LMFDB - Embedded newform 2736.3.m.e.1711.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2736,3,Mod(1711,2736)]

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, 0, 0]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("2736.1711");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 2736 = 2^{4} \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	2736.m (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:	$74.5506003290$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 2 x^{11} - 2 x^{10} - 28 x^{9} - 400 x^{8} - 520 x^{7} + 17067 x^{6} - 3250 x^{5} + \cdots + 2052928 $ x^12 - 2x^11 - 2x^10 - 28x^9 - 400x^8 - 520x^7 + 17067x^6 - 3250x^5 - 195494x^4 + 302996x^3 + 602332x^2 - 2263536*x + 2052928
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{16} $
Twist minimal:	no (minimal twist has level 912)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1711.12
Root			$0.198491 - 5.66831i$ of defining polynomial
Character	$\chi$	$=$	2736.1711
Dual form			2736.3.m.e.1711.11

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+6.54540 q^{5} +0.687253i q^{7} +O(q^{10})$	$q+6.54540 q^{5} +0.687253i q^{7} -12.1735i q^{11} -8.29683 q^{13} +13.7973 q^{17} +4.35890i q^{19} +42.5295i q^{23} +17.8422 q^{25} -53.4629 q^{29} +36.9961i q^{31} +4.49835i q^{35} +24.7234 q^{37} +79.9793 q^{41} -4.56780i q^{43} +78.0044i q^{47} +48.5277 q^{49} +13.8557 q^{53} -79.6806i q^{55} +55.3048i q^{59} +43.2762 q^{61} -54.3061 q^{65} -112.524i q^{67} -8.14089i q^{71} +35.2543 q^{73} +8.36630 q^{77} +4.57702i q^{79} +50.9666i q^{83} +90.3090 q^{85} +107.305 q^{89} -5.70203i q^{91} +28.5307i q^{95} +98.4723 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 10 q^{5}+O(q^{10}) $ 12 * q + 10 * q^5	$ 12 q + 10 q^{5} + 36 q^{13} - 14 q^{17} + 10 q^{25} + 108 q^{29} - 16 q^{37} - 16 q^{41} - 118 q^{49} - 220 q^{53} + 366 q^{61} - 140 q^{65} - 158 q^{73} + 286 q^{77} + 98 q^{85} + 396 q^{89} + 224 q^{97}+O(q^{100}) $ 12 * q + 10 * q^5 + 36 * q^13 - 14 * q^17 + 10 * q^25 + 108 * q^29 - 16 * q^37 - 16 * q^41 - 118 * q^49 - 220 * q^53 + 366 * q^61 - 140 * q^65 - 158 * q^73 + 286 * q^77 + 98 * q^85 + 396 * q^89 + 224 * 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
$5$	6.54540	1.30908	0.654540	−	0.756028i	$-0.272862\pi$
$5$	6.54540	1.30908	0.654540	+	0.756028i	$0.272862\pi$
$6$	0	0
$7$	0.687253i	0.0981791i	0.998794	+	0.0490895i	$0.0156320\pi$
$7$	0.687253i	0.0981791i	−0.998794	+	0.0490895i	$0.984368\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 12.1735i	− 1.10669i	−0.832954	−	0.553343i	$-0.813352\pi$
$11$	− 12.1735i	− 1.10669i	0.832954	−	0.553343i	$-0.186648\pi$
$12$	0	0
$13$	−8.29683	−0.638218	−0.319109	−	0.947718i	$-0.603384\pi$
$13$	−8.29683	−0.638218	−0.319109	+	0.947718i	$0.603384\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	13.7973	0.811608	0.405804	−	0.913960i	$-0.366992\pi$
$17$	13.7973	0.811608	0.405804	+	0.913960i	$0.366992\pi$
$18$	0	0
$19$	4.35890i	0.229416i
$19$	4.35890i	0.229416i
$20$	0	0
$21$	0	0
$22$	0	0
$23$	42.5295i	1.84911i	0.381053	+	0.924553i	$0.375562\pi$
$23$	42.5295i	1.84911i	−0.381053	+	0.924553i	$0.624438\pi$
$24$	0	0
$25$	17.8422	0.713689
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−53.4629	−1.84355	−0.921774	−	0.387729i	$-0.873260\pi$
$29$	−53.4629	−1.84355	−0.921774	+	0.387729i	$0.873260\pi$
$30$	0	0
$31$	36.9961i	1.19342i	0.802456	+	0.596711i	$0.203526\pi$
$31$	36.9961i	1.19342i	−0.802456	+	0.596711i	$0.796474\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	4.49835i	0.128524i
$36$	0	0
$37$	24.7234	0.668201	0.334101	−	0.942537i	$-0.391567\pi$
$37$	24.7234	0.668201	0.334101	+	0.942537i	$0.391567\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	79.9793	1.95071	0.975357	−	0.220633i	$-0.0708122\pi$
$41$	79.9793	1.95071	0.975357	+	0.220633i	$0.0708122\pi$
$42$	0	0
$43$	− 4.56780i	− 0.106228i	−0.998588	−	0.0531140i	$-0.983085\pi$
$43$	− 4.56780i	− 0.106228i	0.998588	−	0.0531140i	$-0.0169147\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	78.0044i	1.65967i	0.558010	+	0.829834i	$0.311565\pi$
$47$	78.0044i	1.65967i	−0.558010	+	0.829834i	$0.688435\pi$
$48$	0	0
$49$	48.5277	0.990361
$50$	0	0
$51$	0	0
$52$	0	0
$53$	13.8557	0.261428	0.130714	−	0.991420i	$-0.458273\pi$
$53$	13.8557	0.261428	0.130714	+	0.991420i	$0.458273\pi$
$54$	0	0
$55$	− 79.6806i	− 1.44874i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	55.3048i	0.937370i	0.883365	+	0.468685i	$0.155272\pi$
$59$	55.3048i	0.937370i	−0.883365	+	0.468685i	$0.844728\pi$
$60$	0	0
$61$	43.2762	0.709446	0.354723	−	0.934972i	$-0.384575\pi$
$61$	43.2762	0.709446	0.354723	+	0.934972i	$0.384575\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−54.3061	−0.835478
