LMFDB - Embedded newform 3240.2.a.r.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3240,2,Mod(1,3240)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3240, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("3240.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3240 = 2^{3} \cdot 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3240.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:	$25.8715302549$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.564.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 5x + 3 $ x^3 - x^2 - 5*x + 3
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 360)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.51414$ of defining polynomial
Character	$\chi$	$=$	3240.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{5} -1.19325 q^{7} +O(q^{10})$	$q+1.00000 q^{5} -1.19325 q^{7} -3.32088 q^{11} +1.70739 q^{13} -6.34916 q^{17} +1.32088 q^{19} +6.86330 q^{23} +1.00000 q^{25} +2.02827 q^{29} -2.67912 q^{31} -1.19325 q^{35} +3.32088 q^{37} +2.32088 q^{41} -6.34916 q^{43} -12.7694 q^{47} -5.57615 q^{49} +1.02827 q^{53} -3.32088 q^{55} -11.6700 q^{59} -9.73566 q^{61} +1.70739 q^{65} -10.5707 q^{67} -1.06562 q^{71} +14.0565 q^{73} +3.96265 q^{77} +1.41478 q^{79} -11.8350 q^{83} -6.34916 q^{85} -11.0000 q^{89} -2.03735 q^{91} +1.32088 q^{95} +16.2553 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{5} - 5 q^{7}+O(q^{10}) $ 3 * q + 3 * q^5 - 5 * q^7	$ 3 q + 3 q^{5} - 5 q^{7} - 2 q^{11} + 2 q^{17} - 4 q^{19} - 7 q^{23} + 3 q^{25} - 7 q^{29} - 16 q^{31} - 5 q^{35} + 2 q^{37} - q^{41} + 2 q^{43} - 13 q^{47} + 10 q^{49} - 10 q^{53} - 2 q^{55} - 6 q^{59} - 11 q^{61} + q^{67} - 14 q^{71} + 16 q^{73} - 12 q^{77} - 6 q^{79} - 21 q^{83} + 2 q^{85} - 33 q^{89} - 30 q^{91} - 4 q^{95} + 30 q^{97}+O(q^{100}) $ 3 * q + 3 * q^5 - 5 * q^7 - 2 * q^11 + 2 * q^17 - 4 * q^19 - 7 * q^23 + 3 * q^25 - 7 * q^29 - 16 * q^31 - 5 * q^35 + 2 * q^37 - q^41 + 2 * q^43 - 13 * q^47 + 10 * q^49 - 10 * q^53 - 2 * q^55 - 6 * q^59 - 11 * q^61 + q^67 - 14 * q^71 + 16 * q^73 - 12 * q^77 - 6 * q^79 - 21 * q^83 + 2 * q^85 - 33 * q^89 - 30 * q^91 - 4 * q^95 + 30 * 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
$7$	−1.19325	−0.451007	−0.225504	−	0.974242i	$-0.572403\pi$
$7$	−1.19325	−0.451007	−0.225504	+	0.974242i	$0.572403\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−3.32088	−1.00128	−0.500642	−	0.865654i	$-0.666903\pi$
$11$	−3.32088	−1.00128	−0.500642	+	0.865654i	$0.666903\pi$
$12$	0	0
$13$	1.70739	0.473545	0.236772	−	0.971565i	$-0.423910\pi$
$13$	1.70739	0.473545	0.236772	+	0.971565i	$0.423910\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−6.34916	−1.53990	−0.769949	−	0.638106i	$-0.779718\pi$
$17$	−6.34916	−1.53990	−0.769949	+	0.638106i	$0.779718\pi$
$18$	0	0
$19$	1.32088	0.303032	0.151516	−	0.988455i	$-0.451585\pi$
$19$	1.32088	0.303032	0.151516	+	0.988455i	$0.451585\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	6.86330	1.43110	0.715548	−	0.698564i	$-0.246177\pi$
$23$	6.86330	1.43110	0.715548	+	0.698564i	$0.246177\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	2.02827	0.376641	0.188321	−	0.982108i	$-0.439696\pi$
$29$	2.02827	0.376641	0.188321	+	0.982108i	$0.439696\pi$
$30$	0	0
$31$	−2.67912	−0.481183	−0.240592	−	0.970626i	$-0.577341\pi$
$31$	−2.67912	−0.481183	−0.240592	+	0.970626i	$0.577341\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−1.19325	−0.201696
$36$	0	0
$37$	3.32088	0.545950	0.272975	−	0.962021i	$-0.411992\pi$
$37$	3.32088	0.545950	0.272975	+	0.962021i	$0.411992\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	2.32088	0.362461	0.181231	−	0.983441i	$-0.441992\pi$
$41$	2.32088	0.362461	0.181231	+	0.983441i	$0.441992\pi$
$42$	0	0
$43$	−6.34916	−0.968238	−0.484119	−	0.875002i	$-0.660860\pi$
$43$	−6.34916	−0.968238	−0.484119	+	0.875002i	$0.660860\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−12.7694	−1.86261	−0.931304	−	0.364242i	$-0.881328\pi$
$47$	−12.7694	−1.86261	−0.931304	+	0.364242i	$0.881328\pi$
$48$	0	0
$49$	−5.57615	−0.796593
$50$	0	0
$51$	0	0
$52$	0	0
$53$	1.02827	0.141244	0.0706221	−	0.997503i	$-0.477502\pi$
$53$	1.02827	0.141244	0.0706221	+	0.997503i	$0.477502\pi$
$54$	0	0
$55$	−3.32088	−0.447788
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−11.6700	−1.51931	−0.759655	−	0.650326i	$-0.774632\pi$
$59$	−11.6700	−1.51931	−0.759655	+	0.650326i	$0.774632\pi$
$60$	0	0
$61$	−9.73566	−1.24652	−0.623262	−	0.782013i	$-0.714193\pi$
$61$	−9.73566	−1.24652	−0.623262	+	0.782013i	$0.714193\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	1.70739	0.211776
$66$	0	0
$67$	−10.5707	−1.29141	−0.645707	−	0.763585i	$-0.723437\pi$
