LMFDB - Embedded newform 9360.2.a.cc.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:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{6}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 6 $ x^2 - 6
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 520)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.44949$ 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)

$f(q)$	$=$	$q-1.00000 q^{5} -2.00000 q^{7} +O(q^{10})$	$q-1.00000 q^{5} -2.00000 q^{7} +0.449490 q^{11} -1.00000 q^{13} +2.89898 q^{17} +4.44949 q^{19} +1.55051 q^{23} +1.00000 q^{25} -4.00000 q^{29} -0.449490 q^{31} +2.00000 q^{35} -4.89898 q^{37} -1.10102 q^{41} +3.34847 q^{43} -2.00000 q^{47} -3.00000 q^{49} -10.8990 q^{53} -0.449490 q^{55} -5.34847 q^{59} +13.7980 q^{61} +1.00000 q^{65} +14.8990 q^{67} -8.44949 q^{71} +14.6969 q^{73} -0.898979 q^{77} +4.89898 q^{79} +2.00000 q^{83} -2.89898 q^{85} -6.00000 q^{89} +2.00000 q^{91} -4.44949 q^{95} -11.7980 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{5} - 4 q^{7}+O(q^{10}) $ 2 * q - 2 * q^5 - 4 * q^7	$ 2 q - 2 q^{5} - 4 q^{7} - 4 q^{11} - 2 q^{13} - 4 q^{17} + 4 q^{19} + 8 q^{23} + 2 q^{25} - 8 q^{29} + 4 q^{31} + 4 q^{35} - 12 q^{41} - 8 q^{43} - 4 q^{47} - 6 q^{49} - 12 q^{53} + 4 q^{55} + 4 q^{59} + 8 q^{61} + 2 q^{65} + 20 q^{67} - 12 q^{71} + 8 q^{77} + 4 q^{83} + 4 q^{85} - 12 q^{89} + 4 q^{91} - 4 q^{95} - 4 q^{97}+O(q^{100}) $ 2 * q - 2 * q^5 - 4 * q^7 - 4 * q^11 - 2 * q^13 - 4 * q^17 + 4 * q^19 + 8 * q^23 + 2 * q^25 - 8 * q^29 + 4 * q^31 + 4 * q^35 - 12 * q^41 - 8 * q^43 - 4 * q^47 - 6 * q^49 - 12 * q^53 + 4 * q^55 + 4 * q^59 + 8 * q^61 + 2 * q^65 + 20 * q^67 - 12 * q^71 + 8 * q^77 + 4 * q^83 + 4 * q^85 - 12 * q^89 + 4 * q^91 - 4 * q^95 - 4 * 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
$7$	−2.00000	−0.755929	−0.377964	−	0.925820i	$-0.623376\pi$
$7$	−2.00000	−0.755929	−0.377964	+	0.925820i	$0.623376\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	0.449490	0.135526	0.0677631	−	0.997701i	$-0.478414\pi$
$11$	0.449490	0.135526	0.0677631	+	0.997701i	$0.478414\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	0	0
$16$	0	0
$17$	2.89898	0.703106	0.351553	−	0.936168i	$-0.385654\pi$
$17$	2.89898	0.703106	0.351553	+	0.936168i	$0.385654\pi$
$18$	0	0
$19$	4.44949	1.02078	0.510391	−	0.859942i	$-0.329501\pi$
$19$	4.44949	1.02078	0.510391	+	0.859942i	$0.329501\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	1.55051	0.323304	0.161652	−	0.986848i	$-0.448318\pi$
$23$	1.55051	0.323304	0.161652	+	0.986848i	$0.448318\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−4.00000	−0.742781	−0.371391	−	0.928477i	$-0.621119\pi$
$29$	−4.00000	−0.742781	−0.371391	+	0.928477i	$0.621119\pi$
$30$	0	0
$31$	−0.449490	−0.0807307	−0.0403654	−	0.999185i	$-0.512852\pi$
$31$	−0.449490	−0.0807307	−0.0403654	+	0.999185i	$0.512852\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	2.00000	0.338062
$36$	0	0
$37$	−4.89898	−0.805387	−0.402694	−	0.915335i	$-0.631926\pi$
$37$	−4.89898	−0.805387	−0.402694	+	0.915335i	$0.631926\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−1.10102	−0.171951	−0.0859753	−	0.996297i	$-0.527401\pi$
$41$	−1.10102	−0.171951	−0.0859753	+	0.996297i	$0.527401\pi$
$42$	0	0
$43$	3.34847	0.510637	0.255318	−	0.966857i	$-0.417820\pi$
$43$	3.34847	0.510637	0.255318	+	0.966857i	$0.417820\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−2.00000	−0.291730	−0.145865	−	0.989305i	$-0.546597\pi$
$47$	−2.00000	−0.291730	−0.145865	+	0.989305i	$0.546597\pi$
$48$	0	0
$49$	−3.00000	−0.428571
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−10.8990	−1.49709	−0.748545	−	0.663084i	$-0.769247\pi$
$53$	−10.8990	−1.49709	−0.748545	+	0.663084i	$0.769247\pi$
$54$	0	0
$55$	−0.449490	−0.0606092
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−5.34847	−0.696311	−0.348156	−	0.937437i	$-0.613192\pi$
$59$	−5.34847	−0.696311	−0.348156	+	0.937437i	$0.613192\pi$
$60$	0	0
$61$	13.7980	1.76665	0.883324	−	0.468763i	$-0.155300\pi$
$61$	13.7980	1.76665	0.883324	+	0.468763i	$0.155300\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	1.00000	0.124035
$66$	0	0
$67$	14.8990	1.82020	0.910100	−	0.414389i	$-0.136005\pi$
$67$	14.8990	1.82020	0.910100	+	0.414389i	$0.136005\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−8.44949	−1.00277	−0.501385	−	0.865224i	$-0.667176\pi$
$71$	−8.44949	−1.00277	−0.501385	+	0.865224i	$0.667176\pi$
$72$	0	0
$73$	14.6969	1.72015	0.860073	−	0.510171i	$-0.170418\pi$
$73$	14.6969	1.72015	0.860073	+	0.510171i	$0.170418\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−0.898979	−0.102448
