi$
$678$	2.00000	0.0768095
$679$	−16.0000	−0.614024
$680$	0	0
$681$	−8.00000	−0.306561
$682$	8.00000	0.306336
$683$	16.0000	0.612223	0.306111	−	0.951996i	$-0.400972\pi$
$683$	16.0000	0.612223	0.306111	+	0.951996i	$0.400972\pi$
$684$	0	0
$685$	0	0
$686$	−40.0000	−1.52721
$687$	−10.0000	−0.381524
$688$	−4.00000	−0.152499
$689$	−11.0000	−0.419067
$690$	0	0
$691$	−8.00000	−0.304334	−0.152167	−	0.988355i	$-0.548625\pi$
$691$	−8.00000	−0.304334	−0.152167	+	0.988355i	$0.548625\pi$
$692$	12.0000	0.456172
$693$	−8.00000	−0.303895
$694$	34.0000	1.29062
$695$	0	0
$696$	0	0
$697$	24.0000	0.909065
$698$	−60.0000	−2.27103
$699$	1.00000	0.0378235
$700$	0	0
$701$	7.00000	0.264386	0.132193	−	0.991224i	$-0.457798\pi$
$701$	7.00000	0.264386	0.132193	+	0.991224i	$0.457798\pi$
$702$	10.0000	0.377426
$703$	0	0
$704$	−16.0000	−0.603023
$705$	0	0
$706$	−48.0000	−1.80650
$707$	−6.00000	−0.225653
$708$	0	0
$709$	−20.0000	−0.751116	−0.375558	−	0.926799i	$-0.622549\pi$
$709$	−20.0000	−0.751116	−0.375558	+	0.926799i	$0.622549\pi$
$710$	0	0
$711$	−30.0000	−1.12509
$712$	0	0
$713$	−18.0000	−0.674105
$714$	8.00000	0.299392
$715$	0	0
$716$	30.0000	1.12115
$717$	0	0
$718$	40.0000	1.49279
$719$	−5.00000	−0.186469	−0.0932343	−	0.995644i	$-0.529721\pi$
$719$	−5.00000	−0.186469	−0.0932343	+	0.995644i	$0.529721\pi$
$720$	0	0
$721$	22.0000	0.819323
$722$	−38.0000	−1.41421
$723$	22.0000	0.818189
$724$	4.00000	0.148659
$725$	0	0
$726$	−14.0000	−0.519589
$727$	−3.00000	−0.111264	−0.0556319	−	0.998451i	$-0.517717\pi$
$727$	−3.00000	−0.111264	−0.0556319	+	0.998451i	$0.517717\pi$
$728$	0	0
$729$	13.0000	0.481481
$730$	0	0
$731$	2.00000	0.0739727
$732$	−26.0000	−0.960988
$733$	16.0000	0.590973	0.295487	−	0.955347i	$-0.404518\pi$
$733$	16.0000	0.590973	0.295487	+	0.955347i	$0.404518\pi$
$734$	−46.0000	−1.69789
$735$	0	0
$736$	72.0000	2.65396
$737$	4.00000	0.147342
$738$	−48.0000	−1.76690
$739$	−30.0000	−1.10357	−0.551784	−	0.833987i	$-0.686053\pi$
$739$	−30.0000	−1.10357	−0.551784	+	0.833987i	$0.686053\pi$
$740$	0	0
$741$	0	0
$742$	44.0000	1.61529
$743$	6.00000	0.220119	0.110059	−	0.993925i	$-0.464896\pi$
$743$	6.00000	0.220119	0.110059	+	0.993925i	$0.464896\pi$
$744$	0	0
$745$	0	0
$746$	2.00000	0.0732252
$747$	8.00000	0.292705
$748$	8.00000	0.292509
$749$	14.0000	0.511549
$750$	0	0
$751$	−13.0000	−0.474377	−0.237188	−	0.971464i	$-0.576226\pi$
$751$	−13.0000	−0.474377	−0.237188	+	0.971464i	$0.576226\pi$
$752$	32.0000	1.16692
$753$	12.0000	0.437304
$754$	−10.0000	−0.364179
$755$	0	0
$756$	−20.0000	−0.727393
$757$	2.00000	0.0726912	0.0363456	−	0.999339i	$-0.488428\pi$
$757$	2.00000	0.0726912	0.0363456	+	0.999339i	$0.488428\pi$
$758$	20.0000	0.726433
$759$	−18.0000	−0.653359
$760$	0	0
$761$	12.0000	0.435000	0.217500	−	0.976060i	$-0.430210\pi$
$761$	12.0000	0.435000	0.217500	+	0.976060i	$0.430210\pi$
$762$	14.0000	0.507166
$763$	20.0000	0.724049
$764$	−6.00000	−0.217072
$765$	0	0
$766$	−48.0000	−1.73431
