LMFDB - Embedded newform 816.2.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [816,2,Mod(1,816)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(816, 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("816.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 816 = 2^{4} \cdot 3 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	816.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:	$6.51579280494$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 408)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	816.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{3} -3.00000 q^{5} +4.00000 q^{7} +1.00000 q^{9} +O(q^{10})$

$q-1.00000 q^{3} -3.00000 q^{5} +4.00000 q^{7} +1.00000 q^{9} -1.00000 q^{11} -5.00000 q^{13} +3.00000 q^{15} +1.00000 q^{17} +7.00000 q^{19} -4.00000 q^{21} -1.00000 q^{23} +4.00000 q^{25} -1.00000 q^{27} +2.00000 q^{29} +6.00000 q^{31} +1.00000 q^{33} -12.0000 q^{35} +8.00000 q^{37} +5.00000 q^{39} +7.00000 q^{41} +1.00000 q^{43} -3.00000 q^{45} +6.00000 q^{47} +9.00000 q^{49} -1.00000 q^{51} -2.00000 q^{53} +3.00000 q^{55} -7.00000 q^{57} +10.0000 q^{59} +8.00000 q^{61} +4.00000 q^{63} +15.0000 q^{65} +12.0000 q^{67} +1.00000 q^{69} -12.0000 q^{71} -14.0000 q^{73} -4.00000 q^{75} -4.00000 q^{77} -10.0000 q^{79} +1.00000 q^{81} +14.0000 q^{83} -3.00000 q^{85} -2.00000 q^{87} +8.00000 q^{89} -20.0000 q^{91} -6.00000 q^{93} -21.0000 q^{95} +12.0000 q^{97} -1.00000 q^{99} +O(q^{100})$

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
$5$	−3.00000	−1.34164	−0.670820	−	0.741620i	$-0.734058\pi$
$5$	−3.00000	−1.34164	−0.670820	+	0.741620i	$0.734058\pi$
$6$	0	0
$7$	4.00000	1.51186	0.755929	−	0.654654i	$-0.227186\pi$
$7$	4.00000	1.51186	0.755929	+	0.654654i	$0.227186\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−1.00000	−0.301511	−0.150756	−	0.988571i	$-0.548171\pi$
$11$	−1.00000	−0.301511	−0.150756	+	0.988571i	$0.548171\pi$
$12$	0	0
$13$	−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
$15$	3.00000	0.774597
$16$	0	0
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	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
$21$	−4.00000	−0.872872
$22$	0	0
$23$	−1.00000	−0.208514	−0.104257	−	0.994550i	$-0.533247\pi$
$23$	−1.00000	−0.208514	−0.104257	+	0.994550i	$0.533247\pi$
$24$	0	0
$25$	4.00000	0.800000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	2.00000	0.371391	0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000	0.371391	0.185695	+	0.982607i	$0.440546\pi$
$30$	0	0
$31$	6.00000	1.07763	0.538816	−	0.842424i	$-0.318872\pi$
$31$	6.00000	1.07763	0.538816	+	0.842424i	$0.318872\pi$
$32$	0	0
$33$	1.00000	0.174078
$34$	0	0
$35$	−12.0000	−2.02837
$36$	0	0
$37$	8.00000	1.31519	0.657596	−	0.753371i	$-0.271573\pi$
$37$	8.00000	1.31519	0.657596	+	0.753371i	$0.271573\pi$
$38$	0	0
$39$	5.00000	0.800641
$40$	0	0
$41$	7.00000	1.09322	0.546608	−	0.837389i	$-0.315919\pi$
$41$	7.00000	1.09322	0.546608	+	0.837389i	$0.315919\pi$
$42$	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
$47$	6.00000	0.875190	0.437595	−	0.899172i	$-0.355830\pi$
$47$	6.00000	0.875190	0.437595	+	0.899172i	$0.355830\pi$
$48$	0	0
$49$	9.00000	1.28571
$50$	0	0
$51$	−1.00000	−0.140028
$52$	0	0
$53$	−2.00000	−0.274721	−0.137361	−	0.990521i	$-0.543862\pi$
$53$	−2.00000	−0.274721	−0.137361	+	0.990521i	$0.543862\pi$
$54$	0	0
$55$	3.00000	0.404520
$56$	0	0
$57$	−7.00000	−0.927173
$58$	0	0
$59$	10.0000	1.30189	0.650945	−	0.759125i	$-0.274373\pi$
$59$	10.0000	1.30189	0.650945	+	0.759125i	$0.274373\pi$
$60$	0	0
$61$	8.00000	1.02430	0.512148	−	0.858898i	$-0.328850\pi$
$61$	8.00000	1.02430	0.512148	+	0.858898i	$0.328850\pi$
$62$	0	0
$63$	4.00000	0.503953
$64$	0	0
$65$	15.0000	1.86052
$66$	0	0
$67$	12.0000	1.46603	0.733017	−	0.680211i	$-0.238112\pi$
$67$	12.0000	1.46603	0.733017	+	0.680211i	$0.238112\pi$
$68$	0	0
$69$	1.00000	0.120386
$70$	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
$73$	−14.0000	−1.63858	−0.819288	−	0.573382i	$-0.805631\pi$
$73$	−14.0000	−1.63858	−0.819288	+	0.573382i	$0.805631\pi$
$74$	0	0
$75$	−4.00000	−0.461880
$76$	0	0
$77$	−4.00000	−0.455842
$78$	0	0
$79$	−10.0000	−1.12509	−0.562544	−	0.826767i	$-0.690177\pi$
