LMFDB - Embedded newform 4368.2.a.l.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:	$1$
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	$\chi$	$=$	4368.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{3} +3.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} -3.00000 q^{11} +1.00000 q^{13} -3.00000 q^{15} -3.00000 q^{17} +7.00000 q^{19} +1.00000 q^{21} -9.00000 q^{23} +4.00000 q^{25} -1.00000 q^{27} -9.00000 q^{29} +4.00000 q^{31} +3.00000 q^{33} -3.00000 q^{35} -7.00000 q^{37} -1.00000 q^{39} +12.0000 q^{41} +1.00000 q^{43} +3.00000 q^{45} +1.00000 q^{49} +3.00000 q^{51} -6.00000 q^{53} -9.00000 q^{55} -7.00000 q^{57} -12.0000 q^{59} -1.00000 q^{61} -1.00000 q^{63} +3.00000 q^{65} -14.0000 q^{67} +9.00000 q^{69} -12.0000 q^{71} -7.00000 q^{73} -4.00000 q^{75} +3.00000 q^{77} +10.0000 q^{79} +1.00000 q^{81} +6.00000 q^{83} -9.00000 q^{85} +9.00000 q^{87} -6.00000 q^{89} -1.00000 q^{91} -4.00000 q^{93} +21.0000 q^{95} -10.0000 q^{97} -3.00000 q^{99} +O(q^{100})$

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
$5$	3.00000	1.34164	0.670820	−	0.741620i	$-0.265942\pi$
$5$	3.00000	1.34164	0.670820	+	0.741620i	$0.265942\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−3.00000	−0.904534	−0.452267	−	0.891883i	$-0.649385\pi$
$11$	−3.00000	−0.904534	−0.452267	+	0.891883i	$0.649385\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	−3.00000	−0.774597
$16$	0	0
$17$	−3.00000	−0.727607	−0.363803	−	0.931476i	$-0.618522\pi$
$17$	−3.00000	−0.727607	−0.363803	+	0.931476i	$0.618522\pi$
$18$	0	0
$19$	7.00000	1.60591	0.802955	−	0.596040i	$-0.203260\pi$
$19$	7.00000	1.60591	0.802955	+	0.596040i	$0.203260\pi$
$20$	0	0
$21$	1.00000	0.218218
$22$	0	0
$23$	−9.00000	−1.87663	−0.938315	−	0.345782i	$-0.887614\pi$
$23$	−9.00000	−1.87663	−0.938315	+	0.345782i	$0.887614\pi$
$24$	0	0
$25$	4.00000	0.800000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−9.00000	−1.67126	−0.835629	−	0.549294i	$-0.814897\pi$
$29$	−9.00000	−1.67126	−0.835629	+	0.549294i	$0.814897\pi$
$30$	0	0
$31$	4.00000	0.718421	0.359211	−	0.933257i	$-0.383046\pi$
$31$	4.00000	0.718421	0.359211	+	0.933257i	$0.383046\pi$
$32$	0	0
$33$	3.00000	0.522233
$34$	0	0
$35$	−3.00000	−0.507093
$36$	0	0
$37$	−7.00000	−1.15079	−0.575396	−	0.817875i	$-0.695152\pi$
$37$	−7.00000	−1.15079	−0.575396	+	0.817875i	$0.695152\pi$
$38$	0	0
$39$	−1.00000	−0.160128
$40$	0	0
$41$	12.0000	1.87409	0.937043	−	0.349215i	$-0.113552\pi$
$41$	12.0000	1.87409	0.937043	+	0.349215i	$0.113552\pi$
$42$	0	0
$43$	1.00000	0.152499	0.0762493	−	0.997089i	$-0.475706\pi$
$43$	1.00000	0.152499	0.0762493	+	0.997089i	$0.475706\pi$
$44$	0	0
$45$	3.00000	0.447214
$46$	0	0
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	3.00000	0.420084
$52$	0	0
$53$	−6.00000	−0.824163	−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000	−0.824163	−0.412082	+	0.911147i	$0.635198\pi$
$54$	0	0
$55$	−9.00000	−1.21356
$56$	0	0
$57$	−7.00000	−0.927173
$58$	0	0
$59$	−12.0000	−1.56227	−0.781133	−	0.624364i	$-0.785358\pi$
$59$	−12.0000	−1.56227	−0.781133	+	0.624364i	$0.785358\pi$
$60$	0	0
$61$	−1.00000	−0.128037	−0.0640184	−	0.997949i	$-0.520392\pi$
$61$	−1.00000	−0.128037	−0.0640184	+	0.997949i	$0.520392\pi$
$62$	0	0
$63$	−1.00000	−0.125988
$64$	0	0
$65$	3.00000	0.372104
$66$	0	0
$67$	−14.0000	−1.71037	−0.855186	−	0.518321i	$-0.826557\pi$
$67$	−14.0000	−1.71037	−0.855186	+	0.518321i	$0.826557\pi$
$68$	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
$73$	−7.00000	−0.819288	−0.409644	−	0.912245i	$-0.634347\pi$
$73$	−7.00000	−0.819288	−0.409644	+	0.912245i	$0.634347\pi$
$74$	0	0
$75$	−4.00000	−0.461880
$76$	0	0
$77$	3.00000	0.341882
$78$	0	0
$79$	10.0000	1.12509	0.562544	−	0.826767i	$-0.309823\pi$
$79$	10.0000	1.12509	0.562544	+	0.826767i	$0.309823\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	6.00000	0.658586	0.329293	−	0.944228i	$-0.393190\pi$
$83$	6.00000	0.658586	0.329293	+	0.944228i	$0.393190\pi$
$84$	0	0
$85$	−9.00000	−0.976187
$86$	0	0
$87$	9.00000	0.964901
$88$	0	0
$89$	−6.00000	−0.635999	−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000	−0.635999	−0.317999	+	0.948091i	$0.603011\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	0	0