$767$	0	0
$768$	16.0000	0.577350
$769$	50.0000	1.80305	0.901523	−	0.432731i	$-0.142450\pi$
$769$	50.0000	1.80305	0.901523	+	0.432731i	$0.142450\pi$
$770$	0	0
$771$	17.0000	0.612240
$772$	32.0000	1.15171
$773$	−14.0000	−0.503545	−0.251773	−	0.967786i	$-0.581013\pi$
$773$	−14.0000	−0.503545	−0.251773	+	0.967786i	$0.581013\pi$
$774$	−4.00000	−0.143777
$775$	0	0
$776$	0	0
$777$	−16.0000	−0.573997
$778$	70.0000	2.50962
$779$	0	0
$780$	0	0
$781$	24.0000	0.858788
$782$	−36.0000	−1.28736
$783$	−25.0000	−0.893427
$784$	12.0000	0.428571
$785$	0	0
$786$	34.0000	1.21274
$787$	32.0000	1.14068	0.570338	−	0.821410i	$-0.306812\pi$
$787$	32.0000	1.14068	0.570338	+	0.821410i	$0.306812\pi$
$788$	24.0000	0.854965
$789$	−4.00000	−0.142404
$790$	0	0
$791$	2.00000	0.0711118
$792$	0	0
$793$	13.0000	0.461644
$794$	−36.0000	−1.27759
$795$	0	0
$796$	−50.0000	−1.77220
$797$	27.0000	0.956389	0.478195	−	0.878254i	$-0.341291\pi$
$797$	27.0000	0.956389	0.478195	+	0.878254i	$0.341291\pi$
$798$	0	0
$799$	−16.0000	−0.566039
$800$	0	0
$801$	20.0000	0.706665
$802$	44.0000	1.55369
$803$	12.0000	0.423471
$804$	4.00000	0.141069
$805$	0	0
$806$	−4.00000	−0.140894
$807$	−15.0000	−0.528025
$808$	0	0
$809$	10.0000	0.351581	0.175791	−	0.984428i	$-0.443752\pi$
$809$	10.0000	0.351581	0.175791	+	0.984428i	$0.443752\pi$
$810$	0	0
$811$	42.0000	1.47482	0.737410	−	0.675446i	$-0.236049\pi$
$811$	42.0000	1.47482	0.737410	+	0.675446i	$0.236049\pi$
$812$	20.0000	0.701862
$813$	−28.0000	−0.982003
$814$	−32.0000	−1.12160
$815$	0	0
$816$	−8.00000	−0.280056
$817$	0	0
$818$	−60.0000	−2.09785
$819$	4.00000	0.139771
$820$	0	0
$821$	2.00000	0.0698005	0.0349002	−	0.999391i	$-0.488889\pi$
$821$	2.00000	0.0698005	0.0349002	+	0.999391i	$0.488889\pi$
$822$	−16.0000	−0.558064
$823$	−19.0000	−0.662298	−0.331149	−	0.943578i	$-0.607436\pi$
$823$	−19.0000	−0.662298	−0.331149	+	0.943578i	$0.607436\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−8.00000	−0.278187	−0.139094	−	0.990279i	$-0.544419\pi$
$827$	−8.00000	−0.278187	−0.139094	+	0.990279i	$0.544419\pi$
$828$	36.0000	1.25109
$829$	25.0000	0.868286	0.434143	−	0.900844i	$-0.357051\pi$
$829$	25.0000	0.868286	0.434143	+	0.900844i	$0.357051\pi$
$830$	0	0
$831$	−23.0000	−0.797861
$832$	8.00000	0.277350
$833$	−6.00000	−0.207888
$834$	30.0000	1.03882
$835$	0	0
$836$	0	0
$837$	−10.0000	−0.345651
$838$	70.0000	2.41811
$839$	−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
$841$	−4.00000	−0.137931
$842$	44.0000	1.51634
$843$	−18.0000	−0.619953
$844$	24.0000	0.826114
$845$	0	0
$846$	32.0000	1.10018
$847$	−14.0000	−0.481046
$848$	−44.0000	−1.51097
$849$	−4.00000	−0.137280
$850$	0	0
$851$	72.0000	2.46813
$852$	24.0000	0.822226
$853$	36.0000	1.23262	0.616308	−	0.787505i	$-0.288628\pi$
$853$	36.0000	1.23262	0.616308	+	0.787505i	$0.288628\pi$
$854$	−52.0000	−1.77940
$855$	0	0
$856$	0	0
$857$	−3.00000	−0.102478	−0.0512390	−	0.998686i	$-0.516317\pi$
