LMFDB - Embedded newform 6480.2.a.ca.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6480,2,Mod(1,6480)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6480, 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("6480.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6480 = 2^{4} \cdot 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6480.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:	$51.7430605098$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.62352.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 11x^{2} + 12x + 24 $ x^4 - 2x^3 - 11x^2 + 12*x + 24
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 3240)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$3.61675$ of defining polynomial
Character	$\chi$	$=$	6480.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{5} -1.88469 q^{7} -1.03090 q^{11} +4.53234 q^{13} -0.816460 q^{17} -6.23349 q^{19} +0.616745 q^{23} +1.00000 q^{25} +2.43321 q^{29} -0.383255 q^{31} -1.88469 q^{35} -5.88113 q^{37} -0.0991303 q^{41} +7.34523 q^{43} -6.89731 q^{47} -3.44793 q^{49} -8.95910 q^{53} -1.03090 q^{55} -1.73205 q^{59} +10.7594 q^{61} +4.53234 q^{65} -12.8664 q^{67} +8.96198 q^{71} +6.41703 q^{73} +1.94292 q^{77} -0.535898 q^{79} -8.04995 q^{83} -0.816460 q^{85} -18.6567 q^{89} -8.54207 q^{91} -6.23349 q^{95} +7.51405 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{5} - 2 q^{7} - 4 q^{11} - 2 q^{17} - 10 q^{23} + 4 q^{25} - 4 q^{29} - 14 q^{31} - 2 q^{35} + 14 q^{37} + 4 q^{41} - 22 q^{43} + 10 q^{49} - 8 q^{53} - 4 q^{55} + 4 q^{61} - 24 q^{67} - 28 q^{71}+ \cdots - 10 q^{97}+O(q^{100}) $ 4 * q + 4 * q^5 - 2 * q^7 - 4 * q^11 - 2 * q^17 - 10 * q^23 + 4 * q^25 - 4 * q^29 - 14 * q^31 - 2 * q^35 + 14 * q^37 + 4 * q^41 - 22 * q^43 + 10 * q^49 - 8 * q^53 - 4 * q^55 + 4 * q^61 - 24 * q^67 - 28 * q^71 + 2 * q^73 - 30 * q^77 - 16 * q^79 - 6 * q^83 - 2 * q^85 - 8 * q^89 - 30 * q^91 - 10 * q^97

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
$7$	−1.88469	−0.712348	−0.356174	−	0.934420i	$-0.615919\pi$
$7$	−1.88469	−0.712348	−0.356174	+	0.934420i	$0.615919\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−1.03090	−0.310827	−0.155413	−	0.987850i	$-0.549671\pi$
$11$	−1.03090	−0.310827	−0.155413	+	0.987850i	$0.549671\pi$
$12$	0	0
$13$	4.53234	1.25704	0.628522	−	0.777792i	$-0.283660\pi$
$13$	4.53234	1.25704	0.628522	+	0.777792i	$0.283660\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−0.816460	−0.198021	−0.0990103	−	0.995086i	$-0.531568\pi$
$17$	−0.816460	−0.198021	−0.0990103	+	0.995086i	$0.531568\pi$
$18$	0	0
$19$	−6.23349	−1.43006	−0.715030	−	0.699093i	$-0.753587\pi$
$19$	−6.23349	−1.43006	−0.715030	+	0.699093i	$0.753587\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	0.616745	0.128600	0.0643001	−	0.997931i	$-0.479518\pi$
$23$	0.616745	0.128600	0.0643001	+	0.997931i	$0.479518\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	2.43321	0.451835	0.225917	−	0.974146i	$-0.427462\pi$
$29$	2.43321	0.451835	0.225917	+	0.974146i	$0.427462\pi$
$30$	0	0
$31$	−0.383255	−0.0688346	−0.0344173	−	0.999408i	$-0.510958\pi$
$31$	−0.383255	−0.0688346	−0.0344173	+	0.999408i	$0.510958\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−1.88469	−0.318572
$36$	0	0
$37$	−5.88113	−0.966852	−0.483426	−	0.875385i	$-0.660608\pi$
$37$	−5.88113	−0.966852	−0.483426	+	0.875385i	$0.660608\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−0.0991303	−0.0154816	−0.00774078	−	0.999970i	$-0.502464\pi$
$41$	−0.0991303	−0.0154816	−0.00774078	+	0.999970i	$0.502464\pi$
$42$	0	0
$43$	7.34523	1.12014	0.560069	−	0.828446i	$-0.310775\pi$
$43$	7.34523	1.12014	0.560069	+	0.828446i	$0.310775\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−6.89731	−1.00608	−0.503038	−	0.864264i	$-0.667784\pi$
$47$	−6.89731	−1.00608	−0.503038	+	0.864264i	$0.667784\pi$
$48$	0	0
$49$	−3.44793	−0.492561
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−8.95910	−1.23063	−0.615313	−	0.788283i	$-0.710971\pi$
$53$	−8.95910	−1.23063	−0.615313	+	0.788283i	$0.710971\pi$
$54$	0	0
$55$	−1.03090	−0.139006
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−1.73205	−0.225494	−0.112747	−	0.993624i	$-0.535965\pi$
$59$	−1.73205	−0.225494	−0.112747	+	0.993624i	$0.535965\pi$
$60$	0	0
$61$	10.7594	1.37760	0.688799	−	0.724952i	$-0.258138\pi$
$61$	10.7594	1.37760	0.688799	+	0.724952i	$0.258138\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	4.53234	0.562167
$66$	0	0
$67$	−12.8664	−1.57188	−0.785941	−	0.618301i	$-0.787821\pi$
