LMFDB - Embedded newform 9576.2.a.bo.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9576,2,Mod(1,9576)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9576, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("9576.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9576 = 2^{3} \cdot 3^{2} \cdot 7 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9576.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:	$76.4647449756$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	9576.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.82843 q^{5} +1.00000 q^{7} +O(q^{10})$	$q+2.82843 q^{5} +1.00000 q^{7} +2.00000 q^{11} +4.82843 q^{13} +4.00000 q^{17} +1.00000 q^{19} -0.828427 q^{23} +3.00000 q^{25} -7.65685 q^{29} +1.17157 q^{31} +2.82843 q^{35} -8.82843 q^{37} +2.00000 q^{41} +1.65685 q^{43} +10.4853 q^{47} +1.00000 q^{49} +6.00000 q^{53} +5.65685 q^{55} +4.00000 q^{59} -6.00000 q^{61} +13.6569 q^{65} +11.3137 q^{67} +13.6569 q^{71} +10.0000 q^{73} +2.00000 q^{77} +14.8284 q^{79} -15.6569 q^{83} +11.3137 q^{85} -11.6569 q^{89} +4.82843 q^{91} +2.82843 q^{95} -11.6569 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{7}+O(q^{10}) $ 2 * q + 2 * q^7	$ 2 q + 2 q^{7} + 4 q^{11} + 4 q^{13} + 8 q^{17} + 2 q^{19} + 4 q^{23} + 6 q^{25} - 4 q^{29} + 8 q^{31} - 12 q^{37} + 4 q^{41} - 8 q^{43} + 4 q^{47} + 2 q^{49} + 12 q^{53} + 8 q^{59} - 12 q^{61} + 16 q^{65} + 16 q^{71} + 20 q^{73} + 4 q^{77} + 24 q^{79} - 20 q^{83} - 12 q^{89} + 4 q^{91} - 12 q^{97}+O(q^{100}) $ 2 * q + 2 * q^7 + 4 * q^11 + 4 * q^13 + 8 * q^17 + 2 * q^19 + 4 * q^23 + 6 * q^25 - 4 * q^29 + 8 * q^31 - 12 * q^37 + 4 * q^41 - 8 * q^43 + 4 * q^47 + 2 * q^49 + 12 * q^53 + 8 * q^59 - 12 * q^61 + 16 * q^65 + 16 * q^71 + 20 * q^73 + 4 * q^77 + 24 * q^79 - 20 * q^83 - 12 * q^89 + 4 * q^91 - 12 * q^97

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	2.82843	1.26491	0.632456	−	0.774597i	$-0.282047\pi$
$5$	2.82843	1.26491	0.632456	+	0.774597i	$0.282047\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	2.00000	0.603023	0.301511	−	0.953463i	$-0.402509\pi$
$11$	2.00000	0.603023	0.301511	+	0.953463i	$0.402509\pi$
$12$	0	0
$13$	4.82843	1.33916	0.669582	−	0.742738i	$-0.266473\pi$
$13$	4.82843	1.33916	0.669582	+	0.742738i	$0.266473\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	4.00000	0.970143	0.485071	−	0.874475i	$-0.338794\pi$
$17$	4.00000	0.970143	0.485071	+	0.874475i	$0.338794\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−0.828427	−0.172739	−0.0863695	−	0.996263i	$-0.527527\pi$
$23$	−0.828427	−0.172739	−0.0863695	+	0.996263i	$0.527527\pi$
$24$	0	0
$25$	3.00000	0.600000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−7.65685	−1.42184	−0.710921	−	0.703272i	$-0.751722\pi$
$29$	−7.65685	−1.42184	−0.710921	+	0.703272i	$0.751722\pi$
$30$	0	0
$31$	1.17157	0.210421	0.105210	−	0.994450i	$-0.466448\pi$
$31$	1.17157	0.210421	0.105210	+	0.994450i	$0.466448\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	2.82843	0.478091
$36$	0	0
$37$	−8.82843	−1.45138	−0.725692	−	0.688019i	$-0.758480\pi$
$37$	−8.82843	−1.45138	−0.725692	+	0.688019i	$0.758480\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	2.00000	0.312348	0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000	0.312348	0.156174	+	0.987730i	$0.450084\pi$
$42$	0	0
$43$	1.65685	0.252668	0.126334	−	0.991988i	$-0.459679\pi$
$43$	1.65685	0.252668	0.126334	+	0.991988i	$0.459679\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	10.4853	1.52944	0.764718	−	0.644365i	$-0.222878\pi$
$47$	10.4853	1.52944	0.764718	+	0.644365i	$0.222878\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	6.00000	0.824163	0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000	0.824163	0.412082	+	0.911147i	$0.364802\pi$
$54$	0	0
$55$	5.65685	0.762770
$56$	0	0
$57$	0	0
$58$	0	0
$59$	4.00000	0.520756	0.260378	−	0.965507i	$-0.416153\pi$
$59$	4.00000	0.520756	0.260378	+	0.965507i	$0.416153\pi$
$60$	0	0
$61$	−6.00000	−0.768221	−0.384111	−	0.923287i	$-0.625492\pi$
$61$	−6.00000	−0.768221	−0.384111	+	0.923287i	$0.625492\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	13.6569	1.69392
$66$	0	0
$67$	11.3137	1.38219	0.691095	−	0.722764i	$-0.257129\pi$
$67$	11.3137	1.38219	0.691095	+	0.722764i	$0.257129\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	13.6569	1.62077	0.810385	−	0.585897i	$-0.199258\pi$
$71$	13.6569	1.62077	0.810385	+	0.585897i	$0.199258\pi$
$72$	0	0
$73$	10.0000	1.17041	0.585206	−	0.810885i	$-0.301014\pi$