$66$	0	0
$67$	− 112.524i	− 1.67947i	−0.542997	−	0.839735i	$-0.682711\pi$
$67$	− 112.524i	− 1.67947i	0.542997	−	0.839735i	$-0.317289\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	− 8.14089i	− 0.114660i	−0.998355	−	0.0573302i	$-0.981741\pi$
$71$	− 8.14089i	− 0.114660i	0.998355	−	0.0573302i	$-0.0182588\pi$
$72$	0	0
$73$	35.2543	0.482936	0.241468	−	0.970409i	$-0.422371\pi$
$73$	35.2543	0.482936	0.241468	+	0.970409i	$0.422371\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	8.36630	0.108653
$78$	0	0
$79$	4.57702i	0.0579369i	0.999580	+	0.0289685i	$0.00922224\pi$
$79$	4.57702i	0.0579369i	−0.999580	+	0.0289685i	$0.990778\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	50.9666i	0.614056i	0.951700	+	0.307028i	$0.0993345\pi$
$83$	50.9666i	0.614056i	−0.951700	+	0.307028i	$0.900665\pi$
$84$	0	0
$85$	90.3090	1.06246
$86$	0	0
$87$	0	0
$88$	0	0
$89$	107.305	1.20567	0.602837	−	0.797865i	$-0.294037\pi$
$89$	107.305	1.20567	0.602837	+	0.797865i	$0.294037\pi$
$90$	0	0
$91$	− 5.70203i	− 0.0626596i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	28.5307i	0.300323i
$96$	0	0
$97$	98.4723	1.01518	0.507589	−	0.861599i	$-0.330537\pi$
$97$	98.4723	1.01518	0.507589	+	0.861599i	$0.330537\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−87.8281	−0.869585	−0.434792	−	0.900531i	$-0.643178\pi$
$101$	−87.8281	−0.869585	−0.434792	+	0.900531i	$0.643178\pi$
$102$	0	0
$103$	− 118.440i	− 1.14991i	−0.818186	−	0.574954i	$-0.805020\pi$
$103$	− 118.440i	− 1.14991i	0.818186	−	0.574954i	$-0.194980\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	52.0889i	0.486812i	0.969924	+	0.243406i	$0.0782648\pi$
$107$	52.0889i	0.486812i	−0.969924	+	0.243406i	$0.921735\pi$
$108$	0	0
$109$	189.534	1.73884	0.869421	−	0.494072i	$-0.164492\pi$
$109$	189.534	1.73884	0.869421	+	0.494072i	$0.164492\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	146.140	1.29327	0.646637	−	0.762798i	$-0.276175\pi$
$113$	146.140	1.29327	0.646637	+	0.762798i	$0.276175\pi$
$114$	0	0
$115$	278.372i	2.42063i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	9.48226i	0.0796829i
$120$	0	0
$121$	−27.1950	−0.224752
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−46.8505	−0.374804
$126$	0	0
$127$	− 32.1666i	− 0.253281i	−0.991949	−	0.126640i	$-0.959581\pi$
$127$	− 32.1666i	− 0.253281i	0.991949	−	0.126640i	$-0.0404194\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 105.773i	− 0.807424i	−0.914886	−	0.403712i	$-0.867720\pi$
$131$	− 105.773i	− 0.807424i	0.914886	−	0.403712i	$-0.132280\pi$
$132$	0	0
$133$	−2.99567	−0.0225238
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−124.502	−0.908771	−0.454385	−	0.890805i	$-0.650141\pi$
$137$	−124.502	−0.908771	−0.454385	+	0.890805i	$0.650141\pi$
$138$	0	0
$139$	232.518i	1.67279i	0.548124	+	0.836397i	$0.315342\pi$
$139$	232.518i	1.67279i	−0.548124	+	0.836397i	$0.684658\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	101.002i	0.706306i
$144$	0	0
$145$	−349.936	−2.41335
$146$	0	0
$147$	0	0
$148$	0	0
$149$	148.922	0.999476	0.499738	−	0.866177i	$-0.333430\pi$
$149$	148.922	0.999476	0.499738	+	0.866177i	$0.333430\pi$
$150$	0	0
$151$	− 194.970i	− 1.29119i	−0.763678	−	0.645597i	$-0.776609\pi$
$151$	− 194.970i	− 1.29119i	0.763678	−	0.645597i	$-0.223391\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	242.154i	1.56229i
$156$	0	0
$157$	−181.166	−1.15392	−0.576961	−	0.816771i	$-0.695762\pi$
$157$	−181.166	−1.15392	−0.576961	+	0.816771i	$0.695762\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−29.2285	−0.181544
$162$	0	0
$163$	− 100.102i	− 0.614123i	−0.951690	−	0.307062i	$-0.900654\pi$
$163$	− 100.102i	− 0.614123i	0.951690	−	0.307062i	$-0.0993458\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 124.012i	− 0.742590i	−0.928515	−	0.371295i	$-0.878914\pi$
$167$	− 124.012i	− 0.742590i	0.928515	−	0.371295i	$-0.121086\pi$
$168$	0	0
$169$	−100.163	−0.592678
$170$	0	0
$171$	0	0
$172$	0	0
$173$	73.6796	0.425894	0.212947	−	0.977064i	$-0.431694\pi$
$173$	73.6796	0.425894	0.212947	+	0.977064i	$0.431694\pi$
$174$	0	0
$175$	12.2621i	0.0700693i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 96.3992i	− 0.538543i	−0.963064	−	0.269272i	$-0.913217\pi$
$179$	− 96.3992i	− 0.538543i	0.963064	−	0.269272i	$-0.0867829\pi$
$180$	0	0
$181$	−301.712	−1.66692	−0.833460	−	0.552580i	$-0.813643\pi$
$181$	−301.712	−1.66692	−0.833460	+	0.552580i	$0.813643\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	161.825	0.874728
$186$	0	0
$187$	− 167.962i	− 0.898194i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	195.895i	1.02563i	0.858500	+	0.512814i	$0.171397\pi$
$191$	195.895i	1.02563i	−0.858500	+	0.512814i	$0.828603\pi$
$192$	0	0
