LMFDB - Embedded newform 2736.2.d.b.2015.4

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			2015.4
Character	$\chi$	$=$	2736.2015
Dual form			2736.2.d.b.2015.21

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.62670i q^{5} +0.815178i q^{7} +O(q^{10})$	$q-2.62670i q^{5} +0.815178i q^{7} +2.98902 q^{11} -1.44280 q^{13} +8.05095i q^{17} -1.00000i q^{19} +2.16106 q^{23} -1.89953 q^{25} +9.88713i q^{29} +10.1936i q^{31} +2.14122 q^{35} -7.24187 q^{37} -3.87898i q^{41} -11.2492i q^{43} +12.5575 q^{47} +6.33549 q^{49} +8.04858i q^{53} -7.85124i q^{55} -9.48367 q^{59} +4.22088 q^{61} +3.78981i q^{65} -11.4294i q^{67} -4.11099 q^{71} +12.1166 q^{73} +2.43658i q^{77} +15.4488i q^{79} -0.745515 q^{83} +21.1474 q^{85} +6.82012i q^{89} -1.17614i q^{91} -2.62670 q^{95} -0.975205 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	− 2.62670i	− 1.17469i	−0.809335	−	0.587347i	$-0.800172\pi$
$5$	− 2.62670i	− 1.17469i	0.809335	−	0.587347i	$-0.199828\pi$
$6$	0	0
$7$	0.815178i	0.308108i	0.988062	+	0.154054i	$0.0492330\pi$
$7$	0.815178i	0.308108i	−0.988062	+	0.154054i	$0.950767\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	2.98902	0.901222	0.450611	−	0.892720i	$-0.351206\pi$
$11$	2.98902	0.901222	0.450611	+	0.892720i	$0.351206\pi$
$12$	0	0
$13$	−1.44280	−0.400162	−0.200081	−	0.979779i	$-0.564121\pi$
$13$	−1.44280	−0.400162	−0.200081	+	0.979779i	$0.564121\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	8.05095i	1.95264i	0.216324	+	0.976322i	$0.430593\pi$
$17$	8.05095i	1.95264i	−0.216324	+	0.976322i	$0.569407\pi$
$18$	0	0
$19$	− 1.00000i	− 0.229416i
$19$	− 1.00000i	− 0.229416i
$20$	0	0
$21$	0	0
$22$	0	0
$23$	2.16106	0.450613	0.225307	−	0.974288i	$-0.427662\pi$
$23$	2.16106	0.450613	0.225307	+	0.974288i	$0.427662\pi$
$24$	0	0
$25$	−1.89953	−0.379906
$26$	0	0
$27$	0	0
$28$	0	0
$29$	9.88713i	1.83599i	0.396586	+	0.917997i	$0.370195\pi$
$29$	9.88713i	1.83599i	−0.396586	+	0.917997i	$0.629805\pi$
$30$	0	0
$31$	10.1936i	1.83082i	0.402523	+	0.915410i	$0.368133\pi$
$31$	10.1936i	1.83082i	−0.402523	+	0.915410i	$0.631867\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	2.14122	0.361933
$36$	0	0
$37$	−7.24187	−1.19056	−0.595278	−	0.803520i	$-0.702958\pi$
$37$	−7.24187	−1.19056	−0.595278	+	0.803520i	$0.702958\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 3.87898i	− 0.605796i	−0.953023	−	0.302898i	$-0.902046\pi$
$41$	− 3.87898i	− 0.605796i	0.953023	−	0.302898i	$-0.0979541\pi$
$42$	0	0
$43$	− 11.2492i	− 1.71549i	−0.514073	−	0.857747i	$-0.671864\pi$
$43$	− 11.2492i	− 1.71549i	0.514073	−	0.857747i	$-0.328136\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	12.5575	1.83170	0.915849	−	0.401522i	$-0.131519\pi$
$47$	12.5575	1.83170	0.915849	+	0.401522i	$0.131519\pi$
$48$	0	0
$49$	6.33549	0.905069
$50$	0	0
$51$	0	0
$52$	0	0
$53$	8.04858i	1.10556i	0.833328	+	0.552779i	$0.186433\pi$
$53$	8.04858i	1.10556i	−0.833328	+	0.552779i	$0.813567\pi$
$54$	0	0
$55$	− 7.85124i	− 1.05866i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−9.48367	−1.23467	−0.617334	−	0.786701i	$-0.711787\pi$
$59$	−9.48367	−1.23467	−0.617334	+	0.786701i	$0.711787\pi$
$60$	0	0
$61$	4.22088	0.540428	0.270214	−	0.962800i	$-0.412905\pi$
$61$	4.22088	0.540428	0.270214	+	0.962800i	$0.412905\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	3.78981i	0.470068i
$66$	0	0
$67$	− 11.4294i	− 1.39633i	−0.715939	−	0.698163i	$-0.754001\pi$
$67$	− 11.4294i	− 1.39633i	0.715939	−	0.698163i	$-0.245999\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−4.11099	−0.487884	−0.243942	−	0.969790i	$-0.578441\pi$
$71$	−4.11099	−0.487884	−0.243942	+	0.969790i	$0.578441\pi$
$72$	0	0
$73$	12.1166	1.41814	0.709070	−	0.705138i	$-0.249115\pi$
$73$	12.1166	1.41814	0.709070	+	0.705138i	$0.249115\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	2.43658i	0.277674i
$78$	0	0
$79$	15.4488i	1.73813i	0.494699	+	0.869064i	$0.335278\pi$
$79$	15.4488i	1.73813i	−0.494699	+	0.869064i	$0.664722\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−0.745515	−0.0818309	−0.0409154	−	0.999163i	$-0.513027\pi$
$83$	−0.745515	−0.0818309	−0.0409154	+	0.999163i	$0.513027\pi$
$84$	0	0
$85$	21.1474	2.29376
$86$	0	0
$87$	0	0
$88$	0	0
$89$	6.82012i	0.722931i	0.932385	+	0.361466i	$0.117724\pi$
$89$	6.82012i	0.722931i	−0.932385	+	0.361466i	$0.882276\pi$
$90$	0	0
$91$	− 1.17614i	− 0.123293i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−2.62670	−0.269493
$96$	0	0
$97$	−0.975205	−0.0990170	−0.0495085	−	0.998774i	$-0.515765\pi$