$73$	10.0000	1.17041	0.585206	+	0.810885i	$0.301014\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	2.00000	0.227921
$78$	0	0
$79$	14.8284	1.66833	0.834164	−	0.551516i	$-0.185951\pi$
$79$	14.8284	1.66833	0.834164	+	0.551516i	$0.185951\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−15.6569	−1.71856	−0.859282	−	0.511503i	$-0.829089\pi$
$83$	−15.6569	−1.71856	−0.859282	+	0.511503i	$0.829089\pi$
$84$	0	0
$85$	11.3137	1.22714
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−11.6569	−1.23562	−0.617812	−	0.786326i	$-0.711981\pi$
$89$	−11.6569	−1.23562	−0.617812	+	0.786326i	$0.711981\pi$
$90$	0	0
$91$	4.82843	0.506157
$92$	0	0
$93$	0	0
$94$	0	0
$95$	2.82843	0.290191
$96$	0	0
$97$	−11.6569	−1.18357	−0.591787	−	0.806094i	$-0.701577\pi$
$97$	−11.6569	−1.18357	−0.591787	+	0.806094i	$0.701577\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−12.4853	−1.24233	−0.621166	−	0.783679i	$-0.713341\pi$
$101$	−12.4853	−1.24233	−0.621166	+	0.783679i	$0.713341\pi$
$102$	0	0
$103$	2.82843	0.278693	0.139347	−	0.990244i	$-0.455500\pi$
$103$	2.82843	0.278693	0.139347	+	0.990244i	$0.455500\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−2.34315	−0.226520	−0.113260	−	0.993565i	$-0.536129\pi$
$107$	−2.34315	−0.226520	−0.113260	+	0.993565i	$0.536129\pi$
$108$	0	0
$109$	12.1421	1.16301	0.581503	−	0.813544i	$-0.302465\pi$
$109$	12.1421	1.16301	0.581503	+	0.813544i	$0.302465\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−2.00000	−0.188144	−0.0940721	−	0.995565i	$-0.529988\pi$
$113$	−2.00000	−0.188144	−0.0940721	+	0.995565i	$0.529988\pi$
$114$	0	0
$115$	−2.34315	−0.218499
$116$	0	0
$117$	0	0
$118$	0	0
$119$	4.00000	0.366679
$120$	0	0
$121$	−7.00000	−0.636364
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−5.65685	−0.505964
$126$	0	0
$127$	8.48528	0.752947	0.376473	−	0.926427i	$-0.377137\pi$
$127$	8.48528	0.752947	0.376473	+	0.926427i	$0.377137\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	15.6569	1.36795	0.683973	−	0.729507i	$-0.260251\pi$
$131$	15.6569	1.36795	0.683973	+	0.729507i	$0.260251\pi$
$132$	0	0
$133$	1.00000	0.0867110
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−12.9706	−1.10815	−0.554075	−	0.832467i	$-0.686928\pi$
$137$	−12.9706	−1.10815	−0.554075	+	0.832467i	$0.686928\pi$
$138$	0	0
$139$	−9.65685	−0.819084	−0.409542	−	0.912291i	$-0.634311\pi$
$139$	−9.65685	−0.819084	−0.409542	+	0.912291i	$0.634311\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	9.65685	0.807547
$144$	0	0
$145$	−21.6569	−1.79850
$146$	0	0
$147$	0	0
$148$	0	0
$149$	1.17157	0.0959790	0.0479895	−	0.998848i	$-0.484719\pi$
$149$	1.17157	0.0959790	0.0479895	+	0.998848i	$0.484719\pi$
$150$	0	0
$151$	−12.4853	−1.01604	−0.508019	−	0.861346i	$-0.669622\pi$
$151$	−12.4853	−1.01604	−0.508019	+	0.861346i	$0.669622\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	3.31371	0.266163
$156$	0	0
$157$	−14.9706	−1.19478	−0.597390	−	0.801950i	$-0.703796\pi$
$157$	−14.9706	−1.19478	−0.597390	+	0.801950i	$0.703796\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−0.828427	−0.0652892
$162$	0	0
$163$	−23.3137	−1.82607	−0.913035	−	0.407881i	$-0.866268\pi$
$163$	−23.3137	−1.82607	−0.913035	+	0.407881i	$0.866268\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−11.3137	−0.875481	−0.437741	−	0.899101i	$-0.644221\pi$
$167$	−11.3137	−0.875481	−0.437741	+	0.899101i	$0.644221\pi$
$168$	0	0
$169$	10.3137	0.793362
$170$	0	0
$171$	0	0
$172$	0	0
$173$	14.0000	1.06440	0.532200	−	0.846619i	$-0.321365\pi$
$173$	14.0000	1.06440	0.532200	+	0.846619i	$0.321365\pi$
$174$	0	0
$175$	3.00000	0.226779
$176$	0	0
$177$	0	0
$178$	0	0
$179$	10.3431	0.773083	0.386542	−	0.922272i	$-0.373670\pi$
$179$	10.3431	0.773083	0.386542	+	0.922272i	$0.373670\pi$
$180$	0	0
$181$	−16.8284	−1.25085	−0.625424	−	0.780285i	$-0.715074\pi$
$181$	−16.8284	−1.25085	−0.625424	+	0.780285i	$0.715074\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−24.9706	−1.83587
$186$	0	0
$187$	8.00000	0.585018
$188$	0	0
$189$	0	0
$190$	0	0
$191$	11.1716	0.808347	0.404173	−	0.914682i	$-0.367559\pi$
$191$	11.1716	0.808347	0.404173	+	0.914682i	$0.367559\pi$
$192$	0	0
$193$	18.9706	1.36553	0.682765	−	0.730638i	$-0.260777\pi$
$193$	18.9706	1.36553	0.682765	+	0.730638i	$0.260777\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	24.4853	1.74450	0.872252	−	0.489057i	$-0.162659\pi$
$197$	24.4853	1.74450	0.872252	+	0.489057i	$0.162659\pi$
$198$	0	0