$193$	233.733	1.21105	0.605527	−	0.795825i	$-0.292962\pi$
$193$	233.733	1.21105	0.605527	+	0.795825i	$0.292962\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−32.3282	−0.164103	−0.0820514	−	0.996628i	$-0.526147\pi$
$197$	−32.3282	−0.164103	−0.0820514	+	0.996628i	$0.526147\pi$
$198$	0	0
$199$	191.303i	0.961324i	0.876906	+	0.480662i	$0.159604\pi$
$199$	191.303i	0.961324i	−0.876906	+	0.480662i	$0.840396\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 36.7425i	− 0.180998i
$204$	0	0
$205$	523.496	2.55364
$206$	0	0
$207$	0	0
$208$	0	0
$209$	53.0632	0.253891
$210$	0	0
$211$	− 196.983i	− 0.933569i	−0.884371	−	0.466785i	$-0.845412\pi$
$211$	− 196.983i	− 0.933569i	0.884371	−	0.466785i	$-0.154588\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	− 29.8981i	− 0.139061i
$216$	0	0
$217$	−25.4257	−0.117169
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−114.474	−0.517982
$222$	0	0
$223$	− 27.4910i	− 0.123278i	−0.998099	−	0.0616390i	$-0.980367\pi$
$223$	− 27.4910i	− 0.123278i	0.998099	−	0.0616390i	$-0.0196327\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	79.3094i	0.349381i	0.984623	+	0.174690i	$0.0558925\pi$
$227$	79.3094i	0.349381i	−0.984623	+	0.174690i	$0.944108\pi$
$228$	0	0
$229$	−228.094	−0.996042	−0.498021	−	0.867165i	$-0.665940\pi$
$229$	−228.094	−0.996042	−0.498021	+	0.867165i	$0.665940\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	309.291	1.32743	0.663715	−	0.747986i	$-0.268979\pi$
$233$	309.291	1.32743	0.663715	+	0.747986i	$0.268979\pi$
$234$	0	0
$235$	510.570i	2.17264i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	118.175i	0.494455i	0.968957	+	0.247228i	$0.0795196\pi$
$239$	118.175i	0.494455i	−0.968957	+	0.247228i	$0.920480\pi$
$240$	0	0
$241$	−165.526	−0.686830	−0.343415	−	0.939184i	$-0.611584\pi$
$241$	−165.526	−0.686830	−0.343415	+	0.939184i	$0.611584\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	317.633	1.29646
$246$	0	0
$247$	− 36.1650i	− 0.146417i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	231.646i	0.922894i	0.887168	+	0.461447i	$0.152669\pi$
$251$	231.646i	0.922894i	−0.887168	+	0.461447i	$0.847331\pi$
$252$	0	0
$253$	517.734	2.04638
$254$	0	0
$255$	0	0
$256$	0	0
$257$	446.625	1.73784	0.868921	−	0.494951i	$-0.164814\pi$
$257$	446.625	1.73784	0.868921	+	0.494951i	$0.164814\pi$
$258$	0	0
$259$	16.9913i	0.0656034i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 113.907i	− 0.433108i	−0.976271	−	0.216554i	$-0.930518\pi$
$263$	− 113.907i	− 0.433108i	0.976271	−	0.216554i	$-0.0694818\pi$
$264$	0	0
$265$	90.6908	0.342229
$266$	0	0
$267$	0	0
$268$	0	0
$269$	286.587	1.06538	0.532689	−	0.846311i	$-0.321182\pi$
$269$	286.587	1.06538	0.532689	+	0.846311i	$0.321182\pi$
$270$	0	0
$271$	− 427.736i	− 1.57836i	−0.614160	−	0.789182i	$-0.710505\pi$
$271$	− 427.736i	− 1.57836i	0.614160	−	0.789182i	$-0.289495\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 217.203i	− 0.789829i
$276$	0	0
$277$	−280.168	−1.01144	−0.505719	−	0.862699i	$-0.668773\pi$
$277$	−280.168	−1.01144	−0.505719	+	0.862699i	$0.668773\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	186.051	0.662104	0.331052	−	0.943613i	$-0.392597\pi$
$281$	186.051	0.662104	0.331052	+	0.943613i	$0.392597\pi$
$282$	0	0
$283$	399.732i	1.41248i	0.707972	+	0.706240i	$0.249610\pi$
$283$	399.732i	1.41248i	−0.707972	+	0.706240i	$0.750390\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	54.9660i	0.191519i
$288$	0	0
$289$	−98.6337	−0.341293
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−262.103	−0.894550	−0.447275	−	0.894397i	$-0.647605\pi$
$293$	−262.103	−0.894550	−0.447275	+	0.894397i	$0.647605\pi$
$294$	0	0
$295$	361.992i	1.22709i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 352.860i	− 1.18013i
$300$	0	0
$301$	3.13924	0.0104294
$302$	0	0
$303$	0	0
$304$	0	0
$305$	283.260	0.928721
$306$	0	0
$307$	48.7469i	0.158785i	0.996843	+	0.0793923i	$0.0252980\pi$
$307$	48.7469i	0.158785i	−0.996843	+	0.0793923i	$0.974702\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	− 162.551i	− 0.522670i	−0.965248	−	0.261335i	$-0.915837\pi$
$311$	− 162.551i	− 0.522670i	0.965248	−	0.261335i	$-0.0841628\pi$
$312$	0	0
$313$	−111.824	−0.357265	−0.178633	−	0.983916i	$-0.557167\pi$
$313$	−111.824	−0.357265	−0.178633	+	0.983916i	$0.557167\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	91.2694	0.287916	0.143958	−	0.989584i	$-0.454017\pi$
$317$	91.2694	0.287916	0.143958	+	0.989584i	$0.454017\pi$
$318$	0	0
$319$	650.832i	2.04023i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	60.1412i	0.186196i
$324$	0	0
$325$	−148.034	−0.455489
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−53.6088	−0.162945
$330$	0	0
$331$	99.7370i	0.301320i	0.988586	+	0.150660i	$0.0481399\pi$
