LMFDB - Embedded newform 3456.2.i.k.1153.3

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			1153.3
Root			$1.19051 - 1.25805i$ of defining polynomial
Character	$\chi$	$=$	3456.1153
Dual form			3456.2.i.k.2305.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.268104 - 0.464369i) q^{5} +(-2.35014 + 4.07056i) q^{7} +O(q^{10})$	$q+(-0.268104 - 0.464369i) q^{5} +(-2.35014 + 4.07056i) q^{7} +(-2.59922 + 4.50198i) q^{11} +(-0.778295 - 1.34805i) q^{13} -0.695781 q^{17} -5.80593 q^{19} +(4.42809 + 7.66967i) q^{23} +(2.35624 - 4.08113i) q^{25} +(-1.92199 + 3.32898i) q^{29} +(-2.77035 - 4.79840i) q^{31} +2.52033 q^{35} +4.09280 q^{37} +(-1.01019 - 1.74970i) q^{41} +(3.71522 - 6.43494i) q^{43} +(-0.186066 + 0.322275i) q^{47} +(-7.54633 - 13.0706i) q^{49} -5.30777 q^{53} +2.78744 q^{55} +(-2.57152 - 4.45401i) q^{59} +(-0.921988 + 1.59693i) q^{61} +(-0.417328 + 0.722833i) q^{65} +(-5.79316 - 10.0340i) q^{67} +10.6289 q^{71} +4.40840 q^{73} +(-12.2171 - 21.1606i) q^{77} +(-3.32244 + 5.75464i) q^{79} +(5.28055 - 9.14617i) q^{83} +(0.186541 + 0.323099i) q^{85} +7.30777 q^{89} +7.31642 q^{91} +(1.55659 + 2.69609i) q^{95} +(-7.81612 + 13.5379i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	−0.268104	−	0.464369i	−0.119900	−	0.207672i	0.799828	−	0.600229i	$-0.204924\pi$
$5$	−0.268104	−	0.464369i	−0.119900	−	0.207672i	−0.919728	+	0.392557i	$0.871591\pi$
$6$		0			0
$7$	−2.35014	+	4.07056i	−0.888270	+	1.53853i	−0.0463510	+	0.998925i	$0.514759\pi$
$7$	−2.35014	+	4.07056i	−0.888270	+	1.53853i	−0.841919	+	0.539604i	$0.818574\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−2.59922	+	4.50198i	−0.783695	+	1.35740i	0.146081	+	0.989273i	$0.453334\pi$
$11$	−2.59922	+	4.50198i	−0.783695	+	1.35740i	−0.929776	+	0.368126i	$0.879999\pi$
$12$		0			0
$13$	−0.778295	−	1.34805i	−0.215860	−	0.373881i	0.737678	−	0.675153i	$-0.235922\pi$
$13$	−0.778295	−	1.34805i	−0.215860	−	0.373881i	−0.953538	+	0.301272i	$0.902589\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−0.695781			−0.168752			−0.0843759	−	0.996434i	$-0.526890\pi$
$17$	−0.695781			−0.168752			−0.0843759	+	0.996434i	$0.526890\pi$
$18$		0			0
$19$	−5.80593			−1.33197			−0.665986	−	0.745965i	$-0.731989\pi$
$19$	−5.80593			−1.33197			−0.665986	+	0.745965i	$0.731989\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	4.42809	+	7.66967i	0.923320	+	1.59924i	0.794241	+	0.607603i	$0.207869\pi$
$23$	4.42809	+	7.66967i	0.923320	+	1.59924i	0.129079	+	0.991634i	$0.458798\pi$
$24$		0			0
$25$	2.35624	−	4.08113i	0.471248	−	0.816226i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−1.92199	+	3.32898i	−0.356904	+	0.618176i	−0.987442	−	0.157982i	$-0.949501\pi$
$29$	−1.92199	+	3.32898i	−0.356904	+	0.618176i	0.630538	+	0.776159i	$0.282834\pi$
$30$		0			0
$31$	−2.77035	−	4.79840i	−0.497570	−	0.861817i	0.502426	−	0.864620i	$-0.332441\pi$
$31$	−2.77035	−	4.79840i	−0.497570	−	0.861817i	−0.999996	+	0.00280317i	$0.999108\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	2.52033			0.426013
$36$		0			0
$37$	4.09280			0.672852			0.336426	−	0.941710i	$-0.390782\pi$
$37$	4.09280			0.672852			0.336426	+	0.941710i	$0.390782\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−1.01019	−	1.74970i	−0.157765	−	0.273258i	0.776297	−	0.630367i	$-0.217096\pi$
$41$	−1.01019	−	1.74970i	−0.157765	−	0.273258i	−0.934063	+	0.357109i	$0.883762\pi$
$42$		0			0
$43$	3.71522	−	6.43494i	0.566565	−	0.981319i	−0.430337	−	0.902668i	$-0.641605\pi$
$43$	3.71522	−	6.43494i	0.566565	−	0.981319i	0.996902	−	0.0786512i	$-0.0250613\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−0.186066	+	0.322275i	−0.0271404	+	0.0470086i	−0.879277	−	0.476311i	$-0.841973\pi$
$47$	−0.186066	+	0.322275i	−0.0271404	+	0.0470086i	0.852136	+	0.523320i	$0.175307\pi$
$48$		0			0
$49$	−7.54633	−	13.0706i	−1.07805	−	1.86723i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−5.30777			−0.729078			−0.364539	−	0.931188i	$-0.618773\pi$
$53$	−5.30777			−0.729078			−0.364539	+	0.931188i	$0.618773\pi$
$54$		0			0
$55$	2.78744			0.375859
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−2.57152	−	4.45401i	−0.334784	−	0.579862i	0.648660	−	0.761079i	$-0.275330\pi$
$59$	−2.57152	−	4.45401i	−0.334784	−	0.579862i	−0.983443	+	0.181216i	$0.941997\pi$
$60$		0			0
$61$	−0.921988	+	1.59693i	−0.118049	+	0.204466i	−0.918994	−	0.394271i	$-0.870997\pi$
$61$	−0.921988	+	1.59693i	−0.118049	+	0.204466i	0.800946	+	0.598737i	$0.204330\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−0.417328	+	0.722833i	−0.0517631	+	0.0896564i
$66$		0			0
$67$	−5.79316	−	10.0340i	−0.707747	−	1.22585i	−0.965691	−	0.259694i	$-0.916378\pi$
$67$	−5.79316	−	10.0340i	−0.707747	−	1.22585i	0.257944	−	0.966160i	$-0.416955\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	10.6289			1.26142			0.630708	−	0.776021i	$-0.282765\pi$
$71$	10.6289			1.26142			0.630708	+	0.776021i	$0.282765\pi$
$72$		0			0
$73$	4.40840			0.515964			0.257982	−	0.966150i	$-0.416943\pi$
$73$	4.40840			0.515964			0.257982	+	0.966150i	$0.416943\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−12.2171	−	21.1606i	−1.39227	−	2.41147i
$78$		0			0
$79$	−3.32244	+	5.75464i	−0.373804	+	0.647448i	−0.990147	−	0.140030i	$-0.955280\pi$
$79$	−3.32244	+	5.75464i	−0.373804	+	0.647448i	0.616343	+	0.787478i	$0.288613\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	5.28055	−	9.14617i	0.579615	−	1.00392i	−0.415908	−	0.909407i	$-0.636536\pi$
$83$	5.28055	−	9.14617i	0.579615	−	1.00392i	0.995523	−	0.0945164i	$-0.0301305\pi$
$84$		0			0
$85$	0.186541	+	0.323099i	0.0202333	+	0.0350450i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	7.30777			0.774622			0.387311	−	0.921949i	$-0.373404\pi$
$89$	7.30777			0.774622			0.387311	+	0.921949i	$0.373404\pi$