$67$	−12.8664	−1.57188	−0.785941	+	0.618301i	$0.787821\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	8.96198	1.06359	0.531796	−	0.846873i	$-0.321517\pi$
$71$	8.96198	1.06359	0.531796	+	0.846873i	$0.321517\pi$
$72$	0	0
$73$	6.41703	0.751057	0.375528	−	0.926811i	$-0.377461\pi$
$73$	6.41703	0.751057	0.375528	+	0.926811i	$0.377461\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	1.94292	0.221417
$78$	0	0
$79$	−0.535898	−0.0602933	−0.0301466	−	0.999545i	$-0.509597\pi$
$79$	−0.535898	−0.0602933	−0.0301466	+	0.999545i	$0.509597\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−8.04995	−0.883597	−0.441798	−	0.897114i	$-0.645659\pi$
$83$	−8.04995	−0.883597	−0.441798	+	0.897114i	$0.645659\pi$
$84$	0	0
$85$	−0.816460	−0.0885575
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−18.6567	−1.97761	−0.988803	−	0.149229i	$-0.952321\pi$
$89$	−18.6567	−1.97761	−0.988803	+	0.149229i	$0.952321\pi$
$90$	0	0
$91$	−8.54207	−0.895452
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−6.23349	−0.639543
$96$	0	0
$97$	7.51405	0.762936	0.381468	−	0.924382i	$-0.375419\pi$
$97$	7.51405	0.762936	0.381468	+	0.924382i	$0.375419\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−1.27439	−0.126807	−0.0634033	−	0.997988i	$-0.520195\pi$
$101$	−1.27439	−0.126807	−0.0634033	+	0.997988i	$0.520195\pi$
$102$	0	0
$103$	−4.22705	−0.416503	−0.208252	−	0.978075i	$-0.566777\pi$
$103$	−4.22705	−0.416503	−0.208252	+	0.978075i	$0.566777\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	14.3923	1.39136	0.695678	−	0.718353i	$-0.255104\pi$
$107$	14.3923	1.39136	0.695678	+	0.718353i	$0.255104\pi$
$108$	0	0
$109$	−1.08085	−0.103526	−0.0517632	−	0.998659i	$-0.516484\pi$
$109$	−1.08085	−0.103526	−0.0517632	+	0.998659i	$0.516484\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	11.1146	1.04558	0.522788	−	0.852463i	$-0.324892\pi$
$113$	11.1146	1.04558	0.522788	+	0.852463i	$0.324892\pi$
$114$	0	0
$115$	0.616745	0.0575118
$116$	0	0
$117$	0	0
$118$	0	0
$119$	1.53878	0.141060
$120$	0	0
$121$	−9.93725	−0.903387
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	1.04351	0.0925964	0.0462982	−	0.998928i	$-0.485258\pi$
$127$	1.04351	0.0925964	0.0462982	+	0.998928i	$0.485258\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−14.4914	−1.26612	−0.633061	−	0.774102i	$-0.718202\pi$
$131$	−14.4914	−1.26612	−0.633061	+	0.774102i	$0.718202\pi$
$132$	0	0
$133$	11.7482	1.01870
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−7.76939	−0.663784	−0.331892	−	0.943317i	$-0.607687\pi$
$137$	−7.76939	−0.663784	−0.331892	+	0.943317i	$0.607687\pi$
$138$	0	0
$139$	−17.5906	−1.49201	−0.746006	−	0.665939i	$-0.768031\pi$
$139$	−17.5906	−1.49201	−0.746006	+	0.665939i	$0.768031\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−4.67237	−0.390723
$144$	0	0
$145$	2.43321	0.202067
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−10.3053	−0.844242	−0.422121	−	0.906539i	$-0.638714\pi$
$149$	−10.3053	−0.844242	−0.422121	+	0.906539i	$0.638714\pi$
$150$	0	0
$151$	−6.91203	−0.562493	−0.281246	−	0.959636i	$-0.590748\pi$
$151$	−6.91203	−0.562493	−0.281246	+	0.959636i	$0.590748\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−0.383255	−0.0307838
$156$	0	0
$157$	−3.82290	−0.305101	−0.152550	−	0.988296i	$-0.548749\pi$
$157$	−3.82290	−0.305101	−0.152550	+	0.988296i	$0.548749\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−1.16238	−0.0916081
$162$	0	0
$163$	−8.28056	−0.648584	−0.324292	−	0.945957i	$-0.605126\pi$
$163$	−8.28056	−0.648584	−0.324292	+	0.945957i	$0.605126\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	4.24821	0.328736	0.164368	−	0.986399i	$-0.447441\pi$
$167$	4.24821	0.328736	0.164368	+	0.986399i	$0.447441\pi$
$168$	0	0
$169$	7.54207	0.580159
$170$	0	0
$171$	0	0
$172$	0	0
$173$	3.54495	0.269517	0.134759	−	0.990878i	$-0.456974\pi$
$173$	3.54495	0.269517	0.134759	+	0.990878i	$0.456974\pi$
$174$	0	0
$175$	−1.88469	−0.142470
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−0.0991303	−0.00740935	−0.00370467	−	0.999993i	$-0.501179\pi$
$179$	−0.0991303	−0.00740935	−0.00370467	+	0.999993i	$0.501179\pi$
$180$	0	0
$181$	12.8502	0.955151	0.477575	−	0.878591i	$-0.341516\pi$
$181$	12.8502	0.955151	0.477575	+	0.878591i	$0.341516\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−5.88113	−0.432389
$186$	0	0
$187$	0.841686	0.0615501
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−27.1308	−1.96312	−0.981558	−	0.191165i	$-0.938773\pi$
$191$	−27.1308	−1.96312	−0.981558	+	0.191165i	$0.938773\pi$
$192$	0	0
$193$	3.29057	0.236860	0.118430	−	0.992962i	$-0.462214\pi$