$67$	−10.5707	−1.29141	−0.645707	+	0.763585i	$0.723437\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−1.06562	−0.126466	−0.0632329	−	0.997999i	$-0.520141\pi$
$71$	−1.06562	−0.126466	−0.0632329	+	0.997999i	$0.520141\pi$
$72$	0	0
$73$	14.0565	1.64519	0.822597	−	0.568624i	$-0.192524\pi$
$73$	14.0565	1.64519	0.822597	+	0.568624i	$0.192524\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	3.96265	0.451586
$78$	0	0
$79$	1.41478	0.159175	0.0795875	−	0.996828i	$-0.474640\pi$
$79$	1.41478	0.159175	0.0795875	+	0.996828i	$0.474640\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−11.8350	−1.29906	−0.649531	−	0.760335i	$-0.725035\pi$
$83$	−11.8350	−1.29906	−0.649531	+	0.760335i	$0.725035\pi$
$84$	0	0
$85$	−6.34916	−0.688663
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−11.0000	−1.16600	−0.582999	−	0.812473i	$-0.698121\pi$
$89$	−11.0000	−1.16600	−0.582999	+	0.812473i	$0.698121\pi$
$90$	0	0
$91$	−2.03735	−0.213572
$92$	0	0
$93$	0	0
$94$	0	0
$95$	1.32088	0.135520
$96$	0	0
$97$	16.2553	1.65047	0.825236	−	0.564788i	$-0.191042\pi$
$97$	16.2553	1.65047	0.825236	+	0.564788i	$0.191042\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−2.38650	−0.237466	−0.118733	−	0.992926i	$-0.537883\pi$
$101$	−2.38650	−0.237466	−0.118733	+	0.992926i	$0.537883\pi$
$102$	0	0
$103$	9.76394	0.962069	0.481035	−	0.876702i	$-0.340261\pi$
$103$	9.76394	0.962069	0.481035	+	0.876702i	$0.340261\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−7.57976	−0.732763	−0.366381	−	0.930465i	$-0.619403\pi$
$107$	−7.57976	−0.732763	−0.366381	+	0.930465i	$0.619403\pi$
$108$	0	0
$109$	16.5761	1.58771	0.793854	−	0.608109i	$-0.208072\pi$
$109$	16.5761	1.58771	0.793854	+	0.608109i	$0.208072\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−21.0848	−1.98349	−0.991747	−	0.128214i	$-0.959076\pi$
$113$	−21.0848	−1.98349	−0.991747	+	0.128214i	$0.959076\pi$
$114$	0	0
$115$	6.86330	0.640006
$116$	0	0
$117$	0	0
$118$	0	0
$119$	7.57615	0.694504
$120$	0	0
$121$	0.0282739	0.00257035
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−15.4112	−1.36752	−0.683760	−	0.729707i	$-0.739657\pi$
$127$	−15.4112	−1.36752	−0.683760	+	0.729707i	$0.739657\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−3.35823	−0.293410	−0.146705	−	0.989180i	$-0.546867\pi$
$131$	−3.35823	−0.293410	−0.146705	+	0.989180i	$0.546867\pi$
$132$	0	0
$133$	−1.57615	−0.136669
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−7.61350	−0.650465	−0.325232	−	0.945634i	$-0.605443\pi$
$137$	−7.61350	−0.650465	−0.325232	+	0.945634i	$0.605443\pi$
$138$	0	0
$139$	13.2835	1.12669	0.563347	−	0.826220i	$-0.309513\pi$
$139$	13.2835	1.12669	0.563347	+	0.826220i	$0.309513\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−5.67004	−0.474153
$144$	0	0
$145$	2.02827	0.168439
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−17.9909	−1.47387	−0.736937	−	0.675961i	$-0.763729\pi$
$149$	−17.9909	−1.47387	−0.736937	+	0.675961i	$0.763729\pi$
$150$	0	0
$151$	−8.67912	−0.706296	−0.353148	−	0.935567i	$-0.614889\pi$
$151$	−8.67912	−0.706296	−0.353148	+	0.935567i	$0.614889\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−2.67912	−0.215192
$156$	0	0
$157$	18.8970	1.50815	0.754074	−	0.656790i	$-0.228086\pi$
$157$	18.8970	1.50815	0.754074	+	0.656790i	$0.228086\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−8.18964	−0.645434
$162$	0	0
$163$	−0.990927	−0.0776154	−0.0388077	−	0.999247i	$-0.512356\pi$
$163$	−0.990927	−0.0776154	−0.0388077	+	0.999247i	$0.512356\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	13.5990	1.05232	0.526160	−	0.850386i	$-0.323631\pi$
$167$	13.5990	1.05232	0.526160	+	0.850386i	$0.323631\pi$
$168$	0	0
$169$	−10.0848	−0.775756
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−0.187788	−0.0142773	−0.00713865	−	0.999975i	$-0.502272\pi$
$173$	−0.187788	−0.0142773	−0.00713865	+	0.999975i	$0.502272\pi$
$174$	0	0
$175$	−1.19325	−0.0902014
$176$	0	0
$177$	0	0
$178$	0	0
$179$	9.76394	0.729791	0.364895	−	0.931048i	$-0.381105\pi$
$179$	9.76394	0.729791	0.364895	+	0.931048i	$0.381105\pi$
$180$	0	0
$181$	−16.0848	−1.19558	−0.597788	−	0.801654i	$-0.703953\pi$
$181$	−16.0848	−1.19558	−0.597788	+	0.801654i	$0.703953\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	3.32088	0.244156
$186$	0	0
$187$	21.0848	1.54187
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−3.06562	−0.221821	−0.110910	−	0.993830i	$-0.535377\pi$
$191$	−3.06562	−0.221821	−0.110910	+	0.993830i	$0.535377\pi$
$192$	0	0
$193$	0.236063	0.0169922	0.00849609	−	0.999964i	$-0.497296\pi$