$79$	−10.0000	−1.12509	−0.562544	+	0.826767i	$0.690177\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	14.0000	1.53670	0.768350	−	0.640030i	$-0.221078\pi$
$83$	14.0000	1.53670	0.768350	+	0.640030i	$0.221078\pi$
$84$	0	0
$85$	−3.00000	−0.325396
$86$	0	0
$87$	−2.00000	−0.214423
$88$	0	0
$89$	8.00000	0.847998	0.423999	−	0.905663i	$-0.360626\pi$
$89$	8.00000	0.847998	0.423999	+	0.905663i	$0.360626\pi$
$90$	0	0
$91$	−20.0000	−2.09657
$92$	0	0
$93$	−6.00000	−0.622171
$94$	0	0
$95$	−21.0000	−2.15455
$96$	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
$99$	−1.00000	−0.100504
$100$	0	0
$101$	0	0		−	1.00000i	$-0.5\pi$
$101$	0	0			1.00000i	$0.5\pi$
$102$	0	0
$103$	−7.00000	−0.689730	−0.344865	−	0.938652i	$-0.612075\pi$
$103$	−7.00000	−0.689730	−0.344865	+	0.938652i	$0.612075\pi$
$104$	0	0
$105$	12.0000	1.17108
$106$	0	0
$107$	−13.0000	−1.25676	−0.628379	−	0.777908i	$-0.716281\pi$
$107$	−13.0000	−1.25676	−0.628379	+	0.777908i	$0.716281\pi$
$108$	0	0
$109$	−20.0000	−1.91565	−0.957826	−	0.287348i	$-0.907226\pi$
$109$	−20.0000	−1.91565	−0.957826	+	0.287348i	$0.907226\pi$
$110$	0	0
$111$	−8.00000	−0.759326
$112$	0	0
$113$	−19.0000	−1.78737	−0.893685	−	0.448695i	$-0.851889\pi$
$113$	−19.0000	−1.78737	−0.893685	+	0.448695i	$0.851889\pi$
$114$	0	0
$115$	3.00000	0.279751
$116$	0	0
$117$	−5.00000	−0.462250
$118$	0	0
$119$	4.00000	0.366679
$120$	0	0
$121$	−10.0000	−0.909091
$122$	0	0
$123$	−7.00000	−0.631169
$124$	0	0
$125$	3.00000	0.268328
$126$	0	0
$127$	7.00000	0.621150	0.310575	−	0.950549i	$-0.399478\pi$
$127$	7.00000	0.621150	0.310575	+	0.950549i	$0.399478\pi$
$128$	0	0
$129$	−1.00000	−0.0880451
$130$	0	0
$131$	1.00000	0.0873704	0.0436852	−	0.999045i	$-0.486090\pi$
$131$	1.00000	0.0873704	0.0436852	+	0.999045i	$0.486090\pi$
$132$	0	0
$133$	28.0000	2.42791
$134$	0	0
$135$	3.00000	0.258199
$136$	0	0
$137$	18.0000	1.53784	0.768922	−	0.639343i	$-0.220793\pi$
$137$	18.0000	1.53784	0.768922	+	0.639343i	$0.220793\pi$
$138$	0	0
$139$	14.0000	1.18746	0.593732	−	0.804663i	$-0.297654\pi$
$139$	14.0000	1.18746	0.593732	+	0.804663i	$0.297654\pi$
$140$	0	0
$141$	−6.00000	−0.505291
$142$	0	0
$143$	5.00000	0.418121
$144$	0	0
$145$	−6.00000	−0.498273
$146$	0	0
$147$	−9.00000	−0.742307
$148$	0	0
$149$	6.00000	0.491539	0.245770	−	0.969328i	$-0.420959\pi$
$149$	6.00000	0.491539	0.245770	+	0.969328i	$0.420959\pi$
$150$	0	0
$151$	0	0		−	1.00000i	$-0.5\pi$
$151$	0	0			1.00000i	$0.5\pi$
$152$	0	0
$153$	1.00000	0.0808452
$154$	0	0
$155$	−18.0000	−1.44579
$156$	0	0
$157$	−9.00000	−0.718278	−0.359139	−	0.933284i	$-0.616930\pi$
$157$	−9.00000	−0.718278	−0.359139	+	0.933284i	$0.616930\pi$
$158$	0	0
$159$	2.00000	0.158610
$160$	0	0
$161$	−4.00000	−0.315244
$162$	0	0
$163$	6.00000	0.469956	0.234978	−	0.972001i	$-0.424498\pi$
$163$	6.00000	0.469956	0.234978	+	0.972001i	$0.424498\pi$
$164$	0	0
$165$	−3.00000	−0.233550
$166$	0	0
$167$	3.00000	0.232147	0.116073	−	0.993241i	$-0.462969\pi$
$167$	3.00000	0.232147	0.116073	+	0.993241i	$0.462969\pi$
$168$	0	0
$169$	12.0000	0.923077
$170$	0	0
$171$	7.00000	0.535303
$172$	0	0
$173$	−23.0000	−1.74866	−0.874329	−	0.485334i	$-0.838698\pi$
$173$	−23.0000	−1.74866	−0.874329	+	0.485334i	$0.838698\pi$
$174$	0	0
$175$	16.0000	1.20949
$176$	0	0
$177$	−10.0000	−0.751646
$178$	0	0
$179$	10.0000	0.747435	0.373718	−	0.927543i	$-0.378083\pi$
$179$	10.0000	0.747435	0.373718	+	0.927543i	$0.378083\pi$
$180$	0	0
$181$	10.0000	0.743294	0.371647	−	0.928374i	$-0.378793\pi$
$181$	10.0000	0.743294	0.371647	+	0.928374i	$0.378793\pi$
$182$	0	0
$183$	−8.00000	−0.591377
$184$	0	0
$185$	−24.0000	−1.76452
$186$	0	0
$187$	−1.00000	−0.0731272
$188$	0	0
$189$	−4.00000	−0.290957
$190$	0	0
$191$	−6.00000	−0.434145	−0.217072	−	0.976156i	$-0.569651\pi$
$191$	−6.00000	−0.434145	−0.217072	+	0.976156i	$0.569651\pi$
$192$	0	0
$193$	6.00000	0.431889	0.215945	−	0.976406i	$-0.430717\pi$
$193$	6.00000	0.431889	0.215945	+	0.976406i	$0.430717\pi$
$194$	0	0
$195$	−15.0000	−1.07417
$196$	0	0
$197$	−27.0000	−1.92367	−0.961835	−	0.273629i	$-0.911776\pi$
$197$	−27.0000	−1.92367	−0.961835	+	0.273629i	$0.911776\pi$
$198$	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
$201$	−12.0000	−0.846415
$202$	0	0
$203$	8.00000	0.561490
$204$	0	0
$205$	−21.0000	−1.46670
$206$	0	0
$207$	−1.00000	−0.0695048
$208$	0	0
