LMFDB - Embedded newform 9360.2.a.cw.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9360,2,Mod(1,9360)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9360, 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("9360.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9360 = 2^{4} \cdot 3^{2} \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9360.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:	$74.7399762919$
Analytic rank:	$1$
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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 585)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.56155$ of defining polynomial
Character	$\chi$	$=$	9360.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{5} +4.56155 q^{7} +O(q^{10})$	$q+1.00000 q^{5} +4.56155 q^{7} -2.56155 q^{11} -1.00000 q^{13} -2.56155 q^{17} -3.12311 q^{19} -6.56155 q^{23} +1.00000 q^{25} +1.12311 q^{29} -6.00000 q^{31} +4.56155 q^{35} +1.68466 q^{37} -0.561553 q^{41} +5.12311 q^{43} -2.87689 q^{47} +13.8078 q^{49} +7.68466 q^{53} -2.56155 q^{55} -12.0000 q^{59} +5.68466 q^{61} -1.00000 q^{65} -13.3693 q^{67} -14.5616 q^{71} -6.00000 q^{73} -11.6847 q^{77} +7.68466 q^{79} -16.4924 q^{83} -2.56155 q^{85} -1.68466 q^{89} -4.56155 q^{91} -3.12311 q^{95} -11.9309 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{5} + 5 q^{7}+O(q^{10}) $ 2 * q + 2 * q^5 + 5 * q^7	$ 2 q + 2 q^{5} + 5 q^{7} - q^{11} - 2 q^{13} - q^{17} + 2 q^{19} - 9 q^{23} + 2 q^{25} - 6 q^{29} - 12 q^{31} + 5 q^{35} - 9 q^{37} + 3 q^{41} + 2 q^{43} - 14 q^{47} + 7 q^{49} + 3 q^{53} - q^{55} - 24 q^{59} - q^{61} - 2 q^{65} - 2 q^{67} - 25 q^{71} - 12 q^{73} - 11 q^{77} + 3 q^{79} - q^{85} + 9 q^{89} - 5 q^{91} + 2 q^{95} + 5 q^{97}+O(q^{100}) $ 2 * q + 2 * q^5 + 5 * q^7 - q^11 - 2 * q^13 - q^17 + 2 * q^19 - 9 * q^23 + 2 * q^25 - 6 * q^29 - 12 * q^31 + 5 * q^35 - 9 * q^37 + 3 * q^41 + 2 * q^43 - 14 * q^47 + 7 * q^49 + 3 * q^53 - q^55 - 24 * q^59 - q^61 - 2 * q^65 - 2 * q^67 - 25 * q^71 - 12 * q^73 - 11 * q^77 + 3 * q^79 - q^85 + 9 * q^89 - 5 * q^91 + 2 * q^95 + 5 * 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
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	4.56155	1.72410	0.862052	−	0.506819i	$-0.169179\pi$
$7$	4.56155	1.72410	0.862052	+	0.506819i	$0.169179\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−2.56155	−0.772337	−0.386169	−	0.922428i	$-0.626202\pi$
$11$	−2.56155	−0.772337	−0.386169	+	0.922428i	$0.626202\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−2.56155	−0.621268	−0.310634	−	0.950530i	$-0.600541\pi$
$17$	−2.56155	−0.621268	−0.310634	+	0.950530i	$0.600541\pi$
$18$	0	0
$19$	−3.12311	−0.716490	−0.358245	−	0.933628i	$-0.616625\pi$
$19$	−3.12311	−0.716490	−0.358245	+	0.933628i	$0.616625\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−6.56155	−1.36818	−0.684089	−	0.729398i	$-0.739800\pi$
$23$	−6.56155	−1.36818	−0.684089	+	0.729398i	$0.739800\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	1.12311	0.208555	0.104278	−	0.994548i	$-0.466747\pi$
$29$	1.12311	0.208555	0.104278	+	0.994548i	$0.466747\pi$
$30$	0	0
$31$	−6.00000	−1.07763	−0.538816	−	0.842424i	$-0.681128\pi$
$31$	−6.00000	−1.07763	−0.538816	+	0.842424i	$0.681128\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	4.56155	0.771043
$36$	0	0
$37$	1.68466	0.276956	0.138478	−	0.990366i	$-0.455779\pi$
$37$	1.68466	0.276956	0.138478	+	0.990366i	$0.455779\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−0.561553	−0.0876998	−0.0438499	−	0.999038i	$-0.513962\pi$
$41$	−0.561553	−0.0876998	−0.0438499	+	0.999038i	$0.513962\pi$
$42$	0	0
$43$	5.12311	0.781266	0.390633	−	0.920546i	$-0.372256\pi$
$43$	5.12311	0.781266	0.390633	+	0.920546i	$0.372256\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−2.87689	−0.419638	−0.209819	−	0.977740i	$-0.567288\pi$
$47$	−2.87689	−0.419638	−0.209819	+	0.977740i	$0.567288\pi$
$48$	0	0
$49$	13.8078	1.97254
$50$	0	0
$51$	0	0
$52$	0	0
$53$	7.68466	1.05557	0.527785	−	0.849378i	$-0.323023\pi$
$53$	7.68466	1.05557	0.527785	+	0.849378i	$0.323023\pi$
$54$	0	0
$55$	−2.56155	−0.345400
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−12.0000	−1.56227	−0.781133	−	0.624364i	$-0.785358\pi$
$59$	−12.0000	−1.56227	−0.781133	+	0.624364i	$0.785358\pi$
$60$	0	0
$61$	5.68466	0.727846	0.363923	−	0.931429i	$-0.381437\pi$
$61$	5.68466	0.727846	0.363923	+	0.931429i	$0.381437\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−1.00000	−0.124035
$66$	0	0
$67$	−13.3693	−1.63332	−0.816661	−	0.577118i	$-0.804177\pi$
$67$	−13.3693	−1.63332	−0.816661	+	0.577118i	$0.804177\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−14.5616	−1.72814	−0.864069	−	0.503373i	$-0.832092\pi$
