LMFDB - Embedded newform 3456.2.i.k.2305.2

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			2305.2
Root			$1.73202 + 0.0102491i$ of defining polynomial
Character	$\chi$	$=$	3456.2305
Dual form			3456.2.i.k.1153.2

$q$-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 $\left(\mathbb{Z}/3456\mathbb{Z}\right)^\times$.

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

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	$-0.912668\pi$
$5$	−0.551563	+	0.955334i	−0.246666	+	0.427238i	0.715933	+	0.698169i	$0.246002\pi$
$6$		0			0
$7$	1.62490	+	2.81442i	0.614156	+	1.06375i	0.990532	+	0.137282i	$0.0438367\pi$
$7$	1.62490	+	2.81442i	0.614156	+	1.06375i	−0.376376	+	0.926467i	$0.622830\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	1.28869	+	2.23208i	0.388555	+	0.672997i	0.992255	−	0.124215i	$-0.0396411\pi$
$11$	1.28869	+	2.23208i	0.388555	+	0.672997i	−0.603701	+	0.797211i	$0.706308\pi$
$12$		0			0
$13$	1.58731	−	2.74930i	0.440241	−	0.762520i	−0.557466	−	0.830200i	$-0.688226\pi$
$13$	1.58731	−	2.74930i	0.440241	−	0.762520i	0.997707	+	0.0676799i	$0.0215597\pi$
$14$		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	$-0.270462\pi$
$19$	5.75569			1.32045			0.660223	+	0.751070i	$0.270462\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	2.35397	−	4.07719i	0.490836	−	0.850152i	−0.509109	−	0.860702i	$-0.670025\pi$
$23$	2.35397	−	4.07719i	0.490836	−	0.850152i	0.999944	+	0.0105499i	$0.00335820\pi$
$24$		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	0.974947	+	0.222435i	$0.0714006\pi$
$29$	3.66250	+	6.34363i	0.680108	+	1.17798i	−0.294839	+	0.955547i	$0.595266\pi$
$30$		0			0
$31$	2.93135	−	5.07724i	0.526485	−	0.911899i	−0.473039	−	0.881042i	$-0.656843\pi$
$31$	2.93135	−	5.07724i	0.526485	−	0.911899i	0.999524	−	0.0308575i	$-0.00982380\pi$
$32$		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	$-0.501871\pi$
$37$	−0.0714979			−0.0117542			−0.00587709	+	0.999983i	$0.501871\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	1.63887	−	2.83861i	0.255949	−	0.443317i	−0.709204	−	0.705004i	$-0.750945\pi$
$41$	1.63887	−	2.83861i	0.255949	−	0.443317i	0.965153	+	0.261687i	$0.0842787\pi$
$42$		0			0
$43$	2.12088	+	3.67347i	0.323431	+	0.560198i	0.981193	−	0.193027i	$-0.0618304\pi$
$43$	2.12088	+	3.67347i	0.323431	+	0.560198i	−0.657763	+	0.753225i	$0.728497\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−4.72803	−	8.18919i	−0.689654	−	1.19452i	−0.971950	−	0.235188i	$-0.924429\pi$
$47$	−4.72803	−	8.18919i	−0.689654	−	1.19452i	0.282296	−	0.959327i	$-0.408904\pi$
$48$		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	$-0.354452\pi$
$53$	6.42812			0.882970			0.441485	+	0.897269i	$0.354452\pi$
$54$		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	0.452244	+	0.891894i	$0.350623\pi$
$59$	−4.19606	+	7.26779i	−0.546281	+	0.946186i	−0.998525	+	0.0542918i	$0.982710\pi$
$60$		0			0
$61$	4.66250	+	8.07568i	0.596971	+	1.03398i	0.993265	+	0.115861i	$0.0369628\pi$
$61$	4.66250	+	8.07568i	0.596971	+	1.03398i	−0.396294	+	0.918124i	$0.629704\pi$
$62$		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	0.204897	+	0.978783i	$0.434314\pi$
$67$	−6.09975	+	10.5651i	−0.745203	+	1.29073i	−0.950100	+	0.311945i	$0.899019\pi$
$68$		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	−0.998493	−	0.0548820i	$-0.982522\pi$
