LMFDB - Embedded newform 1638.2.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1638,2,Mod(1,1638)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1638, 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("1638.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1638 = 2 \cdot 3^{2} \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1638.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:	$13.0794958511$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 546)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	1638.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{2} +1.00000 q^{4} -3.00000 q^{5} -1.00000 q^{7} -1.00000 q^{8} +O(q^{10})$

$q-1.00000 q^{2} +1.00000 q^{4} -3.00000 q^{5} -1.00000 q^{7} -1.00000 q^{8} +3.00000 q^{10} -1.00000 q^{11} -1.00000 q^{13} +1.00000 q^{14} +1.00000 q^{16} -7.00000 q^{17} +1.00000 q^{19} -3.00000 q^{20} +1.00000 q^{22} +7.00000 q^{23} +4.00000 q^{25} +1.00000 q^{26} -1.00000 q^{28} -3.00000 q^{29} -1.00000 q^{32} +7.00000 q^{34} +3.00000 q^{35} -5.00000 q^{37} -1.00000 q^{38} +3.00000 q^{40} -4.00000 q^{41} +11.0000 q^{43} -1.00000 q^{44} -7.00000 q^{46} +1.00000 q^{49} -4.00000 q^{50} -1.00000 q^{52} +14.0000 q^{53} +3.00000 q^{55} +1.00000 q^{56} +3.00000 q^{58} -4.00000 q^{59} +1.00000 q^{61} +1.00000 q^{64} +3.00000 q^{65} -6.00000 q^{67} -7.00000 q^{68} -3.00000 q^{70} +12.0000 q^{71} +5.00000 q^{73} +5.00000 q^{74} +1.00000 q^{76} +1.00000 q^{77} -10.0000 q^{79} -3.00000 q^{80} +4.00000 q^{82} +14.0000 q^{83} +21.0000 q^{85} -11.0000 q^{86} +1.00000 q^{88} +6.00000 q^{89} +1.00000 q^{91} +7.00000 q^{92} -3.00000 q^{95} +6.00000 q^{97} -1.00000 q^{98} +O(q^{100})$

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
$5$	−3.00000	−1.34164	−0.670820	−	0.741620i	$-0.734058\pi$
$5$	−3.00000	−1.34164	−0.670820	+	0.741620i	$0.734058\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−1.00000	−0.353553
$9$	0	0
$10$	3.00000	0.948683
$11$	−1.00000	−0.301511	−0.150756	−	0.988571i	$-0.548171\pi$
$11$	−1.00000	−0.301511	−0.150756	+	0.988571i	$0.548171\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	1.00000	0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	−7.00000	−1.69775	−0.848875	−	0.528594i	$-0.822719\pi$
$17$	−7.00000	−1.69775	−0.848875	+	0.528594i	$0.822719\pi$
$18$	0	0
$19$	1.00000	0.229416	0.114708	−	0.993399i	$-0.463407\pi$
$19$	1.00000	0.229416	0.114708	+	0.993399i	$0.463407\pi$
$20$	−3.00000	−0.670820
$21$	0	0
$22$	1.00000	0.213201
$23$	7.00000	1.45960	0.729800	−	0.683660i	$-0.239613\pi$
$23$	7.00000	1.45960	0.729800	+	0.683660i	$0.239613\pi$
$24$	0	0
$25$	4.00000	0.800000
$26$	1.00000	0.196116
$27$	0	0
$28$	−1.00000	−0.188982
$29$	−3.00000	−0.557086	−0.278543	−	0.960424i	$-0.589851\pi$
$29$	−3.00000	−0.557086	−0.278543	+	0.960424i	$0.589851\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	7.00000	1.20049
$35$	3.00000	0.507093
$36$	0	0
$37$	−5.00000	−0.821995	−0.410997	−	0.911636i	$-0.634819\pi$
$37$	−5.00000	−0.821995	−0.410997	+	0.911636i	$0.634819\pi$
$38$	−1.00000	−0.162221
$39$	0	0
$40$	3.00000	0.474342
$41$	−4.00000	−0.624695	−0.312348	−	0.949968i	$-0.601115\pi$
$41$	−4.00000	−0.624695	−0.312348	+	0.949968i	$0.601115\pi$
$42$	0	0
$43$	11.0000	1.67748	0.838742	−	0.544529i	$-0.183292\pi$
$43$	11.0000	1.67748	0.838742	+	0.544529i	$0.183292\pi$
$44$	−1.00000	−0.150756
$45$	0	0
$46$	−7.00000	−1.03209
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−4.00000	−0.565685
$51$	0	0
$52$	−1.00000	−0.138675
$53$	14.0000	1.92305	0.961524	−	0.274721i	$-0.0885855\pi$
$53$	14.0000	1.92305	0.961524	+	0.274721i	$0.0885855\pi$
$54$	0	0
$55$	3.00000	0.404520
$56$	1.00000	0.133631
$57$	0	0
$58$	3.00000	0.393919
$59$	−4.00000	−0.520756	−0.260378	−	0.965507i	$-0.583847\pi$
$59$	−4.00000	−0.520756	−0.260378	+	0.965507i	$0.583847\pi$
$60$	0	0
$61$	1.00000	0.128037	0.0640184	−	0.997949i	$-0.479608\pi$
$61$	1.00000	0.128037	0.0640184	+	0.997949i	$0.479608\pi$
$62$	0	0
$63$	0	0
$64$	1.00000	0.125000
$65$	3.00000	0.372104
$66$	0	0
$67$	−6.00000	−0.733017	−0.366508	−	0.930415i	$-0.619447\pi$
$67$	−6.00000	−0.733017	−0.366508	+	0.930415i	$0.619447\pi$
$68$	−7.00000	−0.848875
$69$	0	0
$70$	−3.00000	−0.358569
$71$	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
$73$	5.00000	0.585206	0.292603	−	0.956234i	$-0.405479\pi$
$73$	5.00000	0.585206	0.292603	+	0.956234i	$0.405479\pi$
$74$	5.00000	0.581238
$75$	0	0
$76$	1.00000	0.114708
$77$	1.00000	0.113961
$78$	0	0
$79$	−10.0000	−1.12509	−0.562544	−	0.826767i	$-0.690177\pi$
$79$	−10.0000	−1.12509	−0.562544	+	0.826767i	$0.690177\pi$
