LMFDB - Embedded newform 9576.2.a.bd.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(\sqrt{17}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.56155$ 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)

$f(q)$	$=$	$q-2.00000 q^{5} -1.00000 q^{7} +O(q^{10})$	$q-2.00000 q^{5} -1.00000 q^{7} +2.00000 q^{11} -3.12311 q^{13} -3.12311 q^{17} -1.00000 q^{19} -7.12311 q^{23} -1.00000 q^{25} +9.12311 q^{29} -1.12311 q^{31} +2.00000 q^{35} -0.876894 q^{37} -8.24621 q^{41} +4.00000 q^{43} +1.00000 q^{49} +5.12311 q^{53} -4.00000 q^{55} +6.24621 q^{59} -2.00000 q^{61} +6.24621 q^{65} -14.2462 q^{67} -9.36932 q^{71} -10.0000 q^{73} -2.00000 q^{77} +13.1231 q^{79} -9.12311 q^{83} +6.24621 q^{85} -0.246211 q^{89} +3.12311 q^{91} +2.00000 q^{95} +8.24621 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{5} - 2 q^{7}+O(q^{10}) $ 2 * q - 4 * q^5 - 2 * q^7	$ 2 q - 4 q^{5} - 2 q^{7} + 4 q^{11} + 2 q^{13} + 2 q^{17} - 2 q^{19} - 6 q^{23} - 2 q^{25} + 10 q^{29} + 6 q^{31} + 4 q^{35} - 10 q^{37} + 8 q^{43} + 2 q^{49} + 2 q^{53} - 8 q^{55} - 4 q^{59} - 4 q^{61} - 4 q^{65} - 12 q^{67} + 6 q^{71} - 20 q^{73} - 4 q^{77} + 18 q^{79} - 10 q^{83} - 4 q^{85} + 16 q^{89} - 2 q^{91} + 4 q^{95}+O(q^{100}) $ 2 * q - 4 * q^5 - 2 * q^7 + 4 * q^11 + 2 * q^13 + 2 * q^17 - 2 * q^19 - 6 * q^23 - 2 * q^25 + 10 * q^29 + 6 * q^31 + 4 * q^35 - 10 * q^37 + 8 * q^43 + 2 * q^49 + 2 * q^53 - 8 * q^55 - 4 * q^59 - 4 * q^61 - 4 * q^65 - 12 * q^67 + 6 * q^71 - 20 * q^73 - 4 * q^77 + 18 * q^79 - 10 * q^83 - 4 * q^85 + 16 * q^89 - 2 * q^91 + 4 * q^95

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
$5$	−2.00000	−0.894427	−0.447214	−	0.894427i	$-0.647584\pi$
$5$	−2.00000	−0.894427	−0.447214	+	0.894427i	$0.647584\pi$
$6$	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
$13$	−3.12311	−0.866194	−0.433097	−	0.901347i	$-0.642579\pi$
$13$	−3.12311	−0.866194	−0.433097	+	0.901347i	$0.642579\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−3.12311	−0.757464	−0.378732	−	0.925506i	$-0.623640\pi$
$17$	−3.12311	−0.757464	−0.378732	+	0.925506i	$0.623640\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−7.12311	−1.48527	−0.742635	−	0.669696i	$-0.766424\pi$
$23$	−7.12311	−1.48527	−0.742635	+	0.669696i	$0.766424\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	9.12311	1.69412	0.847059	−	0.531499i	$-0.178371\pi$
$29$	9.12311	1.69412	0.847059	+	0.531499i	$0.178371\pi$
$30$	0	0
$31$	−1.12311	−0.201716	−0.100858	−	0.994901i	$-0.532159\pi$
$31$	−1.12311	−0.201716	−0.100858	+	0.994901i	$0.532159\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	2.00000	0.338062
$36$	0	0
$37$	−0.876894	−0.144161	−0.0720803	−	0.997399i	$-0.522964\pi$
$37$	−0.876894	−0.144161	−0.0720803	+	0.997399i	$0.522964\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−8.24621	−1.28784	−0.643921	−	0.765092i	$-0.722693\pi$
$41$	−8.24621	−1.28784	−0.643921	+	0.765092i	$0.722693\pi$
$42$	0	0
$43$	4.00000	0.609994	0.304997	−	0.952353i	$-0.401344\pi$
$43$	4.00000	0.609994	0.304997	+	0.952353i	$0.401344\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	5.12311	0.703713	0.351856	−	0.936054i	$-0.385551\pi$
$53$	5.12311	0.703713	0.351856	+	0.936054i	$0.385551\pi$
$54$	0	0
$55$	−4.00000	−0.539360
$56$	0	0
$57$	0	0
$58$	0	0
$59$	6.24621	0.813187	0.406594	−	0.913609i	$-0.366716\pi$
$59$	6.24621	0.813187	0.406594	+	0.913609i	$0.366716\pi$
$60$	0	0
$61$	−2.00000	−0.256074	−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000	−0.256074	−0.128037	+	0.991769i	$0.540868\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	6.24621	0.774747
$66$	0	0
$67$	−14.2462	−1.74045	−0.870226	−	0.492653i	$-0.836027\pi$
$67$	−14.2462	−1.74045	−0.870226	+	0.492653i	$0.836027\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−9.36932	−1.11193	−0.555967	−	0.831205i	$-0.687652\pi$
$71$	−9.36932	−1.11193	−0.555967	+	0.831205i	$0.687652\pi$
$72$	0	0
$73$	−10.0000	−1.17041	−0.585206	−	0.810885i	$-0.698986\pi$
$73$	−10.0000	−1.17041	−0.585206	+	0.810885i	$0.698986\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−2.00000	−0.227921
$78$	0	0
$79$	13.1231	1.47646	0.738232	−	0.674546i	$-0.235661\pi$
$79$	13.1231	1.47646	0.738232	+	0.674546i	$0.235661\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−9.12311	−1.00139	−0.500695	−	0.865624i	$-0.666922\pi$
$83$	−9.12311	−1.00139	−0.500695	+	0.865624i	$0.666922\pi$
$84$	0	0
