LMFDB - Embedded newform 5225.2.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5225,2,Mod(1,5225)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5225, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("5225.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5225 = 5^{2} \cdot 11 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5225.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:	$41.7218350561$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1045)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	5225.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{2} -1.00000 q^{4} +3.00000 q^{8} -3.00000 q^{9} +O(q^{10})$

$q-1.00000 q^{2} -1.00000 q^{4} +3.00000 q^{8} -3.00000 q^{9} -1.00000 q^{11} -2.00000 q^{13} -1.00000 q^{16} +6.00000 q^{17} +3.00000 q^{18} +1.00000 q^{19} +1.00000 q^{22} +8.00000 q^{23} +2.00000 q^{26} -6.00000 q^{29} +4.00000 q^{31} -5.00000 q^{32} -6.00000 q^{34} +3.00000 q^{36} +2.00000 q^{37} -1.00000 q^{38} -10.0000 q^{41} -4.00000 q^{43} +1.00000 q^{44} -8.00000 q^{46} -7.00000 q^{49} +2.00000 q^{52} +2.00000 q^{53} +6.00000 q^{58} -8.00000 q^{59} +14.0000 q^{61} -4.00000 q^{62} +7.00000 q^{64} -8.00000 q^{67} -6.00000 q^{68} -4.00000 q^{71} -9.00000 q^{72} -2.00000 q^{73} -2.00000 q^{74} -1.00000 q^{76} -16.0000 q^{79} +9.00000 q^{81} +10.0000 q^{82} +4.00000 q^{83} +4.00000 q^{86} -3.00000 q^{88} +10.0000 q^{89} -8.00000 q^{92} -10.0000 q^{97} +7.00000 q^{98} +3.00000 q^{99} +O(q^{100})$

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 $

$2$	−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		−	1.00000i	$-0.5\pi$
$3$	0	0			1.00000i	$0.5\pi$
$4$	−1.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	0	0		−	1.00000i	$-0.5\pi$
$7$	0	0			1.00000i	$0.5\pi$
$8$	3.00000	1.06066
$9$	−3.00000	−1.00000
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−2.00000	−0.554700	−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−2.00000	−0.554700	−0.277350	+	0.960769i	$0.589456\pi$
$14$	0	0
$15$	0	0
$16$	−1.00000	−0.250000
$17$	6.00000	1.45521	0.727607	−	0.685994i	$-0.240633\pi$
$17$	6.00000	1.45521	0.727607	+	0.685994i	$0.240633\pi$
$18$	3.00000	0.707107
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	0	0
$22$	1.00000	0.213201
$23$	8.00000	1.66812	0.834058	−	0.551677i	$-0.186012\pi$
$23$	8.00000	1.66812	0.834058	+	0.551677i	$0.186012\pi$
$24$	0	0
$25$	0	0
$26$	2.00000	0.392232
$27$	0	0
$28$	0	0
$29$	−6.00000	−1.11417	−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−6.00000	−1.11417	−0.557086	+	0.830455i	$0.688081\pi$
$30$	0	0
$31$	4.00000	0.718421	0.359211	−	0.933257i	$-0.383046\pi$
$31$	4.00000	0.718421	0.359211	+	0.933257i	$0.383046\pi$
$32$	−5.00000	−0.883883
$33$	0	0
$34$	−6.00000	−1.02899
$35$	0	0
$36$	3.00000	0.500000
$37$	2.00000	0.328798	0.164399	−	0.986394i	$-0.447432\pi$
$37$	2.00000	0.328798	0.164399	+	0.986394i	$0.447432\pi$
$38$	−1.00000	−0.162221
$39$	0	0
$40$	0	0
$41$	−10.0000	−1.56174	−0.780869	−	0.624695i	$-0.785223\pi$
$41$	−10.0000	−1.56174	−0.780869	+	0.624695i	$0.785223\pi$
$42$	0	0
$43$	−4.00000	−0.609994	−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000	−0.609994	−0.304997	+	0.952353i	$0.598656\pi$
$44$	1.00000	0.150756
$45$	0	0
$46$	−8.00000	−1.17954
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	0	0
$49$	−7.00000	−1.00000
$50$	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
$56$	0	0
$57$	0	0
$58$	6.00000	0.787839
$59$	−8.00000	−1.04151	−0.520756	−	0.853706i	$-0.674350\pi$
$59$	−8.00000	−1.04151	−0.520756	+	0.853706i	$0.674350\pi$
$60$	0	0
$61$	14.0000	1.79252	0.896258	−	0.443533i	$-0.146275\pi$
$61$	14.0000	1.79252	0.896258	+	0.443533i	$0.146275\pi$
$62$	−4.00000	−0.508001
$63$	0	0
$64$	7.00000	0.875000
$65$	0	0
$66$	0	0
$67$	−8.00000	−0.977356	−0.488678	−	0.872464i	$-0.662521\pi$
$67$	−8.00000	−0.977356	−0.488678	+	0.872464i	$0.662521\pi$
$68$	−6.00000	−0.727607
$69$	0	0
$70$	0	0
$71$	−4.00000	−0.474713	−0.237356	−	0.971423i	$-0.576281\pi$
$71$	−4.00000	−0.474713	−0.237356	+	0.971423i	$0.576281\pi$
$72$	−9.00000	−1.06066
$73$	−2.00000	−0.234082	−0.117041	−	0.993127i	$-0.537341\pi$
$73$	−2.00000	−0.234082	−0.117041	+	0.993127i	$0.537341\pi$
$74$	−2.00000	−0.232495
$75$	0	0
$76$	−1.00000	−0.114708
$77$	0	0
$78$	0	0
$79$	−16.0000	−1.80014	−0.900070	−	0.435745i	$-0.856485\pi$
