LMFDB - Embedded newform 208.4.a.f.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [208,4,Mod(1,208)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(208, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("208.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 208 = 2^{4} \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	208.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:	$12.2723972812$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 52)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	208.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+3.00000 q^{3} -13.0000 q^{5} +11.0000 q^{7} -18.0000 q^{9} +O(q^{10})$

$q+3.00000 q^{3} -13.0000 q^{5} +11.0000 q^{7} -18.0000 q^{9} +2.00000 q^{11} -13.0000 q^{13} -39.0000 q^{15} -51.0000 q^{17} -150.000 q^{19} +33.0000 q^{21} +4.00000 q^{23} +44.0000 q^{25} -135.000 q^{27} -118.000 q^{29} +116.000 q^{31} +6.00000 q^{33} -143.000 q^{35} +63.0000 q^{37} -39.0000 q^{39} -288.000 q^{41} +293.000 q^{43} +234.000 q^{45} +335.000 q^{47} -222.000 q^{49} -153.000 q^{51} -708.000 q^{53} -26.0000 q^{55} -450.000 q^{57} -566.000 q^{59} +904.000 q^{61} -198.000 q^{63} +169.000 q^{65} -382.000 q^{67} +12.0000 q^{69} -7.00000 q^{71} +518.000 q^{73} +132.000 q^{75} +22.0000 q^{77} +100.000 q^{79} +81.0000 q^{81} +1440.00 q^{83} +663.000 q^{85} -354.000 q^{87} +1254.00 q^{89} -143.000 q^{91} +348.000 q^{93} +1950.00 q^{95} +1262.00 q^{97} -36.0000 q^{99} +O(q^{100})$

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
$3$	3.00000	0.577350	0.288675	−	0.957427i	$-0.406785\pi$
$3$	3.00000	0.577350	0.288675	+	0.957427i	$0.406785\pi$
$4$	0	0
$5$	−13.0000	−1.16276	−0.581378	−	0.813634i	$-0.697486\pi$
$5$	−13.0000	−1.16276	−0.581378	+	0.813634i	$0.697486\pi$
$6$	0	0
$7$	11.0000	0.593944	0.296972	−	0.954886i	$-0.404023\pi$
$7$	11.0000	0.593944	0.296972	+	0.954886i	$0.404023\pi$
$8$	0	0
$9$	−18.0000	−0.666667
$10$	0	0
$11$	2.00000	0.0548202	0.0274101	−	0.999624i	$-0.491274\pi$
$11$	2.00000	0.0548202	0.0274101	+	0.999624i	$0.491274\pi$
$12$	0	0
$13$	−13.0000	−0.277350
$13$	−13.0000	−0.277350
$14$	0	0
$15$	−39.0000	−0.671317
$16$	0	0
$17$	−51.0000	−0.727607	−0.363803	−	0.931476i	$-0.618522\pi$
$17$	−51.0000	−0.727607	−0.363803	+	0.931476i	$0.618522\pi$
$18$	0	0
$19$	−150.000	−1.81118	−0.905588	−	0.424158i	$-0.860570\pi$
$19$	−150.000	−1.81118	−0.905588	+	0.424158i	$0.860570\pi$
$20$	0	0
$21$	33.0000	0.342914
$22$	0	0
$23$	4.00000	0.0362634	0.0181317	−	0.999836i	$-0.494228\pi$
$23$	4.00000	0.0362634	0.0181317	+	0.999836i	$0.494228\pi$
$24$	0	0
$25$	44.0000	0.352000
$26$	0	0
$27$	−135.000	−0.962250
$28$	0	0
$29$	−118.000	−0.755588	−0.377794	−	0.925890i	$-0.623317\pi$
$29$	−118.000	−0.755588	−0.377794	+	0.925890i	$0.623317\pi$
$30$	0	0
$31$	116.000	0.672071	0.336036	−	0.941849i	$-0.390914\pi$
$31$	116.000	0.672071	0.336036	+	0.941849i	$0.390914\pi$
$32$	0	0
$33$	6.00000	0.0316505
$34$	0	0
$35$	−143.000	−0.690612
$36$	0	0
$37$	63.0000	0.279923	0.139961	−	0.990157i	$-0.455302\pi$
$37$	63.0000	0.279923	0.139961	+	0.990157i	$0.455302\pi$
$38$	0	0
$39$	−39.0000	−0.160128
$40$	0	0
$41$	−288.000	−1.09703	−0.548513	−	0.836142i	$-0.684806\pi$
$41$	−288.000	−1.09703	−0.548513	+	0.836142i	$0.684806\pi$
$42$	0	0
$43$	293.000	1.03912	0.519559	−	0.854435i	$-0.326096\pi$
$43$	293.000	1.03912	0.519559	+	0.854435i	$0.326096\pi$
$44$	0	0
$45$	234.000	0.775170
$46$	0	0
$47$	335.000	1.03968	0.519838	−	0.854265i	$-0.325992\pi$
$47$	335.000	1.03968	0.519838	+	0.854265i	$0.325992\pi$
$48$	0	0
$49$	−222.000	−0.647230
$50$	0	0
$51$	−153.000	−0.420084
$52$	0	0
$53$	−708.000	−1.83493	−0.917465	−	0.397817i	$-0.869768\pi$
$53$	−708.000	−1.83493	−0.917465	+	0.397817i	$0.869768\pi$
$54$	0	0
$55$	−26.0000	−0.0637425
$56$	0	0
$57$	−450.000	−1.04568
$58$	0	0
$59$	−566.000	−1.24893	−0.624465	−	0.781052i	$-0.714683\pi$
$59$	−566.000	−1.24893	−0.624465	+	0.781052i	$0.714683\pi$
$60$	0	0
$61$	904.000	1.89746	0.948732	−	0.316081i	$-0.102367\pi$
$61$	904.000	1.89746	0.948732	+	0.316081i	$0.102367\pi$
$62$	0	0
$63$	−198.000	−0.395963
$64$	0	0
$65$	169.000	0.322490
$66$	0	0
$67$	−382.000	−0.696548	−0.348274	−	0.937393i	$-0.613232\pi$
$67$	−382.000	−0.696548	−0.348274	+	0.937393i	$0.613232\pi$
$68$	0	0
$69$	12.0000	0.0209367
$70$	0	0
$71$	−7.00000	−0.0117007	−0.00585033	−	0.999983i	$-0.501862\pi$
$71$	−7.00000	−0.0117007	−0.00585033	+	0.999983i	$0.501862\pi$
$72$	0	0
$73$	518.000	0.830511	0.415256	−	0.909705i	$-0.363692\pi$
$73$	518.000	0.830511	0.415256	+	0.909705i	$0.363692\pi$
$74$	0	0
