LMFDB - Embedded newform 1872.2.t.j.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 39)
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.j.1153.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{5} +(1.00000 - 1.73205i) q^{7} +O(q^{10})$	$q+1.00000 q^{5} +(1.00000 - 1.73205i) q^{7} +(1.00000 + 1.73205i) q^{11} +(-3.50000 + 0.866025i) q^{13} +(-3.50000 + 6.06218i) q^{17} +(-3.00000 + 5.19615i) q^{19} +(3.00000 + 5.19615i) q^{23} -4.00000 q^{25} +(-0.500000 - 0.866025i) q^{29} -4.00000 q^{31} +(1.00000 - 1.73205i) q^{35} +(-0.500000 - 0.866025i) q^{37} +(4.50000 + 7.79423i) q^{41} +(3.00000 - 5.19615i) q^{43} +6.00000 q^{47} +(1.50000 + 2.59808i) q^{49} +9.00000 q^{53} +(1.00000 + 1.73205i) q^{55} +(-0.500000 + 0.866025i) q^{61} +(-3.50000 + 0.866025i) q^{65} +(-1.00000 - 1.73205i) q^{67} +(-3.00000 + 5.19615i) q^{71} +11.0000 q^{73} +4.00000 q^{77} +4.00000 q^{79} -14.0000 q^{83} +(-3.50000 + 6.06218i) q^{85} +(-7.00000 - 12.1244i) q^{89} +(-2.00000 + 6.92820i) q^{91} +(-3.00000 + 5.19615i) q^{95} +(1.00000 - 1.73205i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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} - 7 q^{13} - 7 q^{17} - 6 q^{19} + 6 q^{23} - 8 q^{25} - q^{29} - 8 q^{31} + 2 q^{35} - q^{37} + 9 q^{41} + 6 q^{43} + 12 q^{47} + 3 q^{49} + 18 q^{53} + 2 q^{55} - q^{61} - 7 q^{65} - 2 q^{67} - 6 q^{71} + 22 q^{73} + 8 q^{77} + 8 q^{79} - 28 q^{83} - 7 q^{85} - 14 q^{89} - 4 q^{91} - 6 q^{95} + 2 q^{97}+O(q^{100}) $ 2 * q + 2 * q^5 + 2 * q^7 + 2 * q^11 - 7 * q^13 - 7 * q^17 - 6 * q^19 + 6 * q^23 - 8 * q^25 - q^29 - 8 * q^31 + 2 * q^35 - q^37 + 9 * q^41 + 6 * q^43 + 12 * q^47 + 3 * q^49 + 18 * q^53 + 2 * q^55 - q^61 - 7 * q^65 - 2 * q^67 - 6 * q^71 + 22 * q^73 + 8 * q^77 + 8 * q^79 - 28 * q^83 - 7 * q^85 - 14 * q^89 - 4 * q^91 - 6 * 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	−0.612801	−	0.790237i	$-0.709957\pi$
$7$	1.00000	−	1.73205i	0.377964	−	0.654654i	0.990766	+	0.135583i	$0.0432908\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	1.00000	+	1.73205i	0.301511	+	0.522233i	0.976478	−	0.215615i	$-0.0691756\pi$
$11$	1.00000	+	1.73205i	0.301511	+	0.522233i	−0.674967	+	0.737848i	$0.735842\pi$
$12$		0			0
$13$	−3.50000	+	0.866025i	−0.970725	+	0.240192i
$13$	−3.50000	+	0.866025i	−0.970725	+	0.240192i
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−3.50000	+	6.06218i	−0.848875	+	1.47029i	0.0333386	+	0.999444i	$0.489386\pi$
$17$	−3.50000	+	6.06218i	−0.848875	+	1.47029i	−0.882213	+	0.470850i	$0.843947\pi$
$18$		0			0
$19$	−3.00000	+	5.19615i	−0.688247	+	1.19208i	0.284157	+	0.958778i	$0.408286\pi$
$19$	−3.00000	+	5.19615i	−0.688247	+	1.19208i	−0.972404	+	0.233301i	$0.925047\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	3.00000	+	5.19615i	0.625543	+	1.08347i	0.988436	+	0.151642i	$0.0484560\pi$
$23$	3.00000	+	5.19615i	0.625543	+	1.08347i	−0.362892	+	0.931831i	$0.618211\pi$
$24$		0			0
$25$	−4.00000			−0.800000
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−0.500000	−	0.866025i	−0.0928477	−	0.160817i	0.815861	−	0.578249i	$-0.196264\pi$
$29$	−0.500000	−	0.866025i	−0.0928477	−	0.160817i	−0.908708	+	0.417432i	$0.862930\pi$
$30$		0			0
$31$	−4.00000			−0.718421			−0.359211	−	0.933257i	$-0.616954\pi$
$31$	−4.00000			−0.718421			−0.359211	+	0.933257i	$0.616954\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	1.00000	−	1.73205i	0.169031	−	0.292770i
$36$		0			0
$37$	−0.500000	−	0.866025i	−0.0821995	−	0.142374i	0.821995	−	0.569495i	$-0.192861\pi$
$37$	−0.500000	−	0.866025i	−0.0821995	−	0.142374i	−0.904194	+	0.427121i	$0.859528\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	4.50000	+	7.79423i	0.702782	+	1.21725i	0.967486	+	0.252924i	$0.0813924\pi$
$41$	4.50000	+	7.79423i	0.702782	+	1.21725i	−0.264704	+	0.964330i	$0.585274\pi$
$42$		0			0
$43$	3.00000	−	5.19615i	0.457496	−	0.792406i	−0.541332	−	0.840809i	$-0.682080\pi$
$43$	3.00000	−	5.19615i	0.457496	−	0.792406i	0.998828	+	0.0484030i	$0.0154132\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	6.00000			0.875190			0.437595	−	0.899172i	$-0.355830\pi$
$47$	6.00000			0.875190			0.437595	+	0.899172i	$0.355830\pi$
$48$		0			0
$49$	1.50000	+	2.59808i	0.214286	+	0.371154i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	9.00000			1.23625			0.618123	−	0.786082i	$-0.287894\pi$
$53$	9.00000			1.23625			0.618123	+	0.786082i	$0.287894\pi$
$54$		0			0
$55$	1.00000	+	1.73205i	0.134840	+	0.233550i
$56$		0			0
$57$		0			0
$58$		0			0
$59$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$59$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$60$		0			0
$61$	−0.500000	+	0.866025i	−0.0640184	+	0.110883i	−0.896258	−	0.443533i	$-0.853725\pi$
$61$	−0.500000	+	0.866025i	−0.0640184	+	0.110883i	0.832240	+	0.554416i	$0.187058\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−3.50000	+	0.866025i	−0.434122	+	0.107417i
$66$		0			0
$67$	−1.00000	−	1.73205i	−0.122169	−	0.211604i	0.798454	−	0.602056i	$-0.205652\pi$
$67$	−1.00000	−	1.73205i	−0.122169	−	0.211604i	−0.920623	+	0.390453i	$0.872318\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−3.00000	+	5.19615i	−0.356034	+	0.616670i	−0.987294	−	0.158901i	$-0.949205\pi$
$71$	−3.00000	+	5.19615i	−0.356034	+	0.616670i	0.631260	+	0.775571i	$0.282538\pi$
$72$		0			0
$73$	11.0000			1.28745			0.643726	−	0.765256i	$-0.277388\pi$
$73$	11.0000			1.28745			0.643726	+	0.765256i	$0.277388\pi$
$74$		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
$83$	−14.0000			−1.53670			−0.768350	−	0.640030i	$-0.778922\pi$
$83$	−14.0000			−1.53670			−0.768350	+	0.640030i	$0.778922\pi$
$84$		0			0
