LMFDB - Embedded newform 2394.2.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2394,2,Mod(1,2394)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2394, 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("2394.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2394 = 2 \cdot 3^{2} \cdot 7 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2394.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:	$19.1161862439$
Analytic rank:	$1$
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	$\chi$	$=$	2394.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{2} +1.00000 q^{4} -4.00000 q^{5} +1.00000 q^{7} -1.00000 q^{8} +O(q^{10})$

$q-1.00000 q^{2} +1.00000 q^{4} -4.00000 q^{5} +1.00000 q^{7} -1.00000 q^{8} +4.00000 q^{10} +2.00000 q^{11} -2.00000 q^{13} -1.00000 q^{14} +1.00000 q^{16} +1.00000 q^{19} -4.00000 q^{20} -2.00000 q^{22} +2.00000 q^{23} +11.0000 q^{25} +2.00000 q^{26} +1.00000 q^{28} -6.00000 q^{29} +4.00000 q^{31} -1.00000 q^{32} -4.00000 q^{35} +2.00000 q^{37} -1.00000 q^{38} +4.00000 q^{40} -10.0000 q^{41} +4.00000 q^{43} +2.00000 q^{44} -2.00000 q^{46} +6.00000 q^{47} +1.00000 q^{49} -11.0000 q^{50} -2.00000 q^{52} +2.00000 q^{53} -8.00000 q^{55} -1.00000 q^{56} +6.00000 q^{58} +4.00000 q^{59} +2.00000 q^{61} -4.00000 q^{62} +1.00000 q^{64} +8.00000 q^{65} -12.0000 q^{67} +4.00000 q^{70} -14.0000 q^{73} -2.00000 q^{74} +1.00000 q^{76} +2.00000 q^{77} +8.00000 q^{79} -4.00000 q^{80} +10.0000 q^{82} +10.0000 q^{83} -4.00000 q^{86} -2.00000 q^{88} -18.0000 q^{89} -2.00000 q^{91} +2.00000 q^{92} -6.00000 q^{94} -4.00000 q^{95} -18.0000 q^{97} -1.00000 q^{98} +O(q^{100})$

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	−4.00000	−1.78885	−0.894427	−	0.447214i	$-0.852416\pi$
$5$	−4.00000	−1.78885	−0.894427	+	0.447214i	$0.852416\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−1.00000	−0.353553
$9$	0	0
$10$	4.00000	1.26491
$11$	2.00000	0.603023	0.301511	−	0.953463i	$-0.402509\pi$
$11$	2.00000	0.603023	0.301511	+	0.953463i	$0.402509\pi$
$12$	0	0
$13$	−2.00000	−0.554700	−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−2.00000	−0.554700	−0.277350	+	0.960769i	$0.589456\pi$
$14$	−1.00000	−0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	−4.00000	−0.894427
$21$	0	0
$22$	−2.00000	−0.426401
$23$	2.00000	0.417029	0.208514	−	0.978019i	$-0.433137\pi$
$23$	2.00000	0.417029	0.208514	+	0.978019i	$0.433137\pi$
$24$	0	0
$25$	11.0000	2.20000
$26$	2.00000	0.392232
$27$	0	0
$28$	1.00000	0.188982
$29$	−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	$0.383046\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	0	0
$35$	−4.00000	−0.676123
$36$	0	0
$37$	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
$40$	4.00000	0.632456
$41$	−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	$-0.401344\pi$
$43$	4.00000	0.609994	0.304997	+	0.952353i	$0.401344\pi$
$44$	2.00000	0.301511
$45$	0	0
$46$	−2.00000	−0.294884
$47$	6.00000	0.875190	0.437595	−	0.899172i	$-0.355830\pi$
$47$	6.00000	0.875190	0.437595	+	0.899172i	$0.355830\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−11.0000	−1.55563
$51$	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
$55$	−8.00000	−1.07872
$56$	−1.00000	−0.133631
$57$	0	0
$58$	6.00000	0.787839
$59$	4.00000	0.520756	0.260378	−	0.965507i	$-0.416153\pi$
$59$	4.00000	0.520756	0.260378	+	0.965507i	$0.416153\pi$
$60$	0	0
$61$	2.00000	0.256074	0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000	0.256074	0.128037	+	0.991769i	$0.459132\pi$
$62$	−4.00000	−0.508001
$63$	0	0
$64$	1.00000	0.125000
$65$	8.00000	0.992278
$66$	0	0
$67$	−12.0000	−1.46603	−0.733017	−	0.680211i	$-0.761888\pi$
$67$	−12.0000	−1.46603	−0.733017	+	0.680211i	$0.761888\pi$
$68$	0	0
$69$	0	0
$70$	4.00000	0.478091
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	−14.0000	−1.63858	−0.819288	−	0.573382i	$-0.805631\pi$
$73$	−14.0000	−1.63858	−0.819288	+	0.573382i	$0.805631\pi$
$74$	−2.00000	−0.232495
$75$	0	0
$76$	1.00000	0.114708
$77$	2.00000	0.227921
$78$	0	0
$79$	8.00000	0.900070	0.450035	−	0.893011i	$-0.351411\pi$
$79$	8.00000	0.900070	0.450035	+	0.893011i	$0.351411\pi$
$80$	−4.00000	−0.447214
$81$	0	0
$82$	10.0000	1.10432
$83$	10.0000	1.09764	0.548821	−	0.835940i	$-0.315077\pi$
$83$	10.0000	1.09764	0.548821	+	0.835940i	$0.315077\pi$
$84$	0	0
$85$	0	0
$86$	−4.00000	−0.431331
$87$	0	0
$88$	−2.00000	−0.213201