$90$		0			0
$91$	7.31642			0.766969
$92$		0			0
$93$		0			0
$94$		0			0
$95$	1.55659	+	2.69609i	0.159703	+	0.276613i
$96$		0			0
$97$	−7.81612	+	13.5379i	−0.793607	+	1.37457i	0.130113	+	0.991499i	$0.458466\pi$
$97$	−7.81612	+	13.5379i	−0.793607	+	1.37457i	−0.923720	+	0.383068i	$0.874867\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	4.43218	−	7.67676i	0.441018	−	0.763866i	−0.556747	−	0.830682i	$-0.687951\pi$
$101$	4.43218	−	7.67676i	0.441018	−	0.763866i	0.997765	+	0.0668159i	$0.0212840\pi$
$102$		0			0
$103$	−1.64986	−	2.85764i	−0.162565	−	0.281571i	0.773223	−	0.634135i	$-0.218644\pi$
$103$	−1.64986	−	2.85764i	−0.162565	−	0.281571i	−0.935788	+	0.352563i	$0.885310\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−7.12139			−0.688451			−0.344226	−	0.938887i	$-0.611858\pi$
$107$	−7.12139			−0.688451			−0.344226	+	0.938887i	$0.611858\pi$
$108$		0			0
$109$	2.98862			0.286258			0.143129	−	0.989704i	$-0.454284\pi$
$109$	2.98862			0.286258			0.143129	+	0.989704i	$0.454284\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−3.40833	−	5.90340i	−0.320629	−	0.555345i	0.659989	−	0.751275i	$-0.270561\pi$
$113$	−3.40833	−	5.90340i	−0.320629	−	0.555345i	−0.980618	+	0.195930i	$0.937227\pi$
$114$		0			0
$115$	2.37437	−	4.11253i	0.221411	−	0.383496i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	1.63518	−	2.83222i	0.149897	−	0.259629i
$120$		0			0
$121$	−8.01190	−	13.8770i	−0.728355	−	1.26155i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−5.20790			−0.465809
$126$		0			0
$127$	17.7567			1.57565			0.787826	−	0.615899i	$-0.211207\pi$
$127$	17.7567			1.57565			0.787826	+	0.615899i	$0.211207\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−3.51253	−	6.08389i	−0.306892	−	0.531552i	0.670789	−	0.741648i	$-0.265956\pi$
$131$	−3.51253	−	6.08389i	−0.306892	−	0.531552i	−0.977681	+	0.210096i	$0.932622\pi$
$132$		0			0
$133$	13.6448	−	23.6334i	1.18315	−	2.04928i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−5.71048	+	9.89083i	−0.487879	+	0.845031i	−0.999903	−	0.0139402i	$-0.995563\pi$
$137$	−5.71048	+	9.89083i	−0.487879	+	0.845031i	0.512024	+	0.858971i	$0.328896\pi$
$138$		0			0
$139$	−3.10659	−	5.38078i	−0.263498	−	0.456392i	0.703671	−	0.710526i	$-0.251543\pi$
$139$	−3.10659	−	5.38078i	−0.263498	−	0.456392i	−0.967169	+	0.254134i	$0.918210\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	8.09185			0.676674
$144$		0			0
$145$	2.06117			0.171171
$146$		0			0
$147$		0			0
$148$		0			0
$149$	5.47858	+	9.48918i	0.448823	+	0.777384i	0.998310	−	0.0581186i	$-0.0185102\pi$
$149$	5.47858	+	9.48918i	0.448823	+	0.777384i	−0.549487	+	0.835502i	$0.685177\pi$
$150$		0			0
$151$	−7.28439	+	12.6169i	−0.592796	+	1.02675i	0.401058	+	0.916053i	$0.368642\pi$
$151$	−7.28439	+	12.6169i	−0.592796	+	1.02675i	−0.993854	+	0.110700i	$0.964691\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−1.48548	+	2.57293i	−0.119317	+	0.206663i
$156$		0			0
$157$	−7.66049	−	13.2684i	−0.611373	−	1.05893i	−0.991009	−	0.133794i	$-0.957284\pi$
$157$	−7.66049	−	13.2684i	−0.611373	−	1.05893i	0.379636	−	0.925136i	$-0.376049\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−41.6265			−3.28063
$162$		0			0
$163$	11.0724			0.867258			0.433629	−	0.901091i	$-0.357233\pi$
$163$	11.0724			0.867258			0.433629	+	0.901091i	$0.357233\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−4.15607	−	7.19852i	−0.321606	−	0.557039i	0.659213	−	0.751956i	$-0.270889\pi$
$167$	−4.15607	−	7.19852i	−0.321606	−	0.557039i	−0.980820	+	0.194917i	$0.937556\pi$
$168$		0			0
$169$	5.28851	−	9.15997i	0.406809	−	0.704613i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−0.0396656	+	0.0687029i	−0.00301572	+	0.00522338i	−0.867529	−	0.497386i	$-0.834293\pi$
$173$	−0.0396656	+	0.0687029i	−0.00301572	+	0.00522338i	0.864514	+	0.502609i	$0.167627\pi$
$174$		0			0
$175$	11.0750	+	19.1825i	0.837191	+	1.45006i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−20.8011			−1.55475			−0.777375	−	0.629038i	$-0.783449\pi$
$179$	−20.8011			−1.55475			−0.777375	+	0.629038i	$0.783449\pi$
$180$		0			0
$181$	−5.13118			−0.381397			−0.190699	−	0.981649i	$-0.561075\pi$
$181$	−5.13118			−0.381397			−0.190699	+	0.981649i	$0.561075\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−1.09729	−	1.90057i	−0.0806747	−	0.139733i
$186$		0			0
$187$	1.80849	−	3.13240i	0.132250	−	0.229063i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−9.89085	+	17.1315i	−0.715677	+	1.23959i	0.247021	+	0.969010i	$0.420548\pi$
$191$	−9.89085	+	17.1315i	−0.715677	+	1.23959i	−0.962698	+	0.270579i	$0.912785\pi$
$192$		0			0
$193$	1.79574	+	3.11031i	0.129260	+	0.223885i	0.923390	−	0.383863i	$-0.125407\pi$
$193$	1.79574	+	3.11031i	0.129260	+	0.223885i	−0.794130	+	0.607748i	$0.792073\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−1.71261			−0.122019			−0.0610093	−	0.998137i	$-0.519432\pi$
$197$	−1.71261			−0.122019			−0.0610093	+	0.998137i	$0.519432\pi$
$198$		0			0
$199$	−18.1299			−1.28520			−0.642598	−	0.766203i	$-0.722144\pi$
$199$	−18.1299			−1.28520			−0.642598	+	0.766203i	$0.722144\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−9.03389	−	15.6472i	−0.634055	−	1.09822i
$204$		0			0
$205$	−0.541672	+	0.938204i	−0.0378320	+	0.0655270i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	15.0909	−	26.1382i	1.04386	−	1.80802i
$210$		0			0
$211$	−0.734306	−	1.27186i	−0.0505517	−	0.0875581i	0.839642	−	0.543140i	$-0.182765\pi$
$211$	−0.734306	−	1.27186i	−0.0505517	−	0.0875581i	−0.890194	+	0.455582i	$0.849431\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−3.98425			−0.271724
$216$		0			0
$217$	26.0429			1.76791
$218$		0			0
$219$		0			0
$220$		0			0
$221$	0.541523	+	0.937946i	0.0364268	+	0.0630931i
$222$		0			0
