LMFDB - Embedded newform 9360.2.a.dd.1.3

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:	$3$
Coefficient field:	3.3.316.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 4x + 2 $ x^3 - x^2 - 4*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 195)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.470683$ 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)

$f(q)$	$=$	$q+1.00000 q^{5} +2.71982 q^{7} +O(q^{10})$	$q+1.00000 q^{5} +2.71982 q^{7} -2.71982 q^{11} +1.00000 q^{13} +2.83709 q^{17} +3.55691 q^{19} -4.83709 q^{23} +1.00000 q^{25} -6.00000 q^{29} -7.55691 q^{31} +2.71982 q^{35} -4.27674 q^{37} -2.83709 q^{41} -11.1138 q^{43} -11.5569 q^{47} +0.397442 q^{49} -1.16291 q^{53} -2.71982 q^{55} -2.11727 q^{59} +6.60256 q^{61} +1.00000 q^{65} -1.88273 q^{67} -6.71982 q^{71} +9.11383 q^{73} -7.39744 q^{77} -10.2767 q^{79} +2.11727 q^{83} +2.83709 q^{85} -1.16291 q^{89} +2.71982 q^{91} +3.55691 q^{95} -10.8371 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{5} - q^{7}+O(q^{10}) $ 3 * q + 3 * q^5 - q^7	$ 3 q + 3 q^{5} - q^{7} + q^{11} + 3 q^{13} + q^{17} - 6 q^{19} - 7 q^{23} + 3 q^{25} - 18 q^{29} - 6 q^{31} - q^{35} + 13 q^{37} - q^{41} - 18 q^{47} + 12 q^{49} - 11 q^{53} + q^{55} - 8 q^{59} + 9 q^{61} + 3 q^{65} - 4 q^{67} - 11 q^{71} - 6 q^{73} - 33 q^{77} - 5 q^{79} + 8 q^{83} + q^{85} - 11 q^{89} - q^{91} - 6 q^{95} - 25 q^{97}+O(q^{100}) $ 3 * q + 3 * q^5 - q^7 + q^11 + 3 * q^13 + q^17 - 6 * q^19 - 7 * q^23 + 3 * q^25 - 18 * q^29 - 6 * q^31 - q^35 + 13 * q^37 - q^41 - 18 * q^47 + 12 * q^49 - 11 * q^53 + q^55 - 8 * q^59 + 9 * q^61 + 3 * q^65 - 4 * q^67 - 11 * q^71 - 6 * q^73 - 33 * q^77 - 5 * q^79 + 8 * q^83 + q^85 - 11 * q^89 - q^91 - 6 * q^95 - 25 * 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
$7$	2.71982	1.02800	0.513998	−	0.857791i	$-0.328164\pi$
$7$	2.71982	1.02800	0.513998	+	0.857791i	$0.328164\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−2.71982	−0.820058	−0.410029	−	0.912073i	$-0.634481\pi$
$11$	−2.71982	−0.820058	−0.410029	+	0.912073i	$0.634481\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	0	0
$16$	0	0
$17$	2.83709	0.688095	0.344048	−	0.938952i	$-0.388202\pi$
$17$	2.83709	0.688095	0.344048	+	0.938952i	$0.388202\pi$
$18$	0	0
$19$	3.55691	0.816012	0.408006	−	0.912979i	$-0.366224\pi$
$19$	3.55691	0.816012	0.408006	+	0.912979i	$0.366224\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−4.83709	−1.00860	−0.504302	−	0.863528i	$-0.668250\pi$
$23$	−4.83709	−1.00860	−0.504302	+	0.863528i	$0.668250\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−6.00000	−1.11417	−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−6.00000	−1.11417	−0.557086	+	0.830455i	$0.688081\pi$
$30$	0	0
$31$	−7.55691	−1.35726	−0.678631	−	0.734479i	$-0.737426\pi$
$31$	−7.55691	−1.35726	−0.678631	+	0.734479i	$0.737426\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	2.71982	0.459734
$36$	0	0
$37$	−4.27674	−0.703091	−0.351546	−	0.936171i	$-0.614344\pi$
$37$	−4.27674	−0.703091	−0.351546	+	0.936171i	$0.614344\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−2.83709	−0.443079	−0.221540	−	0.975151i	$-0.571108\pi$
$41$	−2.83709	−0.443079	−0.221540	+	0.975151i	$0.571108\pi$
$42$	0	0
$43$	−11.1138	−1.69484	−0.847421	−	0.530921i	$-0.821846\pi$
$43$	−11.1138	−1.69484	−0.847421	+	0.530921i	$0.821846\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−11.5569	−1.68575	−0.842875	−	0.538110i	$-0.819138\pi$
$47$	−11.5569	−1.68575	−0.842875	+	0.538110i	$0.819138\pi$
$48$	0	0
$49$	0.397442	0.0567775
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−1.16291	−0.159738	−0.0798690	−	0.996805i	$-0.525450\pi$
$53$	−1.16291	−0.159738	−0.0798690	+	0.996805i	$0.525450\pi$
$54$	0	0
$55$	−2.71982	−0.366741
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−2.11727	−0.275645	−0.137822	−	0.990457i	$-0.544010\pi$
$59$	−2.11727	−0.275645	−0.137822	+	0.990457i	$0.544010\pi$
$60$	0	0
$61$	6.60256	0.845371	0.422685	−	0.906276i	$-0.361088\pi$
$61$	6.60256	0.845371	0.422685	+	0.906276i	$0.361088\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	1.00000	0.124035
$66$	0	0
$67$	−1.88273	−0.230013	−0.115006	−	0.993365i	$-0.536689\pi$
$67$	−1.88273	−0.230013	−0.115006	+	0.993365i	$0.536689\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−6.71982	−0.797496	−0.398748	−	0.917060i	$-0.630555\pi$
$71$	−6.71982	−0.797496	−0.398748	+	0.917060i	$0.630555\pi$
$72$	0	0
$73$	9.11383	1.06669	0.533346	−	0.845897i	$-0.320934\pi$
$73$	9.11383	1.06669	0.533346	+	0.845897i	$0.320934\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−7.39744	−0.843017