$199$	−5.65685	−0.401004	−0.200502	−	0.979693i	$-0.564257\pi$
$199$	−5.65685	−0.401004	−0.200502	+	0.979693i	$0.564257\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−7.65685	−0.537406
$204$	0	0
$205$	5.65685	0.395092
$206$	0	0
$207$	0	0
$208$	0	0
$209$	2.00000	0.138343
$210$	0	0
$211$	−8.97056	−0.617559	−0.308780	−	0.951134i	$-0.599921\pi$
$211$	−8.97056	−0.617559	−0.308780	+	0.951134i	$0.599921\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	4.68629	0.319602
$216$	0	0
$217$	1.17157	0.0795315
$218$	0	0
$219$	0	0
$220$	0	0
$221$	19.3137	1.29918
$222$	0	0
$223$	5.17157	0.346314	0.173157	−	0.984894i	$-0.444603\pi$
$223$	5.17157	0.346314	0.173157	+	0.984894i	$0.444603\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	11.3137	0.750917	0.375459	−	0.926839i	$-0.377485\pi$
$227$	11.3137	0.750917	0.375459	+	0.926839i	$0.377485\pi$
$228$	0	0
$229$	2.00000	0.132164	0.0660819	−	0.997814i	$-0.478950\pi$
$229$	2.00000	0.132164	0.0660819	+	0.997814i	$0.478950\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	11.3137	0.741186	0.370593	−	0.928795i	$-0.379155\pi$
$233$	11.3137	0.741186	0.370593	+	0.928795i	$0.379155\pi$
$234$	0	0
$235$	29.6569	1.93460
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−18.4853	−1.19571	−0.597857	−	0.801603i	$-0.703981\pi$
$239$	−18.4853	−1.19571	−0.597857	+	0.801603i	$0.703981\pi$
$240$	0	0
$241$	−6.97056	−0.449013	−0.224507	−	0.974473i	$-0.572077\pi$
$241$	−6.97056	−0.449013	−0.224507	+	0.974473i	$0.572077\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	2.82843	0.180702
$246$	0	0
$247$	4.82843	0.307225
$248$	0	0
$249$	0	0
$250$	0	0
$251$	22.9706	1.44989	0.724945	−	0.688807i	$-0.241865\pi$
$251$	22.9706	1.44989	0.724945	+	0.688807i	$0.241865\pi$
$252$	0	0
$253$	−1.65685	−0.104166
$254$	0	0
$255$	0	0
$256$	0	0
$257$	11.6569	0.727135	0.363567	−	0.931568i	$-0.381559\pi$
$257$	11.6569	0.727135	0.363567	+	0.931568i	$0.381559\pi$
$258$	0	0
$259$	−8.82843	−0.548572
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−3.85786	−0.237886	−0.118943	−	0.992901i	$-0.537951\pi$
$263$	−3.85786	−0.237886	−0.118943	+	0.992901i	$0.537951\pi$
$264$	0	0
$265$	16.9706	1.04249
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−10.0000	−0.609711	−0.304855	−	0.952399i	$-0.598608\pi$
$269$	−10.0000	−0.609711	−0.304855	+	0.952399i	$0.598608\pi$
$270$	0	0
$271$	−21.6569	−1.31556	−0.657780	−	0.753210i	$-0.728504\pi$
$271$	−21.6569	−1.31556	−0.657780	+	0.753210i	$0.728504\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	6.00000	0.361814
$276$	0	0
$277$	−3.65685	−0.219719	−0.109860	−	0.993947i	$-0.535040\pi$
$277$	−3.65685	−0.219719	−0.109860	+	0.993947i	$0.535040\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−1.31371	−0.0783693	−0.0391846	−	0.999232i	$-0.512476\pi$
$281$	−1.31371	−0.0783693	−0.0391846	+	0.999232i	$0.512476\pi$
$282$	0	0
$283$	−9.65685	−0.574040	−0.287020	−	0.957925i	$-0.592665\pi$
$283$	−9.65685	−0.574040	−0.287020	+	0.957925i	$0.592665\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	2.00000	0.118056
$288$	0	0
$289$	−1.00000	−0.0588235
$290$	0	0
$291$	0	0
$292$	0	0
$293$	14.0000	0.817889	0.408944	−	0.912559i	$-0.365897\pi$
$293$	14.0000	0.817889	0.408944	+	0.912559i	$0.365897\pi$
$294$	0	0
$295$	11.3137	0.658710
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−4.00000	−0.231326
$300$	0	0
$301$	1.65685	0.0954995
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−16.9706	−0.971732
$306$	0	0
$307$	16.9706	0.968561	0.484281	−	0.874913i	$-0.339081\pi$
$307$	16.9706	0.968561	0.484281	+	0.874913i	$0.339081\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−12.1421	−0.688517	−0.344259	−	0.938875i	$-0.611870\pi$
$311$	−12.1421	−0.688517	−0.344259	+	0.938875i	$0.611870\pi$
$312$	0	0
$313$	−16.6274	−0.939837	−0.469919	−	0.882710i	$-0.655717\pi$
$313$	−16.6274	−0.939837	−0.469919	+	0.882710i	$0.655717\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−0.343146	−0.0192730	−0.00963649	−	0.999954i	$-0.503067\pi$
$317$	−0.343146	−0.0192730	−0.00963649	+	0.999954i	$0.503067\pi$
$318$	0	0
$319$	−15.3137	−0.857403
$320$	0	0
$321$	0	0
$322$	0	0
$323$	4.00000	0.222566
$324$	0	0
$325$	14.4853	0.803499
$326$	0	0
$327$	0	0
$328$	0	0
$329$	10.4853	0.578072
$330$	0	0
$331$	8.68629	0.477442	0.238721	−	0.971088i	$-0.423272\pi$
$331$	8.68629	0.477442	0.238721	+	0.971088i	$0.423272\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	32.0000	1.74835
$336$	0	0