$80$	−3.00000	−0.335410
$81$	0	0
$82$	4.00000	0.441726
$83$	14.0000	1.53670	0.768350	−	0.640030i	$-0.221078\pi$
$83$	14.0000	1.53670	0.768350	+	0.640030i	$0.221078\pi$
$84$	0	0
$85$	21.0000	2.27777
$86$	−11.0000	−1.18616
$87$	0	0
$88$	1.00000	0.106600
$89$	6.00000	0.635999	0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	7.00000	0.729800
$93$	0	0
$94$	0	0
$95$	−3.00000	−0.307794
$96$	0	0
$97$	6.00000	0.609208	0.304604	−	0.952479i	$-0.401476\pi$
$97$	6.00000	0.609208	0.304604	+	0.952479i	$0.401476\pi$
$98$	−1.00000	−0.101015
$99$	0	0
$100$	4.00000	0.400000
$101$	12.0000	1.19404	0.597022	−	0.802225i	$-0.296350\pi$
$101$	12.0000	1.19404	0.597022	+	0.802225i	$0.296350\pi$
$102$	0	0
$103$	−7.00000	−0.689730	−0.344865	−	0.938652i	$-0.612075\pi$
$103$	−7.00000	−0.689730	−0.344865	+	0.938652i	$0.612075\pi$
$104$	1.00000	0.0980581
$105$	0	0
$106$	−14.0000	−1.35980
$107$	18.0000	1.74013	0.870063	−	0.492941i	$-0.164078\pi$
$107$	18.0000	1.74013	0.870063	+	0.492941i	$0.164078\pi$
$108$	0	0
$109$	−7.00000	−0.670478	−0.335239	−	0.942133i	$-0.608817\pi$
$109$	−7.00000	−0.670478	−0.335239	+	0.942133i	$0.608817\pi$
$110$	−3.00000	−0.286039
$111$	0	0
$112$	−1.00000	−0.0944911
$113$	14.0000	1.31701	0.658505	−	0.752577i	$-0.271189\pi$
$113$	14.0000	1.31701	0.658505	+	0.752577i	$0.271189\pi$
$114$	0	0
$115$	−21.0000	−1.95826
$116$	−3.00000	−0.278543
$117$	0	0
$118$	4.00000	0.368230
$119$	7.00000	0.641689
$120$	0	0
$121$	−10.0000	−0.909091
$122$	−1.00000	−0.0905357
$123$	0	0
$124$	0	0
$125$	3.00000	0.268328
$126$	0	0
$127$	10.0000	0.887357	0.443678	−	0.896186i	$-0.353673\pi$
$127$	10.0000	0.887357	0.443678	+	0.896186i	$0.353673\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	−3.00000	−0.263117
$131$	−15.0000	−1.31056	−0.655278	−	0.755388i	$-0.727449\pi$
$131$	−15.0000	−1.31056	−0.655278	+	0.755388i	$0.727449\pi$
$132$	0	0
$133$	−1.00000	−0.0867110
$134$	6.00000	0.518321
$135$	0	0
$136$	7.00000	0.600245
$137$	−15.0000	−1.28154	−0.640768	−	0.767734i	$-0.721384\pi$
$137$	−15.0000	−1.28154	−0.640768	+	0.767734i	$0.721384\pi$
$138$	0	0
$139$	4.00000	0.339276	0.169638	−	0.985506i	$-0.445740\pi$
$139$	4.00000	0.339276	0.169638	+	0.985506i	$0.445740\pi$
$140$	3.00000	0.253546
$141$	0	0
$142$	−12.0000	−1.00702
$143$	1.00000	0.0836242
$144$	0	0
$145$	9.00000	0.747409
$146$	−5.00000	−0.413803
$147$	0	0
$148$	−5.00000	−0.410997
$149$	−16.0000	−1.31077	−0.655386	−	0.755295i	$-0.727494\pi$
$149$	−16.0000	−1.31077	−0.655386	+	0.755295i	$0.727494\pi$
$150$	0	0
$151$	7.00000	0.569652	0.284826	−	0.958579i	$-0.408064\pi$
$151$	7.00000	0.569652	0.284826	+	0.958579i	$0.408064\pi$
$152$	−1.00000	−0.0811107
$153$	0	0
$154$	−1.00000	−0.0805823
$155$	0	0
$156$	0	0
$157$	5.00000	0.399043	0.199522	−	0.979893i	$-0.436061\pi$
$157$	5.00000	0.399043	0.199522	+	0.979893i	$0.436061\pi$
$158$	10.0000	0.795557
$159$	0	0
$160$	3.00000	0.237171
$161$	−7.00000	−0.551677
$162$	0	0
$163$	10.0000	0.783260	0.391630	−	0.920123i	$-0.371911\pi$
$163$	10.0000	0.783260	0.391630	+	0.920123i	$0.371911\pi$
$164$	−4.00000	−0.312348
$165$	0	0
$166$	−14.0000	−1.08661
$167$	−7.00000	−0.541676	−0.270838	−	0.962625i	$-0.587301\pi$
$167$	−7.00000	−0.541676	−0.270838	+	0.962625i	$0.587301\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	−21.0000	−1.61063
$171$	0	0
$172$	11.0000	0.838742
$173$	−2.00000	−0.152057	−0.0760286	−	0.997106i	$-0.524224\pi$
$173$	−2.00000	−0.152057	−0.0760286	+	0.997106i	$0.524224\pi$
$174$	0	0
$175$	−4.00000	−0.302372
$176$	−1.00000	−0.0753778
$177$	0	0
$178$	−6.00000	−0.449719
$179$	−8.00000	−0.597948	−0.298974	−	0.954261i	$-0.596644\pi$
$179$	−8.00000	−0.597948	−0.298974	+	0.954261i	$0.596644\pi$
$180$	0	0
$181$	22.0000	1.63525	0.817624	−	0.575753i	$-0.195291\pi$
$181$	22.0000	1.63525	0.817624	+	0.575753i	$0.195291\pi$
$182$	−1.00000	−0.0741249
$183$	0	0
$184$	−7.00000	−0.516047
$185$	15.0000	1.10282
$186$	0	0
$187$	7.00000	0.511891
$188$	0	0
$189$	0	0
$190$	3.00000	0.217643
$191$	15.0000	1.08536	0.542681	−	0.839939i	$-0.317409\pi$
$191$	15.0000	1.08536	0.542681	+	0.839939i	$0.317409\pi$
$192$	0	0
$193$	−8.00000	−0.575853	−0.287926	−	0.957653i	$-0.592966\pi$
$193$	−8.00000	−0.575853	−0.287926	+	0.957653i	$0.592966\pi$
$194$	−6.00000	−0.430775
$195$	0	0
$196$	1.00000	0.0714286
$197$	20.0000	1.42494	0.712470	−	0.701702i	$-0.247576\pi$
$197$	20.0000	1.42494	0.712470	+	0.701702i	$0.247576\pi$
$198$	0	0
$199$	3.00000	0.212664	0.106332	−	0.994331i	$-0.466089\pi$
$199$	3.00000	0.212664	0.106332	+	0.994331i	$0.466089\pi$
$200$	−4.00000	−0.282843
$201$	0	0
$202$	−12.0000	−0.844317