$857$	−3.00000	−0.102478	−0.0512390	+	0.998686i	$0.516317\pi$
$858$	−4.00000	−0.136558
$859$	−20.0000	−0.682391	−0.341196	−	0.939992i	$-0.610832\pi$
$859$	−20.0000	−0.682391	−0.341196	+	0.939992i	$0.610832\pi$
$860$	0	0
$861$	24.0000	0.817918
$862$	4.00000	0.136241
$863$	6.00000	0.204242	0.102121	−	0.994772i	$-0.467437\pi$
$863$	6.00000	0.204242	0.102121	+	0.994772i	$0.467437\pi$
$864$	40.0000	1.36083
$865$	0	0
$866$	2.00000	0.0679628
$867$	−13.0000	−0.441503
$868$	8.00000	0.271538
$869$	30.0000	1.01768
$870$	0	0
$871$	−2.00000	−0.0677674
$872$	0	0
$873$	16.0000	0.541518
$874$	0	0
$875$	0	0
$876$	12.0000	0.405442
$877$	−18.0000	−0.607817	−0.303908	−	0.952701i	$-0.598292\pi$
$877$	−18.0000	−0.607817	−0.303908	+	0.952701i	$0.598292\pi$
$878$	0	0
$879$	−14.0000	−0.472208
$880$	0	0
$881$	−43.0000	−1.44871	−0.724353	−	0.689429i	$-0.757862\pi$
$881$	−43.0000	−1.44871	−0.724353	+	0.689429i	$0.757862\pi$
$882$	12.0000	0.404061
$883$	41.0000	1.37976	0.689880	−	0.723924i	$-0.257663\pi$
$883$	41.0000	1.37976	0.689880	+	0.723924i	$0.257663\pi$
$884$	−4.00000	−0.134535
$885$	0	0
$886$	−8.00000	−0.268765
$887$	−8.00000	−0.268614	−0.134307	−	0.990940i	$-0.542881\pi$
$887$	−8.00000	−0.268614	−0.134307	+	0.990940i	$0.542881\pi$
$888$	0	0
$889$	14.0000	0.469545
$890$	0	0
$891$	2.00000	0.0670025
$892$	−28.0000	−0.937509
$893$	0	0
$894$	0	0
$895$	0	0
$896$	0	0
$897$	9.00000	0.300501
$898$	0	0
$899$	10.0000	0.333519
$900$	0	0
$901$	22.0000	0.732926
$902$	48.0000	1.59823
$903$	2.00000	0.0665558
$904$	0	0
$905$	0	0
$906$	−16.0000	−0.531564
$907$	17.0000	0.564476	0.282238	−	0.959344i	$-0.408923\pi$
$907$	17.0000	0.564476	0.282238	+	0.959344i	$0.408923\pi$
$908$	−16.0000	−0.530979
$909$	6.00000	0.199007
$910$	0	0
$911$	27.0000	0.894550	0.447275	−	0.894397i	$-0.352395\pi$
$911$	27.0000	0.894550	0.447275	+	0.894397i	$0.352395\pi$
$912$	0	0
$913$	−8.00000	−0.264761
$914$	64.0000	2.11693
$915$	0	0
$916$	−20.0000	−0.660819
$917$	34.0000	1.12278
$918$	−20.0000	−0.660098
$919$	20.0000	0.659739	0.329870	−	0.944027i	$-0.392995\pi$
$919$	20.0000	0.659739	0.329870	+	0.944027i	$0.392995\pi$
$920$	0	0
$921$	2.00000	0.0659022
$922$	−76.0000	−2.50293
$923$	−12.0000	−0.394985
$924$	8.00000	0.263181
$925$	0	0
$926$	12.0000	0.394344
$927$	−22.0000	−0.722575
$928$	−40.0000	−1.31306
$929$	40.0000	1.31236	0.656179	−	0.754606i	$-0.272172\pi$
$929$	40.0000	1.31236	0.656179	+	0.754606i	$0.272172\pi$
$930$	0	0
$931$	0	0
$932$	2.00000	0.0655122
$933$	−13.0000	−0.425601
$934$	34.0000	1.11251
$935$	0	0
$936$	0	0
$937$	−33.0000	−1.07806	−0.539032	−	0.842286i	$-0.681210\pi$
$937$	−33.0000	−1.07806	−0.539032	+	0.842286i	$0.681210\pi$
$938$	8.00000	0.261209
$939$	−19.0000	−0.620042
$940$	0	0
$941$	−18.0000	−0.586783	−0.293392	−	0.955992i	$-0.594784\pi$
$941$	−18.0000	−0.586783	−0.293392	+	0.955992i	$0.594784\pi$
$942$	−36.0000	−1.17294
$943$	−108.000	−3.51696
$944$	0	0
$945$	0	0
$946$	4.00000	0.130051