$85$	6.24621	0.677497
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−0.246211	−0.0260983	−0.0130492	−	0.999915i	$-0.504154\pi$
$89$	−0.246211	−0.0260983	−0.0130492	+	0.999915i	$0.504154\pi$
$90$	0	0
$91$	3.12311	0.327390
$92$	0	0
$93$	0	0
$94$	0	0
$95$	2.00000	0.205196
$96$	0	0
$97$	8.24621	0.837276	0.418638	−	0.908153i	$-0.362508\pi$
$97$	8.24621	0.837276	0.418638	+	0.908153i	$0.362508\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	10.0000	0.995037	0.497519	−	0.867453i	$-0.334245\pi$
$101$	10.0000	0.995037	0.497519	+	0.867453i	$0.334245\pi$
$102$	0	0
$103$	−17.1231	−1.68719	−0.843595	−	0.536980i	$-0.819565\pi$
$103$	−17.1231	−1.68719	−0.843595	+	0.536980i	$0.819565\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	3.12311	0.301922	0.150961	−	0.988540i	$-0.451763\pi$
$107$	3.12311	0.301922	0.150961	+	0.988540i	$0.451763\pi$
$108$	0	0
$109$	−0.876894	−0.0839912	−0.0419956	−	0.999118i	$-0.513372\pi$
$109$	−0.876894	−0.0839912	−0.0419956	+	0.999118i	$0.513372\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	9.12311	0.858230	0.429115	−	0.903250i	$-0.358826\pi$
$113$	9.12311	0.858230	0.429115	+	0.903250i	$0.358826\pi$
$114$	0	0
$115$	14.2462	1.32847
$116$	0	0
$117$	0	0
$118$	0	0
$119$	3.12311	0.286295
$120$	0	0
$121$	−7.00000	−0.636364
$122$	0	0
$123$	0	0
$124$	0	0
$125$	12.0000	1.07331
$126$	0	0
$127$	21.1231	1.87437	0.937186	−	0.348829i	$-0.113421\pi$
$127$	21.1231	1.87437	0.937186	+	0.348829i	$0.113421\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	21.6155	1.88856	0.944279	−	0.329147i	$-0.106761\pi$
$131$	21.6155	1.88856	0.944279	+	0.329147i	$0.106761\pi$
$132$	0	0
$133$	1.00000	0.0867110
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−1.75379	−0.149836	−0.0749181	−	0.997190i	$-0.523870\pi$
$137$	−1.75379	−0.149836	−0.0749181	+	0.997190i	$0.523870\pi$
$138$	0	0
$139$	−2.24621	−0.190521	−0.0952606	−	0.995452i	$-0.530368\pi$
$139$	−2.24621	−0.190521	−0.0952606	+	0.995452i	$0.530368\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−6.24621	−0.522334
$144$	0	0
$145$	−18.2462	−1.51527
$146$	0	0
$147$	0	0
$148$	0	0
$149$	17.1231	1.40278	0.701390	−	0.712778i	$-0.252563\pi$
$149$	17.1231	1.40278	0.701390	+	0.712778i	$0.252563\pi$
$150$	0	0
$151$	13.1231	1.06794	0.533972	−	0.845502i	$-0.320699\pi$
$151$	13.1231	1.06794	0.533972	+	0.845502i	$0.320699\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	2.24621	0.180420
$156$	0	0
$157$	−7.75379	−0.618820	−0.309410	−	0.950929i	$-0.600131\pi$
$157$	−7.75379	−0.618820	−0.309410	+	0.950929i	$0.600131\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	7.12311	0.561379
$162$	0	0
$163$	−24.4924	−1.91839	−0.959197	−	0.282738i	$-0.908757\pi$
$163$	−24.4924	−1.91839	−0.959197	+	0.282738i	$0.908757\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	1.75379	0.135712	0.0678561	−	0.997695i	$-0.478384\pi$
$167$	1.75379	0.135712	0.0678561	+	0.997695i	$0.478384\pi$
$168$	0	0
$169$	−3.24621	−0.249709
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−4.24621	−0.322833	−0.161417	−	0.986886i	$-0.551606\pi$
$173$	−4.24621	−0.322833	−0.161417	+	0.986886i	$0.551606\pi$
$174$	0	0
$175$	1.00000	0.0755929
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−13.3693	−0.999270	−0.499635	−	0.866236i	$-0.666532\pi$
$179$	−13.3693	−0.999270	−0.499635	+	0.866236i	$0.666532\pi$
$180$	0	0
$181$	−5.36932	−0.399098	−0.199549	−	0.979888i	$-0.563948\pi$
$181$	−5.36932	−0.399098	−0.199549	+	0.979888i	$0.563948\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	1.75379	0.128941
$186$	0	0
$187$	−6.24621	−0.456768
$188$	0	0
$189$	0	0
$190$	0	0
$191$	1.36932	0.0990803	0.0495401	−	0.998772i	$-0.484224\pi$
$191$	1.36932	0.0990803	0.0495401	+	0.998772i	$0.484224\pi$
$192$	0	0
$193$	−2.00000	−0.143963	−0.0719816	−	0.997406i	$-0.522932\pi$
$193$	−2.00000	−0.143963	−0.0719816	+	0.997406i	$0.522932\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	27.3693	1.94998	0.974992	−	0.222242i	$-0.0713375\pi$
$197$	27.3693	1.94998	0.974992	+	0.222242i	$0.0713375\pi$
$198$	0	0
$199$	−5.75379	−0.407875	−0.203938	−	0.978984i	$-0.565374\pi$
$199$	−5.75379	−0.407875	−0.203938	+	0.978984i	$0.565374\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−9.12311	−0.640316
$204$	0	0
$205$	16.4924	1.15188
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−2.00000	−0.138343
$210$	0	0