$209$	−7.00000	−0.484200
$210$	0	0
$211$	14.0000	0.963800	0.481900	−	0.876226i	$-0.339947\pi$
$211$	14.0000	0.963800	0.481900	+	0.876226i	$0.339947\pi$
$212$	0	0
$213$	12.0000	0.822226
$214$	0	0
$215$	−3.00000	−0.204598
$216$	0	0
$217$	24.0000	1.62923
$218$	0	0
$219$	14.0000	0.946032
$220$	0	0
$221$	−5.00000	−0.336336
$222$	0	0
$223$	−5.00000	−0.334825	−0.167412	−	0.985887i	$-0.553541\pi$
$223$	−5.00000	−0.334825	−0.167412	+	0.985887i	$0.553541\pi$
$224$	0	0
$225$	4.00000	0.266667
$226$	0	0
$227$	9.00000	0.597351	0.298675	−	0.954355i	$-0.403455\pi$
$227$	9.00000	0.597351	0.298675	+	0.954355i	$0.403455\pi$
$228$	0	0
$229$	−10.0000	−0.660819	−0.330409	−	0.943838i	$-0.607187\pi$
$229$	−10.0000	−0.660819	−0.330409	+	0.943838i	$0.607187\pi$
$230$	0	0
$231$	4.00000	0.263181
$232$	0	0
$233$	−1.00000	−0.0655122	−0.0327561	−	0.999463i	$-0.510428\pi$
$233$	−1.00000	−0.0655122	−0.0327561	+	0.999463i	$0.510428\pi$
$234$	0	0
$235$	−18.0000	−1.17419
$236$	0	0
$237$	10.0000	0.649570
$238$	0	0
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	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
$245$	−27.0000	−1.72497
$246$	0	0
$247$	−35.0000	−2.22700
$248$	0	0
$249$	−14.0000	−0.887214
$250$	0	0
$251$	−20.0000	−1.26239	−0.631194	−	0.775625i	$-0.717435\pi$
$251$	−20.0000	−1.26239	−0.631194	+	0.775625i	$0.717435\pi$
$252$	0	0
$253$	1.00000	0.0628695
$254$	0	0
$255$	3.00000	0.187867
$256$	0	0
$257$	−20.0000	−1.24757	−0.623783	−	0.781598i	$-0.714405\pi$
$257$	−20.0000	−1.24757	−0.623783	+	0.781598i	$0.714405\pi$
$258$	0	0
$259$	32.0000	1.98838
$260$	0	0
$261$	2.00000	0.123797
$262$	0	0
$263$	32.0000	1.97320	0.986602	−	0.163144i	$-0.0521635\pi$
$263$	32.0000	1.97320	0.986602	+	0.163144i	$0.0521635\pi$
$264$	0	0
$265$	6.00000	0.368577
$266$	0	0
$267$	−8.00000	−0.489592
$268$	0	0
$269$	15.0000	0.914566	0.457283	−	0.889321i	$-0.348823\pi$
$269$	15.0000	0.914566	0.457283	+	0.889321i	$0.348823\pi$
$270$	0	0
$271$	−25.0000	−1.51864	−0.759321	−	0.650716i	$-0.774469\pi$
$271$	−25.0000	−1.51864	−0.759321	+	0.650716i	$0.774469\pi$
$272$	0	0
$273$	20.0000	1.21046
$274$	0	0
$275$	−4.00000	−0.241209
$276$	0	0
$277$	−14.0000	−0.841178	−0.420589	−	0.907251i	$-0.638177\pi$
$277$	−14.0000	−0.841178	−0.420589	+	0.907251i	$0.638177\pi$
$278$	0	0
$279$	6.00000	0.359211
$280$	0	0
$281$	12.0000	0.715860	0.357930	−	0.933748i	$-0.383483\pi$
$281$	12.0000	0.715860	0.357930	+	0.933748i	$0.383483\pi$
$282$	0	0
$283$	−26.0000	−1.54554	−0.772770	−	0.634686i	$-0.781129\pi$
$283$	−26.0000	−1.54554	−0.772770	+	0.634686i	$0.781129\pi$
$284$	0	0
$285$	21.0000	1.24393
$286$	0	0
$287$	28.0000	1.65279
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	−12.0000	−0.703452
$292$	0	0
$293$	20.0000	1.16841	0.584206	−	0.811605i	$-0.301406\pi$
$293$	20.0000	1.16841	0.584206	+	0.811605i	$0.301406\pi$
$294$	0	0
$295$	−30.0000	−1.74667
$296$	0	0
$297$	1.00000	0.0580259
$298$	0	0
$299$	5.00000	0.289157
$300$	0	0
$301$	4.00000	0.230556
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−24.0000	−1.37424
$306$	0	0
$307$	12.0000	0.684876	0.342438	−	0.939540i	$-0.388747\pi$
$307$	12.0000	0.684876	0.342438	+	0.939540i	$0.388747\pi$
$308$	0	0
$309$	7.00000	0.398216
$310$	0	0
$311$	32.0000	1.81455	0.907277	−	0.420534i	$-0.138157\pi$
$311$	32.0000	1.81455	0.907277	+	0.420534i	$0.138157\pi$
$312$	0	0
$313$	20.0000	1.13047	0.565233	−	0.824931i	$-0.308786\pi$
$313$	20.0000	1.13047	0.565233	+	0.824931i	$0.308786\pi$
$314$	0	0
$315$	−12.0000	−0.676123
$316$	0	0
$317$	−18.0000	−1.01098	−0.505490	−	0.862832i	$-0.668688\pi$
$317$	−18.0000	−1.01098	−0.505490	+	0.862832i	$0.668688\pi$
$318$	0	0
$319$	−2.00000	−0.111979
$320$	0	0
$321$	13.0000	0.725589
$322$	0	0
$323$	7.00000	0.389490
$324$	0	0
$325$	−20.0000	−1.10940
$326$	0	0
$327$	20.0000	1.10600
$328$	0	0
$329$	24.0000	1.32316
$330$	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
$335$	−36.0000	−1.96689
$336$	0	0
$337$	−6.00000	−0.326841	−0.163420	−	0.986557i	$-0.552253\pi$
$337$	−6.00000	−0.326841	−0.163420	+	0.986557i	$0.552253\pi$
$338$	0	0
$339$	19.0000	1.03194
$340$	0	0
$341$	−6.00000	−0.324918
$342$	0	0
$343$	8.00000	0.431959
$344$	0	0
$345$	−3.00000	−0.161515
$346$	0	0
$347$	−28.0000	−1.50312	−0.751559	−	0.659665i	$-0.770698\pi$