$71$	−14.5616	−1.72814	−0.864069	+	0.503373i	$0.832092\pi$
$72$	0	0
$73$	−6.00000	−0.702247	−0.351123	−	0.936329i	$-0.614200\pi$
$73$	−6.00000	−0.702247	−0.351123	+	0.936329i	$0.614200\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−11.6847	−1.33159
$78$	0	0
$79$	7.68466	0.864592	0.432296	−	0.901732i	$-0.357704\pi$
$79$	7.68466	0.864592	0.432296	+	0.901732i	$0.357704\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−16.4924	−1.81028	−0.905139	−	0.425115i	$-0.860234\pi$
$83$	−16.4924	−1.81028	−0.905139	+	0.425115i	$0.860234\pi$
$84$	0	0
$85$	−2.56155	−0.277839
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−1.68466	−0.178573	−0.0892867	−	0.996006i	$-0.528459\pi$
$89$	−1.68466	−0.178573	−0.0892867	+	0.996006i	$0.528459\pi$
$90$	0	0
$91$	−4.56155	−0.478181
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−3.12311	−0.320424
$96$	0	0
$97$	−11.9309	−1.21140	−0.605698	−	0.795695i	$-0.707106\pi$
$97$	−11.9309	−1.21140	−0.605698	+	0.795695i	$0.707106\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−6.24621	−0.621521	−0.310761	−	0.950488i	$-0.600584\pi$
$101$	−6.24621	−0.621521	−0.310761	+	0.950488i	$0.600584\pi$
$102$	0	0
$103$	6.87689	0.677601	0.338800	−	0.940858i	$-0.389979\pi$
$103$	6.87689	0.677601	0.338800	+	0.940858i	$0.389979\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	17.9309	1.73344	0.866721	−	0.498793i	$-0.166223\pi$
$107$	17.9309	1.73344	0.866721	+	0.498793i	$0.166223\pi$
$108$	0	0
$109$	−2.00000	−0.191565	−0.0957826	−	0.995402i	$-0.530535\pi$
$109$	−2.00000	−0.191565	−0.0957826	+	0.995402i	$0.530535\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−13.1231	−1.23452	−0.617259	−	0.786760i	$-0.711757\pi$
$113$	−13.1231	−1.23452	−0.617259	+	0.786760i	$0.711757\pi$
$114$	0	0
$115$	−6.56155	−0.611868
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−11.6847	−1.07113
$120$	0	0
$121$	−4.43845	−0.403495
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−18.2462	−1.61909	−0.809545	−	0.587058i	$-0.800286\pi$
$127$	−18.2462	−1.61909	−0.809545	+	0.587058i	$0.800286\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	0	0
$133$	−14.2462	−1.23530
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−7.12311	−0.608568	−0.304284	−	0.952581i	$-0.598417\pi$
$137$	−7.12311	−0.608568	−0.304284	+	0.952581i	$0.598417\pi$
$138$	0	0
$139$	−15.6847	−1.33036	−0.665178	−	0.746685i	$-0.731644\pi$
$139$	−15.6847	−1.33036	−0.665178	+	0.746685i	$0.731644\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	2.56155	0.214208
$144$	0	0
$145$	1.12311	0.0932688
$146$	0	0
$147$	0	0
$148$	0	0
$149$	2.80776	0.230021	0.115010	−	0.993364i	$-0.463310\pi$
$149$	2.80776	0.230021	0.115010	+	0.993364i	$0.463310\pi$
$150$	0	0
$151$	13.3693	1.08798	0.543990	−	0.839092i	$-0.316913\pi$
$151$	13.3693	1.08798	0.543990	+	0.839092i	$0.316913\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−6.00000	−0.481932
$156$	0	0
$157$	−21.3693	−1.70546	−0.852729	−	0.522354i	$-0.825054\pi$
$157$	−21.3693	−1.70546	−0.852729	+	0.522354i	$0.825054\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−29.9309	−2.35888
$162$	0	0
$163$	21.0540	1.64907	0.824537	−	0.565808i	$-0.191436\pi$
$163$	21.0540	1.64907	0.824537	+	0.565808i	$0.191436\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−13.1231	−1.01550	−0.507748	−	0.861506i	$-0.669522\pi$
$167$	−13.1231	−1.01550	−0.507748	+	0.861506i	$0.669522\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	0	0
$172$	0	0
$173$	21.1231	1.60596	0.802980	−	0.596006i	$-0.203247\pi$
$173$	21.1231	1.60596	0.802980	+	0.596006i	$0.203247\pi$
$174$	0	0
$175$	4.56155	0.344821
$176$	0	0
$177$	0	0
$178$	0	0
$179$	13.1231	0.980867	0.490433	−	0.871479i	$-0.336838\pi$
$179$	13.1231	0.980867	0.490433	+	0.871479i	$0.336838\pi$
$180$	0	0
$181$	25.6847	1.90913	0.954563	−	0.298010i	$-0.0963228\pi$
$181$	25.6847	1.90913	0.954563	+	0.298010i	$0.0963228\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	1.68466	0.123859
$186$	0	0
$187$	6.56155	0.479828
$188$	0	0
$189$	0	0
$190$	0	0
$191$	21.6155	1.56404	0.782022	−	0.623250i	$-0.214188\pi$
$191$	21.6155	1.56404	0.782022	+	0.623250i	$0.214188\pi$
$192$	0	0
$193$	15.4384	1.11128	0.555642	−	0.831422i	$-0.312473\pi$
$193$	15.4384	1.11128	0.555642	+	0.831422i	$0.312473\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	21.3693	1.52250	0.761250	−	0.648458i	$-0.224586\pi$
$197$	21.3693	1.52250	0.761250	+	0.648458i	$0.224586\pi$
$198$	0	0
$199$	8.00000	0.567105	0.283552	−	0.958957i	$-0.408487\pi$