$193$	3.29057	0.236860	0.118430	+	0.992962i	$0.462214\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−23.4508	−1.67080	−0.835400	−	0.549642i	$-0.814764\pi$
$197$	−23.4508	−1.67080	−0.835400	+	0.549642i	$0.814764\pi$
$198$	0	0
$199$	9.99288	0.708376	0.354188	−	0.935174i	$-0.384757\pi$
$199$	9.99288	0.708376	0.354188	+	0.935174i	$0.384757\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−4.58585	−0.321864
$204$	0	0
$205$	−0.0991303	−0.00692356
$206$	0	0
$207$	0	0
$208$	0	0
$209$	6.42608	0.444501
$210$	0	0
$211$	22.4570	1.54600	0.773001	−	0.634405i	$-0.218755\pi$
$211$	22.4570	1.54600	0.773001	+	0.634405i	$0.218755\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	7.34523	0.500941
$216$	0	0
$217$	0.722318	0.0490341
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−3.70047	−0.248921
$222$	0	0
$223$	5.52589	0.370041	0.185021	−	0.982735i	$-0.440765\pi$
$223$	5.52589	0.370041	0.185021	+	0.982735i	$0.440765\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−3.34523	−0.222031	−0.111015	−	0.993819i	$-0.535410\pi$
$227$	−3.34523	−0.222031	−0.111015	+	0.993819i	$0.535410\pi$
$228$	0	0
$229$	22.4670	1.48466	0.742330	−	0.670034i	$-0.233721\pi$
$229$	22.4670	1.48466	0.742330	+	0.670034i	$0.233721\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−8.28056	−0.542478	−0.271239	−	0.962512i	$-0.587433\pi$
$233$	−8.28056	−0.542478	−0.271239	+	0.962512i	$0.587433\pi$
$234$	0	0
$235$	−6.89731	−0.449931
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−8.53590	−0.552141	−0.276071	−	0.961137i	$-0.589032\pi$
$239$	−8.53590	−0.552141	−0.276071	+	0.961137i	$0.589032\pi$
$240$	0	0
$241$	27.1807	1.75087	0.875433	−	0.483340i	$-0.160576\pi$
$241$	27.1807	1.75087	0.875433	+	0.483340i	$0.160576\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−3.44793	−0.220280
$246$	0	0
$247$	−28.2523	−1.79765
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−13.0064	−0.820959	−0.410480	−	0.911870i	$-0.634639\pi$
$251$	−13.0064	−0.820959	−0.410480	+	0.911870i	$0.634639\pi$
$252$	0	0
$253$	−0.635800	−0.0399724
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−19.2953	−1.20361	−0.601803	−	0.798644i	$-0.705551\pi$
$257$	−19.2953	−1.20361	−0.601803	+	0.798644i	$0.705551\pi$
$258$	0	0
$259$	11.0841	0.688735
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−16.2791	−1.00381	−0.501906	−	0.864922i	$-0.667368\pi$
$263$	−16.2791	−1.00381	−0.501906	+	0.864922i	$0.667368\pi$
$264$	0	0
$265$	−8.95910	−0.550353
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−16.1279	−0.983337	−0.491668	−	0.870783i	$-0.663613\pi$
$269$	−16.1279	−0.983337	−0.491668	+	0.870783i	$0.663613\pi$
$270$	0	0
$271$	−8.16169	−0.495788	−0.247894	−	0.968787i	$-0.579738\pi$
$271$	−8.16169	−0.495788	−0.247894	+	0.968787i	$0.579738\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−1.03090	−0.0621654
$276$	0	0
$277$	−14.2899	−0.858596	−0.429298	−	0.903163i	$-0.641239\pi$
$277$	−14.2899	−0.858596	−0.429298	+	0.903163i	$0.641239\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	17.3370	1.03424	0.517118	−	0.855914i	$-0.327005\pi$
$281$	17.3370	1.03424	0.517118	+	0.855914i	$0.327005\pi$
$282$	0	0
$283$	−12.6976	−0.754794	−0.377397	−	0.926052i	$-0.623181\pi$
$283$	−12.6976	−0.754794	−0.377397	+	0.926052i	$0.623181\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0.186830	0.0110282
$288$	0	0
$289$	−16.3334	−0.960788
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−20.9267	−1.22255	−0.611277	−	0.791417i	$-0.709344\pi$
$293$	−20.9267	−1.22255	−0.611277	+	0.791417i	$0.709344\pi$
$294$	0	0
$295$	−1.73205	−0.100844
$296$	0	0
$297$	0	0
$298$	0	0
$299$	2.79530	0.161656
$300$	0	0
$301$	−13.8435	−0.797927
$302$	0	0
$303$	0	0
$304$	0	0
$305$	10.7594	0.616081
$306$	0	0
$307$	−25.7123	−1.46748	−0.733740	−	0.679431i	$-0.762227\pi$
$307$	−25.7123	−1.46748	−0.733740	+	0.679431i	$0.762227\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−25.8650	−1.46667	−0.733334	−	0.679869i	$-0.762037\pi$
$311$	−25.8650	−1.46667	−0.733334	+	0.679869i	$0.762037\pi$
$312$	0	0
$313$	10.4217	0.589072	0.294536	−	0.955640i	$-0.404835\pi$
$313$	10.4217	0.589072	0.294536	+	0.955640i	$0.404835\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−28.1308	−1.57998	−0.789992	−	0.613118i	$-0.789915\pi$
$317$	−28.1308	−1.57998	−0.789992	+	0.613118i	$0.789915\pi$
$318$	0	0
$319$	−2.50838	−0.140442
$320$	0	0
$321$	0	0
$322$	0	0
$323$	5.08940	0.283182
$324$	0	0
$325$	4.53234	0.251409
$326$	0	0
$327$	0	0
$328$	0	0
$329$	12.9993	0.716676
$330$	0	0
$331$	30.1384	1.65656	0.828278	−	0.560317i	$-0.189321\pi$
$331$	30.1384	1.65656	0.828278	+	0.560317i	$0.189321\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−12.8664	−0.702967