$79$	−16.0000	−1.80014	−0.900070	+	0.435745i	$0.856485\pi$
$80$	0	0
$81$	9.00000	1.00000
$82$	10.0000	1.10432
$83$	4.00000	0.439057	0.219529	−	0.975606i	$-0.429548\pi$
$83$	4.00000	0.439057	0.219529	+	0.975606i	$0.429548\pi$
$84$	0	0
$85$	0	0
$86$	4.00000	0.431331
$87$	0	0
$88$	−3.00000	−0.319801
$89$	10.0000	1.06000	0.529999	−	0.847998i	$-0.322192\pi$
$89$	10.0000	1.06000	0.529999	+	0.847998i	$0.322192\pi$
$90$	0	0
$91$	0	0
$92$	−8.00000	−0.834058
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−10.0000	−1.01535	−0.507673	−	0.861550i	$-0.669494\pi$
$97$	−10.0000	−1.01535	−0.507673	+	0.861550i	$0.669494\pi$
$98$	7.00000	0.707107
$99$	3.00000	0.301511
$100$	0	0
$101$	14.0000	1.39305	0.696526	−	0.717532i	$-0.254728\pi$
$101$	14.0000	1.39305	0.696526	+	0.717532i	$0.254728\pi$
$102$	0	0
$103$	4.00000	0.394132	0.197066	−	0.980390i	$-0.436859\pi$
$103$	4.00000	0.394132	0.197066	+	0.980390i	$0.436859\pi$
$104$	−6.00000	−0.588348
$105$	0	0
$106$	−2.00000	−0.194257
$107$	12.0000	1.16008	0.580042	−	0.814587i	$-0.303036\pi$
$107$	12.0000	1.16008	0.580042	+	0.814587i	$0.303036\pi$
$108$	0	0
$109$	−6.00000	−0.574696	−0.287348	−	0.957826i	$-0.592774\pi$
$109$	−6.00000	−0.574696	−0.287348	+	0.957826i	$0.592774\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	6.00000	0.564433	0.282216	−	0.959351i	$-0.408930\pi$
$113$	6.00000	0.564433	0.282216	+	0.959351i	$0.408930\pi$
$114$	0	0
$115$	0	0
$116$	6.00000	0.557086
$117$	6.00000	0.554700
$118$	8.00000	0.736460
$119$	0	0
$120$	0	0
$121$	1.00000	0.0909091
$122$	−14.0000	−1.26750
$123$	0	0
$124$	−4.00000	−0.359211
$125$	0	0
$126$	0	0
$127$	8.00000	0.709885	0.354943	−	0.934888i	$-0.384500\pi$
$127$	8.00000	0.709885	0.354943	+	0.934888i	$0.384500\pi$
$128$	3.00000	0.265165
$129$	0	0
$130$	0	0
$131$	12.0000	1.04844	0.524222	−	0.851581i	$-0.324356\pi$
$131$	12.0000	1.04844	0.524222	+	0.851581i	$0.324356\pi$
$132$	0	0
$133$	0	0
$134$	8.00000	0.691095
$135$	0	0
$136$	18.0000	1.54349
$137$	6.00000	0.512615	0.256307	−	0.966595i	$-0.417494\pi$
$137$	6.00000	0.512615	0.256307	+	0.966595i	$0.417494\pi$
$138$	0	0
$139$	12.0000	1.01783	0.508913	−	0.860818i	$-0.330047\pi$
$139$	12.0000	1.01783	0.508913	+	0.860818i	$0.330047\pi$
$140$	0	0
$141$	0	0
$142$	4.00000	0.335673
$143$	2.00000	0.167248
$144$	3.00000	0.250000
$145$	0	0
$146$	2.00000	0.165521
$147$	0	0
$148$	−2.00000	−0.164399
$149$	14.0000	1.14692	0.573462	−	0.819232i	$-0.305600\pi$
$149$	14.0000	1.14692	0.573462	+	0.819232i	$0.305600\pi$
$150$	0	0
$151$	−16.0000	−1.30206	−0.651031	−	0.759051i	$-0.725663\pi$
$151$	−16.0000	−1.30206	−0.651031	+	0.759051i	$0.725663\pi$
$152$	3.00000	0.243332
$153$	−18.0000	−1.45521
$154$	0	0
$155$	0	0
$156$	0	0
$157$	2.00000	0.159617	0.0798087	−	0.996810i	$-0.474569\pi$
$157$	2.00000	0.159617	0.0798087	+	0.996810i	$0.474569\pi$
$158$	16.0000	1.27289
$159$	0	0
$160$	0	0
$161$	0	0
$162$	−9.00000	−0.707107
$163$	−4.00000	−0.313304	−0.156652	−	0.987654i	$-0.550070\pi$
$163$	−4.00000	−0.313304	−0.156652	+	0.987654i	$0.550070\pi$
$164$	10.0000	0.780869
$165$	0	0
$166$	−4.00000	−0.310460
$167$	−16.0000	−1.23812	−0.619059	−	0.785345i	$-0.712486\pi$
$167$	−16.0000	−1.23812	−0.619059	+	0.785345i	$0.712486\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	0	0
$171$	−3.00000	−0.229416
$172$	4.00000	0.304997
$173$	−18.0000	−1.36851	−0.684257	−	0.729241i	$-0.739873\pi$
$173$	−18.0000	−1.36851	−0.684257	+	0.729241i	$0.739873\pi$
$174$	0	0
$175$	0	0
$176$	1.00000	0.0753778
$177$	0	0
$178$	−10.0000	−0.749532
$179$	16.0000	1.19590	0.597948	−	0.801535i	$-0.295983\pi$
$179$	16.0000	1.19590	0.597948	+	0.801535i	$0.295983\pi$
$180$	0	0
$181$	14.0000	1.04061	0.520306	−	0.853980i	$-0.325818\pi$
$181$	14.0000	1.04061	0.520306	+	0.853980i	$0.325818\pi$
$182$	0	0
$183$	0	0
$184$	24.0000	1.76930
$185$	0	0
$186$	0	0
$187$	−6.00000	−0.438763
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−16.0000	−1.15772	−0.578860	−	0.815427i	$-0.696502\pi$
$191$	−16.0000	−1.15772	−0.578860	+	0.815427i	$0.696502\pi$
$192$	0	0
$193$	2.00000	0.143963	0.0719816	−	0.997406i	$-0.477068\pi$
$193$	2.00000	0.143963	0.0719816	+	0.997406i	$0.477068\pi$
$194$	10.0000	0.717958
$195$	0	0
$196$	7.00000	0.500000
$197$	10.0000	0.712470	0.356235	−	0.934396i	$-0.384060\pi$
$197$	10.0000	0.712470	0.356235	+	0.934396i	$0.384060\pi$
