LMFDB - Embedded newform 1872.2.t.i.289.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1872,2,Mod(289,1872)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1872, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 0, 2]))

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

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

chi := DirichletCharacter("1872.289");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1872 = 2^{4} \cdot 3^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1872.t (of order $3$, degree $2$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$14.9479952584$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 78)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			289.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	1872.289
Dual form			1872.2.t.i.1153.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{5} +(-1.00000 + 1.73205i) q^{7} +O(q^{10})$	$q+1.00000 q^{5} +(-1.00000 + 1.73205i) q^{7} +(-1.00000 - 1.73205i) q^{11} +(2.50000 - 2.59808i) q^{13} +(2.50000 - 4.33013i) q^{17} +(-1.00000 + 1.73205i) q^{19} +(-3.00000 - 5.19615i) q^{23} -4.00000 q^{25} +(-4.50000 - 7.79423i) q^{29} +4.00000 q^{31} +(-1.00000 + 1.73205i) q^{35} +(5.50000 + 9.52628i) q^{37} +(2.50000 + 4.33013i) q^{41} +(5.00000 - 8.66025i) q^{43} +2.00000 q^{47} +(1.50000 + 2.59808i) q^{49} +1.00000 q^{53} +(-1.00000 - 1.73205i) q^{55} +(4.00000 - 6.92820i) q^{59} +(5.50000 - 9.52628i) q^{61} +(2.50000 - 2.59808i) q^{65} +(1.00000 + 1.73205i) q^{67} +(7.00000 - 12.1244i) q^{71} -13.0000 q^{73} +4.00000 q^{77} +4.00000 q^{79} +6.00000 q^{83} +(2.50000 - 4.33013i) q^{85} +(1.00000 + 1.73205i) q^{89} +(2.00000 + 6.92820i) q^{91} +(-1.00000 + 1.73205i) q^{95} +(1.00000 - 1.73205i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{5} - 2 q^{7}+O(q^{10}) $ 2 * q + 2 * q^5 - 2 * q^7	$ 2 q + 2 q^{5} - 2 q^{7} - 2 q^{11} + 5 q^{13} + 5 q^{17} - 2 q^{19} - 6 q^{23} - 8 q^{25} - 9 q^{29} + 8 q^{31} - 2 q^{35} + 11 q^{37} + 5 q^{41} + 10 q^{43} + 4 q^{47} + 3 q^{49} + 2 q^{53} - 2 q^{55} + 8 q^{59} + 11 q^{61} + 5 q^{65} + 2 q^{67} + 14 q^{71} - 26 q^{73} + 8 q^{77} + 8 q^{79} + 12 q^{83} + 5 q^{85} + 2 q^{89} + 4 q^{91} - 2 q^{95} + 2 q^{97}+O(q^{100}) $ 2 * q + 2 * q^5 - 2 * q^7 - 2 * q^11 + 5 * q^13 + 5 * q^17 - 2 * q^19 - 6 * q^23 - 8 * q^25 - 9 * q^29 + 8 * q^31 - 2 * q^35 + 11 * q^37 + 5 * q^41 + 10 * q^43 + 4 * q^47 + 3 * q^49 + 2 * q^53 - 2 * q^55 + 8 * q^59 + 11 * q^61 + 5 * q^65 + 2 * q^67 + 14 * q^71 - 26 * q^73 + 8 * q^77 + 8 * q^79 + 12 * q^83 + 5 * q^85 + 2 * q^89 + 4 * q^91 - 2 * q^95 + 2 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1872\mathbb{Z}\right)^\times$.

$n$	$145$	$209$	$469$	$703$
$\chi(n)$	$e\left(\frac{1}{3}\right)$	$1$	$1$	$1$

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
$5$	1.00000			0.447214			0.223607	−	0.974679i	$-0.428217\pi$
$5$	1.00000			0.447214			0.223607	+	0.974679i	$0.428217\pi$
$6$		0			0
$7$	−1.00000	+	1.73205i	−0.377964	+	0.654654i	−0.990766	−	0.135583i	$-0.956709\pi$
$7$	−1.00000	+	1.73205i	−0.377964	+	0.654654i	0.612801	+	0.790237i	$0.290043\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−1.00000	−	1.73205i	−0.301511	−	0.522233i	0.674967	−	0.737848i	$-0.264158\pi$
$11$	−1.00000	−	1.73205i	−0.301511	−	0.522233i	−0.976478	+	0.215615i	$0.930824\pi$
$12$		0			0
$13$	2.50000	−	2.59808i	0.693375	−	0.720577i
$13$	2.50000	−	2.59808i	0.693375	−	0.720577i
$14$		0			0
$15$		0			0
$16$		0			0
$17$	2.50000	−	4.33013i	0.606339	−	1.05021i	−0.385499	−	0.922708i	$-0.625971\pi$
$17$	2.50000	−	4.33013i	0.606339	−	1.05021i	0.991838	−	0.127502i	$-0.0406959\pi$
$18$		0			0
$19$	−1.00000	+	1.73205i	−0.229416	+	0.397360i	−0.957635	−	0.287984i	$-0.907015\pi$
$19$	−1.00000	+	1.73205i	−0.229416	+	0.397360i	0.728219	+	0.685344i	$0.240348\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−3.00000	−	5.19615i	−0.625543	−	1.08347i	−0.988436	−	0.151642i	$-0.951544\pi$
$23$	−3.00000	−	5.19615i	−0.625543	−	1.08347i	0.362892	−	0.931831i	$-0.381789\pi$
$24$		0			0
$25$	−4.00000			−0.800000
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−4.50000	−	7.79423i	−0.835629	−	1.44735i	−0.893517	−	0.449029i	$-0.851770\pi$
$29$	−4.50000	−	7.79423i	−0.835629	−	1.44735i	0.0578882	−	0.998323i	$-0.481563\pi$
$30$		0			0
$31$	4.00000			0.718421			0.359211	−	0.933257i	$-0.383046\pi$
$31$	4.00000			0.718421			0.359211	+	0.933257i	$0.383046\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−1.00000	+	1.73205i	−0.169031	+	0.292770i
$36$		0			0
$37$	5.50000	+	9.52628i	0.904194	+	1.56611i	0.821995	+	0.569495i	$0.192861\pi$
$37$	5.50000	+	9.52628i	0.904194	+	1.56611i	0.0821995	+	0.996616i	$0.473806\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	2.50000	+	4.33013i	0.390434	+	0.676252i	0.992507	−	0.122189i	$-0.0389915\pi$
$41$	2.50000	+	4.33013i	0.390434	+	0.676252i	−0.602072	+	0.798441i	$0.705658\pi$
$42$		0			0
$43$	5.00000	−	8.66025i	0.762493	−	1.32068i	−0.179069	−	0.983836i	$-0.557309\pi$
$43$	5.00000	−	8.66025i	0.762493	−	1.32068i	0.941562	−	0.336840i	$-0.109358\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	2.00000			0.291730			0.145865	−	0.989305i	$-0.453403\pi$
$47$	2.00000			0.291730			0.145865	+	0.989305i	$0.453403\pi$
$48$		0			0
$49$	1.50000	+	2.59808i	0.214286	+	0.371154i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	1.00000			0.137361			0.0686803	−	0.997639i	$-0.478121\pi$
$53$	1.00000			0.137361			0.0686803	+	0.997639i	$0.478121\pi$
$54$		0			0
$55$	−1.00000	−	1.73205i	−0.134840	−	0.233550i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	4.00000	−	6.92820i	0.520756	−	0.901975i	−0.478953	−	0.877841i	$-0.658984\pi$
$59$	4.00000	−	6.92820i	0.520756	−	0.901975i	0.999709	−	0.0241347i	$-0.00768307\pi$
$60$		0			0
$61$	5.50000	−	9.52628i	0.704203	−	1.21972i	−0.262776	−	0.964857i	$-0.584638\pi$
$61$	5.50000	−	9.52628i	0.704203	−	1.21972i	0.966978	−	0.254858i	$-0.0820288\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	2.50000	−	2.59808i	0.310087	−	0.322252i
$66$		0			0
$67$	1.00000	+	1.73205i	0.122169	+	0.211604i	0.920623	−	0.390453i	$-0.127682\pi$
