LMFDB - Embedded newform 1944.2.a.q.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1944,2,Mod(1,1944)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1944, 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("1944.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1944 = 2^{3} \cdot 3^{5} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1944.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:	$15.5229181529$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.9926793.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 12x^{4} - x^{3} + 36x^{2} + 9x - 9 $ x^6 - 12x^4 - x^3 + 36x^2 + 9*x - 9
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 3^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.58232$ of defining polynomial
Character	$\chi$	$=$	1944.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.27086 q^{5} +3.12719 q^{7} +O(q^{10})$	$q-1.27086 q^{5} +3.12719 q^{7} -4.75001 q^{11} +5.26781 q^{13} -2.31199 q^{17} -0.294360 q^{19} +4.60635 q^{23} -3.38493 q^{25} +6.63740 q^{29} +7.93556 q^{31} -3.97421 q^{35} +3.16641 q^{37} -1.26781 q^{41} -9.42929 q^{43} +5.73430 q^{47} +2.77934 q^{49} +11.1357 q^{53} +6.03657 q^{55} +2.71414 q^{59} -8.99755 q^{61} -6.69463 q^{65} +9.40919 q^{67} -10.5134 q^{71} +13.1342 q^{73} -14.8542 q^{77} +1.79682 q^{79} -6.54551 q^{83} +2.93820 q^{85} +13.5974 q^{89} +16.4735 q^{91} +0.374089 q^{95} +4.23980 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 3 q^{7}+O(q^{10}) $ 6 * q + 3 * q^7	$ 6 q + 3 q^{7} - 3 q^{11} + 3 q^{13} + 9 q^{17} + 9 q^{19} - 6 q^{23} + 18 q^{25} - 3 q^{29} + 24 q^{31} - 15 q^{35} + 15 q^{37} + 21 q^{41} + 18 q^{43} - 6 q^{47} + 21 q^{49} + 12 q^{53} + 18 q^{55} - 12 q^{59} + 18 q^{61} + 33 q^{65} + 24 q^{67} - 15 q^{71} + 6 q^{73} + 3 q^{77} + 21 q^{79} - 18 q^{83} + 24 q^{85} + 18 q^{89} + 30 q^{91} - 39 q^{95} + 27 q^{97}+O(q^{100}) $ 6 * q + 3 * q^7 - 3 * q^11 + 3 * q^13 + 9 * q^17 + 9 * q^19 - 6 * q^23 + 18 * q^25 - 3 * q^29 + 24 * q^31 - 15 * q^35 + 15 * q^37 + 21 * q^41 + 18 * q^43 - 6 * q^47 + 21 * q^49 + 12 * q^53 + 18 * q^55 - 12 * q^59 + 18 * q^61 + 33 * q^65 + 24 * q^67 - 15 * q^71 + 6 * q^73 + 3 * q^77 + 21 * q^79 - 18 * q^83 + 24 * q^85 + 18 * q^89 + 30 * q^91 - 39 * q^95 + 27 * 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
$5$	−1.27086	−0.568344	−0.284172	−	0.958773i	$-0.591719\pi$
$5$	−1.27086	−0.568344	−0.284172	+	0.958773i	$0.591719\pi$
$6$	0	0
$7$	3.12719	1.18197	0.590984	−	0.806683i	$-0.298740\pi$
$7$	3.12719	1.18197	0.590984	+	0.806683i	$0.298740\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−4.75001	−1.43218	−0.716091	−	0.698007i	$-0.754070\pi$
$11$	−4.75001	−1.43218	−0.716091	+	0.698007i	$0.754070\pi$
$12$	0	0
$13$	5.26781	1.46103	0.730514	−	0.682898i	$-0.239280\pi$
$13$	5.26781	1.46103	0.730514	+	0.682898i	$0.239280\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−2.31199	−0.560739	−0.280369	−	0.959892i	$-0.590457\pi$
$17$	−2.31199	−0.560739	−0.280369	+	0.959892i	$0.590457\pi$
$18$	0	0
$19$	−0.294360	−0.0675308	−0.0337654	−	0.999430i	$-0.510750\pi$
$19$	−0.294360	−0.0675308	−0.0337654	+	0.999430i	$0.510750\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	4.60635	0.960490	0.480245	−	0.877135i	$-0.340548\pi$
$23$	4.60635	0.960490	0.480245	+	0.877135i	$0.340548\pi$
$24$	0	0
$25$	−3.38493	−0.676985
$26$	0	0
$27$	0	0
$28$	0	0
$29$	6.63740	1.23253	0.616267	−	0.787537i	$-0.288644\pi$
$29$	6.63740	1.23253	0.616267	+	0.787537i	$0.288644\pi$
$30$	0	0
$31$	7.93556	1.42527	0.712634	−	0.701536i	$-0.247502\pi$
$31$	7.93556	1.42527	0.712634	+	0.701536i	$0.247502\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−3.97421	−0.671764
$36$	0	0
$37$	3.16641	0.520554	0.260277	−	0.965534i	$-0.416186\pi$
$37$	3.16641	0.520554	0.260277	+	0.965534i	$0.416186\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−1.26781	−0.197999	−0.0989995	−	0.995087i	$-0.531564\pi$
$41$	−1.26781	−0.197999	−0.0989995	+	0.995087i	$0.531564\pi$
$42$	0	0
$43$	−9.42929	−1.43795	−0.718977	−	0.695034i	$-0.755389\pi$
$43$	−9.42929	−1.43795	−0.718977	+	0.695034i	$0.755389\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	5.73430	0.836433	0.418217	−	0.908347i	$-0.362655\pi$
$47$	5.73430	0.836433	0.418217	+	0.908347i	$0.362655\pi$
$48$	0	0
$49$	2.77934	0.397048
$50$	0	0
$51$	0	0
$52$	0	0
$53$	11.1357	1.52961	0.764803	−	0.644265i	$-0.222837\pi$
$53$	11.1357	1.52961	0.764803	+	0.644265i	$0.222837\pi$
$54$	0	0
$55$	6.03657	0.813971
$56$	0	0
$57$	0	0
$58$	0	0
$59$	2.71414	0.353351	0.176675	−	0.984269i	$-0.443466\pi$
$59$	2.71414	0.353351	0.176675	+	0.984269i	$0.443466\pi$
$60$	0	0
$61$	−8.99755	−1.15202	−0.576009	−	0.817443i	$-0.695391\pi$
