LMFDB - Embedded newform 208.4.a.g.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:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 13)
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)

$f(q)$

$=$

$q+7.00000 q^{3} -7.00000 q^{5} +13.0000 q^{7} +22.0000 q^{9} +O(q^{10})$

$q+7.00000 q^{3} -7.00000 q^{5} +13.0000 q^{7} +22.0000 q^{9} +26.0000 q^{11} +13.0000 q^{13} -49.0000 q^{15} +77.0000 q^{17} +126.000 q^{19} +91.0000 q^{21} +96.0000 q^{23} -76.0000 q^{25} -35.0000 q^{27} -82.0000 q^{29} -196.000 q^{31} +182.000 q^{33} -91.0000 q^{35} -131.000 q^{37} +91.0000 q^{39} +336.000 q^{41} +201.000 q^{43} -154.000 q^{45} +105.000 q^{47} -174.000 q^{49} +539.000 q^{51} -432.000 q^{53} -182.000 q^{55} +882.000 q^{57} +294.000 q^{59} -56.0000 q^{61} +286.000 q^{63} -91.0000 q^{65} -478.000 q^{67} +672.000 q^{69} -9.00000 q^{71} +98.0000 q^{73} -532.000 q^{75} +338.000 q^{77} -1304.00 q^{79} -839.000 q^{81} +308.000 q^{83} -539.000 q^{85} -574.000 q^{87} -1190.00 q^{89} +169.000 q^{91} -1372.00 q^{93} -882.000 q^{95} +70.0000 q^{97} +572.000 q^{99} +O(q^{100})$

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
$3$	7.00000	1.34715	0.673575	−	0.739119i	$-0.264758\pi$
$3$	7.00000	1.34715	0.673575	+	0.739119i	$0.264758\pi$
$4$	0	0
$5$	−7.00000	−0.626099	−0.313050	−	0.949737i	$-0.601351\pi$
$5$	−7.00000	−0.626099	−0.313050	+	0.949737i	$0.601351\pi$
$6$	0	0
$7$	13.0000	0.701934	0.350967	−	0.936388i	$-0.385853\pi$
$7$	13.0000	0.701934	0.350967	+	0.936388i	$0.385853\pi$
$8$	0	0
$9$	22.0000	0.814815
$10$	0	0
$11$	26.0000	0.712663	0.356332	−	0.934360i	$-0.384027\pi$
$11$	26.0000	0.712663	0.356332	+	0.934360i	$0.384027\pi$
$12$	0	0
$13$	13.0000	0.277350
$13$	13.0000	0.277350
$14$	0	0
$15$	−49.0000	−0.843450
$16$	0	0
$17$	77.0000	1.09854	0.549272	−	0.835644i	$-0.314905\pi$
$17$	77.0000	1.09854	0.549272	+	0.835644i	$0.314905\pi$
$18$	0	0
$19$	126.000	1.52139	0.760694	−	0.649110i	$-0.224859\pi$
$19$	126.000	1.52139	0.760694	+	0.649110i	$0.224859\pi$
$20$	0	0
$21$	91.0000	0.945611
$22$	0	0
$23$	96.0000	0.870321	0.435161	−	0.900353i	$-0.356692\pi$
$23$	96.0000	0.870321	0.435161	+	0.900353i	$0.356692\pi$
$24$	0	0
$25$	−76.0000	−0.608000
$26$	0	0
$27$	−35.0000	−0.249472
$28$	0	0
$29$	−82.0000	−0.525070	−0.262535	−	0.964923i	$-0.584558\pi$
$29$	−82.0000	−0.525070	−0.262535	+	0.964923i	$0.584558\pi$
$30$	0	0
$31$	−196.000	−1.13557	−0.567785	−	0.823177i	$-0.692199\pi$
$31$	−196.000	−1.13557	−0.567785	+	0.823177i	$0.692199\pi$
$32$	0	0
$33$	182.000	0.960065
$34$	0	0
$35$	−91.0000	−0.439480
$36$	0	0
$37$	−131.000	−0.582061	−0.291031	−	0.956714i	$-0.593998\pi$
$37$	−131.000	−0.582061	−0.291031	+	0.956714i	$0.593998\pi$
$38$	0	0
$39$	91.0000	0.373632
$40$	0	0
$41$	336.000	1.27986	0.639932	−	0.768432i	$-0.278963\pi$
$41$	336.000	1.27986	0.639932	+	0.768432i	$0.278963\pi$
$42$	0	0
$43$	201.000	0.712842	0.356421	−	0.934325i	$-0.383997\pi$
$43$	201.000	0.712842	0.356421	+	0.934325i	$0.383997\pi$
$44$	0	0
$45$	−154.000	−0.510155
$46$	0	0
$47$	105.000	0.325869	0.162934	−	0.986637i	$-0.447904\pi$
$47$	105.000	0.325869	0.162934	+	0.986637i	$0.447904\pi$
$48$	0	0
$49$	−174.000	−0.507289
$50$	0	0
$51$	539.000	1.47990
$52$	0	0
$53$	−432.000	−1.11962	−0.559809	−	0.828622i	$-0.689126\pi$
$53$	−432.000	−1.11962	−0.559809	+	0.828622i	$0.689126\pi$
$54$	0	0
$55$	−182.000	−0.446198
$56$	0	0
$57$	882.000	2.04954
$58$	0	0
$59$	294.000	0.648738	0.324369	−	0.945931i	$-0.394848\pi$
$59$	294.000	0.648738	0.324369	+	0.945931i	$0.394848\pi$
$60$	0	0
$61$	−56.0000	−0.117542	−0.0587710	−	0.998271i	$-0.518718\pi$
$61$	−56.0000	−0.117542	−0.0587710	+	0.998271i	$0.518718\pi$
$62$	0	0
$63$	286.000	0.571946
$64$	0	0
$65$	−91.0000	−0.173649
$66$	0	0
$67$	−478.000	−0.871597	−0.435798	−	0.900044i	$-0.643534\pi$
$67$	−478.000	−0.871597	−0.435798	+	0.900044i	$0.643534\pi$
$68$	0	0
$69$	672.000	1.17245
$70$	0	0
$71$	−9.00000	−0.0150437	−0.00752186	−	0.999972i	$-0.502394\pi$
$71$	−9.00000	−0.0150437	−0.00752186	+	0.999972i	$0.502394\pi$
$72$	0	0
$73$	98.0000	0.157124	0.0785619	−	0.996909i	$-0.474967\pi$
$73$	98.0000	0.157124	0.0785619	+	0.996909i	$0.474967\pi$
$74$	0	0
$75$	−532.000	−0.819068
$76$	0	0
$77$	338.000	0.500243
$78$	0	0
$79$	−1304.00	−1.85711	−0.928554	−	0.371198i	$-0.878947\pi$
$79$	−1304.00	−1.85711	−0.928554	+	0.371198i	$0.878947\pi$
$80$	0	0
$81$	−839.000	−1.15089
$82$	0	0
$83$	308.000	0.407318	0.203659	−	0.979042i	$-0.434717\pi$
$83$	308.000	0.407318	0.203659	+	0.979042i	$0.434717\pi$