$67$	1.00000	+	1.73205i	0.122169	+	0.211604i	−0.798454	+	0.602056i	$0.794348\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	7.00000	−	12.1244i	0.830747	−	1.43890i	−0.0666994	−	0.997773i	$-0.521247\pi$
$71$	7.00000	−	12.1244i	0.830747	−	1.43890i	0.897447	−	0.441123i	$-0.145420\pi$
$72$		0			0
$73$	−13.0000			−1.52153			−0.760767	−	0.649025i	$-0.775177\pi$
$73$	−13.0000			−1.52153			−0.760767	+	0.649025i	$0.775177\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	4.00000			0.455842
$78$		0			0
$79$	4.00000			0.450035			0.225018	−	0.974355i	$-0.427756\pi$
$79$	4.00000			0.450035			0.225018	+	0.974355i	$0.427756\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	6.00000			0.658586			0.329293	−	0.944228i	$-0.393190\pi$
$83$	6.00000			0.658586			0.329293	+	0.944228i	$0.393190\pi$
$84$		0			0
$85$	2.50000	−	4.33013i	0.271163	−	0.469668i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	1.00000	+	1.73205i	0.106000	+	0.183597i	0.914146	−	0.405385i	$-0.132862\pi$
$89$	1.00000	+	1.73205i	0.106000	+	0.183597i	−0.808146	+	0.588982i	$0.799529\pi$
$90$		0			0
$91$	2.00000	+	6.92820i	0.209657	+	0.726273i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−1.00000	+	1.73205i	−0.102598	+	0.177705i
$96$		0			0
$97$	1.00000	−	1.73205i	0.101535	−	0.175863i	−0.810782	−	0.585348i	$-0.800958\pi$
$97$	1.00000	−	1.73205i	0.101535	−	0.175863i	0.912317	+	0.409484i	$0.134291\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−2.50000	−	4.33013i	−0.248759	−	0.430864i	0.714423	−	0.699715i	$-0.246689\pi$
$101$	−2.50000	−	4.33013i	−0.248759	−	0.430864i	−0.963182	+	0.268851i	$0.913356\pi$
$102$		0			0
$103$	−10.0000			−0.985329			−0.492665	−	0.870219i	$-0.663977\pi$
$103$	−10.0000			−0.985329			−0.492665	+	0.870219i	$0.663977\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	9.00000	+	15.5885i	0.870063	+	1.50699i	0.861931	+	0.507026i	$0.169255\pi$
$107$	9.00000	+	15.5885i	0.870063	+	1.50699i	0.00813215	+	0.999967i	$0.497411\pi$
$108$		0			0
$109$	−2.00000			−0.191565			−0.0957826	−	0.995402i	$-0.530535\pi$
$109$	−2.00000			−0.191565			−0.0957826	+	0.995402i	$0.530535\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−1.50000	+	2.59808i	−0.141108	+	0.244406i	−0.927914	−	0.372794i	$-0.878400\pi$
$113$	−1.50000	+	2.59808i	−0.141108	+	0.244406i	0.786806	+	0.617200i	$0.211733\pi$
$114$		0			0
$115$	−3.00000	−	5.19615i	−0.279751	−	0.484544i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	5.00000	+	8.66025i	0.458349	+	0.793884i
$120$		0			0
$121$	3.50000	−	6.06218i	0.318182	−	0.551107i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−9.00000			−0.804984
$126$		0			0
$127$	−6.00000	−	10.3923i	−0.532414	−	0.922168i	−0.999284	−	0.0378419i	$-0.987952\pi$
$127$	−6.00000	−	10.3923i	−0.532414	−	0.922168i	0.466870	−	0.884326i	$-0.345382\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−8.00000			−0.698963			−0.349482	−	0.936943i	$-0.613642\pi$
$131$	−8.00000			−0.698963			−0.349482	+	0.936943i	$0.613642\pi$
$132$		0			0
$133$	−2.00000	−	3.46410i	−0.173422	−	0.300376i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	8.50000	−	14.7224i	0.726204	−	1.25782i	−0.232273	−	0.972651i	$-0.574616\pi$
$137$	8.50000	−	14.7224i	0.726204	−	1.25782i	0.958477	−	0.285171i	$-0.0920506\pi$
$138$		0			0
$139$	−6.00000	+	10.3923i	−0.508913	+	0.881464i	0.491033	+	0.871141i	$0.336619\pi$
$139$	−6.00000	+	10.3923i	−0.508913	+	0.881464i	−0.999947	+	0.0103230i	$0.996714\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−7.00000	−	1.73205i	−0.585369	−	0.144841i
$144$		0			0
$145$	−4.50000	−	7.79423i	−0.373705	−	0.647275i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	1.50000	−	2.59808i	0.122885	−	0.212843i	−0.798019	−	0.602632i	$-0.794119\pi$
$149$	1.50000	−	2.59808i	0.122885	−	0.212843i	0.920904	+	0.389789i	$0.127452\pi$
$150$		0			0
$151$	6.00000			0.488273			0.244137	−	0.969741i	$-0.421495\pi$
$151$	6.00000			0.488273			0.244137	+	0.969741i	$0.421495\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	4.00000			0.321288
$156$		0			0
$157$	−7.00000			−0.558661			−0.279330	−	0.960195i	$-0.590112\pi$
$157$	−7.00000			−0.558661			−0.279330	+	0.960195i	$0.590112\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	12.0000			0.945732
$162$		0			0
$163$	10.0000	−	17.3205i	0.783260	−	1.35665i	−0.146772	−	0.989170i	$-0.546888\pi$
$163$	10.0000	−	17.3205i	0.783260	−	1.35665i	0.930033	−	0.367477i	$-0.119778\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−12.0000	−	20.7846i	−0.928588	−	1.60836i	−0.785687	−	0.618624i	$-0.787690\pi$
$167$	−12.0000	−	20.7846i	−0.928588	−	1.60836i	−0.142901	−	0.989737i	$-0.545643\pi$
$168$		0			0
$169$	−0.500000	−	12.9904i	−0.0384615	−	0.999260i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−11.0000	+	19.0526i	−0.836315	+	1.44854i	0.0566411	+	0.998395i	$0.481961\pi$
$173$	−11.0000	+	19.0526i	−0.836315	+	1.44854i	−0.892956	+	0.450145i	$0.851372\pi$
$174$		0			0
$175$	4.00000	−	6.92820i	0.302372	−	0.523723i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	3.00000	+	5.19615i	0.224231	+	0.388379i	0.956088	−	0.293079i	$-0.0946798\pi$
$179$	3.00000	+	5.19615i	0.224231	+	0.388379i	−0.731858	+	0.681457i	$0.761346\pi$
$180$		0			0
$181$	5.00000			0.371647			0.185824	−	0.982583i	$-0.440505\pi$
$181$	5.00000			0.371647			0.185824	+	0.982583i	$0.440505\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	5.50000	+	9.52628i	0.404368	+	0.700386i
$186$		0			0
$187$	−10.0000			−0.731272
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−2.00000	+	3.46410i	−0.144715	+	0.250654i	−0.929267	−	0.369410i	$-0.879560\pi$
$191$	−2.00000	+	3.46410i	−0.144715	+	0.250654i	0.784552	+	0.620063i	$0.212893\pi$
$192$		0			0
$193$	8.50000	+	14.7224i	0.611843	+	1.05974i	0.990930	+	0.134382i	$0.0429051\pi$
$193$	8.50000	+	14.7224i	0.611843	+	1.05974i	−0.379086	+	0.925361i	$0.623762\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	3.00000	+	5.19615i	0.213741	+	0.370211i	0.952882	−	0.303340i	$-0.0981018\pi$
$197$	3.00000	+	5.19615i	0.213741	+	0.370211i	−0.739141	+	0.673550i	$0.764768\pi$
$198$		0			0
$199$	5.00000	−	8.66025i	0.354441	−	0.613909i	−0.632581	−	0.774494i	$-0.718005\pi$
