LMFDB - Embedded newform 2736.2.d.b.2015.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2736,2,Mod(2015,2736)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2736, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 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("2736.2015");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2736 = 2^{4} \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2736.d (of order $2$, degree $1$, 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:	$21.8470699930$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2015.1
Character	$\chi$	$=$	2736.2015
Dual form			2736.2.d.b.2015.24

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.95877i q^{5} -0.569970i q^{7} +O(q^{10})$	$q-2.95877i q^{5} -0.569970i q^{7} +1.70925 q^{11} +3.55382 q^{13} -4.91049i q^{17} -1.00000i q^{19} +8.27966 q^{23} -3.75430 q^{25} -2.39518i q^{29} +4.25776i q^{31} -1.68641 q^{35} -5.95478 q^{37} +3.11214i q^{41} +3.81744i q^{43} -5.85841 q^{47} +6.67513 q^{49} -4.30890i q^{53} -5.05728i q^{55} +2.13451 q^{59} +4.19721 q^{61} -10.5149i q^{65} -12.3687i q^{67} +12.7533 q^{71} -10.8387 q^{73} -0.974222i q^{77} +2.29006i q^{79} +1.73885 q^{83} -14.5290 q^{85} -14.6261i q^{89} -2.02557i q^{91} -2.95877 q^{95} -5.53416 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q+O(q^{10}) $ 24 * q	$ 24 q + 8 q^{25} - 32 q^{37} - 32 q^{49} + 8 q^{73} + 40 q^{85} + 16 q^{97}+O(q^{100}) $ 24 * q + 8 * q^25 - 32 * q^37 - 32 * q^49 + 8 * q^73 + 40 * q^85 + 16 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2736\mathbb{Z}\right)^\times$.

$n$	$1009$	$1217$	$1711$	$2053$
$\chi(n)$	$1$	$-1$	$-1$	$1$

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
$5$	− 2.95877i	− 1.32320i	−0.749857	−	0.661600i	$-0.769878\pi$
$5$	− 2.95877i	− 1.32320i	0.749857	−	0.661600i	$-0.230122\pi$
$6$	0	0
$7$	− 0.569970i	− 0.215428i	−0.994182	−	0.107714i	$-0.965647\pi$
$7$	− 0.569970i	− 0.215428i	0.994182	−	0.107714i	$-0.0343531\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	1.70925	0.515359	0.257679	−	0.966230i	$-0.417042\pi$
$11$	1.70925	0.515359	0.257679	+	0.966230i	$0.417042\pi$
$12$	0	0
$13$	3.55382	0.985651	0.492826	−	0.870128i	$-0.335964\pi$
$13$	3.55382	0.985651	0.492826	+	0.870128i	$0.335964\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 4.91049i	− 1.19097i	−0.803366	−	0.595485i	$-0.796960\pi$
$17$	− 4.91049i	− 1.19097i	0.803366	−	0.595485i	$-0.203040\pi$
$18$	0	0
$19$	− 1.00000i	− 0.229416i
$19$	− 1.00000i	− 0.229416i
$20$	0	0
$21$	0	0
$22$	0	0
$23$	8.27966	1.72643	0.863214	−	0.504838i	$-0.168448\pi$
$23$	8.27966	1.72643	0.863214	+	0.504838i	$0.168448\pi$
$24$	0	0
$25$	−3.75430	−0.750859
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 2.39518i	− 0.444775i	−0.974958	−	0.222387i	$-0.928615\pi$
$29$	− 2.39518i	− 0.444775i	0.974958	−	0.222387i	$-0.0713849\pi$
$30$	0	0
$31$	4.25776i	0.764716i	0.924014	+	0.382358i	$0.124888\pi$
$31$	4.25776i	0.764716i	−0.924014	+	0.382358i	$0.875112\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−1.68641	−0.285055
$36$	0	0
$37$	−5.95478	−0.978959	−0.489480	−	0.872015i	$-0.662813\pi$
$37$	−5.95478	−0.978959	−0.489480	+	0.872015i	$0.662813\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	3.11214i	0.486034i	0.970022	+	0.243017i	$0.0781371\pi$
$41$	3.11214i	0.486034i	−0.970022	+	0.243017i	$0.921863\pi$
$42$	0	0
$43$	3.81744i	0.582154i	0.956700	+	0.291077i	$0.0940135\pi$
$43$	3.81744i	0.582154i	−0.956700	+	0.291077i	$0.905986\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−5.85841	−0.854537	−0.427268	−	0.904125i	$-0.640524\pi$
$47$	−5.85841	−0.854537	−0.427268	+	0.904125i	$0.640524\pi$
$48$	0	0
$49$	6.67513	0.953591
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 4.30890i	− 0.591873i	−0.955208	−	0.295937i	$-0.904368\pi$
$53$	− 4.30890i	− 0.591873i	0.955208	−	0.295937i	$-0.0956317\pi$
$54$	0	0
$55$	− 5.05728i	− 0.681923i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	2.13451	0.277889	0.138945	−	0.990300i	$-0.455629\pi$
$59$	2.13451	0.277889	0.138945	+	0.990300i	$0.455629\pi$
$60$	0	0
$61$	4.19721	0.537398	0.268699	−	0.963224i	$-0.413406\pi$
$61$	4.19721	0.537398	0.268699	+	0.963224i	$0.413406\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 10.5149i	− 1.30421i
$66$	0	0
$67$	− 12.3687i	− 1.51107i	−0.655107	−	0.755536i	$-0.727377\pi$
$67$	− 12.3687i	− 1.51107i	0.655107	−	0.755536i	$-0.272623\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	12.7533	1.51353	0.756767	−	0.653685i	$-0.226778\pi$
$71$	12.7533	1.51353	0.756767	+	0.653685i	$0.226778\pi$
$72$	0	0