$79$	−4.85985	−	8.41750i	−0.546776	−	0.947043i	0.451717	−	0.892161i	$-0.350812\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−3.07022	−	5.31778i	−0.337000	−	0.583702i	0.646867	−	0.762603i	$-0.276079\pi$
$83$	−3.07022	−	5.31778i	−0.337000	−	0.583702i	−0.983867	+	0.178901i	$0.942746\pi$
$84$		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	0.983298	+	0.182005i	$0.0582586\pi$
$97$	6.39456	+	11.0757i	0.649270	+	1.12457i	−0.334028	+	0.942563i	$0.608408\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−3.80137	−	6.58417i	−0.378250	−	0.655149i	0.612557	−	0.790426i	$-0.290141\pi$
$101$	−3.80137	−	6.58417i	−0.378250	−	0.655149i	−0.990808	+	0.135277i	$0.956808\pi$
$102$		0			0
$103$	−5.62490	+	9.74262i	−0.554238	+	0.959969i	0.443724	+	0.896163i	$0.353657\pi$
$103$	−5.62490	+	9.74262i	−0.554238	+	0.959969i	−0.997962	+	0.0638053i	$0.979676\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−2.81493			−0.272130			−0.136065	−	0.990700i	$-0.543446\pi$
$107$	−2.81493			−0.272130			−0.136065	+	0.990700i	$0.543446\pi$
$108$		0			0
$109$	−15.6539			−1.49937			−0.749685	−	0.661795i	$-0.769795\pi$
$109$	−15.6539			−1.49937			−0.749685	+	0.661795i	$0.769795\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−10.1828	+	17.6370i	−0.957913	+	1.65915i	−0.230355	+	0.973107i	$0.573989\pi$
$113$	−10.1828	+	17.6370i	−0.957913	+	1.65915i	−0.727557	+	0.686047i	$0.759344\pi$
$114$		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	$-0.826350\pi$
$131$	0.251085	−	0.434893i	0.0219374	−	0.0379968i	0.876786	+	0.480881i	$0.159683\pi$
$132$		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	0.995704	−	0.0925885i	$-0.0295141\pi$
$137$	4.88868	+	8.46744i	0.417668	+	0.723423i	−0.578036	+	0.816011i	$0.696181\pi$
$138$		0			0
$139$	0.188498	−	0.326488i	0.0159882	−	0.0276924i	−0.857921	−	0.513782i	$-0.828244\pi$
$139$	0.188498	−	0.326488i	0.0159882	−	0.0276924i	0.873909	+	0.486090i	$0.161577\pi$
$140$		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	$-0.963030\pi$
$149$	−4.83712	+	8.37814i	−0.396272	+	0.686364i	0.596990	+	0.802248i	$0.296363\pi$
$150$		0			0
$151$	−8.42915	−	14.5997i	−0.685954	−	1.18811i	−0.973136	−	0.230232i	$-0.926052\pi$
$151$	−8.42915	−	14.5997i	−0.685954	−	1.18811i	0.287181	−	0.957876i	$-0.407282\pi$
$152$		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	$-0.946532\pi$
$157$	−4.36262	+	7.55628i	−0.348175	+	0.603057i	0.637750	+	0.770243i	$0.279865\pi$
$158$		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	$-0.341350\pi$
$163$	12.2063			0.956067			0.478034	+	0.878342i	$0.341350\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	11.3806	−	19.7118i	0.880657	−	1.52534i	0.0300447	−	0.999549i	$-0.490435\pi$
$167$	11.3806	−	19.7118i	0.880657	−	1.52534i	0.850612	−	0.525794i	$-0.176232\pi$
$168$		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	0.812939	+	0.582348i	$0.197866\pi$
$173$	11.9797	+	20.7494i	0.910798	+	1.57755i	0.0978588	+	0.995200i	$0.468801\pi$
$174$		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	$-0.365158\pi$
$179$	10.9992			0.822121			0.411061	+	0.911608i	$0.365158\pi$
$180$		0			0
$181$	22.2168			1.65136			0.825679	−	0.564140i	$-0.190792\pi$
$181$	22.2168			1.65136			0.825679	+	0.564140i	$0.190792\pi$
