LMFDB - Embedded newform 1728.2.s.f.1151.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1728,2,Mod(575,1728)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1728, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 0, 1]))

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

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

chi := DirichletCharacter("1728.575");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1728 = 2^{6} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1728.s (of order $6$, 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:	$13.7981494693$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{6})$
Coefficient field:	8.0.170772624.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 3x^{7} + 5x^{6} - 6x^{5} + 6x^{4} - 12x^{3} + 20x^{2} - 24x + 16 $ x^8 - 3x^7 + 5x^6 - 6x^5 + 6x^4 - 12x^3 + 20x^2 - 24*x + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 36)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			1151.4
Root			$1.41203 + 0.0786378i$ of defining polynomial
Character	$\chi$	$=$	1728.1151
Dual form			1728.2.s.f.575.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.686141 + 0.396143i) q^{5} +(2.35143 - 1.35760i) q^{7} +O(q^{10})$	$q+(0.686141 + 0.396143i) q^{5} +(2.35143 - 1.35760i) q^{7} +(-1.71352 - 2.96790i) q^{11} +(1.68614 - 2.92048i) q^{13} +2.52434i q^{17} -2.20979i q^{19} +(1.07561 - 1.86301i) q^{23} +(-2.18614 - 3.78651i) q^{25} +(-0.686141 + 0.396143i) q^{29} +(-1.47603 - 0.852189i) q^{31} +2.15121 q^{35} -4.74456 q^{37} +(-0.127719 - 0.0737384i) q^{41} +(6.01594 - 3.47331i) q^{43} +(-5.77846 - 10.0086i) q^{47} +(0.186141 - 0.322405i) q^{49} +8.51278i q^{53} -2.71519i q^{55} +(2.58891 - 4.48412i) q^{59} +(1.68614 + 2.92048i) q^{61} +(2.31386 - 1.33591i) q^{65} +(6.01594 + 3.47331i) q^{67} +1.75079 q^{71} -2.37228 q^{73} +(-8.05842 - 4.65253i) q^{77} +(8.80507 - 5.08361i) q^{79} +(-3.62725 - 6.28258i) q^{83} +(-1.00000 + 1.73205i) q^{85} +5.34363i q^{89} -9.15640i q^{91} +(0.875393 - 1.51622i) q^{95} +(6.24456 + 10.8159i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 6 q^{5}+O(q^{10}) $ 8 * q - 6 * q^5	$ 8 q - 6 q^{5} + 2 q^{13} - 6 q^{25} + 6 q^{29} + 8 q^{37} - 24 q^{41} - 10 q^{49} + 2 q^{61} + 30 q^{65} + 4 q^{73} - 30 q^{77} - 8 q^{85} + 4 q^{97}+O(q^{100}) $ 8 * q - 6 * q^5 + 2 * q^13 - 6 * q^25 + 6 * q^29 + 8 * q^37 - 24 * q^41 - 10 * q^49 + 2 * q^61 + 30 * q^65 + 4 * q^73 - 30 * q^77 - 8 * q^85 + 4 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1728\mathbb{Z}\right)^\times$.

$n$	$325$	$703$	$1217$
$\chi(n)$	$1$	$-1$	$e\left(\frac{5}{6}\right)$

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
$5$	0.686141	+	0.396143i	0.306851	+	0.177161i	0.645517	−	0.763746i	$-0.276642\pi$
$5$	0.686141	+	0.396143i	0.306851	+	0.177161i	−0.338665	+	0.940907i	$0.609975\pi$
$6$		0			0
$7$	2.35143	−	1.35760i	0.888756	−	0.513124i	0.0152206	−	0.999884i	$-0.495155\pi$
$7$	2.35143	−	1.35760i	0.888756	−	0.513124i	0.873535	+	0.486761i	$0.161822\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−1.71352	−	2.96790i	−0.516645	−	0.894855i	−0.999813	−	0.0193276i	$-0.993847\pi$
$11$	−1.71352	−	2.96790i	−0.516645	−	0.894855i	0.483168	−	0.875527i	$-0.339486\pi$
$12$		0			0
$13$	1.68614	−	2.92048i	0.467651	−	0.809996i	−0.531666	−	0.846954i	$-0.678434\pi$
$13$	1.68614	−	2.92048i	0.467651	−	0.809996i	0.999317	+	0.0369586i	$0.0117670\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			2.52434i			0.612242i	0.951993	+	0.306121i	$0.0990312\pi$
$17$			2.52434i			0.612242i	−0.951993	+	0.306121i	$0.900969\pi$
$18$		0			0
$19$		−	2.20979i		−	0.506960i	−0.967341	−	0.253480i	$-0.918425\pi$
$19$		−	2.20979i		−	0.506960i	0.967341	−	0.253480i	$-0.0815752\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	1.07561	−	1.86301i	0.224279	−	0.388463i	−0.731824	−	0.681494i	$-0.761330\pi$
$23$	1.07561	−	1.86301i	0.224279	−	0.388463i	0.956103	+	0.293031i	$0.0946638\pi$
$24$		0			0
$25$	−2.18614	−	3.78651i	−0.437228	−	0.757301i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−0.686141	+	0.396143i	−0.127413	+	0.0735620i	−0.562352	−	0.826898i	$-0.690103\pi$
$29$	−0.686141	+	0.396143i	−0.127413	+	0.0735620i	0.434939	+	0.900460i	$0.356770\pi$
$30$		0			0
$31$	−1.47603	−	0.852189i	−0.265104	−	0.153058i	0.361557	−	0.932350i	$-0.382245\pi$
$31$	−1.47603	−	0.852189i	−0.265104	−	0.153058i	−0.626660	+	0.779292i	$0.715579\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	2.15121			0.363621
$36$		0			0
$37$	−4.74456			−0.780001			−0.390001	−	0.920815i	$-0.627525\pi$
$37$	−4.74456			−0.780001			−0.390001	+	0.920815i	$0.627525\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−0.127719	−	0.0737384i	−0.0199463	−	0.0115160i	0.489994	−	0.871726i	$-0.336999\pi$
$41$	−0.127719	−	0.0737384i	−0.0199463	−	0.0115160i	−0.509940	+	0.860210i	$0.670332\pi$
$42$		0			0
$43$	6.01594	−	3.47331i	0.917423	−	0.529674i	0.0346108	−	0.999401i	$-0.488981\pi$
$43$	6.01594	−	3.47331i	0.917423	−	0.529674i	0.882812	+	0.469727i	$0.155647\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−5.77846	−	10.0086i	−0.842875	−	1.45990i	−0.887454	−	0.460897i	$-0.847528\pi$
$47$	−5.77846	−	10.0086i	−0.842875	−	1.45990i	0.0445785	−	0.999006i	$-0.485806\pi$
$48$		0			0
$49$	0.186141	−	0.322405i	0.0265915	−	0.0460579i
$50$		0			0
$51$		0			0
$52$		0			0
$53$			8.51278i			1.16932i	0.811278	+	0.584660i	$0.198772\pi$
$53$			8.51278i			1.16932i	−0.811278	+	0.584660i	$0.801228\pi$
$54$		0			0
$55$		−	2.71519i		−	0.366117i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	2.58891	−	4.48412i	0.337047	−	0.583783i	−0.646829	−	0.762635i	$-0.723905\pi$
$59$	2.58891	−	4.48412i	0.337047	−	0.583783i	0.983876	+	0.178852i	$0.0572384\pi$
$60$		0			0