$347$	−28.0000	−1.50312	−0.751559	+	0.659665i	$0.770698\pi$
$348$	0	0
$349$	1.00000	0.0535288	0.0267644	−	0.999642i	$-0.491480\pi$
$349$	1.00000	0.0535288	0.0267644	+	0.999642i	$0.491480\pi$
$350$	0	0
$351$	5.00000	0.266880
$352$	0	0
$353$	34.0000	1.80964	0.904819	−	0.425797i	$-0.140006\pi$
$353$	34.0000	1.80964	0.904819	+	0.425797i	$0.140006\pi$
$354$	0	0
$355$	36.0000	1.91068
$356$	0	0
$357$	−4.00000	−0.211702
$358$	0	0
$359$	12.0000	0.633336	0.316668	−	0.948536i	$-0.397436\pi$
$359$	12.0000	0.633336	0.316668	+	0.948536i	$0.397436\pi$
$360$	0	0
$361$	30.0000	1.57895
$362$	0	0
$363$	10.0000	0.524864
$364$	0	0
$365$	42.0000	2.19838
$366$	0	0
$367$	28.0000	1.46159	0.730794	−	0.682598i	$-0.239150\pi$
$367$	28.0000	1.46159	0.730794	+	0.682598i	$0.239150\pi$
$368$	0	0
$369$	7.00000	0.364405
$370$	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
$375$	−3.00000	−0.154919
$376$	0	0
$377$	−10.0000	−0.515026
$378$	0	0
$379$	−8.00000	−0.410932	−0.205466	−	0.978664i	$-0.565871\pi$
$379$	−8.00000	−0.410932	−0.205466	+	0.978664i	$0.565871\pi$
$380$	0	0
$381$	−7.00000	−0.358621
$382$	0	0
$383$	0	0		−	1.00000i	$-0.5\pi$
$383$	0	0			1.00000i	$0.5\pi$
$384$	0	0
$385$	12.0000	0.611577
$386$	0	0
$387$	1.00000	0.0508329
$388$	0	0
$389$	8.00000	0.405616	0.202808	−	0.979219i	$-0.434993\pi$
$389$	8.00000	0.405616	0.202808	+	0.979219i	$0.434993\pi$
$390$	0	0
$391$	−1.00000	−0.0505722
$392$	0	0
$393$	−1.00000	−0.0504433
$394$	0	0
$395$	30.0000	1.50946
$396$	0	0
$397$	−8.00000	−0.401508	−0.200754	−	0.979642i	$-0.564339\pi$
$397$	−8.00000	−0.401508	−0.200754	+	0.979642i	$0.564339\pi$
$398$	0	0
$399$	−28.0000	−1.40175
$400$	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
$403$	−30.0000	−1.49441
$404$	0	0
$405$	−3.00000	−0.149071
$406$	0	0
$407$	−8.00000	−0.396545
$408$	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
$411$	−18.0000	−0.887875
$412$	0	0
$413$	40.0000	1.96827
$414$	0	0
$415$	−42.0000	−2.06170
$416$	0	0
$417$	−14.0000	−0.685583
$418$	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
$421$	19.0000	0.926003	0.463002	−	0.886357i	$-0.346772\pi$
$421$	19.0000	0.926003	0.463002	+	0.886357i	$0.346772\pi$
$422$	0	0
$423$	6.00000	0.291730
$424$	0	0
$425$	4.00000	0.194029
$426$	0	0
$427$	32.0000	1.54859
$428$	0	0
$429$	−5.00000	−0.241402
$430$	0	0
$431$	−8.00000	−0.385346	−0.192673	−	0.981263i	$-0.561716\pi$
$431$	−8.00000	−0.385346	−0.192673	+	0.981263i	$0.561716\pi$
$432$	0	0
$433$	−9.00000	−0.432512	−0.216256	−	0.976337i	$-0.569385\pi$
$433$	−9.00000	−0.432512	−0.216256	+	0.976337i	$0.569385\pi$
$434$	0	0
$435$	6.00000	0.287678
$436$	0	0
$437$	−7.00000	−0.334855
$438$	0	0
$439$	−20.0000	−0.954548	−0.477274	−	0.878755i	$-0.658375\pi$
$439$	−20.0000	−0.954548	−0.477274	+	0.878755i	$0.658375\pi$
$440$	0	0
$441$	9.00000	0.428571
$442$	0	0
$443$	−6.00000	−0.285069	−0.142534	−	0.989790i	$-0.545525\pi$
$443$	−6.00000	−0.285069	−0.142534	+	0.989790i	$0.545525\pi$
$444$	0	0
$445$	−24.0000	−1.13771
$446$	0	0
$447$	−6.00000	−0.283790
$448$	0	0
$449$	−10.0000	−0.471929	−0.235965	−	0.971762i	$-0.575825\pi$
$449$	−10.0000	−0.471929	−0.235965	+	0.971762i	$0.575825\pi$
$450$	0	0
$451$	−7.00000	−0.329617
$452$	0	0
$453$	0	0
$454$	0	0
$455$	60.0000	2.81284
$456$	0	0
$457$	−11.0000	−0.514558	−0.257279	−	0.966337i	$-0.582826\pi$
$457$	−11.0000	−0.514558	−0.257279	+	0.966337i	$0.582826\pi$
$458$	0	0
$459$	−1.00000	−0.0466760
$460$	0	0
$461$	34.0000	1.58354	0.791769	−	0.610821i	$-0.209160\pi$
$461$	34.0000	1.58354	0.791769	+	0.610821i	$0.209160\pi$
$462$	0	0
$463$	−24.0000	−1.11537	−0.557687	−	0.830051i	$-0.688311\pi$
$463$	−24.0000	−1.11537	−0.557687	+	0.830051i	$0.688311\pi$
$464$	0	0
$465$	18.0000	0.834730
$466$	0	0
$467$	−30.0000	−1.38823	−0.694117	−	0.719862i	$-0.744205\pi$
$467$	−30.0000	−1.38823	−0.694117	+	0.719862i	$0.744205\pi$
$468$	0	0
$469$	48.0000	2.21643
$470$	0	0
$471$	9.00000	0.414698
$472$	0	0
$473$	−1.00000	−0.0459800
$474$	0	0
$475$	28.0000	1.28473
$476$	0	0
$477$	−2.00000	−0.0915737
$478$	0	0
$479$	3.00000	0.137073	0.0685367	−	0.997649i	$-0.478167\pi$
$479$	3.00000	0.137073	0.0685367	+	0.997649i	$0.478167\pi$
$480$	0	0
$481$	−40.0000	−1.82384
$482$	0	0
$483$	4.00000	0.182006
$484$	0	0
$485$	−36.0000	−1.63468
$486$	0	0