$199$	5.00000	−	8.66025i	0.354441	−	0.613909i	0.987022	+	0.160585i	$0.0513380\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	18.0000			1.26335
$204$		0			0
$205$	2.50000	+	4.33013i	0.174608	+	0.302429i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	4.00000			0.276686
$210$		0			0
$211$	12.0000	+	20.7846i	0.826114	+	1.43087i	0.901065	+	0.433684i	$0.142787\pi$
$211$	12.0000	+	20.7846i	0.826114	+	1.43087i	−0.0749508	+	0.997187i	$0.523880\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	5.00000	−	8.66025i	0.340997	−	0.590624i
$216$		0			0
$217$	−4.00000	+	6.92820i	−0.271538	+	0.470317i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−5.00000	−	17.3205i	−0.336336	−	1.16510i
$222$		0			0
$223$	−8.00000	−	13.8564i	−0.535720	−	0.927894i	−0.999128	−	0.0417488i	$-0.986707\pi$
$223$	−8.00000	−	13.8564i	−0.535720	−	0.927894i	0.463409	−	0.886145i	$-0.346626\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−7.00000	+	12.1244i	−0.464606	+	0.804722i	−0.999184	−	0.0403978i	$-0.987137\pi$
$227$	−7.00000	+	12.1244i	−0.464606	+	0.804722i	0.534577	+	0.845120i	$0.320471\pi$
$228$		0			0
$229$	10.0000			0.660819			0.330409	−	0.943838i	$-0.392813\pi$
$229$	10.0000			0.660819			0.330409	+	0.943838i	$0.392813\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	6.00000			0.393073			0.196537	−	0.980497i	$-0.437031\pi$
$233$	6.00000			0.393073			0.196537	+	0.980497i	$0.437031\pi$
$234$		0			0
$235$	2.00000			0.130466
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−6.00000			−0.388108			−0.194054	−	0.980991i	$-0.562164\pi$
$239$	−6.00000			−0.388108			−0.194054	+	0.980991i	$0.562164\pi$
$240$		0			0
$241$	−3.50000	+	6.06218i	−0.225455	+	0.390499i	−0.956456	−	0.291877i	$-0.905720\pi$
$241$	−3.50000	+	6.06218i	−0.225455	+	0.390499i	0.731001	+	0.682376i	$0.239053\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	1.50000	+	2.59808i	0.0958315	+	0.165985i
$246$		0			0
$247$	2.00000	+	6.92820i	0.127257	+	0.440831i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−2.00000	+	3.46410i	−0.126239	+	0.218652i	−0.922217	−	0.386674i	$-0.873624\pi$
$251$	−2.00000	+	3.46410i	−0.126239	+	0.218652i	0.795978	+	0.605326i	$0.206957\pi$
$252$		0			0
$253$	−6.00000	+	10.3923i	−0.377217	+	0.653359i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−1.50000	−	2.59808i	−0.0935674	−	0.162064i	0.815442	−	0.578838i	$-0.196494\pi$
$257$	−1.50000	−	2.59808i	−0.0935674	−	0.162064i	−0.909010	+	0.416775i	$0.863160\pi$
$258$		0			0
$259$	−22.0000			−1.36701
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−7.00000	−	12.1244i	−0.431638	−	0.747620i	0.565376	−	0.824833i	$-0.308731\pi$
$263$	−7.00000	−	12.1244i	−0.431638	−	0.747620i	−0.997015	+	0.0772134i	$0.975398\pi$
$264$		0			0
$265$	1.00000			0.0614295
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−7.00000	+	12.1244i	−0.426798	+	0.739235i	−0.996586	−	0.0825561i	$-0.973692\pi$
$269$	−7.00000	+	12.1244i	−0.426798	+	0.739235i	0.569789	+	0.821791i	$0.307025\pi$
$270$		0			0
$271$	4.00000	+	6.92820i	0.242983	+	0.420858i	0.961563	−	0.274586i	$-0.0885408\pi$
$271$	4.00000	+	6.92820i	0.242983	+	0.420858i	−0.718580	+	0.695444i	$0.755208\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	4.00000	+	6.92820i	0.241209	+	0.417786i
$276$		0			0
$277$	5.50000	−	9.52628i	0.330463	−	0.572379i	−0.652140	−	0.758099i	$-0.726128\pi$
$277$	5.50000	−	9.52628i	0.330463	−	0.572379i	0.982603	+	0.185720i	$0.0594618\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−25.0000			−1.49137			−0.745687	−	0.666296i	$-0.767879\pi$
$281$	−25.0000			−1.49137			−0.745687	+	0.666296i	$0.767879\pi$
$282$		0			0
$283$	13.0000	+	22.5167i	0.772770	+	1.33848i	0.936039	+	0.351895i	$0.114463\pi$
$283$	13.0000	+	22.5167i	0.772770	+	1.33848i	−0.163270	+	0.986581i	$0.552204\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−10.0000			−0.590281
$288$		0			0
$289$	−4.00000	−	6.92820i	−0.235294	−	0.407541i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−0.500000	+	0.866025i	−0.0292103	+	0.0505937i	−0.880261	−	0.474490i	$-0.842633\pi$
$293$	−0.500000	+	0.866025i	−0.0292103	+	0.0505937i	0.851051	+	0.525084i	$0.175966\pi$
$294$		0			0
$295$	4.00000	−	6.92820i	0.232889	−	0.403376i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−21.0000	−	5.19615i	−1.21446	−	0.300501i
$300$		0			0
$301$	10.0000	+	17.3205i	0.576390	+	0.998337i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	5.50000	−	9.52628i	0.314929	−	0.545473i
$306$		0			0
$307$	14.0000			0.799022			0.399511	−	0.916728i	$-0.369180\pi$
$307$	14.0000			0.799022			0.399511	+	0.916728i	$0.369180\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	6.00000			0.340229			0.170114	−	0.985424i	$-0.445586\pi$
$311$	6.00000			0.340229			0.170114	+	0.985424i	$0.445586\pi$
$312$		0			0
$313$	6.00000			0.339140			0.169570	−	0.985518i	$-0.445762\pi$
$313$	6.00000			0.339140			0.169570	+	0.985518i	$0.445762\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	33.0000			1.85346			0.926732	−	0.375722i	$-0.122605\pi$
$317$	33.0000			1.85346			0.926732	+	0.375722i	$0.122605\pi$
$318$		0			0
$319$	−9.00000	+	15.5885i	−0.503903	+	0.872786i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	5.00000	+	8.66025i	0.278207	+	0.481869i
$324$		0			0
$325$	−10.0000	+	10.3923i	−0.554700	+	0.576461i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−2.00000	+	3.46410i	−0.110264	+	0.190982i
$330$		0			0
$331$	−14.0000	+	24.2487i	−0.769510	+	1.33283i	0.168320	+	0.985732i	$0.446166\pi$
$331$	−14.0000	+	24.2487i	−0.769510	+	1.33283i	−0.937829	+	0.347097i	$0.887167\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	1.00000	+	1.73205i	0.0546358	+	0.0946320i
$336$		0			0
$337$	−9.00000			−0.490261			−0.245131	−	0.969490i	$-0.578831\pi$
$337$	−9.00000			−0.490261			−0.245131	+	0.969490i	$0.578831\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−4.00000	−	6.92820i	−0.216612	−	0.375183i
$342$		0			0
$343$	−20.0000			−1.07990
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−3.00000	+	5.19615i	−0.161048	+	0.278944i	−0.935245	−	0.354001i	$-0.884821\pi$