$223$	−9.57845	+	16.5904i	−0.641420	+	1.11097i	0.343696	+	0.939081i	$0.388321\pi$
$223$	−9.57845	+	16.5904i	−0.641420	+	1.11097i	−0.985116	+	0.171891i	$0.945012\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−5.92737	+	10.2665i	−0.393414	+	0.681412i	−0.992897	−	0.118975i	$-0.962039\pi$
$227$	−5.92737	+	10.2665i	−0.393414	+	0.681412i	0.599484	+	0.800387i	$0.295373\pi$
$228$		0			0
$229$	−1.57219	−	2.72311i	−0.103893	−	0.179948i	0.809392	−	0.587268i	$-0.199797\pi$
$229$	−1.57219	−	2.72311i	−0.103893	−	0.179948i	−0.913285	+	0.407320i	$0.866463\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	2.82935			0.185357			0.0926783	−	0.995696i	$-0.470457\pi$
$233$	2.82935			0.185357			0.0926783	+	0.995696i	$0.470457\pi$
$234$		0			0
$235$	0.199539			0.0130165
$236$		0			0
$237$		0			0
$238$		0			0
$239$	1.58766	+	2.74991i	0.102697	+	0.177877i	0.912795	−	0.408418i	$-0.133919\pi$
$239$	1.58766	+	2.74991i	0.102697	+	0.177877i	−0.810098	+	0.586295i	$0.800586\pi$
$240$		0			0
$241$	6.63053	−	11.4844i	0.427110	−	0.739776i	−0.569505	−	0.821988i	$-0.692865\pi$
$241$	6.63053	−	11.4844i	0.427110	−	0.739776i	0.996615	+	0.0822117i	$0.0261984\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−4.04640	+	7.00857i	−0.258515	+	0.447761i
$246$		0			0
$247$	4.51873	+	7.82666i	0.287520	+	0.497999i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−3.06394			−0.193394			−0.0966970	−	0.995314i	$-0.530828\pi$
$251$	−3.06394			−0.193394			−0.0966970	+	0.995314i	$0.530828\pi$
$252$		0			0
$253$	−46.0383			−2.89440
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−8.14120	−	14.1010i	−0.507834	−	0.879595i	−0.999959	−	0.00907003i	$-0.997113\pi$
$257$	−8.14120	−	14.1010i	−0.507834	−	0.879595i	0.492125	−	0.870525i	$-0.336220\pi$
$258$		0			0
$259$	−9.61866	+	16.6600i	−0.597674	+	1.03520i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	11.0730	−	19.1790i	0.682789	−	1.18262i	−0.291337	−	0.956620i	$-0.594100\pi$
$263$	11.0730	−	19.1790i	0.682789	−	1.18262i	0.974126	−	0.226005i	$-0.0725664\pi$
$264$		0			0
$265$	1.42303	+	2.46476i	0.0874162	+	0.151409i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−31.7805			−1.93769			−0.968845	−	0.247667i	$-0.920336\pi$
$269$	−31.7805			−1.93769			−0.968845	+	0.247667i	$0.920336\pi$
$270$		0			0
$271$	−0.794033			−0.0482341			−0.0241170	−	0.999709i	$-0.507677\pi$
$271$	−0.794033			−0.0482341			−0.0241170	+	0.999709i	$0.507677\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	12.2488	+	21.2155i	0.738629	+	1.27934i
$276$		0			0
$277$	12.6085	−	21.8385i	0.757569	−	1.31215i	−0.186519	−	0.982451i	$-0.559720\pi$
$277$	12.6085	−	21.8385i	0.757569	−	1.31215i	0.944087	−	0.329696i	$-0.106946\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	5.06672	−	8.77581i	0.302255	−	0.523521i	−0.674391	−	0.738374i	$-0.735594\pi$
$281$	5.06672	−	8.77581i	0.302255	−	0.523521i	0.976646	+	0.214853i	$0.0689273\pi$
$282$		0			0
$283$	−11.6766	−	20.2245i	−0.694102	−	1.20222i	−0.970482	−	0.241172i	$-0.922468\pi$
$283$	−11.6766	−	20.2245i	−0.694102	−	1.20222i	0.276380	−	0.961048i	$-0.410865\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	9.49637			0.560553
$288$		0			0
$289$	−16.5159			−0.971523
$290$		0			0
$291$		0			0
$292$		0			0
$293$	13.1967	+	22.8573i	0.770959	+	1.33534i	0.937038	+	0.349228i	$0.113556\pi$
$293$	13.1967	+	22.8573i	0.770959	+	1.33534i	−0.166079	+	0.986112i	$0.553111\pi$
$294$		0			0
$295$	−1.37887	+	2.38827i	−0.0802809	+	0.139051i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	6.89272	−	11.9385i	0.398616	−	0.690424i
$300$		0			0
$301$	17.4626	+	30.2461i	1.00653	+	1.74335i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	0.988754			0.0566159
$306$		0			0
$307$	21.6724			1.23691			0.618454	−	0.785821i	$-0.287759\pi$
$307$	21.6724			1.23691			0.618454	+	0.785821i	$0.287759\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	14.5448	+	25.1924i	0.824761	+	1.42853i	0.902102	+	0.431523i	$0.142024\pi$
$311$	14.5448	+	25.1924i	0.824761	+	1.42853i	−0.0773408	+	0.997005i	$0.524643\pi$
$312$		0			0
$313$	3.52393	−	6.10363i	0.199185	−	0.344998i	−0.749080	−	0.662480i	$-0.769504\pi$
$313$	3.52393	−	6.10363i	0.199185	−	0.344998i	0.948264	+	0.317482i	$0.102837\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−0.444372	+	0.769675i	−0.0249584	+	0.0432292i	−0.878235	−	0.478230i	$-0.841279\pi$
$317$	−0.444372	+	0.769675i	−0.0249584	+	0.0432292i	0.853276	+	0.521459i	$0.174612\pi$
$318$		0			0
$319$	−9.99135	−	17.3055i	−0.559408	−	0.968923i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	4.03966			0.224772
$324$		0			0
$325$	−7.33540			−0.406895
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−0.874561	−	1.51478i	−0.0482161	−	0.0835127i
$330$		0			0
$331$	−11.4513	+	19.8342i	−0.629420	+	1.09019i	0.358249	+	0.933626i	$0.383374\pi$
$331$	−11.4513	+	19.8342i	−0.629420	+	1.09019i	−0.987668	+	0.156561i	$0.949959\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−3.10634	+	5.38033i	−0.169717	+	0.293959i
$336$		0			0
$337$	0.415255	+	0.719243i	0.0226204	+	0.0391796i	0.877114	−	0.480282i	$-0.159466\pi$
$337$	0.415255	+	0.719243i	0.0226204	+	0.0391796i	−0.854494	+	0.519462i	$0.826132\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	28.8031			1.55977
$342$		0			0
$343$	38.0378			2.05385
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−8.53131	−	14.7767i	−0.457985	−	0.793253i	0.540870	−	0.841106i	$-0.318095\pi$
$347$	−8.53131	−	14.7767i	−0.457985	−	0.793253i	−0.998854	+	0.0478537i	$0.984762\pi$
$348$		0			0
$349$	10.9443	−	18.9561i	0.585836	−	1.01470i	−0.408935	−	0.912564i	$-0.634100\pi$
$349$	10.9443	−	18.9561i	0.585836	−	1.01470i	0.994771	−	0.102134i	$-0.0325671\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−9.53407	+	16.5135i	−0.507447	+	0.878924i	0.492516	+	0.870304i	$0.336077\pi$
