LMFDB - Embedded newform 9576.2.a.bo.1.1

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			1.1
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)

$f(q)$	$=$	$q-2.82843 q^{5} +1.00000 q^{7} +O(q^{10})$	$q-2.82843 q^{5} +1.00000 q^{7} +2.00000 q^{11} -0.828427 q^{13} +4.00000 q^{17} +1.00000 q^{19} +4.82843 q^{23} +3.00000 q^{25} +3.65685 q^{29} +6.82843 q^{31} -2.82843 q^{35} -3.17157 q^{37} +2.00000 q^{41} -9.65685 q^{43} -6.48528 q^{47} +1.00000 q^{49} +6.00000 q^{53} -5.65685 q^{55} +4.00000 q^{59} -6.00000 q^{61} +2.34315 q^{65} -11.3137 q^{67} +2.34315 q^{71} +10.0000 q^{73} +2.00000 q^{77} +9.17157 q^{79} -4.34315 q^{83} -11.3137 q^{85} -0.343146 q^{89} -0.828427 q^{91} -2.82843 q^{95} -0.343146 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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	$-0.717953\pi$
$5$	−2.82843	−1.26491	−0.632456	+	0.774597i	$0.717953\pi$
$6$	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
$13$	−0.828427	−0.229764	−0.114882	−	0.993379i	$-0.536649\pi$
$13$	−0.828427	−0.229764	−0.114882	+	0.993379i	$0.536649\pi$
$14$	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
$23$	4.82843	1.00680	0.503398	−	0.864054i	$-0.332083\pi$
$23$	4.82843	1.00680	0.503398	+	0.864054i	$0.332083\pi$
$24$	0	0
$25$	3.00000	0.600000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	3.65685	0.679061	0.339530	−	0.940595i	$-0.389732\pi$
$29$	3.65685	0.679061	0.339530	+	0.940595i	$0.389732\pi$
$30$	0	0
$31$	6.82843	1.22642	0.613211	−	0.789919i	$-0.289878\pi$
$31$	6.82843	1.22642	0.613211	+	0.789919i	$0.289878\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−2.82843	−0.478091
$36$	0	0
$37$	−3.17157	−0.521403	−0.260702	−	0.965419i	$-0.583954\pi$
$37$	−3.17157	−0.521403	−0.260702	+	0.965419i	$0.583954\pi$
$38$	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
$43$	−9.65685	−1.47266	−0.736328	−	0.676625i	$-0.763442\pi$
$43$	−9.65685	−1.47266	−0.736328	+	0.676625i	$0.763442\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−6.48528	−0.945976	−0.472988	−	0.881069i	$-0.656825\pi$
$47$	−6.48528	−0.945976	−0.472988	+	0.881069i	$0.656825\pi$
$48$	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
$65$	2.34315	0.290631
$66$	0	0
$67$	−11.3137	−1.38219	−0.691095	−	0.722764i	$-0.742871\pi$
$67$	−11.3137	−1.38219	−0.691095	+	0.722764i	$0.742871\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	2.34315	0.278080	0.139040	−	0.990287i	$-0.455598\pi$
$71$	2.34315	0.278080	0.139040	+	0.990287i	$0.455598\pi$
$72$	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
$79$	9.17157	1.03188	0.515941	−	0.856624i	$-0.327442\pi$
$79$	9.17157	1.03188	0.515941	+	0.856624i	$0.327442\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−4.34315	−0.476722	−0.238361	−	0.971177i	$-0.576610\pi$
$83$	−4.34315	−0.476722	−0.238361	+	0.971177i	$0.576610\pi$
$84$	0	0
$85$	−11.3137	−1.22714
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−0.343146	−0.0363734	−0.0181867	−	0.999835i	$-0.505789\pi$
$89$	−0.343146	−0.0363734	−0.0181867	+	0.999835i	$0.505789\pi$
$90$	0	0
$91$	−0.828427	−0.0868428
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−2.82843	−0.290191
$96$	0	0
$97$	−0.343146	−0.0348412	−0.0174206	−	0.999848i	$-0.505545\pi$
$97$	−0.343146	−0.0348412	−0.0174206	+	0.999848i	$0.505545\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	4.48528	0.446302	0.223151	−	0.974784i	$-0.428366\pi$
$101$	4.48528	0.446302	0.223151	+	0.974784i	$0.428366\pi$
$102$	0	0
$103$	−2.82843	−0.278693	−0.139347	−	0.990244i	$-0.544500\pi$
$103$	−2.82843	−0.278693	−0.139347	+	0.990244i	$0.544500\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−13.6569	−1.32026	−0.660129	−	0.751152i	$-0.729498\pi$
$107$	−13.6569	−1.32026	−0.660129	+	0.751152i	$0.729498\pi$
$108$	0	0
$109$	−16.1421	−1.54614	−0.773068	−	0.634323i	$-0.781279\pi$
