LMFDB - Embedded newform 1296.2.i.s.865.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1296,2,Mod(433,1296)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1296, base_ring=CyclotomicField(6))

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

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

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

chi := DirichletCharacter("1296.433");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1296 = 2^{4} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1296.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:	$10.3486121020$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{3})$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 3 $
Twist minimal:	no (minimal twist has level 81)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			865.2
Root			$-0.866025 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	1296.865
Dual form			1296.2.i.s.433.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.866025 + 1.50000i) q^{5} +(1.00000 - 1.73205i) q^{7} +O(q^{10})$	$q+(0.866025 + 1.50000i) q^{5} +(1.00000 - 1.73205i) q^{7} +(-1.73205 + 3.00000i) q^{11} +(0.500000 + 0.866025i) q^{13} +5.19615 q^{17} -2.00000 q^{19} +(1.73205 + 3.00000i) q^{23} +(1.00000 - 1.73205i) q^{25} +(-0.866025 + 1.50000i) q^{29} +(4.00000 + 6.92820i) q^{31} +3.46410 q^{35} -7.00000 q^{37} +(3.46410 + 6.00000i) q^{41} +(1.00000 - 1.73205i) q^{43} +(3.46410 - 6.00000i) q^{47} +(1.50000 + 2.59808i) q^{49} -6.00000 q^{55} +(6.92820 + 12.0000i) q^{59} +(3.50000 - 6.06218i) q^{61} +(-0.866025 + 1.50000i) q^{65} +(-5.00000 - 8.66025i) q^{67} +10.3923 q^{71} -7.00000 q^{73} +(3.46410 + 6.00000i) q^{77} +(1.00000 - 1.73205i) q^{79} +(-6.92820 + 12.0000i) q^{83} +(4.50000 + 7.79423i) q^{85} -5.19615 q^{89} +2.00000 q^{91} +(-1.73205 - 3.00000i) q^{95} +(-1.00000 + 1.73205i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{7}+O(q^{10}) $ 4 * q + 4 * q^7	$ 4 q + 4 q^{7} + 2 q^{13} - 8 q^{19} + 4 q^{25} + 16 q^{31} - 28 q^{37} + 4 q^{43} + 6 q^{49} - 24 q^{55} + 14 q^{61} - 20 q^{67} - 28 q^{73} + 4 q^{79} + 18 q^{85} + 8 q^{91} - 4 q^{97}+O(q^{100}) $ 4 * q + 4 * q^7 + 2 * q^13 - 8 * q^19 + 4 * q^25 + 16 * q^31 - 28 * q^37 + 4 * q^43 + 6 * q^49 - 24 * q^55 + 14 * q^61 - 20 * q^67 - 28 * q^73 + 4 * q^79 + 18 * q^85 + 8 * q^91 - 4 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1296\mathbb{Z}\right)^\times$.

$n$	$325$	$1135$	$1217$
$\chi(n)$	$1$	$1$	$e\left(\frac{2}{3}\right)$

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
$5$	0.866025	+	1.50000i	0.387298	+	0.670820i	0.992085	−	0.125567i	$-0.0400750\pi$
$5$	0.866025	+	1.50000i	0.387298	+	0.670820i	−0.604787	+	0.796387i	$0.706742\pi$
$6$		0			0
$7$	1.00000	−	1.73205i	0.377964	−	0.654654i	−0.612801	−	0.790237i	$-0.709957\pi$
$7$	1.00000	−	1.73205i	0.377964	−	0.654654i	0.990766	+	0.135583i	$0.0432908\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−1.73205	+	3.00000i	−0.522233	+	0.904534i	0.477432	+	0.878668i	$0.341568\pi$
$11$	−1.73205	+	3.00000i	−0.522233	+	0.904534i	−0.999665	+	0.0258656i	$0.991766\pi$
$12$		0			0
$13$	0.500000	+	0.866025i	0.138675	+	0.240192i	0.926995	−	0.375073i	$-0.122382\pi$
$13$	0.500000	+	0.866025i	0.138675	+	0.240192i	−0.788320	+	0.615265i	$0.789049\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	5.19615			1.26025			0.630126	−	0.776493i	$-0.283003\pi$
$17$	5.19615			1.26025			0.630126	+	0.776493i	$0.283003\pi$
$18$		0			0
$19$	−2.00000			−0.458831			−0.229416	−	0.973329i	$-0.573682\pi$
$19$	−2.00000			−0.458831			−0.229416	+	0.973329i	$0.573682\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	1.73205	+	3.00000i	0.361158	+	0.625543i	0.988152	−	0.153481i	$-0.0490483\pi$
$23$	1.73205	+	3.00000i	0.361158	+	0.625543i	−0.626994	+	0.779024i	$0.715715\pi$
$24$		0			0
$25$	1.00000	−	1.73205i	0.200000	−	0.346410i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−0.866025	+	1.50000i	−0.160817	+	0.278543i	−0.935162	−	0.354221i	$-0.884746\pi$
$29$	−0.866025	+	1.50000i	−0.160817	+	0.278543i	0.774345	+	0.632764i	$0.218080\pi$
$30$		0			0
$31$	4.00000	+	6.92820i	0.718421	+	1.24434i	0.961625	+	0.274367i	$0.0884683\pi$
$31$	4.00000	+	6.92820i	0.718421	+	1.24434i	−0.243204	+	0.969975i	$0.578198\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	3.46410			0.585540
$36$		0			0
$37$	−7.00000			−1.15079			−0.575396	−	0.817875i	$-0.695152\pi$
$37$	−7.00000			−1.15079			−0.575396	+	0.817875i	$0.695152\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	3.46410	+	6.00000i	0.541002	+	0.937043i	0.998847	+	0.0480106i	$0.0152881\pi$
$41$	3.46410	+	6.00000i	0.541002	+	0.937043i	−0.457845	+	0.889032i	$0.651379\pi$
$42$		0			0
$43$	1.00000	−	1.73205i	0.152499	−	0.264135i	−0.779647	−	0.626219i	$-0.784601\pi$
$43$	1.00000	−	1.73205i	0.152499	−	0.264135i	0.932145	+	0.362084i	$0.117935\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	3.46410	−	6.00000i	0.505291	−	0.875190i	−0.494690	−	0.869069i	$-0.664718\pi$
$47$	3.46410	−	6.00000i	0.505291	−	0.875190i	0.999981	−	0.00612051i	$-0.00194823\pi$
$48$		0			0
$49$	1.50000	+	2.59808i	0.214286	+	0.371154i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		0			0			−	1.00000i	$-0.5\pi$
$53$		0			0				1.00000i	$0.5\pi$
$54$		0			0
$55$	−6.00000			−0.809040
$56$		0			0
$57$		0			0
$58$		0			0
$59$	6.92820	+	12.0000i	0.901975	+	1.56227i	0.824927	+	0.565240i	$0.191216\pi$
$59$	6.92820	+	12.0000i	0.901975	+	1.56227i	0.0770484	+	0.997027i	$0.475450\pi$
$60$		0			0
$61$	3.50000	−	6.06218i	0.448129	−	0.776182i	−0.550135	−	0.835076i	$-0.685424\pi$
$61$	3.50000	−	6.06218i	0.448129	−	0.776182i	0.998264	+	0.0588933i	$0.0187572\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−0.866025	+	1.50000i	−0.107417	+	0.186052i
$66$		0			0
$67$	−5.00000	−	8.66025i	−0.610847	−	1.05802i	−0.991098	−	0.133135i	$-0.957496\pi$
$67$	−5.00000	−	8.66025i	−0.610847	−	1.05802i	0.380251	−	0.924883i	$-0.375838\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	10.3923			1.23334			0.616670	−	0.787222i	$-0.288481\pi$