$84$	0	0
$85$	−539.000	−0.687797
$86$	0	0
$87$	−574.000	−0.707348
$88$	0	0
$89$	−1190.00	−1.41730	−0.708650	−	0.705560i	$-0.750696\pi$
$89$	−1190.00	−1.41730	−0.708650	+	0.705560i	$0.750696\pi$
$90$	0	0
$91$	169.000	0.194681
$92$	0	0
$93$	−1372.00	−1.52978
$94$	0	0
$95$	−882.000	−0.952540
$96$	0	0
$97$	70.0000	0.0732724	0.0366362	−	0.999329i	$-0.488336\pi$
$97$	70.0000	0.0732724	0.0366362	+	0.999329i	$0.488336\pi$
$98$	0	0
$99$	572.000	0.580689
$100$	0	0
$101$	420.000	0.413778	0.206889	−	0.978364i	$-0.433666\pi$
$101$	420.000	0.413778	0.206889	+	0.978364i	$0.433666\pi$
$102$	0	0
$103$	−588.000	−0.562499	−0.281249	−	0.959635i	$-0.590749\pi$
$103$	−588.000	−0.562499	−0.281249	+	0.959635i	$0.590749\pi$
$104$	0	0
$105$	−637.000	−0.592046
$106$	0	0
$107$	684.000	0.617989	0.308994	−	0.951064i	$-0.400008\pi$
$107$	684.000	0.617989	0.308994	+	0.951064i	$0.400008\pi$
$108$	0	0
$109$	373.000	0.327770	0.163885	−	0.986479i	$-0.447597\pi$
$109$	373.000	0.327770	0.163885	+	0.986479i	$0.447597\pi$
$110$	0	0
$111$	−917.000	−0.784124
$112$	0	0
$113$	−1734.00	−1.44355	−0.721774	−	0.692128i	$-0.756673\pi$
$113$	−1734.00	−1.44355	−0.721774	+	0.692128i	$0.756673\pi$
$114$	0	0
$115$	−672.000	−0.544907
$116$	0	0
$117$	286.000	0.225989
$118$	0	0
$119$	1001.00	0.771105
$120$	0	0
$121$	−655.000	−0.492111
$122$	0	0
$123$	2352.00	1.72417
$124$	0	0
$125$	1407.00	1.00677
$126$	0	0
$127$	−1892.00	−1.32195	−0.660976	−	0.750407i	$-0.729857\pi$
$127$	−1892.00	−1.32195	−0.660976	+	0.750407i	$0.729857\pi$
$128$	0	0
$129$	1407.00	0.960306
$130$	0	0
$131$	−1435.00	−0.957073	−0.478536	−	0.878068i	$-0.658833\pi$
$131$	−1435.00	−0.957073	−0.478536	+	0.878068i	$0.658833\pi$
$132$	0	0
$133$	1638.00	1.06791
$134$	0	0
$135$	245.000	0.156194
$136$	0	0
$137$	−1776.00	−1.10755	−0.553773	−	0.832667i	$-0.686813\pi$
$137$	−1776.00	−1.10755	−0.553773	+	0.832667i	$0.686813\pi$
$138$	0	0
$139$	1869.00	1.14048	0.570239	−	0.821479i	$-0.306850\pi$
$139$	1869.00	1.14048	0.570239	+	0.821479i	$0.306850\pi$
$140$	0	0
$141$	735.000	0.438994
$142$	0	0
$143$	338.000	0.197657
$144$	0	0
$145$	574.000	0.328746
$146$	0	0
$147$	−1218.00	−0.683394
$148$	0	0
$149$	2466.00	1.35586	0.677928	−	0.735128i	$-0.262878\pi$
$149$	2466.00	1.35586	0.677928	+	0.735128i	$0.262878\pi$
$150$	0	0
$151$	3323.00	1.79087	0.895437	−	0.445189i	$-0.146863\pi$
$151$	3323.00	1.79087	0.895437	+	0.445189i	$0.146863\pi$
$152$	0	0
$153$	1694.00	0.895110
$154$	0	0
$155$	1372.00	0.710979
$156$	0	0
$157$	−2730.00	−1.38776	−0.693878	−	0.720092i	$-0.744099\pi$
$157$	−2730.00	−1.38776	−0.693878	+	0.720092i	$0.744099\pi$
$158$	0	0
$159$	−3024.00	−1.50829
$160$	0	0
$161$	1248.00	0.610908
$162$	0	0
$163$	544.000	0.261407	0.130704	−	0.991421i	$-0.458276\pi$
$163$	544.000	0.261407	0.130704	+	0.991421i	$0.458276\pi$
$164$	0	0
$165$	−1274.00	−0.601096
$166$	0	0
$167$	−1624.00	−0.752508	−0.376254	−	0.926516i	$-0.622788\pi$
$167$	−1624.00	−0.752508	−0.376254	+	0.926516i	$0.622788\pi$
$168$	0	0
$169$	169.000	0.0769231
$170$	0	0
$171$	2772.00	1.23965
$172$	0	0
$173$	−336.000	−0.147662	−0.0738312	−	0.997271i	$-0.523523\pi$
$173$	−336.000	−0.147662	−0.0738312	+	0.997271i	$0.523523\pi$
$174$	0	0
$175$	−988.000	−0.426776
$176$	0	0
$177$	2058.00	0.873948
$178$	0	0
$179$	3029.00	1.26479	0.632397	−	0.774645i	$-0.282071\pi$
$179$	3029.00	1.26479	0.632397	+	0.774645i	$0.282071\pi$
$180$	0	0
$181$	−28.0000	−0.0114985	−0.00574924	−	0.999983i	$-0.501830\pi$
$181$	−28.0000	−0.0114985	−0.00574924	+	0.999983i	$0.501830\pi$
$182$	0	0
$183$	−392.000	−0.158347
$184$	0	0
$185$	917.000	0.364428
$186$	0	0
$187$	2002.00	0.782892
$188$	0	0
$189$	−455.000	−0.175113
$190$	0	0
$191$	−422.000	−0.159868	−0.0799342	−	0.996800i	$-0.525471\pi$
$191$	−422.000	−0.159868	−0.0799342	+	0.996800i	$0.525471\pi$
$192$	0	0
$193$	492.000	0.183497	0.0917485	−	0.995782i	$-0.470754\pi$
$193$	492.000	0.183497	0.0917485	+	0.995782i	$0.470754\pi$
$194$	0	0
$195$	−637.000	−0.233931
$196$	0	0
$197$	2991.00	1.08173	0.540863	−	0.841111i	$-0.318098\pi$
$197$	2991.00	1.08173	0.540863	+	0.841111i	$0.318098\pi$
$198$	0	0
$199$	70.0000	0.0249355	0.0124678	−	0.999922i	$-0.496031\pi$
$199$	70.0000	0.0249355	0.0124678	+	0.999922i	$0.496031\pi$
$200$	0	0
$201$	−3346.00	−1.17417
$202$	0	0
$203$	−1066.00	−0.368564
$204$	0	0
$205$	−2352.00	−0.801321
$206$	0	0
$207$	2112.00	0.709150
$208$	0	0
$209$	3276.00	1.08424
$210$	0	0
$211$	−2851.00	−0.930194	−0.465097	−	0.885260i	$-0.653981\pi$