$347$	−3.00000	+	5.19615i	−0.161048	+	0.278944i	0.774197	+	0.632945i	$0.218154\pi$
$348$		0			0
$349$	−3.00000	−	5.19615i	−0.160586	−	0.278144i	0.774493	−	0.632583i	$-0.218005\pi$
$349$	−3.00000	−	5.19615i	−0.160586	−	0.278144i	−0.935079	+	0.354439i	$0.884672\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	8.50000	+	14.7224i	0.452409	+	0.783596i	0.998535	−	0.0541072i	$-0.0172313\pi$
$353$	8.50000	+	14.7224i	0.452409	+	0.783596i	−0.546126	+	0.837703i	$0.683898\pi$
$354$		0			0
$355$	7.00000	−	12.1244i	0.371521	−	0.643494i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−30.0000			−1.58334			−0.791670	−	0.610949i	$-0.790788\pi$
$359$	−30.0000			−1.58334			−0.791670	+	0.610949i	$0.790788\pi$
$360$		0			0
$361$	7.50000	+	12.9904i	0.394737	+	0.683704i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−13.0000			−0.680451
$366$		0			0
$367$	−1.00000	−	1.73205i	−0.0521996	−	0.0904123i	0.838745	−	0.544524i	$-0.183290\pi$
$367$	−1.00000	−	1.73205i	−0.0521996	−	0.0904123i	−0.890945	+	0.454112i	$0.849957\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−1.00000	+	1.73205i	−0.0519174	+	0.0899236i
$372$		0			0
$373$	−4.50000	+	7.79423i	−0.233001	+	0.403570i	−0.958690	−	0.284453i	$-0.908188\pi$
$373$	−4.50000	+	7.79423i	−0.233001	+	0.403570i	0.725689	+	0.688023i	$0.241521\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−31.5000	−	7.79423i	−1.62233	−	0.401423i
$378$		0			0
$379$	6.00000	+	10.3923i	0.308199	+	0.533817i	0.977969	−	0.208752i	$-0.0669403\pi$
$379$	6.00000	+	10.3923i	0.308199	+	0.533817i	−0.669769	+	0.742569i	$0.733607\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−12.0000	+	20.7846i	−0.613171	+	1.06204i	0.377531	+	0.925997i	$0.376773\pi$
$383$	−12.0000	+	20.7846i	−0.613171	+	1.06204i	−0.990702	+	0.136047i	$0.956560\pi$
$384$		0			0
$385$	4.00000			0.203859
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−19.0000			−0.963338			−0.481669	−	0.876353i	$-0.659969\pi$
$389$	−19.0000			−0.963338			−0.481669	+	0.876353i	$0.659969\pi$
$390$		0			0
$391$	−30.0000			−1.51717
$392$		0			0
$393$		0			0
$394$		0			0
$395$	4.00000			0.201262
$396$		0			0
$397$	9.00000	−	15.5885i	0.451697	−	0.782362i	−0.546795	−	0.837267i	$-0.684152\pi$
$397$	9.00000	−	15.5885i	0.451697	−	0.782362i	0.998492	+	0.0549046i	$0.0174855\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−13.5000	−	23.3827i	−0.674158	−	1.16768i	−0.976714	−	0.214544i	$-0.931173\pi$
$401$	−13.5000	−	23.3827i	−0.674158	−	1.16768i	0.302556	−	0.953131i	$-0.402160\pi$
$402$		0			0
$403$	10.0000	−	10.3923i	0.498135	−	0.517678i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	11.0000	−	19.0526i	0.545250	−	0.944400i
$408$		0			0
$409$	−11.5000	+	19.9186i	−0.568638	+	0.984911i	0.428063	+	0.903749i	$0.359196\pi$
$409$	−11.5000	+	19.9186i	−0.568638	+	0.984911i	−0.996701	+	0.0811615i	$0.974137\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	8.00000	+	13.8564i	0.393654	+	0.681829i
$414$		0			0
$415$	6.00000			0.294528
$416$		0			0
$417$		0			0
$418$		0			0
$419$	16.0000	+	27.7128i	0.781651	+	1.35386i	0.930979	+	0.365072i	$0.118956\pi$
$419$	16.0000	+	27.7128i	0.781651	+	1.35386i	−0.149328	+	0.988788i	$0.547711\pi$
$420$		0			0
$421$	−23.0000			−1.12095			−0.560476	−	0.828171i	$-0.689382\pi$
$421$	−23.0000			−1.12095			−0.560476	+	0.828171i	$0.689382\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−10.0000	+	17.3205i	−0.485071	+	0.840168i
$426$		0			0
$427$	11.0000	+	19.0526i	0.532327	+	0.922018i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	1.00000	+	1.73205i	0.0481683	+	0.0834300i	0.889104	−	0.457705i	$-0.151328\pi$
$431$	1.00000	+	1.73205i	0.0481683	+	0.0834300i	−0.840936	+	0.541135i	$0.817995\pi$
$432$		0			0
$433$	10.5000	−	18.1865i	0.504598	−	0.873989i	−0.495388	−	0.868672i	$-0.664974\pi$
$433$	10.5000	−	18.1865i	0.504598	−	0.873989i	0.999986	−	0.00531724i	$-0.00169254\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	12.0000			0.574038
$438$		0			0
$439$	5.00000	+	8.66025i	0.238637	+	0.413331i	0.960323	−	0.278889i	$-0.0899661\pi$
$439$	5.00000	+	8.66025i	0.238637	+	0.413331i	−0.721686	+	0.692220i	$0.756633\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	20.0000			0.950229			0.475114	−	0.879924i	$-0.342407\pi$
$443$	20.0000			0.950229			0.475114	+	0.879924i	$0.342407\pi$
$444$		0			0
$445$	1.00000	+	1.73205i	0.0474045	+	0.0821071i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−15.0000	+	25.9808i	−0.707894	+	1.22611i	0.257743	+	0.966213i	$0.417021\pi$
$449$	−15.0000	+	25.9808i	−0.707894	+	1.22611i	−0.965637	+	0.259895i	$0.916312\pi$
$450$		0			0
$451$	5.00000	−	8.66025i	0.235441	−	0.407795i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	2.00000	+	6.92820i	0.0937614	+	0.324799i
$456$		0			0
$457$	−1.50000	−	2.59808i	−0.0701670	−	0.121533i	0.828807	−	0.559534i	$-0.189020\pi$
$457$	−1.50000	−	2.59808i	−0.0701670	−	0.121533i	−0.898974	+	0.438001i	$0.855687\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	1.50000	−	2.59808i	0.0698620	−	0.121004i	−0.828978	−	0.559281i	$-0.811077\pi$
$461$	1.50000	−	2.59808i	0.0698620	−	0.121004i	0.898840	+	0.438276i	$0.144411\pi$
$462$		0			0
$463$	14.0000			0.650635			0.325318	−	0.945605i	$-0.394529\pi$
$463$	14.0000			0.650635			0.325318	+	0.945605i	$0.394529\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−22.0000			−1.01804			−0.509019	−	0.860755i	$-0.669992\pi$
$467$	−22.0000			−1.01804			−0.509019	+	0.860755i	$0.669992\pi$
$468$		0			0
$469$	−4.00000			−0.184703
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−20.0000			−0.919601
$474$		0			0
$475$	4.00000	−	6.92820i	0.183533	−	0.317888i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−16.0000	−	27.7128i	−0.731059	−	1.26623i	−0.956431	−	0.291958i	$-0.905693\pi$
$479$	−16.0000	−	27.7128i	−0.731059	−	1.26623i	0.225372	−	0.974273i	$-0.427640\pi$
$480$		0			0
$481$	38.5000	+	9.52628i	1.75545	+	0.434361i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	1.00000	−	1.73205i	0.0454077	−	0.0786484i
$486$		0			0
$487$	−13.0000	+	22.5167i	−0.589086	+	1.02033i	0.405266	+	0.914199i	$0.367179\pi$