$89$	−18.0000	−1.90800	−0.953998	−	0.299813i	$-0.903076\pi$
$89$	−18.0000	−1.90800	−0.953998	+	0.299813i	$0.903076\pi$
$90$	0	0
$91$	−2.00000	−0.209657
$92$	2.00000	0.208514
$93$	0	0
$94$	−6.00000	−0.618853
$95$	−4.00000	−0.410391
$96$	0	0
$97$	−18.0000	−1.82762	−0.913812	−	0.406138i	$-0.866875\pi$
$97$	−18.0000	−1.82762	−0.913812	+	0.406138i	$0.866875\pi$
$98$	−1.00000	−0.101015
$99$	0	0
$100$	11.0000	1.10000
$101$	0	0		−	1.00000i	$-0.5\pi$
$101$	0	0			1.00000i	$0.5\pi$
$102$	0	0
$103$	0	0		−	1.00000i	$-0.5\pi$
$103$	0	0			1.00000i	$0.5\pi$
$104$	2.00000	0.196116
$105$	0	0
$106$	−2.00000	−0.194257
$107$	0	0		−	1.00000i	$-0.5\pi$
$107$	0	0			1.00000i	$0.5\pi$
$108$	0	0
$109$	−10.0000	−0.957826	−0.478913	−	0.877862i	$-0.658969\pi$
$109$	−10.0000	−0.957826	−0.478913	+	0.877862i	$0.658969\pi$
$110$	8.00000	0.762770
$111$	0	0
$112$	1.00000	0.0944911
$113$	2.00000	0.188144	0.0940721	−	0.995565i	$-0.470012\pi$
$113$	2.00000	0.188144	0.0940721	+	0.995565i	$0.470012\pi$
$114$	0	0
$115$	−8.00000	−0.746004
$116$	−6.00000	−0.557086
$117$	0	0
$118$	−4.00000	−0.368230
$119$	0	0
$120$	0	0
$121$	−7.00000	−0.636364
$122$	−2.00000	−0.181071
$123$	0	0
$124$	4.00000	0.359211
$125$	−24.0000	−2.14663
$126$	0	0
$127$	4.00000	0.354943	0.177471	−	0.984126i	$-0.443208\pi$
$127$	4.00000	0.354943	0.177471	+	0.984126i	$0.443208\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	−8.00000	−0.701646
$131$	−18.0000	−1.57267	−0.786334	−	0.617802i	$-0.788023\pi$
$131$	−18.0000	−1.57267	−0.786334	+	0.617802i	$0.788023\pi$
$132$	0	0
$133$	1.00000	0.0867110
$134$	12.0000	1.03664
$135$	0	0
$136$	0	0
$137$	−12.0000	−1.02523	−0.512615	−	0.858619i	$-0.671323\pi$
$137$	−12.0000	−1.02523	−0.512615	+	0.858619i	$0.671323\pi$
$138$	0	0
$139$	−12.0000	−1.01783	−0.508913	−	0.860818i	$-0.669953\pi$
$139$	−12.0000	−1.01783	−0.508913	+	0.860818i	$0.669953\pi$
$140$	−4.00000	−0.338062
$141$	0	0
$142$	0	0
$143$	−4.00000	−0.334497
$144$	0	0
$145$	24.0000	1.99309
$146$	14.0000	1.15865
$147$	0	0
$148$	2.00000	0.164399
$149$	4.00000	0.327693	0.163846	−	0.986486i	$-0.447610\pi$
$149$	4.00000	0.327693	0.163846	+	0.986486i	$0.447610\pi$
$150$	0	0
$151$	−8.00000	−0.651031	−0.325515	−	0.945537i	$-0.605538\pi$
$151$	−8.00000	−0.651031	−0.325515	+	0.945537i	$0.605538\pi$
$152$	−1.00000	−0.0811107
$153$	0	0
$154$	−2.00000	−0.161165
$155$	−16.0000	−1.28515
$156$	0	0
$157$	−14.0000	−1.11732	−0.558661	−	0.829396i	$-0.688685\pi$
$157$	−14.0000	−1.11732	−0.558661	+	0.829396i	$0.688685\pi$
$158$	−8.00000	−0.636446
$159$	0	0
$160$	4.00000	0.316228
$161$	2.00000	0.157622
$162$	0	0
$163$	−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
$166$	−10.0000	−0.776151
$167$	16.0000	1.23812	0.619059	−	0.785345i	$-0.287514\pi$
$167$	16.0000	1.23812	0.619059	+	0.785345i	$0.287514\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	0	0
$171$	0	0
$172$	4.00000	0.304997
$173$	−14.0000	−1.06440	−0.532200	−	0.846619i	$-0.678635\pi$
$173$	−14.0000	−1.06440	−0.532200	+	0.846619i	$0.678635\pi$
$174$	0	0
$175$	11.0000	0.831522
$176$	2.00000	0.150756
$177$	0	0
$178$	18.0000	1.34916
$179$	0	0		−	1.00000i	$-0.5\pi$
$179$	0	0			1.00000i	$0.5\pi$
$180$	0	0
$181$	−10.0000	−0.743294	−0.371647	−	0.928374i	$-0.621207\pi$
$181$	−10.0000	−0.743294	−0.371647	+	0.928374i	$0.621207\pi$
$182$	2.00000	0.148250
$183$	0	0
$184$	−2.00000	−0.147442
$185$	−8.00000	−0.588172
$186$	0	0
$187$	0	0
$188$	6.00000	0.437595
$189$	0	0
$190$	4.00000	0.290191
$191$	22.0000	1.59186	0.795932	−	0.605386i	$-0.206981\pi$
$191$	22.0000	1.59186	0.795932	+	0.605386i	$0.206981\pi$
$192$	0	0
$193$	18.0000	1.29567	0.647834	−	0.761781i	$-0.275675\pi$
$193$	18.0000	1.29567	0.647834	+	0.761781i	$0.275675\pi$
$194$	18.0000	1.29232
$195$	0	0
$196$	1.00000	0.0714286
$197$	−8.00000	−0.569976	−0.284988	−	0.958531i	$-0.591990\pi$
$197$	−8.00000	−0.569976	−0.284988	+	0.958531i	$0.591990\pi$
$198$	0	0
$199$	8.00000	0.567105	0.283552	−	0.958957i	$-0.408487\pi$
$199$	8.00000	0.567105	0.283552	+	0.958957i	$0.408487\pi$
$200$	−11.0000	−0.777817
$201$	0	0
$202$	0	0
$203$	−6.00000	−0.421117
$204$	0	0
$205$	40.0000	2.79372
$206$	0	0
$207$	0	0
$208$	−2.00000	−0.138675
$209$	2.00000	0.138343
$210$	0	0
$211$	−20.0000	−1.37686	−0.688428	−	0.725304i	$-0.741699\pi$
$211$	−20.0000	−1.37686	−0.688428	+	0.725304i	$0.741699\pi$
$212$	2.00000	0.137361
$213$	0	0
$214$	0	0
$215$	−16.0000	−1.09119
$216$	0	0