$211$	−2851.00	−0.930194	−0.465097	+	0.885260i	$0.653981\pi$
$212$	0	0
$213$	−63.0000	−0.0202661
$214$	0	0
$215$	−1407.00	−0.446310
$216$	0	0
$217$	−2548.00	−0.797095
$218$	0	0
$219$	686.000	0.211669
$220$	0	0
$221$	1001.00	0.304681
$222$	0	0
$223$	−217.000	−0.0651632	−0.0325816	−	0.999469i	$-0.510373\pi$
$223$	−217.000	−0.0651632	−0.0325816	+	0.999469i	$0.510373\pi$
$224$	0	0
$225$	−1672.00	−0.495407
$226$	0	0
$227$	2576.00	0.753194	0.376597	−	0.926377i	$-0.377094\pi$
$227$	2576.00	0.753194	0.376597	+	0.926377i	$0.377094\pi$
$228$	0	0
$229$	455.000	0.131298	0.0656490	−	0.997843i	$-0.479088\pi$
$229$	455.000	0.131298	0.0656490	+	0.997843i	$0.479088\pi$
$230$	0	0
$231$	2366.00	0.673902
$232$	0	0
$233$	3061.00	0.860656	0.430328	−	0.902673i	$-0.358398\pi$
$233$	3061.00	0.860656	0.430328	+	0.902673i	$0.358398\pi$
$234$	0	0
$235$	−735.000	−0.204026
$236$	0	0
$237$	−9128.00	−2.50180
$238$	0	0
$239$	3477.00	0.941039	0.470520	−	0.882389i	$-0.344066\pi$
$239$	3477.00	0.941039	0.470520	+	0.882389i	$0.344066\pi$
$240$	0	0
$241$	−1610.00	−0.430329	−0.215164	−	0.976578i	$-0.569029\pi$
$241$	−1610.00	−0.430329	−0.215164	+	0.976578i	$0.569029\pi$
$242$	0	0
$243$	−4928.00	−1.30095
$244$	0	0
$245$	1218.00	0.317613
$246$	0	0
$247$	1638.00	0.421957
$248$	0	0
$249$	2156.00	0.548719
$250$	0	0
$251$	−1008.00	−0.253484	−0.126742	−	0.991936i	$-0.540452\pi$
$251$	−1008.00	−0.253484	−0.126742	+	0.991936i	$0.540452\pi$
$252$	0	0
$253$	2496.00	0.620246
$254$	0	0
$255$	−3773.00	−0.926566
$256$	0	0
$257$	6041.00	1.46625	0.733127	−	0.680092i	$-0.238060\pi$
$257$	6041.00	1.46625	0.733127	+	0.680092i	$0.238060\pi$
$258$	0	0
$259$	−1703.00	−0.408569
$260$	0	0
$261$	−1804.00	−0.427834
$262$	0	0
$263$	3708.00	0.869373	0.434686	−	0.900582i	$-0.356859\pi$
$263$	3708.00	0.869373	0.434686	+	0.900582i	$0.356859\pi$
$264$	0	0
$265$	3024.00	0.700992
$266$	0	0
$267$	−8330.00	−1.90932
$268$	0	0
$269$	8344.00	1.89124	0.945618	−	0.325278i	$-0.105458\pi$
$269$	8344.00	1.89124	0.945618	+	0.325278i	$0.105458\pi$
$270$	0	0
$271$	1617.00	0.362457	0.181228	−	0.983441i	$-0.441993\pi$
$271$	1617.00	0.362457	0.181228	+	0.983441i	$0.441993\pi$
$272$	0	0
$273$	1183.00	0.262265
$274$	0	0
$275$	−1976.00	−0.433299
$276$	0	0
$277$	−3820.00	−0.828598	−0.414299	−	0.910141i	$-0.635973\pi$
$277$	−3820.00	−0.828598	−0.414299	+	0.910141i	$0.635973\pi$
$278$	0	0
$279$	−4312.00	−0.925278
$280$	0	0
$281$	−6214.00	−1.31920	−0.659602	−	0.751615i	$-0.729275\pi$
$281$	−6214.00	−1.31920	−0.659602	+	0.751615i	$0.729275\pi$
$282$	0	0
$283$	5292.00	1.11158	0.555789	−	0.831323i	$-0.312416\pi$
$283$	5292.00	1.11158	0.555789	+	0.831323i	$0.312416\pi$
$284$	0	0
$285$	−6174.00	−1.28321
$286$	0	0
$287$	4368.00	0.898379
$288$	0	0
$289$	1016.00	0.206798
$290$	0	0
$291$	490.000	0.0987090
$292$	0	0
$293$	−903.000	−0.180047	−0.0900236	−	0.995940i	$-0.528694\pi$
$293$	−903.000	−0.180047	−0.0900236	+	0.995940i	$0.528694\pi$
$294$	0	0
$295$	−2058.00	−0.406174
$296$	0	0
$297$	−910.000	−0.177790
$298$	0	0
$299$	1248.00	0.241384
$300$	0	0
$301$	2613.00	0.500368
$302$	0	0
$303$	2940.00	0.557421
$304$	0	0
$305$	392.000	0.0735930
$306$	0	0
$307$	−2114.00	−0.393004	−0.196502	−	0.980503i	$-0.562958\pi$
$307$	−2114.00	−0.393004	−0.196502	+	0.980503i	$0.562958\pi$
$308$	0	0
$309$	−4116.00	−0.757770
$310$	0	0
$311$	−3402.00	−0.620288	−0.310144	−	0.950690i	$-0.600377\pi$
$311$	−3402.00	−0.620288	−0.310144	+	0.950690i	$0.600377\pi$
$312$	0	0
$313$	−10689.0	−1.93028	−0.965141	−	0.261732i	$-0.915706\pi$
$313$	−10689.0	−1.93028	−0.965141	+	0.261732i	$0.915706\pi$
$314$	0	0
$315$	−2002.00	−0.358095
$316$	0	0
$317$	−7054.00	−1.24982	−0.624909	−	0.780698i	$-0.714864\pi$
$317$	−7054.00	−1.24982	−0.624909	+	0.780698i	$0.714864\pi$
$318$	0	0
$319$	−2132.00	−0.374198
$320$	0	0
$321$	4788.00	0.832524
$322$	0	0
$323$	9702.00	1.67131
$324$	0	0
$325$	−988.000	−0.168629
$326$	0	0
$327$	2611.00	0.441555
$328$	0	0
$329$	1365.00	0.228738
$330$	0	0
$331$	−9704.00	−1.61142	−0.805710	−	0.592310i	$-0.798216\pi$
$331$	−9704.00	−1.61142	−0.805710	+	0.592310i	$0.798216\pi$
$332$	0	0
$333$	−2882.00	−0.474272
$334$	0	0
$335$	3346.00	0.545706
$336$	0	0
$337$	−10449.0	−1.68900	−0.844500	−	0.535555i	$-0.820103\pi$
$337$	−10449.0	−1.68900	−0.844500	+	0.535555i	$0.820103\pi$
$338$	0	0
$339$	−12138.0	−1.94468
$340$	0	0
$341$	−5096.00	−0.809278
$342$	0	0
$343$	−6721.00	−1.05802
$344$	0	0
$345$	−4704.00	−0.734072
$346$	0	0
$347$	621.000	0.0960721	0.0480361	−	0.998846i	$-0.484704\pi$