$97$	−0.975205	−0.0990170	−0.0495085	+	0.998774i	$0.515765\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	3.91044i	0.389103i	0.980892	+	0.194552i	$0.0623252\pi$
$101$	3.91044i	0.389103i	−0.980892	+	0.194552i	$0.937675\pi$
$102$	0	0
$103$	− 9.64976i	− 0.950819i	−0.879765	−	0.475410i	$-0.842300\pi$
$103$	− 9.64976i	− 0.950819i	0.879765	−	0.475410i	$-0.157700\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	17.6478	1.70607	0.853037	−	0.521850i	$-0.174758\pi$
$107$	17.6478	1.70607	0.853037	+	0.521850i	$0.174758\pi$
$108$	0	0
$109$	5.17205	0.495392	0.247696	−	0.968838i	$-0.420327\pi$
$109$	5.17205	0.495392	0.247696	+	0.968838i	$0.420327\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 7.02859i	− 0.661194i	−0.943772	−	0.330597i	$-0.892750\pi$
$113$	− 7.02859i	− 0.661194i	0.943772	−	0.330597i	$-0.107250\pi$
$114$	0	0
$115$	− 5.67646i	− 0.529332i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−6.56296	−0.601625
$120$	0	0
$121$	−2.06578	−0.187799
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 8.14399i	− 0.728420i
$126$	0	0
$127$	9.16207i	0.813002i	0.913650	+	0.406501i	$0.133251\pi$
$127$	9.16207i	0.813002i	−0.913650	+	0.406501i	$0.866749\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	4.31020	0.376584	0.188292	−	0.982113i	$-0.439705\pi$
$131$	4.31020	0.376584	0.188292	+	0.982113i	$0.439705\pi$
$132$	0	0
$133$	0.815178	0.0706849
$134$	0	0
$135$	0	0
$136$	0	0
$137$	14.1028i	1.20488i	0.798164	+	0.602440i	$0.205805\pi$
$137$	14.1028i	1.20488i	−0.798164	+	0.602440i	$0.794195\pi$
$138$	0	0
$139$	− 16.9257i	− 1.43562i	−0.696239	−	0.717810i	$-0.745145\pi$
$139$	− 16.9257i	− 1.43562i	0.696239	−	0.717810i	$-0.254855\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−4.31257	−0.360635
$144$	0	0
$145$	25.9705	2.15673
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 4.94464i	− 0.405081i	−0.979274	−	0.202540i	$-0.935080\pi$
$149$	− 4.94464i	− 0.405081i	0.979274	−	0.202540i	$-0.0649198\pi$
$150$	0	0
$151$	− 18.8312i	− 1.53246i	−0.642564	−	0.766232i	$-0.722129\pi$
$151$	− 18.8312i	− 1.53246i	0.642564	−	0.766232i	$-0.277871\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	26.7754	2.15065
$156$	0	0
$157$	0.840497	0.0670790	0.0335395	−	0.999437i	$-0.489322\pi$
$157$	0.840497	0.0670790	0.0335395	+	0.999437i	$0.489322\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	1.76165i	0.138838i
$162$	0	0
$163$	16.9323i	1.32624i	0.748512	+	0.663121i	$0.230769\pi$
$163$	16.9323i	1.32624i	−0.748512	+	0.663121i	$0.769231\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	8.59532	0.665126	0.332563	−	0.943081i	$-0.392087\pi$
$167$	8.59532	0.665126	0.332563	+	0.943081i	$0.392087\pi$
$168$	0	0
$169$	−10.9183	−0.839870
$170$	0	0
$171$	0	0
$172$	0	0
$173$	14.1459i	1.07549i	0.843106	+	0.537747i	$0.180724\pi$
$173$	14.1459i	1.07549i	−0.843106	+	0.537747i	$0.819276\pi$
$174$	0	0
$175$	− 1.54846i	− 0.117052i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−2.62685	−0.196340	−0.0981699	−	0.995170i	$-0.531299\pi$
$179$	−2.62685	−0.196340	−0.0981699	+	0.995170i	$0.531299\pi$
$180$	0	0
$181$	8.58001	0.637747	0.318874	−	0.947797i	$-0.396695\pi$
$181$	8.58001	0.637747	0.318874	+	0.947797i	$0.396695\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	19.0222i	1.39854i
$186$	0	0
$187$	24.0644i	1.75977i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	2.11622	0.153124	0.0765620	−	0.997065i	$-0.475606\pi$
$191$	2.11622	0.153124	0.0765620	+	0.997065i	$0.475606\pi$
$192$	0	0
$193$	5.89795	0.424544	0.212272	−	0.977211i	$-0.431914\pi$
$193$	5.89795	0.424544	0.212272	+	0.977211i	$0.431914\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	15.6364i	1.11405i	0.830496	+	0.557025i	$0.188057\pi$
$197$	15.6364i	1.11405i	−0.830496	+	0.557025i	$0.811943\pi$
$198$	0	0
$199$	12.5687i	0.890972i	0.895289	+	0.445486i	$0.146969\pi$
$199$	12.5687i	0.890972i	−0.895289	+	0.445486i	$0.853031\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−8.05977	−0.565685
$204$	0	0
$205$	−10.1889	−0.711625
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 2.98902i	− 0.206755i
$210$	0	0
$211$	− 23.7537i	− 1.63527i	−0.575734	−	0.817637i	$-0.695283\pi$
$211$	− 23.7537i	− 1.63527i	0.575734	−	0.817637i	$-0.304717\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−29.5483	−2.01518
$216$	0	0
$217$	−8.30957	−0.564090
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 11.6160i	− 0.781374i
$222$	0	0
$223$	− 3.35923i	− 0.224951i	−0.993655	−	0.112475i	$-0.964122\pi$