$109$	−16.1421	−1.54614	−0.773068	+	0.634323i	$0.781279\pi$
$110$	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
$115$	−13.6569	−1.27351
$116$	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	$-0.622863\pi$
$127$	−8.48528	−0.752947	−0.376473	+	0.926427i	$0.622863\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	4.34315	0.379462	0.189731	−	0.981836i	$-0.439238\pi$
$131$	4.34315	0.379462	0.189731	+	0.981836i	$0.439238\pi$
$132$	0	0
$133$	1.00000	0.0867110
$134$	0	0
$135$	0	0
$136$	0	0
$137$	20.9706	1.79164	0.895818	−	0.444421i	$-0.146591\pi$
$137$	20.9706	1.79164	0.895818	+	0.444421i	$0.146591\pi$
$138$	0	0
$139$	1.65685	0.140533	0.0702663	−	0.997528i	$-0.477615\pi$
$139$	1.65685	0.140533	0.0702663	+	0.997528i	$0.477615\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−1.65685	−0.138553
$144$	0	0
$145$	−10.3431	−0.858952
$146$	0	0
$147$	0	0
$148$	0	0
$149$	6.82843	0.559407	0.279703	−	0.960086i	$-0.409764\pi$
$149$	6.82843	0.559407	0.279703	+	0.960086i	$0.409764\pi$
$150$	0	0
$151$	4.48528	0.365007	0.182504	−	0.983205i	$-0.441580\pi$
$151$	4.48528	0.365007	0.182504	+	0.983205i	$0.441580\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−19.3137	−1.55131
$156$	0	0
$157$	18.9706	1.51402	0.757008	−	0.653406i	$-0.226660\pi$
$157$	18.9706	1.51402	0.757008	+	0.653406i	$0.226660\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	4.82843	0.380533
$162$	0	0
$163$	−0.686292	−0.0537545	−0.0268772	−	0.999639i	$-0.508556\pi$
$163$	−0.686292	−0.0537545	−0.0268772	+	0.999639i	$0.508556\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	11.3137	0.875481	0.437741	−	0.899101i	$-0.355779\pi$
$167$	11.3137	0.875481	0.437741	+	0.899101i	$0.355779\pi$
$168$	0	0
$169$	−12.3137	−0.947208
$170$	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
$179$	21.6569	1.61871	0.809355	−	0.587320i	$-0.199817\pi$
$179$	21.6569	1.61871	0.809355	+	0.587320i	$0.199817\pi$
$180$	0	0
$181$	−11.1716	−0.830376	−0.415188	−	0.909736i	$-0.636284\pi$
$181$	−11.1716	−0.830376	−0.415188	+	0.909736i	$0.636284\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	8.97056	0.659529
$186$	0	0
$187$	8.00000	0.585018
$188$	0	0
$189$	0	0
$190$	0	0
$191$	16.8284	1.21766	0.608831	−	0.793300i	$-0.291639\pi$
$191$	16.8284	1.21766	0.608831	+	0.793300i	$0.291639\pi$
$192$	0	0
$193$	−14.9706	−1.07760	−0.538802	−	0.842432i	$-0.681123\pi$
$193$	−14.9706	−1.07760	−0.538802	+	0.842432i	$0.681123\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	7.51472	0.535402	0.267701	−	0.963502i	$-0.413736\pi$
$197$	7.51472	0.535402	0.267701	+	0.963502i	$0.413736\pi$
$198$	0	0
$199$	5.65685	0.401004	0.200502	−	0.979693i	$-0.435743\pi$
$199$	5.65685	0.401004	0.200502	+	0.979693i	$0.435743\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	3.65685	0.256661
$204$	0	0
$205$	−5.65685	−0.395092
$206$	0	0
$207$	0	0
$208$	0	0
$209$	2.00000	0.138343
$210$	0	0
$211$	24.9706	1.71904	0.859522	−	0.511098i	$-0.170761\pi$
$211$	24.9706	1.71904	0.859522	+	0.511098i	$0.170761\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	27.3137	1.86278
$216$	0	0
$217$	6.82843	0.463544
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−3.31371	−0.222904
$222$	0	0
$223$	10.8284	0.725125	0.362563	−	0.931959i	$-0.381902\pi$
$223$	10.8284	0.725125	0.362563	+	0.931959i	$0.381902\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−11.3137	−0.750917	−0.375459	−	0.926839i	$-0.622515\pi$
$227$	−11.3137	−0.750917	−0.375459	+	0.926839i	$0.622515\pi$
$228$	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	$-0.620845\pi$
$233$	−11.3137	−0.741186	−0.370593	+	0.928795i	$0.620845\pi$
$234$	0	0
$235$	18.3431	1.19657
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−1.51472	−0.0979790	−0.0489895	−	0.998799i	$-0.515600\pi$
$239$	−1.51472	−0.0979790	−0.0489895	+	0.998799i	$0.515600\pi$
$240$	0	0
