LMFDB - Embedded newform 6048.2.j.d.5615.43

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6048,2,Mod(5615,6048)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6048, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1, 1, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("6048.5615");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6048 = 2^{5} \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6048.j (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$48.2935231425$
Analytic rank:	$0$
Dimension:	$48$
Twist minimal:	no (minimal twist has level 1512)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			5615.43
Character	$\chi$	$=$	6048.5615
Dual form			6048.2.j.d.5615.44

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.99711 q^{5} +1.00000i q^{7} +O(q^{10})$	$q+2.99711 q^{5} +1.00000i q^{7} -5.36291i q^{11} -4.55519i q^{13} +0.958700i q^{17} -0.532227 q^{19} -2.78680 q^{23} +3.98265 q^{25} +5.15616 q^{29} -4.66569i q^{31} +2.99711i q^{35} +6.10293i q^{37} -12.3255i q^{41} -4.45583 q^{43} -1.40285 q^{47} -1.00000 q^{49} +9.57019 q^{53} -16.0732i q^{55} -0.0356973i q^{59} +5.13661i q^{61} -13.6524i q^{65} -8.59546 q^{67} -13.3269 q^{71} +6.50808 q^{73} +5.36291 q^{77} +1.90581i q^{79} -3.13371i q^{83} +2.87332i q^{85} +5.91323i q^{89} +4.55519 q^{91} -1.59514 q^{95} -16.0057 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q+O(q^{10}) $ 48 * q	$ 48 q - 16 q^{19} + 48 q^{25} - 64 q^{43} - 48 q^{49} - 32 q^{67}+O(q^{100}) $ 48 * q - 16 * q^19 + 48 * q^25 - 64 * q^43 - 48 * q^49 - 32 * q^67

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/6048\mathbb{Z}\right)^\times$.

$n$	$2593$	$3781$	$3809$	$4159$
$\chi(n)$	$1$	$-1$	$-1$	$-1$

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
$5$	2.99711	1.34035	0.670173	−	0.742205i	$-0.266220\pi$
$5$	2.99711	1.34035	0.670173	+	0.742205i	$0.266220\pi$
$6$	0	0
$7$	1.00000i	0.377964i
$7$	1.00000i	0.377964i
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 5.36291i	− 1.61698i	−0.588512	−	0.808488i	$-0.700286\pi$
$11$	− 5.36291i	− 1.61698i	0.588512	−	0.808488i	$-0.299714\pi$
$12$	0	0
$13$	− 4.55519i	− 1.26338i	−0.775221	−	0.631691i	$-0.782361\pi$
$13$	− 4.55519i	− 1.26338i	0.775221	−	0.631691i	$-0.217639\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	0.958700i	0.232519i	0.993219	+	0.116259i	$0.0370904\pi$
$17$	0.958700i	0.232519i	−0.993219	+	0.116259i	$0.962910\pi$
$18$	0	0
$19$	−0.532227	−0.122101	−0.0610506	−	0.998135i	$-0.519445\pi$
$19$	−0.532227	−0.122101	−0.0610506	+	0.998135i	$0.519445\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−2.78680	−0.581088	−0.290544	−	0.956862i	$-0.593836\pi$
$23$	−2.78680	−0.581088	−0.290544	+	0.956862i	$0.593836\pi$
$24$	0	0
$25$	3.98265	0.796529
$26$	0	0
$27$	0	0
$28$	0	0
$29$	5.15616	0.957475	0.478737	−	0.877958i	$-0.341095\pi$
$29$	5.15616	0.957475	0.478737	+	0.877958i	$0.341095\pi$
$30$	0	0
$31$	− 4.66569i	− 0.837983i	−0.907990	−	0.418992i	$-0.862384\pi$
$31$	− 4.66569i	− 0.837983i	0.907990	−	0.418992i	$-0.137616\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	2.99711i	0.506603i
$36$	0	0
$37$	6.10293i	1.00332i	0.865066	+	0.501658i	$0.167276\pi$
$37$	6.10293i	1.00332i	−0.865066	+	0.501658i	$0.832724\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 12.3255i	− 1.92492i	−0.271423	−	0.962460i	$-0.587494\pi$
$41$	− 12.3255i	− 1.92492i	0.271423	−	0.962460i	$-0.412506\pi$
$42$	0	0
$43$	−4.45583	−0.679508	−0.339754	−	0.940514i	$-0.610344\pi$
$43$	−4.45583	−0.679508	−0.339754	+	0.940514i	$0.610344\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−1.40285	−0.204627	−0.102314	−	0.994752i	$-0.532625\pi$
$47$	−1.40285	−0.204627	−0.102314	+	0.994752i	$0.532625\pi$
$48$	0	0
$49$	−1.00000	−0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	9.57019	1.31457	0.657283	−	0.753644i	$-0.271705\pi$
$53$	9.57019	1.31457	0.657283	+	0.753644i	$0.271705\pi$
$54$	0	0
$55$	− 16.0732i	− 2.16731i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 0.0356973i	− 0.00464739i	−0.999997	−	0.00232369i	$-0.999260\pi$
$59$	− 0.0356973i	− 0.00464739i	0.999997	−	0.00232369i	$-0.000739655\pi$
$60$	0	0
$61$	5.13661i	0.657675i	0.944386	+	0.328838i	$0.106657\pi$
$61$	5.13661i	0.657675i	−0.944386	+	0.328838i	$0.893343\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 13.6524i	− 1.69337i
$66$	0	0
$67$	−8.59546	−1.05010	−0.525051	−	0.851071i	$-0.675954\pi$
$67$	−8.59546	−1.05010	−0.525051	+	0.851071i	$0.675954\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−13.3269	−1.58161	−0.790804	−	0.612070i	$-0.790337\pi$
$71$	−13.3269	−1.58161	−0.790804	+	0.612070i	$0.790337\pi$
$72$	0	0