$93$	−4.00000	−0.414781
$94$	0	0
$95$	21.0000	2.15455
$96$	0	0
$97$	−10.0000	−1.01535	−0.507673	−	0.861550i	$-0.669494\pi$
$97$	−10.0000	−1.01535	−0.507673	+	0.861550i	$0.669494\pi$
$98$	0	0
$99$	−3.00000	−0.301511
$100$	0	0
$101$	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
$103$	13.0000	1.28093	0.640464	−	0.767988i	$-0.278742\pi$
$103$	13.0000	1.28093	0.640464	+	0.767988i	$0.278742\pi$
$104$	0	0
$105$	3.00000	0.292770
$106$	0	0
$107$	6.00000	0.580042	0.290021	−	0.957020i	$-0.406338\pi$
$107$	6.00000	0.580042	0.290021	+	0.957020i	$0.406338\pi$
$108$	0	0
$109$	11.0000	1.05361	0.526804	−	0.849987i	$-0.323390\pi$
$109$	11.0000	1.05361	0.526804	+	0.849987i	$0.323390\pi$
$110$	0	0
$111$	7.00000	0.664411
$112$	0	0
$113$	−18.0000	−1.69330	−0.846649	−	0.532152i	$-0.821383\pi$
$113$	−18.0000	−1.69330	−0.846649	+	0.532152i	$0.821383\pi$
$114$	0	0
$115$	−27.0000	−2.51776
$116$	0	0
$117$	1.00000	0.0924500
$118$	0	0
$119$	3.00000	0.275010
$120$	0	0
$121$	−2.00000	−0.181818
$122$	0	0
$123$	−12.0000	−1.08200
$124$	0	0
$125$	−3.00000	−0.268328
$126$	0	0
$127$	−2.00000	−0.177471	−0.0887357	−	0.996055i	$-0.528283\pi$
$127$	−2.00000	−0.177471	−0.0887357	+	0.996055i	$0.528283\pi$
$128$	0	0
$129$	−1.00000	−0.0880451
$130$	0	0
$131$	15.0000	1.31056	0.655278	−	0.755388i	$-0.272551\pi$
$131$	15.0000	1.31056	0.655278	+	0.755388i	$0.272551\pi$
$132$	0	0
$133$	−7.00000	−0.606977
$134$	0	0
$135$	−3.00000	−0.258199
$136$	0	0
$137$	9.00000	0.768922	0.384461	−	0.923141i	$-0.374387\pi$
$137$	9.00000	0.768922	0.384461	+	0.923141i	$0.374387\pi$
$138$	0	0
$139$	4.00000	0.339276	0.169638	−	0.985506i	$-0.445740\pi$
$139$	4.00000	0.339276	0.169638	+	0.985506i	$0.445740\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−3.00000	−0.250873
$144$	0	0
$145$	−27.0000	−2.24223
$146$	0	0
$147$	−1.00000	−0.0824786
$148$	0	0
$149$	−12.0000	−0.983078	−0.491539	−	0.870855i	$-0.663566\pi$
$149$	−12.0000	−0.983078	−0.491539	+	0.870855i	$0.663566\pi$
$150$	0	0
$151$	−17.0000	−1.38344	−0.691720	−	0.722166i	$-0.743147\pi$
$151$	−17.0000	−1.38344	−0.691720	+	0.722166i	$0.743147\pi$
$152$	0	0
$153$	−3.00000	−0.242536
$154$	0	0
$155$	12.0000	0.963863
$156$	0	0
$157$	−13.0000	−1.03751	−0.518756	−	0.854922i	$-0.673605\pi$
$157$	−13.0000	−1.03751	−0.518756	+	0.854922i	$0.673605\pi$
$158$	0	0
$159$	6.00000	0.475831
$160$	0	0
$161$	9.00000	0.709299
$162$	0	0
$163$	−2.00000	−0.156652	−0.0783260	−	0.996928i	$-0.524958\pi$
$163$	−2.00000	−0.156652	−0.0783260	+	0.996928i	$0.524958\pi$
$164$	0	0
$165$	9.00000	0.700649
$166$	0	0
$167$	−3.00000	−0.232147	−0.116073	−	0.993241i	$-0.537031\pi$
$167$	−3.00000	−0.232147	−0.116073	+	0.993241i	$0.537031\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	7.00000	0.535303
$172$	0	0
$173$	−6.00000	−0.456172	−0.228086	−	0.973641i	$-0.573247\pi$
$173$	−6.00000	−0.456172	−0.228086	+	0.973641i	$0.573247\pi$
$174$	0	0
$175$	−4.00000	−0.302372
$176$	0	0
$177$	12.0000	0.901975
$178$	0	0
$179$	0	0		−	1.00000i	$-0.5\pi$
$179$	0	0			1.00000i	$0.5\pi$
$180$	0	0
$181$	2.00000	0.148659	0.0743294	−	0.997234i	$-0.476318\pi$
$181$	2.00000	0.148659	0.0743294	+	0.997234i	$0.476318\pi$
$182$	0	0
$183$	1.00000	0.0739221
$184$	0	0
$185$	−21.0000	−1.54395
$186$	0	0
$187$	9.00000	0.658145
$188$	0	0
$189$	1.00000	0.0727393
$190$	0	0
$191$	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
$193$	−4.00000	−0.287926	−0.143963	−	0.989583i	$-0.545985\pi$
$193$	−4.00000	−0.287926	−0.143963	+	0.989583i	$0.545985\pi$
$194$	0	0
$195$	−3.00000	−0.214834
$196$	0	0
$197$	−12.0000	−0.854965	−0.427482	−	0.904024i	$-0.640599\pi$
$197$	−12.0000	−0.854965	−0.427482	+	0.904024i	$0.640599\pi$
$198$	0	0
$199$	7.00000	0.496217	0.248108	−	0.968732i	$-0.420191\pi$
$199$	7.00000	0.496217	0.248108	+	0.968732i	$0.420191\pi$
$200$	0	0
$201$	14.0000	0.987484
$202$	0	0
$203$	9.00000	0.631676
$204$	0	0
$205$	36.0000	2.51435
$206$	0	0
$207$	−9.00000	−0.625543
$208$	0	0
$209$	−21.0000	−1.45260
$210$	0	0
$211$	−5.00000	−0.344214	−0.172107	−	0.985078i	$-0.555058\pi$
$211$	−5.00000	−0.344214	−0.172107	+	0.985078i	$0.555058\pi$
$212$	0	0
$213$	12.0000	0.822226