$217$	4.00000	0.271538
$218$	10.0000	0.677285
$219$	0	0
$220$	−8.00000	−0.539360
$221$	0	0
$222$	0	0
$223$	16.0000	1.07144	0.535720	−	0.844396i	$-0.320040\pi$
$223$	16.0000	1.07144	0.535720	+	0.844396i	$0.320040\pi$
$224$	−1.00000	−0.0668153
$225$	0	0
$226$	−2.00000	−0.133038
$227$	0	0		−	1.00000i	$-0.5\pi$
$227$	0	0			1.00000i	$0.5\pi$
$228$	0	0
$229$	−6.00000	−0.396491	−0.198246	−	0.980152i	$-0.563524\pi$
$229$	−6.00000	−0.396491	−0.198246	+	0.980152i	$0.563524\pi$
$230$	8.00000	0.527504
$231$	0	0
$232$	6.00000	0.393919
$233$	8.00000	0.524097	0.262049	−	0.965055i	$-0.415602\pi$
$233$	8.00000	0.524097	0.262049	+	0.965055i	$0.415602\pi$
$234$	0	0
$235$	−24.0000	−1.56559
$236$	4.00000	0.260378
$237$	0	0
$238$	0	0
$239$	14.0000	0.905585	0.452792	−	0.891616i	$-0.350428\pi$
$239$	14.0000	0.905585	0.452792	+	0.891616i	$0.350428\pi$
$240$	0	0
$241$	−10.0000	−0.644157	−0.322078	−	0.946713i	$-0.604381\pi$
$241$	−10.0000	−0.644157	−0.322078	+	0.946713i	$0.604381\pi$
$242$	7.00000	0.449977
$243$	0	0
$244$	2.00000	0.128037
$245$	−4.00000	−0.255551
$246$	0	0
$247$	−2.00000	−0.127257
$248$	−4.00000	−0.254000
$249$	0	0
$250$	24.0000	1.51789
$251$	−30.0000	−1.89358	−0.946792	−	0.321847i	$-0.895696\pi$
$251$	−30.0000	−1.89358	−0.946792	+	0.321847i	$0.895696\pi$
$252$	0	0
$253$	4.00000	0.251478
$254$	−4.00000	−0.250982
$255$	0	0
$256$	1.00000	0.0625000
$257$	10.0000	0.623783	0.311891	−	0.950118i	$-0.399037\pi$
$257$	10.0000	0.623783	0.311891	+	0.950118i	$0.399037\pi$
$258$	0	0
$259$	2.00000	0.124274
$260$	8.00000	0.496139
$261$	0	0
$262$	18.0000	1.11204
$263$	30.0000	1.84988	0.924940	−	0.380114i	$-0.124115\pi$
$263$	30.0000	1.84988	0.924940	+	0.380114i	$0.124115\pi$
$264$	0	0
$265$	−8.00000	−0.491436
$266$	−1.00000	−0.0613139
$267$	0	0
$268$	−12.0000	−0.733017
$269$	−14.0000	−0.853595	−0.426798	−	0.904347i	$-0.640358\pi$
$269$	−14.0000	−0.853595	−0.426798	+	0.904347i	$0.640358\pi$
$270$	0	0
$271$	16.0000	0.971931	0.485965	−	0.873978i	$-0.338468\pi$
$271$	16.0000	0.971931	0.485965	+	0.873978i	$0.338468\pi$
$272$	0	0
$273$	0	0
$274$	12.0000	0.724947
$275$	22.0000	1.32665
$276$	0	0
$277$	−14.0000	−0.841178	−0.420589	−	0.907251i	$-0.638177\pi$
$277$	−14.0000	−0.841178	−0.420589	+	0.907251i	$0.638177\pi$
$278$	12.0000	0.719712
$279$	0	0
$280$	4.00000	0.239046
$281$	−26.0000	−1.55103	−0.775515	−	0.631329i	$-0.782510\pi$
$281$	−26.0000	−1.55103	−0.775515	+	0.631329i	$0.782510\pi$
$282$	0	0
$283$	−28.0000	−1.66443	−0.832214	−	0.554455i	$-0.812927\pi$
$283$	−28.0000	−1.66443	−0.832214	+	0.554455i	$0.812927\pi$
$284$	0	0
$285$	0	0
$286$	4.00000	0.236525
$287$	−10.0000	−0.590281
$288$	0	0
$289$	−17.0000	−1.00000
$290$	−24.0000	−1.40933
$291$	0	0
$292$	−14.0000	−0.819288
$293$	−6.00000	−0.350524	−0.175262	−	0.984522i	$-0.556077\pi$
$293$	−6.00000	−0.350524	−0.175262	+	0.984522i	$0.556077\pi$
$294$	0	0
$295$	−16.0000	−0.931556
$296$	−2.00000	−0.116248
$297$	0	0
$298$	−4.00000	−0.231714
$299$	−4.00000	−0.231326
$300$	0	0
$301$	4.00000	0.230556
$302$	8.00000	0.460348
$303$	0	0
$304$	1.00000	0.0573539
$305$	−8.00000	−0.458079
$306$	0	0
$307$	−8.00000	−0.456584	−0.228292	−	0.973593i	$-0.573314\pi$
$307$	−8.00000	−0.456584	−0.228292	+	0.973593i	$0.573314\pi$
$308$	2.00000	0.113961
$309$	0	0
$310$	16.0000	0.908739
$311$	−18.0000	−1.02069	−0.510343	−	0.859971i	$-0.670482\pi$
$311$	−18.0000	−1.02069	−0.510343	+	0.859971i	$0.670482\pi$
$312$	0	0
$313$	22.0000	1.24351	0.621757	−	0.783210i	$-0.286419\pi$
$313$	22.0000	1.24351	0.621757	+	0.783210i	$0.286419\pi$
$314$	14.0000	0.790066
$315$	0	0
$316$	8.00000	0.450035
$317$	−10.0000	−0.561656	−0.280828	−	0.959758i	$-0.590609\pi$
$317$	−10.0000	−0.561656	−0.280828	+	0.959758i	$0.590609\pi$
$318$	0	0
$319$	−12.0000	−0.671871
$320$	−4.00000	−0.223607
$321$	0	0
$322$	−2.00000	−0.111456
$323$	0	0
$324$	0	0
$325$	−22.0000	−1.22034
$326$	4.00000	0.221540
$327$	0	0
$328$	10.0000	0.552158
$329$	6.00000	0.330791
$330$	0	0
$331$	0	0		−	1.00000i	$-0.5\pi$
$331$	0	0			1.00000i	$0.5\pi$
$332$	10.0000	0.548821
$333$	0	0
$334$	−16.0000	−0.875481
$335$	48.0000	2.62252
$336$	0	0
$337$	−2.00000	−0.108947	−0.0544735	−	0.998515i	$-0.517348\pi$
$337$	−2.00000	−0.108947	−0.0544735	+	0.998515i	$0.517348\pi$
$338$	9.00000	0.489535
$339$	0	0
$340$	0	0
$341$	8.00000	0.433224
$342$	0	0
$343$	1.00000	0.0539949
$344$	−4.00000	−0.215666
$345$	0	0
$346$	14.0000	0.752645