$73$	6.50808	0.761713	0.380857	−	0.924634i	$-0.375629\pi$
$73$	6.50808	0.761713	0.380857	+	0.924634i	$0.375629\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	5.36291	0.611160
$78$	0	0
$79$	1.90581i	0.214420i	0.994236	+	0.107210i	$0.0341917\pi$
$79$	1.90581i	0.214420i	−0.994236	+	0.107210i	$0.965808\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 3.13371i	− 0.343970i	−0.985100	−	0.171985i	$-0.944982\pi$
$83$	− 3.13371i	− 0.343970i	0.985100	−	0.171985i	$-0.0550180\pi$
$84$	0	0
$85$	2.87332i	0.311656i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	5.91323i	0.626801i	0.949621	+	0.313401i	$0.101468\pi$
$89$	5.91323i	0.626801i	−0.949621	+	0.313401i	$0.898532\pi$
$90$	0	0
$91$	4.55519	0.477513
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−1.59514	−0.163658
$96$	0	0
$97$	−16.0057	−1.62514	−0.812568	−	0.582867i	$-0.801931\pi$
$97$	−16.0057	−1.62514	−0.812568	+	0.582867i	$0.801931\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−16.1028	−1.60229	−0.801143	−	0.598473i	$-0.795774\pi$
$101$	−16.1028	−1.60229	−0.801143	+	0.598473i	$0.795774\pi$
$102$	0	0
$103$	− 18.0349i	− 1.77703i	−0.458850	−	0.888514i	$-0.651739\pi$
$103$	− 18.0349i	− 1.77703i	0.458850	−	0.888514i	$-0.348261\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 9.08345i	− 0.878131i	−0.898455	−	0.439065i	$-0.855310\pi$
$107$	− 9.08345i	− 0.878131i	0.898455	−	0.439065i	$-0.144690\pi$
$108$	0	0
$109$	− 17.2591i	− 1.65312i	−0.562848	−	0.826560i	$-0.690294\pi$
$109$	− 17.2591i	− 1.65312i	0.562848	−	0.826560i	$-0.309706\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 1.27590i	− 0.120026i	−0.998198	−	0.0600131i	$-0.980886\pi$
$113$	− 1.27590i	− 0.120026i	0.998198	−	0.0600131i	$-0.0191142\pi$
$114$	0	0
$115$	−8.35233	−0.778859
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−0.958700	−0.0878838
$120$	0	0
$121$	−17.7608	−1.61461
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−3.04912	−0.272721
$126$	0	0
$127$	− 12.7265i	− 1.12929i	−0.825333	−	0.564646i	$-0.809013\pi$
$127$	− 12.7265i	− 1.12929i	0.825333	−	0.564646i	$-0.190987\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	13.3457i	1.16602i	0.812465	+	0.583011i	$0.198125\pi$
$131$	13.3457i	1.16602i	−0.812465	+	0.583011i	$0.801875\pi$
$132$	0	0
$133$	− 0.532227i	− 0.0461499i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	17.0901i	1.46011i	0.683390	+	0.730053i	$0.260505\pi$
$137$	17.0901i	1.46011i	−0.683390	+	0.730053i	$0.739495\pi$
$138$	0	0
$139$	16.0609	1.36227	0.681134	−	0.732158i	$-0.261487\pi$
$139$	16.0609	1.36227	0.681134	+	0.732158i	$0.261487\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−24.4290	−2.04286
$144$	0	0
$145$	15.4536	1.28335
$146$	0	0
$147$	0	0
$148$	0	0
$149$	1.37457	0.112609	0.0563047	−	0.998414i	$-0.482068\pi$
$149$	1.37457	0.112609	0.0563047	+	0.998414i	$0.482068\pi$
$150$	0	0
$151$	5.50504i	0.447994i	0.974590	+	0.223997i	$0.0719105\pi$
$151$	5.50504i	0.447994i	−0.974590	+	0.223997i	$0.928089\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 13.9836i	− 1.12319i
$156$	0	0
$157$	− 18.7801i	− 1.49881i	−0.662110	−	0.749407i	$-0.730339\pi$
$157$	− 18.7801i	− 1.49881i	0.662110	−	0.749407i	$-0.269661\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 2.78680i	− 0.219630i
$162$	0	0
$163$	17.7671	1.39163	0.695813	−	0.718223i	$-0.255044\pi$
$163$	17.7671	1.39163	0.695813	+	0.718223i	$0.255044\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	18.1777	1.40663	0.703315	−	0.710878i	$-0.251702\pi$
$167$	18.1777	1.40663	0.703315	+	0.710878i	$0.251702\pi$
$168$	0	0
$169$	−7.74971	−0.596132
$170$	0	0
$171$	0	0
$172$	0	0
$173$	2.38643	0.181437	0.0907186	−	0.995877i	$-0.471084\pi$
$173$	2.38643	0.181437	0.0907186	+	0.995877i	$0.471084\pi$
$174$	0	0
$175$	3.98265i	0.301060i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	6.73152i	0.503137i	0.967839	+	0.251569i	$0.0809464\pi$
$179$	6.73152i	0.503137i	−0.967839	+	0.251569i	$0.919054\pi$
$180$	0	0
$181$	1.20606i	0.0896460i	0.998995	+	0.0448230i	$0.0142724\pi$
$181$	1.20606i	0.0896460i	−0.998995	+	0.0448230i	$0.985728\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	18.2911i	1.34479i
$186$	0	0
$187$	5.14142	0.375978
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−15.5066	−1.12202	−0.561008	−	0.827810i	$-0.689586\pi$
$191$	−15.5066	−1.12202	−0.561008	+	0.827810i	$0.689586\pi$
$192$	0	0
$193$	7.67734	0.552627	0.276314	−	0.961068i	$-0.410887\pi$
$193$	7.67734	0.552627	0.276314	+	0.961068i	$0.410887\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	23.1208	1.64729	0.823644	−	0.567107i	$-0.191938\pi$
$197$	23.1208	1.64729	0.823644	+	0.567107i	$0.191938\pi$