$214$	0	0
$215$	3.00000	0.204598
$216$	0	0
$217$	−4.00000	−0.271538
$218$	0	0
$219$	7.00000	0.473016
$220$	0	0
$221$	−3.00000	−0.201802
$222$	0	0
$223$	−26.0000	−1.74109	−0.870544	−	0.492090i	$-0.836233\pi$
$223$	−26.0000	−1.74109	−0.870544	+	0.492090i	$0.836233\pi$
$224$	0	0
$225$	4.00000	0.266667
$226$	0	0
$227$	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
$229$	14.0000	0.925146	0.462573	−	0.886581i	$-0.346926\pi$
$229$	14.0000	0.925146	0.462573	+	0.886581i	$0.346926\pi$
$230$	0	0
$231$	−3.00000	−0.197386
$232$	0	0
$233$	24.0000	1.57229	0.786146	−	0.618041i	$-0.212073\pi$
$233$	24.0000	1.57229	0.786146	+	0.618041i	$0.212073\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−10.0000	−0.649570
$238$	0	0
$239$	−24.0000	−1.55243	−0.776215	−	0.630468i	$-0.782863\pi$
$239$	−24.0000	−1.55243	−0.776215	+	0.630468i	$0.782863\pi$
$240$	0	0
$241$	−10.0000	−0.644157	−0.322078	−	0.946713i	$-0.604381\pi$
$241$	−10.0000	−0.644157	−0.322078	+	0.946713i	$0.604381\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	3.00000	0.191663
$246$	0	0
$247$	7.00000	0.445399
$248$	0	0
$249$	−6.00000	−0.380235
$250$	0	0
$251$	−21.0000	−1.32551	−0.662754	−	0.748837i	$-0.730613\pi$
$251$	−21.0000	−1.32551	−0.662754	+	0.748837i	$0.730613\pi$
$252$	0	0
$253$	27.0000	1.69748
$254$	0	0
$255$	9.00000	0.563602
$256$	0	0
$257$	−18.0000	−1.12281	−0.561405	−	0.827541i	$-0.689739\pi$
$257$	−18.0000	−1.12281	−0.561405	+	0.827541i	$0.689739\pi$
$258$	0	0
$259$	7.00000	0.434959
$260$	0	0
$261$	−9.00000	−0.557086
$262$	0	0
$263$	−12.0000	−0.739952	−0.369976	−	0.929041i	$-0.620634\pi$
$263$	−12.0000	−0.739952	−0.369976	+	0.929041i	$0.620634\pi$
$264$	0	0
$265$	−18.0000	−1.10573
$266$	0	0
$267$	6.00000	0.367194
$268$	0	0
$269$	6.00000	0.365826	0.182913	−	0.983129i	$-0.441447\pi$
$269$	6.00000	0.365826	0.182913	+	0.983129i	$0.441447\pi$
$270$	0	0
$271$	−2.00000	−0.121491	−0.0607457	−	0.998153i	$-0.519348\pi$
$271$	−2.00000	−0.121491	−0.0607457	+	0.998153i	$0.519348\pi$
$272$	0	0
$273$	1.00000	0.0605228
$274$	0	0
$275$	−12.0000	−0.723627
$276$	0	0
$277$	26.0000	1.56219	0.781094	−	0.624413i	$-0.214662\pi$
$277$	26.0000	1.56219	0.781094	+	0.624413i	$0.214662\pi$
$278$	0	0
$279$	4.00000	0.239474
$280$	0	0
$281$	−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
$283$	4.00000	0.237775	0.118888	−	0.992908i	$-0.462067\pi$
$283$	4.00000	0.237775	0.118888	+	0.992908i	$0.462067\pi$
$284$	0	0
$285$	−21.0000	−1.24393
$286$	0	0
$287$	−12.0000	−0.708338
$288$	0	0
$289$	−8.00000	−0.470588
$290$	0	0
$291$	10.0000	0.586210
$292$	0	0
$293$	−18.0000	−1.05157	−0.525786	−	0.850617i	$-0.676229\pi$
$293$	−18.0000	−1.05157	−0.525786	+	0.850617i	$0.676229\pi$
$294$	0	0
$295$	−36.0000	−2.09600
$296$	0	0
$297$	3.00000	0.174078
$298$	0	0
$299$	−9.00000	−0.520483
$300$	0	0
$301$	−1.00000	−0.0576390
$302$	0	0
$303$	−12.0000	−0.689382
$304$	0	0
$305$	−3.00000	−0.171780
$306$	0	0
$307$	16.0000	0.913168	0.456584	−	0.889680i	$-0.349073\pi$
$307$	16.0000	0.913168	0.456584	+	0.889680i	$0.349073\pi$
$308$	0	0
$309$	−13.0000	−0.739544
$310$	0	0
$311$	−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
$313$	8.00000	0.452187	0.226093	−	0.974106i	$-0.427405\pi$
$313$	8.00000	0.452187	0.226093	+	0.974106i	$0.427405\pi$
$314$	0	0
$315$	−3.00000	−0.169031
$316$	0	0
$317$	0	0		−	1.00000i	$-0.5\pi$
$317$	0	0			1.00000i	$0.5\pi$
$318$	0	0
$319$	27.0000	1.51171
$320$	0	0
$321$	−6.00000	−0.334887
$322$	0	0
$323$	−21.0000	−1.16847
$324$	0	0
$325$	4.00000	0.221880
$326$	0	0
$327$	−11.0000	−0.608301
$328$	0	0
$329$	0	0
$330$	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
$333$	−7.00000	−0.383598
$334$	0	0
$335$	−42.0000	−2.29471
$336$	0	0
$337$	−13.0000	−0.708155	−0.354078	−	0.935216i	$-0.615205\pi$
$337$	−13.0000	−0.708155	−0.354078	+	0.935216i	$0.615205\pi$
$338$	0	0
$339$	18.0000	0.977626
$340$	0	0
$341$	−12.0000	−0.649836
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	27.0000	1.45363
$346$	0	0
$347$	−12.0000	−0.644194	−0.322097	−	0.946707i	$-0.604388\pi$
$347$	−12.0000	−0.644194	−0.322097	+	0.946707i	$0.604388\pi$
$348$	0	0
$349$	26.0000	1.39175	0.695874	−	0.718164i	$-0.255017\pi$
