LMFDB - Embedded newform 3456.2.i.i.1153.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3456,2,Mod(1153,3456)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3456, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 4]))

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

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

chi := DirichletCharacter("3456.1153");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3456 = 2^{7} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3456.i (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:	$27.5962989386$
Analytic rank:	$0$
Dimension:	$12$
Relative dimension:	$6$ over $\Q(\zeta_{3})$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 2 x^{11} + 3 x^{10} - 8 x^{9} + 22 x^{8} - 42 x^{7} + 51 x^{6} - 126 x^{5} + 198 x^{4} + \cdots + 729 $ x^12 - 2x^11 + 3x^10 - 8x^9 + 22x^8 - 42x^7 + 51x^6 - 126x^5 + 198x^4 - 216x^3 + 243x^2 - 486*x + 729
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 3^{4} $
Twist minimal:	no (minimal twist has level 1152)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			1153.5
Root			$1.73202 - 0.0102491i$ of defining polynomial
Character	$\chi$	$=$	3456.1153
Dual form			3456.2.i.i.2305.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.551563 + 0.955334i) q^{5} +(1.62490 - 2.81442i) q^{7} +O(q^{10})$	$q+(0.551563 + 0.955334i) q^{5} +(1.62490 - 2.81442i) q^{7} +(-1.28869 + 2.23208i) q^{11} +(-1.58731 - 2.74930i) q^{13} -4.71601 q^{17} -5.75569 q^{19} +(2.35397 + 4.07719i) q^{23} +(1.89156 - 3.27627i) q^{25} +(-3.66250 + 6.34363i) q^{29} +(2.93135 + 5.07724i) q^{31} +3.58494 q^{35} +0.0714979 q^{37} +(1.63887 + 2.83861i) q^{41} +(-2.12088 + 3.67347i) q^{43} +(-4.72803 + 8.18919i) q^{47} +(-1.78062 - 3.08413i) q^{49} -6.42812 q^{53} -2.84317 q^{55} +(4.19606 + 7.26779i) q^{59} +(-4.66250 + 8.07568i) q^{61} +(1.75100 - 3.03283i) q^{65} +(6.09975 + 10.5651i) q^{67} -0.335627 q^{71} +14.8664 q^{73} +(4.18800 + 7.25382i) q^{77} +(-4.85985 + 8.41750i) q^{79} +(3.07022 - 5.31778i) q^{83} +(-2.60117 - 4.50537i) q^{85} -4.42812 q^{89} -10.3169 q^{91} +(-3.17462 - 5.49861i) q^{95} +(6.39456 - 11.0757i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 2 q^{5} - 6 q^{7}+O(q^{10}) $ 12 * q - 2 * q^5 - 6 * q^7	$ 12 q - 2 q^{5} - 6 q^{7} - 4 q^{11} - 10 q^{13} - 4 q^{17} + 4 q^{19} + 8 q^{23} - 14 q^{25} - 2 q^{29} - 8 q^{31} - 8 q^{35} + 2 q^{41} - 2 q^{43} - 14 q^{47} - 18 q^{49} + 24 q^{53} + 16 q^{55} - 6 q^{59} - 14 q^{61} + 8 q^{65} + 4 q^{67} - 28 q^{71} + 60 q^{73} + 2 q^{77} - 16 q^{79} - 24 q^{83} - 16 q^{85} + 48 q^{89} - 52 q^{91} - 20 q^{95} - 14 q^{97}+O(q^{100}) $ 12 * q - 2 * q^5 - 6 * q^7 - 4 * q^11 - 10 * q^13 - 4 * q^17 + 4 * q^19 + 8 * q^23 - 14 * q^25 - 2 * q^29 - 8 * q^31 - 8 * q^35 + 2 * q^41 - 2 * q^43 - 14 * q^47 - 18 * q^49 + 24 * q^53 + 16 * q^55 - 6 * q^59 - 14 * q^61 + 8 * q^65 + 4 * q^67 - 28 * q^71 + 60 * q^73 + 2 * q^77 - 16 * q^79 - 24 * q^83 - 16 * q^85 + 48 * q^89 - 52 * q^91 - 20 * q^95 - 14 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3456\mathbb{Z}\right)^\times$.

$n$	$2053$	$2431$	$2945$
$\chi(n)$	$1$	$1$	$e\left(\frac{2}{3}\right)$

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
$5$	0.551563	+	0.955334i	0.246666	+	0.427238i	0.962599	−	0.270931i	$-0.0873315\pi$
$5$	0.551563	+	0.955334i	0.246666	+	0.427238i	−0.715933	+	0.698169i	$0.753998\pi$
$6$		0			0
$7$	1.62490	−	2.81442i	0.614156	−	1.06375i	−0.376376	−	0.926467i	$-0.622830\pi$
$7$	1.62490	−	2.81442i	0.614156	−	1.06375i	0.990532	−	0.137282i	$-0.0438367\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−1.28869	+	2.23208i	−0.388555	+	0.672997i	−0.992255	−	0.124215i	$-0.960359\pi$
$11$	−1.28869	+	2.23208i	−0.388555	+	0.672997i	0.603701	+	0.797211i	$0.293692\pi$
$12$		0			0
$13$	−1.58731	−	2.74930i	−0.440241	−	0.762520i	0.557466	−	0.830200i	$-0.311774\pi$
$13$	−1.58731	−	2.74930i	−0.440241	−	0.762520i	−0.997707	+	0.0676799i	$0.978440\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−4.71601			−1.14380			−0.571900	−	0.820323i	$-0.693794\pi$
$17$	−4.71601			−1.14380			−0.571900	+	0.820323i	$0.693794\pi$
$18$		0			0
$19$	−5.75569			−1.32045			−0.660223	−	0.751070i	$-0.729538\pi$
$19$	−5.75569			−1.32045			−0.660223	+	0.751070i	$0.729538\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	2.35397	+	4.07719i	0.490836	+	0.850152i	0.999944	−	0.0105499i	$-0.00335820\pi$
$23$	2.35397	+	4.07719i	0.490836	+	0.850152i	−0.509109	+	0.860702i	$0.670025\pi$
$24$		0			0
$25$	1.89156	−	3.27627i	0.378312	−	0.655255i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−3.66250	+	6.34363i	−0.680108	+	1.17798i	0.294839	+	0.955547i	$0.404734\pi$
$29$	−3.66250	+	6.34363i	−0.680108	+	1.17798i	−0.974947	+	0.222435i	$0.928599\pi$
$30$		0			0
$31$	2.93135	+	5.07724i	0.526485	+	0.911899i	0.999524	+	0.0308575i	$0.00982380\pi$
$31$	2.93135	+	5.07724i	0.526485	+	0.911899i	−0.473039	+	0.881042i	$0.656843\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	3.58494			0.605966
$36$		0			0
$37$	0.0714979			0.0117542			0.00587709	−	0.999983i	$-0.498129\pi$
$37$	0.0714979			0.0117542			0.00587709	+	0.999983i	$0.498129\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	1.63887	+	2.83861i	0.255949	+	0.443317i	0.965153	−	0.261687i	$-0.0842787\pi$
$41$	1.63887	+	2.83861i	0.255949	+	0.443317i	−0.709204	+	0.705004i	$0.750945\pi$
$42$		0			0
$43$	−2.12088	+	3.67347i	−0.323431	+	0.560198i	−0.981193	−	0.193027i	$-0.938170\pi$
$43$	−2.12088	+	3.67347i	−0.323431	+	0.560198i	0.657763	+	0.753225i	$0.271503\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−4.72803	+	8.18919i	−0.689654	+	1.19452i	0.282296	+	0.959327i	$0.408904\pi$
$47$	−4.72803	+	8.18919i	−0.689654	+	1.19452i	−0.971950	+	0.235188i	$0.924429\pi$
$48$		0			0
$49$	−1.78062	−	3.08413i	−0.254375	−	0.440590i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−6.42812			−0.882970			−0.441485	−	0.897269i	$-0.645548\pi$
$53$	−6.42812			−0.882970			−0.441485	+	0.897269i	$0.645548\pi$
$54$		0			0
$55$	−2.84317			−0.383373
$56$		0			0
$57$		0			0
$58$		0			0