$198$	0	0
$199$	1.14987i	0.0815122i	0.999169	+	0.0407561i	$0.0129767\pi$
$199$	1.14987i	0.0815122i	−0.999169	+	0.0407561i	$0.987023\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	5.15616i	0.361891i
$204$	0	0
$205$	− 36.9408i	− 2.58006i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	2.85428i	0.197435i
$210$	0	0
$211$	−15.6643	−1.07837	−0.539186	−	0.842187i	$-0.681268\pi$
$211$	−15.6643	−1.07837	−0.539186	+	0.842187i	$0.681268\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−13.3546	−0.910776
$216$	0	0
$217$	4.66569	0.316728
$218$	0	0
$219$	0	0
$220$	0	0
$221$	4.36705	0.293760
$222$	0	0
$223$	− 13.9655i	− 0.935198i	−0.883941	−	0.467599i	$-0.845119\pi$
$223$	− 13.9655i	− 0.935198i	0.883941	−	0.467599i	$-0.154881\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	26.1607i	1.73635i	0.496262	+	0.868173i	$0.334705\pi$
$227$	26.1607i	1.73635i	−0.496262	+	0.868173i	$0.665295\pi$
$228$	0	0
$229$	6.01986i	0.397803i	0.980019	+	0.198902i	$0.0637374\pi$
$229$	6.01986i	0.397803i	−0.980019	+	0.198902i	$0.936263\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 13.4333i	− 0.880046i	−0.897986	−	0.440023i	$-0.854970\pi$
$233$	− 13.4333i	− 0.880046i	0.897986	−	0.440023i	$-0.145030\pi$
$234$	0	0
$235$	−4.20450	−0.274271
$236$	0	0
$237$	0	0
$238$	0	0
$239$	26.3499	1.70443	0.852215	−	0.523191i	$-0.175259\pi$
$239$	26.3499	1.70443	0.852215	+	0.523191i	$0.175259\pi$
$240$	0	0
$241$	−12.9481	−0.834063	−0.417031	−	0.908892i	$-0.636929\pi$
$241$	−12.9481	−0.834063	−0.417031	+	0.908892i	$0.636929\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−2.99711	−0.191478
$246$	0	0
$247$	2.42439i	0.154260i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	20.5607i	1.29778i	0.760881	+	0.648891i	$0.224767\pi$
$251$	20.5607i	1.29778i	−0.760881	+	0.648891i	$0.775233\pi$
$252$	0	0
$253$	14.9453i	0.939605i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	1.25071i	0.0780174i	0.999239	+	0.0390087i	$0.0124200\pi$
$257$	1.25071i	0.0780174i	−0.999239	+	0.0390087i	$0.987580\pi$
$258$	0	0
$259$	−6.10293	−0.379218
$260$	0	0
$261$	0	0
$262$	0	0
$263$	25.0372	1.54386	0.771931	−	0.635706i	$-0.219291\pi$
$263$	25.0372	1.54386	0.771931	+	0.635706i	$0.219291\pi$
$264$	0	0
$265$	28.6829	1.76197
$266$	0	0
$267$	0	0
$268$	0	0
$269$	0.520900	0.0317598	0.0158799	−	0.999874i	$-0.494945\pi$
$269$	0.520900	0.0317598	0.0158799	+	0.999874i	$0.494945\pi$
$270$	0	0
$271$	− 7.99100i	− 0.485419i	−0.970099	−	0.242709i	$-0.921964\pi$
$271$	− 7.99100i	− 0.485419i	0.970099	−	0.242709i	$-0.0780361\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 21.3586i	− 1.28797i
$276$	0	0
$277$	− 32.8049i	− 1.97106i	−0.169509	−	0.985529i	$-0.554218\pi$
$277$	− 32.8049i	− 1.97106i	0.169509	−	0.985529i	$-0.445782\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	7.13786i	0.425809i	0.977073	+	0.212905i	$0.0682924\pi$
$281$	7.13786i	0.425809i	−0.977073	+	0.212905i	$0.931708\pi$
$282$	0	0
$283$	4.47861	0.266226	0.133113	−	0.991101i	$-0.457503\pi$
$283$	4.47861	0.266226	0.133113	+	0.991101i	$0.457503\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	12.3255	0.727551
$288$	0	0
$289$	16.0809	0.945935
$290$	0	0
$291$	0	0
$292$	0	0
$293$	23.4454	1.36969	0.684846	−	0.728688i	$-0.259869\pi$
$293$	23.4454	1.36969	0.684846	+	0.728688i	$0.259869\pi$
$294$	0	0
$295$	− 0.106988i	− 0.00622911i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	12.6944i	0.734135i
$300$	0	0
$301$	− 4.45583i	− 0.256830i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	15.3950i	0.881513i
$306$	0	0
$307$	5.07164	0.289454	0.144727	−	0.989472i	$-0.453770\pi$
$307$	5.07164	0.289454	0.144727	+	0.989472i	$0.453770\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−5.43039	−0.307929	−0.153964	−	0.988076i	$-0.549204\pi$
$311$	−5.43039	−0.307929	−0.153964	+	0.988076i	$0.549204\pi$
$312$	0	0
$313$	12.5124	0.707245	0.353622	−	0.935388i	$-0.384950\pi$
$313$	12.5124	0.707245	0.353622	+	0.935388i	$0.384950\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−9.29939	−0.522306	−0.261153	−	0.965297i	$-0.584103\pi$
$317$	−9.29939	−0.522306	−0.261153	+	0.965297i	$0.584103\pi$
$318$	0	0
$319$	− 27.6520i	− 1.54821i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 0.510246i	− 0.0283908i
$324$	0	0
$325$	− 18.1417i	− 1.00632i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 1.40285i	− 0.0773418i
$330$	0	0
$331$	21.4879	1.18108	0.590541	−	0.807008i	$-0.298914\pi$
$331$	21.4879	1.18108	0.590541	+	0.807008i	$0.298914\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−25.7615	−1.40750
$336$	0	0
$337$	−28.1982	−1.53605	−0.768026	−	0.640419i	$-0.778761\pi$