$85$	−3.50000	+	6.06218i	−0.379628	+	0.657536i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−7.00000	−	12.1244i	−0.741999	−	1.28518i	−0.951584	−	0.307389i	$-0.900545\pi$
$89$	−7.00000	−	12.1244i	−0.741999	−	1.28518i	0.209585	−	0.977790i	$-0.432789\pi$
$90$		0			0
$91$	−2.00000	+	6.92820i	−0.209657	+	0.726273i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−3.00000	+	5.19615i	−0.307794	+	0.533114i
$96$		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
$101$	1.50000	+	2.59808i	0.149256	+	0.258518i	0.930953	−	0.365140i	$-0.118979\pi$
$101$	1.50000	+	2.59808i	0.149256	+	0.258518i	−0.781697	+	0.623658i	$0.785646\pi$
$102$		0			0
$103$	−6.00000			−0.591198			−0.295599	−	0.955312i	$-0.595519\pi$
$103$	−6.00000			−0.591198			−0.295599	+	0.955312i	$0.595519\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	3.00000	+	5.19615i	0.290021	+	0.502331i	0.973814	−	0.227345i	$-0.0730044\pi$
$107$	3.00000	+	5.19615i	0.290021	+	0.502331i	−0.683793	+	0.729676i	$0.739671\pi$
$108$		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
$113$	−7.50000	+	12.9904i	−0.705541	+	1.22203i	0.260955	+	0.965351i	$0.415962\pi$
$113$	−7.50000	+	12.9904i	−0.705541	+	1.22203i	−0.966496	+	0.256681i	$0.917371\pi$
$114$		0			0
$115$	3.00000	+	5.19615i	0.279751	+	0.484544i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	7.00000	+	12.1244i	0.641689	+	1.11144i
$120$		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
$127$	10.0000	+	17.3205i	0.887357	+	1.53695i	0.842989	+	0.537931i	$0.180794\pi$
$127$	10.0000	+	17.3205i	0.887357	+	1.53695i	0.0443678	+	0.999015i	$0.485873\pi$
$128$		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
$133$	6.00000	+	10.3923i	0.520266	+	0.901127i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−1.50000	+	2.59808i	−0.128154	+	0.221969i	−0.922961	−	0.384893i	$-0.874238\pi$
$137$	−1.50000	+	2.59808i	−0.128154	+	0.221969i	0.794808	+	0.606861i	$0.207572\pi$
$138$		0			0
$139$	6.00000	−	10.3923i	0.508913	−	0.881464i	−0.491033	−	0.871141i	$-0.663381\pi$
$139$	6.00000	−	10.3923i	0.508913	−	0.881464i	0.999947	−	0.0103230i	$-0.00328598\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−5.00000	−	5.19615i	−0.418121	−	0.434524i
$144$		0			0
$145$	−0.500000	−	0.866025i	−0.0415227	−	0.0719195i
$146$		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
$151$	2.00000			0.162758			0.0813788	−	0.996683i	$-0.474068\pi$
$151$	2.00000			0.162758			0.0813788	+	0.996683i	$0.474068\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−4.00000			−0.321288
$156$		0			0
$157$	−3.00000			−0.239426			−0.119713	−	0.992809i	$-0.538197\pi$
$157$	−3.00000			−0.239426			−0.119713	+	0.992809i	$0.538197\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	12.0000			0.945732
$162$		0			0
$163$	−2.00000	+	3.46410i	−0.156652	+	0.271329i	−0.933659	−	0.358162i	$-0.883403\pi$
$163$	−2.00000	+	3.46410i	−0.156652	+	0.271329i	0.777007	+	0.629492i	$0.216737\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−8.00000	−	13.8564i	−0.619059	−	1.07224i	−0.989658	−	0.143448i	$-0.954181\pi$
$167$	−8.00000	−	13.8564i	−0.619059	−	1.07224i	0.370599	−	0.928793i	$-0.379152\pi$
$168$		0			0
$169$	11.5000	−	6.06218i	0.884615	−	0.466321i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−3.00000	+	5.19615i	−0.228086	+	0.395056i	−0.957241	−	0.289292i	$-0.906580\pi$
$173$	−3.00000	+	5.19615i	−0.228086	+	0.395056i	0.729155	+	0.684349i	$0.239913\pi$
$174$		0			0
$175$	−4.00000	+	6.92820i	−0.302372	+	0.523723i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	1.00000	+	1.73205i	0.0747435	+	0.129460i	0.900975	−	0.433872i	$-0.142853\pi$
$179$	1.00000	+	1.73205i	0.0747435	+	0.129460i	−0.826231	+	0.563331i	$0.809520\pi$
$180$		0			0
$181$	−7.00000			−0.520306			−0.260153	−	0.965567i	$-0.583773\pi$
$181$	−7.00000			−0.520306			−0.260153	+	0.965567i	$0.583773\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−0.500000	−	0.866025i	−0.0367607	−	0.0636715i
$186$		0			0
$187$	−14.0000			−1.02378
$188$		0			0
$189$		0			0
$190$		0			0
$191$	2.00000	−	3.46410i	0.144715	−	0.250654i	−0.784552	−	0.620063i	$-0.787107\pi$
$191$	2.00000	−	3.46410i	0.144715	−	0.250654i	0.929267	+	0.369410i	$0.120440\pi$
$192$		0			0
$193$	4.50000	+	7.79423i	0.323917	+	0.561041i	0.981293	−	0.192522i	$-0.0616668\pi$
$193$	4.50000	+	7.79423i	0.323917	+	0.561041i	−0.657376	+	0.753563i	$0.728333\pi$
$194$		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
$199$	7.00000	−	12.1244i	0.496217	−	0.859473i	−0.503774	−	0.863836i	$-0.668055\pi$
$199$	7.00000	−	12.1244i	0.496217	−	0.859473i	0.999990	+	0.00436292i	$0.00138876\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−2.00000			−0.140372
$204$		0			0
$205$	4.50000	+	7.79423i	0.314294	+	0.544373i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−12.0000			−0.830057
$210$		0			0
$211$	−4.00000	−	6.92820i	−0.275371	−	0.476957i	0.694857	−	0.719148i	$-0.255467\pi$
$211$	−4.00000	−	6.92820i	−0.275371	−	0.476957i	−0.970229	+	0.242190i	$0.922134\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	3.00000	−	5.19615i	0.204598	−	0.354375i
$216$		0			0
$217$	−4.00000	+	6.92820i	−0.271538	+	0.470317i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	7.00000	−	24.2487i	0.470871	−	1.63114i
$222$		0			0
$223$	8.00000	+	13.8564i	0.535720	+	0.927894i	0.999128	+	0.0417488i	$0.0132929\pi$
$223$	8.00000	+	13.8564i	0.535720	+	0.927894i	−0.463409	+	0.886145i	$0.653374\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	7.00000	−	12.1244i	0.464606	−	0.804722i	−0.534577	−	0.845120i	$-0.679529\pi$
$227$	7.00000	−	12.1244i	0.464606	−	0.804722i	0.999184	+	0.0403978i	$0.0128625\pi$
$228$		0			0