$78$	0	0
$79$	4.89898	0.551178	0.275589	−	0.961276i	$-0.411127\pi$
$79$	4.89898	0.551178	0.275589	+	0.961276i	$0.411127\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	2.00000	0.219529	0.109764	−	0.993958i	$-0.464990\pi$
$83$	2.00000	0.219529	0.109764	+	0.993958i	$0.464990\pi$
$84$	0	0
$85$	−2.89898	−0.314438
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−6.00000	−0.635999	−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000	−0.635999	−0.317999	+	0.948091i	$0.603011\pi$
$90$	0	0
$91$	2.00000	0.209657
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−4.44949	−0.456508
$96$	0	0
$97$	−11.7980	−1.19790	−0.598951	−	0.800786i	$-0.704415\pi$
$97$	−11.7980	−1.19790	−0.598951	+	0.800786i	$0.704415\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−3.79796	−0.377911	−0.188956	−	0.981986i	$-0.560510\pi$
$101$	−3.79796	−0.377911	−0.188956	+	0.981986i	$0.560510\pi$
$102$	0	0
$103$	3.34847	0.329934	0.164967	−	0.986299i	$-0.447248\pi$
$103$	3.34847	0.329934	0.164967	+	0.986299i	$0.447248\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	12.2474	1.18401	0.592003	−	0.805936i	$-0.298337\pi$
$107$	12.2474	1.18401	0.592003	+	0.805936i	$0.298337\pi$
$108$	0	0
$109$	−10.0000	−0.957826	−0.478913	−	0.877862i	$-0.658969\pi$
$109$	−10.0000	−0.957826	−0.478913	+	0.877862i	$0.658969\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−10.8990	−1.02529	−0.512645	−	0.858601i	$-0.671334\pi$
$113$	−10.8990	−1.02529	−0.512645	+	0.858601i	$0.671334\pi$
$114$	0	0
$115$	−1.55051	−0.144586
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−5.79796	−0.531498
$120$	0	0
$121$	−10.7980	−0.981633
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−19.3485	−1.71690	−0.858450	−	0.512898i	$-0.828572\pi$
$127$	−19.3485	−1.71690	−0.858450	+	0.512898i	$0.828572\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−19.5959	−1.71210	−0.856052	−	0.516890i	$-0.827090\pi$
$131$	−19.5959	−1.71210	−0.856052	+	0.516890i	$0.827090\pi$
$132$	0	0
$133$	−8.89898	−0.771639
$134$	0	0
$135$	0	0
$136$	0	0
$137$	14.0000	1.19610	0.598050	−	0.801459i	$-0.295942\pi$
$137$	14.0000	1.19610	0.598050	+	0.801459i	$0.295942\pi$
$138$	0	0
$139$	7.10102	0.602301	0.301150	−	0.953577i	$-0.402629\pi$
$139$	7.10102	0.602301	0.301150	+	0.953577i	$0.402629\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−0.449490	−0.0375882
$144$	0	0
$145$	4.00000	0.332182
$146$	0	0
$147$	0	0
$148$	0	0
$149$	7.79796	0.638834	0.319417	−	0.947614i	$-0.396513\pi$
$149$	7.79796	0.638834	0.319417	+	0.947614i	$0.396513\pi$
$150$	0	0
$151$	17.3485	1.41180	0.705899	−	0.708312i	$-0.250543\pi$
$151$	17.3485	1.41180	0.705899	+	0.708312i	$0.250543\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0.449490	0.0361039
$156$	0	0
$157$	10.0000	0.798087	0.399043	−	0.916932i	$-0.369342\pi$
$157$	10.0000	0.798087	0.399043	+	0.916932i	$0.369342\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−3.10102	−0.244395
$162$	0	0
$163$	18.8990	1.48028	0.740141	−	0.672452i	$-0.234759\pi$
$163$	18.8990	1.48028	0.740141	+	0.672452i	$0.234759\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	18.0000	1.39288	0.696441	−	0.717614i	$-0.254766\pi$
$167$	18.0000	1.39288	0.696441	+	0.717614i	$0.254766\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	0	0
$172$	0	0
$173$	2.89898	0.220405	0.110203	−	0.993909i	$-0.464850\pi$
$173$	2.89898	0.220405	0.110203	+	0.993909i	$0.464850\pi$
$174$	0	0
$175$	−2.00000	−0.151186
$176$	0	0
$177$	0	0
$178$	0	0
$179$	8.00000	0.597948	0.298974	−	0.954261i	$-0.403356\pi$
$179$	8.00000	0.597948	0.298974	+	0.954261i	$0.403356\pi$
$180$	0	0
$181$	−13.7980	−1.02559	−0.512797	−	0.858510i	$-0.671391\pi$
$181$	−13.7980	−1.02559	−0.512797	+	0.858510i	$0.671391\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	4.89898	0.360180
$186$	0	0
$187$	1.30306	0.0952893
$188$	0	0
$189$	0	0
$190$	0	0
$191$	17.7980	1.28782	0.643908	−	0.765103i	$-0.277312\pi$
$191$	17.7980	1.28782	0.643908	+	0.765103i	$0.277312\pi$
$192$	0	0
$193$	19.7980	1.42509	0.712544	−	0.701627i	$-0.247543\pi$
$193$	19.7980	1.42509	0.712544	+	0.701627i	$0.247543\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	4.20204	0.299383	0.149692	−	0.988733i	$-0.452172\pi$
$197$	4.20204	0.299383	0.149692	+	0.988733i	$0.452172\pi$
$198$	0	0
$199$	13.7980	0.978111	0.489056	−	0.872253i	$-0.337341\pi$
$199$	13.7980	0.978111	0.489056	+	0.872253i	$0.337341\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	8.00000	0.561490
$204$	0	0
$205$	1.10102	0.0768986
$206$	0	0
$207$	0	0
$208$	0	0