$353$	−9.53407	+	16.5135i	−0.507447	+	0.878924i	−0.999963	+	0.00862082i	$0.997256\pi$
$354$		0			0
$355$	−2.84964	−	4.93572i	−0.151243	−	0.261961i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−13.1482			−0.693938			−0.346969	−	0.937877i	$-0.612789\pi$
$359$	−13.1482			−0.693938			−0.346969	+	0.937877i	$0.612789\pi$
$360$		0			0
$361$	14.7088			0.774147
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−1.18191	−	2.04712i	−0.0618638	−	0.107151i
$366$		0			0
$367$	7.79061	−	13.4937i	0.406666	−	0.704367i	−0.587848	−	0.808972i	$-0.700025\pi$
$367$	7.79061	−	13.4937i	0.406666	−	0.704367i	0.994514	+	0.104605i	$0.0333578\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	12.4740	−	21.6056i	0.647618	−	1.12171i
$372$		0			0
$373$	9.92199	+	17.1854i	0.513741	+	0.889826i	0.999873	+	0.0159402i	$0.00507413\pi$
$373$	9.92199	+	17.1854i	0.513741	+	0.889826i	−0.486132	+	0.873885i	$0.661593\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	5.98350			0.308166
$378$		0			0
$379$	15.0470			0.772914			0.386457	−	0.922307i	$-0.373699\pi$
$379$	15.0470			0.772914			0.386457	+	0.922307i	$0.373699\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−16.4288	−	28.4556i	−0.839474	−	1.45401i	−0.890335	−	0.455305i	$-0.849530\pi$
$383$	−16.4288	−	28.4556i	−0.839474	−	1.45401i	0.0508616	−	0.998706i	$-0.483803\pi$
$384$		0			0
$385$	−6.55089	+	11.3465i	−0.333864	+	0.578270i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	1.87559	−	3.24862i	0.0950962	−	0.164711i	−0.814553	−	0.580090i	$-0.803017\pi$
$389$	1.87559	−	3.24862i	0.0950962	−	0.164711i	0.909649	+	0.415378i	$0.136351\pi$
$390$		0			0
$391$	−3.08098	−	5.33641i	−0.155812	−	0.269874i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	3.56304			0.179276
$396$		0			0
$397$	−23.5495			−1.18192			−0.590958	−	0.806702i	$-0.701250\pi$
$397$	−23.5495			−1.18192			−0.590958	+	0.806702i	$0.701250\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−7.28105	−	12.6111i	−0.363598	−	0.629771i	0.624952	−	0.780663i	$-0.285119\pi$
$401$	−7.28105	−	12.6111i	−0.363598	−	0.629771i	−0.988550	+	0.150893i	$0.951785\pi$
$402$		0			0
$403$	−4.31231	+	7.46914i	−0.214811	+	0.372064i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−10.6381	+	18.4257i	−0.527310	+	0.913328i
$408$		0			0
$409$	3.23655	+	5.60586i	0.160037	+	0.277192i	0.934882	−	0.354959i	$-0.115505\pi$
$409$	3.23655	+	5.60586i	0.160037	+	0.277192i	−0.774845	+	0.632152i	$0.782172\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	24.1738			1.18951
$414$		0			0
$415$	−5.66294			−0.277983
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−13.4378	−	23.2750i	−0.656481	−	1.13706i	−0.981520	−	0.191357i	$-0.938711\pi$
$419$	−13.4378	−	23.2750i	−0.656481	−	1.13706i	0.325040	−	0.945700i	$-0.394622\pi$
$420$		0			0
$421$	−3.85521	+	6.67742i	−0.187892	+	0.325438i	−0.944547	−	0.328376i	$-0.893499\pi$
$421$	−3.85521	+	6.67742i	−0.187892	+	0.325438i	0.756656	+	0.653814i	$0.226832\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−1.63943	+	2.83957i	−0.0795239	+	0.137740i
$426$		0			0
$427$	−4.33361	−	7.50603i	−0.209718	−	0.363242i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	3.45993			0.166659			0.0833294	−	0.996522i	$-0.473445\pi$
$431$	3.45993			0.166659			0.0833294	+	0.996522i	$0.473445\pi$
$432$		0			0
$433$	−12.7863			−0.614471			−0.307236	−	0.951633i	$-0.599404\pi$
$433$	−12.7863			−0.614471			−0.307236	+	0.951633i	$0.599404\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−25.7092	−	44.5296i	−1.22984	−	2.13014i
$438$		0			0
$439$	2.76458	−	4.78840i	0.131946	−	0.228538i	−0.792480	−	0.609897i	$-0.791211\pi$
$439$	2.76458	−	4.78840i	0.131946	−	0.228538i	0.924427	+	0.381359i	$0.124544\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−13.2718	+	22.9874i	−0.630563	+	1.09217i	0.356874	+	0.934152i	$0.383842\pi$
$443$	−13.2718	+	22.9874i	−0.630563	+	1.09217i	−0.987437	+	0.158014i	$0.949491\pi$
$444$		0			0
$445$	−1.95924	−	3.39350i	−0.0928769	−	0.160867i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	29.9625			1.41402			0.707010	−	0.707204i	$-0.250044\pi$
$449$	29.9625			1.41402			0.707010	+	0.707204i	$0.250044\pi$
$450$		0			0
$451$	10.5028			0.494560
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−1.96156	−	3.39752i	−0.0919593	−	0.159278i
$456$		0			0
$457$	−12.3904	+	21.4608i	−0.579597	+	1.00389i	0.415928	+	0.909398i	$0.363457\pi$
$457$	−12.3904	+	21.4608i	−0.579597	+	1.00389i	−0.995525	+	0.0944945i	$0.969877\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	5.30632	−	9.19081i	0.247140	−	0.428059i	−0.715591	−	0.698519i	$-0.753843\pi$
$461$	5.30632	−	9.19081i	0.247140	−	0.428059i	0.962731	+	0.270461i	$0.0871760\pi$
$462$		0			0
$463$	−5.31762	−	9.21040i	−0.247131	−	0.428043i	0.715598	−	0.698513i	$-0.246154\pi$
$463$	−5.31762	−	9.21040i	−0.247131	−	0.428043i	−0.962729	+	0.270469i	$0.912821\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−27.3831			−1.26714			−0.633569	−	0.773686i	$-0.718411\pi$
$467$	−27.3831			−1.26714			−0.633569	+	0.773686i	$0.718411\pi$
$468$		0			0
$469$	54.4590			2.51468
$470$		0			0
$471$		0			0
$472$		0			0
$473$	19.3133	+	33.4517i	0.888028	+	1.53811i
$474$		0			0
$475$	−13.6802	+	23.6947i	−0.627689	+	1.08719i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	20.3026	−	35.1651i	0.927649	−	1.60674i	0.140404	−	0.990094i	$-0.455160\pi$
$479$	20.3026	−	35.1651i	0.927649	−	1.60674i	0.787245	−	0.616641i	$-0.211507\pi$
$480$		0			0
$481$	−3.18541	−	5.51728i	−0.145242	−	0.251566i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	8.38212			0.380613
$486$		0			0
$487$	4.99658			0.226417			0.113208	−	0.993571i	$-0.463887\pi$