$487$	−13.0000	+	22.5167i	−0.589086	+	1.02033i	−0.994352	+	0.106129i	$0.966154\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	15.0000	+	25.9808i	0.676941	+	1.17250i	0.975898	+	0.218229i	$0.0700279\pi$
$491$	15.0000	+	25.9808i	0.676941	+	1.17250i	−0.298957	+	0.954267i	$0.596639\pi$
$492$		0			0
$493$	−45.0000			−2.02670
$494$		0			0
$495$		0			0
$496$		0			0
$497$	14.0000	+	24.2487i	0.627986	+	1.08770i
$498$		0			0
$499$		0			0			−	1.00000i	$-0.5\pi$
$499$		0			0				1.00000i	$0.5\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	7.00000	−	12.1244i	0.312115	−	0.540598i	−0.666705	−	0.745321i	$-0.732296\pi$
$503$	7.00000	−	12.1244i	0.312115	−	0.540598i	0.978820	+	0.204723i	$0.0656294\pi$
$504$		0			0
$505$	−2.50000	−	4.33013i	−0.111249	−	0.192688i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	7.50000	+	12.9904i	0.332432	+	0.575789i	0.982988	−	0.183669i	$-0.0587976\pi$
$509$	7.50000	+	12.9904i	0.332432	+	0.575789i	−0.650556	+	0.759458i	$0.725464\pi$
$510$		0			0
$511$	13.0000	−	22.5167i	0.575086	−	0.996078i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−10.0000			−0.440653
$516$		0			0
$517$	−2.00000	−	3.46410i	−0.0879599	−	0.152351i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−25.0000			−1.09527			−0.547635	−	0.836717i	$-0.684472\pi$
$521$	−25.0000			−1.09527			−0.547635	+	0.836717i	$0.684472\pi$
$522$		0			0
$523$	−19.0000	−	32.9090i	−0.830812	−	1.43901i	−0.897395	−	0.441228i	$-0.854543\pi$
$523$	−19.0000	−	32.9090i	−0.830812	−	1.43901i	0.0665832	−	0.997781i	$-0.478790\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	10.0000	−	17.3205i	0.435607	−	0.754493i
$528$		0			0
$529$	−6.50000	+	11.2583i	−0.282609	+	0.489493i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	17.5000	+	4.33013i	0.758009	+	0.187559i
$534$		0			0
$535$	9.00000	+	15.5885i	0.389104	+	0.673948i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	3.00000	−	5.19615i	0.129219	−	0.223814i
$540$		0			0
$541$	−7.00000			−0.300954			−0.150477	−	0.988614i	$-0.548081\pi$
$541$	−7.00000			−0.300954			−0.150477	+	0.988614i	$0.548081\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−2.00000			−0.0856706
$546$		0			0
$547$	−2.00000			−0.0855138			−0.0427569	−	0.999086i	$-0.513614\pi$
$547$	−2.00000			−0.0855138			−0.0427569	+	0.999086i	$0.513614\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	18.0000			0.766826
$552$		0			0
$553$	−4.00000	+	6.92820i	−0.170097	+	0.294617i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−4.50000	−	7.79423i	−0.190671	−	0.330252i	0.754802	−	0.655953i	$-0.227733\pi$
$557$	−4.50000	−	7.79423i	−0.190671	−	0.330252i	−0.945473	+	0.325701i	$0.894400\pi$
$558$		0			0
$559$	−10.0000	−	34.6410i	−0.422955	−	1.46516i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−20.0000	+	34.6410i	−0.842900	+	1.45994i	0.0445334	+	0.999008i	$0.485820\pi$
$563$	−20.0000	+	34.6410i	−0.842900	+	1.45994i	−0.887433	+	0.460937i	$0.847513\pi$
$564$		0			0
$565$	−1.50000	+	2.59808i	−0.0631055	+	0.109302i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	3.00000	+	5.19615i	0.125767	+	0.217834i	0.922032	−	0.387113i	$-0.126528\pi$
$569$	3.00000	+	5.19615i	0.125767	+	0.217834i	−0.796266	+	0.604947i	$0.793194\pi$
$570$		0			0
$571$	2.00000			0.0836974			0.0418487	−	0.999124i	$-0.486675\pi$
$571$	2.00000			0.0836974			0.0418487	+	0.999124i	$0.486675\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	12.0000	+	20.7846i	0.500435	+	0.866778i
$576$		0			0
$577$	27.0000			1.12402			0.562012	−	0.827129i	$-0.310027\pi$
$577$	27.0000			1.12402			0.562012	+	0.827129i	$0.310027\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−6.00000	+	10.3923i	−0.248922	+	0.431145i
$582$		0			0
$583$	−1.00000	−	1.73205i	−0.0414158	−	0.0717342i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	16.0000	+	27.7128i	0.660391	+	1.14383i	0.980513	+	0.196454i	$0.0629426\pi$
$587$	16.0000	+	27.7128i	0.660391	+	1.14383i	−0.320122	+	0.947376i	$0.603724\pi$
$588$		0			0
$589$	−4.00000	+	6.92820i	−0.164817	+	0.285472i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	39.0000			1.60154			0.800769	−	0.598973i	$-0.204424\pi$
$593$	39.0000			1.60154			0.800769	+	0.598973i	$0.204424\pi$
$594$		0			0
$595$	5.00000	+	8.66025i	0.204980	+	0.355036i
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0			−	1.00000i	$-0.5\pi$
$599$		0			0				1.00000i	$0.5\pi$
$600$		0			0
$601$	−5.50000	−	9.52628i	−0.224350	−	0.388585i	0.731774	−	0.681547i	$-0.238692\pi$
$601$	−5.50000	−	9.52628i	−0.224350	−	0.388585i	−0.956124	+	0.292962i	$0.905359\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	3.50000	−	6.06218i	0.142295	−	0.246463i
$606$		0			0
$607$	16.0000	−	27.7128i	0.649420	−	1.12483i	−0.333842	−	0.942629i	$-0.608345\pi$
$607$	16.0000	−	27.7128i	0.649420	−	1.12483i	0.983262	−	0.182199i	$-0.0583216\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	5.00000	−	5.19615i	0.202278	−	0.210214i
$612$		0			0
$613$	−6.50000	−	11.2583i	−0.262533	−	0.454720i	0.704382	−	0.709821i	$-0.251224\pi$
$613$	−6.50000	−	11.2583i	−0.262533	−	0.454720i	−0.966914	+	0.255102i	$0.917891\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−7.50000	+	12.9904i	−0.301939	+	0.522973i	−0.976575	−	0.215177i	$-0.930967\pi$
$617$	−7.50000	+	12.9904i	−0.301939	+	0.522973i	0.674636	+	0.738150i	$0.264300\pi$
$618$		0			0
$619$	−32.0000			−1.28619			−0.643094	−	0.765787i	$-0.722350\pi$
$619$	−32.0000			−1.28619			−0.643094	+	0.765787i	$0.722350\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−4.00000			−0.160257
$624$		0			0
$625$	11.0000			0.440000
$626$		0			0
$627$		0			0
$628$		0			0
$629$	55.0000			2.19299
$630$		0			0
$631$	6.00000	−	10.3923i	0.238856	−	0.413711i	−0.721530	−	0.692383i	$-0.756561\pi$
$631$	6.00000	−	10.3923i	0.238856	−	0.413711i	0.960386	+	0.278672i	$0.0898942\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−6.00000	−	10.3923i	−0.238103	−	0.412406i
$636$		0			0
$637$	10.5000	+	2.59808i	0.416025	+	0.102940i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	2.50000	−	4.33013i	0.0987441	−	0.171030i	−0.812421	−	0.583071i	$-0.801851\pi$
$641$	2.50000	−	4.33013i	0.0987441	−	0.171030i	0.911165	+	0.412042i	$0.135184\pi$
$642$		0			0