$347$	18.0000	0.966291	0.483145	−	0.875540i	$-0.339494\pi$
$347$	18.0000	0.966291	0.483145	+	0.875540i	$0.339494\pi$
$348$	0	0
$349$	−2.00000	−0.107058	−0.0535288	−	0.998566i	$-0.517047\pi$
$349$	−2.00000	−0.107058	−0.0535288	+	0.998566i	$0.517047\pi$
$350$	−11.0000	−0.587975
$351$	0	0
$352$	−2.00000	−0.106600
$353$	−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
$356$	−18.0000	−0.953998
$357$	0	0
$358$	0	0
$359$	−2.00000	−0.105556	−0.0527780	−	0.998606i	$-0.516808\pi$
$359$	−2.00000	−0.105556	−0.0527780	+	0.998606i	$0.516808\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	10.0000	0.525588
$363$	0	0
$364$	−2.00000	−0.104828
$365$	56.0000	2.93117
$366$	0	0
$367$	0	0		−	1.00000i	$-0.5\pi$
$367$	0	0			1.00000i	$0.5\pi$
$368$	2.00000	0.104257
$369$	0	0
$370$	8.00000	0.415900
$371$	2.00000	0.103835
$372$	0	0
$373$	26.0000	1.34623	0.673114	−	0.739538i	$-0.264956\pi$
$373$	26.0000	1.34623	0.673114	+	0.739538i	$0.264956\pi$
$374$	0	0
$375$	0	0
$376$	−6.00000	−0.309426
$377$	12.0000	0.618031
$378$	0	0
$379$	−24.0000	−1.23280	−0.616399	−	0.787434i	$-0.711409\pi$
$379$	−24.0000	−1.23280	−0.616399	+	0.787434i	$0.711409\pi$
$380$	−4.00000	−0.205196
$381$	0	0
$382$	−22.0000	−1.12562
$383$	−12.0000	−0.613171	−0.306586	−	0.951843i	$-0.599187\pi$
$383$	−12.0000	−0.613171	−0.306586	+	0.951843i	$0.599187\pi$
$384$	0	0
$385$	−8.00000	−0.407718
$386$	−18.0000	−0.916176
$387$	0	0
$388$	−18.0000	−0.913812
$389$	24.0000	1.21685	0.608424	−	0.793612i	$-0.291802\pi$
$389$	24.0000	1.21685	0.608424	+	0.793612i	$0.291802\pi$
$390$	0	0
$391$	0	0
$392$	−1.00000	−0.0505076
$393$	0	0
$394$	8.00000	0.403034
$395$	−32.0000	−1.61009
$396$	0	0
$397$	−18.0000	−0.903394	−0.451697	−	0.892171i	$-0.649181\pi$
$397$	−18.0000	−0.903394	−0.451697	+	0.892171i	$0.649181\pi$
$398$	−8.00000	−0.401004
$399$	0	0
$400$	11.0000	0.550000
$401$	10.0000	0.499376	0.249688	−	0.968326i	$-0.419672\pi$
$401$	10.0000	0.499376	0.249688	+	0.968326i	$0.419672\pi$
$402$	0	0
$403$	−8.00000	−0.398508
$404$	0	0
$405$	0	0
$406$	6.00000	0.297775
$407$	4.00000	0.198273
$408$	0	0
$409$	38.0000	1.87898	0.939490	−	0.342578i	$-0.111300\pi$
$409$	38.0000	1.87898	0.939490	+	0.342578i	$0.111300\pi$
$410$	−40.0000	−1.97546
$411$	0	0
$412$	0	0
$413$	4.00000	0.196827
$414$	0	0
$415$	−40.0000	−1.96352
$416$	2.00000	0.0980581
$417$	0	0
$418$	−2.00000	−0.0978232
$419$	−26.0000	−1.27018	−0.635092	−	0.772437i	$-0.719038\pi$
$419$	−26.0000	−1.27018	−0.635092	+	0.772437i	$0.719038\pi$
$420$	0	0
$421$	−26.0000	−1.26716	−0.633581	−	0.773676i	$-0.718416\pi$
$421$	−26.0000	−1.26716	−0.633581	+	0.773676i	$0.718416\pi$
$422$	20.0000	0.973585
$423$	0	0
$424$	−2.00000	−0.0971286
$425$	0	0
$426$	0	0
$427$	2.00000	0.0967868
$428$	0	0
$429$	0	0
$430$	16.0000	0.771589
$431$	24.0000	1.15604	0.578020	−	0.816023i	$-0.303826\pi$
$431$	24.0000	1.15604	0.578020	+	0.816023i	$0.303826\pi$
$432$	0	0
$433$	26.0000	1.24948	0.624740	−	0.780833i	$-0.285205\pi$
$433$	26.0000	1.24948	0.624740	+	0.780833i	$0.285205\pi$
$434$	−4.00000	−0.192006
$435$	0	0
$436$	−10.0000	−0.478913
$437$	2.00000	0.0956730
$438$	0	0
$439$	−32.0000	−1.52728	−0.763638	−	0.645644i	$-0.776589\pi$
$439$	−32.0000	−1.52728	−0.763638	+	0.645644i	$0.776589\pi$
$440$	8.00000	0.381385
$441$	0	0
$442$	0	0
$443$	10.0000	0.475114	0.237557	−	0.971374i	$-0.423653\pi$
$443$	10.0000	0.475114	0.237557	+	0.971374i	$0.423653\pi$
$444$	0	0
$445$	72.0000	3.41313
$446$	−16.0000	−0.757622
$447$	0	0
$448$	1.00000	0.0472456
$449$	−10.0000	−0.471929	−0.235965	−	0.971762i	$-0.575825\pi$
$449$	−10.0000	−0.471929	−0.235965	+	0.971762i	$0.575825\pi$
$450$	0	0
$451$	−20.0000	−0.941763
$452$	2.00000	0.0940721
$453$	0	0
$454$	0	0
$455$	8.00000	0.375046
$456$	0	0
$457$	18.0000	0.842004	0.421002	−	0.907060i	$-0.361678\pi$
$457$	18.0000	0.842004	0.421002	+	0.907060i	$0.361678\pi$
$458$	6.00000	0.280362
$459$	0	0
$460$	−8.00000	−0.373002
$461$	16.0000	0.745194	0.372597	−	0.927993i	$-0.378467\pi$
$461$	16.0000	0.745194	0.372597	+	0.927993i	$0.378467\pi$
$462$	0	0
$463$	−32.0000	−1.48717	−0.743583	−	0.668644i	$-0.766875\pi$
$463$	−32.0000	−1.48717	−0.743583	+	0.668644i	$0.766875\pi$
$464$	−6.00000	−0.278543
$465$	0	0
$466$	−8.00000	−0.370593
$467$	−6.00000	−0.277647	−0.138823	−	0.990317i	$-0.544332\pi$
$467$	−6.00000	−0.277647	−0.138823	+	0.990317i	$0.544332\pi$
$468$	0	0
$469$	−12.0000	−0.554109