$223$	− 3.35923i	− 0.224951i	0.993655	−	0.112475i	$-0.0358779\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	2.95039	0.195824	0.0979122	−	0.995195i	$-0.468784\pi$
$227$	2.95039	0.195824	0.0979122	+	0.995195i	$0.468784\pi$
$228$	0	0
$229$	−12.8160	−0.846902	−0.423451	−	0.905919i	$-0.639181\pi$
$229$	−12.8160	−0.846902	−0.423451	+	0.905919i	$0.639181\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	19.5210i	1.27886i	0.768849	+	0.639431i	$0.220830\pi$
$233$	19.5210i	1.27886i	−0.768849	+	0.639431i	$0.779170\pi$
$234$	0	0
$235$	− 32.9847i	− 2.15169i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	19.7166	1.27536	0.637679	−	0.770302i	$-0.279895\pi$
$239$	19.7166	1.27536	0.637679	+	0.770302i	$0.279895\pi$
$240$	0	0
$241$	−4.12716	−0.265854	−0.132927	−	0.991126i	$-0.542438\pi$
$241$	−4.12716	−0.265854	−0.132927	+	0.991126i	$0.542438\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 16.6414i	− 1.06318i
$246$	0	0
$247$	1.44280i	0.0918035i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	21.9478	1.38534	0.692668	−	0.721257i	$-0.256435\pi$
$251$	21.9478	1.38534	0.692668	+	0.721257i	$0.256435\pi$
$252$	0	0
$253$	6.45945	0.406102
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 11.6748i	− 0.728253i	−0.931349	−	0.364127i	$-0.881368\pi$
$257$	− 11.6748i	− 0.728253i	0.931349	−	0.364127i	$-0.118632\pi$
$258$	0	0
$259$	− 5.90341i	− 0.366820i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−1.65873	−0.102281	−0.0511407	−	0.998691i	$-0.516286\pi$
$263$	−1.65873	−0.102281	−0.0511407	+	0.998691i	$0.516286\pi$
$264$	0	0
$265$	21.1412	1.29869
$266$	0	0
$267$	0	0
$268$	0	0
$269$	13.7028i	0.835472i	0.908568	+	0.417736i	$0.137176\pi$
$269$	13.7028i	0.835472i	−0.908568	+	0.417736i	$0.862824\pi$
$270$	0	0
$271$	− 17.9008i	− 1.08740i	−0.839280	−	0.543699i	$-0.817023\pi$
$271$	− 17.9008i	− 1.08740i	0.839280	−	0.543699i	$-0.182977\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−5.67773	−0.342380
$276$	0	0
$277$	23.0519	1.38505	0.692526	−	0.721393i	$-0.256498\pi$
$277$	23.0519	1.38505	0.692526	+	0.721393i	$0.256498\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 8.20853i	− 0.489680i	−0.969564	−	0.244840i	$-0.921265\pi$
$281$	− 8.20853i	− 0.489680i	0.969564	−	0.244840i	$-0.0787354\pi$
$282$	0	0
$283$	13.1878i	0.783932i	0.919980	+	0.391966i	$0.128205\pi$
$283$	13.1878i	0.783932i	−0.919980	+	0.391966i	$0.871795\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	3.16206	0.186651
$288$	0	0
$289$	−47.8179	−2.81282
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 0.489522i	− 0.0285982i	−0.999898	−	0.0142991i	$-0.995448\pi$
$293$	− 0.489522i	− 0.0285982i	0.999898	−	0.0142991i	$-0.00455169\pi$
$294$	0	0
$295$	24.9107i	1.45036i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−3.11799	−0.180318
$300$	0	0
$301$	9.17013	0.528558
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 11.0870i	− 0.634838i
$306$	0	0
$307$	8.90843i	0.508431i	0.967148	+	0.254215i	$0.0818172\pi$
$307$	8.90843i	0.508431i	−0.967148	+	0.254215i	$0.918183\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−6.96237	−0.394800	−0.197400	−	0.980323i	$-0.563250\pi$
$311$	−6.96237	−0.394800	−0.197400	+	0.980323i	$0.563250\pi$
$312$	0	0
$313$	−22.8625	−1.29227	−0.646133	−	0.763225i	$-0.723615\pi$
$313$	−22.8625	−1.29227	−0.646133	+	0.763225i	$0.723615\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	24.9416i	1.40086i	0.713721	+	0.700430i	$0.247008\pi$
$317$	24.9416i	1.40086i	−0.713721	+	0.700430i	$0.752992\pi$
$318$	0	0
$319$	29.5528i	1.65464i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	8.05095	0.447967
$324$	0	0
$325$	2.74065	0.152024
$326$	0	0
$327$	0	0
$328$	0	0
$329$	10.2366i	0.564361i
$330$	0	0
$331$	1.69211i	0.0930066i	0.998918	+	0.0465033i	$0.0148078\pi$
$331$	1.69211i	0.0930066i	−0.998918	+	0.0465033i	$0.985192\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−30.0216	−1.64026
$336$	0	0
$337$	−20.1584	−1.09810	−0.549048	−	0.835791i	$-0.685010\pi$
$337$	−20.1584	−1.09810	−0.549048	+	0.835791i	$0.685010\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	30.4687i	1.64998i
$342$	0	0
$343$	10.8708i	0.586967i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	0.673741	0.0361683	0.0180842	−	0.999836i	$-0.494243\pi$
$347$	0.673741	0.0361683	0.0180842	+	0.999836i	$0.494243\pi$
$348$	0	0
$349$	−27.4224	−1.46788	−0.733942	−	0.679212i	$-0.762322\pi$
