LMFDB - Embedded newform 4368.2.a.q.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4368,2,Mod(1,4368)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4368, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("4368.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4368 = 2^{4} \cdot 3 \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4368.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:	$34.8786556029$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 273)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	4368.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+1.00000 q^{3} -1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$

$q+1.00000 q^{3} -1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} +2.00000 q^{11} +1.00000 q^{13} -1.00000 q^{15} -4.00000 q^{17} -3.00000 q^{19} -1.00000 q^{21} +9.00000 q^{23} -4.00000 q^{25} +1.00000 q^{27} -1.00000 q^{29} +5.00000 q^{31} +2.00000 q^{33} +1.00000 q^{35} -8.00000 q^{37} +1.00000 q^{39} +6.00000 q^{41} +9.00000 q^{43} -1.00000 q^{45} +3.00000 q^{47} +1.00000 q^{49} -4.00000 q^{51} +3.00000 q^{53} -2.00000 q^{55} -3.00000 q^{57} +10.0000 q^{61} -1.00000 q^{63} -1.00000 q^{65} +2.00000 q^{67} +9.00000 q^{69} -12.0000 q^{71} +5.00000 q^{73} -4.00000 q^{75} -2.00000 q^{77} +13.0000 q^{79} +1.00000 q^{81} +11.0000 q^{83} +4.00000 q^{85} -1.00000 q^{87} +1.00000 q^{89} -1.00000 q^{91} +5.00000 q^{93} +3.00000 q^{95} +1.00000 q^{97} +2.00000 q^{99} +O(q^{100})$

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 $

$2$	0	0
$2$	0	0
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	−1.00000	−0.447214	−0.223607	−	0.974679i	$-0.571783\pi$
$5$	−1.00000	−0.447214	−0.223607	+	0.974679i	$0.571783\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	2.00000	0.603023	0.301511	−	0.953463i	$-0.402509\pi$
$11$	2.00000	0.603023	0.301511	+	0.953463i	$0.402509\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	−1.00000	−0.258199
$16$	0	0
$17$	−4.00000	−0.970143	−0.485071	−	0.874475i	$-0.661206\pi$
$17$	−4.00000	−0.970143	−0.485071	+	0.874475i	$0.661206\pi$
$18$	0	0
$19$	−3.00000	−0.688247	−0.344124	−	0.938924i	$-0.611824\pi$
$19$	−3.00000	−0.688247	−0.344124	+	0.938924i	$0.611824\pi$
$20$	0	0
$21$	−1.00000	−0.218218
$22$	0	0
$23$	9.00000	1.87663	0.938315	−	0.345782i	$-0.112386\pi$
$23$	9.00000	1.87663	0.938315	+	0.345782i	$0.112386\pi$
$24$	0	0
$25$	−4.00000	−0.800000
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−1.00000	−0.185695	−0.0928477	−	0.995680i	$-0.529597\pi$
$29$	−1.00000	−0.185695	−0.0928477	+	0.995680i	$0.529597\pi$
$30$	0	0
$31$	5.00000	0.898027	0.449013	−	0.893525i	$-0.351776\pi$
$31$	5.00000	0.898027	0.449013	+	0.893525i	$0.351776\pi$
$32$	0	0
$33$	2.00000	0.348155
$34$	0	0
$35$	1.00000	0.169031
$36$	0	0
$37$	−8.00000	−1.31519	−0.657596	−	0.753371i	$-0.728427\pi$
$37$	−8.00000	−1.31519	−0.657596	+	0.753371i	$0.728427\pi$
$38$	0	0
$39$	1.00000	0.160128
$40$	0	0
$41$	6.00000	0.937043	0.468521	−	0.883452i	$-0.344787\pi$
$41$	6.00000	0.937043	0.468521	+	0.883452i	$0.344787\pi$
$42$	0	0
$43$	9.00000	1.37249	0.686244	−	0.727372i	$-0.259258\pi$
$43$	9.00000	1.37249	0.686244	+	0.727372i	$0.259258\pi$
$44$	0	0
$45$	−1.00000	−0.149071
$46$	0	0
$47$	3.00000	0.437595	0.218797	−	0.975770i	$-0.429787\pi$
$47$	3.00000	0.437595	0.218797	+	0.975770i	$0.429787\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−4.00000	−0.560112
$52$	0	0
$53$	3.00000	0.412082	0.206041	−	0.978543i	$-0.433942\pi$
$53$	3.00000	0.412082	0.206041	+	0.978543i	$0.433942\pi$
$54$	0	0
$55$	−2.00000	−0.269680
$56$	0	0
$57$	−3.00000	−0.397360
$58$	0	0
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	10.0000	1.28037	0.640184	−	0.768221i	$-0.278858\pi$
$61$	10.0000	1.28037	0.640184	+	0.768221i	$0.278858\pi$
$62$	0	0
$63$	−1.00000	−0.125988
$64$	0	0
$65$	−1.00000	−0.124035
$66$	0	0
$67$	2.00000	0.244339	0.122169	−	0.992509i	$-0.461015\pi$
$67$	2.00000	0.244339	0.122169	+	0.992509i	$0.461015\pi$
$68$	0	0
$69$	9.00000	1.08347
$70$	0	0
$71$	−12.0000	−1.42414	−0.712069	−	0.702109i	$-0.752242\pi$
$71$	−12.0000	−1.42414	−0.712069	+	0.702109i	$0.752242\pi$
$72$	0	0
$73$	5.00000	0.585206	0.292603	−	0.956234i	$-0.405479\pi$
$73$	5.00000	0.585206	0.292603	+	0.956234i	$0.405479\pi$
$74$	0	0
$75$	−4.00000	−0.461880