$193$	0.236063	0.0169922	0.00849609	+	0.999964i	$0.497296\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−1.74474	−0.124307	−0.0621536	−	0.998067i	$-0.519797\pi$
$197$	−1.74474	−0.124307	−0.0621536	+	0.998067i	$0.519797\pi$
$198$	0	0
$199$	−22.7922	−1.61570	−0.807848	−	0.589390i	$-0.799368\pi$
$199$	−22.7922	−1.61570	−0.807848	+	0.589390i	$0.799368\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−2.42024	−0.169868
$204$	0	0
$205$	2.32088	0.162098
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−4.38650	−0.303421
$210$	0	0
$211$	−13.3774	−0.920940	−0.460470	−	0.887675i	$-0.652319\pi$
$211$	−13.3774	−0.920940	−0.460470	+	0.887675i	$0.652319\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−6.34916	−0.433009
$216$	0	0
$217$	3.19686	0.217017
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−10.8405	−0.729210
$222$	0	0
$223$	−9.48586	−0.635220	−0.317610	−	0.948221i	$-0.602880\pi$
$223$	−9.48586	−0.635220	−0.317610	+	0.948221i	$0.602880\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	3.50867	0.232879	0.116439	−	0.993198i	$-0.462852\pi$
$227$	3.50867	0.232879	0.116439	+	0.993198i	$0.462852\pi$
$228$	0	0
$229$	18.7266	1.23749	0.618744	−	0.785593i	$-0.287642\pi$
$229$	18.7266	1.23749	0.618744	+	0.785593i	$0.287642\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	2.15044	0.140880	0.0704401	−	0.997516i	$-0.477560\pi$
$233$	2.15044	0.140880	0.0704401	+	0.997516i	$0.477560\pi$
$234$	0	0
$235$	−12.7694	−0.832984
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−25.5953	−1.65563	−0.827813	−	0.561004i	$-0.810415\pi$
$239$	−25.5953	−1.65563	−0.827813	+	0.561004i	$0.810415\pi$
$240$	0	0
$241$	4.76394	0.306872	0.153436	−	0.988159i	$-0.450966\pi$
$241$	4.76394	0.306872	0.153436	+	0.988159i	$0.450966\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−5.57615	−0.356247
$246$	0	0
$247$	2.25526	0.143499
$248$	0	0
$249$	0	0
$250$	0	0
$251$	7.76394	0.490055	0.245028	−	0.969516i	$-0.421203\pi$
$251$	7.76394	0.490055	0.245028	+	0.969516i	$0.421203\pi$
$252$	0	0
$253$	−22.7922	−1.43293
$254$	0	0
$255$	0	0
$256$	0	0
$257$	19.2835	1.20287	0.601437	−	0.798920i	$-0.294595\pi$
$257$	19.2835	1.20287	0.601437	+	0.798920i	$0.294595\pi$
$258$	0	0
$259$	−3.96265	−0.246227
$260$	0	0
$261$	0	0
$262$	0	0
$263$	7.70739	0.475258	0.237629	−	0.971356i	$-0.423630\pi$
$263$	7.70739	0.475258	0.237629	+	0.971356i	$0.423630\pi$
$264$	0	0
$265$	1.02827	0.0631664
$266$	0	0
$267$	0	0
$268$	0	0
$269$	14.1222	0.861044	0.430522	−	0.902580i	$-0.358330\pi$
$269$	14.1222	0.861044	0.430522	+	0.902580i	$0.358330\pi$
$270$	0	0
$271$	−3.26434	−0.198294	−0.0991472	−	0.995073i	$-0.531611\pi$
$271$	−3.26434	−0.198294	−0.0991472	+	0.995073i	$0.531611\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−3.32088	−0.200257
$276$	0	0
$277$	26.4623	1.58996	0.794981	−	0.606634i	$-0.207481\pi$
$277$	26.4623	1.58996	0.794981	+	0.606634i	$0.207481\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	19.4057	1.15765	0.578824	−	0.815453i	$-0.303512\pi$
$281$	19.4057	1.15765	0.578824	+	0.815453i	$0.303512\pi$
$282$	0	0
$283$	−5.39197	−0.320519	−0.160260	−	0.987075i	$-0.551233\pi$
$283$	−5.39197	−0.320519	−0.160260	+	0.987075i	$0.551233\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−2.76940	−0.163473
$288$	0	0
$289$	23.3118	1.37128
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−32.9162	−1.92299	−0.961493	−	0.274828i	$-0.911379\pi$
$293$	−32.9162	−1.92299	−0.961493	+	0.274828i	$0.911379\pi$
$294$	0	0
$295$	−11.6700	−0.679456
$296$	0	0
$297$	0	0
$298$	0	0
$299$	11.7183	0.677688
$300$	0	0
$301$	7.57615	0.436682
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−9.73566	−0.557462
$306$	0	0
$307$	6.27807	0.358309	0.179154	−	0.983821i	$-0.442664\pi$
$307$	6.27807	0.358309	0.179154	+	0.983821i	$0.442664\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−20.4623	−1.16031	−0.580154	−	0.814507i	$-0.697008\pi$
$311$	−20.4623	−1.16031	−0.580154	+	0.814507i	$0.697008\pi$
$312$	0	0
$313$	23.3774	1.32137	0.660685	−	0.750663i	$-0.270266\pi$
$313$	23.3774	1.32137	0.660685	+	0.750663i	$0.270266\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−2.09389	−0.117605	−0.0588024	−	0.998270i	$-0.518728\pi$
$317$	−2.09389	−0.117605	−0.0588024	+	0.998270i	$0.518728\pi$
$318$	0	0
$319$	−6.73566	−0.377125
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−8.38650	−0.466638
$324$	0	0
$325$	1.70739	0.0947089
$326$	0	0
$327$	0	0
$328$	0	0
$329$	15.2371	0.840050
$330$	0	0
$331$	−26.3118	−1.44623	−0.723114	−	0.690729i	$-0.757290\pi$
$331$	−26.3118	−1.44623	−0.723114	+	0.690729i	$0.757290\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−10.5707	−0.577538
$336$	0	0