$337$	−26.0000	−1.41631	−0.708155	−	0.706057i	$-0.750472\pi$
$337$	−26.0000	−1.41631	−0.708155	+	0.706057i	$0.750472\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	2.34315	0.126888
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−10.6863	−0.573670	−0.286835	−	0.957980i	$-0.592603\pi$
$347$	−10.6863	−0.573670	−0.286835	+	0.957980i	$0.592603\pi$
$348$	0	0
$349$	−32.6274	−1.74651	−0.873253	−	0.487267i	$-0.837994\pi$
$349$	−32.6274	−1.74651	−0.873253	+	0.487267i	$0.837994\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	28.9706	1.54195	0.770974	−	0.636867i	$-0.219770\pi$
$353$	28.9706	1.54195	0.770974	+	0.636867i	$0.219770\pi$
$354$	0	0
$355$	38.6274	2.05013
$356$	0	0
$357$	0	0
$358$	0	0
$359$	11.1716	0.589613	0.294807	−	0.955557i	$-0.404745\pi$
$359$	11.1716	0.589613	0.294807	+	0.955557i	$0.404745\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	0	0
$364$	0	0
$365$	28.2843	1.48047
$366$	0	0
$367$	−28.2843	−1.47643	−0.738213	−	0.674567i	$-0.764330\pi$
$367$	−28.2843	−1.47643	−0.738213	+	0.674567i	$0.764330\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	6.00000	0.311504
$372$	0	0
$373$	7.17157	0.371330	0.185665	−	0.982613i	$-0.440556\pi$
$373$	7.17157	0.371330	0.185665	+	0.982613i	$0.440556\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−36.9706	−1.90408
$378$	0	0
$379$	23.3137	1.19754	0.598772	−	0.800919i	$-0.295655\pi$
$379$	23.3137	1.19754	0.598772	+	0.800919i	$0.295655\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	6.34315	0.324120	0.162060	−	0.986781i	$-0.448186\pi$
$383$	6.34315	0.324120	0.162060	+	0.986781i	$0.448186\pi$
$384$	0	0
$385$	5.65685	0.288300
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−11.7990	−0.598233	−0.299116	−	0.954217i	$-0.596692\pi$
$389$	−11.7990	−0.598233	−0.299116	+	0.954217i	$0.596692\pi$
$390$	0	0
$391$	−3.31371	−0.167581
$392$	0	0
$393$	0	0
$394$	0	0
$395$	41.9411	2.11029
$396$	0	0
$397$	17.3137	0.868950	0.434475	−	0.900684i	$-0.356934\pi$
$397$	17.3137	0.868950	0.434475	+	0.900684i	$0.356934\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	25.3137	1.26411	0.632053	−	0.774925i	$-0.282212\pi$
$401$	25.3137	1.26411	0.632053	+	0.774925i	$0.282212\pi$
$402$	0	0
$403$	5.65685	0.281788
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−17.6569	−0.875218
$408$	0	0
$409$	18.9706	0.938034	0.469017	−	0.883189i	$-0.344608\pi$
$409$	18.9706	0.938034	0.469017	+	0.883189i	$0.344608\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	4.00000	0.196827
$414$	0	0
$415$	−44.2843	−2.17383
$416$	0	0
$417$	0	0
$418$	0	0
$419$	29.3137	1.43207	0.716034	−	0.698065i	$-0.245955\pi$
$419$	29.3137	1.43207	0.716034	+	0.698065i	$0.245955\pi$
$420$	0	0
$421$	−40.1421	−1.95641	−0.978204	−	0.207646i	$-0.933420\pi$
$421$	−40.1421	−1.95641	−0.978204	+	0.207646i	$0.933420\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	12.0000	0.582086
$426$	0	0
$427$	−6.00000	−0.290360
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−10.3431	−0.498212	−0.249106	−	0.968476i	$-0.580137\pi$
$431$	−10.3431	−0.498212	−0.249106	+	0.968476i	$0.580137\pi$
$432$	0	0
$433$	38.9706	1.87281	0.936403	−	0.350927i	$-0.114133\pi$
$433$	38.9706	1.87281	0.936403	+	0.350927i	$0.114133\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−0.828427	−0.0396290
$438$	0	0
$439$	13.1716	0.628645	0.314322	−	0.949316i	$-0.398223\pi$
$439$	13.1716	0.628645	0.314322	+	0.949316i	$0.398223\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−12.6274	−0.599947	−0.299973	−	0.953948i	$-0.596978\pi$
$443$	−12.6274	−0.599947	−0.299973	+	0.953948i	$0.596978\pi$
$444$	0	0
$445$	−32.9706	−1.56295
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−3.65685	−0.172578	−0.0862888	−	0.996270i	$-0.527501\pi$
$449$	−3.65685	−0.172578	−0.0862888	+	0.996270i	$0.527501\pi$
$450$	0	0
$451$	4.00000	0.188353
$452$	0	0
$453$	0	0
$454$	0	0
$455$	13.6569	0.640243
$456$	0	0
$457$	−25.3137	−1.18413	−0.592063	−	0.805892i	$-0.701686\pi$
$457$	−25.3137	−1.18413	−0.592063	+	0.805892i	$0.701686\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−13.4558	−0.626701	−0.313351	−	0.949638i	$-0.601451\pi$
$461$	−13.4558	−0.626701	−0.313351	+	0.949638i	$0.601451\pi$
$462$	0	0
$463$	2.34315	0.108895	0.0544476	−	0.998517i	$-0.482660\pi$
$463$	2.34315	0.108895	0.0544476	+	0.998517i	$0.482660\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−15.6569	−0.724513	−0.362256	−	0.932078i	$-0.617994\pi$
$467$	−15.6569	−0.724513	−0.362256	+	0.932078i	$0.617994\pi$