$349$	26.0000	1.39175	0.695874	+	0.718164i	$0.255017\pi$
$350$	0	0
$351$	−1.00000	−0.0533761
$352$	0	0
$353$	−6.00000	−0.319348	−0.159674	−	0.987170i	$-0.551044\pi$
$353$	−6.00000	−0.319348	−0.159674	+	0.987170i	$0.551044\pi$
$354$	0	0
$355$	−36.0000	−1.91068
$356$	0	0
$357$	−3.00000	−0.158777
$358$	0	0
$359$	−24.0000	−1.26667	−0.633336	−	0.773877i	$-0.718315\pi$
$359$	−24.0000	−1.26667	−0.633336	+	0.773877i	$0.718315\pi$
$360$	0	0
$361$	30.0000	1.57895
$362$	0	0
$363$	2.00000	0.104973
$364$	0	0
$365$	−21.0000	−1.09919
$366$	0	0
$367$	−8.00000	−0.417597	−0.208798	−	0.977959i	$-0.566955\pi$
$367$	−8.00000	−0.417597	−0.208798	+	0.977959i	$0.566955\pi$
$368$	0	0
$369$	12.0000	0.624695
$370$	0	0
$371$	6.00000	0.311504
$372$	0	0
$373$	−4.00000	−0.207112	−0.103556	−	0.994624i	$-0.533022\pi$
$373$	−4.00000	−0.207112	−0.103556	+	0.994624i	$0.533022\pi$
$374$	0	0
$375$	3.00000	0.154919
$376$	0	0
$377$	−9.00000	−0.463524
$378$	0	0
$379$	16.0000	0.821865	0.410932	−	0.911666i	$-0.365203\pi$
$379$	16.0000	0.821865	0.410932	+	0.911666i	$0.365203\pi$
$380$	0	0
$381$	2.00000	0.102463
$382$	0	0
$383$	21.0000	1.07305	0.536525	−	0.843884i	$-0.319737\pi$
$383$	21.0000	1.07305	0.536525	+	0.843884i	$0.319737\pi$
$384$	0	0
$385$	9.00000	0.458682
$386$	0	0
$387$	1.00000	0.0508329
$388$	0	0
$389$	−30.0000	−1.52106	−0.760530	−	0.649303i	$-0.775061\pi$
$389$	−30.0000	−1.52106	−0.760530	+	0.649303i	$0.775061\pi$
$390$	0	0
$391$	27.0000	1.36545
$392$	0	0
$393$	−15.0000	−0.756650
$394$	0	0
$395$	30.0000	1.50946
$396$	0	0
$397$	−34.0000	−1.70641	−0.853206	−	0.521575i	$-0.825345\pi$
$397$	−34.0000	−1.70641	−0.853206	+	0.521575i	$0.825345\pi$
$398$	0	0
$399$	7.00000	0.350438
$400$	0	0
$401$	18.0000	0.898877	0.449439	−	0.893311i	$-0.351624\pi$
$401$	18.0000	0.898877	0.449439	+	0.893311i	$0.351624\pi$
$402$	0	0
$403$	4.00000	0.199254
$404$	0	0
$405$	3.00000	0.149071
$406$	0	0
$407$	21.0000	1.04093
$408$	0	0
$409$	−13.0000	−0.642809	−0.321404	−	0.946942i	$-0.604155\pi$
$409$	−13.0000	−0.642809	−0.321404	+	0.946942i	$0.604155\pi$
$410$	0	0
$411$	−9.00000	−0.443937
$412$	0	0
$413$	12.0000	0.590481
$414$	0	0
$415$	18.0000	0.883585
$416$	0	0
$417$	−4.00000	−0.195881
$418$	0	0
$419$	−3.00000	−0.146560	−0.0732798	−	0.997311i	$-0.523347\pi$
$419$	−3.00000	−0.146560	−0.0732798	+	0.997311i	$0.523347\pi$
$420$	0	0
$421$	−10.0000	−0.487370	−0.243685	−	0.969854i	$-0.578356\pi$
$421$	−10.0000	−0.487370	−0.243685	+	0.969854i	$0.578356\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−12.0000	−0.582086
$426$	0	0
$427$	1.00000	0.0483934
$428$	0	0
$429$	3.00000	0.144841
$430$	0	0
$431$	−18.0000	−0.867029	−0.433515	−	0.901146i	$-0.642727\pi$
$431$	−18.0000	−0.867029	−0.433515	+	0.901146i	$0.642727\pi$
$432$	0	0
$433$	−16.0000	−0.768911	−0.384455	−	0.923144i	$-0.625611\pi$
$433$	−16.0000	−0.768911	−0.384455	+	0.923144i	$0.625611\pi$
$434$	0	0
$435$	27.0000	1.29455
$436$	0	0
$437$	−63.0000	−3.01370
$438$	0	0
$439$	1.00000	0.0477274	0.0238637	−	0.999715i	$-0.492403\pi$
$439$	1.00000	0.0477274	0.0238637	+	0.999715i	$0.492403\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	−18.0000	−0.855206	−0.427603	−	0.903967i	$-0.640642\pi$
$443$	−18.0000	−0.855206	−0.427603	+	0.903967i	$0.640642\pi$
$444$	0	0
$445$	−18.0000	−0.853282
$446$	0	0
$447$	12.0000	0.567581
$448$	0	0
$449$	21.0000	0.991051	0.495526	−	0.868593i	$-0.334975\pi$
$449$	21.0000	0.991051	0.495526	+	0.868593i	$0.334975\pi$
$450$	0	0
$451$	−36.0000	−1.69517
$452$	0	0
$453$	17.0000	0.798730
$454$	0	0
$455$	−3.00000	−0.140642
$456$	0	0
$457$	−28.0000	−1.30978	−0.654892	−	0.755722i	$-0.727286\pi$
$457$	−28.0000	−1.30978	−0.654892	+	0.755722i	$0.727286\pi$
$458$	0	0
$459$	3.00000	0.140028
$460$	0	0
$461$	33.0000	1.53696	0.768482	−	0.639872i	$-0.221013\pi$
$461$	33.0000	1.53696	0.768482	+	0.639872i	$0.221013\pi$
$462$	0	0
$463$	−41.0000	−1.90543	−0.952716	−	0.303863i	$-0.901724\pi$
$463$	−41.0000	−1.90543	−0.952716	+	0.303863i	$0.901724\pi$
$464$	0	0
$465$	−12.0000	−0.556487
$466$	0	0
$467$	3.00000	0.138823	0.0694117	−	0.997588i	$-0.477888\pi$
$467$	3.00000	0.138823	0.0694117	+	0.997588i	$0.477888\pi$
$468$	0	0