$337$	−28.1982	−1.53605	−0.768026	+	0.640419i	$0.778761\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−25.0217	−1.35500
$342$	0	0
$343$	− 1.00000i	− 0.0539949i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 21.2134i	− 1.13880i	−0.822062	−	0.569398i	$-0.807176\pi$
$347$	− 21.2134i	− 1.13880i	0.822062	−	0.569398i	$-0.192824\pi$
$348$	0	0
$349$	− 31.7541i	− 1.69976i	−0.526979	−	0.849878i	$-0.676675\pi$
$349$	− 31.7541i	− 1.69976i	0.526979	−	0.849878i	$-0.323325\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 7.37308i	− 0.392429i	−0.980561	−	0.196215i	$-0.937135\pi$
$353$	− 7.37308i	− 0.392429i	0.980561	−	0.196215i	$-0.0628650\pi$
$354$	0	0
$355$	−39.9420	−2.11990
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−14.8384	−0.783142	−0.391571	−	0.920148i	$-0.628068\pi$
$359$	−14.8384	−0.783142	−0.391571	+	0.920148i	$0.628068\pi$
$360$	0	0
$361$	−18.7167	−0.985091
$362$	0	0
$363$	0	0
$364$	0	0
$365$	19.5054	1.02096
$366$	0	0
$367$	13.3215i	0.695379i	0.937610	+	0.347690i	$0.113034\pi$
$367$	13.3215i	0.695379i	−0.937610	+	0.347690i	$0.886966\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	9.57019i	0.496859i
$372$	0	0
$373$	24.7297i	1.28046i	0.768185	+	0.640228i	$0.221160\pi$
$373$	24.7297i	1.28046i	−0.768185	+	0.640228i	$0.778840\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 23.4873i	− 1.20966i
$378$	0	0
$379$	20.7764	1.06721	0.533606	−	0.845733i	$-0.320837\pi$
$379$	20.7764	1.06721	0.533606	+	0.845733i	$0.320837\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	4.92022	0.251411	0.125706	−	0.992068i	$-0.459881\pi$
$383$	4.92022	0.251411	0.125706	+	0.992068i	$0.459881\pi$
$384$	0	0
$385$	16.0732	0.819166
$386$	0	0
$387$	0	0
$388$	0	0
$389$	34.4298	1.74566	0.872830	−	0.488025i	$-0.162283\pi$
$389$	34.4298	1.74566	0.872830	+	0.488025i	$0.162283\pi$
$390$	0	0
$391$	− 2.67170i	− 0.135114i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	5.71191i	0.287397i
$396$	0	0
$397$	− 21.8751i	− 1.09788i	−0.835861	−	0.548941i	$-0.815031\pi$
$397$	− 21.8751i	− 1.09788i	0.835861	−	0.548941i	$-0.184969\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	29.0637i	1.45137i	0.688027	+	0.725685i	$0.258477\pi$
$401$	29.0637i	1.45137i	−0.688027	+	0.725685i	$0.741523\pi$
$402$	0	0
$403$	−21.2531	−1.05869
$404$	0	0
$405$	0	0
$406$	0	0
$407$	32.7295	1.62234
$408$	0	0
$409$	−4.15951	−0.205674	−0.102837	−	0.994698i	$-0.532792\pi$
$409$	−4.15951	−0.205674	−0.102837	+	0.994698i	$0.532792\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0.0356973	0.00175655
$414$	0	0
$415$	− 9.39207i	− 0.461039i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	22.5241i	1.10038i	0.835041	+	0.550188i	$0.185444\pi$
$419$	22.5241i	1.10038i	−0.835041	+	0.550188i	$0.814556\pi$
$420$	0	0
$421$	− 32.7375i	− 1.59553i	−0.602970	−	0.797764i	$-0.706016\pi$
$421$	− 32.7375i	− 1.59553i	0.602970	−	0.797764i	$-0.293984\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	3.81816i	0.185208i
$426$	0	0
$427$	−5.13661	−0.248578
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−28.1991	−1.35830	−0.679152	−	0.733998i	$-0.737652\pi$
$431$	−28.1991	−1.35830	−0.679152	+	0.733998i	$0.737652\pi$
$432$	0	0
$433$	−22.1414	−1.06405	−0.532024	−	0.846729i	$-0.678569\pi$
$433$	−22.1414	−1.06405	−0.532024	+	0.846729i	$0.678569\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	1.48321	0.0709515
$438$	0	0
$439$	24.6966i	1.17870i	0.807876	+	0.589352i	$0.200617\pi$
$439$	24.6966i	1.17870i	−0.807876	+	0.589352i	$0.799383\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	24.6550i	1.17139i	0.810531	+	0.585696i	$0.199179\pi$
$443$	24.6550i	1.17139i	−0.810531	+	0.585696i	$0.800821\pi$
$444$	0	0
$445$	17.7226i	0.840131i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	4.80960i	0.226979i	0.993539	+	0.113489i	$0.0362028\pi$
$449$	4.80960i	0.226979i	−0.993539	+	0.113489i	$0.963797\pi$
$450$	0	0
$451$	−66.1005	−3.11255
$452$	0	0
$453$	0	0
$454$	0	0
$455$	13.6524	0.640033
$456$	0	0
$457$	−20.5555	−0.961548	−0.480774	−	0.876845i	$-0.659644\pi$
$457$	−20.5555	−0.961548	−0.480774	+	0.876845i	$0.659644\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−6.96146	−0.324228	−0.162114	−	0.986772i	$-0.551831\pi$
$461$	−6.96146	−0.324228	−0.162114	+	0.986772i	$0.551831\pi$
$462$	0	0
$463$	0.214819i	0.00998347i	0.999988	+	0.00499174i	$0.00158893\pi$
$463$	0.214819i	0.00998347i	−0.999988	+	0.00499174i	$0.998411\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 30.4215i	− 1.40774i	−0.710328	−	0.703870i	$-0.751454\pi$
$467$	− 30.4215i	− 1.40774i	0.710328	−	0.703870i	$-0.248546\pi$
$468$	0	0
$469$	− 8.59546i	− 0.396901i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	23.8962i	1.09875i