$73$	−10.8387	−1.26858	−0.634288	−	0.773097i	$-0.718707\pi$
$73$	−10.8387	−1.26858	−0.634288	+	0.773097i	$0.718707\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 0.974222i	− 0.111023i
$78$	0	0
$79$	2.29006i	0.257652i	0.991667	+	0.128826i	$0.0411209\pi$
$79$	2.29006i	0.257652i	−0.991667	+	0.128826i	$0.958879\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	1.73885	0.190863	0.0954316	−	0.995436i	$-0.469577\pi$
$83$	1.73885	0.190863	0.0954316	+	0.995436i	$0.469577\pi$
$84$	0	0
$85$	−14.5290	−1.57589
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 14.6261i	− 1.55036i	−0.631741	−	0.775179i	$-0.717659\pi$
$89$	− 14.6261i	− 1.55036i	0.631741	−	0.775179i	$-0.282341\pi$
$90$	0	0
$91$	− 2.02557i	− 0.212337i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−2.95877	−0.303563
$96$	0	0
$97$	−5.53416	−0.561909	−0.280954	−	0.959721i	$-0.590651\pi$
$97$	−5.53416	−0.561909	−0.280954	+	0.959721i	$0.590651\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	2.79060i	0.277675i	0.990315	+	0.138838i	$0.0443366\pi$
$101$	2.79060i	0.277675i	−0.990315	+	0.138838i	$0.955663\pi$
$102$	0	0
$103$	7.21853i	0.711263i	0.934626	+	0.355631i	$0.115734\pi$
$103$	7.21853i	0.711263i	−0.934626	+	0.355631i	$0.884266\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	8.58913	0.830342	0.415171	−	0.909743i	$-0.363722\pi$
$107$	8.58913	0.830342	0.415171	+	0.909743i	$0.363722\pi$
$108$	0	0
$109$	−17.2842	−1.65553	−0.827763	−	0.561077i	$-0.810387\pi$
$109$	−17.2842	−1.65553	−0.827763	+	0.561077i	$0.810387\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	2.52206i	0.237255i	0.992939	+	0.118628i	$0.0378495\pi$
$113$	2.52206i	0.237255i	−0.992939	+	0.118628i	$0.962150\pi$
$114$	0	0
$115$	− 24.4976i	− 2.28441i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−2.79883	−0.256569
$120$	0	0
$121$	−8.07846	−0.734405
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 3.68574i	− 0.329663i
$126$	0	0
$127$	9.49347i	0.842410i	0.906966	+	0.421205i	$0.138393\pi$
$127$	9.49347i	0.842410i	−0.906966	+	0.421205i	$0.861607\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−5.47277	−0.478159	−0.239079	−	0.971000i	$-0.576846\pi$
$131$	−5.47277	−0.478159	−0.239079	+	0.971000i	$0.576846\pi$
$132$	0	0
$133$	−0.569970	−0.0494226
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 10.6155i	− 0.906946i	−0.891270	−	0.453473i	$-0.850185\pi$
$137$	− 10.6155i	− 0.906946i	0.891270	−	0.453473i	$-0.149815\pi$
$138$	0	0
$139$	− 20.6801i	− 1.75407i	−0.480430	−	0.877033i	$-0.659519\pi$
$139$	− 20.6801i	− 1.75407i	0.480430	−	0.877033i	$-0.340481\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	6.07437	0.507964
$144$	0	0
$145$	−7.08679	−0.588526
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 2.11655i	− 0.173395i	−0.996235	−	0.0866973i	$-0.972369\pi$
$149$	− 2.11655i	− 0.173395i	0.996235	−	0.0866973i	$-0.0276313\pi$
$150$	0	0
$151$	11.8036i	0.960567i	0.877113	+	0.480284i	$0.159466\pi$
$151$	11.8036i	0.960567i	−0.877113	+	0.480284i	$0.840534\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	12.5977	1.01187
$156$	0	0
$157$	−15.4017	−1.22919	−0.614594	−	0.788843i	$-0.710680\pi$
$157$	−15.4017	−1.22919	−0.614594	+	0.788843i	$0.710680\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 4.71915i	− 0.371921i
$162$	0	0
$163$	− 17.2526i	− 1.35133i	−0.737209	−	0.675665i	$-0.763857\pi$
$163$	− 17.2526i	− 1.35133i	0.737209	−	0.675665i	$-0.236143\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−23.0656	−1.78487	−0.892437	−	0.451173i	$-0.851006\pi$
$167$	−23.0656	−1.78487	−0.892437	+	0.451173i	$0.851006\pi$
$168$	0	0
$169$	−0.370392	−0.0284917
$170$	0	0
$171$	0	0
$172$	0	0
$173$	3.81082i	0.289731i	0.989451	+	0.144866i	$0.0462750\pi$
$173$	3.81082i	0.289731i	−0.989451	+	0.144866i	$0.953725\pi$
$174$	0	0
$175$	2.13984i	0.161756i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	3.85047	0.287798	0.143899	−	0.989592i	$-0.454036\pi$
$179$	3.85047	0.287798	0.143899	+	0.989592i	$0.454036\pi$
$180$	0	0
$181$	16.3304	1.21383	0.606916	−	0.794766i	$-0.292406\pi$
$181$	16.3304	1.21383	0.606916	+	0.794766i	$0.292406\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	17.6188i	1.29536i
$186$	0	0
$187$	− 8.39327i	− 0.613777i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−0.711877	−0.0515096	−0.0257548	−	0.999668i	$-0.508199\pi$
$191$	−0.711877	−0.0515096	−0.0257548	+	0.999668i	$0.508199\pi$
$192$	0	0
$193$	−5.08186	−0.365800	−0.182900	−	0.983132i	$-0.558548\pi$
$193$	−5.08186	−0.365800	−0.182900	+	0.983132i	$0.558548\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 11.3093i	− 0.805757i	−0.915254	−	0.402878i	$-0.868010\pi$