$336$	0	0
$337$	13.9182	0.758173	0.379086	−	0.925361i	$-0.376238\pi$
$337$	13.9182	0.758173	0.379086	+	0.925361i	$0.376238\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	0.395096	0.0213956
$342$	0	0
$343$	19.6912	1.06322
$344$	0	0
$345$	0	0
$346$	0	0
$347$	0.0865176	0.00464451	0.00232225	−	0.999997i	$-0.499261\pi$
$347$	0.0865176	0.00464451	0.00232225	+	0.999997i	$0.499261\pi$
$348$	0	0
$349$	29.5478	1.58166	0.790829	−	0.612037i	$-0.209649\pi$
$349$	29.5478	1.58166	0.790829	+	0.612037i	$0.209649\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−12.8840	−0.685747	−0.342873	−	0.939382i	$-0.611400\pi$
$353$	−12.8840	−0.685747	−0.342873	+	0.939382i	$0.611400\pi$
$354$	0	0
$355$	8.96198	0.475652
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−28.0590	−1.48090	−0.740449	−	0.672113i	$-0.765387\pi$
$359$	−28.0590	−1.48090	−0.740449	+	0.672113i	$0.765387\pi$
$360$	0	0
$361$	19.8564	1.04507
$362$	0	0
$363$	0	0
$364$	0	0
$365$	6.41703	0.335883
$366$	0	0
$367$	24.5216	1.28002	0.640010	−	0.768367i	$-0.278930\pi$
$367$	24.5216	1.28002	0.640010	+	0.768367i	$0.278930\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	16.8852	0.876634
$372$	0	0
$373$	−16.9910	−0.879763	−0.439881	−	0.898056i	$-0.644979\pi$
$373$	−16.9910	−0.879763	−0.439881	+	0.898056i	$0.644979\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	11.0281	0.567976
$378$	0	0
$379$	−1.07084	−0.0550055	−0.0275027	−	0.999622i	$-0.508755\pi$
$379$	−1.07084	−0.0550055	−0.0275027	+	0.999622i	$0.508755\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−15.9738	−0.816224	−0.408112	−	0.912932i	$-0.633813\pi$
$383$	−15.9738	−0.816224	−0.408112	+	0.912932i	$0.633813\pi$
$384$	0	0
$385$	1.94292	0.0990206
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−12.2364	−0.620409	−0.310204	−	0.950670i	$-0.600397\pi$
$389$	−12.2364	−0.620409	−0.310204	+	0.950670i	$0.600397\pi$
$390$	0	0
$391$	−0.503548	−0.0254655
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−0.535898	−0.0269640
$396$	0	0
$397$	−35.4570	−1.77953	−0.889767	−	0.456414i	$-0.849133\pi$
$397$	−35.4570	−1.77953	−0.889767	+	0.456414i	$0.849133\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	8.43819	0.421383	0.210692	−	0.977553i	$-0.432428\pi$
$401$	8.43819	0.421383	0.210692	+	0.977553i	$0.432428\pi$
$402$	0	0
$403$	−1.73704	−0.0865281
$404$	0	0
$405$	0	0
$406$	0	0
$407$	6.06284	0.300524
$408$	0	0
$409$	−7.49604	−0.370655	−0.185328	−	0.982677i	$-0.559335\pi$
$409$	−7.49604	−0.370655	−0.185328	+	0.982677i	$0.559335\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	3.26439	0.160630
$414$	0	0
$415$	−8.04995	−0.395157
$416$	0	0
$417$	0	0
$418$	0	0
$419$	22.6976	1.10885	0.554425	−	0.832234i	$-0.312938\pi$
$419$	22.6976	1.10885	0.554425	+	0.832234i	$0.312938\pi$
$420$	0	0
$421$	−5.05562	−0.246396	−0.123198	−	0.992382i	$-0.539315\pi$
$421$	−5.05562	−0.246396	−0.123198	+	0.992382i	$0.539315\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−0.816460	−0.0396041
$426$	0	0
$427$	−20.2782	−0.981329
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−36.0209	−1.73507	−0.867533	−	0.497380i	$-0.834295\pi$
$431$	−36.0209	−1.73507	−0.867533	+	0.497380i	$0.834295\pi$
$432$	0	0
$433$	−23.1193	−1.11104	−0.555522	−	0.831502i	$-0.687482\pi$
$433$	−23.1193	−1.11104	−0.555522	+	0.831502i	$0.687482\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−3.84448	−0.183906
$438$	0	0
$439$	22.0219	1.05105	0.525525	−	0.850778i	$-0.323869\pi$
$439$	22.0219	1.05105	0.525525	+	0.850778i	$0.323869\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	34.3481	1.63193	0.815964	−	0.578103i	$-0.196207\pi$
$443$	34.3481	1.63193	0.815964	+	0.578103i	$0.196207\pi$
$444$	0	0
$445$	−18.6567	−0.884412
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−5.39441	−0.254578	−0.127289	−	0.991866i	$-0.540628\pi$
$449$	−5.39441	−0.254578	−0.127289	+	0.991866i	$0.540628\pi$
$450$	0	0
$451$	0.102193	0.00481208
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−8.54207	−0.400458
$456$	0	0
$457$	−4.41703	−0.206620	−0.103310	−	0.994649i	$-0.532943\pi$
$457$	−4.41703	−0.206620	−0.103310	+	0.994649i	$0.532943\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	35.2349	1.64105	0.820527	−	0.571608i	$-0.193680\pi$
$461$	35.2349	1.64105	0.820527	+	0.571608i	$0.193680\pi$
$462$	0	0
$463$	−5.62936	−0.261618	−0.130809	−	0.991408i	$-0.541758\pi$
$463$	−5.62936	−0.261618	−0.130809	+	0.991408i	$0.541758\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	40.3862	1.86885	0.934426	−	0.356158i	$-0.115913\pi$