$349$	−27.4224	−1.46788	−0.733942	+	0.679212i	$0.762322\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 9.57862i	− 0.509819i	−0.966965	−	0.254909i	$-0.917954\pi$
$353$	− 9.57862i	− 0.509819i	0.966965	−	0.254909i	$-0.0820456\pi$
$354$	0	0
$355$	10.7983i	0.573115i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	18.7692	0.990598	0.495299	−	0.868723i	$-0.335059\pi$
$359$	18.7692	0.990598	0.495299	+	0.868723i	$0.335059\pi$
$360$	0	0
$361$	−1.00000	−0.0526316
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 31.8266i	− 1.66588i
$366$	0	0
$367$	31.3903i	1.63856i	0.573392	+	0.819281i	$0.305627\pi$
$367$	31.3903i	1.63856i	−0.573392	+	0.819281i	$0.694373\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−6.56103	−0.340631
$372$	0	0
$373$	−26.7527	−1.38520	−0.692601	−	0.721321i	$-0.743535\pi$
$373$	−26.7527	−1.38520	−0.692601	+	0.721321i	$0.743535\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 14.2652i	− 0.734695i
$378$	0	0
$379$	− 17.6222i	− 0.905190i	−0.891716	−	0.452595i	$-0.850498\pi$
$379$	− 17.6222i	− 0.905190i	0.891716	−	0.452595i	$-0.149502\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−8.04666	−0.411165	−0.205582	−	0.978640i	$-0.565909\pi$
$383$	−8.04666	−0.411165	−0.205582	+	0.978640i	$0.565909\pi$
$384$	0	0
$385$	6.40015	0.326182
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 28.7529i	− 1.45783i	−0.684605	−	0.728914i	$-0.740025\pi$
$389$	− 28.7529i	− 1.45783i	0.684605	−	0.728914i	$-0.259975\pi$
$390$	0	0
$391$	17.3986i	0.879886i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	40.5794	2.04177
$396$	0	0
$397$	5.56594	0.279347	0.139673	−	0.990198i	$-0.455395\pi$
$397$	5.56594	0.279347	0.139673	+	0.990198i	$0.455395\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	28.1170i	1.40410i	0.712129	+	0.702049i	$0.247731\pi$
$401$	28.1170i	1.40410i	−0.712129	+	0.702049i	$0.752269\pi$
$402$	0	0
$403$	− 14.7073i	− 0.732624i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−21.6461	−1.07296
$408$	0	0
$409$	18.8681	0.932969	0.466484	−	0.884529i	$-0.345520\pi$
$409$	18.8681	0.932969	0.466484	+	0.884529i	$0.345520\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 7.73088i	− 0.380412i
$414$	0	0
$415$	1.95824i	0.0961262i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−4.59047	−0.224259	−0.112129	−	0.993694i	$-0.535767\pi$
$419$	−4.59047	−0.224259	−0.112129	+	0.993694i	$0.535767\pi$
$420$	0	0
$421$	−4.44038	−0.216411	−0.108205	−	0.994129i	$-0.534510\pi$
$421$	−4.44038	−0.216411	−0.108205	+	0.994129i	$0.534510\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 15.2930i	− 0.741821i
$426$	0	0
$427$	3.44077i	0.166510i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−28.0492	−1.35108	−0.675542	−	0.737321i	$-0.736090\pi$
$431$	−28.0492	−1.35108	−0.675542	+	0.737321i	$0.736090\pi$
$432$	0	0
$433$	34.3499	1.65075	0.825375	−	0.564585i	$-0.190964\pi$
$433$	34.3499	1.65075	0.825375	+	0.564585i	$0.190964\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 2.16106i	− 0.103378i
$438$	0	0
$439$	− 15.6394i	− 0.746429i	−0.927745	−	0.373214i	$-0.878256\pi$
$439$	− 15.6394i	− 0.746429i	0.927745	−	0.373214i	$-0.121744\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	16.9846	0.806961	0.403481	−	0.914988i	$-0.367800\pi$
$443$	16.9846	0.806961	0.403481	+	0.914988i	$0.367800\pi$
$444$	0	0
$445$	17.9144	0.849223
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 18.0721i	− 0.852875i	−0.904517	−	0.426438i	$-0.859768\pi$
$449$	− 18.0721i	− 0.852875i	0.904517	−	0.426438i	$-0.140232\pi$
$450$	0	0
$451$	− 11.5943i	− 0.545956i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−3.08937	−0.144832
$456$	0	0
$457$	21.7270	1.01635	0.508174	−	0.861254i	$-0.330321\pi$
$457$	21.7270	1.01635	0.508174	+	0.861254i	$0.330321\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 30.8407i	− 1.43640i	−0.695839	−	0.718198i	$-0.744967\pi$
$461$	− 30.8407i	− 1.43640i	0.695839	−	0.718198i	$-0.255033\pi$
$462$	0	0
$463$	− 13.7137i	− 0.637328i	−0.947868	−	0.318664i	$-0.896766\pi$
$463$	− 13.7137i	− 0.637328i	0.947868	−	0.318664i	$-0.103234\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	17.7056	0.819319	0.409660	−	0.912239i	$-0.365648\pi$
$467$	17.7056	0.819319	0.409660	+	0.912239i	$0.365648\pi$
$468$	0	0
$469$	9.31701	0.430219
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 33.6242i	− 1.54604i
$474$	0	0
$475$	1.89953i	0.0871565i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−41.3976	−1.89151	−0.945753	−	0.324886i	$-0.894674\pi$
$479$	−41.3976	−1.89151	−0.945753	+	0.324886i	$0.894674\pi$
