LMFDB - Embedded newform 6288.2.a.c.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6288,2,Mod(1,6288)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6288, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("6288.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6288 = 2^{4} \cdot 3 \cdot 131 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6288.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:	$50.2099327910$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 786)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	6288.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{3} -2.00000 q^{5} +2.00000 q^{7} +1.00000 q^{9} +O(q^{10})$

$q-1.00000 q^{3} -2.00000 q^{5} +2.00000 q^{7} +1.00000 q^{9} -3.00000 q^{11} -5.00000 q^{13} +2.00000 q^{15} +7.00000 q^{17} +5.00000 q^{19} -2.00000 q^{21} +4.00000 q^{23} -1.00000 q^{25} -1.00000 q^{27} -3.00000 q^{29} +7.00000 q^{31} +3.00000 q^{33} -4.00000 q^{35} -8.00000 q^{37} +5.00000 q^{39} -12.0000 q^{41} +2.00000 q^{43} -2.00000 q^{45} -8.00000 q^{47} -3.00000 q^{49} -7.00000 q^{51} +4.00000 q^{53} +6.00000 q^{55} -5.00000 q^{57} +3.00000 q^{59} +13.0000 q^{61} +2.00000 q^{63} +10.0000 q^{65} -8.00000 q^{67} -4.00000 q^{69} +16.0000 q^{71} -2.00000 q^{73} +1.00000 q^{75} -6.00000 q^{77} +1.00000 q^{81} -6.00000 q^{83} -14.0000 q^{85} +3.00000 q^{87} -14.0000 q^{89} -10.0000 q^{91} -7.00000 q^{93} -10.0000 q^{95} +12.0000 q^{97} -3.00000 q^{99} +O(q^{100})$

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	−2.00000	−0.894427	−0.447214	−	0.894427i	$-0.647584\pi$
$5$	−2.00000	−0.894427	−0.447214	+	0.894427i	$0.647584\pi$
$6$	0	0
$7$	2.00000	0.755929	0.377964	−	0.925820i	$-0.376624\pi$
$7$	2.00000	0.755929	0.377964	+	0.925820i	$0.376624\pi$
$8$	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
$13$	−5.00000	−1.38675	−0.693375	−	0.720577i	$-0.743877\pi$
$13$	−5.00000	−1.38675	−0.693375	+	0.720577i	$0.743877\pi$
$14$	0	0
$15$	2.00000	0.516398
$16$	0	0
$17$	7.00000	1.69775	0.848875	−	0.528594i	$-0.177281\pi$
$17$	7.00000	1.69775	0.848875	+	0.528594i	$0.177281\pi$
$18$	0	0
$19$	5.00000	1.14708	0.573539	−	0.819178i	$-0.305570\pi$
$19$	5.00000	1.14708	0.573539	+	0.819178i	$0.305570\pi$
$20$	0	0
$21$	−2.00000	−0.436436
$22$	0	0
$23$	4.00000	0.834058	0.417029	−	0.908893i	$-0.363071\pi$
$23$	4.00000	0.834058	0.417029	+	0.908893i	$0.363071\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−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
$31$	7.00000	1.25724	0.628619	−	0.777714i	$-0.283621\pi$
$31$	7.00000	1.25724	0.628619	+	0.777714i	$0.283621\pi$
$32$	0	0
$33$	3.00000	0.522233
$34$	0	0
$35$	−4.00000	−0.676123
$36$	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
$39$	5.00000	0.800641
$40$	0	0
$41$	−12.0000	−1.87409	−0.937043	−	0.349215i	$-0.886448\pi$
$41$	−12.0000	−1.87409	−0.937043	+	0.349215i	$0.886448\pi$
$42$	0	0
$43$	2.00000	0.304997	0.152499	−	0.988304i	$-0.451268\pi$
$43$	2.00000	0.304997	0.152499	+	0.988304i	$0.451268\pi$
$44$	0	0
$45$	−2.00000	−0.298142
$46$	0	0
$47$	−8.00000	−1.16692	−0.583460	−	0.812142i	$-0.698301\pi$
$47$	−8.00000	−1.16692	−0.583460	+	0.812142i	$0.698301\pi$
$48$	0	0
$49$	−3.00000	−0.428571
$50$	0	0
$51$	−7.00000	−0.980196
$52$	0	0
$53$	4.00000	0.549442	0.274721	−	0.961524i	$-0.411414\pi$
$53$	4.00000	0.549442	0.274721	+	0.961524i	$0.411414\pi$
$54$	0	0
$55$	6.00000	0.809040
$56$	0	0
$57$	−5.00000	−0.662266
$58$	0	0
$59$	3.00000	0.390567	0.195283	−	0.980747i	$-0.437437\pi$
$59$	3.00000	0.390567	0.195283	+	0.980747i	$0.437437\pi$
$60$	0	0
$61$	13.0000	1.66448	0.832240	−	0.554416i	$-0.187058\pi$
$61$	13.0000	1.66448	0.832240	+	0.554416i	$0.187058\pi$
$62$	0	0
$63$	2.00000	0.251976
$64$	0	0
$65$	10.0000	1.24035
$66$	0	0
$67$	−8.00000	−0.977356	−0.488678	−	0.872464i	$-0.662521\pi$
$67$	−8.00000	−0.977356	−0.488678	+	0.872464i	$0.662521\pi$
$68$	0	0
$69$	−4.00000	−0.481543
$70$	0	0
$71$	16.0000	1.89885	0.949425	−	0.313993i	$-0.101667\pi$
$71$	16.0000	1.89885	0.949425	+	0.313993i	$0.101667\pi$
$72$	0	0
$73$	−2.00000	−0.234082	−0.117041	−	0.993127i	$-0.537341\pi$
$73$	−2.00000	−0.234082	−0.117041	+	0.993127i	$0.537341\pi$
$74$	0	0
$75$	1.00000	0.115470
$76$	0	0
$77$	−6.00000	−0.683763
