LMFDB - Embedded newform 308.4.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [308,4,Mod(1,308)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(308, 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("308.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 308 = 2^{2} \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	308.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:	$18.1725882818$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	308.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+4.00000 q^{3} -12.0000 q^{5} +7.00000 q^{7} -11.0000 q^{9} +11.0000 q^{11} -10.0000 q^{13} -48.0000 q^{15} +24.0000 q^{17} -94.0000 q^{19} +28.0000 q^{21} -180.000 q^{23} +19.0000 q^{25} -152.000 q^{27} +30.0000 q^{29} -94.0000 q^{31} +44.0000 q^{33} -84.0000 q^{35} -214.000 q^{37} -40.0000 q^{39} -48.0000 q^{41} +8.00000 q^{43} +132.000 q^{45} +30.0000 q^{47} +49.0000 q^{49} +96.0000 q^{51} +54.0000 q^{53} -132.000 q^{55} -376.000 q^{57} +360.000 q^{59} -178.000 q^{61} -77.0000 q^{63} +120.000 q^{65} -292.000 q^{67} -720.000 q^{69} +312.000 q^{71} +728.000 q^{73} +76.0000 q^{75} +77.0000 q^{77} -1288.00 q^{79} -311.000 q^{81} +66.0000 q^{83} -288.000 q^{85} +120.000 q^{87} -390.000 q^{89} -70.0000 q^{91} -376.000 q^{93} +1128.00 q^{95} -1510.00 q^{97} -121.000 q^{99} +O(q^{100})$

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	4.00000	0.769800	0.384900	−	0.922958i	$-0.374236\pi$
$3$	4.00000	0.769800	0.384900	+	0.922958i	$0.374236\pi$
$4$	0	0
$5$	−12.0000	−1.07331	−0.536656	−	0.843801i	$-0.680313\pi$
$5$	−12.0000	−1.07331	−0.536656	+	0.843801i	$0.680313\pi$
$6$	0	0
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	0	0
$9$	−11.0000	−0.407407
$10$	0	0
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	0	0
$13$	−10.0000	−0.213346	−0.106673	−	0.994294i	$-0.534020\pi$
$13$	−10.0000	−0.213346	−0.106673	+	0.994294i	$0.534020\pi$
$14$	0	0
$15$	−48.0000	−0.826236
$16$	0	0
$17$	24.0000	0.342403	0.171202	−	0.985236i	$-0.445235\pi$
$17$	24.0000	0.342403	0.171202	+	0.985236i	$0.445235\pi$
$18$	0	0
$19$	−94.0000	−1.13500	−0.567502	−	0.823372i	$-0.692090\pi$
$19$	−94.0000	−1.13500	−0.567502	+	0.823372i	$0.692090\pi$
$20$	0	0
$21$	28.0000	0.290957
$22$	0	0
$23$	−180.000	−1.63185	−0.815926	−	0.578156i	$-0.803772\pi$
$23$	−180.000	−1.63185	−0.815926	+	0.578156i	$0.803772\pi$
$24$	0	0
$25$	19.0000	0.152000
$26$	0	0
$27$	−152.000	−1.08342
$28$	0	0
$29$	30.0000	0.192099	0.0960493	−	0.995377i	$-0.469379\pi$
$29$	30.0000	0.192099	0.0960493	+	0.995377i	$0.469379\pi$
$30$	0	0
$31$	−94.0000	−0.544610	−0.272305	−	0.962211i	$-0.587786\pi$
$31$	−94.0000	−0.544610	−0.272305	+	0.962211i	$0.587786\pi$
$32$	0	0
$33$	44.0000	0.232104
$34$	0	0
$35$	−84.0000	−0.405674
$36$	0	0
$37$	−214.000	−0.950848	−0.475424	−	0.879757i	$-0.657705\pi$
$37$	−214.000	−0.950848	−0.475424	+	0.879757i	$0.657705\pi$
$38$	0	0
$39$	−40.0000	−0.164234
$40$	0	0
$41$	−48.0000	−0.182838	−0.0914188	−	0.995813i	$-0.529140\pi$
$41$	−48.0000	−0.182838	−0.0914188	+	0.995813i	$0.529140\pi$
$42$	0	0
$43$	8.00000	0.0283718	0.0141859	−	0.999899i	$-0.495484\pi$
$43$	8.00000	0.0283718	0.0141859	+	0.999899i	$0.495484\pi$
$44$	0	0
$45$	132.000	0.437276
$46$	0	0
$47$	30.0000	0.0931053	0.0465527	−	0.998916i	$-0.485176\pi$
$47$	30.0000	0.0931053	0.0465527	+	0.998916i	$0.485176\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	96.0000	0.263582
$52$	0	0
$53$	54.0000	0.139952	0.0699761	−	0.997549i	$-0.477708\pi$
$53$	54.0000	0.139952	0.0699761	+	0.997549i	$0.477708\pi$
$54$	0	0
$55$	−132.000	−0.323616
$56$	0	0
$57$	−376.000	−0.873727
$58$	0	0
$59$	360.000	0.794373	0.397187	−	0.917738i	$-0.369987\pi$
$59$	360.000	0.794373	0.397187	+	0.917738i	$0.369987\pi$
$60$	0	0
$61$	−178.000	−0.373616	−0.186808	−	0.982396i	$-0.559814\pi$
$61$	−178.000	−0.373616	−0.186808	+	0.982396i	$0.559814\pi$
$62$	0	0
$63$	−77.0000	−0.153986
$64$	0	0
$65$	120.000	0.228987
$66$	0	0
$67$	−292.000	−0.532440	−0.266220	−	0.963912i	$-0.585775\pi$
$67$	−292.000	−0.532440	−0.266220	+	0.963912i	$0.585775\pi$
$68$	0	0
$69$	−720.000	−1.25620
$70$	0	0
$71$	312.000	0.521515	0.260758	−	0.965404i	$-0.416028\pi$
$71$	312.000	0.521515	0.260758	+	0.965404i	$0.416028\pi$
$72$	0	0
$73$	728.000	1.16720	0.583602	−	0.812040i	$-0.301643\pi$
$73$	728.000	1.16720	0.583602	+	0.812040i	$0.301643\pi$
$74$	0	0
$75$	76.0000	0.117010
$76$	0	0
$77$	77.0000	0.113961