$199$	8.00000	0.567105	0.283552	+	0.958957i	$0.408487\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	5.12311	0.359572
$204$	0	0
$205$	−0.561553	−0.0392205
$206$	0	0
$207$	0	0
$208$	0	0
$209$	8.00000	0.553372
$210$	0	0
$211$	10.2462	0.705378	0.352689	−	0.935741i	$-0.385267\pi$
$211$	10.2462	0.705378	0.352689	+	0.935741i	$0.385267\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	5.12311	0.349393
$216$	0	0
$217$	−27.3693	−1.85795
$218$	0	0
$219$	0	0
$220$	0	0
$221$	2.56155	0.172309
$222$	0	0
$223$	−9.36932	−0.627416	−0.313708	−	0.949520i	$-0.601571\pi$
$223$	−9.36932	−0.627416	−0.313708	+	0.949520i	$0.601571\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−22.2462	−1.47653	−0.738266	−	0.674509i	$-0.764355\pi$
$227$	−22.2462	−1.47653	−0.738266	+	0.674509i	$0.764355\pi$
$228$	0	0
$229$	8.87689	0.586602	0.293301	−	0.956020i	$-0.405246\pi$
$229$	8.87689	0.586602	0.293301	+	0.956020i	$0.405246\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−7.19224	−0.471179	−0.235590	−	0.971853i	$-0.575702\pi$
$233$	−7.19224	−0.471179	−0.235590	+	0.971853i	$0.575702\pi$
$234$	0	0
$235$	−2.87689	−0.187668
$236$	0	0
$237$	0	0
$238$	0	0
$239$	5.93087	0.383636	0.191818	−	0.981431i	$-0.438562\pi$
$239$	5.93087	0.383636	0.191818	+	0.981431i	$0.438562\pi$
$240$	0	0
$241$	−24.7386	−1.59356	−0.796778	−	0.604272i	$-0.793464\pi$
$241$	−24.7386	−1.59356	−0.796778	+	0.604272i	$0.793464\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	13.8078	0.882146
$246$	0	0
$247$	3.12311	0.198718
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−9.75379	−0.615654	−0.307827	−	0.951442i	$-0.599602\pi$
$251$	−9.75379	−0.615654	−0.307827	+	0.951442i	$0.599602\pi$
$252$	0	0
$253$	16.8078	1.05670
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−21.1231	−1.31762	−0.658812	−	0.752308i	$-0.728941\pi$
$257$	−21.1231	−1.31762	−0.658812	+	0.752308i	$0.728941\pi$
$258$	0	0
$259$	7.68466	0.477501
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−14.2462	−0.878459	−0.439230	−	0.898375i	$-0.644749\pi$
$263$	−14.2462	−0.878459	−0.439230	+	0.898375i	$0.644749\pi$
$264$	0	0
$265$	7.68466	0.472065
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−9.12311	−0.556246	−0.278123	−	0.960546i	$-0.589712\pi$
$269$	−9.12311	−0.556246	−0.278123	+	0.960546i	$0.589712\pi$
$270$	0	0
$271$	−5.36932	−0.326163	−0.163081	−	0.986613i	$-0.552143\pi$
$271$	−5.36932	−0.326163	−0.163081	+	0.986613i	$0.552143\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−2.56155	−0.154467
$276$	0	0
$277$	4.24621	0.255130	0.127565	−	0.991830i	$-0.459284\pi$
$277$	4.24621	0.255130	0.127565	+	0.991830i	$0.459284\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−12.2462	−0.730548	−0.365274	−	0.930900i	$-0.619025\pi$
$281$	−12.2462	−0.730548	−0.365274	+	0.930900i	$0.619025\pi$
$282$	0	0
$283$	4.00000	0.237775	0.118888	−	0.992908i	$-0.462067\pi$
$283$	4.00000	0.237775	0.118888	+	0.992908i	$0.462067\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−2.56155	−0.151204
$288$	0	0
$289$	−10.4384	−0.614026
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−3.75379	−0.219299	−0.109649	−	0.993970i	$-0.534973\pi$
$293$	−3.75379	−0.219299	−0.109649	+	0.993970i	$0.534973\pi$
$294$	0	0
$295$	−12.0000	−0.698667
$296$	0	0
$297$	0	0
$298$	0	0
$299$	6.56155	0.379464
$300$	0	0
$301$	23.3693	1.34699
$302$	0	0
$303$	0	0
$304$	0	0
$305$	5.68466	0.325503
$306$	0	0
$307$	−3.43845	−0.196243	−0.0981213	−	0.995174i	$-0.531283\pi$
$307$	−3.43845	−0.196243	−0.0981213	+	0.995174i	$0.531283\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−27.3693	−1.55197	−0.775986	−	0.630750i	$-0.782747\pi$
$311$	−27.3693	−1.55197	−0.775986	+	0.630750i	$0.782747\pi$
$312$	0	0
$313$	20.8769	1.18003	0.590016	−	0.807392i	$-0.299121\pi$
$313$	20.8769	1.18003	0.590016	+	0.807392i	$0.299121\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	34.4924	1.93729	0.968644	−	0.248454i	$-0.0799224\pi$
$317$	34.4924	1.93729	0.968644	+	0.248454i	$0.0799224\pi$
$318$	0	0
$319$	−2.87689	−0.161075
$320$	0	0
$321$	0	0
$322$	0	0
$323$	8.00000	0.445132
$324$	0	0
$325$	−1.00000	−0.0554700
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−13.1231	−0.723500
$330$	0	0
$331$	−20.8769	−1.14750	−0.573749	−	0.819031i	$-0.694511\pi$
$331$	−20.8769	−1.14750	−0.573749	+	0.819031i	$0.694511\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−13.3693	−0.730444
$336$	0	0
$337$	2.49242	0.135771	0.0678855	−	0.997693i	$-0.478375\pi$
$337$	2.49242	0.135771	0.0678855	+	0.997693i	$0.478375\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	15.3693	0.832295
$342$	0	0
$343$	31.0540	1.67676