$75$	132.000	0.203227
$76$	0	0
$77$	22.0000	0.0325602
$78$	0	0
$79$	100.000	0.142416	0.0712081	−	0.997461i	$-0.477315\pi$
$79$	100.000	0.142416	0.0712081	+	0.997461i	$0.477315\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	1440.00	1.90434	0.952172	−	0.305563i	$-0.0988446\pi$
$83$	1440.00	1.90434	0.952172	+	0.305563i	$0.0988446\pi$
$84$	0	0
$85$	663.000	0.846029
$86$	0	0
$87$	−354.000	−0.436239
$88$	0	0
$89$	1254.00	1.49353	0.746763	−	0.665091i	$-0.231607\pi$
$89$	1254.00	1.49353	0.746763	+	0.665091i	$0.231607\pi$
$90$	0	0
$91$	−143.000	−0.164730
$92$	0	0
$93$	348.000	0.388021
$94$	0	0
$95$	1950.00	2.10596
$96$	0	0
$97$	1262.00	1.32100	0.660498	−	0.750827i	$-0.270345\pi$
$97$	1262.00	1.32100	0.660498	+	0.750827i	$0.270345\pi$
$98$	0	0
$99$	−36.0000	−0.0365468
$100$	0	0
$101$	−1772.00	−1.74575	−0.872874	−	0.487945i	$-0.837747\pi$
$101$	−1772.00	−1.74575	−0.872874	+	0.487945i	$0.837747\pi$
$102$	0	0
$103$	1160.00	1.10969	0.554846	−	0.831953i	$-0.312777\pi$
$103$	1160.00	1.10969	0.554846	+	0.831953i	$0.312777\pi$
$104$	0	0
$105$	−429.000	−0.398725
$106$	0	0
$107$	−500.000	−0.451746	−0.225873	−	0.974157i	$-0.572523\pi$
$107$	−500.000	−0.451746	−0.225873	+	0.974157i	$0.572523\pi$
$108$	0	0
$109$	−329.000	−0.289105	−0.144553	−	0.989497i	$-0.546174\pi$
$109$	−329.000	−0.289105	−0.144553	+	0.989497i	$0.546174\pi$
$110$	0	0
$111$	189.000	0.161613
$112$	0	0
$113$	−766.000	−0.637692	−0.318846	−	0.947807i	$-0.603295\pi$
$113$	−766.000	−0.637692	−0.318846	+	0.947807i	$0.603295\pi$
$114$	0	0
$115$	−52.0000	−0.0421654
$116$	0	0
$117$	234.000	0.184900
$118$	0	0
$119$	−561.000	−0.432158
$120$	0	0
$121$	−1327.00	−0.996995
$122$	0	0
$123$	−864.000	−0.633368
$124$	0	0
$125$	1053.00	0.753465
$126$	0	0
$127$	2144.00	1.49803	0.749013	−	0.662556i	$-0.230528\pi$
$127$	2144.00	1.49803	0.749013	+	0.662556i	$0.230528\pi$
$128$	0	0
$129$	879.000	0.599935
$130$	0	0
$131$	−1847.00	−1.23186	−0.615928	−	0.787802i	$-0.711219\pi$
$131$	−1847.00	−1.23186	−0.615928	+	0.787802i	$0.711219\pi$
$132$	0	0
$133$	−1650.00	−1.07574
$134$	0	0
$135$	1755.00	1.11886
$136$	0	0
$137$	−564.000	−0.351721	−0.175860	−	0.984415i	$-0.556271\pi$
$137$	−564.000	−0.351721	−0.175860	+	0.984415i	$0.556271\pi$
$138$	0	0
$139$	−2719.00	−1.65916	−0.829578	−	0.558391i	$-0.811419\pi$
$139$	−2719.00	−1.65916	−0.829578	+	0.558391i	$0.811419\pi$
$140$	0	0
$141$	1005.00	0.600257
$142$	0	0
$143$	−26.0000	−0.0152044
$144$	0	0
$145$	1534.00	0.878564
$146$	0	0
$147$	−666.000	−0.373679
$148$	0	0
$149$	1254.00	0.689474	0.344737	−	0.938699i	$-0.387968\pi$
$149$	1254.00	0.689474	0.344737	+	0.938699i	$0.387968\pi$
$150$	0	0
$151$	1173.00	0.632168	0.316084	−	0.948731i	$-0.397632\pi$
$151$	1173.00	0.632168	0.316084	+	0.948731i	$0.397632\pi$
$152$	0	0
$153$	918.000	0.485071
$154$	0	0
$155$	−1508.00	−0.781455
$156$	0	0
$157$	−1714.00	−0.871287	−0.435644	−	0.900119i	$-0.643479\pi$
$157$	−1714.00	−0.871287	−0.435644	+	0.900119i	$0.643479\pi$
$158$	0	0
$159$	−2124.00	−1.05940
$160$	0	0
$161$	44.0000	0.0215384
$162$	0	0
$163$	−2300.00	−1.10521	−0.552607	−	0.833442i	$-0.686367\pi$
$163$	−2300.00	−1.10521	−0.552607	+	0.833442i	$0.686367\pi$
$164$	0	0
$165$	−78.0000	−0.0368018
$166$	0	0
$167$	−1176.00	−0.544920	−0.272460	−	0.962167i	$-0.587837\pi$
$167$	−1176.00	−0.544920	−0.272460	+	0.962167i	$0.587837\pi$
$168$	0	0
$169$	169.000	0.0769231
$170$	0	0
$171$	2700.00	1.20745
$172$	0	0
$173$	−844.000	−0.370914	−0.185457	−	0.982652i	$-0.559377\pi$
$173$	−844.000	−0.370914	−0.185457	+	0.982652i	$0.559377\pi$
$174$	0	0
$175$	484.000	0.209068
$176$	0	0
$177$	−1698.00	−0.721071
$178$	0	0
$179$	−1527.00	−0.637616	−0.318808	−	0.947819i	$-0.603283\pi$
$179$	−1527.00	−0.637616	−0.318808	+	0.947819i	$0.603283\pi$
$180$	0	0
$181$	−960.000	−0.394233	−0.197117	−	0.980380i	$-0.563158\pi$
$181$	−960.000	−0.394233	−0.197117	+	0.980380i	$0.563158\pi$
$182$	0	0
$183$	2712.00	1.09550
$184$	0	0
$185$	−819.000	−0.325482
$186$	0	0
$187$	−102.000	−0.0398876
$188$	0	0
$189$	−1485.00	−0.571523
$190$	0	0
$191$	110.000	0.0416718	0.0208359	−	0.999783i	$-0.493367\pi$
$191$	110.000	0.0416718	0.0208359	+	0.999783i	$0.493367\pi$
$192$	0	0
$193$	−4096.00	−1.52765	−0.763825	−	0.645423i	$-0.776681\pi$
$193$	−4096.00	−1.52765	−0.763825	+	0.645423i	$0.776681\pi$
$194$	0	0
$195$	507.000	0.186190
$196$	0	0
$197$	−1635.00	−0.591314	−0.295657	−	0.955294i	$-0.595539\pi$
$197$	−1635.00	−0.591314	−0.295657	+	0.955294i	$0.595539\pi$
$198$	0	0
$199$	−2414.00	−0.859919	−0.429960	−	0.902848i	$-0.641472\pi$