$337$	−23.0667	−1.25652	−0.628261	−	0.778003i	$-0.716233\pi$
$337$	−23.0667	−1.25652	−0.628261	+	0.778003i	$0.716233\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	8.89703	0.481801
$342$	0	0
$343$	15.0065	0.810276
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−14.5369	−0.780384	−0.390192	−	0.920733i	$-0.627591\pi$
$347$	−14.5369	−0.780384	−0.390192	+	0.920733i	$0.627591\pi$
$348$	0	0
$349$	−22.9819	−1.23019	−0.615095	−	0.788453i	$-0.710882\pi$
$349$	−22.9819	−1.23019	−0.615095	+	0.788453i	$0.710882\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	21.3966	1.13883	0.569414	−	0.822051i	$-0.307170\pi$
$353$	21.3966	1.13883	0.569414	+	0.822051i	$0.307170\pi$
$354$	0	0
$355$	−1.06562	−0.0565573
$356$	0	0
$357$	0	0
$358$	0	0
$359$	22.5935	1.19244	0.596220	−	0.802821i	$-0.296669\pi$
$359$	22.5935	1.19244	0.596220	+	0.802821i	$0.296669\pi$
$360$	0	0
$361$	−17.2553	−0.908172
$362$	0	0
$363$	0	0
$364$	0	0
$365$	14.0565	0.735753
$366$	0	0
$367$	8.29261	0.432871	0.216435	−	0.976297i	$-0.430557\pi$
$367$	8.29261	0.432871	0.216435	+	0.976297i	$0.430557\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−1.22699	−0.0637022
$372$	0	0
$373$	19.4823	1.00875	0.504376	−	0.863484i	$-0.331722\pi$
$373$	19.4823	1.00875	0.504376	+	0.863484i	$0.331722\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	3.46305	0.178356
$378$	0	0
$379$	22.0000	1.13006	0.565032	−	0.825069i	$-0.308864\pi$
$379$	22.0000	1.13006	0.565032	+	0.825069i	$0.308864\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	11.5196	0.588624	0.294312	−	0.955709i	$-0.404910\pi$
$383$	11.5196	0.588624	0.294312	+	0.955709i	$0.404910\pi$
$384$	0	0
$385$	3.96265	0.201956
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−22.3774	−1.13458	−0.567290	−	0.823518i	$-0.692008\pi$
$389$	−22.3774	−1.13458	−0.567290	+	0.823518i	$0.692008\pi$
$390$	0	0
$391$	−43.5761	−2.20374
$392$	0	0
$393$	0	0
$394$	0	0
$395$	1.41478	0.0711852
$396$	0	0
$397$	−9.34009	−0.468765	−0.234383	−	0.972144i	$-0.575307\pi$
$397$	−9.34009	−0.468765	−0.234383	+	0.972144i	$0.575307\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−9.72659	−0.485723	−0.242861	−	0.970061i	$-0.578086\pi$
$401$	−9.72659	−0.485723	−0.242861	+	0.970061i	$0.578086\pi$
$402$	0	0
$403$	−4.57429	−0.227862
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−11.0283	−0.546651
$408$	0	0
$409$	10.1878	0.503754	0.251877	−	0.967759i	$-0.418952\pi$
$409$	10.1878	0.503754	0.251877	+	0.967759i	$0.418952\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	13.9253	0.685220
$414$	0	0
$415$	−11.8350	−0.580958
$416$	0	0
$417$	0	0
$418$	0	0
$419$	12.0192	0.587176	0.293588	−	0.955932i	$-0.405151\pi$
$419$	12.0192	0.587176	0.293588	+	0.955932i	$0.405151\pi$
$420$	0	0
$421$	19.4713	0.948974	0.474487	−	0.880262i	$-0.342634\pi$
$421$	19.4713	0.948974	0.474487	+	0.880262i	$0.342634\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−6.34916	−0.307979
$426$	0	0
$427$	11.6171	0.562191
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−16.8296	−0.810651	−0.405326	−	0.914172i	$-0.632842\pi$
$431$	−16.8296	−0.810651	−0.405326	+	0.914172i	$0.632842\pi$
$432$	0	0
$433$	−8.40571	−0.403952	−0.201976	−	0.979390i	$-0.564736\pi$
$433$	−8.40571	−0.403952	−0.201976	+	0.979390i	$0.564736\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	9.06562	0.433667
$438$	0	0
$439$	−29.0848	−1.38814	−0.694071	−	0.719906i	$-0.744185\pi$
$439$	−29.0848	−1.38814	−0.694071	+	0.719906i	$0.744185\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	14.1842	0.673911	0.336955	−	0.941521i	$-0.390603\pi$
$443$	14.1842	0.673911	0.336955	+	0.941521i	$0.390603\pi$
$444$	0	0
$445$	−11.0000	−0.521450
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−26.6874	−1.25946	−0.629728	−	0.776816i	$-0.716834\pi$
$449$	−26.6874	−1.25946	−0.629728	+	0.776816i	$0.716834\pi$
$450$	0	0
$451$	−7.70739	−0.362927
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−2.03735	−0.0955123
$456$	0	0
$457$	15.9061	0.744056	0.372028	−	0.928221i	$-0.378662\pi$
$457$	15.9061	0.744056	0.372028	+	0.928221i	$0.378662\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−38.0667	−1.77294	−0.886471	−	0.462784i	$-0.846850\pi$
$461$	−38.0667	−1.77294	−0.886471	+	0.462784i	$0.846850\pi$
$462$	0	0
$463$	−35.9445	−1.67048	−0.835241	−	0.549883i	$-0.814672\pi$
$463$	−35.9445	−1.67048	−0.835241	+	0.549883i	$0.814672\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−26.4623	−1.22453	−0.612264	−	0.790654i	$-0.709741\pi$
$467$	−26.4623	−1.22453	−0.612264	+	0.790654i	$0.709741\pi$
$468$	0	0
$469$	12.6135	0.582437
$470$	0	0
$471$	0	0
$472$	0	0
$473$	21.0848	0.969481
$474$	0	0
$475$	1.32088	0.0606063