$203$	3.00000	0.210559
$204$	0	0
$205$	12.0000	0.838116
$206$	7.00000	0.487713
$207$	0	0
$208$	−1.00000	−0.0693375
$209$	−1.00000	−0.0691714
$210$	0	0
$211$	25.0000	1.72107	0.860535	−	0.509390i	$-0.170129\pi$
$211$	25.0000	1.72107	0.860535	+	0.509390i	$0.170129\pi$
$212$	14.0000	0.961524
$213$	0	0
$214$	−18.0000	−1.23045
$215$	−33.0000	−2.25058
$216$	0	0
$217$	0	0
$218$	7.00000	0.474100
$219$	0	0
$220$	3.00000	0.202260
$221$	7.00000	0.470871
$222$	0	0
$223$	14.0000	0.937509	0.468755	−	0.883328i	$-0.344703\pi$
$223$	14.0000	0.937509	0.468755	+	0.883328i	$0.344703\pi$
$224$	1.00000	0.0668153
$225$	0	0
$226$	−14.0000	−0.931266
$227$	18.0000	1.19470	0.597351	−	0.801980i	$-0.296220\pi$
$227$	18.0000	1.19470	0.597351	+	0.801980i	$0.296220\pi$
$228$	0	0
$229$	18.0000	1.18947	0.594737	−	0.803921i	$-0.297256\pi$
$229$	18.0000	1.18947	0.594737	+	0.803921i	$0.297256\pi$
$230$	21.0000	1.38470
$231$	0	0
$232$	3.00000	0.196960
$233$	12.0000	0.786146	0.393073	−	0.919507i	$-0.371412\pi$
$233$	12.0000	0.786146	0.393073	+	0.919507i	$0.371412\pi$
$234$	0	0
$235$	0	0
$236$	−4.00000	−0.260378
$237$	0	0
$238$	−7.00000	−0.453743
$239$	−20.0000	−1.29369	−0.646846	−	0.762620i	$-0.723912\pi$
$239$	−20.0000	−1.29369	−0.646846	+	0.762620i	$0.723912\pi$
$240$	0	0
$241$	−18.0000	−1.15948	−0.579741	−	0.814801i	$-0.696846\pi$
$241$	−18.0000	−1.15948	−0.579741	+	0.814801i	$0.696846\pi$
$242$	10.0000	0.642824
$243$	0	0
$244$	1.00000	0.0640184
$245$	−3.00000	−0.191663
$246$	0	0
$247$	−1.00000	−0.0636285
$248$	0	0
$249$	0	0
$250$	−3.00000	−0.189737
$251$	−3.00000	−0.189358	−0.0946792	−	0.995508i	$-0.530183\pi$
$251$	−3.00000	−0.189358	−0.0946792	+	0.995508i	$0.530183\pi$
$252$	0	0
$253$	−7.00000	−0.440086
$254$	−10.0000	−0.627456
$255$	0	0
$256$	1.00000	0.0625000
$257$	−10.0000	−0.623783	−0.311891	−	0.950118i	$-0.600963\pi$
$257$	−10.0000	−0.623783	−0.311891	+	0.950118i	$0.600963\pi$
$258$	0	0
$259$	5.00000	0.310685
$260$	3.00000	0.186052
$261$	0	0
$262$	15.0000	0.926703
$263$	4.00000	0.246651	0.123325	−	0.992366i	$-0.460644\pi$
$263$	4.00000	0.246651	0.123325	+	0.992366i	$0.460644\pi$
$264$	0	0
$265$	−42.0000	−2.58004
$266$	1.00000	0.0613139
$267$	0	0
$268$	−6.00000	−0.366508
$269$	−2.00000	−0.121942	−0.0609711	−	0.998140i	$-0.519420\pi$
$269$	−2.00000	−0.121942	−0.0609711	+	0.998140i	$0.519420\pi$
$270$	0	0
$271$	−30.0000	−1.82237	−0.911185	−	0.411997i	$-0.864831\pi$
$271$	−30.0000	−1.82237	−0.911185	+	0.411997i	$0.864831\pi$
$272$	−7.00000	−0.424437
$273$	0	0
$274$	15.0000	0.906183
$275$	−4.00000	−0.241209
$276$	0	0
$277$	10.0000	0.600842	0.300421	−	0.953807i	$-0.402873\pi$
$277$	10.0000	0.600842	0.300421	+	0.953807i	$0.402873\pi$
$278$	−4.00000	−0.239904
$279$	0	0
$280$	−3.00000	−0.179284
$281$	−6.00000	−0.357930	−0.178965	−	0.983855i	$-0.557275\pi$
$281$	−6.00000	−0.357930	−0.178965	+	0.983855i	$0.557275\pi$
$282$	0	0
$283$	−16.0000	−0.951101	−0.475551	−	0.879688i	$-0.657751\pi$
$283$	−16.0000	−0.951101	−0.475551	+	0.879688i	$0.657751\pi$
$284$	12.0000	0.712069
$285$	0	0
$286$	−1.00000	−0.0591312
$287$	4.00000	0.236113
$288$	0	0
$289$	32.0000	1.88235
$290$	−9.00000	−0.528498
$291$	0	0
$292$	5.00000	0.292603
$293$	−30.0000	−1.75262	−0.876309	−	0.481749i	$-0.840002\pi$
$293$	−30.0000	−1.75262	−0.876309	+	0.481749i	$0.840002\pi$
$294$	0	0
$295$	12.0000	0.698667
$296$	5.00000	0.290619
$297$	0	0
$298$	16.0000	0.926855
$299$	−7.00000	−0.404820
$300$	0	0
$301$	−11.0000	−0.634029
$302$	−7.00000	−0.402805
$303$	0	0
$304$	1.00000	0.0573539
$305$	−3.00000	−0.171780
$306$	0	0
$307$	−32.0000	−1.82634	−0.913168	−	0.407583i	$-0.866372\pi$
$307$	−32.0000	−1.82634	−0.913168	+	0.407583i	$0.866372\pi$
$308$	1.00000	0.0569803
$309$	0	0
$310$	0	0
$311$	−2.00000	−0.113410	−0.0567048	−	0.998391i	$-0.518059\pi$
$311$	−2.00000	−0.113410	−0.0567048	+	0.998391i	$0.518059\pi$
$312$	0	0
$313$	0	0		−	1.00000i	$-0.5\pi$
$313$	0	0			1.00000i	$0.5\pi$
$314$	−5.00000	−0.282166
$315$	0	0
$316$	−10.0000	−0.562544
$317$	28.0000	1.57264	0.786318	−	0.617822i	$-0.211985\pi$
$317$	28.0000	1.57264	0.786318	+	0.617822i	$0.211985\pi$
$318$	0	0
$319$	3.00000	0.167968
$320$	−3.00000	−0.167705
$321$	0	0
$322$	7.00000	0.390095
$323$	−7.00000	−0.389490
$324$	0	0
$325$	−4.00000	−0.221880
$326$	−10.0000	−0.553849
$327$	0	0
$328$	4.00000	0.220863
$329$	0	0
$330$	0	0
$331$	−14.0000	−0.769510	−0.384755	−	0.923019i	$-0.625714\pi$
$331$	−14.0000	−0.769510	−0.384755	+	0.923019i	$0.625714\pi$
$332$	14.0000	0.768350
$333$	0	0
$334$	7.00000	0.383023
$335$	18.0000	0.983445
$336$	0	0
$337$	3.00000	0.163420	0.0817102	−	0.996656i	$-0.473962\pi$