$469$	14.0000	0.646460
$470$	0	0
$471$	13.0000	0.599008
$472$	0	0
$473$	−3.00000	−0.137940
$474$	0	0
$475$	28.0000	1.28473
$476$	0	0
$477$	−6.00000	−0.274721
$478$	0	0
$479$	15.0000	0.685367	0.342684	−	0.939451i	$-0.388664\pi$
$479$	15.0000	0.685367	0.342684	+	0.939451i	$0.388664\pi$
$480$	0	0
$481$	−7.00000	−0.319173
$482$	0	0
$483$	−9.00000	−0.409514
$484$	0	0
$485$	−30.0000	−1.36223
$486$	0	0
$487$	16.0000	0.725029	0.362515	−	0.931978i	$-0.381918\pi$
$487$	16.0000	0.725029	0.362515	+	0.931978i	$0.381918\pi$
$488$	0	0
$489$	2.00000	0.0904431
$490$	0	0
$491$	−30.0000	−1.35388	−0.676941	−	0.736038i	$-0.736695\pi$
$491$	−30.0000	−1.35388	−0.676941	+	0.736038i	$0.736695\pi$
$492$	0	0
$493$	27.0000	1.21602
$494$	0	0
$495$	−9.00000	−0.404520
$496$	0	0
$497$	12.0000	0.538274
$498$	0	0
$499$	−32.0000	−1.43252	−0.716258	−	0.697835i	$-0.754147\pi$
$499$	−32.0000	−1.43252	−0.716258	+	0.697835i	$0.754147\pi$
$500$	0	0
$501$	3.00000	0.134030
$502$	0	0
$503$	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
$506$	0	0
$507$	−1.00000	−0.0444116
$508$	0	0
$509$	9.00000	0.398918	0.199459	−	0.979906i	$-0.436082\pi$
$509$	9.00000	0.398918	0.199459	+	0.979906i	$0.436082\pi$
$510$	0	0
$511$	7.00000	0.309662
$512$	0	0
$513$	−7.00000	−0.309058
$514$	0	0
$515$	39.0000	1.71855
$516$	0	0
$517$	0	0
$518$	0	0
$519$	6.00000	0.263371
$520$	0	0
$521$	−33.0000	−1.44576	−0.722878	−	0.690976i	$-0.757181\pi$
$521$	−33.0000	−1.44576	−0.722878	+	0.690976i	$0.757181\pi$
$522$	0	0
$523$	16.0000	0.699631	0.349816	−	0.936819i	$-0.386244\pi$
$523$	16.0000	0.699631	0.349816	+	0.936819i	$0.386244\pi$
$524$	0	0
$525$	4.00000	0.174574
$526$	0	0
$527$	−12.0000	−0.522728
$528$	0	0
$529$	58.0000	2.52174
$530$	0	0
$531$	−12.0000	−0.520756
$532$	0	0
$533$	12.0000	0.519778
$534$	0	0
$535$	18.0000	0.778208
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−3.00000	−0.129219
$540$	0	0
$541$	11.0000	0.472927	0.236463	−	0.971640i	$-0.424012\pi$
$541$	11.0000	0.472927	0.236463	+	0.971640i	$0.424012\pi$
$542$	0	0
$543$	−2.00000	−0.0858282
$544$	0	0
$545$	33.0000	1.41356
$546$	0	0
$547$	28.0000	1.19719	0.598597	−	0.801050i	$-0.295725\pi$
$547$	28.0000	1.19719	0.598597	+	0.801050i	$0.295725\pi$
$548$	0	0
$549$	−1.00000	−0.0426790
$550$	0	0
$551$	−63.0000	−2.68389
$552$	0	0
$553$	−10.0000	−0.425243
$554$	0	0
$555$	21.0000	0.891400
$556$	0	0
$557$	0	0		−	1.00000i	$-0.5\pi$
$557$	0	0			1.00000i	$0.5\pi$
$558$	0	0
$559$	1.00000	0.0422955
$560$	0	0
$561$	−9.00000	−0.379980
$562$	0	0
$563$	15.0000	0.632175	0.316087	−	0.948730i	$-0.397631\pi$
$563$	15.0000	0.632175	0.316087	+	0.948730i	$0.397631\pi$
$564$	0	0
$565$	−54.0000	−2.27180
$566$	0	0
$567$	−1.00000	−0.0419961
$568$	0	0
$569$	6.00000	0.251533	0.125767	−	0.992060i	$-0.459861\pi$
$569$	6.00000	0.251533	0.125767	+	0.992060i	$0.459861\pi$
$570$	0	0
$571$	−32.0000	−1.33916	−0.669579	−	0.742741i	$-0.733526\pi$
$571$	−32.0000	−1.33916	−0.669579	+	0.742741i	$0.733526\pi$
$572$	0	0
$573$	−15.0000	−0.626634
$574$	0	0
$575$	−36.0000	−1.50130
$576$	0	0
$577$	38.0000	1.58196	0.790980	−	0.611842i	$-0.209571\pi$
$577$	38.0000	1.58196	0.790980	+	0.611842i	$0.209571\pi$
$578$	0	0
$579$	4.00000	0.166234
$580$	0	0
$581$	−6.00000	−0.248922
$582$	0	0
$583$	18.0000	0.745484
$584$	0	0
$585$	3.00000	0.124035
$586$	0	0
$587$	18.0000	0.742940	0.371470	−	0.928445i	$-0.378854\pi$
$587$	18.0000	0.742940	0.371470	+	0.928445i	$0.378854\pi$
$588$	0	0
$589$	28.0000	1.15372
$590$	0	0
$591$	12.0000	0.493614
$592$	0	0
$593$	6.00000	0.246390	0.123195	−	0.992382i	$-0.460686\pi$
$593$	6.00000	0.246390	0.123195	+	0.992382i	$0.460686\pi$
$594$	0	0
$595$	9.00000	0.368964
$596$	0	0
$597$	−7.00000	−0.286491
$598$	0	0
$599$	9.00000	0.367730	0.183865	−	0.982952i	$-0.441139\pi$
$599$	9.00000	0.367730	0.183865	+	0.982952i	$0.441139\pi$
$600$	0	0
$601$	8.00000	0.326327	0.163163	−	0.986599i	$-0.447830\pi$
$601$	8.00000	0.326327	0.163163	+	0.986599i	$0.447830\pi$
$602$	0	0
$603$	−14.0000	−0.570124
$604$	0	0
$605$	−6.00000	−0.243935
$606$	0	0
$607$	−23.0000	−0.933541	−0.466771	−	0.884378i	$-0.654583\pi$
$607$	−23.0000	−0.933541	−0.466771	+	0.884378i	$0.654583\pi$
$608$	0	0