$71$	10.3923			1.23334			0.616670	+	0.787222i	$0.288481\pi$
$72$		0			0
$73$	−7.00000			−0.819288			−0.409644	−	0.912245i	$-0.634347\pi$
$73$	−7.00000			−0.819288			−0.409644	+	0.912245i	$0.634347\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	3.46410	+	6.00000i	0.394771	+	0.683763i
$78$		0			0
$79$	1.00000	−	1.73205i	0.112509	−	0.194871i	−0.804272	−	0.594261i	$-0.797445\pi$
$79$	1.00000	−	1.73205i	0.112509	−	0.194871i	0.916781	+	0.399390i	$0.130778\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−6.92820	+	12.0000i	−0.760469	+	1.31717i	0.182140	+	0.983273i	$0.441698\pi$
$83$	−6.92820	+	12.0000i	−0.760469	+	1.31717i	−0.942609	+	0.333899i	$0.891636\pi$
$84$		0			0
$85$	4.50000	+	7.79423i	0.488094	+	0.845403i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−5.19615			−0.550791			−0.275396	−	0.961331i	$-0.588809\pi$
$89$	−5.19615			−0.550791			−0.275396	+	0.961331i	$0.588809\pi$
$90$		0			0
$91$	2.00000			0.209657
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−1.73205	−	3.00000i	−0.177705	−	0.307794i
$96$		0			0
$97$	−1.00000	+	1.73205i	−0.101535	+	0.175863i	−0.912317	−	0.409484i	$-0.865709\pi$
$97$	−1.00000	+	1.73205i	−0.101535	+	0.175863i	0.810782	+	0.585348i	$0.199042\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−3.46410	+	6.00000i	−0.344691	+	0.597022i	−0.985298	−	0.170847i	$-0.945350\pi$
$101$	−3.46410	+	6.00000i	−0.344691	+	0.597022i	0.640607	+	0.767869i	$0.278683\pi$
$102$		0			0
$103$	4.00000	+	6.92820i	0.394132	+	0.682656i	0.992990	−	0.118199i	$-0.0377120\pi$
$103$	4.00000	+	6.92820i	0.394132	+	0.682656i	−0.598858	+	0.800855i	$0.704379\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0			−	1.00000i	$-0.5\pi$
$107$		0			0				1.00000i	$0.5\pi$
$108$		0			0
$109$	11.0000			1.05361			0.526804	−	0.849987i	$-0.323390\pi$
$109$	11.0000			1.05361			0.526804	+	0.849987i	$0.323390\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	0.866025	+	1.50000i	0.0814688	+	0.141108i	0.903881	−	0.427784i	$-0.140706\pi$
$113$	0.866025	+	1.50000i	0.0814688	+	0.141108i	−0.822412	+	0.568892i	$0.807372\pi$
$114$		0			0
$115$	−3.00000	+	5.19615i	−0.279751	+	0.484544i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	5.19615	−	9.00000i	0.476331	−	0.825029i
$120$		0			0
$121$	−0.500000	−	0.866025i	−0.0454545	−	0.0787296i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	12.1244			1.08444
$126$		0			0
$127$	−2.00000			−0.177471			−0.0887357	−	0.996055i	$-0.528283\pi$
$127$	−2.00000			−0.177471			−0.0887357	+	0.996055i	$0.528283\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	1.73205	+	3.00000i	0.151330	+	0.262111i	0.931717	−	0.363186i	$-0.118311\pi$
$131$	1.73205	+	3.00000i	0.151330	+	0.262111i	−0.780387	+	0.625297i	$0.784978\pi$
$132$		0			0
$133$	−2.00000	+	3.46410i	−0.173422	+	0.300376i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−0.866025	+	1.50000i	−0.0739895	+	0.128154i	−0.900646	−	0.434553i	$-0.856906\pi$
$137$	−0.866025	+	1.50000i	−0.0739895	+	0.128154i	0.826657	+	0.562706i	$0.190240\pi$
$138$		0			0
$139$	4.00000	+	6.92820i	0.339276	+	0.587643i	0.984297	−	0.176522i	$-0.0564848\pi$
$139$	4.00000	+	6.92820i	0.339276	+	0.587643i	−0.645021	+	0.764165i	$0.723151\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−3.46410			−0.289683
$144$		0			0
$145$	−3.00000			−0.249136
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−4.33013	−	7.50000i	−0.354738	−	0.614424i	0.632335	−	0.774695i	$-0.282097\pi$
$149$	−4.33013	−	7.50000i	−0.354738	−	0.614424i	−0.987073	+	0.160271i	$0.948763\pi$
$150$		0			0
$151$	10.0000	−	17.3205i	0.813788	−	1.40952i	−0.0964061	−	0.995342i	$-0.530735\pi$
$151$	10.0000	−	17.3205i	0.813788	−	1.40952i	0.910195	−	0.414181i	$-0.135932\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−6.92820	+	12.0000i	−0.556487	+	0.963863i
$156$		0			0
$157$	−8.50000	−	14.7224i	−0.678374	−	1.17498i	−0.975470	−	0.220131i	$-0.929352\pi$
$157$	−8.50000	−	14.7224i	−0.678374	−	1.17498i	0.297097	−	0.954847i	$-0.403982\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	6.92820			0.546019
$162$		0			0
$163$	16.0000			1.25322			0.626608	−	0.779334i	$-0.284443\pi$
$163$	16.0000			1.25322			0.626608	+	0.779334i	$0.284443\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−8.66025	−	15.0000i	−0.670151	−	1.16073i	−0.977861	−	0.209255i	$-0.932896\pi$
$167$	−8.66025	−	15.0000i	−0.670151	−	1.16073i	0.307711	−	0.951480i	$-0.400437\pi$
$168$		0			0
$169$	6.00000	−	10.3923i	0.461538	−	0.799408i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	9.52628	−	16.5000i	0.724270	−	1.25447i	−0.235004	−	0.971994i	$-0.575510\pi$
$173$	9.52628	−	16.5000i	0.724270	−	1.25447i	0.959274	−	0.282477i	$-0.0911562\pi$
$174$		0			0
$175$	−2.00000	−	3.46410i	−0.151186	−	0.261861i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−20.7846			−1.55351			−0.776757	−	0.629800i	$-0.783137\pi$
$179$	−20.7846			−1.55351			−0.776757	+	0.629800i	$0.783137\pi$
$180$		0			0
$181$	2.00000			0.148659			0.0743294	−	0.997234i	$-0.476318\pi$
$181$	2.00000			0.148659			0.0743294	+	0.997234i	$0.476318\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−6.06218	−	10.5000i	−0.445700	−	0.771975i
$186$		0			0
$187$	−9.00000	+	15.5885i	−0.658145	+	1.13994i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	8.66025	−	15.0000i	0.626634	−	1.08536i	−0.361588	−	0.932338i	$-0.617765\pi$
$191$	8.66025	−	15.0000i	0.626634	−	1.08536i	0.988222	−	0.153024i	$-0.0489012\pi$
$192$		0			0
$193$	0.500000	+	0.866025i	0.0359908	+	0.0623379i	0.883460	−	0.468507i	$-0.155208\pi$
$193$	0.500000	+	0.866025i	0.0359908	+	0.0623379i	−0.847469	+	0.530845i	$0.821875\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−5.19615			−0.370211			−0.185105	−	0.982719i	$-0.559263\pi$
$197$	−5.19615			−0.370211			−0.185105	+	0.982719i	$0.559263\pi$
$198$		0			0
$199$	−20.0000			−1.41776			−0.708881	−	0.705328i	$-0.750800\pi$
$199$	−20.0000			−1.41776			−0.708881	+	0.705328i	$0.750800\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	1.73205	+	3.00000i	0.121566	+	0.210559i
$204$		0			0
