LMFDB - Embedded newform 3564.2.a.o.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3564,2,Mod(1,3564)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3564, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("3564.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3564 = 2^{2} \cdot 3^{4} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3564.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:	$28.4586832804$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	6.6.1166692032.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 20x^{4} + 52x^{2} - 24x - 3 $ x^6 - 20x^4 + 52x^2 - 24*x - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.98594$ of defining polynomial
Character	$\chi$	$=$	3564.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.98594 q^{5} -0.575360 q^{7} +O(q^{10})$	$q-1.98594 q^{5} -0.575360 q^{7} -1.00000 q^{11} +2.71799 q^{13} -5.85033 q^{17} +7.40818 q^{19} +3.26900 q^{23} -1.05605 q^{25} -4.01763 q^{29} +4.72144 q^{31} +1.14263 q^{35} -8.38727 q^{37} +11.1447 q^{41} -0.916864 q^{43} -0.142507 q^{47} -6.66896 q^{49} +3.52288 q^{53} +1.98594 q^{55} -12.3916 q^{59} -5.89651 q^{61} -5.39776 q^{65} +9.87573 q^{67} -9.86439 q^{71} +9.04993 q^{73} +0.575360 q^{77} -9.48266 q^{79} -5.96498 q^{83} +11.6184 q^{85} -6.94155 q^{89} -1.56382 q^{91} -14.7122 q^{95} -18.9022 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 2 q^{7}+O(q^{10}) $ 6 * q - 2 * q^7	$ 6 q - 2 q^{7} - 6 q^{11} - 6 q^{13} - 4 q^{17} - 4 q^{23} + 10 q^{25} + 4 q^{29} + 6 q^{31} - 14 q^{35} - 6 q^{37} - 22 q^{41} - 10 q^{43} - 20 q^{47} + 6 q^{49} - 4 q^{53} - 24 q^{59} - 10 q^{61} - 40 q^{65} - 6 q^{67} - 40 q^{71} - 8 q^{73} + 2 q^{77} - 14 q^{79} - 12 q^{83} + 6 q^{85} - 24 q^{89} + 10 q^{91} - 44 q^{95} + 20 q^{97}+O(q^{100}) $ 6 * q - 2 * q^7 - 6 * q^11 - 6 * q^13 - 4 * q^17 - 4 * q^23 + 10 * q^25 + 4 * q^29 + 6 * q^31 - 14 * q^35 - 6 * q^37 - 22 * q^41 - 10 * q^43 - 20 * q^47 + 6 * q^49 - 4 * q^53 - 24 * q^59 - 10 * q^61 - 40 * q^65 - 6 * q^67 - 40 * q^71 - 8 * q^73 + 2 * q^77 - 14 * q^79 - 12 * q^83 + 6 * q^85 - 24 * q^89 + 10 * q^91 - 44 * q^95 + 20 * 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
$5$	−1.98594	−0.888139	−0.444069	−	0.895992i	$-0.646466\pi$
$5$	−1.98594	−0.888139	−0.444069	+	0.895992i	$0.646466\pi$
$6$	0	0
$7$	−0.575360	−0.217466	−0.108733	−	0.994071i	$-0.534679\pi$
$7$	−0.575360	−0.217466	−0.108733	+	0.994071i	$0.534679\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	2.71799	0.753835	0.376917	−	0.926247i	$-0.376984\pi$
$13$	2.71799	0.753835	0.376917	+	0.926247i	$0.376984\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−5.85033	−1.41891	−0.709456	−	0.704749i	$-0.751059\pi$
$17$	−5.85033	−1.41891	−0.709456	+	0.704749i	$0.751059\pi$
$18$	0	0
$19$	7.40818	1.69955	0.849776	−	0.527143i	$-0.176737\pi$
$19$	7.40818	1.69955	0.849776	+	0.527143i	$0.176737\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	3.26900	0.681633	0.340816	−	0.940130i	$-0.389297\pi$
$23$	3.26900	0.681633	0.340816	+	0.940130i	$0.389297\pi$
$24$	0	0
$25$	−1.05605	−0.211209
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−4.01763	−0.746055	−0.373028	−	0.927820i	$-0.621680\pi$
$29$	−4.01763	−0.746055	−0.373028	+	0.927820i	$0.621680\pi$
$30$	0	0
$31$	4.72144	0.847995	0.423998	−	0.905663i	$-0.360627\pi$
$31$	4.72144	0.847995	0.423998	+	0.905663i	$0.360627\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	1.14263	0.193140
$36$	0	0
$37$	−8.38727	−1.37886	−0.689429	−	0.724353i	$-0.742139\pi$
$37$	−8.38727	−1.37886	−0.689429	+	0.724353i	$0.742139\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	11.1447	1.74051	0.870257	−	0.492599i	$-0.163953\pi$
$41$	11.1447	1.74051	0.870257	+	0.492599i	$0.163953\pi$
$42$	0	0
$43$	−0.916864	−0.139820	−0.0699102	−	0.997553i	$-0.522271\pi$
$43$	−0.916864	−0.139820	−0.0699102	+	0.997553i	$0.522271\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−0.142507	−0.0207868	−0.0103934	−	0.999946i	$-0.503308\pi$
$47$	−0.142507	−0.0207868	−0.0103934	+	0.999946i	$0.503308\pi$
$48$	0	0
$49$	−6.66896	−0.952709
$50$	0	0
$51$	0	0
$52$	0	0
$53$	3.52288	0.483905	0.241953	−	0.970288i	$-0.422212\pi$
$53$	3.52288	0.483905	0.241953	+	0.970288i	$0.422212\pi$
$54$	0	0
$55$	1.98594	0.267784
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−12.3916	−1.61325	−0.806624	−	0.591065i	$-0.798708\pi$
$59$	−12.3916	−1.61325	−0.806624	+	0.591065i	$0.798708\pi$
$60$	0	0
$61$	−5.89651	−0.754971	−0.377485	−	0.926016i	$-0.623211\pi$
$61$	−5.89651	−0.754971	−0.377485	+	0.926016i	$0.623211\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−5.39776	−0.669510