$197$	− 11.3093i	− 0.805757i	0.915254	−	0.402878i	$-0.131990\pi$
$198$	0	0
$199$	− 16.5428i	− 1.17269i	−0.810062	−	0.586345i	$-0.800566\pi$
$199$	− 16.5428i	− 1.17269i	0.810062	−	0.586345i	$-0.199434\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−1.36518	−0.0958170
$204$	0	0
$205$	9.20808	0.643120
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 1.70925i	− 0.118231i
$210$	0	0
$211$	13.6026i	0.936439i	0.883612	+	0.468220i	$0.155104\pi$
$211$	13.6026i	0.936439i	−0.883612	+	0.468220i	$0.844896\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	11.2949	0.770306
$216$	0	0
$217$	2.42679	0.164741
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 17.4510i	− 1.17388i
$222$	0	0
$223$	24.3517i	1.63071i	0.578960	+	0.815356i	$0.303459\pi$
$223$	24.3517i	1.63071i	−0.578960	+	0.815356i	$0.696541\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	24.7382	1.64193	0.820967	−	0.570976i	$-0.193435\pi$
$227$	24.7382	1.64193	0.820967	+	0.570976i	$0.193435\pi$
$228$	0	0
$229$	8.04362	0.531537	0.265769	−	0.964037i	$-0.414374\pi$
$229$	8.04362	0.531537	0.265769	+	0.964037i	$0.414374\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	13.8280i	0.905901i	0.891536	+	0.452951i	$0.149629\pi$
$233$	13.8280i	0.905901i	−0.891536	+	0.452951i	$0.850371\pi$
$234$	0	0
$235$	17.3337i	1.13072i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	1.99484	0.129036	0.0645178	−	0.997917i	$-0.479449\pi$
$239$	1.99484	0.129036	0.0645178	+	0.997917i	$0.479449\pi$
$240$	0	0
$241$	6.12602	0.394611	0.197306	−	0.980342i	$-0.436781\pi$
$241$	6.12602	0.394611	0.197306	+	0.980342i	$0.436781\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 19.7502i	− 1.26179i
$246$	0	0
$247$	− 3.55382i	− 0.226124i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	8.97473	0.566480	0.283240	−	0.959049i	$-0.408591\pi$
$251$	8.97473	0.566480	0.283240	+	0.959049i	$0.408591\pi$
$252$	0	0
$253$	14.1520	0.889730
$254$	0	0
$255$	0	0
$256$	0	0
$257$	9.65103i	0.602015i	0.953622	+	0.301007i	$0.0973229\pi$
$257$	9.65103i	0.602015i	−0.953622	+	0.301007i	$0.902677\pi$
$258$	0	0
$259$	3.39404i	0.210896i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	2.74147	0.169046	0.0845231	−	0.996422i	$-0.473063\pi$
$263$	2.74147	0.169046	0.0845231	+	0.996422i	$0.473063\pi$
$264$	0	0
$265$	−12.7490	−0.783167
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 15.8606i	− 0.967040i	−0.875333	−	0.483520i	$-0.839358\pi$
$269$	− 15.8606i	− 0.967040i	0.875333	−	0.483520i	$-0.160642\pi$
$270$	0	0
$271$	10.0169i	0.608484i	0.952595	+	0.304242i	$0.0984031\pi$
$271$	10.0169i	0.608484i	−0.952595	+	0.304242i	$0.901597\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−6.41704	−0.386962
$276$	0	0
$277$	1.21067	0.0727419	0.0363709	−	0.999338i	$-0.488420\pi$
$277$	1.21067	0.0727419	0.0363709	+	0.999338i	$0.488420\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 29.4526i	− 1.75699i	−0.477748	−	0.878497i	$-0.658547\pi$
$281$	− 29.4526i	− 1.75699i	0.477748	−	0.878497i	$-0.341453\pi$
$282$	0	0
$283$	17.8603i	1.06168i	0.847471	+	0.530841i	$0.178124\pi$
$283$	17.8603i	1.06168i	−0.847471	+	0.530841i	$0.821876\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	1.77382	0.104705
$288$	0	0
$289$	−7.11296	−0.418409
$290$	0	0
$291$	0	0
$292$	0	0
$293$	8.61432i	0.503254i	0.967824	+	0.251627i	$0.0809656\pi$
$293$	8.61432i	0.503254i	−0.967824	+	0.251627i	$0.919034\pi$
$294$	0	0
$295$	− 6.31551i	− 0.367703i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	29.4244	1.70166
$300$	0	0
$301$	2.17582	0.125412
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 12.4186i	− 0.711086i
$306$	0	0
$307$	1.66899i	0.0952541i	0.998865	+	0.0476271i	$0.0151659\pi$
$307$	1.66899i	0.0952541i	−0.998865	+	0.0476271i	$0.984834\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−15.7976	−0.895802	−0.447901	−	0.894083i	$-0.647828\pi$
$311$	−15.7976	−0.895802	−0.447901	+	0.894083i	$0.647828\pi$
$312$	0	0
$313$	32.4916	1.83653	0.918267	−	0.395961i	$-0.129589\pi$
$313$	32.4916	1.83653	0.918267	+	0.395961i	$0.129589\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	9.48080i	0.532495i	0.963905	+	0.266247i	$0.0857838\pi$
$317$	9.48080i	0.532495i	−0.963905	+	0.266247i	$0.914216\pi$
$318$	0	0
$319$	− 4.09397i	− 0.229218i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−4.91049	−0.273227
$324$	0	0
$325$	−13.3421	−0.740085
$326$	0	0
$327$	0	0
$328$	0	0
$329$	3.33911i	0.184091i
$330$	0	0
$331$	− 10.2220i	− 0.561852i	−0.959729	−	0.280926i	$-0.909358\pi$