$211$	8.49242	0.584642	0.292321	−	0.956320i	$-0.405572\pi$
$211$	8.49242	0.584642	0.292321	+	0.956320i	$0.405572\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−8.00000	−0.545595
$216$	0	0
$217$	1.12311	0.0762414
$218$	0	0
$219$	0	0
$220$	0	0
$221$	9.75379	0.656111
$222$	0	0
$223$	15.3693	1.02921	0.514603	−	0.857429i	$-0.327939\pi$
$223$	15.3693	1.02921	0.514603	+	0.857429i	$0.327939\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	14.2462	0.945554	0.472777	−	0.881182i	$-0.343252\pi$
$227$	14.2462	0.945554	0.472777	+	0.881182i	$0.343252\pi$
$228$	0	0
$229$	−20.7386	−1.37045	−0.685224	−	0.728333i	$-0.740296\pi$
$229$	−20.7386	−1.37045	−0.685224	+	0.728333i	$0.740296\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	14.2462	0.933300	0.466650	−	0.884442i	$-0.345461\pi$
$233$	14.2462	0.933300	0.466650	+	0.884442i	$0.345461\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	8.87689	0.574199	0.287099	−	0.957901i	$-0.407309\pi$
$239$	8.87689	0.574199	0.287099	+	0.957901i	$0.407309\pi$
$240$	0	0
$241$	20.7386	1.33589	0.667946	−	0.744209i	$-0.267174\pi$
$241$	20.7386	1.33589	0.667946	+	0.744209i	$0.267174\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−2.00000	−0.127775
$246$	0	0
$247$	3.12311	0.198718
$248$	0	0
$249$	0	0
$250$	0	0
$251$	0.630683	0.0398084	0.0199042	−	0.999802i	$-0.493664\pi$
$251$	0.630683	0.0398084	0.0199042	+	0.999802i	$0.493664\pi$
$252$	0	0
$253$	−14.2462	−0.895652
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−14.0000	−0.873296	−0.436648	−	0.899632i	$-0.643834\pi$
$257$	−14.0000	−0.873296	−0.436648	+	0.899632i	$0.643834\pi$
$258$	0	0
$259$	0.876894	0.0544876
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−11.6155	−0.716244	−0.358122	−	0.933675i	$-0.616583\pi$
$263$	−11.6155	−0.716244	−0.358122	+	0.933675i	$0.616583\pi$
$264$	0	0
$265$	−10.2462	−0.629420
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−14.0000	−0.853595	−0.426798	−	0.904347i	$-0.640358\pi$
$269$	−14.0000	−0.853595	−0.426798	+	0.904347i	$0.640358\pi$
$270$	0	0
$271$	12.0000	0.728948	0.364474	−	0.931214i	$-0.381249\pi$
$271$	12.0000	0.728948	0.364474	+	0.931214i	$0.381249\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−2.00000	−0.120605
$276$	0	0
$277$	−2.00000	−0.120168	−0.0600842	−	0.998193i	$-0.519137\pi$
$277$	−2.00000	−0.120168	−0.0600842	+	0.998193i	$0.519137\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	21.1231	1.26010	0.630049	−	0.776555i	$-0.283035\pi$
$281$	21.1231	1.26010	0.630049	+	0.776555i	$0.283035\pi$
$282$	0	0
$283$	−8.49242	−0.504822	−0.252411	−	0.967620i	$-0.581224\pi$
$283$	−8.49242	−0.504822	−0.252411	+	0.967620i	$0.581224\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	8.24621	0.486758
$288$	0	0
$289$	−7.24621	−0.426248
$290$	0	0
$291$	0	0
$292$	0	0
$293$	16.2462	0.949114	0.474557	−	0.880225i	$-0.342608\pi$
$293$	16.2462	0.949114	0.474557	+	0.880225i	$0.342608\pi$
$294$	0	0
$295$	−12.4924	−0.727337
$296$	0	0
$297$	0	0
$298$	0	0
$299$	22.2462	1.28653
$300$	0	0
$301$	−4.00000	−0.230556
$302$	0	0
$303$	0	0
$304$	0	0
$305$	4.00000	0.229039
$306$	0	0
$307$	24.4924	1.39786	0.698928	−	0.715192i	$-0.253661\pi$
$307$	24.4924	1.39786	0.698928	+	0.715192i	$0.253661\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	16.0000	0.907277	0.453638	−	0.891186i	$-0.350126\pi$
$311$	16.0000	0.907277	0.453638	+	0.891186i	$0.350126\pi$
$312$	0	0
$313$	−30.9848	−1.75137	−0.875683	−	0.482886i	$-0.839589\pi$
$313$	−30.9848	−1.75137	−0.875683	+	0.482886i	$0.839589\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−15.3693	−0.863227	−0.431613	−	0.902059i	$-0.642056\pi$
$317$	−15.3693	−0.863227	−0.431613	+	0.902059i	$0.642056\pi$
$318$	0	0
$319$	18.2462	1.02159
$320$	0	0
$321$	0	0
$322$	0	0
$323$	3.12311	0.173774
$324$	0	0
$325$	3.12311	0.173239
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−4.00000	−0.219860	−0.109930	−	0.993939i	$-0.535063\pi$
$331$	−4.00000	−0.219860	−0.109930	+	0.993939i	$0.535063\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	28.4924	1.55671
$336$	0	0
$337$	−6.49242	−0.353665	−0.176832	−	0.984241i	$-0.556585\pi$
$337$	−6.49242	−0.353665	−0.176832	+	0.984241i	$0.556585\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−2.24621	−0.121639
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	24.2462	1.30160	0.650802	−	0.759247i	$-0.274433\pi$
$347$	24.2462	1.30160	0.650802	+	0.759247i	$0.274433\pi$