$61$	1.68614	+	2.92048i	0.215888	+	0.373929i	0.953547	−	0.301244i	$-0.0974020\pi$
$61$	1.68614	+	2.92048i	0.215888	+	0.373929i	−0.737659	+	0.675174i	$0.764069\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	2.31386	−	1.33591i	0.286999	−	0.165699i
$66$		0			0
$67$	6.01594	+	3.47331i	0.734964	+	0.424332i	0.820236	−	0.572026i	$-0.193842\pi$
$67$	6.01594	+	3.47331i	0.734964	+	0.424332i	−0.0852711	+	0.996358i	$0.527176\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	1.75079			0.207780			0.103890	−	0.994589i	$-0.466871\pi$
$71$	1.75079			0.207780			0.103890	+	0.994589i	$0.466871\pi$
$72$		0			0
$73$	−2.37228			−0.277655			−0.138827	−	0.990317i	$-0.544333\pi$
$73$	−2.37228			−0.277655			−0.138827	+	0.990317i	$0.544333\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−8.05842	−	4.65253i	−0.918342	−	0.530205i
$78$		0			0
$79$	8.80507	−	5.08361i	0.990647	−	0.571951i	0.0851797	−	0.996366i	$-0.472854\pi$
$79$	8.80507	−	5.08361i	0.990647	−	0.571951i	0.905468	+	0.424415i	$0.139520\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−3.62725	−	6.28258i	−0.398142	−	0.689603i	0.595355	−	0.803463i	$-0.297012\pi$
$83$	−3.62725	−	6.28258i	−0.398142	−	0.689603i	−0.993497	+	0.113861i	$0.963678\pi$
$84$		0			0
$85$	−1.00000	+	1.73205i	−0.108465	+	0.187867i
$86$		0			0
$87$		0			0
$88$		0			0
$89$			5.34363i			0.566424i	0.959057	+	0.283212i	$0.0913999\pi$
$89$			5.34363i			0.566424i	−0.959057	+	0.283212i	$0.908600\pi$
$90$		0			0
$91$		−	9.15640i		−	0.959852i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	0.875393	−	1.51622i	0.0898134	−	0.155561i
$96$		0			0
$97$	6.24456	+	10.8159i	0.634039	+	1.09819i	0.986718	+	0.162444i	$0.0519376\pi$
$97$	6.24456	+	10.8159i	0.634039	+	1.09819i	−0.352679	+	0.935745i	$0.614729\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	15.4307	−	8.90892i	1.53541	−	0.886471i	0.536314	−	0.844019i	$-0.319816\pi$
$101$	15.4307	−	8.90892i	1.53541	−	0.886471i	0.999099	−	0.0424521i	$-0.0135170\pi$
$102$		0			0
$103$	12.6325	+	7.29339i	1.24472	+	0.718640i	0.970051	−	0.242899i	$-0.0780985\pi$
$103$	12.6325	+	7.29339i	1.24472	+	0.718640i	0.274669	+	0.961539i	$0.411432\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−13.2332			−1.27930			−0.639650	−	0.768667i	$-0.720920\pi$
$107$	−13.2332			−1.27930			−0.639650	+	0.768667i	$0.720920\pi$
$108$		0			0
$109$	9.48913			0.908893			0.454447	−	0.890774i	$-0.349837\pi$
$109$	9.48913			0.908893			0.454447	+	0.890774i	$0.349837\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−13.8030	−	7.96916i	−1.29848	−	0.749675i	−0.318335	−	0.947978i	$-0.603124\pi$
$113$	−13.8030	−	7.96916i	−1.29848	−	0.749675i	−0.980141	+	0.198303i	$0.936457\pi$
$114$		0			0
$115$	1.47603	−	0.852189i	0.137641	−	0.0794670i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	3.42703	+	5.93580i	0.314156	+	0.544134i
$120$		0			0
$121$	−0.372281	+	0.644810i	−0.0338438	+	0.0586191i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		−	7.42554i		−	0.664160i
$126$		0			0
$127$		−	16.2912i		−	1.44561i	−0.691053	−	0.722804i	$-0.742853\pi$
$127$		−	16.2912i		−	1.44561i	0.691053	−	0.722804i	$-0.257147\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	3.62725	−	6.28258i	0.316914	−	0.548911i	−0.662928	−	0.748683i	$-0.730687\pi$
$131$	3.62725	−	6.28258i	0.316914	−	0.548911i	0.979843	+	0.199771i	$0.0640200\pi$
$132$		0			0
$133$	−3.00000	−	5.19615i	−0.260133	−	0.450564i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	16.2446	−	9.37880i	1.38787	−	0.801285i	0.394792	−	0.918771i	$-0.370817\pi$
$137$	16.2446	−	9.37880i	1.38787	−	0.801285i	0.993075	+	0.117485i	$0.0374833\pi$
$138$		0			0
$139$	−8.09262	−	4.67228i	−0.686407	−	0.396297i	0.115858	−	0.993266i	$-0.463038\pi$
$139$	−8.09262	−	4.67228i	−0.686407	−	0.396297i	−0.802265	+	0.596968i	$0.796372\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−11.5569			−0.966438
$144$		0			0
$145$	−0.627719			−0.0521292
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−8.31386	−	4.80001i	−0.681098	−	0.393232i	0.119171	−	0.992874i	$-0.461976\pi$
$149$	−8.31386	−	4.80001i	−0.681098	−	0.393232i	−0.800269	+	0.599642i	$0.795310\pi$
$150$		0			0
$151$	−7.92967	+	4.57820i	−0.645308	+	0.372569i	−0.786656	−	0.617391i	$-0.788190\pi$
$151$	−7.92967	+	4.57820i	−0.645308	+	0.372569i	0.141348	+	0.989960i	$0.454856\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−0.675178	−	1.16944i	−0.0542316	−	0.0939319i
$156$		0			0
$157$	−8.43070	+	14.6024i	−0.672843	+	1.16540i	0.304251	+	0.952592i	$0.401594\pi$
$157$	−8.43070	+	14.6024i	−0.672843	+	1.16540i	−0.977094	+	0.212807i	$0.931739\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		−	5.84096i		−	0.460332i
$162$		0			0
$163$			11.8716i			0.929855i	0.885349	+	0.464928i	$0.153920\pi$
$163$			11.8716i			0.929855i	−0.885349	+	0.464928i	$0.846080\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	9.60592	−	16.6379i	0.743329	−	1.28748i	−0.207643	−	0.978205i	$-0.566579\pi$
$167$	9.60592	−	16.6379i	0.743329	−	1.28748i	0.950972	−	0.309278i	$-0.100087\pi$
$168$		0			0
$169$	0.813859	+	1.40965i	0.0626046	+	0.108434i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−4.80298	+	2.77300i	−0.365164	+	0.210828i	−0.671344	−	0.741146i	$-0.734283\pi$
$173$	−4.80298	+	2.77300i	−0.365164	+	0.210828i	0.306180	+	0.951974i	$0.400949\pi$
$174$		0			0
$175$	−10.2811	−	5.93580i	−0.777178	−	0.448704i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	18.8114			1.40603			0.703016	−	0.711174i	$-0.251836\pi$
$179$	18.8114			1.40603			0.703016	+	0.711174i	$0.251836\pi$
$180$		0			0
$181$	4.00000			0.297318			0.148659	−	0.988889i	$-0.452504\pi$
$181$	4.00000			0.297318			0.148659	+	0.988889i	$0.452504\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−3.25544	−	1.87953i	−0.239345	−	0.138186i
$186$		0			0