$487$	−6.00000	−0.271886	−0.135943	−	0.990717i	$-0.543406\pi$
$487$	−6.00000	−0.271886	−0.135943	+	0.990717i	$0.543406\pi$
$488$	0	0
$489$	−6.00000	−0.271329
$490$	0	0
$491$	14.0000	0.631811	0.315906	−	0.948791i	$-0.397692\pi$
$491$	14.0000	0.631811	0.315906	+	0.948791i	$0.397692\pi$
$492$	0	0
$493$	2.00000	0.0900755
$494$	0	0
$495$	3.00000	0.134840
$496$	0	0
$497$	−48.0000	−2.15309
$498$	0	0
$499$	2.00000	0.0895323	0.0447661	−	0.998997i	$-0.485746\pi$
$499$	2.00000	0.0895323	0.0447661	+	0.998997i	$0.485746\pi$
$500$	0	0
$501$	−3.00000	−0.134030
$502$	0	0
$503$	9.00000	0.401290	0.200645	−	0.979664i	$-0.435696\pi$
$503$	9.00000	0.401290	0.200645	+	0.979664i	$0.435696\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−12.0000	−0.532939
$508$	0	0
$509$	−12.0000	−0.531891	−0.265945	−	0.963988i	$-0.585684\pi$
$509$	−12.0000	−0.531891	−0.265945	+	0.963988i	$0.585684\pi$
$510$	0	0
$511$	−56.0000	−2.47729
$512$	0	0
$513$	−7.00000	−0.309058
$514$	0	0
$515$	21.0000	0.925371
$516$	0	0
$517$	−6.00000	−0.263880
$518$	0	0
$519$	23.0000	1.00959
$520$	0	0
$521$	−1.00000	−0.0438108	−0.0219054	−	0.999760i	$-0.506973\pi$
$521$	−1.00000	−0.0438108	−0.0219054	+	0.999760i	$0.506973\pi$
$522$	0	0
$523$	−4.00000	−0.174908	−0.0874539	−	0.996169i	$-0.527873\pi$
$523$	−4.00000	−0.174908	−0.0874539	+	0.996169i	$0.527873\pi$
$524$	0	0
$525$	−16.0000	−0.698297
$526$	0	0
$527$	6.00000	0.261364
$528$	0	0
$529$	−22.0000	−0.956522
$530$	0	0
$531$	10.0000	0.433963
$532$	0	0
$533$	−35.0000	−1.51602
$534$	0	0
$535$	39.0000	1.68612
$536$	0	0
$537$	−10.0000	−0.431532
$538$	0	0
$539$	−9.00000	−0.387657
$540$	0	0
$541$	20.0000	0.859867	0.429934	−	0.902861i	$-0.358537\pi$
$541$	20.0000	0.859867	0.429934	+	0.902861i	$0.358537\pi$
$542$	0	0
$543$	−10.0000	−0.429141
$544$	0	0
$545$	60.0000	2.57012
$546$	0	0
$547$	0	0		−	1.00000i	$-0.5\pi$
$547$	0	0			1.00000i	$0.5\pi$
$548$	0	0
$549$	8.00000	0.341432
$550$	0	0
$551$	14.0000	0.596420
$552$	0	0
$553$	−40.0000	−1.70097
$554$	0	0
$555$	24.0000	1.01874
$556$	0	0
$557$	38.0000	1.61011	0.805056	−	0.593199i	$-0.202135\pi$
$557$	38.0000	1.61011	0.805056	+	0.593199i	$0.202135\pi$
$558$	0	0
$559$	−5.00000	−0.211477
$560$	0	0
$561$	1.00000	0.0422200
$562$	0	0
$563$	2.00000	0.0842900	0.0421450	−	0.999112i	$-0.486581\pi$
$563$	2.00000	0.0842900	0.0421450	+	0.999112i	$0.486581\pi$
$564$	0	0
$565$	57.0000	2.39801
$566$	0	0
$567$	4.00000	0.167984
$568$	0	0
$569$	28.0000	1.17382	0.586911	−	0.809652i	$-0.300344\pi$
$569$	28.0000	1.17382	0.586911	+	0.809652i	$0.300344\pi$
$570$	0	0
$571$	44.0000	1.84134	0.920671	−	0.390339i	$-0.127642\pi$
$571$	44.0000	1.84134	0.920671	+	0.390339i	$0.127642\pi$
$572$	0	0
$573$	6.00000	0.250654
$574$	0	0
$575$	−4.00000	−0.166812
$576$	0	0
$577$	−39.0000	−1.62359	−0.811796	−	0.583942i	$-0.801510\pi$
$577$	−39.0000	−1.62359	−0.811796	+	0.583942i	$0.801510\pi$
$578$	0	0
$579$	−6.00000	−0.249351
$580$	0	0
$581$	56.0000	2.32327
$582$	0	0
$583$	2.00000	0.0828315
$584$	0	0
$585$	15.0000	0.620174
$586$	0	0
$587$	−20.0000	−0.825488	−0.412744	−	0.910847i	$-0.635430\pi$
$587$	−20.0000	−0.825488	−0.412744	+	0.910847i	$0.635430\pi$
$588$	0	0
$589$	42.0000	1.73058
$590$	0	0
$591$	27.0000	1.11063
$592$	0	0
$593$	−46.0000	−1.88899	−0.944497	−	0.328521i	$-0.893450\pi$
$593$	−46.0000	−1.88899	−0.944497	+	0.328521i	$0.893450\pi$
$594$	0	0
$595$	−12.0000	−0.491952
$596$	0	0
$597$	−4.00000	−0.163709
$598$	0	0
$599$	10.0000	0.408589	0.204294	−	0.978909i	$-0.434510\pi$
$599$	10.0000	0.408589	0.204294	+	0.978909i	$0.434510\pi$
$600$	0	0
$601$	10.0000	0.407909	0.203954	−	0.978980i	$-0.434621\pi$
$601$	10.0000	0.407909	0.203954	+	0.978980i	$0.434621\pi$
$602$	0	0
$603$	12.0000	0.488678
$604$	0	0
$605$	30.0000	1.21967
$606$	0	0
$607$	−22.0000	−0.892952	−0.446476	−	0.894795i	$-0.647321\pi$
$607$	−22.0000	−0.892952	−0.446476	+	0.894795i	$0.647321\pi$
$608$	0	0
$609$	−8.00000	−0.324176
$610$	0	0
$611$	−30.0000	−1.21367
$612$	0	0
$613$	−9.00000	−0.363507	−0.181753	−	0.983344i	$-0.558177\pi$
$613$	−9.00000	−0.363507	−0.181753	+	0.983344i	$0.558177\pi$
$614$	0	0
$615$	21.0000	0.846802
$616$	0	0
$617$	14.0000	0.563619	0.281809	−	0.959470i	$-0.409065\pi$
$617$	14.0000	0.563619	0.281809	+	0.959470i	$0.409065\pi$
$618$	0	0