$76$	0	0
$77$	−2.00000	−0.227921
$78$	0	0
$79$	13.0000	1.46261	0.731307	−	0.682048i	$-0.238911\pi$
$79$	13.0000	1.46261	0.731307	+	0.682048i	$0.238911\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	11.0000	1.20741	0.603703	−	0.797209i	$-0.293691\pi$
$83$	11.0000	1.20741	0.603703	+	0.797209i	$0.293691\pi$
$84$	0	0
$85$	4.00000	0.433861
$86$	0	0
$87$	−1.00000	−0.107211
$88$	0	0
$89$	1.00000	0.106000	0.0529999	−	0.998595i	$-0.483122\pi$
$89$	1.00000	0.106000	0.0529999	+	0.998595i	$0.483122\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	0	0
$93$	5.00000	0.518476
$94$	0	0
$95$	3.00000	0.307794
$96$	0	0
$97$	1.00000	0.101535	0.0507673	−	0.998711i	$-0.483833\pi$
$97$	1.00000	0.101535	0.0507673	+	0.998711i	$0.483833\pi$
$98$	0	0
$99$	2.00000	0.201008
$100$	0	0
$101$	−6.00000	−0.597022	−0.298511	−	0.954406i	$-0.596490\pi$
$101$	−6.00000	−0.597022	−0.298511	+	0.954406i	$0.596490\pi$
$102$	0	0
$103$	12.0000	1.18240	0.591198	−	0.806527i	$-0.298655\pi$
$103$	12.0000	1.18240	0.591198	+	0.806527i	$0.298655\pi$
$104$	0	0
$105$	1.00000	0.0975900
$106$	0	0
$107$	−12.0000	−1.16008	−0.580042	−	0.814587i	$-0.696964\pi$
$107$	−12.0000	−1.16008	−0.580042	+	0.814587i	$0.696964\pi$
$108$	0	0
$109$	−2.00000	−0.191565	−0.0957826	−	0.995402i	$-0.530535\pi$
$109$	−2.00000	−0.191565	−0.0957826	+	0.995402i	$0.530535\pi$
$110$	0	0
$111$	−8.00000	−0.759326
$112$	0	0
$113$	−15.0000	−1.41108	−0.705541	−	0.708669i	$-0.749296\pi$
$113$	−15.0000	−1.41108	−0.705541	+	0.708669i	$0.749296\pi$
$114$	0	0
$115$	−9.00000	−0.839254
$116$	0	0
$117$	1.00000	0.0924500
$118$	0	0
$119$	4.00000	0.366679
$120$	0	0
$121$	−7.00000	−0.636364
$122$	0	0
$123$	6.00000	0.541002
$124$	0	0
$125$	9.00000	0.804984
$126$	0	0
$127$	4.00000	0.354943	0.177471	−	0.984126i	$-0.443208\pi$
$127$	4.00000	0.354943	0.177471	+	0.984126i	$0.443208\pi$
$128$	0	0
$129$	9.00000	0.792406
$130$	0	0
$131$	−4.00000	−0.349482	−0.174741	−	0.984614i	$-0.555909\pi$
$131$	−4.00000	−0.349482	−0.174741	+	0.984614i	$0.555909\pi$
$132$	0	0
$133$	3.00000	0.260133
$134$	0	0
$135$	−1.00000	−0.0860663
$136$	0	0
$137$	0	0		−	1.00000i	$-0.5\pi$
$137$	0	0			1.00000i	$0.5\pi$
$138$	0	0
$139$	18.0000	1.52674	0.763370	−	0.645961i	$-0.223543\pi$
$139$	18.0000	1.52674	0.763370	+	0.645961i	$0.223543\pi$
$140$	0	0
$141$	3.00000	0.252646
$142$	0	0
$143$	2.00000	0.167248
$144$	0	0
$145$	1.00000	0.0830455
$146$	0	0
$147$	1.00000	0.0824786
$148$	0	0
$149$	18.0000	1.47462	0.737309	−	0.675556i	$-0.236096\pi$
$149$	18.0000	1.47462	0.737309	+	0.675556i	$0.236096\pi$
$150$	0	0
$151$	−4.00000	−0.325515	−0.162758	−	0.986666i	$-0.552039\pi$
$151$	−4.00000	−0.325515	−0.162758	+	0.986666i	$0.552039\pi$
$152$	0	0
$153$	−4.00000	−0.323381
$154$	0	0
$155$	−5.00000	−0.401610
$156$	0	0
$157$	24.0000	1.91541	0.957704	−	0.287754i	$-0.0929087\pi$
$157$	24.0000	1.91541	0.957704	+	0.287754i	$0.0929087\pi$
$158$	0	0
$159$	3.00000	0.237915
$160$	0	0
$161$	−9.00000	−0.709299
$162$	0	0
$163$	−8.00000	−0.626608	−0.313304	−	0.949653i	$-0.601436\pi$
$163$	−8.00000	−0.626608	−0.313304	+	0.949653i	$0.601436\pi$
$164$	0	0
$165$	−2.00000	−0.155700
$166$	0	0
$167$	−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
$170$	0	0
$171$	−3.00000	−0.229416
$172$	0	0
$173$	12.0000	0.912343	0.456172	−	0.889892i	$-0.349220\pi$
$173$	12.0000	0.912343	0.456172	+	0.889892i	$0.349220\pi$
$174$	0	0
$175$	4.00000	0.302372
$176$	0	0
$177$	0	0
$178$	0	0
$179$	5.00000	0.373718	0.186859	−	0.982387i	$-0.440169\pi$
$179$	5.00000	0.373718	0.186859	+	0.982387i	$0.440169\pi$
$180$	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
$183$	10.0000	0.739221
$184$	0	0
$185$	8.00000	0.588172
$186$	0	0
$187$	−8.00000	−0.585018
$188$	0	0
$189$	−1.00000	−0.0727393
$190$	0	0
$191$	16.0000	1.15772	0.578860	−	0.815427i	$-0.303498\pi$
$191$	16.0000	1.15772	0.578860	+	0.815427i	$0.303498\pi$
$192$	0	0
$193$	−6.00000	−0.431889	−0.215945	−	0.976406i	$-0.569283\pi$
$193$	−6.00000	−0.431889	−0.215945	+	0.976406i	$0.569283\pi$
$194$	0	0
$195$	−1.00000	−0.0716115
$196$	0	0
$197$	−6.00000	−0.427482	−0.213741	−	0.976890i	$-0.568565\pi$
$197$	−6.00000	−0.427482	−0.213741	+	0.976890i	$0.568565\pi$
$198$	0	0
$199$	4.00000	0.283552	0.141776	−	0.989899i	$-0.454719\pi$
$199$	4.00000	0.283552	0.141776	+	0.989899i	$0.454719\pi$