$474$	0	0
$475$	−2.11967	−0.0972572
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−36.9888	−1.69006	−0.845030	−	0.534719i	$-0.820418\pi$
$479$	−36.9888	−1.69006	−0.845030	+	0.534719i	$0.820418\pi$
$480$	0	0
$481$	27.8000	1.26757
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−47.9709	−2.17824
$486$	0	0
$487$	10.2932i	0.466431i	0.972425	+	0.233215i	$0.0749248\pi$
$487$	10.2932i	0.466431i	−0.972425	+	0.233215i	$0.925075\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 1.77521i	− 0.0801142i	−0.999197	−	0.0400571i	$-0.987246\pi$
$491$	− 1.77521i	− 0.0801142i	0.999197	−	0.0400571i	$-0.0127540\pi$
$492$	0	0
$493$	4.94321i	0.222631i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 13.3269i	− 0.597791i
$498$	0	0
$499$	27.4254	1.22773	0.613864	−	0.789412i	$-0.289614\pi$
$499$	27.4254	1.22773	0.613864	+	0.789412i	$0.289614\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	31.1546	1.38911	0.694557	−	0.719437i	$-0.255600\pi$
$503$	31.1546	1.38911	0.694557	+	0.719437i	$0.255600\pi$
$504$	0	0
$505$	−48.2617	−2.14762
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−3.20979	−0.142271	−0.0711356	−	0.997467i	$-0.522662\pi$
$509$	−3.20979	−0.142271	−0.0711356	+	0.997467i	$0.522662\pi$
$510$	0	0
$511$	6.50808i	0.287901i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 54.0524i	− 2.38183i
$516$	0	0
$517$	7.52337i	0.330877i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 34.3057i	− 1.50296i	−0.659756	−	0.751480i	$-0.729340\pi$
$521$	− 34.3057i	− 1.50296i	0.659756	−	0.751480i	$-0.270660\pi$
$522$	0	0
$523$	37.4283	1.63662	0.818312	−	0.574774i	$-0.194910\pi$
$523$	37.4283	1.63662	0.818312	+	0.574774i	$0.194910\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	4.47300	0.194847
$528$	0	0
$529$	−15.2338	−0.662337
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−56.1449	−2.43191
$534$	0	0
$535$	− 27.2241i	− 1.17700i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	5.36291i	0.230997i
$540$	0	0
$541$	− 1.04111i	− 0.0447610i	−0.999750	−	0.0223805i	$-0.992875\pi$
$541$	− 1.04111i	− 0.0447610i	0.999750	−	0.0223805i	$-0.00712453\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 51.7273i	− 2.21576i
$546$	0	0
$547$	−3.89746	−0.166643	−0.0833216	−	0.996523i	$-0.526553\pi$
$547$	−3.89746	−0.166643	−0.0833216	+	0.996523i	$0.526553\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−2.74425	−0.116909
$552$	0	0
$553$	−1.90581	−0.0810432
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−32.1046	−1.36031	−0.680157	−	0.733067i	$-0.738088\pi$
$557$	−32.1046	−1.36031	−0.680157	+	0.733067i	$0.738088\pi$
$558$	0	0
$559$	20.2971i	0.858477i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 9.40478i	− 0.396364i	−0.980165	−	0.198182i	$-0.936496\pi$
$563$	− 9.40478i	− 0.396364i	0.980165	−	0.198182i	$-0.0635037\pi$
$564$	0	0
$565$	− 3.82399i	− 0.160877i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	5.32036i	0.223041i	0.993762	+	0.111521i	$0.0355721\pi$
$569$	5.32036i	0.223041i	−0.993762	+	0.111521i	$0.964428\pi$
$570$	0	0
$571$	47.3101	1.97986	0.989932	−	0.141542i	$-0.0452059\pi$
$571$	47.3101	1.97986	0.989932	+	0.141542i	$0.0452059\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−11.0988	−0.462853
$576$	0	0
$577$	21.0280	0.875407	0.437704	−	0.899119i	$-0.355792\pi$
$577$	21.0280	0.875407	0.437704	+	0.899119i	$0.355792\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	3.13371	0.130008
$582$	0	0
$583$	− 51.3240i	− 2.12562i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 18.8579i	− 0.778348i	−0.921164	−	0.389174i	$-0.872761\pi$
$587$	− 18.8579i	− 0.778348i	0.921164	−	0.389174i	$-0.127239\pi$
$588$	0	0
$589$	2.48321i	0.102319i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	15.7984i	0.648764i	0.945926	+	0.324382i	$0.105156\pi$
$593$	15.7984i	0.648764i	−0.945926	+	0.324382i	$0.894844\pi$
$594$	0	0
$595$	−2.87332	−0.117795
$596$	0	0
$597$	0	0
$598$	0	0
$599$	24.1841	0.988136	0.494068	−	0.869423i	$-0.335509\pi$
$599$	24.1841	0.988136	0.494068	+	0.869423i	$0.335509\pi$
$600$	0	0
$601$	31.5679	1.28768	0.643842	−	0.765159i	$-0.277340\pi$
$601$	31.5679	1.28768	0.643842	+	0.765159i	$0.277340\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−53.2309	−2.16414
$606$	0	0
$607$	− 23.4349i	− 0.951194i	−0.879663	−	0.475597i	$-0.842232\pi$
$607$	− 23.4349i	− 0.951194i	0.879663	−	0.475597i	$-0.157768\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	6.39025i	0.258522i
$612$	0	0
$613$	21.4865i	0.867832i	0.900953	+	0.433916i	$0.142868\pi$