$198$	−3.00000	−0.213201
$199$	0	0		−	1.00000i	$-0.5\pi$
$199$	0	0			1.00000i	$0.5\pi$
$200$	0	0
$201$	0	0
$202$	−14.0000	−0.985037
$203$	0	0
$204$	0	0
$205$	0	0
$206$	−4.00000	−0.278693
$207$	−24.0000	−1.66812
$208$	2.00000	0.138675
$209$	−1.00000	−0.0691714
$210$	0	0
$211$	4.00000	0.275371	0.137686	−	0.990476i	$-0.456034\pi$
$211$	4.00000	0.275371	0.137686	+	0.990476i	$0.456034\pi$
$212$	−2.00000	−0.137361
$213$	0	0
$214$	−12.0000	−0.820303
$215$	0	0
$216$	0	0
$217$	0	0
$218$	6.00000	0.406371
$219$	0	0
$220$	0	0
$221$	−12.0000	−0.807207
$222$	0	0
$223$	4.00000	0.267860	0.133930	−	0.990991i	$-0.457240\pi$
$223$	4.00000	0.267860	0.133930	+	0.990991i	$0.457240\pi$
$224$	0	0
$225$	0	0
$226$	−6.00000	−0.399114
$227$	−4.00000	−0.265489	−0.132745	−	0.991150i	$-0.542379\pi$
$227$	−4.00000	−0.265489	−0.132745	+	0.991150i	$0.542379\pi$
$228$	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
$231$	0	0
$232$	−18.0000	−1.18176
$233$	6.00000	0.393073	0.196537	−	0.980497i	$-0.437031\pi$
$233$	6.00000	0.393073	0.196537	+	0.980497i	$0.437031\pi$
$234$	−6.00000	−0.392232
$235$	0	0
$236$	8.00000	0.520756
$237$	0	0
$238$	0	0
$239$	−16.0000	−1.03495	−0.517477	−	0.855697i	$-0.673129\pi$
$239$	−16.0000	−1.03495	−0.517477	+	0.855697i	$0.673129\pi$
$240$	0	0
$241$	22.0000	1.41714	0.708572	−	0.705638i	$-0.249340\pi$
$241$	22.0000	1.41714	0.708572	+	0.705638i	$0.249340\pi$
$242$	−1.00000	−0.0642824
$243$	0	0
$244$	−14.0000	−0.896258
$245$	0	0
$246$	0	0
$247$	−2.00000	−0.127257
$248$	12.0000	0.762001
$249$	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
$253$	−8.00000	−0.502956
$254$	−8.00000	−0.501965
$255$	0	0
$256$	−17.0000	−1.06250
$257$	30.0000	1.87135	0.935674	−	0.352865i	$-0.114792\pi$
$257$	30.0000	1.87135	0.935674	+	0.352865i	$0.114792\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	18.0000	1.11417
$262$	−12.0000	−0.741362
$263$	24.0000	1.47990	0.739952	−	0.672660i	$-0.234848\pi$
$263$	24.0000	1.47990	0.739952	+	0.672660i	$0.234848\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	8.00000	0.488678
$269$	6.00000	0.365826	0.182913	−	0.983129i	$-0.441447\pi$
$269$	6.00000	0.365826	0.182913	+	0.983129i	$0.441447\pi$
$270$	0	0
$271$	0	0		−	1.00000i	$-0.5\pi$
$271$	0	0			1.00000i	$0.5\pi$
$272$	−6.00000	−0.363803
$273$	0	0
$274$	−6.00000	−0.362473
$275$	0	0
$276$	0	0
$277$	2.00000	0.120168	0.0600842	−	0.998193i	$-0.480863\pi$
$277$	2.00000	0.120168	0.0600842	+	0.998193i	$0.480863\pi$
$278$	−12.0000	−0.719712
$279$	−12.0000	−0.718421
$280$	0	0
$281$	30.0000	1.78965	0.894825	−	0.446417i	$-0.147300\pi$
$281$	30.0000	1.78965	0.894825	+	0.446417i	$0.147300\pi$
$282$	0	0
$283$	20.0000	1.18888	0.594438	−	0.804141i	$-0.297374\pi$
$283$	20.0000	1.18888	0.594438	+	0.804141i	$0.297374\pi$
$284$	4.00000	0.237356
$285$	0	0
$286$	−2.00000	−0.118262
$287$	0	0
$288$	15.0000	0.883883
$289$	19.0000	1.11765
$290$	0	0
$291$	0	0
$292$	2.00000	0.117041
$293$	14.0000	0.817889	0.408944	−	0.912559i	$-0.365897\pi$
$293$	14.0000	0.817889	0.408944	+	0.912559i	$0.365897\pi$
$294$	0	0
$295$	0	0
$296$	6.00000	0.348743
$297$	0	0
$298$	−14.0000	−0.810998
$299$	−16.0000	−0.925304
$300$	0	0
$301$	0	0
$302$	16.0000	0.920697
$303$	0	0
$304$	−1.00000	−0.0573539
$305$	0	0
$306$	18.0000	1.02899
$307$	4.00000	0.228292	0.114146	−	0.993464i	$-0.463587\pi$
$307$	4.00000	0.228292	0.114146	+	0.993464i	$0.463587\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	0	0
$313$	22.0000	1.24351	0.621757	−	0.783210i	$-0.286419\pi$
$313$	22.0000	1.24351	0.621757	+	0.783210i	$0.286419\pi$
$314$	−2.00000	−0.112867
$315$	0	0
$316$	16.0000	0.900070
$317$	−30.0000	−1.68497	−0.842484	−	0.538721i	$-0.818908\pi$
$317$	−30.0000	−1.68497	−0.842484	+	0.538721i	$0.818908\pi$
$318$	0	0
$319$	6.00000	0.335936
$320$	0	0
$321$	0	0
$322$	0	0
$323$	6.00000	0.333849
$324$	−9.00000	−0.500000
$325$	0	0
$326$	4.00000	0.221540
$327$	0	0
$328$	−30.0000	−1.65647
$329$	0	0
$330$	0	0
$331$	−32.0000	−1.75888	−0.879440	−	0.476011i	$-0.842082\pi$
$331$	−32.0000	−1.75888	−0.879440	+	0.476011i	$0.842082\pi$
$332$	−4.00000	−0.219529
$333$	−6.00000	−0.328798
$334$	16.0000	0.875481
$335$	0	0
$336$	0	0
$337$	18.0000	0.980522	0.490261	−	0.871576i	$-0.336901\pi$
$337$	18.0000	0.980522	0.490261	+	0.871576i	$0.336901\pi$
$338$	9.00000	0.489535