$348$	0	0
$349$	3.75379	0.200936	0.100468	−	0.994940i	$-0.467966\pi$
$349$	3.75379	0.200936	0.100468	+	0.994940i	$0.467966\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	2.63068	0.140017	0.0700086	−	0.997546i	$-0.477697\pi$
$353$	2.63068	0.140017	0.0700086	+	0.997546i	$0.477697\pi$
$354$	0	0
$355$	18.7386	0.994543
$356$	0	0
$357$	0	0
$358$	0	0
$359$	17.3693	0.916717	0.458359	−	0.888767i	$-0.348437\pi$
$359$	17.3693	0.916717	0.458359	+	0.888767i	$0.348437\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	0	0
$364$	0	0
$365$	20.0000	1.04685
$366$	0	0
$367$	−26.2462	−1.37004	−0.685021	−	0.728524i	$-0.740207\pi$
$367$	−26.2462	−1.37004	−0.685021	+	0.728524i	$0.740207\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−5.12311	−0.265978
$372$	0	0
$373$	−11.6155	−0.601429	−0.300715	−	0.953714i	$-0.597225\pi$
$373$	−11.6155	−0.601429	−0.300715	+	0.953714i	$0.597225\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−28.4924	−1.46743
$378$	0	0
$379$	−8.49242	−0.436226	−0.218113	−	0.975923i	$-0.569990\pi$
$379$	−8.49242	−0.436226	−0.218113	+	0.975923i	$0.569990\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	32.0000	1.63512	0.817562	−	0.575841i	$-0.195325\pi$
$383$	32.0000	1.63512	0.817562	+	0.575841i	$0.195325\pi$
$384$	0	0
$385$	4.00000	0.203859
$386$	0	0
$387$	0	0
$388$	0	0
$389$	2.87689	0.145864	0.0729322	−	0.997337i	$-0.476764\pi$
$389$	2.87689	0.145864	0.0729322	+	0.997337i	$0.476764\pi$
$390$	0	0
$391$	22.2462	1.12504
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−26.2462	−1.32059
$396$	0	0
$397$	20.7386	1.04084	0.520421	−	0.853910i	$-0.325775\pi$
$397$	20.7386	1.04084	0.520421	+	0.853910i	$0.325775\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−1.61553	−0.0806756	−0.0403378	−	0.999186i	$-0.512843\pi$
$401$	−1.61553	−0.0806756	−0.0403378	+	0.999186i	$0.512843\pi$
$402$	0	0
$403$	3.50758	0.174725
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−1.75379	−0.0869321
$408$	0	0
$409$	−20.2462	−1.00111	−0.500555	−	0.865705i	$-0.666871\pi$
$409$	−20.2462	−1.00111	−0.500555	+	0.865705i	$0.666871\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−6.24621	−0.307356
$414$	0	0
$415$	18.2462	0.895671
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−23.3693	−1.14167	−0.570833	−	0.821066i	$-0.693380\pi$
$419$	−23.3693	−1.14167	−0.570833	+	0.821066i	$0.693380\pi$
$420$	0	0
$421$	35.1231	1.71180	0.855898	−	0.517145i	$-0.173005\pi$
$421$	35.1231	1.71180	0.855898	+	0.517145i	$0.173005\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	3.12311	0.151493
$426$	0	0
$427$	2.00000	0.0967868
$428$	0	0
$429$	0	0
$430$	0	0
$431$	35.1231	1.69182	0.845910	−	0.533325i	$-0.179058\pi$
$431$	35.1231	1.69182	0.845910	+	0.533325i	$0.179058\pi$
$432$	0	0
$433$	24.7386	1.18886	0.594431	−	0.804146i	$-0.297377\pi$
$433$	24.7386	1.18886	0.594431	+	0.804146i	$0.297377\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	7.12311	0.340744
$438$	0	0
$439$	39.3693	1.87899	0.939497	−	0.342556i	$-0.111293\pi$
$439$	39.3693	1.87899	0.939497	+	0.342556i	$0.111293\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−4.24621	−0.201744	−0.100872	−	0.994899i	$-0.532163\pi$
$443$	−4.24621	−0.201744	−0.100872	+	0.994899i	$0.532163\pi$
$444$	0	0
$445$	0.492423	0.0233431
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−33.1231	−1.56318	−0.781588	−	0.623795i	$-0.785590\pi$
$449$	−33.1231	−1.56318	−0.781588	+	0.623795i	$0.785590\pi$
$450$	0	0
$451$	−16.4924	−0.776598
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−6.24621	−0.292827
$456$	0	0
$457$	32.2462	1.50841	0.754207	−	0.656637i	$-0.228021\pi$
$457$	32.2462	1.50841	0.754207	+	0.656637i	$0.228021\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−32.2462	−1.50186	−0.750928	−	0.660384i	$-0.770393\pi$
$461$	−32.2462	−1.50186	−0.750928	+	0.660384i	$0.770393\pi$
$462$	0	0
$463$	30.7386	1.42855	0.714273	−	0.699867i	$-0.246758\pi$
$463$	30.7386	1.42855	0.714273	+	0.699867i	$0.246758\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−23.3693	−1.08140	−0.540702	−	0.841215i	$-0.681841\pi$
$467$	−23.3693	−1.08140	−0.540702	+	0.841215i	$0.681841\pi$
$468$	0	0
$469$	14.2462	0.657829
$470$	0	0
$471$	0	0
$472$	0	0
$473$	8.00000	0.367840
$474$	0	0
$475$	1.00000	0.0458831
$476$	0	0
$477$	0	0
$478$	0	0
$479$	36.4924	1.66738	0.833691	−	0.552232i	$-0.186224\pi$