$480$	0	0
$481$	10.4486	0.476415
$482$	0	0
$483$	0	0
$484$	0	0
$485$	2.56157i	0.116315i
$486$	0	0
$487$	− 29.8467i	− 1.35249i	−0.736679	−	0.676243i	$-0.763607\pi$
$487$	− 29.8467i	− 1.35249i	0.736679	−	0.676243i	$-0.236393\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−21.7538	−0.981736	−0.490868	−	0.871234i	$-0.663320\pi$
$491$	−21.7538	−0.981736	−0.490868	+	0.871234i	$0.663320\pi$
$492$	0	0
$493$	−79.6009	−3.58504
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 3.35118i	− 0.150321i
$498$	0	0
$499$	− 27.1040i	− 1.21334i	−0.794954	−	0.606670i	$-0.792505\pi$
$499$	− 27.1040i	− 1.21334i	0.794954	−	0.606670i	$-0.207495\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−15.6118	−0.696096	−0.348048	−	0.937477i	$-0.613155\pi$
$503$	−15.6118	−0.696096	−0.348048	+	0.937477i	$0.613155\pi$
$504$	0	0
$505$	10.2715	0.457077
$506$	0	0
$507$	0	0
$508$	0	0
$509$	2.87470i	0.127419i	0.997968	+	0.0637093i	$0.0202930\pi$
$509$	2.87470i	0.127419i	−0.997968	+	0.0637093i	$0.979707\pi$
$510$	0	0
$511$	9.87717i	0.436940i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−25.3470	−1.11692
$516$	0	0
$517$	37.5346	1.65077
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 36.1305i	− 1.58291i	−0.611229	−	0.791454i	$-0.709325\pi$
$521$	− 36.1305i	− 1.58291i	0.611229	−	0.791454i	$-0.290675\pi$
$522$	0	0
$523$	25.1009i	1.09759i	0.835959	+	0.548793i	$0.184912\pi$
$523$	25.1009i	1.09759i	−0.835959	+	0.548793i	$0.815088\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−82.0680	−3.57494
$528$	0	0
$529$	−18.3298	−0.796948
$530$	0	0
$531$	0	0
$532$	0	0
$533$	5.59662i	0.242416i
$534$	0	0
$535$	− 46.3553i	− 2.00412i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	18.9369	0.815669
$540$	0	0
$541$	2.51424	0.108096	0.0540478	−	0.998538i	$-0.482788\pi$
$541$	2.51424	0.108096	0.0540478	+	0.998538i	$0.482788\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 13.5854i	− 0.581935i
$546$	0	0
$547$	21.0986i	0.902110i	0.892496	+	0.451055i	$0.148952\pi$
$547$	21.0986i	0.902110i	−0.892496	+	0.451055i	$0.851048\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	9.88713	0.421206
$552$	0	0
$553$	−12.5935	−0.535532
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 9.86835i	− 0.418136i	−0.977901	−	0.209068i	$-0.932957\pi$
$557$	− 9.86835i	− 0.418136i	0.977901	−	0.209068i	$-0.0670429\pi$
$558$	0	0
$559$	16.2305i	0.686475i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	30.1845	1.27212	0.636062	−	0.771638i	$-0.280562\pi$
$563$	30.1845	1.27212	0.636062	+	0.771638i	$0.280562\pi$
$564$	0	0
$565$	−18.4620	−0.776701
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 35.9090i	− 1.50538i	−0.658374	−	0.752691i	$-0.728755\pi$
$569$	− 35.9090i	− 1.50538i	0.658374	−	0.752691i	$-0.271245\pi$
$570$	0	0
$571$	30.2305i	1.26511i	0.774517	+	0.632553i	$0.217993\pi$
$571$	30.2305i	1.26511i	−0.774517	+	0.632553i	$0.782007\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−4.10501	−0.171191
$576$	0	0
$577$	−4.96144	−0.206547	−0.103274	−	0.994653i	$-0.532932\pi$
$577$	−4.96144	−0.206547	−0.103274	+	0.994653i	$0.532932\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 0.607727i	− 0.0252128i
$582$	0	0
$583$	24.0573i	0.996353i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	30.9544	1.27762	0.638812	−	0.769363i	$-0.279426\pi$
$587$	30.9544	1.27762	0.638812	+	0.769363i	$0.279426\pi$
$588$	0	0
$589$	10.1936	0.420019
$590$	0	0
$591$	0	0
$592$	0	0
$593$	10.1038i	0.414911i	0.978244	+	0.207456i	$0.0665183\pi$
$593$	10.1038i	0.414911i	−0.978244	+	0.207456i	$0.933482\pi$
$594$	0	0
$595$	17.2389i	0.706726i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	13.0183	0.531911	0.265956	−	0.963985i	$-0.414313\pi$
$599$	13.0183	0.531911	0.265956	+	0.963985i	$0.414313\pi$
$600$	0	0
$601$	−29.1678	−1.18978	−0.594889	−	0.803808i	$-0.702804\pi$
$601$	−29.1678	−1.18978	−0.594889	+	0.803808i	$0.702804\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	5.42619i	0.220606i
$606$	0	0
$607$	− 24.6729i	− 1.00144i	−0.865609	−	0.500721i	$-0.833068\pi$
$607$	− 24.6729i	− 1.00144i	0.865609	−	0.500721i	$-0.166932\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−18.1180	−0.732976
$612$	0	0
$613$	−23.1751	−0.936032	−0.468016	−	0.883720i	$-0.655031\pi$
$613$	−23.1751	−0.936032	−0.468016	+	0.883720i	$0.655031\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	13.3422i	0.537138i	0.963260	+	0.268569i	$0.0865507\pi$
$617$	13.3422i	0.537138i	−0.963260	+	0.268569i	$0.913449\pi$