$66$	0	0
$67$	9.87573	1.20651	0.603256	−	0.797548i	$-0.293870\pi$
$67$	9.87573	1.20651	0.603256	+	0.797548i	$0.293870\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−9.86439	−1.17069	−0.585344	−	0.810785i	$-0.699040\pi$
$71$	−9.86439	−1.17069	−0.585344	+	0.810785i	$0.699040\pi$
$72$	0	0
$73$	9.04993	1.05921	0.529607	−	0.848243i	$-0.322339\pi$
$73$	9.04993	1.05921	0.529607	+	0.848243i	$0.322339\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0.575360	0.0655684
$78$	0	0
$79$	−9.48266	−1.06688	−0.533441	−	0.845837i	$-0.679101\pi$
$79$	−9.48266	−1.06688	−0.533441	+	0.845837i	$0.679101\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−5.96498	−0.654742	−0.327371	−	0.944896i	$-0.606163\pi$
$83$	−5.96498	−0.654742	−0.327371	+	0.944896i	$0.606163\pi$
$84$	0	0
$85$	11.6184	1.26019
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−6.94155	−0.735803	−0.367902	−	0.929865i	$-0.619924\pi$
$89$	−6.94155	−0.735803	−0.367902	+	0.929865i	$0.619924\pi$
$90$	0	0
$91$	−1.56382	−0.163933
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−14.7122	−1.50944
$96$	0	0
$97$	−18.9022	−1.91922	−0.959612	−	0.281325i	$-0.909226\pi$
$97$	−18.9022	−1.91922	−0.959612	+	0.281325i	$0.909226\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	2.97533	0.296056	0.148028	−	0.988983i	$-0.452707\pi$
$101$	2.97533	0.296056	0.148028	+	0.988983i	$0.452707\pi$
$102$	0	0
$103$	12.0894	1.19120	0.595601	−	0.803280i	$-0.296914\pi$
$103$	12.0894	1.19120	0.595601	+	0.803280i	$0.296914\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−12.5867	−1.21680	−0.608401	−	0.793630i	$-0.708189\pi$
$107$	−12.5867	−1.21680	−0.608401	+	0.793630i	$0.708189\pi$
$108$	0	0
$109$	8.20065	0.785479	0.392740	−	0.919650i	$-0.371527\pi$
$109$	8.20065	0.785479	0.392740	+	0.919650i	$0.371527\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−6.99415	−0.657954	−0.328977	−	0.944338i	$-0.606704\pi$
$113$	−6.99415	−0.657954	−0.328977	+	0.944338i	$0.606704\pi$
$114$	0	0
$115$	−6.49203	−0.605385
$116$	0	0
$117$	0	0
$118$	0	0
$119$	3.36604	0.308565
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	0	0
$124$	0	0
$125$	12.0269	1.07572
$126$	0	0
$127$	−15.4384	−1.36994	−0.684968	−	0.728573i	$-0.740184\pi$
$127$	−15.4384	−1.36994	−0.684968	+	0.728573i	$0.740184\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−14.5109	−1.26782	−0.633912	−	0.773405i	$-0.718552\pi$
$131$	−14.5109	−1.26782	−0.633912	+	0.773405i	$0.718552\pi$
$132$	0	0
$133$	−4.26237	−0.369594
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−0.441828	−0.0377479	−0.0188740	−	0.999822i	$-0.506008\pi$
$137$	−0.441828	−0.0377479	−0.0188740	+	0.999822i	$0.506008\pi$
$138$	0	0
$139$	−14.4608	−1.22655	−0.613276	−	0.789869i	$-0.710149\pi$
$139$	−14.4608	−1.22655	−0.613276	+	0.789869i	$0.710149\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−2.71799	−0.227290
$144$	0	0
$145$	7.97877	0.662601
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−17.5866	−1.44075	−0.720373	−	0.693587i	$-0.756029\pi$
$149$	−17.5866	−1.44075	−0.720373	+	0.693587i	$0.756029\pi$
$150$	0	0
$151$	5.63069	0.458219	0.229110	−	0.973401i	$-0.426419\pi$
$151$	5.63069	0.458219	0.229110	+	0.973401i	$0.426419\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−9.37649	−0.753138
$156$	0	0
$157$	−2.36063	−0.188398	−0.0941992	−	0.995553i	$-0.530029\pi$
$157$	−2.36063	−0.188398	−0.0941992	+	0.995553i	$0.530029\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−1.88085	−0.148232
$162$	0	0
$163$	7.00734	0.548857	0.274429	−	0.961608i	$-0.411511\pi$
$163$	7.00734	0.548857	0.274429	+	0.961608i	$0.411511\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	6.86348	0.531112	0.265556	−	0.964095i	$-0.414444\pi$
$167$	6.86348	0.531112	0.265556	+	0.964095i	$0.414444\pi$
$168$	0	0
$169$	−5.61253	−0.431733
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−12.4720	−0.948232	−0.474116	−	0.880462i	$-0.657232\pi$
$173$	−12.4720	−0.948232	−0.474116	+	0.880462i	$0.657232\pi$
$174$	0	0
$175$	0.607607	0.0459307
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−4.87051	−0.364039	−0.182019	−	0.983295i	$-0.558263\pi$
$179$	−4.87051	−0.364039	−0.182019	+	0.983295i	$0.558263\pi$
$180$	0	0
$181$	15.0264	1.11691	0.558454	−	0.829536i	$-0.311395\pi$
$181$	15.0264	1.11691	0.558454	+	0.829536i	$0.311395\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	16.6566	1.22462
$186$	0	0
$187$	5.85033	0.427818
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−9.08194	−0.657146	−0.328573	−	0.944479i	$-0.606568\pi$
$191$	−9.08194	−0.657146	−0.328573	+	0.944479i	$0.606568\pi$