$199$	−2414.00	−0.859919	−0.429960	+	0.902848i	$0.641472\pi$
$200$	0	0
$201$	−1146.00	−0.402152
$202$	0	0
$203$	−1298.00	−0.448777
$204$	0	0
$205$	3744.00	1.27557
$206$	0	0
$207$	−72.0000	−0.0241756
$208$	0	0
$209$	−300.000	−0.0992892
$210$	0	0
$211$	3697.00	1.20622	0.603109	−	0.797659i	$-0.293928\pi$
$211$	3697.00	1.20622	0.603109	+	0.797659i	$0.293928\pi$
$212$	0	0
$213$	−21.0000	−0.00675538
$214$	0	0
$215$	−3809.00	−1.20824
$216$	0	0
$217$	1276.00	0.399173
$218$	0	0
$219$	1554.00	0.479496
$220$	0	0
$221$	663.000	0.201802
$222$	0	0
$223$	2841.00	0.853127	0.426564	−	0.904457i	$-0.359724\pi$
$223$	2841.00	0.853127	0.426564	+	0.904457i	$0.359724\pi$
$224$	0	0
$225$	−792.000	−0.234667
$226$	0	0
$227$	2424.00	0.708751	0.354376	−	0.935103i	$-0.384693\pi$
$227$	2424.00	0.708751	0.354376	+	0.935103i	$0.384693\pi$
$228$	0	0
$229$	429.000	0.123795	0.0618976	−	0.998083i	$-0.480285\pi$
$229$	429.000	0.123795	0.0618976	+	0.998083i	$0.480285\pi$
$230$	0	0
$231$	66.0000	0.0187986
$232$	0	0
$233$	−683.000	−0.192038	−0.0960189	−	0.995380i	$-0.530611\pi$
$233$	−683.000	−0.192038	−0.0960189	+	0.995380i	$0.530611\pi$
$234$	0	0
$235$	−4355.00	−1.20889
$236$	0	0
$237$	300.000	0.0822240
$238$	0	0
$239$	−4053.00	−1.09693	−0.548466	−	0.836173i	$-0.684788\pi$
$239$	−4053.00	−1.09693	−0.548466	+	0.836173i	$0.684788\pi$
$240$	0	0
$241$	−4206.00	−1.12420	−0.562100	−	0.827069i	$-0.690006\pi$
$241$	−4206.00	−1.12420	−0.562100	+	0.827069i	$0.690006\pi$
$242$	0	0
$243$	3888.00	1.02640
$244$	0	0
$245$	2886.00	0.752571
$246$	0	0
$247$	1950.00	0.502330
$248$	0	0
$249$	4320.00	1.09947
$250$	0	0
$251$	−6312.00	−1.58729	−0.793645	−	0.608381i	$-0.791819\pi$
$251$	−6312.00	−1.58729	−0.793645	+	0.608381i	$0.791819\pi$
$252$	0	0
$253$	8.00000	0.00198797
$254$	0	0
$255$	1989.00	0.488455
$256$	0	0
$257$	4065.00	0.986645	0.493322	−	0.869847i	$-0.335782\pi$
$257$	4065.00	0.986645	0.493322	+	0.869847i	$0.335782\pi$
$258$	0	0
$259$	693.000	0.166258
$260$	0	0
$261$	2124.00	0.503725
$262$	0	0
$263$	−7308.00	−1.71342	−0.856712	−	0.515795i	$-0.827497\pi$
$263$	−7308.00	−1.71342	−0.856712	+	0.515795i	$0.827497\pi$
$264$	0	0
$265$	9204.00	2.13357
$266$	0	0
$267$	3762.00	0.862287
$268$	0	0
$269$	−4488.00	−1.01724	−0.508621	−	0.860990i	$-0.669845\pi$
$269$	−4488.00	−1.01724	−0.508621	+	0.860990i	$0.669845\pi$
$270$	0	0
$271$	3935.00	0.882045	0.441023	−	0.897496i	$-0.354616\pi$
$271$	3935.00	0.882045	0.441023	+	0.897496i	$0.354616\pi$
$272$	0	0
$273$	−429.000	−0.0951072
$274$	0	0
$275$	88.0000	0.0192967
$276$	0	0
$277$	−1140.00	−0.247278	−0.123639	−	0.992327i	$-0.539456\pi$
$277$	−1140.00	−0.247278	−0.123639	+	0.992327i	$0.539456\pi$
$278$	0	0
$279$	−2088.00	−0.448048
$280$	0	0
$281$	838.000	0.177904	0.0889518	−	0.996036i	$-0.471648\pi$
$281$	838.000	0.177904	0.0889518	+	0.996036i	$0.471648\pi$
$282$	0	0
$283$	5972.00	1.25441	0.627206	−	0.778853i	$-0.284198\pi$
$283$	5972.00	1.25441	0.627206	+	0.778853i	$0.284198\pi$
$284$	0	0
$285$	5850.00	1.21587
$286$	0	0
$287$	−3168.00	−0.651572
$288$	0	0
$289$	−2312.00	−0.470588
$290$	0	0
$291$	3786.00	0.762678
$292$	0	0
$293$	−8093.00	−1.61365	−0.806823	−	0.590794i	$-0.798815\pi$
$293$	−8093.00	−1.61365	−0.806823	+	0.590794i	$0.798815\pi$
$294$	0	0
$295$	7358.00	1.45220
$296$	0	0
$297$	−270.000	−0.0527508
$298$	0	0
$299$	−52.0000	−0.0100577
$300$	0	0
$301$	3223.00	0.617178
$302$	0	0
$303$	−5316.00	−1.00791
$304$	0	0
$305$	−11752.0	−2.20629
$306$	0	0
$307$	−2894.00	−0.538011	−0.269005	−	0.963139i	$-0.586695\pi$
$307$	−2894.00	−0.538011	−0.269005	+	0.963139i	$0.586695\pi$
$308$	0	0
$309$	3480.00	0.640681
$310$	0	0
$311$	6006.00	1.09508	0.547539	−	0.836780i	$-0.315565\pi$
$311$	6006.00	1.09508	0.547539	+	0.836780i	$0.315565\pi$
$312$	0	0
$313$	2063.00	0.372548	0.186274	−	0.982498i	$-0.440359\pi$
$313$	2063.00	0.372548	0.186274	+	0.982498i	$0.440359\pi$
$314$	0	0
$315$	2574.00	0.460408
$316$	0	0
$317$	5622.00	0.996098	0.498049	−	0.867149i	$-0.334050\pi$
$317$	5622.00	0.996098	0.498049	+	0.867149i	$0.334050\pi$
$318$	0	0
$319$	−236.000	−0.0414215
$320$	0	0
$321$	−1500.00	−0.260816
$322$	0	0
$323$	7650.00	1.31782
$324$	0	0
$325$	−572.000	−0.0976272
$326$	0	0
$327$	−987.000	−0.166915
$328$	0	0
$329$	3685.00	0.617510
$330$	0	0
$331$	5188.00	0.861505	0.430753	−	0.902470i	$-0.358248\pi$
$331$	5188.00	0.861505	0.430753	+	0.902470i	$0.358248\pi$
$332$	0	0
$333$	−1134.00	−0.186615
$334$	0	0
$335$	4966.00	0.809915
$336$	0	0
$337$	−5761.00	−0.931222	−0.465611	−	0.884990i	$-0.654165\pi$
$337$	−5761.00	−0.931222	−0.465611	+	0.884990i	$0.654165\pi$
$338$	0	0