$187$	7.49198	−	4.32550i	0.547868	−	0.316312i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−1.95100	−	3.37923i	−0.141169	−	0.244512i	0.786768	−	0.617249i	$-0.211753\pi$
$191$	−1.95100	−	3.37923i	−0.141169	−	0.244512i	−0.927937	+	0.372736i	$0.878420\pi$
$192$		0			0
$193$	1.87228	−	3.24289i	0.134770	−	0.233428i	−0.790740	−	0.612153i	$-0.790304\pi$
$193$	1.87228	−	3.24289i	0.134770	−	0.233428i	0.925509	+	0.378724i	$0.123637\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			10.6873i			0.761436i	0.924691	+	0.380718i	$0.124323\pi$
$197$			10.6873i			0.761436i	−0.924691	+	0.380718i	$0.875677\pi$
$198$		0			0
$199$			19.3236i			1.36981i	0.728630	+	0.684907i	$0.240157\pi$
$199$			19.3236i			1.36981i	−0.728630	+	0.684907i	$0.759843\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−1.07561	+	1.86301i	−0.0754928	+	0.130757i
$204$		0			0
$205$	−0.0584220	−	0.101190i	−0.00408037	−	0.00706741i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−6.55842	+	3.78651i	−0.453656	+	0.261918i
$210$		0			0
$211$	5.30350	+	3.06198i	0.365108	+	0.210795i	0.671319	−	0.741169i	$-0.265728\pi$
$211$	5.30350	+	3.06198i	0.365108	+	0.210795i	−0.306211	+	0.951964i	$0.599061\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	5.50371			0.375350
$216$		0			0
$217$	−4.62772			−0.314150
$218$		0			0
$219$		0			0
$220$		0			0
$221$	7.37228	+	4.25639i	0.495913	+	0.286316i
$222$		0			0
$223$	−1.47603	+	0.852189i	−0.0988426	+	0.0570668i	−0.548606	−	0.836081i	$-0.684841\pi$
$223$	−1.47603	+	0.852189i	−0.0988426	+	0.0570668i	0.449764	+	0.893147i	$0.351508\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	9.44298	+	16.3557i	0.626752	+	1.08557i	0.988199	+	0.153174i	$0.0489496\pi$
$227$	9.44298	+	16.3557i	0.626752	+	1.08557i	−0.361447	+	0.932393i	$0.617717\pi$
$228$		0			0
$229$	4.68614	−	8.11663i	0.309669	−	0.536362i	−0.668621	−	0.743603i	$-0.733115\pi$
$229$	4.68614	−	8.11663i	0.309669	−	0.536362i	0.978290	+	0.207241i	$0.0664485\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			28.0627i			1.83845i	0.393737	+	0.919223i	$0.371182\pi$
$233$			28.0627i			1.83845i	−0.393737	+	0.919223i	$0.628818\pi$
$234$		0			0
$235$		−	9.15640i		−	0.597298i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−8.33010	+	14.4282i	−0.538830	+	0.933280i	0.460138	+	0.887847i	$0.347800\pi$
$239$	−8.33010	+	14.4282i	−0.538830	+	0.933280i	−0.998967	+	0.0454327i	$0.985533\pi$
$240$		0			0
$241$	−8.24456	−	14.2800i	−0.531079	−	0.919856i	−0.999342	−	0.0362667i	$-0.988453\pi$
$241$	−8.24456	−	14.2800i	−0.531079	−	0.919856i	0.468263	−	0.883589i	$-0.344880\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	0.255437	−	0.147477i	0.0163193	−	0.00942195i
$246$		0			0
$247$	−6.45364	−	3.72601i	−0.410635	−	0.237080i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−16.7347			−1.05629			−0.528144	−	0.849155i	$-0.677112\pi$
$251$	−16.7347			−1.05629			−0.528144	+	0.849155i	$0.677112\pi$
$252$		0			0
$253$	−7.37228			−0.463491
$254$		0			0
$255$		0			0
$256$		0			0
$257$	9.98913	+	5.76722i	0.623105	+	0.359750i	0.778077	−	0.628169i	$-0.216195\pi$
$257$	9.98913	+	5.76722i	0.623105	+	0.359750i	−0.154972	+	0.987919i	$0.549529\pi$
$258$		0			0
$259$	−11.1565	+	6.44121i	−0.693231	+	0.400237i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	6.25343	+	10.8313i	0.385603	+	0.667884i	0.991853	−	0.127391i	$-0.0406602\pi$
$263$	6.25343	+	10.8313i	0.385603	+	0.667884i	−0.606250	+	0.795274i	$0.707327\pi$
$264$		0			0
$265$	−3.37228	+	5.84096i	−0.207158	+	0.358807i
$266$		0			0
$267$		0			0
$268$		0			0
$269$			24.9484i			1.52113i	0.649260	+	0.760566i	$0.275079\pi$
$269$			24.9484i			1.52113i	−0.649260	+	0.760566i	$0.724921\pi$
$270$		0			0
$271$			1.38712i			0.0842618i	0.999112	+	0.0421309i	$0.0134147\pi$
$271$			1.38712i			0.0842618i	−0.999112	+	0.0421309i	$0.986585\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−7.49198	+	12.9765i	−0.451783	+	0.782512i
$276$		0			0
$277$	0.313859	+	0.543620i	0.0188580	+	0.0326630i	0.875300	−	0.483580i	$-0.160664\pi$
$277$	0.313859	+	0.543620i	0.0188580	+	0.0326630i	−0.856442	+	0.516243i	$0.827330\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	5.56930	−	3.21543i	0.332236	−	0.191817i	−0.324597	−	0.945852i	$-0.605229\pi$
$281$	5.56930	−	3.21543i	0.332236	−	0.191817i	0.656834	+	0.754036i	$0.271895\pi$
$282$		0			0
$283$	−8.80507	−	5.08361i	−0.523407	−	0.302189i	0.214921	−	0.976632i	$-0.431051\pi$
$283$	−8.80507	−	5.08361i	−0.523407	−	0.302189i	−0.738327	+	0.674442i	$0.764384\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−0.400428			−0.0236365
$288$		0			0
$289$	10.6277			0.625160
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−17.3139	−	9.99616i	−1.01149	−	0.583982i	−0.0998599	−	0.995002i	$-0.531839\pi$
$293$	−17.3139	−	9.99616i	−1.01149	−	0.583982i	−0.911627	+	0.411020i	$0.865173\pi$
$294$		0			0
$295$	3.55271	−	2.05116i	0.206847	−	0.119423i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−3.62725	−	6.28258i	−0.209769	−	0.363331i
$300$		0			0
$301$	9.43070	−	16.3345i	0.543577	−	0.941502i
$302$		0			0
$303$		0			0
$304$		0			0
$305$			2.67181i			0.152988i
$306$		0			0
$307$			25.9530i			1.48121i	0.671938	+	0.740607i	$0.265462\pi$
$307$			25.9530i			1.48121i	−0.671938	+	0.740607i	$0.734538\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−15.6591	+	27.1224i	−0.887948	+	1.53797i	−0.0456512	+	0.998957i	$0.514536\pi$
$311$	−15.6591	+	27.1224i	−0.887948	+	1.53797i	−0.842297	+	0.539014i	$0.818797\pi$
$312$		0			0
$313$	−2.24456	−	3.88770i	−0.126870	−	0.219746i	0.795592	−	0.605832i	$-0.207160\pi$
$313$	−2.24456	−	3.88770i	−0.126870	−	0.219746i	−0.922462	+	0.386087i	$0.873826\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	20.6644	−	11.9306i	1.16063	−	0.670089i	0.209173	−	0.977879i	$-0.432923\pi$