$241$	26.9706	1.73733	0.868663	−	0.495403i	$-0.164980\pi$
$241$	26.9706	1.73733	0.868663	+	0.495403i	$0.164980\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−2.82843	−0.180702
$246$	0	0
$247$	−0.828427	−0.0527116
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−10.9706	−0.692456	−0.346228	−	0.938150i	$-0.612538\pi$
$251$	−10.9706	−0.692456	−0.346228	+	0.938150i	$0.612538\pi$
$252$	0	0
$253$	9.65685	0.607121
$254$	0	0
$255$	0	0
$256$	0	0
$257$	0.343146	0.0214048	0.0107024	−	0.999943i	$-0.496593\pi$
$257$	0.343146	0.0214048	0.0107024	+	0.999943i	$0.496593\pi$
$258$	0	0
$259$	−3.17157	−0.197072
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−32.1421	−1.98197	−0.990984	−	0.133977i	$-0.957225\pi$
$263$	−32.1421	−1.98197	−0.990984	+	0.133977i	$0.957225\pi$
$264$	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
$271$	−10.3431	−0.628301	−0.314151	−	0.949373i	$-0.601720\pi$
$271$	−10.3431	−0.628301	−0.314151	+	0.949373i	$0.601720\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	6.00000	0.361814
$276$	0	0
$277$	7.65685	0.460056	0.230028	−	0.973184i	$-0.426118\pi$
$277$	7.65685	0.460056	0.230028	+	0.973184i	$0.426118\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	21.3137	1.27147	0.635735	−	0.771908i	$-0.280697\pi$
$281$	21.3137	1.27147	0.635735	+	0.771908i	$0.280697\pi$
$282$	0	0
$283$	1.65685	0.0984898	0.0492449	−	0.998787i	$-0.484319\pi$
$283$	1.65685	0.0984898	0.0492449	+	0.998787i	$0.484319\pi$
$284$	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
$301$	−9.65685	−0.556612
$302$	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	$-0.660919\pi$
$307$	−16.9706	−0.968561	−0.484281	+	0.874913i	$0.660919\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	16.1421	0.915337	0.457668	−	0.889123i	$-0.348685\pi$
$311$	16.1421	0.915337	0.457668	+	0.889123i	$0.348685\pi$
$312$	0	0
$313$	28.6274	1.61812	0.809059	−	0.587728i	$-0.199977\pi$
$313$	28.6274	1.61812	0.809059	+	0.587728i	$0.199977\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−11.6569	−0.654714	−0.327357	−	0.944901i	$-0.606158\pi$
$317$	−11.6569	−0.654714	−0.327357	+	0.944901i	$0.606158\pi$
$318$	0	0
$319$	7.31371	0.409489
$320$	0	0
$321$	0	0
$322$	0	0
$323$	4.00000	0.222566
$324$	0	0
$325$	−2.48528	−0.137859
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−6.48528	−0.357545
$330$	0	0
$331$	31.3137	1.72116	0.860579	−	0.509318i	$-0.170102\pi$
$331$	31.3137	1.72116	0.860579	+	0.509318i	$0.170102\pi$
$332$	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
$341$	13.6569	0.739560
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−33.3137	−1.78837	−0.894187	−	0.447694i	$-0.852245\pi$
$347$	−33.3137	−1.78837	−0.894187	+	0.447694i	$0.852245\pi$
$348$	0	0
$349$	12.6274	0.675930	0.337965	−	0.941159i	$-0.390261\pi$
$349$	12.6274	0.675930	0.337965	+	0.941159i	$0.390261\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−4.97056	−0.264556	−0.132278	−	0.991213i	$-0.542229\pi$
$353$	−4.97056	−0.264556	−0.132278	+	0.991213i	$0.542229\pi$
$354$	0	0
$355$	−6.62742	−0.351747
$356$	0	0
$357$	0	0
$358$	0	0
$359$	16.8284	0.888170	0.444085	−	0.895985i	$-0.353529\pi$
$359$	16.8284	0.888170	0.444085	+	0.895985i	$0.353529\pi$
$360$	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	$-0.235670\pi$
$367$	28.2843	1.47643	0.738213	+	0.674567i	$0.235670\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	6.00000	0.311504
$372$	0	0
$373$	12.8284	0.664231	0.332115	−	0.943239i	$-0.392238\pi$
$373$	12.8284	0.664231	0.332115	+	0.943239i	$0.392238\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−3.02944	−0.156024
$378$	0	0
$379$	0.686292	0.0352524	0.0176262	−	0.999845i	$-0.494389\pi$
$379$	0.686292	0.0352524	0.0176262	+	0.999845i	$0.494389\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	17.6569	0.902223	0.451112	−	0.892468i	$-0.351028\pi$
$383$	17.6569	0.902223	0.451112	+	0.892468i	$0.351028\pi$