$467$	40.3862	1.86885	0.934426	+	0.356158i	$0.115913\pi$
$468$	0	0
$469$	24.2493	1.11973
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−7.57217	−0.348169
$474$	0	0
$475$	−6.23349	−0.286012
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−9.34907	−0.427170	−0.213585	−	0.976925i	$-0.568514\pi$
$479$	−9.34907	−0.427170	−0.213585	+	0.976925i	$0.568514\pi$
$480$	0	0
$481$	−26.6553	−1.21538
$482$	0	0
$483$	0	0
$484$	0	0
$485$	7.51405	0.341196
$486$	0	0
$487$	35.7839	1.62152	0.810762	−	0.585376i	$-0.199053\pi$
$487$	35.7839	1.62152	0.810762	+	0.585376i	$0.199053\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	5.39441	0.243446	0.121723	−	0.992564i	$-0.461158\pi$
$491$	5.39441	0.243446	0.121723	+	0.992564i	$0.461158\pi$
$492$	0	0
$493$	−1.98662	−0.0894727
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−16.8906	−0.757647
$498$	0	0
$499$	17.4346	0.780481	0.390241	−	0.920713i	$-0.372392\pi$
$499$	17.4346	0.780481	0.390241	+	0.920713i	$0.372392\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	2.97765	0.132767	0.0663835	−	0.997794i	$-0.478854\pi$
$503$	2.97765	0.132767	0.0663835	+	0.997794i	$0.478854\pi$
$504$	0	0
$505$	−1.27439	−0.0567097
$506$	0	0
$507$	0	0
$508$	0	0
$509$	14.9340	0.661936	0.330968	−	0.943642i	$-0.392625\pi$
$509$	14.9340	0.661936	0.330968	+	0.943642i	$0.392625\pi$
$510$	0	0
$511$	−12.0941	−0.535013
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−4.22705	−0.186266
$516$	0	0
$517$	7.11041	0.312715
$518$	0	0
$519$	0	0
$520$	0	0
$521$	12.2723	0.537658	0.268829	−	0.963188i	$-0.413363\pi$
$521$	12.2723	0.537658	0.268829	+	0.963188i	$0.413363\pi$
$522$	0	0
$523$	−29.5806	−1.29347	−0.646734	−	0.762716i	$-0.723866\pi$
$523$	−29.5806	−1.29347	−0.646734	+	0.762716i	$0.723866\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0.312912	0.0136307
$528$	0	0
$529$	−22.6196	−0.983462
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−0.449292	−0.0194610
$534$	0	0
$535$	14.3923	0.622234
$536$	0	0
$537$	0	0
$538$	0	0
$539$	3.55445	0.153101
$540$	0	0
$541$	2.10319	0.0904233	0.0452117	−	0.998977i	$-0.485604\pi$
$541$	2.10319	0.0904233	0.0452117	+	0.998977i	$0.485604\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−1.08085	−0.0462984
$546$	0	0
$547$	−0.318172	−0.0136041	−0.00680203	−	0.999977i	$-0.502165\pi$
$547$	−0.318172	−0.0136041	−0.00680203	+	0.999977i	$0.502165\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−15.1674	−0.646151
$552$	0	0
$553$	1.01000	0.0429498
$554$	0	0
$555$	0	0
$556$	0	0
$557$	36.7965	1.55912	0.779560	−	0.626328i	$-0.215443\pi$
$557$	36.7965	1.55912	0.779560	+	0.626328i	$0.215443\pi$
$558$	0	0
$559$	33.2911	1.40806
$560$	0	0
$561$	0	0
$562$	0	0
$563$	37.3381	1.57361	0.786807	−	0.617199i	$-0.211733\pi$
$563$	37.3381	1.57361	0.786807	+	0.617199i	$0.211733\pi$
$564$	0	0
$565$	11.1146	0.467596
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−8.74206	−0.366486	−0.183243	−	0.983068i	$-0.558660\pi$
$569$	−8.74206	−0.366486	−0.183243	+	0.983068i	$0.558660\pi$
$570$	0	0
$571$	39.1607	1.63883	0.819413	−	0.573204i	$-0.194300\pi$
$571$	39.1607	1.63883	0.819413	+	0.573204i	$0.194300\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0.616745	0.0257201
$576$	0	0
$577$	−38.2369	−1.59182	−0.795911	−	0.605414i	$-0.793008\pi$
$577$	−38.2369	−1.59182	−0.795911	+	0.605414i	$0.793008\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	15.1717	0.629428
$582$	0	0
$583$	9.23590	0.382512
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−6.51693	−0.268983	−0.134491	−	0.990915i	$-0.542940\pi$
$587$	−6.51693	−0.268983	−0.134491	+	0.990915i	$0.542940\pi$
$588$	0	0
$589$	2.38901	0.0984376
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−13.2464	−0.543963	−0.271982	−	0.962302i	$-0.587679\pi$
$593$	−13.2464	−0.543963	−0.271982	+	0.962302i	$0.587679\pi$
$594$	0	0
$595$	1.53878	0.0630838
$596$	0	0
$597$	0	0
$598$	0	0
$599$	31.5525	1.28920	0.644601	−	0.764519i	$-0.277024\pi$
$599$	31.5525	1.28920	0.644601	+	0.764519i	$0.277024\pi$
$600$	0	0
$601$	3.89876	0.159034	0.0795169	−	0.996834i	$-0.474662\pi$
$601$	3.89876	0.159034	0.0795169	+	0.996834i	$0.474662\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−9.93725	−0.404007
$606$	0	0
$607$	−30.8964	−1.25404	−0.627022	−	0.779001i	$-0.715726\pi$
$607$	−30.8964	−1.25404	−0.627022	+	0.779001i	$0.715726\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−31.2609	−1.26468
$612$	0	0
$613$	−26.0582	−1.05248	−0.526241	−	0.850335i	$-0.676399\pi$
$613$	−26.0582	−1.05248	−0.526241	+	0.850335i	$0.676399\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−38.9211	−1.56690	−0.783452	−	0.621453i	$-0.786543\pi$