$59$	4.19606	+	7.26779i	0.546281	+	0.946186i	0.998525	+	0.0542918i	$0.0172901\pi$
$59$	4.19606	+	7.26779i	0.546281	+	0.946186i	−0.452244	+	0.891894i	$0.649377\pi$
$60$		0			0
$61$	−4.66250	+	8.07568i	−0.596971	+	1.03398i	0.396294	+	0.918124i	$0.370296\pi$
$61$	−4.66250	+	8.07568i	−0.596971	+	1.03398i	−0.993265	+	0.115861i	$0.963037\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	1.75100	−	3.03283i	0.217185	−	0.376176i
$66$		0			0
$67$	6.09975	+	10.5651i	0.745203	+	1.29073i	0.950100	+	0.311945i	$0.100981\pi$
$67$	6.09975	+	10.5651i	0.745203	+	1.29073i	−0.204897	+	0.978783i	$0.565686\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−0.335627			−0.0398316			−0.0199158	−	0.999802i	$-0.506340\pi$
$71$	−0.335627			−0.0398316			−0.0199158	+	0.999802i	$0.506340\pi$
$72$		0			0
$73$	14.8664			1.73998			0.869989	−	0.493071i	$-0.164126\pi$
$73$	14.8664			1.73998			0.869989	+	0.493071i	$0.164126\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	4.18800	+	7.25382i	0.477266	+	0.826650i
$78$		0			0
$79$	−4.85985	+	8.41750i	−0.546776	+	0.947043i	0.451717	+	0.892161i	$0.350812\pi$
$79$	−4.85985	+	8.41750i	−0.546776	+	0.947043i	−0.998493	+	0.0548820i	$0.982522\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	3.07022	−	5.31778i	0.337000	−	0.583702i	−0.646867	−	0.762603i	$-0.723921\pi$
$83$	3.07022	−	5.31778i	0.337000	−	0.583702i	0.983867	+	0.178901i	$0.0572543\pi$
$84$		0			0
$85$	−2.60117	−	4.50537i	−0.282137	−	0.488676i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−4.42812			−0.469379			−0.234690	−	0.972070i	$-0.575407\pi$
$89$	−4.42812			−0.469379			−0.234690	+	0.972070i	$0.575407\pi$
$90$		0			0
$91$	−10.3169			−1.08151
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−3.17462	−	5.49861i	−0.325709	−	0.564145i
$96$		0			0
$97$	6.39456	−	11.0757i	0.649270	−	1.12457i	−0.334028	−	0.942563i	$-0.608408\pi$
$97$	6.39456	−	11.0757i	0.649270	−	1.12457i	0.983298	−	0.182005i	$-0.0582586\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	3.80137	−	6.58417i	0.378250	−	0.655149i	−0.612557	−	0.790426i	$-0.709859\pi$
$101$	3.80137	−	6.58417i	0.378250	−	0.655149i	0.990808	+	0.135277i	$0.0431925\pi$
$102$		0			0
$103$	−5.62490	−	9.74262i	−0.554238	−	0.959969i	−0.997962	−	0.0638053i	$-0.979676\pi$
$103$	−5.62490	−	9.74262i	−0.554238	−	0.959969i	0.443724	−	0.896163i	$-0.353657\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	2.81493			0.272130			0.136065	−	0.990700i	$-0.456554\pi$
$107$	2.81493			0.272130			0.136065	+	0.990700i	$0.456554\pi$
$108$		0			0
$109$	15.6539			1.49937			0.749685	−	0.661795i	$-0.230205\pi$
$109$	15.6539			1.49937			0.749685	+	0.661795i	$0.230205\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−10.1828	−	17.6370i	−0.957913	−	1.65915i	−0.727557	−	0.686047i	$-0.759344\pi$
$113$	−10.1828	−	17.6370i	−0.957913	−	1.65915i	−0.230355	−	0.973107i	$-0.573989\pi$
$114$		0			0
$115$	−2.59672	+	4.49765i	−0.242145	+	0.419408i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−7.66306	+	13.2728i	−0.702472	+	1.21672i
$120$		0			0
$121$	2.17855	+	3.77337i	0.198050	+	0.343033i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	9.68887			0.866599
$126$		0			0
$127$	−3.09888			−0.274981			−0.137491	−	0.990503i	$-0.543904\pi$
$127$	−3.09888			−0.274981			−0.137491	+	0.990503i	$0.543904\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−0.251085	−	0.434893i	−0.0219374	−	0.0379968i	0.854848	−	0.518878i	$-0.173650\pi$
$131$	−0.251085	−	0.434893i	−0.0219374	−	0.0379968i	−0.876786	+	0.480881i	$0.840317\pi$
$132$		0			0
$133$	−9.35244	+	16.1989i	−0.810960	+	1.40462i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	4.88868	−	8.46744i	0.417668	−	0.723423i	−0.578036	−	0.816011i	$-0.696181\pi$
$137$	4.88868	−	8.46744i	0.417668	−	0.723423i	0.995704	+	0.0925885i	$0.0295141\pi$
$138$		0			0
$139$	−0.188498	−	0.326488i	−0.0159882	−	0.0276924i	0.857921	−	0.513782i	$-0.171756\pi$
$139$	−0.188498	−	0.326488i	−0.0159882	−	0.0276924i	−0.873909	+	0.486090i	$0.838423\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	8.18221			0.684231
$144$		0			0
$145$	−8.08038			−0.671039
$146$		0			0
$147$		0			0
$148$		0			0
$149$	4.83712	+	8.37814i	0.396272	+	0.686364i	0.993263	−	0.115885i	$-0.0369703\pi$
$149$	4.83712	+	8.37814i	0.396272	+	0.686364i	−0.596990	+	0.802248i	$0.703637\pi$
$150$		0			0
$151$	−8.42915	+	14.5997i	−0.685954	+	1.18811i	0.287181	+	0.957876i	$0.407282\pi$
$151$	−8.42915	+	14.5997i	−0.685954	+	1.18811i	−0.973136	+	0.230232i	$0.926052\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−3.23364	+	5.60083i	−0.259732	+	0.449870i
$156$		0			0
$157$	4.36262	+	7.55628i	0.348175	+	0.603057i	0.985925	−	0.167187i	$-0.0534682\pi$
$157$	4.36262	+	7.55628i	0.348175	+	0.603057i	−0.637750	+	0.770243i	$0.720135\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	15.2999			1.20580
$162$		0			0
$163$	−12.2063			−0.956067			−0.478034	−	0.878342i	$-0.658650\pi$
$163$	−12.2063			−0.956067			−0.478034	+	0.878342i	$0.658650\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	11.3806	+	19.7118i	0.880657	+	1.52534i	0.850612	+	0.525794i	$0.176232\pi$
$167$	11.3806	+	19.7118i	0.880657	+	1.52534i	0.0300447	+	0.999549i	$0.490435\pi$
$168$		0			0
$169$	1.46088	−	2.53033i	0.112376	−	0.194640i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−11.9797	+	20.7494i	−0.910798	+	1.57755i	−0.0978588	+	0.995200i	$0.531199\pi$
$173$	−11.9797	+	20.7494i	−0.910798	+	1.57755i	−0.812939	+	0.582348i	$0.802134\pi$
$174$		0			0
$175$	−6.14720	−	10.6473i	−0.464684	−	0.804857i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−10.9992			−0.822121			−0.411061	−	0.911608i	$-0.634842\pi$
$179$	−10.9992			−0.822121			−0.411061	+	0.911608i	$0.634842\pi$
$180$		0			0
$181$	−22.2168			−1.65136			−0.825679	−	0.564140i	$-0.809208\pi$
$181$	−22.2168			−1.65136			−0.825679	+	0.564140i	$0.809208\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	0.0394356	+	0.0683044i	0.00289936	+	0.00502184i
$186$		0			0
$187$	6.07748	−	10.5265i	0.444429	−	0.769774i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	5.48760	−	9.50479i	0.397068	−	0.687743i	−0.596294	−	0.802766i	$-0.703361\pi$