$384$	0	0
$385$	−5.65685	−0.288300
$386$	0	0
$387$	0	0
$388$	0	0
$389$	27.7990	1.40946	0.704732	−	0.709473i	$-0.251067\pi$
$389$	27.7990	1.40946	0.704732	+	0.709473i	$0.251067\pi$
$390$	0	0
$391$	19.3137	0.976736
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−25.9411	−1.30524
$396$	0	0
$397$	−5.31371	−0.266687	−0.133344	−	0.991070i	$-0.542571\pi$
$397$	−5.31371	−0.266687	−0.133344	+	0.991070i	$0.542571\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	2.68629	0.134147	0.0670735	−	0.997748i	$-0.478634\pi$
$401$	2.68629	0.134147	0.0670735	+	0.997748i	$0.478634\pi$
$402$	0	0
$403$	−5.65685	−0.281788
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−6.34315	−0.314418
$408$	0	0
$409$	−14.9706	−0.740247	−0.370123	−	0.928983i	$-0.620685\pi$
$409$	−14.9706	−0.740247	−0.370123	+	0.928983i	$0.620685\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	4.00000	0.196827
$414$	0	0
$415$	12.2843	0.603011
$416$	0	0
$417$	0	0
$418$	0	0
$419$	6.68629	0.326647	0.163323	−	0.986573i	$-0.447779\pi$
$419$	6.68629	0.326647	0.163323	+	0.986573i	$0.447779\pi$
$420$	0	0
$421$	−11.8579	−0.577917	−0.288958	−	0.957342i	$-0.593309\pi$
$421$	−11.8579	−0.577917	−0.288958	+	0.957342i	$0.593309\pi$
$422$	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
$431$	−21.6569	−1.04317	−0.521587	−	0.853198i	$-0.674660\pi$
$431$	−21.6569	−1.04317	−0.521587	+	0.853198i	$0.674660\pi$
$432$	0	0
$433$	5.02944	0.241699	0.120850	−	0.992671i	$-0.461438\pi$
$433$	5.02944	0.241699	0.120850	+	0.992671i	$0.461438\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	4.82843	0.230975
$438$	0	0
$439$	18.8284	0.898632	0.449316	−	0.893373i	$-0.351668\pi$
$439$	18.8284	0.898632	0.449316	+	0.893373i	$0.351668\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	32.6274	1.55018	0.775088	−	0.631854i	$-0.217706\pi$
$443$	32.6274	1.55018	0.775088	+	0.631854i	$0.217706\pi$
$444$	0	0
$445$	0.970563	0.0460091
$446$	0	0
$447$	0	0
$448$	0	0
$449$	7.65685	0.361349	0.180675	−	0.983543i	$-0.442172\pi$
$449$	7.65685	0.361349	0.180675	+	0.983543i	$0.442172\pi$
$450$	0	0
$451$	4.00000	0.188353
$452$	0	0
$453$	0	0
$454$	0	0
$455$	2.34315	0.109848
$456$	0	0
$457$	−2.68629	−0.125659	−0.0628297	−	0.998024i	$-0.520012\pi$
$457$	−2.68629	−0.125659	−0.0628297	+	0.998024i	$0.520012\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	37.4558	1.74449	0.872246	−	0.489067i	$-0.162663\pi$
$461$	37.4558	1.74449	0.872246	+	0.489067i	$0.162663\pi$
$462$	0	0
$463$	13.6569	0.634688	0.317344	−	0.948311i	$-0.397209\pi$
$463$	13.6569	0.634688	0.317344	+	0.948311i	$0.397209\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−4.34315	−0.200977	−0.100488	−	0.994938i	$-0.532041\pi$
$467$	−4.34315	−0.200977	−0.100488	+	0.994938i	$0.532041\pi$
$468$	0	0
$469$	−11.3137	−0.522419
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−19.3137	−0.888045
$474$	0	0
$475$	3.00000	0.137649
$476$	0	0
$477$	0	0
$478$	0	0
$479$	33.7990	1.54432	0.772158	−	0.635431i	$-0.219178\pi$
$479$	33.7990	1.54432	0.772158	+	0.635431i	$0.219178\pi$
$480$	0	0
$481$	2.62742	0.119800
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0.970563	0.0440710
$486$	0	0
$487$	21.1716	0.959376	0.479688	−	0.877439i	$-0.340750\pi$
$487$	21.1716	0.959376	0.479688	+	0.877439i	$0.340750\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−3.65685	−0.165032	−0.0825158	−	0.996590i	$-0.526295\pi$
$491$	−3.65685	−0.165032	−0.0825158	+	0.996590i	$0.526295\pi$
$492$	0	0
$493$	14.6274	0.658786
$494$	0	0
$495$	0	0
$496$	0	0
$497$	2.34315	0.105104
$498$	0	0
$499$	−9.65685	−0.432300	−0.216150	−	0.976360i	$-0.569350\pi$
$499$	−9.65685	−0.432300	−0.216150	+	0.976360i	$0.569350\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−2.48528	−0.110813	−0.0554066	−	0.998464i	$-0.517646\pi$
$503$	−2.48528	−0.110813	−0.0554066	+	0.998464i	$0.517646\pi$