$209$	2.00000	0.138343
$210$	0	0
$211$	6.20204	0.426966	0.213483	−	0.976947i	$-0.431519\pi$
$211$	6.20204	0.426966	0.213483	+	0.976947i	$0.431519\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−3.34847	−0.228364
$216$	0	0
$217$	0.898979	0.0610267
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−2.89898	−0.195006
$222$	0	0
$223$	−21.5959	−1.44617	−0.723085	−	0.690759i	$-0.757276\pi$
$223$	−21.5959	−1.44617	−0.723085	+	0.690759i	$0.757276\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	13.1010	0.869545	0.434773	−	0.900540i	$-0.356829\pi$
$227$	13.1010	0.869545	0.434773	+	0.900540i	$0.356829\pi$
$228$	0	0
$229$	−9.10102	−0.601412	−0.300706	−	0.953717i	$-0.597222\pi$
$229$	−9.10102	−0.601412	−0.300706	+	0.953717i	$0.597222\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−13.5959	−0.890698	−0.445349	−	0.895357i	$-0.646920\pi$
$233$	−13.5959	−0.890698	−0.445349	+	0.895357i	$0.646920\pi$
$234$	0	0
$235$	2.00000	0.130466
$236$	0	0
$237$	0	0
$238$	0	0
$239$	23.1464	1.49722	0.748609	−	0.663012i	$-0.230722\pi$
$239$	23.1464	1.49722	0.748609	+	0.663012i	$0.230722\pi$
$240$	0	0
$241$	28.6969	1.84853	0.924266	−	0.381749i	$-0.124678\pi$
$241$	28.6969	1.84853	0.924266	+	0.381749i	$0.124678\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	3.00000	0.191663
$246$	0	0
$247$	−4.44949	−0.283114
$248$	0	0
$249$	0	0
$250$	0	0
$251$	14.6969	0.927663	0.463831	−	0.885924i	$-0.346474\pi$
$251$	14.6969	0.927663	0.463831	+	0.885924i	$0.346474\pi$
$252$	0	0
$253$	0.696938	0.0438161
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−15.7980	−0.985450	−0.492725	−	0.870185i	$-0.663999\pi$
$257$	−15.7980	−0.985450	−0.492725	+	0.870185i	$0.663999\pi$
$258$	0	0
$259$	9.79796	0.608816
$260$	0	0
$261$	0	0
$262$	0	0
$263$	21.5505	1.32886	0.664431	−	0.747350i	$-0.268674\pi$
$263$	21.5505	1.32886	0.664431	+	0.747350i	$0.268674\pi$
$264$	0	0
$265$	10.8990	0.669519
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−26.0000	−1.58525	−0.792624	−	0.609711i	$-0.791286\pi$
$269$	−26.0000	−1.58525	−0.792624	+	0.609711i	$0.791286\pi$
$270$	0	0
$271$	−8.44949	−0.513270	−0.256635	−	0.966508i	$-0.582614\pi$
$271$	−8.44949	−0.513270	−0.256635	+	0.966508i	$0.582614\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0.449490	0.0271053
$276$	0	0
$277$	26.4949	1.59192	0.795962	−	0.605347i	$-0.206965\pi$
$277$	26.4949	1.59192	0.795962	+	0.605347i	$0.206965\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−2.89898	−0.172939	−0.0864693	−	0.996255i	$-0.527558\pi$
$281$	−2.89898	−0.172939	−0.0864693	+	0.996255i	$0.527558\pi$
$282$	0	0
$283$	1.55051	0.0921683	0.0460841	−	0.998938i	$-0.485326\pi$
$283$	1.55051	0.0921683	0.0460841	+	0.998938i	$0.485326\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	2.20204	0.129982
$288$	0	0
$289$	−8.59592	−0.505642
$290$	0	0
$291$	0	0
$292$	0	0
$293$	20.4949	1.19732	0.598662	−	0.801001i	$-0.295699\pi$
$293$	20.4949	1.19732	0.598662	+	0.801001i	$0.295699\pi$
$294$	0	0
$295$	5.34847	0.311400
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−1.55051	−0.0896683
$300$	0	0
$301$	−6.69694	−0.386005
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−13.7980	−0.790069
$306$	0	0
$307$	11.7980	0.673345	0.336673	−	0.941622i	$-0.390698\pi$
$307$	11.7980	0.673345	0.336673	+	0.941622i	$0.390698\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	12.8990	0.731434	0.365717	−	0.930726i	$-0.380824\pi$
$311$	12.8990	0.731434	0.365717	+	0.930726i	$0.380824\pi$
$312$	0	0
$313$	26.4949	1.49758	0.748790	−	0.662807i	$-0.230635\pi$
$313$	26.4949	1.49758	0.748790	+	0.662807i	$0.230635\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	2.69694	0.151475	0.0757376	−	0.997128i	$-0.475869\pi$
$317$	2.69694	0.151475	0.0757376	+	0.997128i	$0.475869\pi$
$318$	0	0
$319$	−1.79796	−0.100666
$320$	0	0
$321$	0	0
$322$	0	0
$323$	12.8990	0.717718
$324$	0	0
$325$	−1.00000	−0.0554700
$326$	0	0
$327$	0	0
$328$	0	0
$329$	4.00000	0.220527
$330$	0	0
$331$	8.44949	0.464426	0.232213	−	0.972665i	$-0.425403\pi$
$331$	8.44949	0.464426	0.232213	+	0.972665i	$0.425403\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−14.8990	−0.814018
$336$	0	0
$337$	12.6969	0.691646	0.345823	−	0.938300i	$-0.387600\pi$
$337$	12.6969	0.691646	0.345823	+	0.938300i	$0.387600\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−0.202041	−0.0109411
$342$	0	0
$343$	20.0000	1.07990
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−4.24745	−0.228015	−0.114007	−	0.993480i	$-0.536369\pi$
$347$	−4.24745	−0.228015	−0.114007	+	0.993480i	$0.536369\pi$
$348$	0	0