$191$	5.48760	−	9.50479i	0.397068	−	0.687743i	0.993363	+	0.115023i	$0.0366942\pi$
$192$		0			0
$193$	−7.11682	−	12.3267i	−0.512280	−	0.887294i	−0.999899	−	0.0142378i	$-0.995468\pi$
$193$	−7.11682	−	12.3267i	−0.512280	−	0.887294i	0.487619	−	0.873057i	$-0.337866\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	8.15037			0.580690			0.290345	−	0.956922i	$-0.406230\pi$
$197$	8.15037			0.580690			0.290345	+	0.956922i	$0.406230\pi$
$198$		0			0
$199$	−6.09200			−0.431850			−0.215925	−	0.976410i	$-0.569277\pi$
$199$	−6.09200			−0.431850			−0.215925	+	0.976410i	$0.569277\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	11.9024	+	20.6156i	0.835385	+	1.44693i
$204$		0			0
$205$	−1.80788	+	3.13135i	−0.126268	+	0.218703i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	7.41730	−	12.8471i	0.513066	−	0.888656i
$210$		0			0
$211$	3.01985	+	5.23054i	0.207895	+	0.360085i	0.951051	−	0.309033i	$-0.100005\pi$
$211$	3.01985	+	5.23054i	0.207895	+	0.360085i	−0.743156	+	0.669118i	$0.766672\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−4.67918			−0.319118
$216$		0			0
$217$	19.0526			1.29338
$218$		0			0
$219$		0			0
$220$		0			0
$221$	7.48578	+	12.9657i	0.503548	+	0.872170i
$222$		0			0
$223$	−10.5391	+	18.2542i	−0.705749	+	1.22239i	0.260671	+	0.965428i	$0.416056\pi$
$223$	−10.5391	+	18.2542i	−0.705749	+	1.22239i	−0.966420	+	0.256966i	$0.917277\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−14.9946	+	25.9713i	−0.995224	+	1.72378i	−0.413069	+	0.910700i	$0.635543\pi$
$227$	−14.9946	+	25.9713i	−0.995224	+	1.72378i	−0.582155	+	0.813078i	$0.697790\pi$
$228$		0			0
$229$	−9.53170	−	16.5094i	−0.629873	−	1.09097i	−0.987577	−	0.157136i	$-0.949774\pi$
$229$	−9.53170	−	16.5094i	−0.629873	−	1.09097i	0.357704	−	0.933835i	$-0.383560\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−7.91098			−0.518266			−0.259133	−	0.965842i	$-0.583437\pi$
$233$	−7.91098			−0.518266			−0.259133	+	0.965842i	$0.583437\pi$
$234$		0			0
$235$	−10.4312			−0.680457
$236$		0			0
$237$		0			0
$238$		0			0
$239$	2.96685	+	5.13873i	0.191910	+	0.332397i	0.945883	−	0.324508i	$-0.105199\pi$
$239$	2.96685	+	5.13873i	0.191910	+	0.332397i	−0.753973	+	0.656905i	$0.771865\pi$
$240$		0			0
$241$	−14.2494	+	24.6808i	−0.917888	+	1.58983i	−0.115270	+	0.993334i	$0.536773\pi$
$241$	−14.2494	+	24.6808i	−0.917888	+	1.58983i	−0.802618	+	0.596494i	$0.796560\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	1.96425	−	3.40218i	0.125491	−	0.217357i
$246$		0			0
$247$	9.13607	+	15.8241i	0.581314	+	1.00687i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−15.6924			−0.990498			−0.495249	−	0.868751i	$-0.664923\pi$
$251$	−15.6924			−0.990498			−0.495249	+	0.868751i	$0.664923\pi$
$252$		0			0
$253$	−12.1341			−0.762866
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−11.5645	−	20.0304i	−0.721377	−	1.24946i	−0.960448	−	0.278459i	$-0.910176\pi$
$257$	−11.5645	−	20.0304i	−0.721377	−	1.24946i	0.239071	−	0.971002i	$-0.423157\pi$
$258$		0			0
$259$	0.116177	−	0.201225i	0.00721890	−	0.0125035i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	12.0737	−	20.9122i	0.744494	−	1.28950i	−0.205937	−	0.978565i	$-0.566024\pi$
$263$	12.0737	−	20.9122i	0.744494	−	1.28950i	0.950431	−	0.310936i	$-0.100642\pi$
$264$		0			0
$265$	−3.54551	−	6.14100i	−0.217799	−	0.377239i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−9.45599			−0.576542			−0.288271	−	0.957549i	$-0.593080\pi$
$269$	−9.45599			−0.576542			−0.288271	+	0.957549i	$0.593080\pi$
$270$		0			0
$271$	15.5750			0.946115			0.473057	−	0.881032i	$-0.343150\pi$
$271$	15.5750			0.946115			0.473057	+	0.881032i	$0.343150\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	4.87526	+	8.44420i	0.293989	+	0.509205i
$276$		0			0
$277$	−2.87862	+	4.98592i	−0.172960	+	0.299575i	−0.939453	−	0.342677i	$-0.888666\pi$
$277$	−2.87862	+	4.98592i	−0.172960	+	0.299575i	0.766494	+	0.642252i	$0.222000\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−5.99712	+	10.3873i	−0.357758	+	0.619656i	−0.987586	−	0.157079i	$-0.949792\pi$
$281$	−5.99712	+	10.3873i	−0.357758	+	0.619656i	0.629828	+	0.776735i	$0.283126\pi$
$282$		0			0
$283$	−0.604018	−	1.04619i	−0.0359051	−	0.0621895i	0.847514	−	0.530772i	$-0.178098\pi$
$283$	−0.604018	−	1.04619i	−0.0359051	−	0.0621895i	−0.883420	+	0.468583i	$0.844765\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	10.6520			0.628771
$288$		0			0
$289$	5.24075			0.308279
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−10.4657	−	18.1272i	−0.611415	−	1.05900i	−0.991002	−	0.133846i	$-0.957267\pi$
$293$	−10.4657	−	18.1272i	−0.611415	−	1.05900i	0.379587	−	0.925156i	$-0.376066\pi$
$294$		0			0
$295$	−4.62878	+	8.01728i	−0.269498	+	0.466784i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	7.47295	−	12.9435i	0.432172	−	0.748544i
$300$		0			0
$301$	6.89244	+	11.9381i	0.397274	+	0.688098i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−10.2866			−0.589011
$306$		0			0
$307$	5.12445			0.292468			0.146234	−	0.989250i	$-0.453285\pi$
$307$	5.12445			0.292468			0.146234	+	0.989250i	$0.453285\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−4.70739	−	8.15344i	−0.266931	−	0.462339i	0.701136	−	0.713027i	$-0.252676\pi$
$311$	−4.70739	−	8.15344i	−0.266931	−	0.462339i	−0.968068	+	0.250688i	$0.919343\pi$
$312$		0			0
$313$	−9.48986	+	16.4369i	−0.536398	+	0.929069i	0.462696	+	0.886517i	$0.346882\pi$
$313$	−9.48986	+	16.4369i	−0.536398	+	0.929069i	−0.999094	+	0.0425521i	$0.986451\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	14.2294	−	24.6461i	0.799205	−	1.38426i	−0.120930	−	0.992661i	$-0.538588\pi$
$317$	14.2294	−	24.6461i	0.799205	−	1.38426i	0.920135	−	0.391602i	$-0.128079\pi$
$318$		0			0
$319$	−9.43965	−	16.3499i	−0.528519	−	0.915421i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	27.1439			1.51033
$324$		0			0
$325$	−12.0100			−0.666193
$326$		0			0
$327$		0			0
$328$		0			0
$329$	15.3652	+	26.6133i	0.847110	+	1.46724i
$330$		0			0
$331$	0.837151	−	1.44999i	0.0460140	−	0.0796986i	−0.842101	−	0.539320i	$-0.818681\pi$