$205$	−6.00000	+	10.3923i	−0.419058	+	0.725830i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	3.46410	−	6.00000i	0.239617	−	0.415029i
$210$		0			0
$211$	−5.00000	−	8.66025i	−0.344214	−	0.596196i	0.640996	−	0.767544i	$-0.278521\pi$
$211$	−5.00000	−	8.66025i	−0.344214	−	0.596196i	−0.985211	+	0.171347i	$0.945188\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	3.46410			0.236250
$216$		0			0
$217$	16.0000			1.08615
$218$		0			0
$219$		0			0
$220$		0			0
$221$	2.59808	+	4.50000i	0.174766	+	0.302703i
$222$		0			0
$223$	1.00000	−	1.73205i	0.0669650	−	0.115987i	−0.830599	−	0.556871i	$-0.812002\pi$
$223$	1.00000	−	1.73205i	0.0669650	−	0.115987i	0.897564	+	0.440884i	$0.145335\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−1.73205	+	3.00000i	−0.114960	+	0.199117i	−0.917764	−	0.397127i	$-0.870007\pi$
$227$	−1.73205	+	3.00000i	−0.114960	+	0.199117i	0.802804	+	0.596244i	$0.203341\pi$
$228$		0			0
$229$	0.500000	+	0.866025i	0.0330409	+	0.0572286i	0.882073	−	0.471113i	$-0.156147\pi$
$229$	0.500000	+	0.866025i	0.0330409	+	0.0572286i	−0.849032	+	0.528341i	$0.822814\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−25.9808			−1.70206			−0.851028	−	0.525120i	$-0.824020\pi$
$233$	−25.9808			−1.70206			−0.851028	+	0.525120i	$0.824020\pi$
$234$		0			0
$235$	12.0000			0.782794
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−13.8564	−	24.0000i	−0.896296	−	1.55243i	−0.832192	−	0.554488i	$-0.812914\pi$
$239$	−13.8564	−	24.0000i	−0.896296	−	1.55243i	−0.0641045	−	0.997943i	$-0.520419\pi$
$240$		0			0
$241$	−14.5000	+	25.1147i	−0.934027	+	1.61778i	−0.157667	+	0.987492i	$0.550397\pi$
$241$	−14.5000	+	25.1147i	−0.934027	+	1.61778i	−0.776360	+	0.630290i	$0.782936\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−2.59808	+	4.50000i	−0.165985	+	0.287494i
$246$		0			0
$247$	−1.00000	−	1.73205i	−0.0636285	−	0.110208i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−10.3923			−0.655956			−0.327978	−	0.944685i	$-0.606367\pi$
$251$	−10.3923			−0.655956			−0.327978	+	0.944685i	$0.606367\pi$
$252$		0			0
$253$	−12.0000			−0.754434
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−4.33013	−	7.50000i	−0.270106	−	0.467837i	0.698783	−	0.715334i	$-0.253725\pi$
$257$	−4.33013	−	7.50000i	−0.270106	−	0.467837i	−0.968889	+	0.247497i	$0.920392\pi$
$258$		0			0
$259$	−7.00000	+	12.1244i	−0.434959	+	0.753371i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	3.46410	−	6.00000i	0.213606	−	0.369976i	−0.739235	−	0.673448i	$-0.764813\pi$
$263$	3.46410	−	6.00000i	0.213606	−	0.369976i	0.952840	+	0.303472i	$0.0981459\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−15.5885			−0.950445			−0.475223	−	0.879866i	$-0.657632\pi$
$269$	−15.5885			−0.950445			−0.475223	+	0.879866i	$0.657632\pi$
$270$		0			0
$271$	−2.00000			−0.121491			−0.0607457	−	0.998153i	$-0.519348\pi$
$271$	−2.00000			−0.121491			−0.0607457	+	0.998153i	$0.519348\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	3.46410	+	6.00000i	0.208893	+	0.361814i
$276$		0			0
$277$	−1.00000	+	1.73205i	−0.0600842	+	0.104069i	−0.894503	−	0.447062i	$-0.852470\pi$
$277$	−1.00000	+	1.73205i	−0.0600842	+	0.104069i	0.834419	+	0.551131i	$0.185804\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−6.06218	+	10.5000i	−0.361639	+	0.626377i	−0.988231	−	0.152970i	$-0.951116\pi$
$281$	−6.06218	+	10.5000i	−0.361639	+	0.626377i	0.626592	+	0.779348i	$0.284449\pi$
$282$		0			0
$283$	−14.0000	−	24.2487i	−0.832214	−	1.44144i	−0.896279	−	0.443491i	$-0.853740\pi$
$283$	−14.0000	−	24.2487i	−0.832214	−	1.44144i	0.0640654	−	0.997946i	$-0.479593\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	13.8564			0.817918
$288$		0			0
$289$	10.0000			0.588235
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−9.52628	−	16.5000i	−0.556531	−	0.963940i	−0.997783	−	0.0665568i	$-0.978799\pi$
$293$	−9.52628	−	16.5000i	−0.556531	−	0.963940i	0.441251	−	0.897384i	$-0.354535\pi$
$294$		0			0
$295$	−12.0000	+	20.7846i	−0.698667	+	1.21013i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−1.73205	+	3.00000i	−0.100167	+	0.173494i
$300$		0			0
$301$	−2.00000	−	3.46410i	−0.115278	−	0.199667i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	12.1244			0.694239
$306$		0			0
$307$	16.0000			0.913168			0.456584	−	0.889680i	$-0.349073\pi$
$307$	16.0000			0.913168			0.456584	+	0.889680i	$0.349073\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−3.46410	−	6.00000i	−0.196431	−	0.340229i	0.750938	−	0.660373i	$-0.229602\pi$
$311$	−3.46410	−	6.00000i	−0.196431	−	0.340229i	−0.947369	+	0.320144i	$0.896269\pi$
$312$		0			0
$313$	12.5000	−	21.6506i	0.706542	−	1.22377i	−0.259590	−	0.965719i	$-0.583588\pi$
$313$	12.5000	−	21.6506i	0.706542	−	1.22377i	0.966132	−	0.258047i	$-0.0830791\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	4.33013	−	7.50000i	0.243204	−	0.421242i	−0.718421	−	0.695609i	$-0.755135\pi$
$317$	4.33013	−	7.50000i	0.243204	−	0.421242i	0.961625	+	0.274367i	$0.0884682\pi$
$318$		0			0
$319$	−3.00000	−	5.19615i	−0.167968	−	0.290929i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−10.3923			−0.578243
$324$		0			0
$325$	2.00000			0.110940
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−6.92820	−	12.0000i	−0.381964	−	0.661581i
$330$		0			0
$331$	1.00000	−	1.73205i	0.0549650	−	0.0952021i	−0.837234	−	0.546845i	$-0.815829\pi$
$331$	1.00000	−	1.73205i	0.0549650	−	0.0952021i	0.892199	+	0.451643i	$0.149162\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	8.66025	−	15.0000i	0.473160	−	0.819538i
$336$		0			0
$337$	−13.0000	−	22.5167i	−0.708155	−	1.22656i	−0.965541	−	0.260252i	$-0.916194\pi$
$337$	−13.0000	−	22.5167i	−0.708155	−	1.22656i	0.257386	−	0.966309i	$-0.417139\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−27.7128			−1.50073
$342$		0			0
$343$	20.0000			1.07990
$344$		0			0
$345$		0			0
$346$		0			0
$347$	1.73205	+	3.00000i	0.0929814	+	0.161048i	0.908764	−	0.417310i	$-0.137027\pi$
$347$	1.73205	+	3.00000i	0.0929814	+	0.161048i	−0.815783	+	0.578358i	$0.803694\pi$
$348$		0			0
$349$	−1.00000	+	1.73205i	−0.0535288	+	0.0927146i	−0.891548	−	0.452926i	$-0.850380\pi$