$331$	99.7370i	0.301320i	−0.988586	+	0.150660i	$0.951860\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	− 736.517i	− 2.19856i
$336$	0	0
$337$	341.182	1.01241	0.506205	−	0.862413i	$-0.331048\pi$
$337$	341.182	1.01241	0.506205	+	0.862413i	$0.331048\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	450.373	1.32074
$342$	0	0
$343$	67.0262i	0.195412i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 184.889i	− 0.532823i	−0.963859	−	0.266411i	$-0.914162\pi$
$347$	− 184.889i	− 0.532823i	0.963859	−	0.266411i	$-0.0858379\pi$
$348$	0	0
$349$	673.919	1.93100	0.965500	−	0.260402i	$-0.0838552\pi$
$349$	673.919	1.93100	0.965500	+	0.260402i	$0.0838552\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−517.052	−1.46474	−0.732369	−	0.680908i	$-0.761585\pi$
$353$	−517.052	−1.46474	−0.732369	+	0.680908i	$0.761585\pi$
$354$	0	0
$355$	− 53.2854i	− 0.150100i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	662.385i	1.84508i	0.385896	+	0.922542i	$0.373892\pi$
$359$	662.385i	1.84508i	−0.385896	+	0.922542i	$0.626108\pi$
$360$	0	0
$361$	−19.0000	−0.0526316
$362$	0	0
$363$	0	0
$364$	0	0
$365$	230.753	0.632201
$366$	0	0
$367$	173.077i	0.471598i	0.971802	+	0.235799i	$0.0757708\pi$
$367$	173.077i	0.471598i	−0.971802	+	0.235799i	$0.924229\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	9.52235i	0.0256667i
$372$	0	0
$373$	506.294	1.35736	0.678679	−	0.734435i	$-0.262553\pi$
$373$	506.294	1.35736	0.678679	+	0.734435i	$0.262553\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	443.572	1.17658
$378$	0	0
$379$	205.418i	0.542000i	0.962579	+	0.271000i	$0.0873544\pi$
$379$	205.418i	0.542000i	−0.962579	+	0.271000i	$0.912646\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	80.2868i	0.209626i	0.994492	+	0.104813i	$0.0334244\pi$
$383$	80.2868i	0.209626i	−0.994492	+	0.104813i	$0.966576\pi$
$384$	0	0
$385$	54.7608	0.142236
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−93.7423	−0.240983	−0.120491	−	0.992714i	$-0.538447\pi$
$389$	−93.7423	−0.240983	−0.120491	+	0.992714i	$0.538447\pi$
$390$	0	0
$391$	586.793i	1.50075i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	29.9584i	0.0758441i
$396$	0	0
$397$	81.7947	0.206032	0.103016	−	0.994680i	$-0.467151\pi$
$397$	81.7947	0.206032	0.103016	+	0.994680i	$0.467151\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	125.065	0.311884	0.155942	−	0.987766i	$-0.450159\pi$
$401$	125.065	0.311884	0.155942	+	0.987766i	$0.450159\pi$
$402$	0	0
$403$	− 306.950i	− 0.761664i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 300.972i	− 0.739488i
$408$	0	0
$409$	−533.175	−1.30361	−0.651803	−	0.758389i	$-0.725987\pi$
$409$	−533.175	−1.30361	−0.651803	+	0.758389i	$0.725987\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−38.0084	−0.0920301
$414$	0	0
$415$	333.597i	0.803848i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	681.090i	1.62551i	0.582604	+	0.812756i	$0.302034\pi$
$419$	681.090i	1.62551i	−0.582604	+	0.812756i	$0.697966\pi$
$420$	0	0
$421$	−360.626	−0.856593	−0.428297	−	0.903638i	$-0.640886\pi$
$421$	−360.626	−0.856593	−0.428297	+	0.903638i	$0.640886\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	246.175	0.579236
$426$	0	0
$427$	29.7417i	0.0696527i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	611.707i	1.41927i	0.704568	+	0.709636i	$0.251141\pi$
$431$	611.707i	1.41927i	−0.704568	+	0.709636i	$0.748859\pi$
$432$	0	0
$433$	95.0757	0.219574	0.109787	−	0.993955i	$-0.464983\pi$
$433$	95.0757	0.219574	0.109787	+	0.993955i	$0.464983\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−185.382	−0.424214
$438$	0	0
$439$	− 40.7763i	− 0.0928845i	−0.998921	−	0.0464423i	$-0.985212\pi$
$439$	− 40.7763i	− 0.0928845i	0.998921	−	0.0464423i	$-0.0147883\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 372.478i	− 0.840808i	−0.907337	−	0.420404i	$-0.861888\pi$
$443$	− 372.478i	− 0.840808i	0.907337	−	0.420404i	$-0.138112\pi$
$444$	0	0
$445$	702.353	1.57832
$446$	0	0
$447$	0	0
$448$	0	0
$449$	214.582	0.477912	0.238956	−	0.971030i	$-0.423195\pi$
$449$	214.582	0.477912	0.238956	+	0.971030i	$0.423195\pi$
$450$	0	0
$451$	− 973.631i	− 2.15883i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	− 37.3220i	− 0.0820264i
$456$	0	0
$457$	496.667	1.08680	0.543399	−	0.839474i	$-0.317137\pi$
$457$	496.667	1.08680	0.543399	+	0.839474i	$0.317137\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−891.986	−1.93489	−0.967447	−	0.253075i	$-0.918558\pi$
$461$	−891.986	−1.93489	−0.967447	+	0.253075i	$0.918558\pi$
$462$	0	0
$463$	− 259.065i	− 0.559535i	−0.960068	−	0.279767i	$-0.909743\pi$
$463$	− 259.065i	− 0.559535i	0.960068	−	0.279767i	$-0.0902573\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	57.7642i	0.123692i	0.998086	+	0.0618461i	$0.0196988\pi$