$339$	0	0
$340$	0	0
$341$	−4.00000	−0.216612
$342$	3.00000	0.162221
$343$	0	0
$344$	−12.0000	−0.646997
$345$	0	0
$346$	18.0000	0.967686
$347$	4.00000	0.214731	0.107366	−	0.994220i	$-0.465758\pi$
$347$	4.00000	0.214731	0.107366	+	0.994220i	$0.465758\pi$
$348$	0	0
$349$	−10.0000	−0.535288	−0.267644	−	0.963518i	$-0.586245\pi$
$349$	−10.0000	−0.535288	−0.267644	+	0.963518i	$0.586245\pi$
$350$	0	0
$351$	0	0
$352$	5.00000	0.266501
$353$	−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
$356$	−10.0000	−0.529999
$357$	0	0
$358$	−16.0000	−0.845626
$359$	−8.00000	−0.422224	−0.211112	−	0.977462i	$-0.567708\pi$
$359$	−8.00000	−0.422224	−0.211112	+	0.977462i	$0.567708\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	−14.0000	−0.735824
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	32.0000	1.67039	0.835193	−	0.549957i	$-0.185356\pi$
$367$	32.0000	1.67039	0.835193	+	0.549957i	$0.185356\pi$
$368$	−8.00000	−0.417029
$369$	30.0000	1.56174
$370$	0	0
$371$	0	0
$372$	0	0
$373$	6.00000	0.310668	0.155334	−	0.987862i	$-0.450355\pi$
$373$	6.00000	0.310668	0.155334	+	0.987862i	$0.450355\pi$
$374$	6.00000	0.310253
$375$	0	0
$376$	0	0
$377$	12.0000	0.618031
$378$	0	0
$379$	32.0000	1.64373	0.821865	−	0.569683i	$-0.192934\pi$
$379$	32.0000	1.64373	0.821865	+	0.569683i	$0.192934\pi$
$380$	0	0
$381$	0	0
$382$	16.0000	0.818631
$383$	36.0000	1.83951	0.919757	−	0.392488i	$-0.128386\pi$
$383$	36.0000	1.83951	0.919757	+	0.392488i	$0.128386\pi$
$384$	0	0
$385$	0	0
$386$	−2.00000	−0.101797
$387$	12.0000	0.609994
$388$	10.0000	0.507673
$389$	6.00000	0.304212	0.152106	−	0.988364i	$-0.451394\pi$
$389$	6.00000	0.304212	0.152106	+	0.988364i	$0.451394\pi$
$390$	0	0
$391$	48.0000	2.42746
$392$	−21.0000	−1.06066
$393$	0	0
$394$	−10.0000	−0.503793
$395$	0	0
$396$	−3.00000	−0.150756
$397$	18.0000	0.903394	0.451697	−	0.892171i	$-0.350819\pi$
$397$	18.0000	0.903394	0.451697	+	0.892171i	$0.350819\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−22.0000	−1.09863	−0.549314	−	0.835616i	$-0.685111\pi$
$401$	−22.0000	−1.09863	−0.549314	+	0.835616i	$0.685111\pi$
$402$	0	0
$403$	−8.00000	−0.398508
$404$	−14.0000	−0.696526
$405$	0	0
$406$	0	0
$407$	−2.00000	−0.0991363
$408$	0	0
$409$	30.0000	1.48340	0.741702	−	0.670729i	$-0.234019\pi$
$409$	30.0000	1.48340	0.741702	+	0.670729i	$0.234019\pi$
$410$	0	0
$411$	0	0
$412$	−4.00000	−0.197066
$413$	0	0
$414$	24.0000	1.17954
$415$	0	0
$416$	10.0000	0.490290
$417$	0	0
$418$	1.00000	0.0489116
$419$	−28.0000	−1.36789	−0.683945	−	0.729534i	$-0.739737\pi$
$419$	−28.0000	−1.36789	−0.683945	+	0.729534i	$0.739737\pi$
$420$	0	0
$421$	6.00000	0.292422	0.146211	−	0.989253i	$-0.453292\pi$
$421$	6.00000	0.292422	0.146211	+	0.989253i	$0.453292\pi$
$422$	−4.00000	−0.194717
$423$	0	0
$424$	6.00000	0.291386
$425$	0	0
$426$	0	0
$427$	0	0
$428$	−12.0000	−0.580042
$429$	0	0
$430$	0	0
$431$	0	0		−	1.00000i	$-0.5\pi$
$431$	0	0			1.00000i	$0.5\pi$
$432$	0	0
$433$	22.0000	1.05725	0.528626	−	0.848855i	$-0.322707\pi$
$433$	22.0000	1.05725	0.528626	+	0.848855i	$0.322707\pi$
$434$	0	0
$435$	0	0
$436$	6.00000	0.287348
$437$	8.00000	0.382692
$438$	0	0
$439$	−8.00000	−0.381819	−0.190910	−	0.981608i	$-0.561144\pi$
$439$	−8.00000	−0.381819	−0.190910	+	0.981608i	$0.561144\pi$
$440$	0	0
$441$	21.0000	1.00000
$442$	12.0000	0.570782
$443$	28.0000	1.33032	0.665160	−	0.746701i	$-0.268363\pi$
$443$	28.0000	1.33032	0.665160	+	0.746701i	$0.268363\pi$
$444$	0	0
$445$	0	0
$446$	−4.00000	−0.189405
$447$	0	0
$448$	0	0
$449$	2.00000	0.0943858	0.0471929	−	0.998886i	$-0.484972\pi$
$449$	2.00000	0.0943858	0.0471929	+	0.998886i	$0.484972\pi$
$450$	0	0
$451$	10.0000	0.470882
$452$	−6.00000	−0.282216
$453$	0	0
$454$	4.00000	0.187729
$455$	0	0
$456$	0	0
$457$	−10.0000	−0.467780	−0.233890	−	0.972263i	$-0.575146\pi$
$457$	−10.0000	−0.467780	−0.233890	+	0.972263i	$0.575146\pi$
$458$	10.0000	0.467269
$459$	0	0
$460$	0	0
$461$	−10.0000	−0.465746	−0.232873	−	0.972507i	$-0.574813\pi$
$461$	−10.0000	−0.465746	−0.232873	+	0.972507i	$0.574813\pi$
$462$	0	0
$463$	24.0000	1.11537	0.557687	−	0.830051i	$-0.311689\pi$
$463$	24.0000	1.11537	0.557687	+	0.830051i	$0.311689\pi$
$464$	6.00000	0.278543
$465$	0	0
$466$	−6.00000	−0.277945
$467$	12.0000	0.555294	0.277647	−	0.960683i	$-0.410445\pi$
$467$	12.0000	0.555294	0.277647	+	0.960683i	$0.410445\pi$