$78$	0	0
$79$	−10.2767	−1.15622	−0.578112	−	0.815958i	$-0.696210\pi$
$79$	−10.2767	−1.15622	−0.578112	+	0.815958i	$0.696210\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	2.11727	0.232400	0.116200	−	0.993226i	$-0.462929\pi$
$83$	2.11727	0.232400	0.116200	+	0.993226i	$0.462929\pi$
$84$	0	0
$85$	2.83709	0.307726
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−1.16291	−0.123268	−0.0616341	−	0.998099i	$-0.519631\pi$
$89$	−1.16291	−0.123268	−0.0616341	+	0.998099i	$0.519631\pi$
$90$	0	0
$91$	2.71982	0.285115
$92$	0	0
$93$	0	0
$94$	0	0
$95$	3.55691	0.364932
$96$	0	0
$97$	−10.8371	−1.10034	−0.550170	−	0.835053i	$-0.685437\pi$
$97$	−10.8371	−1.10034	−0.550170	+	0.835053i	$0.685437\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−7.67418	−0.763610	−0.381805	−	0.924243i	$-0.624697\pi$
$101$	−7.67418	−0.763610	−0.381805	+	0.924243i	$0.624697\pi$
$102$	0	0
$103$	−3.76547	−0.371023	−0.185511	−	0.982642i	$-0.559394\pi$
$103$	−3.76547	−0.371023	−0.185511	+	0.982642i	$0.559394\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−12.6026	−1.21834	−0.609168	−	0.793041i	$-0.708496\pi$
$107$	−12.6026	−1.21834	−0.609168	+	0.793041i	$0.708496\pi$
$108$	0	0
$109$	11.4396	1.09572	0.547860	−	0.836570i	$-0.315443\pi$
$109$	11.4396	1.09572	0.547860	+	0.836570i	$0.315443\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	13.1138	1.23365	0.616823	−	0.787102i	$-0.288420\pi$
$113$	13.1138	1.23365	0.616823	+	0.787102i	$0.288420\pi$
$114$	0	0
$115$	−4.83709	−0.451061
$116$	0	0
$117$	0	0
$118$	0	0
$119$	7.71639	0.707360
$120$	0	0
$121$	−3.60256	−0.327505
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	13.4396	1.19258	0.596288	−	0.802771i	$-0.296642\pi$
$127$	13.4396	1.19258	0.596288	+	0.802771i	$0.296642\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−9.43965	−0.824746	−0.412373	−	0.911015i	$-0.635300\pi$
$131$	−9.43965	−0.824746	−0.412373	+	0.911015i	$0.635300\pi$
$132$	0	0
$133$	9.67418	0.838858
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−1.76547	−0.150834	−0.0754170	−	0.997152i	$-0.524029\pi$
$137$	−1.76547	−0.150834	−0.0754170	+	0.997152i	$0.524029\pi$
$138$	0	0
$139$	6.27674	0.532386	0.266193	−	0.963920i	$-0.414234\pi$
$139$	6.27674	0.532386	0.266193	+	0.963920i	$0.414234\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−2.71982	−0.227443
$144$	0	0
$145$	−6.00000	−0.498273
$146$	0	0
$147$	0	0
$148$	0	0
$149$	20.8302	1.70648	0.853239	−	0.521520i	$-0.174635\pi$
$149$	20.8302	1.70648	0.853239	+	0.521520i	$0.174635\pi$
$150$	0	0
$151$	−4.99656	−0.406614	−0.203307	−	0.979115i	$-0.565169\pi$
$151$	−4.99656	−0.406614	−0.203307	+	0.979115i	$0.565169\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−7.55691	−0.606986
$156$	0	0
$157$	8.87930	0.708645	0.354322	−	0.935123i	$-0.384712\pi$
$157$	8.87930	0.708645	0.354322	+	0.935123i	$0.384712\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−13.1560	−1.03684
$162$	0	0
$163$	13.8337	1.08354	0.541768	−	0.840528i	$-0.317755\pi$
$163$	13.8337	1.08354	0.541768	+	0.840528i	$0.317755\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−9.88273	−0.764749	−0.382374	−	0.924007i	$-0.624894\pi$
$167$	−9.88273	−0.764749	−0.382374	+	0.924007i	$0.624894\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−13.1138	−0.997026	−0.498513	−	0.866882i	$-0.666120\pi$
$173$	−13.1138	−0.997026	−0.498513	+	0.866882i	$0.666120\pi$
$174$	0	0
$175$	2.71982	0.205599
$176$	0	0
$177$	0	0
$178$	0	0
$179$	8.55348	0.639317	0.319658	−	0.947533i	$-0.396432\pi$
$179$	8.55348	0.639317	0.319658	+	0.947533i	$0.396432\pi$
$180$	0	0
$181$	3.72326	0.276748	0.138374	−	0.990380i	$-0.455812\pi$
$181$	3.72326	0.276748	0.138374	+	0.990380i	$0.455812\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−4.27674	−0.314432
$186$	0	0
$187$	−7.71639	−0.564278
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−4.23453	−0.306400	−0.153200	−	0.988195i	$-0.548958\pi$
$191$	−4.23453	−0.306400	−0.153200	+	0.988195i	$0.548958\pi$
$192$	0	0
$193$	−23.3906	−1.68369	−0.841845	−	0.539719i	$-0.818530\pi$
$193$	−23.3906	−1.68369	−0.841845	+	0.539719i	$0.818530\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−14.5535	−1.03689	−0.518446	−	0.855110i	$-0.673489\pi$
$197$	−14.5535	−1.03689	−0.518446	+	0.855110i	$0.673489\pi$
$198$	0	0
$199$	15.1138	1.07139	0.535695	−	0.844411i	$-0.320050\pi$
$199$	15.1138	1.07139	0.535695	+	0.844411i	$0.320050\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−16.3189	−1.14537
$204$	0	0
$205$	−2.83709	−0.198151
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−9.67418	−0.669177
$210$	0	0