$476$	0	0
$477$	0	0
$478$	0	0
$479$	38.1131	1.74143	0.870716	−	0.491786i	$-0.163656\pi$
$479$	38.1131	1.74143	0.870716	+	0.491786i	$0.163656\pi$
$480$	0	0
$481$	5.67004	0.258532
$482$	0	0
$483$	0	0
$484$	0	0
$485$	16.2553	0.738114
$486$	0	0
$487$	38.6610	1.75190	0.875948	−	0.482406i	$-0.160237\pi$
$487$	38.6610	1.75190	0.875948	+	0.482406i	$0.160237\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	31.7266	1.43180	0.715900	−	0.698202i	$-0.246016\pi$
$491$	31.7266	1.43180	0.715900	+	0.698202i	$0.246016\pi$
$492$	0	0
$493$	−12.8778	−0.579988
$494$	0	0
$495$	0	0
$496$	0	0
$497$	1.27155	0.0570370
$498$	0	0
$499$	38.4815	1.72267	0.861333	−	0.508040i	$-0.169630\pi$
$499$	38.4815	1.72267	0.861333	+	0.508040i	$0.169630\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	11.0994	0.494896	0.247448	−	0.968901i	$-0.420408\pi$
$503$	11.0994	0.494896	0.247448	+	0.968901i	$0.420408\pi$
$504$	0	0
$505$	−2.38650	−0.106198
$506$	0	0
$507$	0	0
$508$	0	0
$509$	26.5279	1.17583	0.587914	−	0.808924i	$-0.299949\pi$
$509$	26.5279	1.17583	0.587914	+	0.808924i	$0.299949\pi$
$510$	0	0
$511$	−16.7730	−0.741994
$512$	0	0
$513$	0	0
$514$	0	0
$515$	9.76394	0.430250
$516$	0	0
$517$	42.4057	1.86500
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−9.89703	−0.433597	−0.216798	−	0.976216i	$-0.569561\pi$
$521$	−9.89703	−0.433597	−0.216798	+	0.976216i	$0.569561\pi$
$522$	0	0
$523$	28.4394	1.24357	0.621785	−	0.783188i	$-0.286408\pi$
$523$	28.4394	1.24357	0.621785	+	0.783188i	$0.286408\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	17.0101	0.740973
$528$	0	0
$529$	24.1048	1.04804
$530$	0	0
$531$	0	0
$532$	0	0
$533$	3.96265	0.171642
$534$	0	0
$535$	−7.57976	−0.327701
$536$	0	0
$537$	0	0
$538$	0	0
$539$	18.5177	0.797616
$540$	0	0
$541$	13.1131	0.563776	0.281888	−	0.959447i	$-0.409039\pi$
$541$	13.1131	0.563776	0.281888	+	0.959447i	$0.409039\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	16.5761	0.710044
$546$	0	0
$547$	24.7321	1.05747	0.528733	−	0.848788i	$-0.322667\pi$
$547$	24.7321	1.05747	0.528733	+	0.848788i	$0.322667\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	2.67912	0.114134
$552$	0	0
$553$	−1.68819	−0.0717891
$554$	0	0
$555$	0	0
$556$	0	0
$557$	22.7730	0.964923	0.482462	−	0.875917i	$-0.339743\pi$
$557$	22.7730	0.964923	0.482462	+	0.875917i	$0.339743\pi$
$558$	0	0
$559$	−10.8405	−0.458504
$560$	0	0
$561$	0	0
$562$	0	0
$563$	29.4112	1.23953	0.619767	−	0.784786i	$-0.287227\pi$
$563$	29.4112	1.23953	0.619767	+	0.784786i	$0.287227\pi$
$564$	0	0
$565$	−21.0848	−0.887045
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−36.5671	−1.53297	−0.766486	−	0.642261i	$-0.777996\pi$
$569$	−36.5671	−1.53297	−0.766486	+	0.642261i	$0.777996\pi$
$570$	0	0
$571$	4.37558	0.183112	0.0915561	−	0.995800i	$-0.470816\pi$
$571$	4.37558	0.183112	0.0915561	+	0.995800i	$0.470816\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	6.86330	0.286219
$576$	0	0
$577$	9.02827	0.375852	0.187926	−	0.982183i	$-0.439823\pi$
$577$	9.02827	0.375852	0.187926	+	0.982183i	$0.439823\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	14.1222	0.585886
$582$	0	0
$583$	−3.41478	−0.141426
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−3.26795	−0.134883	−0.0674413	−	0.997723i	$-0.521484\pi$
$587$	−3.26795	−0.134883	−0.0674413	+	0.997723i	$0.521484\pi$
$588$	0	0
$589$	−3.53880	−0.145814
$590$	0	0
$591$	0	0
$592$	0	0
$593$	3.39558	0.139440	0.0697198	−	0.997567i	$-0.477789\pi$
$593$	3.39558	0.139440	0.0697198	+	0.997567i	$0.477789\pi$
$594$	0	0
$595$	7.57615	0.310592
$596$	0	0
$597$	0	0
$598$	0	0
$599$	0.603367	0.0246529	0.0123264	−	0.999924i	$-0.496076\pi$
$599$	0.603367	0.0246529	0.0123264	+	0.999924i	$0.496076\pi$
$600$	0	0
$601$	−41.9253	−1.71017	−0.855084	−	0.518489i	$-0.826495\pi$
$601$	−41.9253	−1.71017	−0.855084	+	0.518489i	$0.826495\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0.0282739	0.00114950
$606$	0	0
$607$	−5.42932	−0.220369	−0.110185	−	0.993911i	$-0.535144\pi$
$607$	−5.42932	−0.220369	−0.110185	+	0.993911i	$0.535144\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−21.8023	−0.882028
$612$	0	0
$613$	2.42385	0.0978984	0.0489492	−	0.998801i	$-0.484413\pi$
$613$	2.42385	0.0978984	0.0489492	+	0.998801i	$0.484413\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	29.0848	1.17091	0.585455	−	0.810705i	$-0.300916\pi$
$617$	29.0848	1.17091	0.585455	+	0.810705i	$0.300916\pi$
$618$	0	0
$619$	7.45213	0.299526	0.149763	−	0.988722i	$-0.452149\pi$
$619$	7.45213	0.299526	0.149763	+	0.988722i	$0.452149\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	13.1258	0.525873