$61$	−8.99755	−1.15202	−0.576009	+	0.817443i	$0.695391\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−6.69463	−0.830366
$66$	0	0
$67$	9.40919	1.14952	0.574758	−	0.818324i	$-0.305096\pi$
$67$	9.40919	1.14952	0.574758	+	0.818324i	$0.305096\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−10.5134	−1.24772	−0.623858	−	0.781538i	$-0.714436\pi$
$71$	−10.5134	−1.24772	−0.623858	+	0.781538i	$0.714436\pi$
$72$	0	0
$73$	13.1342	1.53724	0.768620	−	0.639706i	$-0.220944\pi$
$73$	13.1342	1.53724	0.768620	+	0.639706i	$0.220944\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−14.8542	−1.69279
$78$	0	0
$79$	1.79682	0.202158	0.101079	−	0.994878i	$-0.467770\pi$
$79$	1.79682	0.202158	0.101079	+	0.994878i	$0.467770\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−6.54551	−0.718463	−0.359232	−	0.933248i	$-0.616961\pi$
$83$	−6.54551	−0.718463	−0.359232	+	0.933248i	$0.616961\pi$
$84$	0	0
$85$	2.93820	0.318693
$86$	0	0
$87$	0	0
$88$	0	0
$89$	13.5974	1.44132	0.720662	−	0.693287i	$-0.243838\pi$
$89$	13.5974	1.44132	0.720662	+	0.693287i	$0.243838\pi$
$90$	0	0
$91$	16.4735	1.72689
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0.374089	0.0383807
$96$	0	0
$97$	4.23980	0.430487	0.215243	−	0.976560i	$-0.430946\pi$
$97$	4.23980	0.430487	0.215243	+	0.976560i	$0.430946\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	4.30267	0.428131	0.214066	−	0.976819i	$-0.431329\pi$
$101$	4.30267	0.428131	0.214066	+	0.976819i	$0.431329\pi$
$102$	0	0
$103$	14.0773	1.38708	0.693541	−	0.720418i	$-0.256050\pi$
$103$	14.0773	1.38708	0.693541	+	0.720418i	$0.256050\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	2.90825	0.281151	0.140576	−	0.990070i	$-0.455105\pi$
$107$	2.90825	0.281151	0.140576	+	0.990070i	$0.455105\pi$
$108$	0	0
$109$	−8.18614	−0.784090	−0.392045	−	0.919946i	$-0.628232\pi$
$109$	−8.18614	−0.784090	−0.392045	+	0.919946i	$0.628232\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	20.4996	1.92844	0.964222	−	0.265095i	$-0.0854033\pi$
$113$	20.4996	1.92844	0.964222	+	0.265095i	$0.0854033\pi$
$114$	0	0
$115$	−5.85400	−0.545888
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−7.23003	−0.662776
$120$	0	0
$121$	11.5626	1.05114
$122$	0	0
$123$	0	0
$124$	0	0
$125$	10.6560	0.953104
$126$	0	0
$127$	−8.92987	−0.792398	−0.396199	−	0.918165i	$-0.629671\pi$
$127$	−8.92987	−0.792398	−0.396199	+	0.918165i	$0.629671\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−20.4305	−1.78502	−0.892508	−	0.451031i	$-0.851056\pi$
$131$	−20.4305	−1.78502	−0.892508	+	0.451031i	$0.851056\pi$
$132$	0	0
$133$	−0.920520	−0.0798192
$134$	0	0
$135$	0	0
$136$	0	0
$137$	17.6550	1.50837	0.754185	−	0.656662i	$-0.228032\pi$
$137$	17.6550	1.50837	0.754185	+	0.656662i	$0.228032\pi$
$138$	0	0
$139$	4.96033	0.420730	0.210365	−	0.977623i	$-0.432535\pi$
$139$	4.96033	0.420730	0.210365	+	0.977623i	$0.432535\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−25.0221	−2.09246
$144$	0	0
$145$	−8.43517	−0.700503
$146$	0	0
$147$	0	0
$148$	0	0
$149$	0.222946	0.0182644	0.00913221	−	0.999958i	$-0.497093\pi$
$149$	0.222946	0.0182644	0.00913221	+	0.999958i	$0.497093\pi$
$150$	0	0
$151$	−2.50792	−0.204092	−0.102046	−	0.994780i	$-0.532539\pi$
$151$	−2.50792	−0.204092	−0.102046	+	0.994780i	$0.532539\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−10.0849	−0.810043
$156$	0	0
$157$	23.2858	1.85841	0.929206	−	0.369563i	$-0.120493\pi$
$157$	23.2858	1.85841	0.929206	+	0.369563i	$0.120493\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	14.4049	1.13527
$162$	0	0
$163$	−4.25135	−0.332991	−0.166496	−	0.986042i	$-0.553245\pi$
$163$	−4.25135	−0.332991	−0.166496	+	0.986042i	$0.553245\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	15.4202	1.19325	0.596624	−	0.802521i	$-0.296508\pi$
$167$	15.4202	1.19325	0.596624	+	0.802521i	$0.296508\pi$
$168$	0	0
$169$	14.7498	1.13460
$170$	0	0
$171$	0	0
$172$	0	0
$173$	5.47764	0.416457	0.208228	−	0.978080i	$-0.433230\pi$
$173$	5.47764	0.416457	0.208228	+	0.978080i	$0.433230\pi$
$174$	0	0
$175$	−10.5853	−0.800175
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−20.6262	−1.54167	−0.770837	−	0.637032i	$-0.780162\pi$
$179$	−20.6262	−1.54167	−0.770837	+	0.637032i	$0.780162\pi$
$180$	0	0
$181$	11.9463	0.887965	0.443982	−	0.896036i	$-0.353565\pi$
$181$	11.9463	0.887965	0.443982	+	0.896036i	$0.353565\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−4.02405	−0.295854
$186$	0	0
$187$	10.9820	0.803080
$188$	0	0
$189$	0	0
$190$	0	0
$191$	14.2098	1.02818	0.514092	−	0.857735i	$-0.328129\pi$
$191$	14.2098	1.02818	0.514092	+	0.857735i	$0.328129\pi$
$192$	0	0
$193$	6.06695	0.436709	0.218354	−	0.975870i	$-0.429931\pi$