$504$	0	0
$505$	−12.6863	−0.564533
$506$	0	0
$507$	0	0
$508$	0	0
$509$	8.34315	0.369803	0.184902	−	0.982757i	$-0.440803\pi$
$509$	8.34315	0.369803	0.184902	+	0.982757i	$0.440803\pi$
$510$	0	0
$511$	10.0000	0.442374
$512$	0	0
$513$	0	0
$514$	0	0
$515$	8.00000	0.352522
$516$	0	0
$517$	−12.9706	−0.570445
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−35.6569	−1.56216	−0.781078	−	0.624434i	$-0.785330\pi$
$521$	−35.6569	−1.56216	−0.781078	+	0.624434i	$0.785330\pi$
$522$	0	0
$523$	1.65685	0.0724492	0.0362246	−	0.999344i	$-0.488467\pi$
$523$	1.65685	0.0724492	0.0362246	+	0.999344i	$0.488467\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	27.3137	1.18980
$528$	0	0
$529$	0.313708	0.0136395
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−1.65685	−0.0717663
$534$	0	0
$535$	38.6274	1.67001
$536$	0	0
$537$	0	0
$538$	0	0
$539$	2.00000	0.0861461
$540$	0	0
$541$	−12.3431	−0.530673	−0.265337	−	0.964156i	$-0.585483\pi$
$541$	−12.3431	−0.530673	−0.265337	+	0.964156i	$0.585483\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	45.6569	1.95572
$546$	0	0
$547$	26.6274	1.13851	0.569253	−	0.822162i	$-0.307232\pi$
$547$	26.6274	1.13851	0.569253	+	0.822162i	$0.307232\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	3.65685	0.155787
$552$	0	0
$553$	9.17157	0.390015
$554$	0	0
$555$	0	0
$556$	0	0
$557$	0.201010	0.00851707	0.00425854	−	0.999991i	$-0.498644\pi$
$557$	0.201010	0.00851707	0.00425854	+	0.999991i	$0.498644\pi$
$558$	0	0
$559$	8.00000	0.338364
$560$	0	0
$561$	0	0
$562$	0	0
$563$	20.6863	0.871823	0.435912	−	0.899989i	$-0.356426\pi$
$563$	20.6863	0.871823	0.435912	+	0.899989i	$0.356426\pi$
$564$	0	0
$565$	5.65685	0.237986
$566$	0	0
$567$	0	0
$568$	0	0
$569$	20.3431	0.852829	0.426415	−	0.904528i	$-0.359776\pi$
$569$	20.3431	0.852829	0.426415	+	0.904528i	$0.359776\pi$
$570$	0	0
$571$	−11.0294	−0.461568	−0.230784	−	0.973005i	$-0.574129\pi$
$571$	−11.0294	−0.461568	−0.230784	+	0.973005i	$0.574129\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	14.4853	0.604078
$576$	0	0
$577$	12.6274	0.525686	0.262843	−	0.964839i	$-0.415340\pi$
$577$	12.6274	0.525686	0.262843	+	0.964839i	$0.415340\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−4.34315	−0.180184
$582$	0	0
$583$	12.0000	0.496989
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−0.627417	−0.0258963	−0.0129481	−	0.999916i	$-0.504122\pi$
$587$	−0.627417	−0.0258963	−0.0129481	+	0.999916i	$0.504122\pi$
$588$	0	0
$589$	6.82843	0.281360
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−44.2843	−1.81854	−0.909269	−	0.416210i	$-0.863358\pi$
$593$	−44.2843	−1.81854	−0.909269	+	0.416210i	$0.863358\pi$
$594$	0	0
$595$	−11.3137	−0.463817
$596$	0	0
$597$	0	0
$598$	0	0
$599$	29.9411	1.22336	0.611681	−	0.791105i	$-0.290494\pi$
$599$	29.9411	1.22336	0.611681	+	0.791105i	$0.290494\pi$
$600$	0	0
$601$	−32.6274	−1.33090	−0.665450	−	0.746442i	$-0.731760\pi$
$601$	−32.6274	−1.33090	−0.665450	+	0.746442i	$0.731760\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	19.7990	0.804943
$606$	0	0
$607$	13.8579	0.562473	0.281237	−	0.959638i	$-0.409255\pi$
$607$	13.8579	0.562473	0.281237	+	0.959638i	$0.409255\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	5.37258	0.217351
$612$	0	0
$613$	−28.6274	−1.15625	−0.578125	−	0.815948i	$-0.696215\pi$
$613$	−28.6274	−1.15625	−0.578125	+	0.815948i	$0.696215\pi$
$614$	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
$619$	37.9411	1.52498	0.762491	−	0.646998i	$-0.223976\pi$
$619$	37.9411	1.52498	0.762491	+	0.646998i	$0.223976\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−0.343146	−0.0137478
$624$	0	0
$625$	−31.0000	−1.24000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−12.6863	−0.505836
$630$	0	0
$631$	−47.5980	−1.89485	−0.947423	−	0.319984i	$-0.896322\pi$
$631$	−47.5980	−1.89485	−0.947423	+	0.319984i	$0.896322\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	24.0000	0.952411