$337$	3.00000	0.163420	0.0817102	+	0.996656i	$0.473962\pi$
$338$	−1.00000	−0.0543928
$339$	0	0
$340$	21.0000	1.13888
$341$	0	0
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	−11.0000	−0.593080
$345$	0	0
$346$	2.00000	0.107521
$347$	−32.0000	−1.71785	−0.858925	−	0.512101i	$-0.828867\pi$
$347$	−32.0000	−1.71785	−0.858925	+	0.512101i	$0.828867\pi$
$348$	0	0
$349$	2.00000	0.107058	0.0535288	−	0.998566i	$-0.482953\pi$
$349$	2.00000	0.107058	0.0535288	+	0.998566i	$0.482953\pi$
$350$	4.00000	0.213809
$351$	0	0
$352$	1.00000	0.0533002
$353$	10.0000	0.532246	0.266123	−	0.963939i	$-0.414257\pi$
$353$	10.0000	0.532246	0.266123	+	0.963939i	$0.414257\pi$
$354$	0	0
$355$	−36.0000	−1.91068
$356$	6.00000	0.317999
$357$	0	0
$358$	8.00000	0.422813
$359$	−12.0000	−0.633336	−0.316668	−	0.948536i	$-0.602564\pi$
$359$	−12.0000	−0.633336	−0.316668	+	0.948536i	$0.602564\pi$
$360$	0	0
$361$	−18.0000	−0.947368
$362$	−22.0000	−1.15629
$363$	0	0
$364$	1.00000	0.0524142
$365$	−15.0000	−0.785136
$366$	0	0
$367$	32.0000	1.67039	0.835193	−	0.549957i	$-0.185356\pi$
$367$	32.0000	1.67039	0.835193	+	0.549957i	$0.185356\pi$
$368$	7.00000	0.364900
$369$	0	0
$370$	−15.0000	−0.779813
$371$	−14.0000	−0.726844
$372$	0	0
$373$	12.0000	0.621336	0.310668	−	0.950518i	$-0.399447\pi$
$373$	12.0000	0.621336	0.310668	+	0.950518i	$0.399447\pi$
$374$	−7.00000	−0.361961
$375$	0	0
$376$	0	0
$377$	3.00000	0.154508
$378$	0	0
$379$	20.0000	1.02733	0.513665	−	0.857991i	$-0.328287\pi$
$379$	20.0000	1.02733	0.513665	+	0.857991i	$0.328287\pi$
$380$	−3.00000	−0.153897
$381$	0	0
$382$	−15.0000	−0.767467
$383$	1.00000	0.0510976	0.0255488	−	0.999674i	$-0.491867\pi$
$383$	1.00000	0.0510976	0.0255488	+	0.999674i	$0.491867\pi$
$384$	0	0
$385$	−3.00000	−0.152894
$386$	8.00000	0.407189
$387$	0	0
$388$	6.00000	0.304604
$389$	−34.0000	−1.72387	−0.861934	−	0.507020i	$-0.830747\pi$
$389$	−34.0000	−1.72387	−0.861934	+	0.507020i	$0.830747\pi$
$390$	0	0
$391$	−49.0000	−2.47804
$392$	−1.00000	−0.0505076
$393$	0	0
$394$	−20.0000	−1.00759
$395$	30.0000	1.50946
$396$	0	0
$397$	−14.0000	−0.702640	−0.351320	−	0.936255i	$-0.614267\pi$
$397$	−14.0000	−0.702640	−0.351320	+	0.936255i	$0.614267\pi$
$398$	−3.00000	−0.150376
$399$	0	0
$400$	4.00000	0.200000
$401$	2.00000	0.0998752	0.0499376	−	0.998752i	$-0.484098\pi$
$401$	2.00000	0.0998752	0.0499376	+	0.998752i	$0.484098\pi$
$402$	0	0
$403$	0	0
$404$	12.0000	0.597022
$405$	0	0
$406$	−3.00000	−0.148888
$407$	5.00000	0.247841
$408$	0	0
$409$	23.0000	1.13728	0.568638	−	0.822588i	$-0.307470\pi$
$409$	23.0000	1.13728	0.568638	+	0.822588i	$0.307470\pi$
$410$	−12.0000	−0.592638
$411$	0	0
$412$	−7.00000	−0.344865
$413$	4.00000	0.196827
$414$	0	0
$415$	−42.0000	−2.06170
$416$	1.00000	0.0490290
$417$	0	0
$418$	1.00000	0.0489116
$419$	−29.0000	−1.41674	−0.708371	−	0.705840i	$-0.750570\pi$
$419$	−29.0000	−1.41674	−0.708371	+	0.705840i	$0.750570\pi$
$420$	0	0
$421$	26.0000	1.26716	0.633581	−	0.773676i	$-0.281584\pi$
$421$	26.0000	1.26716	0.633581	+	0.773676i	$0.281584\pi$
$422$	−25.0000	−1.21698
$423$	0	0
$424$	−14.0000	−0.679900
$425$	−28.0000	−1.35820
$426$	0	0
$427$	−1.00000	−0.0483934
$428$	18.0000	0.870063
$429$	0	0
$430$	33.0000	1.59140
$431$	10.0000	0.481683	0.240842	−	0.970564i	$-0.422577\pi$
$431$	10.0000	0.481683	0.240842	+	0.970564i	$0.422577\pi$
$432$	0	0
$433$	−8.00000	−0.384455	−0.192228	−	0.981350i	$-0.561571\pi$
$433$	−8.00000	−0.384455	−0.192228	+	0.981350i	$0.561571\pi$
$434$	0	0
$435$	0	0
$436$	−7.00000	−0.335239
$437$	7.00000	0.334855
$438$	0	0
$439$	−19.0000	−0.906821	−0.453410	−	0.891302i	$-0.649793\pi$
$439$	−19.0000	−0.906821	−0.453410	+	0.891302i	$0.649793\pi$
$440$	−3.00000	−0.143019
$441$	0	0
$442$	−7.00000	−0.332956
$443$	−26.0000	−1.23530	−0.617649	−	0.786454i	$-0.711915\pi$
$443$	−26.0000	−1.23530	−0.617649	+	0.786454i	$0.711915\pi$
$444$	0	0
$445$	−18.0000	−0.853282
$446$	−14.0000	−0.662919
$447$	0	0
$448$	−1.00000	−0.0472456
$449$	−19.0000	−0.896665	−0.448333	−	0.893867i	$-0.647982\pi$
$449$	−19.0000	−0.896665	−0.448333	+	0.893867i	$0.647982\pi$
$450$	0	0
$451$	4.00000	0.188353
$452$	14.0000	0.658505
$453$	0	0
$454$	−18.0000	−0.844782
$455$	−3.00000	−0.140642
$456$	0	0
$457$	−40.0000	−1.87112	−0.935561	−	0.353166i	$-0.885105\pi$
$457$	−40.0000	−1.87112	−0.935561	+	0.353166i	$0.885105\pi$
$458$	−18.0000	−0.841085
$459$	0	0
$460$	−21.0000	−0.979130
$461$	23.0000	1.07122	0.535608	−	0.844466i	$-0.320082\pi$
$461$	23.0000	1.07122	0.535608	+	0.844466i	$0.320082\pi$
$462$	0	0
$463$	23.0000	1.06890	0.534450	−	0.845200i	$-0.320519\pi$