$617$	−38.9211	−1.56690	−0.783452	+	0.621453i	$0.786543\pi$
$618$	0	0
$619$	40.3719	1.62268	0.811342	−	0.584572i	$-0.198738\pi$
$619$	40.3719	1.62268	0.811342	+	0.584572i	$0.198738\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	35.1622	1.40874
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	4.80171	0.191457
$630$	0	0
$631$	−37.5773	−1.49593	−0.747964	−	0.663740i	$-0.768968\pi$
$631$	−37.5773	−1.49593	−0.747964	+	0.663740i	$0.768968\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	1.04351	0.0414104
$636$	0	0
$637$	−15.6272	−0.619171
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−32.3572	−1.27803	−0.639016	−	0.769194i	$-0.720658\pi$
$641$	−32.3572	−1.27803	−0.639016	+	0.769194i	$0.720658\pi$
$642$	0	0
$643$	18.1735	0.716694	0.358347	−	0.933588i	$-0.383340\pi$
$643$	18.1735	0.716694	0.358347	+	0.933588i	$0.383340\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−23.7299	−0.932919	−0.466460	−	0.884543i	$-0.654471\pi$
$647$	−23.7299	−0.932919	−0.466460	+	0.884543i	$0.654471\pi$
$648$	0	0
$649$	1.78556	0.0700895
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−9.11934	−0.356867	−0.178434	−	0.983952i	$-0.557103\pi$
$653$	−9.11934	−0.356867	−0.178434	+	0.983952i	$0.557103\pi$
$654$	0	0
$655$	−14.4914	−0.566227
$656$	0	0
$657$	0	0
$658$	0	0
$659$	40.8176	1.59003	0.795014	−	0.606591i	$-0.207463\pi$
$659$	40.8176	1.59003	0.795014	+	0.606591i	$0.207463\pi$
$660$	0	0
$661$	−38.0695	−1.48073	−0.740366	−	0.672205i	$-0.765348\pi$
$661$	−38.0695	−1.48073	−0.740366	+	0.672205i	$0.765348\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	11.7482	0.455577
$666$	0	0
$667$	1.50067	0.0581061
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−11.0918	−0.428194
$672$	0	0
$673$	1.73128	0.0667359	0.0333680	−	0.999443i	$-0.489377\pi$
$673$	1.73128	0.0667359	0.0333680	+	0.999443i	$0.489377\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−20.0680	−0.771274	−0.385637	−	0.922650i	$-0.626018\pi$
$677$	−20.0680	−0.771274	−0.385637	+	0.922650i	$0.626018\pi$
$678$	0	0
$679$	−14.1617	−0.543476
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−48.2473	−1.84613	−0.923067	−	0.384640i	$-0.874326\pi$
$683$	−48.2473	−1.84613	−0.923067	+	0.384640i	$0.874326\pi$
$684$	0	0
$685$	−7.76939	−0.296853
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−40.6056	−1.54695
$690$	0	0
$691$	6.25679	0.238020	0.119010	−	0.992893i	$-0.462028\pi$
$691$	6.25679	0.238020	0.119010	+	0.992893i	$0.462028\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−17.5906	−0.667248
$696$	0	0
$697$	0.0809359	0.00306567
$698$	0	0
$699$	0	0
$700$	0	0
$701$	38.4261	1.45133	0.725667	−	0.688047i	$-0.241532\pi$
$701$	38.4261	1.45133	0.725667	+	0.688047i	$0.241532\pi$
$702$	0	0
$703$	36.6600	1.38266
$704$	0	0
$705$	0	0
$706$	0	0
$707$	2.40184	0.0903304
$708$	0	0
$709$	16.0423	0.602482	0.301241	−	0.953548i	$-0.402599\pi$
$709$	16.0423	0.602482	0.301241	+	0.953548i	$0.402599\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−0.236371	−0.00885215
$714$	0	0
$715$	−4.67237	−0.174737
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−24.9125	−0.929081	−0.464540	−	0.885552i	$-0.653780\pi$
$719$	−24.9125	−0.929081	−0.464540	+	0.885552i	$0.653780\pi$
$720$	0	0
$721$	7.96670	0.296695
$722$	0	0
$723$	0	0
$724$	0	0
$725$	2.43321	0.0903670
$726$	0	0
$727$	−39.7911	−1.47577	−0.737884	−	0.674927i	$-0.764175\pi$
$727$	−39.7911	−1.47577	−0.737884	+	0.674927i	$0.764175\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−5.99709	−0.221810
$732$	0	0
$733$	−10.1617	−0.375331	−0.187665	−	0.982233i	$-0.560092\pi$
$733$	−10.1617	−0.375331	−0.187665	+	0.982233i	$0.560092\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	13.2639	0.488583
$738$	0	0
$739$	9.93859	0.365597	0.182798	−	0.983150i	$-0.441484\pi$
$739$	9.93859	0.365597	0.182798	+	0.983150i	$0.441484\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−13.0234	−0.477782	−0.238891	−	0.971046i	$-0.576784\pi$
$743$	−13.0234	−0.477782	−0.238891	+	0.971046i	$0.576784\pi$
$744$	0	0
$745$	−10.3053	−0.377557
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−27.1251	−0.991130
$750$	0	0
$751$	19.8888	0.725751	0.362876	−	0.931838i	$-0.381795\pi$
$751$	19.8888	0.725751	0.362876	+	0.931838i	$0.381795\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−6.91203	−0.251554
$756$	0	0
$757$	−25.6617	−0.932691	−0.466345	−	0.884603i	$-0.654430\pi$
$757$	−25.6617	−0.932691	−0.466345	+	0.884603i	$0.654430\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	26.8291	0.972553	0.486277	−	0.873805i	$-0.338355\pi$