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−11.0540	−0.593408	−0.296704	−	0.954969i	$-0.595888\pi$
$347$	−11.0540	−0.593408	−0.296704	+	0.954969i	$0.595888\pi$
$348$	0	0
$349$	1.36932	0.0732979	0.0366489	−	0.999328i	$-0.488332\pi$
$349$	1.36932	0.0732979	0.0366489	+	0.999328i	$0.488332\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	10.4924	0.558455	0.279228	−	0.960225i	$-0.409922\pi$
$353$	10.4924	0.558455	0.279228	+	0.960225i	$0.409922\pi$
$354$	0	0
$355$	−14.5616	−0.772847
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−2.24621	−0.118550	−0.0592752	−	0.998242i	$-0.518879\pi$
$359$	−2.24621	−0.118550	−0.0592752	+	0.998242i	$0.518879\pi$
$360$	0	0
$361$	−9.24621	−0.486643
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−6.00000	−0.314054
$366$	0	0
$367$	18.2462	0.952444	0.476222	−	0.879325i	$-0.342006\pi$
$367$	18.2462	0.952444	0.476222	+	0.879325i	$0.342006\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	35.0540	1.81991
$372$	0	0
$373$	−4.24621	−0.219860	−0.109930	−	0.993939i	$-0.535063\pi$
$373$	−4.24621	−0.219860	−0.109930	+	0.993939i	$0.535063\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−1.12311	−0.0578429
$378$	0	0
$379$	−2.49242	−0.128027	−0.0640136	−	0.997949i	$-0.520390\pi$
$379$	−2.49242	−0.128027	−0.0640136	+	0.997949i	$0.520390\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−36.4924	−1.86468	−0.932338	−	0.361588i	$-0.882235\pi$
$383$	−36.4924	−1.86468	−0.932338	+	0.361588i	$0.882235\pi$
$384$	0	0
$385$	−11.6847	−0.595505
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−19.8617	−1.00703	−0.503515	−	0.863986i	$-0.667960\pi$
$389$	−19.8617	−1.00703	−0.503515	+	0.863986i	$0.667960\pi$
$390$	0	0
$391$	16.8078	0.850005
$392$	0	0
$393$	0	0
$394$	0	0
$395$	7.68466	0.386657
$396$	0	0
$397$	−8.56155	−0.429692	−0.214846	−	0.976648i	$-0.568925\pi$
$397$	−8.56155	−0.429692	−0.214846	+	0.976648i	$0.568925\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	20.2462	1.01105	0.505524	−	0.862813i	$-0.331299\pi$
$401$	20.2462	1.01105	0.505524	+	0.862813i	$0.331299\pi$
$402$	0	0
$403$	6.00000	0.298881
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−4.31534	−0.213904
$408$	0	0
$409$	−15.1231	−0.747789	−0.373895	−	0.927471i	$-0.621978\pi$
$409$	−15.1231	−0.747789	−0.373895	+	0.927471i	$0.621978\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−54.7386	−2.69351
$414$	0	0
$415$	−16.4924	−0.809581
$416$	0	0
$417$	0	0
$418$	0	0
$419$	33.6155	1.64223	0.821113	−	0.570766i	$-0.193354\pi$
$419$	33.6155	1.64223	0.821113	+	0.570766i	$0.193354\pi$
$420$	0	0
$421$	14.6307	0.713056	0.356528	−	0.934285i	$-0.383960\pi$
$421$	14.6307	0.713056	0.356528	+	0.934285i	$0.383960\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−2.56155	−0.124254
$426$	0	0
$427$	25.9309	1.25488
$428$	0	0
$429$	0	0
$430$	0	0
$431$	36.4924	1.75778	0.878889	−	0.477026i	$-0.158285\pi$
$431$	36.4924	1.75778	0.878889	+	0.477026i	$0.158285\pi$
$432$	0	0
$433$	31.6155	1.51935	0.759673	−	0.650306i	$-0.225359\pi$
$433$	31.6155	1.51935	0.759673	+	0.650306i	$0.225359\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	20.4924	0.980286
$438$	0	0
$439$	15.0540	0.718487	0.359244	−	0.933244i	$-0.383035\pi$
$439$	15.0540	0.718487	0.359244	+	0.933244i	$0.383035\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−12.8078	−0.608515	−0.304258	−	0.952590i	$-0.598408\pi$
$443$	−12.8078	−0.608515	−0.304258	+	0.952590i	$0.598408\pi$
$444$	0	0
$445$	−1.68466	−0.0798605
$446$	0	0
$447$	0	0
$448$	0	0
$449$	10.1771	0.480286	0.240143	−	0.970738i	$-0.422806\pi$
$449$	10.1771	0.480286	0.240143	+	0.970738i	$0.422806\pi$
$450$	0	0
$451$	1.43845	0.0677338
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−4.56155	−0.213849
$456$	0	0
$457$	−25.6847	−1.20148	−0.600739	−	0.799445i	$-0.705127\pi$
$457$	−25.6847	−1.20148	−0.600739	+	0.799445i	$0.705127\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−29.0540	−1.35318	−0.676589	−	0.736361i	$-0.736543\pi$
$461$	−29.0540	−1.35318	−0.676589	+	0.736361i	$0.736543\pi$
$462$	0	0
$463$	0.0691303	0.00321276	0.00160638	−	0.999999i	$-0.499489\pi$
$463$	0.0691303	0.00321276	0.00160638	+	0.999999i	$0.499489\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	21.9309	1.01484	0.507420	−	0.861699i	$-0.330599\pi$
$467$	21.9309	1.01484	0.507420	+	0.861699i	$0.330599\pi$
$468$	0	0
$469$	−60.9848	−2.81602
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−13.1231	−0.603401
$474$	0	0
$475$	−3.12311	−0.143298
$476$	0	0
$477$	0	0
$478$	0	0
$479$	11.0540	0.505069	0.252535	−	0.967588i	$-0.418736\pi$
$479$	11.0540	0.505069	0.252535	+	0.967588i	$0.418736\pi$