$78$	0	0
$79$	−1288.00	−1.83432	−0.917160	−	0.398519i	$-0.869524\pi$
$79$	−1288.00	−1.83432	−0.917160	+	0.398519i	$0.869524\pi$
$80$	0	0
$81$	−311.000	−0.426612
$82$	0	0
$83$	66.0000	0.0872824	0.0436412	−	0.999047i	$-0.486104\pi$
$83$	66.0000	0.0872824	0.0436412	+	0.999047i	$0.486104\pi$
$84$	0	0
$85$	−288.000	−0.367506
$86$	0	0
$87$	120.000	0.147878
$88$	0	0
$89$	−390.000	−0.464493	−0.232247	−	0.972657i	$-0.574608\pi$
$89$	−390.000	−0.464493	−0.232247	+	0.972657i	$0.574608\pi$
$90$	0	0
$91$	−70.0000	−0.0806373
$92$	0	0
$93$	−376.000	−0.419241
$94$	0	0
$95$	1128.00	1.21821
$96$	0	0
$97$	−1510.00	−1.58059	−0.790295	−	0.612726i	$-0.790073\pi$
$97$	−1510.00	−1.58059	−0.790295	+	0.612726i	$0.790073\pi$
$98$	0	0
$99$	−121.000	−0.122838
$100$	0	0
$101$	90.0000	0.0886667	0.0443333	−	0.999017i	$-0.485884\pi$
$101$	90.0000	0.0886667	0.0443333	+	0.999017i	$0.485884\pi$
$102$	0	0
$103$	302.000	0.288902	0.144451	−	0.989512i	$-0.453858\pi$
$103$	302.000	0.288902	0.144451	+	0.989512i	$0.453858\pi$
$104$	0	0
$105$	−336.000	−0.312288
$106$	0	0
$107$	240.000	0.216838	0.108419	−	0.994105i	$-0.465421\pi$
$107$	240.000	0.216838	0.108419	+	0.994105i	$0.465421\pi$
$108$	0	0
$109$	−166.000	−0.145871	−0.0729354	−	0.997337i	$-0.523237\pi$
$109$	−166.000	−0.145871	−0.0729354	+	0.997337i	$0.523237\pi$
$110$	0	0
$111$	−856.000	−0.731963
$112$	0	0
$113$	1662.00	1.38361	0.691804	−	0.722085i	$-0.256816\pi$
$113$	1662.00	1.38361	0.691804	+	0.722085i	$0.256816\pi$
$114$	0	0
$115$	2160.00	1.75149
$116$	0	0
$117$	110.000	0.0869188
$118$	0	0
$119$	168.000	0.129416
$120$	0	0
$121$	121.000	0.0909091
$122$	0	0
$123$	−192.000	−0.140748
$124$	0	0
$125$	1272.00	0.910169
$126$	0	0
$127$	1544.00	1.07880	0.539401	−	0.842049i	$-0.318651\pi$
$127$	1544.00	1.07880	0.539401	+	0.842049i	$0.318651\pi$
$128$	0	0
$129$	32.0000	0.0218406
$130$	0	0
$131$	2358.00	1.57267	0.786334	−	0.617802i	$-0.211977\pi$
$131$	2358.00	1.57267	0.786334	+	0.617802i	$0.211977\pi$
$132$	0	0
$133$	−658.000	−0.428991
$134$	0	0
$135$	1824.00	1.16285
$136$	0	0
$137$	1206.00	0.752084	0.376042	−	0.926603i	$-0.377285\pi$
$137$	1206.00	0.752084	0.376042	+	0.926603i	$0.377285\pi$
$138$	0	0
$139$	2630.00	1.60485	0.802423	−	0.596755i	$-0.203544\pi$
$139$	2630.00	1.60485	0.802423	+	0.596755i	$0.203544\pi$
$140$	0	0
$141$	120.000	0.0716725
$142$	0	0
$143$	−110.000	−0.0643263
$144$	0	0
$145$	−360.000	−0.206182
$146$	0	0
$147$	196.000	0.109971
$148$	0	0
$149$	−126.000	−0.0692773	−0.0346387	−	0.999400i	$-0.511028\pi$
$149$	−126.000	−0.0692773	−0.0346387	+	0.999400i	$0.511028\pi$
$150$	0	0
$151$	−1456.00	−0.784686	−0.392343	−	0.919819i	$-0.628335\pi$
$151$	−1456.00	−0.784686	−0.392343	+	0.919819i	$0.628335\pi$
$152$	0	0
$153$	−264.000	−0.139498
$154$	0	0
$155$	1128.00	0.584536
$156$	0	0
$157$	992.000	0.504269	0.252134	−	0.967692i	$-0.418867\pi$
$157$	992.000	0.504269	0.252134	+	0.967692i	$0.418867\pi$
$158$	0	0
$159$	216.000	0.107735
$160$	0	0
$161$	−1260.00	−0.616782
$162$	0	0
$163$	3188.00	1.53192	0.765961	−	0.642887i	$-0.222263\pi$
$163$	3188.00	1.53192	0.765961	+	0.642887i	$0.222263\pi$
$164$	0	0
$165$	−528.000	−0.249120
$166$	0	0
$167$	−252.000	−0.116769	−0.0583843	−	0.998294i	$-0.518595\pi$
$167$	−252.000	−0.116769	−0.0583843	+	0.998294i	$0.518595\pi$
$168$	0	0
$169$	−2097.00	−0.954483
$170$	0	0
$171$	1034.00	0.462409
$172$	0	0
$173$	−3882.00	−1.70603	−0.853014	−	0.521887i	$-0.825228\pi$
$173$	−3882.00	−1.70603	−0.853014	+	0.521887i	$0.825228\pi$
$174$	0	0
$175$	133.000	0.0574506
$176$	0	0
$177$	1440.00	0.611509
$178$	0	0
$179$	−3300.00	−1.37795	−0.688976	−	0.724784i	$-0.741940\pi$
$179$	−3300.00	−1.37795	−0.688976	+	0.724784i	$0.741940\pi$
$180$	0	0
$181$	−124.000	−0.0509218	−0.0254609	−	0.999676i	$-0.508105\pi$
$181$	−124.000	−0.0509218	−0.0254609	+	0.999676i	$0.508105\pi$
$182$	0	0
$183$	−712.000	−0.287610
$184$	0	0
$185$	2568.00	1.02056
$186$	0	0
$187$	264.000	0.103238
$188$	0	0
$189$	−1064.00	−0.409495
$190$	0	0
$191$	1860.00	0.704633	0.352316	−	0.935881i	$-0.385394\pi$
$191$	1860.00	0.704633	0.352316	+	0.935881i	$0.385394\pi$
$192$	0	0
$193$	−1318.00	−0.491563	−0.245782	−	0.969325i	$-0.579045\pi$
$193$	−1318.00	−0.491563	−0.245782	+	0.969325i	$0.579045\pi$
$194$	0	0
$195$	480.000	0.176274
$196$	0	0
$197$	−1266.00	−0.457862	−0.228931	−	0.973443i	$-0.573523\pi$
$197$	−1266.00	−0.457862	−0.228931	+	0.973443i	$0.573523\pi$