$636$	0	0
$637$	−0.828427	−0.0328235
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−4.34315	−0.171544	−0.0857720	−	0.996315i	$-0.527336\pi$
$641$	−4.34315	−0.171544	−0.0857720	+	0.996315i	$0.527336\pi$
$642$	0	0
$643$	−24.2843	−0.957678	−0.478839	−	0.877903i	$-0.658942\pi$
$643$	−24.2843	−0.957678	−0.478839	+	0.877903i	$0.658942\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	21.7990	0.857007	0.428503	−	0.903540i	$-0.359041\pi$
$647$	21.7990	0.857007	0.428503	+	0.903540i	$0.359041\pi$
$648$	0	0
$649$	8.00000	0.314027
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−19.1127	−0.747938	−0.373969	−	0.927441i	$-0.622003\pi$
$653$	−19.1127	−0.747938	−0.373969	+	0.927441i	$0.622003\pi$
$654$	0	0
$655$	−12.2843	−0.479986
$656$	0	0
$657$	0	0
$658$	0	0
$659$	16.2843	0.634345	0.317173	−	0.948368i	$-0.397267\pi$
$659$	16.2843	0.634345	0.317173	+	0.948368i	$0.397267\pi$
$660$	0	0
$661$	9.79899	0.381137	0.190568	−	0.981674i	$-0.438967\pi$
$661$	9.79899	0.381137	0.190568	+	0.981674i	$0.438967\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−2.82843	−0.109682
$666$	0	0
$667$	17.6569	0.683676
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−12.0000	−0.463255
$672$	0	0
$673$	9.02944	0.348059	0.174030	−	0.984740i	$-0.444321\pi$
$673$	9.02944	0.348059	0.174030	+	0.984740i	$0.444321\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	31.6569	1.21667	0.608336	−	0.793680i	$-0.291837\pi$
$677$	31.6569	1.21667	0.608336	+	0.793680i	$0.291837\pi$
$678$	0	0
$679$	−0.343146	−0.0131687
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−5.65685	−0.216454	−0.108227	−	0.994126i	$-0.534517\pi$
$683$	−5.65685	−0.216454	−0.108227	+	0.994126i	$0.534517\pi$
$684$	0	0
$685$	−59.3137	−2.26626
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−4.97056	−0.189363
$690$	0	0
$691$	15.3137	0.582561	0.291280	−	0.956638i	$-0.405919\pi$
$691$	15.3137	0.582561	0.291280	+	0.956638i	$0.405919\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−4.68629	−0.177761
$696$	0	0
$697$	8.00000	0.303022
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−45.4558	−1.71684	−0.858422	−	0.512945i	$-0.828555\pi$
$701$	−45.4558	−1.71684	−0.858422	+	0.512945i	$0.828555\pi$
$702$	0	0
$703$	−3.17157	−0.119618
$704$	0	0
$705$	0	0
$706$	0	0
$707$	4.48528	0.168686
$708$	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
$713$	32.9706	1.23476
$714$	0	0
$715$	4.68629	0.175257
$716$	0	0
$717$	0	0
$718$	0	0
$719$	24.1421	0.900350	0.450175	−	0.892940i	$-0.351362\pi$
$719$	24.1421	0.900350	0.450175	+	0.892940i	$0.351362\pi$
$720$	0	0
$721$	−2.82843	−0.105336
$722$	0	0
$723$	0	0
$724$	0	0
$725$	10.9706	0.407436
$726$	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
$731$	−38.6274	−1.42869
$732$	0	0
$733$	12.6274	0.466404	0.233202	−	0.972428i	$-0.425080\pi$
$733$	12.6274	0.466404	0.233202	+	0.972428i	$0.425080\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−22.6274	−0.833492
$738$	0	0
$739$	−18.6274	−0.685221	−0.342610	−	0.939478i	$-0.611311\pi$
$739$	−18.6274	−0.685221	−0.342610	+	0.939478i	$0.611311\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	28.2843	1.03765	0.518825	−	0.854881i	$-0.326370\pi$
$743$	28.2843	1.03765	0.518825	+	0.854881i	$0.326370\pi$
$744$	0	0
$745$	−19.3137	−0.707600
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−13.6569	−0.499011
$750$	0	0
$751$	5.85786	0.213757	0.106878	−	0.994272i	$-0.465914\pi$
$751$	5.85786	0.213757	0.106878	+	0.994272i	$0.465914\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−12.6863	−0.461701
$756$	0	0
$757$	−12.6274	−0.458951	−0.229476	−	0.973314i	$-0.573701\pi$
$757$	−12.6274	−0.458951	−0.229476	+	0.973314i	$0.573701\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−15.3137	−0.555121	−0.277561	−	0.960708i	$-0.589526\pi$