$947$	−48.0000	−1.55979	−0.779895	−	0.625910i	$-0.784728\pi$
$947$	−48.0000	−1.55979	−0.779895	+	0.625910i	$0.784728\pi$
$948$	30.0000	0.974355
$949$	−6.00000	−0.194768
$950$	0	0
$951$	−18.0000	−0.583690
$952$	0	0
$953$	−54.0000	−1.74923	−0.874616	−	0.484817i	$-0.838886\pi$
$953$	−54.0000	−1.74923	−0.874616	+	0.484817i	$0.838886\pi$
$954$	−44.0000	−1.42455
$955$	0	0
$956$	0	0
$957$	10.0000	0.323254
$958$	−60.0000	−1.93851
$959$	−16.0000	−0.516667
$960$	0	0
$961$	−27.0000	−0.870968
$962$	16.0000	0.515861
$963$	−14.0000	−0.451144
$964$	44.0000	1.41714
$965$	0	0
$966$	−36.0000	−1.15828
$967$	52.0000	1.67221	0.836104	−	0.548572i	$-0.184828\pi$
$967$	52.0000	1.67221	0.836104	+	0.548572i	$0.184828\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−28.0000	−0.898563	−0.449281	−	0.893390i	$-0.648320\pi$
$971$	−28.0000	−0.898563	−0.449281	+	0.893390i	$0.648320\pi$
$972$	32.0000	1.02640
$973$	30.0000	0.961756
$974$	64.0000	2.05069
$975$	0	0
$976$	52.0000	1.66448
$977$	−18.0000	−0.575871	−0.287936	−	0.957650i	$-0.592969\pi$
$977$	−18.0000	−0.575871	−0.287936	+	0.957650i	$0.592969\pi$
$978$	−48.0000	−1.53487
$979$	−20.0000	−0.639203
$980$	0	0
$981$	−20.0000	−0.638551
$982$	54.0000	1.72321
$983$	−4.00000	−0.127580	−0.0637901	−	0.997963i	$-0.520319\pi$
$983$	−4.00000	−0.127580	−0.0637901	+	0.997963i	$0.520319\pi$
$984$	0	0
$985$	0	0
$986$	20.0000	0.636930
$987$	−16.0000	−0.509286
$988$	0	0
$989$	−9.00000	−0.286183
$990$	0	0
$991$	−33.0000	−1.04828	−0.524140	−	0.851632i	$-0.675613\pi$
$991$	−33.0000	−1.04828	−0.524140	+	0.851632i	$0.675613\pi$
$992$	−16.0000	−0.508001
$993$	−18.0000	−0.571213
$994$	48.0000	1.52247
$995$	0	0
$996$	−8.00000	−0.253490
$997$	47.0000	1.48850	0.744252	−	0.667898i	$-0.232806\pi$
$997$	47.0000	1.48850	0.744252	+	0.667898i	$0.232806\pi$
$998$	20.0000	0.633089
$999$	40.0000	1.26554

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	325.2.a.e.1.1	yes	1
3.2	odd	2		2925.2.a.a.1.1		1
4.3	odd	2		5200.2.a.k.1.1		1
5.2	odd	4		325.2.b.a.274.2		2
5.3	odd	4		325.2.b.a.274.1		2
5.4	even	2		325.2.a.a.1.1	✓	1
13.12	even	2		4225.2.a.b.1.1		1
15.2	even	4		2925.2.c.b.2224.1		2
15.8	even	4		2925.2.c.b.2224.2		2
15.14	odd	2		2925.2.a.r.1.1		1
20.19	odd	2		5200.2.a.z.1.1		1
65.64	even	2		4225.2.a.p.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
325.2.a.a.1.1	✓	1	5.4	even	2
325.2.a.e.1.1	yes	1	1.1	even	1	trivial
325.2.b.a.274.1		2	5.3	odd	4
325.2.b.a.274.2		2	5.2	odd	4
2925.2.a.a.1.1		1	3.2	odd	2
2925.2.a.r.1.1		1	15.14	odd	2
2925.2.c.b.2224.1		2	15.2	even	4
2925.2.c.b.2224.2		2	15.8	even	4
4225.2.a.b.1.1		1	13.12	even	2
4225.2.a.p.1.1		1	65.64	even	2
5200.2.a.k.1.1		1	4.3	odd	2
5200.2.a.z.1.1		1	20.19	odd	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:	\( N \)	\(=\)	\( 325 = 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	325.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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