$468$	−6.00000	−0.277350
$469$	0	0
$470$	0	0
$471$	0	0
$472$	−24.0000	−1.10469
$473$	4.00000	0.183920
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−6.00000	−0.274721
$478$	16.0000	0.731823
$479$	24.0000	1.09659	0.548294	−	0.836286i	$-0.315277\pi$
$479$	24.0000	1.09659	0.548294	+	0.836286i	$0.315277\pi$
$480$	0	0
$481$	−4.00000	−0.182384
$482$	−22.0000	−1.00207
$483$	0	0
$484$	−1.00000	−0.0454545
$485$	0	0
$486$	0	0
$487$	−4.00000	−0.181257	−0.0906287	−	0.995885i	$-0.528888\pi$
$487$	−4.00000	−0.181257	−0.0906287	+	0.995885i	$0.528888\pi$
$488$	42.0000	1.90125
$489$	0	0
$490$	0	0
$491$	−28.0000	−1.26362	−0.631811	−	0.775122i	$-0.717688\pi$
$491$	−28.0000	−1.26362	−0.631811	+	0.775122i	$0.717688\pi$
$492$	0	0
$493$	−36.0000	−1.62136
$494$	2.00000	0.0899843
$495$	0	0
$496$	−4.00000	−0.179605
$497$	0	0
$498$	0	0
$499$	36.0000	1.61158	0.805791	−	0.592200i	$-0.201741\pi$
$499$	36.0000	1.61158	0.805791	+	0.592200i	$0.201741\pi$
$500$	0	0
$501$	0	0
$502$	−12.0000	−0.535586
$503$	16.0000	0.713405	0.356702	−	0.934218i	$-0.383901\pi$
$503$	16.0000	0.713405	0.356702	+	0.934218i	$0.383901\pi$
$504$	0	0
$505$	0	0
$506$	8.00000	0.355643
$507$	0	0
$508$	−8.00000	−0.354943
$509$	−10.0000	−0.443242	−0.221621	−	0.975133i	$-0.571135\pi$
$509$	−10.0000	−0.443242	−0.221621	+	0.975133i	$0.571135\pi$
$510$	0	0
$511$	0	0
$512$	11.0000	0.486136
$513$	0	0
$514$	−30.0000	−1.32324
$515$	0	0
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−30.0000	−1.31432	−0.657162	−	0.753749i	$-0.728243\pi$
$521$	−30.0000	−1.31432	−0.657162	+	0.753749i	$0.728243\pi$
$522$	−18.0000	−0.787839
$523$	20.0000	0.874539	0.437269	−	0.899331i	$-0.355946\pi$
$523$	20.0000	0.874539	0.437269	+	0.899331i	$0.355946\pi$
$524$	−12.0000	−0.524222
$525$	0	0
$526$	−24.0000	−1.04645
$527$	24.0000	1.04546
$528$	0	0
$529$	41.0000	1.78261
$530$	0	0
$531$	24.0000	1.04151
$532$	0	0
$533$	20.0000	0.866296
$534$	0	0
$535$	0	0
$536$	−24.0000	−1.03664
$537$	0	0
$538$	−6.00000	−0.258678
$539$	7.00000	0.301511
$540$	0	0
$541$	−34.0000	−1.46177	−0.730887	−	0.682498i	$-0.760893\pi$
$541$	−34.0000	−1.46177	−0.730887	+	0.682498i	$0.760893\pi$
$542$	0	0
$543$	0	0
$544$	−30.0000	−1.28624
$545$	0	0
$546$	0	0
$547$	36.0000	1.53925	0.769624	−	0.638497i	$-0.220443\pi$
$547$	36.0000	1.53925	0.769624	+	0.638497i	$0.220443\pi$
$548$	−6.00000	−0.256307
$549$	−42.0000	−1.79252
$550$	0	0
$551$	−6.00000	−0.255609
$552$	0	0
$553$	0	0
$554$	−2.00000	−0.0849719
$555$	0	0
$556$	−12.0000	−0.508913
$557$	−22.0000	−0.932170	−0.466085	−	0.884740i	$-0.654336\pi$
$557$	−22.0000	−0.932170	−0.466085	+	0.884740i	$0.654336\pi$
$558$	12.0000	0.508001
$559$	8.00000	0.338364
$560$	0	0
$561$	0	0
$562$	−30.0000	−1.26547
$563$	−36.0000	−1.51722	−0.758610	−	0.651546i	$-0.774121\pi$
$563$	−36.0000	−1.51722	−0.758610	+	0.651546i	$0.774121\pi$
$564$	0	0
$565$	0	0
$566$	−20.0000	−0.840663
$567$	0	0
$568$	−12.0000	−0.503509
$569$	−10.0000	−0.419222	−0.209611	−	0.977785i	$-0.567220\pi$
$569$	−10.0000	−0.419222	−0.209611	+	0.977785i	$0.567220\pi$
$570$	0	0
$571$	12.0000	0.502184	0.251092	−	0.967963i	$-0.419210\pi$
$571$	12.0000	0.502184	0.251092	+	0.967963i	$0.419210\pi$
$572$	−2.00000	−0.0836242
$573$	0	0
$574$	0	0
$575$	0	0
$576$	−21.0000	−0.875000
$577$	46.0000	1.91501	0.957503	−	0.288425i	$-0.0931316\pi$
$577$	46.0000	1.91501	0.957503	+	0.288425i	$0.0931316\pi$
$578$	−19.0000	−0.790296
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−2.00000	−0.0828315
$584$	−6.00000	−0.248282
$585$	0	0
$586$	−14.0000	−0.578335
$587$	−36.0000	−1.48588	−0.742940	−	0.669359i	$-0.766569\pi$
$587$	−36.0000	−1.48588	−0.742940	+	0.669359i	$0.766569\pi$
$588$	0	0
$589$	4.00000	0.164817
$590$	0	0
$591$	0	0
$592$	−2.00000	−0.0821995
$593$	−34.0000	−1.39621	−0.698106	−	0.715994i	$-0.745974\pi$
$593$	−34.0000	−1.39621	−0.698106	+	0.715994i	$0.745974\pi$
$594$	0	0
$595$	0	0
$596$	−14.0000	−0.573462
$597$	0	0
$598$	16.0000	0.654289
$599$	12.0000	0.490307	0.245153	−	0.969484i	$-0.421162\pi$
$599$	12.0000	0.490307	0.245153	+	0.969484i	$0.421162\pi$
$600$	0	0
$601$	14.0000	0.571072	0.285536	−	0.958368i	$-0.407828\pi$
$601$	14.0000	0.571072	0.285536	+	0.958368i	$0.407828\pi$
$602$	0	0
$603$	24.0000	0.977356
$604$	16.0000	0.651031
$605$	0	0
$606$	0	0
$607$	32.0000	1.29884	0.649420	−	0.760430i	$-0.275012\pi$