$480$	0	0
$481$	−1.68466	−0.0768138
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−11.9309	−0.541753
$486$	0	0
$487$	−5.05398	−0.229017	−0.114509	−	0.993422i	$-0.536529\pi$
$487$	−5.05398	−0.229017	−0.114509	+	0.993422i	$0.536529\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−13.6155	−0.614460	−0.307230	−	0.951635i	$-0.599402\pi$
$491$	−13.6155	−0.614460	−0.307230	+	0.951635i	$0.599402\pi$
$492$	0	0
$493$	−2.87689	−0.129569
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−66.4233	−2.97949
$498$	0	0
$499$	30.9848	1.38707	0.693536	−	0.720422i	$-0.256052\pi$
$499$	30.9848	1.38707	0.693536	+	0.720422i	$0.256052\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−30.2462	−1.34861	−0.674306	−	0.738452i	$-0.735557\pi$
$503$	−30.2462	−1.34861	−0.674306	+	0.738452i	$0.735557\pi$
$504$	0	0
$505$	−6.24621	−0.277953
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−19.4384	−0.861594	−0.430797	−	0.902449i	$-0.641768\pi$
$509$	−19.4384	−0.861594	−0.430797	+	0.902449i	$0.641768\pi$
$510$	0	0
$511$	−27.3693	−1.21075
$512$	0	0
$513$	0	0
$514$	0	0
$515$	6.87689	0.303032
$516$	0	0
$517$	7.36932	0.324102
$518$	0	0
$519$	0	0
$520$	0	0
$521$	31.8617	1.39589	0.697944	−	0.716152i	$-0.254098\pi$
$521$	31.8617	1.39589	0.697944	+	0.716152i	$0.254098\pi$
$522$	0	0
$523$	−18.7386	−0.819383	−0.409692	−	0.912224i	$-0.634364\pi$
$523$	−18.7386	−0.819383	−0.409692	+	0.912224i	$0.634364\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	15.3693	0.669498
$528$	0	0
$529$	20.0540	0.871912
$530$	0	0
$531$	0	0
$532$	0	0
$533$	0.561553	0.0243236
$534$	0	0
$535$	17.9309	0.775219
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−35.3693	−1.52346
$540$	0	0
$541$	27.1231	1.16611	0.583057	−	0.812431i	$-0.301857\pi$
$541$	27.1231	1.16611	0.583057	+	0.812431i	$0.301857\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−2.00000	−0.0856706
$546$	0	0
$547$	−19.3693	−0.828172	−0.414086	−	0.910238i	$-0.635899\pi$
$547$	−19.3693	−0.828172	−0.414086	+	0.910238i	$0.635899\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−3.50758	−0.149428
$552$	0	0
$553$	35.0540	1.49065
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−26.4924	−1.12252	−0.561260	−	0.827640i	$-0.689683\pi$
$557$	−26.4924	−1.12252	−0.561260	+	0.827640i	$0.689683\pi$
$558$	0	0
$559$	−5.12311	−0.216684
$560$	0	0
$561$	0	0
$562$	0	0
$563$	31.6847	1.33535	0.667675	−	0.744453i	$-0.267290\pi$
$563$	31.6847	1.33535	0.667675	+	0.744453i	$0.267290\pi$
$564$	0	0
$565$	−13.1231	−0.552093
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−41.1231	−1.72397	−0.861985	−	0.506934i	$-0.830779\pi$
$569$	−41.1231	−1.72397	−0.861985	+	0.506934i	$0.830779\pi$
$570$	0	0
$571$	−15.0540	−0.629989	−0.314995	−	0.949093i	$-0.602003\pi$
$571$	−15.0540	−0.629989	−0.314995	+	0.949093i	$0.602003\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−6.56155	−0.273636
$576$	0	0
$577$	28.5616	1.18903	0.594517	−	0.804083i	$-0.297343\pi$
$577$	28.5616	1.18903	0.594517	+	0.804083i	$0.297343\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−75.2311	−3.12111
$582$	0	0
$583$	−19.6847	−0.815255
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−0.492423	−0.0203245	−0.0101622	−	0.999948i	$-0.503235\pi$
$587$	−0.492423	−0.0203245	−0.0101622	+	0.999948i	$0.503235\pi$
$588$	0	0
$589$	18.7386	0.772112
$590$	0	0
$591$	0	0
$592$	0	0
$593$	7.75379	0.318410	0.159205	−	0.987246i	$-0.449107\pi$
$593$	7.75379	0.318410	0.159205	+	0.987246i	$0.449107\pi$
$594$	0	0
$595$	−11.6847	−0.479024
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−15.3693	−0.627973	−0.313987	−	0.949427i	$-0.601665\pi$
$599$	−15.3693	−0.627973	−0.313987	+	0.949427i	$0.601665\pi$
$600$	0	0
$601$	41.5464	1.69471	0.847356	−	0.531025i	$-0.178193\pi$
$601$	41.5464	1.69471	0.847356	+	0.531025i	$0.178193\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−4.43845	−0.180449
$606$	0	0
$607$	24.0000	0.974130	0.487065	−	0.873366i	$-0.338067\pi$
$607$	24.0000	0.974130	0.487065	+	0.873366i	$0.338067\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	2.87689	0.116387
$612$	0	0
$613$	37.5464	1.51648	0.758242	−	0.651973i	$-0.226058\pi$
$613$	37.5464	1.51648	0.758242	+	0.651973i	$0.226058\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−48.7386	−1.96214	−0.981072	−	0.193645i	$-0.937969\pi$
$617$	−48.7386	−1.96214	−0.981072	+	0.193645i	$0.937969\pi$
$618$	0	0
$619$	33.3693	1.34123	0.670613	−	0.741807i	$-0.266031\pi$
$619$	33.3693	1.34123	0.670613	+	0.741807i	$0.266031\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−7.68466	−0.307879
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−4.31534	−0.172064