$229$	−22.0000			−1.45380			−0.726900	−	0.686743i	$-0.759040\pi$
$229$	−22.0000			−1.45380			−0.726900	+	0.686743i	$0.759040\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−10.0000			−0.655122			−0.327561	−	0.944830i	$-0.606227\pi$
$233$	−10.0000			−0.655122			−0.327561	+	0.944830i	$0.606227\pi$
$234$		0			0
$235$	6.00000			0.391397
$236$		0			0
$237$		0			0
$238$		0			0
$239$	30.0000			1.94054			0.970269	−	0.242028i	$-0.0778125\pi$
$239$	30.0000			1.94054			0.970269	+	0.242028i	$0.0778125\pi$
$240$		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
$247$	6.00000	−	20.7846i	0.381771	−	1.32249i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−6.00000	+	10.3923i	−0.378717	+	0.655956i	−0.990876	−	0.134778i	$-0.956968\pi$
$251$	−6.00000	+	10.3923i	−0.378717	+	0.655956i	0.612159	+	0.790735i	$0.290301\pi$
$252$		0			0
$253$	−6.00000	+	10.3923i	−0.377217	+	0.653359i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−3.50000	−	6.06218i	−0.218324	−	0.378148i	0.735972	−	0.677012i	$-0.236726\pi$
$257$	−3.50000	−	6.06218i	−0.218324	−	0.378148i	−0.954296	+	0.298864i	$0.903392\pi$
$258$		0			0
$259$	−2.00000			−0.124274
$260$		0			0
$261$		0			0
$262$		0			0
$263$	15.0000	+	25.9808i	0.924940	+	1.60204i	0.791658	+	0.610964i	$0.209218\pi$
$263$	15.0000	+	25.9808i	0.924940	+	1.60204i	0.133281	+	0.991078i	$0.457449\pi$
$264$		0			0
$265$	9.00000			0.552866
$266$		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
$271$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$271$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−4.00000	−	6.92820i	−0.241209	−	0.417786i
$276$		0			0
$277$	15.5000	−	26.8468i	0.931305	−	1.61307i	0.150210	−	0.988654i	$-0.452005\pi$
$277$	15.5000	−	26.8468i	0.931305	−	1.61307i	0.781094	−	0.624413i	$-0.214662\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	19.0000			1.13344			0.566722	−	0.823909i	$-0.308211\pi$
$281$	19.0000			1.13344			0.566722	+	0.823909i	$0.308211\pi$
$282$		0			0
$283$	−9.00000	−	15.5885i	−0.534994	−	0.926638i	−0.999164	−	0.0408910i	$-0.986980\pi$
$283$	−9.00000	−	15.5885i	−0.534994	−	0.926638i	0.464169	−	0.885747i	$-0.346353\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	18.0000			1.06251
$288$		0			0
$289$	−16.0000	−	27.7128i	−0.941176	−	1.63017i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−4.50000	+	7.79423i	−0.262893	+	0.455344i	−0.967009	−	0.254741i	$-0.918010\pi$
$293$	−4.50000	+	7.79423i	−0.262893	+	0.455344i	0.704117	+	0.710084i	$0.251343\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−15.0000	−	15.5885i	−0.867472	−	0.901504i
$300$		0			0
$301$	−6.00000	−	10.3923i	−0.345834	−	0.599002i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−0.500000	+	0.866025i	−0.0286299	+	0.0495885i
$306$		0			0
$307$	−14.0000			−0.799022			−0.399511	−	0.916728i	$-0.630820\pi$
$307$	−14.0000			−0.799022			−0.399511	+	0.916728i	$0.630820\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	18.0000			1.02069			0.510343	−	0.859971i	$-0.329518\pi$
$311$	18.0000			1.02069			0.510343	+	0.859971i	$0.329518\pi$
$312$		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
$317$	25.0000			1.40414			0.702070	−	0.712108i	$-0.252259\pi$
$317$	25.0000			1.40414			0.702070	+	0.712108i	$0.252259\pi$
$318$		0			0
$319$	1.00000	−	1.73205i	0.0559893	−	0.0969762i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−21.0000	−	36.3731i	−1.16847	−	2.02385i
$324$		0			0
$325$	14.0000	−	3.46410i	0.776580	−	0.192154i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	6.00000	−	10.3923i	0.330791	−	0.572946i
$330$		0			0
$331$	−2.00000	+	3.46410i	−0.109930	+	0.190404i	−0.915742	−	0.401768i	$-0.868396\pi$
$331$	−2.00000	+	3.46410i	−0.109930	+	0.190404i	0.805812	+	0.592172i	$0.201729\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−1.00000	−	1.73205i	−0.0546358	−	0.0946320i
$336$		0			0
$337$	−33.0000			−1.79762			−0.898812	−	0.438334i	$-0.855569\pi$
$337$	−33.0000			−1.79762			−0.898812	+	0.438334i	$0.855569\pi$
$338$		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
$347$	−9.00000	+	15.5885i	−0.483145	+	0.836832i	−0.999813	−	0.0193540i	$-0.993839\pi$
$347$	−9.00000	+	15.5885i	−0.483145	+	0.836832i	0.516667	+	0.856186i	$0.327172\pi$
$348$		0			0
$349$	13.0000	+	22.5167i	0.695874	+	1.20529i	0.969885	+	0.243563i	$0.0783162\pi$
$349$	13.0000	+	22.5167i	0.695874	+	1.20529i	−0.274011	+	0.961727i	$0.588351\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−5.50000	−	9.52628i	−0.292735	−	0.507033i	0.681720	−	0.731613i	$-0.261232\pi$
$353$	−5.50000	−	9.52628i	−0.292735	−	0.507033i	−0.974456	+	0.224580i	$0.927899\pi$
$354$		0			0
$355$	−3.00000	+	5.19615i	−0.159223	+	0.275783i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−18.0000			−0.950004			−0.475002	−	0.879985i	$-0.657553\pi$
$359$	−18.0000			−0.950004			−0.475002	+	0.879985i	$0.657553\pi$
$360$		0			0
$361$	−8.50000	−	14.7224i	−0.447368	−	0.774865i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	11.0000			0.575766
$366$		0			0
$367$	5.00000	+	8.66025i	0.260998	+	0.452062i	0.966507	−	0.256639i	$-0.0826151\pi$
$367$	5.00000	+	8.66025i	0.260998	+	0.452062i	−0.705509	+	0.708700i	$0.749282\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	9.00000	−	15.5885i	0.467257	−	0.809312i
$372$		0			0
$373$	5.50000	−	9.52628i	0.284779	−	0.493252i	−0.687776	−	0.725923i	$-0.741413\pi$
$373$	5.50000	−	9.52628i	0.284779	−	0.493252i	0.972556	+	0.232671i	$0.0747464\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	2.50000	+	2.59808i	0.128757	+	0.133808i
$378$		0			0
$379$	−18.0000	−	31.1769i	−0.924598	−	1.60145i	−0.792207	−	0.610253i	$-0.791068\pi$