$331$	− 10.2220i	− 0.561852i	0.959729	−	0.280926i	$-0.0906416\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−36.5960	−1.99945
$336$	0	0
$337$	11.6998	0.637326	0.318663	−	0.947868i	$-0.396766\pi$
$337$	11.6998	0.637326	0.318663	+	0.947868i	$0.396766\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	7.27758i	0.394103i
$342$	0	0
$343$	− 7.79441i	− 0.420859i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	19.7600	1.06078	0.530388	−	0.847755i	$-0.322046\pi$
$347$	19.7600	1.06078	0.530388	+	0.847755i	$0.322046\pi$
$348$	0	0
$349$	2.61480	0.139967	0.0699836	−	0.997548i	$-0.477705\pi$
$349$	2.61480	0.139967	0.0699836	+	0.997548i	$0.477705\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	1.48062i	0.0788056i	0.999223	+	0.0394028i	$0.0125456\pi$
$353$	1.48062i	0.0788056i	−0.999223	+	0.0394028i	$0.987454\pi$
$354$	0	0
$355$	− 37.7339i	− 2.00271i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−29.0406	−1.53271	−0.766353	−	0.642420i	$-0.777931\pi$
$359$	−29.0406	−1.53271	−0.766353	+	0.642420i	$0.777931\pi$
$360$	0	0
$361$	−1.00000	−0.0526316
$362$	0	0
$363$	0	0
$364$	0	0
$365$	32.0693i	1.67858i
$366$	0	0
$367$	17.9185i	0.935337i	0.883904	+	0.467668i	$0.154906\pi$
$367$	17.9185i	0.935337i	−0.883904	+	0.467668i	$0.845094\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−2.45594	−0.127506
$372$	0	0
$373$	5.65432	0.292770	0.146385	−	0.989228i	$-0.453236\pi$
$373$	5.65432	0.292770	0.146385	+	0.989228i	$0.453236\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 8.51204i	− 0.438393i
$378$	0	0
$379$	31.7595i	1.63138i	0.578493	+	0.815688i	$0.303641\pi$
$379$	31.7595i	1.63138i	−0.578493	+	0.815688i	$0.696359\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−26.2692	−1.34229	−0.671146	−	0.741325i	$-0.734198\pi$
$383$	−26.2692	−1.34229	−0.671146	+	0.741325i	$0.734198\pi$
$384$	0	0
$385$	−2.88249	−0.146905
$386$	0	0
$387$	0	0
$388$	0	0
$389$	8.02764i	0.407017i	0.979073	+	0.203509i	$0.0652345\pi$
$389$	8.02764i	0.407017i	−0.979073	+	0.203509i	$0.934765\pi$
$390$	0	0
$391$	− 40.6572i	− 2.05612i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	6.77576	0.340926
$396$	0	0
$397$	3.24160	0.162691	0.0813457	−	0.996686i	$-0.474078\pi$
$397$	3.24160	0.162691	0.0813457	+	0.996686i	$0.474078\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	7.12034i	0.355573i	0.984069	+	0.177786i	$0.0568936\pi$
$401$	7.12034i	0.355573i	−0.984069	+	0.177786i	$0.943106\pi$
$402$	0	0
$403$	15.1313i	0.753743i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−10.1782	−0.504515
$408$	0	0
$409$	−8.49493	−0.420047	−0.210024	−	0.977696i	$-0.567354\pi$
$409$	−8.49493	−0.420047	−0.210024	+	0.977696i	$0.567354\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 1.21660i	− 0.0598652i
$414$	0	0
$415$	− 5.14484i	− 0.252550i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	3.98442	0.194652	0.0973258	−	0.995253i	$-0.468971\pi$
$419$	3.98442	0.194652	0.0973258	+	0.995253i	$0.468971\pi$
$420$	0	0
$421$	30.1475	1.46930	0.734650	−	0.678446i	$-0.237346\pi$
$421$	30.1475	1.46930	0.734650	+	0.678446i	$0.237346\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	18.4355i	0.894251i
$426$	0	0
$427$	− 2.39229i	− 0.115771i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	28.6558	1.38030	0.690152	−	0.723665i	$-0.257544\pi$
$431$	28.6558	1.38030	0.690152	+	0.723665i	$0.257544\pi$
$432$	0	0
$433$	11.7254	0.563487	0.281743	−	0.959490i	$-0.409087\pi$
$433$	11.7254	0.563487	0.281743	+	0.959490i	$0.409087\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 8.27966i	− 0.396070i
$438$	0	0
$439$	39.5668i	1.88842i	0.329343	+	0.944211i	$0.393173\pi$
$439$	39.5668i	1.88842i	−0.329343	+	0.944211i	$0.606827\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	20.5909	0.978303	0.489151	−	0.872199i	$-0.337307\pi$
$443$	20.5909	0.978303	0.489151	+	0.872199i	$0.337307\pi$
$444$	0	0
$445$	−43.2751	−2.05144
$446$	0	0
$447$	0	0
$448$	0	0
$449$	30.6253i	1.44530i	0.691217	+	0.722648i	$0.257075\pi$
$449$	30.6253i	1.44530i	−0.691217	+	0.722648i	$0.742925\pi$
$450$	0	0
$451$	5.31942i	0.250482i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−5.99318	−0.280965
$456$	0	0
$457$	32.4736	1.51905	0.759525	−	0.650478i	$-0.225431\pi$
$457$	32.4736	1.51905	0.759525	+	0.650478i	$0.225431\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	7.17109i	0.333991i	0.985958	+	0.166995i	$0.0534065\pi$
$461$	7.17109i	0.333991i	−0.985958	+	0.166995i	$0.946594\pi$
$462$	0	0
$463$	24.9553i	1.15977i	0.814699	+	0.579884i	$0.196902\pi$
$463$	24.9553i	1.15977i	−0.814699	+	0.579884i	$0.803098\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−36.4340	−1.68596	−0.842981	−	0.537943i	$-0.819202\pi$