$193$	6.06695	0.436709	0.218354	+	0.975870i	$0.429931\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−26.7457	−1.90555	−0.952775	−	0.303676i	$-0.901786\pi$
$197$	−26.7457	−1.90555	−0.952775	+	0.303676i	$0.901786\pi$
$198$	0	0
$199$	4.50868	0.319612	0.159806	−	0.987148i	$-0.448913\pi$
$199$	4.50868	0.319612	0.159806	+	0.987148i	$0.448913\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	20.7564	1.45682
$204$	0	0
$205$	1.61121	0.112532
$206$	0	0
$207$	0	0
$208$	0	0
$209$	1.39821	0.0967163
$210$	0	0
$211$	−23.3847	−1.60987	−0.804935	−	0.593364i	$-0.797800\pi$
$211$	−23.3847	−1.60987	−0.804935	+	0.593364i	$0.797800\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	11.9833	0.817252
$216$	0	0
$217$	24.8160	1.68462
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−12.1791	−0.819255
$222$	0	0
$223$	−18.1858	−1.21781	−0.608905	−	0.793243i	$-0.708391\pi$
$223$	−18.1858	−1.21781	−0.608905	+	0.793243i	$0.708391\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−26.1283	−1.73420	−0.867098	−	0.498138i	$-0.834017\pi$
$227$	−26.1283	−1.73420	−0.867098	+	0.498138i	$0.834017\pi$
$228$	0	0
$229$	−21.9572	−1.45097	−0.725487	−	0.688236i	$-0.758385\pi$
$229$	−21.9572	−1.45097	−0.725487	+	0.688236i	$0.758385\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	27.1121	1.77617	0.888086	−	0.459678i	$-0.152035\pi$
$233$	27.1121	1.77617	0.888086	+	0.459678i	$0.152035\pi$
$234$	0	0
$235$	−7.28746	−0.475382
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−9.11027	−0.589295	−0.294647	−	0.955606i	$-0.595202\pi$
$239$	−9.11027	−0.589295	−0.294647	+	0.955606i	$0.595202\pi$
$240$	0	0
$241$	−26.0438	−1.67763	−0.838815	−	0.544416i	$-0.816751\pi$
$241$	−26.0438	−1.67763	−0.838815	+	0.544416i	$0.816751\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−3.53214	−0.225660
$246$	0	0
$247$	−1.55063	−0.0986644
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−8.73156	−0.551131	−0.275566	−	0.961282i	$-0.588865\pi$
$251$	−8.73156	−0.551131	−0.275566	+	0.961282i	$0.588865\pi$
$252$	0	0
$253$	−21.8802	−1.37560
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−7.34237	−0.458004	−0.229002	−	0.973426i	$-0.573546\pi$
$257$	−7.34237	−0.458004	−0.229002	+	0.973426i	$0.573546\pi$
$258$	0	0
$259$	9.90197	0.615278
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−16.7663	−1.03386	−0.516928	−	0.856029i	$-0.672924\pi$
$263$	−16.7663	−1.03386	−0.516928	+	0.856029i	$0.672924\pi$
$264$	0	0
$265$	−14.1519	−0.869342
$266$	0	0
$267$	0	0
$268$	0	0
$269$	10.5498	0.643235	0.321618	−	0.946870i	$-0.395773\pi$
$269$	10.5498	0.643235	0.321618	+	0.946870i	$0.395773\pi$
$270$	0	0
$271$	14.2767	0.867248	0.433624	−	0.901094i	$-0.357235\pi$
$271$	14.2767	0.867248	0.433624	+	0.901094i	$0.357235\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	16.0784	0.969566
$276$	0	0
$277$	11.0350	0.663028	0.331514	−	0.943450i	$-0.392441\pi$
$277$	11.0350	0.663028	0.331514	+	0.943450i	$0.392441\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−18.8399	−1.12390	−0.561948	−	0.827172i	$-0.689948\pi$
$281$	−18.8399	−1.12390	−0.561948	+	0.827172i	$0.689948\pi$
$282$	0	0
$283$	18.4767	1.09833	0.549164	−	0.835715i	$-0.314946\pi$
$283$	18.4767	1.09833	0.549164	+	0.835715i	$0.314946\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−3.96469	−0.234028
$288$	0	0
$289$	−11.6547	−0.685572
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−32.2829	−1.88599	−0.942994	−	0.332809i	$-0.892004\pi$
$293$	−32.2829	−1.88599	−0.942994	+	0.332809i	$0.892004\pi$
$294$	0	0
$295$	−3.44928	−0.200825
$296$	0	0
$297$	0	0
$298$	0	0
$299$	24.2654	1.40330
$300$	0	0
$301$	−29.4872	−1.69962
$302$	0	0
$303$	0	0
$304$	0	0
$305$	11.4346	0.654743
$306$	0	0
$307$	−12.6003	−0.719139	−0.359569	−	0.933118i	$-0.617076\pi$
$307$	−12.6003	−0.719139	−0.359569	+	0.933118i	$0.617076\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	4.49470	0.254871	0.127436	−	0.991847i	$-0.459325\pi$
$311$	4.49470	0.254871	0.127436	+	0.991847i	$0.459325\pi$
$312$	0	0
$313$	5.06953	0.286547	0.143273	−	0.989683i	$-0.454237\pi$
$313$	5.06953	0.286547	0.143273	+	0.989683i	$0.454237\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	20.2947	1.13987	0.569933	−	0.821691i	$-0.306969\pi$
$317$	20.2947	1.13987	0.569933	+	0.821691i	$0.306969\pi$
$318$	0	0
$319$	−31.5277	−1.76521
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0.680556	0.0378671
$324$	0	0
$325$	−17.8312	−0.989095
$326$	0	0
$327$	0	0
$328$	0	0
$329$	17.9323	0.988637
$330$	0	0
$331$	0.771436	0.0424019	0.0212010	−	0.999775i	$-0.493251\pi$
$331$	0.771436	0.0424019	0.0212010	+	0.999775i	$0.493251\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−11.9577	−0.653320