$379$	−18.0000	−	31.1769i	−0.924598	−	1.60145i	−0.132391	−	0.991198i	$-0.542266\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−4.00000	+	6.92820i	−0.204390	+	0.354015i	−0.949938	−	0.312437i	$-0.898855\pi$
$383$	−4.00000	+	6.92820i	−0.204390	+	0.354015i	0.745548	+	0.666452i	$0.232188\pi$
$384$		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
$391$	−42.0000			−2.12403
$392$		0			0
$393$		0			0
$394$		0			0
$395$	4.00000			0.201262
$396$		0			0
$397$	17.0000	−	29.4449i	0.853206	−	1.47780i	−0.0250943	−	0.999685i	$-0.507989\pi$
$397$	17.0000	−	29.4449i	0.853206	−	1.47780i	0.878300	−	0.478110i	$-0.158678\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	0.500000	+	0.866025i	0.0249688	+	0.0432472i	0.878240	−	0.478220i	$-0.158718\pi$
$401$	0.500000	+	0.866025i	0.0249688	+	0.0432472i	−0.853271	+	0.521468i	$0.825385\pi$
$402$		0			0
$403$	14.0000	−	3.46410i	0.697390	−	0.172559i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	1.00000	−	1.73205i	0.0495682	−	0.0858546i
$408$		0			0
$409$	−3.50000	+	6.06218i	−0.173064	+	0.299755i	−0.939490	−	0.342578i	$-0.888700\pi$
$409$	−3.50000	+	6.06218i	−0.173064	+	0.299755i	0.766426	+	0.642333i	$0.222033\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$	−14.0000			−0.687233
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−8.00000	−	13.8564i	−0.390826	−	0.676930i	0.601733	−	0.798697i	$-0.294477\pi$
$419$	−8.00000	−	13.8564i	−0.390826	−	0.676930i	−0.992559	+	0.121768i	$0.961144\pi$
$420$		0			0
$421$	−19.0000			−0.926003			−0.463002	−	0.886357i	$-0.653228\pi$
$421$	−19.0000			−0.926003			−0.463002	+	0.886357i	$0.653228\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	14.0000	−	24.2487i	0.679100	−	1.17624i
$426$		0			0
$427$	1.00000	+	1.73205i	0.0483934	+	0.0838198i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	15.0000	+	25.9808i	0.722525	+	1.25145i	0.959985	+	0.280052i	$0.0903517\pi$
$431$	15.0000	+	25.9808i	0.722525	+	1.25145i	−0.237460	+	0.971397i	$0.576315\pi$
$432$		0			0
$433$	−9.50000	+	16.4545i	−0.456541	+	0.790752i	−0.998775	−	0.0494752i	$-0.984245\pi$
$433$	−9.50000	+	16.4545i	−0.456541	+	0.790752i	0.542234	+	0.840227i	$0.317578\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−36.0000			−1.72211
$438$		0			0
$439$	7.00000	+	12.1244i	0.334092	+	0.578664i	0.983310	−	0.181938i	$-0.0582371\pi$
$439$	7.00000	+	12.1244i	0.334092	+	0.578664i	−0.649218	+	0.760602i	$0.724904\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−4.00000			−0.190046			−0.0950229	−	0.995475i	$-0.530292\pi$
$443$	−4.00000			−0.190046			−0.0950229	+	0.995475i	$0.530292\pi$
$444$		0			0
$445$	−7.00000	−	12.1244i	−0.331832	−	0.574750i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	17.0000	−	29.4449i	0.802280	−	1.38959i	−0.115833	−	0.993269i	$-0.536954\pi$
$449$	17.0000	−	29.4449i	0.802280	−	1.38959i	0.918112	−	0.396320i	$-0.129713\pi$
$450$		0			0
$451$	−9.00000	+	15.5885i	−0.423793	+	0.734032i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−2.00000	+	6.92820i	−0.0937614	+	0.324799i
$456$		0			0
$457$	6.50000	+	11.2583i	0.304057	+	0.526642i	0.977051	−	0.213006i	$-0.0683253\pi$
$457$	6.50000	+	11.2583i	0.304057	+	0.526642i	−0.672994	+	0.739648i	$0.734992\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	9.50000	−	16.4545i	0.442459	−	0.766362i	−0.555412	−	0.831575i	$-0.687440\pi$
$461$	9.50000	−	16.4545i	0.442459	−	0.766362i	0.997871	+	0.0652135i	$0.0207728\pi$
$462$		0			0
$463$	26.0000			1.20832			0.604161	−	0.796862i	$-0.293508\pi$
$463$	26.0000			1.20832			0.604161	+	0.796862i	$0.293508\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	6.00000			0.277647			0.138823	−	0.990317i	$-0.455668\pi$
$467$	6.00000			0.277647			0.138823	+	0.990317i	$0.455668\pi$
$468$		0			0
$469$	−4.00000			−0.184703
$470$		0			0
$471$		0			0
$472$		0			0
$473$	12.0000			0.551761
$474$		0			0
$475$	12.0000	−	20.7846i	0.550598	−	0.953663i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−12.0000	−	20.7846i	−0.548294	−	0.949673i	−0.998392	−	0.0566937i	$-0.981944\pi$
$479$	−12.0000	−	20.7846i	−0.548294	−	0.949673i	0.450098	−	0.892979i	$-0.351389\pi$
$480$		0			0
$481$	2.50000	+	2.59808i	0.113990	+	0.118462i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	1.00000	−	1.73205i	0.0454077	−	0.0786484i
$486$		0			0
$487$	9.00000	−	15.5885i	0.407829	−	0.706380i	−0.586817	−	0.809719i	$-0.699619\pi$
$487$	9.00000	−	15.5885i	0.407829	−	0.706380i	0.994646	+	0.103339i	$0.0329526\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−3.00000	−	5.19615i	−0.135388	−	0.234499i	0.790358	−	0.612646i	$-0.209895\pi$
$491$	−3.00000	−	5.19615i	−0.135388	−	0.234499i	−0.925746	+	0.378147i	$0.876561\pi$
$492$		0			0
$493$	7.00000			0.315264
$494$		0			0
$495$		0			0
$496$		0			0
$497$	6.00000	+	10.3923i	0.269137	+	0.466159i
$498$		0			0
$499$	−24.0000			−1.07439			−0.537194	−	0.843459i	$-0.680516\pi$
$499$	−24.0000			−1.07439			−0.537194	+	0.843459i	$0.680516\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	1.00000	−	1.73205i	0.0445878	−	0.0772283i	−0.842870	−	0.538117i	$-0.819136\pi$
$503$	1.00000	−	1.73205i	0.0445878	−	0.0772283i	0.887458	+	0.460889i	$0.152469\pi$
$504$		0			0
$505$	1.50000	+	2.59808i	0.0667491	+	0.115613i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	3.50000	+	6.06218i	0.155135	+	0.268701i	0.933108	−	0.359596i	$-0.117085\pi$
$509$	3.50000	+	6.06218i	0.155135	+	0.268701i	−0.777973	+	0.628297i	$0.783752\pi$
$510$		0			0
$511$	11.0000	−	19.0526i	0.486611	−	0.842836i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−6.00000			−0.264392
$516$		0			0