$347$	621.000	0.0960721	0.0480361	+	0.998846i	$0.484704\pi$
$348$	0	0
$349$	12481.0	1.91431	0.957153	−	0.289584i	$-0.0935168\pi$
$349$	12481.0	1.91431	0.957153	+	0.289584i	$0.0935168\pi$
$350$	0	0
$351$	−455.000	−0.0691912
$352$	0	0
$353$	−1400.00	−0.211089	−0.105545	−	0.994415i	$-0.533659\pi$
$353$	−1400.00	−0.211089	−0.105545	+	0.994415i	$0.533659\pi$
$354$	0	0
$355$	63.0000	0.00941885
$356$	0	0
$357$	7007.00	1.03879
$358$	0	0
$359$	4968.00	0.730365	0.365182	−	0.930936i	$-0.381007\pi$
$359$	4968.00	0.730365	0.365182	+	0.930936i	$0.381007\pi$
$360$	0	0
$361$	9017.00	1.31462
$362$	0	0
$363$	−4585.00	−0.662948
$364$	0	0
$365$	−686.000	−0.0983750
$366$	0	0
$367$	−8722.00	−1.24056	−0.620279	−	0.784381i	$-0.712981\pi$
$367$	−8722.00	−1.24056	−0.620279	+	0.784381i	$0.712981\pi$
$368$	0	0
$369$	7392.00	1.04285
$370$	0	0
$371$	−5616.00	−0.785898
$372$	0	0
$373$	10012.0	1.38982	0.694908	−	0.719098i	$-0.255445\pi$
$373$	10012.0	1.38982	0.694908	+	0.719098i	$0.255445\pi$
$374$	0	0
$375$	9849.00	1.35627
$376$	0	0
$377$	−1066.00	−0.145628
$378$	0	0
$379$	3372.00	0.457013	0.228507	−	0.973542i	$-0.426616\pi$
$379$	3372.00	0.457013	0.228507	+	0.973542i	$0.426616\pi$
$380$	0	0
$381$	−13244.0	−1.78087
$382$	0	0
$383$	847.000	0.113002	0.0565009	−	0.998403i	$-0.482006\pi$
$383$	847.000	0.113002	0.0565009	+	0.998403i	$0.482006\pi$
$384$	0	0
$385$	−2366.00	−0.313201
$386$	0	0
$387$	4422.00	0.580834
$388$	0	0
$389$	11314.0	1.47466	0.737330	−	0.675533i	$-0.236086\pi$
$389$	11314.0	1.47466	0.737330	+	0.675533i	$0.236086\pi$
$390$	0	0
$391$	7392.00	0.956086
$392$	0	0
$393$	−10045.0	−1.28932
$394$	0	0
$395$	9128.00	1.16273
$396$	0	0
$397$	1862.00	0.235393	0.117697	−	0.993050i	$-0.462449\pi$
$397$	1862.00	0.235393	0.117697	+	0.993050i	$0.462449\pi$
$398$	0	0
$399$	11466.0	1.43864
$400$	0	0
$401$	6820.00	0.849313	0.424657	−	0.905355i	$-0.360395\pi$
$401$	6820.00	0.849313	0.424657	+	0.905355i	$0.360395\pi$
$402$	0	0
$403$	−2548.00	−0.314950
$404$	0	0
$405$	5873.00	0.720572
$406$	0	0
$407$	−3406.00	−0.414814
$408$	0	0
$409$	−12992.0	−1.57069	−0.785346	−	0.619057i	$-0.787515\pi$
$409$	−12992.0	−1.57069	−0.785346	+	0.619057i	$0.787515\pi$
$410$	0	0
$411$	−12432.0	−1.49203
$412$	0	0
$413$	3822.00	0.455371
$414$	0	0
$415$	−2156.00	−0.255021
$416$	0	0
$417$	13083.0	1.53640
$418$	0	0
$419$	7343.00	0.856155	0.428078	−	0.903742i	$-0.359191\pi$
$419$	7343.00	0.856155	0.428078	+	0.903742i	$0.359191\pi$
$420$	0	0
$421$	−5059.00	−0.585655	−0.292827	−	0.956165i	$-0.594596\pi$
$421$	−5059.00	−0.585655	−0.292827	+	0.956165i	$0.594596\pi$
$422$	0	0
$423$	2310.00	0.265523
$424$	0	0
$425$	−5852.00	−0.667915
$426$	0	0
$427$	−728.000	−0.0825068
$428$	0	0
$429$	2366.00	0.266274
$430$	0	0
$431$	−3243.00	−0.362436	−0.181218	−	0.983443i	$-0.558004\pi$
$431$	−3243.00	−0.362436	−0.181218	+	0.983443i	$0.558004\pi$
$432$	0	0
$433$	11599.0	1.28733	0.643663	−	0.765309i	$-0.277414\pi$
$433$	11599.0	1.28733	0.643663	+	0.765309i	$0.277414\pi$
$434$	0	0
$435$	4018.00	0.442870
$436$	0	0
$437$	12096.0	1.32410
$438$	0	0
$439$	17374.0	1.88887	0.944437	−	0.328692i	$-0.106608\pi$
$439$	17374.0	1.88887	0.944437	+	0.328692i	$0.106608\pi$
$440$	0	0
$441$	−3828.00	−0.413346
$442$	0	0
$443$	−989.000	−0.106070	−0.0530348	−	0.998593i	$-0.516889\pi$
$443$	−989.000	−0.106070	−0.0530348	+	0.998593i	$0.516889\pi$
$444$	0	0
$445$	8330.00	0.887370
$446$	0	0
$447$	17262.0	1.82654
$448$	0	0
$449$	−14474.0	−1.52131	−0.760657	−	0.649154i	$-0.775123\pi$
$449$	−14474.0	−1.52131	−0.760657	+	0.649154i	$0.775123\pi$
$450$	0	0
$451$	8736.00	0.912111
$452$	0	0
$453$	23261.0	2.41258
$454$	0	0
$455$	−1183.00	−0.121890
$456$	0	0
$457$	−1594.00	−0.163160	−0.0815801	−	0.996667i	$-0.525997\pi$
$457$	−1594.00	−0.163160	−0.0815801	+	0.996667i	$0.525997\pi$
$458$	0	0
$459$	−2695.00	−0.274056
$460$	0	0
$461$	−5915.00	−0.597590	−0.298795	−	0.954317i	$-0.596585\pi$
$461$	−5915.00	−0.597590	−0.298795	+	0.954317i	$0.596585\pi$
$462$	0	0
$463$	11072.0	1.11136	0.555680	−	0.831396i	$-0.312458\pi$
$463$	11072.0	1.11136	0.555680	+	0.831396i	$0.312458\pi$
$464$	0	0
$465$	9604.00	0.957795
$466$	0	0
$467$	−1260.00	−0.124852	−0.0624260	−	0.998050i	$-0.519884\pi$
$467$	−1260.00	−0.124852	−0.0624260	+	0.998050i	$0.519884\pi$
$468$	0	0
$469$	−6214.00	−0.611804
$470$	0	0
$471$	−19110.0	−1.86952
$472$	0	0
$473$	5226.00	0.508016
$474$	0	0
$475$	−9576.00	−0.925004
$476$	0	0
$477$	−9504.00	−0.912281
$478$	0	0
$479$	12033.0	1.14781	0.573906	−	0.818921i	$-0.305428\pi$