$192$	0	0
$193$	4.78827	0.344667	0.172334	−	0.985039i	$-0.444869\pi$
$193$	4.78827	0.344667	0.172334	+	0.985039i	$0.444869\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	13.4686	0.959598	0.479799	−	0.877379i	$-0.340710\pi$
$197$	13.4686	0.959598	0.479799	+	0.877379i	$0.340710\pi$
$198$	0	0
$199$	−20.5485	−1.45664	−0.728321	−	0.685236i	$-0.759699\pi$
$199$	−20.5485	−1.45664	−0.728321	+	0.685236i	$0.759699\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	2.31158	0.162241
$204$	0	0
$205$	−22.1327	−1.54582
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−7.40818	−0.512434
$210$	0	0
$211$	3.77748	0.260053	0.130026	−	0.991511i	$-0.458494\pi$
$211$	3.77748	0.260053	0.130026	+	0.991511i	$0.458494\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	1.82084	0.124180
$216$	0	0
$217$	−2.71653	−0.184410
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−15.9011	−1.06963
$222$	0	0
$223$	−23.9246	−1.60211	−0.801056	−	0.598590i	$-0.795728\pi$
$223$	−23.9246	−1.60211	−0.801056	+	0.598590i	$0.795728\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	4.30738	0.285891	0.142946	−	0.989731i	$-0.454343\pi$
$227$	4.30738	0.285891	0.142946	+	0.989731i	$0.454343\pi$
$228$	0	0
$229$	7.18702	0.474931	0.237466	−	0.971396i	$-0.423683\pi$
$229$	7.18702	0.474931	0.237466	+	0.971396i	$0.423683\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	2.58221	0.169166	0.0845830	−	0.996416i	$-0.473044\pi$
$233$	2.58221	0.169166	0.0845830	+	0.996416i	$0.473044\pi$
$234$	0	0
$235$	0.283011	0.0184616
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−18.4181	−1.19136	−0.595682	−	0.803220i	$-0.703118\pi$
$239$	−18.4181	−1.19136	−0.595682	+	0.803220i	$0.703118\pi$
$240$	0	0
$241$	−19.3236	−1.24474	−0.622370	−	0.782723i	$-0.713830\pi$
$241$	−19.3236	−1.24474	−0.622370	+	0.782723i	$0.713830\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	13.2441	0.846138
$246$	0	0
$247$	20.1354	1.28118
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−6.11394	−0.385908	−0.192954	−	0.981208i	$-0.561807\pi$
$251$	−6.11394	−0.385908	−0.192954	+	0.981208i	$0.561807\pi$
$252$	0	0
$253$	−3.26900	−0.205520
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−6.12660	−0.382167	−0.191083	−	0.981574i	$-0.561200\pi$
$257$	−6.12660	−0.382167	−0.191083	+	0.981574i	$0.561200\pi$
$258$	0	0
$259$	4.82570	0.299854
$260$	0	0
$261$	0	0
$262$	0	0
$263$	4.79608	0.295739	0.147869	−	0.989007i	$-0.452758\pi$
$263$	4.79608	0.295739	0.147869	+	0.989007i	$0.452758\pi$
$264$	0	0
$265$	−6.99623	−0.429775
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−17.6991	−1.07913	−0.539567	−	0.841942i	$-0.681412\pi$
$269$	−17.6991	−1.07913	−0.539567	+	0.841942i	$0.681412\pi$
$270$	0	0
$271$	8.24853	0.501063	0.250531	−	0.968108i	$-0.419395\pi$
$271$	8.24853	0.501063	0.250531	+	0.968108i	$0.419395\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	1.05605	0.0636820
$276$	0	0
$277$	−19.0029	−1.14178	−0.570888	−	0.821028i	$-0.693401\pi$
$277$	−19.0029	−1.14178	−0.570888	+	0.821028i	$0.693401\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−4.49639	−0.268232	−0.134116	−	0.990966i	$-0.542819\pi$
$281$	−4.49639	−0.268232	−0.134116	+	0.990966i	$0.542819\pi$
$282$	0	0
$283$	10.8648	0.645846	0.322923	−	0.946425i	$-0.395334\pi$
$283$	10.8648	0.645846	0.322923	+	0.946425i	$0.395334\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−6.41223	−0.378502
$288$	0	0
$289$	17.2263	1.01331
$290$	0	0
$291$	0	0
$292$	0	0
$293$	30.6817	1.79244	0.896221	−	0.443608i	$-0.146302\pi$
$293$	30.6817	1.79244	0.896221	+	0.443608i	$0.146302\pi$
$294$	0	0
$295$	24.6089	1.43279
$296$	0	0
$297$	0	0
$298$	0	0
$299$	8.88510	0.513838
$300$	0	0
$301$	0.527527	0.0304061
$302$	0	0
$303$	0	0
$304$	0	0
$305$	11.7101	0.670519
$306$	0	0
$307$	3.68451	0.210286	0.105143	−	0.994457i	$-0.466470\pi$
$307$	3.68451	0.210286	0.105143	+	0.994457i	$0.466470\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	1.21231	0.0687436	0.0343718	−	0.999409i	$-0.489057\pi$
$311$	1.21231	0.0687436	0.0343718	+	0.999409i	$0.489057\pi$
$312$	0	0
$313$	14.3839	0.813029	0.406514	−	0.913644i	$-0.366744\pi$
$313$	14.3839	0.813029	0.406514	+	0.913644i	$0.366744\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−15.3096	−0.859874	−0.429937	−	0.902859i	$-0.641464\pi$
$317$	−15.3096	−0.859874	−0.429937	+	0.902859i	$0.641464\pi$
$318$	0	0
$319$	4.01763	0.224944
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−43.3403	−2.41152
$324$	0	0
$325$	−2.87032	−0.159217
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0.0819929	0.00452042
$330$	0	0