$467$	−36.4340	−1.68596	−0.842981	+	0.537943i	$0.819202\pi$
$468$	0	0
$469$	−7.04976	−0.325528
$470$	0	0
$471$	0	0
$472$	0	0
$473$	6.52496i	0.300018i
$474$	0	0
$475$	3.75430i	0.172259i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−36.0402	−1.64672	−0.823360	−	0.567520i	$-0.807903\pi$
$479$	−36.0402	−1.64672	−0.823360	+	0.567520i	$0.807903\pi$
$480$	0	0
$481$	−21.1622	−0.964913
$482$	0	0
$483$	0	0
$484$	0	0
$485$	16.3743i	0.743518i
$486$	0	0
$487$	− 23.8944i	− 1.08276i	−0.840778	−	0.541380i	$-0.817902\pi$
$487$	− 23.8944i	− 1.08276i	0.840778	−	0.541380i	$-0.182098\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	0.453416	0.0204624	0.0102312	−	0.999948i	$-0.496743\pi$
$491$	0.453416	0.0204624	0.0102312	+	0.999948i	$0.496743\pi$
$492$	0	0
$493$	−11.7615	−0.529713
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 7.26897i	− 0.326058i
$498$	0	0
$499$	37.4719i	1.67747i	0.544539	+	0.838735i	$0.316705\pi$
$499$	37.4719i	1.67747i	−0.544539	+	0.838735i	$0.683295\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−37.9757	−1.69325	−0.846626	−	0.532188i	$-0.821370\pi$
$503$	−37.9757	−1.69325	−0.846626	+	0.532188i	$0.821370\pi$
$504$	0	0
$505$	8.25674	0.367420
$506$	0	0
$507$	0	0
$508$	0	0
$509$	41.8041i	1.85294i	0.376374	+	0.926468i	$0.377171\pi$
$509$	41.8041i	1.85294i	−0.376374	+	0.926468i	$0.622829\pi$
$510$	0	0
$511$	6.17774i	0.273287i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	21.3579	0.941143
$516$	0	0
$517$	−10.0135	−0.440393
$518$	0	0
$519$	0	0
$520$	0	0
$521$	30.4902i	1.33580i	0.744252	+	0.667899i	$0.232806\pi$
$521$	30.4902i	1.33580i	−0.744252	+	0.667899i	$0.767194\pi$
$522$	0	0
$523$	− 28.9130i	− 1.26428i	−0.774856	−	0.632138i	$-0.782178\pi$
$523$	− 28.9130i	− 1.26428i	0.774856	−	0.632138i	$-0.217822\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	20.9077	0.910753
$528$	0	0
$529$	45.5528	1.98055
$530$	0	0
$531$	0	0
$532$	0	0
$533$	11.0600i	0.479060i
$534$	0	0
$535$	− 25.4132i	− 1.09871i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	11.4095	0.491441
$540$	0	0
$541$	32.3339	1.39014	0.695072	−	0.718940i	$-0.255372\pi$
$541$	32.3339	1.39014	0.695072	+	0.718940i	$0.255372\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	51.1399i	2.19059i
$546$	0	0
$547$	− 8.92847i	− 0.381754i	−0.981614	−	0.190877i	$-0.938867\pi$
$547$	− 8.92847i	− 0.381754i	0.981614	−	0.190877i	$-0.0611331\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−2.39518	−0.102038
$552$	0	0
$553$	1.30527	0.0555056
$554$	0	0
$555$	0	0
$556$	0	0
$557$	0.828782i	0.0351166i	0.999846	+	0.0175583i	$0.00558927\pi$
$557$	0.828782i	0.0351166i	−0.999846	+	0.0175583i	$0.994411\pi$
$558$	0	0
$559$	13.5665i	0.573800i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−16.3635	−0.689638	−0.344819	−	0.938669i	$-0.612060\pi$
$563$	−16.3635	−0.689638	−0.344819	+	0.938669i	$0.612060\pi$
$564$	0	0
$565$	7.46218	0.313936
$566$	0	0
$567$	0	0
$568$	0	0
$569$	0.104329i	0.00437368i	0.999998	+	0.00218684i	$0.000696093\pi$
$569$	0.104329i	0.00437368i	−0.999998	+	0.00218684i	$0.999304\pi$
$570$	0	0
$571$	− 4.44526i	− 0.186028i	−0.995665	−	0.0930142i	$-0.970350\pi$
$571$	− 4.44526i	− 0.186028i	0.995665	−	0.0930142i	$-0.0296502\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−31.0843	−1.29630
$576$	0	0
$577$	−23.9811	−0.998346	−0.499173	−	0.866502i	$-0.666363\pi$
$577$	−23.9811	−0.998346	−0.499173	+	0.866502i	$0.666363\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 0.991089i	− 0.0411173i
$582$	0	0
$583$	− 7.36500i	− 0.305027i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−10.8155	−0.446401	−0.223201	−	0.974773i	$-0.571651\pi$
$587$	−10.8155	−0.446401	−0.223201	+	0.974773i	$0.571651\pi$
$588$	0	0
$589$	4.25776	0.175438
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 29.0703i	− 1.19377i	−0.802325	−	0.596887i	$-0.796404\pi$
$593$	− 29.0703i	− 1.19377i	0.802325	−	0.596887i	$-0.203596\pi$
$594$	0	0
$595$	8.28109i	0.339492i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	5.11836	0.209131	0.104565	−	0.994518i	$-0.466655\pi$
$599$	5.11836	0.209131	0.104565	+	0.994518i	$0.466655\pi$
$600$	0	0
$601$	−22.3642	−0.912254	−0.456127	−	0.889915i	$-0.650764\pi$
$601$	−22.3642	−0.912254	−0.456127	+	0.889915i	$0.650764\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	23.9023i	0.971765i
$606$	0	0
$607$	6.11562i	0.248226i	0.992268	+	0.124113i	$0.0396085\pi$
$607$	6.11562i	0.248226i	−0.992268	+	0.124113i	$0.960392\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−20.8197	−0.842275
$612$	0	0