$339$	−2298.00	−0.368172
$340$	0	0
$341$	232.000	0.0368431
$342$	0	0
$343$	−6215.00	−0.978363
$344$	0	0
$345$	−156.000	−0.0243442
$346$	0	0
$347$	4161.00	0.643730	0.321865	−	0.946786i	$-0.395690\pi$
$347$	4161.00	0.643730	0.321865	+	0.946786i	$0.395690\pi$
$348$	0	0
$349$	6211.00	0.952628	0.476314	−	0.879275i	$-0.341973\pi$
$349$	6211.00	0.952628	0.476314	+	0.879275i	$0.341973\pi$
$350$	0	0
$351$	1755.00	0.266880
$352$	0	0
$353$	−332.000	−0.0500583	−0.0250291	−	0.999687i	$-0.507968\pi$
$353$	−332.000	−0.0500583	−0.0250291	+	0.999687i	$0.507968\pi$
$354$	0	0
$355$	91.0000	0.0136050
$356$	0	0
$357$	−1683.00	−0.249506
$358$	0	0
$359$	−1296.00	−0.190530	−0.0952650	−	0.995452i	$-0.530370\pi$
$359$	−1296.00	−0.190530	−0.0952650	+	0.995452i	$0.530370\pi$
$360$	0	0
$361$	15641.0	2.28036
$362$	0	0
$363$	−3981.00	−0.575615
$364$	0	0
$365$	−6734.00	−0.965681
$366$	0	0
$367$	−11726.0	−1.66783	−0.833913	−	0.551896i	$-0.813905\pi$
$367$	−11726.0	−1.66783	−0.833913	+	0.551896i	$0.813905\pi$
$368$	0	0
$369$	5184.00	0.731350
$370$	0	0
$371$	−7788.00	−1.08985
$372$	0	0
$373$	2780.00	0.385906	0.192953	−	0.981208i	$-0.438194\pi$
$373$	2780.00	0.385906	0.192953	+	0.981208i	$0.438194\pi$
$374$	0	0
$375$	3159.00	0.435013
$376$	0	0
$377$	1534.00	0.209562
$378$	0	0
$379$	9920.00	1.34448	0.672238	−	0.740335i	$-0.265333\pi$
$379$	9920.00	1.34448	0.672238	+	0.740335i	$0.265333\pi$
$380$	0	0
$381$	6432.00	0.864885
$382$	0	0
$383$	3033.00	0.404645	0.202323	−	0.979319i	$-0.435151\pi$
$383$	3033.00	0.404645	0.202323	+	0.979319i	$0.435151\pi$
$384$	0	0
$385$	−286.000	−0.0378595
$386$	0	0
$387$	−5274.00	−0.692745
$388$	0	0
$389$	−12630.0	−1.64619	−0.823093	−	0.567906i	$-0.807754\pi$
$389$	−12630.0	−1.64619	−0.823093	+	0.567906i	$0.807754\pi$
$390$	0	0
$391$	−204.000	−0.0263855
$392$	0	0
$393$	−5541.00	−0.711212
$394$	0	0
$395$	−1300.00	−0.165595
$396$	0	0
$397$	674.000	0.0852068	0.0426034	−	0.999092i	$-0.486435\pi$
$397$	674.000	0.0852068	0.0426034	+	0.999092i	$0.486435\pi$
$398$	0	0
$399$	−4950.00	−0.621078
$400$	0	0
$401$	3144.00	0.391531	0.195765	−	0.980651i	$-0.437281\pi$
$401$	3144.00	0.391531	0.195765	+	0.980651i	$0.437281\pi$
$402$	0	0
$403$	−1508.00	−0.186399
$404$	0	0
$405$	−1053.00	−0.129195
$406$	0	0
$407$	126.000	0.0153454
$408$	0	0
$409$	1108.00	0.133954	0.0669769	−	0.997755i	$-0.478665\pi$
$409$	1108.00	0.133954	0.0669769	+	0.997755i	$0.478665\pi$
$410$	0	0
$411$	−1692.00	−0.203066
$412$	0	0
$413$	−6226.00	−0.741795
$414$	0	0
$415$	−18720.0	−2.21429
$416$	0	0
$417$	−8157.00	−0.957914
$418$	0	0
$419$	−2277.00	−0.265486	−0.132743	−	0.991150i	$-0.542379\pi$
$419$	−2277.00	−0.265486	−0.132743	+	0.991150i	$0.542379\pi$
$420$	0	0
$421$	−8225.00	−0.952166	−0.476083	−	0.879400i	$-0.657944\pi$
$421$	−8225.00	−0.952166	−0.476083	+	0.879400i	$0.657944\pi$
$422$	0	0
$423$	−6030.00	−0.693117
$424$	0	0
$425$	−2244.00	−0.256118
$426$	0	0
$427$	9944.00	1.12699
$428$	0	0
$429$	−78.0000	−0.00877826
$430$	0	0
$431$	−7005.00	−0.782875	−0.391437	−	0.920205i	$-0.628022\pi$
$431$	−7005.00	−0.782875	−0.391437	+	0.920205i	$0.628022\pi$
$432$	0	0
$433$	5215.00	0.578792	0.289396	−	0.957209i	$-0.406546\pi$
$433$	5215.00	0.578792	0.289396	+	0.957209i	$0.406546\pi$
$434$	0	0
$435$	4602.00	0.507239
$436$	0	0
$437$	−600.000	−0.0656794
$438$	0	0
$439$	−4658.00	−0.506411	−0.253205	−	0.967413i	$-0.581485\pi$
$439$	−4658.00	−0.506411	−0.253205	+	0.967413i	$0.581485\pi$
$440$	0	0
$441$	3996.00	0.431487
$442$	0	0
$443$	−8217.00	−0.881267	−0.440634	−	0.897687i	$-0.645246\pi$
$443$	−8217.00	−0.881267	−0.440634	+	0.897687i	$0.645246\pi$
$444$	0	0
$445$	−16302.0	−1.73660
$446$	0	0
$447$	3762.00	0.398068
$448$	0	0
$449$	−16634.0	−1.74835	−0.874173	−	0.485615i	$-0.838596\pi$
$449$	−16634.0	−1.74835	−0.874173	+	0.485615i	$0.838596\pi$
$450$	0	0
$451$	−576.000	−0.0601392
$452$	0	0
$453$	3519.00	0.364982
$454$	0	0
$455$	1859.00	0.191541
$456$	0	0
$457$	−9254.00	−0.947229	−0.473615	−	0.880732i	$-0.657051\pi$
$457$	−9254.00	−0.947229	−0.473615	+	0.880732i	$0.657051\pi$
$458$	0	0
$459$	6885.00	0.700140
$460$	0	0
$461$	5295.00	0.534952	0.267476	−	0.963565i	$-0.413810\pi$
$461$	5295.00	0.534952	0.267476	+	0.963565i	$0.413810\pi$
$462$	0	0
$463$	14984.0	1.50403	0.752015	−	0.659146i	$-0.229082\pi$
$463$	14984.0	1.50403	0.752015	+	0.659146i	$0.229082\pi$
$464$	0	0
$465$	−4524.00	−0.451173
$466$	0	0
$467$	9076.00	0.899330	0.449665	−	0.893197i	$-0.351543\pi$
$467$	9076.00	0.899330	0.449665	+	0.893197i	$0.351543\pi$
$468$	0	0
$469$	−4202.00	−0.413711
$470$	0	0
$471$	−5142.00	−0.503038