$517$	6.00000	+	10.3923i	0.263880	+	0.457053i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	3.00000			0.131432			0.0657162	−	0.997838i	$-0.479067\pi$
$521$	3.00000			0.131432			0.0657162	+	0.997838i	$0.479067\pi$
$522$		0			0
$523$	7.00000	+	12.1244i	0.306089	+	0.530161i	0.977503	−	0.210921i	$-0.0676463\pi$
$523$	7.00000	+	12.1244i	0.306089	+	0.530161i	−0.671414	+	0.741082i	$0.734313\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	14.0000	−	24.2487i	0.609850	−	1.05629i
$528$		0			0
$529$	−6.50000	+	11.2583i	−0.282609	+	0.489493i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−22.5000	−	23.3827i	−0.974583	−	1.01282i
$534$		0			0
$535$	3.00000	+	5.19615i	0.129701	+	0.224649i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−3.00000	+	5.19615i	−0.129219	+	0.223814i
$540$		0			0
$541$	45.0000			1.93470			0.967351	−	0.253442i	$-0.0815627\pi$
$541$	45.0000			1.93470			0.967351	+	0.253442i	$0.0815627\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−2.00000			−0.0856706
$546$		0			0
$547$	26.0000			1.11168			0.555840	−	0.831289i	$-0.312397\pi$
$547$	26.0000			1.11168			0.555840	+	0.831289i	$0.312397\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	6.00000			0.255609
$552$		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
$559$	−6.00000	+	20.7846i	−0.253773	+	0.879095i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$563$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$564$		0			0
$565$	−7.50000	+	12.9904i	−0.315527	+	0.546509i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	11.0000	+	19.0526i	0.461144	+	0.798725i	0.999018	−	0.0443003i	$-0.0141058\pi$
$569$	11.0000	+	19.0526i	0.461144	+	0.798725i	−0.537874	+	0.843025i	$0.680772\pi$
$570$		0			0
$571$	−26.0000			−1.08807			−0.544033	−	0.839064i	$-0.683103\pi$
$571$	−26.0000			−1.08807			−0.544033	+	0.839064i	$0.683103\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−12.0000	−	20.7846i	−0.500435	−	0.866778i
$576$		0			0
$577$	11.0000			0.457936			0.228968	−	0.973434i	$-0.426465\pi$
$577$	11.0000			0.457936			0.228968	+	0.973434i	$0.426465\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−14.0000	+	24.2487i	−0.580818	+	1.00601i
$582$		0			0
$583$	9.00000	+	15.5885i	0.372742	+	0.645608i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−8.00000	−	13.8564i	−0.330195	−	0.571915i	0.652355	−	0.757914i	$-0.273781\pi$
$587$	−8.00000	−	13.8564i	−0.330195	−	0.571915i	−0.982550	+	0.185999i	$0.940448\pi$
$588$		0			0
$589$	12.0000	−	20.7846i	0.494451	−	0.856415i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−13.0000			−0.533846			−0.266923	−	0.963718i	$-0.586007\pi$
$593$	−13.0000			−0.533846			−0.266923	+	0.963718i	$0.586007\pi$
$594$		0			0
$595$	7.00000	+	12.1244i	0.286972	+	0.497050i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−16.0000			−0.653742			−0.326871	−	0.945069i	$-0.605994\pi$
$599$	−16.0000			−0.653742			−0.326871	+	0.945069i	$0.605994\pi$
$600$		0			0
$601$	2.50000	+	4.33013i	0.101977	+	0.176630i	0.912499	−	0.409079i	$-0.134150\pi$
$601$	2.50000	+	4.33013i	0.101977	+	0.176630i	−0.810522	+	0.585708i	$0.800816\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	3.50000	−	6.06218i	0.142295	−	0.246463i
$606$		0			0
$607$	4.00000	−	6.92820i	0.162355	−	0.281207i	−0.773358	−	0.633970i	$-0.781424\pi$
$607$	4.00000	−	6.92820i	0.162355	−	0.281207i	0.935713	+	0.352763i	$0.114758\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−21.0000	+	5.19615i	−0.849569	+	0.210214i
$612$		0			0
$613$	11.5000	+	19.9186i	0.464481	+	0.804504i	0.999178	−	0.0405396i	$-0.0129077\pi$
$613$	11.5000	+	19.9186i	0.464481	+	0.804504i	−0.534697	+	0.845044i	$0.679574\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	6.50000	−	11.2583i	0.261680	−	0.453243i	−0.705008	−	0.709199i	$-0.749057\pi$
$617$	6.50000	−	11.2583i	0.261680	−	0.453243i	0.966689	+	0.255956i	$0.0823901\pi$
$618$		0			0
$619$	24.0000			0.964641			0.482321	−	0.875995i	$-0.339794\pi$
$619$	24.0000			0.964641			0.482321	+	0.875995i	$0.339794\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−28.0000			−1.12180
$624$		0			0
$625$	11.0000			0.440000
$626$		0			0
$627$		0			0
$628$		0			0
$629$	7.00000			0.279108
$630$		0			0
$631$	10.0000	−	17.3205i	0.398094	−	0.689519i	−0.595397	−	0.803432i	$-0.703005\pi$
$631$	10.0000	−	17.3205i	0.398094	−	0.689519i	0.993491	+	0.113913i	$0.0363385\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	10.0000	+	17.3205i	0.396838	+	0.687343i
$636$		0			0
$637$	−7.50000	−	7.79423i	−0.297161	−	0.308819i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−15.5000	+	26.8468i	−0.612213	+	1.06038i	0.378653	+	0.925539i	$0.376387\pi$
$641$	−15.5000	+	26.8468i	−0.612213	+	1.06038i	−0.990867	+	0.134846i	$0.956946\pi$
$642$		0			0
$643$	−8.00000	+	13.8564i	−0.315489	+	0.546443i	−0.979541	−	0.201243i	$-0.935502\pi$
$643$	−8.00000	+	13.8564i	−0.315489	+	0.546443i	0.664052	+	0.747686i	$0.268835\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	16.0000	+	27.7128i	0.629025	+	1.08950i	0.987748	+	0.156059i	$0.0498790\pi$
$647$	16.0000	+	27.7128i	0.629025	+	1.08950i	−0.358723	+	0.933444i	$0.616788\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−3.00000	−	5.19615i	−0.117399	−	0.203341i	0.801337	−	0.598213i	$-0.204122\pi$
$653$	−3.00000	−	5.19615i	−0.117399	−	0.203341i	−0.918736	+	0.394872i	$0.870789\pi$
$654$		0			0
$655$	−8.00000			−0.312586
$656$		0			0
$657$		0			0
$658$		0			0
$659$	4.00000	−	6.92820i	0.155818	−	0.269884i	−0.777539	−	0.628835i	$-0.783532\pi$
$659$	4.00000	−	6.92820i	0.155818	−	0.269884i	0.933357	+	0.358951i	$0.116865\pi$
$660$		0			0
$661$	−22.5000	−	38.9711i	−0.875149	−	1.51580i	−0.856604	−	0.515974i	$-0.827430\pi$