$613$	−37.7179	−1.52341	−0.761707	−	0.647922i	$-0.775638\pi$
$613$	−37.7179	−1.52341	−0.761707	+	0.647922i	$0.775638\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 38.3799i	− 1.54512i	−0.634943	−	0.772559i	$-0.718976\pi$
$617$	− 38.3799i	− 1.54512i	0.634943	−	0.772559i	$-0.281024\pi$
$618$	0	0
$619$	− 5.86517i	− 0.235741i	−0.993029	−	0.117871i	$-0.962393\pi$
$619$	− 5.86517i	− 0.235741i	0.993029	−	0.117871i	$-0.0376068\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−8.33641	−0.333991
$624$	0	0
$625$	−29.6767	−1.18707
$626$	0	0
$627$	0	0
$628$	0	0
$629$	29.2409i	1.16591i
$630$	0	0
$631$	11.1860i	0.445307i	0.974898	+	0.222654i	$0.0714719\pi$
$631$	11.1860i	0.445307i	−0.974898	+	0.222654i	$0.928528\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	28.0890	1.11468
$636$	0	0
$637$	23.7222	0.939908
$638$	0	0
$639$	0	0
$640$	0	0
$641$	39.4688i	1.55892i	0.626450	+	0.779462i	$0.284507\pi$
$641$	39.4688i	1.55892i	−0.626450	+	0.779462i	$0.715493\pi$
$642$	0	0
$643$	− 37.3994i	− 1.47489i	−0.675409	−	0.737443i	$-0.736033\pi$
$643$	− 37.3994i	− 1.47489i	0.675409	−	0.737443i	$-0.263967\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−1.71980	−0.0676122	−0.0338061	−	0.999428i	$-0.510763\pi$
$647$	−1.71980	−0.0676122	−0.0338061	+	0.999428i	$0.510763\pi$
$648$	0	0
$649$	3.64841	0.143213
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 18.8547i	− 0.737840i	−0.929461	−	0.368920i	$-0.879728\pi$
$653$	− 18.8547i	− 0.737840i	0.929461	−	0.368920i	$-0.120272\pi$
$654$	0	0
$655$	16.1927i	0.632700i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	10.6461	0.414714	0.207357	−	0.978265i	$-0.433514\pi$
$659$	10.6461	0.414714	0.207357	+	0.978265i	$0.433514\pi$
$660$	0	0
$661$	13.9227	0.541532	0.270766	−	0.962645i	$-0.412723\pi$
$661$	13.9227	0.541532	0.270766	+	0.962645i	$0.412723\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	1.68641i	0.0653961i
$666$	0	0
$667$	− 19.8313i	− 0.767871i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	7.17410	0.276953
$672$	0	0
$673$	36.2932	1.39900	0.699500	−	0.714633i	$-0.253406\pi$
$673$	36.2932	1.39900	0.699500	+	0.714633i	$0.253406\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 7.72096i	− 0.296741i	−0.988932	−	0.148370i	$-0.952597\pi$
$677$	− 7.72096i	− 0.296741i	0.988932	−	0.148370i	$-0.0474028\pi$
$678$	0	0
$679$	3.15430i	0.121051i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−18.9531	−0.725220	−0.362610	−	0.931941i	$-0.618114\pi$
$683$	−18.9531	−0.725220	−0.362610	+	0.931941i	$0.618114\pi$
$684$	0	0
$685$	−31.4089	−1.20007
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 15.3131i	− 0.583381i
$690$	0	0
$691$	32.5480i	1.23818i	0.785319	+	0.619091i	$0.212499\pi$
$691$	32.5480i	1.23818i	−0.785319	+	0.619091i	$0.787501\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−61.1877	−2.32098
$696$	0	0
$697$	15.2821	0.578852
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 4.55361i	− 0.171987i	−0.996296	−	0.0859937i	$-0.972593\pi$
$701$	− 4.55361i	− 0.171987i	0.996296	−	0.0859937i	$-0.0274065\pi$
$702$	0	0
$703$	5.95478i	0.224589i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	1.59056	0.0598191
$708$	0	0
$709$	49.6482	1.86458	0.932289	−	0.361715i	$-0.117809\pi$
$709$	49.6482	1.86458	0.932289	+	0.361715i	$0.117809\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	35.2528i	1.32023i
$714$	0	0
$715$	− 17.9726i	− 0.672138i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	1.55893	0.0581381	0.0290691	−	0.999577i	$-0.490746\pi$
$719$	1.55893	0.0581381	0.0290691	+	0.999577i	$0.490746\pi$
$720$	0	0
$721$	4.11434	0.153226
$722$	0	0
$723$	0	0
$724$	0	0
$725$	8.99223i	0.333963i
$726$	0	0
$727$	39.8464i	1.47782i	0.673803	+	0.738911i	$0.264660\pi$
$727$	39.8464i	1.47782i	−0.673803	+	0.738911i	$0.735340\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	18.7455	0.693328
$732$	0	0
$733$	33.6913	1.24442	0.622209	−	0.782851i	$-0.286235\pi$
$733$	33.6913	1.24442	0.622209	+	0.782851i	$0.286235\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 21.1411i	− 0.778744i
$738$	0	0
$739$	6.29537i	0.231579i	0.993274	+	0.115789i	$0.0369398\pi$
$739$	6.29537i	0.231579i	−0.993274	+	0.115789i	$0.963060\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	46.2166	1.69552	0.847760	−	0.530379i	$-0.177950\pi$
$743$	46.2166	1.69552	0.847760	+	0.530379i	$0.177950\pi$
$744$	0	0
$745$	−6.26238	−0.229436
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 4.89554i	− 0.178879i
$750$	0	0
$751$	33.6850i	1.22918i	0.788846	+	0.614591i	$0.210679\pi$
$751$	33.6850i	1.22918i	−0.788846	+	0.614591i	$0.789321\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	34.9242	1.27102