$607$	32.0000	1.29884	0.649420	+	0.760430i	$0.275012\pi$
$608$	−5.00000	−0.202777
$609$	0	0
$610$	0	0
$611$	0	0
$612$	18.0000	0.727607
$613$	−46.0000	−1.85792	−0.928961	−	0.370177i	$-0.879297\pi$
$613$	−46.0000	−1.85792	−0.928961	+	0.370177i	$0.879297\pi$
$614$	−4.00000	−0.161427
$615$	0	0
$616$	0	0
$617$	22.0000	0.885687	0.442843	−	0.896599i	$-0.353970\pi$
$617$	22.0000	0.885687	0.442843	+	0.896599i	$0.353970\pi$
$618$	0	0
$619$	4.00000	0.160774	0.0803868	−	0.996764i	$-0.474384\pi$
$619$	4.00000	0.160774	0.0803868	+	0.996764i	$0.474384\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	0	0
$626$	−22.0000	−0.879297
$627$	0	0
$628$	−2.00000	−0.0798087
$629$	12.0000	0.478471
$630$	0	0
$631$	16.0000	0.636950	0.318475	−	0.947931i	$-0.396829\pi$
$631$	16.0000	0.636950	0.318475	+	0.947931i	$0.396829\pi$
$632$	−48.0000	−1.90934
$633$	0	0
$634$	30.0000	1.19145
$635$	0	0
$636$	0	0
$637$	14.0000	0.554700
$638$	−6.00000	−0.237542
$639$	12.0000	0.474713
$640$	0	0
$641$	−14.0000	−0.552967	−0.276483	−	0.961019i	$-0.589169\pi$
$641$	−14.0000	−0.552967	−0.276483	+	0.961019i	$0.589169\pi$
$642$	0	0
$643$	−20.0000	−0.788723	−0.394362	−	0.918955i	$-0.629034\pi$
$643$	−20.0000	−0.788723	−0.394362	+	0.918955i	$0.629034\pi$
$644$	0	0
$645$	0	0
$646$	−6.00000	−0.236067
$647$	16.0000	0.629025	0.314512	−	0.949253i	$-0.398159\pi$
$647$	16.0000	0.629025	0.314512	+	0.949253i	$0.398159\pi$
$648$	27.0000	1.06066
$649$	8.00000	0.314027
$650$	0	0
$651$	0	0
$652$	4.00000	0.156652
$653$	34.0000	1.33052	0.665261	−	0.746611i	$-0.268320\pi$
$653$	34.0000	1.33052	0.665261	+	0.746611i	$0.268320\pi$
$654$	0	0
$655$	0	0
$656$	10.0000	0.390434
$657$	6.00000	0.234082
$658$	0	0
$659$	12.0000	0.467454	0.233727	−	0.972302i	$-0.424908\pi$
$659$	12.0000	0.467454	0.233727	+	0.972302i	$0.424908\pi$
$660$	0	0
$661$	−10.0000	−0.388955	−0.194477	−	0.980907i	$-0.562301\pi$
$661$	−10.0000	−0.388955	−0.194477	+	0.980907i	$0.562301\pi$
$662$	32.0000	1.24372
$663$	0	0
$664$	12.0000	0.465690
$665$	0	0
$666$	6.00000	0.232495
$667$	−48.0000	−1.85857
$668$	16.0000	0.619059
$669$	0	0
$670$	0	0
$671$	−14.0000	−0.540464
$672$	0	0
$673$	−22.0000	−0.848038	−0.424019	−	0.905653i	$-0.639381\pi$
$673$	−22.0000	−0.848038	−0.424019	+	0.905653i	$0.639381\pi$
$674$	−18.0000	−0.693334
$675$	0	0
$676$	9.00000	0.346154
$677$	−34.0000	−1.30673	−0.653363	−	0.757045i	$-0.726642\pi$
$677$	−34.0000	−1.30673	−0.653363	+	0.757045i	$0.726642\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	4.00000	0.153168
$683$	40.0000	1.53056	0.765279	−	0.643699i	$-0.222601\pi$
$683$	40.0000	1.53056	0.765279	+	0.643699i	$0.222601\pi$
$684$	3.00000	0.114708
$685$	0	0
$686$	0	0
$687$	0	0
$688$	4.00000	0.152499
$689$	−4.00000	−0.152388
$690$	0	0
$691$	20.0000	0.760836	0.380418	−	0.924815i	$-0.375780\pi$
$691$	20.0000	0.760836	0.380418	+	0.924815i	$0.375780\pi$
$692$	18.0000	0.684257
$693$	0	0
$694$	−4.00000	−0.151838
$695$	0	0
$696$	0	0
$697$	−60.0000	−2.27266
$698$	10.0000	0.378506
$699$	0	0
$700$	0	0
$701$	−26.0000	−0.982006	−0.491003	−	0.871158i	$-0.663370\pi$
$701$	−26.0000	−0.982006	−0.491003	+	0.871158i	$0.663370\pi$
$702$	0	0
$703$	2.00000	0.0754314
$704$	−7.00000	−0.263822
$705$	0	0
$706$	18.0000	0.677439
$707$	0	0
$708$	0	0
$709$	−10.0000	−0.375558	−0.187779	−	0.982211i	$-0.560129\pi$
$709$	−10.0000	−0.375558	−0.187779	+	0.982211i	$0.560129\pi$
$710$	0	0
$711$	48.0000	1.80014
$712$	30.0000	1.12430
$713$	32.0000	1.19841
$714$	0	0
$715$	0	0
$716$	−16.0000	−0.597948
$717$	0	0
$718$	8.00000	0.298557
$719$	−24.0000	−0.895049	−0.447524	−	0.894272i	$-0.647694\pi$
$719$	−24.0000	−0.895049	−0.447524	+	0.894272i	$0.647694\pi$
$720$	0	0
$721$	0	0
$722$	−1.00000	−0.0372161
$723$	0	0
$724$	−14.0000	−0.520306
$725$	0	0
$726$	0	0
$727$	32.0000	1.18681	0.593407	−	0.804902i	$-0.297782\pi$
$727$	32.0000	1.18681	0.593407	+	0.804902i	$0.297782\pi$
$728$	0	0
$729$	−27.0000	−1.00000
$730$	0	0
$731$	−24.0000	−0.887672
$732$	0	0
$733$	−30.0000	−1.10808	−0.554038	−	0.832492i	$-0.686914\pi$
$733$	−30.0000	−1.10808	−0.554038	+	0.832492i	$0.686914\pi$
$734$	−32.0000	−1.18114
$735$	0	0
$736$	−40.0000	−1.47442
$737$	8.00000	0.294684
$738$	−30.0000	−1.10432
$739$	−20.0000	−0.735712	−0.367856	−	0.929883i	$-0.619908\pi$