$182$		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	0.993363	−	0.115023i	$-0.0366942\pi$
$191$	5.48760	+	9.50479i	0.397068	+	0.687743i	−0.596294	+	0.802766i	$0.703361\pi$
$192$		0			0
$193$	−7.11682	+	12.3267i	−0.512280	+	0.887294i	0.487619	+	0.873057i	$0.337866\pi$
$193$	−7.11682	+	12.3267i	−0.512280	+	0.887294i	−0.999899	+	0.0142378i	$0.995468\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−8.15037			−0.580690			−0.290345	−	0.956922i	$-0.593770\pi$
$197$	−8.15037			−0.580690			−0.290345	+	0.956922i	$0.593770\pi$
$198$		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	$-0.899995\pi$
$211$	−3.01985	+	5.23054i	−0.207895	+	0.360085i	0.743156	+	0.669118i	$0.233328\pi$
$212$		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	−0.966420	−	0.256966i	$-0.917277\pi$
$223$	−10.5391	−	18.2542i	−0.705749	−	1.22239i	0.260671	−	0.965428i	$-0.416056\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	14.9946	+	25.9713i	0.995224	+	1.72378i	0.582155	+	0.813078i	$0.302210\pi$
$227$	14.9946	+	25.9713i	0.995224	+	1.72378i	0.413069	+	0.910700i	$0.364457\pi$
$228$		0			0
$229$	9.53170	−	16.5094i	0.629873	−	1.09097i	−0.357704	−	0.933835i	$-0.616440\pi$
$229$	9.53170	−	16.5094i	0.629873	−	1.09097i	0.987577	−	0.157136i	$-0.0502262\pi$
$230$		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	−0.753973	−	0.656905i	$-0.771865\pi$
$239$	2.96685	−	5.13873i	0.191910	−	0.332397i	0.945883	+	0.324508i	$0.105199\pi$
$240$		0			0
$241$	−14.2494	−	24.6808i	−0.917888	−	1.58983i	−0.802618	−	0.596494i	$-0.796560\pi$
$241$	−14.2494	−	24.6808i	−0.917888	−	1.58983i	−0.115270	−	0.993334i	$-0.536773\pi$
$242$		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	$-0.335077\pi$
$251$	15.6924			0.990498			0.495249	+	0.868751i	$0.335077\pi$
$252$		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	0.239071	+	0.971002i	$0.423157\pi$
$257$	−11.5645	+	20.0304i	−0.721377	+	1.24946i	−0.960448	+	0.278459i	$0.910176\pi$
$258$		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	0.950431	+	0.310936i	$0.100642\pi$
$263$	12.0737	+	20.9122i	0.744494	+	1.28950i	−0.205937	+	0.978565i	$0.566024\pi$
$264$		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	$-0.406920\pi$
$269$	9.45599			0.576542			0.288271	+	0.957549i	$0.406920\pi$
$270$		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	$-0.111334\pi$
$277$	2.87862	+	4.98592i	0.172960	+	0.299575i	−0.766494	+	0.642252i	$0.778000\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−5.99712	−	10.3873i	−0.357758	−	0.619656i	0.629828	−	0.776735i	$-0.283126\pi$
$281$	−5.99712	−	10.3873i	−0.357758	−	0.619656i	−0.987586	+	0.157079i	$0.949792\pi$
$282$		0			0
$283$	0.604018	−	1.04619i	0.0359051	−	0.0621895i	−0.847514	−	0.530772i	$-0.821902\pi$
$283$	0.604018	−	1.04619i	0.0359051	−	0.0621895i	0.883420	+	0.468583i	$0.155235\pi$
$284$		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	−0.379587	−	0.925156i	$-0.623934\pi$
$293$	10.4657	−	18.1272i	0.611415	−	1.05900i	0.991002	−	0.133846i	$-0.0427328\pi$
$294$		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	$-0.546715\pi$
$307$	−5.12445			−0.292468			−0.146234	+	0.989250i	$0.546715\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−4.70739	+	8.15344i	−0.266931	+	0.462339i	−0.968068	−	0.250688i	$-0.919343\pi$