$317$	20.6644	−	11.9306i	1.16063	−	0.670089i	0.951454	+	0.307790i	$0.0995893\pi$
$318$		0			0
$319$	2.35143	+	1.35760i	0.131655	+	0.0760109i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	5.57825			0.310382
$324$		0			0
$325$	−14.7446			−0.817881
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−27.1753	−	15.6896i	−1.49822	−	0.864998i
$330$		0			0
$331$	−24.1149	+	13.9228i	−1.32548	+	0.765264i	−0.984596	−	0.174843i	$-0.944058\pi$
$331$	−24.1149	+	13.9228i	−1.32548	+	0.765264i	−0.340879	+	0.940107i	$0.610725\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	2.75186	+	4.76635i	0.150350	+	0.260414i
$336$		0			0
$337$	2.12772	−	3.68532i	0.115904	−	0.200752i	−0.802237	−	0.597006i	$-0.796357\pi$
$337$	2.12772	−	3.68532i	0.115904	−	0.200752i	0.918141	+	0.396254i	$0.129690\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			5.84096i			0.316306i
$342$		0			0
$343$			17.9955i			0.971668i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−3.86473	+	6.69391i	−0.207470	+	0.359348i	−0.950917	−	0.309447i	$-0.899856\pi$
$347$	−3.86473	+	6.69391i	−0.207470	+	0.359348i	0.743447	+	0.668795i	$0.233189\pi$
$348$		0			0
$349$	−5.68614	−	9.84868i	−0.304372	−	0.527188i	0.672749	−	0.739871i	$-0.265113\pi$
$349$	−5.68614	−	9.84868i	−0.304372	−	0.527188i	−0.977121	+	0.212683i	$0.931780\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	4.24456	−	2.45060i	0.225915	−	0.130432i	−0.382771	−	0.923843i	$-0.625030\pi$
$353$	4.24456	−	2.45060i	0.225915	−	0.130432i	0.608686	+	0.793411i	$0.291697\pi$
$354$		0			0
$355$	1.20128	+	0.693562i	0.0637576	+	0.0368105i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	29.9679			1.58165			0.790823	−	0.612045i	$-0.209653\pi$
$359$	29.9679			1.58165			0.790823	+	0.612045i	$0.209653\pi$
$360$		0			0
$361$	14.1168			0.742992
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−1.62772	−	0.939764i	−0.0851987	−	0.0491895i
$366$		0			0
$367$	−11.7571	+	6.78799i	−0.613718	+	0.354330i	−0.774419	−	0.632673i	$-0.781958\pi$
$367$	−11.7571	+	6.78799i	−0.613718	+	0.354330i	0.160701	+	0.987003i	$0.448624\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	11.5569	+	20.0172i	0.600006	+	1.03924i
$372$		0			0
$373$	4.43070	−	7.67420i	0.229413	−	0.397355i	−0.728221	−	0.685342i	$-0.759653\pi$
$373$	4.43070	−	7.67420i	0.229413	−	0.397355i	0.957634	+	0.287987i	$0.0929860\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			2.67181i			0.137605i
$378$		0			0
$379$		−	9.66181i		−	0.496294i	−0.968722	−	0.248147i	$-0.920178\pi$
$379$		−	9.66181i		−	0.496294i	0.968722	−	0.248147i	$-0.0798216\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	0.200214	−	0.346781i	0.0102305	−	0.0177197i	−0.860865	−	0.508834i	$-0.830077\pi$
$383$	0.200214	−	0.346781i	0.0102305	−	0.0177197i	0.871095	+	0.491114i	$0.163410\pi$
$384$		0			0
$385$	−3.68614	−	6.38458i	−0.187863	−	0.325388i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	7.19702	−	4.15520i	0.364903	−	0.210677i	−0.306326	−	0.951927i	$-0.599100\pi$
$389$	7.19702	−	4.15520i	0.364903	−	0.210677i	0.671229	+	0.741250i	$0.265767\pi$
$390$		0			0
$391$	4.70285	+	2.71519i	0.237834	+	0.137313i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	8.05535			0.405309
$396$		0			0
$397$	7.25544			0.364140			0.182070	−	0.983286i	$-0.441720\pi$
$397$	7.25544			0.364140			0.182070	+	0.983286i	$0.441720\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	18.9891	+	10.9634i	0.948272	+	0.547485i	0.892544	−	0.450961i	$-0.148919\pi$
$401$	18.9891	+	10.9634i	0.948272	+	0.547485i	0.0557281	+	0.998446i	$0.482252\pi$
$402$		0			0
$403$	−4.97760	+	2.87382i	−0.247952	+	0.143155i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	8.12989	+	14.0814i	0.402984	+	0.697988i
$408$		0			0
$409$	−1.12772	+	1.95327i	−0.0557621	+	0.0965828i	−0.892559	−	0.450931i	$-0.851092\pi$
$409$	−1.12772	+	1.95327i	−0.0557621	+	0.0965828i	0.836797	+	0.547513i	$0.184426\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		−	14.0588i		−	0.691788i
$414$		0			0
$415$		−	5.74764i		−	0.282141i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−2.82639	+	4.89545i	−0.138078	+	0.239159i	−0.926769	−	0.375631i	$-0.877426\pi$
$419$	−2.82639	+	4.89545i	−0.138078	+	0.239159i	0.788691	+	0.614790i	$0.210759\pi$
$420$		0			0
$421$	14.8030	+	25.6395i	0.721453	+	1.24959i	0.960417	+	0.278565i	$0.0898589\pi$
$421$	14.8030	+	25.6395i	0.721453	+	1.24959i	−0.238964	+	0.971028i	$0.576808\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	9.55842	−	5.51856i	0.463652	−	0.267689i
$426$		0			0
$427$	7.92967	+	4.57820i	0.383744	+	0.221555i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−14.6581			−0.706054			−0.353027	−	0.935613i	$-0.614848\pi$
$431$	−14.6581			−0.706054			−0.353027	+	0.935613i	$0.614848\pi$
$432$		0			0
$433$	−14.3723			−0.690688			−0.345344	−	0.938476i	$-0.612238\pi$
$433$	−14.3723			−0.690688			−0.345344	+	0.938476i	$0.612238\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−4.11684	−	2.37686i	−0.196935	−	0.113701i
$438$		0			0
$439$	−28.4919	+	16.4498i	−1.35984	+	0.785106i	−0.989602	−	0.143830i	$-0.954058\pi$
$439$	−28.4919	+	16.4498i	−1.35984	+	0.785106i	−0.370241	+	0.928936i	$0.620725\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−2.58891	−	4.48412i	−0.123003	−	0.213047i	0.797948	−	0.602727i	$-0.205919\pi$
$443$	−2.58891	−	4.48412i	−0.123003	−	0.213047i	−0.920950	+	0.389680i	$0.872586\pi$
$444$		0			0
$445$	−2.11684	+	3.66648i	−0.100348	+	0.173808i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			3.81396i			0.179992i	0.995942	+	0.0899959i	$0.0286854\pi$
$449$			3.81396i			0.179992i	−0.995942	+	0.0899959i	$0.971315\pi$
$450$		0			0
$451$			0.505408i			0.0237987i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	3.62725	−	6.28258i	0.170048	−	0.294532i