$739$	−20.0000	−0.735712	−0.367856	+	0.929883i	$0.619908\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	8.00000	0.293492	0.146746	−	0.989174i	$-0.453120\pi$
$743$	8.00000	0.293492	0.146746	+	0.989174i	$0.453120\pi$
$744$	0	0
$745$	0	0
$746$	−6.00000	−0.219676
$747$	−12.0000	−0.439057
$748$	6.00000	0.219382
$749$	0	0
$750$	0	0
$751$	4.00000	0.145962	0.0729810	−	0.997333i	$-0.476749\pi$
$751$	4.00000	0.145962	0.0729810	+	0.997333i	$0.476749\pi$
$752$	0	0
$753$	0	0
$754$	−12.0000	−0.437014
$755$	0	0
$756$	0	0
$757$	42.0000	1.52652	0.763258	−	0.646094i	$-0.223599\pi$
$757$	42.0000	1.52652	0.763258	+	0.646094i	$0.223599\pi$
$758$	−32.0000	−1.16229
$759$	0	0
$760$	0	0
$761$	34.0000	1.23250	0.616250	−	0.787551i	$-0.288651\pi$
$761$	34.0000	1.23250	0.616250	+	0.787551i	$0.288651\pi$
$762$	0	0
$763$	0	0
$764$	16.0000	0.578860
$765$	0	0
$766$	−36.0000	−1.30073
$767$	16.0000	0.577727
$768$	0	0
$769$	2.00000	0.0721218	0.0360609	−	0.999350i	$-0.488519\pi$
$769$	2.00000	0.0721218	0.0360609	+	0.999350i	$0.488519\pi$
$770$	0	0
$771$	0	0
$772$	−2.00000	−0.0719816
$773$	−14.0000	−0.503545	−0.251773	−	0.967786i	$-0.581013\pi$
$773$	−14.0000	−0.503545	−0.251773	+	0.967786i	$0.581013\pi$
$774$	−12.0000	−0.431331
$775$	0	0
$776$	−30.0000	−1.07694
$777$	0	0
$778$	−6.00000	−0.215110
$779$	−10.0000	−0.358287
$780$	0	0
$781$	4.00000	0.143131
$782$	−48.0000	−1.71648
$783$	0	0
$784$	7.00000	0.250000
$785$	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	$0.522712\pi$
$788$	−10.0000	−0.356235
$789$	0	0
$790$	0	0
$791$	0	0
$792$	9.00000	0.319801
$793$	−28.0000	−0.994309
$794$	−18.0000	−0.638796
$795$	0	0
$796$	0	0
$797$	2.00000	0.0708436	0.0354218	−	0.999372i	$-0.488723\pi$
$797$	2.00000	0.0708436	0.0354218	+	0.999372i	$0.488723\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	−30.0000	−1.06000
$802$	22.0000	0.776847
$803$	2.00000	0.0705785
$804$	0	0
$805$	0	0
$806$	8.00000	0.281788
$807$	0	0
$808$	42.0000	1.47755
$809$	42.0000	1.47664	0.738321	−	0.674450i	$-0.235619\pi$
$809$	42.0000	1.47664	0.738321	+	0.674450i	$0.235619\pi$
$810$	0	0
$811$	12.0000	0.421377	0.210688	−	0.977553i	$-0.432429\pi$
$811$	12.0000	0.421377	0.210688	+	0.977553i	$0.432429\pi$
$812$	0	0
$813$	0	0
$814$	2.00000	0.0701000
$815$	0	0
$816$	0	0
$817$	−4.00000	−0.139942
$818$	−30.0000	−1.04893
$819$	0	0
$820$	0	0
$821$	30.0000	1.04701	0.523504	−	0.852023i	$-0.324625\pi$
$821$	30.0000	1.04701	0.523504	+	0.852023i	$0.324625\pi$
$822$	0	0
$823$	−48.0000	−1.67317	−0.836587	−	0.547833i	$-0.815453\pi$
$823$	−48.0000	−1.67317	−0.836587	+	0.547833i	$0.815453\pi$
$824$	12.0000	0.418040
$825$	0	0
$826$	0	0
$827$	−36.0000	−1.25184	−0.625921	−	0.779886i	$-0.715277\pi$
$827$	−36.0000	−1.25184	−0.625921	+	0.779886i	$0.715277\pi$
$828$	24.0000	0.834058
$829$	46.0000	1.59765	0.798823	−	0.601566i	$-0.205456\pi$
$829$	46.0000	1.59765	0.798823	+	0.601566i	$0.205456\pi$
$830$	0	0
$831$	0	0
$832$	−14.0000	−0.485363
$833$	−42.0000	−1.45521
$834$	0	0
$835$	0	0
$836$	1.00000	0.0345857
$837$	0	0
$838$	28.0000	0.967244
$839$	4.00000	0.138095	0.0690477	−	0.997613i	$-0.478004\pi$
$839$	4.00000	0.138095	0.0690477	+	0.997613i	$0.478004\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	−6.00000	−0.206774
$843$	0	0
$844$	−4.00000	−0.137686
$845$	0	0
$846$	0	0
$847$	0	0
$848$	−2.00000	−0.0686803
$849$	0	0
$850$	0	0
$851$	16.0000	0.548473
$852$	0	0
$853$	−38.0000	−1.30110	−0.650548	−	0.759465i	$-0.725461\pi$
$853$	−38.0000	−1.30110	−0.650548	+	0.759465i	$0.725461\pi$
$854$	0	0
$855$	0	0
$856$	36.0000	1.23045
$857$	−30.0000	−1.02478	−0.512390	−	0.858753i	$-0.671240\pi$
$857$	−30.0000	−1.02478	−0.512390	+	0.858753i	$0.671240\pi$
$858$	0	0
$859$	−44.0000	−1.50126	−0.750630	−	0.660722i	$-0.770250\pi$
$859$	−44.0000	−1.50126	−0.750630	+	0.660722i	$0.770250\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−4.00000	−0.136162	−0.0680808	−	0.997680i	$-0.521688\pi$
$863$	−4.00000	−0.136162	−0.0680808	+	0.997680i	$0.521688\pi$
$864$	0	0
$865$	0	0
$866$	−22.0000	−0.747590
$867$	0	0
$868$	0	0
$869$	16.0000	0.542763
$870$	0	0
$871$	16.0000	0.542139
$872$	−18.0000	−0.609557
$873$	30.0000	1.01535
$874$	−8.00000	−0.270604
$875$	0	0
$876$	0	0
$877$	−2.00000	−0.0675352	−0.0337676	−	0.999430i	$-0.510751\pi$
$877$	−2.00000	−0.0675352	−0.0337676	+	0.999430i	$0.510751\pi$
$878$	8.00000	0.269987
$879$	0	0
$880$	0	0