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−21.0848	−0.840707
$630$	0	0
$631$	−27.8013	−1.10675	−0.553376	−	0.832932i	$-0.686661\pi$
$631$	−27.8013	−1.10675	−0.553376	+	0.832932i	$0.686661\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−15.4112	−0.611574
$636$	0	0
$637$	−9.52066	−0.377222
$638$	0	0
$639$	0	0
$640$	0	0
$641$	34.0475	1.34479	0.672397	−	0.740191i	$-0.265265\pi$
$641$	34.0475	1.34479	0.672397	+	0.740191i	$0.265265\pi$
$642$	0	0
$643$	−13.8542	−0.546357	−0.273179	−	0.961963i	$-0.588075\pi$
$643$	−13.8542	−0.546357	−0.273179	+	0.961963i	$0.588075\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−5.11750	−0.201190	−0.100595	−	0.994927i	$-0.532075\pi$
$647$	−5.11750	−0.201190	−0.100595	+	0.994927i	$0.532075\pi$
$648$	0	0
$649$	38.7549	1.52126
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−7.39558	−0.289411	−0.144706	−	0.989475i	$-0.546224\pi$
$653$	−7.39558	−0.289411	−0.144706	+	0.989475i	$0.546224\pi$
$654$	0	0
$655$	−3.35823	−0.131217
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−17.0848	−0.665530	−0.332765	−	0.943010i	$-0.607982\pi$
$659$	−17.0848	−0.665530	−0.332765	+	0.943010i	$0.607982\pi$
$660$	0	0
$661$	−38.6236	−1.50228	−0.751142	−	0.660140i	$-0.770497\pi$
$661$	−38.6236	−1.50228	−0.751142	+	0.660140i	$0.770497\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−1.57615	−0.0611204
$666$	0	0
$667$	13.9206	0.539009
$668$	0	0
$669$	0	0
$670$	0	0
$671$	32.3310	1.24812
$672$	0	0
$673$	26.5297	1.02265	0.511323	−	0.859389i	$-0.329156\pi$
$673$	26.5297	1.02265	0.511323	+	0.859389i	$0.329156\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	38.0192	1.46120	0.730598	−	0.682808i	$-0.239241\pi$
$677$	38.0192	1.46120	0.730598	+	0.682808i	$0.239241\pi$
$678$	0	0
$679$	−19.3966	−0.744374
$680$	0	0
$681$	0	0
$682$	0	0
$683$	6.23606	0.238616	0.119308	−	0.992857i	$-0.461932\pi$
$683$	6.23606	0.238616	0.119308	+	0.992857i	$0.461932\pi$
$684$	0	0
$685$	−7.61350	−0.290897
$686$	0	0
$687$	0	0
$688$	0	0
$689$	1.75566	0.0668855
$690$	0	0
$691$	3.17044	0.120609	0.0603047	−	0.998180i	$-0.480793\pi$
$691$	3.17044	0.120609	0.0603047	+	0.998180i	$0.480793\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	13.2835	0.503873
$696$	0	0
$697$	−14.7357	−0.558153
$698$	0	0
$699$	0	0
$700$	0	0
$701$	51.5863	1.94839	0.974193	−	0.225715i	$-0.0724718\pi$
$701$	51.5863	1.94839	0.974193	+	0.225715i	$0.0724718\pi$
$702$	0	0
$703$	4.38650	0.165440
$704$	0	0
$705$	0	0
$706$	0	0
$707$	2.84770	0.107099
$708$	0	0
$709$	12.8578	0.482886	0.241443	−	0.970415i	$-0.422379\pi$
$709$	12.8578	0.482886	0.241443	+	0.970415i	$0.422379\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−18.3876	−0.688620
$714$	0	0
$715$	−5.67004	−0.212048
$716$	0	0
$717$	0	0
$718$	0	0
$719$	4.84049	0.180520	0.0902598	−	0.995918i	$-0.471230\pi$
$719$	4.84049	0.180520	0.0902598	+	0.995918i	$0.471230\pi$
$720$	0	0
$721$	−11.6508	−0.433900
$722$	0	0
$723$	0	0
$724$	0	0
$725$	2.02827	0.0753282
$726$	0	0
$727$	−41.0703	−1.52321	−0.761606	−	0.648040i	$-0.775589\pi$
$727$	−41.0703	−1.52321	−0.761606	+	0.648040i	$0.775589\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	40.3118	1.49099
$732$	0	0
$733$	35.3966	1.30740	0.653702	−	0.756752i	$-0.273215\pi$
$733$	35.3966	1.30740	0.653702	+	0.756752i	$0.273215\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	35.1040	1.29307
$738$	0	0
$739$	−19.4823	−0.716666	−0.358333	−	0.933594i	$-0.616655\pi$
$739$	−19.4823	−0.716666	−0.358333	+	0.933594i	$0.616655\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−16.2034	−0.594444	−0.297222	−	0.954808i	$-0.596060\pi$
$743$	−16.2034	−0.594444	−0.297222	+	0.954808i	$0.596060\pi$
$744$	0	0
$745$	−17.9909	−0.659137
$746$	0	0
$747$	0	0
$748$	0	0
$749$	9.04456	0.330481
$750$	0	0
$751$	14.3492	0.523608	0.261804	−	0.965121i	$-0.415683\pi$
$751$	14.3492	0.523608	0.261804	+	0.965121i	$0.415683\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−8.67912	−0.315865
$756$	0	0
$757$	16.7466	0.608665	0.304333	−	0.952566i	$-0.401567\pi$
$757$	16.7466	0.608665	0.304333	+	0.952566i	$0.401567\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−19.6308	−0.711617	−0.355809	−	0.934559i	$-0.615795\pi$
$761$	−19.6308	−0.711617	−0.355809	+	0.934559i	$0.615795\pi$
$762$	0	0
$763$	−19.7795	−0.716067
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−19.9253	−0.719461
$768$	0	0
$769$	−42.9144	−1.54753	−0.773766	−	0.633471i	$-0.781629\pi$
$769$	−42.9144	−1.54753	−0.773766	+	0.633471i	$0.781629\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−7.22699	−0.259937	−0.129968	−	0.991518i	$-0.541488\pi$
$773$	−7.22699	−0.259937	−0.129968	+	0.991518i	$0.541488\pi$