$630$	0	0
$631$	−16.7386	−0.666354	−0.333177	−	0.942864i	$-0.608121\pi$
$631$	−16.7386	−0.666354	−0.333177	+	0.942864i	$0.608121\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−18.2462	−0.724079
$636$	0	0
$637$	−13.8078	−0.547084
$638$	0	0
$639$	0	0
$640$	0	0
$641$	32.9848	1.30282	0.651412	−	0.758725i	$-0.274177\pi$
$641$	32.9848	1.30282	0.651412	+	0.758725i	$0.274177\pi$
$642$	0	0
$643$	−1.68466	−0.0664364	−0.0332182	−	0.999448i	$-0.510576\pi$
$643$	−1.68466	−0.0664364	−0.0332182	+	0.999448i	$0.510576\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−11.1922	−0.440012	−0.220006	−	0.975498i	$-0.570608\pi$
$647$	−11.1922	−0.440012	−0.220006	+	0.975498i	$0.570608\pi$
$648$	0	0
$649$	30.7386	1.20660
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−8.63068	−0.337745	−0.168872	−	0.985638i	$-0.554013\pi$
$653$	−8.63068	−0.337745	−0.168872	+	0.985638i	$0.554013\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−9.12311	−0.355386	−0.177693	−	0.984086i	$-0.556863\pi$
$659$	−9.12311	−0.355386	−0.177693	+	0.984086i	$0.556863\pi$
$660$	0	0
$661$	−26.4924	−1.03044	−0.515218	−	0.857059i	$-0.672289\pi$
$661$	−26.4924	−1.03044	−0.515218	+	0.857059i	$0.672289\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−14.2462	−0.552444
$666$	0	0
$667$	−7.36932	−0.285341
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−14.5616	−0.562143
$672$	0	0
$673$	−32.7386	−1.26198	−0.630991	−	0.775790i	$-0.717351\pi$
$673$	−32.7386	−1.26198	−0.630991	+	0.775790i	$0.717351\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−18.5616	−0.713378	−0.356689	−	0.934223i	$-0.616094\pi$
$677$	−18.5616	−0.713378	−0.356689	+	0.934223i	$0.616094\pi$
$678$	0	0
$679$	−54.4233	−2.08857
$680$	0	0
$681$	0	0
$682$	0	0
$683$	0.492423	0.0188420	0.00942101	−	0.999956i	$-0.497001\pi$
$683$	0.492423	0.0188420	0.00942101	+	0.999956i	$0.497001\pi$
$684$	0	0
$685$	−7.12311	−0.272160
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−7.68466	−0.292762
$690$	0	0
$691$	−19.6155	−0.746210	−0.373105	−	0.927789i	$-0.621707\pi$
$691$	−19.6155	−0.746210	−0.373105	+	0.927789i	$0.621707\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−15.6847	−0.594953
$696$	0	0
$697$	1.43845	0.0544851
$698$	0	0
$699$	0	0
$700$	0	0
$701$	16.9848	0.641509	0.320754	−	0.947162i	$-0.396064\pi$
$701$	16.9848	0.641509	0.320754	+	0.947162i	$0.396064\pi$
$702$	0	0
$703$	−5.26137	−0.198436
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−28.4924	−1.07157
$708$	0	0
$709$	0.876894	0.0329325	0.0164662	−	0.999864i	$-0.494758\pi$
$709$	0.876894	0.0329325	0.0164662	+	0.999864i	$0.494758\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	39.3693	1.47439
$714$	0	0
$715$	2.56155	0.0957966
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−36.0000	−1.34257	−0.671287	−	0.741198i	$-0.734258\pi$
$719$	−36.0000	−1.34257	−0.671287	+	0.741198i	$0.734258\pi$
$720$	0	0
$721$	31.3693	1.16825
$722$	0	0
$723$	0	0
$724$	0	0
$725$	1.12311	0.0417111
$726$	0	0
$727$	−21.1231	−0.783413	−0.391706	−	0.920090i	$-0.628115\pi$
$727$	−21.1231	−0.783413	−0.391706	+	0.920090i	$0.628115\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−13.1231	−0.485376
$732$	0	0
$733$	−20.5616	−0.759458	−0.379729	−	0.925098i	$-0.623983\pi$
$733$	−20.5616	−0.759458	−0.379729	+	0.925098i	$0.623983\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	34.2462	1.26148
$738$	0	0
$739$	0.738634	0.0271711	0.0135855	−	0.999908i	$-0.495675\pi$
$739$	0.738634	0.0271711	0.0135855	+	0.999908i	$0.495675\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	30.7386	1.12769	0.563846	−	0.825880i	$-0.309321\pi$
$743$	30.7386	1.12769	0.563846	+	0.825880i	$0.309321\pi$
$744$	0	0
$745$	2.80776	0.102869
$746$	0	0
$747$	0	0
$748$	0	0
$749$	81.7926	2.98864
$750$	0	0
$751$	23.0540	0.841252	0.420626	−	0.907234i	$-0.361811\pi$
$751$	23.0540	0.841252	0.420626	+	0.907234i	$0.361811\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	13.3693	0.486559
$756$	0	0
$757$	18.4924	0.672119	0.336059	−	0.941841i	$-0.390906\pi$
$757$	18.4924	0.672119	0.336059	+	0.941841i	$0.390906\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−37.2311	−1.34962	−0.674812	−	0.737989i	$-0.735775\pi$
$761$	−37.2311	−1.34962	−0.674812	+	0.737989i	$0.735775\pi$
$762$	0	0
$763$	−9.12311	−0.330279
$764$	0	0
$765$	0	0
$766$	0	0
$767$	12.0000	0.433295
$768$	0	0
$769$	−30.0000	−1.08183	−0.540914	−	0.841078i	$-0.681921\pi$
$769$	−30.0000	−1.08183	−0.540914	+	0.841078i	$0.681921\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−0.876894	−0.0315397	−0.0157698	−	0.999876i	$-0.505020\pi$