$349$	−1.00000	+	1.73205i	−0.0535288	+	0.0927146i	0.838019	+	0.545640i	$0.183714\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	6.92820	−	12.0000i	0.368751	−	0.638696i	−0.620620	−	0.784112i	$-0.713119\pi$
$353$	6.92820	−	12.0000i	0.368751	−	0.638696i	0.989371	+	0.145416i	$0.0464522\pi$
$354$		0			0
$355$	9.00000	+	15.5885i	0.477670	+	0.827349i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−10.3923			−0.548485			−0.274242	−	0.961661i	$-0.588427\pi$
$359$	−10.3923			−0.548485			−0.274242	+	0.961661i	$0.588427\pi$
$360$		0			0
$361$	−15.0000			−0.789474
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−6.06218	−	10.5000i	−0.317309	−	0.549595i
$366$		0			0
$367$	10.0000	−	17.3205i	0.521996	−	0.904123i	−0.477677	−	0.878536i	$-0.658521\pi$
$367$	10.0000	−	17.3205i	0.521996	−	0.904123i	0.999673	−	0.0255875i	$-0.00814566\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	5.00000	+	8.66025i	0.258890	+	0.448411i	0.965945	−	0.258748i	$-0.0833099\pi$
$373$	5.00000	+	8.66025i	0.258890	+	0.448411i	−0.707055	+	0.707159i	$0.749977\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−1.73205			−0.0892052
$378$		0			0
$379$	16.0000			0.821865			0.410932	−	0.911666i	$-0.365203\pi$
$379$	16.0000			0.821865			0.410932	+	0.911666i	$0.365203\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−8.66025	−	15.0000i	−0.442518	−	0.766464i	0.555357	−	0.831612i	$-0.312581\pi$
$383$	−8.66025	−	15.0000i	−0.442518	−	0.766464i	−0.997876	+	0.0651476i	$0.979248\pi$
$384$		0			0
$385$	−6.00000	+	10.3923i	−0.305788	+	0.529641i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−13.8564	+	24.0000i	−0.702548	+	1.21685i	0.265022	+	0.964242i	$0.414621\pi$
$389$	−13.8564	+	24.0000i	−0.702548	+	1.21685i	−0.967569	+	0.252606i	$0.918712\pi$
$390$		0			0
$391$	9.00000	+	15.5885i	0.455150	+	0.788342i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	3.46410			0.174298
$396$		0			0
$397$	29.0000			1.45547			0.727734	−	0.685859i	$-0.240573\pi$
$397$	29.0000			1.45547			0.727734	+	0.685859i	$0.240573\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	6.06218	+	10.5000i	0.302731	+	0.524345i	0.976753	−	0.214366i	$-0.0687684\pi$
$401$	6.06218	+	10.5000i	0.302731	+	0.524345i	−0.674023	+	0.738711i	$0.735435\pi$
$402$		0			0
$403$	−4.00000	+	6.92820i	−0.199254	+	0.345118i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	12.1244	−	21.0000i	0.600982	−	1.04093i
$408$		0			0
$409$	9.50000	+	16.4545i	0.469745	+	0.813622i	0.999402	−	0.0345902i	$-0.0110126\pi$
$409$	9.50000	+	16.4545i	0.469745	+	0.813622i	−0.529657	+	0.848212i	$0.677679\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	27.7128			1.36366
$414$		0			0
$415$	−24.0000			−1.17811
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−3.46410	−	6.00000i	−0.169232	−	0.293119i	0.768918	−	0.639348i	$-0.220796\pi$
$419$	−3.46410	−	6.00000i	−0.169232	−	0.293119i	−0.938150	+	0.346228i	$0.887462\pi$
$420$		0			0
$421$	12.5000	−	21.6506i	0.609213	−	1.05519i	−0.382158	−	0.924097i	$-0.624819\pi$
$421$	12.5000	−	21.6506i	0.609213	−	1.05519i	0.991370	−	0.131090i	$-0.0418478\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	5.19615	−	9.00000i	0.252050	−	0.436564i
$426$		0			0
$427$	−7.00000	−	12.1244i	−0.338754	−	0.586739i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0			−	1.00000i	$-0.5\pi$
$431$		0			0				1.00000i	$0.5\pi$
$432$		0			0
$433$	11.0000			0.528626			0.264313	−	0.964437i	$-0.414855\pi$
$433$	11.0000			0.528626			0.264313	+	0.964437i	$0.414855\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−3.46410	−	6.00000i	−0.165710	−	0.287019i
$438$		0			0
$439$	10.0000	−	17.3205i	0.477274	−	0.826663i	−0.522387	−	0.852709i	$-0.674958\pi$
$439$	10.0000	−	17.3205i	0.477274	−	0.826663i	0.999661	+	0.0260459i	$0.00829161\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−17.3205	+	30.0000i	−0.822922	+	1.42534i	0.0805748	+	0.996749i	$0.474324\pi$
$443$	−17.3205	+	30.0000i	−0.822922	+	1.42534i	−0.903497	+	0.428594i	$0.859009\pi$
$444$		0			0
$445$	−4.50000	−	7.79423i	−0.213320	−	0.369482i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	20.7846			0.980886			0.490443	−	0.871473i	$-0.336835\pi$
$449$	20.7846			0.980886			0.490443	+	0.871473i	$0.336835\pi$
$450$		0			0
$451$	−24.0000			−1.13012
$452$		0			0
$453$		0			0
$454$		0			0
$455$	1.73205	+	3.00000i	0.0811998	+	0.140642i
$456$		0			0
$457$	−14.5000	+	25.1147i	−0.678281	+	1.17482i	0.297217	+	0.954810i	$0.403942\pi$
$457$	−14.5000	+	25.1147i	−0.678281	+	1.17482i	−0.975498	+	0.220008i	$0.929392\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	6.92820	−	12.0000i	0.322679	−	0.558896i	−0.658361	−	0.752702i	$-0.728750\pi$
$461$	6.92820	−	12.0000i	0.322679	−	0.558896i	0.981040	+	0.193806i	$0.0620834\pi$
$462$		0			0
$463$	4.00000	+	6.92820i	0.185896	+	0.321981i	0.943878	−	0.330294i	$-0.107148\pi$
$463$	4.00000	+	6.92820i	0.185896	+	0.321981i	−0.757982	+	0.652275i	$0.773815\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	20.7846			0.961797			0.480899	−	0.876776i	$-0.340311\pi$
$467$	20.7846			0.961797			0.480899	+	0.876776i	$0.340311\pi$
$468$		0			0
$469$	−20.0000			−0.923514
$470$		0			0
$471$		0			0
$472$		0			0
$473$	3.46410	+	6.00000i	0.159280	+	0.275880i
$474$		0			0
$475$	−2.00000	+	3.46410i	−0.0917663	+	0.158944i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−12.1244	+	21.0000i	−0.553976	+	0.959514i	0.444006	+	0.896024i	$0.353557\pi$
$479$	−12.1244	+	21.0000i	−0.553976	+	0.959514i	−0.997982	+	0.0634909i	$0.979777\pi$
$480$		0			0
$481$	−3.50000	−	6.06218i	−0.159586	−	0.276412i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−3.46410			−0.157297
$486$		0			0
$487$	16.0000			0.725029			0.362515	−	0.931978i	$-0.381918\pi$
$487$	16.0000			0.725029			0.362515	+	0.931978i	$0.381918\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−8.66025	−	15.0000i	−0.390832	−	0.676941i	0.601728	−	0.798701i	$-0.294479\pi$
$491$	−8.66025	−	15.0000i	−0.390832	−	0.676941i	−0.992559	+	0.121761i	$0.961146\pi$
$492$		0			0