$774$	0	0
$775$	−2.67912	−0.0962367
$776$	0	0
$777$	0	0
$778$	0	0
$779$	3.06562	0.109837
$780$	0	0
$781$	3.53880	0.126628
$782$	0	0
$783$	0	0
$784$	0	0
$785$	18.8970	0.674464
$786$	0	0
$787$	30.2361	1.07780	0.538900	−	0.842370i	$-0.318840\pi$
$787$	30.2361	1.07780	0.538900	+	0.842370i	$0.318840\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	25.1595	0.894569
$792$	0	0
$793$	−16.6226	−0.590285
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−27.4158	−0.971119	−0.485559	−	0.874204i	$-0.661384\pi$
$797$	−27.4158	−0.971119	−0.485559	+	0.874204i	$0.661384\pi$
$798$	0	0
$799$	81.0749	2.86823
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−46.6802	−1.64731
$804$	0	0
$805$	−8.18964	−0.288647
$806$	0	0
$807$	0	0
$808$	0	0
$809$	28.1806	0.990776	0.495388	−	0.868672i	$-0.335026\pi$
$809$	28.1806	0.990776	0.495388	+	0.868672i	$0.335026\pi$
$810$	0	0
$811$	20.1312	0.706903	0.353452	−	0.935453i	$-0.385008\pi$
$811$	20.1312	0.706903	0.353452	+	0.935453i	$0.385008\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−0.990927	−0.0347107
$816$	0	0
$817$	−8.38650	−0.293407
$818$	0	0
$819$	0	0
$820$	0	0
$821$	12.2462	0.427395	0.213698	−	0.976900i	$-0.431449\pi$
$821$	12.2462	0.427395	0.213698	+	0.976900i	$0.431449\pi$
$822$	0	0
$823$	−27.9289	−0.973541	−0.486770	−	0.873530i	$-0.661825\pi$
$823$	−27.9289	−0.973541	−0.486770	+	0.873530i	$0.661825\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	8.57068	0.298032	0.149016	−	0.988835i	$-0.452389\pi$
$827$	8.57068	0.298032	0.149016	+	0.988835i	$0.452389\pi$
$828$	0	0
$829$	−19.7357	−0.685448	−0.342724	−	0.939436i	$-0.611350\pi$
$829$	−19.7357	−0.685448	−0.342724	+	0.939436i	$0.611350\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	35.4039	1.22667
$834$	0	0
$835$	13.5990	0.470611
$836$	0	0
$837$	0	0
$838$	0	0
$839$	46.7175	1.61287	0.806434	−	0.591324i	$-0.201395\pi$
$839$	46.7175	1.61287	0.806434	+	0.591324i	$0.201395\pi$
$840$	0	0
$841$	−24.8861	−0.858142
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−10.0848	−0.346928
$846$	0	0
$847$	−0.0337379	−0.00115925
$848$	0	0
$849$	0	0
$850$	0	0
$851$	22.7922	0.781307
$852$	0	0
$853$	5.39558	0.184741	0.0923705	−	0.995725i	$-0.470556\pi$
$853$	5.39558	0.184741	0.0923705	+	0.995725i	$0.470556\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	28.6719	0.979413	0.489707	−	0.871887i	$-0.337104\pi$
$857$	28.6719	0.979413	0.489707	+	0.871887i	$0.337104\pi$
$858$	0	0
$859$	22.7466	0.776104	0.388052	−	0.921638i	$-0.373148\pi$
$859$	22.7466	0.776104	0.388052	+	0.921638i	$0.373148\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−0.202325	−0.00688723	−0.00344361	−	0.999994i	$-0.501096\pi$
$863$	−0.202325	−0.00688723	−0.00344361	+	0.999994i	$0.501096\pi$
$864$	0	0
$865$	−0.187788	−0.00638500
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−4.69832	−0.159379
$870$	0	0
$871$	−18.0483	−0.611542
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−1.19325	−0.0403393
$876$	0	0
$877$	12.9728	0.438060	0.219030	−	0.975718i	$-0.429711\pi$
$877$	12.9728	0.438060	0.219030	+	0.975718i	$0.429711\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	16.4905	0.555580	0.277790	−	0.960642i	$-0.410398\pi$
$881$	16.4905	0.555580	0.277790	+	0.960642i	$0.410398\pi$
$882$	0	0
$883$	−8.38290	−0.282107	−0.141053	−	0.990002i	$-0.545049\pi$
$883$	−8.38290	−0.282107	−0.141053	+	0.990002i	$0.545049\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−58.4623	−1.96297	−0.981485	−	0.191538i	$-0.938652\pi$
$887$	−58.4623	−1.96297	−0.981485	+	0.191538i	$0.938652\pi$
$888$	0	0
$889$	18.3894	0.616761
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−16.8669	−0.564429
$894$	0	0
$895$	9.76394	0.326372
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−5.43398	−0.181233
$900$	0	0
$901$	−6.52867	−0.217502
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−16.0848	−0.534678
$906$	0	0
$907$	30.7019	1.01944	0.509720	−	0.860340i	$-0.329749\pi$
$907$	30.7019	1.01944	0.509720	+	0.860340i	$0.329749\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−34.3118	−1.13680	−0.568401	−	0.822752i	$-0.692438\pi$
$911$	−34.3118	−1.13680	−0.568401	+	0.822752i	$0.692438\pi$
$912$	0	0
$913$	39.3027	1.30073
$914$	0	0
$915$	0	0
$916$	0	0
$917$	4.00722	0.132330
$918$	0	0
$919$	−18.8861	−0.622995	−0.311498	−	0.950247i	$-0.600831\pi$
$919$	−18.8861	−0.622995	−0.311498	+	0.950247i	$0.600831\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−1.81943	−0.0598872
$924$	0	0
$925$	3.32088	0.109190
$926$	0	0
$927$	0	0
$928$	0	0
$929$	8.09495	0.265587	0.132793	−	0.991144i	$-0.457605\pi$
$929$	8.09495	0.265587	0.132793	+	0.991144i	$0.457605\pi$