$467$	57.7642i	0.123692i	−0.998086	+	0.0618461i	$0.980301\pi$
$468$	0	0
$469$	77.3328	0.164889
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−55.6063	−0.117561
$474$	0	0
$475$	77.7725i	0.163732i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	367.638i	0.767511i	0.923435	+	0.383756i	$0.125370\pi$
$479$	367.638i	0.767511i	−0.923435	+	0.383756i	$0.874630\pi$
$480$	0	0
$481$	−205.126	−0.426458
$482$	0	0
$483$	0	0
$484$	0	0
$485$	644.540	1.32895
$486$	0	0
$487$	− 242.495i	− 0.497936i	−0.968512	−	0.248968i	$-0.919909\pi$
$487$	− 242.495i	− 0.497936i	0.968512	−	0.248968i	$-0.0800914\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	310.627i	0.632642i	0.948652	+	0.316321i	$0.102448\pi$
$491$	310.627i	0.632642i	−0.948652	+	0.316321i	$0.897552\pi$
$492$	0	0
$493$	−737.645	−1.49624
$494$	0	0
$495$	0	0
$496$	0	0
$497$	5.59486	0.0112573
$498$	0	0
$499$	− 549.682i	− 1.10157i	−0.834648	−	0.550784i	$-0.814329\pi$
$499$	− 549.682i	− 1.10157i	0.834648	−	0.550784i	$-0.185671\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	730.644i	1.45257i	0.687393	+	0.726286i	$0.258755\pi$
$503$	730.644i	1.45257i	−0.687393	+	0.726286i	$0.741245\pi$
$504$	0	0
$505$	−574.870	−1.13836
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−674.278	−1.32471	−0.662355	−	0.749190i	$-0.730443\pi$
$509$	−674.278	−1.32471	−0.662355	+	0.749190i	$0.730443\pi$
$510$	0	0
$511$	24.2286i	0.0474142i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 775.240i	− 1.50532i
$516$	0	0
$517$	949.590	1.83673
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−178.003	−0.341657	−0.170828	−	0.985301i	$-0.554644\pi$
$521$	−178.003	−0.341657	−0.170828	+	0.985301i	$0.554644\pi$
$522$	0	0
$523$	− 12.0969i	− 0.0231299i	−0.999933	−	0.0115650i	$-0.996319\pi$
$523$	− 12.0969i	− 0.0231299i	0.999933	−	0.0115650i	$-0.00368132\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	510.447i	0.968591i
$528$	0	0
$529$	−1279.75	−2.41919
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−663.575	−1.24498
$534$	0	0
$535$	340.943i	0.637276i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 590.753i	− 1.09602i
$540$	0	0
$541$	281.263	0.519894	0.259947	−	0.965623i	$-0.416295\pi$
$541$	281.263	0.519894	0.259947	+	0.965623i	$0.416295\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	1240.57	2.27628
$546$	0	0
$547$	587.696i	1.07440i	0.843456	+	0.537199i	$0.180518\pi$
$547$	587.696i	1.07440i	−0.843456	+	0.537199i	$0.819482\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	− 233.039i	− 0.422939i
$552$	0	0
$553$	−3.14557	−0.00568819
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−291.671	−0.523647	−0.261823	−	0.965116i	$-0.584324\pi$
$557$	−291.671	−0.523647	−0.261823	+	0.965116i	$0.584324\pi$
$558$	0	0
$559$	37.8983i	0.0677966i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	122.609i	0.217778i	0.994054	+	0.108889i	$0.0347293\pi$
$563$	122.609i	0.217778i	−0.994054	+	0.108889i	$0.965271\pi$
$564$	0	0
$565$	956.544	1.69300
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−284.130	−0.499350	−0.249675	−	0.968330i	$-0.580324\pi$
$569$	−284.130	−0.499350	−0.249675	+	0.968330i	$0.580324\pi$
$570$	0	0
$571$	− 538.393i	− 0.942895i	−0.881894	−	0.471447i	$-0.843732\pi$
$571$	− 538.393i	− 0.942895i	0.881894	−	0.471447i	$-0.156268\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	758.820i	1.31969i
$576$	0	0
$577$	−391.498	−0.678506	−0.339253	−	0.940695i	$-0.610174\pi$
$577$	−391.498	−0.678506	−0.339253	+	0.940695i	$0.610174\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−35.0270	−0.0602874
$582$	0	0
$583$	− 168.672i	− 0.289318i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	781.146i	1.33074i	0.746512	+	0.665372i	$0.231727\pi$
$587$	781.146i	1.33074i	−0.746512	+	0.665372i	$0.768273\pi$
$588$	0	0
$589$	−161.262	−0.273790
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−259.511	−0.437624	−0.218812	−	0.975767i	$-0.570218\pi$
$593$	−259.511	−0.437624	−0.218812	+	0.975767i	$0.570218\pi$
$594$	0	0
$595$	62.0652i	0.104311i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	− 22.8441i	− 0.0381371i	−0.999818	−	0.0190686i	$-0.993930\pi$
$599$	− 22.8441i	− 0.0381371i	0.999818	−	0.0190686i	$-0.00607008\pi$
$600$	0	0
$601$	523.174	0.870506	0.435253	−	0.900308i	$-0.356659\pi$
$601$	523.174	0.870506	0.435253	+	0.900308i	$0.356659\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−178.002	−0.294218
$606$	0	0
$607$	− 375.022i	− 0.617829i	−0.951090	−	0.308915i	$-0.900034\pi$
$607$	− 375.022i	− 0.617829i	0.951090	−	0.308915i	$-0.0999657\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 647.190i	− 1.05923i
$612$	0	0