$609$	−9.00000	−0.364698
$610$	0	0
$611$	0	0
$612$	0	0
$613$	11.0000	0.444286	0.222143	−	0.975014i	$-0.428695\pi$
$613$	11.0000	0.444286	0.222143	+	0.975014i	$0.428695\pi$
$614$	0	0
$615$	−36.0000	−1.45166
$616$	0	0
$617$	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
$619$	−17.0000	−0.683288	−0.341644	−	0.939829i	$-0.610984\pi$
$619$	−17.0000	−0.683288	−0.341644	+	0.939829i	$0.610984\pi$
$620$	0	0
$621$	9.00000	0.361158
$622$	0	0
$623$	6.00000	0.240385
$624$	0	0
$625$	−29.0000	−1.16000
$626$	0	0
$627$	21.0000	0.838659
$628$	0	0
$629$	21.0000	0.837325
$630$	0	0
$631$	−29.0000	−1.15447	−0.577236	−	0.816577i	$-0.695869\pi$
$631$	−29.0000	−1.15447	−0.577236	+	0.816577i	$0.695869\pi$
$632$	0	0
$633$	5.00000	0.198732
$634$	0	0
$635$	−6.00000	−0.238103
$636$	0	0
$637$	1.00000	0.0396214
$638$	0	0
$639$	−12.0000	−0.474713
$640$	0	0
$641$	18.0000	0.710957	0.355479	−	0.934684i	$-0.384318\pi$
$641$	18.0000	0.710957	0.355479	+	0.934684i	$0.384318\pi$
$642$	0	0
$643$	49.0000	1.93237	0.966186	−	0.257847i	$-0.0830131\pi$
$643$	49.0000	1.93237	0.966186	+	0.257847i	$0.0830131\pi$
$644$	0	0
$645$	−3.00000	−0.118125
$646$	0	0
$647$	0	0		−	1.00000i	$-0.5\pi$
$647$	0	0			1.00000i	$0.5\pi$
$648$	0	0
$649$	36.0000	1.41312
$650$	0	0
$651$	4.00000	0.156772
$652$	0	0
$653$	9.00000	0.352197	0.176099	−	0.984373i	$-0.443652\pi$
$653$	9.00000	0.352197	0.176099	+	0.984373i	$0.443652\pi$
$654$	0	0
$655$	45.0000	1.75830
$656$	0	0
$657$	−7.00000	−0.273096
$658$	0	0
$659$	−6.00000	−0.233727	−0.116863	−	0.993148i	$-0.537284\pi$
$659$	−6.00000	−0.233727	−0.116863	+	0.993148i	$0.537284\pi$
$660$	0	0
$661$	14.0000	0.544537	0.272268	−	0.962221i	$-0.412226\pi$
$661$	14.0000	0.544537	0.272268	+	0.962221i	$0.412226\pi$
$662$	0	0
$663$	3.00000	0.116510
$664$	0	0
$665$	−21.0000	−0.814345
$666$	0	0
$667$	81.0000	3.13633
$668$	0	0
$669$	26.0000	1.00522
$670$	0	0
$671$	3.00000	0.115814
$672$	0	0
$673$	−1.00000	−0.0385472	−0.0192736	−	0.999814i	$-0.506135\pi$
$673$	−1.00000	−0.0385472	−0.0192736	+	0.999814i	$0.506135\pi$
$674$	0	0
$675$	−4.00000	−0.153960
$676$	0	0
$677$	18.0000	0.691796	0.345898	−	0.938272i	$-0.387574\pi$
$677$	18.0000	0.691796	0.345898	+	0.938272i	$0.387574\pi$
$678$	0	0
$679$	10.0000	0.383765
$680$	0	0
$681$	−18.0000	−0.689761
$682$	0	0
$683$	−33.0000	−1.26271	−0.631355	−	0.775494i	$-0.717501\pi$
$683$	−33.0000	−1.26271	−0.631355	+	0.775494i	$0.717501\pi$
$684$	0	0
$685$	27.0000	1.03162
$686$	0	0
$687$	−14.0000	−0.534133
$688$	0	0
$689$	−6.00000	−0.228582
$690$	0	0
$691$	28.0000	1.06517	0.532585	−	0.846376i	$-0.321221\pi$
$691$	28.0000	1.06517	0.532585	+	0.846376i	$0.321221\pi$
$692$	0	0
$693$	3.00000	0.113961
$694$	0	0
$695$	12.0000	0.455186
$696$	0	0
$697$	−36.0000	−1.36360
$698$	0	0
$699$	−24.0000	−0.907763
$700$	0	0
$701$	−18.0000	−0.679851	−0.339925	−	0.940452i	$-0.610402\pi$
$701$	−18.0000	−0.679851	−0.339925	+	0.940452i	$0.610402\pi$
$702$	0	0
$703$	−49.0000	−1.84807
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−12.0000	−0.451306
$708$	0	0
$709$	−10.0000	−0.375558	−0.187779	−	0.982211i	$-0.560129\pi$
$709$	−10.0000	−0.375558	−0.187779	+	0.982211i	$0.560129\pi$
$710$	0	0
$711$	10.0000	0.375029
$712$	0	0
$713$	−36.0000	−1.34821
$714$	0	0
$715$	−9.00000	−0.336581
$716$	0	0
$717$	24.0000	0.896296
$718$	0	0
$719$	−18.0000	−0.671287	−0.335643	−	0.941989i	$-0.608954\pi$
$719$	−18.0000	−0.671287	−0.335643	+	0.941989i	$0.608954\pi$
$720$	0	0
$721$	−13.0000	−0.484145
$722$	0	0
$723$	10.0000	0.371904
$724$	0	0
$725$	−36.0000	−1.33701
$726$	0	0
$727$	1.00000	0.0370879	0.0185440	−	0.999828i	$-0.494097\pi$
$727$	1.00000	0.0370879	0.0185440	+	0.999828i	$0.494097\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−3.00000	−0.110959
$732$	0	0
$733$	32.0000	1.18195	0.590973	−	0.806691i	$-0.298744\pi$
$733$	32.0000	1.18195	0.590973	+	0.806691i	$0.298744\pi$
$734$	0	0
$735$	−3.00000	−0.110657
$736$	0	0
$737$	42.0000	1.54709
$738$	0	0
$739$	34.0000	1.25071	0.625355	−	0.780340i	$-0.284954\pi$
$739$	34.0000	1.25071	0.625355	+	0.780340i	$0.284954\pi$
$740$	0	0
$741$	−7.00000	−0.257151
$742$	0	0
$743$	−30.0000	−1.10059	−0.550297	−	0.834969i	$-0.685485\pi$