$456$		0			0
$457$	2.98913	+	5.17732i	0.139825	+	0.242185i	0.927430	−	0.373996i	$-0.122013\pi$
$457$	2.98913	+	5.17732i	0.139825	+	0.242185i	−0.787605	+	0.616180i	$0.788679\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−19.8030	+	11.4333i	−0.922317	+	0.532500i	−0.884373	−	0.466780i	$-0.845414\pi$
$461$	−19.8030	+	11.4333i	−0.922317	+	0.532500i	−0.0379435	+	0.999280i	$0.512081\pi$
$462$		0			0
$463$	−18.2108	−	10.5140i	−0.846327	−	0.488627i	0.0130831	−	0.999914i	$-0.495835\pi$
$463$	−18.2108	−	10.5140i	−0.846327	−	0.488627i	−0.859410	+	0.511288i	$0.829169\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	24.3897			1.12862			0.564310	−	0.825563i	$-0.309142\pi$
$467$	24.3897			1.12862			0.564310	+	0.825563i	$0.309142\pi$
$468$		0			0
$469$	18.8614			0.870939
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−20.6168	−	11.9031i	−0.947963	−	0.547307i
$474$		0			0
$475$	−8.36737	+	4.83090i	−0.383921	+	0.221657i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−4.90307	−	8.49236i	−0.224027	−	0.388026i	0.732000	−	0.681304i	$-0.238587\pi$
$479$	−4.90307	−	8.49236i	−0.224027	−	0.388026i	−0.956027	+	0.293278i	$0.905254\pi$
$480$		0			0
$481$	−8.00000	+	13.8564i	−0.364769	+	0.631798i
$482$		0			0
$483$		0			0
$484$		0			0
$485$			9.89497i			0.449308i
$486$		0			0
$487$			17.9365i			0.812780i	0.913700	+	0.406390i	$0.133213\pi$
$487$			17.9365i			0.812780i	−0.913700	+	0.406390i	$0.866787\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−12.3950	+	21.4689i	−0.559381	+	0.968876i	0.438168	+	0.898893i	$0.355628\pi$
$491$	−12.3950	+	21.4689i	−0.559381	+	0.968876i	−0.997548	+	0.0699824i	$0.977706\pi$
$492$		0			0
$493$	−1.00000	−	1.73205i	−0.0450377	−	0.0780076i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	4.11684	−	2.37686i	0.184666	−	0.106617i
$498$		0			0
$499$	−0.437696	−	0.252704i	−0.0195940	−	0.0113126i	0.490171	−	0.871626i	$-0.336934\pi$
$499$	−0.437696	−	0.252704i	−0.0195940	−	0.0113126i	−0.509765	+	0.860314i	$0.670268\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−12.9073			−0.575507			−0.287754	−	0.957704i	$-0.592908\pi$
$503$	−12.9073			−0.575507			−0.287754	+	0.957704i	$0.592908\pi$
$504$		0			0
$505$	14.1168			0.628191
$506$		0			0
$507$		0			0
$508$		0			0
$509$	15.6861	+	9.05640i	0.695276	+	0.401418i	0.805586	−	0.592480i	$-0.201851\pi$
$509$	15.6861	+	9.05640i	0.695276	+	0.401418i	−0.110310	+	0.993897i	$0.535184\pi$
$510$		0			0
$511$	−5.57825	+	3.22060i	−0.246767	+	0.142471i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	5.77846	+	10.0086i	0.254629	+	0.441031i
$516$		0			0
$517$	−19.8030	+	34.2998i	−0.870934	+	1.50850i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	10.4472i		−	0.457700i	−0.973462	−	0.228850i	$-0.926503\pi$
$521$		−	10.4472i		−	0.457700i	0.973462	−	0.228850i	$-0.0734966\pi$
$522$		0			0
$523$			4.41957i			0.193254i	0.995321	+	0.0966272i	$0.0308055\pi$
$523$			4.41957i			0.193254i	−0.995321	+	0.0966272i	$0.969195\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	2.15121	−	3.72601i	0.0937083	−	0.162308i
$528$		0			0
$529$	9.18614	+	15.9109i	0.399397	+	0.691777i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−0.430703	+	0.248667i	−0.0186558	+	0.0107709i
$534$		0			0
$535$	−9.07982	−	5.24224i	−0.392555	−	0.226642i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−1.27582			−0.0549535
$540$		0			0
$541$	−36.9783			−1.58982			−0.794910	−	0.606728i	$-0.792482\pi$
$541$	−36.9783			−1.58982			−0.794910	+	0.606728i	$0.792482\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	6.51087	+	3.75906i	0.278895	+	0.161020i
$546$		0			0
$547$	7.21723	−	4.16687i	0.308586	−	0.178162i	−0.337707	−	0.941251i	$-0.609651\pi$
$547$	7.21723	−	4.16687i	0.308586	−	0.178162i	0.646294	+	0.763089i	$0.276318\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	0.875393	+	1.51622i	0.0372930	+	0.0645933i
$552$		0			0
$553$	13.8030	−	23.9075i	0.586963	−	1.01665i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	20.4897i		−	0.868175i	−0.900871	−	0.434087i	$-0.857071\pi$
$557$		−	20.4897i		−	0.868175i	0.900871	−	0.434087i	$-0.142929\pi$
$558$		0			0
$559$		−	23.4259i		−	0.990811i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	5.54098	−	9.59726i	0.233524	−	0.404476i	−0.725318	−	0.688414i	$-0.758307\pi$
$563$	5.54098	−	9.59726i	0.233524	−	0.404476i	0.958843	+	0.283937i	$0.0916408\pi$
$564$		0			0
$565$	−6.31386	−	10.9359i	−0.265626	−	0.460078i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−0.989125	+	0.571072i	−0.0414663	+	0.0239406i	−0.520590	−	0.853807i	$-0.674288\pi$
$569$	−0.989125	+	0.571072i	−0.0414663	+	0.0239406i	0.479124	+	0.877747i	$0.340955\pi$
$570$		0			0
$571$	−14.5463	−	8.39829i	−0.608742	−	0.351457i	0.163731	−	0.986505i	$-0.447647\pi$
$571$	−14.5463	−	8.39829i	−0.608742	−	0.351457i	−0.772473	+	0.635048i	$0.780980\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−9.40571			−0.392245
$576$		0			0
$577$	−42.8397			−1.78344			−0.891719	−	0.452589i	$-0.850500\pi$
$577$	−42.8397			−1.78344			−0.891719	+	0.452589i	$0.850500\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−17.0584	−	9.84868i	−0.707703	−	0.408592i
$582$		0			0
$583$	25.2651	−	14.5868i	1.04637	−	0.604123i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−6.41637	−	11.1135i	−0.264832	−	0.458702i	0.702688	−	0.711499i	$-0.251983\pi$
$587$	−6.41637	−	11.1135i	−0.264832	−	0.458702i	−0.967520	+	0.252796i	$0.918650\pi$
$588$		0			0
$589$	−1.88316	+	3.26172i	−0.0775941	+	0.134397i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	15.4410i		−	0.634085i	−0.948411	−	0.317043i	$-0.897310\pi$
$593$		−	15.4410i		−	0.634085i	0.948411	−	0.317043i	$-0.102690\pi$
$594$		0			0