$470$	24.0000	1.10704
$471$	0	0
$472$	−4.00000	−0.184115
$473$	8.00000	0.367840
$474$	0	0
$475$	11.0000	0.504715
$476$	0	0
$477$	0	0
$478$	−14.0000	−0.640345
$479$	18.0000	0.822441	0.411220	−	0.911536i	$-0.365103\pi$
$479$	18.0000	0.822441	0.411220	+	0.911536i	$0.365103\pi$
$480$	0	0
$481$	−4.00000	−0.182384
$482$	10.0000	0.455488
$483$	0	0
$484$	−7.00000	−0.318182
$485$	72.0000	3.26935
$486$	0	0
$487$	−4.00000	−0.181257	−0.0906287	−	0.995885i	$-0.528888\pi$
$487$	−4.00000	−0.181257	−0.0906287	+	0.995885i	$0.528888\pi$
$488$	−2.00000	−0.0905357
$489$	0	0
$490$	4.00000	0.180702
$491$	10.0000	0.451294	0.225647	−	0.974209i	$-0.427550\pi$
$491$	10.0000	0.451294	0.225647	+	0.974209i	$0.427550\pi$
$492$	0	0
$493$	0	0
$494$	2.00000	0.0899843
$495$	0	0
$496$	4.00000	0.179605
$497$	0	0
$498$	0	0
$499$	12.0000	0.537194	0.268597	−	0.963253i	$-0.413440\pi$
$499$	12.0000	0.537194	0.268597	+	0.963253i	$0.413440\pi$
$500$	−24.0000	−1.07331
$501$	0	0
$502$	30.0000	1.33897
$503$	26.0000	1.15928	0.579641	−	0.814872i	$-0.303193\pi$
$503$	26.0000	1.15928	0.579641	+	0.814872i	$0.303193\pi$
$504$	0	0
$505$	0	0
$506$	−4.00000	−0.177822
$507$	0	0
$508$	4.00000	0.177471
$509$	−14.0000	−0.620539	−0.310270	−	0.950649i	$-0.600419\pi$
$509$	−14.0000	−0.620539	−0.310270	+	0.950649i	$0.600419\pi$
$510$	0	0
$511$	−14.0000	−0.619324
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	−10.0000	−0.441081
$515$	0	0
$516$	0	0
$517$	12.0000	0.527759
$518$	−2.00000	−0.0878750
$519$	0	0
$520$	−8.00000	−0.350823
$521$	14.0000	0.613351	0.306676	−	0.951814i	$-0.400783\pi$
$521$	14.0000	0.613351	0.306676	+	0.951814i	$0.400783\pi$
$522$	0	0
$523$	20.0000	0.874539	0.437269	−	0.899331i	$-0.355946\pi$
$523$	20.0000	0.874539	0.437269	+	0.899331i	$0.355946\pi$
$524$	−18.0000	−0.786334
$525$	0	0
$526$	−30.0000	−1.30806
$527$	0	0
$528$	0	0
$529$	−19.0000	−0.826087
$530$	8.00000	0.347498
$531$	0	0
$532$	1.00000	0.0433555
$533$	20.0000	0.866296
$534$	0	0
$535$	0	0
$536$	12.0000	0.518321
$537$	0	0
$538$	14.0000	0.603583
$539$	2.00000	0.0861461
$540$	0	0
$541$	−18.0000	−0.773880	−0.386940	−	0.922105i	$-0.626468\pi$
$541$	−18.0000	−0.773880	−0.386940	+	0.922105i	$0.626468\pi$
$542$	−16.0000	−0.687259
$543$	0	0
$544$	0	0
$545$	40.0000	1.71341
$546$	0	0
$547$	8.00000	0.342055	0.171028	−	0.985266i	$-0.445291\pi$
$547$	8.00000	0.342055	0.171028	+	0.985266i	$0.445291\pi$
$548$	−12.0000	−0.512615
$549$	0	0
$550$	−22.0000	−0.938083
$551$	−6.00000	−0.255609
$552$	0	0
$553$	8.00000	0.340195
$554$	14.0000	0.594803
$555$	0	0
$556$	−12.0000	−0.508913
$557$	−28.0000	−1.18640	−0.593199	−	0.805056i	$-0.702135\pi$
$557$	−28.0000	−1.18640	−0.593199	+	0.805056i	$0.702135\pi$
$558$	0	0
$559$	−8.00000	−0.338364
$560$	−4.00000	−0.169031
$561$	0	0
$562$	26.0000	1.09674
$563$	24.0000	1.01148	0.505740	−	0.862686i	$-0.331220\pi$
$563$	24.0000	1.01148	0.505740	+	0.862686i	$0.331220\pi$
$564$	0	0
$565$	−8.00000	−0.336563
$566$	28.0000	1.17693
$567$	0	0
$568$	0	0
$569$	38.0000	1.59304	0.796521	−	0.604610i	$-0.206671\pi$
$569$	38.0000	1.59304	0.796521	+	0.604610i	$0.206671\pi$
$570$	0	0
$571$	36.0000	1.50655	0.753277	−	0.657704i	$-0.228472\pi$
$571$	36.0000	1.50655	0.753277	+	0.657704i	$0.228472\pi$
$572$	−4.00000	−0.167248
$573$	0	0
$574$	10.0000	0.417392
$575$	22.0000	0.917463
$576$	0	0
$577$	14.0000	0.582828	0.291414	−	0.956597i	$-0.405874\pi$
$577$	14.0000	0.582828	0.291414	+	0.956597i	$0.405874\pi$
$578$	17.0000	0.707107
$579$	0	0
$580$	24.0000	0.996546
$581$	10.0000	0.414870
$582$	0	0
$583$	4.00000	0.165663
$584$	14.0000	0.579324
$585$	0	0
$586$	6.00000	0.247858
$587$	42.0000	1.73353	0.866763	−	0.498721i	$-0.166197\pi$
$587$	42.0000	1.73353	0.866763	+	0.498721i	$0.166197\pi$
$588$	0	0
$589$	4.00000	0.164817
$590$	16.0000	0.658710
$591$	0	0
$592$	2.00000	0.0821995
$593$	−36.0000	−1.47834	−0.739171	−	0.673517i	$-0.764783\pi$
$593$	−36.0000	−1.47834	−0.739171	+	0.673517i	$0.764783\pi$
$594$	0	0
$595$	0	0
$596$	4.00000	0.163846
$597$	0	0
$598$	4.00000	0.163572
$599$	4.00000	0.163436	0.0817178	−	0.996656i	$-0.473959\pi$
$599$	4.00000	0.163436	0.0817178	+	0.996656i	$0.473959\pi$
$600$	0	0
$601$	42.0000	1.71322	0.856608	−	0.515968i	$-0.172568\pi$
$601$	42.0000	1.71322	0.856608	+	0.515968i	$0.172568\pi$
$602$	−4.00000	−0.163028
$603$	0	0
$604$	−8.00000	−0.325515
$605$	28.0000	1.13836
$606$	0	0
$607$	4.00000	0.162355	0.0811775	−	0.996700i	$-0.474132\pi$