$349$	−8.69694	−0.465536	−0.232768	−	0.972532i	$-0.574778\pi$
$349$	−8.69694	−0.465536	−0.232768	+	0.972532i	$0.574778\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−20.4949	−1.09083	−0.545417	−	0.838165i	$-0.683629\pi$
$353$	−20.4949	−1.09083	−0.545417	+	0.838165i	$0.683629\pi$
$354$	0	0
$355$	8.44949	0.448452
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−30.2474	−1.59640	−0.798200	−	0.602393i	$-0.794214\pi$
$359$	−30.2474	−1.59640	−0.798200	+	0.602393i	$0.794214\pi$
$360$	0	0
$361$	0.797959	0.0419978
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−14.6969	−0.769273
$366$	0	0
$367$	18.9444	0.988889	0.494444	−	0.869209i	$-0.335372\pi$
$367$	18.9444	0.988889	0.494444	+	0.869209i	$0.335372\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	21.7980	1.13169
$372$	0	0
$373$	−9.59592	−0.496858	−0.248429	−	0.968650i	$-0.579914\pi$
$373$	−9.59592	−0.496858	−0.248429	+	0.968650i	$0.579914\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	4.00000	0.206010
$378$	0	0
$379$	11.5505	0.593310	0.296655	−	0.954985i	$-0.404129\pi$
$379$	11.5505	0.593310	0.296655	+	0.954985i	$0.404129\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	13.5959	0.694719	0.347359	−	0.937732i	$-0.387078\pi$
$383$	13.5959	0.694719	0.347359	+	0.937732i	$0.387078\pi$
$384$	0	0
$385$	0.898979	0.0458162
$386$	0	0
$387$	0	0
$388$	0	0
$389$	37.5959	1.90619	0.953094	−	0.302673i	$-0.0978791\pi$
$389$	37.5959	1.90619	0.953094	+	0.302673i	$0.0978791\pi$
$390$	0	0
$391$	4.49490	0.227317
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−4.89898	−0.246494
$396$	0	0
$397$	1.30306	0.0653988	0.0326994	−	0.999465i	$-0.489590\pi$
$397$	1.30306	0.0653988	0.0326994	+	0.999465i	$0.489590\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−6.00000	−0.299626	−0.149813	−	0.988714i	$-0.547867\pi$
$401$	−6.00000	−0.299626	−0.149813	+	0.988714i	$0.547867\pi$
$402$	0	0
$403$	0.449490	0.0223907
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−2.20204	−0.109151
$408$	0	0
$409$	2.89898	0.143345	0.0716727	−	0.997428i	$-0.477166\pi$
$409$	2.89898	0.143345	0.0716727	+	0.997428i	$0.477166\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	10.6969	0.526362
$414$	0	0
$415$	−2.00000	−0.0981761
$416$	0	0
$417$	0	0
$418$	0	0
$419$	9.30306	0.454484	0.227242	−	0.973838i	$-0.427029\pi$
$419$	9.30306	0.454484	0.227242	+	0.973838i	$0.427029\pi$
$420$	0	0
$421$	15.7980	0.769945	0.384973	−	0.922928i	$-0.374211\pi$
$421$	15.7980	0.769945	0.384973	+	0.922928i	$0.374211\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	2.89898	0.140621
$426$	0	0
$427$	−27.5959	−1.33546
$428$	0	0
$429$	0	0
$430$	0	0
$431$	12.0454	0.580207	0.290103	−	0.956995i	$-0.406310\pi$
$431$	12.0454	0.580207	0.290103	+	0.956995i	$0.406310\pi$
$432$	0	0
$433$	−7.79796	−0.374746	−0.187373	−	0.982289i	$-0.559997\pi$
$433$	−7.79796	−0.374746	−0.187373	+	0.982289i	$0.559997\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	6.89898	0.330023
$438$	0	0
$439$	−1.79796	−0.0858119	−0.0429059	−	0.999079i	$-0.513662\pi$
$439$	−1.79796	−0.0858119	−0.0429059	+	0.999079i	$0.513662\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	22.0454	1.04741	0.523704	−	0.851900i	$-0.324550\pi$
$443$	22.0454	1.04741	0.523704	+	0.851900i	$0.324550\pi$
$444$	0	0
$445$	6.00000	0.284427
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−8.69694	−0.410434	−0.205217	−	0.978717i	$-0.565790\pi$
$449$	−8.69694	−0.410434	−0.205217	+	0.978717i	$0.565790\pi$
$450$	0	0
$451$	−0.494897	−0.0233038
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−2.00000	−0.0937614
$456$	0	0
$457$	22.0000	1.02912	0.514558	−	0.857455i	$-0.327956\pi$
$457$	22.0000	1.02912	0.514558	+	0.857455i	$0.327956\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	3.30306	0.153839	0.0769195	−	0.997037i	$-0.475492\pi$
$461$	3.30306	0.153839	0.0769195	+	0.997037i	$0.475492\pi$
$462$	0	0
$463$	13.1010	0.608856	0.304428	−	0.952535i	$-0.401535\pi$
$463$	13.1010	0.608856	0.304428	+	0.952535i	$0.401535\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−30.9444	−1.43194	−0.715968	−	0.698133i	$-0.754014\pi$
$467$	−30.9444	−1.43194	−0.715968	+	0.698133i	$0.754014\pi$
$468$	0	0
$469$	−29.7980	−1.37594
$470$	0	0
$471$	0	0
$472$	0	0
$473$	1.50510	0.0692047
$474$	0	0
$475$	4.44949	0.204157
$476$	0	0
$477$	0	0
$478$	0	0
$479$	5.34847	0.244378	0.122189	−	0.992507i	$-0.461009\pi$
$479$	5.34847	0.244378	0.122189	+	0.992507i	$0.461009\pi$
$480$	0	0
$481$	4.89898	0.223374
$482$	0	0
$483$	0	0
$484$	0	0
$485$	11.7980	0.535718
$486$	0	0