$487$	4.99658			0.226417			0.113208	+	0.993571i	$0.463887\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	19.2049	+	33.2639i	0.866705	+	1.50118i	0.865344	+	0.501178i	$0.167100\pi$
$491$	19.2049	+	33.2639i	0.866705	+	1.50118i	0.00136059	+	0.999999i	$0.499567\pi$
$492$		0			0
$493$	1.33728	−	2.31624i	0.0602282	−	0.104318i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−24.9794	+	43.2655i	−1.12048	+	1.94072i
$498$		0			0
$499$	−11.3616	−	19.6788i	−0.508614	−	0.880945i	−0.999950	−	0.00997497i	$-0.996825\pi$
$499$	−11.3616	−	19.6788i	−0.508614	−	0.880945i	0.491337	−	0.870970i	$-0.336509\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−40.2323			−1.79387			−0.896935	−	0.442161i	$-0.854212\pi$
$503$	−40.2323			−1.79387			−0.896935	+	0.442161i	$0.854212\pi$
$504$		0			0
$505$	−4.75313			−0.211512
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−4.30968	−	7.46459i	−0.191023	−	0.330862i	0.754566	−	0.656224i	$-0.227847\pi$
$509$	−4.30968	−	7.46459i	−0.191023	−	0.330862i	−0.945590	+	0.325362i	$0.894514\pi$
$510$		0			0
$511$	−10.3604	+	17.9447i	−0.458315	+	0.793825i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−0.884666	+	1.53229i	−0.0389830	+	0.0675206i
$516$		0			0
$517$	−0.967251	−	1.67533i	−0.0425396	−	0.0736808i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	20.5770			0.901496			0.450748	−	0.892651i	$-0.351157\pi$
$521$	20.5770			0.901496			0.450748	+	0.892651i	$0.351157\pi$
$522$		0			0
$523$	15.6990			0.686470			0.343235	−	0.939250i	$-0.388477\pi$
$523$	15.6990			0.686470			0.343235	+	0.939250i	$0.388477\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	1.92756	+	3.33863i	0.0839659	+	0.145433i
$528$		0			0
$529$	−27.7159	+	48.0054i	−1.20504	+	2.08719i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−1.57245	+	2.72357i	−0.0681106	+	0.117971i
$534$		0			0
$535$	1.90927	+	3.30696i	0.0825450	+	0.142972i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	78.4584			3.37944
$540$		0			0
$541$	4.79886			0.206319			0.103160	−	0.994665i	$-0.467105\pi$
$541$	4.79886			0.206319			0.103160	+	0.994665i	$0.467105\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−0.801260	−	1.38782i	−0.0343222	−	0.0594478i
$546$		0			0
$547$	8.26596	−	14.3171i	0.353427	−	0.612153i	−0.633421	−	0.773808i	$-0.718350\pi$
$547$	8.26596	−	14.3171i	0.353427	−	0.612153i	0.986847	+	0.161654i	$0.0516830\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	11.1589	−	19.3278i	0.475386	−	0.823393i
$552$		0			0
$553$	−15.6164	−	27.0484i	−0.664078	−	1.15022i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−14.2466			−0.603649			−0.301825	−	0.953363i	$-0.597596\pi$
$557$	−14.2466			−0.603649			−0.301825	+	0.953363i	$0.597596\pi$
$558$		0			0
$559$	−11.5661			−0.489196
$560$		0			0
$561$		0			0
$562$		0			0
$563$	13.4981	+	23.3794i	0.568876	+	0.985322i	0.996677	+	0.0814497i	$0.0259550\pi$
$563$	13.4981	+	23.3794i	0.568876	+	0.985322i	−0.427801	+	0.903873i	$0.640712\pi$
$564$		0			0
$565$	−1.82757	+	3.16545i	−0.0768865	+	0.133171i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−4.82124	+	8.35063i	−0.202117	+	0.350076i	−0.949210	−	0.314643i	$-0.898115\pi$
$569$	−4.82124	+	8.35063i	−0.202117	+	0.350076i	0.747094	+	0.664719i	$0.231449\pi$
$570$		0			0
$571$	15.0536	+	26.0736i	0.629973	+	1.09115i	0.987557	+	0.157264i	$0.0502674\pi$
$571$	15.0536	+	26.0736i	0.629973	+	1.09115i	−0.357584	+	0.933881i	$0.616399\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	41.7346			1.74045
$576$		0			0
$577$	14.9642			0.622968			0.311484	−	0.950251i	$-0.399174\pi$
$577$	14.9642			0.622968			0.311484	+	0.950251i	$0.399174\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	24.8201	+	42.9896i	1.02971	+	1.78351i
$582$		0			0
$583$	13.7961	−	23.8955i	0.571375	−	0.989650i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−12.0511	+	20.8731i	−0.497401	+	0.861523i	−0.999996	−	0.00299890i	$-0.999045\pi$
$587$	−12.0511	+	20.8731i	−0.497401	+	0.861523i	0.502595	+	0.864522i	$0.332379\pi$
$588$		0			0
$589$	16.0845	+	27.8591i	0.662749	+	1.14792i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−5.62882			−0.231148			−0.115574	−	0.993299i	$-0.536871\pi$
$593$	−5.62882			−0.231148			−0.115574	+	0.993299i	$0.536871\pi$
$594$		0			0
$595$	−1.75360			−0.0718904
$596$		0			0
$597$		0			0
$598$		0			0
$599$	9.75087	+	16.8890i	0.398410	+	0.690066i	0.993530	−	0.113571i	$-0.0362289\pi$
$599$	9.75087	+	16.8890i	0.398410	+	0.690066i	−0.595120	+	0.803637i	$0.702896\pi$
$600$		0			0
$601$	−1.36834	+	2.37003i	−0.0558158	+	0.0966757i	−0.892583	−	0.450883i	$-0.851109\pi$
$601$	−1.36834	+	2.37003i	−0.0558158	+	0.0966757i	0.836767	+	0.547558i	$0.184443\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−4.29604	+	7.44096i	−0.174659	+	0.302518i
$606$		0			0
$607$	−23.6876	−	41.0282i	−0.961452	−	1.66528i	−0.718860	−	0.695155i	$-0.755336\pi$
$607$	−23.6876	−	41.0282i	−0.961452	−	1.66528i	−0.242592	−	0.970128i	$-0.577998\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	0.579256			0.0234342
$612$		0			0
$613$	−23.1963			−0.936891			−0.468446	−	0.883492i	$-0.655186\pi$
$613$	−23.1963			−0.936891			−0.468446	+	0.883492i	$0.655186\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	19.8253	+	34.3384i	0.798137	+	1.38241i	0.920828	+	0.389969i	$0.127514\pi$
$617$	19.8253	+	34.3384i	0.798137	+	1.38241i	−0.122691	+	0.992445i	$0.539152\pi$
$618$		0			0
$619$	0.813544	−	1.40910i	0.0326991	−	0.0566365i	−0.849213	−	0.528051i	$-0.822923\pi$
$619$	0.813544	−	1.40910i	0.0326991	−	0.0566365i	0.881912	+	0.471414i	$0.156256\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−17.1743	+	29.7467i	−0.688074	+	1.19178i
$624$		0			0
$625$	−10.3849	−	17.9873i	−0.415398	−	0.719490i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−2.84769			−0.113545