$78$	0	0
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−6.00000	−0.658586	−0.329293	−	0.944228i	$-0.606810\pi$
$83$	−6.00000	−0.658586	−0.329293	+	0.944228i	$0.606810\pi$
$84$	0	0
$85$	−14.0000	−1.51851
$86$	0	0
$87$	3.00000	0.321634
$88$	0	0
$89$	−14.0000	−1.48400	−0.741999	−	0.670402i	$-0.766122\pi$
$89$	−14.0000	−1.48400	−0.741999	+	0.670402i	$0.766122\pi$
$90$	0	0
$91$	−10.0000	−1.04828
$92$	0	0
$93$	−7.00000	−0.725866
$94$	0	0
$95$	−10.0000	−1.02598
$96$	0	0
$97$	12.0000	1.21842	0.609208	−	0.793011i	$-0.291488\pi$
$97$	12.0000	1.21842	0.609208	+	0.793011i	$0.291488\pi$
$98$	0	0
$99$	−3.00000	−0.301511
$100$	0	0
$101$	−18.0000	−1.79107	−0.895533	−	0.444994i	$-0.853206\pi$
$101$	−18.0000	−1.79107	−0.895533	+	0.444994i	$0.853206\pi$
$102$	0	0
$103$	0	0		−	1.00000i	$-0.5\pi$
$103$	0	0			1.00000i	$0.5\pi$
$104$	0	0
$105$	4.00000	0.390360
$106$	0	0
$107$	−5.00000	−0.483368	−0.241684	−	0.970355i	$-0.577700\pi$
$107$	−5.00000	−0.483368	−0.241684	+	0.970355i	$0.577700\pi$
$108$	0	0
$109$	3.00000	0.287348	0.143674	−	0.989625i	$-0.454108\pi$
$109$	3.00000	0.287348	0.143674	+	0.989625i	$0.454108\pi$
$110$	0	0
$111$	8.00000	0.759326
$112$	0	0
$113$	12.0000	1.12887	0.564433	−	0.825479i	$-0.309095\pi$
$113$	12.0000	1.12887	0.564433	+	0.825479i	$0.309095\pi$
$114$	0	0
$115$	−8.00000	−0.746004
$116$	0	0
$117$	−5.00000	−0.462250
$118$	0	0
$119$	14.0000	1.28338
$120$	0	0
$121$	−2.00000	−0.181818
$122$	0	0
$123$	12.0000	1.08200
$124$	0	0
$125$	12.0000	1.07331
$126$	0	0
$127$	19.0000	1.68598	0.842989	−	0.537931i	$-0.180794\pi$
$127$	19.0000	1.68598	0.842989	+	0.537931i	$0.180794\pi$
$128$	0	0
$129$	−2.00000	−0.176090
$130$	0	0
$131$	−1.00000	−0.0873704
$131$	−1.00000	−0.0873704
$132$	0	0
$133$	10.0000	0.867110
$134$	0	0
$135$	2.00000	0.172133
$136$	0	0
$137$	5.00000	0.427179	0.213589	−	0.976924i	$-0.431485\pi$
$137$	5.00000	0.427179	0.213589	+	0.976924i	$0.431485\pi$
$138$	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
$141$	8.00000	0.673722
$142$	0	0
$143$	15.0000	1.25436
$144$	0	0
$145$	6.00000	0.498273
$146$	0	0
$147$	3.00000	0.247436
$148$	0	0
$149$	21.0000	1.72039	0.860194	−	0.509968i	$-0.170343\pi$
$149$	21.0000	1.72039	0.860194	+	0.509968i	$0.170343\pi$
$150$	0	0
$151$	20.0000	1.62758	0.813788	−	0.581161i	$-0.197401\pi$
$151$	20.0000	1.62758	0.813788	+	0.581161i	$0.197401\pi$
$152$	0	0
$153$	7.00000	0.565916
$154$	0	0
$155$	−14.0000	−1.12451
$156$	0	0
$157$	18.0000	1.43656	0.718278	−	0.695756i	$-0.244931\pi$
$157$	18.0000	1.43656	0.718278	+	0.695756i	$0.244931\pi$
$158$	0	0
$159$	−4.00000	−0.317221
$160$	0	0
$161$	8.00000	0.630488
$162$	0	0
$163$	3.00000	0.234978	0.117489	−	0.993074i	$-0.462515\pi$
$163$	3.00000	0.234978	0.117489	+	0.993074i	$0.462515\pi$
$164$	0	0
$165$	−6.00000	−0.467099
$166$	0	0
$167$	−11.0000	−0.851206	−0.425603	−	0.904910i	$-0.639938\pi$
$167$	−11.0000	−0.851206	−0.425603	+	0.904910i	$0.639938\pi$
$168$	0	0
$169$	12.0000	0.923077
$170$	0	0
$171$	5.00000	0.382360
$172$	0	0
$173$	1.00000	0.0760286	0.0380143	−	0.999277i	$-0.487897\pi$
$173$	1.00000	0.0760286	0.0380143	+	0.999277i	$0.487897\pi$
$174$	0	0
$175$	−2.00000	−0.151186
$176$	0	0
$177$	−3.00000	−0.225494
$178$	0	0
$179$	−20.0000	−1.49487	−0.747435	−	0.664335i	$-0.768715\pi$
$179$	−20.0000	−1.49487	−0.747435	+	0.664335i	$0.768715\pi$
$180$	0	0
$181$	−4.00000	−0.297318	−0.148659	−	0.988889i	$-0.547496\pi$
$181$	−4.00000	−0.297318	−0.148659	+	0.988889i	$0.547496\pi$
$182$	0	0
$183$	−13.0000	−0.960988
$184$	0	0
$185$	16.0000	1.17634
$186$	0	0
$187$	−21.0000	−1.53567
$188$	0	0
$189$	−2.00000	−0.145479
$190$	0	0
$191$	−13.0000	−0.940647	−0.470323	−	0.882494i	$-0.655863\pi$
$191$	−13.0000	−0.940647	−0.470323	+	0.882494i	$0.655863\pi$
$192$	0	0
$193$	11.0000	0.791797	0.395899	−	0.918294i	$-0.370433\pi$
$193$	11.0000	0.791797	0.395899	+	0.918294i	$0.370433\pi$
$194$	0	0
$195$	−10.0000	−0.716115
$196$	0	0
$197$	6.00000	0.427482	0.213741	−	0.976890i	$-0.431435\pi$
$197$	6.00000	0.427482	0.213741	+	0.976890i	$0.431435\pi$
$198$	0	0
$199$	−13.0000	−0.921546	−0.460773	−	0.887518i	$-0.652428\pi$
$199$	−13.0000	−0.921546	−0.460773	+	0.887518i	$0.652428\pi$
$200$	0	0
$201$	8.00000	0.564276
$202$	0	0