$619$	−2.00000	−0.0803868	−0.0401934	−	0.999192i	$-0.512797\pi$
$619$	−2.00000	−0.0803868	−0.0401934	+	0.999192i	$0.512797\pi$
$620$	0	0
$621$	1.00000	0.0401286
$622$	0	0
$623$	32.0000	1.28205
$624$	0	0
$625$	−29.0000	−1.16000
$626$	0	0
$627$	7.00000	0.279553
$628$	0	0
$629$	8.00000	0.318981
$630$	0	0
$631$	23.0000	0.915616	0.457808	−	0.889051i	$-0.348635\pi$
$631$	23.0000	0.915616	0.457808	+	0.889051i	$0.348635\pi$
$632$	0	0
$633$	−14.0000	−0.556450
$634$	0	0
$635$	−21.0000	−0.833360
$636$	0	0
$637$	−45.0000	−1.78296
$638$	0	0
$639$	−12.0000	−0.474713
$640$	0	0
$641$	5.00000	0.197488	0.0987441	−	0.995113i	$-0.468517\pi$
$641$	5.00000	0.197488	0.0987441	+	0.995113i	$0.468517\pi$
$642$	0	0
$643$	20.0000	0.788723	0.394362	−	0.918955i	$-0.370966\pi$
$643$	20.0000	0.788723	0.394362	+	0.918955i	$0.370966\pi$
$644$	0	0
$645$	3.00000	0.118125
$646$	0	0
$647$	18.0000	0.707653	0.353827	−	0.935311i	$-0.384880\pi$
$647$	18.0000	0.707653	0.353827	+	0.935311i	$0.384880\pi$
$648$	0	0
$649$	−10.0000	−0.392534
$650$	0	0
$651$	−24.0000	−0.940634
$652$	0	0
$653$	−19.0000	−0.743527	−0.371764	−	0.928327i	$-0.621247\pi$
$653$	−19.0000	−0.743527	−0.371764	+	0.928327i	$0.621247\pi$
$654$	0	0
$655$	−3.00000	−0.117220
$656$	0	0
$657$	−14.0000	−0.546192
$658$	0	0
$659$	−2.00000	−0.0779089	−0.0389545	−	0.999241i	$-0.512403\pi$
$659$	−2.00000	−0.0779089	−0.0389545	+	0.999241i	$0.512403\pi$
$660$	0	0
$661$	−3.00000	−0.116686	−0.0583432	−	0.998297i	$-0.518582\pi$
$661$	−3.00000	−0.116686	−0.0583432	+	0.998297i	$0.518582\pi$
$662$	0	0
$663$	5.00000	0.194184
$664$	0	0
$665$	−84.0000	−3.25738
$666$	0	0
$667$	−2.00000	−0.0774403
$668$	0	0
$669$	5.00000	0.193311
$670$	0	0
$671$	−8.00000	−0.308837
$672$	0	0
$673$	2.00000	0.0770943	0.0385472	−	0.999257i	$-0.487727\pi$
$673$	2.00000	0.0770943	0.0385472	+	0.999257i	$0.487727\pi$
$674$	0	0
$675$	−4.00000	−0.153960
$676$	0	0
$677$	11.0000	0.422764	0.211382	−	0.977403i	$-0.432204\pi$
$677$	11.0000	0.422764	0.211382	+	0.977403i	$0.432204\pi$
$678$	0	0
$679$	48.0000	1.84207
$680$	0	0
$681$	−9.00000	−0.344881
$682$	0	0
$683$	−39.0000	−1.49229	−0.746147	−	0.665782i	$-0.768098\pi$
$683$	−39.0000	−1.49229	−0.746147	+	0.665782i	$0.768098\pi$
$684$	0	0
$685$	−54.0000	−2.06323
$686$	0	0
$687$	10.0000	0.381524
$688$	0	0
$689$	10.0000	0.380970
$690$	0	0
$691$	−28.0000	−1.06517	−0.532585	−	0.846376i	$-0.678779\pi$
$691$	−28.0000	−1.06517	−0.532585	+	0.846376i	$0.678779\pi$
$692$	0	0
$693$	−4.00000	−0.151947
$694$	0	0
$695$	−42.0000	−1.59315
$696$	0	0
$697$	7.00000	0.265144
$698$	0	0
$699$	1.00000	0.0378235
$700$	0	0
$701$	−8.00000	−0.302156	−0.151078	−	0.988522i	$-0.548274\pi$
$701$	−8.00000	−0.302156	−0.151078	+	0.988522i	$0.548274\pi$
$702$	0	0
$703$	56.0000	2.11208
$704$	0	0
$705$	18.0000	0.677919
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−34.0000	−1.27690	−0.638448	−	0.769665i	$-0.720423\pi$
$709$	−34.0000	−1.27690	−0.638448	+	0.769665i	$0.720423\pi$
$710$	0	0
$711$	−10.0000	−0.375029
$712$	0	0
$713$	−6.00000	−0.224702
$714$	0	0
$715$	−15.0000	−0.560968
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−27.0000	−1.00693	−0.503465	−	0.864016i	$-0.667942\pi$
$719$	−27.0000	−1.00693	−0.503465	+	0.864016i	$0.667942\pi$
$720$	0	0
$721$	−28.0000	−1.04277
$722$	0	0
$723$	0	0
$724$	0	0
$725$	8.00000	0.297113
$726$	0	0
$727$	8.00000	0.296704	0.148352	−	0.988935i	$-0.452603\pi$
$727$	8.00000	0.296704	0.148352	+	0.988935i	$0.452603\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	1.00000	0.0369863
$732$	0	0
$733$	42.0000	1.55131	0.775653	−	0.631160i	$-0.217421\pi$
$733$	42.0000	1.55131	0.775653	+	0.631160i	$0.217421\pi$
$734$	0	0
$735$	27.0000	0.995910
$736$	0	0
$737$	−12.0000	−0.442026
$738$	0	0
$739$	15.0000	0.551784	0.275892	−	0.961189i	$-0.411027\pi$
$739$	15.0000	0.551784	0.275892	+	0.961189i	$0.411027\pi$
$740$	0	0
$741$	35.0000	1.28576
$742$	0	0
$743$	8.00000	0.293492	0.146746	−	0.989174i	$-0.453120\pi$
$743$	8.00000	0.293492	0.146746	+	0.989174i	$0.453120\pi$
$744$	0	0
$745$	−18.0000	−0.659469
$746$	0	0
$747$	14.0000	0.512233
$748$	0	0
$749$	−52.0000	−1.90004
$750$	0	0
$751$	−26.0000	−0.948753	−0.474377	−	0.880322i	$-0.657327\pi$
$751$	−26.0000	−0.948753	−0.474377	+	0.880322i	$0.657327\pi$
$752$	0	0