$618$	0	0
$619$	5.93322i	0.238476i	0.992866	+	0.119238i	$0.0380452\pi$
$619$	5.93322i	0.238476i	−0.992866	+	0.119238i	$0.961955\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−5.55961	−0.222741
$624$	0	0
$625$	−30.8894	−1.23558
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 58.3039i	− 2.32473i
$630$	0	0
$631$	− 5.73792i	− 0.228423i	−0.993456	−	0.114212i	$-0.963566\pi$
$631$	− 5.73792i	− 0.228423i	0.993456	−	0.114212i	$-0.0364342\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	24.0660	0.955029
$636$	0	0
$637$	−9.14087	−0.362174
$638$	0	0
$639$	0	0
$640$	0	0
$641$	21.7901i	0.860656i	0.902673	+	0.430328i	$0.141602\pi$
$641$	21.7901i	0.860656i	−0.902673	+	0.430328i	$0.858398\pi$
$642$	0	0
$643$	− 6.09045i	− 0.240184i	−0.992763	−	0.120092i	$-0.961681\pi$
$643$	− 6.09045i	− 0.240184i	0.992763	−	0.120092i	$-0.0383189\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	19.0718	0.749791	0.374896	−	0.927067i	$-0.377679\pi$
$647$	19.0718	0.749791	0.374896	+	0.927067i	$0.377679\pi$
$648$	0	0
$649$	−28.3468	−1.11271
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 20.9126i	− 0.818374i	−0.912451	−	0.409187i	$-0.865812\pi$
$653$	− 20.9126i	− 0.818374i	0.912451	−	0.409187i	$-0.134188\pi$
$654$	0	0
$655$	− 11.3216i	− 0.442370i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−24.5980	−0.958201	−0.479100	−	0.877760i	$-0.659037\pi$
$659$	−24.5980	−0.958201	−0.479100	+	0.877760i	$0.659037\pi$
$660$	0	0
$661$	−8.18377	−0.318312	−0.159156	−	0.987253i	$-0.550877\pi$
$661$	−8.18377	−0.318312	−0.159156	+	0.987253i	$0.550877\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 2.14122i	− 0.0830331i
$666$	0	0
$667$	21.3667i	0.827323i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	12.6163	0.487046
$672$	0	0
$673$	−8.98555	−0.346367	−0.173184	−	0.984890i	$-0.555405\pi$
$673$	−8.98555	−0.346367	−0.173184	+	0.984890i	$0.555405\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	41.0215i	1.57658i	0.615302	+	0.788291i	$0.289034\pi$
$677$	41.0215i	1.57658i	−0.615302	+	0.788291i	$0.710966\pi$
$678$	0	0
$679$	− 0.794965i	− 0.0305080i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−23.6744	−0.905878	−0.452939	−	0.891542i	$-0.649624\pi$
$683$	−23.6744	−0.905878	−0.452939	+	0.891542i	$0.649624\pi$
$684$	0	0
$685$	37.0437	1.41537
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 11.6125i	− 0.442402i
$690$	0	0
$691$	25.5371i	0.971476i	0.874105	+	0.485738i	$0.161449\pi$
$691$	25.5371i	0.971476i	−0.874105	+	0.485738i	$0.838551\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−44.4587	−1.68641
$696$	0	0
$697$	31.2295	1.18290
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 16.8830i	− 0.637663i	−0.947811	−	0.318832i	$-0.896710\pi$
$701$	− 16.8830i	− 0.637663i	0.947811	−	0.318832i	$-0.103290\pi$
$702$	0	0
$703$	7.24187i	0.273132i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−3.18770	−0.119886
$708$	0	0
$709$	−20.2233	−0.759503	−0.379751	−	0.925089i	$-0.623990\pi$
$709$	−20.2233	−0.759503	−0.379751	+	0.925089i	$0.623990\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	22.0290i	0.824991i
$714$	0	0
$715$	11.3278i	0.423636i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−32.5986	−1.21572	−0.607862	−	0.794043i	$-0.707972\pi$
$719$	−32.5986	−1.21572	−0.607862	+	0.794043i	$0.707972\pi$
$720$	0	0
$721$	7.86627	0.292955
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 18.7809i	− 0.697506i
$726$	0	0
$727$	4.01163i	0.148783i	0.997229	+	0.0743915i	$0.0237014\pi$
$727$	4.01163i	0.148783i	−0.997229	+	0.0743915i	$0.976299\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	90.5671	3.34975
$732$	0	0
$733$	0.936390	0.0345863	0.0172932	−	0.999850i	$-0.494495\pi$
$733$	0.936390	0.0345863	0.0172932	+	0.999850i	$0.494495\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 34.1627i	− 1.25840i
$738$	0	0
$739$	− 43.4973i	− 1.60008i	−0.599949	−	0.800038i	$-0.704813\pi$
$739$	− 43.4973i	− 1.60008i	0.599949	−	0.800038i	$-0.295187\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−33.3942	−1.22512	−0.612558	−	0.790426i	$-0.709859\pi$
$743$	−33.3942	−1.22512	−0.612558	+	0.790426i	$0.709859\pi$
$744$	0	0
$745$	−12.9881	−0.475846
$746$	0	0
$747$	0	0
$748$	0	0
$749$	14.3861i	0.525656i
$750$	0	0
$751$	44.3664i	1.61895i	0.587153	+	0.809476i	$0.300249\pi$
$751$	44.3664i	1.61895i	−0.587153	+	0.809476i	$0.699751\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−49.4639	−1.80018
$756$	0	0