$203$	−6.00000	−0.421117
$204$	0	0
$205$	24.0000	1.67623
$206$	0	0
$207$	4.00000	0.278019
$208$	0	0
$209$	−15.0000	−1.03757
$210$	0	0
$211$	10.0000	0.688428	0.344214	−	0.938891i	$-0.388145\pi$
$211$	10.0000	0.688428	0.344214	+	0.938891i	$0.388145\pi$
$212$	0	0
$213$	−16.0000	−1.09630
$214$	0	0
$215$	−4.00000	−0.272798
$216$	0	0
$217$	14.0000	0.950382
$218$	0	0
$219$	2.00000	0.135147
$220$	0	0
$221$	−35.0000	−2.35435
$222$	0	0
$223$	17.0000	1.13840	0.569202	−	0.822198i	$-0.307252\pi$
$223$	17.0000	1.13840	0.569202	+	0.822198i	$0.307252\pi$
$224$	0	0
$225$	−1.00000	−0.0666667
$226$	0	0
$227$	26.0000	1.72568	0.862840	−	0.505477i	$-0.168683\pi$
$227$	26.0000	1.72568	0.862840	+	0.505477i	$0.168683\pi$
$228$	0	0
$229$	4.00000	0.264327	0.132164	−	0.991228i	$-0.457808\pi$
$229$	4.00000	0.264327	0.132164	+	0.991228i	$0.457808\pi$
$230$	0	0
$231$	6.00000	0.394771
$232$	0	0
$233$	−12.0000	−0.786146	−0.393073	−	0.919507i	$-0.628588\pi$
$233$	−12.0000	−0.786146	−0.393073	+	0.919507i	$0.628588\pi$
$234$	0	0
$235$	16.0000	1.04372
$236$	0	0
$237$	0	0
$238$	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
$241$	0	0		−	1.00000i	$-0.5\pi$
$241$	0	0			1.00000i	$0.5\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	6.00000	0.383326
$246$	0	0
$247$	−25.0000	−1.59071
$248$	0	0
$249$	6.00000	0.380235
$250$	0	0
$251$	18.0000	1.13615	0.568075	−	0.822977i	$-0.307688\pi$
$251$	18.0000	1.13615	0.568075	+	0.822977i	$0.307688\pi$
$252$	0	0
$253$	−12.0000	−0.754434
$254$	0	0
$255$	14.0000	0.876714
$256$	0	0
$257$	−6.00000	−0.374270	−0.187135	−	0.982334i	$-0.559920\pi$
$257$	−6.00000	−0.374270	−0.187135	+	0.982334i	$0.559920\pi$
$258$	0	0
$259$	−16.0000	−0.994192
$260$	0	0
$261$	−3.00000	−0.185695
$262$	0	0
$263$	−21.0000	−1.29492	−0.647458	−	0.762101i	$-0.724168\pi$
$263$	−21.0000	−1.29492	−0.647458	+	0.762101i	$0.724168\pi$
$264$	0	0
$265$	−8.00000	−0.491436
$266$	0	0
$267$	14.0000	0.856786
$268$	0	0
$269$	16.0000	0.975537	0.487769	−	0.872973i	$-0.337811\pi$
$269$	16.0000	0.975537	0.487769	+	0.872973i	$0.337811\pi$
$270$	0	0
$271$	20.0000	1.21491	0.607457	−	0.794353i	$-0.292190\pi$
$271$	20.0000	1.21491	0.607457	+	0.794353i	$0.292190\pi$
$272$	0	0
$273$	10.0000	0.605228
$274$	0	0
$275$	3.00000	0.180907
$276$	0	0
$277$	19.0000	1.14160	0.570800	−	0.821089i	$-0.306633\pi$
$277$	19.0000	1.14160	0.570800	+	0.821089i	$0.306633\pi$
$278$	0	0
$279$	7.00000	0.419079
$280$	0	0
$281$	6.00000	0.357930	0.178965	−	0.983855i	$-0.442725\pi$
$281$	6.00000	0.357930	0.178965	+	0.983855i	$0.442725\pi$
$282$	0	0
$283$	26.0000	1.54554	0.772770	−	0.634686i	$-0.218871\pi$
$283$	26.0000	1.54554	0.772770	+	0.634686i	$0.218871\pi$
$284$	0	0
$285$	10.0000	0.592349
$286$	0	0
$287$	−24.0000	−1.41668
$288$	0	0
$289$	32.0000	1.88235
$290$	0	0
$291$	−12.0000	−0.703452
$292$	0	0
$293$	−15.0000	−0.876309	−0.438155	−	0.898900i	$-0.644368\pi$
$293$	−15.0000	−0.876309	−0.438155	+	0.898900i	$0.644368\pi$
$294$	0	0
$295$	−6.00000	−0.349334
$296$	0	0
$297$	3.00000	0.174078
$298$	0	0
$299$	−20.0000	−1.15663
$300$	0	0
$301$	4.00000	0.230556
$302$	0	0
$303$	18.0000	1.03407
$304$	0	0
$305$	−26.0000	−1.48876
$306$	0	0
$307$	26.0000	1.48390	0.741949	−	0.670456i	$-0.233902\pi$
$307$	26.0000	1.48390	0.741949	+	0.670456i	$0.233902\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	7.00000	0.396934	0.198467	−	0.980108i	$-0.436404\pi$
$311$	7.00000	0.396934	0.198467	+	0.980108i	$0.436404\pi$
$312$	0	0
$313$	24.0000	1.35656	0.678280	−	0.734803i	$-0.262726\pi$
$313$	24.0000	1.35656	0.678280	+	0.734803i	$0.262726\pi$
$314$	0	0
$315$	−4.00000	−0.225374
$316$	0	0
$317$	2.00000	0.112331	0.0561656	−	0.998421i	$-0.482113\pi$
$317$	2.00000	0.112331	0.0561656	+	0.998421i	$0.482113\pi$
$318$	0	0
$319$	9.00000	0.503903
$320$	0	0
$321$	5.00000	0.279073
$322$	0	0
$323$	35.0000	1.94745
$324$	0	0
$325$	5.00000	0.277350
$326$	0	0
$327$	−3.00000	−0.165900
$328$	0	0
$329$	−16.0000	−0.882109
$330$	0	0
$331$	11.0000	0.604615	0.302307	−	0.953211i	$-0.402243\pi$
$331$	11.0000	0.604615	0.302307	+	0.953211i	$0.402243\pi$
$332$	0	0
$333$	−8.00000	−0.438397
$334$	0	0
$335$	16.0000	0.874173
$336$	0	0
$337$	−31.0000	−1.68868	−0.844339	−	0.535810i	$-0.820006\pi$