$630$		0			0
$631$	30.9685			1.23283			0.616417	−	0.787420i	$-0.288583\pi$
$631$	30.9685			1.23283			0.616417	+	0.787420i	$0.288583\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−4.76063	−	8.24566i	−0.188920	−	0.327219i
$636$		0			0
$637$	−11.7466	+	20.3456i	−0.465415	+	0.806123i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	16.1499	−	27.9725i	0.637883	−	1.10485i	−0.348014	−	0.937489i	$-0.613144\pi$
$641$	16.1499	−	27.9725i	0.637883	−	1.10485i	0.985897	−	0.167356i	$-0.0535230\pi$
$642$		0			0
$643$	−3.28376	−	5.68763i	−0.129499	−	0.224298i	0.793984	−	0.607939i	$-0.208003\pi$
$643$	−3.28376	−	5.68763i	−0.129499	−	0.224298i	−0.923482	+	0.383641i	$0.874670\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−40.0823			−1.57580			−0.787899	−	0.615805i	$-0.788831\pi$
$647$	−40.0823			−1.57580			−0.787899	+	0.615805i	$0.788831\pi$
$648$		0			0
$649$	26.7358			1.04947
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−1.62772	−	2.81929i	−0.0636976	−	0.110327i	0.832418	−	0.554148i	$-0.186956\pi$
$653$	−1.62772	−	2.81929i	−0.0636976	−	0.110327i	−0.896116	+	0.443821i	$0.853623\pi$
$654$		0			0
$655$	−1.88345	+	3.26223i	−0.0735924	+	0.127466i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	7.58278	−	13.1338i	0.295383	−	0.511619i	−0.679691	−	0.733499i	$-0.737886\pi$
$659$	7.58278	−	13.1338i	0.295383	−	0.511619i	0.975074	+	0.221880i	$0.0712193\pi$
$660$		0			0
$661$	10.1447	+	17.5711i	0.394582	+	0.683435i	0.993048	−	0.117713i	$-0.0375562\pi$
$661$	10.1447	+	17.5711i	0.394582	+	0.683435i	−0.598466	+	0.801148i	$0.704223\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−14.6328			−0.567437
$666$		0			0
$667$	−34.0429			−1.31815
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−4.79290	−	8.30155i	−0.185028	−	0.320478i
$672$		0			0
$673$	11.6256	−	20.1361i	0.448134	−	0.776191i	−0.550130	−	0.835079i	$-0.685422\pi$
$673$	11.6256	−	20.1361i	0.448134	−	0.776191i	0.998265	+	0.0588875i	$0.0187553\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−15.6380	+	27.0859i	−0.601018	+	1.04099i	0.391649	+	0.920115i	$0.371905\pi$
$677$	−15.6380	+	27.0859i	−0.601018	+	1.04099i	−0.992667	+	0.120879i	$0.961429\pi$
$678$		0			0
$679$	−36.7380	−	63.6320i	−1.40987	−	2.44197i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	24.1812			0.925269			0.462635	−	0.886549i	$-0.346904\pi$
$683$	24.1812			0.925269			0.462635	+	0.886549i	$0.346904\pi$
$684$		0			0
$685$	6.12400			0.233986
$686$		0			0
$687$		0			0
$688$		0			0
$689$	4.13101	+	7.15512i	0.157379	+	0.272588i
$690$		0			0
$691$	−9.03942	+	15.6567i	−0.343876	+	0.595610i	−0.985149	−	0.171702i	$-0.945073\pi$
$691$	−9.03942	+	15.6567i	−0.343876	+	0.595610i	0.641273	+	0.767313i	$0.278407\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−1.66578	+	2.88521i	−0.0631866	+	0.109442i
$696$		0			0
$697$	0.702872	+	1.21741i	0.0266232	+	0.0461127i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−17.2240			−0.650541			−0.325271	−	0.945621i	$-0.605455\pi$
$701$	−17.2240			−0.650541			−0.325271	+	0.945621i	$0.605455\pi$
$702$		0			0
$703$	−23.7625			−0.896219
$704$		0			0
$705$		0			0
$706$		0			0
$707$	20.8325	+	36.0830i	0.783487	+	1.35704i
$708$		0			0
$709$	−13.3258	+	23.0809i	−0.500459	+	0.866821i	0.499541	+	0.866290i	$0.333502\pi$
$709$	−13.3258	+	23.0809i	−0.500459	+	0.866821i	−1.00000		0.000530358i	$0.999831\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	24.5347	−	42.4954i	0.918833	−	1.59147i
$714$		0			0
$715$	−2.16945	−	3.75760i	−0.0811330	−	0.140526i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−10.4273			−0.388875			−0.194437	−	0.980915i	$-0.562288\pi$
$719$	−10.4273			−0.388875			−0.194437	+	0.980915i	$0.562288\pi$
$720$		0			0
$721$	15.5096			0.577608
$722$		0			0
$723$		0			0
$724$		0			0
$725$	9.05733	+	15.6878i	0.336381	+	0.582629i
$726$		0			0
$727$	−10.5682	+	18.3047i	−0.391954	+	0.678884i	−0.992707	−	0.120550i	$-0.961534\pi$
$727$	−10.5682	+	18.3047i	−0.391954	+	0.678884i	0.600753	+	0.799434i	$0.294867\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−2.58498	+	4.47731i	−0.0956088	+	0.165599i
$732$		0			0
$733$	15.6473	+	27.1020i	0.577948	+	1.00103i	0.995714	+	0.0924806i	$0.0294796\pi$
$733$	15.6473	+	27.1020i	0.577948	+	1.00103i	−0.417767	+	0.908554i	$0.637187\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	60.2308			2.21863
$738$		0			0
$739$	−31.0455			−1.14203			−0.571014	−	0.820940i	$-0.693450\pi$
$739$	−31.0455			−1.14203			−0.571014	+	0.820940i	$0.693450\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−8.93131	−	15.4695i	−0.327658	−	0.567520i	0.654389	−	0.756158i	$-0.272926\pi$
$743$	−8.93131	−	15.4695i	−0.327658	−	0.567520i	−0.982047	+	0.188638i	$0.939593\pi$
$744$		0			0
$745$	2.93765	−	5.08817i	0.107627	−	0.186416i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	16.7363	−	28.9881i	0.611530	−	1.05920i
$750$		0			0
$751$	−4.70046	−	8.14144i	−0.171522	−	0.297085i	0.767430	−	0.641133i	$-0.221535\pi$
$751$	−4.70046	−	8.14144i	−0.171522	−	0.297085i	−0.938952	+	0.344047i	$0.888202\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	7.81189			0.284304
$756$		0			0
$757$	−49.7959			−1.80986			−0.904931	−	0.425558i	$-0.860078\pi$
$757$	−49.7959			−1.80986			−0.904931	+	0.425558i	$0.860078\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−5.42633	−	9.39868i	−0.196704	−	0.340702i	0.750754	−	0.660582i	$-0.229691\pi$
$761$	−5.42633	−	9.39868i	−0.196704	−	0.340702i	−0.947458	+	0.319880i	$0.896357\pi$
$762$		0			0
$763$	−7.02368	+	12.1654i	−0.254274	+	0.440416i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−4.00281	+	6.93307i	−0.144533	+	0.250338i
$768$		0			0