$198$	0	0
$199$	830.000	0.295664	0.147832	−	0.989012i	$-0.452771\pi$
$199$	830.000	0.295664	0.147832	+	0.989012i	$0.452771\pi$
$200$	0	0
$201$	−1168.00	−0.409872
$202$	0	0
$203$	210.000	0.0726065
$204$	0	0
$205$	576.000	0.196242
$206$	0	0
$207$	1980.00	0.664829
$208$	0	0
$209$	−1034.00	−0.342217
$210$	0	0
$211$	−4408.00	−1.43820	−0.719098	−	0.694909i	$-0.755445\pi$
$211$	−4408.00	−1.43820	−0.719098	+	0.694909i	$0.755445\pi$
$212$	0	0
$213$	1248.00	0.401463
$214$	0	0
$215$	−96.0000	−0.0304518
$216$	0	0
$217$	−658.000	−0.205843
$218$	0	0
$219$	2912.00	0.898515
$220$	0	0
$221$	−240.000	−0.0730504
$222$	0	0
$223$	−4246.00	−1.27504	−0.637518	−	0.770435i	$-0.720039\pi$
$223$	−4246.00	−1.27504	−0.637518	+	0.770435i	$0.720039\pi$
$224$	0	0
$225$	−209.000	−0.0619259
$226$	0	0
$227$	1710.00	0.499985	0.249993	−	0.968248i	$-0.419572\pi$
$227$	1710.00	0.499985	0.249993	+	0.968248i	$0.419572\pi$
$228$	0	0
$229$	−2044.00	−0.589831	−0.294916	−	0.955523i	$-0.595292\pi$
$229$	−2044.00	−0.589831	−0.294916	+	0.955523i	$0.595292\pi$
$230$	0	0
$231$	308.000	0.0877269
$232$	0	0
$233$	54.0000	0.0151831	0.00759154	−	0.999971i	$-0.497584\pi$
$233$	54.0000	0.0151831	0.00759154	+	0.999971i	$0.497584\pi$
$234$	0	0
$235$	−360.000	−0.0999311
$236$	0	0
$237$	−5152.00	−1.41206
$238$	0	0
$239$	−4416.00	−1.19518	−0.597588	−	0.801803i	$-0.703874\pi$
$239$	−4416.00	−1.19518	−0.597588	+	0.801803i	$0.703874\pi$
$240$	0	0
$241$	−808.000	−0.215966	−0.107983	−	0.994153i	$-0.534439\pi$
$241$	−808.000	−0.215966	−0.107983	+	0.994153i	$0.534439\pi$
$242$	0	0
$243$	2860.00	0.755017
$244$	0	0
$245$	−588.000	−0.153330
$246$	0	0
$247$	940.000	0.242149
$248$	0	0
$249$	264.000	0.0671900
$250$	0	0
$251$	5400.00	1.35795	0.678974	−	0.734162i	$-0.262425\pi$
$251$	5400.00	1.35795	0.678974	+	0.734162i	$0.262425\pi$
$252$	0	0
$253$	−1980.00	−0.492022
$254$	0	0
$255$	−1152.00	−0.282906
$256$	0	0
$257$	3882.00	0.942228	0.471114	−	0.882072i	$-0.343852\pi$
$257$	3882.00	0.942228	0.471114	+	0.882072i	$0.343852\pi$
$258$	0	0
$259$	−1498.00	−0.359387
$260$	0	0
$261$	−330.000	−0.0782624
$262$	0	0
$263$	−7536.00	−1.76688	−0.883440	−	0.468543i	$-0.844779\pi$
$263$	−7536.00	−1.76688	−0.883440	+	0.468543i	$0.844779\pi$
$264$	0	0
$265$	−648.000	−0.150213
$266$	0	0
$267$	−1560.00	−0.357567
$268$	0	0
$269$	1596.00	0.361747	0.180873	−	0.983506i	$-0.442108\pi$
$269$	1596.00	0.361747	0.180873	+	0.983506i	$0.442108\pi$
$270$	0	0
$271$	20.0000	0.00448308	0.00224154	−	0.999997i	$-0.499286\pi$
$271$	20.0000	0.00448308	0.00224154	+	0.999997i	$0.499286\pi$
$272$	0	0
$273$	−280.000	−0.0620746
$274$	0	0
$275$	209.000	0.0458297
$276$	0	0
$277$	−5374.00	−1.16568	−0.582838	−	0.812588i	$-0.698058\pi$
$277$	−5374.00	−1.16568	−0.582838	+	0.812588i	$0.698058\pi$
$278$	0	0
$279$	1034.00	0.221878
$280$	0	0
$281$	−786.000	−0.166864	−0.0834321	−	0.996513i	$-0.526588\pi$
$281$	−786.000	−0.166864	−0.0834321	+	0.996513i	$0.526588\pi$
$282$	0	0
$283$	3878.00	0.814570	0.407285	−	0.913301i	$-0.366476\pi$
$283$	3878.00	0.814570	0.407285	+	0.913301i	$0.366476\pi$
$284$	0	0
$285$	4512.00	0.937782
$286$	0	0
$287$	−336.000	−0.0691061
$288$	0	0
$289$	−4337.00	−0.882760
$290$	0	0
$291$	−6040.00	−1.21674
$292$	0	0
$293$	−8226.00	−1.64016	−0.820082	−	0.572246i	$-0.806072\pi$
$293$	−8226.00	−1.64016	−0.820082	+	0.572246i	$0.806072\pi$
$294$	0	0
$295$	−4320.00	−0.852611
$296$	0	0
$297$	−1672.00	−0.326664
$298$	0	0
$299$	1800.00	0.348149
$300$	0	0
$301$	56.0000	0.0107235
$302$	0	0
$303$	360.000	0.0682556
$304$	0	0
$305$	2136.00	0.401007
$306$	0	0
$307$	3926.00	0.729865	0.364933	−	0.931034i	$-0.381092\pi$
$307$	3926.00	0.729865	0.364933	+	0.931034i	$0.381092\pi$
$308$	0	0
$309$	1208.00	0.222397
$310$	0	0
$311$	618.000	0.112680	0.0563401	−	0.998412i	$-0.482057\pi$
$311$	618.000	0.112680	0.0563401	+	0.998412i	$0.482057\pi$
$312$	0	0
$313$	686.000	0.123882	0.0619409	−	0.998080i	$-0.480271\pi$
$313$	686.000	0.123882	0.0619409	+	0.998080i	$0.480271\pi$
$314$	0	0
$315$	924.000	0.165275
$316$	0	0
$317$	2094.00	0.371012	0.185506	−	0.982643i	$-0.440608\pi$
$317$	2094.00	0.371012	0.185506	+	0.982643i	$0.440608\pi$
$318$	0	0
$319$	330.000	0.0579199
$320$	0	0
$321$	960.000	0.166922
$322$	0	0
$323$	−2256.00	−0.388629
$324$	0	0
$325$	−190.000	−0.0324286
$326$	0	0
$327$	−664.000	−0.112291
$328$	0	0
$329$	210.000	0.0351905
$330$	0	0
$331$	2444.00	0.405844	0.202922	−	0.979195i	$-0.434956\pi$