$336$	0	0
$337$	−23.5481	−1.28275	−0.641374	−	0.767228i	$-0.721635\pi$
$337$	−23.5481	−1.28275	−0.641374	+	0.767228i	$0.721635\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−37.6940	−2.04124
$342$	0	0
$343$	−13.1988	−0.712669
$344$	0	0
$345$	0	0
$346$	0	0
$347$	9.92938	0.533037	0.266518	−	0.963830i	$-0.414127\pi$
$347$	9.92938	0.533037	0.266518	+	0.963830i	$0.414127\pi$
$348$	0	0
$349$	−4.95335	−0.265147	−0.132573	−	0.991173i	$-0.542324\pi$
$349$	−4.95335	−0.265147	−0.132573	+	0.991173i	$0.542324\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	11.3410	0.603621	0.301811	−	0.953368i	$-0.402409\pi$
$353$	11.3410	0.603621	0.301811	+	0.953368i	$0.402409\pi$
$354$	0	0
$355$	13.3611	0.709132
$356$	0	0
$357$	0	0
$358$	0	0
$359$	5.59535	0.295311	0.147656	−	0.989039i	$-0.452827\pi$
$359$	5.59535	0.295311	0.147656	+	0.989039i	$0.452827\pi$
$360$	0	0
$361$	−18.9134	−0.995440
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−16.6916	−0.873680
$366$	0	0
$367$	0.216859	0.0113200	0.00565998	−	0.999984i	$-0.498198\pi$
$367$	0.216859	0.0113200	0.00565998	+	0.999984i	$0.498198\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	34.8235	1.80794
$372$	0	0
$373$	20.8387	1.07899	0.539494	−	0.841989i	$-0.318616\pi$
$373$	20.8387	1.07899	0.539494	+	0.841989i	$0.318616\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	34.9646	1.80077
$378$	0	0
$379$	−20.8224	−1.06957	−0.534787	−	0.844987i	$-0.679608\pi$
$379$	−20.8224	−1.06957	−0.534787	+	0.844987i	$0.679608\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	1.74505	0.0891679	0.0445840	−	0.999006i	$-0.485804\pi$
$383$	1.74505	0.0891679	0.0445840	+	0.999006i	$0.485804\pi$
$384$	0	0
$385$	18.8775	0.962088
$386$	0	0
$387$	0	0
$388$	0	0
$389$	2.02917	0.102883	0.0514415	−	0.998676i	$-0.483618\pi$
$389$	2.02917	0.102883	0.0514415	+	0.998676i	$0.483618\pi$
$390$	0	0
$391$	−10.6498	−0.538584
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−2.28350	−0.114895
$396$	0	0
$397$	8.96888	0.450135	0.225068	−	0.974343i	$-0.427740\pi$
$397$	8.96888	0.450135	0.225068	+	0.974343i	$0.427740\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	23.8799	1.19251	0.596253	−	0.802797i	$-0.296656\pi$
$401$	23.8799	1.19251	0.596253	+	0.802797i	$0.296656\pi$
$402$	0	0
$403$	41.8030	2.08236
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−15.0405	−0.745528
$408$	0	0
$409$	−32.8546	−1.62456	−0.812278	−	0.583270i	$-0.801773\pi$
$409$	−32.8546	−1.62456	−0.812278	+	0.583270i	$0.801773\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	8.48764	0.417649
$414$	0	0
$415$	8.31840	0.408334
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−23.4249	−1.14438	−0.572191	−	0.820120i	$-0.693907\pi$
$419$	−23.4249	−1.14438	−0.572191	+	0.820120i	$0.693907\pi$
$420$	0	0
$421$	19.6675	0.958536	0.479268	−	0.877669i	$-0.340902\pi$
$421$	19.6675	0.958536	0.479268	+	0.877669i	$0.340902\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	7.82590	0.379612
$426$	0	0
$427$	−28.1371	−1.36165
$428$	0	0
$429$	0	0
$430$	0	0
$431$	11.5348	0.555610	0.277805	−	0.960638i	$-0.410393\pi$
$431$	11.5348	0.555610	0.277805	+	0.960638i	$0.410393\pi$
$432$	0	0
$433$	−17.6449	−0.847962	−0.423981	−	0.905671i	$-0.639368\pi$
$433$	−17.6449	−0.847962	−0.423981	+	0.905671i	$0.639368\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−1.35592	−0.0648626
$438$	0	0
$439$	18.7701	0.895847	0.447924	−	0.894072i	$-0.352164\pi$
$439$	18.7701	0.895847	0.447924	+	0.894072i	$0.352164\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	1.77564	0.0843630	0.0421815	−	0.999110i	$-0.486569\pi$
$443$	1.77564	0.0843630	0.0421815	+	0.999110i	$0.486569\pi$
$444$	0	0
$445$	−17.2804	−0.819168
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−27.4245	−1.29424	−0.647120	−	0.762388i	$-0.724027\pi$
$449$	−27.4245	−1.29424	−0.647120	+	0.762388i	$0.724027\pi$
$450$	0	0
$451$	6.02212	0.283570
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−20.9354	−0.981467
$456$	0	0
$457$	4.08530	0.191102	0.0955512	−	0.995425i	$-0.469539\pi$
$457$	4.08530	0.191102	0.0955512	+	0.995425i	$0.469539\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	15.6605	0.729381	0.364691	−	0.931129i	$-0.381175\pi$
$461$	15.6605	0.729381	0.364691	+	0.931129i	$0.381175\pi$
$462$	0	0
$463$	−5.36128	−0.249160	−0.124580	−	0.992210i	$-0.539758\pi$
$463$	−5.36128	−0.249160	−0.124580	+	0.992210i	$0.539758\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	16.7820	0.776576	0.388288	−	0.921538i	$-0.373067\pi$
$467$	16.7820	0.776576	0.388288	+	0.921538i	$0.373067\pi$
$468$	0	0
$469$	29.4244	1.35869
$470$	0	0
$471$	0	0
$472$	0	0
$473$	44.7892	2.05941
$474$	0	0