$613$	21.4865i	0.867832i	−0.900953	+	0.433916i	$0.857132\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 12.4334i	− 0.500549i	−0.968175	−	0.250275i	$-0.919479\pi$
$617$	− 12.4334i	− 0.500549i	0.968175	−	0.250275i	$-0.0805209\pi$
$618$	0	0
$619$	20.4500	0.821956	0.410978	−	0.911645i	$-0.365187\pi$
$619$	20.4500	0.821956	0.410978	+	0.911645i	$0.365187\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−5.91323	−0.236909
$624$	0	0
$625$	−29.0518	−1.16207
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−5.85088	−0.233290
$630$	0	0
$631$	− 14.3682i	− 0.571990i	−0.958231	−	0.285995i	$-0.907676\pi$
$631$	− 14.3682i	− 0.571990i	0.958231	−	0.285995i	$-0.0923240\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 38.1426i	− 1.51364i
$636$	0	0
$637$	4.55519i	0.180483i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	13.7983i	0.544998i	0.962156	+	0.272499i	$0.0878502\pi$
$641$	13.7983i	0.544998i	−0.962156	+	0.272499i	$0.912150\pi$
$642$	0	0
$643$	36.7203	1.44811	0.724053	−	0.689745i	$-0.242277\pi$
$643$	36.7203	1.44811	0.724053	+	0.689745i	$0.242277\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	1.59099	0.0625483	0.0312742	−	0.999511i	$-0.490044\pi$
$647$	1.59099	0.0625483	0.0312742	+	0.999511i	$0.490044\pi$
$648$	0	0
$649$	−0.191441	−0.00751472
$650$	0	0
$651$	0	0
$652$	0	0
$653$	22.5953	0.884223	0.442111	−	0.896960i	$-0.354230\pi$
$653$	22.5953	0.884223	0.442111	+	0.896960i	$0.354230\pi$
$654$	0	0
$655$	39.9986i	1.56287i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	24.8372i	0.967520i	0.875201	+	0.483760i	$0.160729\pi$
$659$	24.8372i	0.967520i	−0.875201	+	0.483760i	$0.839271\pi$
$660$	0	0
$661$	28.3986i	1.10458i	0.833653	+	0.552288i	$0.186245\pi$
$661$	28.3986i	1.10458i	−0.833653	+	0.552288i	$0.813755\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 1.59514i	− 0.0618569i
$666$	0	0
$667$	−14.3692	−0.556377
$668$	0	0
$669$	0	0
$670$	0	0
$671$	27.5471	1.06345
$672$	0	0
$673$	3.94402	0.152031	0.0760153	−	0.997107i	$-0.475780\pi$
$673$	3.94402	0.152031	0.0760153	+	0.997107i	$0.475780\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	8.72694	0.335403	0.167702	−	0.985838i	$-0.446365\pi$
$677$	8.72694	0.335403	0.167702	+	0.985838i	$0.446365\pi$
$678$	0	0
$679$	− 16.0057i	− 0.614243i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 23.0246i	− 0.881014i	−0.897749	−	0.440507i	$-0.854799\pi$
$683$	− 23.0246i	− 0.881014i	0.897749	−	0.440507i	$-0.145201\pi$
$684$	0	0
$685$	51.2209i	1.95705i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 43.5940i	− 1.66080i
$690$	0	0
$691$	19.5528	0.743823	0.371912	−	0.928268i	$-0.378702\pi$
$691$	19.5528	0.743823	0.371912	+	0.928268i	$0.378702\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	48.1363	1.82591
$696$	0	0
$697$	11.8165	0.447580
$698$	0	0
$699$	0	0
$700$	0	0
$701$	22.1070	0.834971	0.417486	−	0.908684i	$-0.362911\pi$
$701$	22.1070	0.834971	0.417486	+	0.908684i	$0.362911\pi$
$702$	0	0
$703$	− 3.24814i	− 0.122506i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 16.1028i	− 0.605607i
$708$	0	0
$709$	− 15.3427i	− 0.576207i	−0.957599	−	0.288104i	$-0.906975\pi$
$709$	− 15.3427i	− 0.576207i	0.957599	−	0.288104i	$-0.0930247\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	13.0024i	0.486942i
$714$	0	0
$715$	−73.2164	−2.73814
$716$	0	0
$717$	0	0
$718$	0	0
$719$	15.1069	0.563392	0.281696	−	0.959504i	$-0.409103\pi$
$719$	15.1069	0.563392	0.281696	+	0.959504i	$0.409103\pi$
$720$	0	0
$721$	18.0349	0.671653
$722$	0	0
$723$	0	0
$724$	0	0
$725$	20.5352	0.762657
$726$	0	0
$727$	20.4051i	0.756784i	0.925645	+	0.378392i	$0.123523\pi$
$727$	20.4051i	0.756784i	−0.925645	+	0.378392i	$0.876477\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 4.27180i	− 0.157998i
$732$	0	0
$733$	− 39.8824i	− 1.47309i	−0.676389	−	0.736544i	$-0.736456\pi$
$733$	− 39.8824i	− 1.47309i	0.676389	−	0.736544i	$-0.263544\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	46.0966i	1.69799i
$738$	0	0
$739$	−31.8675	−1.17226	−0.586132	−	0.810216i	$-0.699350\pi$
$739$	−31.8675	−1.17226	−0.586132	+	0.810216i	$0.699350\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	36.1243	1.32527	0.662636	−	0.748942i	$-0.269438\pi$
$743$	36.1243	1.32527	0.662636	+	0.748942i	$0.269438\pi$
$744$	0	0
$745$	4.11974	0.150936
$746$	0	0
$747$	0	0
$748$	0	0
$749$	9.08345	0.331902
$750$	0	0
$751$	− 7.07701i	− 0.258244i	−0.991629	−	0.129122i	$-0.958784\pi$
$751$	− 7.07701i	− 0.258244i	0.991629	−	0.129122i	$-0.0412159\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	16.4992i	0.600467i
$756$	0	0
$757$	− 25.8712i	− 0.940304i	−0.882585	−	0.470152i	$-0.844199\pi$
$757$	− 25.8712i	− 0.940304i	0.882585	−	0.470152i	$-0.155801\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	3.07176i	0.111351i	0.998449	+	0.0556757i	$0.0177313\pi$