$761$	−15.3137	−0.555121	−0.277561	+	0.960708i	$0.589526\pi$
$762$	0	0
$763$	−16.1421	−0.584385
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−3.31371	−0.119651
$768$	0	0
$769$	20.6274	0.743844	0.371922	−	0.928264i	$-0.378699\pi$
$769$	20.6274	0.743844	0.371922	+	0.928264i	$0.378699\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	26.2843	0.945380	0.472690	−	0.881229i	$-0.343283\pi$
$773$	26.2843	0.945380	0.472690	+	0.881229i	$0.343283\pi$
$774$	0	0
$775$	20.4853	0.735853
$776$	0	0
$777$	0	0
$778$	0	0
$779$	2.00000	0.0716574
$780$	0	0
$781$	4.68629	0.167689
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−53.6569	−1.91510
$786$	0	0
$787$	41.2548	1.47058	0.735288	−	0.677755i	$-0.237047\pi$
$787$	41.2548	1.47058	0.735288	+	0.677755i	$0.237047\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−2.00000	−0.0711118
$792$	0	0
$793$	4.97056	0.176510
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−4.34315	−0.153842	−0.0769211	−	0.997037i	$-0.524509\pi$
$797$	−4.34315	−0.153842	−0.0769211	+	0.997037i	$0.524509\pi$
$798$	0	0
$799$	−25.9411	−0.917731
$800$	0	0
$801$	0	0
$802$	0	0
$803$	20.0000	0.705785
$804$	0	0
$805$	−13.6569	−0.481341
$806$	0	0
$807$	0	0
$808$	0	0
$809$	26.3431	0.926176	0.463088	−	0.886312i	$-0.346741\pi$
$809$	26.3431	0.926176	0.463088	+	0.886312i	$0.346741\pi$
$810$	0	0
$811$	−0.970563	−0.0340811	−0.0170405	−	0.999855i	$-0.505424\pi$
$811$	−0.970563	−0.0340811	−0.0170405	+	0.999855i	$0.505424\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	1.94113	0.0679947
$816$	0	0
$817$	−9.65685	−0.337851
$818$	0	0
$819$	0	0
$820$	0	0
$821$	44.4853	1.55255	0.776274	−	0.630396i	$-0.217108\pi$
$821$	44.4853	1.55255	0.776274	+	0.630396i	$0.217108\pi$
$822$	0	0
$823$	0.970563	0.0338317	0.0169158	−	0.999857i	$-0.494615\pi$
$823$	0.970563	0.0338317	0.0169158	+	0.999857i	$0.494615\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−10.3431	−0.359666	−0.179833	−	0.983697i	$-0.557556\pi$
$827$	−10.3431	−0.359666	−0.179833	+	0.983697i	$0.557556\pi$
$828$	0	0
$829$	−46.4853	−1.61450	−0.807250	−	0.590209i	$-0.799045\pi$
$829$	−46.4853	−1.61450	−0.807250	+	0.590209i	$0.799045\pi$
$830$	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
$839$	−16.2843	−0.562195	−0.281098	−	0.959679i	$-0.590699\pi$
$839$	−16.2843	−0.562195	−0.281098	+	0.959679i	$0.590699\pi$
$840$	0	0
$841$	−15.6274	−0.538876
$842$	0	0
$843$	0	0
$844$	0	0
$845$	34.8284	1.19813
$846$	0	0
$847$	−7.00000	−0.240523
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−15.3137	−0.524947
$852$	0	0
$853$	10.2843	0.352127	0.176063	−	0.984379i	$-0.443664\pi$
$853$	10.2843	0.352127	0.176063	+	0.984379i	$0.443664\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−46.9706	−1.60448	−0.802242	−	0.596999i	$-0.796360\pi$
$857$	−46.9706	−1.60448	−0.802242	+	0.596999i	$0.796360\pi$
$858$	0	0
$859$	2.62742	0.0896463	0.0448232	−	0.998995i	$-0.485728\pi$
$859$	2.62742	0.0896463	0.0448232	+	0.998995i	$0.485728\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	20.2843	0.690485	0.345242	−	0.938514i	$-0.387797\pi$
$863$	20.2843	0.690485	0.345242	+	0.938514i	$0.387797\pi$
$864$	0	0
$865$	−39.5980	−1.34637
$866$	0	0
$867$	0	0
$868$	0	0
$869$	18.3431	0.622249
$870$	0	0
$871$	9.37258	0.317578
$872$	0	0
$873$	0	0
$874$	0	0
$875$	5.65685	0.191237
$876$	0	0
$877$	−30.4853	−1.02941	−0.514707	−	0.857366i	$-0.672099\pi$
$877$	−30.4853	−1.02941	−0.514707	+	0.857366i	$0.672099\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−20.9706	−0.706516	−0.353258	−	0.935526i	$-0.614926\pi$
$881$	−20.9706	−0.706516	−0.353258	+	0.935526i	$0.614926\pi$
$882$	0	0
$883$	11.0294	0.371170	0.185585	−	0.982628i	$-0.440582\pi$
$883$	11.0294	0.371170	0.185585	+	0.982628i	$0.440582\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	38.6274	1.29698	0.648491	−	0.761222i	$-0.275400\pi$
$887$	38.6274	1.29698	0.648491	+	0.761222i	$0.275400\pi$