$475$	0.996386	0.0457173
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−18.2770	−0.835099	−0.417549	−	0.908654i	$-0.637111\pi$
$479$	−18.2770	−0.835099	−0.417549	+	0.908654i	$0.637111\pi$
$480$	0	0
$481$	16.6800	0.760544
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−5.38818	−0.244665
$486$	0	0
$487$	−18.5419	−0.840215	−0.420108	−	0.907474i	$-0.638008\pi$
$487$	−18.5419	−0.840215	−0.420108	+	0.907474i	$0.638008\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	6.07975	0.274375	0.137188	−	0.990545i	$-0.456194\pi$
$491$	6.07975	0.274375	0.137188	+	0.990545i	$0.456194\pi$
$492$	0	0
$493$	−15.3456	−0.691130
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−32.8776	−1.47476
$498$	0	0
$499$	−22.9161	−1.02587	−0.512933	−	0.858429i	$-0.671441\pi$
$499$	−22.9161	−1.02587	−0.512933	+	0.858429i	$0.671441\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−0.761473	−0.0339524	−0.0169762	−	0.999856i	$-0.505404\pi$
$503$	−0.761473	−0.0339524	−0.0169762	+	0.999856i	$0.505404\pi$
$504$	0	0
$505$	−5.46807	−0.243326
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−12.1358	−0.537911	−0.268956	−	0.963153i	$-0.586679\pi$
$509$	−12.1358	−0.537911	−0.268956	+	0.963153i	$0.586679\pi$
$510$	0	0
$511$	41.0731	1.81697
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−17.8903	−0.788339
$516$	0	0
$517$	−27.2380	−1.19792
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−7.58587	−0.332343	−0.166172	−	0.986097i	$-0.553141\pi$
$521$	−7.58587	−0.332343	−0.166172	+	0.986097i	$0.553141\pi$
$522$	0	0
$523$	−14.7951	−0.646944	−0.323472	−	0.946238i	$-0.604850\pi$
$523$	−14.7951	−0.646944	−0.323472	+	0.946238i	$0.604850\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−18.3469	−0.799204
$528$	0	0
$529$	−1.78158	−0.0774599
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−6.67860	−0.289282
$534$	0	0
$535$	−3.69597	−0.159791
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−13.2019	−0.568645
$540$	0	0
$541$	−0.487824	−0.0209732	−0.0104866	−	0.999945i	$-0.503338\pi$
$541$	−0.487824	−0.0209732	−0.0104866	+	0.999945i	$0.503338\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	10.4034	0.445633
$546$	0	0
$547$	−38.5230	−1.64712	−0.823562	−	0.567227i	$-0.808016\pi$
$547$	−38.5230	−1.64712	−0.823562	+	0.567227i	$0.808016\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−1.95378	−0.0832340
$552$	0	0
$553$	5.61901	0.238945
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−4.84276	−0.205194	−0.102597	−	0.994723i	$-0.532715\pi$
$557$	−4.84276	−0.205194	−0.102597	+	0.994723i	$0.532715\pi$
$558$	0	0
$559$	−49.6718	−2.10089
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−14.5379	−0.612700	−0.306350	−	0.951919i	$-0.599108\pi$
$563$	−14.5379	−0.612700	−0.306350	+	0.951919i	$0.599108\pi$
$564$	0	0
$565$	−26.0521	−1.09602
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−29.4080	−1.23285	−0.616424	−	0.787415i	$-0.711419\pi$
$569$	−29.4080	−1.23285	−0.616424	+	0.787415i	$0.711419\pi$
$570$	0	0
$571$	−2.12758	−0.0890366	−0.0445183	−	0.999009i	$-0.514175\pi$
$571$	−2.12758	−0.0890366	−0.0445183	+	0.999009i	$0.514175\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−15.5921	−0.650237
$576$	0	0
$577$	−9.49317	−0.395206	−0.197603	−	0.980282i	$-0.563316\pi$
$577$	−9.49317	−0.395206	−0.197603	+	0.980282i	$0.563316\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−20.4691	−0.849200
$582$	0	0
$583$	−52.8946	−2.19067
$584$	0	0
$585$	0	0
$586$	0	0
$587$	33.6829	1.39024	0.695121	−	0.718892i	$-0.255351\pi$
$587$	33.6829	1.39024	0.695121	+	0.718892i	$0.255351\pi$
$588$	0	0
$589$	−2.33591	−0.0962495
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−12.8992	−0.529706	−0.264853	−	0.964289i	$-0.585323\pi$
$593$	−12.8992	−0.529706	−0.264853	+	0.964289i	$0.585323\pi$
$594$	0	0
$595$	9.18832	0.376684
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−19.1349	−0.781830	−0.390915	−	0.920427i	$-0.627841\pi$
$599$	−19.1349	−0.781830	−0.390915	+	0.920427i	$0.627841\pi$
$600$	0	0
$601$	1.30050	0.0530485	0.0265242	−	0.999648i	$-0.491556\pi$
$601$	1.30050	0.0530485	0.0265242	+	0.999648i	$0.491556\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−14.6944	−0.597411
$606$	0	0
$607$	−5.18645	−0.210511	−0.105256	−	0.994445i	$-0.533566\pi$
$607$	−5.18645	−0.210511	−0.105256	+	0.994445i	$0.533566\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	30.2072	1.22205
$612$	0	0
$613$	−0.526327	−0.0212581	−0.0106291	−	0.999944i	$-0.503383\pi$
$613$	−0.526327	−0.0212581	−0.0106291	+	0.999944i	$0.503383\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	30.5128	1.22840	0.614200	−	0.789151i	$-0.289479\pi$