$311$	−4.70739	+	8.15344i	−0.266931	+	0.462339i	0.701136	+	0.713027i	$0.252676\pi$
$312$		0			0
$313$	−9.48986	−	16.4369i	−0.536398	−	0.929069i	−0.999094	−	0.0425521i	$-0.986451\pi$
$313$	−9.48986	−	16.4369i	−0.536398	−	0.929069i	0.462696	−	0.886517i	$-0.346882\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−14.2294	−	24.6461i	−0.799205	−	1.38426i	−0.920135	−	0.391602i	$-0.871921\pi$
$317$	−14.2294	−	24.6461i	−0.799205	−	1.38426i	0.120930	−	0.992661i	$-0.461412\pi$
$318$		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	$-0.181319\pi$
$331$	−0.837151	−	1.44999i	−0.0460140	−	0.0796986i	−0.888115	+	0.459621i	$0.847985\pi$
$332$		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	0.0806146	+	0.996745i	$0.474312\pi$
$337$	−15.1064	+	26.1651i	−0.822899	+	1.42530i	−0.903514	+	0.428558i	$0.859022\pi$
$338$		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	$-0.683259\pi$
$347$	8.46076	−	14.6545i	0.454197	−	0.786693i	0.998642	+	0.0521042i	$0.0165928\pi$
$348$		0			0
$349$	8.92436	+	15.4574i	0.477710	+	0.827418i	0.999674	−	0.0255500i	$-0.00813369\pi$
$349$	8.92436	+	15.4574i	0.477710	+	0.827418i	−0.521964	+	0.852968i	$0.674800\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	6.93593	+	12.0134i	0.369162	+	0.639407i	0.989435	−	0.144979i	$-0.0463114\pi$
$353$	6.93593	+	12.0134i	0.369162	+	0.639407i	−0.620273	+	0.784386i	$0.712978\pi$
$354$		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	−0.998124	−	0.0612208i	$-0.980501\pi$
$367$	−10.5763	−	18.3188i	−0.552081	−	0.956232i	0.446043	−	0.895011i	$-0.352833\pi$
$368$		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	$-0.761230\pi$
$373$	4.33750	−	7.51278i	0.224587	−	0.388997i	0.956196	+	0.292729i	$0.0945632\pi$
$374$		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	$-0.619302\pi$
$379$	−14.2538			−0.732168			−0.366084	+	0.930582i	$0.619302\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	5.11696	−	8.86283i	0.261464	−	0.452869i	−0.705167	−	0.709041i	$-0.749128\pi$
$383$	5.11696	−	8.86283i	0.261464	−	0.452869i	0.966631	+	0.256172i	$0.0824613\pi$
$384$		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	$-0.192951\pi$
$389$	−1.62675	−	2.81761i	−0.0824793	−	0.142858i	−0.904314	+	0.426867i	$0.859617\pi$
$390$		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	$-0.782090\pi$
$397$	−30.8709			−1.54936			−0.774682	+	0.632351i	$0.782090\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	2.01000	−	3.48143i	0.100375	−	0.173854i	−0.811464	−	0.584402i	$-0.801329\pi$
$401$	2.01000	−	3.48143i	0.100375	−	0.173854i	0.911839	+	0.410548i	$0.134662\pi$
$402$		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	−0.936700	−	0.350132i	$-0.886137\pi$
$409$	−3.33949	+	5.78416i	−0.165127	+	0.286008i	0.771573	+	0.636140i	$0.219470\pi$
$410$		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	−0.0626815	−	0.998034i	$-0.519965\pi$
$419$	17.0507	−	29.5327i	0.832982	−	1.44277i	0.895663	−	0.444733i	$-0.146701\pi$
$420$		0			0
$421$	9.34688	+	16.1893i	0.455539	+	0.789017i	0.998719	−	0.0505996i	$-0.0161132\pi$
$421$	9.34688	+	16.1893i	0.455539	+	0.789017i	−0.543180	+	0.839616i	$0.682780\pi$
$422$		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	0.760336	−	0.649530i	$-0.225034\pi$