$487$	−2.89898	−0.131365	−0.0656826	−	0.997841i	$-0.520922\pi$
$487$	−2.89898	−0.131365	−0.0656826	+	0.997841i	$0.520922\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	34.2929	1.54761	0.773807	−	0.633421i	$-0.218350\pi$
$491$	34.2929	1.54761	0.773807	+	0.633421i	$0.218350\pi$
$492$	0	0
$493$	−11.5959	−0.522254
$494$	0	0
$495$	0	0
$496$	0	0
$497$	16.8990	0.758023
$498$	0	0
$499$	−21.3485	−0.955689	−0.477844	−	0.878445i	$-0.658582\pi$
$499$	−21.3485	−0.955689	−0.477844	+	0.878445i	$0.658582\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−26.9444	−1.20139	−0.600695	−	0.799478i	$-0.705110\pi$
$503$	−26.9444	−1.20139	−0.600695	+	0.799478i	$0.705110\pi$
$504$	0	0
$505$	3.79796	0.169007
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−26.8990	−1.19228	−0.596138	−	0.802882i	$-0.703299\pi$
$509$	−26.8990	−1.19228	−0.596138	+	0.802882i	$0.703299\pi$
$510$	0	0
$511$	−29.3939	−1.30031
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−3.34847	−0.147551
$516$	0	0
$517$	−0.898979	−0.0395371
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−6.20204	−0.271716	−0.135858	−	0.990728i	$-0.543379\pi$
$521$	−6.20204	−0.271716	−0.135858	+	0.990728i	$0.543379\pi$
$522$	0	0
$523$	39.3485	1.72059	0.860294	−	0.509798i	$-0.170280\pi$
$523$	39.3485	1.72059	0.860294	+	0.509798i	$0.170280\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−1.30306	−0.0567623
$528$	0	0
$529$	−20.5959	−0.895475
$530$	0	0
$531$	0	0
$532$	0	0
$533$	1.10102	0.0476905
$534$	0	0
$535$	−12.2474	−0.529503
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−1.34847	−0.0580827
$540$	0	0
$541$	5.10102	0.219310	0.109655	−	0.993970i	$-0.465025\pi$
$541$	5.10102	0.219310	0.109655	+	0.993970i	$0.465025\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	10.0000	0.428353
$546$	0	0
$547$	−0.651531	−0.0278574	−0.0139287	−	0.999903i	$-0.504434\pi$
$547$	−0.651531	−0.0278574	−0.0139287	+	0.999903i	$0.504434\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−17.7980	−0.758219
$552$	0	0
$553$	−9.79796	−0.416652
$554$	0	0
$555$	0	0
$556$	0	0
$557$	1.30306	0.0552125	0.0276062	−	0.999619i	$-0.491212\pi$
$557$	1.30306	0.0552125	0.0276062	+	0.999619i	$0.491212\pi$
$558$	0	0
$559$	−3.34847	−0.141625
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−7.75255	−0.326731	−0.163366	−	0.986566i	$-0.552235\pi$
$563$	−7.75255	−0.326731	−0.163366	+	0.986566i	$0.552235\pi$
$564$	0	0
$565$	10.8990	0.458524
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−10.2020	−0.427692	−0.213846	−	0.976867i	$-0.568599\pi$
$569$	−10.2020	−0.427692	−0.213846	+	0.976867i	$0.568599\pi$
$570$	0	0
$571$	−28.4949	−1.19247	−0.596237	−	0.802808i	$-0.703338\pi$
$571$	−28.4949	−1.19247	−0.596237	+	0.802808i	$0.703338\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	1.55051	0.0646607
$576$	0	0
$577$	27.1010	1.12823	0.564115	−	0.825696i	$-0.309217\pi$
$577$	27.1010	1.12823	0.564115	+	0.825696i	$0.309217\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−4.00000	−0.165948
$582$	0	0
$583$	−4.89898	−0.202895
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−10.8990	−0.449849	−0.224925	−	0.974376i	$-0.572214\pi$
$587$	−10.8990	−0.449849	−0.224925	+	0.974376i	$0.572214\pi$
$588$	0	0
$589$	−2.00000	−0.0824086
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−6.00000	−0.246390	−0.123195	−	0.992382i	$-0.539314\pi$
$593$	−6.00000	−0.246390	−0.123195	+	0.992382i	$0.539314\pi$
$594$	0	0
$595$	5.79796	0.237693
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−28.8990	−1.18078	−0.590390	−	0.807118i	$-0.701026\pi$
$599$	−28.8990	−1.18078	−0.590390	+	0.807118i	$0.701026\pi$
$600$	0	0
$601$	−33.5959	−1.37041	−0.685203	−	0.728352i	$-0.740287\pi$
$601$	−33.5959	−1.37041	−0.685203	+	0.728352i	$0.740287\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	10.7980	0.438999
$606$	0	0
$607$	−21.5505	−0.874708	−0.437354	−	0.899289i	$-0.644084\pi$
$607$	−21.5505	−0.874708	−0.437354	+	0.899289i	$0.644084\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	2.00000	0.0809113
$612$	0	0
$613$	45.1918	1.82528	0.912641	−	0.408763i	$-0.134040\pi$
$613$	45.1918	1.82528	0.912641	+	0.408763i	$0.134040\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	35.3939	1.42490	0.712452	−	0.701721i	$-0.247585\pi$
$617$	35.3939	1.42490	0.712452	+	0.701721i	$0.247585\pi$
$618$	0	0
$619$	5.34847	0.214973	0.107487	−	0.994207i	$-0.465720\pi$
$619$	5.34847	0.214973	0.107487	+	0.994207i	$0.465720\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	12.0000	0.480770
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−14.2020	−0.566272
$630$	0	0