$761$	26.8291	0.972553	0.486277	+	0.873805i	$0.338355\pi$
$762$	0	0
$763$	2.03707	0.0737468
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−7.85024	−0.283456
$768$	0	0
$769$	44.0566	1.58872	0.794361	−	0.607446i	$-0.207806\pi$
$769$	44.0566	1.58872	0.794361	+	0.607446i	$0.207806\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−24.8388	−0.893389	−0.446694	−	0.894687i	$-0.647399\pi$
$773$	−24.8388	−0.893389	−0.446694	+	0.894687i	$0.647399\pi$
$774$	0	0
$775$	−0.383255	−0.0137669
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0.617928	0.0221396
$780$	0	0
$781$	−9.23887	−0.330593
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−3.82290	−0.136445
$786$	0	0
$787$	−25.1764	−0.897442	−0.448721	−	0.893672i	$-0.648120\pi$
$787$	−25.1764	−0.897442	−0.448721	+	0.893672i	$0.648120\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−20.9477	−0.744813
$792$	0	0
$793$	48.7651	1.73170
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−8.84640	−0.313356	−0.156678	−	0.987650i	$-0.550078\pi$
$797$	−8.84640	−0.313356	−0.156678	+	0.987650i	$0.550078\pi$
$798$	0	0
$799$	5.63138	0.199224
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−6.61529	−0.233449
$804$	0	0
$805$	−1.16238	−0.0409684
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−39.5324	−1.38989	−0.694943	−	0.719065i	$-0.744570\pi$
$809$	−39.5324	−1.38989	−0.694943	+	0.719065i	$0.744570\pi$
$810$	0	0
$811$	−35.9948	−1.26395	−0.631974	−	0.774989i	$-0.717755\pi$
$811$	−35.9948	−1.26395	−0.631974	+	0.774989i	$0.717755\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−8.28056	−0.290055
$816$	0	0
$817$	−45.7864	−1.60186
$818$	0	0
$819$	0	0
$820$	0	0
$821$	5.79741	0.202331	0.101165	−	0.994870i	$-0.467743\pi$
$821$	5.79741	0.202331	0.101165	+	0.994870i	$0.467743\pi$
$822$	0	0
$823$	31.3757	1.09369	0.546844	−	0.837234i	$-0.315829\pi$
$823$	31.3757	1.09369	0.546844	+	0.837234i	$0.315829\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	43.4441	1.51070	0.755350	−	0.655322i	$-0.227467\pi$
$827$	43.4441	1.51070	0.755350	+	0.655322i	$0.227467\pi$
$828$	0	0
$829$	−6.91915	−0.240312	−0.120156	−	0.992755i	$-0.538339\pi$
$829$	−6.91915	−0.240312	−0.120156	+	0.992755i	$0.538339\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	2.81509	0.0975373
$834$	0	0
$835$	4.24821	0.147015
$836$	0	0
$837$	0	0
$838$	0	0
$839$	40.9420	1.41347	0.706737	−	0.707476i	$-0.250166\pi$
$839$	40.9420	1.41347	0.706737	+	0.707476i	$0.250166\pi$
$840$	0	0
$841$	−23.0795	−0.795845
$842$	0	0
$843$	0	0
$844$	0	0
$845$	7.54207	0.259455
$846$	0	0
$847$	18.7287	0.643525
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−3.62716	−0.124337
$852$	0	0
$853$	−17.2382	−0.590225	−0.295112	−	0.955463i	$-0.595357\pi$
$853$	−17.2382	−0.590225	−0.295112	+	0.955463i	$0.595357\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−34.7523	−1.18711	−0.593557	−	0.804792i	$-0.702277\pi$
$857$	−34.7523	−1.18711	−0.593557	+	0.804792i	$0.702277\pi$
$858$	0	0
$859$	28.9809	0.988817	0.494409	−	0.869230i	$-0.335385\pi$
$859$	28.9809	0.988817	0.494409	+	0.869230i	$0.335385\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−42.1926	−1.43625	−0.718126	−	0.695913i	$-0.755000\pi$
$863$	−42.1926	−1.43625	−0.718126	+	0.695913i	$0.755000\pi$
$864$	0	0
$865$	3.54495	0.120532
$866$	0	0
$867$	0	0
$868$	0	0
$869$	0.552456	0.0187408
$870$	0	0
$871$	−58.3149	−1.97592
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−1.88469	−0.0637143
$876$	0	0
$877$	13.3241	0.449922	0.224961	−	0.974368i	$-0.427775\pi$
$877$	13.3241	0.449922	0.224961	+	0.974368i	$0.427775\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	28.1897	0.949735	0.474868	−	0.880057i	$-0.342496\pi$
$881$	28.1897	0.949735	0.474868	+	0.880057i	$0.342496\pi$
$882$	0	0
$883$	6.46648	0.217614	0.108807	−	0.994063i	$-0.465297\pi$
$883$	6.46648	0.217614	0.108807	+	0.994063i	$0.465297\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−4.77740	−0.160409	−0.0802046	−	0.996778i	$-0.525557\pi$
$887$	−4.77740	−0.160409	−0.0802046	+	0.996778i	$0.525557\pi$
$888$	0	0
$889$	−1.96670	−0.0659608
$890$	0	0
$891$	0	0
$892$	0	0
$893$	42.9943	1.43875
$894$	0	0
$895$	−0.0991303	−0.00331356
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−0.932537	−0.0311019
$900$	0	0
$901$	7.31475	0.243690
$902$	0	0
$903$	0	0
$904$	0	0
$905$	12.8502	0.427156
$906$	0	0
$907$	45.4009	1.50751	0.753757	−	0.657153i	$-0.228240\pi$
$907$	45.4009	1.50751	0.753757	+	0.657153i	$0.228240\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−30.1702	−0.999585	−0.499792	−	0.866145i	$-0.666590\pi$