$607$	4.00000	0.162355	0.0811775	+	0.996700i	$0.474132\pi$
$608$	−1.00000	−0.0405554
$609$	0	0
$610$	8.00000	0.323911
$611$	−12.0000	−0.485468
$612$	0	0
$613$	−6.00000	−0.242338	−0.121169	−	0.992632i	$-0.538664\pi$
$613$	−6.00000	−0.242338	−0.121169	+	0.992632i	$0.538664\pi$
$614$	8.00000	0.322854
$615$	0	0
$616$	−2.00000	−0.0805823
$617$	20.0000	0.805170	0.402585	−	0.915383i	$-0.368112\pi$
$617$	20.0000	0.805170	0.402585	+	0.915383i	$0.368112\pi$
$618$	0	0
$619$	−20.0000	−0.803868	−0.401934	−	0.915669i	$-0.631662\pi$
$619$	−20.0000	−0.803868	−0.401934	+	0.915669i	$0.631662\pi$
$620$	−16.0000	−0.642575
$621$	0	0
$622$	18.0000	0.721734
$623$	−18.0000	−0.721155
$624$	0	0
$625$	41.0000	1.64000
$626$	−22.0000	−0.879297
$627$	0	0
$628$	−14.0000	−0.558661
$629$	0	0
$630$	0	0
$631$	16.0000	0.636950	0.318475	−	0.947931i	$-0.396829\pi$
$631$	16.0000	0.636950	0.318475	+	0.947931i	$0.396829\pi$
$632$	−8.00000	−0.318223
$633$	0	0
$634$	10.0000	0.397151
$635$	−16.0000	−0.634941
$636$	0	0
$637$	−2.00000	−0.0792429
$638$	12.0000	0.475085
$639$	0	0
$640$	4.00000	0.158114
$641$	−30.0000	−1.18493	−0.592464	−	0.805597i	$-0.701845\pi$
$641$	−30.0000	−1.18493	−0.592464	+	0.805597i	$0.701845\pi$
$642$	0	0
$643$	−12.0000	−0.473234	−0.236617	−	0.971603i	$-0.576039\pi$
$643$	−12.0000	−0.473234	−0.236617	+	0.971603i	$0.576039\pi$
$644$	2.00000	0.0788110
$645$	0	0
$646$	0	0
$647$	−18.0000	−0.707653	−0.353827	−	0.935311i	$-0.615120\pi$
$647$	−18.0000	−0.707653	−0.353827	+	0.935311i	$0.615120\pi$
$648$	0	0
$649$	8.00000	0.314027
$650$	22.0000	0.862911
$651$	0	0
$652$	−4.00000	−0.156652
$653$	36.0000	1.40879	0.704394	−	0.709809i	$-0.251219\pi$
$653$	36.0000	1.40879	0.704394	+	0.709809i	$0.251219\pi$
$654$	0	0
$655$	72.0000	2.81327
$656$	−10.0000	−0.390434
$657$	0	0
$658$	−6.00000	−0.233904
$659$	−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
$661$	26.0000	1.01128	0.505641	−	0.862744i	$-0.331256\pi$
$661$	26.0000	1.01128	0.505641	+	0.862744i	$0.331256\pi$
$662$	0	0
$663$	0	0
$664$	−10.0000	−0.388075
$665$	−4.00000	−0.155113
$666$	0	0
$667$	−12.0000	−0.464642
$668$	16.0000	0.619059
$669$	0	0
$670$	−48.0000	−1.85440
$671$	4.00000	0.154418
$672$	0	0
$673$	−46.0000	−1.77317	−0.886585	−	0.462566i	$-0.846929\pi$
$673$	−46.0000	−1.77317	−0.886585	+	0.462566i	$0.846929\pi$
$674$	2.00000	0.0770371
$675$	0	0
$676$	−9.00000	−0.346154
$677$	14.0000	0.538064	0.269032	−	0.963131i	$-0.413296\pi$
$677$	14.0000	0.538064	0.269032	+	0.963131i	$0.413296\pi$
$678$	0	0
$679$	−18.0000	−0.690777
$680$	0	0
$681$	0	0
$682$	−8.00000	−0.306336
$683$	−48.0000	−1.83667	−0.918334	−	0.395805i	$-0.870466\pi$
$683$	−48.0000	−1.83667	−0.918334	+	0.395805i	$0.870466\pi$
$684$	0	0
$685$	48.0000	1.83399
$686$	−1.00000	−0.0381802
$687$	0	0
$688$	4.00000	0.152499
$689$	−4.00000	−0.152388
$690$	0	0
$691$	4.00000	0.152167	0.0760836	−	0.997101i	$-0.475758\pi$
$691$	4.00000	0.152167	0.0760836	+	0.997101i	$0.475758\pi$
$692$	−14.0000	−0.532200
$693$	0	0
$694$	−18.0000	−0.683271
$695$	48.0000	1.82074
$696$	0	0
$697$	0	0
$698$	2.00000	0.0757011
$699$	0	0
$700$	11.0000	0.415761
$701$	−12.0000	−0.453234	−0.226617	−	0.973984i	$-0.572767\pi$
$701$	−12.0000	−0.453234	−0.226617	+	0.973984i	$0.572767\pi$
$702$	0	0
$703$	2.00000	0.0754314
$704$	2.00000	0.0753778
$705$	0	0
$706$	24.0000	0.903252
$707$	0	0
$708$	0	0
$709$	38.0000	1.42712	0.713560	−	0.700594i	$-0.247082\pi$
$709$	38.0000	1.42712	0.713560	+	0.700594i	$0.247082\pi$
$710$	0	0
$711$	0	0
$712$	18.0000	0.674579
$713$	8.00000	0.299602
$714$	0	0
$715$	16.0000	0.598366
$716$	0	0
$717$	0	0
$718$	2.00000	0.0746393
$719$	−26.0000	−0.969636	−0.484818	−	0.874615i	$-0.661114\pi$
$719$	−26.0000	−0.969636	−0.484818	+	0.874615i	$0.661114\pi$
$720$	0	0
$721$	0	0
$722$	−1.00000	−0.0372161
$723$	0	0
$724$	−10.0000	−0.371647
$725$	−66.0000	−2.45118
$726$	0	0
$727$	16.0000	0.593407	0.296704	−	0.954970i	$-0.404113\pi$
$727$	16.0000	0.593407	0.296704	+	0.954970i	$0.404113\pi$
$728$	2.00000	0.0741249
$729$	0	0
$730$	−56.0000	−2.07265
$731$	0	0
$732$	0	0
$733$	−2.00000	−0.0738717	−0.0369358	−	0.999318i	$-0.511760\pi$
$733$	−2.00000	−0.0738717	−0.0369358	+	0.999318i	$0.511760\pi$
$734$	0	0
$735$	0	0
$736$	−2.00000	−0.0737210
$737$	−24.0000	−0.884051
$738$	0	0
$739$	4.00000	0.147142	0.0735712	−	0.997290i	$-0.476560\pi$
$739$	4.00000	0.147142	0.0735712	+	0.997290i	$0.476560\pi$