$613$	−460.402	−0.751063	−0.375531	−	0.926810i	$-0.622540\pi$
$613$	−460.402	−0.751063	−0.375531	+	0.926810i	$0.622540\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−309.856	−0.502198	−0.251099	−	0.967961i	$-0.580792\pi$
$617$	−309.856	−0.502198	−0.251099	+	0.967961i	$0.580792\pi$
$618$	0	0
$619$	94.1201i	0.152052i	0.997106	+	0.0760259i	$0.0242232\pi$
$619$	94.1201i	0.152052i	−0.997106	+	0.0760259i	$0.975777\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	73.7457i	0.118372i
$624$	0	0
$625$	−752.711	−1.20434
$626$	0	0
$627$	0	0
$628$	0	0
$629$	341.117	0.542317
$630$	0	0
$631$	122.501i	0.194137i	0.995278	+	0.0970687i	$0.0309466\pi$
$631$	122.501i	0.194137i	−0.995278	+	0.0970687i	$0.969053\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 210.543i	− 0.331565i
$636$	0	0
$637$	−402.626	−0.632066
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−333.744	−0.520662	−0.260331	−	0.965519i	$-0.583832\pi$
$641$	−333.744	−0.520662	−0.260331	+	0.965519i	$0.583832\pi$
$642$	0	0
$643$	− 1152.17i	− 1.79187i	−0.444182	−	0.895937i	$-0.646506\pi$
$643$	− 1152.17i	− 1.79187i	0.444182	−	0.895937i	$-0.353494\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 727.806i	− 1.12489i	−0.826834	−	0.562446i	$-0.809860\pi$
$647$	− 727.806i	− 1.12489i	0.826834	−	0.562446i	$-0.190140\pi$
$648$	0	0
$649$	673.255	1.03737
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−67.8357	−0.103883	−0.0519415	−	0.998650i	$-0.516541\pi$
$653$	−67.8357	−0.103883	−0.0519415	+	0.998650i	$0.516541\pi$
$654$	0	0
$655$	− 692.324i	− 1.05698i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 1094.38i	− 1.66067i	−0.557267	−	0.830334i	$-0.688150\pi$
$659$	− 1094.38i	− 1.66067i	0.557267	−	0.830334i	$-0.311850\pi$
$660$	0	0
$661$	−446.796	−0.675939	−0.337970	−	0.941157i	$-0.609740\pi$
$661$	−446.796	−0.675939	−0.337970	+	0.941157i	$0.609740\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−19.6078	−0.0294855
$666$	0	0
$667$	− 2273.75i	− 3.40891i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	− 526.824i	− 0.785133i
$672$	0	0
$673$	147.894	0.219753	0.109877	−	0.993945i	$-0.464954\pi$
$673$	147.894	0.219753	0.109877	+	0.993945i	$0.464954\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−361.747	−0.534338	−0.267169	−	0.963650i	$-0.586088\pi$
$677$	−361.747	−0.534338	−0.267169	+	0.963650i	$0.586088\pi$
$678$	0	0
$679$	67.6754i	0.0996692i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 703.800i	− 1.03045i	−0.857054	−	0.515227i	$-0.827708\pi$
$683$	− 703.800i	− 1.03045i	0.857054	−	0.515227i	$-0.172292\pi$
$684$	0	0
$685$	−814.913	−1.18965
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−114.958	−0.166848
$690$	0	0
$691$	− 1251.94i	− 1.81178i	−0.423512	−	0.905891i	$-0.639203\pi$
$691$	− 1251.94i	− 1.81178i	0.423512	−	0.905891i	$-0.360797\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	1521.92i	2.18982i
$696$	0	0
$697$	1103.50	1.58321
$698$	0	0
$699$	0	0
$700$	0	0
$701$	436.940	0.623309	0.311655	−	0.950195i	$-0.399117\pi$
$701$	436.940	0.623309	0.311655	+	0.950195i	$0.399117\pi$
$702$	0	0
$703$	107.767i	0.153296i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 60.3601i	− 0.0853750i
$708$	0	0
$709$	−430.576	−0.607301	−0.303650	−	0.952784i	$-0.598205\pi$
$709$	−430.576	−0.607301	−0.303650	+	0.952784i	$0.598205\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−1573.42	−2.20677
$714$	0	0
$715$	661.097i	0.924611i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	745.980i	1.03752i	0.854918	+	0.518762i	$0.173607\pi$
$719$	745.980i	1.03752i	−0.854918	+	0.518762i	$0.826393\pi$
$720$	0	0
$721$	81.3986	0.112897
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−953.897	−1.31572
$726$	0	0
$727$	− 829.865i	− 1.14149i	−0.821127	−	0.570746i	$-0.806654\pi$
$727$	− 829.865i	− 1.14149i	0.821127	−	0.570746i	$-0.193346\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 63.0235i	− 0.0862154i
$732$	0	0
$733$	−528.169	−0.720557	−0.360279	−	0.932845i	$-0.617318\pi$
$733$	−528.169	−0.720557	−0.360279	+	0.932845i	$0.617318\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−1369.82	−1.85864
$738$	0	0
$739$	− 49.7757i	− 0.0673554i	−0.999433	−	0.0336777i	$-0.989278\pi$
$739$	− 49.7757i	− 0.0673554i	0.999433	−	0.0336777i	$-0.0107220\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	653.784i	0.879924i	0.898016	+	0.439962i	$0.145008\pi$
$743$	653.784i	0.879924i	−0.898016	+	0.439962i	$0.854992\pi$
$744$	0	0
$745$	974.753	1.30839
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−35.7983	−0.0477948
$750$	0	0
$751$	1037.59i	1.38162i	0.723039	+	0.690808i	$0.242745\pi$
$751$	1037.59i	1.38162i	−0.723039	+	0.690808i	$0.757255\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 1276.16i	− 1.69028i