$479$	12033.0	1.14781	0.573906	+	0.818921i	$0.305428\pi$
$480$	0	0
$481$	−1703.00	−0.161435
$482$	0	0
$483$	8736.00	0.822985
$484$	0	0
$485$	−490.000	−0.0458758
$486$	0	0
$487$	2280.00	0.212149	0.106075	−	0.994358i	$-0.466172\pi$
$487$	2280.00	0.212149	0.106075	+	0.994358i	$0.466172\pi$
$488$	0	0
$489$	3808.00	0.352155
$490$	0	0
$491$	−16767.0	−1.54111	−0.770554	−	0.637375i	$-0.780020\pi$
$491$	−16767.0	−1.54111	−0.770554	+	0.637375i	$0.780020\pi$
$492$	0	0
$493$	−6314.00	−0.576812
$494$	0	0
$495$	−4004.00	−0.363569
$496$	0	0
$497$	−117.000	−0.0105597
$498$	0	0
$499$	−12840.0	−1.15190	−0.575949	−	0.817485i	$-0.695367\pi$
$499$	−12840.0	−1.15190	−0.575949	+	0.817485i	$0.695367\pi$
$500$	0	0
$501$	−11368.0	−1.01374
$502$	0	0
$503$	2198.00	0.194839	0.0974195	−	0.995243i	$-0.468941\pi$
$503$	2198.00	0.194839	0.0974195	+	0.995243i	$0.468941\pi$
$504$	0	0
$505$	−2940.00	−0.259066
$506$	0	0
$507$	1183.00	0.103627
$508$	0	0
$509$	−17066.0	−1.48612	−0.743062	−	0.669223i	$-0.766627\pi$
$509$	−17066.0	−1.48612	−0.743062	+	0.669223i	$0.766627\pi$
$510$	0	0
$511$	1274.00	0.110290
$512$	0	0
$513$	−4410.00	−0.379544
$514$	0	0
$515$	4116.00	0.352180
$516$	0	0
$517$	2730.00	0.232235
$518$	0	0
$519$	−2352.00	−0.198924
$520$	0	0
$521$	2583.00	0.217204	0.108602	−	0.994085i	$-0.465363\pi$
$521$	2583.00	0.217204	0.108602	+	0.994085i	$0.465363\pi$
$522$	0	0
$523$	−18620.0	−1.55678	−0.778390	−	0.627781i	$-0.783963\pi$
$523$	−18620.0	−1.55678	−0.778390	+	0.627781i	$0.783963\pi$
$524$	0	0
$525$	−6916.00	−0.574931
$526$	0	0
$527$	−15092.0	−1.24747
$528$	0	0
$529$	−2951.00	−0.242541
$530$	0	0
$531$	6468.00	0.528601
$532$	0	0
$533$	4368.00	0.354970
$534$	0	0
$535$	−4788.00	−0.386922
$536$	0	0
$537$	21203.0	1.70387
$538$	0	0
$539$	−4524.00	−0.361526
$540$	0	0
$541$	−16833.0	−1.33772	−0.668861	−	0.743388i	$-0.733218\pi$
$541$	−16833.0	−1.33772	−0.668861	+	0.743388i	$0.733218\pi$
$542$	0	0
$543$	−196.000	−0.0154902
$544$	0	0
$545$	−2611.00	−0.205216
$546$	0	0
$547$	8615.00	0.673402	0.336701	−	0.941612i	$-0.390689\pi$
$547$	8615.00	0.673402	0.336701	+	0.941612i	$0.390689\pi$
$548$	0	0
$549$	−1232.00	−0.0957750
$550$	0	0
$551$	−10332.0	−0.798835
$552$	0	0
$553$	−16952.0	−1.30357
$554$	0	0
$555$	6419.00	0.490939
$556$	0	0
$557$	8535.00	0.649263	0.324632	−	0.945841i	$-0.394760\pi$
$557$	8535.00	0.649263	0.324632	+	0.945841i	$0.394760\pi$
$558$	0	0
$559$	2613.00	0.197707
$560$	0	0
$561$	14014.0	1.05467
$562$	0	0
$563$	4641.00	0.347415	0.173708	−	0.984797i	$-0.444425\pi$
$563$	4641.00	0.347415	0.173708	+	0.984797i	$0.444425\pi$
$564$	0	0
$565$	12138.0	0.903804
$566$	0	0
$567$	−10907.0	−0.807850
$568$	0	0
$569$	−4793.00	−0.353134	−0.176567	−	0.984289i	$-0.556499\pi$
$569$	−4793.00	−0.353134	−0.176567	+	0.984289i	$0.556499\pi$
$570$	0	0
$571$	5563.00	0.407713	0.203857	−	0.979001i	$-0.434652\pi$
$571$	5563.00	0.407713	0.203857	+	0.979001i	$0.434652\pi$
$572$	0	0
$573$	−2954.00	−0.215367
$574$	0	0
$575$	−7296.00	−0.529155
$576$	0	0
$577$	24038.0	1.73434	0.867171	−	0.498011i	$-0.165936\pi$
$577$	24038.0	1.73434	0.867171	+	0.498011i	$0.165936\pi$
$578$	0	0
$579$	3444.00	0.247198
$580$	0	0
$581$	4004.00	0.285910
$582$	0	0
$583$	−11232.0	−0.797911
$584$	0	0
$585$	−2002.00	−0.141491
$586$	0	0
$587$	21224.0	1.49235	0.746174	−	0.665751i	$-0.231889\pi$
$587$	21224.0	1.49235	0.746174	+	0.665751i	$0.231889\pi$
$588$	0	0
$589$	−24696.0	−1.72764
$590$	0	0
$591$	20937.0	1.45725
$592$	0	0
$593$	4354.00	0.301513	0.150757	−	0.988571i	$-0.451829\pi$
$593$	4354.00	0.301513	0.150757	+	0.988571i	$0.451829\pi$
$594$	0	0
$595$	−7007.00	−0.482788
$596$	0	0
$597$	490.000	0.0335919
$598$	0	0
$599$	−7310.00	−0.498629	−0.249314	−	0.968423i	$-0.580205\pi$
$599$	−7310.00	−0.498629	−0.249314	+	0.968423i	$0.580205\pi$
$600$	0	0
$601$	−7595.00	−0.515485	−0.257743	−	0.966214i	$-0.582979\pi$
$601$	−7595.00	−0.515485	−0.257743	+	0.966214i	$0.582979\pi$
$602$	0	0
$603$	−10516.0	−0.710190
$604$	0	0
$605$	4585.00	0.308110
$606$	0	0
$607$	826.000	0.0552328	0.0276164	−	0.999619i	$-0.491208\pi$
$607$	826.000	0.0552328	0.0276164	+	0.999619i	$0.491208\pi$
$608$	0	0
$609$	−7462.00	−0.496511
$610$	0	0
$611$	1365.00	0.0903797
$612$	0	0
$613$	14590.0	0.961312	0.480656	−	0.876909i	$-0.340398\pi$
$613$	14590.0	0.961312	0.480656	+	0.876909i	$0.340398\pi$
$614$	0	0
$615$	−16464.0	−1.07950
$616$	0	0
$617$	4888.00	0.318936	0.159468	−	0.987203i	$-0.449022\pi$
$617$	4888.00	0.318936	0.159468	+	0.987203i	$0.449022\pi$
$618$	0	0
$619$	11004.0	0.714520	0.357260	−	0.934005i	$-0.383711\pi$