$463$	23.0000	1.06890	0.534450	+	0.845200i	$0.320519\pi$
$464$	−3.00000	−0.139272
$465$	0	0
$466$	−12.0000	−0.555889
$467$	−3.00000	−0.138823	−0.0694117	−	0.997588i	$-0.522112\pi$
$467$	−3.00000	−0.138823	−0.0694117	+	0.997588i	$0.522112\pi$
$468$	0	0
$469$	6.00000	0.277054
$470$	0	0
$471$	0	0
$472$	4.00000	0.184115
$473$	−11.0000	−0.505781
$474$	0	0
$475$	4.00000	0.183533
$476$	7.00000	0.320844
$477$	0	0
$478$	20.0000	0.914779
$479$	11.0000	0.502603	0.251301	−	0.967909i	$-0.419141\pi$
$479$	11.0000	0.502603	0.251301	+	0.967909i	$0.419141\pi$
$480$	0	0
$481$	5.00000	0.227980
$482$	18.0000	0.819878
$483$	0	0
$484$	−10.0000	−0.454545
$485$	−18.0000	−0.817338
$486$	0	0
$487$	−32.0000	−1.45006	−0.725029	−	0.688718i	$-0.758174\pi$
$487$	−32.0000	−1.45006	−0.725029	+	0.688718i	$0.758174\pi$
$488$	−1.00000	−0.0452679
$489$	0	0
$490$	3.00000	0.135526
$491$	−2.00000	−0.0902587	−0.0451294	−	0.998981i	$-0.514370\pi$
$491$	−2.00000	−0.0902587	−0.0451294	+	0.998981i	$0.514370\pi$
$492$	0	0
$493$	21.0000	0.945792
$494$	1.00000	0.0449921
$495$	0	0
$496$	0	0
$497$	−12.0000	−0.538274
$498$	0	0
$499$	16.0000	0.716258	0.358129	−	0.933672i	$-0.383415\pi$
$499$	16.0000	0.716258	0.358129	+	0.933672i	$0.383415\pi$
$500$	3.00000	0.134164
$501$	0	0
$502$	3.00000	0.133897
$503$	6.00000	0.267527	0.133763	−	0.991013i	$-0.457294\pi$
$503$	6.00000	0.267527	0.133763	+	0.991013i	$0.457294\pi$
$504$	0	0
$505$	−36.0000	−1.60198
$506$	7.00000	0.311188
$507$	0	0
$508$	10.0000	0.443678
$509$	39.0000	1.72864	0.864322	−	0.502938i	$-0.167748\pi$
$509$	39.0000	1.72864	0.864322	+	0.502938i	$0.167748\pi$
$510$	0	0
$511$	−5.00000	−0.221187
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	10.0000	0.441081
$515$	21.0000	0.925371
$516$	0	0
$517$	0	0
$518$	−5.00000	−0.219687
$519$	0	0
$520$	−3.00000	−0.131559
$521$	11.0000	0.481919	0.240959	−	0.970535i	$-0.422538\pi$
$521$	11.0000	0.481919	0.240959	+	0.970535i	$0.422538\pi$
$522$	0	0
$523$	20.0000	0.874539	0.437269	−	0.899331i	$-0.355946\pi$
$523$	20.0000	0.874539	0.437269	+	0.899331i	$0.355946\pi$
$524$	−15.0000	−0.655278
$525$	0	0
$526$	−4.00000	−0.174408
$527$	0	0
$528$	0	0
$529$	26.0000	1.13043
$530$	42.0000	1.82436
$531$	0	0
$532$	−1.00000	−0.0433555
$533$	4.00000	0.173259
$534$	0	0
$535$	−54.0000	−2.33462
$536$	6.00000	0.259161
$537$	0	0
$538$	2.00000	0.0862261
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	25.0000	1.07483	0.537417	−	0.843317i	$-0.319400\pi$
$541$	25.0000	1.07483	0.537417	+	0.843317i	$0.319400\pi$
$542$	30.0000	1.28861
$543$	0	0
$544$	7.00000	0.300123
$545$	21.0000	0.899541
$546$	0	0
$547$	28.0000	1.19719	0.598597	−	0.801050i	$-0.295725\pi$
$547$	28.0000	1.19719	0.598597	+	0.801050i	$0.295725\pi$
$548$	−15.0000	−0.640768
$549$	0	0
$550$	4.00000	0.170561
$551$	−3.00000	−0.127804
$552$	0	0
$553$	10.0000	0.425243
$554$	−10.0000	−0.424859
$555$	0	0
$556$	4.00000	0.169638
$557$	12.0000	0.508456	0.254228	−	0.967144i	$-0.418179\pi$
$557$	12.0000	0.508456	0.254228	+	0.967144i	$0.418179\pi$
$558$	0	0
$559$	−11.0000	−0.465250
$560$	3.00000	0.126773
$561$	0	0
$562$	6.00000	0.253095
$563$	−39.0000	−1.64365	−0.821827	−	0.569737i	$-0.807045\pi$
$563$	−39.0000	−1.64365	−0.821827	+	0.569737i	$0.807045\pi$
$564$	0	0
$565$	−42.0000	−1.76695
$566$	16.0000	0.672530
$567$	0	0
$568$	−12.0000	−0.503509
$569$	−14.0000	−0.586911	−0.293455	−	0.955973i	$-0.594805\pi$
$569$	−14.0000	−0.586911	−0.293455	+	0.955973i	$0.594805\pi$
$570$	0	0
$571$	−8.00000	−0.334790	−0.167395	−	0.985890i	$-0.553535\pi$
$571$	−8.00000	−0.334790	−0.167395	+	0.985890i	$0.553535\pi$
$572$	1.00000	0.0418121
$573$	0	0
$574$	−4.00000	−0.166957
$575$	28.0000	1.16768
$576$	0	0
$577$	−26.0000	−1.08239	−0.541197	−	0.840896i	$-0.682029\pi$
$577$	−26.0000	−1.08239	−0.541197	+	0.840896i	$0.682029\pi$
$578$	−32.0000	−1.33102
$579$	0	0
$580$	9.00000	0.373705
$581$	−14.0000	−0.580818
$582$	0	0
$583$	−14.0000	−0.579821
$584$	−5.00000	−0.206901
$585$	0	0
$586$	30.0000	1.23929
$587$	−42.0000	−1.73353	−0.866763	−	0.498721i	$-0.833803\pi$
$587$	−42.0000	−1.73353	−0.866763	+	0.498721i	$0.833803\pi$
$588$	0	0
$589$	0	0
$590$	−12.0000	−0.494032
$591$	0	0
$592$	−5.00000	−0.205499
$593$	42.0000	1.72473	0.862367	−	0.506284i	$-0.168981\pi$
$593$	42.0000	1.72473	0.862367	+	0.506284i	$0.168981\pi$
$594$	0	0
$595$	−21.0000	−0.860916
$596$	−16.0000	−0.655386
$597$	0	0
$598$	7.00000	0.286251
$599$	1.00000	0.0408589	0.0204294	−	0.999791i	$-0.493497\pi$
$599$	1.00000	0.0408589	0.0204294	+	0.999791i	$0.493497\pi$
$600$	0	0
$601$	16.0000	0.652654	0.326327	−	0.945257i	$-0.394189\pi$