$337$	−31.0000	−1.68868	−0.844339	+	0.535810i	$0.820006\pi$
$338$	0	0
$339$	−12.0000	−0.651751
$340$	0	0
$341$	−21.0000	−1.13721
$342$	0	0
$343$	−20.0000	−1.07990
$344$	0	0
$345$	8.00000	0.430706
$346$	0	0
$347$	−6.00000	−0.322097	−0.161048	−	0.986947i	$-0.551488\pi$
$347$	−6.00000	−0.322097	−0.161048	+	0.986947i	$0.551488\pi$
$348$	0	0
$349$	10.0000	0.535288	0.267644	−	0.963518i	$-0.413755\pi$
$349$	10.0000	0.535288	0.267644	+	0.963518i	$0.413755\pi$
$350$	0	0
$351$	5.00000	0.266880
$352$	0	0
$353$	−4.00000	−0.212899	−0.106449	−	0.994318i	$-0.533948\pi$
$353$	−4.00000	−0.212899	−0.106449	+	0.994318i	$0.533948\pi$
$354$	0	0
$355$	−32.0000	−1.69838
$356$	0	0
$357$	−14.0000	−0.740959
$358$	0	0
$359$	18.0000	0.950004	0.475002	−	0.879985i	$-0.342447\pi$
$359$	18.0000	0.950004	0.475002	+	0.879985i	$0.342447\pi$
$360$	0	0
$361$	6.00000	0.315789
$362$	0	0
$363$	2.00000	0.104973
$364$	0	0
$365$	4.00000	0.209370
$366$	0	0
$367$	20.0000	1.04399	0.521996	−	0.852948i	$-0.325188\pi$
$367$	20.0000	1.04399	0.521996	+	0.852948i	$0.325188\pi$
$368$	0	0
$369$	−12.0000	−0.624695
$370$	0	0
$371$	8.00000	0.415339
$372$	0	0
$373$	−22.0000	−1.13912	−0.569558	−	0.821951i	$-0.692886\pi$
$373$	−22.0000	−1.13912	−0.569558	+	0.821951i	$0.692886\pi$
$374$	0	0
$375$	−12.0000	−0.619677
$376$	0	0
$377$	15.0000	0.772539
$378$	0	0
$379$	−14.0000	−0.719132	−0.359566	−	0.933120i	$-0.617075\pi$
$379$	−14.0000	−0.719132	−0.359566	+	0.933120i	$0.617075\pi$
$380$	0	0
$381$	−19.0000	−0.973399
$382$	0	0
$383$	23.0000	1.17525	0.587623	−	0.809135i	$-0.300064\pi$
$383$	23.0000	1.17525	0.587623	+	0.809135i	$0.300064\pi$
$384$	0	0
$385$	12.0000	0.611577
$386$	0	0
$387$	2.00000	0.101666
$388$	0	0
$389$	31.0000	1.57176	0.785881	−	0.618378i	$-0.212210\pi$
$389$	31.0000	1.57176	0.785881	+	0.618378i	$0.212210\pi$
$390$	0	0
$391$	28.0000	1.41602
$392$	0	0
$393$	1.00000	0.0504433
$394$	0	0
$395$	0	0
$396$	0	0
$397$	2.00000	0.100377	0.0501886	−	0.998740i	$-0.484018\pi$
$397$	2.00000	0.100377	0.0501886	+	0.998740i	$0.484018\pi$
$398$	0	0
$399$	−10.0000	−0.500626
$400$	0	0
$401$	27.0000	1.34832	0.674158	−	0.738587i	$-0.264507\pi$
$401$	27.0000	1.34832	0.674158	+	0.738587i	$0.264507\pi$
$402$	0	0
$403$	−35.0000	−1.74347
$404$	0	0
$405$	−2.00000	−0.0993808
$406$	0	0
$407$	24.0000	1.18964
$408$	0	0
$409$	−22.0000	−1.08783	−0.543915	−	0.839140i	$-0.683059\pi$
$409$	−22.0000	−1.08783	−0.543915	+	0.839140i	$0.683059\pi$
$410$	0	0
$411$	−5.00000	−0.246632
$412$	0	0
$413$	6.00000	0.295241
$414$	0	0
$415$	12.0000	0.589057
$416$	0	0
$417$	−4.00000	−0.195881
$418$	0	0
$419$	−12.0000	−0.586238	−0.293119	−	0.956076i	$-0.594693\pi$
$419$	−12.0000	−0.586238	−0.293119	+	0.956076i	$0.594693\pi$
$420$	0	0
$421$	37.0000	1.80327	0.901635	−	0.432498i	$-0.142368\pi$
$421$	37.0000	1.80327	0.901635	+	0.432498i	$0.142368\pi$
$422$	0	0
$423$	−8.00000	−0.388973
$424$	0	0
$425$	−7.00000	−0.339550
$426$	0	0
$427$	26.0000	1.25823
$428$	0	0
$429$	−15.0000	−0.724207
$430$	0	0
$431$	12.0000	0.578020	0.289010	−	0.957326i	$-0.406674\pi$
$431$	12.0000	0.578020	0.289010	+	0.957326i	$0.406674\pi$
$432$	0	0
$433$	−10.0000	−0.480569	−0.240285	−	0.970702i	$-0.577241\pi$
$433$	−10.0000	−0.480569	−0.240285	+	0.970702i	$0.577241\pi$
$434$	0	0
$435$	−6.00000	−0.287678
$436$	0	0
$437$	20.0000	0.956730
$438$	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
$441$	−3.00000	−0.142857
$442$	0	0
$443$	14.0000	0.665160	0.332580	−	0.943075i	$-0.392081\pi$
$443$	14.0000	0.665160	0.332580	+	0.943075i	$0.392081\pi$
$444$	0	0
$445$	28.0000	1.32733
$446$	0	0
$447$	−21.0000	−0.993266
$448$	0	0
$449$	11.0000	0.519122	0.259561	−	0.965727i	$-0.416422\pi$
$449$	11.0000	0.519122	0.259561	+	0.965727i	$0.416422\pi$
$450$	0	0
$451$	36.0000	1.69517
$452$	0	0
$453$	−20.0000	−0.939682
$454$	0	0
$455$	20.0000	0.937614
$456$	0	0
$457$	6.00000	0.280668	0.140334	−	0.990104i	$-0.455182\pi$
$457$	6.00000	0.280668	0.140334	+	0.990104i	$0.455182\pi$
$458$	0	0
$459$	−7.00000	−0.326732
$460$	0	0
$461$	−6.00000	−0.279448	−0.139724	−	0.990190i	$-0.544622\pi$
$461$	−6.00000	−0.279448	−0.139724	+	0.990190i	$0.544622\pi$
$462$	0	0
$463$	−7.00000	−0.325318	−0.162659	−	0.986682i	$-0.552007\pi$
$463$	−7.00000	−0.325318	−0.162659	+	0.986682i	$0.552007\pi$