$331$	2444.00	0.405844	0.202922	+	0.979195i	$0.434956\pi$
$332$	0	0
$333$	2354.00	0.387383
$334$	0	0
$335$	3504.00	0.571475
$336$	0	0
$337$	−7378.00	−1.19260	−0.596299	−	0.802763i	$-0.703363\pi$
$337$	−7378.00	−1.19260	−0.596299	+	0.802763i	$0.703363\pi$
$338$	0	0
$339$	6648.00	1.06510
$340$	0	0
$341$	−1034.00	−0.164206
$342$	0	0
$343$	343.000	0.0539949
$344$	0	0
$345$	8640.00	1.34830
$346$	0	0
$347$	−6840.00	−1.05819	−0.529093	−	0.848564i	$-0.677468\pi$
$347$	−6840.00	−1.05819	−0.529093	+	0.848564i	$0.677468\pi$
$348$	0	0
$349$	−5614.00	−0.861062	−0.430531	−	0.902576i	$-0.641674\pi$
$349$	−5614.00	−0.861062	−0.430531	+	0.902576i	$0.641674\pi$
$350$	0	0
$351$	1520.00	0.231144
$352$	0	0
$353$	9066.00	1.36695	0.683477	−	0.729972i	$-0.260467\pi$
$353$	9066.00	1.36695	0.683477	+	0.729972i	$0.260467\pi$
$354$	0	0
$355$	−3744.00	−0.559749
$356$	0	0
$357$	672.000	0.0996247
$358$	0	0
$359$	−5256.00	−0.772705	−0.386352	−	0.922351i	$-0.626265\pi$
$359$	−5256.00	−0.772705	−0.386352	+	0.922351i	$0.626265\pi$
$360$	0	0
$361$	1977.00	0.288234
$362$	0	0
$363$	484.000	0.0699819
$364$	0	0
$365$	−8736.00	−1.25278
$366$	0	0
$367$	−2350.00	−0.334248	−0.167124	−	0.985936i	$-0.553448\pi$
$367$	−2350.00	−0.334248	−0.167124	+	0.985936i	$0.553448\pi$
$368$	0	0
$369$	528.000	0.0744894
$370$	0	0
$371$	378.000	0.0528970
$372$	0	0
$373$	−4306.00	−0.597738	−0.298869	−	0.954294i	$-0.596609\pi$
$373$	−4306.00	−0.597738	−0.298869	+	0.954294i	$0.596609\pi$
$374$	0	0
$375$	5088.00	0.700649
$376$	0	0
$377$	−300.000	−0.0409835
$378$	0	0
$379$	−14380.0	−1.94895	−0.974474	−	0.224502i	$-0.927924\pi$
$379$	−14380.0	−1.94895	−0.974474	+	0.224502i	$0.927924\pi$
$380$	0	0
$381$	6176.00	0.830462
$382$	0	0
$383$	10746.0	1.43367	0.716834	−	0.697244i	$-0.245590\pi$
$383$	10746.0	1.43367	0.716834	+	0.697244i	$0.245590\pi$
$384$	0	0
$385$	−924.000	−0.122315
$386$	0	0
$387$	−88.0000	−0.0115589
$388$	0	0
$389$	−6870.00	−0.895431	−0.447716	−	0.894176i	$-0.647762\pi$
$389$	−6870.00	−0.895431	−0.447716	+	0.894176i	$0.647762\pi$
$390$	0	0
$391$	−4320.00	−0.558751
$392$	0	0
$393$	9432.00	1.21064
$394$	0	0
$395$	15456.0	1.96880
$396$	0	0
$397$	15728.0	1.98833	0.994163	−	0.107885i	$-0.0344079\pi$
$397$	15728.0	1.98833	0.994163	+	0.107885i	$0.0344079\pi$
$398$	0	0
$399$	−2632.00	−0.330238
$400$	0	0
$401$	12846.0	1.59975	0.799874	−	0.600168i	$-0.204900\pi$
$401$	12846.0	1.59975	0.799874	+	0.600168i	$0.204900\pi$
$402$	0	0
$403$	940.000	0.116190
$404$	0	0
$405$	3732.00	0.457888
$406$	0	0
$407$	−2354.00	−0.286692
$408$	0	0
$409$	−5740.00	−0.693948	−0.346974	−	0.937875i	$-0.612791\pi$
$409$	−5740.00	−0.693948	−0.346974	+	0.937875i	$0.612791\pi$
$410$	0	0
$411$	4824.00	0.578955
$412$	0	0
$413$	2520.00	0.300245
$414$	0	0
$415$	−792.000	−0.0936813
$416$	0	0
$417$	10520.0	1.23541
$418$	0	0
$419$	2292.00	0.267235	0.133618	−	0.991033i	$-0.457341\pi$
$419$	2292.00	0.267235	0.133618	+	0.991033i	$0.457341\pi$
$420$	0	0
$421$	−10834.0	−1.25420	−0.627098	−	0.778940i	$-0.715758\pi$
$421$	−10834.0	−1.25420	−0.627098	+	0.778940i	$0.715758\pi$
$422$	0	0
$423$	−330.000	−0.0379318
$424$	0	0
$425$	456.000	0.0520453
$426$	0	0
$427$	−1246.00	−0.141214
$428$	0	0
$429$	−440.000	−0.0495184
$430$	0	0
$431$	−11400.0	−1.27406	−0.637029	−	0.770840i	$-0.719837\pi$
$431$	−11400.0	−1.27406	−0.637029	+	0.770840i	$0.719837\pi$
$432$	0	0
$433$	−15958.0	−1.77111	−0.885557	−	0.464530i	$-0.846223\pi$
$433$	−15958.0	−1.77111	−0.885557	+	0.464530i	$0.846223\pi$
$434$	0	0
$435$	−1440.00	−0.158719
$436$	0	0
$437$	16920.0	1.85216
$438$	0	0
$439$	−268.000	−0.0291365	−0.0145683	−	0.999894i	$-0.504637\pi$
$439$	−268.000	−0.0291365	−0.0145683	+	0.999894i	$0.504637\pi$
$440$	0	0
$441$	−539.000	−0.0582011
$442$	0	0
$443$	−15324.0	−1.64349	−0.821744	−	0.569857i	$-0.806999\pi$
$443$	−15324.0	−1.64349	−0.821744	+	0.569857i	$0.806999\pi$
$444$	0	0
$445$	4680.00	0.498547
$446$	0	0
$447$	−504.000	−0.0533297
$448$	0	0
$449$	2238.00	0.235229	0.117614	−	0.993059i	$-0.462475\pi$
$449$	2238.00	0.235229	0.117614	+	0.993059i	$0.462475\pi$
$450$	0	0
$451$	−528.000	−0.0551276
$452$	0	0
$453$	−5824.00	−0.604052
$454$	0	0
$455$	840.000	0.0865490
$456$	0	0
$457$	−9046.00	−0.925939	−0.462969	−	0.886374i	$-0.653216\pi$
$457$	−9046.00	−0.925939	−0.462969	+	0.886374i	$0.653216\pi$
$458$	0	0
$459$	−3648.00	−0.370967
$460$	0	0
$461$	11898.0	1.20205	0.601025	−	0.799230i	$-0.294759\pi$