$200$	0	0
$201$	2.00000	0.141069
$202$	0	0
$203$	1.00000	0.0701862
$204$	0	0
$205$	−6.00000	−0.419058
$206$	0	0
$207$	9.00000	0.625543
$208$	0	0
$209$	−6.00000	−0.415029
$210$	0	0
$211$	−19.0000	−1.30801	−0.654007	−	0.756489i	$-0.726913\pi$
$211$	−19.0000	−1.30801	−0.654007	+	0.756489i	$0.726913\pi$
$212$	0	0
$213$	−12.0000	−0.822226
$214$	0	0
$215$	−9.00000	−0.613795
$216$	0	0
$217$	−5.00000	−0.339422
$218$	0	0
$219$	5.00000	0.337869
$220$	0	0
$221$	−4.00000	−0.269069
$222$	0	0
$223$	23.0000	1.54019	0.770097	−	0.637927i	$-0.220208\pi$
$223$	23.0000	1.54019	0.770097	+	0.637927i	$0.220208\pi$
$224$	0	0
$225$	−4.00000	−0.266667
$226$	0	0
$227$	−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
$229$	26.0000	1.71813	0.859064	−	0.511868i	$-0.171046\pi$
$229$	26.0000	1.71813	0.859064	+	0.511868i	$0.171046\pi$
$230$	0	0
$231$	−2.00000	−0.131590
$232$	0	0
$233$	19.0000	1.24473	0.622366	−	0.782727i	$-0.286172\pi$
$233$	19.0000	1.24473	0.622366	+	0.782727i	$0.286172\pi$
$234$	0	0
$235$	−3.00000	−0.195698
$236$	0	0
$237$	13.0000	0.844441
$238$	0	0
$239$	24.0000	1.55243	0.776215	−	0.630468i	$-0.217137\pi$
$239$	24.0000	1.55243	0.776215	+	0.630468i	$0.217137\pi$
$240$	0	0
$241$	25.0000	1.61039	0.805196	−	0.593009i	$-0.202060\pi$
$241$	25.0000	1.61039	0.805196	+	0.593009i	$0.202060\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−1.00000	−0.0638877
$246$	0	0
$247$	−3.00000	−0.190885
$248$	0	0
$249$	11.0000	0.697097
$250$	0	0
$251$	−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
$253$	18.0000	1.13165
$254$	0	0
$255$	4.00000	0.250490
$256$	0	0
$257$	14.0000	0.873296	0.436648	−	0.899632i	$-0.356166\pi$
$257$	14.0000	0.873296	0.436648	+	0.899632i	$0.356166\pi$
$258$	0	0
$259$	8.00000	0.497096
$260$	0	0
$261$	−1.00000	−0.0618984
$262$	0	0
$263$	27.0000	1.66489	0.832446	−	0.554107i	$-0.186940\pi$
$263$	27.0000	1.66489	0.832446	+	0.554107i	$0.186940\pi$
$264$	0	0
$265$	−3.00000	−0.184289
$266$	0	0
$267$	1.00000	0.0611990
$268$	0	0
$269$	−20.0000	−1.21942	−0.609711	−	0.792624i	$-0.708714\pi$
$269$	−20.0000	−1.21942	−0.609711	+	0.792624i	$0.708714\pi$
$270$	0	0
$271$	0	0		−	1.00000i	$-0.5\pi$
$271$	0	0			1.00000i	$0.5\pi$
$272$	0	0
$273$	−1.00000	−0.0605228
$274$	0	0
$275$	−8.00000	−0.482418
$276$	0	0
$277$	−7.00000	−0.420589	−0.210295	−	0.977638i	$-0.567442\pi$
$277$	−7.00000	−0.420589	−0.210295	+	0.977638i	$0.567442\pi$
$278$	0	0
$279$	5.00000	0.299342
$280$	0	0
$281$	−22.0000	−1.31241	−0.656205	−	0.754583i	$-0.727839\pi$
$281$	−22.0000	−1.31241	−0.656205	+	0.754583i	$0.727839\pi$
$282$	0	0
$283$	−24.0000	−1.42665	−0.713326	−	0.700832i	$-0.752812\pi$
$283$	−24.0000	−1.42665	−0.713326	+	0.700832i	$0.752812\pi$
$284$	0	0
$285$	3.00000	0.177705
$286$	0	0
$287$	−6.00000	−0.354169
$288$	0	0
$289$	−1.00000	−0.0588235
$290$	0	0
$291$	1.00000	0.0586210
$292$	0	0
$293$	−33.0000	−1.92788	−0.963940	−	0.266119i	$-0.914259\pi$
$293$	−33.0000	−1.92788	−0.963940	+	0.266119i	$0.914259\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	2.00000	0.116052
$298$	0	0
$299$	9.00000	0.520483
$300$	0	0
$301$	−9.00000	−0.518751
$302$	0	0
$303$	−6.00000	−0.344691
$304$	0	0
$305$	−10.0000	−0.572598
$306$	0	0
$307$	7.00000	0.399511	0.199756	−	0.979846i	$-0.435985\pi$
$307$	7.00000	0.399511	0.199756	+	0.979846i	$0.435985\pi$
$308$	0	0
$309$	12.0000	0.682656
$310$	0	0
$311$	18.0000	1.02069	0.510343	−	0.859971i	$-0.329518\pi$
$311$	18.0000	1.02069	0.510343	+	0.859971i	$0.329518\pi$
$312$	0	0
$313$	30.0000	1.69570	0.847850	−	0.530236i	$-0.177897\pi$
$313$	30.0000	1.69570	0.847850	+	0.530236i	$0.177897\pi$
$314$	0	0
$315$	1.00000	0.0563436
$316$	0	0
$317$	20.0000	1.12331	0.561656	−	0.827371i	$-0.310164\pi$
$317$	20.0000	1.12331	0.561656	+	0.827371i	$0.310164\pi$
$318$	0	0
$319$	−2.00000	−0.111979
$320$	0	0
$321$	−12.0000	−0.669775
$322$	0	0
$323$	12.0000	0.667698
$324$	0	0
$325$	−4.00000	−0.221880
$326$	0	0
$327$	−2.00000	−0.110600
$328$	0	0
$329$	−3.00000	−0.165395
$330$	0	0
$331$	10.0000	0.549650	0.274825	−	0.961494i	$-0.411380\pi$
$331$	10.0000	0.549650	0.274825	+	0.961494i	$0.411380\pi$
$332$	0	0
$333$	−8.00000	−0.438397
$334$	0	0
$335$	−2.00000	−0.109272
$336$	0	0
$337$	9.00000	0.490261	0.245131	−	0.969490i	$-0.421169\pi$
$337$	9.00000	0.490261	0.245131	+	0.969490i	$0.421169\pi$