$743$	−30.0000	−1.10059	−0.550297	+	0.834969i	$0.685485\pi$
$744$	0	0
$745$	−36.0000	−1.31894
$746$	0	0
$747$	6.00000	0.219529
$748$	0	0
$749$	−6.00000	−0.219235
$750$	0	0
$751$	4.00000	0.145962	0.0729810	−	0.997333i	$-0.476749\pi$
$751$	4.00000	0.145962	0.0729810	+	0.997333i	$0.476749\pi$
$752$	0	0
$753$	21.0000	0.765283
$754$	0	0
$755$	−51.0000	−1.85608
$756$	0	0
$757$	38.0000	1.38113	0.690567	−	0.723269i	$-0.257361\pi$
$757$	38.0000	1.38113	0.690567	+	0.723269i	$0.257361\pi$
$758$	0	0
$759$	−27.0000	−0.980038
$760$	0	0
$761$	−12.0000	−0.435000	−0.217500	−	0.976060i	$-0.569790\pi$
$761$	−12.0000	−0.435000	−0.217500	+	0.976060i	$0.569790\pi$
$762$	0	0
$763$	−11.0000	−0.398227
$764$	0	0
$765$	−9.00000	−0.325396
$766$	0	0
$767$	−12.0000	−0.433295
$768$	0	0
$769$	−13.0000	−0.468792	−0.234396	−	0.972141i	$-0.575311\pi$
$769$	−13.0000	−0.468792	−0.234396	+	0.972141i	$0.575311\pi$
$770$	0	0
$771$	18.0000	0.648254
$772$	0	0
$773$	33.0000	1.18693	0.593464	−	0.804861i	$-0.297760\pi$
$773$	33.0000	1.18693	0.593464	+	0.804861i	$0.297760\pi$
$774$	0	0
$775$	16.0000	0.574737
$776$	0	0
$777$	−7.00000	−0.251124
$778$	0	0
$779$	84.0000	3.00961
$780$	0	0
$781$	36.0000	1.28818
$782$	0	0
$783$	9.00000	0.321634
$784$	0	0
$785$	−39.0000	−1.39197
$786$	0	0
$787$	31.0000	1.10503	0.552515	−	0.833503i	$-0.313668\pi$
$787$	31.0000	1.10503	0.552515	+	0.833503i	$0.313668\pi$
$788$	0	0
$789$	12.0000	0.427211
$790$	0	0
$791$	18.0000	0.640006
$792$	0	0
$793$	−1.00000	−0.0355110
$794$	0	0
$795$	18.0000	0.638394
$796$	0	0
$797$	36.0000	1.27519	0.637593	−	0.770374i	$-0.279930\pi$
$797$	36.0000	1.27519	0.637593	+	0.770374i	$0.279930\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	−6.00000	−0.212000
$802$	0	0
$803$	21.0000	0.741074
$804$	0	0
$805$	27.0000	0.951625
$806$	0	0
$807$	−6.00000	−0.211210
$808$	0	0
$809$	54.0000	1.89854	0.949269	−	0.314464i	$-0.101825\pi$
$809$	54.0000	1.89854	0.949269	+	0.314464i	$0.101825\pi$
$810$	0	0
$811$	43.0000	1.50993	0.754967	−	0.655763i	$-0.227653\pi$
$811$	43.0000	1.50993	0.754967	+	0.655763i	$0.227653\pi$
$812$	0	0
$813$	2.00000	0.0701431
$814$	0	0
$815$	−6.00000	−0.210171
$816$	0	0
$817$	7.00000	0.244899
$818$	0	0
$819$	−1.00000	−0.0349428
$820$	0	0
$821$	24.0000	0.837606	0.418803	−	0.908077i	$-0.362450\pi$
$821$	24.0000	0.837606	0.418803	+	0.908077i	$0.362450\pi$
$822$	0	0
$823$	40.0000	1.39431	0.697156	−	0.716919i	$-0.254448\pi$
$823$	40.0000	1.39431	0.697156	+	0.716919i	$0.254448\pi$
$824$	0	0
$825$	12.0000	0.417786
$826$	0	0
$827$	−27.0000	−0.938882	−0.469441	−	0.882964i	$-0.655545\pi$
$827$	−27.0000	−0.938882	−0.469441	+	0.882964i	$0.655545\pi$
$828$	0	0
$829$	−25.0000	−0.868286	−0.434143	−	0.900844i	$-0.642949\pi$
$829$	−25.0000	−0.868286	−0.434143	+	0.900844i	$0.642949\pi$
$830$	0	0
$831$	−26.0000	−0.901930
$832$	0	0
$833$	−3.00000	−0.103944
$834$	0	0
$835$	−9.00000	−0.311458
$836$	0	0
$837$	−4.00000	−0.138260
$838$	0	0
$839$	12.0000	0.414286	0.207143	−	0.978311i	$-0.433583\pi$
$839$	12.0000	0.414286	0.207143	+	0.978311i	$0.433583\pi$
$840$	0	0
$841$	52.0000	1.79310
$842$	0	0
$843$	6.00000	0.206651
$844$	0	0
$845$	3.00000	0.103203
$846$	0	0
$847$	2.00000	0.0687208
$848$	0	0
$849$	−4.00000	−0.137280
$850$	0	0
$851$	63.0000	2.15961
$852$	0	0
$853$	44.0000	1.50653	0.753266	−	0.657716i	$-0.228477\pi$
$853$	44.0000	1.50653	0.753266	+	0.657716i	$0.228477\pi$
$854$	0	0
$855$	21.0000	0.718185
$856$	0	0
$857$	42.0000	1.43469	0.717346	−	0.696717i	$-0.245357\pi$
$857$	42.0000	1.43469	0.717346	+	0.696717i	$0.245357\pi$
$858$	0	0
$859$	4.00000	0.136478	0.0682391	−	0.997669i	$-0.478262\pi$
$859$	4.00000	0.136478	0.0682391	+	0.997669i	$0.478262\pi$
$860$	0	0
$861$	12.0000	0.408959
$862$	0	0
$863$	−6.00000	−0.204242	−0.102121	−	0.994772i	$-0.532563\pi$
$863$	−6.00000	−0.204242	−0.102121	+	0.994772i	$0.532563\pi$
$864$	0	0
$865$	−18.0000	−0.612018
$866$	0	0
$867$	8.00000	0.271694
$868$	0	0
$869$	−30.0000	−1.01768
$870$	0	0
$871$	−14.0000	−0.474372
$872$	0	0
$873$	−10.0000	−0.338449
$874$	0	0
$875$	3.00000	0.101419
$876$	0	0
$877$	14.0000	0.472746	0.236373	−	0.971662i	$-0.424041\pi$
$877$	14.0000	0.472746	0.236373	+	0.971662i	$0.424041\pi$