$773$	−0.876894	−0.0315397	−0.0157698	+	0.999876i	$0.505020\pi$
$774$	0	0
$775$	−6.00000	−0.215526
$776$	0	0
$777$	0	0
$778$	0	0
$779$	1.75379	0.0628360
$780$	0	0
$781$	37.3002	1.33471
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−21.3693	−0.762704
$786$	0	0
$787$	13.3693	0.476565	0.238282	−	0.971196i	$-0.423416\pi$
$787$	13.3693	0.476565	0.238282	+	0.971196i	$0.423416\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−59.8617	−2.12844
$792$	0	0
$793$	−5.68466	−0.201868
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−7.68466	−0.272205	−0.136102	−	0.990695i	$-0.543458\pi$
$797$	−7.68466	−0.272205	−0.136102	+	0.990695i	$0.543458\pi$
$798$	0	0
$799$	7.36932	0.260708
$800$	0	0
$801$	0	0
$802$	0	0
$803$	15.3693	0.542371
$804$	0	0
$805$	−29.9309	−1.05492
$806$	0	0
$807$	0	0
$808$	0	0
$809$	40.4924	1.42364	0.711819	−	0.702363i	$-0.247872\pi$
$809$	40.4924	1.42364	0.711819	+	0.702363i	$0.247872\pi$
$810$	0	0
$811$	12.2462	0.430023	0.215011	−	0.976612i	$-0.431021\pi$
$811$	12.2462	0.430023	0.215011	+	0.976612i	$0.431021\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	21.0540	0.737489
$816$	0	0
$817$	−16.0000	−0.559769
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−21.5464	−0.751974	−0.375987	−	0.926625i	$-0.622696\pi$
$821$	−21.5464	−0.751974	−0.375987	+	0.926625i	$0.622696\pi$
$822$	0	0
$823$	46.2462	1.61204	0.806021	−	0.591887i	$-0.201617\pi$
$823$	46.2462	1.61204	0.806021	+	0.591887i	$0.201617\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	1.12311	0.0390542	0.0195271	−	0.999809i	$-0.493784\pi$
$827$	1.12311	0.0390542	0.0195271	+	0.999809i	$0.493784\pi$
$828$	0	0
$829$	−48.7386	−1.69276	−0.846381	−	0.532577i	$-0.821224\pi$
$829$	−48.7386	−1.69276	−0.846381	+	0.532577i	$0.821224\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−35.3693	−1.22547
$834$	0	0
$835$	−13.1231	−0.454144
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−41.7926	−1.44284	−0.721421	−	0.692497i	$-0.756510\pi$
$839$	−41.7926	−1.44284	−0.721421	+	0.692497i	$0.756510\pi$
$840$	0	0
$841$	−27.7386	−0.956505
$842$	0	0
$843$	0	0
$844$	0	0
$845$	1.00000	0.0344010
$846$	0	0
$847$	−20.2462	−0.695668
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−11.0540	−0.378925
$852$	0	0
$853$	−20.4233	−0.699280	−0.349640	−	0.936884i	$-0.613696\pi$
$853$	−20.4233	−0.699280	−0.349640	+	0.936884i	$0.613696\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	6.56155	0.224138	0.112069	−	0.993700i	$-0.464252\pi$
$857$	6.56155	0.224138	0.112069	+	0.993700i	$0.464252\pi$
$858$	0	0
$859$	16.8078	0.573474	0.286737	−	0.958009i	$-0.407429\pi$
$859$	16.8078	0.573474	0.286737	+	0.958009i	$0.407429\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−15.3693	−0.523178	−0.261589	−	0.965179i	$-0.584246\pi$
$863$	−15.3693	−0.523178	−0.261589	+	0.965179i	$0.584246\pi$
$864$	0	0
$865$	21.1231	0.718207
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−19.6847	−0.667756
$870$	0	0
$871$	13.3693	0.453002
$872$	0	0
$873$	0	0
$874$	0	0
$875$	4.56155	0.154209
$876$	0	0
$877$	−18.9848	−0.641073	−0.320536	−	0.947236i	$-0.603863\pi$
$877$	−18.9848	−0.641073	−0.320536	+	0.947236i	$0.603863\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	3.36932	0.113515	0.0567576	−	0.998388i	$-0.481924\pi$
$881$	3.36932	0.113515	0.0567576	+	0.998388i	$0.481924\pi$
$882$	0	0
$883$	−18.1080	−0.609381	−0.304691	−	0.952451i	$-0.598553\pi$
$883$	−18.1080	−0.609381	−0.304691	+	0.952451i	$0.598553\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−1.43845	−0.0482983	−0.0241492	−	0.999708i	$-0.507688\pi$
$887$	−1.43845	−0.0482983	−0.0241492	+	0.999708i	$0.507688\pi$
$888$	0	0
$889$	−83.2311	−2.79148
$890$	0	0
$891$	0	0
$892$	0	0
$893$	8.98485	0.300666
$894$	0	0
$895$	13.1231	0.438657
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−6.73863	−0.224746
$900$	0	0
$901$	−19.6847	−0.655791
$902$	0	0
$903$	0	0
$904$	0	0
$905$	25.6847	0.853787
$906$	0	0
$907$	27.3693	0.908783	0.454392	−	0.890802i	$-0.349857\pi$
$907$	27.3693	0.908783	0.454392	+	0.890802i	$0.349857\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	12.0000	0.397578	0.198789	−	0.980042i	$-0.436299\pi$
$911$	12.0000	0.397578	0.198789	+	0.980042i	$0.436299\pi$
$912$	0	0
$913$	42.2462	1.39815
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−39.0540	−1.28827	−0.644136	−	0.764911i	$-0.722783\pi$
$919$	−39.0540	−1.28827	−0.644136	+	0.764911i	$0.722783\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	14.5616	0.479299
$924$	0	0
$925$	1.68466	0.0553912
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−38.6695	−1.26871	−0.634353	−	0.773044i	$-0.718733\pi$