$479$	36.4924	1.66738	0.833691	+	0.552232i	$0.186224\pi$
$480$	0	0
$481$	2.73863	0.124871
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−16.4924	−0.748882
$486$	0	0
$487$	35.8617	1.62505	0.812525	−	0.582926i	$-0.198092\pi$
$487$	35.8617	1.62505	0.812525	+	0.582926i	$0.198092\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−2.00000	−0.0902587	−0.0451294	−	0.998981i	$-0.514370\pi$
$491$	−2.00000	−0.0902587	−0.0451294	+	0.998981i	$0.514370\pi$
$492$	0	0
$493$	−28.4924	−1.28323
$494$	0	0
$495$	0	0
$496$	0	0
$497$	9.36932	0.420271
$498$	0	0
$499$	28.0000	1.25345	0.626726	−	0.779240i	$-0.284395\pi$
$499$	28.0000	1.25345	0.626726	+	0.779240i	$0.284395\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	12.0000	0.535054	0.267527	−	0.963550i	$-0.413794\pi$
$503$	12.0000	0.535054	0.267527	+	0.963550i	$0.413794\pi$
$504$	0	0
$505$	−20.0000	−0.889988
$506$	0	0
$507$	0	0
$508$	0	0
$509$	8.73863	0.387333	0.193667	−	0.981067i	$-0.437962\pi$
$509$	8.73863	0.387333	0.193667	+	0.981067i	$0.437962\pi$
$510$	0	0
$511$	10.0000	0.442374
$512$	0	0
$513$	0	0
$514$	0	0
$515$	34.2462	1.50907
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	34.4924	1.51114	0.755570	−	0.655068i	$-0.227360\pi$
$521$	34.4924	1.51114	0.755570	+	0.655068i	$0.227360\pi$
$522$	0	0
$523$	40.9848	1.79214	0.896071	−	0.443911i	$-0.146409\pi$
$523$	40.9848	1.79214	0.896071	+	0.443911i	$0.146409\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	3.50758	0.152792
$528$	0	0
$529$	27.7386	1.20603
$530$	0	0
$531$	0	0
$532$	0	0
$533$	25.7538	1.11552
$534$	0	0
$535$	−6.24621	−0.270047
$536$	0	0
$537$	0	0
$538$	0	0
$539$	2.00000	0.0861461
$540$	0	0
$541$	−15.7538	−0.677308	−0.338654	−	0.940911i	$-0.609972\pi$
$541$	−15.7538	−0.677308	−0.338654	+	0.940911i	$0.609972\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	1.75379	0.0751241
$546$	0	0
$547$	−20.9848	−0.897247	−0.448624	−	0.893721i	$-0.648086\pi$
$547$	−20.9848	−0.897247	−0.448624	+	0.893721i	$0.648086\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−9.12311	−0.388657
$552$	0	0
$553$	−13.1231	−0.558051
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−27.3693	−1.15968	−0.579838	−	0.814732i	$-0.696884\pi$
$557$	−27.3693	−1.15968	−0.579838	+	0.814732i	$0.696884\pi$
$558$	0	0
$559$	−12.4924	−0.528373
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−8.49242	−0.357913	−0.178956	−	0.983857i	$-0.557272\pi$
$563$	−8.49242	−0.357913	−0.178956	+	0.983857i	$0.557272\pi$
$564$	0	0
$565$	−18.2462	−0.767624
$566$	0	0
$567$	0	0
$568$	0	0
$569$	41.6155	1.74461	0.872307	−	0.488959i	$-0.162623\pi$
$569$	41.6155	1.74461	0.872307	+	0.488959i	$0.162623\pi$
$570$	0	0
$571$	20.0000	0.836974	0.418487	−	0.908223i	$-0.362561\pi$
$571$	20.0000	0.836974	0.418487	+	0.908223i	$0.362561\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	7.12311	0.297054
$576$	0	0
$577$	−16.2462	−0.676339	−0.338169	−	0.941085i	$-0.609808\pi$
$577$	−16.2462	−0.676339	−0.338169	+	0.941085i	$0.609808\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	9.12311	0.378490
$582$	0	0
$583$	10.2462	0.424355
$584$	0	0
$585$	0	0
$586$	0	0
$587$	7.36932	0.304164	0.152082	−	0.988368i	$-0.451402\pi$
$587$	7.36932	0.304164	0.152082	+	0.988368i	$0.451402\pi$
$588$	0	0
$589$	1.12311	0.0462768
$590$	0	0
$591$	0	0
$592$	0	0
$593$	29.8617	1.22627	0.613137	−	0.789976i	$-0.289907\pi$
$593$	29.8617	1.22627	0.613137	+	0.789976i	$0.289907\pi$
$594$	0	0
$595$	−6.24621	−0.256070
$596$	0	0
$597$	0	0
$598$	0	0
$599$	14.6307	0.597794	0.298897	−	0.954285i	$-0.403381\pi$
$599$	14.6307	0.597794	0.298897	+	0.954285i	$0.403381\pi$
$600$	0	0
$601$	14.4924	0.591158	0.295579	−	0.955318i	$-0.404487\pi$
$601$	14.4924	0.591158	0.295579	+	0.955318i	$0.404487\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	14.0000	0.569181
$606$	0	0
$607$	6.38447	0.259138	0.129569	−	0.991570i	$-0.458641\pi$
$607$	6.38447	0.259138	0.129569	+	0.991570i	$0.458641\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	−24.7386	−0.999184	−0.499592	−	0.866261i	$-0.666517\pi$
$613$	−24.7386	−0.999184	−0.499592	+	0.866261i	$0.666517\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	32.4924	1.30810	0.654048	−	0.756453i	$-0.273069\pi$
$617$	32.4924	1.30810	0.654048	+	0.756453i	$0.273069\pi$
$618$	0	0
$619$	−5.75379	−0.231264	−0.115632	−	0.993292i	$-0.536889\pi$
$619$	−5.75379	−0.231264	−0.115632	+	0.993292i	$0.536889\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0.246211	0.00986425