$930$	0	0
$931$	−7.36545	−0.241393
$932$	0	0
$933$	0	0
$934$	0	0
$935$	21.0848	0.689547
$936$	0	0
$937$	29.7084	0.970533	0.485266	−	0.874366i	$-0.338723\pi$
$937$	29.7084	0.970533	0.485266	+	0.874366i	$0.338723\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	10.4823	0.341712	0.170856	−	0.985296i	$-0.445347\pi$
$941$	10.4823	0.341712	0.170856	+	0.985296i	$0.445347\pi$
$942$	0	0
$943$	15.9289	0.518717
$944$	0	0
$945$	0	0
$946$	0	0
$947$	4.65631	0.151310	0.0756548	−	0.997134i	$-0.475895\pi$
$947$	4.65631	0.151310	0.0756548	+	0.997134i	$0.475895\pi$
$948$	0	0
$949$	24.0000	0.779073
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−21.9445	−0.710852	−0.355426	−	0.934704i	$-0.615664\pi$
$953$	−21.9445	−0.710852	−0.355426	+	0.934704i	$0.615664\pi$
$954$	0	0
$955$	−3.06562	−0.0992011
$956$	0	0
$957$	0	0
$958$	0	0
$959$	9.08482	0.293364
$960$	0	0
$961$	−23.8223	−0.768463
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0.236063	0.00759913
$966$	0	0
$967$	60.7030	1.95208	0.976038	−	0.217600i	$-0.0698227\pi$
$967$	60.7030	1.95208	0.976038	+	0.217600i	$0.0698227\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	38.9053	1.24853	0.624265	−	0.781212i	$-0.285398\pi$
$971$	38.9053	1.24853	0.624265	+	0.781212i	$0.285398\pi$
$972$	0	0
$973$	−15.8506	−0.508147
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−14.4166	−0.461229	−0.230614	−	0.973045i	$-0.574074\pi$
$977$	−14.4166	−0.461229	−0.230614	+	0.973045i	$0.574074\pi$
$978$	0	0
$979$	36.5297	1.16750
$980$	0	0
$981$	0	0
$982$	0	0
$983$	25.9025	0.826161	0.413081	−	0.910694i	$-0.364453\pi$
$983$	25.9025	0.826161	0.413081	+	0.910694i	$0.364453\pi$
$984$	0	0
$985$	−1.74474	−0.0555919
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−43.5761	−1.38564
$990$	0	0
$991$	−26.1987	−0.832230	−0.416115	−	0.909312i	$-0.636609\pi$
$991$	−26.1987	−0.832230	−0.416115	+	0.909312i	$0.636609\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−22.7922	−0.722562
$996$	0	0
$997$	−24.7658	−0.784341	−0.392170	−	0.919893i	$-0.628276\pi$
$997$	−24.7658	−0.784341	−0.392170	+	0.919893i	$0.628276\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3240.2.a.r.1.2		3
3.2	odd	2		3240.2.a.q.1.2		3
4.3	odd	2		6480.2.a.bx.1.2		3
9.2	odd	6		1080.2.q.d.361.2		6
9.4	even	3		360.2.q.d.241.2	yes	6
9.5	odd	6		1080.2.q.d.721.2		6
9.7	even	3		360.2.q.d.121.2	✓	6
12.11	even	2		6480.2.a.bu.1.2		3
36.7	odd	6		720.2.q.j.481.2		6
36.11	even	6		2160.2.q.j.1441.2		6
36.23	even	6		2160.2.q.j.721.2		6
36.31	odd	6		720.2.q.j.241.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
360.2.q.d.121.2	✓	6	9.7	even	3
360.2.q.d.241.2	yes	6	9.4	even	3
720.2.q.j.241.2		6	36.31	odd	6
720.2.q.j.481.2		6	36.7	odd	6
1080.2.q.d.361.2		6	9.2	odd	6
1080.2.q.d.721.2		6	9.5	odd	6
2160.2.q.j.721.2		6	36.23	even	6
2160.2.q.j.1441.2		6	36.11	even	6
3240.2.a.q.1.2		3	3.2	odd	2
3240.2.a.r.1.2		3	1.1	even	1	trivial
6480.2.a.bu.1.2		3	12.11	even	2
6480.2.a.bx.1.2		3	4.3	odd	2

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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:	\( N \)	\(=\)	\( 3240 = 2^{3} \cdot 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3240.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{5} -1.19325 q^{7} +O(q^{10})\)	\(q+1.00000 q^{5} -1.19325 q^{7} -3.32088 q^{11} +1.70739 q^{13} -6.34916 q^{17} +1.32088 q^{19} +6.86330 q^{23} +1.00000 q^{25} +2.02827 q^{29} -2.67912 q^{31} -1.19325 q^{35} +3.32088 q^{37} +2.32088 q^{41} -6.34916 q^{43} -12.7694 q^{47} -5.57615 q^{49} +1.02827 q^{53} -3.32088 q^{55} -11.6700 q^{59} -9.73566 q^{61} +1.70739 q^{65} -10.5707 q^{67} -1.06562 q^{71} +14.0565 q^{73} +3.96265 q^{77} +1.41478 q^{79} -11.8350 q^{83} -6.34916 q^{85} -11.0000 q^{89} -2.03735 q^{91} +1.32088 q^{95} +16.2553 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{5} - 5 q^{7}+O(q^{10}) \) 3 * q + 3 * q^5 - 5 * q^7	\( 3 q + 3 q^{5} - 5 q^{7} - 2 q^{11} + 2 q^{17} - 4 q^{19} - 7 q^{23} + 3 q^{25} - 7 q^{29} - 16 q^{31} - 5 q^{35} + 2 q^{37} - q^{41} + 2 q^{43} - 13 q^{47} + 10 q^{49} - 10 q^{53} - 2 q^{55} - 6 q^{59} - 11 q^{61} + q^{67} - 14 q^{71} + 16 q^{73} - 12 q^{77} - 6 q^{79} - 21 q^{83} + 2 q^{85} - 33 q^{89} - 30 q^{91} - 4 q^{95} + 30 q^{97}+O(q^{100}) \) 3 * q + 3 * q^5 - 5 * q^7 - 2 * q^11 + 2 * q^17 - 4 * q^19 - 7 * q^23 + 3 * q^25 - 7 * q^29 - 16 * q^31 - 5 * q^35 + 2 * q^37 - q^41 + 2 * q^43 - 13 * q^47 + 10 * q^49 - 10 * q^53 - 2 * q^55 - 6 * q^59 - 11 * q^61 + q^67 - 14 * q^71 + 16 * q^73 - 12 * q^77 - 6 * q^79 - 21 * q^83 + 2 * q^85 - 33 * q^89 - 30 * q^91 - 4 * q^95 + 30 * q^97

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