$211$	18.2277	1.25484	0.627422	−	0.778680i	$-0.284110\pi$
$211$	18.2277	1.25484	0.627422	+	0.778680i	$0.284110\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−11.1138	−0.757957
$216$	0	0
$217$	−20.5535	−1.39526
$218$	0	0
$219$	0	0
$220$	0	0
$221$	2.83709	0.190843
$222$	0	0
$223$	−10.1173	−0.677502	−0.338751	−	0.940876i	$-0.610004\pi$
$223$	−10.1173	−0.677502	−0.338751	+	0.940876i	$0.610004\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	11.3224	0.751493	0.375746	−	0.926723i	$-0.377386\pi$
$227$	11.3224	0.751493	0.375746	+	0.926723i	$0.377386\pi$
$228$	0	0
$229$	−6.23453	−0.411990	−0.205995	−	0.978553i	$-0.566043\pi$
$229$	−6.23453	−0.411990	−0.205995	+	0.978553i	$0.566043\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−6.83709	−0.447913	−0.223956	−	0.974599i	$-0.571897\pi$
$233$	−6.83709	−0.447913	−0.223956	+	0.974599i	$0.571897\pi$
$234$	0	0
$235$	−11.5569	−0.753890
$236$	0	0
$237$	0	0
$238$	0	0
$239$	1.28018	0.0828077	0.0414039	−	0.999142i	$-0.486817\pi$
$239$	1.28018	0.0828077	0.0414039	+	0.999142i	$0.486817\pi$
$240$	0	0
$241$	−6.00000	−0.386494	−0.193247	−	0.981150i	$-0.561902\pi$
$241$	−6.00000	−0.386494	−0.193247	+	0.981150i	$0.561902\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0.397442	0.0253917
$246$	0	0
$247$	3.55691	0.226321
$248$	0	0
$249$	0	0
$250$	0	0
$251$	18.2277	1.15052	0.575260	−	0.817971i	$-0.304901\pi$
$251$	18.2277	1.15052	0.575260	+	0.817971i	$0.304901\pi$
$252$	0	0
$253$	13.1560	0.827113
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−1.11383	−0.0694787	−0.0347394	−	0.999396i	$-0.511060\pi$
$257$	−1.11383	−0.0694787	−0.0347394	+	0.999396i	$0.511060\pi$
$258$	0	0
$259$	−11.6320	−0.722776
$260$	0	0
$261$	0	0
$262$	0	0
$263$	8.00000	0.493301	0.246651	−	0.969104i	$-0.420670\pi$
$263$	8.00000	0.493301	0.246651	+	0.969104i	$0.420670\pi$
$264$	0	0
$265$	−1.16291	−0.0714370
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−15.6742	−0.955672	−0.477836	−	0.878449i	$-0.658579\pi$
$269$	−15.6742	−0.955672	−0.477836	+	0.878449i	$0.658579\pi$
$270$	0	0
$271$	−0.443086	−0.0269155	−0.0134578	−	0.999909i	$-0.504284\pi$
$271$	−0.443086	−0.0269155	−0.0134578	+	0.999909i	$0.504284\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−2.71982	−0.164012
$276$	0	0
$277$	−4.87930	−0.293168	−0.146584	−	0.989198i	$-0.546828\pi$
$277$	−4.87930	−0.293168	−0.146584	+	0.989198i	$0.546828\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	9.11383	0.543685	0.271843	−	0.962342i	$-0.412367\pi$
$281$	9.11383	0.543685	0.271843	+	0.962342i	$0.412367\pi$
$282$	0	0
$283$	−33.3415	−1.98195	−0.990973	−	0.134063i	$-0.957198\pi$
$283$	−33.3415	−1.98195	−0.990973	+	0.134063i	$0.957198\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−7.71639	−0.455484
$288$	0	0
$289$	−8.95092	−0.526525
$290$	0	0
$291$	0	0
$292$	0	0
$293$	29.4328	1.71948	0.859740	−	0.510731i	$-0.170625\pi$
$293$	29.4328	1.71948	0.859740	+	0.510731i	$0.170625\pi$
$294$	0	0
$295$	−2.11727	−0.123272
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−4.83709	−0.279736
$300$	0	0
$301$	−30.2277	−1.74229
$302$	0	0
$303$	0	0
$304$	0	0
$305$	6.60256	0.378061
$306$	0	0
$307$	21.8337	1.24611	0.623056	−	0.782177i	$-0.285891\pi$
$307$	21.8337	1.24611	0.623056	+	0.782177i	$0.285891\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	25.1070	1.42368	0.711842	−	0.702339i	$-0.247861\pi$
$311$	25.1070	1.42368	0.711842	+	0.702339i	$0.247861\pi$
$312$	0	0
$313$	8.22766	0.465055	0.232527	−	0.972590i	$-0.425300\pi$
$313$	8.22766	0.465055	0.232527	+	0.972590i	$0.425300\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−27.6742	−1.55434	−0.777168	−	0.629293i	$-0.783345\pi$
$317$	−27.6742	−1.55434	−0.777168	+	0.629293i	$0.783345\pi$
$318$	0	0
$319$	16.3189	0.913685
$320$	0	0
$321$	0	0
$322$	0	0
$323$	10.0913	0.561494
$324$	0	0
$325$	1.00000	0.0554700
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−31.4328	−1.73294
$330$	0	0
$331$	13.2311	0.727247	0.363623	−	0.931546i	$-0.381540\pi$
$331$	13.2311	0.727247	0.363623	+	0.931546i	$0.381540\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−1.88273	−0.102865
$336$	0	0
$337$	−4.32582	−0.235642	−0.117821	−	0.993035i	$-0.537591\pi$
$337$	−4.32582	−0.235642	−0.117821	+	0.993035i	$0.537591\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	20.5535	1.11303
$342$	0	0
$343$	−17.9578	−0.969630
$344$	0	0
$345$	0	0
$346$	0	0
$347$	6.27674	0.336953	0.168476	−	0.985706i	$-0.446115\pi$
$347$	6.27674	0.336953	0.168476	+	0.985706i	$0.446115\pi$
$348$	0	0
$349$	17.6673	0.945709	0.472855	−	0.881140i	$-0.343224\pi$