$757$	−25.6121	−0.930888	−0.465444	−	0.885077i	$-0.654105\pi$
$757$	−25.6121	−0.930888	−0.465444	+	0.885077i	$0.654105\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 14.4372i	− 0.523348i	−0.965156	−	0.261674i	$-0.915726\pi$
$761$	− 14.4372i	− 0.523348i	0.965156	−	0.261674i	$-0.0842745\pi$
$762$	0	0
$763$	4.21614i	0.152634i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	13.6831	0.494068
$768$	0	0
$769$	29.6580	1.06949	0.534747	−	0.845012i	$-0.320407\pi$
$769$	29.6580	1.06949	0.534747	+	0.845012i	$0.320407\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 16.4950i	− 0.593282i	−0.954989	−	0.296641i	$-0.904133\pi$
$773$	− 16.4950i	− 0.593282i	0.954989	−	0.296641i	$-0.0958665\pi$
$774$	0	0
$775$	− 19.3630i	− 0.695540i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−3.87898	−0.138979
$780$	0	0
$781$	−12.2878	−0.439692
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 2.20773i	− 0.0787973i
$786$	0	0
$787$	− 33.6571i	− 1.19975i	−0.800095	−	0.599874i	$-0.795217\pi$
$787$	− 33.6571i	− 1.19975i	0.800095	−	0.599874i	$-0.204783\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	5.72955	0.203719
$792$	0	0
$793$	−6.08991	−0.216259
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 17.4632i	− 0.618578i	−0.950968	−	0.309289i	$-0.899909\pi$
$797$	− 17.4632i	− 0.618578i	0.950968	−	0.309289i	$-0.100091\pi$
$798$	0	0
$799$	101.100i	3.57665i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	36.2167	1.27806
$804$	0	0
$805$	4.62732	0.163092
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 0.726707i	− 0.0255496i	−0.999918	−	0.0127748i	$-0.995934\pi$
$809$	− 0.726707i	− 0.0255496i	0.999918	−	0.0127748i	$-0.00406646\pi$
$810$	0	0
$811$	6.24583i	0.219321i	0.993969	+	0.109660i	$0.0349763\pi$
$811$	6.24583i	0.219321i	−0.993969	+	0.109660i	$0.965024\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	44.4761	1.55793
$816$	0	0
$817$	−11.2492	−0.393561
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 24.0958i	− 0.840948i	−0.907305	−	0.420474i	$-0.861864\pi$
$821$	− 24.0958i	− 0.840948i	0.907305	−	0.420474i	$-0.138136\pi$
$822$	0	0
$823$	29.8946i	1.04206i	0.853538	+	0.521030i	$0.174452\pi$
$823$	29.8946i	1.04206i	−0.853538	+	0.521030i	$0.825548\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−32.0577	−1.11476	−0.557378	−	0.830259i	$-0.688193\pi$
$827$	−32.0577	−1.11476	−0.557378	+	0.830259i	$0.688193\pi$
$828$	0	0
$829$	14.2197	0.493870	0.246935	−	0.969032i	$-0.420577\pi$
$829$	14.2197	0.493870	0.246935	+	0.969032i	$0.420577\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	51.0067i	1.76728i
$834$	0	0
$835$	− 22.5773i	− 0.781319i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	6.93757	0.239511	0.119756	−	0.992803i	$-0.461789\pi$
$839$	6.93757	0.239511	0.119756	+	0.992803i	$0.461789\pi$
$840$	0	0
$841$	−68.7554	−2.37088
$842$	0	0
$843$	0	0
$844$	0	0
$845$	28.6791i	0.986591i
$846$	0	0
$847$	− 1.68398i	− 0.0578623i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−15.6501	−0.536480
$852$	0	0
$853$	15.3000	0.523861	0.261931	−	0.965087i	$-0.415641\pi$
$853$	15.3000	0.523861	0.261931	+	0.965087i	$0.415641\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	40.5820i	1.38626i	0.720815	+	0.693128i	$0.243768\pi$
$857$	40.5820i	1.38626i	−0.720815	+	0.693128i	$0.756232\pi$
$858$	0	0
$859$	− 23.3837i	− 0.797842i	−0.916985	−	0.398921i	$-0.869385\pi$
$859$	− 23.3837i	− 0.797842i	0.916985	−	0.398921i	$-0.130615\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	15.5016	0.527681	0.263840	−	0.964566i	$-0.415011\pi$
$863$	15.5016	0.527681	0.263840	+	0.964566i	$0.415011\pi$
$864$	0	0
$865$	37.1570	1.26338
$866$	0	0
$867$	0	0
$868$	0	0
$869$	46.1768i	1.56644i
$870$	0	0
$871$	16.4904i	0.558756i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	6.63880	0.224432
$876$	0	0
$877$	17.0316	0.575117	0.287558	−	0.957763i	$-0.407156\pi$
$877$	17.0316	0.575117	0.287558	+	0.957763i	$0.407156\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 6.42144i	− 0.216344i	−0.994132	−	0.108172i	$-0.965500\pi$
$881$	− 6.42144i	− 0.216344i	0.994132	−	0.108172i	$-0.0344997\pi$
$882$	0	0
$883$	10.2267i	0.344157i	0.985083	+	0.172078i	$0.0550482\pi$
$883$	10.2267i	0.344157i	−0.985083	+	0.172078i	$0.944952\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	8.61770	0.289354	0.144677	−	0.989479i	$-0.453786\pi$
$887$	8.61770	0.289354	0.144677	+	0.989479i	$0.453786\pi$
$888$	0	0
$889$	−7.46872	−0.250493
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 12.5575i	− 0.420221i