$472$	0	0
$473$	586.000	0.0569647
$474$	0	0
$475$	−6600.00	−0.637534
$476$	0	0
$477$	12744.0	1.22329
$478$	0	0
$479$	4575.00	0.436403	0.218202	−	0.975904i	$-0.429981\pi$
$479$	4575.00	0.436403	0.218202	+	0.975904i	$0.429981\pi$
$480$	0	0
$481$	−819.000	−0.0776366
$482$	0	0
$483$	132.000	0.0124352
$484$	0	0
$485$	−16406.0	−1.53600
$486$	0	0
$487$	−20504.0	−1.90785	−0.953927	−	0.300039i	$-0.903000\pi$
$487$	−20504.0	−1.90785	−0.953927	+	0.300039i	$0.903000\pi$
$488$	0	0
$489$	−6900.00	−0.638096
$490$	0	0
$491$	16949.0	1.55784	0.778918	−	0.627126i	$-0.215769\pi$
$491$	16949.0	1.55784	0.778918	+	0.627126i	$0.215769\pi$
$492$	0	0
$493$	6018.00	0.549771
$494$	0	0
$495$	468.000	0.0424950
$496$	0	0
$497$	−77.0000	−0.00694954
$498$	0	0
$499$	11072.0	0.993288	0.496644	−	0.867954i	$-0.334565\pi$
$499$	11072.0	0.993288	0.496644	+	0.867954i	$0.334565\pi$
$500$	0	0
$501$	−3528.00	−0.314610
$502$	0	0
$503$	−5194.00	−0.460416	−0.230208	−	0.973142i	$-0.573941\pi$
$503$	−5194.00	−0.460416	−0.230208	+	0.973142i	$0.573941\pi$
$504$	0	0
$505$	23036.0	2.02988
$506$	0	0
$507$	507.000	0.0444116
$508$	0	0
$509$	16186.0	1.40949	0.704746	−	0.709459i	$-0.251061\pi$
$509$	16186.0	1.40949	0.704746	+	0.709459i	$0.251061\pi$
$510$	0	0
$511$	5698.00	0.493277
$512$	0	0
$513$	20250.0	1.74281
$514$	0	0
$515$	−15080.0	−1.29030
$516$	0	0
$517$	670.000	0.0569953
$518$	0	0
$519$	−2532.00	−0.214147
$520$	0	0
$521$	8055.00	0.677343	0.338672	−	0.940905i	$-0.390022\pi$
$521$	8055.00	0.677343	0.338672	+	0.940905i	$0.390022\pi$
$522$	0	0
$523$	−7092.00	−0.592947	−0.296474	−	0.955041i	$-0.595811\pi$
$523$	−7092.00	−0.592947	−0.296474	+	0.955041i	$0.595811\pi$
$524$	0	0
$525$	1452.00	0.120706
$526$	0	0
$527$	−5916.00	−0.489004
$528$	0	0
$529$	−12151.0	−0.998685
$530$	0	0
$531$	10188.0	0.832621
$532$	0	0
$533$	3744.00	0.304260
$534$	0	0
$535$	6500.00	0.525270
$536$	0	0
$537$	−4581.00	−0.368128
$538$	0	0
$539$	−444.000	−0.0354813
$540$	0	0
$541$	11741.0	0.933059	0.466530	−	0.884506i	$-0.345504\pi$
$541$	11741.0	0.933059	0.466530	+	0.884506i	$0.345504\pi$
$542$	0	0
$543$	−2880.00	−0.227611
$544$	0	0
$545$	4277.00	0.336159
$546$	0	0
$547$	−11605.0	−0.907119	−0.453559	−	0.891226i	$-0.649846\pi$
$547$	−11605.0	−0.907119	−0.453559	+	0.891226i	$0.649846\pi$
$548$	0	0
$549$	−16272.0	−1.26498
$550$	0	0
$551$	17700.0	1.36850
$552$	0	0
$553$	1100.00	0.0845873
$554$	0	0
$555$	−2457.00	−0.187917
$556$	0	0
$557$	13773.0	1.04772	0.523861	−	0.851804i	$-0.324491\pi$
$557$	13773.0	1.04772	0.523861	+	0.851804i	$0.324491\pi$
$558$	0	0
$559$	−3809.00	−0.288200
$560$	0	0
$561$	−306.000	−0.0230291
$562$	0	0
$563$	4165.00	0.311783	0.155891	−	0.987774i	$-0.450175\pi$
$563$	4165.00	0.311783	0.155891	+	0.987774i	$0.450175\pi$
$564$	0	0
$565$	9958.00	0.741480
$566$	0	0
$567$	891.000	0.0659938
$568$	0	0
$569$	−10505.0	−0.773976	−0.386988	−	0.922085i	$-0.626485\pi$
$569$	−10505.0	−0.773976	−0.386988	+	0.922085i	$0.626485\pi$
$570$	0	0
$571$	3831.00	0.280775	0.140387	−	0.990097i	$-0.455165\pi$
$571$	3831.00	0.280775	0.140387	+	0.990097i	$0.455165\pi$
$572$	0	0
$573$	330.000	0.0240592
$574$	0	0
$575$	176.000	0.0127647
$576$	0	0
$577$	−13566.0	−0.978787	−0.489393	−	0.872063i	$-0.662782\pi$
$577$	−13566.0	−0.978787	−0.489393	+	0.872063i	$0.662782\pi$
$578$	0	0
$579$	−12288.0	−0.881989
$580$	0	0
$581$	15840.0	1.13107
$582$	0	0
$583$	−1416.00	−0.100591
$584$	0	0
$585$	−3042.00	−0.214994
$586$	0	0
$587$	−6932.00	−0.487418	−0.243709	−	0.969848i	$-0.578364\pi$
$587$	−6932.00	−0.487418	−0.243709	+	0.969848i	$0.578364\pi$
$588$	0	0
$589$	−17400.0	−1.21724
$590$	0	0
$591$	−4905.00	−0.341395
$592$	0	0
$593$	14762.0	1.02226	0.511132	−	0.859502i	$-0.329226\pi$
$593$	14762.0	1.02226	0.511132	+	0.859502i	$0.329226\pi$
$594$	0	0
$595$	7293.00	0.502494
$596$	0	0
$597$	−7242.00	−0.496475
$598$	0	0
$599$	27962.0	1.90734	0.953670	−	0.300855i	$-0.0972722\pi$
$599$	27962.0	1.90734	0.953670	+	0.300855i	$0.0972722\pi$
$600$	0	0
$601$	4165.00	0.282685	0.141343	−	0.989961i	$-0.454858\pi$
$601$	4165.00	0.282685	0.141343	+	0.989961i	$0.454858\pi$
$602$	0	0
$603$	6876.00	0.464365
$604$	0	0
$605$	17251.0	1.15926
$606$	0	0
$607$	−8286.00	−0.554066	−0.277033	−	0.960860i	$-0.589351\pi$
$607$	−8286.00	−0.554066	−0.277033	+	0.960860i	$0.589351\pi$
$608$	0	0
$609$	−3894.00	−0.259102
$610$	0	0
$611$	−4355.00	−0.288354
$612$	0	0
$613$	−1534.00	−0.101073	−0.0505364	−	0.998722i	$-0.516093\pi$
$613$	−1534.00	−0.101073	−0.0505364	+	0.998722i	$0.516093\pi$
$614$	0	0
$615$	11232.0	0.736452
$616$	0	0