$338$	0	0
$339$	−15.0000	−0.814688
$340$	0	0
$341$	10.0000	0.541530
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−9.00000	−0.484544
$346$	0	0
$347$	−24.0000	−1.28839	−0.644194	−	0.764862i	$-0.722807\pi$
$347$	−24.0000	−1.28839	−0.644194	+	0.764862i	$0.722807\pi$
$348$	0	0
$349$	−35.0000	−1.87351	−0.936754	−	0.349990i	$-0.886185\pi$
$349$	−35.0000	−1.87351	−0.936754	+	0.349990i	$0.886185\pi$
$350$	0	0
$351$	1.00000	0.0533761
$352$	0	0
$353$	−14.0000	−0.745145	−0.372572	−	0.928003i	$-0.621524\pi$
$353$	−14.0000	−0.745145	−0.372572	+	0.928003i	$0.621524\pi$
$354$	0	0
$355$	12.0000	0.636894
$356$	0	0
$357$	4.00000	0.211702
$358$	0	0
$359$	−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
$361$	−10.0000	−0.526316
$362$	0	0
$363$	−7.00000	−0.367405
$364$	0	0
$365$	−5.00000	−0.261712
$366$	0	0
$367$	−10.0000	−0.521996	−0.260998	−	0.965339i	$-0.584052\pi$
$367$	−10.0000	−0.521996	−0.260998	+	0.965339i	$0.584052\pi$
$368$	0	0
$369$	6.00000	0.312348
$370$	0	0
$371$	−3.00000	−0.155752
$372$	0	0
$373$	−2.00000	−0.103556	−0.0517780	−	0.998659i	$-0.516489\pi$
$373$	−2.00000	−0.103556	−0.0517780	+	0.998659i	$0.516489\pi$
$374$	0	0
$375$	9.00000	0.464758
$376$	0	0
$377$	−1.00000	−0.0515026
$378$	0	0
$379$	18.0000	0.924598	0.462299	−	0.886724i	$-0.347025\pi$
$379$	18.0000	0.924598	0.462299	+	0.886724i	$0.347025\pi$
$380$	0	0
$381$	4.00000	0.204926
$382$	0	0
$383$	4.00000	0.204390	0.102195	−	0.994764i	$-0.467413\pi$
$383$	4.00000	0.204390	0.102195	+	0.994764i	$0.467413\pi$
$384$	0	0
$385$	2.00000	0.101929
$386$	0	0
$387$	9.00000	0.457496
$388$	0	0
$389$	−26.0000	−1.31825	−0.659126	−	0.752032i	$-0.729074\pi$
$389$	−26.0000	−1.31825	−0.659126	+	0.752032i	$0.729074\pi$
$390$	0	0
$391$	−36.0000	−1.82060
$392$	0	0
$393$	−4.00000	−0.201773
$394$	0	0
$395$	−13.0000	−0.654101
$396$	0	0
$397$	−19.0000	−0.953583	−0.476791	−	0.879017i	$-0.658200\pi$
$397$	−19.0000	−0.953583	−0.476791	+	0.879017i	$0.658200\pi$
$398$	0	0
$399$	3.00000	0.150188
$400$	0	0
$401$	−20.0000	−0.998752	−0.499376	−	0.866385i	$-0.666437\pi$
$401$	−20.0000	−0.998752	−0.499376	+	0.866385i	$0.666437\pi$
$402$	0	0
$403$	5.00000	0.249068
$404$	0	0
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	−16.0000	−0.793091
$408$	0	0
$409$	−11.0000	−0.543915	−0.271957	−	0.962309i	$-0.587671\pi$
$409$	−11.0000	−0.543915	−0.271957	+	0.962309i	$0.587671\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−11.0000	−0.539969
$416$	0	0
$417$	18.0000	0.881464
$418$	0	0
$419$	2.00000	0.0977064	0.0488532	−	0.998806i	$-0.484443\pi$
$419$	2.00000	0.0977064	0.0488532	+	0.998806i	$0.484443\pi$
$420$	0	0
$421$	20.0000	0.974740	0.487370	−	0.873195i	$-0.337956\pi$
$421$	20.0000	0.974740	0.487370	+	0.873195i	$0.337956\pi$
$422$	0	0
$423$	3.00000	0.145865
$424$	0	0
$425$	16.0000	0.776114
$426$	0	0
$427$	−10.0000	−0.483934
$428$	0	0
$429$	2.00000	0.0965609
$430$	0	0
$431$	−30.0000	−1.44505	−0.722525	−	0.691345i	$-0.757018\pi$
$431$	−30.0000	−1.44505	−0.722525	+	0.691345i	$0.757018\pi$
$432$	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
$435$	1.00000	0.0479463
$436$	0	0
$437$	−27.0000	−1.29159
$438$	0	0
$439$	34.0000	1.62273	0.811366	−	0.584539i	$-0.198725\pi$
$439$	34.0000	1.62273	0.811366	+	0.584539i	$0.198725\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	−23.0000	−1.09276	−0.546381	−	0.837536i	$-0.683995\pi$
$443$	−23.0000	−1.09276	−0.546381	+	0.837536i	$0.683995\pi$
$444$	0	0
$445$	−1.00000	−0.0474045
$446$	0	0
$447$	18.0000	0.851371
$448$	0	0
$449$	16.0000	0.755087	0.377543	−	0.925992i	$-0.376769\pi$
$449$	16.0000	0.755087	0.377543	+	0.925992i	$0.376769\pi$
$450$	0	0
$451$	12.0000	0.565058
$452$	0	0
$453$	−4.00000	−0.187936
$454$	0	0
$455$	1.00000	0.0468807
$456$	0	0
$457$	8.00000	0.374224	0.187112	−	0.982339i	$-0.440087\pi$
$457$	8.00000	0.374224	0.187112	+	0.982339i	$0.440087\pi$
$458$	0	0
$459$	−4.00000	−0.186704
$460$	0	0
$461$	−2.00000	−0.0931493	−0.0465746	−	0.998915i	$-0.514831\pi$
$461$	−2.00000	−0.0931493	−0.0465746	+	0.998915i	$0.514831\pi$
$462$	0	0
$463$	−34.0000	−1.58011	−0.790057	−	0.613033i	$-0.789949\pi$
$463$	−34.0000	−1.58011	−0.790057	+	0.613033i	$0.789949\pi$
$464$	0	0
$465$	−5.00000	−0.231869
$466$	0	0
$467$	18.0000	0.832941	0.416470	−	0.909149i	$-0.363267\pi$