$493$	−4.50000	+	7.79423i	−0.202670	+	0.351034i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	10.3923	−	18.0000i	0.466159	−	0.807410i
$498$		0			0
$499$	−5.00000	−	8.66025i	−0.223831	−	0.387686i	0.732137	−	0.681157i	$-0.238523\pi$
$499$	−5.00000	−	8.66025i	−0.223831	−	0.387686i	−0.955968	+	0.293471i	$0.905190\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−20.7846			−0.926740			−0.463370	−	0.886165i	$-0.653360\pi$
$503$	−20.7846			−0.926740			−0.463370	+	0.886165i	$0.653360\pi$
$504$		0			0
$505$	−12.0000			−0.533993
$506$		0			0
$507$		0			0
$508$		0			0
$509$	13.8564	+	24.0000i	0.614174	+	1.06378i	0.990529	+	0.137305i	$0.0438442\pi$
$509$	13.8564	+	24.0000i	0.614174	+	1.06378i	−0.376354	+	0.926476i	$0.622822\pi$
$510$		0			0
$511$	−7.00000	+	12.1244i	−0.309662	+	0.536350i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−6.92820	+	12.0000i	−0.305293	+	0.528783i
$516$		0			0
$517$	12.0000	+	20.7846i	0.527759	+	0.914106i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−20.7846			−0.910590			−0.455295	−	0.890341i	$-0.650466\pi$
$521$	−20.7846			−0.910590			−0.455295	+	0.890341i	$0.650466\pi$
$522$		0			0
$523$	−38.0000			−1.66162			−0.830812	−	0.556553i	$-0.812124\pi$
$523$	−38.0000			−1.66162			−0.830812	+	0.556553i	$0.812124\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	20.7846	+	36.0000i	0.905392	+	1.56818i
$528$		0			0
$529$	5.50000	−	9.52628i	0.239130	−	0.414186i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−3.46410	+	6.00000i	−0.150047	+	0.259889i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−10.3923			−0.447628
$540$		0			0
$541$	11.0000			0.472927			0.236463	−	0.971640i	$-0.424012\pi$
$541$	11.0000			0.472927			0.236463	+	0.971640i	$0.424012\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	9.52628	+	16.5000i	0.408061	+	0.706782i
$546$		0			0
$547$	10.0000	−	17.3205i	0.427569	−	0.740571i	−0.569087	−	0.822277i	$-0.692703\pi$
$547$	10.0000	−	17.3205i	0.427569	−	0.740571i	0.996657	+	0.0817056i	$0.0260367\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	1.73205	−	3.00000i	0.0737878	−	0.127804i
$552$		0			0
$553$	−2.00000	−	3.46410i	−0.0850487	−	0.147309i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	36.3731			1.54118			0.770588	−	0.637333i	$-0.219963\pi$
$557$	36.3731			1.54118			0.770588	+	0.637333i	$0.219963\pi$
$558$		0			0
$559$	2.00000			0.0845910
$560$		0			0
$561$		0			0
$562$		0			0
$563$	17.3205	+	30.0000i	0.729972	+	1.26435i	0.956894	+	0.290436i	$0.0938004\pi$
$563$	17.3205	+	30.0000i	0.729972	+	1.26435i	−0.226922	+	0.973913i	$0.572866\pi$
$564$		0			0
$565$	−1.50000	+	2.59808i	−0.0631055	+	0.109302i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−16.4545	+	28.5000i	−0.689808	+	1.19478i	0.282092	+	0.959387i	$0.408972\pi$
$569$	−16.4545	+	28.5000i	−0.689808	+	1.19478i	−0.971900	+	0.235395i	$0.924362\pi$
$570$		0			0
$571$	4.00000	+	6.92820i	0.167395	+	0.289936i	0.937503	−	0.347977i	$-0.113131\pi$
$571$	4.00000	+	6.92820i	0.167395	+	0.289936i	−0.770108	+	0.637913i	$0.779798\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	6.92820			0.288926
$576$		0			0
$577$	11.0000			0.457936			0.228968	−	0.973434i	$-0.426465\pi$
$577$	11.0000			0.457936			0.228968	+	0.973434i	$0.426465\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	13.8564	+	24.0000i	0.574861	+	0.995688i
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$	19.0526	−	33.0000i	0.786383	−	1.36206i	−0.141786	−	0.989897i	$-0.545284\pi$
$587$	19.0526	−	33.0000i	0.786383	−	1.36206i	0.928169	−	0.372158i	$-0.121382\pi$
$588$		0			0
$589$	−8.00000	−	13.8564i	−0.329634	−	0.570943i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	15.5885			0.640141			0.320071	−	0.947394i	$-0.396293\pi$
$593$	15.5885			0.640141			0.320071	+	0.947394i	$0.396293\pi$
$594$		0			0
$595$	18.0000			0.737928
$596$		0			0
$597$		0			0
$598$		0			0
$599$	6.92820	+	12.0000i	0.283079	+	0.490307i	0.972141	−	0.234395i	$-0.0753109\pi$
$599$	6.92820	+	12.0000i	0.283079	+	0.490307i	−0.689063	+	0.724702i	$0.741978\pi$
$600$		0			0
$601$	12.5000	−	21.6506i	0.509886	−	0.883148i	−0.490049	−	0.871695i	$-0.663021\pi$
$601$	12.5000	−	21.6506i	0.509886	−	0.883148i	0.999934	−	0.0114528i	$-0.00364562\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	0.866025	−	1.50000i	0.0352089	−	0.0609837i
$606$		0			0
$607$	13.0000	+	22.5167i	0.527654	+	0.913923i	0.999480	+	0.0322317i	$0.0102615\pi$
$607$	13.0000	+	22.5167i	0.527654	+	0.913923i	−0.471827	+	0.881691i	$0.656405\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	6.92820			0.280285
$612$		0			0
$613$	−34.0000			−1.37325			−0.686624	−	0.727013i	$-0.740908\pi$
$613$	−34.0000			−1.37325			−0.686624	+	0.727013i	$0.740908\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	6.06218	+	10.5000i	0.244054	+	0.422714i	0.961865	−	0.273524i	$-0.0881892\pi$
$617$	6.06218	+	10.5000i	0.244054	+	0.422714i	−0.717811	+	0.696238i	$0.754856\pi$
$618$		0			0
$619$	10.0000	−	17.3205i	0.401934	−	0.696170i	−0.592025	−	0.805919i	$-0.701671\pi$
$619$	10.0000	−	17.3205i	0.401934	−	0.696170i	0.993959	+	0.109749i	$0.0350048\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−5.19615	+	9.00000i	−0.208179	+	0.360577i
$624$		0			0
$625$	5.50000	+	9.52628i	0.220000	+	0.381051i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−36.3731			−1.45029
$630$		0			0
$631$	−20.0000			−0.796187			−0.398094	−	0.917345i	$-0.630328\pi$
$631$	−20.0000			−0.796187			−0.398094	+	0.917345i	$0.630328\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−1.73205	−	3.00000i	−0.0687343	−	0.119051i
$636$		0			0
$637$	−1.50000	+	2.59808i	−0.0594322	+	0.102940i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−11.2583	+	19.5000i	−0.444677	+	0.770204i	−0.998030	−	0.0627436i	$-0.980015\pi$
$641$	−11.2583	+	19.5000i	−0.444677	+	0.770204i	0.553352	+	0.832947i	$0.313348\pi$
$642$		0			0
$643$	4.00000	+	6.92820i	0.157745	+	0.273222i	0.934055	−	0.357129i	$-0.116244\pi$