$439$	−3.82047	−	6.61724i	−0.182341	−	0.315824i	−0.942677	+	0.333706i	$0.891701\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−6.94625	−	12.0313i	−0.330026	−	0.571623i	0.652490	−	0.757797i	$-0.273724\pi$
$443$	−6.94625	−	12.0313i	−0.330026	−	0.571623i	−0.982517	+	0.186175i	$0.940391\pi$
$444$		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	0.885455	−	0.464725i	$-0.153847\pi$
$457$	0.860741	+	1.49085i	0.0402638	+	0.0697389i	−0.845191	+	0.534464i	$0.820514\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	15.8265	+	27.4123i	0.737113	+	1.27672i	0.953790	+	0.300474i	$0.0971447\pi$
$461$	15.8265	+	27.4123i	0.737113	+	1.27672i	−0.216677	+	0.976243i	$0.569522\pi$
$462$		0			0
$463$	−1.71702	+	2.97396i	−0.0797966	+	0.138212i	−0.903162	−	0.429300i	$-0.858760\pi$
$463$	−1.71702	+	2.97396i	−0.0797966	+	0.138212i	0.823366	+	0.567511i	$0.192094\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−15.5333			−0.718797			−0.359398	−	0.933184i	$-0.617018\pi$
$467$	−15.5333			−0.718797			−0.359398	+	0.933184i	$0.617018\pi$
$468$		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	−0.941449	−	0.337156i	$-0.890535\pi$
$479$	−16.6927	−	28.9126i	−0.762710	−	1.32105i	0.178739	−	0.983897i	$-0.442798\pi$
$480$		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	$-0.863623\pi$
$491$	−2.10538	+	3.64663i	−0.0950146	+	0.164570i	0.814600	+	0.580023i	$0.196956\pi$
$492$		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	$-0.757851\pi$
$499$	5.24770	−	9.08928i	0.234919	−	0.406892i	0.959249	+	0.282561i	$0.0911841\pi$
$500$		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	$-0.796185\pi$
$509$	2.62702	−	4.55013i	0.116440	−	0.201681i	0.918355	+	0.395758i	$0.129518\pi$
$510$		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	$-0.535600\pi$
$523$	−5.10475			−0.223215			−0.111607	+	0.993752i	$0.535600\pi$
$524$		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	$-0.803994\pi$
$541$	−37.9746			−1.63266			−0.816328	+	0.577589i	$0.803994\pi$
$542$		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	−0.974575	−	0.224063i	$-0.928068\pi$
$547$	−15.9350	−	27.6003i	−0.681332	−	1.18010i	0.293243	−	0.956038i	$-0.405266\pi$
$548$		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	$-0.421030\pi$
$557$	11.5906			0.491111			0.245555	+	0.969383i	$0.421030\pi$
$558$		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	$-0.816538\pi$
$563$	1.25138	−	2.16745i	0.0527392	−	0.0913470i	0.891190	+	0.453631i	$0.149871\pi$
$564$		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	−0.998712	−	0.0507432i	$-0.983841\pi$
$569$	−12.9597	−	22.4469i	−0.543301	−	0.941024i	0.455411	−	0.890281i	$-0.349492\pi$
$570$		0			0
$571$	−5.03679	+	8.72398i	−0.210783	+	0.365087i	−0.951960	−	0.306223i	$-0.900935\pi$
$571$	−5.03679	+	8.72398i	−0.210783	+	0.365087i	0.741177	+	0.671310i	$0.234268\pi$
$572$		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	−0.999897	−	0.0143435i	$-0.995434\pi$
$587$	−12.4138	−	21.5012i	−0.512370	−	0.887451i	0.487527	−	0.873108i	$-0.337899\pi$
$588$		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	−0.391274	−	0.920274i	$-0.627965\pi$
$599$	14.7176	−	25.4916i	0.601344	−	1.04156i	0.992618	−	0.121284i	$-0.0387013\pi$
$600$		0			0
$601$	−1.76388	−	3.05514i	−0.0719503	−	0.124622i	0.827806	−	0.561015i	$-0.189589\pi$