$461$	11898.0	1.20205	0.601025	+	0.799230i	$0.294759\pi$
$462$	0	0
$463$	6512.00	0.653646	0.326823	−	0.945085i	$-0.394022\pi$
$463$	6512.00	0.653646	0.326823	+	0.945085i	$0.394022\pi$
$464$	0	0
$465$	4512.00	0.449976
$466$	0	0
$467$	11676.0	1.15696	0.578481	−	0.815696i	$-0.303646\pi$
$467$	11676.0	1.15696	0.578481	+	0.815696i	$0.303646\pi$
$468$	0	0
$469$	−2044.00	−0.201243
$470$	0	0
$471$	3968.00	0.388186
$472$	0	0
$473$	88.0000	0.00855443
$474$	0	0
$475$	−1786.00	−0.172521
$476$	0	0
$477$	−594.000	−0.0570176
$478$	0	0
$479$	16920.0	1.61398	0.806988	−	0.590568i	$-0.201096\pi$
$479$	16920.0	1.61398	0.806988	+	0.590568i	$0.201096\pi$
$480$	0	0
$481$	2140.00	0.202860
$482$	0	0
$483$	−5040.00	−0.474799
$484$	0	0
$485$	18120.0	1.69647
$486$	0	0
$487$	−5188.00	−0.482732	−0.241366	−	0.970434i	$-0.577596\pi$
$487$	−5188.00	−0.482732	−0.241366	+	0.970434i	$0.577596\pi$
$488$	0	0
$489$	12752.0	1.17927
$490$	0	0
$491$	21036.0	1.93349	0.966743	−	0.255751i	$-0.0823226\pi$
$491$	21036.0	1.93349	0.966743	+	0.255751i	$0.0823226\pi$
$492$	0	0
$493$	720.000	0.0657752
$494$	0	0
$495$	1452.00	0.131844
$496$	0	0
$497$	2184.00	0.197114
$498$	0	0
$499$	−2284.00	−0.204902	−0.102451	−	0.994738i	$-0.532668\pi$
$499$	−2284.00	−0.204902	−0.102451	+	0.994738i	$0.532668\pi$
$500$	0	0
$501$	−1008.00	−0.0898885
$502$	0	0
$503$	−11004.0	−0.975436	−0.487718	−	0.873001i	$-0.662170\pi$
$503$	−11004.0	−0.975436	−0.487718	+	0.873001i	$0.662170\pi$
$504$	0	0
$505$	−1080.00	−0.0951671
$506$	0	0
$507$	−8388.00	−0.734762
$508$	0	0
$509$	−5244.00	−0.456653	−0.228326	−	0.973585i	$-0.573325\pi$
$509$	−5244.00	−0.456653	−0.228326	+	0.973585i	$0.573325\pi$
$510$	0	0
$511$	5096.00	0.441162
$512$	0	0
$513$	14288.0	1.22969
$514$	0	0
$515$	−3624.00	−0.310083
$516$	0	0
$517$	330.000	0.0280723
$518$	0	0
$519$	−15528.0	−1.31330
$520$	0	0
$521$	−1902.00	−0.159939	−0.0799694	−	0.996797i	$-0.525482\pi$
$521$	−1902.00	−0.159939	−0.0799694	+	0.996797i	$0.525482\pi$
$522$	0	0
$523$	−8194.00	−0.685083	−0.342542	−	0.939503i	$-0.611288\pi$
$523$	−8194.00	−0.685083	−0.342542	+	0.939503i	$0.611288\pi$
$524$	0	0
$525$	532.000	0.0442255
$526$	0	0
$527$	−2256.00	−0.186476
$528$	0	0
$529$	20233.0	1.66294
$530$	0	0
$531$	−3960.00	−0.323633
$532$	0	0
$533$	480.000	0.0390077
$534$	0	0
$535$	−2880.00	−0.232735
$536$	0	0
$537$	−13200.0	−1.06075
$538$	0	0
$539$	539.000	0.0430730
$540$	0	0
$541$	10406.0	0.826967	0.413483	−	0.910512i	$-0.364312\pi$
$541$	10406.0	0.826967	0.413483	+	0.910512i	$0.364312\pi$
$542$	0	0
$543$	−496.000	−0.0391996
$544$	0	0
$545$	1992.00	0.156565
$546$	0	0
$547$	20900.0	1.63367	0.816837	−	0.576869i	$-0.195726\pi$
$547$	20900.0	1.63367	0.816837	+	0.576869i	$0.195726\pi$
$548$	0	0
$549$	1958.00	0.152214
$550$	0	0
$551$	−2820.00	−0.218033
$552$	0	0
$553$	−9016.00	−0.693308
$554$	0	0
$555$	10272.0	0.785625
$556$	0	0
$557$	10038.0	0.763597	0.381799	−	0.924246i	$-0.375305\pi$
$557$	10038.0	0.763597	0.381799	+	0.924246i	$0.375305\pi$
$558$	0	0
$559$	−80.0000	−0.00605302
$560$	0	0
$561$	1056.00	0.0794730
$562$	0	0
$563$	−8154.00	−0.610391	−0.305195	−	0.952290i	$-0.598722\pi$
$563$	−8154.00	−0.610391	−0.305195	+	0.952290i	$0.598722\pi$
$564$	0	0
$565$	−19944.0	−1.48504
$566$	0	0
$567$	−2177.00	−0.161244
$568$	0	0
$569$	−2730.00	−0.201138	−0.100569	−	0.994930i	$-0.532066\pi$
$569$	−2730.00	−0.201138	−0.100569	+	0.994930i	$0.532066\pi$
$570$	0	0
$571$	3692.00	0.270587	0.135294	−	0.990806i	$-0.456802\pi$
$571$	3692.00	0.270587	0.135294	+	0.990806i	$0.456802\pi$
$572$	0	0
$573$	7440.00	0.542427
$574$	0	0
$575$	−3420.00	−0.248041
$576$	0	0
$577$	−3946.00	−0.284704	−0.142352	−	0.989816i	$-0.545466\pi$
$577$	−3946.00	−0.284704	−0.142352	+	0.989816i	$0.545466\pi$
$578$	0	0
$579$	−5272.00	−0.378406
$580$	0	0
$581$	462.000	0.0329897
$582$	0	0
$583$	594.000	0.0421972
$584$	0	0
$585$	−1320.00	−0.0932911
$586$	0	0
$587$	−3312.00	−0.232881	−0.116440	−	0.993198i	$-0.537148\pi$
$587$	−3312.00	−0.232881	−0.116440	+	0.993198i	$0.537148\pi$
$588$	0	0
$589$	8836.00	0.618134
$590$	0	0
$591$	−5064.00	−0.352462
$592$	0	0
$593$	−10920.0	−0.756207	−0.378103	−	0.925763i	$-0.623424\pi$
$593$	−10920.0	−0.756207	−0.378103	+	0.925763i	$0.623424\pi$
$594$	0	0
$595$	−2016.00	−0.138904
$596$	0	0
$597$	3320.00	0.227602
$598$	0	0
$599$	−25800.0	−1.75987	−0.879933	−	0.475098i	$-0.842413\pi$
$599$	−25800.0	−1.75987	−0.879933	+	0.475098i	$0.842413\pi$