$601$	16.0000	0.652654	0.326327	+	0.945257i	$0.394189\pi$
$602$	11.0000	0.448327
$603$	0	0
$604$	7.00000	0.284826
$605$	30.0000	1.21967
$606$	0	0
$607$	13.0000	0.527654	0.263827	−	0.964570i	$-0.415015\pi$
$607$	13.0000	0.527654	0.263827	+	0.964570i	$0.415015\pi$
$608$	−1.00000	−0.0405554
$609$	0	0
$610$	3.00000	0.121466
$611$	0	0
$612$	0	0
$613$	−7.00000	−0.282727	−0.141364	−	0.989958i	$-0.545149\pi$
$613$	−7.00000	−0.282727	−0.141364	+	0.989958i	$0.545149\pi$
$614$	32.0000	1.29141
$615$	0	0
$616$	−1.00000	−0.0402911
$617$	27.0000	1.08698	0.543490	−	0.839416i	$-0.317103\pi$
$617$	27.0000	1.08698	0.543490	+	0.839416i	$0.317103\pi$
$618$	0	0
$619$	1.00000	0.0401934	0.0200967	−	0.999798i	$-0.493603\pi$
$619$	1.00000	0.0401934	0.0200967	+	0.999798i	$0.493603\pi$
$620$	0	0
$621$	0	0
$622$	2.00000	0.0801927
$623$	−6.00000	−0.240385
$624$	0	0
$625$	−29.0000	−1.16000
$626$	0	0
$627$	0	0
$628$	5.00000	0.199522
$629$	35.0000	1.39554
$630$	0	0
$631$	−37.0000	−1.47295	−0.736473	−	0.676467i	$-0.763510\pi$
$631$	−37.0000	−1.47295	−0.736473	+	0.676467i	$0.763510\pi$
$632$	10.0000	0.397779
$633$	0	0
$634$	−28.0000	−1.11202
$635$	−30.0000	−1.19051
$636$	0	0
$637$	−1.00000	−0.0396214
$638$	−3.00000	−0.118771
$639$	0	0
$640$	3.00000	0.118585
$641$	−42.0000	−1.65890	−0.829450	−	0.558581i	$-0.811346\pi$
$641$	−42.0000	−1.65890	−0.829450	+	0.558581i	$0.811346\pi$
$642$	0	0
$643$	−9.00000	−0.354925	−0.177463	−	0.984128i	$-0.556789\pi$
$643$	−9.00000	−0.354925	−0.177463	+	0.984128i	$0.556789\pi$
$644$	−7.00000	−0.275839
$645$	0	0
$646$	7.00000	0.275411
$647$	36.0000	1.41531	0.707653	−	0.706560i	$-0.249754\pi$
$647$	36.0000	1.41531	0.707653	+	0.706560i	$0.249754\pi$
$648$	0	0
$649$	4.00000	0.157014
$650$	4.00000	0.156893
$651$	0	0
$652$	10.0000	0.391630
$653$	43.0000	1.68272	0.841360	−	0.540475i	$-0.181755\pi$
$653$	43.0000	1.68272	0.841360	+	0.540475i	$0.181755\pi$
$654$	0	0
$655$	45.0000	1.75830
$656$	−4.00000	−0.156174
$657$	0	0
$658$	0	0
$659$	18.0000	0.701180	0.350590	−	0.936529i	$-0.385981\pi$
$659$	18.0000	0.701180	0.350590	+	0.936529i	$0.385981\pi$
$660$	0	0
$661$	38.0000	1.47803	0.739014	−	0.673690i	$-0.235292\pi$
$661$	38.0000	1.47803	0.739014	+	0.673690i	$0.235292\pi$
$662$	14.0000	0.544125
$663$	0	0
$664$	−14.0000	−0.543305
$665$	3.00000	0.116335
$666$	0	0
$667$	−21.0000	−0.813123
$668$	−7.00000	−0.270838
$669$	0	0
$670$	−18.0000	−0.695401
$671$	−1.00000	−0.0386046
$672$	0	0
$673$	7.00000	0.269830	0.134915	−	0.990857i	$-0.456924\pi$
$673$	7.00000	0.269830	0.134915	+	0.990857i	$0.456924\pi$
$674$	−3.00000	−0.115556
$675$	0	0
$676$	1.00000	0.0384615
$677$	38.0000	1.46046	0.730229	−	0.683202i	$-0.239413\pi$
$677$	38.0000	1.46046	0.730229	+	0.683202i	$0.239413\pi$
$678$	0	0
$679$	−6.00000	−0.230259
$680$	−21.0000	−0.805313
$681$	0	0
$682$	0	0
$683$	13.0000	0.497431	0.248716	−	0.968577i	$-0.419992\pi$
$683$	13.0000	0.497431	0.248716	+	0.968577i	$0.419992\pi$
$684$	0	0
$685$	45.0000	1.71936
$686$	1.00000	0.0381802
$687$	0	0
$688$	11.0000	0.419371
$689$	−14.0000	−0.533358
$690$	0	0
$691$	−4.00000	−0.152167	−0.0760836	−	0.997101i	$-0.524242\pi$
$691$	−4.00000	−0.152167	−0.0760836	+	0.997101i	$0.524242\pi$
$692$	−2.00000	−0.0760286
$693$	0	0
$694$	32.0000	1.21470
$695$	−12.0000	−0.455186
$696$	0	0
$697$	28.0000	1.06058
$698$	−2.00000	−0.0757011
$699$	0	0
$700$	−4.00000	−0.151186
$701$	42.0000	1.58632	0.793159	−	0.609015i	$-0.208435\pi$
$701$	42.0000	1.58632	0.793159	+	0.609015i	$0.208435\pi$
$702$	0	0
$703$	−5.00000	−0.188579
$704$	−1.00000	−0.0376889
$705$	0	0
$706$	−10.0000	−0.376355
$707$	−12.0000	−0.451306
$708$	0	0
$709$	50.0000	1.87779	0.938895	−	0.344204i	$-0.111851\pi$
$709$	50.0000	1.87779	0.938895	+	0.344204i	$0.111851\pi$
$710$	36.0000	1.35106
$711$	0	0
$712$	−6.00000	−0.224860
$713$	0	0
$714$	0	0
$715$	−3.00000	−0.112194
$716$	−8.00000	−0.298974
$717$	0	0
$718$	12.0000	0.447836
$719$	−30.0000	−1.11881	−0.559406	−	0.828894i	$-0.688971\pi$
$719$	−30.0000	−1.11881	−0.559406	+	0.828894i	$0.688971\pi$
$720$	0	0
$721$	7.00000	0.260694
$722$	18.0000	0.669891
$723$	0	0
$724$	22.0000	0.817624
$725$	−12.0000	−0.445669
$726$	0	0
$727$	37.0000	1.37225	0.686127	−	0.727482i	$-0.259309\pi$
$727$	37.0000	1.37225	0.686127	+	0.727482i	$0.259309\pi$
$728$	−1.00000	−0.0370625
$729$	0	0
$730$	15.0000	0.555175
$731$	−77.0000	−2.84795
$732$	0	0
$733$	−4.00000	−0.147743	−0.0738717	−	0.997268i	$-0.523536\pi$
$733$	−4.00000	−0.147743	−0.0738717	+	0.997268i	$0.523536\pi$
$734$	−32.0000	−1.18114
$735$	0	0
$736$	−7.00000	−0.258023
$737$	6.00000	0.221013
$738$	0	0