$467$	18.0000	0.832941	0.416470	+	0.909149i	$0.363267\pi$
$468$	0	0
$469$	−2.00000	−0.0923514
$470$	0	0
$471$	24.0000	1.10586
$472$	0	0
$473$	18.0000	0.827641
$474$	0	0
$475$	12.0000	0.550598
$476$	0	0
$477$	3.00000	0.137361
$478$	0	0
$479$	−39.0000	−1.78196	−0.890978	−	0.454047i	$-0.849980\pi$
$479$	−39.0000	−1.78196	−0.890978	+	0.454047i	$0.849980\pi$
$480$	0	0
$481$	−8.00000	−0.364769
$482$	0	0
$483$	−9.00000	−0.409514
$484$	0	0
$485$	−1.00000	−0.0454077
$486$	0	0
$487$	18.0000	0.815658	0.407829	−	0.913058i	$-0.366286\pi$
$487$	18.0000	0.815658	0.407829	+	0.913058i	$0.366286\pi$
$488$	0	0
$489$	−8.00000	−0.361773
$490$	0	0
$491$	36.0000	1.62466	0.812329	−	0.583200i	$-0.198200\pi$
$491$	36.0000	1.62466	0.812329	+	0.583200i	$0.198200\pi$
$492$	0	0
$493$	4.00000	0.180151
$494$	0	0
$495$	−2.00000	−0.0898933
$496$	0	0
$497$	12.0000	0.538274
$498$	0	0
$499$	−24.0000	−1.07439	−0.537194	−	0.843459i	$-0.680516\pi$
$499$	−24.0000	−1.07439	−0.537194	+	0.843459i	$0.680516\pi$
$500$	0	0
$501$	−7.00000	−0.312737
$502$	0	0
$503$	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
$505$	6.00000	0.266996
$506$	0	0
$507$	1.00000	0.0444116
$508$	0	0
$509$	−17.0000	−0.753512	−0.376756	−	0.926313i	$-0.622960\pi$
$509$	−17.0000	−0.753512	−0.376756	+	0.926313i	$0.622960\pi$
$510$	0	0
$511$	−5.00000	−0.221187
$512$	0	0
$513$	−3.00000	−0.132453
$514$	0	0
$515$	−12.0000	−0.528783
$516$	0	0
$517$	6.00000	0.263880
$518$	0	0
$519$	12.0000	0.526742
$520$	0	0
$521$	0	0		−	1.00000i	$-0.5\pi$
$521$	0	0			1.00000i	$0.5\pi$
$522$	0	0
$523$	34.0000	1.48672	0.743358	−	0.668894i	$-0.233232\pi$
$523$	34.0000	1.48672	0.743358	+	0.668894i	$0.233232\pi$
$524$	0	0
$525$	4.00000	0.174574
$526$	0	0
$527$	−20.0000	−0.871214
$528$	0	0
$529$	58.0000	2.52174
$530$	0	0
$531$	0	0
$532$	0	0
$533$	6.00000	0.259889
$534$	0	0
$535$	12.0000	0.518805
$536$	0	0
$537$	5.00000	0.215766
$538$	0	0
$539$	2.00000	0.0861461
$540$	0	0
$541$	12.0000	0.515920	0.257960	−	0.966156i	$-0.416950\pi$
$541$	12.0000	0.515920	0.257960	+	0.966156i	$0.416950\pi$
$542$	0	0
$543$	14.0000	0.600798
$544$	0	0
$545$	2.00000	0.0856706
$546$	0	0
$547$	−25.0000	−1.06892	−0.534461	−	0.845193i	$-0.679486\pi$
$547$	−25.0000	−1.06892	−0.534461	+	0.845193i	$0.679486\pi$
$548$	0	0
$549$	10.0000	0.426790
$550$	0	0
$551$	3.00000	0.127804
$552$	0	0
$553$	−13.0000	−0.552816
$554$	0	0
$555$	8.00000	0.339581
$556$	0	0
$557$	−12.0000	−0.508456	−0.254228	−	0.967144i	$-0.581821\pi$
$557$	−12.0000	−0.508456	−0.254228	+	0.967144i	$0.581821\pi$
$558$	0	0
$559$	9.00000	0.380659
$560$	0	0
$561$	−8.00000	−0.337760
$562$	0	0
$563$	0	0		−	1.00000i	$-0.5\pi$
$563$	0	0			1.00000i	$0.5\pi$
$564$	0	0
$565$	15.0000	0.631055
$566$	0	0
$567$	−1.00000	−0.0419961
$568$	0	0
$569$	−29.0000	−1.21574	−0.607872	−	0.794035i	$-0.707976\pi$
$569$	−29.0000	−1.21574	−0.607872	+	0.794035i	$0.707976\pi$
$570$	0	0
$571$	1.00000	0.0418487	0.0209243	−	0.999781i	$-0.493339\pi$
$571$	1.00000	0.0418487	0.0209243	+	0.999781i	$0.493339\pi$
$572$	0	0
$573$	16.0000	0.668410
$574$	0	0
$575$	−36.0000	−1.50130
$576$	0	0
$577$	−34.0000	−1.41544	−0.707719	−	0.706494i	$-0.750276\pi$
$577$	−34.0000	−1.41544	−0.707719	+	0.706494i	$0.750276\pi$
$578$	0	0
$579$	−6.00000	−0.249351
$580$	0	0
$581$	−11.0000	−0.456357
$582$	0	0
$583$	6.00000	0.248495
$584$	0	0
$585$	−1.00000	−0.0413449
$586$	0	0
$587$	−13.0000	−0.536567	−0.268284	−	0.963340i	$-0.586456\pi$
$587$	−13.0000	−0.536567	−0.268284	+	0.963340i	$0.586456\pi$
$588$	0	0
$589$	−15.0000	−0.618064
$590$	0	0
$591$	−6.00000	−0.246807
$592$	0	0
$593$	−17.0000	−0.698106	−0.349053	−	0.937103i	$-0.613497\pi$
$593$	−17.0000	−0.698106	−0.349053	+	0.937103i	$0.613497\pi$
$594$	0	0
$595$	−4.00000	−0.163984
$596$	0	0
$597$	4.00000	0.163709
$598$	0	0
$599$	25.0000	1.02147	0.510736	−	0.859738i	$-0.329373\pi$
$599$	25.0000	1.02147	0.510736	+	0.859738i	$0.329373\pi$
$600$	0	0
$601$	22.0000	0.897399	0.448699	−	0.893683i	$-0.351887\pi$
$601$	22.0000	0.897399	0.448699	+	0.893683i	$0.351887\pi$
$602$	0	0
$603$	2.00000	0.0814463
$604$	0	0
$605$	7.00000	0.284590
$606$	0	0
$607$	46.0000	1.86708	0.933541	−	0.358470i	$-0.116702\pi$
$607$	46.0000	1.86708	0.933541	+	0.358470i	$0.116702\pi$
$608$	0	0
$609$	1.00000	0.0405220
$610$	0	0
$611$	3.00000	0.121367
$612$	0	0