$619$	11004.0	0.714520	0.357260	+	0.934005i	$0.383711\pi$
$620$	0	0
$621$	−3360.00	−0.217121
$622$	0	0
$623$	−15470.0	−0.994851
$624$	0	0
$625$	−349.000	−0.0223360
$626$	0	0
$627$	22932.0	1.46063
$628$	0	0
$629$	−10087.0	−0.639420
$630$	0	0
$631$	4975.00	0.313869	0.156935	−	0.987609i	$-0.449839\pi$
$631$	4975.00	0.313869	0.156935	+	0.987609i	$0.449839\pi$
$632$	0	0
$633$	−19957.0	−1.25311
$634$	0	0
$635$	13244.0	0.827673
$636$	0	0
$637$	−2262.00	−0.140697
$638$	0	0
$639$	−198.000	−0.0122578
$640$	0	0
$641$	3950.00	0.243394	0.121697	−	0.992567i	$-0.461166\pi$
$641$	3950.00	0.243394	0.121697	+	0.992567i	$0.461166\pi$
$642$	0	0
$643$	3682.00	0.225823	0.112911	−	0.993605i	$-0.463982\pi$
$643$	3682.00	0.225823	0.112911	+	0.993605i	$0.463982\pi$
$644$	0	0
$645$	−9849.00	−0.601247
$646$	0	0
$647$	−10402.0	−0.632063	−0.316032	−	0.948749i	$-0.602351\pi$
$647$	−10402.0	−0.632063	−0.316032	+	0.948749i	$0.602351\pi$
$648$	0	0
$649$	7644.00	0.462332
$650$	0	0
$651$	−17836.0	−1.07381
$652$	0	0
$653$	−31680.0	−1.89852	−0.949260	−	0.314491i	$-0.898166\pi$
$653$	−31680.0	−1.89852	−0.949260	+	0.314491i	$0.898166\pi$
$654$	0	0
$655$	10045.0	0.599222
$656$	0	0
$657$	2156.00	0.128027
$658$	0	0
$659$	−21940.0	−1.29691	−0.648453	−	0.761255i	$-0.724584\pi$
$659$	−21940.0	−1.29691	−0.648453	+	0.761255i	$0.724584\pi$
$660$	0	0
$661$	−31374.0	−1.84615	−0.923077	−	0.384616i	$-0.874334\pi$
$661$	−31374.0	−1.84615	−0.923077	+	0.384616i	$0.874334\pi$
$662$	0	0
$663$	7007.00	0.410451
$664$	0	0
$665$	−11466.0	−0.668620
$666$	0	0
$667$	−7872.00	−0.456979
$668$	0	0
$669$	−1519.00	−0.0877847
$670$	0	0
$671$	−1456.00	−0.0837679
$672$	0	0
$673$	18013.0	1.03172	0.515862	−	0.856672i	$-0.327472\pi$
$673$	18013.0	1.03172	0.515862	+	0.856672i	$0.327472\pi$
$674$	0	0
$675$	2660.00	0.151679
$676$	0	0
$677$	−10640.0	−0.604030	−0.302015	−	0.953303i	$-0.597659\pi$
$677$	−10640.0	−0.604030	−0.302015	+	0.953303i	$0.597659\pi$
$678$	0	0
$679$	910.000	0.0514324
$680$	0	0
$681$	18032.0	1.01467
$682$	0	0
$683$	9336.00	0.523034	0.261517	−	0.965199i	$-0.415777\pi$
$683$	9336.00	0.523034	0.261517	+	0.965199i	$0.415777\pi$
$684$	0	0
$685$	12432.0	0.693434
$686$	0	0
$687$	3185.00	0.176878
$688$	0	0
$689$	−5616.00	−0.310526
$690$	0	0
$691$	−4200.00	−0.231224	−0.115612	−	0.993294i	$-0.536883\pi$
$691$	−4200.00	−0.231224	−0.115612	+	0.993294i	$0.536883\pi$
$692$	0	0
$693$	7436.00	0.407605
$694$	0	0
$695$	−13083.0	−0.714052
$696$	0	0
$697$	25872.0	1.40599
$698$	0	0
$699$	21427.0	1.15943
$700$	0	0
$701$	9872.00	0.531898	0.265949	−	0.963987i	$-0.414315\pi$
$701$	9872.00	0.531898	0.265949	+	0.963987i	$0.414315\pi$
$702$	0	0
$703$	−16506.0	−0.885541
$704$	0	0
$705$	−5145.00	−0.274854
$706$	0	0
$707$	5460.00	0.290445
$708$	0	0
$709$	28450.0	1.50700	0.753499	−	0.657449i	$-0.228364\pi$
$709$	28450.0	1.50700	0.753499	+	0.657449i	$0.228364\pi$
$710$	0	0
$711$	−28688.0	−1.51320
$712$	0	0
$713$	−18816.0	−0.988310
$714$	0	0
$715$	−2366.00	−0.123753
$716$	0	0
$717$	24339.0	1.26772
$718$	0	0
$719$	−32718.0	−1.69705	−0.848523	−	0.529159i	$-0.822507\pi$
$719$	−32718.0	−1.69705	−0.848523	+	0.529159i	$0.822507\pi$
$720$	0	0
$721$	−7644.00	−0.394837
$722$	0	0
$723$	−11270.0	−0.579718
$724$	0	0
$725$	6232.00	0.319242
$726$	0	0
$727$	22834.0	1.16488	0.582439	−	0.812874i	$-0.302099\pi$
$727$	22834.0	1.16488	0.582439	+	0.812874i	$0.302099\pi$
$728$	0	0
$729$	−11843.0	−0.601687
$730$	0	0
$731$	15477.0	0.783088
$732$	0	0
$733$	7875.00	0.396821	0.198410	−	0.980119i	$-0.436422\pi$
$733$	7875.00	0.396821	0.198410	+	0.980119i	$0.436422\pi$
$734$	0	0
$735$	8526.00	0.427872
$736$	0	0
$737$	−12428.0	−0.621155
$738$	0	0
$739$	2140.00	0.106524	0.0532620	−	0.998581i	$-0.483038\pi$
$739$	2140.00	0.106524	0.0532620	+	0.998581i	$0.483038\pi$
$740$	0	0
$741$	11466.0	0.568440
$742$	0	0
$743$	−31971.0	−1.57860	−0.789302	−	0.614006i	$-0.789557\pi$
$743$	−31971.0	−1.57860	−0.789302	+	0.614006i	$0.789557\pi$
$744$	0	0
$745$	−17262.0	−0.848900
$746$	0	0
$747$	6776.00	0.331889
$748$	0	0
$749$	8892.00	0.433787
$750$	0	0
$751$	7432.00	0.361115	0.180558	−	0.983564i	$-0.442210\pi$
$751$	7432.00	0.361115	0.180558	+	0.983564i	$0.442210\pi$
$752$	0	0
$753$	−7056.00	−0.341481
$754$	0	0
$755$	−23261.0	−1.12126
$756$	0	0
$757$	20176.0	0.968704	0.484352	−	0.874873i	$-0.339055\pi$
$757$	20176.0	0.968704	0.484352	+	0.874873i	$0.339055\pi$
$758$	0	0
$759$	17472.0	0.835564
$760$	0	0
$761$	−9478.00	−0.451481	−0.225741	−	0.974187i	$-0.572480\pi$