$617$	30.5128	1.22840	0.614200	+	0.789151i	$0.289479\pi$
$618$	0	0
$619$	−18.5294	−0.744761	−0.372381	−	0.928080i	$-0.621458\pi$
$619$	−18.5294	−0.744761	−0.372381	+	0.928080i	$0.621458\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	42.5218	1.70360
$624$	0	0
$625$	3.38236	0.135294
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−7.32069	−0.291895
$630$	0	0
$631$	45.3063	1.80361	0.901807	−	0.432139i	$-0.142241\pi$
$631$	45.3063	1.80361	0.901807	+	0.432139i	$0.142241\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	11.3486	0.450355
$636$	0	0
$637$	14.6410	0.580099
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−42.9101	−1.69485	−0.847424	−	0.530917i	$-0.821847\pi$
$641$	−42.9101	−1.69485	−0.847424	+	0.530917i	$0.821847\pi$
$642$	0	0
$643$	−46.4968	−1.83366	−0.916828	−	0.399282i	$-0.869260\pi$
$643$	−46.4968	−1.83366	−0.916828	+	0.399282i	$0.869260\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	14.2301	0.559441	0.279721	−	0.960081i	$-0.409758\pi$
$647$	14.2301	0.559441	0.279721	+	0.960081i	$0.409758\pi$
$648$	0	0
$649$	−12.8922	−0.506062
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−25.2128	−0.986654	−0.493327	−	0.869844i	$-0.664219\pi$
$653$	−25.2128	−0.986654	−0.493327	+	0.869844i	$0.664219\pi$
$654$	0	0
$655$	25.9642	1.01450
$656$	0	0
$657$	0	0
$658$	0	0
$659$	29.3565	1.14357	0.571783	−	0.820405i	$-0.306252\pi$
$659$	29.3565	1.14357	0.571783	+	0.820405i	$0.306252\pi$
$660$	0	0
$661$	19.1858	0.746243	0.373121	−	0.927783i	$-0.378288\pi$
$661$	19.1858	0.746243	0.373121	+	0.927783i	$0.378288\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	1.16985	0.0453648
$666$	0	0
$667$	30.5741	1.18384
$668$	0	0
$669$	0	0
$670$	0	0
$671$	42.7384	1.64990
$672$	0	0
$673$	−23.9749	−0.924165	−0.462082	−	0.886837i	$-0.652898\pi$
$673$	−23.9749	−0.924165	−0.462082	+	0.886837i	$0.652898\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	12.8806	0.495040	0.247520	−	0.968883i	$-0.420384\pi$
$677$	12.8806	0.495040	0.247520	+	0.968883i	$0.420384\pi$
$678$	0	0
$679$	13.2587	0.508822
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−18.5377	−0.709327	−0.354664	−	0.934994i	$-0.615405\pi$
$683$	−18.5377	−0.709327	−0.354664	+	0.934994i	$0.615405\pi$
$684$	0	0
$685$	−22.4370	−0.857273
$686$	0	0
$687$	0	0
$688$	0	0
$689$	58.6607	2.23480
$690$	0	0
$691$	15.1802	0.577482	0.288741	−	0.957407i	$-0.406763\pi$
$691$	15.1802	0.577482	0.288741	+	0.957407i	$0.406763\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−6.30386	−0.239119
$696$	0	0
$697$	2.93116	0.111026
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−28.0910	−1.06098	−0.530490	−	0.847691i	$-0.677992\pi$
$701$	−28.0910	−1.06098	−0.530490	+	0.847691i	$0.677992\pi$
$702$	0	0
$703$	−0.932063	−0.0351534
$704$	0	0
$705$	0	0
$706$	0	0
$707$	13.4553	0.506037
$708$	0	0
$709$	34.4843	1.29508	0.647542	−	0.762030i	$-0.275797\pi$
$709$	34.4843	1.29508	0.647542	+	0.762030i	$0.275797\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	36.5539	1.36896
$714$	0	0
$715$	31.7995	1.18924
$716$	0	0
$717$	0	0
$718$	0	0
$719$	38.3351	1.42966	0.714828	−	0.699300i	$-0.246505\pi$
$719$	38.3351	1.42966	0.714828	+	0.699300i	$0.246505\pi$
$720$	0	0
$721$	44.0226	1.63949
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−22.4671	−0.834407
$726$	0	0
$727$	−0.795330	−0.0294972	−0.0147486	−	0.999891i	$-0.504695\pi$
$727$	−0.795330	−0.0294972	−0.0147486	+	0.999891i	$0.504695\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	21.8004	0.806317
$732$	0	0
$733$	7.25165	0.267846	0.133923	−	0.990992i	$-0.457243\pi$
$733$	7.25165	0.267846	0.133923	+	0.990992i	$0.457243\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−44.6937	−1.64631
$738$	0	0
$739$	17.7042	0.651259	0.325629	−	0.945498i	$-0.394424\pi$
$739$	17.7042	0.651259	0.325629	+	0.945498i	$0.394424\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−30.6186	−1.12329	−0.561643	−	0.827380i	$-0.689831\pi$
$743$	−30.6186	−1.12329	−0.561643	+	0.827380i	$0.689831\pi$
$744$	0	0
$745$	−0.283332	−0.0103805
$746$	0	0
$747$	0	0
$748$	0	0
$749$	9.09467	0.332312
$750$	0	0
$751$	31.9488	1.16583	0.582914	−	0.812534i	$-0.301912\pi$
$751$	31.9488	1.16583	0.582914	+	0.812534i	$0.301912\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	3.18720	0.115994
$756$	0	0
$757$	20.3502	0.739641	0.369821	−	0.929103i	$-0.379419\pi$
$757$	20.3502	0.739641	0.369821	+	0.929103i	$0.379419\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−8.13592	−0.294927	−0.147463	−	0.989068i	$-0.547111\pi$
$761$	−8.13592	−0.294927	−0.147463	+	0.989068i	$0.547111\pi$
$762$	0	0
$763$	−25.5997	−0.926770
$764$	0	0
$765$	0	0
$766$	0	0
$767$	14.2976	0.516256
$768$	0	0