$613$	28.0000	1.13091	0.565455	−	0.824779i	$-0.308701\pi$
$613$	28.0000	1.13091	0.565455	+	0.824779i	$0.308701\pi$
$614$	0	0
$615$	−6.00000	−0.241943
$616$	0	0
$617$	10.0000	0.402585	0.201292	−	0.979531i	$-0.435486\pi$
$617$	10.0000	0.402585	0.201292	+	0.979531i	$0.435486\pi$
$618$	0	0
$619$	−36.0000	−1.44696	−0.723481	−	0.690344i	$-0.757459\pi$
$619$	−36.0000	−1.44696	−0.723481	+	0.690344i	$0.757459\pi$
$620$	0	0
$621$	9.00000	0.361158
$622$	0	0
$623$	−1.00000	−0.0400642
$624$	0	0
$625$	11.0000	0.440000
$626$	0	0
$627$	−6.00000	−0.239617
$628$	0	0
$629$	32.0000	1.27592
$630$	0	0
$631$	34.0000	1.35352	0.676759	−	0.736204i	$-0.263384\pi$
$631$	34.0000	1.35352	0.676759	+	0.736204i	$0.263384\pi$
$632$	0	0
$633$	−19.0000	−0.755182
$634$	0	0
$635$	−4.00000	−0.158735
$636$	0	0
$637$	1.00000	0.0396214
$638$	0	0
$639$	−12.0000	−0.474713
$640$	0	0
$641$	−19.0000	−0.750455	−0.375227	−	0.926933i	$-0.622435\pi$
$641$	−19.0000	−0.750455	−0.375227	+	0.926933i	$0.622435\pi$
$642$	0	0
$643$	−8.00000	−0.315489	−0.157745	−	0.987480i	$-0.550422\pi$
$643$	−8.00000	−0.315489	−0.157745	+	0.987480i	$0.550422\pi$
$644$	0	0
$645$	−9.00000	−0.354375
$646$	0	0
$647$	−22.0000	−0.864909	−0.432455	−	0.901656i	$-0.642352\pi$
$647$	−22.0000	−0.864909	−0.432455	+	0.901656i	$0.642352\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	−5.00000	−0.195965
$652$	0	0
$653$	−6.00000	−0.234798	−0.117399	−	0.993085i	$-0.537456\pi$
$653$	−6.00000	−0.234798	−0.117399	+	0.993085i	$0.537456\pi$
$654$	0	0
$655$	4.00000	0.156293
$656$	0	0
$657$	5.00000	0.195069
$658$	0	0
$659$	−37.0000	−1.44132	−0.720658	−	0.693291i	$-0.756160\pi$
$659$	−37.0000	−1.44132	−0.720658	+	0.693291i	$0.756160\pi$
$660$	0	0
$661$	9.00000	0.350059	0.175030	−	0.984563i	$-0.443998\pi$
$661$	9.00000	0.350059	0.175030	+	0.984563i	$0.443998\pi$
$662$	0	0
$663$	−4.00000	−0.155347
$664$	0	0
$665$	−3.00000	−0.116335
$666$	0	0
$667$	−9.00000	−0.348481
$668$	0	0
$669$	23.0000	0.889231
$670$	0	0
$671$	20.0000	0.772091
$672$	0	0
$673$	−23.0000	−0.886585	−0.443292	−	0.896377i	$-0.646190\pi$
$673$	−23.0000	−0.886585	−0.443292	+	0.896377i	$0.646190\pi$
$674$	0	0
$675$	−4.00000	−0.153960
$676$	0	0
$677$	−14.0000	−0.538064	−0.269032	−	0.963131i	$-0.586704\pi$
$677$	−14.0000	−0.538064	−0.269032	+	0.963131i	$0.586704\pi$
$678$	0	0
$679$	−1.00000	−0.0383765
$680$	0	0
$681$	−4.00000	−0.153280
$682$	0	0
$683$	12.0000	0.459167	0.229584	−	0.973289i	$-0.426264\pi$
$683$	12.0000	0.459167	0.229584	+	0.973289i	$0.426264\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	26.0000	0.991962
$688$	0	0
$689$	3.00000	0.114291
$690$	0	0
$691$	−45.0000	−1.71188	−0.855940	−	0.517075i	$-0.827021\pi$
$691$	−45.0000	−1.71188	−0.855940	+	0.517075i	$0.827021\pi$
$692$	0	0
$693$	−2.00000	−0.0759737
$694$	0	0
$695$	−18.0000	−0.682779
$696$	0	0
$697$	−24.0000	−0.909065
$698$	0	0
$699$	19.0000	0.718646
$700$	0	0
$701$	−7.00000	−0.264386	−0.132193	−	0.991224i	$-0.542202\pi$
$701$	−7.00000	−0.264386	−0.132193	+	0.991224i	$0.542202\pi$
$702$	0	0
$703$	24.0000	0.905177
$704$	0	0
$705$	−3.00000	−0.112987
$706$	0	0
$707$	6.00000	0.225653
$708$	0	0
$709$	−38.0000	−1.42712	−0.713560	−	0.700594i	$-0.752918\pi$
$709$	−38.0000	−1.42712	−0.713560	+	0.700594i	$0.752918\pi$
$710$	0	0
$711$	13.0000	0.487538
$712$	0	0
$713$	45.0000	1.68526
$714$	0	0
$715$	−2.00000	−0.0747958
$716$	0	0
$717$	24.0000	0.896296
$718$	0	0
$719$	−6.00000	−0.223762	−0.111881	−	0.993722i	$-0.535688\pi$
$719$	−6.00000	−0.223762	−0.111881	+	0.993722i	$0.535688\pi$
$720$	0	0
$721$	−12.0000	−0.446903
$722$	0	0
$723$	25.0000	0.929760
$724$	0	0
$725$	4.00000	0.148556
$726$	0	0
$727$	−14.0000	−0.519231	−0.259616	−	0.965712i	$-0.583596\pi$
$727$	−14.0000	−0.519231	−0.259616	+	0.965712i	$0.583596\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−36.0000	−1.33151
$732$	0	0
$733$	−27.0000	−0.997268	−0.498634	−	0.866813i	$-0.666165\pi$
$733$	−27.0000	−0.997268	−0.498634	+	0.866813i	$0.666165\pi$
$734$	0	0
$735$	−1.00000	−0.0368856
$736$	0	0
$737$	4.00000	0.147342
$738$	0	0
$739$	22.0000	0.809283	0.404642	−	0.914475i	$-0.367396\pi$
$739$	22.0000	0.809283	0.404642	+	0.914475i	$0.367396\pi$
$740$	0	0
$741$	−3.00000	−0.110208
$742$	0	0
$743$	24.0000	0.880475	0.440237	−	0.897881i	$-0.354894\pi$
$743$	24.0000	0.880475	0.440237	+	0.897881i	$0.354894\pi$