$464$	0	0
$465$	14.0000	0.649234
$466$	0	0
$467$	17.0000	0.786666	0.393333	−	0.919396i	$-0.371322\pi$
$467$	17.0000	0.786666	0.393333	+	0.919396i	$0.371322\pi$
$468$	0	0
$469$	−16.0000	−0.738811
$470$	0	0
$471$	−18.0000	−0.829396
$472$	0	0
$473$	−6.00000	−0.275880
$474$	0	0
$475$	−5.00000	−0.229416
$476$	0	0
$477$	4.00000	0.183147
$478$	0	0
$479$	28.0000	1.27935	0.639676	−	0.768644i	$-0.279068\pi$
$479$	28.0000	1.27935	0.639676	+	0.768644i	$0.279068\pi$
$480$	0	0
$481$	40.0000	1.82384
$482$	0	0
$483$	−8.00000	−0.364013
$484$	0	0
$485$	−24.0000	−1.08978
$486$	0	0
$487$	−28.0000	−1.26880	−0.634401	−	0.773004i	$-0.718753\pi$
$487$	−28.0000	−1.26880	−0.634401	+	0.773004i	$0.718753\pi$
$488$	0	0
$489$	−3.00000	−0.135665
$490$	0	0
$491$	34.0000	1.53440	0.767199	−	0.641409i	$-0.221650\pi$
$491$	34.0000	1.53440	0.767199	+	0.641409i	$0.221650\pi$
$492$	0	0
$493$	−21.0000	−0.945792
$494$	0	0
$495$	6.00000	0.269680
$496$	0	0
$497$	32.0000	1.43540
$498$	0	0
$499$	−16.0000	−0.716258	−0.358129	−	0.933672i	$-0.616585\pi$
$499$	−16.0000	−0.716258	−0.358129	+	0.933672i	$0.616585\pi$
$500$	0	0
$501$	11.0000	0.491444
$502$	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
$507$	−12.0000	−0.532939
$508$	0	0
$509$	−6.00000	−0.265945	−0.132973	−	0.991120i	$-0.542452\pi$
$509$	−6.00000	−0.265945	−0.132973	+	0.991120i	$0.542452\pi$
$510$	0	0
$511$	−4.00000	−0.176950
$512$	0	0
$513$	−5.00000	−0.220755
$514$	0	0
$515$	0	0
$516$	0	0
$517$	24.0000	1.05552
$518$	0	0
$519$	−1.00000	−0.0438951
$520$	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
$523$	−11.0000	−0.480996	−0.240498	−	0.970650i	$-0.577311\pi$
$523$	−11.0000	−0.480996	−0.240498	+	0.970650i	$0.577311\pi$
$524$	0	0
$525$	2.00000	0.0872872
$526$	0	0
$527$	49.0000	2.13447
$528$	0	0
$529$	−7.00000	−0.304348
$530$	0	0
$531$	3.00000	0.130189
$532$	0	0
$533$	60.0000	2.59889
$534$	0	0
$535$	10.0000	0.432338
$536$	0	0
$537$	20.0000	0.863064
$538$	0	0
$539$	9.00000	0.387657
$540$	0	0
$541$	10.0000	0.429934	0.214967	−	0.976621i	$-0.431036\pi$
$541$	10.0000	0.429934	0.214967	+	0.976621i	$0.431036\pi$
$542$	0	0
$543$	4.00000	0.171656
$544$	0	0
$545$	−6.00000	−0.257012
$546$	0	0
$547$	35.0000	1.49649	0.748246	−	0.663421i	$-0.230896\pi$
$547$	35.0000	1.49649	0.748246	+	0.663421i	$0.230896\pi$
$548$	0	0
$549$	13.0000	0.554826
$550$	0	0
$551$	−15.0000	−0.639021
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−16.0000	−0.679162
$556$	0	0
$557$	16.0000	0.677942	0.338971	−	0.940797i	$-0.389921\pi$
$557$	16.0000	0.677942	0.338971	+	0.940797i	$0.389921\pi$
$558$	0	0
$559$	−10.0000	−0.422955
$560$	0	0
$561$	21.0000	0.886621
$562$	0	0
$563$	−24.0000	−1.01148	−0.505740	−	0.862686i	$-0.668780\pi$
$563$	−24.0000	−1.01148	−0.505740	+	0.862686i	$0.668780\pi$
$564$	0	0
$565$	−24.0000	−1.00969
$566$	0	0
$567$	2.00000	0.0839921
$568$	0	0
$569$	−22.0000	−0.922288	−0.461144	−	0.887325i	$-0.652561\pi$
$569$	−22.0000	−0.922288	−0.461144	+	0.887325i	$0.652561\pi$
$570$	0	0
$571$	11.0000	0.460336	0.230168	−	0.973151i	$-0.426072\pi$
$571$	11.0000	0.460336	0.230168	+	0.973151i	$0.426072\pi$
$572$	0	0
$573$	13.0000	0.543083
$574$	0	0
$575$	−4.00000	−0.166812
$576$	0	0
$577$	−11.0000	−0.457936	−0.228968	−	0.973434i	$-0.573535\pi$
$577$	−11.0000	−0.457936	−0.228968	+	0.973434i	$0.573535\pi$
$578$	0	0
$579$	−11.0000	−0.457144
$580$	0	0
$581$	−12.0000	−0.497844
$582$	0	0
$583$	−12.0000	−0.496989
$584$	0	0
$585$	10.0000	0.413449
$586$	0	0
$587$	−17.0000	−0.701665	−0.350833	−	0.936438i	$-0.614101\pi$
$587$	−17.0000	−0.701665	−0.350833	+	0.936438i	$0.614101\pi$
$588$	0	0
$589$	35.0000	1.44215
$590$	0	0
$591$	−6.00000	−0.246807
$592$	0	0
$593$	−9.00000	−0.369586	−0.184793	−	0.982777i	$-0.559161\pi$
$593$	−9.00000	−0.369586	−0.184793	+	0.982777i	$0.559161\pi$
$594$	0	0
$595$	−28.0000	−1.14789
$596$	0	0
$597$	13.0000	0.532055
$598$	0	0
$599$	12.0000	0.490307	0.245153	−	0.969484i	$-0.421162\pi$
$599$	12.0000	0.490307	0.245153	+	0.969484i	$0.421162\pi$
$600$	0	0
$601$	45.0000	1.83559	0.917794	−	0.397057i	$-0.129968\pi$
$601$	45.0000	1.83559	0.917794	+	0.397057i	$0.129968\pi$
$602$	0	0
$603$	−8.00000	−0.325785
$604$	0	0
$605$	4.00000	0.162623
$606$	0	0