$739$	38.0000	1.39785	0.698926	−	0.715194i	$-0.253662\pi$
$739$	38.0000	1.39785	0.698926	+	0.715194i	$0.253662\pi$
$740$	15.0000	0.551411
$741$	0	0
$742$	14.0000	0.513956
$743$	26.0000	0.953847	0.476924	−	0.878945i	$-0.341752\pi$
$743$	26.0000	0.953847	0.476924	+	0.878945i	$0.341752\pi$
$744$	0	0
$745$	48.0000	1.75858
$746$	−12.0000	−0.439351
$747$	0	0
$748$	7.00000	0.255945
$749$	−18.0000	−0.657706
$750$	0	0
$751$	8.00000	0.291924	0.145962	−	0.989290i	$-0.453372\pi$
$751$	8.00000	0.291924	0.145962	+	0.989290i	$0.453372\pi$
$752$	0	0
$753$	0	0
$754$	−3.00000	−0.109254
$755$	−21.0000	−0.764268
$756$	0	0
$757$	26.0000	0.944986	0.472493	−	0.881334i	$-0.343354\pi$
$757$	26.0000	0.944986	0.472493	+	0.881334i	$0.343354\pi$
$758$	−20.0000	−0.726433
$759$	0	0
$760$	3.00000	0.108821
$761$	32.0000	1.16000	0.580000	−	0.814617i	$-0.303053\pi$
$761$	32.0000	1.16000	0.580000	+	0.814617i	$0.303053\pi$
$762$	0	0
$763$	7.00000	0.253417
$764$	15.0000	0.542681
$765$	0	0
$766$	−1.00000	−0.0361315
$767$	4.00000	0.144432
$768$	0	0
$769$	−49.0000	−1.76699	−0.883493	−	0.468445i	$-0.844814\pi$
$769$	−49.0000	−1.76699	−0.883493	+	0.468445i	$0.844814\pi$
$770$	3.00000	0.108112
$771$	0	0
$772$	−8.00000	−0.287926
$773$	7.00000	0.251773	0.125886	−	0.992045i	$-0.459823\pi$
$773$	7.00000	0.251773	0.125886	+	0.992045i	$0.459823\pi$
$774$	0	0
$775$	0	0
$776$	−6.00000	−0.215387
$777$	0	0
$778$	34.0000	1.21896
$779$	−4.00000	−0.143315
$780$	0	0
$781$	−12.0000	−0.429394
$782$	49.0000	1.75224
$783$	0	0
$784$	1.00000	0.0357143
$785$	−15.0000	−0.535373
$786$	0	0
$787$	−31.0000	−1.10503	−0.552515	−	0.833503i	$-0.686332\pi$
$787$	−31.0000	−1.10503	−0.552515	+	0.833503i	$0.686332\pi$
$788$	20.0000	0.712470
$789$	0	0
$790$	−30.0000	−1.06735
$791$	−14.0000	−0.497783
$792$	0	0
$793$	−1.00000	−0.0355110
$794$	14.0000	0.496841
$795$	0	0
$796$	3.00000	0.106332
$797$	−36.0000	−1.27519	−0.637593	−	0.770374i	$-0.720070\pi$
$797$	−36.0000	−1.27519	−0.637593	+	0.770374i	$0.720070\pi$
$798$	0	0
$799$	0	0
$800$	−4.00000	−0.141421
$801$	0	0
$802$	−2.00000	−0.0706225
$803$	−5.00000	−0.176446
$804$	0	0
$805$	21.0000	0.740153
$806$	0	0
$807$	0	0
$808$	−12.0000	−0.422159
$809$	50.0000	1.75791	0.878953	−	0.476908i	$-0.158243\pi$
$809$	50.0000	1.75791	0.878953	+	0.476908i	$0.158243\pi$
$810$	0	0
$811$	21.0000	0.737410	0.368705	−	0.929547i	$-0.379801\pi$
$811$	21.0000	0.737410	0.368705	+	0.929547i	$0.379801\pi$
$812$	3.00000	0.105279
$813$	0	0
$814$	−5.00000	−0.175250
$815$	−30.0000	−1.05085
$816$	0	0
$817$	11.0000	0.384841
$818$	−23.0000	−0.804176
$819$	0	0
$820$	12.0000	0.419058
$821$	32.0000	1.11681	0.558404	−	0.829569i	$-0.311414\pi$
$821$	32.0000	1.11681	0.558404	+	0.829569i	$0.311414\pi$
$822$	0	0
$823$	4.00000	0.139431	0.0697156	−	0.997567i	$-0.477791\pi$
$823$	4.00000	0.139431	0.0697156	+	0.997567i	$0.477791\pi$
$824$	7.00000	0.243857
$825$	0	0
$826$	−4.00000	−0.139178
$827$	−9.00000	−0.312961	−0.156480	−	0.987681i	$-0.550015\pi$
$827$	−9.00000	−0.312961	−0.156480	+	0.987681i	$0.550015\pi$
$828$	0	0
$829$	49.0000	1.70184	0.850920	−	0.525295i	$-0.176045\pi$
$829$	49.0000	1.70184	0.850920	+	0.525295i	$0.176045\pi$
$830$	42.0000	1.45784
$831$	0	0
$832$	−1.00000	−0.0346688
$833$	−7.00000	−0.242536
$834$	0	0
$835$	21.0000	0.726735
$836$	−1.00000	−0.0345857
$837$	0	0
$838$	29.0000	1.00179
$839$	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
$841$	−20.0000	−0.689655
$842$	−26.0000	−0.896019
$843$	0	0
$844$	25.0000	0.860535
$845$	−3.00000	−0.103203
$846$	0	0
$847$	10.0000	0.343604
$848$	14.0000	0.480762
$849$	0	0
$850$	28.0000	0.960392
$851$	−35.0000	−1.19978
$852$	0	0
$853$	44.0000	1.50653	0.753266	−	0.657716i	$-0.228477\pi$
$853$	44.0000	1.50653	0.753266	+	0.657716i	$0.228477\pi$
$854$	1.00000	0.0342193
$855$	0	0
$856$	−18.0000	−0.615227
$857$	−22.0000	−0.751506	−0.375753	−	0.926720i	$-0.622616\pi$
$857$	−22.0000	−0.751506	−0.375753	+	0.926720i	$0.622616\pi$
$858$	0	0
$859$	40.0000	1.36478	0.682391	−	0.730987i	$-0.260940\pi$
$859$	40.0000	1.36478	0.682391	+	0.730987i	$0.260940\pi$
$860$	−33.0000	−1.12529
$861$	0	0
$862$	−10.0000	−0.340601
$863$	42.0000	1.42970	0.714848	−	0.699280i	$-0.246496\pi$
$863$	42.0000	1.42970	0.714848	+	0.699280i	$0.246496\pi$
$864$	0	0
$865$	6.00000	0.204006
$866$	8.00000	0.271851
$867$	0	0
$868$	0	0
$869$	10.0000	0.339227
$870$	0	0
$871$	6.00000	0.203302
$872$	7.00000	0.237050
$873$	0	0
$874$	−7.00000	−0.236779
$875$	−3.00000	−0.101419
$876$	0	0
$877$	18.0000	0.607817	0.303908	−	0.952701i	$-0.401708\pi$
$877$	18.0000	0.607817	0.303908	+	0.952701i	$0.401708\pi$
$878$	19.0000	0.641219
$879$	0	0