$349$	17.6673	0.945709	0.472855	+	0.881140i	$0.343224\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	13.7655	0.732662	0.366331	−	0.930485i	$-0.380614\pi$
$353$	13.7655	0.732662	0.366331	+	0.930485i	$0.380614\pi$
$354$	0	0
$355$	−6.71982	−0.356651
$356$	0	0
$357$	0	0
$358$	0	0
$359$	0.996562	0.0525965	0.0262983	−	0.999654i	$-0.491628\pi$
$359$	0.996562	0.0525965	0.0262983	+	0.999654i	$0.491628\pi$
$360$	0	0
$361$	−6.34836	−0.334124
$362$	0	0
$363$	0	0
$364$	0	0
$365$	9.11383	0.477040
$366$	0	0
$367$	−14.2277	−0.742678	−0.371339	−	0.928497i	$-0.621101\pi$
$367$	−14.2277	−0.742678	−0.371339	+	0.928497i	$0.621101\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−3.16291	−0.164210
$372$	0	0
$373$	15.6742	0.811578	0.405789	−	0.913967i	$-0.366997\pi$
$373$	15.6742	0.811578	0.405789	+	0.913967i	$0.366997\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−6.00000	−0.309016
$378$	0	0
$379$	−26.2017	−1.34589	−0.672945	−	0.739693i	$-0.734971\pi$
$379$	−26.2017	−1.34589	−0.672945	+	0.739693i	$0.734971\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	22.4362	1.14644	0.573218	−	0.819403i	$-0.305695\pi$
$383$	22.4362	1.14644	0.573218	+	0.819403i	$0.305695\pi$
$384$	0	0
$385$	−7.39744	−0.377009
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−31.6742	−1.60594	−0.802972	−	0.596016i	$-0.796749\pi$
$389$	−31.6742	−1.60594	−0.802972	+	0.596016i	$0.796749\pi$
$390$	0	0
$391$	−13.7233	−0.694015
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−10.2767	−0.517079
$396$	0	0
$397$	17.9509	0.900931	0.450465	−	0.892794i	$-0.351258\pi$
$397$	17.9509	0.900931	0.450465	+	0.892794i	$0.351258\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−13.5829	−0.678297	−0.339149	−	0.940733i	$-0.610139\pi$
$401$	−13.5829	−0.678297	−0.339149	+	0.940733i	$0.610139\pi$
$402$	0	0
$403$	−7.55691	−0.376437
$404$	0	0
$405$	0	0
$406$	0	0
$407$	11.6320	0.576576
$408$	0	0
$409$	−14.0000	−0.692255	−0.346128	−	0.938187i	$-0.612504\pi$
$409$	−14.0000	−0.692255	−0.346128	+	0.938187i	$0.612504\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−5.75859	−0.283362
$414$	0	0
$415$	2.11727	0.103933
$416$	0	0
$417$	0	0
$418$	0	0
$419$	12.3189	0.601820	0.300910	−	0.953653i	$-0.402710\pi$
$419$	12.3189	0.601820	0.300910	+	0.953653i	$0.402710\pi$
$420$	0	0
$421$	22.7880	1.11062	0.555310	−	0.831644i	$-0.312600\pi$
$421$	22.7880	1.11062	0.555310	+	0.831644i	$0.312600\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	2.83709	0.137619
$426$	0	0
$427$	17.9578	0.869039
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−8.99656	−0.433349	−0.216675	−	0.976244i	$-0.569521\pi$
$431$	−8.99656	−0.433349	−0.216675	+	0.976244i	$0.569521\pi$
$432$	0	0
$433$	−20.3258	−0.976797	−0.488398	−	0.872621i	$-0.662419\pi$
$433$	−20.3258	−0.976797	−0.488398	+	0.872621i	$0.662419\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−17.2051	−0.823032
$438$	0	0
$439$	25.3906	1.21183	0.605913	−	0.795531i	$-0.292808\pi$
$439$	25.3906	1.21183	0.605913	+	0.795531i	$0.292808\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	10.9284	0.519223	0.259611	−	0.965713i	$-0.416406\pi$
$443$	10.9284	0.519223	0.259611	+	0.965713i	$0.416406\pi$
$444$	0	0
$445$	−1.16291	−0.0551272
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−2.83709	−0.133891	−0.0669453	−	0.997757i	$-0.521325\pi$
$449$	−2.83709	−0.133891	−0.0669453	+	0.997757i	$0.521325\pi$
$450$	0	0
$451$	7.71639	0.363350
$452$	0	0
$453$	0	0
$454$	0	0
$455$	2.71982	0.127507
$456$	0	0
$457$	−13.7164	−0.641625	−0.320813	−	0.947143i	$-0.603956\pi$
$457$	−13.7164	−0.641625	−0.320813	+	0.947143i	$0.603956\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	19.6251	0.914032	0.457016	−	0.889458i	$-0.348918\pi$
$461$	19.6251	0.914032	0.457016	+	0.889458i	$0.348918\pi$
$462$	0	0
$463$	−27.0388	−1.25660	−0.628299	−	0.777972i	$-0.716249\pi$
$463$	−27.0388	−1.25660	−0.628299	+	0.777972i	$0.716249\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	28.9215	1.33833	0.669164	−	0.743115i	$-0.266652\pi$
$467$	28.9215	1.33833	0.669164	+	0.743115i	$0.266652\pi$
$468$	0	0
$469$	−5.12070	−0.236452
$470$	0	0
$471$	0	0
$472$	0	0
$473$	30.2277	1.38987
$474$	0	0
$475$	3.55691	0.163202
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−12.1595	−0.555580	−0.277790	−	0.960642i	$-0.589602\pi$
$479$	−12.1595	−0.555580	−0.277790	+	0.960642i	$0.589602\pi$
$480$	0	0
$481$	−4.27674	−0.195002
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−10.8371	−0.492087
$486$	0	0
$487$	0.159472	0.00722636	0.00361318	−	0.999993i	$-0.498850\pi$