$617$	−21056.0	−1.37388	−0.686939	−	0.726715i	$-0.741046\pi$
$617$	−21056.0	−1.37388	−0.686939	+	0.726715i	$0.741046\pi$
$618$	0	0
$619$	19916.0	1.29320	0.646601	−	0.762829i	$-0.276190\pi$
$619$	19916.0	1.29320	0.646601	+	0.762829i	$0.276190\pi$
$620$	0	0
$621$	−540.000	−0.0348945
$622$	0	0
$623$	13794.0	0.887071
$624$	0	0
$625$	−19189.0	−1.22810
$626$	0	0
$627$	−900.000	−0.0573246
$628$	0	0
$629$	−3213.00	−0.203674
$630$	0	0
$631$	2993.00	0.188826	0.0944132	−	0.995533i	$-0.469903\pi$
$631$	2993.00	0.188826	0.0944132	+	0.995533i	$0.469903\pi$
$632$	0	0
$633$	11091.0	0.696410
$634$	0	0
$635$	−27872.0	−1.74184
$636$	0	0
$637$	2886.00	0.179509
$638$	0	0
$639$	126.000	0.00780044
$640$	0	0
$641$	5950.00	0.366632	0.183316	−	0.983054i	$-0.441317\pi$
$641$	5950.00	0.366632	0.183316	+	0.983054i	$0.441317\pi$
$642$	0	0
$643$	−5198.00	−0.318801	−0.159401	−	0.987214i	$-0.550956\pi$
$643$	−5198.00	−0.318801	−0.159401	+	0.987214i	$0.550956\pi$
$644$	0	0
$645$	−11427.0	−0.697578
$646$	0	0
$647$	25982.0	1.57876	0.789380	−	0.613905i	$-0.210402\pi$
$647$	25982.0	1.57876	0.789380	+	0.613905i	$0.210402\pi$
$648$	0	0
$649$	−1132.00	−0.0684667
$650$	0	0
$651$	3828.00	0.230463
$652$	0	0
$653$	28824.0	1.72737	0.863683	−	0.504035i	$-0.168152\pi$
$653$	28824.0	1.72737	0.863683	+	0.504035i	$0.168152\pi$
$654$	0	0
$655$	24011.0	1.43235
$656$	0	0
$657$	−9324.00	−0.553674
$658$	0	0
$659$	−8220.00	−0.485896	−0.242948	−	0.970039i	$-0.578115\pi$
$659$	−8220.00	−0.485896	−0.242948	+	0.970039i	$0.578115\pi$
$660$	0	0
$661$	−8242.00	−0.484987	−0.242494	−	0.970153i	$-0.577965\pi$
$661$	−8242.00	−0.484987	−0.242494	+	0.970153i	$0.577965\pi$
$662$	0	0
$663$	1989.00	0.116510
$664$	0	0
$665$	21450.0	1.25082
$666$	0	0
$667$	−472.000	−0.0274002
$668$	0	0
$669$	8523.00	0.492553
$670$	0	0
$671$	1808.00	0.104019
$672$	0	0
$673$	9013.00	0.516234	0.258117	−	0.966114i	$-0.416898\pi$
$673$	9013.00	0.516234	0.258117	+	0.966114i	$0.416898\pi$
$674$	0	0
$675$	−5940.00	−0.338712
$676$	0	0
$677$	−14100.0	−0.800454	−0.400227	−	0.916416i	$-0.631069\pi$
$677$	−14100.0	−0.800454	−0.400227	+	0.916416i	$0.631069\pi$
$678$	0	0
$679$	13882.0	0.784598
$680$	0	0
$681$	7272.00	0.409198
$682$	0	0
$683$	−32260.0	−1.80731	−0.903656	−	0.428258i	$-0.859127\pi$
$683$	−32260.0	−1.80731	−0.903656	+	0.428258i	$0.859127\pi$
$684$	0	0
$685$	7332.00	0.408965
$686$	0	0
$687$	1287.00	0.0714732
$688$	0	0
$689$	9204.00	0.508918
$690$	0	0
$691$	20840.0	1.14731	0.573655	−	0.819097i	$-0.305525\pi$
$691$	20840.0	1.14731	0.573655	+	0.819097i	$0.305525\pi$
$692$	0	0
$693$	−396.000	−0.0217068
$694$	0	0
$695$	35347.0	1.92919
$696$	0	0
$697$	14688.0	0.798203
$698$	0	0
$699$	−2049.00	−0.110873
$700$	0	0
$701$	15492.0	0.834700	0.417350	−	0.908746i	$-0.362959\pi$
$701$	15492.0	0.834700	0.417350	+	0.908746i	$0.362959\pi$
$702$	0	0
$703$	−9450.00	−0.506989
$704$	0	0
$705$	−13065.0	−0.697952
$706$	0	0
$707$	−19492.0	−1.03688
$708$	0	0
$709$	−8770.00	−0.464548	−0.232274	−	0.972650i	$-0.574617\pi$
$709$	−8770.00	−0.464548	−0.232274	+	0.972650i	$0.574617\pi$
$710$	0	0
$711$	−1800.00	−0.0949441
$712$	0	0
$713$	464.000	0.0243716
$714$	0	0
$715$	338.000	0.0176790
$716$	0	0
$717$	−12159.0	−0.633314
$718$	0	0
$719$	−24626.0	−1.27732	−0.638661	−	0.769488i	$-0.720511\pi$
$719$	−24626.0	−1.27732	−0.638661	+	0.769488i	$0.720511\pi$
$720$	0	0
$721$	12760.0	0.659095
$722$	0	0
$723$	−12618.0	−0.649057
$724$	0	0
$725$	−5192.00	−0.265967
$726$	0	0
$727$	−31162.0	−1.58973	−0.794866	−	0.606786i	$-0.792459\pi$
$727$	−31162.0	−1.58973	−0.794866	+	0.606786i	$0.792459\pi$
$728$	0	0
$729$	9477.00	0.481481
$730$	0	0
$731$	−14943.0	−0.756070
$732$	0	0
$733$	−21031.0	−1.05975	−0.529876	−	0.848075i	$-0.677761\pi$
$733$	−21031.0	−1.05975	−0.529876	+	0.848075i	$0.677761\pi$
$734$	0	0
$735$	8658.00	0.434497
$736$	0	0
$737$	−764.000	−0.0381849
$738$	0	0
$739$	−19116.0	−0.951547	−0.475774	−	0.879568i	$-0.657832\pi$
$739$	−19116.0	−0.951547	−0.475774	+	0.879568i	$0.657832\pi$
$740$	0	0
$741$	5850.00	0.290020
$742$	0	0
$743$	−25309.0	−1.24966	−0.624830	−	0.780761i	$-0.714832\pi$
$743$	−25309.0	−1.24966	−0.624830	+	0.780761i	$0.714832\pi$
$744$	0	0
$745$	−16302.0	−0.801690
$746$	0	0
$747$	−25920.0	−1.26956
$748$	0	0
$749$	−5500.00	−0.268312
$750$	0	0
$751$	6552.00	0.318357	0.159178	−	0.987250i	$-0.449116\pi$
$751$	6552.00	0.318357	0.159178	+	0.987250i	$0.449116\pi$
$752$	0	0
$753$	−18936.0	−0.916423
$754$	0	0
$755$	−15249.0	−0.735057
$756$	0	0
$757$	9684.00	0.464955	0.232478	−	0.972602i	$-0.425317\pi$
$757$	9684.00	0.464955	0.232478	+	0.972602i	$0.425317\pi$