$911$	−30.1702	−0.999585	−0.499792	+	0.866145i	$0.666590\pi$
$912$	0	0
$913$	8.29866	0.274646
$914$	0	0
$915$	0	0
$916$	0	0
$917$	27.3119	0.901919
$918$	0	0
$919$	49.0666	1.61856	0.809279	−	0.587425i	$-0.199858\pi$
$919$	49.0666	1.61856	0.809279	+	0.587425i	$0.199858\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	40.6187	1.33698
$924$	0	0
$925$	−5.88113	−0.193370
$926$	0	0
$927$	0	0
$928$	0	0
$929$	48.2430	1.58280	0.791401	−	0.611297i	$-0.209352\pi$
$929$	48.2430	1.58280	0.791401	+	0.611297i	$0.209352\pi$
$930$	0	0
$931$	21.4926	0.704392
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0.841686	0.0275261
$936$	0	0
$937$	−34.8959	−1.14000	−0.569999	−	0.821645i	$-0.693056\pi$
$937$	−34.8959	−1.14000	−0.569999	+	0.821645i	$0.693056\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	10.5159	0.342808	0.171404	−	0.985201i	$-0.445170\pi$
$941$	10.5159	0.342808	0.171404	+	0.985201i	$0.445170\pi$
$942$	0	0
$943$	−0.0611381	−0.00199093
$944$	0	0
$945$	0	0
$946$	0	0
$947$	13.8188	0.449052	0.224526	−	0.974468i	$-0.427917\pi$
$947$	13.8188	0.449052	0.224526	+	0.974468i	$0.427917\pi$
$948$	0	0
$949$	29.0841	0.944111
$950$	0	0
$951$	0	0
$952$	0	0
$953$	28.3276	0.917621	0.458811	−	0.888534i	$-0.348276\pi$
$953$	28.3276	0.917621	0.458811	+	0.888534i	$0.348276\pi$
$954$	0	0
$955$	−27.1308	−0.877932
$956$	0	0
$957$	0	0
$958$	0	0
$959$	14.6429	0.472845
$960$	0	0
$961$	−30.8531	−0.995262
$962$	0	0
$963$	0	0
$964$	0	0
$965$	3.29057	0.105927
$966$	0	0
$967$	10.6900	0.343766	0.171883	−	0.985117i	$-0.445015\pi$
$967$	10.6900	0.343766	0.171883	+	0.985117i	$0.445015\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	13.9555	0.447854	0.223927	−	0.974606i	$-0.428112\pi$
$971$	13.9555	0.447854	0.223927	+	0.974606i	$0.428112\pi$
$972$	0	0
$973$	33.1528	1.06283
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−13.3334	−0.426573	−0.213287	−	0.976990i	$-0.568417\pi$
$977$	−13.3334	−0.426573	−0.213287	+	0.976990i	$0.568417\pi$
$978$	0	0
$979$	19.2331	0.614693
$980$	0	0
$981$	0	0
$982$	0	0
$983$	17.7318	0.565556	0.282778	−	0.959185i	$-0.408744\pi$
$983$	17.7318	0.565556	0.282778	+	0.959185i	$0.408744\pi$
$984$	0	0
$985$	−23.4508	−0.747205
$986$	0	0
$987$	0	0
$988$	0	0
$989$	4.53014	0.144050
$990$	0	0
$991$	52.7053	1.67424	0.837119	−	0.547021i	$-0.184238\pi$
$991$	52.7053	1.67424	0.837119	+	0.547021i	$0.184238\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	9.99288	0.316795
$996$	0	0
$997$	−26.4387	−0.837322	−0.418661	−	0.908143i	$-0.637500\pi$
$997$	−26.4387	−0.837322	−0.418661	+	0.908143i	$0.637500\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6480.2.a.ca.1.2		4
3.2	odd	2		6480.2.a.by.1.2		4
4.3	odd	2		3240.2.a.v.1.3	yes	4
12.11	even	2		3240.2.a.t.1.3	✓	4
36.7	odd	6		3240.2.q.bg.1081.2		8
36.11	even	6		3240.2.q.bh.1081.2		8
36.23	even	6		3240.2.q.bh.2161.2		8
36.31	odd	6		3240.2.q.bg.2161.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3240.2.a.t.1.3	✓	4	12.11	even	2
3240.2.a.v.1.3	yes	4	4.3	odd	2
3240.2.q.bg.1081.2		8	36.7	odd	6
3240.2.q.bg.2161.2		8	36.31	odd	6
3240.2.q.bh.1081.2		8	36.11	even	6
3240.2.q.bh.2161.2		8	36.23	even	6
6480.2.a.by.1.2		4	3.2	odd	2
6480.2.a.ca.1.2		4	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 6480 = 2^{4} \cdot 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6480.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{5} -1.88469 q^{7} -1.03090 q^{11} +4.53234 q^{13} -0.816460 q^{17} -6.23349 q^{19} +0.616745 q^{23} +1.00000 q^{25} +2.43321 q^{29} -0.383255 q^{31} -1.88469 q^{35} -5.88113 q^{37} -0.0991303 q^{41} +7.34523 q^{43} -6.89731 q^{47} -3.44793 q^{49} -8.95910 q^{53} -1.03090 q^{55} -1.73205 q^{59} +10.7594 q^{61} +4.53234 q^{65} -12.8664 q^{67} +8.96198 q^{71} +6.41703 q^{73} +1.94292 q^{77} -0.535898 q^{79} -8.04995 q^{83} -0.816460 q^{85} -18.6567 q^{89} -8.54207 q^{91} -6.23349 q^{95} +7.51405 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{5} - 2 q^{7} - 4 q^{11} - 2 q^{17} - 10 q^{23} + 4 q^{25} - 4 q^{29} - 14 q^{31} - 2 q^{35} + 14 q^{37} + 4 q^{41} - 22 q^{43} + 10 q^{49} - 8 q^{53} - 4 q^{55} + 4 q^{61} - 24 q^{67} - 28 q^{71}+ \cdots - 10 q^{97}+O(q^{100}) \) 4 * q + 4 * q^5 - 2 * q^7 - 4 * q^11 - 2 * q^17 - 10 * q^23 + 4 * q^25 - 4 * q^29 - 14 * q^31 - 2 * q^35 + 14 * q^37 + 4 * q^41 - 22 * q^43 + 10 * q^49 - 8 * q^53 - 4 * q^55 + 4 * q^61 - 24 * q^67 - 28 * q^71 + 2 * q^73 - 30 * q^77 - 16 * q^79 - 6 * q^83 - 2 * q^85 - 8 * q^89 - 30 * q^91 - 10 * q^97

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