$487$	0.159472	0.00722636	0.00361318	+	0.999993i	$0.498850\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−42.2277	−1.90571	−0.952854	−	0.303430i	$-0.901868\pi$
$491$	−42.2277	−1.90571	−0.952854	+	0.303430i	$0.901868\pi$
$492$	0	0
$493$	−17.0225	−0.766657
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−18.2767	−0.819824
$498$	0	0
$499$	−7.79145	−0.348793	−0.174397	−	0.984676i	$-0.555797\pi$
$499$	−7.79145	−0.348793	−0.174397	+	0.984676i	$0.555797\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	27.3484	1.21940	0.609702	−	0.792631i	$-0.291289\pi$
$503$	27.3484	1.21940	0.609702	+	0.792631i	$0.291289\pi$
$504$	0	0
$505$	−7.67418	−0.341497
$506$	0	0
$507$	0	0
$508$	0	0
$509$	33.4819	1.48406	0.742029	−	0.670368i	$-0.233864\pi$
$509$	33.4819	1.48406	0.742029	+	0.670368i	$0.233864\pi$
$510$	0	0
$511$	24.7880	1.09656
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−3.76547	−0.165926
$516$	0	0
$517$	31.4328	1.38241
$518$	0	0
$519$	0	0
$520$	0	0
$521$	17.3484	0.760045	0.380023	−	0.924977i	$-0.375916\pi$
$521$	17.3484	0.760045	0.380023	+	0.924977i	$0.375916\pi$
$522$	0	0
$523$	4.00000	0.174908	0.0874539	−	0.996169i	$-0.472127\pi$
$523$	4.00000	0.174908	0.0874539	+	0.996169i	$0.472127\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−21.4396	−0.933926
$528$	0	0
$529$	0.397442	0.0172801
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−2.83709	−0.122888
$534$	0	0
$535$	−12.6026	−0.544856
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−1.08097	−0.0465608
$540$	0	0
$541$	−32.6448	−1.40351	−0.701754	−	0.712419i	$-0.747599\pi$
$541$	−32.6448	−1.40351	−0.701754	+	0.712419i	$0.747599\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	11.4396	0.490021
$546$	0	0
$547$	−34.2277	−1.46347	−0.731734	−	0.681590i	$-0.761289\pi$
$547$	−34.2277	−1.46347	−0.731734	+	0.681590i	$0.761289\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−21.3415	−0.909178
$552$	0	0
$553$	−27.9509	−1.18859
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−6.65164	−0.281839	−0.140919	−	0.990021i	$-0.545006\pi$
$557$	−6.65164	−0.281839	−0.140919	+	0.990021i	$0.545006\pi$
$558$	0	0
$559$	−11.1138	−0.470065
$560$	0	0
$561$	0	0
$562$	0	0
$563$	40.2699	1.69717	0.848586	−	0.529057i	$-0.177454\pi$
$563$	40.2699	1.69717	0.848586	+	0.529057i	$0.177454\pi$
$564$	0	0
$565$	13.1138	0.551703
$566$	0	0
$567$	0	0
$568$	0	0
$569$	13.4328	0.563131	0.281566	−	0.959542i	$-0.409146\pi$
$569$	13.4328	0.563131	0.281566	+	0.959542i	$0.409146\pi$
$570$	0	0
$571$	35.7164	1.49468	0.747342	−	0.664439i	$-0.231330\pi$
$571$	35.7164	1.49468	0.747342	+	0.664439i	$0.231330\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−4.83709	−0.201721
$576$	0	0
$577$	−13.7164	−0.571021	−0.285510	−	0.958376i	$-0.592163\pi$
$577$	−13.7164	−0.571021	−0.285510	+	0.958376i	$0.592163\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	5.75859	0.238907
$582$	0	0
$583$	3.16291	0.130994
$584$	0	0
$585$	0	0
$586$	0	0
$587$	30.6707	1.26592	0.632959	−	0.774186i	$-0.281840\pi$
$587$	30.6707	1.26592	0.632959	+	0.774186i	$0.281840\pi$
$588$	0	0
$589$	−26.8793	−1.10754
$590$	0	0
$591$	0	0
$592$	0	0
$593$	45.6673	1.87533	0.937666	−	0.347538i	$-0.112982\pi$
$593$	45.6673	1.87533	0.937666	+	0.347538i	$0.112982\pi$
$594$	0	0
$595$	7.71639	0.316341
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−40.2208	−1.64338	−0.821688	−	0.569937i	$-0.806968\pi$
$599$	−40.2208	−1.64338	−0.821688	+	0.569937i	$0.806968\pi$
$600$	0	0
$601$	17.3974	0.709656	0.354828	−	0.934932i	$-0.384539\pi$
$601$	17.3974	0.709656	0.354828	+	0.934932i	$0.384539\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−3.60256	−0.146465
$606$	0	0
$607$	14.2277	0.577483	0.288741	−	0.957407i	$-0.406763\pi$
$607$	14.2277	0.577483	0.288741	+	0.957407i	$0.406763\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−11.5569	−0.467543
$612$	0	0
$613$	−40.8302	−1.64912	−0.824558	−	0.565777i	$-0.808576\pi$
$613$	−40.8302	−1.64912	−0.824558	+	0.565777i	$0.808576\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−5.11383	−0.205875	−0.102937	−	0.994688i	$-0.532824\pi$
$617$	−5.11383	−0.205875	−0.102937	+	0.994688i	$0.532824\pi$
$618$	0	0
$619$	−11.5569	−0.464512	−0.232256	−	0.972655i	$-0.574611\pi$
$619$	−11.5569	−0.464512	−0.232256	+	0.972655i	$0.574611\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−3.16291	−0.126719
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−12.1335	−0.483794
$630$	0	0
$631$	−35.2242	−1.40225	−0.701127	−	0.713036i	$-0.747319\pi$
$631$	−35.2242	−1.40225	−0.701127	+	0.713036i	$0.747319\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	13.4396	0.533336
$636$	0	0
$637$	0.397442	0.0157472