$595$			5.43039i			0.222624i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−9.20550	+	15.9444i	−0.376126	+	0.651470i	−0.990495	−	0.137549i	$-0.956077\pi$
$599$	−9.20550	+	15.9444i	−0.376126	+	0.651470i	0.614369	+	0.789019i	$0.289411\pi$
$600$		0			0
$601$	14.9891	+	25.9619i	0.611419	+	1.05901i	0.991001	+	0.133851i	$0.0427344\pi$
$601$	14.9891	+	25.9619i	0.611419	+	1.05901i	−0.379582	+	0.925158i	$0.623932\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−0.510875	+	0.294954i	−0.0207700	+	0.0119916i
$606$		0			0
$607$	33.1947	+	19.1650i	1.34733	+	0.777883i	0.987871	−	0.155277i	$-0.0496270\pi$
$607$	33.1947	+	19.1650i	1.34733	+	0.777883i	0.359462	+	0.933160i	$0.382960\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−38.9732			−1.57669
$612$		0			0
$613$	30.2337			1.22113			0.610564	−	0.791967i	$-0.290943\pi$
$613$	30.2337			1.22113			0.610564	+	0.791967i	$0.290943\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−7.24456	−	4.18265i	−0.291655	−	0.168387i	0.347033	−	0.937853i	$-0.387189\pi$
$617$	−7.24456	−	4.18265i	−0.291655	−	0.168387i	−0.638688	+	0.769466i	$0.720523\pi$
$618$		0			0
$619$	23.9520	−	13.8287i	0.962711	−	0.555821i	0.0657046	−	0.997839i	$-0.479070\pi$
$619$	23.9520	−	13.8287i	0.962711	−	0.555821i	0.897006	+	0.442018i	$0.145737\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	7.25450	+	12.5652i	0.290645	+	0.503412i
$624$		0			0
$625$	−7.98913	+	13.8376i	−0.319565	+	0.553503i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	11.9769i		−	0.477549i
$630$		0			0
$631$		−	17.9365i		−	0.714040i	−0.934097	−	0.357020i	$-0.883793\pi$
$631$		−	17.9365i		−	0.714040i	0.934097	−	0.357020i	$-0.116207\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	6.45364	−	11.1780i	0.256105	−	0.443587i
$636$		0			0
$637$	−0.627719	−	1.08724i	−0.0248711	−	0.0430780i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−23.8723	+	13.7827i	−0.942898	+	0.544383i	−0.890868	−	0.454263i	$-0.849903\pi$
$641$	−23.8723	+	13.7827i	−0.942898	+	0.544383i	−0.0520307	+	0.998645i	$0.516569\pi$
$642$		0			0
$643$	−5.46644	−	3.15605i	−0.215575	−	0.124463i	0.388324	−	0.921523i	$-0.373054\pi$
$643$	−5.46644	−	3.15605i	−0.215575	−	0.124463i	−0.603900	+	0.797060i	$0.706387\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	36.9711			1.45348			0.726741	−	0.686912i	$-0.241034\pi$
$647$	36.9711			1.45348			0.726741	+	0.686912i	$0.241034\pi$
$648$		0			0
$649$	−17.7446			−0.696535
$650$		0			0
$651$		0			0
$652$		0			0
$653$	6.68614	+	3.86025i	0.261649	+	0.151063i	0.625087	−	0.780555i	$-0.285064\pi$
$653$	6.68614	+	3.86025i	0.261649	+	0.151063i	−0.363438	+	0.931619i	$0.618397\pi$
$654$		0			0
$655$	4.97760	−	2.87382i	0.194491	−	0.112290i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−8.33010	−	14.4282i	−0.324495	−	0.562041i	0.656915	−	0.753964i	$-0.271861\pi$
$659$	−8.33010	−	14.4282i	−0.324495	−	0.562041i	−0.981410	+	0.191923i	$0.938528\pi$
$660$		0			0
$661$	10.6861	−	18.5089i	0.415643	−	0.719914i	−0.579853	−	0.814721i	$-0.696890\pi$
$661$	10.6861	−	18.5089i	0.415643	−	0.719914i	0.995496	+	0.0948069i	$0.0302234\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		−	4.75372i		−	0.184341i
$666$		0			0
$667$			1.70438i			0.0659938i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	5.77846	−	10.0086i	0.223075	−	0.386377i
$672$		0			0
$673$	−0.569297	−	0.986051i	−0.0219448	−	0.0380095i	0.854844	−	0.518884i	$-0.173652\pi$
$673$	−0.569297	−	0.986051i	−0.0219448	−	0.0380095i	−0.876789	+	0.480875i	$0.840319\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	28.5475	−	16.4819i	1.09717	−	0.633452i	0.161695	−	0.986841i	$-0.448304\pi$
$677$	28.5475	−	16.4819i	1.09717	−	0.633452i	0.935477	+	0.353389i	$0.114971\pi$
$678$		0			0
$679$	29.3673	+	16.9552i	1.12701	+	0.650681i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−2.07668			−0.0794618			−0.0397309	−	0.999210i	$-0.512650\pi$
$683$	−2.07668			−0.0794618			−0.0397309	+	0.999210i	$0.512650\pi$
$684$		0			0
$685$	14.8614			0.567825
$686$		0			0
$687$		0			0
$688$		0			0
$689$	24.8614	+	14.3537i	0.947144	+	0.546834i
$690$		0			0
$691$	−10.0064	+	5.77717i	−0.380660	+	0.219774i	−0.678105	−	0.734965i	$-0.737199\pi$
$691$	−10.0064	+	5.77717i	−0.380660	+	0.219774i	0.297446	+	0.954739i	$0.403865\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−3.70178	−	6.41168i	−0.140417	−	0.243209i
$696$		0			0
$697$	0.186141	−	0.322405i	0.00705058	−	0.0122120i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	26.1282i		−	0.986850i	−0.869788	−	0.493425i	$-0.835745\pi$
$701$		−	26.1282i		−	0.986850i	0.869788	−	0.493425i	$-0.164255\pi$
$702$		0			0
$703$			10.4845i			0.395429i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	24.1895	−	41.8974i	0.909738	−	1.57571i
$708$		0			0
$709$	−2.43070	−	4.21010i	−0.0912870	−	0.158114i	0.816766	−	0.576969i	$-0.195765\pi$
$709$	−2.43070	−	4.21010i	−0.0912870	−	0.158114i	−0.908053	+	0.418855i	$0.862431\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−3.17527	+	1.83324i	−0.118915	+	0.0686554i
$714$		0			0
$715$	−7.92967	−	4.57820i	−0.296553	−	0.171215i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−7.65492			−0.285481			−0.142740	−	0.989760i	$-0.545591\pi$
$719$	−7.65492			−0.285481			−0.142740	+	0.989760i	$0.545591\pi$
$720$		0			0
$721$	39.6060			1.47500
$722$		0			0
$723$		0			0
$724$		0			0
$725$	3.00000	+	1.73205i	0.111417	+	0.0643268i
$726$		0			0
$727$	−2.67732	+	1.54575i	−0.0992963	+	0.0573287i	−0.548826	−	0.835937i	$-0.684925\pi$
$727$	−2.67732	+	1.54575i	−0.0992963	+	0.0573287i	0.449530	+	0.893265i	$0.351592\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	8.76780	+	15.1863i	0.324289	+	0.561685i
$732$		0			0
$733$	−2.68614	+	4.65253i	−0.0992149	+	0.171845i	−0.911360	−	0.411610i	$-0.864966\pi$