$881$	−30.0000	−1.01073	−0.505363	−	0.862907i	$-0.668641\pi$
$881$	−30.0000	−1.01073	−0.505363	+	0.862907i	$0.668641\pi$
$882$	−21.0000	−0.707107
$883$	52.0000	1.74994	0.874970	−	0.484178i	$-0.160881\pi$
$883$	52.0000	1.74994	0.874970	+	0.484178i	$0.160881\pi$
$884$	12.0000	0.403604
$885$	0	0
$886$	−28.0000	−0.940678
$887$	40.0000	1.34307	0.671534	−	0.740973i	$-0.265636\pi$
$887$	40.0000	1.34307	0.671534	+	0.740973i	$0.265636\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−9.00000	−0.301511
$892$	−4.00000	−0.133930
$893$	0	0
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	−2.00000	−0.0667409
$899$	−24.0000	−0.800445
$900$	0	0
$901$	12.0000	0.399778
$902$	−10.0000	−0.332964
$903$	0	0
$904$	18.0000	0.598671
$905$	0	0
$906$	0	0
$907$	40.0000	1.32818	0.664089	−	0.747653i	$-0.268820\pi$
$907$	40.0000	1.32818	0.664089	+	0.747653i	$0.268820\pi$
$908$	4.00000	0.132745
$909$	−42.0000	−1.39305
$910$	0	0
$911$	20.0000	0.662630	0.331315	−	0.943520i	$-0.392508\pi$
$911$	20.0000	0.662630	0.331315	+	0.943520i	$0.392508\pi$
$912$	0	0
$913$	−4.00000	−0.132381
$914$	10.0000	0.330771
$915$	0	0
$916$	10.0000	0.330409
$917$	0	0
$918$	0	0
$919$	24.0000	0.791687	0.395843	−	0.918318i	$-0.370452\pi$
$919$	24.0000	0.791687	0.395843	+	0.918318i	$0.370452\pi$
$920$	0	0
$921$	0	0
$922$	10.0000	0.329332
$923$	8.00000	0.263323
$924$	0	0
$925$	0	0
$926$	−24.0000	−0.788689
$927$	−12.0000	−0.394132
$928$	30.0000	0.984798
$929$	34.0000	1.11550	0.557752	−	0.830008i	$-0.311664\pi$
$929$	34.0000	1.11550	0.557752	+	0.830008i	$0.311664\pi$
$930$	0	0
$931$	−7.00000	−0.229416
$932$	−6.00000	−0.196537
$933$	0	0
$934$	−12.0000	−0.392652
$935$	0	0
$936$	18.0000	0.588348
$937$	30.0000	0.980057	0.490029	−	0.871706i	$-0.336986\pi$
$937$	30.0000	0.980057	0.490029	+	0.871706i	$0.336986\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−14.0000	−0.456387	−0.228193	−	0.973616i	$-0.573282\pi$
$941$	−14.0000	−0.456387	−0.228193	+	0.973616i	$0.573282\pi$
$942$	0	0
$943$	−80.0000	−2.60516
$944$	8.00000	0.260378
$945$	0	0
$946$	−4.00000	−0.130051
$947$	36.0000	1.16984	0.584921	−	0.811090i	$-0.301125\pi$
$947$	36.0000	1.16984	0.584921	+	0.811090i	$0.301125\pi$
$948$	0	0
$949$	4.00000	0.129845
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−6.00000	−0.194359	−0.0971795	−	0.995267i	$-0.530982\pi$
$953$	−6.00000	−0.194359	−0.0971795	+	0.995267i	$0.530982\pi$
$954$	6.00000	0.194257
$955$	0	0
$956$	16.0000	0.517477
$957$	0	0
$958$	−24.0000	−0.775405
$959$	0	0
$960$	0	0
$961$	−15.0000	−0.483871
$962$	4.00000	0.128965
$963$	−36.0000	−1.16008
$964$	−22.0000	−0.708572
$965$	0	0
$966$	0	0
$967$	−32.0000	−1.02905	−0.514525	−	0.857475i	$-0.672032\pi$
$967$	−32.0000	−1.02905	−0.514525	+	0.857475i	$0.672032\pi$
$968$	3.00000	0.0964237
$969$	0	0
$970$	0	0
$971$	−24.0000	−0.770197	−0.385098	−	0.922876i	$-0.625832\pi$
$971$	−24.0000	−0.770197	−0.385098	+	0.922876i	$0.625832\pi$
$972$	0	0
$973$	0	0
$974$	4.00000	0.128168
$975$	0	0
$976$	−14.0000	−0.448129
$977$	−58.0000	−1.85558	−0.927792	−	0.373097i	$-0.878296\pi$
$977$	−58.0000	−1.85558	−0.927792	+	0.373097i	$0.878296\pi$
$978$	0	0
$979$	−10.0000	−0.319601
$980$	0	0
$981$	18.0000	0.574696
$982$	28.0000	0.893516
$983$	4.00000	0.127580	0.0637901	−	0.997963i	$-0.479681\pi$
$983$	4.00000	0.127580	0.0637901	+	0.997963i	$0.479681\pi$
$984$	0	0
$985$	0	0
$986$	36.0000	1.14647
$987$	0	0
$988$	2.00000	0.0636285
$989$	−32.0000	−1.01754
$990$	0	0
$991$	−4.00000	−0.127064	−0.0635321	−	0.997980i	$-0.520237\pi$
$991$	−4.00000	−0.127064	−0.0635321	+	0.997980i	$0.520237\pi$
$992$	−20.0000	−0.635001
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−14.0000	−0.443384	−0.221692	−	0.975117i	$-0.571158\pi$
$997$	−14.0000	−0.443384	−0.221692	+	0.975117i	$0.571158\pi$
$998$	−36.0000	−1.13956
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5225.2.a.a.1.1		1
5.4	even	2		1045.2.a.b.1.1	✓	1
15.14	odd	2		9405.2.a.d.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1045.2.a.b.1.1	✓	1	5.4	even	2
5225.2.a.a.1.1		1	1.1	even	1	trivial
9405.2.a.d.1.1		1	15.14	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:	\( N \)	\(=\)	\( 5225 = 5^{2} \cdot 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5225.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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