$758$	0	0
$759$	24.0000	0.00114775
$760$	0	0
$761$	8118.00	0.386698	0.193349	−	0.981130i	$-0.438065\pi$
$761$	8118.00	0.386698	0.193349	+	0.981130i	$0.438065\pi$
$762$	0	0
$763$	−3619.00	−0.171712
$764$	0	0
$765$	−11934.0	−0.564019
$766$	0	0
$767$	7358.00	0.346391
$768$	0	0
$769$	768.000	0.0360140	0.0180070	−	0.999838i	$-0.494268\pi$
$769$	768.000	0.0360140	0.0180070	+	0.999838i	$0.494268\pi$
$770$	0	0
$771$	12195.0	0.569640
$772$	0	0
$773$	−11137.0	−0.518202	−0.259101	−	0.965850i	$-0.583426\pi$
$773$	−11137.0	−0.518202	−0.259101	+	0.965850i	$0.583426\pi$
$774$	0	0
$775$	5104.00	0.236569
$776$	0	0
$777$	2079.00	0.0959893
$778$	0	0
$779$	43200.0	1.98691
$780$	0	0
$781$	−14.0000	−0.000641433 0
$782$	0	0
$783$	15930.0	0.727065
$784$	0	0
$785$	22282.0	1.01309
$786$	0	0
$787$	28000.0	1.26822	0.634112	−	0.773241i	$-0.281366\pi$
$787$	28000.0	1.26822	0.634112	+	0.773241i	$0.281366\pi$
$788$	0	0
$789$	−21924.0	−0.989246
$790$	0	0
$791$	−8426.00	−0.378754
$792$	0	0
$793$	−11752.0	−0.526262
$794$	0	0
$795$	27612.0	1.23182
$796$	0	0
$797$	−26082.0	−1.15919	−0.579593	−	0.814906i	$-0.696789\pi$
$797$	−26082.0	−1.15919	−0.579593	+	0.814906i	$0.696789\pi$
$798$	0	0
$799$	−17085.0	−0.756475
$800$	0	0
$801$	−22572.0	−0.995683
$802$	0	0
$803$	1036.00	0.0455288
$804$	0	0
$805$	−572.000	−0.0250439
$806$	0	0
$807$	−13464.0	−0.587305
$808$	0	0
$809$	−8409.00	−0.365445	−0.182722	−	0.983165i	$-0.558491\pi$
$809$	−8409.00	−0.365445	−0.182722	+	0.983165i	$0.558491\pi$
$810$	0	0
$811$	−41444.0	−1.79445	−0.897223	−	0.441578i	$-0.854419\pi$
$811$	−41444.0	−1.79445	−0.897223	+	0.441578i	$0.854419\pi$
$812$	0	0
$813$	11805.0	0.509249
$814$	0	0
$815$	29900.0	1.28509
$816$	0	0
$817$	−43950.0	−1.88203
$818$	0	0
$819$	2574.00	0.109820
$820$	0	0
$821$	10715.0	0.455489	0.227744	−	0.973721i	$-0.426865\pi$
$821$	10715.0	0.455489	0.227744	+	0.973721i	$0.426865\pi$
$822$	0	0
$823$	−32622.0	−1.38169	−0.690845	−	0.723003i	$-0.742761\pi$
$823$	−32622.0	−1.38169	−0.690845	+	0.723003i	$0.742761\pi$
$824$	0	0
$825$	264.000	0.0111410
$826$	0	0
$827$	−26718.0	−1.12343	−0.561715	−	0.827331i	$-0.689858\pi$
$827$	−26718.0	−1.12343	−0.561715	+	0.827331i	$0.689858\pi$
$828$	0	0
$829$	23266.0	0.974743	0.487371	−	0.873195i	$-0.337956\pi$
$829$	23266.0	0.974743	0.487371	+	0.873195i	$0.337956\pi$
$830$	0	0
$831$	−3420.00	−0.142766
$832$	0	0
$833$	11322.0	0.470929
$834$	0	0
$835$	15288.0	0.633608
$836$	0	0
$837$	−15660.0	−0.646701
$838$	0	0
$839$	44080.0	1.81384	0.906919	−	0.421304i	$-0.138428\pi$
$839$	44080.0	1.81384	0.906919	+	0.421304i	$0.138428\pi$
$840$	0	0
$841$	−10465.0	−0.429087
$842$	0	0
$843$	2514.00	0.102713
$844$	0	0
$845$	−2197.00	−0.0894427
$846$	0	0
$847$	−14597.0	−0.592159
$848$	0	0
$849$	17916.0	0.724235
$850$	0	0
$851$	252.000	0.0101509
$852$	0	0
$853$	−16787.0	−0.673829	−0.336914	−	0.941535i	$-0.609383\pi$
$853$	−16787.0	−0.673829	−0.336914	+	0.941535i	$0.609383\pi$
$854$	0	0
$855$	−35100.0	−1.40397
$856$	0	0
$857$	28030.0	1.11725	0.558627	−	0.829419i	$-0.311328\pi$
$857$	28030.0	1.11725	0.558627	+	0.829419i	$0.311328\pi$
$858$	0	0
$859$	1684.00	0.0668886	0.0334443	−	0.999441i	$-0.489352\pi$
$859$	1684.00	0.0668886	0.0334443	+	0.999441i	$0.489352\pi$
$860$	0	0
$861$	−9504.00	−0.376185
$862$	0	0
$863$	41135.0	1.62254	0.811270	−	0.584672i	$-0.198777\pi$
$863$	41135.0	1.62254	0.811270	+	0.584672i	$0.198777\pi$
$864$	0	0
$865$	10972.0	0.431282
$866$	0	0
$867$	−6936.00	−0.271694
$868$	0	0
$869$	200.000	0.00780729
$870$	0	0
$871$	4966.00	0.193188
$872$	0	0
$873$	−22716.0	−0.880665
$874$	0	0
$875$	11583.0	0.447516
$876$	0	0
$877$	−12483.0	−0.480640	−0.240320	−	0.970694i	$-0.577252\pi$
$877$	−12483.0	−0.480640	−0.240320	+	0.970694i	$0.577252\pi$
$878$	0	0
$879$	−24279.0	−0.931639
$880$	0	0
$881$	−22299.0	−0.852750	−0.426375	−	0.904547i	$-0.640210\pi$
$881$	−22299.0	−0.852750	−0.426375	+	0.904547i	$0.640210\pi$
$882$	0	0
$883$	2663.00	0.101492	0.0507458	−	0.998712i	$-0.483840\pi$
$883$	2663.00	0.101492	0.0507458	+	0.998712i	$0.483840\pi$
$884$	0	0
$885$	22074.0	0.838429
$886$	0	0
$887$	−14488.0	−0.548432	−0.274216	−	0.961668i	$-0.588418\pi$
$887$	−14488.0	−0.548432	−0.274216	+	0.961668i	$0.588418\pi$
$888$	0	0
$889$	23584.0	0.889744
$890$	0	0
$891$	162.000	0.00609114
$892$	0	0
$893$	−50250.0	−1.88304
$894$	0	0
$895$	19851.0	0.741392
$896$	0	0
$897$	−156.000	−0.00580679
$898$	0	0
$899$	−13688.0	−0.507809
$900$	0	0
$901$	36108.0	1.33511
$902$	0	0
$903$	9669.00	0.356328
$904$	0	0
$905$	12480.0	0.458397
$906$	0	0