$761$	−9478.00	−0.451481	−0.225741	+	0.974187i	$0.572480\pi$
$762$	0	0
$763$	4849.00	0.230073
$764$	0	0
$765$	−11858.0	−0.560427
$766$	0	0
$767$	3822.00	0.179928
$768$	0	0
$769$	−12096.0	−0.567221	−0.283610	−	0.958940i	$-0.591532\pi$
$769$	−12096.0	−0.567221	−0.283610	+	0.958940i	$0.591532\pi$
$770$	0	0
$771$	42287.0	1.97526
$772$	0	0
$773$	17941.0	0.834790	0.417395	−	0.908725i	$-0.362943\pi$
$773$	17941.0	0.834790	0.417395	+	0.908725i	$0.362943\pi$
$774$	0	0
$775$	14896.0	0.690426
$776$	0	0
$777$	−11921.0	−0.550403
$778$	0	0
$779$	42336.0	1.94717
$780$	0	0
$781$	−234.000	−0.0107211
$782$	0	0
$783$	2870.00	0.130990
$784$	0	0
$785$	19110.0	0.868873
$786$	0	0
$787$	−6664.00	−0.301837	−0.150919	−	0.988546i	$-0.548223\pi$
$787$	−6664.00	−0.301837	−0.150919	+	0.988546i	$0.548223\pi$
$788$	0	0
$789$	25956.0	1.17118
$790$	0	0
$791$	−22542.0	−1.01328
$792$	0	0
$793$	−728.000	−0.0326003
$794$	0	0
$795$	21168.0	0.944342
$796$	0	0
$797$	−1442.00	−0.0640882	−0.0320441	−	0.999486i	$-0.510202\pi$
$797$	−1442.00	−0.0640882	−0.0320441	+	0.999486i	$0.510202\pi$
$798$	0	0
$799$	8085.00	0.357981
$800$	0	0
$801$	−26180.0	−1.15484
$802$	0	0
$803$	2548.00	0.111976
$804$	0	0
$805$	−8736.00	−0.382489
$806$	0	0
$807$	58408.0	2.54778
$808$	0	0
$809$	30207.0	1.31276	0.656379	−	0.754431i	$-0.272087\pi$
$809$	30207.0	1.31276	0.656379	+	0.754431i	$0.272087\pi$
$810$	0	0
$811$	−21140.0	−0.915322	−0.457661	−	0.889127i	$-0.651313\pi$
$811$	−21140.0	−0.915322	−0.457661	+	0.889127i	$0.651313\pi$
$812$	0	0
$813$	11319.0	0.488284
$814$	0	0
$815$	−3808.00	−0.163667
$816$	0	0
$817$	25326.0	1.08451
$818$	0	0
$819$	3718.00	0.158629
$820$	0	0
$821$	569.000	0.0241879	0.0120939	−	0.999927i	$-0.496150\pi$
$821$	569.000	0.0241879	0.0120939	+	0.999927i	$0.496150\pi$
$822$	0	0
$823$	8538.00	0.361623	0.180812	−	0.983518i	$-0.442128\pi$
$823$	8538.00	0.361623	0.180812	+	0.983518i	$0.442128\pi$
$824$	0	0
$825$	−13832.0	−0.583719
$826$	0	0
$827$	32702.0	1.37504	0.687521	−	0.726164i	$-0.258699\pi$
$827$	32702.0	1.37504	0.687521	+	0.726164i	$0.258699\pi$
$828$	0	0
$829$	−21154.0	−0.886259	−0.443130	−	0.896458i	$-0.646132\pi$
$829$	−21154.0	−0.886259	−0.443130	+	0.896458i	$0.646132\pi$
$830$	0	0
$831$	−26740.0	−1.11625
$832$	0	0
$833$	−13398.0	−0.557279
$834$	0	0
$835$	11368.0	0.471145
$836$	0	0
$837$	6860.00	0.283293
$838$	0	0
$839$	2184.00	0.0898690	0.0449345	−	0.998990i	$-0.485692\pi$
$839$	2184.00	0.0898690	0.0449345	+	0.998990i	$0.485692\pi$
$840$	0	0
$841$	−17665.0	−0.724302
$842$	0	0
$843$	−43498.0	−1.77717
$844$	0	0
$845$	−1183.00	−0.0481615
$846$	0	0
$847$	−8515.00	−0.345430
$848$	0	0
$849$	37044.0	1.49746
$850$	0	0
$851$	−12576.0	−0.506580
$852$	0	0
$853$	36687.0	1.47261	0.736307	−	0.676648i	$-0.236568\pi$
$853$	36687.0	1.47261	0.736307	+	0.676648i	$0.236568\pi$
$854$	0	0
$855$	−19404.0	−0.776144
$856$	0	0
$857$	36806.0	1.46706	0.733529	−	0.679658i	$-0.237872\pi$
$857$	36806.0	1.46706	0.733529	+	0.679658i	$0.237872\pi$
$858$	0	0
$859$	−4900.00	−0.194628	−0.0973142	−	0.995254i	$-0.531025\pi$
$859$	−4900.00	−0.194628	−0.0973142	+	0.995254i	$0.531025\pi$
$860$	0	0
$861$	30576.0	1.21025
$862$	0	0
$863$	13697.0	0.540268	0.270134	−	0.962823i	$-0.412932\pi$
$863$	13697.0	0.540268	0.270134	+	0.962823i	$0.412932\pi$
$864$	0	0
$865$	2352.00	0.0924513
$866$	0	0
$867$	7112.00	0.278588
$868$	0	0
$869$	−33904.0	−1.32349
$870$	0	0
$871$	−6214.00	−0.241737
$872$	0	0
$873$	1540.00	0.0597034
$874$	0	0
$875$	18291.0	0.706684
$876$	0	0
$877$	6239.00	0.240224	0.120112	−	0.992760i	$-0.461675\pi$
$877$	6239.00	0.240224	0.120112	+	0.992760i	$0.461675\pi$
$878$	0	0
$879$	−6321.00	−0.242551
$880$	0	0
$881$	133.000	0.00508613	0.00254307	−	0.999997i	$-0.499191\pi$
$881$	133.000	0.00508613	0.00254307	+	0.999997i	$0.499191\pi$
$882$	0	0
$883$	26003.0	0.991020	0.495510	−	0.868602i	$-0.334981\pi$
$883$	26003.0	0.991020	0.495510	+	0.868602i	$0.334981\pi$
$884$	0	0
$885$	−14406.0	−0.547178
$886$	0	0
$887$	31248.0	1.18287	0.591435	−	0.806353i	$-0.298562\pi$
$887$	31248.0	1.18287	0.591435	+	0.806353i	$0.298562\pi$
$888$	0	0
$889$	−24596.0	−0.927923
$890$	0	0
$891$	−21814.0	−0.820198
$892$	0	0
$893$	13230.0	0.495773
$894$	0	0
$895$	−21203.0	−0.791886
$896$	0	0
$897$	8736.00	0.325180
$898$	0	0
$899$	16072.0	0.596253
$900$	0	0
$901$	−33264.0	−1.22995
$902$	0	0
$903$	18291.0	0.674071
$904$	0	0
$905$	196.000	0.00719918
$906$	0	0
$907$	38253.0	1.40041	0.700204	−	0.713943i	$-0.253092\pi$
$907$	38253.0	1.40041	0.700204	+	0.713943i	$0.253092\pi$