$740$	−8.00000	−0.294086
$741$	0	0
$742$	−2.00000	−0.0734223
$743$	−8.00000	−0.293492	−0.146746	−	0.989174i	$-0.546880\pi$
$743$	−8.00000	−0.293492	−0.146746	+	0.989174i	$0.546880\pi$
$744$	0	0
$745$	−16.0000	−0.586195
$746$	−26.0000	−0.951928
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−24.0000	−0.875772	−0.437886	−	0.899030i	$-0.644273\pi$
$751$	−24.0000	−0.875772	−0.437886	+	0.899030i	$0.644273\pi$
$752$	6.00000	0.218797
$753$	0	0
$754$	−12.0000	−0.437014
$755$	32.0000	1.16460
$756$	0	0
$757$	10.0000	0.363456	0.181728	−	0.983349i	$-0.441831\pi$
$757$	10.0000	0.363456	0.181728	+	0.983349i	$0.441831\pi$
$758$	24.0000	0.871719
$759$	0	0
$760$	4.00000	0.145095
$761$	0	0		−	1.00000i	$-0.5\pi$
$761$	0	0			1.00000i	$0.5\pi$
$762$	0	0
$763$	−10.0000	−0.362024
$764$	22.0000	0.795932
$765$	0	0
$766$	12.0000	0.433578
$767$	−8.00000	−0.288863
$768$	0	0
$769$	−18.0000	−0.649097	−0.324548	−	0.945869i	$-0.605212\pi$
$769$	−18.0000	−0.649097	−0.324548	+	0.945869i	$0.605212\pi$
$770$	8.00000	0.288300
$771$	0	0
$772$	18.0000	0.647834
$773$	−14.0000	−0.503545	−0.251773	−	0.967786i	$-0.581013\pi$
$773$	−14.0000	−0.503545	−0.251773	+	0.967786i	$0.581013\pi$
$774$	0	0
$775$	44.0000	1.58053
$776$	18.0000	0.646162
$777$	0	0
$778$	−24.0000	−0.860442
$779$	−10.0000	−0.358287
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	1.00000	0.0357143
$785$	56.0000	1.99873
$786$	0	0
$787$	52.0000	1.85360	0.926800	−	0.375555i	$-0.122548\pi$
$787$	52.0000	1.85360	0.926800	+	0.375555i	$0.122548\pi$
$788$	−8.00000	−0.284988
$789$	0	0
$790$	32.0000	1.13851
$791$	2.00000	0.0711118
$792$	0	0
$793$	−4.00000	−0.142044
$794$	18.0000	0.638796
$795$	0	0
$796$	8.00000	0.283552
$797$	−46.0000	−1.62940	−0.814702	−	0.579880i	$-0.803099\pi$
$797$	−46.0000	−1.62940	−0.814702	+	0.579880i	$0.803099\pi$
$798$	0	0
$799$	0	0
$800$	−11.0000	−0.388909
$801$	0	0
$802$	−10.0000	−0.353112
$803$	−28.0000	−0.988099
$804$	0	0
$805$	−8.00000	−0.281963
$806$	8.00000	0.281788
$807$	0	0
$808$	0	0
$809$	0	0		−	1.00000i	$-0.5\pi$
$809$	0	0			1.00000i	$0.5\pi$
$810$	0	0
$811$	24.0000	0.842754	0.421377	−	0.906886i	$-0.361547\pi$
$811$	24.0000	0.842754	0.421377	+	0.906886i	$0.361547\pi$
$812$	−6.00000	−0.210559
$813$	0	0
$814$	−4.00000	−0.140200
$815$	16.0000	0.560456
$816$	0	0
$817$	4.00000	0.139942
$818$	−38.0000	−1.32864
$819$	0	0
$820$	40.0000	1.39686
$821$	36.0000	1.25641	0.628204	−	0.778048i	$-0.283790\pi$
$821$	36.0000	1.25641	0.628204	+	0.778048i	$0.283790\pi$
$822$	0	0
$823$	32.0000	1.11545	0.557725	−	0.830026i	$-0.311674\pi$
$823$	32.0000	1.11545	0.557725	+	0.830026i	$0.311674\pi$
$824$	0	0
$825$	0	0
$826$	−4.00000	−0.139178
$827$	−48.0000	−1.66912	−0.834562	−	0.550914i	$-0.814279\pi$
$827$	−48.0000	−1.66912	−0.834562	+	0.550914i	$0.814279\pi$
$828$	0	0
$829$	−10.0000	−0.347314	−0.173657	−	0.984806i	$-0.555558\pi$
$829$	−10.0000	−0.347314	−0.173657	+	0.984806i	$0.555558\pi$
$830$	40.0000	1.38842
$831$	0	0
$832$	−2.00000	−0.0693375
$833$	0	0
$834$	0	0
$835$	−64.0000	−2.21481
$836$	2.00000	0.0691714
$837$	0	0
$838$	26.0000	0.898155
$839$	−12.0000	−0.414286	−0.207143	−	0.978311i	$-0.566417\pi$
$839$	−12.0000	−0.414286	−0.207143	+	0.978311i	$0.566417\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	26.0000	0.896019
$843$	0	0
$844$	−20.0000	−0.688428
$845$	36.0000	1.23844
$846$	0	0
$847$	−7.00000	−0.240523
$848$	2.00000	0.0686803
$849$	0	0
$850$	0	0
$851$	4.00000	0.137118
$852$	0	0
$853$	14.0000	0.479351	0.239675	−	0.970853i	$-0.422959\pi$
$853$	14.0000	0.479351	0.239675	+	0.970853i	$0.422959\pi$
$854$	−2.00000	−0.0684386
$855$	0	0
$856$	0	0
$857$	22.0000	0.751506	0.375753	−	0.926720i	$-0.377384\pi$
$857$	22.0000	0.751506	0.375753	+	0.926720i	$0.377384\pi$
$858$	0	0
$859$	52.0000	1.77422	0.887109	−	0.461561i	$-0.152710\pi$
$859$	52.0000	1.77422	0.887109	+	0.461561i	$0.152710\pi$
$860$	−16.0000	−0.545595
$861$	0	0
$862$	−24.0000	−0.817443
$863$	32.0000	1.08929	0.544646	−	0.838666i	$-0.316664\pi$
$863$	32.0000	1.08929	0.544646	+	0.838666i	$0.316664\pi$
$864$	0	0
$865$	56.0000	1.90406
$866$	−26.0000	−0.883516
$867$	0	0
$868$	4.00000	0.135769
$869$	16.0000	0.542763
$870$	0	0
$871$	24.0000	0.813209
$872$	10.0000	0.338643
$873$	0	0
$874$	−2.00000	−0.0676510
$875$	−24.0000	−0.811348
$876$	0	0
$877$	2.00000	0.0675352	0.0337676	−	0.999430i	$-0.489249\pi$