$878$	0	0
$879$	18.0000	0.607125
$880$	0	0
$881$	−3.00000	−0.101073	−0.0505363	−	0.998722i	$-0.516093\pi$
$881$	−3.00000	−0.101073	−0.0505363	+	0.998722i	$0.516093\pi$
$882$	0	0
$883$	25.0000	0.841317	0.420658	−	0.907219i	$-0.361799\pi$
$883$	25.0000	0.841317	0.420658	+	0.907219i	$0.361799\pi$
$884$	0	0
$885$	36.0000	1.21013
$886$	0	0
$887$	−36.0000	−1.20876	−0.604381	−	0.796696i	$-0.706579\pi$
$887$	−36.0000	−1.20876	−0.604381	+	0.796696i	$0.706579\pi$
$888$	0	0
$889$	2.00000	0.0670778
$890$	0	0
$891$	−3.00000	−0.100504
$892$	0	0
$893$	0	0
$894$	0	0
$895$	0	0
$896$	0	0
$897$	9.00000	0.300501
$898$	0	0
$899$	−36.0000	−1.20067
$900$	0	0
$901$	18.0000	0.599667
$902$	0	0
$903$	1.00000	0.0332779
$904$	0	0
$905$	6.00000	0.199447
$906$	0	0
$907$	28.0000	0.929725	0.464862	−	0.885383i	$-0.346104\pi$
$907$	28.0000	0.929725	0.464862	+	0.885383i	$0.346104\pi$
$908$	0	0
$909$	12.0000	0.398015
$910$	0	0
$911$	33.0000	1.09334	0.546669	−	0.837349i	$-0.315895\pi$
$911$	33.0000	1.09334	0.546669	+	0.837349i	$0.315895\pi$
$912$	0	0
$913$	−18.0000	−0.595713
$914$	0	0
$915$	3.00000	0.0991769
$916$	0	0
$917$	−15.0000	−0.495344
$918$	0	0
$919$	34.0000	1.12156	0.560778	−	0.827966i	$-0.310502\pi$
$919$	34.0000	1.12156	0.560778	+	0.827966i	$0.310502\pi$
$920$	0	0
$921$	−16.0000	−0.527218
$922$	0	0
$923$	−12.0000	−0.394985
$924$	0	0
$925$	−28.0000	−0.920634
$926$	0	0
$927$	13.0000	0.426976
$928$	0	0
$929$	12.0000	0.393707	0.196854	−	0.980433i	$-0.436928\pi$
$929$	12.0000	0.393707	0.196854	+	0.980433i	$0.436928\pi$
$930$	0	0
$931$	7.00000	0.229416
$932$	0	0
$933$	18.0000	0.589294
$934$	0	0
$935$	27.0000	0.882994
$936$	0	0
$937$	−34.0000	−1.11073	−0.555366	−	0.831606i	$-0.687422\pi$
$937$	−34.0000	−1.11073	−0.555366	+	0.831606i	$0.687422\pi$
$938$	0	0
$939$	−8.00000	−0.261070
$940$	0	0
$941$	18.0000	0.586783	0.293392	−	0.955992i	$-0.405216\pi$
$941$	18.0000	0.586783	0.293392	+	0.955992i	$0.405216\pi$
$942$	0	0
$943$	−108.000	−3.51696
$944$	0	0
$945$	3.00000	0.0975900
$946$	0	0
$947$	57.0000	1.85225	0.926126	−	0.377215i	$-0.123118\pi$
$947$	57.0000	1.85225	0.926126	+	0.377215i	$0.123118\pi$
$948$	0	0
$949$	−7.00000	−0.227230
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−24.0000	−0.777436	−0.388718	−	0.921357i	$-0.627082\pi$
$953$	−24.0000	−0.777436	−0.388718	+	0.921357i	$0.627082\pi$
$954$	0	0
$955$	45.0000	1.45617
$956$	0	0
$957$	−27.0000	−0.872786
$958$	0	0
$959$	−9.00000	−0.290625
$960$	0	0
$961$	−15.0000	−0.483871
$962$	0	0
$963$	6.00000	0.193347
$964$	0	0
$965$	−12.0000	−0.386294
$966$	0	0
$967$	13.0000	0.418052	0.209026	−	0.977910i	$-0.432971\pi$
$967$	13.0000	0.418052	0.209026	+	0.977910i	$0.432971\pi$
$968$	0	0
$969$	21.0000	0.674617
$970$	0	0
$971$	0	0		−	1.00000i	$-0.5\pi$
$971$	0	0			1.00000i	$0.5\pi$
$972$	0	0
$973$	−4.00000	−0.128234
$974$	0	0
$975$	−4.00000	−0.128103
$976$	0	0
$977$	15.0000	0.479893	0.239946	−	0.970786i	$-0.422870\pi$
$977$	15.0000	0.479893	0.239946	+	0.970786i	$0.422870\pi$
$978$	0	0
$979$	18.0000	0.575282
$980$	0	0
$981$	11.0000	0.351203
$982$	0	0
$983$	39.0000	1.24391	0.621953	−	0.783054i	$-0.286339\pi$
$983$	39.0000	1.24391	0.621953	+	0.783054i	$0.286339\pi$
$984$	0	0
$985$	−36.0000	−1.14706
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−9.00000	−0.286183
$990$	0	0
$991$	−38.0000	−1.20711	−0.603555	−	0.797321i	$-0.706250\pi$
$991$	−38.0000	−1.20711	−0.603555	+	0.797321i	$0.706250\pi$
$992$	0	0
$993$	−10.0000	−0.317340
$994$	0	0
$995$	21.0000	0.665745
$996$	0	0
$997$	−46.0000	−1.45683	−0.728417	−	0.685134i	$-0.759744\pi$
$997$	−46.0000	−1.45683	−0.728417	+	0.685134i	$0.759744\pi$
$998$	0	0
$999$	7.00000	0.221470

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4368.2.a.l.1.1		1
4.3	odd	2		546.2.a.d.1.1	✓	1
12.11	even	2		1638.2.a.l.1.1		1
28.27	even	2		3822.2.a.a.1.1		1
52.51	odd	2		7098.2.a.w.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
546.2.a.d.1.1	✓	1	4.3	odd	2
1638.2.a.l.1.1		1	12.11	even	2
3822.2.a.a.1.1		1	28.27	even	2
4368.2.a.l.1.1		1	1.1	even	1	trivial
7098.2.a.w.1.1		1	52.51	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

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