$331$	0.837151	−	1.44999i	0.0460140	−	0.0796986i	0.888115	+	0.459621i	$0.152015\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−6.72878	+	11.6546i	−0.367633	+	0.636758i
$336$		0			0
$337$	−15.1064	−	26.1651i	−0.822899	−	1.42530i	−0.903514	−	0.428558i	$-0.859022\pi$
$337$	−15.1064	−	26.1651i	−0.822899	−	1.42530i	0.0806146	−	0.996745i	$-0.474312\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−15.1104			−0.818273
$342$		0			0
$343$	11.1753			0.603408
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−8.46076	−	14.6545i	−0.454197	−	0.786693i	0.544444	−	0.838797i	$-0.316741\pi$
$347$	−8.46076	−	14.6545i	−0.454197	−	0.786693i	−0.998642	+	0.0521042i	$0.983407\pi$
$348$		0			0
$349$	−8.92436	+	15.4574i	−0.477710	+	0.827418i	−0.999674	−	0.0255500i	$-0.991866\pi$
$349$	−8.92436	+	15.4574i	−0.477710	+	0.827418i	0.521964	+	0.852968i	$0.325200\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	6.93593	−	12.0134i	0.369162	−	0.639407i	−0.620273	−	0.784386i	$-0.712978\pi$
$353$	6.93593	−	12.0134i	0.369162	−	0.639407i	0.989435	+	0.144979i	$0.0463114\pi$
$354$		0			0
$355$	−0.185119	−	0.320636i	−0.00982510	−	0.0170176i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−0.333139			−0.0175824			−0.00879120	−	0.999961i	$-0.502798\pi$
$359$	−0.333139			−0.0175824			−0.00879120	+	0.999961i	$0.502798\pi$
$360$		0			0
$361$	14.1280			0.743577
$362$		0			0
$363$		0			0
$364$		0			0
$365$	8.19974	+	14.2024i	0.429194	+	0.743386i
$366$		0			0
$367$	−10.5763	+	18.3188i	−0.552081	+	0.956232i	0.446043	+	0.895011i	$0.352833\pi$
$367$	−10.5763	+	18.3188i	−0.552081	+	0.956232i	−0.998124	+	0.0612208i	$0.980501\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−10.4451	+	18.0914i	−0.542281	+	0.939258i
$372$		0			0
$373$	−4.33750	−	7.51278i	−0.224587	−	0.388997i	0.731608	−	0.681725i	$-0.238770\pi$
$373$	−4.33750	−	7.51278i	−0.224587	−	0.388997i	−0.956196	+	0.292729i	$0.905437\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	23.2541			1.19765
$378$		0			0
$379$	14.2538			0.732168			0.366084	−	0.930582i	$-0.380698\pi$
$379$	14.2538			0.732168			0.366084	+	0.930582i	$0.380698\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	5.11696	+	8.86283i	0.261464	+	0.452869i	0.966631	−	0.256172i	$-0.0824613\pi$
$383$	5.11696	+	8.86283i	0.261464	+	0.452869i	−0.705167	+	0.709041i	$0.749128\pi$
$384$		0			0
$385$	−4.61988	+	8.00187i	−0.235451	+	0.407813i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	1.62675	−	2.81761i	0.0824793	−	0.142858i	−0.821835	−	0.569726i	$-0.807049\pi$
$389$	1.62675	−	2.81761i	0.0824793	−	0.142858i	0.904314	+	0.426867i	$0.140383\pi$
$390$		0			0
$391$	−11.1013	−	19.2281i	−0.561418	−	0.972405i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−10.7220			−0.539484
$396$		0			0
$397$	30.8709			1.54936			0.774682	−	0.632351i	$-0.217910\pi$
$397$	30.8709			1.54936			0.774682	+	0.632351i	$0.217910\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	2.01000	+	3.48143i	0.100375	+	0.173854i	0.911839	−	0.410548i	$-0.134662\pi$
$401$	2.01000	+	3.48143i	0.100375	+	0.173854i	−0.811464	+	0.584402i	$0.801329\pi$
$402$		0			0
$403$	9.30592	−	16.1183i	0.463561	−	0.802911i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−0.0921386	+	0.159589i	−0.00456714	+	0.00791052i
$408$		0			0
$409$	−3.33949	−	5.78416i	−0.165127	−	0.286008i	0.771573	−	0.636140i	$-0.219470\pi$
$409$	−3.33949	−	5.78416i	−0.165127	−	0.286008i	−0.936700	+	0.350132i	$0.886137\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	27.2728			1.34201
$414$		0			0
$415$	6.77367			0.332507
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−17.0507	−	29.5327i	−0.832982	−	1.44277i	−0.895663	−	0.444733i	$-0.853299\pi$
$419$	−17.0507	−	29.5327i	−0.832982	−	1.44277i	0.0626815	−	0.998034i	$-0.480035\pi$
$420$		0			0
$421$	−9.34688	+	16.1893i	−0.455539	+	0.789017i	−0.998719	−	0.0505996i	$-0.983887\pi$
$421$	−9.34688	+	16.1893i	−0.455539	+	0.789017i	0.543180	+	0.839616i	$0.317220\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−8.92060	+	15.4509i	−0.432713	+	0.749481i
$426$		0			0
$427$	15.1522	+	26.2444i	0.733267	+	1.27006i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−6.49967			−0.313078			−0.156539	−	0.987672i	$-0.550034\pi$
$431$	−6.49967			−0.313078			−0.156539	+	0.987672i	$0.550034\pi$
$432$		0			0
$433$	28.3266			1.36129			0.680645	−	0.732613i	$-0.261700\pi$
$433$	28.3266			1.36129			0.680645	+	0.732613i	$0.261700\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−13.5487	−	23.4670i	−0.648122	−	1.12258i
$438$		0			0
$439$	−3.82047	+	6.61724i	−0.182341	+	0.315824i	−0.942677	−	0.333706i	$-0.891701\pi$
$439$	−3.82047	+	6.61724i	−0.182341	+	0.315824i	0.760336	+	0.649530i	$0.225034\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	6.94625	−	12.0313i	0.330026	−	0.571623i	−0.652490	−	0.757797i	$-0.726276\pi$
$443$	6.94625	−	12.0313i	0.330026	−	0.571623i	0.982517	+	0.186175i	$0.0596090\pi$
$444$		0			0
$445$	−2.44238	−	4.23033i	−0.115780	−	0.200537i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−11.8869			−0.560976			−0.280488	−	0.959857i	$-0.590496\pi$
$449$	−11.8869			−0.560976			−0.280488	+	0.959857i	$0.590496\pi$
$450$		0			0
$451$	−8.44800			−0.397801
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−5.69042	−	9.85610i	−0.266771	−	0.462061i
$456$		0			0
$457$	0.860741	−	1.49085i	0.0402638	−	0.0697389i	−0.845191	−	0.534464i	$-0.820514\pi$
$457$	0.860741	−	1.49085i	0.0402638	−	0.0697389i	0.885455	+	0.464725i	$0.153847\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−15.8265	+	27.4123i	−0.737113	+	1.27672i	0.216677	+	0.976243i	$0.430478\pi$
$461$	−15.8265	+	27.4123i	−0.737113	+	1.27672i	−0.953790	+	0.300474i	$0.902855\pi$
$462$		0			0
$463$	−1.71702	−	2.97396i	−0.0797966	−	0.138212i	0.823366	−	0.567511i	$-0.192094\pi$
$463$	−1.71702	−	2.97396i	−0.0797966	−	0.138212i	−0.903162	+	0.429300i	$0.858760\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	15.5333			0.718797			0.359398	−	0.933184i	$-0.382982\pi$
$467$	15.5333			0.718797			0.359398	+	0.933184i	$0.382982\pi$
$468$		0			0
$469$	39.6460			1.83068