$331$	−8.73386	−0.480057	−0.240028	−	0.970766i	$-0.577157\pi$
$331$	−8.73386	−0.480057	−0.240028	+	0.970766i	$0.577157\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−19.6126	−1.07155
$336$	0	0
$337$	17.8292	0.971220	0.485610	−	0.874175i	$-0.338597\pi$
$337$	17.8292	0.971220	0.485610	+	0.874175i	$0.338597\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−4.72144	−0.255680
$342$	0	0
$343$	7.86457	0.424647
$344$	0	0
$345$	0	0
$346$	0	0
$347$	3.52709	0.189344	0.0946722	−	0.995509i	$-0.469820\pi$
$347$	3.52709	0.189344	0.0946722	+	0.995509i	$0.469820\pi$
$348$	0	0
$349$	−8.30858	−0.444748	−0.222374	−	0.974961i	$-0.571381\pi$
$349$	−8.30858	−0.444748	−0.222374	+	0.974961i	$0.571381\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	28.7352	1.52942	0.764710	−	0.644375i	$-0.222882\pi$
$353$	28.7352	1.52942	0.764710	+	0.644375i	$0.222882\pi$
$354$	0	0
$355$	19.5901	1.03973
$356$	0	0
$357$	0	0
$358$	0	0
$359$	9.18975	0.485017	0.242508	−	0.970149i	$-0.422030\pi$
$359$	9.18975	0.485017	0.242508	+	0.970149i	$0.422030\pi$
$360$	0	0
$361$	35.8811	1.88848
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−17.9726	−0.940729
$366$	0	0
$367$	30.3424	1.58386	0.791929	−	0.610613i	$-0.209077\pi$
$367$	30.3424	1.58386	0.791929	+	0.610613i	$0.209077\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−2.02693	−0.105233
$372$	0	0
$373$	−14.7650	−0.764500	−0.382250	−	0.924059i	$-0.624851\pi$
$373$	−14.7650	−0.764500	−0.382250	+	0.924059i	$0.624851\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−10.9199	−0.562402
$378$	0	0
$379$	−31.3491	−1.61029	−0.805147	−	0.593076i	$-0.797913\pi$
$379$	−31.3491	−1.61029	−0.805147	+	0.593076i	$0.797913\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	32.7798	1.67497	0.837485	−	0.546460i	$-0.184025\pi$
$383$	32.7798	1.67497	0.837485	+	0.546460i	$0.184025\pi$
$384$	0	0
$385$	−1.14263	−0.0582338
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−20.6156	−1.04525	−0.522625	−	0.852563i	$-0.675047\pi$
$389$	−20.6156	−1.04525	−0.522625	+	0.852563i	$0.675047\pi$
$390$	0	0
$391$	−19.1247	−0.967177
$392$	0	0
$393$	0	0
$394$	0	0
$395$	18.8320	0.947540
$396$	0	0
$397$	3.22545	0.161880	0.0809402	−	0.996719i	$-0.474208\pi$
$397$	3.22545	0.161880	0.0809402	+	0.996719i	$0.474208\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	12.0256	0.600528	0.300264	−	0.953856i	$-0.402925\pi$
$401$	12.0256	0.600528	0.300264	+	0.953856i	$0.402925\pi$
$402$	0	0
$403$	12.8328	0.639248
$404$	0	0
$405$	0	0
$406$	0	0
$407$	8.38727	0.415742
$408$	0	0
$409$	−35.4288	−1.75184	−0.875920	−	0.482456i	$-0.839745\pi$
$409$	−35.4288	−1.75184	−0.875920	+	0.482456i	$0.839745\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	7.12963	0.350826
$414$	0	0
$415$	11.8461	0.581502
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−25.3685	−1.23933	−0.619666	−	0.784865i	$-0.712732\pi$
$419$	−25.3685	−1.23933	−0.619666	+	0.784865i	$0.712732\pi$
$420$	0	0
$421$	36.4451	1.77622	0.888112	−	0.459628i	$-0.152017\pi$
$421$	36.4451	1.77622	0.888112	+	0.459628i	$0.152017\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	6.17821	0.299687
$426$	0	0
$427$	3.39262	0.164180
$428$	0	0
$429$	0	0
$430$	0	0
$431$	26.9989	1.30049	0.650246	−	0.759724i	$-0.274666\pi$
$431$	26.9989	1.30049	0.650246	+	0.759724i	$0.274666\pi$
$432$	0	0
$433$	−22.1196	−1.06300	−0.531500	−	0.847058i	$-0.678372\pi$
$433$	−22.1196	−1.06300	−0.531500	+	0.847058i	$0.678372\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	24.2173	1.15847
$438$	0	0
$439$	−13.5936	−0.648787	−0.324394	−	0.945922i	$-0.605160\pi$
$439$	−13.5936	−0.648787	−0.324394	+	0.945922i	$0.605160\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−30.8048	−1.46358	−0.731791	−	0.681529i	$-0.761315\pi$
$443$	−30.8048	−1.46358	−0.731791	+	0.681529i	$0.761315\pi$
$444$	0	0
$445$	13.7855	0.653495
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−33.8606	−1.59798	−0.798990	−	0.601344i	$-0.794632\pi$
$449$	−33.8606	−1.59798	−0.798990	+	0.601344i	$0.794632\pi$
$450$	0	0
$451$	−11.1447	−0.524784
$452$	0	0
$453$	0	0
$454$	0	0
$455$	3.10566	0.145595
$456$	0	0
$457$	−1.24912	−0.0584315	−0.0292157	−	0.999573i	$-0.509301\pi$
$457$	−1.24912	−0.0584315	−0.0292157	+	0.999573i	$0.509301\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	32.5234	1.51477	0.757383	−	0.652971i	$-0.226478\pi$
$461$	32.5234	1.51477	0.757383	+	0.652971i	$0.226478\pi$
$462$	0	0
$463$	−32.7183	−1.52055	−0.760274	−	0.649602i	$-0.774935\pi$
$463$	−32.7183	−1.52055	−0.760274	+	0.649602i	$0.774935\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	27.1398	1.25588	0.627940	−	0.778261i	$-0.283898\pi$