$744$	0	0
$745$	−18.0000	−0.659469
$746$	0	0
$747$	11.0000	0.402469
$748$	0	0
$749$	12.0000	0.438470
$750$	0	0
$751$	25.0000	0.912263	0.456131	−	0.889912i	$-0.349235\pi$
$751$	25.0000	0.912263	0.456131	+	0.889912i	$0.349235\pi$
$752$	0	0
$753$	−30.0000	−1.09326
$754$	0	0
$755$	4.00000	0.145575
$756$	0	0
$757$	1.00000	0.0363456	0.0181728	−	0.999835i	$-0.494215\pi$
$757$	1.00000	0.0363456	0.0181728	+	0.999835i	$0.494215\pi$
$758$	0	0
$759$	18.0000	0.653359
$760$	0	0
$761$	35.0000	1.26875	0.634375	−	0.773026i	$-0.281258\pi$
$761$	35.0000	1.26875	0.634375	+	0.773026i	$0.281258\pi$
$762$	0	0
$763$	2.00000	0.0724049
$764$	0	0
$765$	4.00000	0.144620
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−45.0000	−1.62274	−0.811371	−	0.584532i	$-0.801278\pi$
$769$	−45.0000	−1.62274	−0.811371	+	0.584532i	$0.801278\pi$
$770$	0	0
$771$	14.0000	0.504198
$772$	0	0
$773$	10.0000	0.359675	0.179838	−	0.983696i	$-0.442443\pi$
$773$	10.0000	0.359675	0.179838	+	0.983696i	$0.442443\pi$
$774$	0	0
$775$	−20.0000	−0.718421
$776$	0	0
$777$	8.00000	0.286998
$778$	0	0
$779$	−18.0000	−0.644917
$780$	0	0
$781$	−24.0000	−0.858788
$782$	0	0
$783$	−1.00000	−0.0357371
$784$	0	0
$785$	−24.0000	−0.856597
$786$	0	0
$787$	5.00000	0.178231	0.0891154	−	0.996021i	$-0.471596\pi$
$787$	5.00000	0.178231	0.0891154	+	0.996021i	$0.471596\pi$
$788$	0	0
$789$	27.0000	0.961225
$790$	0	0
$791$	15.0000	0.533339
$792$	0	0
$793$	10.0000	0.355110
$794$	0	0
$795$	−3.00000	−0.106399
$796$	0	0
$797$	−38.0000	−1.34603	−0.673015	−	0.739629i	$-0.735001\pi$
$797$	−38.0000	−1.34603	−0.673015	+	0.739629i	$0.735001\pi$
$798$	0	0
$799$	−12.0000	−0.424529
$800$	0	0
$801$	1.00000	0.0353333
$802$	0	0
$803$	10.0000	0.352892
$804$	0	0
$805$	9.00000	0.317208
$806$	0	0
$807$	−20.0000	−0.704033
$808$	0	0
$809$	−27.0000	−0.949269	−0.474635	−	0.880183i	$-0.657420\pi$
$809$	−27.0000	−0.949269	−0.474635	+	0.880183i	$0.657420\pi$
$810$	0	0
$811$	28.0000	0.983213	0.491606	−	0.870817i	$-0.336410\pi$
$811$	28.0000	0.983213	0.491606	+	0.870817i	$0.336410\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	8.00000	0.280228
$816$	0	0
$817$	−27.0000	−0.944610
$818$	0	0
$819$	−1.00000	−0.0349428
$820$	0	0
$821$	−34.0000	−1.18661	−0.593304	−	0.804978i	$-0.702177\pi$
$821$	−34.0000	−1.18661	−0.593304	+	0.804978i	$0.702177\pi$
$822$	0	0
$823$	0	0		−	1.00000i	$-0.5\pi$
$823$	0	0			1.00000i	$0.5\pi$
$824$	0	0
$825$	−8.00000	−0.278524
$826$	0	0
$827$	8.00000	0.278187	0.139094	−	0.990279i	$-0.455581\pi$
$827$	8.00000	0.278187	0.139094	+	0.990279i	$0.455581\pi$
$828$	0	0
$829$	2.00000	0.0694629	0.0347314	−	0.999397i	$-0.488942\pi$
$829$	2.00000	0.0694629	0.0347314	+	0.999397i	$0.488942\pi$
$830$	0	0
$831$	−7.00000	−0.242827
$832$	0	0
$833$	−4.00000	−0.138592
$834$	0	0
$835$	7.00000	0.242245
$836$	0	0
$837$	5.00000	0.172825
$838$	0	0
$839$	0	0		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	0	0
$841$	−28.0000	−0.965517
$842$	0	0
$843$	−22.0000	−0.757720
$844$	0	0
$845$	−1.00000	−0.0344010
$846$	0	0
$847$	7.00000	0.240523
$848$	0	0
$849$	−24.0000	−0.823678
$850$	0	0
$851$	−72.0000	−2.46813
$852$	0	0
$853$	−45.0000	−1.54077	−0.770385	−	0.637579i	$-0.779936\pi$
$853$	−45.0000	−1.54077	−0.770385	+	0.637579i	$0.779936\pi$
$854$	0	0
$855$	3.00000	0.102598
$856$	0	0
$857$	38.0000	1.29806	0.649028	−	0.760765i	$-0.275176\pi$
$857$	38.0000	1.29806	0.649028	+	0.760765i	$0.275176\pi$
$858$	0	0
$859$	26.0000	0.887109	0.443554	−	0.896248i	$-0.353717\pi$
$859$	26.0000	0.887109	0.443554	+	0.896248i	$0.353717\pi$
$860$	0	0
$861$	−6.00000	−0.204479
$862$	0	0
$863$	−40.0000	−1.36162	−0.680808	−	0.732462i	$-0.738371\pi$
$863$	−40.0000	−1.36162	−0.680808	+	0.732462i	$0.738371\pi$
$864$	0	0
$865$	−12.0000	−0.408012
$866$	0	0
$867$	−1.00000	−0.0339618
$868$	0	0
$869$	26.0000	0.881990
$870$	0	0
$871$	2.00000	0.0677674
$872$	0	0
$873$	1.00000	0.0338449
$874$	0	0
$875$	−9.00000	−0.304256
$876$	0	0
$877$	−40.0000	−1.35070	−0.675352	−	0.737496i	$-0.736008\pi$
$877$	−40.0000	−1.35070	−0.675352	+	0.737496i	$0.736008\pi$
$878$	0	0
$879$	−33.0000	−1.11306
$880$	0	0
$881$	10.0000	0.336909	0.168454	−	0.985709i	$-0.446122\pi$
$881$	10.0000	0.336909	0.168454	+	0.985709i	$0.446122\pi$
$882$	0	0
$883$	20.0000	0.673054	0.336527	−	0.941674i	$-0.390748\pi$