$624$	0	0
$625$	−19.0000	−0.760000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	2.73863	0.109196
$630$	0	0
$631$	−5.75379	−0.229055	−0.114527	−	0.993420i	$-0.536535\pi$
$631$	−5.75379	−0.229055	−0.114527	+	0.993420i	$0.536535\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−42.2462	−1.67649
$636$	0	0
$637$	−3.12311	−0.123742
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−29.1231	−1.15029	−0.575147	−	0.818050i	$-0.695055\pi$
$641$	−29.1231	−1.15029	−0.575147	+	0.818050i	$0.695055\pi$
$642$	0	0
$643$	−12.0000	−0.473234	−0.236617	−	0.971603i	$-0.576039\pi$
$643$	−12.0000	−0.473234	−0.236617	+	0.971603i	$0.576039\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	18.2462	0.717333	0.358666	−	0.933466i	$-0.383232\pi$
$647$	18.2462	0.717333	0.358666	+	0.933466i	$0.383232\pi$
$648$	0	0
$649$	12.4924	0.490370
$650$	0	0
$651$	0	0
$652$	0	0
$653$	11.8617	0.464186	0.232093	−	0.972694i	$-0.425443\pi$
$653$	11.8617	0.464186	0.232093	+	0.972694i	$0.425443\pi$
$654$	0	0
$655$	−43.2311	−1.68918
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−13.8617	−0.539977	−0.269988	−	0.962864i	$-0.587020\pi$
$659$	−13.8617	−0.539977	−0.269988	+	0.962864i	$0.587020\pi$
$660$	0	0
$661$	−35.6155	−1.38528	−0.692642	−	0.721282i	$-0.743553\pi$
$661$	−35.6155	−1.38528	−0.692642	+	0.721282i	$0.743553\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−2.00000	−0.0775567
$666$	0	0
$667$	−64.9848	−2.51622
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−4.00000	−0.154418
$672$	0	0
$673$	−46.4924	−1.79215	−0.896076	−	0.443901i	$-0.853594\pi$
$673$	−46.4924	−1.79215	−0.896076	+	0.443901i	$0.853594\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−20.2462	−0.778125	−0.389063	−	0.921211i	$-0.627201\pi$
$677$	−20.2462	−0.778125	−0.389063	+	0.921211i	$0.627201\pi$
$678$	0	0
$679$	−8.24621	−0.316461
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−0.876894	−0.0335534	−0.0167767	−	0.999859i	$-0.505340\pi$
$683$	−0.876894	−0.0335534	−0.0167767	+	0.999859i	$0.505340\pi$
$684$	0	0
$685$	3.50758	0.134018
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−16.0000	−0.609551
$690$	0	0
$691$	−21.7538	−0.827553	−0.413777	−	0.910378i	$-0.635791\pi$
$691$	−21.7538	−0.827553	−0.413777	+	0.910378i	$0.635791\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	4.49242	0.170407
$696$	0	0
$697$	25.7538	0.975494
$698$	0	0
$699$	0	0
$700$	0	0
$701$	33.6155	1.26964	0.634820	−	0.772660i	$-0.281074\pi$
$701$	33.6155	1.26964	0.634820	+	0.772660i	$0.281074\pi$
$702$	0	0
$703$	0.876894	0.0330727
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−10.0000	−0.376089
$708$	0	0
$709$	45.2311	1.69869	0.849344	−	0.527840i	$-0.176998\pi$
$709$	45.2311	1.69869	0.849344	+	0.527840i	$0.176998\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	8.00000	0.299602
$714$	0	0
$715$	12.4924	0.467190
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−6.24621	−0.232944	−0.116472	−	0.993194i	$-0.537159\pi$
$719$	−6.24621	−0.232944	−0.116472	+	0.993194i	$0.537159\pi$
$720$	0	0
$721$	17.1231	0.637698
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−9.12311	−0.338824
$726$	0	0
$727$	−10.7386	−0.398274	−0.199137	−	0.979972i	$-0.563814\pi$
$727$	−10.7386	−0.398274	−0.199137	+	0.979972i	$0.563814\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−12.4924	−0.462049
$732$	0	0
$733$	6.49242	0.239803	0.119902	−	0.992786i	$-0.461742\pi$
$733$	6.49242	0.239803	0.119902	+	0.992786i	$0.461742\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−28.4924	−1.04953
$738$	0	0
$739$	32.4924	1.19525	0.597627	−	0.801775i	$-0.296111\pi$
$739$	32.4924	1.19525	0.597627	+	0.801775i	$0.296111\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	22.6307	0.830239	0.415120	−	0.909767i	$-0.363740\pi$
$743$	22.6307	0.830239	0.415120	+	0.909767i	$0.363740\pi$
$744$	0	0
$745$	−34.2462	−1.25468
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−3.12311	−0.114116
$750$	0	0
$751$	−7.36932	−0.268910	−0.134455	−	0.990920i	$-0.542928\pi$
$751$	−7.36932	−0.268910	−0.134455	+	0.990920i	$0.542928\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−26.2462	−0.955197
$756$	0	0
$757$	−26.4924	−0.962883	−0.481442	−	0.876478i	$-0.659887\pi$
$757$	−26.4924	−0.962883	−0.481442	+	0.876478i	$0.659887\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	27.1231	0.983212	0.491606	−	0.870818i	$-0.336410\pi$