$769$	2.93798	+	5.08873i	0.105946	+	0.183504i	0.914124	−	0.405434i	$-0.132880\pi$
$769$	2.93798	+	5.08873i	0.105946	+	0.183504i	−0.808178	+	0.588938i	$0.799546\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	26.2781			0.945159			0.472579	−	0.881288i	$-0.343323\pi$
$773$	26.2781			0.945159			0.472579	+	0.881288i	$0.343323\pi$
$774$		0			0
$775$	−26.1105			−0.937917
$776$		0			0
$777$		0			0
$778$		0			0
$779$	5.86510	+	10.1587i	0.210139	+	0.363971i
$780$		0			0
$781$	−27.6268	+	47.8510i	−0.988564	+	1.71224i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−4.10761	+	7.11459i	−0.146607	+	0.253930i
$786$		0			0
$787$	16.3824	+	28.3752i	0.583970	+	1.01146i	0.995003	+	0.0998447i	$0.0318346\pi$
$787$	16.3824	+	28.3752i	0.583970	+	1.01146i	−0.411034	+	0.911620i	$0.634832\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	32.0402			1.13922
$792$		0			0
$793$	2.87032			0.101928
$794$		0			0
$795$		0			0
$796$		0			0
$797$	26.8586	+	46.5205i	0.951382	+	1.64784i	0.742440	+	0.669913i	$0.233669\pi$
$797$	26.8586	+	46.5205i	0.951382	+	1.64784i	0.208942	+	0.977928i	$0.432998\pi$
$798$		0			0
$799$	0.129461	−	0.224233i	0.00458000	−	0.00793279i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−11.4584	+	19.8465i	−0.404358	+	0.700368i
$804$		0			0
$805$	11.1602	+	19.3301i	0.393346	+	0.681296i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	5.82729			0.204877			0.102438	−	0.994739i	$-0.467336\pi$
$809$	5.82729			0.204877			0.102438	+	0.994739i	$0.467336\pi$
$810$		0			0
$811$	−25.5700			−0.897883			−0.448942	−	0.893561i	$-0.648199\pi$
$811$	−25.5700			−0.897883			−0.448942	+	0.893561i	$0.648199\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−2.96855	−	5.14169i	−0.103984	−	0.180105i
$816$		0			0
$817$	−21.5703	+	37.3608i	−0.754648	+	1.30709i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	6.62309	−	11.4715i	0.231147	−	0.400359i	−0.726999	−	0.686639i	$-0.759085\pi$
$821$	6.62309	−	11.4715i	0.231147	−	0.400359i	0.958146	+	0.286280i	$0.0924187\pi$
$822$		0			0
$823$	5.17425	+	8.96206i	0.180363	+	0.312398i	0.942004	−	0.335601i	$-0.108939\pi$
$823$	5.17425	+	8.96206i	0.180363	+	0.312398i	−0.761641	+	0.647999i	$0.775606\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−31.2799			−1.08771			−0.543854	−	0.839180i	$-0.683035\pi$
$827$	−31.2799			−1.08771			−0.543854	+	0.839180i	$0.683035\pi$
$828$		0			0
$829$	17.3415			0.602297			0.301148	−	0.953577i	$-0.402630\pi$
$829$	17.3415			0.602297			0.301148	+	0.953577i	$0.402630\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	5.25060	+	9.09430i	0.181922	+	0.315099i
$834$		0			0
$835$	−2.22852	+	3.85990i	−0.0771209	+	0.133577i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	11.3050	−	19.5809i	0.390293	−	0.676008i	−0.602195	−	0.798349i	$-0.705707\pi$
$839$	11.3050	−	19.5809i	0.390293	−	0.676008i	0.992488	+	0.122341i	$0.0390402\pi$
$840$		0			0
$841$	7.11192	+	12.3182i	0.245239	+	0.424766i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−5.67148			−0.195105
$846$		0			0
$847$	75.3164			2.58790
$848$		0			0
$849$		0			0
$850$		0			0
$851$	18.1233	+	31.3904i	0.621258	+	1.07605i
$852$		0			0
$853$	3.98508	−	6.90236i	0.136447	−	0.236332i	−0.789703	−	0.613490i	$-0.789765\pi$
$853$	3.98508	−	6.90236i	0.136447	−	0.236332i	0.926149	+	0.377157i	$0.123098\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	0.886072	−	1.53472i	0.0302676	−	0.0524251i	−0.850495	−	0.525983i	$-0.823697\pi$
$857$	0.886072	−	1.53472i	0.0302676	−	0.0524251i	0.880762	+	0.473558i	$0.157031\pi$
$858$		0			0
$859$	0.441545	+	0.764779i	0.0150653	+	0.0260939i	0.873460	−	0.486896i	$-0.161871\pi$
$859$	0.441545	+	0.764779i	0.0150653	+	0.0260939i	−0.858394	+	0.512990i	$0.828538\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−50.3624			−1.71436			−0.857178	−	0.515021i	$-0.827784\pi$
$863$	−50.3624			−1.71436			−0.857178	+	0.515021i	$0.827784\pi$
$864$		0			0
$865$	0.0425380			0.00144633
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−17.2715	−	29.9152i	−0.585896	−	1.01480i
$870$		0			0
$871$	−9.01758	+	15.6189i	−0.305549	+	0.529226i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	12.2393	−	21.1991i	0.413764	−	0.716661i
$876$		0			0
$877$	−19.6437	−	34.0238i	−0.663319	−	1.14890i	−0.979738	−	0.200283i	$-0.935814\pi$
$877$	−19.6437	−	34.0238i	−0.663319	−	1.14890i	0.316419	−	0.948620i	$-0.397520\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−31.7416			−1.06940			−0.534701	−	0.845041i	$-0.679576\pi$
$881$	−31.7416			−1.06940			−0.534701	+	0.845041i	$0.679576\pi$
$882$		0			0
$883$	29.5613			0.994818			0.497409	−	0.867516i	$-0.334285\pi$
$883$	29.5613			0.994818			0.497409	+	0.867516i	$0.334285\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	2.39404	+	4.14660i	0.0803839	+	0.139229i	0.903415	−	0.428767i	$-0.141052\pi$
$887$	2.39404	+	4.14660i	0.0803839	+	0.139229i	−0.823031	+	0.567997i	$0.807719\pi$
$888$		0			0
$889$	−41.7307	+	72.2797i	−1.39960	+	2.42418i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	1.08028	−	1.87111i	0.0361503	−	0.0626141i
$894$		0			0
$895$	5.57686	+	9.65941i	0.186414	+	0.322878i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	21.2984			0.710340
$900$		0			0
$901$	3.69305			0.123033
$902$		0			0
$903$		0			0
$904$		0			0
$905$	1.37569	+	2.38276i	0.0457294	+	0.0792057i
$906$		0			0
$907$	−7.27487	+	12.6004i	−0.241558	+	0.418391i	−0.961158	−	0.275998i	$-0.910992\pi$
$907$	−7.27487	+	12.6004i	−0.241558	+	0.418391i	0.719600	+	0.694389i	$0.244325\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−24.3323	+	42.1448i	−0.806166	+	1.39632i	0.109336	+	0.994005i	$0.465128\pi$
$911$	−24.3323	+	42.1448i	−0.806166	+	1.39632i	−0.915501	+	0.402315i	$0.868206\pi$
$912$		0			0