$733$	−2.68614	+	4.65253i	−0.0992149	+	0.171845i	0.812145	+	0.583456i	$0.198300\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	23.8063i		−	0.876916i
$738$		0			0
$739$			9.66181i			0.355415i	0.984083	+	0.177708i	$0.0568681\pi$
$739$			9.66181i			0.355415i	−0.984083	+	0.177708i	$0.943132\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	22.5132	−	38.9940i	0.825929	−	1.43055i	−0.0752776	−	0.997163i	$-0.523984\pi$
$743$	22.5132	−	38.9940i	0.825929	−	1.43055i	0.901207	−	0.433389i	$-0.142682\pi$
$744$		0			0
$745$	−3.80298	−	6.58696i	−0.139331	−	0.241328i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−31.1168	+	17.9653i	−1.13698	+	0.656438i
$750$		0			0
$751$	−39.9743	−	23.0792i	−1.45868	−	0.842170i	−0.459735	−	0.888056i	$-0.652056\pi$
$751$	−39.9743	−	23.0792i	−1.45868	−	0.842170i	−0.998947	+	0.0458859i	$0.985389\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−7.25450			−0.264018
$756$		0			0
$757$	−14.0000			−0.508839			−0.254419	−	0.967094i	$-0.581884\pi$
$757$	−14.0000			−0.508839			−0.254419	+	0.967094i	$0.581884\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	16.5475	+	9.55373i	0.599848	+	0.346322i	0.768982	−	0.639271i	$-0.220764\pi$
$761$	16.5475	+	9.55373i	0.599848	+	0.346322i	−0.169134	+	0.985593i	$0.554097\pi$
$762$		0			0
$763$	22.3130	−	12.8824i	0.807784	−	0.466375i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−8.73053	−	15.1217i	−0.315241	−	0.546014i
$768$		0			0
$769$	27.5475	−	47.7138i	0.993390	−	1.72060i	0.397285	−	0.917695i	$-0.369952\pi$
$769$	27.5475	−	47.7138i	0.993390	−	1.72060i	0.596105	−	0.802907i	$-0.296714\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			45.7330i			1.64490i	0.568835	+	0.822451i	$0.307394\pi$
$773$			45.7330i			1.64490i	−0.568835	+	0.822451i	$0.692606\pi$
$774$		0			0
$775$			7.45202i			0.267685i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−0.162946	+	0.282231i	−0.00583815	+	0.0101120i
$780$		0			0
$781$	−3.00000	−	5.19615i	−0.107348	−	0.185933i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−11.5693	+	6.67954i	−0.412926	+	0.238403i
$786$		0			0
$787$	17.0095	+	9.82043i	0.606323	+	0.350061i	0.771525	−	0.636199i	$-0.219494\pi$
$787$	17.0095	+	9.82043i	0.606323	+	0.350061i	−0.165202	+	0.986260i	$0.552828\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−43.2756			−1.53870
$792$		0			0
$793$	11.3723			0.403842
$794$		0			0
$795$		0			0
$796$		0			0
$797$	40.8030	+	23.5576i	1.44532	+	0.834454i	0.998197	−	0.0600211i	$-0.0191168\pi$
$797$	40.8030	+	23.5576i	1.44532	+	0.834454i	0.447119	+	0.894475i	$0.352450\pi$
$798$		0			0
$799$	25.2651	−	14.5868i	0.893814	−	0.516043i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	4.06494	+	7.04069i	0.143449	+	0.248461i
$804$		0			0
$805$	2.31386	−	4.00772i	0.0815528	−	0.141254i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			27.1778i			0.955521i	0.878490	+	0.477760i	$0.158551\pi$
$809$			27.1778i			0.955521i	−0.878490	+	0.477760i	$0.841449\pi$
$810$		0			0
$811$		−	25.9530i		−	0.911332i	−0.890151	−	0.455666i	$-0.849401\pi$
$811$		−	25.9530i		−	0.911332i	0.890151	−	0.455666i	$-0.150599\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−4.70285	+	8.14558i	−0.164734	+	0.285327i
$816$		0			0
$817$	−7.67527	−	13.2940i	−0.268524	−	0.465096i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−18.6861	+	10.7884i	−0.652151	+	0.376519i	−0.789280	−	0.614034i	$-0.789546\pi$
$821$	−18.6861	+	10.7884i	−0.652151	+	0.376519i	0.137129	+	0.990553i	$0.456213\pi$
$822$		0			0
$823$	−25.8657	−	14.9336i	−0.901622	−	0.520552i	−0.0238957	−	0.999714i	$-0.507607\pi$
$823$	−25.8657	−	14.9336i	−0.901622	−	0.520552i	−0.877726	+	0.479163i	$0.840940\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	7.00314			0.243523			0.121762	−	0.992559i	$-0.461146\pi$
$827$	7.00314			0.243523			0.121762	+	0.992559i	$0.461146\pi$
$828$		0			0
$829$	13.7663			0.478124			0.239062	−	0.971004i	$-0.423160\pi$
$829$	13.7663			0.478124			0.239062	+	0.971004i	$0.423160\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	0.813859	+	0.469882i	0.0281986	+	0.0162804i
$834$		0			0
$835$	13.1820	−	7.61065i	0.456183	−	0.263377i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−22.8391	−	39.5585i	−0.788493	−	1.36571i	−0.926890	−	0.375333i	$-0.877528\pi$
$839$	−22.8391	−	39.5585i	−0.788493	−	1.36571i	0.138397	−	0.990377i	$-0.455805\pi$
$840$		0			0
$841$	−14.1861	+	24.5711i	−0.489177	+	0.847280i
$842$		0			0
$843$		0			0
$844$		0			0
$845$			1.28962i			0.0443643i
$846$		0			0
$847$			2.02163i			0.0694641i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−5.10328	+	8.83915i	−0.174938	+	0.303002i
$852$		0			0
$853$	2.19702	+	3.80534i	0.0752244	+	0.130292i	0.901184	−	0.433437i	$-0.142699\pi$
$853$	2.19702	+	3.80534i	0.0752244	+	0.130292i	−0.825959	+	0.563730i	$0.809366\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	22.0367	−	12.7229i	0.752758	−	0.434605i	−0.0739313	−	0.997263i	$-0.523555\pi$
$857$	22.0367	−	12.7229i	0.752758	−	0.434605i	0.826690	+	0.562658i	$0.190221\pi$
$858$		0			0
$859$	30.4056	+	17.5547i	1.03743	+	0.598958i	0.919103	−	0.394017i	$-0.128915\pi$
$859$	30.4056	+	17.5547i	1.03743	+	0.598958i	0.118323	+	0.992975i	$0.462248\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−4.15335			−0.141382			−0.0706909	−	0.997498i	$-0.522520\pi$
$863$	−4.15335			−0.141382			−0.0706909	+	0.997498i	$0.522520\pi$
$864$		0			0
$865$	−4.39403			−0.149402
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−30.1753	−	17.4217i	−1.02363	−	0.590991i
$870$		0			0
$871$	20.2875	−	11.7130i	0.687414	−	0.396879i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−10.0809	−	17.4606i	−0.340796	−	0.590276i
$876$		0			0
$877$	13.6861	−	23.7051i	0.462148	−	0.800464i	−0.536920	−	0.843633i	$-0.680412\pi$