$468$	0	0
$469$	11.3137	0.522419
$470$	0	0
$471$	0	0
$472$	0	0
$473$	3.31371	0.152364
$474$	0	0
$475$	3.00000	0.137649
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−5.79899	−0.264963	−0.132481	−	0.991186i	$-0.542294\pi$
$479$	−5.79899	−0.264963	−0.132481	+	0.991186i	$0.542294\pi$
$480$	0	0
$481$	−42.6274	−1.94364
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−32.9706	−1.49712
$486$	0	0
$487$	26.8284	1.21571	0.607856	−	0.794047i	$-0.292030\pi$
$487$	26.8284	1.21571	0.607856	+	0.794047i	$0.292030\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	7.65685	0.345549	0.172774	−	0.984961i	$-0.444727\pi$
$491$	7.65685	0.345549	0.172774	+	0.984961i	$0.444727\pi$
$492$	0	0
$493$	−30.6274	−1.37939
$494$	0	0
$495$	0	0
$496$	0	0
$497$	13.6569	0.612594
$498$	0	0
$499$	1.65685	0.0741710	0.0370855	−	0.999312i	$-0.488193\pi$
$499$	1.65685	0.0741710	0.0370855	+	0.999312i	$0.488193\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	14.4853	0.645867	0.322933	−	0.946422i	$-0.395331\pi$
$503$	14.4853	0.645867	0.322933	+	0.946422i	$0.395331\pi$
$504$	0	0
$505$	−35.3137	−1.57144
$506$	0	0
$507$	0	0
$508$	0	0
$509$	19.6569	0.871275	0.435637	−	0.900122i	$-0.356523\pi$
$509$	19.6569	0.871275	0.435637	+	0.900122i	$0.356523\pi$
$510$	0	0
$511$	10.0000	0.442374
$512$	0	0
$513$	0	0
$514$	0	0
$515$	8.00000	0.352522
$516$	0	0
$517$	20.9706	0.922284
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−24.3431	−1.06649	−0.533246	−	0.845960i	$-0.679028\pi$
$521$	−24.3431	−1.06649	−0.533246	+	0.845960i	$0.679028\pi$
$522$	0	0
$523$	−9.65685	−0.422265	−0.211132	−	0.977457i	$-0.567715\pi$
$523$	−9.65685	−0.422265	−0.211132	+	0.977457i	$0.567715\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	4.68629	0.204138
$528$	0	0
$529$	−22.3137	−0.970161
$530$	0	0
$531$	0	0
$532$	0	0
$533$	9.65685	0.418285
$534$	0	0
$535$	−6.62742	−0.286528
$536$	0	0
$537$	0	0
$538$	0	0
$539$	2.00000	0.0861461
$540$	0	0
$541$	−23.6569	−1.01709	−0.508544	−	0.861036i	$-0.669816\pi$
$541$	−23.6569	−1.01709	−0.508544	+	0.861036i	$0.669816\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	34.3431	1.47110
$546$	0	0
$547$	−18.6274	−0.796451	−0.398225	−	0.917288i	$-0.630374\pi$
$547$	−18.6274	−0.796451	−0.398225	+	0.917288i	$0.630374\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−7.65685	−0.326193
$552$	0	0
$553$	14.8284	0.630569
$554$	0	0
$555$	0	0
$556$	0	0
$557$	39.7990	1.68634	0.843169	−	0.537649i	$-0.180687\pi$
$557$	39.7990	1.68634	0.843169	+	0.537649i	$0.180687\pi$
$558$	0	0
$559$	8.00000	0.338364
$560$	0	0
$561$	0	0
$562$	0	0
$563$	43.3137	1.82546	0.912728	−	0.408569i	$-0.133972\pi$
$563$	43.3137	1.82546	0.912728	+	0.408569i	$0.133972\pi$
$564$	0	0
$565$	−5.65685	−0.237986
$566$	0	0
$567$	0	0
$568$	0	0
$569$	31.6569	1.32712	0.663562	−	0.748121i	$-0.269044\pi$
$569$	31.6569	1.32712	0.663562	+	0.748121i	$0.269044\pi$
$570$	0	0
$571$	−44.9706	−1.88196	−0.940980	−	0.338463i	$-0.890093\pi$
$571$	−44.9706	−1.88196	−0.940980	+	0.338463i	$0.890093\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−2.48528	−0.103643
$576$	0	0
$577$	−32.6274	−1.35830	−0.679149	−	0.734001i	$-0.737651\pi$
$577$	−32.6274	−1.35830	−0.679149	+	0.734001i	$0.737651\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−15.6569	−0.649556
$582$	0	0
$583$	12.0000	0.496989
$584$	0	0
$585$	0	0
$586$	0	0
$587$	44.6274	1.84197	0.920985	−	0.389597i	$-0.127386\pi$
$587$	44.6274	1.84197	0.920985	+	0.389597i	$0.127386\pi$
$588$	0	0
$589$	1.17157	0.0482738
$590$	0	0
$591$	0	0
$592$	0	0
$593$	12.2843	0.504455	0.252227	−	0.967668i	$-0.418837\pi$
$593$	12.2843	0.504455	0.252227	+	0.967668i	$0.418837\pi$
$594$	0	0
$595$	11.3137	0.463817
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−37.9411	−1.55023	−0.775116	−	0.631819i	$-0.782309\pi$
$599$	−37.9411	−1.55023	−0.775116	+	0.631819i	$0.782309\pi$
$600$	0	0
$601$	12.6274	0.515083	0.257542	−	0.966267i	$-0.417088\pi$
$601$	12.6274	0.515083	0.257542	+	0.966267i	$0.417088\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−19.7990	−0.804943
$606$	0	0
$607$	42.1421	1.71050	0.855248	−	0.518219i	$-0.173405\pi$
$607$	42.1421	1.71050	0.855248	+	0.518219i	$0.173405\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	50.6274	2.04817
$612$	0	0
$613$	16.6274	0.671575	0.335788	−	0.941938i	$-0.390998\pi$
$613$	16.6274	0.671575	0.335788	+	0.941938i	$0.390998\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−12.0000	−0.483102	−0.241551	−	0.970388i	$-0.577656\pi$