$894$	0	0
$895$	6.89993i	0.230639i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−100.785	−3.36138
$900$	0	0
$901$	−64.7988	−2.15876
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 22.5371i	− 0.749158i
$906$	0	0
$907$	− 30.3232i	− 1.00687i	−0.864034	−	0.503433i	$-0.832070\pi$
$907$	− 30.3232i	− 1.00687i	0.864034	−	0.503433i	$-0.167930\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−5.84477	−0.193646	−0.0968229	−	0.995302i	$-0.530868\pi$
$911$	−5.84477	−0.193646	−0.0968229	+	0.995302i	$0.530868\pi$
$912$	0	0
$913$	−2.22835	−0.0737478
$914$	0	0
$915$	0	0
$916$	0	0
$917$	3.51357i	0.116028i
$918$	0	0
$919$	4.02134i	0.132652i	0.997798	+	0.0663259i	$0.0211277\pi$
$919$	4.02134i	0.132652i	−0.997798	+	0.0663259i	$0.978872\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	5.93135	0.195233
$924$	0	0
$925$	13.7562	0.452300
$926$	0	0
$927$	0	0
$928$	0	0
$929$	10.4138i	0.341664i	0.985300	+	0.170832i	$0.0546456\pi$
$929$	10.4138i	0.341664i	−0.985300	+	0.170832i	$0.945354\pi$
$930$	0	0
$931$	− 6.33549i	− 0.207637i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	63.2099	2.06719
$936$	0	0
$937$	14.6965	0.480113	0.240056	−	0.970759i	$-0.422834\pi$
$937$	14.6965	0.480113	0.240056	+	0.970759i	$0.422834\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	25.9949i	0.847411i	0.905800	+	0.423705i	$0.139271\pi$
$941$	25.9949i	0.847411i	−0.905800	+	0.423705i	$0.860729\pi$
$942$	0	0
$943$	− 8.38274i	− 0.272979i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	26.0210	0.845568	0.422784	−	0.906230i	$-0.361053\pi$
$947$	26.0210	0.845568	0.422784	+	0.906230i	$0.361053\pi$
$948$	0	0
$949$	−17.4819	−0.567486
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 25.9572i	− 0.840835i	−0.907331	−	0.420417i	$-0.861884\pi$
$953$	− 25.9572i	− 0.840835i	0.907331	−	0.420417i	$-0.138116\pi$
$954$	0	0
$955$	− 5.55865i	− 0.179874i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−11.4963	−0.371234
$960$	0	0
$961$	−72.9089	−2.35190
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 15.4921i	− 0.498709i
$966$	0	0
$967$	− 36.8551i	− 1.18518i	−0.805504	−	0.592590i	$-0.798105\pi$
$967$	− 36.8551i	− 1.18518i	0.805504	−	0.592590i	$-0.201895\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	25.3533	0.813625	0.406813	−	0.913512i	$-0.366640\pi$
$971$	25.3533	0.813625	0.406813	+	0.913512i	$0.366640\pi$
$972$	0	0
$973$	13.7975	0.442326
$974$	0	0
$975$	0	0
$976$	0	0
$977$	30.3795i	0.971925i	0.873980	+	0.485962i	$0.161531\pi$
$977$	30.3795i	0.971925i	−0.873980	+	0.485962i	$0.838469\pi$
$978$	0	0
$979$	20.3854i	0.651522i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−5.94251	−0.189536	−0.0947682	−	0.995499i	$-0.530211\pi$
$983$	−5.94251	−0.189536	−0.0947682	+	0.995499i	$0.530211\pi$
$984$	0	0
$985$	41.0721	1.30867
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 24.3103i	− 0.773024i
$990$	0	0
$991$	− 4.74708i	− 0.150796i	−0.997154	−	0.0753981i	$-0.975977\pi$
$991$	− 4.74708i	− 0.150796i	0.997154	−	0.0753981i	$-0.0240227\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	33.0142	1.04662
$996$	0	0
$997$	−15.8258	−0.501209	−0.250604	−	0.968090i	$-0.580629\pi$
$997$	−15.8258	−0.501209	−0.250604	+	0.968090i	$0.580629\pi$
$998$	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.2015.4	yes	24
3.2	odd	2	inner	2736.2.d.b.2015.22	yes	24
4.3	odd	2	inner	2736.2.d.b.2015.3	✓	24
12.11	even	2	inner	2736.2.d.b.2015.21	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2736.2.d.b.2015.3	✓	24	4.3	odd	2	inner
2736.2.d.b.2015.4	yes	24	1.1	even	1	trivial
2736.2.d.b.2015.21	yes	24	12.11	even	2	inner
2736.2.d.b.2015.22	yes	24	3.2	odd	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)

\(f(q)\)	\(=\)	\(q-2.62670i q^{5} +0.815178i q^{7} +O(q^{10})\)	\(q-2.62670i q^{5} +0.815178i q^{7} +2.98902 q^{11} -1.44280 q^{13} +8.05095i q^{17} -1.00000i q^{19} +2.16106 q^{23} -1.89953 q^{25} +9.88713i q^{29} +10.1936i q^{31} +2.14122 q^{35} -7.24187 q^{37} -3.87898i q^{41} -11.2492i q^{43} +12.5575 q^{47} +6.33549 q^{49} +8.04858i q^{53} -7.85124i q^{55} -9.48367 q^{59} +4.22088 q^{61} +3.78981i q^{65} -11.4294i q^{67} -4.11099 q^{71} +12.1166 q^{73} +2.43658i q^{77} +15.4488i q^{79} -0.745515 q^{83} +21.1474 q^{85} +6.82012i q^{89} -1.17614i q^{91} -2.62670 q^{95} -0.975205 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q+O(q^{10}) \) 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

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