$661$	−22.5000	−	38.9711i	−0.875149	−	1.51580i	−0.0185442	−	0.999828i	$-0.505903\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	6.00000	+	10.3923i	0.232670	+	0.402996i
$666$		0			0
$667$	3.00000	−	5.19615i	0.116160	−	0.201196i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−2.00000			−0.0772091
$672$		0			0
$673$	14.5000	+	25.1147i	0.558934	+	0.968102i	0.997586	+	0.0694449i	$0.0221228\pi$
$673$	14.5000	+	25.1147i	0.558934	+	0.968102i	−0.438652	+	0.898657i	$0.644544\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−34.0000			−1.30673			−0.653363	−	0.757045i	$-0.726642\pi$
$677$	−34.0000			−1.30673			−0.653363	+	0.757045i	$0.726642\pi$
$678$		0			0
$679$	−2.00000	−	3.46410i	−0.0767530	−	0.132940i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	12.0000	−	20.7846i	0.459167	−	0.795301i	−0.539750	−	0.841825i	$-0.681481\pi$
$683$	12.0000	−	20.7846i	0.459167	−	0.795301i	0.998917	+	0.0465244i	$0.0148145\pi$
$684$		0			0
$685$	−1.50000	+	2.59808i	−0.0573121	+	0.0992674i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−31.5000	+	7.79423i	−1.20005	+	0.296936i
$690$		0			0
$691$	21.0000	+	36.3731i	0.798878	+	1.38370i	0.920348	+	0.391102i	$0.127906\pi$
$691$	21.0000	+	36.3731i	0.798878	+	1.38370i	−0.121470	+	0.992595i	$0.538761\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	6.00000	−	10.3923i	0.227593	−	0.394203i
$696$		0			0
$697$	−63.0000			−2.38630
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−2.00000			−0.0755390			−0.0377695	−	0.999286i	$-0.512025\pi$
$701$	−2.00000			−0.0755390			−0.0377695	+	0.999286i	$0.512025\pi$
$702$		0			0
$703$	6.00000			0.226294
$704$		0			0
$705$		0			0
$706$		0			0
$707$	6.00000			0.225653
$708$		0			0
$709$	5.50000	−	9.52628i	0.206557	−	0.357767i	−0.744071	−	0.668101i	$-0.767108\pi$
$709$	5.50000	−	9.52628i	0.206557	−	0.357767i	0.950628	+	0.310334i	$0.100441\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−12.0000	−	20.7846i	−0.449404	−	0.778390i
$714$		0			0
$715$	−5.00000	−	5.19615i	−0.186989	−	0.194325i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	24.0000	−	41.5692i	0.895049	−	1.55027i	0.0613050	−	0.998119i	$-0.480474\pi$
$719$	24.0000	−	41.5692i	0.895049	−	1.55027i	0.833744	−	0.552151i	$-0.186193\pi$
$720$		0			0
$721$	−6.00000	+	10.3923i	−0.223452	+	0.387030i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	2.00000	+	3.46410i	0.0742781	+	0.128654i
$726$		0			0
$727$	−14.0000			−0.519231			−0.259616	−	0.965712i	$-0.583596\pi$
$727$	−14.0000			−0.519231			−0.259616	+	0.965712i	$0.583596\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	21.0000	+	36.3731i	0.776713	+	1.34531i
$732$		0			0
$733$	−15.0000			−0.554038			−0.277019	−	0.960864i	$-0.589346\pi$
$733$	−15.0000			−0.554038			−0.277019	+	0.960864i	$0.589346\pi$
$734$		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	0.680534	−	0.732717i	$-0.261748\pi$
$739$	−8.00000	−	13.8564i	−0.294285	−	0.509716i	−0.974818	+	0.223001i	$0.928415\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	18.0000	+	31.1769i	0.660356	+	1.14377i	0.980522	+	0.196409i	$0.0629279\pi$
$743$	18.0000	+	31.1769i	0.660356	+	1.14377i	−0.320166	+	0.947361i	$0.603739\pi$
$744$		0			0
$745$	1.50000	−	2.59808i	0.0549557	−	0.0951861i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	12.0000			0.438470
$750$		0			0
$751$	−17.0000	−	29.4449i	−0.620339	−	1.07446i	−0.989423	−	0.145062i	$-0.953662\pi$
$751$	−17.0000	−	29.4449i	−0.620339	−	1.07446i	0.369084	−	0.929396i	$-0.379672\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	2.00000			0.0727875
$756$		0			0
$757$	25.0000	+	43.3013i	0.908640	+	1.57381i	0.815955	+	0.578116i	$0.196212\pi$
$757$	25.0000	+	43.3013i	0.908640	+	1.57381i	0.0926859	+	0.995695i	$0.470455\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	25.0000	−	43.3013i	0.906249	−	1.56967i	0.0870179	−	0.996207i	$-0.472266\pi$
$761$	25.0000	−	43.3013i	0.906249	−	1.56967i	0.819231	−	0.573463i	$-0.194400\pi$
$762$		0			0
$763$	−2.00000	+	3.46410i	−0.0724049	+	0.125409i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	−15.0000	−	25.9808i	−0.540914	−	0.936890i	−0.998852	−	0.0479061i	$-0.984745\pi$
$769$	−15.0000	−	25.9808i	−0.540914	−	0.936890i	0.457938	−	0.888984i	$-0.348588\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−7.00000	+	12.1244i	−0.251773	+	0.436083i	−0.964014	−	0.265852i	$-0.914347\pi$
$773$	−7.00000	+	12.1244i	−0.251773	+	0.436083i	0.712241	+	0.701935i	$0.247680\pi$
$774$		0			0
$775$	16.0000			0.574737
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−54.0000			−1.93475
$780$		0			0
$781$	−12.0000			−0.429394
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−3.00000			−0.107075
$786$		0			0
$787$	14.0000	−	24.2487i	0.499046	−	0.864373i	−0.500953	−	0.865474i	$-0.667017\pi$
$787$	14.0000	−	24.2487i	0.499046	−	0.864373i	0.999999	+	0.00110111i	$0.000350496\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	15.0000	+	25.9808i	0.533339	+	0.923770i
$792$		0			0
$793$	1.00000	−	3.46410i	0.0355110	−	0.123014i
$794$		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
$799$	−21.0000	+	36.3731i	−0.742927	+	1.28679i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	11.0000	+	19.0526i	0.388182	+	0.672350i
$804$		0			0
$805$	12.0000			0.422944
$806$		0			0
$807$		0			0
$808$		0			0
$809$	16.5000	+	28.5788i	0.580109	+	1.00478i	0.995466	+	0.0951198i	$0.0303234\pi$
$809$	16.5000	+	28.5788i	0.580109	+	1.00478i	−0.415357	+	0.909659i	$0.636343\pi$
$810$		0			0
$811$	28.0000			0.983213			0.491606	−	0.870817i	$-0.336410\pi$