$617$	−12.0000	−0.483102	−0.241551	+	0.970388i	$0.577656\pi$
$618$	0	0
$619$	−29.9411	−1.20344	−0.601718	−	0.798709i	$-0.705517\pi$
$619$	−29.9411	−1.20344	−0.601718	+	0.798709i	$0.705517\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−11.6569	−0.467022
$624$	0	0
$625$	−31.0000	−1.24000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−35.3137	−1.40805
$630$	0	0
$631$	31.5980	1.25790	0.628948	−	0.777447i	$-0.283486\pi$
$631$	31.5980	1.25790	0.628948	+	0.777447i	$0.283486\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	24.0000	0.952411
$636$	0	0
$637$	4.82843	0.191309
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−15.6569	−0.618409	−0.309204	−	0.950996i	$-0.600063\pi$
$641$	−15.6569	−0.618409	−0.309204	+	0.950996i	$0.600063\pi$
$642$	0	0
$643$	32.2843	1.27317	0.636584	−	0.771208i	$-0.280347\pi$
$643$	32.2843	1.27317	0.636584	+	0.771208i	$0.280347\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−17.7990	−0.699750	−0.349875	−	0.936796i	$-0.613776\pi$
$647$	−17.7990	−0.699750	−0.349875	+	0.936796i	$0.613776\pi$
$648$	0	0
$649$	8.00000	0.314027
$650$	0	0
$651$	0	0
$652$	0	0
$653$	43.1127	1.68713	0.843565	−	0.537027i	$-0.180453\pi$
$653$	43.1127	1.68713	0.843565	+	0.537027i	$0.180453\pi$
$654$	0	0
$655$	44.2843	1.73033
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−40.2843	−1.56925	−0.784626	−	0.619969i	$-0.787145\pi$
$659$	−40.2843	−1.56925	−0.784626	+	0.619969i	$0.787145\pi$
$660$	0	0
$661$	−29.7990	−1.15905	−0.579523	−	0.814956i	$-0.696761\pi$
$661$	−29.7990	−1.15905	−0.579523	+	0.814956i	$0.696761\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	2.82843	0.109682
$666$	0	0
$667$	6.34315	0.245608
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−12.0000	−0.463255
$672$	0	0
$673$	42.9706	1.65639	0.828197	−	0.560437i	$-0.189367\pi$
$673$	42.9706	1.65639	0.828197	+	0.560437i	$0.189367\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	20.3431	0.781851	0.390925	−	0.920422i	$-0.372155\pi$
$677$	20.3431	0.781851	0.390925	+	0.920422i	$0.372155\pi$
$678$	0	0
$679$	−11.6569	−0.447349
$680$	0	0
$681$	0	0
$682$	0	0
$683$	5.65685	0.216454	0.108227	−	0.994126i	$-0.465483\pi$
$683$	5.65685	0.216454	0.108227	+	0.994126i	$0.465483\pi$
$684$	0	0
$685$	−36.6863	−1.40171
$686$	0	0
$687$	0	0
$688$	0	0
$689$	28.9706	1.10369
$690$	0	0
$691$	−7.31371	−0.278227	−0.139113	−	0.990276i	$-0.544425\pi$
$691$	−7.31371	−0.278227	−0.139113	+	0.990276i	$0.544425\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−27.3137	−1.03607
$696$	0	0
$697$	8.00000	0.303022
$698$	0	0
$699$	0	0
$700$	0	0
$701$	5.45584	0.206064	0.103032	−	0.994678i	$-0.467146\pi$
$701$	5.45584	0.206064	0.103032	+	0.994678i	$0.467146\pi$
$702$	0	0
$703$	−8.82843	−0.332970
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−12.4853	−0.469557
$708$	0	0
$709$	22.0000	0.826227	0.413114	−	0.910679i	$-0.364441\pi$
$709$	22.0000	0.826227	0.413114	+	0.910679i	$0.364441\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−0.970563	−0.0363479
$714$	0	0
$715$	27.3137	1.02147
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−4.14214	−0.154476	−0.0772378	−	0.997013i	$-0.524610\pi$
$719$	−4.14214	−0.154476	−0.0772378	+	0.997013i	$0.524610\pi$
$720$	0	0
$721$	2.82843	0.105336
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−22.9706	−0.853105
$726$	0	0
$727$	16.0000	0.593407	0.296704	−	0.954970i	$-0.404113\pi$
$727$	16.0000	0.593407	0.296704	+	0.954970i	$0.404113\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	6.62742	0.245124
$732$	0	0
$733$	−32.6274	−1.20512	−0.602561	−	0.798073i	$-0.705853\pi$
$733$	−32.6274	−1.20512	−0.602561	+	0.798073i	$0.705853\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	22.6274	0.833492
$738$	0	0
$739$	26.6274	0.979505	0.489753	−	0.871861i	$-0.337087\pi$
$739$	26.6274	0.979505	0.489753	+	0.871861i	$0.337087\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−28.2843	−1.03765	−0.518825	−	0.854881i	$-0.673630\pi$
$743$	−28.2843	−1.03765	−0.518825	+	0.854881i	$0.673630\pi$
$744$	0	0
$745$	3.31371	0.121405
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−2.34315	−0.0856167
$750$	0	0
$751$	34.1421	1.24586	0.622932	−	0.782276i	$-0.285941\pi$
$751$	34.1421	1.24586	0.622932	+	0.782276i	$0.285941\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−35.3137	−1.28520
$756$	0	0
$757$	32.6274	1.18586	0.592932	−	0.805253i	$-0.297970\pi$