$643$	4.00000	−	6.92820i	0.157745	−	0.273222i	−0.776310	−	0.630351i	$-0.782911\pi$
$643$	4.00000	−	6.92820i	0.157745	−	0.273222i	0.934055	+	0.357129i	$0.116244\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$647$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$648$		0			0
$649$	−16.0000			−0.628055
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−11.0000	−	19.0526i	−0.430463	−	0.745584i	0.566450	−	0.824096i	$-0.308316\pi$
$653$	−11.0000	−	19.0526i	−0.430463	−	0.745584i	−0.996913	+	0.0785119i	$0.974983\pi$
$654$		0			0
$655$	−8.00000			−0.312586
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−12.0000	+	20.7846i	−0.467454	+	0.809653i	−0.999309	−	0.0371821i	$-0.988162\pi$
$659$	−12.0000	+	20.7846i	−0.467454	+	0.809653i	0.531855	+	0.846836i	$0.321495\pi$
$660$		0			0
$661$	−12.5000	−	21.6506i	−0.486194	−	0.842112i	0.513680	−	0.857982i	$-0.328282\pi$
$661$	−12.5000	−	21.6506i	−0.486194	−	0.842112i	−0.999874	+	0.0158695i	$0.994948\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−2.00000	−	3.46410i	−0.0775567	−	0.134332i
$666$		0			0
$667$	−27.0000	+	46.7654i	−1.04544	+	1.81076i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−22.0000			−0.849301
$672$		0			0
$673$	−21.5000	−	37.2391i	−0.828764	−	1.43546i	−0.899008	−	0.437932i	$-0.855711\pi$
$673$	−21.5000	−	37.2391i	−0.828764	−	1.43546i	0.0702442	−	0.997530i	$-0.477622\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	46.0000			1.76792			0.883962	−	0.467559i	$-0.154866\pi$
$677$	46.0000			1.76792			0.883962	+	0.467559i	$0.154866\pi$
$678$		0			0
$679$	2.00000	+	3.46410i	0.0767530	+	0.132940i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	20.0000	−	34.6410i	0.765279	−	1.32550i	−0.174820	−	0.984600i	$-0.555934\pi$
$683$	20.0000	−	34.6410i	0.765279	−	1.32550i	0.940099	−	0.340901i	$-0.110732\pi$
$684$		0			0
$685$	8.50000	−	14.7224i	0.324768	−	0.562515i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	2.50000	−	2.59808i	0.0952424	−	0.0989788i
$690$		0			0
$691$	−1.00000	−	1.73205i	−0.0380418	−	0.0658903i	0.846378	−	0.532583i	$-0.178779\pi$
$691$	−1.00000	−	1.73205i	−0.0380418	−	0.0658903i	−0.884419	+	0.466693i	$0.845445\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−6.00000	+	10.3923i	−0.227593	+	0.394203i
$696$		0			0
$697$	25.0000			0.946943
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−34.0000			−1.28416			−0.642081	−	0.766637i	$-0.721929\pi$
$701$	−34.0000			−1.28416			−0.642081	+	0.766637i	$0.721929\pi$
$702$		0			0
$703$	−22.0000			−0.829746
$704$		0			0
$705$		0			0
$706$		0			0
$707$	10.0000			0.376089
$708$		0			0
$709$	7.50000	−	12.9904i	0.281668	−	0.487864i	−0.690127	−	0.723688i	$-0.742446\pi$
$709$	7.50000	−	12.9904i	0.281668	−	0.487864i	0.971796	+	0.235824i	$0.0757789\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−12.0000	−	20.7846i	−0.449404	−	0.778390i
$714$		0			0
$715$	−7.00000	−	1.73205i	−0.261785	−	0.0647750i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−12.0000	+	20.7846i	−0.447524	+	0.775135i	−0.998224	−	0.0595683i	$-0.981028\pi$
$719$	−12.0000	+	20.7846i	−0.447524	+	0.775135i	0.550700	+	0.834703i	$0.314361\pi$
$720$		0			0
$721$	10.0000	−	17.3205i	0.372419	−	0.645049i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	18.0000	+	31.1769i	0.668503	+	1.15788i
$726$		0			0
$727$	−2.00000			−0.0741759			−0.0370879	−	0.999312i	$-0.511808\pi$
$727$	−2.00000			−0.0741759			−0.0370879	+	0.999312i	$0.511808\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−25.0000	−	43.3013i	−0.924658	−	1.60156i
$732$		0			0
$733$	13.0000			0.480166			0.240083	−	0.970752i	$-0.422825\pi$
$733$	13.0000			0.480166			0.240083	+	0.970752i	$0.422825\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	2.00000	−	3.46410i	0.0736709	−	0.127602i
$738$		0			0
$739$	8.00000	+	13.8564i	0.294285	+	0.509716i	0.974818	−	0.223001i	$-0.0715853\pi$
$739$	8.00000	+	13.8564i	0.294285	+	0.509716i	−0.680534	+	0.732717i	$0.738252\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	6.00000	+	10.3923i	0.220119	+	0.381257i	0.954844	−	0.297108i	$-0.0960222\pi$
$743$	6.00000	+	10.3923i	0.220119	+	0.381257i	−0.734725	+	0.678365i	$0.762689\pi$
$744$		0			0
$745$	1.50000	−	2.59808i	0.0549557	−	0.0951861i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−36.0000			−1.31541
$750$		0			0
$751$	13.0000	+	22.5167i	0.474377	+	0.821645i	0.999570	−	0.0293387i	$-0.00934013\pi$
$751$	13.0000	+	22.5167i	0.474377	+	0.821645i	−0.525193	+	0.850983i	$0.676007\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	6.00000			0.218362
$756$		0			0
$757$	9.00000	+	15.5885i	0.327111	+	0.566572i	0.981937	−	0.189207i	$-0.0605917\pi$
$757$	9.00000	+	15.5885i	0.327111	+	0.566572i	−0.654827	+	0.755779i	$0.727258\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	17.0000	−	29.4449i	0.616250	−	1.06738i	−0.373914	−	0.927463i	$-0.621985\pi$
$761$	17.0000	−	29.4449i	0.616250	−	1.06738i	0.990164	−	0.139912i	$-0.0446820\pi$
$762$		0			0
$763$	2.00000	−	3.46410i	0.0724049	−	0.125409i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−8.00000	−	27.7128i	−0.288863	−	1.00065i
$768$		0			0
$769$	17.0000	+	29.4449i	0.613036	+	1.06181i	0.990726	+	0.135877i	$0.0433852\pi$
$769$	17.0000	+	29.4449i	0.613036	+	1.06181i	−0.377690	+	0.925932i	$0.623282\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	9.00000	−	15.5885i	0.323708	−	0.560678i	−0.657542	−	0.753418i	$-0.728404\pi$
$773$	9.00000	−	15.5885i	0.323708	−	0.560678i	0.981250	+	0.192740i	$0.0617373\pi$
$774$		0			0
$775$	−16.0000			−0.574737
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−10.0000			−0.358287
$780$		0			0
$781$	−28.0000			−1.00192
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−7.00000			−0.249841
$786$		0			0
$787$	2.00000	−	3.46410i	0.0712923	−	0.123482i	−0.828176	−	0.560469i	$-0.810621\pi$
$787$	2.00000	−	3.46410i	0.0712923	−	0.123482i	0.899468	+	0.436987i	$0.143954\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−3.00000	−	5.19615i	−0.106668	−	0.184754i
$792$		0			0
$793$	−11.0000	−	38.1051i	−0.390621	−	1.35315i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−1.00000	+	1.73205i	−0.0354218	+	0.0613524i	−0.883193	−	0.469010i	$-0.844611\pi$