$607$	−28.0000	−1.13648	−0.568242	−	0.822861i	$-0.692376\pi$
$607$	−28.0000	−1.13648	−0.568242	+	0.822861i	$0.692376\pi$
$608$	0	0
$609$	6.00000	0.243132
$610$	0	0
$611$	40.0000	1.61823
$612$	0	0
$613$	−11.0000	−0.444286	−0.222143	−	0.975014i	$-0.571305\pi$
$613$	−11.0000	−0.444286	−0.222143	+	0.975014i	$0.571305\pi$
$614$	0	0
$615$	−24.0000	−0.967773
$616$	0	0
$617$	6.00000	0.241551	0.120775	−	0.992680i	$-0.461462\pi$
$617$	6.00000	0.241551	0.120775	+	0.992680i	$0.461462\pi$
$618$	0	0
$619$	45.0000	1.80870	0.904351	−	0.426789i	$-0.140355\pi$
$619$	45.0000	1.80870	0.904351	+	0.426789i	$0.140355\pi$
$620$	0	0
$621$	−4.00000	−0.160514
$622$	0	0
$623$	−28.0000	−1.12180
$624$	0	0
$625$	−19.0000	−0.760000
$626$	0	0
$627$	15.0000	0.599042
$628$	0	0
$629$	−56.0000	−2.23287
$630$	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
$633$	−10.0000	−0.397464
$634$	0	0
$635$	−38.0000	−1.50798
$636$	0	0
$637$	15.0000	0.594322
$638$	0	0
$639$	16.0000	0.632950
$640$	0	0
$641$	10.0000	0.394976	0.197488	−	0.980305i	$-0.436722\pi$
$641$	10.0000	0.394976	0.197488	+	0.980305i	$0.436722\pi$
$642$	0	0
$643$	9.00000	0.354925	0.177463	−	0.984128i	$-0.443211\pi$
$643$	9.00000	0.354925	0.177463	+	0.984128i	$0.443211\pi$
$644$	0	0
$645$	4.00000	0.157500
$646$	0	0
$647$	1.00000	0.0393141	0.0196570	−	0.999807i	$-0.493743\pi$
$647$	1.00000	0.0393141	0.0196570	+	0.999807i	$0.493743\pi$
$648$	0	0
$649$	−9.00000	−0.353281
$650$	0	0
$651$	−14.0000	−0.548703
$652$	0	0
$653$	−14.0000	−0.547862	−0.273931	−	0.961749i	$-0.588324\pi$
$653$	−14.0000	−0.547862	−0.273931	+	0.961749i	$0.588324\pi$
$654$	0	0
$655$	2.00000	0.0781465
$656$	0	0
$657$	−2.00000	−0.0780274
$658$	0	0
$659$	−19.0000	−0.740135	−0.370067	−	0.929005i	$-0.620665\pi$
$659$	−19.0000	−0.740135	−0.370067	+	0.929005i	$0.620665\pi$
$660$	0	0
$661$	−4.00000	−0.155582	−0.0777910	−	0.996970i	$-0.524787\pi$
$661$	−4.00000	−0.155582	−0.0777910	+	0.996970i	$0.524787\pi$
$662$	0	0
$663$	35.0000	1.35929
$664$	0	0
$665$	−20.0000	−0.775567
$666$	0	0
$667$	−12.0000	−0.464642
$668$	0	0
$669$	−17.0000	−0.657258
$670$	0	0
$671$	−39.0000	−1.50558
$672$	0	0
$673$	8.00000	0.308377	0.154189	−	0.988041i	$-0.450724\pi$
$673$	8.00000	0.308377	0.154189	+	0.988041i	$0.450724\pi$
$674$	0	0
$675$	1.00000	0.0384900
$676$	0	0
$677$	−18.0000	−0.691796	−0.345898	−	0.938272i	$-0.612426\pi$
$677$	−18.0000	−0.691796	−0.345898	+	0.938272i	$0.612426\pi$
$678$	0	0
$679$	24.0000	0.921035
$680$	0	0
$681$	−26.0000	−0.996322
$682$	0	0
$683$	9.00000	0.344375	0.172188	−	0.985064i	$-0.444916\pi$
$683$	9.00000	0.344375	0.172188	+	0.985064i	$0.444916\pi$
$684$	0	0
$685$	−10.0000	−0.382080
$686$	0	0
$687$	−4.00000	−0.152610
$688$	0	0
$689$	−20.0000	−0.761939
$690$	0	0
$691$	−48.0000	−1.82601	−0.913003	−	0.407953i	$-0.866243\pi$
$691$	−48.0000	−1.82601	−0.913003	+	0.407953i	$0.866243\pi$
$692$	0	0
$693$	−6.00000	−0.227921
$694$	0	0
$695$	−8.00000	−0.303457
$696$	0	0
$697$	−84.0000	−3.18173
$698$	0	0
$699$	12.0000	0.453882
$700$	0	0
$701$	50.0000	1.88847	0.944237	−	0.329267i	$-0.106802\pi$
$701$	50.0000	1.88847	0.944237	+	0.329267i	$0.106802\pi$
$702$	0	0
$703$	−40.0000	−1.50863
$704$	0	0
$705$	−16.0000	−0.602595
$706$	0	0
$707$	−36.0000	−1.35392
$708$	0	0
$709$	−6.00000	−0.225335	−0.112667	−	0.993633i	$-0.535939\pi$
$709$	−6.00000	−0.225335	−0.112667	+	0.993633i	$0.535939\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	28.0000	1.04861
$714$	0	0
$715$	−30.0000	−1.12194
$716$	0	0
$717$	24.0000	0.896296
$718$	0	0
$719$	24.0000	0.895049	0.447524	−	0.894272i	$-0.352306\pi$
$719$	24.0000	0.895049	0.447524	+	0.894272i	$0.352306\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	0	0
$724$	0	0
$725$	3.00000	0.111417
$726$	0	0
$727$	16.0000	0.593407	0.296704	−	0.954970i	$-0.404113\pi$
$727$	16.0000	0.593407	0.296704	+	0.954970i	$0.404113\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	14.0000	0.517809
$732$	0	0
$733$	−34.0000	−1.25582	−0.627909	−	0.778287i	$-0.716089\pi$
$733$	−34.0000	−1.25582	−0.627909	+	0.778287i	$0.716089\pi$
$734$	0	0
$735$	−6.00000	−0.221313
$736$	0	0
$737$	24.0000	0.884051
$738$	0	0
$739$	−42.0000	−1.54499	−0.772497	−	0.635018i	$-0.780993\pi$
$739$	−42.0000	−1.54499	−0.772497	+	0.635018i	$0.780993\pi$