$757$	32.6274	1.18586	0.592932	+	0.805253i	$0.297970\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	7.31371	0.265122	0.132561	−	0.991175i	$-0.457680\pi$
$761$	7.31371	0.265122	0.132561	+	0.991175i	$0.457680\pi$
$762$	0	0
$763$	12.1421	0.439575
$764$	0	0
$765$	0	0
$766$	0	0
$767$	19.3137	0.697378
$768$	0	0
$769$	−24.6274	−0.888087	−0.444044	−	0.896005i	$-0.646457\pi$
$769$	−24.6274	−0.888087	−0.444044	+	0.896005i	$0.646457\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−30.2843	−1.08925	−0.544625	−	0.838680i	$-0.683328\pi$
$773$	−30.2843	−1.08925	−0.544625	+	0.838680i	$0.683328\pi$
$774$	0	0
$775$	3.51472	0.126252
$776$	0	0
$777$	0	0
$778$	0	0
$779$	2.00000	0.0716574
$780$	0	0
$781$	27.3137	0.977361
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−42.3431	−1.51129
$786$	0	0
$787$	−49.2548	−1.75575	−0.877873	−	0.478894i	$-0.841038\pi$
$787$	−49.2548	−1.75575	−0.877873	+	0.478894i	$0.841038\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−2.00000	−0.0711118
$792$	0	0
$793$	−28.9706	−1.02877
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−15.6569	−0.554594	−0.277297	−	0.960784i	$-0.589439\pi$
$797$	−15.6569	−0.554594	−0.277297	+	0.960784i	$0.589439\pi$
$798$	0	0
$799$	41.9411	1.48377
$800$	0	0
$801$	0	0
$802$	0	0
$803$	20.0000	0.705785
$804$	0	0
$805$	−2.34315	−0.0825850
$806$	0	0
$807$	0	0
$808$	0	0
$809$	37.6569	1.32394	0.661972	−	0.749528i	$-0.269720\pi$
$809$	37.6569	1.32394	0.661972	+	0.749528i	$0.269720\pi$
$810$	0	0
$811$	32.9706	1.15775	0.578877	−	0.815415i	$-0.303491\pi$
$811$	32.9706	1.15775	0.578877	+	0.815415i	$0.303491\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−65.9411	−2.30982
$816$	0	0
$817$	1.65685	0.0579660
$818$	0	0
$819$	0	0
$820$	0	0
$821$	27.5147	0.960270	0.480135	−	0.877195i	$-0.340588\pi$
$821$	27.5147	0.960270	0.480135	+	0.877195i	$0.340588\pi$
$822$	0	0
$823$	−32.9706	−1.14928	−0.574641	−	0.818406i	$-0.694858\pi$
$823$	−32.9706	−1.14928	−0.574641	+	0.818406i	$0.694858\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−21.6569	−0.753083	−0.376541	−	0.926400i	$-0.622887\pi$
$827$	−21.6569	−0.753083	−0.376541	+	0.926400i	$0.622887\pi$
$828$	0	0
$829$	−29.5147	−1.02509	−0.512544	−	0.858661i	$-0.671297\pi$
$829$	−29.5147	−1.02509	−0.512544	+	0.858661i	$0.671297\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	4.00000	0.138592
$834$	0	0
$835$	−32.0000	−1.10741
$836$	0	0
$837$	0	0
$838$	0	0
$839$	40.2843	1.39077	0.695384	−	0.718639i	$-0.255234\pi$
$839$	40.2843	1.39077	0.695384	+	0.718639i	$0.255234\pi$
$840$	0	0
$841$	29.6274	1.02164
$842$	0	0
$843$	0	0
$844$	0	0
$845$	29.1716	1.00353
$846$	0	0
$847$	−7.00000	−0.240523
$848$	0	0
$849$	0	0
$850$	0	0
$851$	7.31371	0.250711
$852$	0	0
$853$	−46.2843	−1.58474	−0.792372	−	0.610039i	$-0.791154\pi$
$853$	−46.2843	−1.58474	−0.792372	+	0.610039i	$0.791154\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−13.0294	−0.445077	−0.222539	−	0.974924i	$-0.571434\pi$
$857$	−13.0294	−0.445077	−0.222539	+	0.974924i	$0.571434\pi$
$858$	0	0
$859$	−42.6274	−1.45443	−0.727214	−	0.686410i	$-0.759185\pi$
$859$	−42.6274	−1.45443	−0.727214	+	0.686410i	$0.759185\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−36.2843	−1.23513	−0.617565	−	0.786519i	$-0.711881\pi$
$863$	−36.2843	−1.23513	−0.617565	+	0.786519i	$0.711881\pi$
$864$	0	0
$865$	39.5980	1.34637
$866$	0	0
$867$	0	0
$868$	0	0
$869$	29.6569	1.00604
$870$	0	0
$871$	54.6274	1.85098
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−5.65685	−0.191237
$876$	0	0
$877$	−13.5147	−0.456360	−0.228180	−	0.973619i	$-0.573277\pi$
$877$	−13.5147	−0.456360	−0.228180	+	0.973619i	$0.573277\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	12.9706	0.436989	0.218495	−	0.975838i	$-0.429885\pi$
$881$	12.9706	0.436989	0.218495	+	0.975838i	$0.429885\pi$
$882$	0	0
$883$	44.9706	1.51338	0.756690	−	0.653774i	$-0.226815\pi$
$883$	44.9706	1.51338	0.756690	+	0.653774i	$0.226815\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−6.62742	−0.222527	−0.111263	−	0.993791i	$-0.535490\pi$
$887$	−6.62742	−0.222527	−0.111263	+	0.993791i	$0.535490\pi$
$888$	0	0
$889$	8.48528	0.284587
$890$	0	0
$891$	0	0
$892$	0	0
$893$	10.4853	0.350877
$894$	0	0
$895$	29.2548	0.977881
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−8.97056	−0.299185
$900$	0	0
$901$	24.0000	0.799556
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−47.5980	−1.58221
$906$	0	0
$907$	−46.6274	−1.54824	−0.774119	−	0.633040i	$-0.781807\pi$