$797$	−1.00000	+	1.73205i	−0.0354218	+	0.0613524i	0.847771	+	0.530362i	$0.177944\pi$
$798$		0			0
$799$	5.00000	−	8.66025i	0.176887	−	0.306378i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	13.0000	+	22.5167i	0.458760	+	0.794596i
$804$		0			0
$805$	12.0000			0.422944
$806$		0			0
$807$		0			0
$808$		0			0
$809$	2.50000	+	4.33013i	0.0878953	+	0.152239i	0.906621	−	0.421945i	$-0.138653\pi$
$809$	2.50000	+	4.33013i	0.0878953	+	0.152239i	−0.818726	+	0.574184i	$0.805319\pi$
$810$		0			0
$811$	36.0000			1.26413			0.632065	−	0.774915i	$-0.282207\pi$
$811$	36.0000			1.26413			0.632065	+	0.774915i	$0.282207\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	10.0000	−	17.3205i	0.350285	−	0.606711i
$816$		0			0
$817$	10.0000	+	17.3205i	0.349856	+	0.605968i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−15.0000	−	25.9808i	−0.523504	−	0.906735i	−0.999626	−	0.0273557i	$-0.991291\pi$
$821$	−15.0000	−	25.9808i	−0.523504	−	0.906735i	0.476122	−	0.879379i	$-0.342042\pi$
$822$		0			0
$823$	−8.00000	+	13.8564i	−0.278862	+	0.483004i	−0.971102	−	0.238664i	$-0.923291\pi$
$823$	−8.00000	+	13.8564i	−0.278862	+	0.483004i	0.692240	+	0.721668i	$0.256624\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−8.00000			−0.278187			−0.139094	−	0.990279i	$-0.544419\pi$
$827$	−8.00000			−0.278187			−0.139094	+	0.990279i	$0.544419\pi$
$828$		0			0
$829$	17.5000	+	30.3109i	0.607800	+	1.05274i	0.991602	+	0.129325i	$0.0412811\pi$
$829$	17.5000	+	30.3109i	0.607800	+	1.05274i	−0.383802	+	0.923415i	$0.625386\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	15.0000			0.519719
$834$		0			0
$835$	−12.0000	−	20.7846i	−0.415277	−	0.719281i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−22.0000	+	38.1051i	−0.759524	+	1.31553i	0.183569	+	0.983007i	$0.441235\pi$
$839$	−22.0000	+	38.1051i	−0.759524	+	1.31553i	−0.943093	+	0.332528i	$0.892098\pi$
$840$		0			0
$841$	−26.0000	+	45.0333i	−0.896552	+	1.55287i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−0.500000	−	12.9904i	−0.0172005	−	0.446883i
$846$		0			0
$847$	7.00000	+	12.1244i	0.240523	+	0.416598i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	33.0000	−	57.1577i	1.13123	−	1.95934i
$852$		0			0
$853$	49.0000			1.67773			0.838864	−	0.544341i	$-0.183220\pi$
$853$	49.0000			1.67773			0.838864	+	0.544341i	$0.183220\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−45.0000			−1.53717			−0.768585	−	0.639747i	$-0.779039\pi$
$857$	−45.0000			−1.53717			−0.768585	+	0.639747i	$0.779039\pi$
$858$		0			0
$859$	50.0000			1.70598			0.852989	−	0.521929i	$-0.174787\pi$
$859$	50.0000			1.70598			0.852989	+	0.521929i	$0.174787\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	46.0000			1.56586			0.782929	−	0.622111i	$-0.213725\pi$
$863$	46.0000			1.56586			0.782929	+	0.622111i	$0.213725\pi$
$864$		0			0
$865$	−11.0000	+	19.0526i	−0.374011	+	0.647806i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−4.00000	−	6.92820i	−0.135691	−	0.235023i
$870$		0			0
$871$	7.00000	+	1.73205i	0.237186	+	0.0586883i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	9.00000	−	15.5885i	0.304256	−	0.526986i
$876$		0			0
$877$	−18.5000	+	32.0429i	−0.624701	+	1.08201i	0.363898	+	0.931439i	$0.381446\pi$
$877$	−18.5000	+	32.0429i	−0.624701	+	1.08201i	−0.988599	+	0.150574i	$0.951888\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	8.50000	+	14.7224i	0.286372	+	0.496011i	0.972941	−	0.231054i	$-0.0742173\pi$
$881$	8.50000	+	14.7224i	0.286372	+	0.496011i	−0.686569	+	0.727065i	$0.740884\pi$
$882$		0			0
$883$	8.00000			0.269221			0.134611	−	0.990899i	$-0.457022\pi$
$883$	8.00000			0.269221			0.134611	+	0.990899i	$0.457022\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−12.0000	−	20.7846i	−0.402921	−	0.697879i	0.591156	−	0.806557i	$-0.298672\pi$
$887$	−12.0000	−	20.7846i	−0.402921	−	0.697879i	−0.994077	+	0.108678i	$0.965338\pi$
$888$		0			0
$889$	24.0000			0.804934
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−2.00000	+	3.46410i	−0.0669274	+	0.115922i
$894$		0			0
$895$	3.00000	+	5.19615i	0.100279	+	0.173688i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−18.0000	−	31.1769i	−0.600334	−	1.03981i
$900$		0			0
$901$	2.50000	−	4.33013i	0.0832871	−	0.144257i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	5.00000			0.166206
$906$		0			0
$907$	−22.0000	−	38.1051i	−0.730498	−	1.26526i	−0.956671	−	0.291172i	$-0.905955\pi$
$907$	−22.0000	−	38.1051i	−0.730498	−	1.26526i	0.226173	−	0.974087i	$-0.427379\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−32.0000			−1.06021			−0.530104	−	0.847933i	$-0.677847\pi$
$911$	−32.0000			−1.06021			−0.530104	+	0.847933i	$0.677847\pi$
$912$		0			0
$913$	−6.00000	−	10.3923i	−0.198571	−	0.343935i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	8.00000	−	13.8564i	0.264183	−	0.457579i
$918$		0			0
$919$	−8.00000	+	13.8564i	−0.263896	+	0.457081i	−0.967274	−	0.253735i	$-0.918341\pi$
$919$	−8.00000	+	13.8564i	−0.263896	+	0.457081i	0.703378	+	0.710816i	$0.251674\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−14.0000	−	48.4974i	−0.460816	−	1.59631i
$924$		0			0
$925$	−22.0000	−	38.1051i	−0.723356	−	1.25289i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−11.5000	+	19.9186i	−0.377303	+	0.653508i	−0.990669	−	0.136291i	$-0.956482\pi$
$929$	−11.5000	+	19.9186i	−0.377303	+	0.653508i	0.613366	+	0.789799i	$0.289815\pi$
$930$		0			0
$931$	−6.00000			−0.196642
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−10.0000			−0.327035
$936$		0			0
$937$	−1.00000			−0.0326686			−0.0163343	−	0.999867i	$-0.505200\pi$
$937$	−1.00000			−0.0326686			−0.0163343	+	0.999867i	$0.505200\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	22.0000			0.717180			0.358590	−	0.933495i	$-0.383258\pi$
$941$	22.0000			0.717180			0.358590	+	0.933495i	$0.383258\pi$
$942$		0			0
$943$	15.0000	−	25.9808i	0.488467	−	0.846050i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−4.00000	−	6.92820i	−0.129983	−	0.225136i	0.793687	−	0.608326i	$-0.208159\pi$
$947$	−4.00000	−	6.92820i	−0.129983	−	0.225136i	−0.923670	+	0.383190i	$0.874825\pi$