$756$	0	0
$757$	−977.709	−1.29156	−0.645779	−	0.763525i	$-0.723467\pi$
$757$	−977.709	−1.29156	−0.645779	+	0.763525i	$0.723467\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−988.753	−1.29928	−0.649640	−	0.760242i	$-0.725081\pi$
$761$	−988.753	−1.29928	−0.649640	+	0.760242i	$0.725081\pi$
$762$	0	0
$763$	130.258i	0.170718i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 458.855i	− 0.598246i
$768$	0	0
$769$	−989.337	−1.28652	−0.643262	−	0.765646i	$-0.722419\pi$
$769$	−989.337	−1.28652	−0.643262	+	0.765646i	$0.722419\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−1046.66	−1.35402	−0.677012	−	0.735972i	$-0.736725\pi$
$773$	−1046.66	−1.35402	−0.677012	+	0.735972i	$0.736725\pi$
$774$	0	0
$775$	660.093i	0.851733i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	348.622i	0.447524i
$780$	0	0
$781$	−99.1034	−0.126893
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−1185.80	−1.51058
$786$	0	0
$787$	858.268i	1.09056i	0.838255	+	0.545278i	$0.183576\pi$
$787$	858.268i	1.09056i	−0.838255	+	0.545278i	$0.816424\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	100.435i	0.126972i
$792$	0	0
$793$	−359.055	−0.452781
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−168.251	−0.211105	−0.105553	−	0.994414i	$-0.533661\pi$
$797$	−168.251	−0.211105	−0.105553	+	0.994414i	$0.533661\pi$
$798$	0	0
$799$	1076.25i	1.34700i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 429.170i	− 0.534458i
$804$	0	0
$805$	−191.312	−0.237655
$806$	0	0
$807$	0	0
$808$	0	0
$809$	1163.63	1.43835	0.719175	−	0.694829i	$-0.244520\pi$
$809$	1163.63	1.43835	0.719175	+	0.694829i	$0.244520\pi$
$810$	0	0
$811$	− 454.683i	− 0.560645i	−0.959906	−	0.280323i	$-0.909559\pi$
$811$	− 454.683i	− 0.560645i	0.959906	−	0.280323i	$-0.0904414\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	− 655.208i	− 0.803936i
$816$	0	0
$817$	19.9106	0.0243704
$818$	0	0
$819$	0	0
$820$	0	0
$821$	663.057	0.807622	0.403811	−	0.914842i	$-0.367685\pi$
$821$	663.057	0.807622	0.403811	+	0.914842i	$0.367685\pi$
$822$	0	0
$823$	− 913.655i	− 1.11015i	−0.831800	−	0.555076i	$-0.812689\pi$
$823$	− 913.655i	− 1.11015i	0.831800	−	0.555076i	$-0.187311\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 771.244i	− 0.932581i	−0.884632	−	0.466290i	$-0.845590\pi$
$827$	− 771.244i	− 0.932581i	0.884632	−	0.466290i	$-0.154410\pi$
$828$	0	0
$829$	−593.218	−0.715583	−0.357791	−	0.933802i	$-0.616470\pi$
$829$	−593.218	−0.715583	−0.357791	+	0.933802i	$0.616470\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	669.552	0.803784
$834$	0	0
$835$	− 811.711i	− 0.972109i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	1179.56i	1.40591i	0.711233	+	0.702956i	$0.248137\pi$
$839$	1179.56i	1.40591i	−0.711233	+	0.702956i	$0.751863\pi$
$840$	0	0
$841$	2017.28	2.39867
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−655.604	−0.775863
$846$	0	0
$847$	− 18.6898i	− 0.0220659i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	1051.47i	1.23558i
$852$	0	0
$853$	−205.334	−0.240719	−0.120360	−	0.992730i	$-0.538405\pi$
$853$	−205.334	−0.240719	−0.120360	+	0.992730i	$0.538405\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	885.437	1.03318	0.516591	−	0.856232i	$-0.327201\pi$
$857$	885.437	1.03318	0.516591	+	0.856232i	$0.327201\pi$
$858$	0	0
$859$	651.131i	0.758011i	0.925394	+	0.379005i	$0.123734\pi$
$859$	651.131i	0.758011i	−0.925394	+	0.379005i	$0.876266\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	1136.22i	1.31659i	0.752759	+	0.658296i	$0.228722\pi$
$863$	1136.22i	1.31659i	−0.752759	+	0.658296i	$0.771278\pi$
$864$	0	0
$865$	482.262	0.557529
$866$	0	0
$867$	0	0
$868$	0	0
$869$	55.7185	0.0641180
$870$	0	0
$871$	933.596i	1.07187i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 32.1981i	− 0.0367979i
$876$	0	0
$877$	−133.033	−0.151692	−0.0758458	−	0.997120i	$-0.524166\pi$
$877$	−133.033	−0.151692	−0.0758458	+	0.997120i	$0.524166\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	47.4893	0.0539039	0.0269520	−	0.999637i	$-0.491420\pi$
$881$	47.4893	0.0539039	0.0269520	+	0.999637i	$0.491420\pi$
$882$	0	0
$883$	1126.12i	1.27533i	0.770314	+	0.637664i	$0.220099\pi$
$883$	1126.12i	1.27533i	−0.770314	+	0.637664i	$0.779901\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 12.9148i	− 0.0145601i	−0.999974	−	0.00728003i	$-0.997683\pi$
$887$	− 12.9148i	− 0.0145601i	0.999974	−	0.00728003i	$-0.00231733\pi$
$888$	0	0
$889$	22.1066	0.0248669
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−340.013	−0.380754
$894$	0	0
$895$	− 630.971i	− 0.704996i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 1977.92i	− 2.20013i
$900$	0	0
$901$	191.171	0.212177
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−1974.83	−2.18213