$753$	20.0000	0.728841
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−23.0000	−0.835949	−0.417975	−	0.908459i	$-0.637260\pi$
$757$	−23.0000	−0.835949	−0.417975	+	0.908459i	$0.637260\pi$
$758$	0	0
$759$	−1.00000	−0.0362977
$760$	0	0
$761$	22.0000	0.797499	0.398750	−	0.917060i	$-0.369444\pi$
$761$	22.0000	0.797499	0.398750	+	0.917060i	$0.369444\pi$
$762$	0	0
$763$	−80.0000	−2.89619
$764$	0	0
$765$	−3.00000	−0.108465
$766$	0	0
$767$	−50.0000	−1.80540
$768$	0	0
$769$	−23.0000	−0.829401	−0.414701	−	0.909958i	$-0.636114\pi$
$769$	−23.0000	−0.829401	−0.414701	+	0.909958i	$0.636114\pi$
$770$	0	0
$771$	20.0000	0.720282
$772$	0	0
$773$	−4.00000	−0.143870	−0.0719350	−	0.997409i	$-0.522917\pi$
$773$	−4.00000	−0.143870	−0.0719350	+	0.997409i	$0.522917\pi$
$774$	0	0
$775$	24.0000	0.862105
$776$	0	0
$777$	−32.0000	−1.14799
$778$	0	0
$779$	49.0000	1.75561
$780$	0	0
$781$	12.0000	0.429394
$782$	0	0
$783$	−2.00000	−0.0714742
$784$	0	0
$785$	27.0000	0.963671
$786$	0	0
$787$	−36.0000	−1.28326	−0.641631	−	0.767014i	$-0.721742\pi$
$787$	−36.0000	−1.28326	−0.641631	+	0.767014i	$0.721742\pi$
$788$	0	0
$789$	−32.0000	−1.13923
$790$	0	0
$791$	−76.0000	−2.70225
$792$	0	0
$793$	−40.0000	−1.42044
$794$	0	0
$795$	−6.00000	−0.212798
$796$	0	0
$797$	48.0000	1.70025	0.850124	−	0.526583i	$-0.176527\pi$
$797$	48.0000	1.70025	0.850124	+	0.526583i	$0.176527\pi$
$798$	0	0
$799$	6.00000	0.212265
$800$	0	0
$801$	8.00000	0.282666
$802$	0	0
$803$	14.0000	0.494049
$804$	0	0
$805$	12.0000	0.422944
$806$	0	0
$807$	−15.0000	−0.528025
$808$	0	0
$809$	−9.00000	−0.316423	−0.158212	−	0.987405i	$-0.550573\pi$
$809$	−9.00000	−0.316423	−0.158212	+	0.987405i	$0.550573\pi$
$810$	0	0
$811$	−2.00000	−0.0702295	−0.0351147	−	0.999383i	$-0.511180\pi$
$811$	−2.00000	−0.0702295	−0.0351147	+	0.999383i	$0.511180\pi$
$812$	0	0
$813$	25.0000	0.876788
$814$	0	0
$815$	−18.0000	−0.630512
$816$	0	0
$817$	7.00000	0.244899
$818$	0	0
$819$	−20.0000	−0.698857
$820$	0	0
$821$	5.00000	0.174501	0.0872506	−	0.996186i	$-0.472192\pi$
$821$	5.00000	0.174501	0.0872506	+	0.996186i	$0.472192\pi$
$822$	0	0
$823$	−48.0000	−1.67317	−0.836587	−	0.547833i	$-0.815453\pi$
$823$	−48.0000	−1.67317	−0.836587	+	0.547833i	$0.815453\pi$
$824$	0	0
$825$	4.00000	0.139262
$826$	0	0
$827$	−47.0000	−1.63435	−0.817175	−	0.576390i	$-0.804461\pi$
$827$	−47.0000	−1.63435	−0.817175	+	0.576390i	$0.804461\pi$
$828$	0	0
$829$	54.0000	1.87550	0.937749	−	0.347314i	$-0.112906\pi$
$829$	54.0000	1.87550	0.937749	+	0.347314i	$0.112906\pi$
$830$	0	0
$831$	14.0000	0.485655
$832$	0	0
$833$	9.00000	0.311832
$834$	0	0
$835$	−9.00000	−0.311458
$836$	0	0
$837$	−6.00000	−0.207390
$838$	0	0
$839$	−7.00000	−0.241667	−0.120833	−	0.992673i	$-0.538557\pi$
$839$	−7.00000	−0.241667	−0.120833	+	0.992673i	$0.538557\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	0	0
$843$	−12.0000	−0.413302
$844$	0	0
$845$	−36.0000	−1.23844
$846$	0	0
$847$	−40.0000	−1.37442
$848$	0	0
$849$	26.0000	0.892318
$850$	0	0
$851$	−8.00000	−0.274236
$852$	0	0
$853$	2.00000	0.0684787	0.0342393	−	0.999414i	$-0.489099\pi$
$853$	2.00000	0.0684787	0.0342393	+	0.999414i	$0.489099\pi$
$854$	0	0
$855$	−21.0000	−0.718185
$856$	0	0
$857$	−30.0000	−1.02478	−0.512390	−	0.858753i	$-0.671240\pi$
$857$	−30.0000	−1.02478	−0.512390	+	0.858753i	$0.671240\pi$
$858$	0	0
$859$	20.0000	0.682391	0.341196	−	0.939992i	$-0.389168\pi$
$859$	20.0000	0.682391	0.341196	+	0.939992i	$0.389168\pi$
$860$	0	0
$861$	−28.0000	−0.954237
$862$	0	0
$863$	24.0000	0.816970	0.408485	−	0.912765i	$-0.366057\pi$
$863$	24.0000	0.816970	0.408485	+	0.912765i	$0.366057\pi$
$864$	0	0
$865$	69.0000	2.34607
$866$	0	0
$867$	−1.00000	−0.0339618
$868$	0	0
$869$	10.0000	0.339227
$870$	0	0
$871$	−60.0000	−2.03302
$872$	0	0
$873$	12.0000	0.406138
$874$	0	0
$875$	12.0000	0.405674
$876$	0	0
$877$	−54.0000	−1.82345	−0.911725	−	0.410801i	$-0.865249\pi$
$877$	−54.0000	−1.82345	−0.911725	+	0.410801i	$0.865249\pi$
$878$	0	0
$879$	−20.0000	−0.674583
$880$	0	0
$881$	30.0000	1.01073	0.505363	−	0.862907i	$-0.331359\pi$
$881$	30.0000	1.01073	0.505363	+	0.862907i	$0.331359\pi$
$882$	0	0
$883$	−5.00000	−0.168263	−0.0841317	−	0.996455i	$-0.526812\pi$
$883$	−5.00000	−0.168263	−0.0841317	+	0.996455i	$0.526812\pi$
$884$	0	0
$885$	30.0000	1.00844