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−5.46631	−	9.46792i	−0.251341	−	0.435335i
$474$		0			0
$475$	−10.8872	+	18.8572i	−0.499540	+	0.865228i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−16.6927	+	28.9126i	−0.762710	+	1.32105i	0.178739	+	0.983897i	$0.442798\pi$
$479$	−16.6927	+	28.9126i	−0.762710	+	1.32105i	−0.941449	+	0.337156i	$0.890535\pi$
$480$		0			0
$481$	−0.113489	−	0.196569i	−0.00517467	−	0.00896280i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	14.1080			0.640612
$486$		0			0
$487$	20.0794			0.909883			0.454941	−	0.890521i	$-0.349660\pi$
$487$	20.0794			0.909883			0.454941	+	0.890521i	$0.349660\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	2.10538	+	3.64663i	0.0950146	+	0.164570i	0.909615	−	0.415453i	$-0.136377\pi$
$491$	2.10538	+	3.64663i	0.0950146	+	0.164570i	−0.814600	+	0.580023i	$0.803044\pi$
$492$		0			0
$493$	17.2724	−	29.9166i	0.777908	−	1.34738i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−0.545361	+	0.944593i	−0.0244628	+	0.0423708i
$498$		0			0
$499$	−5.24770	−	9.08928i	−0.234919	−	0.406892i	0.724330	−	0.689453i	$-0.242149\pi$
$499$	−5.24770	−	9.08928i	−0.234919	−	0.406892i	−0.959249	+	0.282561i	$0.908816\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−34.5118			−1.53881			−0.769403	−	0.638764i	$-0.779446\pi$
$503$	−34.5118			−1.53881			−0.769403	+	0.638764i	$0.779446\pi$
$504$		0			0
$505$	8.38677			0.373206
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−2.62702	−	4.55013i	−0.116440	−	0.201681i	0.801914	−	0.597439i	$-0.203815\pi$
$509$	−2.62702	−	4.55013i	−0.116440	−	0.201681i	−0.918355	+	0.395758i	$0.870482\pi$
$510$		0			0
$511$	24.1564	−	41.8402i	1.06862	−	1.85090i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	6.20497	−	10.7473i	0.273424	−	0.473584i
$516$		0			0
$517$	−12.1859	−	21.1066i	−0.535937	−	0.928269i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	12.9218			0.566113			0.283056	−	0.959103i	$-0.408652\pi$
$521$	12.9218			0.566113			0.283056	+	0.959103i	$0.408652\pi$
$522$		0			0
$523$	5.10475			0.223215			0.111607	−	0.993752i	$-0.464400\pi$
$523$	5.10475			0.223215			0.111607	+	0.993752i	$0.464400\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−13.8243	−	23.9443i	−0.602194	−	1.04303i
$528$		0			0
$529$	0.417694	−	0.723468i	0.0181606	−	0.0314551i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	5.20281	−	9.01153i	0.225359	−	0.390333i
$534$		0			0
$535$	1.55261	+	2.68920i	0.0671252	+	0.116264i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	9.17869			0.395354
$540$		0			0
$541$	37.9746			1.63266			0.816328	−	0.577589i	$-0.196006\pi$
$541$	37.9746			1.63266			0.816328	+	0.577589i	$0.196006\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	8.63410	+	14.9547i	0.369844	+	0.640589i
$546$		0			0
$547$	15.9350	−	27.6003i	0.681332	−	1.18010i	−0.293243	−	0.956038i	$-0.594734\pi$
$547$	15.9350	−	27.6003i	0.681332	−	1.18010i	0.974575	−	0.224063i	$-0.0719323\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	21.0802	−	36.5120i	0.898046	−	1.55546i
$552$		0			0
$553$	15.7936	+	27.3553i	0.671611	+	1.16326i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−11.5906			−0.491111			−0.245555	−	0.969383i	$-0.578970\pi$
$557$	−11.5906			−0.491111			−0.245555	+	0.969383i	$0.578970\pi$
$558$		0			0
$559$	13.4660			0.569550
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−1.25138	−	2.16745i	−0.0527392	−	0.0913470i	0.838451	−	0.544978i	$-0.183462\pi$
$563$	−1.25138	−	2.16745i	−0.0527392	−	0.0913470i	−0.891190	+	0.453631i	$0.850129\pi$
$564$		0			0
$565$	11.2328	−	19.4559i	0.472569	−	0.818514i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−12.9597	+	22.4469i	−0.543301	+	0.941024i	0.455411	+	0.890281i	$0.349492\pi$
$569$	−12.9597	+	22.4469i	−0.543301	+	0.941024i	−0.998712	+	0.0507432i	$0.983841\pi$
$570$		0			0
$571$	5.03679	+	8.72398i	0.210783	+	0.365087i	0.951960	−	0.306223i	$-0.0990653\pi$
$571$	5.03679	+	8.72398i	0.210783	+	0.365087i	−0.741177	+	0.671310i	$0.765732\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	17.8106			0.742755
$576$		0			0
$577$	−23.4726			−0.977177			−0.488588	−	0.872514i	$-0.662488\pi$
$577$	−23.4726			−0.977177			−0.488588	+	0.872514i	$0.662488\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−9.97762	−	17.2818i	−0.413942	−	0.716968i
$582$		0			0
$583$	8.28385	−	14.3481i	0.343082	−	0.594236i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	12.4138	−	21.5012i	0.512370	−	0.887451i	−0.487527	−	0.873108i	$-0.662101\pi$
$587$	12.4138	−	21.5012i	0.512370	−	0.887451i	0.999897	−	0.0143435i	$-0.00456582\pi$
$588$		0			0
$589$	−16.8719	−	29.2230i	−0.695195	−	1.20411i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	7.70977			0.316602			0.158301	−	0.987391i	$-0.449398\pi$
$593$	7.70977			0.316602			0.158301	+	0.987391i	$0.449398\pi$
$594$		0			0
$595$	−16.9066			−0.693104
$596$		0			0
$597$		0			0
$598$		0			0
$599$	14.7176	+	25.4916i	0.601344	+	1.04156i	0.992618	+	0.121284i	$0.0387013\pi$
$599$	14.7176	+	25.4916i	0.601344	+	1.04156i	−0.391274	+	0.920274i	$0.627965\pi$
$600$		0			0
$601$	−1.76388	+	3.05514i	−0.0719503	+	0.124622i	−0.899756	−	0.436393i	$-0.856256\pi$
$601$	−1.76388	+	3.05514i	−0.0719503	+	0.124622i	0.827806	+	0.561015i	$0.189589\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−2.40322	+	4.16250i	−0.0977047	+	0.169230i
$606$		0			0
$607$	−13.3211	−	23.0728i	−0.540687	−	0.936497i	−0.998865	−	0.0476362i	$-0.984831\pi$
$607$	−13.3211	−	23.0728i	−0.540687	−	0.936497i	0.458178	−	0.888860i	$-0.348502\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	30.0194			1.21446
$612$		0			0
$613$	−0.706406			−0.0285315			−0.0142657	−	0.999898i	$-0.504541\pi$
$613$	−0.706406			−0.0285315			−0.0142657	+	0.999898i	$0.504541\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−8.58480	−	14.8693i	−0.345611	−	0.598616i	0.639853	−	0.768497i	$-0.278995\pi$
$617$	−8.58480	−	14.8693i	−0.345611	−	0.598616i	−0.985465	+	0.169881i	$0.945662\pi$
$618$		0			0
$619$	4.17800	−	7.23651i	0.167928	−	0.290860i	−0.769763	−	0.638330i	$-0.779626\pi$