$631$	16.4495	0.654844	0.327422	−	0.944878i	$-0.393820\pi$
$631$	16.4495	0.654844	0.327422	+	0.944878i	$0.393820\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	19.3485	0.767821
$636$	0	0
$637$	3.00000	0.118864
$638$	0	0
$639$	0	0
$640$	0	0
$641$	35.3939	1.39797	0.698987	−	0.715134i	$-0.253634\pi$
$641$	35.3939	1.39797	0.698987	+	0.715134i	$0.253634\pi$
$642$	0	0
$643$	30.0000	1.18308	0.591542	−	0.806274i	$-0.298519\pi$
$643$	30.0000	1.18308	0.591542	+	0.806274i	$0.298519\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	14.9444	0.587524	0.293762	−	0.955879i	$-0.405093\pi$
$647$	14.9444	0.587524	0.293762	+	0.955879i	$0.405093\pi$
$648$	0	0
$649$	−2.40408	−0.0943685
$650$	0	0
$651$	0	0
$652$	0	0
$653$	43.7980	1.71395	0.856973	−	0.515361i	$-0.172342\pi$
$653$	43.7980	1.71395	0.856973	+	0.515361i	$0.172342\pi$
$654$	0	0
$655$	19.5959	0.765676
$656$	0	0
$657$	0	0
$658$	0	0
$659$	14.6969	0.572511	0.286256	−	0.958153i	$-0.407589\pi$
$659$	14.6969	0.572511	0.286256	+	0.958153i	$0.407589\pi$
$660$	0	0
$661$	39.7980	1.54796	0.773981	−	0.633209i	$-0.218263\pi$
$661$	39.7980	1.54796	0.773981	+	0.633209i	$0.218263\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	8.89898	0.345088
$666$	0	0
$667$	−6.20204	−0.240144
$668$	0	0
$669$	0	0
$670$	0	0
$671$	6.20204	0.239427
$672$	0	0
$673$	−1.10102	−0.0424412	−0.0212206	−	0.999775i	$-0.506755\pi$
$673$	−1.10102	−0.0424412	−0.0212206	+	0.999775i	$0.506755\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	40.6969	1.56411	0.782055	−	0.623209i	$-0.214171\pi$
$677$	40.6969	1.56411	0.782055	+	0.623209i	$0.214171\pi$
$678$	0	0
$679$	23.5959	0.905528
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−42.4949	−1.62602	−0.813011	−	0.582248i	$-0.802173\pi$
$683$	−42.4949	−1.62602	−0.813011	+	0.582248i	$0.802173\pi$
$684$	0	0
$685$	−14.0000	−0.534913
$686$	0	0
$687$	0	0
$688$	0	0
$689$	10.8990	0.415218
$690$	0	0
$691$	14.6515	0.557370	0.278685	−	0.960382i	$-0.410101\pi$
$691$	14.6515	0.557370	0.278685	+	0.960382i	$0.410101\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−7.10102	−0.269357
$696$	0	0
$697$	−3.19184	−0.120899
$698$	0	0
$699$	0	0
$700$	0	0
$701$	37.1918	1.40472	0.702358	−	0.711824i	$-0.252131\pi$
$701$	37.1918	1.40472	0.702358	+	0.711824i	$0.252131\pi$
$702$	0	0
$703$	−21.7980	−0.822126
$704$	0	0
$705$	0	0
$706$	0	0
$707$	7.59592	0.285674
$708$	0	0
$709$	−32.2929	−1.21278	−0.606392	−	0.795166i	$-0.707384\pi$
$709$	−32.2929	−1.21278	−0.606392	+	0.795166i	$0.707384\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−0.696938	−0.0261006
$714$	0	0
$715$	0.449490	0.0168100
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−20.0000	−0.745874	−0.372937	−	0.927857i	$-0.621649\pi$
$719$	−20.0000	−0.745874	−0.372937	+	0.927857i	$0.621649\pi$
$720$	0	0
$721$	−6.69694	−0.249407
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−4.00000	−0.148556
$726$	0	0
$727$	−18.0454	−0.669267	−0.334634	−	0.942348i	$-0.608613\pi$
$727$	−18.0454	−0.669267	−0.334634	+	0.942348i	$0.608613\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	9.70714	0.359032
$732$	0	0
$733$	−6.00000	−0.221615	−0.110808	−	0.993842i	$-0.535344\pi$
$733$	−6.00000	−0.221615	−0.110808	+	0.993842i	$0.535344\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	6.69694	0.246685
$738$	0	0
$739$	6.24745	0.229816	0.114908	−	0.993376i	$-0.463343\pi$
$739$	6.24745	0.229816	0.114908	+	0.993376i	$0.463343\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−35.3939	−1.29848	−0.649238	−	0.760586i	$-0.724912\pi$
$743$	−35.3939	−1.29848	−0.649238	+	0.760586i	$0.724912\pi$
$744$	0	0
$745$	−7.79796	−0.285695
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−24.4949	−0.895024
$750$	0	0
$751$	−31.1010	−1.13489	−0.567446	−	0.823410i	$-0.692069\pi$
$751$	−31.1010	−1.13489	−0.567446	+	0.823410i	$0.692069\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−17.3485	−0.631375
$756$	0	0
$757$	26.4949	0.962973	0.481487	−	0.876453i	$-0.340097\pi$
$757$	26.4949	0.962973	0.481487	+	0.876453i	$0.340097\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	10.4041	0.377148	0.188574	−	0.982059i	$-0.439614\pi$
$761$	10.4041	0.377148	0.188574	+	0.982059i	$0.439614\pi$
$762$	0	0
$763$	20.0000	0.724049
$764$	0	0
$765$	0	0
$766$	0	0
$767$	5.34847	0.193122
$768$	0	0
$769$	39.3939	1.42058	0.710290	−	0.703909i	$-0.248564\pi$
$769$	39.3939	1.42058	0.710290	+	0.703909i	$0.248564\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−44.0908	−1.58584	−0.792918	−	0.609328i	$-0.791439\pi$
$773$	−44.0908	−1.58584	−0.792918	+	0.609328i	$0.791439\pi$