$740$	0	0
$741$	25.0000	0.918398
$742$	0	0
$743$	−4.00000	−0.146746	−0.0733729	−	0.997305i	$-0.523376\pi$
$743$	−4.00000	−0.146746	−0.0733729	+	0.997305i	$0.523376\pi$
$744$	0	0
$745$	−42.0000	−1.53876
$746$	0	0
$747$	−6.00000	−0.219529
$748$	0	0
$749$	−10.0000	−0.365392
$750$	0	0
$751$	33.0000	1.20419	0.602094	−	0.798426i	$-0.294333\pi$
$751$	33.0000	1.20419	0.602094	+	0.798426i	$0.294333\pi$
$752$	0	0
$753$	−18.0000	−0.655956
$754$	0	0
$755$	−40.0000	−1.45575
$756$	0	0
$757$	19.0000	0.690567	0.345283	−	0.938498i	$-0.387783\pi$
$757$	19.0000	0.690567	0.345283	+	0.938498i	$0.387783\pi$
$758$	0	0
$759$	12.0000	0.435572
$760$	0	0
$761$	7.00000	0.253750	0.126875	−	0.991919i	$-0.459505\pi$
$761$	7.00000	0.253750	0.126875	+	0.991919i	$0.459505\pi$
$762$	0	0
$763$	6.00000	0.217215
$764$	0	0
$765$	−14.0000	−0.506171
$766$	0	0
$767$	−15.0000	−0.541619
$768$	0	0
$769$	−41.0000	−1.47850	−0.739249	−	0.673432i	$-0.764819\pi$
$769$	−41.0000	−1.47850	−0.739249	+	0.673432i	$0.764819\pi$
$770$	0	0
$771$	6.00000	0.216085
$772$	0	0
$773$	55.0000	1.97821	0.989106	−	0.147203i	$-0.0470272\pi$
$773$	55.0000	1.97821	0.989106	+	0.147203i	$0.0470272\pi$
$774$	0	0
$775$	−7.00000	−0.251447
$776$	0	0
$777$	16.0000	0.573997
$778$	0	0
$779$	−60.0000	−2.14972
$780$	0	0
$781$	−48.0000	−1.71758
$782$	0	0
$783$	3.00000	0.107211
$784$	0	0
$785$	−36.0000	−1.28490
$786$	0	0
$787$	44.0000	1.56843	0.784215	−	0.620489i	$-0.213066\pi$
$787$	44.0000	1.56843	0.784215	+	0.620489i	$0.213066\pi$
$788$	0	0
$789$	21.0000	0.747620
$790$	0	0
$791$	24.0000	0.853342
$792$	0	0
$793$	−65.0000	−2.30822
$794$	0	0
$795$	8.00000	0.283731
$796$	0	0
$797$	−6.00000	−0.212531	−0.106265	−	0.994338i	$-0.533889\pi$
$797$	−6.00000	−0.212531	−0.106265	+	0.994338i	$0.533889\pi$
$798$	0	0
$799$	−56.0000	−1.98114
$800$	0	0
$801$	−14.0000	−0.494666
$802$	0	0
$803$	6.00000	0.211735
$804$	0	0
$805$	−16.0000	−0.563926
$806$	0	0
$807$	−16.0000	−0.563227
$808$	0	0
$809$	−6.00000	−0.210949	−0.105474	−	0.994422i	$-0.533636\pi$
$809$	−6.00000	−0.210949	−0.105474	+	0.994422i	$0.533636\pi$
$810$	0	0
$811$	22.0000	0.772524	0.386262	−	0.922389i	$-0.373766\pi$
$811$	22.0000	0.772524	0.386262	+	0.922389i	$0.373766\pi$
$812$	0	0
$813$	−20.0000	−0.701431
$814$	0	0
$815$	−6.00000	−0.210171
$816$	0	0
$817$	10.0000	0.349856
$818$	0	0
$819$	−10.0000	−0.349428
$820$	0	0
$821$	−12.0000	−0.418803	−0.209401	−	0.977830i	$-0.567152\pi$
$821$	−12.0000	−0.418803	−0.209401	+	0.977830i	$0.567152\pi$
$822$	0	0
$823$	−37.0000	−1.28974	−0.644869	−	0.764293i	$-0.723088\pi$
$823$	−37.0000	−1.28974	−0.644869	+	0.764293i	$0.723088\pi$
$824$	0	0
$825$	−3.00000	−0.104447
$826$	0	0
$827$	1.00000	0.0347734	0.0173867	−	0.999849i	$-0.494465\pi$
$827$	1.00000	0.0347734	0.0173867	+	0.999849i	$0.494465\pi$
$828$	0	0
$829$	25.0000	0.868286	0.434143	−	0.900844i	$-0.357051\pi$
$829$	25.0000	0.868286	0.434143	+	0.900844i	$0.357051\pi$
$830$	0	0
$831$	−19.0000	−0.659103
$832$	0	0
$833$	−21.0000	−0.727607
$834$	0	0
$835$	22.0000	0.761341
$836$	0	0
$837$	−7.00000	−0.241955
$838$	0	0
$839$	3.00000	0.103572	0.0517858	−	0.998658i	$-0.483509\pi$
$839$	3.00000	0.103572	0.0517858	+	0.998658i	$0.483509\pi$
$840$	0	0
$841$	−20.0000	−0.689655
$842$	0	0
$843$	−6.00000	−0.206651
$844$	0	0
$845$	−24.0000	−0.825625
$846$	0	0
$847$	−4.00000	−0.137442
$848$	0	0
$849$	−26.0000	−0.892318
$850$	0	0
$851$	−32.0000	−1.09695
$852$	0	0
$853$	4.00000	0.136957	0.0684787	−	0.997653i	$-0.478185\pi$
$853$	4.00000	0.136957	0.0684787	+	0.997653i	$0.478185\pi$
$854$	0	0
$855$	−10.0000	−0.341993
$856$	0	0
$857$	31.0000	1.05894	0.529470	−	0.848329i	$-0.322391\pi$
$857$	31.0000	1.05894	0.529470	+	0.848329i	$0.322391\pi$
$858$	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
$861$	24.0000	0.817918
$862$	0	0
$863$	−3.00000	−0.102121	−0.0510606	−	0.998696i	$-0.516260\pi$
$863$	−3.00000	−0.102121	−0.0510606	+	0.998696i	$0.516260\pi$
$864$	0	0
$865$	−2.00000	−0.0680020
$866$	0	0
$867$	−32.0000	−1.08678
$868$	0	0
$869$	0	0
$870$	0	0
$871$	40.0000	1.35535
$872$	0	0
$873$	12.0000	0.406138
$874$	0	0
$875$	24.0000	0.811348
$876$	0	0
$877$	−10.0000	−0.337676	−0.168838	−	0.985644i	$-0.554001\pi$
$877$	−10.0000	−0.337676	−0.168838	+	0.985644i	$0.554001\pi$