$906$	0	0
$907$	− 670.830i	− 0.739614i	−0.929109	−	0.369807i	$-0.879424\pi$
$907$	− 670.830i	− 0.739614i	0.929109	−	0.369807i	$-0.120576\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	1205.33i	1.32309i	0.749907	+	0.661544i	$0.230098\pi$
$911$	1205.33i	1.32309i	−0.749907	+	0.661544i	$0.769902\pi$
$912$	0	0
$913$	620.444	0.679567
$914$	0	0
$915$	0	0
$916$	0	0
$917$	72.6926	0.0792721
$918$	0	0
$919$	− 1040.11i	− 1.13178i	−0.824480	−	0.565891i	$-0.808532\pi$
$919$	− 1040.11i	− 1.13178i	0.824480	−	0.565891i	$-0.191468\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	67.5436i	0.0731783i
$924$	0	0
$925$	441.121	0.476888
$926$	0	0
$927$	0	0
$928$	0	0
$929$	1822.92	1.96223	0.981117	−	0.193414i	$-0.0619562\pi$
$929$	1822.92	1.96223	0.981117	+	0.193414i	$0.0619562\pi$
$930$	0	0
$931$	211.527i	0.227204i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	− 1099.38i	− 1.17581i
$936$	0	0
$937$	239.998	0.256135	0.128067	−	0.991765i	$-0.459123\pi$
$937$	239.998	0.256135	0.128067	+	0.991765i	$0.459123\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	337.194	0.358336	0.179168	−	0.983819i	$-0.442659\pi$
$941$	337.194	0.358336	0.179168	+	0.983819i	$0.442659\pi$
$942$	0	0
$943$	3401.47i	3.60708i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 1681.16i	− 1.77525i	−0.460567	−	0.887625i	$-0.652354\pi$
$947$	− 1681.16i	− 1.77525i	0.460567	−	0.887625i	$-0.347646\pi$
$948$	0	0
$949$	−292.499	−0.308218
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−1305.03	−1.36939	−0.684697	−	0.728827i	$-0.740066\pi$
$953$	−1305.03	−1.36939	−0.684697	+	0.728827i	$0.740066\pi$
$954$	0	0
$955$	1282.21i	1.34263i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	− 85.5642i	− 0.0892223i
$960$	0	0
$961$	−407.712	−0.424258
$962$	0	0
$963$	0	0
$964$	0	0
$965$	1529.88	1.58537
$966$	0	0
$967$	− 999.541i	− 1.03365i	−0.856090	−	0.516826i	$-0.827113\pi$
$967$	− 999.541i	− 1.03365i	0.856090	−	0.516826i	$-0.172887\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	888.319i	0.914849i	0.889248	+	0.457425i	$0.151228\pi$
$971$	888.319i	0.914849i	−0.889248	+	0.457425i	$0.848772\pi$
$972$	0	0
$973$	−159.799	−0.164233
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−92.4982	−0.0946758	−0.0473379	−	0.998879i	$-0.515074\pi$
$977$	−92.4982	−0.0946758	−0.0473379	+	0.998879i	$0.515074\pi$
$978$	0	0
$979$	− 1306.28i	− 1.33430i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	− 1893.25i	− 1.92599i	−0.269514	−	0.962996i	$-0.586863\pi$
$983$	− 1893.25i	− 1.92599i	0.269514	−	0.962996i	$-0.413137\pi$
$984$	0	0
$985$	−211.601	−0.214824
$986$	0	0
$987$	0	0
$988$	0	0
$989$	194.266	0.196427
$990$	0	0
$991$	809.380i	0.816730i	0.912819	+	0.408365i	$0.133901\pi$
$991$	809.380i	0.816730i	−0.912819	+	0.408365i	$0.866099\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	1252.16i	1.25845i
$996$	0	0
$997$	−1929.33	−1.93514	−0.967570	−	0.252604i	$-0.918713\pi$
$997$	−1929.33	−1.93514	−0.967570	+	0.252604i	$0.918713\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2736.3.m.e.1711.12		12
3.2	odd	2		912.3.m.b.799.7	yes	12
4.3	odd	2	inner	2736.3.m.e.1711.11		12
12.11	even	2		912.3.m.b.799.1	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
912.3.m.b.799.1	✓	12	12.11	even	2
912.3.m.b.799.7	yes	12	3.2	odd	2
2736.3.m.e.1711.11		12	4.3	odd	2	inner
2736.3.m.e.1711.12		12	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2736 = 2^{4} \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	2736.m (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+6.54540 q^{5} +0.687253i q^{7} +O(q^{10})\)	\(q+6.54540 q^{5} +0.687253i q^{7} -12.1735i q^{11} -8.29683 q^{13} +13.7973 q^{17} +4.35890i q^{19} +42.5295i q^{23} +17.8422 q^{25} -53.4629 q^{29} +36.9961i q^{31} +4.49835i q^{35} +24.7234 q^{37} +79.9793 q^{41} -4.56780i q^{43} +78.0044i q^{47} +48.5277 q^{49} +13.8557 q^{53} -79.6806i q^{55} +55.3048i q^{59} +43.2762 q^{61} -54.3061 q^{65} -112.524i q^{67} -8.14089i q^{71} +35.2543 q^{73} +8.36630 q^{77} +4.57702i q^{79} +50.9666i q^{83} +90.3090 q^{85} +107.305 q^{89} -5.70203i q^{91} +28.5307i q^{95} +98.4723 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 10 q^{5}+O(q^{10}) \) 12 * q + 10 * q^5	\( 12 q + 10 q^{5} + 36 q^{13} - 14 q^{17} + 10 q^{25} + 108 q^{29} - 16 q^{37} - 16 q^{41} - 118 q^{49} - 220 q^{53} + 366 q^{61} - 140 q^{65} - 158 q^{73} + 286 q^{77} + 98 q^{85} + 396 q^{89} + 224 q^{97}+O(q^{100}) \) 12 * q + 10 * q^5 + 36 * q^13 - 14 * q^17 + 10 * q^25 + 108 * q^29 - 16 * q^37 - 16 * q^41 - 118 * q^49 - 220 * q^53 + 366 * q^61 - 140 * q^65 - 158 * q^73 + 286 * q^77 + 98 * q^85 + 396 * q^89 + 224 * q^97

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