$907$	−43863.0	−1.60579	−0.802893	−	0.596124i	$-0.796707\pi$
$907$	−43863.0	−1.60579	−0.802893	+	0.596124i	$0.796707\pi$
$908$	0	0
$909$	31896.0	1.16383
$910$	0	0
$911$	8142.00	0.296110	0.148055	−	0.988979i	$-0.452699\pi$
$911$	8142.00	0.296110	0.148055	+	0.988979i	$0.452699\pi$
$912$	0	0
$913$	2880.00	0.104397
$914$	0	0
$915$	−35256.0	−1.27380
$916$	0	0
$917$	−20317.0	−0.731654
$918$	0	0
$919$	−18504.0	−0.664190	−0.332095	−	0.943246i	$-0.607755\pi$
$919$	−18504.0	−0.664190	−0.332095	+	0.943246i	$0.607755\pi$
$920$	0	0
$921$	−8682.00	−0.310621
$922$	0	0
$923$	91.0000	0.00324518
$924$	0	0
$925$	2772.00	0.0985328
$926$	0	0
$927$	−20880.0	−0.739794
$928$	0	0
$929$	−26484.0	−0.935320	−0.467660	−	0.883909i	$-0.654903\pi$
$929$	−26484.0	−0.935320	−0.467660	+	0.883909i	$0.654903\pi$
$930$	0	0
$931$	33300.0	1.17225
$932$	0	0
$933$	18018.0	0.632243
$934$	0	0
$935$	1326.00	0.0463795
$936$	0	0
$937$	−4890.00	−0.170490	−0.0852451	−	0.996360i	$-0.527167\pi$
$937$	−4890.00	−0.170490	−0.0852451	+	0.996360i	$0.527167\pi$
$938$	0	0
$939$	6189.00	0.215091
$940$	0	0
$941$	39709.0	1.37564	0.687820	−	0.725882i	$-0.258568\pi$
$941$	39709.0	1.37564	0.687820	+	0.725882i	$0.258568\pi$
$942$	0	0
$943$	−1152.00	−0.0397818
$944$	0	0
$945$	19305.0	0.664541
$946$	0	0
$947$	−15582.0	−0.534685	−0.267343	−	0.963602i	$-0.586146\pi$
$947$	−15582.0	−0.534685	−0.267343	+	0.963602i	$0.586146\pi$
$948$	0	0
$949$	−6734.00	−0.230342
$950$	0	0
$951$	16866.0	0.575097
$952$	0	0
$953$	45887.0	1.55973	0.779867	−	0.625946i	$-0.215287\pi$
$953$	45887.0	1.55973	0.779867	+	0.625946i	$0.215287\pi$
$954$	0	0
$955$	−1430.00	−0.0484542
$956$	0	0
$957$	−708.000	−0.0239147
$958$	0	0
$959$	−6204.00	−0.208903
$960$	0	0
$961$	−16335.0	−0.548320
$962$	0	0
$963$	9000.00	0.301164
$964$	0	0
$965$	53248.0	1.77628
$966$	0	0
$967$	−25139.0	−0.836004	−0.418002	−	0.908446i	$-0.637269\pi$
$967$	−25139.0	−0.836004	−0.418002	+	0.908446i	$0.637269\pi$
$968$	0	0
$969$	22950.0	0.760846
$970$	0	0
$971$	36591.0	1.20933	0.604666	−	0.796479i	$-0.293307\pi$
$971$	36591.0	1.20933	0.604666	+	0.796479i	$0.293307\pi$
$972$	0	0
$973$	−29909.0	−0.985446
$974$	0	0
$975$	−1716.00	−0.0563651
$976$	0	0
$977$	−19182.0	−0.628134	−0.314067	−	0.949401i	$-0.601692\pi$
$977$	−19182.0	−0.628134	−0.314067	+	0.949401i	$0.601692\pi$
$978$	0	0
$979$	2508.00	0.0818754
$980$	0	0
$981$	5922.00	0.192737
$982$	0	0
$983$	39835.0	1.29251	0.646256	−	0.763121i	$-0.276334\pi$
$983$	39835.0	1.29251	0.646256	+	0.763121i	$0.276334\pi$
$984$	0	0
$985$	21255.0	0.687554
$986$	0	0
$987$	11055.0	0.356519
$988$	0	0
$989$	1172.00	0.0376819
$990$	0	0
$991$	28662.0	0.918747	0.459374	−	0.888243i	$-0.348074\pi$
$991$	28662.0	0.918747	0.459374	+	0.888243i	$0.348074\pi$
$992$	0	0
$993$	15564.0	0.497390
$994$	0	0
$995$	31382.0	0.999876
$996$	0	0
$997$	39182.0	1.24464	0.622320	−	0.782763i	$-0.286190\pi$
$997$	39182.0	1.24464	0.622320	+	0.782763i	$0.286190\pi$
$998$	0	0
$999$	−8505.00	−0.269356

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	208.4.a.f.1.1		1
3.2	odd	2		1872.4.a.n.1.1		1
4.3	odd	2		52.4.a.a.1.1	✓	1
8.3	odd	2		832.4.a.n.1.1		1
8.5	even	2		832.4.a.f.1.1		1
12.11	even	2		468.4.a.c.1.1		1
20.3	even	4		1300.4.c.b.1249.1		2
20.7	even	4		1300.4.c.b.1249.2		2
20.19	odd	2		1300.4.a.d.1.1		1
52.3	odd	6		676.4.e.a.529.1		2
52.7	even	12		676.4.h.d.361.1		4
52.11	even	12		676.4.h.d.485.1		4
52.15	even	12		676.4.h.d.485.2		4
52.19	even	12		676.4.h.d.361.2		4
52.23	odd	6		676.4.e.b.529.1		2
52.31	even	4		676.4.d.a.337.2		2
52.35	odd	6		676.4.e.a.653.1		2
52.43	odd	6		676.4.e.b.653.1		2
52.47	even	4		676.4.d.a.337.1		2
52.51	odd	2		676.4.a.a.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
52.4.a.a.1.1	✓	1	4.3	odd	2
208.4.a.f.1.1		1	1.1	even	1	trivial
468.4.a.c.1.1		1	12.11	even	2
676.4.a.a.1.1		1	52.51	odd	2
676.4.d.a.337.1		2	52.47	even	4
676.4.d.a.337.2		2	52.31	even	4
676.4.e.a.529.1		2	52.3	odd	6
676.4.e.a.653.1		2	52.35	odd	6
676.4.e.b.529.1		2	52.23	odd	6
676.4.e.b.653.1		2	52.43	odd	6
676.4.h.d.361.1		4	52.7	even	12
676.4.h.d.361.2		4	52.19	even	12
676.4.h.d.485.1		4	52.11	even	12
676.4.h.d.485.2		4	52.15	even	12
832.4.a.f.1.1		1	8.5	even	2
832.4.a.n.1.1		1	8.3	odd	2
1300.4.a.d.1.1		1	20.19	odd	2
1300.4.c.b.1249.1		2	20.3	even	4
1300.4.c.b.1249.2		2	20.7	even	4
1872.4.a.n.1.1		1	3.2	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:	\( N \)	\(=\)	\( 208 = 2^{4} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	208.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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