$929$	−38.6695	−1.26871	−0.634353	+	0.773044i	$0.718733\pi$
$930$	0	0
$931$	−43.1231	−1.41330
$932$	0	0
$933$	0	0
$934$	0	0
$935$	6.56155	0.214586
$936$	0	0
$937$	−2.63068	−0.0859407	−0.0429703	−	0.999076i	$-0.513682\pi$
$937$	−2.63068	−0.0859407	−0.0429703	+	0.999076i	$0.513682\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−21.1922	−0.690847	−0.345424	−	0.938447i	$-0.612265\pi$
$941$	−21.1922	−0.690847	−0.345424	+	0.938447i	$0.612265\pi$
$942$	0	0
$943$	3.68466	0.119989
$944$	0	0
$945$	0	0
$946$	0	0
$947$	46.6004	1.51431	0.757154	−	0.653236i	$-0.226589\pi$
$947$	46.6004	1.51431	0.757154	+	0.653236i	$0.226589\pi$
$948$	0	0
$949$	6.00000	0.194768
$950$	0	0
$951$	0	0
$952$	0	0
$953$	25.4384	0.824032	0.412016	−	0.911177i	$-0.364825\pi$
$953$	25.4384	0.824032	0.412016	+	0.911177i	$0.364825\pi$
$954$	0	0
$955$	21.6155	0.699462
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−32.4924	−1.04924
$960$	0	0
$961$	5.00000	0.161290
$962$	0	0
$963$	0	0
$964$	0	0
$965$	15.4384	0.496981
$966$	0	0
$967$	−17.3693	−0.558560	−0.279280	−	0.960210i	$-0.590096\pi$
$967$	−17.3693	−0.558560	−0.279280	+	0.960210i	$0.590096\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−8.49242	−0.272535	−0.136267	−	0.990672i	$-0.543511\pi$
$971$	−8.49242	−0.272535	−0.136267	+	0.990672i	$0.543511\pi$
$972$	0	0
$973$	−71.5464	−2.29367
$974$	0	0
$975$	0	0
$976$	0	0
$977$	55.4773	1.77488	0.887438	−	0.460928i	$-0.152483\pi$
$977$	55.4773	1.77488	0.887438	+	0.460928i	$0.152483\pi$
$978$	0	0
$979$	4.31534	0.137919
$980$	0	0
$981$	0	0
$982$	0	0
$983$	6.73863	0.214929	0.107465	−	0.994209i	$-0.465727\pi$
$983$	6.73863	0.214929	0.107465	+	0.994209i	$0.465727\pi$
$984$	0	0
$985$	21.3693	0.680883
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−33.6155	−1.06891
$990$	0	0
$991$	27.0540	0.859398	0.429699	−	0.902972i	$-0.358620\pi$
$991$	27.0540	0.859398	0.429699	+	0.902972i	$0.358620\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	8.00000	0.253617
$996$	0	0
$997$	11.7538	0.372246	0.186123	−	0.982526i	$-0.440408\pi$
$997$	11.7538	0.372246	0.186123	+	0.982526i	$0.440408\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9360.2.a.cw.1.2		2
3.2	odd	2		9360.2.a.cl.1.2		2
4.3	odd	2		585.2.a.l.1.1	yes	2
12.11	even	2		585.2.a.j.1.2	✓	2
20.3	even	4		2925.2.c.p.2224.3		4
20.7	even	4		2925.2.c.p.2224.2		4
20.19	odd	2		2925.2.a.x.1.2		2
52.51	odd	2		7605.2.a.bd.1.2		2
60.23	odd	4		2925.2.c.o.2224.2		4
60.47	odd	4		2925.2.c.o.2224.3		4
60.59	even	2		2925.2.a.bc.1.1		2
156.155	even	2		7605.2.a.bi.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
585.2.a.j.1.2	✓	2	12.11	even	2
585.2.a.l.1.1	yes	2	4.3	odd	2
2925.2.a.x.1.2		2	20.19	odd	2
2925.2.a.bc.1.1		2	60.59	even	2
2925.2.c.o.2224.2		4	60.23	odd	4
2925.2.c.o.2224.3		4	60.47	odd	4
2925.2.c.p.2224.2		4	20.7	even	4
2925.2.c.p.2224.3		4	20.3	even	4
7605.2.a.bd.1.2		2	52.51	odd	2
7605.2.a.bi.1.1		2	156.155	even	2
9360.2.a.cl.1.2		2	3.2	odd	2
9360.2.a.cw.1.2		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:	\( N \)	\(=\)	\( 9360 = 2^{4} \cdot 3^{2} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9360.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{5} +4.56155 q^{7} +O(q^{10})\)	\(q+1.00000 q^{5} +4.56155 q^{7} -2.56155 q^{11} -1.00000 q^{13} -2.56155 q^{17} -3.12311 q^{19} -6.56155 q^{23} +1.00000 q^{25} +1.12311 q^{29} -6.00000 q^{31} +4.56155 q^{35} +1.68466 q^{37} -0.561553 q^{41} +5.12311 q^{43} -2.87689 q^{47} +13.8078 q^{49} +7.68466 q^{53} -2.56155 q^{55} -12.0000 q^{59} +5.68466 q^{61} -1.00000 q^{65} -13.3693 q^{67} -14.5616 q^{71} -6.00000 q^{73} -11.6847 q^{77} +7.68466 q^{79} -16.4924 q^{83} -2.56155 q^{85} -1.68466 q^{89} -4.56155 q^{91} -3.12311 q^{95} -11.9309 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{5} + 5 q^{7}+O(q^{10}) \) 2 * q + 2 * q^5 + 5 * q^7	\( 2 q + 2 q^{5} + 5 q^{7} - q^{11} - 2 q^{13} - q^{17} + 2 q^{19} - 9 q^{23} + 2 q^{25} - 6 q^{29} - 12 q^{31} + 5 q^{35} - 9 q^{37} + 3 q^{41} + 2 q^{43} - 14 q^{47} + 7 q^{49} + 3 q^{53} - q^{55} - 24 q^{59} - q^{61} - 2 q^{65} - 2 q^{67} - 25 q^{71} - 12 q^{73} - 11 q^{77} + 3 q^{79} - q^{85} + 9 q^{89} - 5 q^{91} + 2 q^{95} + 5 q^{97}+O(q^{100}) \) 2 * q + 2 * q^5 + 5 * q^7 - q^11 - 2 * q^13 - q^17 + 2 * q^19 - 9 * q^23 + 2 * q^25 - 6 * q^29 - 12 * q^31 + 5 * q^35 - 9 * q^37 + 3 * q^41 + 2 * q^43 - 14 * q^47 + 7 * q^49 + 3 * q^53 - q^55 - 24 * q^59 - q^61 - 2 * q^65 - 2 * q^67 - 25 * q^71 - 12 * q^73 - 11 * q^77 + 3 * q^79 - q^85 + 9 * q^89 - 5 * q^91 + 2 * q^95 + 5 * q^97

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