$761$	27.1231	0.983212	0.491606	+	0.870818i	$0.336410\pi$
$762$	0	0
$763$	0.876894	0.0317457
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−19.5076	−0.704378
$768$	0	0
$769$	−14.0000	−0.504853	−0.252426	−	0.967616i	$-0.581229\pi$
$769$	−14.0000	−0.504853	−0.252426	+	0.967616i	$0.581229\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	42.4924	1.52835	0.764173	−	0.645011i	$-0.223147\pi$
$773$	42.4924	1.52835	0.764173	+	0.645011i	$0.223147\pi$
$774$	0	0
$775$	1.12311	0.0403431
$776$	0	0
$777$	0	0
$778$	0	0
$779$	8.24621	0.295451
$780$	0	0
$781$	−18.7386	−0.670521
$782$	0	0
$783$	0	0
$784$	0	0
$785$	15.5076	0.553489
$786$	0	0
$787$	−1.26137	−0.0449629	−0.0224814	−	0.999747i	$-0.507157\pi$
$787$	−1.26137	−0.0449629	−0.0224814	+	0.999747i	$0.507157\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−9.12311	−0.324380
$792$	0	0
$793$	6.24621	0.221809
$794$	0	0
$795$	0	0
$796$	0	0
$797$	22.9848	0.814165	0.407082	−	0.913391i	$-0.366546\pi$
$797$	22.9848	0.814165	0.407082	+	0.913391i	$0.366546\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−20.0000	−0.705785
$804$	0	0
$805$	−14.2462	−0.502113
$806$	0	0
$807$	0	0
$808$	0	0
$809$	14.7386	0.518183	0.259091	−	0.965853i	$-0.416577\pi$
$809$	14.7386	0.518183	0.259091	+	0.965853i	$0.416577\pi$
$810$	0	0
$811$	−46.7386	−1.64122	−0.820608	−	0.571492i	$-0.806365\pi$
$811$	−46.7386	−1.64122	−0.820608	+	0.571492i	$0.806365\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	48.9848	1.71586
$816$	0	0
$817$	−4.00000	−0.139942
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−50.1080	−1.74878	−0.874390	−	0.485224i	$-0.838738\pi$
$821$	−50.1080	−1.74878	−0.874390	+	0.485224i	$0.838738\pi$
$822$	0	0
$823$	−6.73863	−0.234894	−0.117447	−	0.993079i	$-0.537471\pi$
$823$	−6.73863	−0.234894	−0.117447	+	0.993079i	$0.537471\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	49.8617	1.73386	0.866931	−	0.498428i	$-0.166089\pi$
$827$	49.8617	1.73386	0.866931	+	0.498428i	$0.166089\pi$
$828$	0	0
$829$	3.61553	0.125572	0.0627862	−	0.998027i	$-0.480001\pi$
$829$	3.61553	0.125572	0.0627862	+	0.998027i	$0.480001\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−3.12311	−0.108209
$834$	0	0
$835$	−3.50758	−0.121385
$836$	0	0
$837$	0	0
$838$	0	0
$839$	26.7386	0.923120	0.461560	−	0.887109i	$-0.347290\pi$
$839$	26.7386	0.923120	0.461560	+	0.887109i	$0.347290\pi$
$840$	0	0
$841$	54.2311	1.87004
$842$	0	0
$843$	0	0
$844$	0	0
$845$	6.49242	0.223346
$846$	0	0
$847$	7.00000	0.240523
$848$	0	0
$849$	0	0
$850$	0	0
$851$	6.24621	0.214117
$852$	0	0
$853$	6.00000	0.205436	0.102718	−	0.994711i	$-0.467246\pi$
$853$	6.00000	0.205436	0.102718	+	0.994711i	$0.467246\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−12.7386	−0.435143	−0.217572	−	0.976044i	$-0.569814\pi$
$857$	−12.7386	−0.435143	−0.217572	+	0.976044i	$0.569814\pi$
$858$	0	0
$859$	−4.00000	−0.136478	−0.0682391	−	0.997669i	$-0.521738\pi$
$859$	−4.00000	−0.136478	−0.0682391	+	0.997669i	$0.521738\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−31.6155	−1.07621	−0.538103	−	0.842879i	$-0.680859\pi$
$863$	−31.6155	−1.07621	−0.538103	+	0.842879i	$0.680859\pi$
$864$	0	0
$865$	8.49242	0.288751
$866$	0	0
$867$	0	0
$868$	0	0
$869$	26.2462	0.890342
$870$	0	0
$871$	44.4924	1.50757
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−12.0000	−0.405674
$876$	0	0
$877$	−19.6155	−0.662369	−0.331185	−	0.943566i	$-0.607448\pi$
$877$	−19.6155	−0.662369	−0.331185	+	0.943566i	$0.607448\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−24.8769	−0.838124	−0.419062	−	0.907958i	$-0.637641\pi$
$881$	−24.8769	−0.838124	−0.419062	+	0.907958i	$0.637641\pi$
$882$	0	0
$883$	52.9848	1.78308	0.891541	−	0.452940i	$-0.149625\pi$
$883$	52.9848	1.78308	0.891541	+	0.452940i	$0.149625\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−9.75379	−0.327500	−0.163750	−	0.986502i	$-0.552359\pi$
$887$	−9.75379	−0.327500	−0.163750	+	0.986502i	$0.552359\pi$
$888$	0	0
$889$	−21.1231	−0.708446
$890$	0	0
$891$	0	0
$892$	0	0
$893$	0	0
$894$	0	0
$895$	26.7386	0.893774
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−10.2462	−0.341730
$900$	0	0
$901$	−16.0000	−0.533037
$902$	0	0
$903$	0	0
$904$	0	0
$905$	10.7386	0.356964
$906$	0	0
$907$	−6.24621	−0.207402	−0.103701	−	0.994609i	$-0.533069\pi$