$619$	4.17800	−	7.23651i	0.167928	−	0.290860i	0.937691	+	0.347470i	$0.112959\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−7.19526	+	12.4626i	−0.288272	+	0.499302i
$624$		0			0
$625$	−4.11377	−	7.12526i	−0.164551	−	0.285010i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−0.337185			−0.0134444
$630$		0			0
$631$	23.9865			0.954889			0.477444	−	0.878662i	$-0.341563\pi$
$631$	23.9865			0.954889			0.477444	+	0.878662i	$0.341563\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−1.70923	−	2.96047i	−0.0678286	−	0.117483i
$636$		0			0
$637$	−5.65281	+	9.79095i	−0.223972	+	0.387932i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−6.58068	+	11.3981i	−0.259921	+	0.450197i	−0.966221	−	0.257716i	$-0.917030\pi$
$641$	−6.58068	+	11.3981i	−0.259921	+	0.450197i	0.706299	+	0.707913i	$0.250363\pi$
$642$		0			0
$643$	−7.85931	−	13.6127i	−0.309941	−	0.536834i	0.668408	−	0.743795i	$-0.266976\pi$
$643$	−7.85931	−	13.6127i	−0.309941	−	0.536834i	−0.978349	+	0.206961i	$0.933643\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	23.5146			0.924455			0.462228	−	0.886761i	$-0.347050\pi$
$647$	23.5146			0.924455			0.462228	+	0.886761i	$0.347050\pi$
$648$		0			0
$649$	−21.6297			−0.849040
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−13.1340	−	22.7487i	−0.513971	−	0.890224i	−0.999869	−	0.0162084i	$-0.994840\pi$
$653$	−13.1340	−	22.7487i	−0.513971	−	0.890224i	0.485897	−	0.874016i	$-0.338493\pi$
$654$		0			0
$655$	0.276979	−	0.479741i	0.0108224	−	0.0187450i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	13.2710	−	22.9860i	0.516963	−	0.895406i	−0.482843	−	0.875707i	$-0.660396\pi$
$659$	13.2710	−	22.9860i	0.516963	−	0.895406i	0.999806	−	0.0196993i	$-0.00627088\pi$
$660$		0			0
$661$	−0.981745	−	1.70043i	−0.0381855	−	0.0661392i	0.846301	−	0.532705i	$-0.178824\pi$
$661$	−0.981745	−	1.70043i	−0.0381855	−	0.0661392i	−0.884487	+	0.466566i	$0.845491\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−20.6338			−0.800145
$666$		0			0
$667$	−34.4856			−1.33529
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−12.0170	−	20.8141i	−0.463912	−	0.803519i
$672$		0			0
$673$	−18.9859	+	32.8846i	−0.731854	+	1.26761i	0.224236	+	0.974535i	$0.428011\pi$
$673$	−18.9859	+	32.8846i	−0.731854	+	1.26761i	−0.956090	+	0.293073i	$0.905322\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−13.5894	+	23.5375i	−0.522282	+	0.904619i	0.477382	+	0.878696i	$0.341586\pi$
$677$	−13.5894	+	23.5375i	−0.522282	+	0.904619i	−0.999664	+	0.0259229i	$0.991748\pi$
$678$		0			0
$679$	−20.7811	−	35.9939i	−0.797505	−	1.38132i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−46.9121			−1.79504			−0.897520	−	0.440974i	$-0.854633\pi$
$683$	−46.9121			−1.79504			−0.897520	+	0.440974i	$0.854633\pi$
$684$		0			0
$685$	10.7857			0.412099
$686$		0			0
$687$		0			0
$688$		0			0
$689$	10.2034	+	17.6728i	0.388719	+	0.673282i
$690$		0			0
$691$	12.6750	−	21.9538i	0.482181	−	0.835161i	−0.517610	−	0.855617i	$-0.673178\pi$
$691$	12.6750	−	21.9538i	0.482181	−	0.835161i	0.999791	+	0.0204552i	$0.00651153\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	0.207937	−	0.360157i	0.00788750	−	0.0136616i
$696$		0			0
$697$	−7.72895	−	13.3869i	−0.292755	−	0.507066i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−44.2840			−1.67258			−0.836292	−	0.548284i	$-0.815281\pi$
$701$	−44.2840			−1.67258			−0.836292	+	0.548284i	$0.815281\pi$
$702$		0			0
$703$	−0.411520			−0.0155208
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−12.3537	−	21.3973i	−0.464609	−	0.804727i
$708$		0			0
$709$	7.80457	−	13.5179i	0.293107	−	0.507676i	−0.681436	−	0.731878i	$-0.738644\pi$
$709$	7.80457	−	13.5179i	0.293107	−	0.507676i	0.974543	+	0.224202i	$0.0719776\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−13.8006	+	23.9033i	−0.516836	+	0.895185i
$714$		0			0
$715$	4.51300	+	7.81675i	0.168777	+	0.292330i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−21.1560			−0.788985			−0.394493	−	0.918899i	$-0.629080\pi$
$719$	−21.1560			−0.788985			−0.394493	+	0.918899i	$0.629080\pi$
$720$		0			0
$721$	−36.5597			−1.36155
$722$		0			0
$723$		0			0
$724$		0			0
$725$	13.8556	+	23.9987i	0.514586	+	0.891289i
$726$		0			0
$727$	−12.9909	+	22.5009i	−0.481805	+	0.834511i	−0.999782	−	0.0208834i	$-0.993352\pi$
$727$	−12.9909	+	22.5009i	−0.481805	+	0.834511i	0.517977	+	0.855395i	$0.326685\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	10.0021	−	17.3241i	0.369940	−	0.640755i
$732$		0			0
$733$	5.41447	+	9.37814i	0.199988	+	0.346390i	0.948524	−	0.316704i	$-0.102576\pi$
$733$	5.41447	+	9.37814i	0.199988	+	0.346390i	−0.748536	+	0.663094i	$0.769243\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−31.4427			−1.15821
$738$		0			0
$739$	11.4520			0.421270			0.210635	−	0.977565i	$-0.432447\pi$
$739$	11.4520			0.421270			0.210635	+	0.977565i	$0.432447\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−24.0077	−	41.5826i	−0.880758	−	1.52552i	−0.850500	−	0.525975i	$-0.823701\pi$
$743$	−24.0077	−	41.5826i	−0.880758	−	1.52552i	−0.0302573	−	0.999542i	$-0.509633\pi$
$744$		0			0
$745$	−5.33595	+	9.24213i	−0.195494	+	0.338605i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	4.57399	−	7.92239i	0.167130	−	0.289478i
$750$		0			0
$751$	22.5881	+	39.1238i	0.824253	+	1.42765i	0.902489	+	0.430713i	$0.141738\pi$
$751$	22.5881	+	39.1238i	0.824253	+	1.42765i	−0.0782360	+	0.996935i	$0.524929\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−18.5968			−0.676807
$756$		0			0
$757$	16.5457			0.601365			0.300682	−	0.953724i	$-0.402786\pi$
$757$	16.5457			0.601365			0.300682	+	0.953724i	$0.402786\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	20.6826	+	35.8234i	0.749745	+	1.29860i	0.947945	+	0.318435i	$0.103157\pi$
$761$	20.6826	+	35.8234i	0.749745	+	1.29860i	−0.198200	+	0.980162i	$0.563509\pi$
$762$		0			0
$763$	25.4361	−	44.0565i	0.920847	−	1.59495i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	13.3209	−	23.0725i	0.480990	−	0.833100i
$768$		0			0
$769$	−3.22518	−	5.58617i	−0.116303	−	0.201443i	0.801997	−	0.597328i	$-0.203771\pi$