$600$	0	0
$601$	−24700.0	−1.67643	−0.838215	−	0.545340i	$-0.816400\pi$
$601$	−24700.0	−1.67643	−0.838215	+	0.545340i	$0.816400\pi$
$602$	0	0
$603$	3212.00	0.216920
$604$	0	0
$605$	−1452.00	−0.0975739
$606$	0	0
$607$	−9016.00	−0.602880	−0.301440	−	0.953485i	$-0.597467\pi$
$607$	−9016.00	−0.602880	−0.301440	+	0.953485i	$0.597467\pi$
$608$	0	0
$609$	840.000	0.0558925
$610$	0	0
$611$	−300.000	−0.0198637
$612$	0	0
$613$	−1618.00	−0.106608	−0.0533038	−	0.998578i	$-0.516975\pi$
$613$	−1618.00	−0.106608	−0.0533038	+	0.998578i	$0.516975\pi$
$614$	0	0
$615$	2304.00	0.151067
$616$	0	0
$617$	−11310.0	−0.737963	−0.368982	−	0.929437i	$-0.620294\pi$
$617$	−11310.0	−0.737963	−0.368982	+	0.929437i	$0.620294\pi$
$618$	0	0
$619$	−24700.0	−1.60384	−0.801920	−	0.597432i	$-0.796188\pi$
$619$	−24700.0	−1.60384	−0.801920	+	0.597432i	$0.796188\pi$
$620$	0	0
$621$	27360.0	1.76799
$622$	0	0
$623$	−2730.00	−0.175562
$624$	0	0
$625$	−17639.0	−1.12890
$626$	0	0
$627$	−4136.00	−0.263438
$628$	0	0
$629$	−5136.00	−0.325573
$630$	0	0
$631$	−2500.00	−0.157723	−0.0788617	−	0.996886i	$-0.525129\pi$
$631$	−2500.00	−0.157723	−0.0788617	+	0.996886i	$0.525129\pi$
$632$	0	0
$633$	−17632.0	−1.10712
$634$	0	0
$635$	−18528.0	−1.15789
$636$	0	0
$637$	−490.000	−0.0304780
$638$	0	0
$639$	−3432.00	−0.212469
$640$	0	0
$641$	3450.00	0.212585	0.106292	−	0.994335i	$-0.466102\pi$
$641$	3450.00	0.212585	0.106292	+	0.994335i	$0.466102\pi$
$642$	0	0
$643$	18416.0	1.12948	0.564740	−	0.825269i	$-0.308976\pi$
$643$	18416.0	1.12948	0.564740	+	0.825269i	$0.308976\pi$
$644$	0	0
$645$	−384.000	−0.0234418
$646$	0	0
$647$	−7014.00	−0.426196	−0.213098	−	0.977031i	$-0.568355\pi$
$647$	−7014.00	−0.426196	−0.213098	+	0.977031i	$0.568355\pi$
$648$	0	0
$649$	3960.00	0.239512
$650$	0	0
$651$	−2632.00	−0.158458
$652$	0	0
$653$	−5070.00	−0.303835	−0.151918	−	0.988393i	$-0.548545\pi$
$653$	−5070.00	−0.303835	−0.151918	+	0.988393i	$0.548545\pi$
$654$	0	0
$655$	−28296.0	−1.68796
$656$	0	0
$657$	−8008.00	−0.475528
$658$	0	0
$659$	30600.0	1.80881	0.904406	−	0.426673i	$-0.140315\pi$
$659$	30600.0	1.80881	0.904406	+	0.426673i	$0.140315\pi$
$660$	0	0
$661$	22364.0	1.31597	0.657987	−	0.753029i	$-0.271408\pi$
$661$	22364.0	1.31597	0.657987	+	0.753029i	$0.271408\pi$
$662$	0	0
$663$	−960.000	−0.0562343
$664$	0	0
$665$	7896.00	0.460442
$666$	0	0
$667$	−5400.00	−0.313477
$668$	0	0
$669$	−16984.0	−0.981524
$670$	0	0
$671$	−1958.00	−0.112649
$672$	0	0
$673$	−31522.0	−1.80547	−0.902737	−	0.430193i	$-0.858445\pi$
$673$	−31522.0	−1.80547	−0.902737	+	0.430193i	$0.858445\pi$
$674$	0	0
$675$	−2888.00	−0.164680
$676$	0	0
$677$	−30210.0	−1.71501	−0.857507	−	0.514472i	$-0.827988\pi$
$677$	−30210.0	−1.71501	−0.857507	+	0.514472i	$0.827988\pi$
$678$	0	0
$679$	−10570.0	−0.597407
$680$	0	0
$681$	6840.00	0.384889
$682$	0	0
$683$	12492.0	0.699843	0.349922	−	0.936779i	$-0.386208\pi$
$683$	12492.0	0.699843	0.349922	+	0.936779i	$0.386208\pi$
$684$	0	0
$685$	−14472.0	−0.807221
$686$	0	0
$687$	−8176.00	−0.454052
$688$	0	0
$689$	−540.000	−0.0298583
$690$	0	0
$691$	21608.0	1.18959	0.594795	−	0.803877i	$-0.297233\pi$
$691$	21608.0	1.18959	0.594795	+	0.803877i	$0.297233\pi$
$692$	0	0
$693$	−847.000	−0.0464284
$694$	0	0
$695$	−31560.0	−1.72250
$696$	0	0
$697$	−1152.00	−0.0626042
$698$	0	0
$699$	216.000	0.0116879
$700$	0	0
$701$	28266.0	1.52296	0.761478	−	0.648191i	$-0.224474\pi$
$701$	28266.0	1.52296	0.761478	+	0.648191i	$0.224474\pi$
$702$	0	0
$703$	20116.0	1.07922
$704$	0	0
$705$	−1440.00	−0.0769270
$706$	0	0
$707$	630.000	0.0335129
$708$	0	0
$709$	5978.00	0.316655	0.158328	−	0.987387i	$-0.449390\pi$
$709$	5978.00	0.316655	0.158328	+	0.987387i	$0.449390\pi$
$710$	0	0
$711$	14168.0	0.747316
$712$	0	0
$713$	16920.0	0.888722
$714$	0	0
$715$	1320.00	0.0690422
$716$	0	0
$717$	−17664.0	−0.920048
$718$	0	0
$719$	6330.00	0.328330	0.164165	−	0.986433i	$-0.447507\pi$
$719$	6330.00	0.328330	0.164165	+	0.986433i	$0.447507\pi$
$720$	0	0
$721$	2114.00	0.109195
$722$	0	0
$723$	−3232.00	−0.166251
$724$	0	0
$725$	570.000	0.0291990
$726$	0	0
$727$	−8710.00	−0.444341	−0.222171	−	0.975008i	$-0.571314\pi$
$727$	−8710.00	−0.444341	−0.222171	+	0.975008i	$0.571314\pi$
$728$	0	0
$729$	19837.0	1.00782
$730$	0	0
$731$	192.000	0.00971461
$732$	0	0
$733$	−2506.00	−0.126277	−0.0631386	−	0.998005i	$-0.520111\pi$
$733$	−2506.00	−0.126277	−0.0631386	+	0.998005i	$0.520111\pi$