$756$	0	0
$757$	35.5063	1.29050	0.645249	−	0.763973i	$-0.276754\pi$
$757$	35.5063	1.29050	0.645249	+	0.763973i	$0.276754\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	2.95585i	0.107150i	0.998564	+	0.0535748i	$0.0170616\pi$
$761$	2.95585i	0.107150i	−0.998564	+	0.0535748i	$0.982938\pi$
$762$	0	0
$763$	9.85147i	0.356647i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	7.58564	0.273902
$768$	0	0
$769$	−0.239047	−0.00862026	−0.00431013	−	0.999991i	$-0.501372\pi$
$769$	−0.239047	−0.00862026	−0.00431013	+	0.999991i	$0.501372\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	13.2405i	0.476229i	0.971237	+	0.238114i	$0.0765293\pi$
$773$	13.2405i	0.476229i	−0.971237	+	0.238114i	$0.923471\pi$
$774$	0	0
$775$	− 15.9849i	− 0.574194i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	3.11214	0.111504
$780$	0	0
$781$	21.7985	0.780013
$782$	0	0
$783$	0	0
$784$	0	0
$785$	45.5700i	1.62646i
$786$	0	0
$787$	12.9966i	0.463279i	0.972802	+	0.231640i	$0.0744090\pi$
$787$	12.9966i	0.463279i	−0.972802	+	0.231640i	$0.925591\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	1.43750	0.0511115
$792$	0	0
$793$	14.9161	0.529687
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 24.1127i	− 0.854114i	−0.904225	−	0.427057i	$-0.859550\pi$
$797$	− 24.1127i	− 0.854114i	0.904225	−	0.427057i	$-0.140450\pi$
$798$	0	0
$799$	28.7677i	1.01773i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−18.5261	−0.653772
$804$	0	0
$805$	−13.9629	−0.492127
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 10.5178i	− 0.369788i	−0.982758	−	0.184894i	$-0.940806\pi$
$809$	− 10.5178i	− 0.369788i	0.982758	−	0.184894i	$-0.0591942\pi$
$810$	0	0
$811$	− 25.3137i	− 0.888883i	−0.895808	−	0.444441i	$-0.853402\pi$
$811$	− 25.3137i	− 0.888883i	0.895808	−	0.444441i	$-0.146598\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−51.0465	−1.78808
$816$	0	0
$817$	3.81744	0.133555
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 18.8811i	− 0.658956i	−0.944163	−	0.329478i	$-0.893127\pi$
$821$	− 18.8811i	− 0.658956i	0.944163	−	0.329478i	$-0.106873\pi$
$822$	0	0
$823$	3.76306i	0.131172i	0.997847	+	0.0655860i	$0.0208917\pi$
$823$	3.76306i	0.131172i	−0.997847	+	0.0655860i	$0.979108\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	30.5208	1.06131	0.530656	−	0.847587i	$-0.321946\pi$
$827$	30.5208	1.06131	0.530656	+	0.847587i	$0.321946\pi$
$828$	0	0
$829$	46.1222	1.60189	0.800945	−	0.598738i	$-0.204331\pi$
$829$	46.1222	1.60189	0.800945	+	0.598738i	$0.204331\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 32.7782i	− 1.13570i
$834$	0	0
$835$	68.2459i	2.36175i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−30.1716	−1.04164	−0.520819	−	0.853667i	$-0.674374\pi$
$839$	−30.1716	−1.04164	−0.520819	+	0.853667i	$0.674374\pi$
$840$	0	0
$841$	23.2631	0.802176
$842$	0	0
$843$	0	0
$844$	0	0
$845$	1.09590i	0.0377002i
$846$	0	0
$847$	4.60448i	0.158212i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−49.3035	−1.69010
$852$	0	0
$853$	18.4674	0.632313	0.316156	−	0.948707i	$-0.397608\pi$
$853$	18.4674	0.632313	0.316156	+	0.948707i	$0.397608\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	23.9843i	0.819290i	0.912245	+	0.409645i	$0.134347\pi$
$857$	23.9843i	0.819290i	−0.912245	+	0.409645i	$0.865653\pi$
$858$	0	0
$859$	17.4158i	0.594219i	0.954843	+	0.297110i	$0.0960227\pi$
$859$	17.4158i	0.594219i	−0.954843	+	0.297110i	$0.903977\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−17.4424	−0.593746	−0.296873	−	0.954917i	$-0.595944\pi$
$863$	−17.4424	−0.593746	−0.296873	+	0.954917i	$0.595944\pi$
$864$	0	0
$865$	11.2753	0.383373
$866$	0	0
$867$	0	0
$868$	0	0
$869$	3.91429i	0.132783i
$870$	0	0
$871$	− 43.9559i	− 1.48939i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−2.10076	−0.0710187
$876$	0	0
$877$	−7.46831	−0.252187	−0.126093	−	0.992018i	$-0.540244\pi$
$877$	−7.46831	−0.252187	−0.126093	+	0.992018i	$0.540244\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	23.2123i	0.782042i	0.920382	+	0.391021i	$0.127878\pi$
$881$	23.2123i	0.782042i	−0.920382	+	0.391021i	$0.872122\pi$
$882$	0	0
$883$	− 36.7029i	− 1.23515i	−0.786512	−	0.617575i	$-0.788115\pi$
$883$	− 36.7029i	− 1.23515i	0.786512	−	0.617575i	$-0.211885\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	50.1244	1.68301	0.841507	−	0.540246i	$-0.181669\pi$
$887$	50.1244	1.68301	0.841507	+	0.540246i	$0.181669\pi$
$888$	0	0
$889$	5.41099	0.181479
$890$	0	0
$891$	0	0
$892$	0	0
$893$	5.85841i	0.196044i
$894$	0	0
$895$	− 11.3926i	− 0.380814i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	10.1981	0.340126