$601$	−1.76388	−	3.05514i	−0.0719503	−	0.124622i	−0.899756	+	0.436393i	$0.856256\pi$
$602$		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	0.458178	+	0.888860i	$0.348502\pi$
$607$	−13.3211	+	23.0728i	−0.540687	+	0.936497i	−0.998865	+	0.0476362i	$0.984831\pi$
$608$		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	$-0.495459\pi$
$613$	0.706406			0.0285315			0.0142657	+	0.999898i	$0.495459\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−8.58480	+	14.8693i	−0.345611	+	0.598616i	−0.985465	−	0.169881i	$-0.945662\pi$
$617$	−8.58480	+	14.8693i	−0.345611	+	0.598616i	0.639853	+	0.768497i	$0.278995\pi$
$618$		0			0
$619$	−4.17800	−	7.23651i	−0.167928	−	0.290860i	0.769763	−	0.638330i	$-0.220374\pi$
$619$	−4.17800	−	7.23651i	−0.167928	−	0.290860i	−0.937691	+	0.347470i	$0.887041\pi$
$620$		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	0.706299	−	0.707913i	$-0.250363\pi$
$641$	−6.58068	−	11.3981i	−0.259921	−	0.450197i	−0.966221	+	0.257716i	$0.917030\pi$
$642$		0			0
$643$	7.85931	−	13.6127i	0.309941	−	0.536834i	−0.668408	−	0.743795i	$-0.733024\pi$
$643$	7.85931	−	13.6127i	0.309941	−	0.536834i	0.978349	+	0.206961i	$0.0663573\pi$
$644$		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	−0.485897	−	0.874016i	$-0.661507\pi$
$653$	13.1340	−	22.7487i	0.513971	−	0.890224i	0.999869	−	0.0162084i	$-0.00515952\pi$
$654$		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	−0.999806	−	0.0196993i	$-0.993729\pi$
$659$	−13.2710	−	22.9860i	−0.516963	−	0.895406i	0.482843	−	0.875707i	$-0.339604\pi$
$660$		0			0
$661$	0.981745	−	1.70043i	0.0381855	−	0.0661392i	−0.846301	−	0.532705i	$-0.821176\pi$
$661$	0.981745	−	1.70043i	0.0381855	−	0.0661392i	0.884487	+	0.466566i	$0.154509\pi$
$662$		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	−0.956090	−	0.293073i	$-0.905322\pi$
$673$	−18.9859	−	32.8846i	−0.731854	−	1.26761i	0.224236	−	0.974535i	$-0.428011\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	13.5894	+	23.5375i	0.522282	+	0.904619i	0.999664	+	0.0259229i	$0.00825242\pi$
$677$	13.5894	+	23.5375i	0.522282	+	0.904619i	−0.477382	+	0.878696i	$0.658414\pi$
$678$		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	$-0.145367\pi$
$683$	46.9121			1.79504			0.897520	+	0.440974i	$0.145367\pi$
$684$		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	$-0.326822\pi$
$691$	−12.6750	−	21.9538i	−0.482181	−	0.835161i	−0.999791	+	0.0204552i	$0.993488\pi$
$692$		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	$-0.184719\pi$
$701$	44.2840			1.67258			0.836292	+	0.548284i	$0.184719\pi$
$702$		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	$-0.261356\pi$
$709$	−7.80457	−	13.5179i	−0.293107	−	0.507676i	−0.974543	+	0.224202i	$0.928022\pi$
$710$		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	0.517977	−	0.855395i	$-0.326685\pi$
$727$	−12.9909	−	22.5009i	−0.481805	−	0.834511i	−0.999782	+	0.0208834i	$0.993352\pi$
$728$		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	$-0.897424\pi$
$733$	−5.41447	+	9.37814i	−0.199988	+	0.346390i	0.748536	+	0.663094i	$0.230757\pi$
$734$		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	$-0.567553\pi$
$739$	−11.4520			−0.421270			−0.210635	+	0.977565i	$0.567553\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−24.0077	+	41.5826i	−0.880758	+	1.52552i	−0.0302573	+	0.999542i	$0.509633\pi$