$734$	0	0
$735$	−2352.00	−0.118034
$736$	0	0
$737$	−3212.00	−0.160537
$738$	0	0
$739$	17060.0	0.849205	0.424602	−	0.905380i	$-0.360414\pi$
$739$	17060.0	0.849205	0.424602	+	0.905380i	$0.360414\pi$
$740$	0	0
$741$	3760.00	0.186406
$742$	0	0
$743$	22392.0	1.10563	0.552815	−	0.833304i	$-0.313554\pi$
$743$	22392.0	1.10563	0.552815	+	0.833304i	$0.313554\pi$
$744$	0	0
$745$	1512.00	0.0743562
$746$	0	0
$747$	−726.000	−0.0355595
$748$	0	0
$749$	1680.00	0.0819571
$750$	0	0
$751$	−16828.0	−0.817660	−0.408830	−	0.912611i	$-0.634063\pi$
$751$	−16828.0	−0.817660	−0.408830	+	0.912611i	$0.634063\pi$
$752$	0	0
$753$	21600.0	1.04535
$754$	0	0
$755$	17472.0	0.842213
$756$	0	0
$757$	10514.0	0.504806	0.252403	−	0.967622i	$-0.418779\pi$
$757$	10514.0	0.504806	0.252403	+	0.967622i	$0.418779\pi$
$758$	0	0
$759$	−7920.00	−0.378759
$760$	0	0
$761$	12084.0	0.575617	0.287809	−	0.957688i	$-0.407073\pi$
$761$	12084.0	0.575617	0.287809	+	0.957688i	$0.407073\pi$
$762$	0	0
$763$	−1162.00	−0.0551340
$764$	0	0
$765$	3168.00	0.149725
$766$	0	0
$767$	−3600.00	−0.169476
$768$	0	0
$769$	1940.00	0.0909729	0.0454865	−	0.998965i	$-0.485516\pi$
$769$	1940.00	0.0909729	0.0454865	+	0.998965i	$0.485516\pi$
$770$	0	0
$771$	15528.0	0.725327
$772$	0	0
$773$	−10800.0	−0.502521	−0.251261	−	0.967919i	$-0.580845\pi$
$773$	−10800.0	−0.502521	−0.251261	+	0.967919i	$0.580845\pi$
$774$	0	0
$775$	−1786.00	−0.0827807
$776$	0	0
$777$	−5992.00	−0.276656
$778$	0	0
$779$	4512.00	0.207521
$780$	0	0
$781$	3432.00	0.157243
$782$	0	0
$783$	−4560.00	−0.208124
$784$	0	0
$785$	−11904.0	−0.541238
$786$	0	0
$787$	11378.0	0.515352	0.257676	−	0.966231i	$-0.417043\pi$
$787$	11378.0	0.515352	0.257676	+	0.966231i	$0.417043\pi$
$788$	0	0
$789$	−30144.0	−1.36015
$790$	0	0
$791$	11634.0	0.522955
$792$	0	0
$793$	1780.00	0.0797095
$794$	0	0
$795$	−2592.00	−0.115634
$796$	0	0
$797$	15528.0	0.690125	0.345063	−	0.938580i	$-0.387858\pi$
$797$	15528.0	0.690125	0.345063	+	0.938580i	$0.387858\pi$
$798$	0	0
$799$	720.000	0.0318796
$800$	0	0
$801$	4290.00	0.189238
$802$	0	0
$803$	8008.00	0.351926
$804$	0	0
$805$	15120.0	0.662000
$806$	0	0
$807$	6384.00	0.278473
$808$	0	0
$809$	5082.00	0.220857	0.110429	−	0.993884i	$-0.464778\pi$
$809$	5082.00	0.220857	0.110429	+	0.993884i	$0.464778\pi$
$810$	0	0
$811$	24086.0	1.04288	0.521439	−	0.853289i	$-0.325395\pi$
$811$	24086.0	1.04288	0.521439	+	0.853289i	$0.325395\pi$
$812$	0	0
$813$	80.0000	0.00345107
$814$	0	0
$815$	−38256.0	−1.64423
$816$	0	0
$817$	−752.000	−0.0322021
$818$	0	0
$819$	770.000	0.0328522
$820$	0	0
$821$	6210.00	0.263984	0.131992	−	0.991251i	$-0.457863\pi$
$821$	6210.00	0.263984	0.131992	+	0.991251i	$0.457863\pi$
$822$	0	0
$823$	20612.0	0.873012	0.436506	−	0.899701i	$-0.356216\pi$
$823$	20612.0	0.873012	0.436506	+	0.899701i	$0.356216\pi$
$824$	0	0
$825$	836.000	0.0352797
$826$	0	0
$827$	−3948.00	−0.166004	−0.0830021	−	0.996549i	$-0.526451\pi$
$827$	−3948.00	−0.166004	−0.0830021	+	0.996549i	$0.526451\pi$
$828$	0	0
$829$	−5692.00	−0.238470	−0.119235	−	0.992866i	$-0.538044\pi$
$829$	−5692.00	−0.238470	−0.119235	+	0.992866i	$0.538044\pi$
$830$	0	0
$831$	−21496.0	−0.897338
$832$	0	0
$833$	1176.00	0.0489147
$834$	0	0
$835$	3024.00	0.125329
$836$	0	0
$837$	14288.0	0.590042
$838$	0	0
$839$	−29070.0	−1.19620	−0.598098	−	0.801423i	$-0.704077\pi$
$839$	−29070.0	−1.19620	−0.598098	+	0.801423i	$0.704077\pi$
$840$	0	0
$841$	−23489.0	−0.963098
$842$	0	0
$843$	−3144.00	−0.128452
$844$	0	0
$845$	25164.0	1.02446
$846$	0	0
$847$	847.000	0.0343604
$848$	0	0
$849$	15512.0	0.627056
$850$	0	0
$851$	38520.0	1.55164
$852$	0	0
$853$	20582.0	0.826160	0.413080	−	0.910695i	$-0.364453\pi$
$853$	20582.0	0.826160	0.413080	+	0.910695i	$0.364453\pi$
$854$	0	0
$855$	−12408.0	−0.496310
$856$	0	0
$857$	−36156.0	−1.44115	−0.720575	−	0.693377i	$-0.756122\pi$
$857$	−36156.0	−1.44115	−0.720575	+	0.693377i	$0.756122\pi$
$858$	0	0
$859$	19784.0	0.785822	0.392911	−	0.919576i	$-0.371468\pi$
$859$	19784.0	0.785822	0.392911	+	0.919576i	$0.371468\pi$
$860$	0	0
$861$	−1344.00	−0.0531979
$862$	0	0
$863$	−34068.0	−1.34379	−0.671894	−	0.740648i	$-0.734519\pi$
$863$	−34068.0	−1.34379	−0.671894	+	0.740648i	$0.734519\pi$
$864$	0	0
$865$	46584.0	1.83110
$866$	0	0
$867$	−17348.0	−0.679549
$868$	0	0
$869$	−14168.0	−0.553068
$870$	0	0
$871$	2920.00	0.113594
$872$	0	0
$873$	16610.0	0.643944
$874$	0	0
$875$	8904.00	0.344012
$876$	0	0