$900$	0	0
$901$	−21.1588	−0.704903
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 48.3179i	− 1.60614i
$906$	0	0
$907$	− 35.8688i	− 1.19101i	−0.803353	−	0.595503i	$-0.796953\pi$
$907$	− 35.8688i	− 1.19101i	0.803353	−	0.595503i	$-0.203047\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−43.0961	−1.42784	−0.713919	−	0.700229i	$-0.753081\pi$
$911$	−43.0961	−1.42784	−0.713919	+	0.700229i	$0.753081\pi$
$912$	0	0
$913$	2.97212	0.0983630
$914$	0	0
$915$	0	0
$916$	0	0
$917$	3.11932i	0.103009i
$918$	0	0
$919$	21.5024i	0.709300i	0.934999	+	0.354650i	$0.115400\pi$
$919$	21.5024i	0.709300i	−0.934999	+	0.354650i	$0.884600\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	45.3227	1.49182
$924$	0	0
$925$	22.3560	0.735061
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 19.3969i	− 0.636391i	−0.948025	−	0.318196i	$-0.896923\pi$
$929$	− 19.3969i	− 0.636391i	0.948025	−	0.318196i	$-0.103077\pi$
$930$	0	0
$931$	− 6.67513i	− 0.218769i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−24.8337	−0.812150
$936$	0	0
$937$	−38.4396	−1.25577	−0.627884	−	0.778307i	$-0.716079\pi$
$937$	−38.4396	−1.25577	−0.627884	+	0.778307i	$0.716079\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 46.5127i	− 1.51627i	−0.652098	−	0.758135i	$-0.726111\pi$
$941$	− 46.5127i	− 1.51627i	0.652098	−	0.758135i	$-0.273889\pi$
$942$	0	0
$943$	25.7674i	0.839103i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	43.8937	1.42635	0.713177	−	0.700984i	$-0.247256\pi$
$947$	43.8937	1.42635	0.713177	+	0.700984i	$0.247256\pi$
$948$	0	0
$949$	−38.5188	−1.25037
$950$	0	0
$951$	0	0
$952$	0	0
$953$	4.72133i	0.152939i	0.997072	+	0.0764694i	$0.0243648\pi$
$953$	4.72133i	0.152939i	−0.997072	+	0.0764694i	$0.975635\pi$
$954$	0	0
$955$	2.10628i	0.0681576i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−6.05053	−0.195382
$960$	0	0
$961$	12.8715	0.415210
$962$	0	0
$963$	0	0
$964$	0	0
$965$	15.0360i	0.484027i
$966$	0	0
$967$	33.2550i	1.06941i	0.845039	+	0.534704i	$0.179577\pi$
$967$	33.2550i	1.06941i	−0.845039	+	0.534704i	$0.820423\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	11.3041	0.362766	0.181383	−	0.983413i	$-0.441943\pi$
$971$	11.3041	0.362766	0.181383	+	0.983413i	$0.441943\pi$
$972$	0	0
$973$	−11.7871	−0.377875
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 28.7282i	− 0.919095i	−0.888153	−	0.459548i	$-0.848012\pi$
$977$	− 28.7282i	− 0.919095i	0.888153	−	0.459548i	$-0.151988\pi$
$978$	0	0
$979$	− 24.9996i	− 0.798991i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	28.4309	0.906806	0.453403	−	0.891306i	$-0.350210\pi$
$983$	28.4309	0.906806	0.453403	+	0.891306i	$0.350210\pi$
$984$	0	0
$985$	−33.4617	−1.06618
$986$	0	0
$987$	0	0
$988$	0	0
$989$	31.6071i	1.00505i
$990$	0	0
$991$	6.74590i	0.214290i	0.994243	+	0.107145i	$0.0341710\pi$
$991$	6.74590i	0.214290i	−0.994243	+	0.107145i	$0.965829\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−48.9463	−1.55170
$996$	0	0
$997$	−56.9248	−1.80283	−0.901414	−	0.432959i	$-0.857469\pi$
$997$	−56.9248	−1.80283	−0.901414	+	0.432959i	$0.857469\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2736.2.d.b.2015.1	✓	24
3.2	odd	2	inner	2736.2.d.b.2015.23	yes	24
4.3	odd	2	inner	2736.2.d.b.2015.2	yes	24
12.11	even	2	inner	2736.2.d.b.2015.24	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2736.2.d.b.2015.1	✓	24	1.1	even	1	trivial
2736.2.d.b.2015.2	yes	24	4.3	odd	2	inner
2736.2.d.b.2015.23	yes	24	3.2	odd	2	inner
2736.2.d.b.2015.24	yes	24	12.11	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:	\( N \)	\(=\)	\( 2736 = 2^{4} \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2736.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.95877i q^{5} -0.569970i q^{7} +O(q^{10})\)	\(q-2.95877i q^{5} -0.569970i q^{7} +1.70925 q^{11} +3.55382 q^{13} -4.91049i q^{17} -1.00000i q^{19} +8.27966 q^{23} -3.75430 q^{25} -2.39518i q^{29} +4.25776i q^{31} -1.68641 q^{35} -5.95478 q^{37} +3.11214i q^{41} +3.81744i q^{43} -5.85841 q^{47} +6.67513 q^{49} -4.30890i q^{53} -5.05728i q^{55} +2.13451 q^{59} +4.19721 q^{61} -10.5149i q^{65} -12.3687i q^{67} +12.7533 q^{71} -10.8387 q^{73} -0.974222i q^{77} +2.29006i q^{79} +1.73885 q^{83} -14.5290 q^{85} -14.6261i q^{89} -2.02557i q^{91} -2.95877 q^{95} -5.53416 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q+O(q^{10}) \) 24 * q	\( 24 q + 8 q^{25} - 32 q^{37} - 32 q^{49} + 8 q^{73} + 40 q^{85} + 16 q^{97}+O(q^{100}) \) 24 * q + 8 * q^25 - 32 * q^37 - 32 * q^49 + 8 * q^73 + 40 * q^85 + 16 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more