$913$	27.4506	+	47.5459i	0.908483	+	1.57354i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	33.0198			1.09041
$918$		0			0
$919$	26.6958			0.880612			0.440306	−	0.897848i	$-0.354870\pi$
$919$	26.6958			0.880612			0.440306	+	0.897848i	$0.354870\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−8.27240	−	14.3282i	−0.272289	−	0.471619i
$924$		0			0
$925$	9.64362	−	16.7032i	0.317080	−	0.549199i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	5.27705	−	9.14012i	0.173134	−	0.299878i	−0.766380	−	0.642388i	$-0.777944\pi$
$929$	5.27705	−	9.14012i	0.173134	−	0.299878i	0.939514	+	0.342510i	$0.111277\pi$
$930$		0			0
$931$	43.8135	+	75.8871i	1.43593	+	2.48710i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−1.93945			−0.0634268
$936$		0			0
$937$	−17.1990			−0.561866			−0.280933	−	0.959727i	$-0.590644\pi$
$937$	−17.1990			−0.561866			−0.280933	+	0.959727i	$0.590644\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	19.3051	+	33.4373i	0.629327	+	1.09003i	0.987687	+	0.156443i	$0.0500026\pi$
$941$	19.3051	+	33.4373i	0.629327	+	1.09003i	−0.358360	+	0.933583i	$0.616664\pi$
$942$		0			0
$943$	8.94643	−	15.4957i	0.291336	−	0.504609i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	15.2306	−	26.3802i	0.494929	−	0.857243i	−0.505054	−	0.863088i	$-0.668527\pi$
$947$	15.2306	−	26.3802i	0.494929	−	0.857243i	0.999983	+	0.00584526i	$0.00186061\pi$
$948$		0			0
$949$	−3.43103	−	5.94272i	−0.111376	−	0.192909i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−7.78951			−0.252327			−0.126163	−	0.992009i	$-0.540266\pi$
$953$	−7.78951			−0.252327			−0.126163	+	0.992009i	$0.540266\pi$
$954$		0			0
$955$	10.6071			0.343238
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−26.8409	−	46.4897i	−0.866736	−	1.50123i
$960$		0			0
$961$	0.150268	−	0.260272i	0.00484736	−	0.00839587i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	0.962887	−	1.66777i	0.0309964	−	0.0536874i
$966$		0			0
$967$	−19.3058	−	33.4386i	−0.620832	−	1.07531i	−0.989331	−	0.145685i	$-0.953462\pi$
$967$	−19.3058	−	33.4386i	−0.620832	−	1.07531i	0.368499	−	0.929628i	$-0.379872\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−15.7257			−0.504661			−0.252331	−	0.967641i	$-0.581197\pi$
$971$	−15.7257			−0.504661			−0.252331	+	0.967641i	$0.581197\pi$
$972$		0			0
$973$	29.2037			0.936229
$974$		0			0
$975$		0			0
$976$		0			0
$977$	9.58188	+	16.5963i	0.306551	+	0.530963i	0.977606	−	0.210446i	$-0.0674915\pi$
$977$	9.58188	+	16.5963i	0.306551	+	0.530963i	−0.671054	+	0.741408i	$0.734158\pi$
$978$		0			0
$979$	−18.9945	+	32.8995i	−0.607067	+	1.05147i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−19.5590	+	33.8772i	−0.623835	+	1.08051i	0.364930	+	0.931035i	$0.381093\pi$
$983$	−19.5590	+	33.8772i	−0.623835	+	1.08051i	−0.988765	+	0.149479i	$0.952240\pi$
$984$		0			0
$985$	0.459158	+	0.795285i	0.0146300	+	0.0253399i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	65.8052			2.09248
$990$		0			0
$991$	27.5470			0.875060			0.437530	−	0.899204i	$-0.355853\pi$
$991$	27.5470			0.875060			0.437530	+	0.899204i	$0.355853\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	4.86070	+	8.41898i	0.154095	+	0.266900i
$996$		0			0
$997$	15.6863	−	27.1695i	0.496791	−	0.860467i	−0.503202	−	0.864169i	$-0.667845\pi$
$997$	15.6863	−	27.1695i	0.496791	−	0.860467i	0.999993	+	0.00370166i	$0.00117828\pi$
$998$		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.1153.3		12
3.2	odd	2		1152.2.i.i.385.5	✓	12
4.3	odd	2		3456.2.i.l.1153.3		12
8.3	odd	2		3456.2.i.j.1153.4		12
8.5	even	2		3456.2.i.i.1153.4		12
9.4	even	3	inner	3456.2.i.k.2305.3		12
9.5	odd	6		1152.2.i.i.769.5	yes	12
12.11	even	2		1152.2.i.k.385.2	yes	12
24.5	odd	2		1152.2.i.l.385.2	yes	12
24.11	even	2		1152.2.i.j.385.5	yes	12
36.23	even	6		1152.2.i.k.769.2	yes	12
36.31	odd	6		3456.2.i.l.2305.3		12
72.5	odd	6		1152.2.i.l.769.2	yes	12
72.13	even	6		3456.2.i.i.2305.4		12
72.59	even	6		1152.2.i.j.769.5	yes	12
72.67	odd	6		3456.2.i.j.2305.4		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1152.2.i.i.385.5	✓	12	3.2	odd	2
1152.2.i.i.769.5	yes	12	9.5	odd	6
1152.2.i.j.385.5	yes	12	24.11	even	2
1152.2.i.j.769.5	yes	12	72.59	even	6
1152.2.i.k.385.2	yes	12	12.11	even	2
1152.2.i.k.769.2	yes	12	36.23	even	6
1152.2.i.l.385.2	yes	12	24.5	odd	2
1152.2.i.l.769.2	yes	12	72.5	odd	6
3456.2.i.i.1153.4		12	8.5	even	2
3456.2.i.i.2305.4		12	72.13	even	6
3456.2.i.j.1153.4		12	8.3	odd	2
3456.2.i.j.2305.4		12	72.67	odd	6
3456.2.i.k.1153.3		12	1.1	even	1	trivial
3456.2.i.k.2305.3		12	9.4	even	3	inner
3456.2.i.l.1153.3		12	4.3	odd	2
3456.2.i.l.2305.3		12	36.31	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)

\(f(q)\)	\(=\)	\(q+(-0.268104 - 0.464369i) q^{5} +(-2.35014 + 4.07056i) q^{7} +O(q^{10})\)	\(q+(-0.268104 - 0.464369i) q^{5} +(-2.35014 + 4.07056i) q^{7} +(-2.59922 + 4.50198i) q^{11} +(-0.778295 - 1.34805i) q^{13} -0.695781 q^{17} -5.80593 q^{19} +(4.42809 + 7.66967i) q^{23} +(2.35624 - 4.08113i) q^{25} +(-1.92199 + 3.32898i) q^{29} +(-2.77035 - 4.79840i) q^{31} +2.52033 q^{35} +4.09280 q^{37} +(-1.01019 - 1.74970i) q^{41} +(3.71522 - 6.43494i) q^{43} +(-0.186066 + 0.322275i) q^{47} +(-7.54633 - 13.0706i) q^{49} -5.30777 q^{53} +2.78744 q^{55} +(-2.57152 - 4.45401i) q^{59} +(-0.921988 + 1.59693i) q^{61} +(-0.417328 + 0.722833i) q^{65} +(-5.79316 - 10.0340i) q^{67} +10.6289 q^{71} +4.40840 q^{73} +(-12.2171 - 21.1606i) q^{77} +(-3.32244 + 5.75464i) q^{79} +(5.28055 - 9.14617i) q^{83} +(0.186541 + 0.323099i) q^{85} +7.30777 q^{89} +7.31642 q^{91} +(1.55659 + 2.69609i) q^{95} +(-7.81612 + 13.5379i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 2 q^{5} - 6 q^{7}+O(q^{10}) \) 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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