$948$		0			0
$949$	−32.5000	+	33.7750i	−1.05499	+	1.09638i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	27.0000	−	46.7654i	0.874616	−	1.51488i	0.0174443	−	0.999848i	$-0.494447\pi$
$953$	27.0000	−	46.7654i	0.874616	−	1.51488i	0.857171	−	0.515031i	$-0.172220\pi$
$954$		0			0
$955$	−2.00000	+	3.46410i	−0.0647185	+	0.112096i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	17.0000	+	29.4449i	0.548959	+	0.950824i
$960$		0			0
$961$	−15.0000			−0.483871
$962$		0			0
$963$		0			0
$964$		0			0
$965$	8.50000	+	14.7224i	0.273625	+	0.473932i
$966$		0			0
$967$	50.0000			1.60789			0.803946	−	0.594703i	$-0.202730\pi$
$967$	50.0000			1.60789			0.803946	+	0.594703i	$0.202730\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−10.0000	+	17.3205i	−0.320915	+	0.555842i	−0.980677	−	0.195633i	$-0.937324\pi$
$971$	−10.0000	+	17.3205i	−0.320915	+	0.555842i	0.659762	+	0.751475i	$0.270657\pi$
$972$		0			0
$973$	−12.0000	−	20.7846i	−0.384702	−	0.666324i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	10.5000	+	18.1865i	0.335925	+	0.581839i	0.983662	−	0.180025i	$-0.0576179\pi$
$977$	10.5000	+	18.1865i	0.335925	+	0.581839i	−0.647737	+	0.761864i	$0.724285\pi$
$978$		0			0
$979$	2.00000	−	3.46410i	0.0639203	−	0.110713i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	60.0000			1.91370			0.956851	−	0.290578i	$-0.0938475\pi$
$983$	60.0000			1.91370			0.956851	+	0.290578i	$0.0938475\pi$
$984$		0			0
$985$	3.00000	+	5.19615i	0.0955879	+	0.165563i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−60.0000			−1.90789
$990$		0			0
$991$	9.00000	+	15.5885i	0.285894	+	0.495184i	0.972826	−	0.231539i	$-0.0743760\pi$
$991$	9.00000	+	15.5885i	0.285894	+	0.495184i	−0.686931	+	0.726722i	$0.741043\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	5.00000	−	8.66025i	0.158511	−	0.274549i
$996$		0			0
$997$	11.5000	−	19.9186i	0.364209	−	0.630828i	−0.624440	−	0.781073i	$-0.714673\pi$
$997$	11.5000	−	19.9186i	0.364209	−	0.630828i	0.988649	+	0.150245i	$0.0480062\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1872.2.t.i.289.1		2
3.2	odd	2		624.2.q.b.289.1		2
4.3	odd	2		234.2.h.b.55.1		2
12.11	even	2		78.2.e.b.55.1	✓	2
13.9	even	3	inner	1872.2.t.i.1153.1		2
39.23	odd	6		8112.2.a.bb.1.1		1
39.29	odd	6		8112.2.a.x.1.1		1
39.35	odd	6		624.2.q.b.529.1		2
52.3	odd	6		3042.2.a.m.1.1		1
52.11	even	12		3042.2.b.d.1351.2		2
52.15	even	12		3042.2.b.d.1351.1		2
52.23	odd	6		3042.2.a.d.1.1		1
52.35	odd	6		234.2.h.b.217.1		2
60.23	odd	4		1950.2.z.b.1849.2		4
60.47	odd	4		1950.2.z.b.1849.1		4
60.59	even	2		1950.2.i.b.601.1		2
156.11	odd	12		1014.2.b.a.337.1		2
156.23	even	6		1014.2.a.e.1.1		1
156.35	even	6		78.2.e.b.61.1	yes	2
156.47	odd	4		1014.2.i.e.361.1		4
156.59	odd	12		1014.2.i.e.823.2		4
156.71	odd	12		1014.2.i.e.823.1		4
156.83	odd	4		1014.2.i.e.361.2		4
156.95	even	6		1014.2.e.d.529.1		2
156.107	even	6		1014.2.a.a.1.1		1
156.119	odd	12		1014.2.b.a.337.2		2
156.155	even	2		1014.2.e.d.991.1		2
780.347	odd	12		1950.2.z.b.1699.2		4
780.503	odd	12		1950.2.z.b.1699.1		4
780.659	even	6		1950.2.i.b.451.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
78.2.e.b.55.1	✓	2	12.11	even	2
78.2.e.b.61.1	yes	2	156.35	even	6
234.2.h.b.55.1		2	4.3	odd	2
234.2.h.b.217.1		2	52.35	odd	6
624.2.q.b.289.1		2	3.2	odd	2
624.2.q.b.529.1		2	39.35	odd	6
1014.2.a.a.1.1		1	156.107	even	6
1014.2.a.e.1.1		1	156.23	even	6
1014.2.b.a.337.1		2	156.11	odd	12
1014.2.b.a.337.2		2	156.119	odd	12
1014.2.e.d.529.1		2	156.95	even	6
1014.2.e.d.991.1		2	156.155	even	2
1014.2.i.e.361.1		4	156.47	odd	4
1014.2.i.e.361.2		4	156.83	odd	4
1014.2.i.e.823.1		4	156.71	odd	12
1014.2.i.e.823.2		4	156.59	odd	12
1872.2.t.i.289.1		2	1.1	even	1	trivial
1872.2.t.i.1153.1		2	13.9	even	3	inner
1950.2.i.b.451.1		2	780.659	even	6
1950.2.i.b.601.1		2	60.59	even	2
1950.2.z.b.1699.1		4	780.503	odd	12
1950.2.z.b.1699.2		4	780.347	odd	12
1950.2.z.b.1849.1		4	60.47	odd	4
1950.2.z.b.1849.2		4	60.23	odd	4
3042.2.a.d.1.1		1	52.23	odd	6
3042.2.a.m.1.1		1	52.3	odd	6
3042.2.b.d.1351.1		2	52.15	even	12
3042.2.b.d.1351.2		2	52.11	even	12
8112.2.a.x.1.1		1	39.29	odd	6
8112.2.a.bb.1.1		1	39.23	odd	6

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:	\( N \)	\(=\)	\( 1872 = 2^{4} \cdot 3^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1872.t (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000 q^{5} +(-1.00000 + 1.73205i) q^{7} +O(q^{10})\)	\(q+1.00000 q^{5} +(-1.00000 + 1.73205i) q^{7} +(-1.00000 - 1.73205i) q^{11} +(2.50000 - 2.59808i) q^{13} +(2.50000 - 4.33013i) q^{17} +(-1.00000 + 1.73205i) q^{19} +(-3.00000 - 5.19615i) q^{23} -4.00000 q^{25} +(-4.50000 - 7.79423i) q^{29} +4.00000 q^{31} +(-1.00000 + 1.73205i) q^{35} +(5.50000 + 9.52628i) q^{37} +(2.50000 + 4.33013i) q^{41} +(5.00000 - 8.66025i) q^{43} +2.00000 q^{47} +(1.50000 + 2.59808i) q^{49} +1.00000 q^{53} +(-1.00000 - 1.73205i) q^{55} +(4.00000 - 6.92820i) q^{59} +(5.50000 - 9.52628i) q^{61} +(2.50000 - 2.59808i) q^{65} +(1.00000 + 1.73205i) q^{67} +(7.00000 - 12.1244i) q^{71} -13.0000 q^{73} +4.00000 q^{77} +4.00000 q^{79} +6.00000 q^{83} +(2.50000 - 4.33013i) q^{85} +(1.00000 + 1.73205i) q^{89} +(2.00000 + 6.92820i) q^{91} +(-1.00000 + 1.73205i) q^{95} +(1.00000 - 1.73205i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{5} - 2 q^{7}+O(q^{10}) \) 2 * q + 2 * q^5 - 2 * q^7	\( 2 q + 2 q^{5} - 2 q^{7} - 2 q^{11} + 5 q^{13} + 5 q^{17} - 2 q^{19} - 6 q^{23} - 8 q^{25} - 9 q^{29} + 8 q^{31} - 2 q^{35} + 11 q^{37} + 5 q^{41} + 10 q^{43} + 4 q^{47} + 3 q^{49} + 2 q^{53} - 2 q^{55} + 8 q^{59} + 11 q^{61} + 5 q^{65} + 2 q^{67} + 14 q^{71} - 26 q^{73} + 8 q^{77} + 8 q^{79} + 12 q^{83} + 5 q^{85} + 2 q^{89} + 4 q^{91} - 2 q^{95} + 2 q^{97}+O(q^{100}) \) 2 * q + 2 * q^5 - 2 * q^7 - 2 * q^11 + 5 * q^13 + 5 * q^17 - 2 * q^19 - 6 * q^23 - 8 * q^25 - 9 * q^29 + 8 * q^31 - 2 * q^35 + 11 * q^37 + 5 * q^41 + 10 * q^43 + 4 * q^47 + 3 * q^49 + 2 * q^53 - 2 * q^55 + 8 * q^59 + 11 * q^61 + 5 * q^65 + 2 * q^67 + 14 * q^71 - 26 * q^73 + 8 * q^77 + 8 * q^79 + 12 * q^83 + 5 * q^85 + 2 * q^89 + 4 * q^91 - 2 * q^95 + 2 * q^97

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