$908$	0	0
$909$	9240.00	0.337152
$910$	0	0
$911$	−36374.0	−1.32286	−0.661429	−	0.750007i	$-0.730050\pi$
$911$	−36374.0	−1.32286	−0.661429	+	0.750007i	$0.730050\pi$
$912$	0	0
$913$	8008.00	0.290281
$914$	0	0
$915$	2744.00	0.0991408
$916$	0	0
$917$	−18655.0	−0.671802
$918$	0	0
$919$	27648.0	0.992408	0.496204	−	0.868206i	$-0.334727\pi$
$919$	27648.0	0.992408	0.496204	+	0.868206i	$0.334727\pi$
$920$	0	0
$921$	−14798.0	−0.529436
$922$	0	0
$923$	−117.000	−0.00417237
$924$	0	0
$925$	9956.00	0.353893
$926$	0	0
$927$	−12936.0	−0.458332
$928$	0	0
$929$	756.000	0.0266992	0.0133496	−	0.999911i	$-0.495751\pi$
$929$	756.000	0.0266992	0.0133496	+	0.999911i	$0.495751\pi$
$930$	0	0
$931$	−21924.0	−0.771783
$932$	0	0
$933$	−23814.0	−0.835622
$934$	0	0
$935$	−14014.0	−0.490168
$936$	0	0
$937$	20846.0	0.726797	0.363399	−	0.931634i	$-0.381616\pi$
$937$	20846.0	0.726797	0.363399	+	0.931634i	$0.381616\pi$
$938$	0	0
$939$	−74823.0	−2.60038
$940$	0	0
$941$	−41321.0	−1.43148	−0.715742	−	0.698365i	$-0.753911\pi$
$941$	−41321.0	−1.43148	−0.715742	+	0.698365i	$0.753911\pi$
$942$	0	0
$943$	32256.0	1.11389
$944$	0	0
$945$	3185.00	0.109638
$946$	0	0
$947$	−54966.0	−1.88612	−0.943060	−	0.332624i	$-0.892066\pi$
$947$	−54966.0	−1.88612	−0.943060	+	0.332624i	$0.892066\pi$
$948$	0	0
$949$	1274.00	0.0435783
$950$	0	0
$951$	−49378.0	−1.68369
$952$	0	0
$953$	−44553.0	−1.51439	−0.757195	−	0.653189i	$-0.773431\pi$
$953$	−44553.0	−1.51439	−0.757195	+	0.653189i	$0.773431\pi$
$954$	0	0
$955$	2954.00	0.100093
$956$	0	0
$957$	−14924.0	−0.504101
$958$	0	0
$959$	−23088.0	−0.777425
$960$	0	0
$961$	8625.00	0.289517
$962$	0	0
$963$	15048.0	0.503546
$964$	0	0
$965$	−3444.00	−0.114887
$966$	0	0
$967$	27907.0	0.928054	0.464027	−	0.885821i	$-0.346404\pi$
$967$	27907.0	0.928054	0.464027	+	0.885821i	$0.346404\pi$
$968$	0	0
$969$	67914.0	2.25151
$970$	0	0
$971$	16443.0	0.543441	0.271720	−	0.962376i	$-0.412407\pi$
$971$	16443.0	0.543441	0.271720	+	0.962376i	$0.412407\pi$
$972$	0	0
$973$	24297.0	0.800541
$974$	0	0
$975$	−6916.00	−0.227168
$976$	0	0
$977$	−45414.0	−1.48713	−0.743563	−	0.668666i	$-0.766866\pi$
$977$	−45414.0	−1.48713	−0.743563	+	0.668666i	$0.766866\pi$
$978$	0	0
$979$	−30940.0	−1.01006
$980$	0	0
$981$	8206.00	0.267072
$982$	0	0
$983$	8981.00	0.291403	0.145702	−	0.989329i	$-0.453456\pi$
$983$	8981.00	0.291403	0.145702	+	0.989329i	$0.453456\pi$
$984$	0	0
$985$	−20937.0	−0.677267
$986$	0	0
$987$	9555.00	0.308145
$988$	0	0
$989$	19296.0	0.620402
$990$	0	0
$991$	17414.0	0.558198	0.279099	−	0.960262i	$-0.409964\pi$
$991$	17414.0	0.558198	0.279099	+	0.960262i	$0.409964\pi$
$992$	0	0
$993$	−67928.0	−2.17083
$994$	0	0
$995$	−490.000	−0.0156121
$996$	0	0
$997$	−23702.0	−0.752909	−0.376454	−	0.926435i	$-0.622857\pi$
$997$	−23702.0	−0.752909	−0.376454	+	0.926435i	$0.622857\pi$
$998$	0	0
$999$	4585.00	0.145208

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	208.4.a.g.1.1		1
3.2	odd	2		1872.4.a.k.1.1		1
4.3	odd	2		13.4.a.a.1.1	✓	1
8.3	odd	2		832.4.a.r.1.1		1
8.5	even	2		832.4.a.a.1.1		1
12.11	even	2		117.4.a.b.1.1		1
20.3	even	4		325.4.b.b.274.2		2
20.7	even	4		325.4.b.b.274.1		2
20.19	odd	2		325.4.a.d.1.1		1
28.27	even	2		637.4.a.a.1.1		1
44.43	even	2		1573.4.a.a.1.1		1
52.3	odd	6		169.4.c.e.22.1		2
52.7	even	12		169.4.e.e.23.1		4
52.11	even	12		169.4.e.e.147.2		4
52.15	even	12		169.4.e.e.147.1		4
52.19	even	12		169.4.e.e.23.2		4
52.23	odd	6		169.4.c.a.22.1		2
52.31	even	4		169.4.b.a.168.2		2
52.35	odd	6		169.4.c.e.146.1		2
52.43	odd	6		169.4.c.a.146.1		2
52.47	even	4		169.4.b.a.168.1		2
52.51	odd	2		169.4.a.e.1.1		1
156.155	even	2		1521.4.a.a.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
13.4.a.a.1.1	✓	1	4.3	odd	2
117.4.a.b.1.1		1	12.11	even	2
169.4.a.e.1.1		1	52.51	odd	2
169.4.b.a.168.1		2	52.47	even	4
169.4.b.a.168.2		2	52.31	even	4
169.4.c.a.22.1		2	52.23	odd	6
169.4.c.a.146.1		2	52.43	odd	6
169.4.c.e.22.1		2	52.3	odd	6
169.4.c.e.146.1		2	52.35	odd	6
169.4.e.e.23.1		4	52.7	even	12
169.4.e.e.23.2		4	52.19	even	12
169.4.e.e.147.1		4	52.15	even	12
169.4.e.e.147.2		4	52.11	even	12
208.4.a.g.1.1		1	1.1	even	1	trivial
325.4.a.d.1.1		1	20.19	odd	2
325.4.b.b.274.1		2	20.7	even	4
325.4.b.b.274.2		2	20.3	even	4
637.4.a.a.1.1		1	28.27	even	2
832.4.a.a.1.1		1	8.5	even	2
832.4.a.r.1.1		1	8.3	odd	2
1521.4.a.a.1.1		1	156.155	even	2
1573.4.a.a.1.1		1	44.43	even	2
1872.4.a.k.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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