$774$	0	0
$775$	−0.449490	−0.0161461
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−4.89898	−0.175524
$780$	0	0
$781$	−3.79796	−0.135902
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−10.0000	−0.356915
$786$	0	0
$787$	−38.0000	−1.35455	−0.677277	−	0.735728i	$-0.736840\pi$
$787$	−38.0000	−1.35455	−0.677277	+	0.735728i	$0.736840\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	21.7980	0.775046
$792$	0	0
$793$	−13.7980	−0.489980
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−27.7980	−0.984654	−0.492327	−	0.870410i	$-0.663854\pi$
$797$	−27.7980	−0.984654	−0.492327	+	0.870410i	$0.663854\pi$
$798$	0	0
$799$	−5.79796	−0.205117
$800$	0	0
$801$	0	0
$802$	0	0
$803$	6.60612	0.233125
$804$	0	0
$805$	3.10102	0.109297
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−12.0000	−0.421898	−0.210949	−	0.977497i	$-0.567655\pi$
$809$	−12.0000	−0.421898	−0.210949	+	0.977497i	$0.567655\pi$
$810$	0	0
$811$	53.8434	1.89070	0.945348	−	0.326063i	$-0.105722\pi$
$811$	53.8434	1.89070	0.945348	+	0.326063i	$0.105722\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−18.8990	−0.662002
$816$	0	0
$817$	14.8990	0.521249
$818$	0	0
$819$	0	0
$820$	0	0
$821$	53.1918	1.85641	0.928204	−	0.372072i	$-0.121352\pi$
$821$	53.1918	1.85641	0.928204	+	0.372072i	$0.121352\pi$
$822$	0	0
$823$	−30.4495	−1.06140	−0.530701	−	0.847559i	$-0.678071\pi$
$823$	−30.4495	−1.06140	−0.530701	+	0.847559i	$0.678071\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	57.1918	1.98875	0.994377	−	0.105893i	$-0.0337702\pi$
$827$	57.1918	1.98875	0.994377	+	0.105893i	$0.0337702\pi$
$828$	0	0
$829$	−20.0000	−0.694629	−0.347314	−	0.937749i	$-0.612906\pi$
$829$	−20.0000	−0.694629	−0.347314	+	0.937749i	$0.612906\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−8.69694	−0.301331
$834$	0	0
$835$	−18.0000	−0.622916
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−19.5505	−0.674959	−0.337479	−	0.941333i	$-0.609574\pi$
$839$	−19.5505	−0.674959	−0.337479	+	0.941333i	$0.609574\pi$
$840$	0	0
$841$	−13.0000	−0.448276
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−1.00000	−0.0344010
$846$	0	0
$847$	21.5959	0.742045
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−7.59592	−0.260385
$852$	0	0
$853$	−18.6969	−0.640171	−0.320085	−	0.947389i	$-0.603712\pi$
$853$	−18.6969	−0.640171	−0.320085	+	0.947389i	$0.603712\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−35.3939	−1.20903	−0.604516	−	0.796593i	$-0.706633\pi$
$857$	−35.3939	−1.20903	−0.604516	+	0.796593i	$0.706633\pi$
$858$	0	0
$859$	30.6969	1.04737	0.523683	−	0.851913i	$-0.324558\pi$
$859$	30.6969	1.04737	0.523683	+	0.851913i	$0.324558\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	23.3939	0.796337	0.398168	−	0.917312i	$-0.369646\pi$
$863$	23.3939	0.796337	0.398168	+	0.917312i	$0.369646\pi$
$864$	0	0
$865$	−2.89898	−0.0985683
$866$	0	0
$867$	0	0
$868$	0	0
$869$	2.20204	0.0746991
$870$	0	0
$871$	−14.8990	−0.504833
$872$	0	0
$873$	0	0
$874$	0	0
$875$	2.00000	0.0676123
$876$	0	0
$877$	17.5959	0.594172	0.297086	−	0.954851i	$-0.403985\pi$
$877$	17.5959	0.594172	0.297086	+	0.954851i	$0.403985\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−9.79796	−0.330102	−0.165051	−	0.986285i	$-0.552779\pi$
$881$	−9.79796	−0.330102	−0.165051	+	0.986285i	$0.552779\pi$
$882$	0	0
$883$	11.3485	0.381906	0.190953	−	0.981599i	$-0.438842\pi$
$883$	11.3485	0.381906	0.190953	+	0.981599i	$0.438842\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	4.24745	0.142615	0.0713077	−	0.997454i	$-0.477283\pi$
$887$	4.24745	0.142615	0.0713077	+	0.997454i	$0.477283\pi$
$888$	0	0
$889$	38.6969	1.29785
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−8.89898	−0.297793
$894$	0	0
$895$	−8.00000	−0.267411
$896$	0	0
$897$	0	0
$898$	0	0
$899$	1.79796	0.0599653
$900$	0	0
$901$	−31.5959	−1.05261
$902$	0	0
$903$	0	0
$904$	0	0
$905$	13.7980	0.458660
$906$	0	0
$907$	−43.8434	−1.45580	−0.727898	−	0.685686i	$-0.759502\pi$
$907$	−43.8434	−1.45580	−0.727898	+	0.685686i	$0.759502\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−49.3939	−1.63649	−0.818246	−	0.574868i	$-0.805053\pi$
$911$	−49.3939	−1.63649	−0.818246	+	0.574868i	$0.805053\pi$
$912$	0	0
$913$	0.898979	0.0297519
$914$	0	0
$915$	0	0
$916$	0	0
$917$	39.1918	1.29423
$918$	0	0
$919$	11.1010	0.366189	0.183094	−	0.983095i	$-0.441389\pi$
$919$	11.1010	0.366189	0.183094	+	0.983095i	$0.441389\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	8.44949	0.278118
$924$	0	0
$925$	−4.89898	−0.161077
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−24.6969	−0.810280	−0.405140	−	0.914255i	$-0.632777\pi$