$643$	4.00000	+	6.92820i	0.157745	+	0.273222i	−0.776310	+	0.630351i	$0.782911\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	31.1769			1.22569			0.612845	−	0.790203i	$-0.290025\pi$
$647$	31.1769			1.22569			0.612845	+	0.790203i	$0.290025\pi$
$648$		0			0
$649$	−48.0000			−1.88416
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−6.92820	−	12.0000i	−0.271122	−	0.469596i	0.698028	−	0.716071i	$-0.254061\pi$
$653$	−6.92820	−	12.0000i	−0.271122	−	0.469596i	−0.969149	+	0.246474i	$0.920728\pi$
$654$		0			0
$655$	−3.00000	+	5.19615i	−0.117220	+	0.203030i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−1.73205	+	3.00000i	−0.0674711	+	0.116863i	−0.897787	−	0.440429i	$-0.854826\pi$
$659$	−1.73205	+	3.00000i	−0.0674711	+	0.116863i	0.830316	+	0.557292i	$0.188160\pi$
$660$		0			0
$661$	−8.50000	−	14.7224i	−0.330612	−	0.572636i	0.652020	−	0.758202i	$-0.273922\pi$
$661$	−8.50000	−	14.7224i	−0.330612	−	0.572636i	−0.982632	+	0.185565i	$0.940588\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−6.92820			−0.268664
$666$		0			0
$667$	−6.00000			−0.232321
$668$		0			0
$669$		0			0
$670$		0			0
$671$	12.1244	+	21.0000i	0.468056	+	0.810696i
$672$		0			0
$673$	12.5000	−	21.6506i	0.481840	−	0.834571i	−0.517943	−	0.855415i	$-0.673302\pi$
$673$	12.5000	−	21.6506i	0.481840	−	0.834571i	0.999783	+	0.0208444i	$0.00663546\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	6.92820	−	12.0000i	0.266272	−	0.461197i	−0.701624	−	0.712548i	$-0.747541\pi$
$677$	6.92820	−	12.0000i	0.266272	−	0.461197i	0.967896	+	0.251350i	$0.0808746\pi$
$678$		0			0
$679$	2.00000	+	3.46410i	0.0767530	+	0.132940i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	20.7846			0.795301			0.397650	−	0.917537i	$-0.369826\pi$
$683$	20.7846			0.795301			0.397650	+	0.917537i	$0.369826\pi$
$684$		0			0
$685$	−3.00000			−0.114624
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$	1.00000	−	1.73205i	0.0380418	−	0.0658903i	−0.846378	−	0.532583i	$-0.821221\pi$
$691$	1.00000	−	1.73205i	0.0380418	−	0.0658903i	0.884419	+	0.466693i	$0.154555\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−6.92820	+	12.0000i	−0.262802	+	0.455186i
$696$		0			0
$697$	18.0000	+	31.1769i	0.681799	+	1.18091i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	46.7654			1.76630			0.883152	−	0.469087i	$-0.155417\pi$
$701$	46.7654			1.76630			0.883152	+	0.469087i	$0.155417\pi$
$702$		0			0
$703$	14.0000			0.528020
$704$		0			0
$705$		0			0
$706$		0			0
$707$	6.92820	+	12.0000i	0.260562	+	0.451306i
$708$		0			0
$709$	12.5000	−	21.6506i	0.469447	−	0.813107i	−0.529943	−	0.848034i	$-0.677787\pi$
$709$	12.5000	−	21.6506i	0.469447	−	0.813107i	0.999390	+	0.0349269i	$0.0111198\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−13.8564	+	24.0000i	−0.518927	+	0.898807i
$714$		0			0
$715$	−3.00000	−	5.19615i	−0.112194	−	0.194325i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	10.3923			0.387568			0.193784	−	0.981044i	$-0.437924\pi$
$719$	10.3923			0.387568			0.193784	+	0.981044i	$0.437924\pi$
$720$		0			0
$721$	16.0000			0.595871
$722$		0			0
$723$		0			0
$724$		0			0
$725$	1.73205	+	3.00000i	0.0643268	+	0.111417i
$726$		0			0
$727$	−17.0000	+	29.4449i	−0.630495	+	1.09205i	0.356956	+	0.934121i	$0.383815\pi$
$727$	−17.0000	+	29.4449i	−0.630495	+	1.09205i	−0.987451	+	0.157928i	$0.949519\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	5.19615	−	9.00000i	0.192187	−	0.332877i
$732$		0			0
$733$	23.0000	+	39.8372i	0.849524	+	1.47142i	0.881633	+	0.471935i	$0.156444\pi$
$733$	23.0000	+	39.8372i	0.849524	+	1.47142i	−0.0321090	+	0.999484i	$0.510222\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	34.6410			1.27602
$738$		0			0
$739$	−20.0000			−0.735712			−0.367856	−	0.929883i	$-0.619908\pi$
$739$	−20.0000			−0.735712			−0.367856	+	0.929883i	$0.619908\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−3.46410	−	6.00000i	−0.127086	−	0.220119i	0.795461	−	0.606005i	$-0.207229\pi$
$743$	−3.46410	−	6.00000i	−0.127086	−	0.220119i	−0.922546	+	0.385887i	$0.873896\pi$
$744$		0			0
$745$	7.50000	−	12.9904i	0.274779	−	0.475931i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−5.00000	−	8.66025i	−0.182453	−	0.316017i	0.760263	−	0.649616i	$-0.225070\pi$
$751$	−5.00000	−	8.66025i	−0.182453	−	0.316017i	−0.942715	+	0.333599i	$0.891737\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	34.6410			1.26072
$756$		0			0
$757$	2.00000			0.0726912			0.0363456	−	0.999339i	$-0.488428\pi$
$757$	2.00000			0.0726912			0.0363456	+	0.999339i	$0.488428\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−14.7224	−	25.5000i	−0.533688	−	0.924374i	−0.999226	−	0.0393463i	$-0.987472\pi$
$761$	−14.7224	−	25.5000i	−0.533688	−	0.924374i	0.465538	−	0.885028i	$-0.345861\pi$
$762$		0			0
$763$	11.0000	−	19.0526i	0.398227	−	0.689749i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−6.92820	+	12.0000i	−0.250163	+	0.433295i
$768$		0			0
$769$	0.500000	+	0.866025i	0.0180305	+	0.0312297i	0.874900	−	0.484304i	$-0.160927\pi$
$769$	0.500000	+	0.866025i	0.0180305	+	0.0312297i	−0.856869	+	0.515534i	$0.827594\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−25.9808			−0.934463			−0.467232	−	0.884135i	$-0.654749\pi$
$773$	−25.9808			−0.934463			−0.467232	+	0.884135i	$0.654749\pi$
$774$		0			0
$775$	16.0000			0.574737
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−6.92820	−	12.0000i	−0.248229	−	0.429945i
$780$		0			0
$781$	−18.0000	+	31.1769i	−0.644091	+	1.11560i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	14.7224	−	25.5000i	0.525466	−	0.910134i
$786$		0			0
$787$	13.0000	+	22.5167i	0.463400	+	0.802632i	0.999128	−	0.0417585i	$-0.0132960\pi$
$787$	13.0000	+	22.5167i	0.463400	+	0.802632i	−0.535728	+	0.844391i	$0.679963\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	3.46410			0.123169
$792$		0			0
$793$	7.00000			0.248577
$794$		0			0
$795$		0			0
$796$		0			0
$797$	26.8468	+	46.5000i	0.950962	+	1.64711i	0.743349	+	0.668903i	$0.233236\pi$
$797$	26.8468	+	46.5000i	0.950962	+	1.64711i	0.207613	+	0.978211i	$0.433431\pi$
$798$		0			0
$799$	18.0000	−	31.1769i	0.636794	−	1.10296i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	12.1244	−	21.0000i	0.427859	−	0.741074i