$743$	−24.0077	+	41.5826i	−0.880758	+	1.52552i	−0.850500	+	0.525975i	$0.823701\pi$
$744$		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	−0.0782360	−	0.996935i	$-0.524929\pi$
$751$	22.5881	−	39.1238i	0.824253	−	1.42765i	0.902489	−	0.430713i	$-0.141738\pi$
$752$		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	$-0.597214\pi$
$757$	−16.5457			−0.601365			−0.300682	+	0.953724i	$0.597214\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	20.6826	−	35.8234i	0.749745	−	1.29860i	−0.198200	−	0.980162i	$-0.563509\pi$
$761$	20.6826	−	35.8234i	0.749745	−	1.29860i	0.947945	−	0.318435i	$-0.103157\pi$
$762$		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	−0.918300	−	0.395886i	$-0.870438\pi$
$769$	−3.22518	+	5.58617i	−0.116303	+	0.201443i	0.801997	+	0.597328i	$0.203771\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−0.949001			−0.0341332			−0.0170666	−	0.999854i	$-0.505433\pi$
$773$	−0.949001			−0.0341332			−0.0170666	+	0.999854i	$0.505433\pi$
$774$		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	0.356458	+	0.934312i	$0.383984\pi$
$787$	−17.6992	+	30.6559i	−0.630909	+	1.09277i	−0.987366	+	0.158454i	$0.949349\pi$
$788$		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	$-0.849421\pi$
$797$	−1.42624	+	2.47032i	−0.0505200	+	0.0875032i	0.839660	+	0.543113i	$0.182755\pi$
$798$		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	$-0.457713\pi$
$811$	7.54428			0.264915			0.132458	+	0.991189i	$0.457713\pi$
$812$		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	−0.983879	−	0.178835i	$-0.942767\pi$
$821$	−18.5333	−	32.1006i	−0.646815	−	1.12032i	0.337064	−	0.941482i	$-0.390566\pi$
$822$		0			0
$823$	21.9008	−	37.9334i	0.763416	−	1.32227i	−0.177665	−	0.984091i	$-0.556854\pi$
$823$	21.9008	−	37.9334i	0.763416	−	1.32227i	0.941080	−	0.338184i	$-0.109812\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−12.4192			−0.431859			−0.215929	−	0.976409i	$-0.569278\pi$
$827$	−12.4192			−0.431859			−0.215929	+	0.976409i	$0.569278\pi$
$828$		0			0
$829$	15.1889			0.527532			0.263766	−	0.964587i	$-0.415035\pi$
$829$	15.1889			0.527532			0.263766	+	0.964587i	$0.415035\pi$
$830$		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	0.877372	+	0.479811i	$0.159295\pi$
$839$	24.7428	+	42.8557i	0.854215	+	1.47954i	−0.0231569	+	0.999732i	$0.507372\pi$
$840$		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	$-0.0967157\pi$
$853$	6.36674	+	11.0275i	0.217993	+	0.377575i	−0.736201	+	0.676763i	$0.763382\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	10.6560	+	18.4567i	0.364001	+	0.630467i	0.988615	−	0.150466i	$-0.0480773\pi$
$857$	10.6560	+	18.4567i	0.364001	+	0.630467i	−0.624615	+	0.780933i	$0.714744\pi$
$858$		0			0
$859$	−6.26681	+	10.8544i	−0.213821	+	0.370348i	−0.952907	−	0.303262i	$-0.901924\pi$
$859$	−6.26681	+	10.8544i	−0.213821	+	0.370348i	0.739086	+	0.673611i	$0.235257\pi$
$860$		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	$-0.821834\pi$
$877$	1.06962	−	1.85264i	0.0361185	−	0.0625591i	0.883520	+	0.468394i	$0.155167\pi$
$878$		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	$-0.445749\pi$
$883$	10.0800			0.339221			0.169610	+	0.985511i	$0.445749\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	16.6871	−	28.9028i	0.560297	−	0.970462i	−0.437174	−	0.899377i	$-0.644020\pi$