$769$	6.32778	0.228186	0.114093	−	0.993470i	$-0.463604\pi$
$769$	6.32778	0.228186	0.114093	+	0.993470i	$0.463604\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−6.93000	−0.249255	−0.124627	−	0.992204i	$-0.539774\pi$
$773$	−6.93000	−0.249255	−0.124627	+	0.992204i	$0.539774\pi$
$774$	0	0
$775$	−26.8613	−0.964886
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0.373193	0.0133710
$780$	0	0
$781$	49.9389	1.78696
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−29.5929	−1.05622
$786$	0	0
$787$	18.0862	0.644704	0.322352	−	0.946620i	$-0.395527\pi$
$787$	18.0862	0.644704	0.322352	+	0.946620i	$0.395527\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	64.1064	2.27936
$792$	0	0
$793$	−47.3974	−1.68313
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−31.0723	−1.10064	−0.550319	−	0.834955i	$-0.685494\pi$
$797$	−31.0723	−1.10064	−0.550319	+	0.834955i	$0.685494\pi$
$798$	0	0
$799$	−13.2576	−0.469021
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−62.3874	−2.20160
$804$	0	0
$805$	−18.3066	−0.645223
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−15.5907	−0.548140	−0.274070	−	0.961710i	$-0.588370\pi$
$809$	−15.5907	−0.548140	−0.274070	+	0.961710i	$0.588370\pi$
$810$	0	0
$811$	35.0229	1.22982	0.614910	−	0.788598i	$-0.289192\pi$
$811$	35.0229	1.22982	0.614910	+	0.788598i	$0.289192\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	5.40285	0.189253
$816$	0	0
$817$	2.77561	0.0971061
$818$	0	0
$819$	0	0
$820$	0	0
$821$	8.78185	0.306489	0.153244	−	0.988188i	$-0.451028\pi$
$821$	8.78185	0.306489	0.153244	+	0.988188i	$0.451028\pi$
$822$	0	0
$823$	41.8314	1.45815	0.729076	−	0.684433i	$-0.239950\pi$
$823$	41.8314	1.45815	0.729076	+	0.684433i	$0.239950\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	34.3445	1.19427	0.597137	−	0.802139i	$-0.296305\pi$
$827$	34.3445	1.19427	0.597137	+	0.802139i	$0.296305\pi$
$828$	0	0
$829$	38.5458	1.33875	0.669376	−	0.742924i	$-0.266562\pi$
$829$	38.5458	1.33875	0.669376	+	0.742924i	$0.266562\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−6.42579	−0.222641
$834$	0	0
$835$	−19.5968	−0.678175
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−41.3772	−1.42850	−0.714250	−	0.699891i	$-0.753232\pi$
$839$	−41.3772	−1.42850	−0.714250	+	0.699891i	$0.753232\pi$
$840$	0	0
$841$	15.0550	0.519139
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−18.7449	−0.644845
$846$	0	0
$847$	36.1584	1.24242
$848$	0	0
$849$	0	0
$850$	0	0
$851$	14.5856	0.499987
$852$	0	0
$853$	−18.8033	−0.643813	−0.321906	−	0.946772i	$-0.604324\pi$
$853$	−18.8033	−0.643813	−0.321906	+	0.946772i	$0.604324\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−4.75673	−0.162487	−0.0812434	−	0.996694i	$-0.525889\pi$
$857$	−4.75673	−0.162487	−0.0812434	+	0.996694i	$0.525889\pi$
$858$	0	0
$859$	35.3053	1.20460	0.602301	−	0.798269i	$-0.294251\pi$
$859$	35.3053	1.20460	0.602301	+	0.798269i	$0.294251\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	10.2688	0.349554	0.174777	−	0.984608i	$-0.444080\pi$
$863$	10.2688	0.349554	0.174777	+	0.984608i	$0.444080\pi$
$864$	0	0
$865$	−6.96128	−0.236691
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−8.53492	−0.289527
$870$	0	0
$871$	49.5658	1.67947
$872$	0	0
$873$	0	0
$874$	0	0
$875$	33.3235	1.12654
$876$	0	0
$877$	16.2075	0.547290	0.273645	−	0.961831i	$-0.411771\pi$
$877$	16.2075	0.547290	0.273645	+	0.961831i	$0.411771\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	41.1842	1.38753	0.693765	−	0.720202i	$-0.255951\pi$
$881$	41.1842	1.38753	0.693765	+	0.720202i	$0.255951\pi$
$882$	0	0
$883$	−45.3425	−1.52590	−0.762948	−	0.646460i	$-0.776249\pi$
$883$	−45.3425	−1.52590	−0.762948	+	0.646460i	$0.776249\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−40.1662	−1.34865	−0.674325	−	0.738435i	$-0.735565\pi$
$887$	−40.1662	−1.34865	−0.674325	+	0.738435i	$0.735565\pi$
$888$	0	0
$889$	−27.9254	−0.936589
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−1.68795	−0.0564850
$894$	0	0
$895$	26.2129	0.876201
$896$	0	0
$897$	0	0
$898$	0	0
$899$	52.6715	1.75669
$900$	0	0
$901$	−25.7456	−0.857709
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−15.1821	−0.504669
$906$	0	0
$907$	9.35587	0.310657	0.155328	−	0.987863i	$-0.450356\pi$
$907$	9.35587	0.310657	0.155328	+	0.987863i	$0.450356\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	46.6472	1.54549	0.772746	−	0.634716i	$-0.218883\pi$
$911$	46.6472	1.54549	0.772746	+	0.634716i	$0.218883\pi$
$912$	0	0
$913$	31.0912	1.02897
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−63.8900	−2.10983
$918$	0	0
$919$	15.2079	0.501662	0.250831	−	0.968031i	$-0.419296\pi$