$907$	−46.6274	−1.54824	−0.774119	+	0.633040i	$0.781807\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	31.3137	1.03747	0.518735	−	0.854935i	$-0.326403\pi$
$911$	31.3137	1.03747	0.518735	+	0.854935i	$0.326403\pi$
$912$	0	0
$913$	−31.3137	−1.03633
$914$	0	0
$915$	0	0
$916$	0	0
$917$	15.6569	0.517035
$918$	0	0
$919$	34.3431	1.13288	0.566438	−	0.824104i	$-0.308321\pi$
$919$	34.3431	1.13288	0.566438	+	0.824104i	$0.308321\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	65.9411	2.17048
$924$	0	0
$925$	−26.4853	−0.870831
$926$	0	0
$927$	0	0
$928$	0	0
$929$	11.3137	0.371191	0.185595	−	0.982626i	$-0.440579\pi$
$929$	11.3137	0.371191	0.185595	+	0.982626i	$0.440579\pi$
$930$	0	0
$931$	1.00000	0.0327737
$932$	0	0
$933$	0	0
$934$	0	0
$935$	22.6274	0.739996
$936$	0	0
$937$	−17.3137	−0.565614	−0.282807	−	0.959177i	$-0.591266\pi$
$937$	−17.3137	−0.565614	−0.282807	+	0.959177i	$0.591266\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	53.5980	1.74724	0.873622	−	0.486605i	$-0.161765\pi$
$941$	53.5980	1.74724	0.873622	+	0.486605i	$0.161765\pi$
$942$	0	0
$943$	−1.65685	−0.0539546
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−30.9706	−1.00641	−0.503204	−	0.864168i	$-0.667846\pi$
$947$	−30.9706	−1.00641	−0.503204	+	0.864168i	$0.667846\pi$
$948$	0	0
$949$	48.2843	1.56737
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−36.6274	−1.18648	−0.593239	−	0.805026i	$-0.702151\pi$
$953$	−36.6274	−1.18648	−0.593239	+	0.805026i	$0.702151\pi$
$954$	0	0
$955$	31.5980	1.02249
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−12.9706	−0.418841
$960$	0	0
$961$	−29.6274	−0.955723
$962$	0	0
$963$	0	0
$964$	0	0
$965$	53.6569	1.72728
$966$	0	0
$967$	8.00000	0.257263	0.128631	−	0.991692i	$-0.458942\pi$
$967$	8.00000	0.257263	0.128631	+	0.991692i	$0.458942\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−45.2548	−1.45230	−0.726148	−	0.687538i	$-0.758691\pi$
$971$	−45.2548	−1.45230	−0.726148	+	0.687538i	$0.758691\pi$
$972$	0	0
$973$	−9.65685	−0.309585
$974$	0	0
$975$	0	0
$976$	0	0
$977$	38.0000	1.21573	0.607864	−	0.794041i	$-0.292027\pi$
$977$	38.0000	1.21573	0.607864	+	0.794041i	$0.292027\pi$
$978$	0	0
$979$	−23.3137	−0.745109
$980$	0	0
$981$	0	0
$982$	0	0
$983$	3.31371	0.105691	0.0528454	−	0.998603i	$-0.483171\pi$
$983$	3.31371	0.105691	0.0528454	+	0.998603i	$0.483171\pi$
$984$	0	0
$985$	69.2548	2.20664
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−1.37258	−0.0436456
$990$	0	0
$991$	35.7990	1.13719	0.568596	−	0.822617i	$-0.307487\pi$
$991$	35.7990	1.13719	0.568596	+	0.822617i	$0.307487\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−16.0000	−0.507234
$996$	0	0
$997$	−11.6569	−0.369176	−0.184588	−	0.982816i	$-0.559095\pi$
$997$	−11.6569	−0.369176	−0.184588	+	0.982816i	$0.559095\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9576.2.a.bo.1.2	yes	2
3.2	odd	2		9576.2.a.bl.1.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
9576.2.a.bl.1.1	✓	2	3.2	odd	2
9576.2.a.bo.1.2	yes	2	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 9576 = 2^{3} \cdot 3^{2} \cdot 7 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9576.a (trivial)

\(f(q)\)	\(=\)	\(q+2.82843 q^{5} +1.00000 q^{7} +O(q^{10})\)	\(q+2.82843 q^{5} +1.00000 q^{7} +2.00000 q^{11} +4.82843 q^{13} +4.00000 q^{17} +1.00000 q^{19} -0.828427 q^{23} +3.00000 q^{25} -7.65685 q^{29} +1.17157 q^{31} +2.82843 q^{35} -8.82843 q^{37} +2.00000 q^{41} +1.65685 q^{43} +10.4853 q^{47} +1.00000 q^{49} +6.00000 q^{53} +5.65685 q^{55} +4.00000 q^{59} -6.00000 q^{61} +13.6569 q^{65} +11.3137 q^{67} +13.6569 q^{71} +10.0000 q^{73} +2.00000 q^{77} +14.8284 q^{79} -15.6569 q^{83} +11.3137 q^{85} -11.6569 q^{89} +4.82843 q^{91} +2.82843 q^{95} -11.6569 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{7}+O(q^{10}) \) 2 * q + 2 * q^7	\( 2 q + 2 q^{7} + 4 q^{11} + 4 q^{13} + 8 q^{17} + 2 q^{19} + 4 q^{23} + 6 q^{25} - 4 q^{29} + 8 q^{31} - 12 q^{37} + 4 q^{41} - 8 q^{43} + 4 q^{47} + 2 q^{49} + 12 q^{53} + 8 q^{59} - 12 q^{61} + 16 q^{65} + 16 q^{71} + 20 q^{73} + 4 q^{77} + 24 q^{79} - 20 q^{83} - 12 q^{89} + 4 q^{91} - 12 q^{97}+O(q^{100}) \) 2 * q + 2 * q^7 + 4 * q^11 + 4 * q^13 + 8 * q^17 + 2 * q^19 + 4 * q^23 + 6 * q^25 - 4 * q^29 + 8 * q^31 - 12 * q^37 + 4 * q^41 - 8 * q^43 + 4 * q^47 + 2 * q^49 + 12 * q^53 + 8 * q^59 - 12 * q^61 + 16 * q^65 + 16 * q^71 + 20 * q^73 + 4 * q^77 + 24 * q^79 - 20 * q^83 - 12 * q^89 + 4 * q^91 - 12 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more