$887$	16.6871	−	28.9028i	0.560297	−	0.970462i	0.997470	−	0.0710852i	$-0.0226462\pi$
$888$		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	−0.909360	−	0.416010i	$-0.863428\pi$
$907$	−24.5436	−	42.5107i	−0.814955	−	1.41154i	0.0944045	−	0.995534i	$-0.469905\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	7.31861	+	12.6762i	0.242476	+	0.419981i	0.961419	−	0.275088i	$-0.0887070\pi$
$911$	7.31861	+	12.6762i	0.242476	+	0.419981i	−0.718943	+	0.695069i	$0.755374\pi$
$912$		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	0.685904	−	0.727692i	$-0.259407\pi$
$929$	−8.75518	−	15.1644i	−0.287248	−	0.497528i	−0.973152	+	0.230164i	$0.926074\pi$
$930$		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	$-0.812839\pi$
$941$	1.97367	−	3.41849i	0.0643397	−	0.111440i	0.896401	+	0.443244i	$0.146173\pi$
$942$		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	$-0.223411\pi$
$947$	−5.45686	−	9.45155i	−0.177324	−	0.307134i	−0.940963	+	0.338509i	$0.890077\pi$
$948$		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	−0.996586	−	0.0825607i	$-0.973690\pi$
$967$	−13.2718	+	22.9875i	−0.426793	+	0.739228i	0.569793	+	0.821788i	$0.307024\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−19.9365			−0.639792			−0.319896	−	0.947453i	$-0.603648\pi$
$971$	−19.9365			−0.639792			−0.319896	+	0.947453i	$0.603648\pi$
$972$		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	−0.219787	−	0.975548i	$-0.570536\pi$
$977$	22.9725	−	39.7895i	0.734956	−	1.27298i	0.954743	−	0.297433i	$-0.0961303\pi$
$978$		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	−0.999884	−	0.0151995i	$-0.995162\pi$
$983$	−16.0873	−	27.8640i	−0.513105	−	0.888725i	0.486779	−	0.873525i	$-0.338172\pi$
$984$		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	$-0.260088\pi$
$997$	−9.13466	−	15.8217i	−0.289297	−	0.501078i	−0.973642	+	0.228081i	$0.926755\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3456.2.i.k.2305.2		12
3.2	odd	2		1152.2.i.i.769.3	yes	12
4.3	odd	2		3456.2.i.l.2305.2		12
8.3	odd	2		3456.2.i.j.2305.5		12
8.5	even	2		3456.2.i.i.2305.5		12
9.2	odd	6		1152.2.i.i.385.3	✓	12
9.7	even	3	inner	3456.2.i.k.1153.2		12
12.11	even	2		1152.2.i.k.769.4	yes	12
24.5	odd	2		1152.2.i.l.769.4	yes	12
24.11	even	2		1152.2.i.j.769.3	yes	12
36.7	odd	6		3456.2.i.l.1153.2		12
36.11	even	6		1152.2.i.k.385.4	yes	12
72.11	even	6		1152.2.i.j.385.3	yes	12
72.29	odd	6		1152.2.i.l.385.4	yes	12
72.43	odd	6		3456.2.i.j.1153.5		12
72.61	even	6		3456.2.i.i.1153.5		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1152.2.i.i.385.3	✓	12	9.2	odd	6
1152.2.i.i.769.3	yes	12	3.2	odd	2
1152.2.i.j.385.3	yes	12	72.11	even	6
1152.2.i.j.769.3	yes	12	24.11	even	2
1152.2.i.k.385.4	yes	12	36.11	even	6
1152.2.i.k.769.4	yes	12	12.11	even	2
1152.2.i.l.385.4	yes	12	72.29	odd	6
1152.2.i.l.769.4	yes	12	24.5	odd	2
3456.2.i.i.1153.5		12	72.61	even	6
3456.2.i.i.2305.5		12	8.5	even	2
3456.2.i.j.1153.5		12	72.43	odd	6
3456.2.i.j.2305.5		12	8.3	odd	2
3456.2.i.k.1153.2		12	9.7	even	3	inner
3456.2.i.k.2305.2		12	1.1	even	1	trivial
3456.2.i.l.1153.2		12	36.7	odd	6
3456.2.i.l.2305.2		12	4.3	odd	2

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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