$919$	15.2079	0.501662	0.250831	+	0.968031i	$0.419296\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−55.3828	−1.82295
$924$	0	0
$925$	−10.7181	−0.352408
$926$	0	0
$927$	0	0
$928$	0	0
$929$	19.1979	0.629862	0.314931	−	0.949115i	$-0.398019\pi$
$929$	19.1979	0.629862	0.314931	+	0.949115i	$0.398019\pi$
$930$	0	0
$931$	−0.818126	−0.0268130
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−13.9565	−0.456425
$936$	0	0
$937$	25.5026	0.833135	0.416567	−	0.909105i	$-0.363233\pi$
$937$	25.5026	0.833135	0.416567	+	0.909105i	$0.363233\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	30.4753	0.993466	0.496733	−	0.867903i	$-0.334533\pi$
$941$	30.4753	0.993466	0.496733	+	0.867903i	$0.334533\pi$
$942$	0	0
$943$	−5.83998	−0.190176
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−40.2149	−1.30681	−0.653404	−	0.757009i	$-0.726660\pi$
$947$	−40.2149	−1.30681	−0.653404	+	0.757009i	$0.726660\pi$
$948$	0	0
$949$	69.1884	2.24595
$950$	0	0
$951$	0	0
$952$	0	0
$953$	26.6008	0.861685	0.430843	−	0.902427i	$-0.358216\pi$
$953$	26.6008	0.861685	0.430843	+	0.902427i	$0.358216\pi$
$954$	0	0
$955$	−18.0586	−0.584362
$956$	0	0
$957$	0	0
$958$	0	0
$959$	55.2107	1.78285
$960$	0	0
$961$	31.9731	1.03139
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−7.71022	−0.248201
$966$	0	0
$967$	7.36957	0.236990	0.118495	−	0.992955i	$-0.462193\pi$
$967$	7.36957	0.236990	0.118495	+	0.992955i	$0.462193\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	2.04075	0.0654909	0.0327454	−	0.999464i	$-0.489575\pi$
$971$	2.04075	0.0654909	0.0327454	+	0.999464i	$0.489575\pi$
$972$	0	0
$973$	15.5119	0.497289
$974$	0	0
$975$	0	0
$976$	0	0
$977$	61.6934	1.97375	0.986874	−	0.161493i	$-0.0516309\pi$
$977$	61.6934	1.97375	0.986874	+	0.161493i	$0.0516309\pi$
$978$	0	0
$979$	−64.5879	−2.06424
$980$	0	0
$981$	0	0
$982$	0	0
$983$	32.3935	1.03319	0.516596	−	0.856230i	$-0.327199\pi$
$983$	32.3935	1.03319	0.516596	+	0.856230i	$0.327199\pi$
$984$	0	0
$985$	33.9899	1.08301
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−43.4346	−1.38114
$990$	0	0
$991$	−21.1323	−0.671288	−0.335644	−	0.941989i	$-0.608954\pi$
$991$	−21.1323	−0.671288	−0.335644	+	0.941989i	$0.608954\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−5.72988	−0.181649
$996$	0	0
$997$	−59.1520	−1.87336	−0.936682	−	0.350181i	$-0.886120\pi$
$997$	−59.1520	−1.87336	−0.936682	+	0.350181i	$0.886120\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1944.2.a.q.1.3	✓	6
3.2	odd	2		1944.2.a.r.1.4	yes	6
4.3	odd	2		3888.2.a.bm.1.3		6
9.2	odd	6		1944.2.i.q.1297.3		12
9.4	even	3		1944.2.i.r.649.4		12
9.5	odd	6		1944.2.i.q.649.3		12
9.7	even	3		1944.2.i.r.1297.4		12
12.11	even	2		3888.2.a.bl.1.4		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1944.2.a.q.1.3	✓	6	1.1	even	1	trivial
1944.2.a.r.1.4	yes	6	3.2	odd	2
1944.2.i.q.649.3		12	9.5	odd	6
1944.2.i.q.1297.3		12	9.2	odd	6
1944.2.i.r.649.4		12	9.4	even	3
1944.2.i.r.1297.4		12	9.7	even	3
3888.2.a.bl.1.4		6	12.11	even	2
3888.2.a.bm.1.3		6	4.3	odd	2

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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:	\( N \)	\(=\)	\( 1944 = 2^{3} \cdot 3^{5} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1944.a (trivial)

\(f(q)\)	\(=\)	\(q-1.27086 q^{5} +3.12719 q^{7} +O(q^{10})\)	\(q-1.27086 q^{5} +3.12719 q^{7} -4.75001 q^{11} +5.26781 q^{13} -2.31199 q^{17} -0.294360 q^{19} +4.60635 q^{23} -3.38493 q^{25} +6.63740 q^{29} +7.93556 q^{31} -3.97421 q^{35} +3.16641 q^{37} -1.26781 q^{41} -9.42929 q^{43} +5.73430 q^{47} +2.77934 q^{49} +11.1357 q^{53} +6.03657 q^{55} +2.71414 q^{59} -8.99755 q^{61} -6.69463 q^{65} +9.40919 q^{67} -10.5134 q^{71} +13.1342 q^{73} -14.8542 q^{77} +1.79682 q^{79} -6.54551 q^{83} +2.93820 q^{85} +13.5974 q^{89} +16.4735 q^{91} +0.374089 q^{95} +4.23980 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 3 q^{7}+O(q^{10}) \) 6 * q + 3 * q^7	\( 6 q + 3 q^{7} - 3 q^{11} + 3 q^{13} + 9 q^{17} + 9 q^{19} - 6 q^{23} + 18 q^{25} - 3 q^{29} + 24 q^{31} - 15 q^{35} + 15 q^{37} + 21 q^{41} + 18 q^{43} - 6 q^{47} + 21 q^{49} + 12 q^{53} + 18 q^{55} - 12 q^{59} + 18 q^{61} + 33 q^{65} + 24 q^{67} - 15 q^{71} + 6 q^{73} + 3 q^{77} + 21 q^{79} - 18 q^{83} + 24 q^{85} + 18 q^{89} + 30 q^{91} - 39 q^{95} + 27 q^{97}+O(q^{100}) \) 6 * q + 3 * q^7 - 3 * q^11 + 3 * q^13 + 9 * q^17 + 9 * q^19 - 6 * q^23 + 18 * q^25 - 3 * q^29 + 24 * q^31 - 15 * q^35 + 15 * q^37 + 21 * q^41 + 18 * q^43 - 6 * q^47 + 21 * q^49 + 12 * q^53 + 18 * q^55 - 12 * q^59 + 18 * q^61 + 33 * q^65 + 24 * q^67 - 15 * q^71 + 6 * q^73 + 3 * q^77 + 21 * q^79 - 18 * q^83 + 24 * q^85 + 18 * q^89 + 30 * q^91 - 39 * q^95 + 27 * q^97

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