$880$	3.00000	0.101130
$881$	17.0000	0.572745	0.286372	−	0.958118i	$-0.407551\pi$
$881$	17.0000	0.572745	0.286372	+	0.958118i	$0.407551\pi$
$882$	0	0
$883$	35.0000	1.17784	0.588922	−	0.808190i	$-0.299553\pi$
$883$	35.0000	1.17784	0.588922	+	0.808190i	$0.299553\pi$
$884$	7.00000	0.235435
$885$	0	0
$886$	26.0000	0.873487
$887$	24.0000	0.805841	0.402921	−	0.915235i	$-0.367995\pi$
$887$	24.0000	0.805841	0.402921	+	0.915235i	$0.367995\pi$
$888$	0	0
$889$	−10.0000	−0.335389
$890$	18.0000	0.603361
$891$	0	0
$892$	14.0000	0.468755
$893$	0	0
$894$	0	0
$895$	24.0000	0.802232
$896$	1.00000	0.0334077
$897$	0	0
$898$	19.0000	0.634038
$899$	0	0
$900$	0	0
$901$	−98.0000	−3.26485
$902$	−4.00000	−0.133185
$903$	0	0
$904$	−14.0000	−0.465633
$905$	−66.0000	−2.19391
$906$	0	0
$907$	12.0000	0.398453	0.199227	−	0.979953i	$-0.436157\pi$
$907$	12.0000	0.398453	0.199227	+	0.979953i	$0.436157\pi$
$908$	18.0000	0.597351
$909$	0	0
$910$	3.00000	0.0994490
$911$	−15.0000	−0.496972	−0.248486	−	0.968635i	$-0.579933\pi$
$911$	−15.0000	−0.496972	−0.248486	+	0.968635i	$0.579933\pi$
$912$	0	0
$913$	−14.0000	−0.463332
$914$	40.0000	1.32308
$915$	0	0
$916$	18.0000	0.594737
$917$	15.0000	0.495344
$918$	0	0
$919$	−50.0000	−1.64935	−0.824674	−	0.565608i	$-0.808641\pi$
$919$	−50.0000	−1.64935	−0.824674	+	0.565608i	$0.808641\pi$
$920$	21.0000	0.692349
$921$	0	0
$922$	−23.0000	−0.757465
$923$	−12.0000	−0.394985
$924$	0	0
$925$	−20.0000	−0.657596
$926$	−23.0000	−0.755827
$927$	0	0
$928$	3.00000	0.0984798
$929$	−48.0000	−1.57483	−0.787414	−	0.616424i	$-0.788581\pi$
$929$	−48.0000	−1.57483	−0.787414	+	0.616424i	$0.788581\pi$
$930$	0	0
$931$	1.00000	0.0327737
$932$	12.0000	0.393073
$933$	0	0
$934$	3.00000	0.0981630
$935$	−21.0000	−0.686773
$936$	0	0
$937$	−22.0000	−0.718709	−0.359354	−	0.933201i	$-0.617003\pi$
$937$	−22.0000	−0.718709	−0.359354	+	0.933201i	$0.617003\pi$
$938$	−6.00000	−0.195907
$939$	0	0
$940$	0	0
$941$	−34.0000	−1.10837	−0.554184	−	0.832394i	$-0.686970\pi$
$941$	−34.0000	−1.10837	−0.554184	+	0.832394i	$0.686970\pi$
$942$	0	0
$943$	−28.0000	−0.911805
$944$	−4.00000	−0.130189
$945$	0	0
$946$	11.0000	0.357641
$947$	11.0000	0.357452	0.178726	−	0.983899i	$-0.442802\pi$
$947$	11.0000	0.357452	0.178726	+	0.983899i	$0.442802\pi$
$948$	0	0
$949$	−5.00000	−0.162307
$950$	−4.00000	−0.129777
$951$	0	0
$952$	−7.00000	−0.226871
$953$	−36.0000	−1.16615	−0.583077	−	0.812417i	$-0.698151\pi$
$953$	−36.0000	−1.16615	−0.583077	+	0.812417i	$0.698151\pi$
$954$	0	0
$955$	−45.0000	−1.45617
$956$	−20.0000	−0.646846
$957$	0	0
$958$	−11.0000	−0.355394
$959$	15.0000	0.484375
$960$	0	0
$961$	−31.0000	−1.00000
$962$	−5.00000	−0.161206
$963$	0	0
$964$	−18.0000	−0.579741
$965$	24.0000	0.772587
$966$	0	0
$967$	13.0000	0.418052	0.209026	−	0.977910i	$-0.432971\pi$
$967$	13.0000	0.418052	0.209026	+	0.977910i	$0.432971\pi$
$968$	10.0000	0.321412
$969$	0	0
$970$	18.0000	0.577945
$971$	−48.0000	−1.54039	−0.770197	−	0.637806i	$-0.779842\pi$
$971$	−48.0000	−1.54039	−0.770197	+	0.637806i	$0.779842\pi$
$972$	0	0
$973$	−4.00000	−0.128234
$974$	32.0000	1.02535
$975$	0	0
$976$	1.00000	0.0320092
$977$	7.00000	0.223950	0.111975	−	0.993711i	$-0.464282\pi$
$977$	7.00000	0.223950	0.111975	+	0.993711i	$0.464282\pi$
$978$	0	0
$979$	−6.00000	−0.191761
$980$	−3.00000	−0.0958315
$981$	0	0
$982$	2.00000	0.0638226
$983$	19.0000	0.606006	0.303003	−	0.952990i	$-0.402011\pi$
$983$	19.0000	0.606006	0.303003	+	0.952990i	$0.402011\pi$
$984$	0	0
$985$	−60.0000	−1.91176
$986$	−21.0000	−0.668776
$987$	0	0
$988$	−1.00000	−0.0318142
$989$	77.0000	2.44846
$990$	0	0
$991$	34.0000	1.08005	0.540023	−	0.841650i	$-0.318416\pi$
$991$	34.0000	1.08005	0.540023	+	0.841650i	$0.318416\pi$
$992$	0	0
$993$	0	0
$994$	12.0000	0.380617
$995$	−9.00000	−0.285319
$996$	0	0
$997$	54.0000	1.71020	0.855099	−	0.518465i	$-0.173497\pi$
$997$	54.0000	1.71020	0.855099	+	0.518465i	$0.173497\pi$
$998$	−16.0000	−0.506471
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1638.2.a.a.1.1		1
3.2	odd	2		546.2.a.e.1.1	✓	1
12.11	even	2		4368.2.a.z.1.1		1
21.20	even	2		3822.2.a.bc.1.1		1
39.38	odd	2		7098.2.a.b.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
546.2.a.e.1.1	✓	1	3.2	odd	2
1638.2.a.a.1.1		1	1.1	even	1	trivial
3822.2.a.bc.1.1		1	21.20	even	2
4368.2.a.z.1.1		1	12.11	even	2
7098.2.a.b.1.1		1	39.38	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:	\( N \)	\(=\)	\( 1638 = 2 \cdot 3^{2} \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1638.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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