$804$		0			0
$805$	6.00000	+	10.3923i	0.211472	+	0.366281i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−46.7654			−1.64418			−0.822091	−	0.569355i	$-0.807193\pi$
$809$	−46.7654			−1.64418			−0.822091	+	0.569355i	$0.807193\pi$
$810$		0			0
$811$	16.0000			0.561836			0.280918	−	0.959732i	$-0.409361\pi$
$811$	16.0000			0.561836			0.280918	+	0.959732i	$0.409361\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	13.8564	+	24.0000i	0.485369	+	0.840683i
$816$		0			0
$817$	−2.00000	+	3.46410i	−0.0699711	+	0.121194i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−6.06218	+	10.5000i	−0.211571	+	0.366453i	−0.952207	−	0.305455i	$-0.901191\pi$
$821$	−6.06218	+	10.5000i	−0.211571	+	0.366453i	0.740635	+	0.671908i	$0.234525\pi$
$822$		0			0
$823$	−14.0000	−	24.2487i	−0.488009	−	0.845257i	0.511896	−	0.859048i	$-0.328943\pi$
$823$	−14.0000	−	24.2487i	−0.488009	−	0.845257i	−0.999905	+	0.0137907i	$0.995610\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	10.3923			0.361376			0.180688	−	0.983540i	$-0.442168\pi$
$827$	10.3923			0.361376			0.180688	+	0.983540i	$0.442168\pi$
$828$		0			0
$829$	2.00000			0.0694629			0.0347314	−	0.999397i	$-0.488942\pi$
$829$	2.00000			0.0694629			0.0347314	+	0.999397i	$0.488942\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	7.79423	+	13.5000i	0.270054	+	0.467747i
$834$		0			0
$835$	15.0000	−	25.9808i	0.519096	−	0.899101i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−22.5167	+	39.0000i	−0.777361	+	1.34643i	0.156096	+	0.987742i	$0.450109\pi$
$839$	−22.5167	+	39.0000i	−0.777361	+	1.34643i	−0.933458	+	0.358688i	$0.883224\pi$
$840$		0			0
$841$	13.0000	+	22.5167i	0.448276	+	0.776437i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	20.7846			0.715012
$846$		0			0
$847$	−2.00000			−0.0687208
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−12.1244	−	21.0000i	−0.415618	−	0.719871i
$852$		0			0
$853$	17.0000	−	29.4449i	0.582069	−	1.00817i	−0.413165	−	0.910656i	$-0.635577\pi$
$853$	17.0000	−	29.4449i	0.582069	−	1.00817i	0.995234	−	0.0975167i	$-0.0310899\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−11.2583	+	19.5000i	−0.384577	+	0.666107i	−0.991710	−	0.128493i	$-0.958986\pi$
$857$	−11.2583	+	19.5000i	−0.384577	+	0.666107i	0.607133	+	0.794600i	$0.292319\pi$
$858$		0			0
$859$	22.0000	+	38.1051i	0.750630	+	1.30013i	0.947518	+	0.319704i	$0.103583\pi$
$859$	22.0000	+	38.1051i	0.750630	+	1.30013i	−0.196887	+	0.980426i	$0.563083\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−31.1769			−1.06127			−0.530637	−	0.847599i	$-0.678047\pi$
$863$	−31.1769			−1.06127			−0.530637	+	0.847599i	$0.678047\pi$
$864$		0			0
$865$	33.0000			1.12203
$866$		0			0
$867$		0			0
$868$		0			0
$869$	3.46410	+	6.00000i	0.117512	+	0.203536i
$870$		0			0
$871$	5.00000	−	8.66025i	0.169419	−	0.293442i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	12.1244	−	21.0000i	0.409878	−	0.709930i
$876$		0			0
$877$	−26.5000	−	45.8993i	−0.894841	−	1.54991i	−0.834001	−	0.551763i	$-0.813955\pi$
$877$	−26.5000	−	45.8993i	−0.894841	−	1.54991i	−0.0608407	−	0.998147i	$-0.519378\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	20.7846			0.700251			0.350126	−	0.936703i	$-0.386139\pi$
$881$	20.7846			0.700251			0.350126	+	0.936703i	$0.386139\pi$
$882$		0			0
$883$	−56.0000			−1.88455			−0.942275	−	0.334840i	$-0.891318\pi$
$883$	−56.0000			−1.88455			−0.942275	+	0.334840i	$0.891318\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	1.73205	+	3.00000i	0.0581566	+	0.100730i	0.893638	−	0.448789i	$-0.148144\pi$
$887$	1.73205	+	3.00000i	0.0581566	+	0.100730i	−0.835481	+	0.549519i	$0.814811\pi$
$888$		0			0
$889$	−2.00000	+	3.46410i	−0.0670778	+	0.116182i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−6.92820	+	12.0000i	−0.231843	+	0.401565i
$894$		0			0
$895$	−18.0000	−	31.1769i	−0.601674	−	1.04213i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−13.8564			−0.462137
$900$		0			0
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$	1.73205	+	3.00000i	0.0575753	+	0.0997234i
$906$		0			0
$907$	−26.0000	+	45.0333i	−0.863316	+	1.49531i	0.00539395	+	0.999985i	$0.498283\pi$
$907$	−26.0000	+	45.0333i	−0.863316	+	1.49531i	−0.868710	+	0.495321i	$0.835050\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−12.1244	+	21.0000i	−0.401698	+	0.695761i	−0.993931	−	0.110006i	$-0.964913\pi$
$911$	−12.1244	+	21.0000i	−0.401698	+	0.695761i	0.592233	+	0.805767i	$0.298246\pi$
$912$		0			0
$913$	−24.0000	−	41.5692i	−0.794284	−	1.37574i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	6.92820			0.228789
$918$		0			0
$919$	−2.00000			−0.0659739			−0.0329870	−	0.999456i	$-0.510502\pi$
$919$	−2.00000			−0.0659739			−0.0329870	+	0.999456i	$0.510502\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	5.19615	+	9.00000i	0.171033	+	0.296239i
$924$		0			0
$925$	−7.00000	+	12.1244i	−0.230159	+	0.398646i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	25.1147	−	43.5000i	0.823988	−	1.42719i	−0.0787027	−	0.996898i	$-0.525078\pi$
$929$	25.1147	−	43.5000i	0.823988	−	1.42719i	0.902690	−	0.430291i	$-0.141589\pi$
$930$		0			0
$931$	−3.00000	−	5.19615i	−0.0983210	−	0.170297i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−31.1769			−1.01959
$936$		0			0
$937$	−25.0000			−0.816714			−0.408357	−	0.912822i	$-0.633898\pi$
$937$	−25.0000			−0.816714			−0.408357	+	0.912822i	$0.633898\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−25.1147	−	43.5000i	−0.818717	−	1.41806i	−0.906628	−	0.421931i	$-0.861352\pi$
$941$	−25.1147	−	43.5000i	−0.818717	−	1.41806i	0.0879109	−	0.996128i	$-0.471981\pi$
$942$		0			0
$943$	−12.0000	+	20.7846i	−0.390774	+	0.676840i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	8.66025	−	15.0000i	0.281420	−	0.487435i	−0.690314	−	0.723510i	$-0.742528\pi$
$947$	8.66025	−	15.0000i	0.281420	−	0.487435i	0.971735	+	0.236075i	$0.0758611\pi$
$948$		0			0
$949$	−3.50000	−	6.06218i	−0.113615	−	0.196787i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−5.19615			−0.168320			−0.0841599	−	0.996452i	$-0.526821\pi$