$877$	2.00000	0.0675352	0.0337676	+	0.999430i	$0.489249\pi$
$878$	32.0000	1.07995
$879$	0	0
$880$	−8.00000	−0.269680
$881$	24.0000	0.808581	0.404290	−	0.914631i	$-0.367519\pi$
$881$	24.0000	0.808581	0.404290	+	0.914631i	$0.367519\pi$
$882$	0	0
$883$	−20.0000	−0.673054	−0.336527	−	0.941674i	$-0.609252\pi$
$883$	−20.0000	−0.673054	−0.336527	+	0.941674i	$0.609252\pi$
$884$	0	0
$885$	0	0
$886$	−10.0000	−0.335957
$887$	−48.0000	−1.61168	−0.805841	−	0.592132i	$-0.798286\pi$
$887$	−48.0000	−1.61168	−0.805841	+	0.592132i	$0.798286\pi$
$888$	0	0
$889$	4.00000	0.134156
$890$	−72.0000	−2.41345
$891$	0	0
$892$	16.0000	0.535720
$893$	6.00000	0.200782
$894$	0	0
$895$	0	0
$896$	−1.00000	−0.0334077
$897$	0	0
$898$	10.0000	0.333704
$899$	−24.0000	−0.800445
$900$	0	0
$901$	0	0
$902$	20.0000	0.665927
$903$	0	0
$904$	−2.00000	−0.0665190
$905$	40.0000	1.32964
$906$	0	0
$907$	−28.0000	−0.929725	−0.464862	−	0.885383i	$-0.653896\pi$
$907$	−28.0000	−0.929725	−0.464862	+	0.885383i	$0.653896\pi$
$908$	0	0
$909$	0	0
$910$	−8.00000	−0.265197
$911$	−36.0000	−1.19273	−0.596367	−	0.802712i	$-0.703390\pi$
$911$	−36.0000	−1.19273	−0.596367	+	0.802712i	$0.703390\pi$
$912$	0	0
$913$	20.0000	0.661903
$914$	−18.0000	−0.595387
$915$	0	0
$916$	−6.00000	−0.198246
$917$	−18.0000	−0.594412
$918$	0	0
$919$	−24.0000	−0.791687	−0.395843	−	0.918318i	$-0.629548\pi$
$919$	−24.0000	−0.791687	−0.395843	+	0.918318i	$0.629548\pi$
$920$	8.00000	0.263752
$921$	0	0
$922$	−16.0000	−0.526932
$923$	0	0
$924$	0	0
$925$	22.0000	0.723356
$926$	32.0000	1.05159
$927$	0	0
$928$	6.00000	0.196960
$929$	36.0000	1.18112	0.590561	−	0.806993i	$-0.298907\pi$
$929$	36.0000	1.18112	0.590561	+	0.806993i	$0.298907\pi$
$930$	0	0
$931$	1.00000	0.0327737
$932$	8.00000	0.262049
$933$	0	0
$934$	6.00000	0.196326
$935$	0	0
$936$	0	0
$937$	−38.0000	−1.24141	−0.620703	−	0.784046i	$-0.713153\pi$
$937$	−38.0000	−1.24141	−0.620703	+	0.784046i	$0.713153\pi$
$938$	12.0000	0.391814
$939$	0	0
$940$	−24.0000	−0.782794
$941$	−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
$943$	−20.0000	−0.651290
$944$	4.00000	0.130189
$945$	0	0
$946$	−8.00000	−0.260102
$947$	−6.00000	−0.194974	−0.0974869	−	0.995237i	$-0.531080\pi$
$947$	−6.00000	−0.194974	−0.0974869	+	0.995237i	$0.531080\pi$
$948$	0	0
$949$	28.0000	0.908918
$950$	−11.0000	−0.356887
$951$	0	0
$952$	0	0
$953$	−2.00000	−0.0647864	−0.0323932	−	0.999475i	$-0.510313\pi$
$953$	−2.00000	−0.0647864	−0.0323932	+	0.999475i	$0.510313\pi$
$954$	0	0
$955$	−88.0000	−2.84761
$956$	14.0000	0.452792
$957$	0	0
$958$	−18.0000	−0.581554
$959$	−12.0000	−0.387500
$960$	0	0
$961$	−15.0000	−0.483871
$962$	4.00000	0.128965
$963$	0	0
$964$	−10.0000	−0.322078
$965$	−72.0000	−2.31776
$966$	0	0
$967$	−48.0000	−1.54358	−0.771788	−	0.635880i	$-0.780637\pi$
$967$	−48.0000	−1.54358	−0.771788	+	0.635880i	$0.780637\pi$
$968$	7.00000	0.224989
$969$	0	0
$970$	−72.0000	−2.31178
$971$	−16.0000	−0.513464	−0.256732	−	0.966483i	$-0.582646\pi$
$971$	−16.0000	−0.513464	−0.256732	+	0.966483i	$0.582646\pi$
$972$	0	0
$973$	−12.0000	−0.384702
$974$	4.00000	0.128168
$975$	0	0
$976$	2.00000	0.0640184
$977$	−14.0000	−0.447900	−0.223950	−	0.974601i	$-0.571895\pi$
$977$	−14.0000	−0.447900	−0.223950	+	0.974601i	$0.571895\pi$
$978$	0	0
$979$	−36.0000	−1.15056
$980$	−4.00000	−0.127775
$981$	0	0
$982$	−10.0000	−0.319113
$983$	24.0000	0.765481	0.382741	−	0.923856i	$-0.374980\pi$
$983$	24.0000	0.765481	0.382741	+	0.923856i	$0.374980\pi$
$984$	0	0
$985$	32.0000	1.01960
$986$	0	0
$987$	0	0
$988$	−2.00000	−0.0636285
$989$	8.00000	0.254385
$990$	0	0
$991$	−4.00000	−0.127064	−0.0635321	−	0.997980i	$-0.520237\pi$
$991$	−4.00000	−0.127064	−0.0635321	+	0.997980i	$0.520237\pi$
$992$	−4.00000	−0.127000
$993$	0	0
$994$	0	0
$995$	−32.0000	−1.01447
$996$	0	0
$997$	−22.0000	−0.696747	−0.348373	−	0.937356i	$-0.613266\pi$
$997$	−22.0000	−0.696747	−0.348373	+	0.937356i	$0.613266\pi$
$998$	−12.0000	−0.379853
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2394.2.a.a.1.1	✓	1
3.2	odd	2		2394.2.a.o.1.1	yes	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2394.2.a.a.1.1	✓	1	1.1	even	1	trivial
2394.2.a.o.1.1	yes	1	3.2	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:	\( N \)	\(=\)	\( 2394 = 2 \cdot 3^{2} \cdot 7 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2394.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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