$878$	0	0
$879$	15.0000	0.505937
$880$	0	0
$881$	−2.00000	−0.0673817	−0.0336909	−	0.999432i	$-0.510726\pi$
$881$	−2.00000	−0.0673817	−0.0336909	+	0.999432i	$0.510726\pi$
$882$	0	0
$883$	−13.0000	−0.437485	−0.218742	−	0.975783i	$-0.570195\pi$
$883$	−13.0000	−0.437485	−0.218742	+	0.975783i	$0.570195\pi$
$884$	0	0
$885$	6.00000	0.201688
$886$	0	0
$887$	8.00000	0.268614	0.134307	−	0.990940i	$-0.457119\pi$
$887$	8.00000	0.268614	0.134307	+	0.990940i	$0.457119\pi$
$888$	0	0
$889$	38.0000	1.27448
$890$	0	0
$891$	−3.00000	−0.100504
$892$	0	0
$893$	−40.0000	−1.33855
$894$	0	0
$895$	40.0000	1.33705
$896$	0	0
$897$	20.0000	0.667781
$898$	0	0
$899$	−21.0000	−0.700389
$900$	0	0
$901$	28.0000	0.932815
$902$	0	0
$903$	−4.00000	−0.133112
$904$	0	0
$905$	8.00000	0.265929
$906$	0	0
$907$	−2.00000	−0.0664089	−0.0332045	−	0.999449i	$-0.510571\pi$
$907$	−2.00000	−0.0664089	−0.0332045	+	0.999449i	$0.510571\pi$
$908$	0	0
$909$	−18.0000	−0.597022
$910$	0	0
$911$	−1.00000	−0.0331315	−0.0165657	−	0.999863i	$-0.505273\pi$
$911$	−1.00000	−0.0331315	−0.0165657	+	0.999863i	$0.505273\pi$
$912$	0	0
$913$	18.0000	0.595713
$914$	0	0
$915$	26.0000	0.859533
$916$	0	0
$917$	−2.00000	−0.0660458
$918$	0	0
$919$	13.0000	0.428830	0.214415	−	0.976743i	$-0.431215\pi$
$919$	13.0000	0.428830	0.214415	+	0.976743i	$0.431215\pi$
$920$	0	0
$921$	−26.0000	−0.856729
$922$	0	0
$923$	−80.0000	−2.63323
$924$	0	0
$925$	8.00000	0.263038
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−42.0000	−1.37798	−0.688988	−	0.724773i	$-0.741945\pi$
$929$	−42.0000	−1.37798	−0.688988	+	0.724773i	$0.741945\pi$
$930$	0	0
$931$	−15.0000	−0.491605
$932$	0	0
$933$	−7.00000	−0.229170
$934$	0	0
$935$	42.0000	1.37355
$936$	0	0
$937$	49.0000	1.60076	0.800380	−	0.599493i	$-0.204631\pi$
$937$	49.0000	1.60076	0.800380	+	0.599493i	$0.204631\pi$
$938$	0	0
$939$	−24.0000	−0.783210
$940$	0	0
$941$	−45.0000	−1.46696	−0.733479	−	0.679712i	$-0.762105\pi$
$941$	−45.0000	−1.46696	−0.733479	+	0.679712i	$0.762105\pi$
$942$	0	0
$943$	−48.0000	−1.56310
$944$	0	0
$945$	4.00000	0.130120
$946$	0	0
$947$	−8.00000	−0.259965	−0.129983	−	0.991516i	$-0.541492\pi$
$947$	−8.00000	−0.259965	−0.129983	+	0.991516i	$0.541492\pi$
$948$	0	0
$949$	10.0000	0.324614
$950$	0	0
$951$	−2.00000	−0.0648544
$952$	0	0
$953$	−48.0000	−1.55487	−0.777436	−	0.628962i	$-0.783480\pi$
$953$	−48.0000	−1.55487	−0.777436	+	0.628962i	$0.783480\pi$
$954$	0	0
$955$	26.0000	0.841340
$956$	0	0
$957$	−9.00000	−0.290929
$958$	0	0
$959$	10.0000	0.322917
$960$	0	0
$961$	18.0000	0.580645
$962$	0	0
$963$	−5.00000	−0.161123
$964$	0	0
$965$	−22.0000	−0.708205
$966$	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
$969$	−35.0000	−1.12436
$970$	0	0
$971$	−12.0000	−0.385098	−0.192549	−	0.981287i	$-0.561675\pi$
$971$	−12.0000	−0.385098	−0.192549	+	0.981287i	$0.561675\pi$
$972$	0	0
$973$	8.00000	0.256468
$974$	0	0
$975$	−5.00000	−0.160128
$976$	0	0
$977$	36.0000	1.15174	0.575871	−	0.817541i	$-0.304663\pi$
$977$	36.0000	1.15174	0.575871	+	0.817541i	$0.304663\pi$
$978$	0	0
$979$	42.0000	1.34233
$980$	0	0
$981$	3.00000	0.0957826
$982$	0	0
$983$	16.0000	0.510321	0.255160	−	0.966899i	$-0.417872\pi$
$983$	16.0000	0.510321	0.255160	+	0.966899i	$0.417872\pi$
$984$	0	0
$985$	−12.0000	−0.382352
$986$	0	0
$987$	16.0000	0.509286
$988$	0	0
$989$	8.00000	0.254385
$990$	0	0
$991$	26.0000	0.825917	0.412959	−	0.910750i	$-0.364495\pi$
$991$	26.0000	0.825917	0.412959	+	0.910750i	$0.364495\pi$
$992$	0	0
$993$	−11.0000	−0.349074
$994$	0	0
$995$	26.0000	0.824255
$996$	0	0
$997$	−13.0000	−0.411714	−0.205857	−	0.978582i	$-0.565998\pi$
$997$	−13.0000	−0.411714	−0.205857	+	0.978582i	$0.565998\pi$
$998$	0	0
$999$	8.00000	0.253109

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6288.2.a.c.1.1		1
4.3	odd	2		786.2.a.g.1.1	✓	1
12.11	even	2		2358.2.a.v.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
786.2.a.g.1.1	✓	1	4.3	odd	2
2358.2.a.v.1.1		1	12.11	even	2
6288.2.a.c.1.1		1	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 6288 = 2^{4} \cdot 3 \cdot 131 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6288.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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