$769$	−3.22518	−	5.58617i	−0.116303	−	0.201443i	−0.918300	+	0.395886i	$0.870438\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	0.949001			0.0341332			0.0170666	−	0.999854i	$-0.494567\pi$
$773$	0.949001			0.0341332			0.0170666	+	0.999854i	$0.494567\pi$
$774$		0			0
$775$	22.1792			0.796702
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−9.43285	−	16.3382i	−0.337967	−	0.585376i
$780$		0			0
$781$	0.432519	−	0.749145i	0.0154767	−	0.0268065i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−4.81251	+	8.33552i	−0.171766	+	0.297507i
$786$		0			0
$787$	17.6992	+	30.6559i	0.630909	+	1.09277i	0.987366	+	0.158454i	$0.0506511\pi$
$787$	17.6992	+	30.6559i	0.630909	+	1.09277i	−0.356458	+	0.934312i	$0.616016\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−66.1840			−2.35323
$792$		0			0
$793$	29.6033			1.05125
$794$		0			0
$795$		0			0
$796$		0			0
$797$	1.42624	+	2.47032i	0.0505200	+	0.0875032i	0.890180	−	0.455610i	$-0.150579\pi$
$797$	1.42624	+	2.47032i	0.0505200	+	0.0875032i	−0.839660	+	0.543113i	$0.817245\pi$
$798$		0			0
$799$	22.2974	−	38.6203i	0.788826	−	1.36629i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−19.1582	+	33.1829i	−0.676077	+	1.17100i
$804$		0			0
$805$	8.43883	+	14.6165i	0.297430	+	0.515164i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	2.18770			0.0769155			0.0384577	−	0.999260i	$-0.487756\pi$
$809$	2.18770			0.0769155			0.0384577	+	0.999260i	$0.487756\pi$
$810$		0			0
$811$	−7.54428			−0.264915			−0.132458	−	0.991189i	$-0.542287\pi$
$811$	−7.54428			−0.264915			−0.132458	+	0.991189i	$0.542287\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−6.73251	−	11.6610i	−0.235830	−	0.408469i
$816$		0			0
$817$	12.2071	−	21.1433i	0.427073	−	0.739712i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	18.5333	−	32.1006i	0.646815	−	1.12032i	−0.337064	−	0.941482i	$-0.609434\pi$
$821$	18.5333	−	32.1006i	0.646815	−	1.12032i	0.983879	−	0.178835i	$-0.0572329\pi$
$822$		0			0
$823$	21.9008	+	37.9334i	0.763416	+	1.32227i	0.941080	+	0.338184i	$0.109812\pi$
$823$	21.9008	+	37.9334i	0.763416	+	1.32227i	−0.177665	+	0.984091i	$0.556854\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	12.4192			0.431859			0.215929	−	0.976409i	$-0.430722\pi$
$827$	12.4192			0.431859			0.215929	+	0.976409i	$0.430722\pi$
$828$		0			0
$829$	−15.1889			−0.527532			−0.263766	−	0.964587i	$-0.584965\pi$
$829$	−15.1889			−0.527532			−0.263766	+	0.964587i	$0.584965\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	8.39744	+	14.5448i	0.290954	+	0.503947i
$834$		0			0
$835$	−12.5542	+	21.7445i	−0.434457	+	0.752501i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	24.7428	−	42.8557i	0.854215	−	1.47954i	−0.0231569	−	0.999732i	$-0.507372\pi$
$839$	24.7428	−	42.8557i	0.854215	−	1.47954i	0.877372	−	0.479811i	$-0.159295\pi$
$840$		0			0
$841$	−12.3278	−	21.3523i	−0.425095	−	0.736286i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	3.22308			0.110877
$846$		0			0
$847$	14.1598			0.486535
$848$		0			0
$849$		0			0
$850$		0			0
$851$	0.168304	+	0.291510i	0.00576937	+	0.00999284i
$852$		0			0
$853$	−6.36674	+	11.0275i	−0.217993	+	0.377575i	−0.954194	−	0.299188i	$-0.903284\pi$
$853$	−6.36674	+	11.0275i	−0.217993	+	0.377575i	0.736201	+	0.676763i	$0.236618\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	10.6560	−	18.4567i	0.364001	−	0.630467i	−0.624615	−	0.780933i	$-0.714744\pi$
$857$	10.6560	−	18.4567i	0.364001	−	0.630467i	0.988615	+	0.150466i	$0.0480773\pi$
$858$		0			0
$859$	6.26681	+	10.8544i	0.213821	+	0.370348i	0.952907	−	0.303262i	$-0.0980759\pi$
$859$	6.26681	+	10.8544i	0.213821	+	0.370348i	−0.739086	+	0.673611i	$0.764743\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	49.1767			1.67399			0.836997	−	0.547207i	$-0.184309\pi$
$863$	49.1767			1.67399			0.836997	+	0.547207i	$0.184309\pi$
$864$		0			0
$865$	−26.4302			−0.898653
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−12.5257	−	21.6951i	−0.424905	−	0.735956i
$870$		0			0
$871$	19.3644	−	33.5401i	0.656137	−	1.13646i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	15.7435	−	27.2685i	0.532227	−	0.921844i
$876$		0			0
$877$	−1.06962	−	1.85264i	−0.0361185	−	0.0625591i	0.847401	−	0.530953i	$-0.178166\pi$
$877$	−1.06962	−	1.85264i	−0.0361185	−	0.0625591i	−0.883520	+	0.468394i	$0.844833\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	7.40791			0.249579			0.124789	−	0.992183i	$-0.460174\pi$
$881$	7.40791			0.249579			0.124789	+	0.992183i	$0.460174\pi$
$882$		0			0
$883$	−10.0800			−0.339221			−0.169610	−	0.985511i	$-0.554251\pi$
$883$	−10.0800			−0.339221			−0.169610	+	0.985511i	$0.554251\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	16.6871	+	28.9028i	0.560297	+	0.970462i	0.997470	+	0.0710852i	$0.0226462\pi$
$887$	16.6871	+	28.9028i	0.560297	+	0.970462i	−0.437174	+	0.899377i	$0.644020\pi$
$888$		0			0
$889$	−5.03538	+	8.72154i	−0.168881	+	0.292511i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	27.2131	−	47.1344i	0.910651	−	1.57729i
$894$		0			0
$895$	−6.06676	−	10.5079i	−0.202790	−	0.351242i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−42.9442			−1.43227
$900$		0			0
$901$	30.3151			1.00994
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−12.2539	−	21.2244i	−0.407334	−	0.705524i
$906$		0			0
$907$	24.5436	−	42.5107i	0.814955	−	1.41154i	−0.0944045	−	0.995534i	$-0.530095\pi$
$907$	24.5436	−	42.5107i	0.814955	−	1.41154i	0.909360	−	0.416010i	$-0.136572\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	7.31861	−	12.6762i	0.242476	−	0.419981i	−0.718943	−	0.695069i	$-0.755374\pi$
$911$	7.31861	−	12.6762i	0.242476	−	0.419981i	0.961419	+	0.275088i	$0.0887070\pi$
$912$		0			0
$913$	7.91313	+	13.7059i	0.261886	+	0.453600i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−1.63196			−0.0538920
$918$		0			0
$919$	−5.03261			−0.166011			−0.0830053	−	0.996549i	$-0.526452\pi$
$919$	−5.03261			−0.166011			−0.0830053	+	0.996549i	$0.526452\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	0.532744	+	0.922740i	0.0175355	+	0.0303724i
$924$		0			0
$925$	0.135242	−	0.234247i	0.00444674	−	0.00770198i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−8.75518	+	15.1644i	−0.287248	+	0.497528i	−0.973152	−	0.230164i	$-0.926074\pi$