$467$	27.1398	1.25588	0.627940	+	0.778261i	$0.283898\pi$
$468$	0	0
$469$	−5.68210	−0.262375
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0.916864	0.0421574
$474$	0	0
$475$	−7.82338	−0.358961
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−6.75297	−0.308551	−0.154276	−	0.988028i	$-0.549304\pi$
$479$	−6.75297	−0.308551	−0.154276	+	0.988028i	$0.549304\pi$
$480$	0	0
$481$	−22.7965	−1.03943
$482$	0	0
$483$	0	0
$484$	0	0
$485$	37.5386	1.70454
$486$	0	0
$487$	−3.85480	−0.174678	−0.0873388	−	0.996179i	$-0.527836\pi$
$487$	−3.85480	−0.174678	−0.0873388	+	0.996179i	$0.527836\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	17.4571	0.787827	0.393913	−	0.919148i	$-0.371121\pi$
$491$	17.4571	0.787827	0.393913	+	0.919148i	$0.371121\pi$
$492$	0	0
$493$	23.5045	1.05859
$494$	0	0
$495$	0	0
$496$	0	0
$497$	5.67557	0.254584
$498$	0	0
$499$	15.9746	0.715122	0.357561	−	0.933890i	$-0.383608\pi$
$499$	15.9746	0.715122	0.357561	+	0.933890i	$0.383608\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−32.9141	−1.46757	−0.733784	−	0.679383i	$-0.762248\pi$
$503$	−32.9141	−1.46757	−0.733784	+	0.679383i	$0.762248\pi$
$504$	0	0
$505$	−5.90881	−0.262939
$506$	0	0
$507$	0	0
$508$	0	0
$509$	35.6163	1.57866	0.789332	−	0.613967i	$-0.210427\pi$
$509$	35.6163	1.57866	0.789332	+	0.613967i	$0.210427\pi$
$510$	0	0
$511$	−5.20697	−0.230343
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−24.0088	−1.05795
$516$	0	0
$517$	0.142507	0.00626746
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−11.4481	−0.501552	−0.250776	−	0.968045i	$-0.580686\pi$
$521$	−11.4481	−0.501552	−0.250776	+	0.968045i	$0.580686\pi$
$522$	0	0
$523$	−33.9589	−1.48492	−0.742460	−	0.669890i	$-0.766341\pi$
$523$	−33.9589	−1.48492	−0.742460	+	0.669890i	$0.766341\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−27.6219	−1.20323
$528$	0	0
$529$	−12.3137	−0.535377
$530$	0	0
$531$	0	0
$532$	0	0
$533$	30.2912	1.31206
$534$	0	0
$535$	24.9964	1.08069
$536$	0	0
$537$	0	0
$538$	0	0
$539$	6.66896	0.287252
$540$	0	0
$541$	23.0541	0.991174	0.495587	−	0.868558i	$-0.334953\pi$
$541$	23.0541	0.991174	0.495587	+	0.868558i	$0.334953\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−16.2860	−0.697615
$546$	0	0
$547$	16.8295	0.719578	0.359789	−	0.933034i	$-0.382849\pi$
$547$	16.8295	0.719578	0.359789	+	0.933034i	$0.382849\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−29.7633	−1.26796
$552$	0	0
$553$	5.45594	0.232010
$554$	0	0
$555$	0	0
$556$	0	0
$557$	5.61763	0.238026	0.119013	−	0.992893i	$-0.462027\pi$
$557$	5.61763	0.238026	0.119013	+	0.992893i	$0.462027\pi$
$558$	0	0
$559$	−2.49203	−0.105401
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−27.1693	−1.14505	−0.572525	−	0.819887i	$-0.694036\pi$
$563$	−27.1693	−1.14505	−0.572525	+	0.819887i	$0.694036\pi$
$564$	0	0
$565$	13.8900	0.584355
$566$	0	0
$567$	0	0
$568$	0	0
$569$	28.7201	1.20401	0.602006	−	0.798492i	$-0.294369\pi$
$569$	28.7201	1.20401	0.602006	+	0.798492i	$0.294369\pi$
$570$	0	0
$571$	−10.4389	−0.436855	−0.218428	−	0.975853i	$-0.570093\pi$
$571$	−10.4389	−0.436855	−0.218428	+	0.975853i	$0.570093\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−3.45221	−0.143967
$576$	0	0
$577$	26.3489	1.09692	0.548459	−	0.836177i	$-0.315215\pi$
$577$	26.3489	1.09692	0.548459	+	0.836177i	$0.315215\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	3.43201	0.142384
$582$	0	0
$583$	−3.52288	−0.145903
$584$	0	0
$585$	0	0
$586$	0	0
$587$	15.9982	0.660316	0.330158	−	0.943926i	$-0.392898\pi$
$587$	15.9982	0.660316	0.330158	+	0.943926i	$0.392898\pi$
$588$	0	0
$589$	34.9772	1.44121
$590$	0	0
$591$	0	0
$592$	0	0
$593$	15.4966	0.636370	0.318185	−	0.948029i	$-0.396927\pi$
$593$	15.4966	0.636370	0.318185	+	0.948029i	$0.396927\pi$
$594$	0	0
$595$	−6.68476	−0.274048
$596$	0	0
$597$	0	0
$598$	0	0
$599$	4.79914	0.196088	0.0980438	−	0.995182i	$-0.468741\pi$
$599$	4.79914	0.196088	0.0980438	+	0.995182i	$0.468741\pi$
$600$	0	0
$601$	7.42635	0.302927	0.151464	−	0.988463i	$-0.451601\pi$
$601$	7.42635	0.302927	0.151464	+	0.988463i	$0.451601\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−1.98594	−0.0807399
$606$	0	0
$607$	−40.5322	−1.64515	−0.822576	−	0.568655i	$-0.807464\pi$
$607$	−40.5322	−1.64515	−0.822576	+	0.568655i	$0.807464\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−0.387333	−0.0156698
$612$	0	0
$613$	−0.298767	−0.0120671	−0.00603355	−	0.999982i	$-0.501921\pi$
$613$	−0.298767	−0.0120671	−0.00603355	+	0.999982i	$0.501921\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−32.7135	−1.31700	−0.658498	−	0.752583i	$-0.728808\pi$