$761$	3.07176i	0.111351i	−0.998449	+	0.0556757i	$0.982269\pi$
$762$	0	0
$763$	17.2591	0.624821
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−0.162608	−0.00587142
$768$	0	0
$769$	39.3610	1.41940	0.709698	−	0.704506i	$-0.248831\pi$
$769$	39.3610	1.41940	0.709698	+	0.704506i	$0.248831\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−48.9129	−1.75928	−0.879638	−	0.475644i	$-0.842215\pi$
$773$	−48.9129	−1.75928	−0.879638	+	0.475644i	$0.842215\pi$
$774$	0	0
$775$	− 18.5818i	− 0.667478i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	6.55996i	0.235035i
$780$	0	0
$781$	71.4707i	2.55742i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 56.2859i	− 2.00893i
$786$	0	0
$787$	−16.5486	−0.589893	−0.294946	−	0.955514i	$-0.595302\pi$
$787$	−16.5486	−0.589893	−0.294946	+	0.955514i	$0.595302\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	1.27590	0.0453656
$792$	0	0
$793$	23.3982	0.830895
$794$	0	0
$795$	0	0
$796$	0	0
$797$	6.15737	0.218105	0.109053	−	0.994036i	$-0.465218\pi$
$797$	6.15737	0.218105	0.109053	+	0.994036i	$0.465218\pi$
$798$	0	0
$799$	− 1.34491i	− 0.0475797i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 34.9022i	− 1.23167i
$804$	0	0
$805$	− 8.35233i	− 0.294381i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	10.7152i	0.376728i	0.982099	+	0.188364i	$0.0603185\pi$
$809$	10.7152i	0.376728i	−0.982099	+	0.188364i	$0.939682\pi$
$810$	0	0
$811$	−1.60272	−0.0562791	−0.0281395	−	0.999604i	$-0.508958\pi$
$811$	−1.60272	−0.0562791	−0.0281395	+	0.999604i	$0.508958\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	53.2498	1.86526
$816$	0	0
$817$	2.37151	0.0829687
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−5.62139	−0.196188	−0.0980940	−	0.995177i	$-0.531275\pi$
$821$	−5.62139	−0.196188	−0.0980940	+	0.995177i	$0.531275\pi$
$822$	0	0
$823$	− 10.5605i	− 0.368114i	−0.982916	−	0.184057i	$-0.941077\pi$
$823$	− 10.5605i	− 0.368114i	0.982916	−	0.184057i	$-0.0589232\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	37.6285i	1.30847i	0.756290	+	0.654236i	$0.227010\pi$
$827$	37.6285i	1.30847i	−0.756290	+	0.654236i	$0.772990\pi$
$828$	0	0
$829$	2.85388i	0.0991193i	0.998771	+	0.0495596i	$0.0157818\pi$
$829$	2.85388i	0.0991193i	−0.998771	+	0.0495596i	$0.984218\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 0.958700i	− 0.0332170i
$834$	0	0
$835$	54.4804	1.88537
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−9.63628	−0.332681	−0.166341	−	0.986068i	$-0.553195\pi$
$839$	−9.63628	−0.332681	−0.166341	+	0.986068i	$0.553195\pi$
$840$	0	0
$841$	−2.41401	−0.0832418
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−23.2267	−0.799023
$846$	0	0
$847$	− 17.7608i	− 0.610267i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 17.0076i	− 0.583015i
$852$	0	0
$853$	4.13385i	0.141540i	0.997493	+	0.0707701i	$0.0225457\pi$
$853$	4.13385i	0.141540i	−0.997493	+	0.0707701i	$0.977454\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	32.0416i	1.09452i	0.836963	+	0.547260i	$0.184329\pi$
$857$	32.0416i	1.09452i	−0.836963	+	0.547260i	$0.815671\pi$
$858$	0	0
$859$	17.8213	0.608055	0.304027	−	0.952663i	$-0.401669\pi$
$859$	17.8213	0.608055	0.304027	+	0.952663i	$0.401669\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−2.53777	−0.0863868	−0.0431934	−	0.999067i	$-0.513753\pi$
$863$	−2.53777	−0.0863868	−0.0431934	+	0.999067i	$0.513753\pi$
$864$	0	0
$865$	7.15239	0.243189
$866$	0	0
$867$	0	0
$868$	0	0
$869$	10.2207	0.346713
$870$	0	0
$871$	39.1539i	1.32668i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 3.04912i	− 0.103079i
$876$	0	0
$877$	46.9723i	1.58614i	0.609129	+	0.793072i	$0.291519\pi$
$877$	46.9723i	1.58614i	−0.609129	+	0.793072i	$0.708481\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	45.8003i	1.54305i	0.636198	+	0.771525i	$0.280506\pi$
$881$	45.8003i	1.54305i	−0.636198	+	0.771525i	$0.719494\pi$
$882$	0	0
$883$	−35.1176	−1.18180	−0.590901	−	0.806744i	$-0.701228\pi$
$883$	−35.1176	−1.18180	−0.590901	+	0.806744i	$0.701228\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	10.1586	0.341091	0.170546	−	0.985350i	$-0.445447\pi$
$887$	10.1586	0.341091	0.170546	+	0.985350i	$0.445447\pi$
$888$	0	0
$889$	12.7265	0.426833
$890$	0	0
$891$	0	0
$892$	0	0
$893$	0.746636	0.0249852
$894$	0	0
$895$	20.1751i	0.674378i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 24.0571i	− 0.802348i
$900$	0	0
$901$	9.17494i	0.305661i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	3.61470i	0.120157i
$906$	0	0
$907$	−35.4243	−1.17625	−0.588123	−	0.808772i	$-0.700133\pi$