$953$	−5.19615			−0.168320			−0.0841599	+	0.996452i	$0.526821\pi$
$954$		0			0
$955$	30.0000			0.970777
$956$		0			0
$957$		0			0
$958$		0			0
$959$	1.73205	+	3.00000i	0.0559308	+	0.0968751i
$960$		0			0
$961$	−16.5000	+	28.5788i	−0.532258	+	0.921898i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−0.866025	+	1.50000i	−0.0278783	+	0.0482867i
$966$		0			0
$967$	−23.0000	−	39.8372i	−0.739630	−	1.28108i	−0.952662	−	0.304032i	$-0.901667\pi$
$967$	−23.0000	−	39.8372i	−0.739630	−	1.28108i	0.213032	−	0.977045i	$-0.431666\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−31.1769			−1.00051			−0.500257	−	0.865877i	$-0.666761\pi$
$971$	−31.1769			−1.00051			−0.500257	+	0.865877i	$0.666761\pi$
$972$		0			0
$973$	16.0000			0.512936
$974$		0			0
$975$		0			0
$976$		0			0
$977$	24.2487	+	42.0000i	0.775785	+	1.34370i	0.934352	+	0.356351i	$0.115979\pi$
$977$	24.2487	+	42.0000i	0.775785	+	1.34370i	−0.158567	+	0.987348i	$0.550687\pi$
$978$		0			0
$979$	9.00000	−	15.5885i	0.287641	−	0.498209i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−17.3205	+	30.0000i	−0.552438	+	0.956851i	0.445659	+	0.895203i	$0.352969\pi$
$983$	−17.3205	+	30.0000i	−0.552438	+	0.956851i	−0.998098	+	0.0616488i	$0.980364\pi$
$984$		0			0
$985$	−4.50000	−	7.79423i	−0.143382	−	0.248345i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	6.92820			0.220304
$990$		0			0
$991$	34.0000			1.08005			0.540023	−	0.841650i	$-0.318416\pi$
$991$	34.0000			1.08005			0.540023	+	0.841650i	$0.318416\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−17.3205	−	30.0000i	−0.549097	−	0.951064i
$996$		0			0
$997$	3.50000	−	6.06218i	0.110846	−	0.191991i	−0.805266	−	0.592914i	$-0.797977\pi$
$997$	3.50000	−	6.06218i	0.110846	−	0.191991i	0.916112	+	0.400923i	$0.131311\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1296.2.i.s.865.2		4
3.2	odd	2	inner	1296.2.i.s.865.1		4
4.3	odd	2		81.2.c.b.55.1		4
9.2	odd	6		1296.2.a.o.1.2		2
9.4	even	3	inner	1296.2.i.s.433.2		4
9.5	odd	6	inner	1296.2.i.s.433.1		4
9.7	even	3		1296.2.a.o.1.1		2
12.11	even	2		81.2.c.b.55.2		4
36.7	odd	6		81.2.a.a.1.2	yes	2
36.11	even	6		81.2.a.a.1.1	✓	2
36.23	even	6		81.2.c.b.28.2		4
36.31	odd	6		81.2.c.b.28.1		4
72.11	even	6		5184.2.a.br.1.1		2
72.29	odd	6		5184.2.a.bq.1.1		2
72.43	odd	6		5184.2.a.br.1.2		2
72.61	even	6		5184.2.a.bq.1.2		2
108.7	odd	18		729.2.e.o.82.2		12
108.11	even	18		729.2.e.o.568.2		12
108.23	even	18		729.2.e.o.163.2		12
108.31	odd	18		729.2.e.o.163.1		12
108.43	odd	18		729.2.e.o.568.1		12
108.47	even	18		729.2.e.o.82.1		12
108.59	even	18		729.2.e.o.406.1		12
108.67	odd	18		729.2.e.o.649.2		12
108.79	odd	18		729.2.e.o.325.2		12
108.83	even	18		729.2.e.o.325.1		12
108.95	even	18		729.2.e.o.649.1		12
108.103	odd	18		729.2.e.o.406.2		12
180.7	even	12		2025.2.b.k.649.4		4
180.43	even	12		2025.2.b.k.649.1		4
180.47	odd	12		2025.2.b.k.649.2		4
180.79	odd	6		2025.2.a.j.1.1		2
180.83	odd	12		2025.2.b.k.649.3		4
180.119	even	6		2025.2.a.j.1.2		2
252.83	odd	6		3969.2.a.i.1.1		2
252.223	even	6		3969.2.a.i.1.2		2
396.43	even	6		9801.2.a.v.1.1		2
396.263	odd	6		9801.2.a.v.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
81.2.a.a.1.1	✓	2	36.11	even	6
81.2.a.a.1.2	yes	2	36.7	odd	6
81.2.c.b.28.1		4	36.31	odd	6
81.2.c.b.28.2		4	36.23	even	6
81.2.c.b.55.1		4	4.3	odd	2
81.2.c.b.55.2		4	12.11	even	2
729.2.e.o.82.1		12	108.47	even	18
729.2.e.o.82.2		12	108.7	odd	18
729.2.e.o.163.1		12	108.31	odd	18
729.2.e.o.163.2		12	108.23	even	18
729.2.e.o.325.1		12	108.83	even	18
729.2.e.o.325.2		12	108.79	odd	18
729.2.e.o.406.1		12	108.59	even	18
729.2.e.o.406.2		12	108.103	odd	18
729.2.e.o.568.1		12	108.43	odd	18
729.2.e.o.568.2		12	108.11	even	18
729.2.e.o.649.1		12	108.95	even	18
729.2.e.o.649.2		12	108.67	odd	18
1296.2.a.o.1.1		2	9.7	even	3
1296.2.a.o.1.2		2	9.2	odd	6
1296.2.i.s.433.1		4	9.5	odd	6	inner
1296.2.i.s.433.2		4	9.4	even	3	inner
1296.2.i.s.865.1		4	3.2	odd	2	inner
1296.2.i.s.865.2		4	1.1	even	1	trivial
2025.2.a.j.1.1		2	180.79	odd	6
2025.2.a.j.1.2		2	180.119	even	6
2025.2.b.k.649.1		4	180.43	even	12
2025.2.b.k.649.2		4	180.47	odd	12
2025.2.b.k.649.3		4	180.83	odd	12
2025.2.b.k.649.4		4	180.7	even	12
3969.2.a.i.1.1		2	252.83	odd	6
3969.2.a.i.1.2		2	252.223	even	6
5184.2.a.bq.1.1		2	72.29	odd	6
5184.2.a.bq.1.2		2	72.61	even	6
5184.2.a.br.1.1		2	72.11	even	6
5184.2.a.br.1.2		2	72.43	odd	6
9801.2.a.v.1.1		2	396.43	even	6
9801.2.a.v.1.2		2	396.263	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:	\( N \)	\(=\)	\( 1296 = 2^{4} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1296.i (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.866025 + 1.50000i) q^{5} +(1.00000 - 1.73205i) q^{7} +O(q^{10})\)	\(q+(0.866025 + 1.50000i) q^{5} +(1.00000 - 1.73205i) q^{7} +(-1.73205 + 3.00000i) q^{11} +(0.500000 + 0.866025i) q^{13} +5.19615 q^{17} -2.00000 q^{19} +(1.73205 + 3.00000i) q^{23} +(1.00000 - 1.73205i) q^{25} +(-0.866025 + 1.50000i) q^{29} +(4.00000 + 6.92820i) q^{31} +3.46410 q^{35} -7.00000 q^{37} +(3.46410 + 6.00000i) q^{41} +(1.00000 - 1.73205i) q^{43} +(3.46410 - 6.00000i) q^{47} +(1.50000 + 2.59808i) q^{49} -6.00000 q^{55} +(6.92820 + 12.0000i) q^{59} +(3.50000 - 6.06218i) q^{61} +(-0.866025 + 1.50000i) q^{65} +(-5.00000 - 8.66025i) q^{67} +10.3923 q^{71} -7.00000 q^{73} +(3.46410 + 6.00000i) q^{77} +(1.00000 - 1.73205i) q^{79} +(-6.92820 + 12.0000i) q^{83} +(4.50000 + 7.79423i) q^{85} -5.19615 q^{89} +2.00000 q^{91} +(-1.73205 - 3.00000i) q^{95} +(-1.00000 + 1.73205i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{7}+O(q^{10}) \) 4 * q + 4 * q^7	\( 4 q + 4 q^{7} + 2 q^{13} - 8 q^{19} + 4 q^{25} + 16 q^{31} - 28 q^{37} + 4 q^{43} + 6 q^{49} - 24 q^{55} + 14 q^{61} - 20 q^{67} - 28 q^{73} + 4 q^{79} + 18 q^{85} + 8 q^{91} - 4 q^{97}+O(q^{100}) \) 4 * q + 4 * q^7 + 2 * q^13 - 8 * q^19 + 4 * q^25 + 16 * q^31 - 28 * q^37 + 4 * q^43 + 6 * q^49 - 24 * q^55 + 14 * q^61 - 20 * q^67 - 28 * q^73 + 4 * q^79 + 18 * q^85 + 8 * q^91 - 4 * q^97

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