$617$	−32.7135	−1.31700	−0.658498	+	0.752583i	$0.728808\pi$
$618$	0	0
$619$	22.6830	0.911706	0.455853	−	0.890055i	$-0.349334\pi$
$619$	22.6830	0.911706	0.455853	+	0.890055i	$0.349334\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	3.99389	0.160012
$624$	0	0
$625$	−18.6045	−0.744181
$626$	0	0
$627$	0	0
$628$	0	0
$629$	49.0683	1.95648
$630$	0	0
$631$	−21.0600	−0.838385	−0.419192	−	0.907897i	$-0.637687\pi$
$631$	−21.0600	−0.838385	−0.419192	+	0.907897i	$0.637687\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	30.6597	1.21669
$636$	0	0
$637$	−18.1262	−0.718185
$638$	0	0
$639$	0	0
$640$	0	0
$641$	25.8953	1.02280	0.511401	−	0.859342i	$-0.329127\pi$
$641$	25.8953	1.02280	0.511401	+	0.859342i	$0.329127\pi$
$642$	0	0
$643$	0.916418	0.0361400	0.0180700	−	0.999837i	$-0.494248\pi$
$643$	0.916418	0.0361400	0.0180700	+	0.999837i	$0.494248\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−26.2401	−1.03160	−0.515802	−	0.856708i	$-0.672506\pi$
$647$	−26.2401	−1.03160	−0.515802	+	0.856708i	$0.672506\pi$
$648$	0	0
$649$	12.3916	0.486413
$650$	0	0
$651$	0	0
$652$	0	0
$653$	7.05872	0.276229	0.138115	−	0.990416i	$-0.455896\pi$
$653$	7.05872	0.276229	0.138115	+	0.990416i	$0.455896\pi$
$654$	0	0
$655$	28.8178	1.12600
$656$	0	0
$657$	0	0
$658$	0	0
$659$	33.5928	1.30859	0.654295	−	0.756239i	$-0.272965\pi$
$659$	33.5928	1.30859	0.654295	+	0.756239i	$0.272965\pi$
$660$	0	0
$661$	−3.84614	−0.149597	−0.0747987	−	0.997199i	$-0.523831\pi$
$661$	−3.84614	−0.149597	−0.0747987	+	0.997199i	$0.523831\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	8.46481	0.328251
$666$	0	0
$667$	−13.1336	−0.508536
$668$	0	0
$669$	0	0
$670$	0	0
$671$	5.89651	0.227632
$672$	0	0
$673$	−12.5784	−0.484861	−0.242430	−	0.970169i	$-0.577945\pi$
$673$	−12.5784	−0.484861	−0.242430	+	0.970169i	$0.577945\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	19.1794	0.737124	0.368562	−	0.929603i	$-0.379850\pi$
$677$	19.1794	0.737124	0.368562	+	0.929603i	$0.379850\pi$
$678$	0	0
$679$	10.8756	0.417365
$680$	0	0
$681$	0	0
$682$	0	0
$683$	38.6508	1.47893	0.739466	−	0.673194i	$-0.235078\pi$
$683$	38.6508	1.47893	0.739466	+	0.673194i	$0.235078\pi$
$684$	0	0
$685$	0.877443	0.0335254
$686$	0	0
$687$	0	0
$688$	0	0
$689$	9.57516	0.364785
$690$	0	0
$691$	6.79528	0.258505	0.129252	−	0.991612i	$-0.458742\pi$
$691$	6.79528	0.258505	0.129252	+	0.991612i	$0.458742\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	28.7183	1.08935
$696$	0	0
$697$	−65.2003	−2.46964
$698$	0	0
$699$	0	0
$700$	0	0
$701$	3.22013	0.121623	0.0608113	−	0.998149i	$-0.480631\pi$
$701$	3.22013	0.121623	0.0608113	+	0.998149i	$0.480631\pi$
$702$	0	0
$703$	−62.1344	−2.34344
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−1.71188	−0.0643820
$708$	0	0
$709$	25.5547	0.959727	0.479864	−	0.877343i	$-0.340686\pi$
$709$	25.5547	0.959727	0.479864	+	0.877343i	$0.340686\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	15.4344	0.578021
$714$	0	0
$715$	5.39776	0.201865
$716$	0	0
$717$	0	0
$718$	0	0
$719$	27.5706	1.02821	0.514105	−	0.857727i	$-0.328124\pi$
$719$	27.5706	1.02821	0.514105	+	0.857727i	$0.328124\pi$
$720$	0	0
$721$	−6.95575	−0.259046
$722$	0	0
$723$	0	0
$724$	0	0
$725$	4.24280	0.157574
$726$	0	0
$727$	44.8308	1.66268	0.831341	−	0.555763i	$-0.187574\pi$
$727$	44.8308	1.66268	0.831341	+	0.555763i	$0.187574\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	5.36395	0.198393
$732$	0	0
$733$	−7.79697	−0.287988	−0.143994	−	0.989579i	$-0.545995\pi$
$733$	−7.79697	−0.287988	−0.143994	+	0.989579i	$0.545995\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−9.87573	−0.363777
$738$	0	0
$739$	−42.7823	−1.57377	−0.786886	−	0.617098i	$-0.788308\pi$
$739$	−42.7823	−1.57377	−0.786886	+	0.617098i	$0.788308\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−35.4749	−1.30145	−0.650724	−	0.759314i	$-0.725535\pi$
$743$	−35.4749	−1.30145	−0.650724	+	0.759314i	$0.725535\pi$
$744$	0	0
$745$	34.9258	1.27958
$746$	0	0
$747$	0	0
$748$	0	0
$749$	7.24188	0.264613
$750$	0	0
$751$	−42.5527	−1.55277	−0.776385	−	0.630259i	$-0.782949\pi$
$751$	−42.5527	−1.55277	−0.776385	+	0.630259i	$0.782949\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−11.1822	−0.406963
$756$	0	0
$757$	47.2389	1.71693	0.858464	−	0.512874i	$-0.171419\pi$
$757$	47.2389	1.71693	0.858464	+	0.512874i	$0.171419\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−2.15815	−0.0782330	−0.0391165	−	0.999235i	$-0.512454\pi$
$761$	−2.15815	−0.0782330	−0.0391165	+	0.999235i	$0.512454\pi$
$762$	0	0
$763$	−4.71832	−0.170815
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−33.6802	−1.21612