$638$	0	0
$639$	0	0
$640$	0	0
$641$	21.9018	0.865071	0.432535	−	0.901617i	$-0.357619\pi$
$641$	21.9018	0.865071	0.432535	+	0.901617i	$0.357619\pi$
$642$	0	0
$643$	−7.50783	−0.296080	−0.148040	−	0.988981i	$-0.547296\pi$
$643$	−7.50783	−0.296080	−0.148040	+	0.988981i	$0.547296\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−14.0422	−0.552056	−0.276028	−	0.961150i	$-0.589018\pi$
$647$	−14.0422	−0.552056	−0.276028	+	0.961150i	$0.589018\pi$
$648$	0	0
$649$	5.75859	0.226044
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−7.99312	−0.312795	−0.156398	−	0.987694i	$-0.549988\pi$
$653$	−7.99312	−0.312795	−0.156398	+	0.987694i	$0.549988\pi$
$654$	0	0
$655$	−9.43965	−0.368838
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−25.3415	−0.987164	−0.493582	−	0.869699i	$-0.664313\pi$
$659$	−25.3415	−0.987164	−0.493582	+	0.869699i	$0.664313\pi$
$660$	0	0
$661$	27.4396	1.06728	0.533639	−	0.845712i	$-0.320824\pi$
$661$	27.4396	1.06728	0.533639	+	0.845712i	$0.320824\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	9.67418	0.375149
$666$	0	0
$667$	29.0225	1.12376
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−17.9578	−0.693253
$672$	0	0
$673$	27.1070	1.04490	0.522448	−	0.852671i	$-0.325019\pi$
$673$	27.1070	1.04490	0.522448	+	0.852671i	$0.325019\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	36.5957	1.40649	0.703243	−	0.710949i	$-0.251735\pi$
$677$	36.5957	1.40649	0.703243	+	0.710949i	$0.251735\pi$
$678$	0	0
$679$	−29.4750	−1.13115
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−13.4656	−0.515248	−0.257624	−	0.966245i	$-0.582940\pi$
$683$	−13.4656	−0.515248	−0.257624	+	0.966245i	$0.582940\pi$
$684$	0	0
$685$	−1.76547	−0.0674550
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−1.16291	−0.0443033
$690$	0	0
$691$	−29.5500	−1.12414	−0.562068	−	0.827091i	$-0.689994\pi$
$691$	−29.5500	−1.12414	−0.562068	+	0.827091i	$0.689994\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	6.27674	0.238090
$696$	0	0
$697$	−8.04908	−0.304881
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−43.6604	−1.64903	−0.824516	−	0.565839i	$-0.808552\pi$
$701$	−43.6604	−1.64903	−0.824516	+	0.565839i	$0.808552\pi$
$702$	0	0
$703$	−15.2120	−0.573731
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−20.8724	−0.784988
$708$	0	0
$709$	−26.7880	−1.00604	−0.503022	−	0.864273i	$-0.667779\pi$
$709$	−26.7880	−1.00604	−0.503022	+	0.864273i	$0.667779\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	36.5535	1.36894
$714$	0	0
$715$	−2.71982	−0.101716
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−34.8793	−1.30078	−0.650389	−	0.759601i	$-0.725394\pi$
$719$	−34.8793	−1.30078	−0.650389	+	0.759601i	$0.725394\pi$
$720$	0	0
$721$	−10.2414	−0.381410
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−6.00000	−0.222834
$726$	0	0
$727$	−37.4396	−1.38856	−0.694280	−	0.719705i	$-0.744277\pi$
$727$	−37.4396	−1.38856	−0.694280	+	0.719705i	$0.744277\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−31.5309	−1.16621
$732$	0	0
$733$	−47.1560	−1.74175	−0.870874	−	0.491506i	$-0.836446\pi$
$733$	−47.1560	−1.74175	−0.870874	+	0.491506i	$0.836446\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	5.12070	0.188624
$738$	0	0
$739$	31.7914	1.16947	0.584734	−	0.811225i	$-0.301199\pi$
$739$	31.7914	1.16947	0.584734	+	0.811225i	$0.301199\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	16.6776	0.611842	0.305921	−	0.952057i	$-0.401036\pi$
$743$	16.6776	0.611842	0.305921	+	0.952057i	$0.401036\pi$
$744$	0	0
$745$	20.8302	0.763160
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−34.2767	−1.25244
$750$	0	0
$751$	−16.1855	−0.590616	−0.295308	−	0.955402i	$-0.595422\pi$
$751$	−16.1855	−0.590616	−0.295308	+	0.955402i	$0.595422\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−4.99656	−0.181844
$756$	0	0
$757$	12.3258	0.447990	0.223995	−	0.974590i	$-0.428090\pi$
$757$	12.3258	0.447990	0.223995	+	0.974590i	$0.428090\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	0.00687569	0.000249244 0	0.000124622	−	1.00000i	$-0.499960\pi$
$761$	0.00687569	0.000249244 0	0.000124622		1.00000i	$0.499960\pi$
$762$	0	0
$763$	31.1138	1.12640
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−2.11727	−0.0764501
$768$	0	0
$769$	−20.3258	−0.732968	−0.366484	−	0.930424i	$-0.619439\pi$
$769$	−20.3258	−0.732968	−0.366484	+	0.930424i	$0.619439\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−9.90184	−0.356144	−0.178072	−	0.984017i	$-0.556986\pi$
$773$	−9.90184	−0.356144	−0.178072	+	0.984017i	$0.556986\pi$
$774$	0	0
$775$	−7.55691	−0.271452