$907$	−35.4243	−1.17625	−0.588123	+	0.808772i	$0.700133\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−37.6389	−1.24703	−0.623517	−	0.781810i	$-0.714297\pi$
$911$	−37.6389	−1.24703	−0.623517	+	0.781810i	$0.714297\pi$
$912$	0	0
$913$	−16.8058	−0.556191
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−13.3457	−0.440715
$918$	0	0
$919$	39.1964i	1.29297i	0.762927	+	0.646485i	$0.223762\pi$
$919$	39.1964i	1.29297i	−0.762927	+	0.646485i	$0.776238\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	60.7063i	1.99817i
$924$	0	0
$925$	24.3058i	0.799171i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 9.19387i	− 0.301641i	−0.988561	−	0.150821i	$-0.951808\pi$
$929$	− 9.19387i	− 0.301641i	0.988561	−	0.150821i	$-0.0481916\pi$
$930$	0	0
$931$	0.532227	0.0174430
$932$	0	0
$933$	0	0
$934$	0	0
$935$	15.4094	0.503940
$936$	0	0
$937$	−48.7873	−1.59381	−0.796905	−	0.604104i	$-0.793531\pi$
$937$	−48.7873	−1.59381	−0.796905	+	0.604104i	$0.793531\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	0.0818594	0.00266854	0.00133427	−	0.999999i	$-0.499575\pi$
$941$	0.0818594	0.00266854	0.00133427	+	0.999999i	$0.499575\pi$
$942$	0	0
$943$	34.3487i	1.11855i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	21.5632i	0.700711i	0.936617	+	0.350356i	$0.113939\pi$
$947$	21.5632i	0.700711i	−0.936617	+	0.350356i	$0.886061\pi$
$948$	0	0
$949$	− 29.6455i	− 0.962334i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	13.2536i	0.429325i	0.976688	+	0.214662i	$0.0688651\pi$
$953$	13.2536i	0.429325i	−0.976688	+	0.214662i	$0.931135\pi$
$954$	0	0
$955$	−46.4748	−1.50389
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−17.0901	−0.551868
$960$	0	0
$961$	9.23129	0.297784
$962$	0	0
$963$	0	0
$964$	0	0
$965$	23.0098	0.740712
$966$	0	0
$967$	6.74105i	0.216778i	0.994109	+	0.108389i	$0.0345691\pi$
$967$	6.74105i	0.216778i	−0.994109	+	0.108389i	$0.965431\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 23.5075i	− 0.754391i	−0.926134	−	0.377196i	$-0.876888\pi$
$971$	− 23.5075i	− 0.754391i	0.926134	−	0.377196i	$-0.123112\pi$
$972$	0	0
$973$	16.0609i	0.514889i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 13.3757i	− 0.427927i	−0.976842	−	0.213964i	$-0.931363\pi$
$977$	− 13.3757i	− 0.427927i	0.976842	−	0.213964i	$-0.0686374\pi$
$978$	0	0
$979$	31.7121	1.01352
$980$	0	0
$981$	0	0
$982$	0	0
$983$	18.3093	0.583975	0.291987	−	0.956422i	$-0.405683\pi$
$983$	18.3093	0.583975	0.291987	+	0.956422i	$0.405683\pi$
$984$	0	0
$985$	69.2955	2.20794
$986$	0	0
$987$	0	0
$988$	0	0
$989$	12.4175	0.394853
$990$	0	0
$991$	− 58.0580i	− 1.84427i	−0.386865	−	0.922136i	$-0.626442\pi$
$991$	− 58.0580i	− 1.84427i	0.386865	−	0.922136i	$-0.373558\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	3.44629i	0.109255i
$996$	0	0
$997$	53.3202i	1.68867i	0.535818	+	0.844334i	$0.320003\pi$
$997$	53.3202i	1.68867i	−0.535818	+	0.844334i	$0.679997\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6048.2.j.d.5615.43		48
3.2	odd	2	inner	6048.2.j.d.5615.6		48
4.3	odd	2		1512.2.j.d.323.22	yes	48
8.3	odd	2	inner	6048.2.j.d.5615.5		48
8.5	even	2		1512.2.j.d.323.28	yes	48
12.11	even	2		1512.2.j.d.323.27	yes	48
24.5	odd	2		1512.2.j.d.323.21	✓	48
24.11	even	2	inner	6048.2.j.d.5615.44		48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1512.2.j.d.323.21	✓	48	24.5	odd	2
1512.2.j.d.323.22	yes	48	4.3	odd	2
1512.2.j.d.323.27	yes	48	12.11	even	2
1512.2.j.d.323.28	yes	48	8.5	even	2
6048.2.j.d.5615.5		48	8.3	odd	2	inner
6048.2.j.d.5615.6		48	3.2	odd	2	inner
6048.2.j.d.5615.43		48	1.1	even	1	trivial
6048.2.j.d.5615.44		48	24.11	even	2	inner

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:	\( N \)	\(=\)	\( 6048 = 2^{5} \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6048.j (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.99711 q^{5} +1.00000i q^{7} +O(q^{10})\)	\(q+2.99711 q^{5} +1.00000i q^{7} -5.36291i q^{11} -4.55519i q^{13} +0.958700i q^{17} -0.532227 q^{19} -2.78680 q^{23} +3.98265 q^{25} +5.15616 q^{29} -4.66569i q^{31} +2.99711i q^{35} +6.10293i q^{37} -12.3255i q^{41} -4.45583 q^{43} -1.40285 q^{47} -1.00000 q^{49} +9.57019 q^{53} -16.0732i q^{55} -0.0356973i q^{59} +5.13661i q^{61} -13.6524i q^{65} -8.59546 q^{67} -13.3269 q^{71} +6.50808 q^{73} +5.36291 q^{77} +1.90581i q^{79} -3.13371i q^{83} +2.87332i q^{85} +5.91323i q^{89} +4.55519 q^{91} -1.59514 q^{95} -16.0057 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q+O(q^{10}) \) 48 * q	\( 48 q - 16 q^{19} + 48 q^{25} - 64 q^{43} - 48 q^{49} - 32 q^{67}+O(q^{100}) \) 48 * q - 16 * q^19 + 48 * q^25 - 64 * q^43 - 48 * q^49 - 32 * q^67

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more