$886$	0	0
$887$	9.00000	0.302190	0.151095	−	0.988519i	$-0.451720\pi$
$887$	9.00000	0.302190	0.151095	+	0.988519i	$0.451720\pi$
$888$	0	0
$889$	28.0000	0.939090
$890$	0	0
$891$	−1.00000	−0.0335013
$892$	0	0
$893$	42.0000	1.40548
$894$	0	0
$895$	−30.0000	−1.00279
$896$	0	0
$897$	−5.00000	−0.166945
$898$	0	0
$899$	12.0000	0.400222
$900$	0	0
$901$	−2.00000	−0.0666297
$902$	0	0
$903$	−4.00000	−0.133112
$904$	0	0
$905$	−30.0000	−0.997234
$906$	0	0
$907$	−26.0000	−0.863316	−0.431658	−	0.902037i	$-0.642071\pi$
$907$	−26.0000	−0.863316	−0.431658	+	0.902037i	$0.642071\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−21.0000	−0.695761	−0.347881	−	0.937539i	$-0.613099\pi$
$911$	−21.0000	−0.695761	−0.347881	+	0.937539i	$0.613099\pi$
$912$	0	0
$913$	−14.0000	−0.463332
$914$	0	0
$915$	24.0000	0.793416
$916$	0	0
$917$	4.00000	0.132092
$918$	0	0
$919$	39.0000	1.28649	0.643246	−	0.765660i	$-0.277587\pi$
$919$	39.0000	1.28649	0.643246	+	0.765660i	$0.277587\pi$
$920$	0	0
$921$	−12.0000	−0.395413
$922$	0	0
$923$	60.0000	1.97492
$924$	0	0
$925$	32.0000	1.05215
$926$	0	0
$927$	−7.00000	−0.229910
$928$	0	0
$929$	−3.00000	−0.0984268	−0.0492134	−	0.998788i	$-0.515671\pi$
$929$	−3.00000	−0.0984268	−0.0492134	+	0.998788i	$0.515671\pi$
$930$	0	0
$931$	63.0000	2.06474
$932$	0	0
$933$	−32.0000	−1.04763
$934$	0	0
$935$	3.00000	0.0981105
$936$	0	0
$937$	−14.0000	−0.457360	−0.228680	−	0.973502i	$-0.573441\pi$
$937$	−14.0000	−0.457360	−0.228680	+	0.973502i	$0.573441\pi$
$938$	0	0
$939$	−20.0000	−0.652675
$940$	0	0
$941$	−50.0000	−1.62995	−0.814977	−	0.579494i	$-0.803250\pi$
$941$	−50.0000	−1.62995	−0.814977	+	0.579494i	$0.803250\pi$
$942$	0	0
$943$	−7.00000	−0.227951
$944$	0	0
$945$	12.0000	0.390360
$946$	0	0
$947$	36.0000	1.16984	0.584921	−	0.811090i	$-0.301125\pi$
$947$	36.0000	1.16984	0.584921	+	0.811090i	$0.301125\pi$
$948$	0	0
$949$	70.0000	2.27230
$950$	0	0
$951$	18.0000	0.583690
$952$	0	0
$953$	−28.0000	−0.907009	−0.453504	−	0.891254i	$-0.649826\pi$
$953$	−28.0000	−0.907009	−0.453504	+	0.891254i	$0.649826\pi$
$954$	0	0
$955$	18.0000	0.582466
$956$	0	0
$957$	2.00000	0.0646508
$958$	0	0
$959$	72.0000	2.32500
$960$	0	0
$961$	5.00000	0.161290
$962$	0	0
$963$	−13.0000	−0.418919
$964$	0	0
$965$	−18.0000	−0.579441
$966$	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
$969$	−7.00000	−0.224872
$970$	0	0
$971$	−34.0000	−1.09111	−0.545556	−	0.838074i	$-0.683681\pi$
$971$	−34.0000	−1.09111	−0.545556	+	0.838074i	$0.683681\pi$
$972$	0	0
$973$	56.0000	1.79528
$974$	0	0
$975$	20.0000	0.640513
$976$	0	0
$977$	−26.0000	−0.831814	−0.415907	−	0.909407i	$-0.636536\pi$
$977$	−26.0000	−0.831814	−0.415907	+	0.909407i	$0.636536\pi$
$978$	0	0
$979$	−8.00000	−0.255681
$980$	0	0
$981$	−20.0000	−0.638551
$982$	0	0
$983$	9.00000	0.287055	0.143528	−	0.989646i	$-0.454155\pi$
$983$	9.00000	0.287055	0.143528	+	0.989646i	$0.454155\pi$
$984$	0	0
$985$	81.0000	2.58087
$986$	0	0
$987$	−24.0000	−0.763928
$988$	0	0
$989$	−1.00000	−0.0317982
$990$	0	0
$991$	40.0000	1.27064	0.635321	−	0.772248i	$-0.280868\pi$
$991$	40.0000	1.27064	0.635321	+	0.772248i	$0.280868\pi$
$992$	0	0
$993$	−11.0000	−0.349074
$994$	0	0
$995$	−12.0000	−0.380426
$996$	0	0
$997$	−2.00000	−0.0633406	−0.0316703	−	0.999498i	$-0.510083\pi$
$997$	−2.00000	−0.0633406	−0.0316703	+	0.999498i	$0.510083\pi$
$998$	0	0
$999$	−8.00000	−0.253109

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	816.2.a.a.1.1		1
3.2	odd	2		2448.2.a.s.1.1		1
4.3	odd	2		408.2.a.b.1.1	✓	1
8.3	odd	2		3264.2.a.n.1.1		1
8.5	even	2		3264.2.a.be.1.1		1
12.11	even	2		1224.2.a.h.1.1		1
24.5	odd	2		9792.2.a.h.1.1		1
24.11	even	2		9792.2.a.c.1.1		1
68.67	odd	2		6936.2.a.h.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
408.2.a.b.1.1	✓	1	4.3	odd	2
816.2.a.a.1.1		1	1.1	even	1	trivial
1224.2.a.h.1.1		1	12.11	even	2
2448.2.a.s.1.1		1	3.2	odd	2
3264.2.a.n.1.1		1	8.3	odd	2
3264.2.a.be.1.1		1	8.5	even	2
6936.2.a.h.1.1		1	68.67	odd	2
9792.2.a.c.1.1		1	24.11	even	2
9792.2.a.h.1.1		1	24.5	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:	\( N \)	\(=\)	\( 816 = 2^{4} \cdot 3 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	816.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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