$811$	28.0000			0.983213			0.491606	+	0.870817i	$0.336410\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−2.00000	+	3.46410i	−0.0700569	+	0.121342i
$816$		0			0
$817$	18.0000	+	31.1769i	0.629740	+	1.09074i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	25.0000	+	43.3013i	0.872506	+	1.51122i	0.859396	+	0.511311i	$0.170840\pi$
$821$	25.0000	+	43.3013i	0.872506	+	1.51122i	0.0131101	+	0.999914i	$0.495827\pi$
$822$		0			0
$823$	−12.0000	+	20.7846i	−0.418294	+	0.724506i	−0.995768	−	0.0919029i	$-0.970705\pi$
$823$	−12.0000	+	20.7846i	−0.418294	+	0.724506i	0.577474	+	0.816409i	$0.304038\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	16.0000			0.556375			0.278187	−	0.960527i	$-0.410266\pi$
$827$	16.0000			0.556375			0.278187	+	0.960527i	$0.410266\pi$
$828$		0			0
$829$	−8.50000	−	14.7224i	−0.295217	−	0.511331i	0.679818	−	0.733381i	$-0.262059\pi$
$829$	−8.50000	−	14.7224i	−0.295217	−	0.511331i	−0.975035	+	0.222049i	$0.928725\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−21.0000			−0.727607
$834$		0			0
$835$	−8.00000	−	13.8564i	−0.276851	−	0.479521i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−6.00000	+	10.3923i	−0.207143	+	0.358782i	−0.950813	−	0.309764i	$-0.899750\pi$
$839$	−6.00000	+	10.3923i	−0.207143	+	0.358782i	0.743670	+	0.668546i	$0.233083\pi$
$840$		0			0
$841$	14.0000	−	24.2487i	0.482759	−	0.836162i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	11.5000	−	6.06218i	0.395612	−	0.208545i
$846$		0			0
$847$	−7.00000	−	12.1244i	−0.240523	−	0.416598i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	3.00000	−	5.19615i	0.102839	−	0.178122i
$852$		0			0
$853$	21.0000			0.719026			0.359513	−	0.933140i	$-0.382943\pi$
$853$	21.0000			0.719026			0.359513	+	0.933140i	$0.382943\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	31.0000			1.05894			0.529470	−	0.848329i	$-0.322391\pi$
$857$	31.0000			1.05894			0.529470	+	0.848329i	$0.322391\pi$
$858$		0			0
$859$	−34.0000			−1.16007			−0.580033	−	0.814593i	$-0.696960\pi$
$859$	−34.0000			−1.16007			−0.580033	+	0.814593i	$0.696960\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	10.0000			0.340404			0.170202	−	0.985409i	$-0.445558\pi$
$863$	10.0000			0.340404			0.170202	+	0.985409i	$0.445558\pi$
$864$		0			0
$865$	−3.00000	+	5.19615i	−0.102003	+	0.176674i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	4.00000	+	6.92820i	0.135691	+	0.235023i
$870$		0			0
$871$	5.00000	+	5.19615i	0.169419	+	0.176065i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−9.00000	+	15.5885i	−0.304256	+	0.526986i
$876$		0			0
$877$	−8.50000	+	14.7224i	−0.287025	+	0.497141i	−0.973098	−	0.230391i	$-0.925999\pi$
$877$	−8.50000	+	14.7224i	−0.287025	+	0.497141i	0.686074	+	0.727532i	$0.259333\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	18.5000	+	32.0429i	0.623281	+	1.07955i	0.988871	+	0.148778i	$0.0475340\pi$
$881$	18.5000	+	32.0429i	0.623281	+	1.07955i	−0.365590	+	0.930776i	$0.619133\pi$
$882$		0			0
$883$	−16.0000			−0.538443			−0.269221	−	0.963078i	$-0.586766\pi$
$883$	−16.0000			−0.538443			−0.269221	+	0.963078i	$0.586766\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$887$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$888$		0			0
$889$	40.0000			1.34156
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−18.0000	+	31.1769i	−0.602347	+	1.04330i
$894$		0			0
$895$	1.00000	+	1.73205i	0.0334263	+	0.0578961i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	2.00000	+	3.46410i	0.0667037	+	0.115534i
$900$		0			0
$901$	−31.5000	+	54.5596i	−1.04942	+	1.81764i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−7.00000			−0.232688
$906$		0			0
$907$	6.00000	+	10.3923i	0.199227	+	0.345071i	0.948278	−	0.317441i	$-0.102824\pi$
$907$	6.00000	+	10.3923i	0.199227	+	0.345071i	−0.749051	+	0.662512i	$0.769490\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−40.0000			−1.32526			−0.662630	−	0.748947i	$-0.730560\pi$
$911$	−40.0000			−1.32526			−0.662630	+	0.748947i	$0.730560\pi$
$912$		0			0
$913$	−14.0000	−	24.2487i	−0.463332	−	0.802515i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−8.00000	+	13.8564i	−0.264183	+	0.457579i
$918$		0			0
$919$	12.0000	−	20.7846i	0.395843	−	0.685621i	−0.597365	−	0.801970i	$-0.703786\pi$
$919$	12.0000	−	20.7846i	0.395843	−	0.685621i	0.993208	+	0.116348i	$0.0371189\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	6.00000	−	20.7846i	0.197492	−	0.684134i
$924$		0			0
$925$	2.00000	+	3.46410i	0.0657596	+	0.113899i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−13.5000	+	23.3827i	−0.442921	+	0.767161i	−0.997905	−	0.0646999i	$-0.979391\pi$
$929$	−13.5000	+	23.3827i	−0.442921	+	0.767161i	0.554984	+	0.831861i	$0.312724\pi$
$930$		0			0
$931$	−18.0000			−0.589926
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−14.0000			−0.457849
$936$		0			0
$937$	−49.0000			−1.60076			−0.800380	−	0.599493i	$-0.795369\pi$
$937$	−49.0000			−1.60076			−0.800380	+	0.599493i	$0.795369\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	38.0000			1.23876			0.619382	−	0.785090i	$-0.287383\pi$
$941$	38.0000			1.23876			0.619382	+	0.785090i	$0.287383\pi$
$942$		0			0
$943$	−27.0000	+	46.7654i	−0.879241	+	1.52289i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	24.0000	+	41.5692i	0.779895	+	1.35082i	0.932002	+	0.362454i	$0.118061\pi$
$947$	24.0000	+	41.5692i	0.779895	+	1.35082i	−0.152106	+	0.988364i	$0.548606\pi$
$948$		0			0
$949$	−38.5000	+	9.52628i	−1.24976	+	0.309236i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	3.00000	−	5.19615i	0.0971795	−	0.168320i	−0.813337	−	0.581793i	$-0.802351\pi$
$953$	3.00000	−	5.19615i	0.0971795	−	0.168320i	0.910516	+	0.413473i	$0.135685\pi$
$954$		0			0