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−10.0913	−0.361558
$780$	0	0
$781$	18.2767	0.653993
$782$	0	0
$783$	0	0
$784$	0	0
$785$	8.87930	0.316916
$786$	0	0
$787$	−36.3449	−1.29556	−0.647778	−	0.761829i	$-0.724302\pi$
$787$	−36.3449	−1.29556	−0.647778	+	0.761829i	$0.724302\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	35.6673	1.26818
$792$	0	0
$793$	6.60256	0.234464
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−18.8371	−0.667244	−0.333622	−	0.942707i	$-0.608271\pi$
$797$	−18.8371	−0.667244	−0.333622	+	0.942707i	$0.608271\pi$
$798$	0	0
$799$	−32.7880	−1.15996
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−24.7880	−0.874750
$804$	0	0
$805$	−13.1560	−0.463689
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−32.2277	−1.13306	−0.566532	−	0.824040i	$-0.691715\pi$
$809$	−32.2277	−1.13306	−0.566532	+	0.824040i	$0.691715\pi$
$810$	0	0
$811$	23.0034	0.807760	0.403880	−	0.914812i	$-0.367661\pi$
$811$	23.0034	0.807760	0.403880	+	0.914812i	$0.367661\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	13.8337	0.484572
$816$	0	0
$817$	−39.5309	−1.38301
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−49.9372	−1.74282	−0.871410	−	0.490556i	$-0.836794\pi$
$821$	−49.9372	−1.74282	−0.871410	+	0.490556i	$0.836794\pi$
$822$	0	0
$823$	−28.2345	−0.984194	−0.492097	−	0.870540i	$-0.663769\pi$
$823$	−28.2345	−0.984194	−0.492097	+	0.870540i	$0.663769\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−9.55004	−0.332087	−0.166044	−	0.986118i	$-0.553099\pi$
$827$	−9.55004	−0.332087	−0.166044	+	0.986118i	$0.553099\pi$
$828$	0	0
$829$	−37.9862	−1.31932	−0.659658	−	0.751565i	$-0.729299\pi$
$829$	−37.9862	−1.31932	−0.659658	+	0.751565i	$0.729299\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	1.12758	0.0390683
$834$	0	0
$835$	−9.88273	−0.342006
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−4.72670	−0.163184	−0.0815919	−	0.996666i	$-0.526000\pi$
$839$	−4.72670	−0.163184	−0.0815919	+	0.996666i	$0.526000\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	0	0
$843$	0	0
$844$	0	0
$845$	1.00000	0.0344010
$846$	0	0
$847$	−9.79832	−0.336674
$848$	0	0
$849$	0	0
$850$	0	0
$851$	20.6870	0.709140
$852$	0	0
$853$	−48.3611	−1.65585	−0.827927	−	0.560836i	$-0.810480\pi$
$853$	−48.3611	−1.65585	−0.827927	+	0.560836i	$0.810480\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−6.83709	−0.233551	−0.116775	−	0.993158i	$-0.537256\pi$
$857$	−6.83709	−0.233551	−0.116775	+	0.993158i	$0.537256\pi$
$858$	0	0
$859$	−12.6026	−0.429994	−0.214997	−	0.976615i	$-0.568974\pi$
$859$	−12.6026	−0.429994	−0.214997	+	0.976615i	$0.568974\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	8.20855	0.279422	0.139711	−	0.990192i	$-0.455383\pi$
$863$	8.20855	0.279422	0.139711	+	0.990192i	$0.455383\pi$
$864$	0	0
$865$	−13.1138	−0.445884
$866$	0	0
$867$	0	0
$868$	0	0
$869$	27.9509	0.948170
$870$	0	0
$871$	−1.88273	−0.0637940
$872$	0	0
$873$	0	0
$874$	0	0
$875$	2.71982	0.0919468
$876$	0	0
$877$	13.5309	0.456907	0.228454	−	0.973555i	$-0.426633\pi$
$877$	13.5309	0.456907	0.228454	+	0.973555i	$0.426633\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	9.34836	0.314954	0.157477	−	0.987523i	$-0.449664\pi$
$881$	9.34836	0.314954	0.157477	+	0.987523i	$0.449664\pi$
$882$	0	0
$883$	55.1001	1.85427	0.927133	−	0.374733i	$-0.122266\pi$
$883$	55.1001	1.85427	0.927133	+	0.374733i	$0.122266\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	0.133492	0.00448223	0.00224112	−	0.999997i	$-0.499287\pi$
$887$	0.133492	0.00448223	0.00224112	+	0.999997i	$0.499287\pi$
$888$	0	0
$889$	36.5535	1.22596
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−41.1070	−1.37559
$894$	0	0
$895$	8.55348	0.285911
$896$	0	0
$897$	0	0
$898$	0	0
$899$	45.3415	1.51222
$900$	0	0
$901$	−3.29928	−0.109915
$902$	0	0
$903$	0	0
$904$	0	0
$905$	3.72326	0.123765
$906$	0	0
$907$	58.5466	1.94401	0.972004	−	0.234964i	$-0.0754973\pi$
$907$	58.5466	1.94401	0.972004	+	0.234964i	$0.0754973\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	50.4622	1.67189	0.835943	−	0.548816i	$-0.184921\pi$
$911$	50.4622	1.67189	0.835943	+	0.548816i	$0.184921\pi$
$912$	0	0
$913$	−5.75859	−0.190582
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−25.6742	−0.847836
$918$	0	0
$919$	−56.9735	−1.87938	−0.939691	−	0.342026i	$-0.888887\pi$
$919$	−56.9735	−1.87938	−0.939691	+	0.342026i	$0.888887\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−6.71982	−0.221186
$924$	0	0
$925$	−4.27674	−0.140618
$926$	0	0
$927$	0	0
$928$	0	0
$929$	36.5957	1.20067	0.600333	−	0.799750i	$-0.295035\pi$
$929$	36.5957	1.20067	0.600333	+	0.799750i	$0.295035\pi$