$883$	20.0000	0.673054	0.336527	+	0.941674i	$0.390748\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	48.0000	1.61168	0.805841	−	0.592132i	$-0.201714\pi$
$887$	48.0000	1.61168	0.805841	+	0.592132i	$0.201714\pi$
$888$	0	0
$889$	−4.00000	−0.134156
$890$	0	0
$891$	2.00000	0.0670025
$892$	0	0
$893$	−9.00000	−0.301174
$894$	0	0
$895$	−5.00000	−0.167132
$896$	0	0
$897$	9.00000	0.300501
$898$	0	0
$899$	−5.00000	−0.166759
$900$	0	0
$901$	−12.0000	−0.399778
$902$	0	0
$903$	−9.00000	−0.299501
$904$	0	0
$905$	−14.0000	−0.465376
$906$	0	0
$907$	−39.0000	−1.29497	−0.647487	−	0.762077i	$-0.724180\pi$
$907$	−39.0000	−1.29497	−0.647487	+	0.762077i	$0.724180\pi$
$908$	0	0
$909$	−6.00000	−0.199007
$910$	0	0
$911$	−5.00000	−0.165657	−0.0828287	−	0.996564i	$-0.526395\pi$
$911$	−5.00000	−0.165657	−0.0828287	+	0.996564i	$0.526395\pi$
$912$	0	0
$913$	22.0000	0.728094
$914$	0	0
$915$	−10.0000	−0.330590
$916$	0	0
$917$	4.00000	0.132092
$918$	0	0
$919$	−48.0000	−1.58337	−0.791687	−	0.610927i	$-0.790797\pi$
$919$	−48.0000	−1.58337	−0.791687	+	0.610927i	$0.790797\pi$
$920$	0	0
$921$	7.00000	0.230658
$922$	0	0
$923$	−12.0000	−0.394985
$924$	0	0
$925$	32.0000	1.05215
$926$	0	0
$927$	12.0000	0.394132
$928$	0	0
$929$	−9.00000	−0.295280	−0.147640	−	0.989041i	$-0.547168\pi$
$929$	−9.00000	−0.295280	−0.147640	+	0.989041i	$0.547168\pi$
$930$	0	0
$931$	−3.00000	−0.0983210
$932$	0	0
$933$	18.0000	0.589294
$934$	0	0
$935$	8.00000	0.261628
$936$	0	0
$937$	44.0000	1.43742	0.718709	−	0.695311i	$-0.244734\pi$
$937$	44.0000	1.43742	0.718709	+	0.695311i	$0.244734\pi$
$938$	0	0
$939$	30.0000	0.979013
$940$	0	0
$941$	−35.0000	−1.14097	−0.570484	−	0.821309i	$-0.693244\pi$
$941$	−35.0000	−1.14097	−0.570484	+	0.821309i	$0.693244\pi$
$942$	0	0
$943$	54.0000	1.75848
$944$	0	0
$945$	1.00000	0.0325300
$946$	0	0
$947$	−18.0000	−0.584921	−0.292461	−	0.956278i	$-0.594474\pi$
$947$	−18.0000	−0.584921	−0.292461	+	0.956278i	$0.594474\pi$
$948$	0	0
$949$	5.00000	0.162307
$950$	0	0
$951$	20.0000	0.648544
$952$	0	0
$953$	−3.00000	−0.0971795	−0.0485898	−	0.998819i	$-0.515473\pi$
$953$	−3.00000	−0.0971795	−0.0485898	+	0.998819i	$0.515473\pi$
$954$	0	0
$955$	−16.0000	−0.517748
$956$	0	0
$957$	−2.00000	−0.0646508
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−6.00000	−0.193548
$962$	0	0
$963$	−12.0000	−0.386695
$964$	0	0
$965$	6.00000	0.193147
$966$	0	0
$967$	46.0000	1.47926	0.739630	−	0.673014i	$-0.235000\pi$
$967$	46.0000	1.47926	0.739630	+	0.673014i	$0.235000\pi$
$968$	0	0
$969$	12.0000	0.385496
$970$	0	0
$971$	30.0000	0.962746	0.481373	−	0.876516i	$-0.340138\pi$
$971$	30.0000	0.962746	0.481373	+	0.876516i	$0.340138\pi$
$972$	0	0
$973$	−18.0000	−0.577054
$974$	0	0
$975$	−4.00000	−0.128103
$976$	0	0
$977$	54.0000	1.72761	0.863807	−	0.503824i	$-0.168074\pi$
$977$	54.0000	1.72761	0.863807	+	0.503824i	$0.168074\pi$
$978$	0	0
$979$	2.00000	0.0639203
$980$	0	0
$981$	−2.00000	−0.0638551
$982$	0	0
$983$	−43.0000	−1.37149	−0.685744	−	0.727843i	$-0.740523\pi$
$983$	−43.0000	−1.37149	−0.685744	+	0.727843i	$0.740523\pi$
$984$	0	0
$985$	6.00000	0.191176
$986$	0	0
$987$	−3.00000	−0.0954911
$988$	0	0
$989$	81.0000	2.57565
$990$	0	0
$991$	−4.00000	−0.127064	−0.0635321	−	0.997980i	$-0.520237\pi$
$991$	−4.00000	−0.127064	−0.0635321	+	0.997980i	$0.520237\pi$
$992$	0	0
$993$	10.0000	0.317340
$994$	0	0
$995$	−4.00000	−0.126809
$996$	0	0
$997$	52.0000	1.64686	0.823428	−	0.567420i	$-0.192059\pi$
$997$	52.0000	1.64686	0.823428	+	0.567420i	$0.192059\pi$
$998$	0	0
$999$	−8.00000	−0.253109

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4368.2.a.q.1.1		1
4.3	odd	2		273.2.a.a.1.1	✓	1
12.11	even	2		819.2.a.e.1.1		1
20.19	odd	2		6825.2.a.l.1.1		1
28.27	even	2		1911.2.a.a.1.1		1
52.51	odd	2		3549.2.a.d.1.1		1
84.83	odd	2		5733.2.a.m.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
273.2.a.a.1.1	✓	1	4.3	odd	2
819.2.a.e.1.1		1	12.11	even	2
1911.2.a.a.1.1		1	28.27	even	2
3549.2.a.d.1.1		1	52.51	odd	2
4368.2.a.q.1.1		1	1.1	even	1	trivial
5733.2.a.m.1.1		1	84.83	odd	2
6825.2.a.l.1.1		1	20.19	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 4368 = 2^{4} \cdot 3 \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4368.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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