$877$	−19510.0	−0.751204	−0.375602	−	0.926781i	$-0.622564\pi$
$877$	−19510.0	−0.751204	−0.375602	+	0.926781i	$0.622564\pi$
$878$	0	0
$879$	−32904.0	−1.26260
$880$	0	0
$881$	−24474.0	−0.935925	−0.467963	−	0.883748i	$-0.655012\pi$
$881$	−24474.0	−0.935925	−0.467963	+	0.883748i	$0.655012\pi$
$882$	0	0
$883$	49388.0	1.88226	0.941132	−	0.338040i	$-0.109764\pi$
$883$	49388.0	1.88226	0.941132	+	0.338040i	$0.109764\pi$
$884$	0	0
$885$	−17280.0	−0.656340
$886$	0	0
$887$	−30216.0	−1.14380	−0.571902	−	0.820322i	$-0.693794\pi$
$887$	−30216.0	−1.14380	−0.571902	+	0.820322i	$0.693794\pi$
$888$	0	0
$889$	10808.0	0.407749
$890$	0	0
$891$	−3421.00	−0.128628
$892$	0	0
$893$	−2820.00	−0.105675
$894$	0	0
$895$	39600.0	1.47897
$896$	0	0
$897$	7200.00	0.268006
$898$	0	0
$899$	−2820.00	−0.104619
$900$	0	0
$901$	1296.00	0.0479201
$902$	0	0
$903$	224.000	0.00825499
$904$	0	0
$905$	1488.00	0.0546550
$906$	0	0
$907$	−4492.00	−0.164448	−0.0822240	−	0.996614i	$-0.526202\pi$
$907$	−4492.00	−0.164448	−0.0822240	+	0.996614i	$0.526202\pi$
$908$	0	0
$909$	−990.000	−0.0361235
$910$	0	0
$911$	28932.0	1.05221	0.526103	−	0.850421i	$-0.323653\pi$
$911$	28932.0	1.05221	0.526103	+	0.850421i	$0.323653\pi$
$912$	0	0
$913$	726.000	0.0263166
$914$	0	0
$915$	8544.00	0.308695
$916$	0	0
$917$	16506.0	0.594412
$918$	0	0
$919$	−32056.0	−1.15063	−0.575315	−	0.817932i	$-0.695121\pi$
$919$	−32056.0	−1.15063	−0.575315	+	0.817932i	$0.695121\pi$
$920$	0	0
$921$	15704.0	0.561851
$922$	0	0
$923$	−3120.00	−0.111263
$924$	0	0
$925$	−4066.00	−0.144529
$926$	0	0
$927$	−3322.00	−0.117701
$928$	0	0
$929$	−22182.0	−0.783388	−0.391694	−	0.920095i	$-0.628111\pi$
$929$	−22182.0	−0.783388	−0.391694	+	0.920095i	$0.628111\pi$
$930$	0	0
$931$	−4606.00	−0.162143
$932$	0	0
$933$	2472.00	0.0867413
$934$	0	0
$935$	−3168.00	−0.110807
$936$	0	0
$937$	29792.0	1.03870	0.519350	−	0.854562i	$-0.326174\pi$
$937$	29792.0	1.03870	0.519350	+	0.854562i	$0.326174\pi$
$938$	0	0
$939$	2744.00	0.0953643
$940$	0	0
$941$	30702.0	1.06361	0.531805	−	0.846867i	$-0.321514\pi$
$941$	30702.0	1.06361	0.531805	+	0.846867i	$0.321514\pi$
$942$	0	0
$943$	8640.00	0.298364
$944$	0	0
$945$	12768.0	0.439516
$946$	0	0
$947$	5244.00	0.179944	0.0899721	−	0.995944i	$-0.471322\pi$
$947$	5244.00	0.179944	0.0899721	+	0.995944i	$0.471322\pi$
$948$	0	0
$949$	−7280.00	−0.249019
$950$	0	0
$951$	8376.00	0.285605
$952$	0	0
$953$	−24150.0	−0.820876	−0.410438	−	0.911888i	$-0.634624\pi$
$953$	−24150.0	−0.820876	−0.410438	+	0.911888i	$0.634624\pi$
$954$	0	0
$955$	−22320.0	−0.756291
$956$	0	0
$957$	1320.00	0.0445868
$958$	0	0
$959$	8442.00	0.284261
$960$	0	0
$961$	−20955.0	−0.703400
$962$	0	0
$963$	−2640.00	−0.0883414
$964$	0	0
$965$	15816.0	0.527601
$966$	0	0
$967$	15512.0	0.515856	0.257928	−	0.966164i	$-0.416960\pi$
$967$	15512.0	0.515856	0.257928	+	0.966164i	$0.416960\pi$
$968$	0	0
$969$	−9024.00	−0.299167
$970$	0	0
$971$	52404.0	1.73195	0.865975	−	0.500086i	$-0.166698\pi$
$971$	52404.0	1.73195	0.865975	+	0.500086i	$0.166698\pi$
$972$	0	0
$973$	18410.0	0.606575
$974$	0	0
$975$	−760.000	−0.0249636
$976$	0	0
$977$	−5070.00	−0.166022	−0.0830111	−	0.996549i	$-0.526454\pi$
$977$	−5070.00	−0.166022	−0.0830111	+	0.996549i	$0.526454\pi$
$978$	0	0
$979$	−4290.00	−0.140050
$980$	0	0
$981$	1826.00	0.0594288
$982$	0	0
$983$	18702.0	0.606817	0.303409	−	0.952861i	$-0.401875\pi$
$983$	18702.0	0.606817	0.303409	+	0.952861i	$0.401875\pi$
$984$	0	0
$985$	15192.0	0.491429
$986$	0	0
$987$	840.000	0.0270897
$988$	0	0
$989$	−1440.00	−0.0462986
$990$	0	0
$991$	40496.0	1.29808	0.649040	−	0.760754i	$-0.275171\pi$
$991$	40496.0	1.29808	0.649040	+	0.760754i	$0.275171\pi$
$992$	0	0
$993$	9776.00	0.312419
$994$	0	0
$995$	−9960.00	−0.317340
$996$	0	0
$997$	−23686.0	−0.752400	−0.376200	−	0.926538i	$-0.622770\pi$
$997$	−23686.0	−0.752400	−0.376200	+	0.926538i	$0.622770\pi$
$998$	0	0
$999$	32528.0	1.03017

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	308.4.a.b.1.1	✓	1
4.3	odd	2		1232.4.a.c.1.1		1
7.6	odd	2		2156.4.a.a.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
308.4.a.b.1.1	✓	1	1.1	even	1	trivial
1232.4.a.c.1.1		1	4.3	odd	2
2156.4.a.a.1.1		1	7.6	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:	\( N \)	\(=\)	\( 308 = 2^{2} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	308.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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