$888$	0	0
$889$	−8.48528	−0.284587
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−6.48528	−0.217022
$894$	0	0
$895$	−61.2548	−2.04752
$896$	0	0
$897$	0	0
$898$	0	0
$899$	24.9706	0.832815
$900$	0	0
$901$	24.0000	0.799556
$902$	0	0
$903$	0	0
$904$	0	0
$905$	31.5980	1.05035
$906$	0	0
$907$	−1.37258	−0.0455759	−0.0227879	−	0.999740i	$-0.507254\pi$
$907$	−1.37258	−0.0455759	−0.0227879	+	0.999740i	$0.507254\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	8.68629	0.287790	0.143895	−	0.989593i	$-0.454037\pi$
$911$	8.68629	0.287790	0.143895	+	0.989593i	$0.454037\pi$
$912$	0	0
$913$	−8.68629	−0.287474
$914$	0	0
$915$	0	0
$916$	0	0
$917$	4.34315	0.143423
$918$	0	0
$919$	45.6569	1.50608	0.753040	−	0.657974i	$-0.228586\pi$
$919$	45.6569	1.50608	0.753040	+	0.657974i	$0.228586\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−1.94113	−0.0638929
$924$	0	0
$925$	−9.51472	−0.312842
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−11.3137	−0.371191	−0.185595	−	0.982626i	$-0.559421\pi$
$929$	−11.3137	−0.371191	−0.185595	+	0.982626i	$0.559421\pi$
$930$	0	0
$931$	1.00000	0.0327737
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−22.6274	−0.739996
$936$	0	0
$937$	5.31371	0.173591	0.0867956	−	0.996226i	$-0.472337\pi$
$937$	5.31371	0.173591	0.0867956	+	0.996226i	$0.472337\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−25.5980	−0.834470	−0.417235	−	0.908799i	$-0.637001\pi$
$941$	−25.5980	−0.834470	−0.417235	+	0.908799i	$0.637001\pi$
$942$	0	0
$943$	9.65685	0.314470
$944$	0	0
$945$	0	0
$946$	0	0
$947$	2.97056	0.0965303	0.0482652	−	0.998835i	$-0.484631\pi$
$947$	2.97056	0.0965303	0.0482652	+	0.998835i	$0.484631\pi$
$948$	0	0
$949$	−8.28427	−0.268919
$950$	0	0
$951$	0	0
$952$	0	0
$953$	8.62742	0.279469	0.139735	−	0.990189i	$-0.455375\pi$
$953$	8.62742	0.279469	0.139735	+	0.990189i	$0.455375\pi$
$954$	0	0
$955$	−47.5980	−1.54023
$956$	0	0
$957$	0	0
$958$	0	0
$959$	20.9706	0.677175
$960$	0	0
$961$	15.6274	0.504110
$962$	0	0
$963$	0	0
$964$	0	0
$965$	42.3431	1.36307
$966$	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	$-0.241309\pi$
$971$	45.2548	1.45230	0.726148	+	0.687538i	$0.241309\pi$
$972$	0	0
$973$	1.65685	0.0531163
$974$	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
$979$	−0.686292	−0.0219340
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−19.3137	−0.616012	−0.308006	−	0.951384i	$-0.599662\pi$
$983$	−19.3137	−0.616012	−0.308006	+	0.951384i	$0.599662\pi$
$984$	0	0
$985$	−21.2548	−0.677235
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−46.6274	−1.48267
$990$	0	0
$991$	−3.79899	−0.120679	−0.0603394	−	0.998178i	$-0.519218\pi$
$991$	−3.79899	−0.120679	−0.0603394	+	0.998178i	$0.519218\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−16.0000	−0.507234
$996$	0	0
$997$	−0.343146	−0.0108675	−0.00543377	−	0.999985i	$-0.501730\pi$
$997$	−0.343146	−0.0108675	−0.00543377	+	0.999985i	$0.501730\pi$
$998$	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.1.1	yes	2
3.2	odd	2		9576.2.a.bl.1.2	✓	2

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

\(f(q)\)	\(=\)	\(q-2.82843 q^{5} +1.00000 q^{7} +O(q^{10})\)	\(q-2.82843 q^{5} +1.00000 q^{7} +2.00000 q^{11} -0.828427 q^{13} +4.00000 q^{17} +1.00000 q^{19} +4.82843 q^{23} +3.00000 q^{25} +3.65685 q^{29} +6.82843 q^{31} -2.82843 q^{35} -3.17157 q^{37} +2.00000 q^{41} -9.65685 q^{43} -6.48528 q^{47} +1.00000 q^{49} +6.00000 q^{53} -5.65685 q^{55} +4.00000 q^{59} -6.00000 q^{61} +2.34315 q^{65} -11.3137 q^{67} +2.34315 q^{71} +10.0000 q^{73} +2.00000 q^{77} +9.17157 q^{79} -4.34315 q^{83} -11.3137 q^{85} -0.343146 q^{89} -0.828427 q^{91} -2.82843 q^{95} -0.343146 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{7}+O(q^{10}) \) 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

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