$768$	0	0
$769$	−44.3546	−1.59947	−0.799733	−	0.600355i	$-0.795026\pi$
$769$	−44.3546	−1.59947	−0.799733	+	0.600355i	$0.795026\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	48.5617	1.74664	0.873322	−	0.487143i	$-0.161961\pi$
$773$	48.5617	1.74664	0.873322	+	0.487143i	$0.161961\pi$
$774$	0	0
$775$	−4.98606	−0.179104
$776$	0	0
$777$	0	0
$778$	0	0
$779$	82.5621	2.95809
$780$	0	0
$781$	9.86439	0.352976
$782$	0	0
$783$	0	0
$784$	0	0
$785$	4.68806	0.167324
$786$	0	0
$787$	−41.4733	−1.47836	−0.739182	−	0.673505i	$-0.764788\pi$
$787$	−41.4733	−1.47836	−0.739182	+	0.673505i	$0.764788\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	4.02416	0.143082
$792$	0	0
$793$	−16.0267	−0.569123
$794$	0	0
$795$	0	0
$796$	0	0
$797$	28.3254	1.00334	0.501669	−	0.865059i	$-0.332719\pi$
$797$	28.3254	1.00334	0.501669	+	0.865059i	$0.332719\pi$
$798$	0	0
$799$	0.833714	0.0294947
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−9.04993	−0.319365
$804$	0	0
$805$	3.73525	0.131650
$806$	0	0
$807$	0	0
$808$	0	0
$809$	26.4597	0.930274	0.465137	−	0.885239i	$-0.346005\pi$
$809$	26.4597	0.930274	0.465137	+	0.885239i	$0.346005\pi$
$810$	0	0
$811$	−3.34212	−0.117358	−0.0586789	−	0.998277i	$-0.518689\pi$
$811$	−3.34212	−0.117358	−0.0586789	+	0.998277i	$0.518689\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−13.9161	−0.487461
$816$	0	0
$817$	−6.79229	−0.237632
$818$	0	0
$819$	0	0
$820$	0	0
$821$	11.6144	0.405347	0.202673	−	0.979246i	$-0.435037\pi$
$821$	11.6144	0.405347	0.202673	+	0.979246i	$0.435037\pi$
$822$	0	0
$823$	21.4851	0.748922	0.374461	−	0.927243i	$-0.377828\pi$
$823$	21.4851	0.748922	0.374461	+	0.927243i	$0.377828\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−43.3480	−1.50736	−0.753678	−	0.657244i	$-0.771722\pi$
$827$	−43.3480	−1.50736	−0.753678	+	0.657244i	$0.771722\pi$
$828$	0	0
$829$	25.6941	0.892393	0.446196	−	0.894935i	$-0.352778\pi$
$829$	25.6941	0.892393	0.446196	+	0.894935i	$0.352778\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	39.0156	1.35181
$834$	0	0
$835$	−13.6304	−0.471701
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−16.8196	−0.580676	−0.290338	−	0.956924i	$-0.593768\pi$
$839$	−16.8196	−0.580676	−0.290338	+	0.956924i	$0.593768\pi$
$840$	0	0
$841$	−12.8586	−0.443401
$842$	0	0
$843$	0	0
$844$	0	0
$845$	11.1461	0.383439
$846$	0	0
$847$	−0.575360	−0.0197696
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−27.4180	−0.939875
$852$	0	0
$853$	−39.6145	−1.35637	−0.678187	−	0.734889i	$-0.737234\pi$
$853$	−39.6145	−1.35637	−0.678187	+	0.734889i	$0.737234\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−7.08137	−0.241895	−0.120947	−	0.992659i	$-0.538593\pi$
$857$	−7.08137	−0.241895	−0.120947	+	0.992659i	$0.538593\pi$
$858$	0	0
$859$	4.67109	0.159376	0.0796879	−	0.996820i	$-0.474608\pi$
$859$	4.67109	0.159376	0.0796879	+	0.996820i	$0.474608\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−3.72380	−0.126760	−0.0633798	−	0.997989i	$-0.520188\pi$
$863$	−3.72380	−0.126760	−0.0633798	+	0.997989i	$0.520188\pi$
$864$	0	0
$865$	24.7687	0.842162
$866$	0	0
$867$	0	0
$868$	0	0
$869$	9.48266	0.321677
$870$	0	0
$871$	26.8421	0.909511
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−6.91982	−0.233933
$876$	0	0
$877$	26.7162	0.902142	0.451071	−	0.892488i	$-0.351042\pi$
$877$	26.7162	0.902142	0.451071	+	0.892488i	$0.351042\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−21.8872	−0.737400	−0.368700	−	0.929548i	$-0.620197\pi$
$881$	−21.8872	−0.737400	−0.368700	+	0.929548i	$0.620197\pi$
$882$	0	0
$883$	−0.675656	−0.0227376	−0.0113688	−	0.999935i	$-0.503619\pi$
$883$	−0.675656	−0.0227376	−0.0113688	+	0.999935i	$0.503619\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−35.6541	−1.19715	−0.598574	−	0.801068i	$-0.704266\pi$
$887$	−35.6541	−1.19715	−0.598574	+	0.801068i	$0.704266\pi$
$888$	0	0
$889$	8.88263	0.297914
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−1.05572	−0.0353283
$894$	0	0
$895$	9.67253	0.323317
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−18.9690	−0.632651
$900$	0	0
$901$	−20.6100	−0.686619
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−29.8416	−0.991969
$906$	0	0
$907$	32.6990	1.08575	0.542877	−	0.839812i	$-0.317335\pi$
$907$	32.6990	1.08575	0.542877	+	0.839812i	$0.317335\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−3.00050	−0.0994111	−0.0497055	−	0.998764i	$-0.515828\pi$
$911$	−3.00050	−0.0994111	−0.0497055	+	0.998764i	$0.515828\pi$
$912$	0	0
$913$	5.96498	0.197412
$914$	0	0
$915$	0	0
$916$	0	0
$917$	8.34899	0.275708
$918$	0	0