$929$	−8.75518	+	15.1644i	−0.287248	+	0.497528i	0.685904	+	0.727692i	$0.259407\pi$
$930$		0			0
$931$	10.2487	+	17.7513i	0.335888	+	0.581776i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	13.4084			0.438503
$936$		0			0
$937$	−23.7518			−0.775936			−0.387968	−	0.921673i	$-0.626823\pi$
$937$	−23.7518			−0.775936			−0.387968	+	0.921673i	$0.626823\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−1.97367	−	3.41849i	−0.0643397	−	0.111440i	0.832061	−	0.554684i	$-0.187161\pi$
$941$	−1.97367	−	3.41849i	−0.0643397	−	0.111440i	−0.896401	+	0.443244i	$0.853827\pi$
$942$		0			0
$943$	−7.71571	+	13.3640i	−0.251258	+	0.435191i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	5.45686	−	9.45155i	0.177324	−	0.307134i	−0.763639	−	0.645643i	$-0.776589\pi$
$947$	5.45686	−	9.45155i	0.177324	−	0.307134i	0.940963	+	0.338509i	$0.109923\pi$
$948$		0			0
$949$	−23.5976	−	40.8722i	−0.766010	−	1.32677i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	51.6348			1.67261			0.836307	−	0.548261i	$-0.184710\pi$
$953$	51.6348			1.67261			0.836307	+	0.548261i	$0.184710\pi$
$954$		0			0
$955$	12.1070			0.391774
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−15.8873	−	27.5176i	−0.513027	−	0.888588i
$960$		0			0
$961$	−1.68558	+	2.91951i	−0.0543735	+	0.0941776i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	7.85074	−	13.5979i	0.252724	−	0.437731i
$966$		0			0
$967$	−13.2718	−	22.9875i	−0.426793	−	0.739228i	0.569793	−	0.821788i	$-0.307024\pi$
$967$	−13.2718	−	22.9875i	−0.426793	−	0.739228i	−0.996586	+	0.0825607i	$0.973690\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	19.9365			0.639792			0.319896	−	0.947453i	$-0.396352\pi$
$971$	19.9365			0.639792			0.319896	+	0.947453i	$0.396352\pi$
$972$		0			0
$973$	−1.22517			−0.0392770
$974$		0			0
$975$		0			0
$976$		0			0
$977$	22.9725	+	39.7895i	0.734956	+	1.27298i	0.954743	+	0.297433i	$0.0961303\pi$
$977$	22.9725	+	39.7895i	0.734956	+	1.27298i	−0.219787	+	0.975548i	$0.570536\pi$
$978$		0			0
$979$	5.70647	−	9.88390i	0.182380	−	0.315891i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−16.0873	+	27.8640i	−0.513105	+	0.888725i	0.486779	+	0.873525i	$0.338172\pi$
$983$	−16.0873	+	27.8640i	−0.513105	+	0.888725i	−0.999884	+	0.0151995i	$0.995162\pi$
$984$		0			0
$985$	4.49544	+	7.78633i	0.143237	+	0.248093i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−19.9699			−0.635005
$990$		0			0
$991$	−9.32904			−0.296347			−0.148173	−	0.988961i	$-0.547339\pi$
$991$	−9.32904			−0.296347			−0.148173	+	0.988961i	$0.547339\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−3.36012	−	5.81990i	−0.106523	−	0.184503i
$996$		0			0
$997$	9.13466	−	15.8217i	0.289297	−	0.501078i	−0.684345	−	0.729159i	$-0.739912\pi$
$997$	9.13466	−	15.8217i	0.289297	−	0.501078i	0.973642	+	0.228081i	$0.0732450\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3456.2.i.i.1153.5		12
3.2	odd	2		1152.2.i.l.385.4	yes	12
4.3	odd	2		3456.2.i.j.1153.5		12
8.3	odd	2		3456.2.i.l.1153.2		12
8.5	even	2		3456.2.i.k.1153.2		12
9.4	even	3	inner	3456.2.i.i.2305.5		12
9.5	odd	6		1152.2.i.l.769.4	yes	12
12.11	even	2		1152.2.i.j.385.3	yes	12
24.5	odd	2		1152.2.i.i.385.3	✓	12
24.11	even	2		1152.2.i.k.385.4	yes	12
36.23	even	6		1152.2.i.j.769.3	yes	12
36.31	odd	6		3456.2.i.j.2305.5		12
72.5	odd	6		1152.2.i.i.769.3	yes	12
72.13	even	6		3456.2.i.k.2305.2		12
72.59	even	6		1152.2.i.k.769.4	yes	12
72.67	odd	6		3456.2.i.l.2305.2		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1152.2.i.i.385.3	✓	12	24.5	odd	2
1152.2.i.i.769.3	yes	12	72.5	odd	6
1152.2.i.j.385.3	yes	12	12.11	even	2
1152.2.i.j.769.3	yes	12	36.23	even	6
1152.2.i.k.385.4	yes	12	24.11	even	2
1152.2.i.k.769.4	yes	12	72.59	even	6
1152.2.i.l.385.4	yes	12	3.2	odd	2
1152.2.i.l.769.4	yes	12	9.5	odd	6
3456.2.i.i.1153.5		12	1.1	even	1	trivial
3456.2.i.i.2305.5		12	9.4	even	3	inner
3456.2.i.j.1153.5		12	4.3	odd	2
3456.2.i.j.2305.5		12	36.31	odd	6
3456.2.i.k.1153.2		12	8.5	even	2
3456.2.i.k.2305.2		12	72.13	even	6
3456.2.i.l.1153.2		12	8.3	odd	2
3456.2.i.l.2305.2		12	72.67	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:	\( N \)	\(=\)	\( 3456 = 2^{7} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3456.i (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.551563 + 0.955334i) q^{5} +(1.62490 - 2.81442i) q^{7} +O(q^{10})\)	\(q+(0.551563 + 0.955334i) q^{5} +(1.62490 - 2.81442i) q^{7} +(-1.28869 + 2.23208i) q^{11} +(-1.58731 - 2.74930i) q^{13} -4.71601 q^{17} -5.75569 q^{19} +(2.35397 + 4.07719i) q^{23} +(1.89156 - 3.27627i) q^{25} +(-3.66250 + 6.34363i) q^{29} +(2.93135 + 5.07724i) q^{31} +3.58494 q^{35} +0.0714979 q^{37} +(1.63887 + 2.83861i) q^{41} +(-2.12088 + 3.67347i) q^{43} +(-4.72803 + 8.18919i) q^{47} +(-1.78062 - 3.08413i) q^{49} -6.42812 q^{53} -2.84317 q^{55} +(4.19606 + 7.26779i) q^{59} +(-4.66250 + 8.07568i) q^{61} +(1.75100 - 3.03283i) q^{65} +(6.09975 + 10.5651i) q^{67} -0.335627 q^{71} +14.8664 q^{73} +(4.18800 + 7.25382i) q^{77} +(-4.85985 + 8.41750i) q^{79} +(3.07022 - 5.31778i) q^{83} +(-2.60117 - 4.50537i) q^{85} -4.42812 q^{89} -10.3169 q^{91} +(-3.17462 - 5.49861i) q^{95} +(6.39456 - 11.0757i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 2 q^{5} - 6 q^{7}+O(q^{10}) \) 12 * q - 2 * q^5 - 6 * q^7	\( 12 q - 2 q^{5} - 6 q^{7} - 4 q^{11} - 10 q^{13} - 4 q^{17} + 4 q^{19} + 8 q^{23} - 14 q^{25} - 2 q^{29} - 8 q^{31} - 8 q^{35} + 2 q^{41} - 2 q^{43} - 14 q^{47} - 18 q^{49} + 24 q^{53} + 16 q^{55} - 6 q^{59} - 14 q^{61} + 8 q^{65} + 4 q^{67} - 28 q^{71} + 60 q^{73} + 2 q^{77} - 16 q^{79} - 24 q^{83} - 16 q^{85} + 48 q^{89} - 52 q^{91} - 20 q^{95} - 14 q^{97}+O(q^{100}) \) 12 * q - 2 * q^5 - 6 * q^7 - 4 * q^11 - 10 * q^13 - 4 * q^17 + 4 * q^19 + 8 * q^23 - 14 * q^25 - 2 * q^29 - 8 * q^31 - 8 * q^35 + 2 * q^41 - 2 * q^43 - 14 * q^47 - 18 * q^49 + 24 * q^53 + 16 * q^55 - 6 * q^59 - 14 * q^61 + 8 * q^65 + 4 * q^67 - 28 * q^71 + 60 * q^73 + 2 * q^77 - 16 * q^79 - 24 * q^83 - 16 * q^85 + 48 * q^89 - 52 * q^91 - 20 * q^95 - 14 * q^97

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