$929$	−24.6969	−0.810280	−0.405140	+	0.914255i	$0.632777\pi$
$930$	0	0
$931$	−13.3485	−0.437478
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−1.30306	−0.0426147
$936$	0	0
$937$	−19.3939	−0.633570	−0.316785	−	0.948497i	$-0.602603\pi$
$937$	−19.3939	−0.633570	−0.316785	+	0.948497i	$0.602603\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	23.3031	0.759658	0.379829	−	0.925057i	$-0.375983\pi$
$941$	23.3031	0.759658	0.379829	+	0.925057i	$0.375983\pi$
$942$	0	0
$943$	−1.70714	−0.0555922
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−47.7980	−1.55322	−0.776612	−	0.629979i	$-0.783064\pi$
$947$	−47.7980	−1.55322	−0.776612	+	0.629979i	$0.783064\pi$
$948$	0	0
$949$	−14.6969	−0.477083
$950$	0	0
$951$	0	0
$952$	0	0
$953$	57.1918	1.85263	0.926313	−	0.376756i	$-0.122960\pi$
$953$	57.1918	1.85263	0.926313	+	0.376756i	$0.122960\pi$
$954$	0	0
$955$	−17.7980	−0.575928
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−28.0000	−0.904167
$960$	0	0
$961$	−30.7980	−0.993483
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−19.7980	−0.637319
$966$	0	0
$967$	−25.1010	−0.807194	−0.403597	−	0.914937i	$-0.632240\pi$
$967$	−25.1010	−0.807194	−0.403597	+	0.914937i	$0.632240\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	27.5959	0.885595	0.442798	−	0.896622i	$-0.353986\pi$
$971$	27.5959	0.885595	0.442798	+	0.896622i	$0.353986\pi$
$972$	0	0
$973$	−14.2020	−0.455297
$974$	0	0
$975$	0	0
$976$	0	0
$977$	1.30306	0.0416886	0.0208443	−	0.999783i	$-0.493365\pi$
$977$	1.30306	0.0416886	0.0208443	+	0.999783i	$0.493365\pi$
$978$	0	0
$979$	−2.69694	−0.0861945
$980$	0	0
$981$	0	0
$982$	0	0
$983$	58.8990	1.87859	0.939293	−	0.343117i	$-0.111483\pi$
$983$	58.8990	1.87859	0.939293	+	0.343117i	$0.111483\pi$
$984$	0	0
$985$	−4.20204	−0.133888
$986$	0	0
$987$	0	0
$988$	0	0
$989$	5.19184	0.165091
$990$	0	0
$991$	30.2020	0.959399	0.479700	−	0.877433i	$-0.340746\pi$
$991$	30.2020	0.959399	0.479700	+	0.877433i	$0.340746\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−13.7980	−0.437425
$996$	0	0
$997$	25.1010	0.794957	0.397479	−	0.917611i	$-0.369885\pi$
$997$	25.1010	0.794957	0.397479	+	0.917611i	$0.369885\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9360.2.a.cc.1.2		2
3.2	odd	2		1040.2.a.k.1.1		2
4.3	odd	2		4680.2.a.z.1.1		2
12.11	even	2		520.2.a.f.1.2	✓	2
15.14	odd	2		5200.2.a.bv.1.2		2
24.5	odd	2		4160.2.a.ba.1.2		2
24.11	even	2		4160.2.a.bg.1.1		2
60.23	odd	4		2600.2.d.h.1249.3		4
60.47	odd	4		2600.2.d.h.1249.2		4
60.59	even	2		2600.2.a.q.1.1		2
156.155	even	2		6760.2.a.s.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
520.2.a.f.1.2	✓	2	12.11	even	2
1040.2.a.k.1.1		2	3.2	odd	2
2600.2.a.q.1.1		2	60.59	even	2
2600.2.d.h.1249.2		4	60.47	odd	4
2600.2.d.h.1249.3		4	60.23	odd	4
4160.2.a.ba.1.2		2	24.5	odd	2
4160.2.a.bg.1.1		2	24.11	even	2
4680.2.a.z.1.1		2	4.3	odd	2
5200.2.a.bv.1.2		2	15.14	odd	2
6760.2.a.s.1.2		2	156.155	even	2
9360.2.a.cc.1.2		2	1.1	even	1	trivial

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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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{5} -2.00000 q^{7} +O(q^{10})\)	\(q-1.00000 q^{5} -2.00000 q^{7} +0.449490 q^{11} -1.00000 q^{13} +2.89898 q^{17} +4.44949 q^{19} +1.55051 q^{23} +1.00000 q^{25} -4.00000 q^{29} -0.449490 q^{31} +2.00000 q^{35} -4.89898 q^{37} -1.10102 q^{41} +3.34847 q^{43} -2.00000 q^{47} -3.00000 q^{49} -10.8990 q^{53} -0.449490 q^{55} -5.34847 q^{59} +13.7980 q^{61} +1.00000 q^{65} +14.8990 q^{67} -8.44949 q^{71} +14.6969 q^{73} -0.898979 q^{77} +4.89898 q^{79} +2.00000 q^{83} -2.89898 q^{85} -6.00000 q^{89} +2.00000 q^{91} -4.44949 q^{95} -11.7980 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{5} - 4 q^{7}+O(q^{10}) \) 2 * q - 2 * q^5 - 4 * q^7	\( 2 q - 2 q^{5} - 4 q^{7} - 4 q^{11} - 2 q^{13} - 4 q^{17} + 4 q^{19} + 8 q^{23} + 2 q^{25} - 8 q^{29} + 4 q^{31} + 4 q^{35} - 12 q^{41} - 8 q^{43} - 4 q^{47} - 6 q^{49} - 12 q^{53} + 4 q^{55} + 4 q^{59} + 8 q^{61} + 2 q^{65} + 20 q^{67} - 12 q^{71} + 8 q^{77} + 4 q^{83} + 4 q^{85} - 12 q^{89} + 4 q^{91} - 4 q^{95} - 4 q^{97}+O(q^{100}) \) 2 * q - 2 * q^5 - 4 * q^7 - 4 * q^11 - 2 * q^13 - 4 * q^17 + 4 * q^19 + 8 * q^23 + 2 * q^25 - 8 * q^29 + 4 * q^31 + 4 * q^35 - 12 * q^41 - 8 * q^43 - 4 * q^47 - 6 * q^49 - 12 * q^53 + 4 * q^55 + 4 * q^59 + 8 * q^61 + 2 * q^65 + 20 * q^67 - 12 * q^71 + 8 * q^77 + 4 * q^83 + 4 * q^85 - 12 * q^89 + 4 * q^91 - 4 * q^95 - 4 * q^97

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