$930$	0	0
$931$	1.41367	0.0463311
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−7.71639	−0.252353
$936$	0	0
$937$	−47.1070	−1.53892	−0.769459	−	0.638697i	$-0.779474\pi$
$937$	−47.1070	−1.53892	−0.769459	+	0.638697i	$0.779474\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	36.3611	1.18534	0.592670	−	0.805446i	$-0.298074\pi$
$941$	36.3611	1.18534	0.592670	+	0.805446i	$0.298074\pi$
$942$	0	0
$943$	13.7233	0.446891
$944$	0	0
$945$	0	0
$946$	0	0
$947$	44.5795	1.44864	0.724319	−	0.689465i	$-0.242154\pi$
$947$	44.5795	1.44864	0.724319	+	0.689465i	$0.242154\pi$
$948$	0	0
$949$	9.11383	0.295847
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−19.8596	−0.643317	−0.321658	−	0.946856i	$-0.604240\pi$
$953$	−19.8596	−0.643317	−0.321658	+	0.946856i	$0.604240\pi$
$954$	0	0
$955$	−4.23453	−0.137026
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−4.80176	−0.155057
$960$	0	0
$961$	26.1070	0.842160
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−23.3906	−0.752969
$966$	0	0
$967$	−47.4068	−1.52450	−0.762250	−	0.647283i	$-0.775905\pi$
$967$	−47.4068	−1.52450	−0.762250	+	0.647283i	$0.775905\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−10.6448	−0.341607	−0.170803	−	0.985305i	$-0.554636\pi$
$971$	−10.6448	−0.341607	−0.170803	+	0.985305i	$0.554636\pi$
$972$	0	0
$973$	17.0716	0.547291
$974$	0	0
$975$	0	0
$976$	0	0
$977$	1.21199	0.0387750	0.0193875	−	0.999812i	$-0.493828\pi$
$977$	1.21199	0.0387750	0.0193875	+	0.999812i	$0.493828\pi$
$978$	0	0
$979$	3.16291	0.101087
$980$	0	0
$981$	0	0
$982$	0	0
$983$	51.8759	1.65458	0.827291	−	0.561773i	$-0.189881\pi$
$983$	51.8759	1.65458	0.827291	+	0.561773i	$0.189881\pi$
$984$	0	0
$985$	−14.5535	−0.463712
$986$	0	0
$987$	0	0
$988$	0	0
$989$	53.7586	1.70942
$990$	0	0
$991$	21.6251	0.686944	0.343472	−	0.939163i	$-0.388397\pi$
$991$	21.6251	0.686944	0.343472	+	0.939163i	$0.388397\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	15.1138	0.479141
$996$	0	0
$997$	23.2051	0.734913	0.367457	−	0.930041i	$-0.380229\pi$
$997$	23.2051	0.734913	0.367457	+	0.930041i	$0.380229\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9360.2.a.dd.1.3		3
3.2	odd	2		3120.2.a.bj.1.3		3
4.3	odd	2		585.2.a.n.1.1		3
12.11	even	2		195.2.a.e.1.3	✓	3
20.3	even	4		2925.2.c.w.2224.6		6
20.7	even	4		2925.2.c.w.2224.1		6
20.19	odd	2		2925.2.a.bh.1.3		3
52.51	odd	2		7605.2.a.bx.1.3		3
60.23	odd	4		975.2.c.i.274.1		6
60.47	odd	4		975.2.c.i.274.6		6
60.59	even	2		975.2.a.o.1.1		3
84.83	odd	2		9555.2.a.bq.1.3		3
156.155	even	2		2535.2.a.bc.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
195.2.a.e.1.3	✓	3	12.11	even	2
585.2.a.n.1.1		3	4.3	odd	2
975.2.a.o.1.1		3	60.59	even	2
975.2.c.i.274.1		6	60.23	odd	4
975.2.c.i.274.6		6	60.47	odd	4
2535.2.a.bc.1.1		3	156.155	even	2
2925.2.a.bh.1.3		3	20.19	odd	2
2925.2.c.w.2224.1		6	20.7	even	4
2925.2.c.w.2224.6		6	20.3	even	4
3120.2.a.bj.1.3		3	3.2	odd	2
7605.2.a.bx.1.3		3	52.51	odd	2
9360.2.a.dd.1.3		3	1.1	even	1	trivial
9555.2.a.bq.1.3		3	84.83	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:	\( N \)	\(=\)	\( 9360 = 2^{4} \cdot 3^{2} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9360.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{5} +2.71982 q^{7} +O(q^{10})\)	\(q+1.00000 q^{5} +2.71982 q^{7} -2.71982 q^{11} +1.00000 q^{13} +2.83709 q^{17} +3.55691 q^{19} -4.83709 q^{23} +1.00000 q^{25} -6.00000 q^{29} -7.55691 q^{31} +2.71982 q^{35} -4.27674 q^{37} -2.83709 q^{41} -11.1138 q^{43} -11.5569 q^{47} +0.397442 q^{49} -1.16291 q^{53} -2.71982 q^{55} -2.11727 q^{59} +6.60256 q^{61} +1.00000 q^{65} -1.88273 q^{67} -6.71982 q^{71} +9.11383 q^{73} -7.39744 q^{77} -10.2767 q^{79} +2.11727 q^{83} +2.83709 q^{85} -1.16291 q^{89} +2.71982 q^{91} +3.55691 q^{95} -10.8371 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{5} - q^{7}+O(q^{10}) \) 3 * q + 3 * q^5 - q^7	\( 3 q + 3 q^{5} - q^{7} + q^{11} + 3 q^{13} + q^{17} - 6 q^{19} - 7 q^{23} + 3 q^{25} - 18 q^{29} - 6 q^{31} - q^{35} + 13 q^{37} - q^{41} - 18 q^{47} + 12 q^{49} - 11 q^{53} + q^{55} - 8 q^{59} + 9 q^{61} + 3 q^{65} - 4 q^{67} - 11 q^{71} - 6 q^{73} - 33 q^{77} - 5 q^{79} + 8 q^{83} + q^{85} - 11 q^{89} - q^{91} - 6 q^{95} - 25 q^{97}+O(q^{100}) \) 3 * q + 3 * q^5 - q^7 + q^11 + 3 * q^13 + q^17 - 6 * q^19 - 7 * q^23 + 3 * q^25 - 18 * q^29 - 6 * q^31 - q^35 + 13 * q^37 - q^41 - 18 * q^47 + 12 * q^49 - 11 * q^53 + q^55 - 8 * q^59 + 9 * q^61 + 3 * q^65 - 4 * q^67 - 11 * q^71 - 6 * q^73 - 33 * q^77 - 5 * q^79 + 8 * q^83 + q^85 - 11 * q^89 - q^91 - 6 * q^95 - 25 * q^97

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