$919$	22.2784	0.734895	0.367447	−	0.930044i	$-0.380232\pi$
$919$	22.2784	0.734895	0.367447	+	0.930044i	$0.380232\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−26.8113	−0.882505
$924$	0	0
$925$	8.85735	0.291228
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−50.3330	−1.65137	−0.825687	−	0.564129i	$-0.809212\pi$
$929$	−50.3330	−1.65137	−0.825687	+	0.564129i	$0.809212\pi$
$930$	0	0
$931$	−49.4049	−1.61918
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−11.6184	−0.379962
$936$	0	0
$937$	−31.9868	−1.04496	−0.522482	−	0.852651i	$-0.674994\pi$
$937$	−31.9868	−1.04496	−0.522482	+	0.852651i	$0.674994\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	11.8254	0.385498	0.192749	−	0.981248i	$-0.438260\pi$
$941$	11.8254	0.385498	0.192749	+	0.981248i	$0.438260\pi$
$942$	0	0
$943$	36.4320	1.18639
$944$	0	0
$945$	0	0
$946$	0	0
$947$	29.7022	0.965190	0.482595	−	0.875844i	$-0.339694\pi$
$947$	29.7022	0.965190	0.482595	+	0.875844i	$0.339694\pi$
$948$	0	0
$949$	24.5976	0.798472
$950$	0	0
$951$	0	0
$952$	0	0
$953$	50.8712	1.64788	0.823939	−	0.566678i	$-0.191772\pi$
$953$	50.8712	1.64788	0.823939	+	0.566678i	$0.191772\pi$
$954$	0	0
$955$	18.0362	0.583637
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0.254210	0.00820887
$960$	0	0
$961$	−8.70803	−0.280904
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−9.50921	−0.306112
$966$	0	0
$967$	−20.7219	−0.666370	−0.333185	−	0.942861i	$-0.608123\pi$
$967$	−20.7219	−0.666370	−0.333185	+	0.942861i	$0.608123\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−0.799831	−0.0256678	−0.0128339	−	0.999918i	$-0.504085\pi$
$971$	−0.799831	−0.0256678	−0.0128339	+	0.999918i	$0.504085\pi$
$972$	0	0
$973$	8.32019	0.266733
$974$	0	0
$975$	0	0
$976$	0	0
$977$	23.9487	0.766186	0.383093	−	0.923710i	$-0.374859\pi$
$977$	23.9487	0.766186	0.383093	+	0.923710i	$0.374859\pi$
$978$	0	0
$979$	6.94155	0.221853
$980$	0	0
$981$	0	0
$982$	0	0
$983$	35.3753	1.12830	0.564149	−	0.825673i	$-0.309204\pi$
$983$	35.3753	1.12830	0.564149	+	0.825673i	$0.309204\pi$
$984$	0	0
$985$	−26.7478	−0.852256
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−2.99722	−0.0953061
$990$	0	0
$991$	−17.9943	−0.571607	−0.285804	−	0.958288i	$-0.592261\pi$
$991$	−17.9943	−0.571607	−0.285804	+	0.958288i	$0.592261\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	40.8080	1.29370
$996$	0	0
$997$	−10.3861	−0.328930	−0.164465	−	0.986383i	$-0.552590\pi$
$997$	−10.3861	−0.328930	−0.164465	+	0.986383i	$0.552590\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3564.2.a.o.1.2	✓	6
3.2	odd	2		3564.2.a.p.1.5	yes	6
9.2	odd	6		3564.2.i.s.2377.2		12
9.4	even	3		3564.2.i.t.1189.5		12
9.5	odd	6		3564.2.i.s.1189.2		12
9.7	even	3		3564.2.i.t.2377.5		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3564.2.a.o.1.2	✓	6	1.1	even	1	trivial
3564.2.a.p.1.5	yes	6	3.2	odd	2
3564.2.i.s.1189.2		12	9.5	odd	6
3564.2.i.s.2377.2		12	9.2	odd	6
3564.2.i.t.1189.5		12	9.4	even	3
3564.2.i.t.2377.5		12	9.7	even	3

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:	\( N \)	\(=\)	\( 3564 = 2^{2} \cdot 3^{4} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3564.a (trivial)

\(f(q)\)	\(=\)	\(q-1.98594 q^{5} -0.575360 q^{7} +O(q^{10})\)	\(q-1.98594 q^{5} -0.575360 q^{7} -1.00000 q^{11} +2.71799 q^{13} -5.85033 q^{17} +7.40818 q^{19} +3.26900 q^{23} -1.05605 q^{25} -4.01763 q^{29} +4.72144 q^{31} +1.14263 q^{35} -8.38727 q^{37} +11.1447 q^{41} -0.916864 q^{43} -0.142507 q^{47} -6.66896 q^{49} +3.52288 q^{53} +1.98594 q^{55} -12.3916 q^{59} -5.89651 q^{61} -5.39776 q^{65} +9.87573 q^{67} -9.86439 q^{71} +9.04993 q^{73} +0.575360 q^{77} -9.48266 q^{79} -5.96498 q^{83} +11.6184 q^{85} -6.94155 q^{89} -1.56382 q^{91} -14.7122 q^{95} -18.9022 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 2 q^{7}+O(q^{10}) \) 6 * q - 2 * q^7	\( 6 q - 2 q^{7} - 6 q^{11} - 6 q^{13} - 4 q^{17} - 4 q^{23} + 10 q^{25} + 4 q^{29} + 6 q^{31} - 14 q^{35} - 6 q^{37} - 22 q^{41} - 10 q^{43} - 20 q^{47} + 6 q^{49} - 4 q^{53} - 24 q^{59} - 10 q^{61} - 40 q^{65} - 6 q^{67} - 40 q^{71} - 8 q^{73} + 2 q^{77} - 14 q^{79} - 12 q^{83} + 6 q^{85} - 24 q^{89} + 10 q^{91} - 44 q^{95} + 20 q^{97}+O(q^{100}) \) 6 * q - 2 * q^7 - 6 * q^11 - 6 * q^13 - 4 * q^17 - 4 * q^23 + 10 * q^25 + 4 * q^29 + 6 * q^31 - 14 * q^35 - 6 * q^37 - 22 * q^41 - 10 * q^43 - 20 * q^47 + 6 * q^49 - 4 * q^53 - 24 * q^59 - 10 * q^61 - 40 * q^65 - 6 * q^67 - 40 * q^71 - 8 * q^73 + 2 * q^77 - 14 * q^79 - 12 * q^83 + 6 * q^85 - 24 * q^89 + 10 * q^91 - 44 * q^95 + 20 * q^97

Introduction

L-functions

Modular forms

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