$955$	2.00000	−	3.46410i	0.0647185	−	0.112096i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	3.00000	+	5.19615i	0.0968751	+	0.167793i
$960$		0			0
$961$	−15.0000			−0.483871
$962$		0			0
$963$		0			0
$964$		0			0
$965$	4.50000	+	7.79423i	0.144860	+	0.250905i
$966$		0			0
$967$	−2.00000			−0.0643157			−0.0321578	−	0.999483i	$-0.510238\pi$
$967$	−2.00000			−0.0643157			−0.0321578	+	0.999483i	$0.510238\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−18.0000	+	31.1769i	−0.577647	+	1.00051i	0.418101	+	0.908401i	$0.362696\pi$
$971$	−18.0000	+	31.1769i	−0.577647	+	1.00051i	−0.995748	+	0.0921142i	$0.970638\pi$
$972$		0			0
$973$	−12.0000	−	20.7846i	−0.384702	−	0.666324i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	16.5000	+	28.5788i	0.527882	+	0.914318i	0.999472	+	0.0325001i	$0.0103469\pi$
$977$	16.5000	+	28.5788i	0.527882	+	0.914318i	−0.471590	+	0.881818i	$0.656320\pi$
$978$		0			0
$979$	14.0000	−	24.2487i	0.447442	−	0.774992i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	4.00000			0.127580			0.0637901	−	0.997963i	$-0.479681\pi$
$983$	4.00000			0.127580			0.0637901	+	0.997963i	$0.479681\pi$
$984$		0			0
$985$	3.00000	+	5.19615i	0.0955879	+	0.165563i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	36.0000			1.14473
$990$		0			0
$991$	−1.00000	−	1.73205i	−0.0317660	−	0.0550204i	0.849705	−	0.527258i	$-0.176780\pi$
$991$	−1.00000	−	1.73205i	−0.0317660	−	0.0550204i	−0.881471	+	0.472237i	$0.843446\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	7.00000	−	12.1244i	0.221915	−	0.384368i
$996$		0			0
$997$	17.5000	−	30.3109i	0.554231	−	0.959955i	−0.443732	−	0.896159i	$-0.646346\pi$
$997$	17.5000	−	30.3109i	0.554231	−	0.959955i	0.997963	−	0.0637961i	$-0.0203207\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1872.2.t.j.289.1		2
3.2	odd	2		624.2.q.c.289.1		2
4.3	odd	2		117.2.g.b.55.1		2
12.11	even	2		39.2.e.a.16.1	✓	2
13.9	even	3	inner	1872.2.t.j.1153.1		2
39.23	odd	6		8112.2.a.bc.1.1		1
39.29	odd	6		8112.2.a.w.1.1		1
39.35	odd	6		624.2.q.c.529.1		2
52.3	odd	6		1521.2.a.a.1.1		1
52.11	even	12		1521.2.b.c.1351.1		2
52.15	even	12		1521.2.b.c.1351.2		2
52.23	odd	6		1521.2.a.d.1.1		1
52.35	odd	6		117.2.g.b.100.1		2
60.23	odd	4		975.2.bb.d.874.1		4
60.47	odd	4		975.2.bb.d.874.2		4
60.59	even	2		975.2.i.f.601.1		2
156.11	odd	12		507.2.b.b.337.2		2
156.23	even	6		507.2.a.b.1.1		1
156.35	even	6		39.2.e.a.22.1	yes	2
156.47	odd	4		507.2.j.d.361.2		4
156.59	odd	12		507.2.j.d.316.1		4
156.71	odd	12		507.2.j.d.316.2		4
156.83	odd	4		507.2.j.d.361.1		4
156.95	even	6		507.2.e.c.22.1		2
156.107	even	6		507.2.a.c.1.1		1
156.119	odd	12		507.2.b.b.337.1		2
156.155	even	2		507.2.e.c.484.1		2
780.347	odd	12		975.2.bb.d.724.1		4
780.503	odd	12		975.2.bb.d.724.2		4
780.659	even	6		975.2.i.f.451.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
39.2.e.a.16.1	✓	2	12.11	even	2
39.2.e.a.22.1	yes	2	156.35	even	6
117.2.g.b.55.1		2	4.3	odd	2
117.2.g.b.100.1		2	52.35	odd	6
507.2.a.b.1.1		1	156.23	even	6
507.2.a.c.1.1		1	156.107	even	6
507.2.b.b.337.1		2	156.119	odd	12
507.2.b.b.337.2		2	156.11	odd	12
507.2.e.c.22.1		2	156.95	even	6
507.2.e.c.484.1		2	156.155	even	2
507.2.j.d.316.1		4	156.59	odd	12
507.2.j.d.316.2		4	156.71	odd	12
507.2.j.d.361.1		4	156.83	odd	4
507.2.j.d.361.2		4	156.47	odd	4
624.2.q.c.289.1		2	3.2	odd	2
624.2.q.c.529.1		2	39.35	odd	6
975.2.i.f.451.1		2	780.659	even	6
975.2.i.f.601.1		2	60.59	even	2
975.2.bb.d.724.1		4	780.347	odd	12
975.2.bb.d.724.2		4	780.503	odd	12
975.2.bb.d.874.1		4	60.23	odd	4
975.2.bb.d.874.2		4	60.47	odd	4
1521.2.a.a.1.1		1	52.3	odd	6
1521.2.a.d.1.1		1	52.23	odd	6
1521.2.b.c.1351.1		2	52.11	even	12
1521.2.b.c.1351.2		2	52.15	even	12
1872.2.t.j.289.1		2	1.1	even	1	trivial
1872.2.t.j.1153.1		2	13.9	even	3	inner
8112.2.a.w.1.1		1	39.29	odd	6
8112.2.a.bc.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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{5} +(1.00000 - 1.73205i) q^{7} +O(q^{10})\)	\(q+1.00000 q^{5} +(1.00000 - 1.73205i) q^{7} +(1.00000 + 1.73205i) q^{11} +(-3.50000 + 0.866025i) q^{13} +(-3.50000 + 6.06218i) q^{17} +(-3.00000 + 5.19615i) q^{19} +(3.00000 + 5.19615i) q^{23} -4.00000 q^{25} +(-0.500000 - 0.866025i) q^{29} -4.00000 q^{31} +(1.00000 - 1.73205i) q^{35} +(-0.500000 - 0.866025i) q^{37} +(4.50000 + 7.79423i) q^{41} +(3.00000 - 5.19615i) q^{43} +6.00000 q^{47} +(1.50000 + 2.59808i) q^{49} +9.00000 q^{53} +(1.00000 + 1.73205i) q^{55} +(-0.500000 + 0.866025i) q^{61} +(-3.50000 + 0.866025i) q^{65} +(-1.00000 - 1.73205i) q^{67} +(-3.00000 + 5.19615i) q^{71} +11.0000 q^{73} +4.00000 q^{77} +4.00000 q^{79} -14.0000 q^{83} +(-3.50000 + 6.06218i) q^{85} +(-7.00000 - 12.1244i) q^{89} +(-2.00000 + 6.92820i) q^{91} +(-3.00000 + 5.19615i) q^{95} +(1.00000 - 1.73205i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{5} + 2 q^{7}+O(q^{10}) \) 2 * q + 2 * q^5 + 2 * q^7	\( 2 q + 2 q^{5} + 2 q^{7} + 2 q^{11} - 7 q^{13} - 7 q^{17} - 6 q^{19} + 6 q^{23} - 8 q^{25} - q^{29} - 8 q^{31} + 2 q^{35} - q^{37} + 9 q^{41} + 6 q^{43} + 12 q^{47} + 3 q^{49} + 18 q^{53} + 2 q^{55} - q^{61} - 7 q^{65} - 2 q^{67} - 6 q^{71} + 22 q^{73} + 8 q^{77} + 8 q^{79} - 28 q^{83} - 7 q^{85} - 14 q^{89} - 4 q^{91} - 6 q^{95} + 2 q^{97}+O(q^{100}) \) 2 * q + 2 * q^5 + 2 * q^7 + 2 * q^11 - 7 * q^13 - 7 * q^17 - 6 * q^19 + 6 * q^23 - 8 * q^25 - q^29 - 8 * q^31 + 2 * q^35 - q^37 + 9 * q^41 + 6 * q^43 + 12 * q^47 + 3 * q^49 + 18 * q^53 + 2 * q^55 - q^61 - 7 * q^65 - 2 * q^67 - 6 * q^71 + 22 * q^73 + 8 * q^77 + 8 * q^79 - 28 * q^83 - 7 * q^85 - 14 * q^89 - 4 * q^91 - 6 * q^95 + 2 * q^97

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