$907$	−6.24621	−0.207402	−0.103701	+	0.994609i	$0.533069\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	2.13826	0.0708437	0.0354219	−	0.999372i	$-0.488723\pi$
$911$	2.13826	0.0708437	0.0354219	+	0.999372i	$0.488723\pi$
$912$	0	0
$913$	−18.2462	−0.603861
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−21.6155	−0.713808
$918$	0	0
$919$	−5.75379	−0.189800	−0.0949000	−	0.995487i	$-0.530253\pi$
$919$	−5.75379	−0.189800	−0.0949000	+	0.995487i	$0.530253\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	29.2614	0.963150
$924$	0	0
$925$	0.876894	0.0288321
$926$	0	0
$927$	0	0
$928$	0	0
$929$	4.38447	0.143850	0.0719249	−	0.997410i	$-0.477086\pi$
$929$	4.38447	0.143850	0.0719249	+	0.997410i	$0.477086\pi$
$930$	0	0
$931$	−1.00000	−0.0327737
$932$	0	0
$933$	0	0
$934$	0	0
$935$	12.4924	0.408546
$936$	0	0
$937$	−60.7386	−1.98424	−0.992122	−	0.125273i	$-0.960019\pi$
$937$	−60.7386	−1.98424	−0.992122	+	0.125273i	$0.960019\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−49.2311	−1.60489	−0.802443	−	0.596728i	$-0.796467\pi$
$941$	−49.2311	−1.60489	−0.802443	+	0.596728i	$0.796467\pi$
$942$	0	0
$943$	58.7386	1.91279
$944$	0	0
$945$	0	0
$946$	0	0
$947$	32.7386	1.06386	0.531931	−	0.846787i	$-0.321466\pi$
$947$	32.7386	1.06386	0.531931	+	0.846787i	$0.321466\pi$
$948$	0	0
$949$	31.2311	1.01380
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−42.1080	−1.36401	−0.682005	−	0.731347i	$-0.738892\pi$
$953$	−42.1080	−1.36401	−0.682005	+	0.731347i	$0.738892\pi$
$954$	0	0
$955$	−2.73863	−0.0886201
$956$	0	0
$957$	0	0
$958$	0	0
$959$	1.75379	0.0566328
$960$	0	0
$961$	−29.7386	−0.959311
$962$	0	0
$963$	0	0
$964$	0	0
$965$	4.00000	0.128765
$966$	0	0
$967$	24.0000	0.771788	0.385894	−	0.922543i	$-0.373893\pi$
$967$	24.0000	0.771788	0.385894	+	0.922543i	$0.373893\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−20.4924	−0.657633	−0.328817	−	0.944394i	$-0.606650\pi$
$971$	−20.4924	−0.657633	−0.328817	+	0.944394i	$0.606650\pi$
$972$	0	0
$973$	2.24621	0.0720102
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−17.6155	−0.563571	−0.281785	−	0.959477i	$-0.590927\pi$
$977$	−17.6155	−0.563571	−0.281785	+	0.959477i	$0.590927\pi$
$978$	0	0
$979$	−0.492423	−0.0157379
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−8.00000	−0.255160	−0.127580	−	0.991828i	$-0.540721\pi$
$983$	−8.00000	−0.255160	−0.127580	+	0.991828i	$0.540721\pi$
$984$	0	0
$985$	−54.7386	−1.74412
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−28.4924	−0.906006
$990$	0	0
$991$	−15.3693	−0.488222	−0.244111	−	0.969747i	$-0.578496\pi$
$991$	−15.3693	−0.488222	−0.244111	+	0.969747i	$0.578496\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	11.5076	0.364815
$996$	0	0
$997$	−28.2462	−0.894566	−0.447283	−	0.894392i	$-0.647608\pi$
$997$	−28.2462	−0.894566	−0.447283	+	0.894392i	$0.647608\pi$
$998$	0	0
$999$	0	0

Twists

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

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

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-2.00000 q^{5} -1.00000 q^{7} +O(q^{10})\)	\(q-2.00000 q^{5} -1.00000 q^{7} +2.00000 q^{11} -3.12311 q^{13} -3.12311 q^{17} -1.00000 q^{19} -7.12311 q^{23} -1.00000 q^{25} +9.12311 q^{29} -1.12311 q^{31} +2.00000 q^{35} -0.876894 q^{37} -8.24621 q^{41} +4.00000 q^{43} +1.00000 q^{49} +5.12311 q^{53} -4.00000 q^{55} +6.24621 q^{59} -2.00000 q^{61} +6.24621 q^{65} -14.2462 q^{67} -9.36932 q^{71} -10.0000 q^{73} -2.00000 q^{77} +13.1231 q^{79} -9.12311 q^{83} +6.24621 q^{85} -0.246211 q^{89} +3.12311 q^{91} +2.00000 q^{95} +8.24621 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{5} - 2 q^{7}+O(q^{10}) \) 2 * q - 4 * q^5 - 2 * q^7	\( 2 q - 4 q^{5} - 2 q^{7} + 4 q^{11} + 2 q^{13} + 2 q^{17} - 2 q^{19} - 6 q^{23} - 2 q^{25} + 10 q^{29} + 6 q^{31} + 4 q^{35} - 10 q^{37} + 8 q^{43} + 2 q^{49} + 2 q^{53} - 8 q^{55} - 4 q^{59} - 4 q^{61} - 4 q^{65} - 12 q^{67} + 6 q^{71} - 20 q^{73} - 4 q^{77} + 18 q^{79} - 10 q^{83} - 4 q^{85} + 16 q^{89} - 2 q^{91} + 4 q^{95}+O(q^{100}) \) 2 * q - 4 * q^5 - 2 * q^7 + 4 * q^11 + 2 * q^13 + 2 * q^17 - 2 * q^19 - 6 * q^23 - 2 * q^25 + 10 * q^29 + 6 * q^31 + 4 * q^35 - 10 * q^37 + 8 * q^43 + 2 * q^49 + 2 * q^53 - 8 * q^55 - 4 * q^59 - 4 * q^61 - 4 * q^65 - 12 * q^67 + 6 * q^71 - 20 * q^73 - 4 * q^77 + 18 * q^79 - 10 * q^83 - 4 * q^85 + 16 * q^89 - 2 * q^91 + 4 * q^95

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