$877$	13.6861	−	23.7051i	0.462148	−	0.800464i	0.999068	+	0.0431693i	$0.0137455\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	19.7899i		−	0.666740i	−0.942796	−	0.333370i	$-0.891814\pi$
$881$		−	19.7899i		−	0.666740i	0.942796	−	0.333370i	$-0.108186\pi$
$882$		0			0
$883$		−	34.7921i		−	1.17085i	−0.810727	−	0.585424i	$-0.800928\pi$
$883$		−	34.7921i		−	1.17085i	0.810727	−	0.585424i	$-0.199072\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−2.75186	+	4.76635i	−0.0923983	+	0.160039i	−0.908520	−	0.417842i	$-0.862787\pi$
$887$	−2.75186	+	4.76635i	−0.0923983	+	0.160039i	0.816122	+	0.577880i	$0.196120\pi$
$888$		0			0
$889$	−22.1168	−	38.3075i	−0.741775	−	1.28479i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−22.1168	+	12.7692i	−0.740112	+	0.427304i
$894$		0			0
$895$	12.9073	+	7.45202i	0.431443	+	0.249094i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	1.35036			0.0450369
$900$		0			0
$901$	−21.4891			−0.715907
$902$		0			0
$903$		0			0
$904$		0			0
$905$	2.74456	+	1.58457i	0.0912323	+	0.0526730i
$906$		0			0
$907$	21.3258	−	12.3125i	0.708111	−	0.408828i	−0.102250	−	0.994759i	$-0.532604\pi$
$907$	21.3258	−	12.3125i	0.708111	−	0.408828i	0.810361	+	0.585930i	$0.199271\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	6.57932	+	11.3957i	0.217983	+	0.377557i	0.954191	−	0.299198i	$-0.0967191\pi$
$911$	6.57932	+	11.3957i	0.217983	+	0.377557i	−0.736209	+	0.676755i	$0.763386\pi$
$912$		0			0
$913$	−12.4307	+	21.5306i	−0.411396	+	0.712559i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		−	19.6974i		−	0.650464i
$918$		0			0
$919$		−	46.0993i		−	1.52067i	−0.649529	−	0.760337i	$-0.725034\pi$
$919$		−	46.0993i		−	1.52067i	0.649529	−	0.760337i	$-0.274966\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	2.95207	−	5.11313i	0.0971686	−	0.168301i
$924$		0			0
$925$	10.3723	+	17.9653i	0.341039	+	0.590696i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−30.4307	+	17.5692i	−0.998399	+	0.576426i	−0.907774	−	0.419459i	$-0.862220\pi$
$929$	−30.4307	+	17.5692i	−0.998399	+	0.576426i	−0.0906248	+	0.995885i	$0.528886\pi$
$930$		0			0
$931$	−0.712446	−	0.411331i	−0.0233495	−	0.0134808i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	6.85407			0.224152
$936$		0			0
$937$	−11.7228			−0.382968			−0.191484	−	0.981496i	$-0.561330\pi$
$937$	−11.7228			−0.382968			−0.191484	+	0.981496i	$0.561330\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	31.8030	+	18.3615i	1.03675	+	0.598567i	0.918910	−	0.394466i	$-0.129071\pi$
$941$	31.8030	+	18.3615i	1.03675	+	0.598567i	0.117837	+	0.993033i	$0.462404\pi$
$942$		0			0
$943$	−0.274750	+	0.158627i	−0.00894709	+	0.00516561i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−22.2757	−	38.5827i	−0.723864	−	1.25377i	−0.959440	−	0.281913i	$-0.909031\pi$
$947$	−22.2757	−	38.5827i	−0.723864	−	1.25377i	0.235576	−	0.971856i	$-0.424302\pi$
$948$		0			0
$949$	−4.00000	+	6.92820i	−0.129845	+	0.224899i
$950$		0			0
$951$		0			0
$952$		0			0
$953$			32.8164i			1.06303i	0.847050	+	0.531514i	$0.178377\pi$
$953$			32.8164i			1.06303i	−0.847050	+	0.531514i	$0.821623\pi$
$954$		0			0
$955$		−	3.09150i		−	0.100039i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	25.4653	−	44.1071i	0.822317	−	1.42429i
$960$		0			0
$961$	−14.0475	−	24.3311i	−0.453147	−	0.784873i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	2.56930	−	1.48338i	0.0827086	−	0.0477518i
$966$		0			0
$967$	48.7282	+	28.1332i	1.56699	+	0.904704i	0.996517	+	0.0833895i	$0.0265746\pi$
$967$	48.7282	+	28.1332i	1.56699	+	0.904704i	0.570476	+	0.821314i	$0.306759\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	24.7156			0.793160			0.396580	−	0.918000i	$-0.370197\pi$
$971$	24.7156			0.793160			0.396580	+	0.918000i	$0.370197\pi$
$972$		0			0
$973$	−25.3723			−0.813398
$974$		0			0
$975$		0			0
$976$		0			0
$977$	18.6386	+	10.7610i	0.596301	+	0.344275i	0.767585	−	0.640947i	$-0.221458\pi$
$977$	18.6386	+	10.7610i	0.596301	+	0.344275i	−0.171284	+	0.985222i	$0.554791\pi$
$978$		0			0
$979$	15.8593	−	9.15640i	0.506867	−	0.292640i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−8.40464	−	14.5573i	−0.268066	−	0.464305i	0.700296	−	0.713852i	$-0.253051\pi$
$983$	−8.40464	−	14.5573i	−0.268066	−	0.464305i	−0.968362	+	0.249548i	$0.919718\pi$
$984$		0			0
$985$	−4.23369	+	7.33296i	−0.134897	+	0.233648i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	14.9436i		−	0.475180i
$990$		0			0
$991$			16.2912i			0.517506i	0.965944	+	0.258753i	$0.0833116\pi$
$991$			16.2912i			0.517506i	−0.965944	+	0.258753i	$0.916688\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−7.65492	+	13.2587i	−0.242677	+	0.420330i
$996$		0			0
$997$	−26.4307	−	45.7793i	−0.837069	−	1.44985i	−0.892335	−	0.451374i	$-0.850934\pi$
$997$	−26.4307	−	45.7793i	−0.837069	−	1.44985i	0.0552661	−	0.998472i	$-0.482399\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1728.2.s.f.1151.4		8
3.2	odd	2		576.2.s.f.383.3		8
4.3	odd	2	inner	1728.2.s.f.1151.3		8
8.3	odd	2		108.2.h.a.71.4		8
8.5	even	2		108.2.h.a.71.3		8
9.2	odd	6	inner	1728.2.s.f.575.3		8
9.4	even	3		5184.2.c.j.5183.4		8
9.5	odd	6		5184.2.c.j.5183.6		8
9.7	even	3		576.2.s.f.191.2		8
12.11	even	2		576.2.s.f.383.2		8
24.5	odd	2		36.2.h.a.23.2	yes	8
24.11	even	2		36.2.h.a.23.1	yes	8
36.7	odd	6		576.2.s.f.191.3		8
36.11	even	6	inner	1728.2.s.f.575.4		8
36.23	even	6		5184.2.c.j.5183.5		8
36.31	odd	6		5184.2.c.j.5183.3		8
72.5	odd	6		324.2.b.b.323.3		8
72.11	even	6		108.2.h.a.35.3		8
72.13	even	6		324.2.b.b.323.6		8
72.29	odd	6		108.2.h.a.35.4		8
72.43	odd	6		36.2.h.a.11.2	yes	8
72.59	even	6		324.2.b.b.323.5		8
72.61	even	6		36.2.h.a.11.1	✓	8
72.67	odd	6		324.2.b.b.323.4		8
120.29	odd	2		900.2.r.c.851.3		8
120.53	even	4		900.2.o.a.599.7		16
120.59	even	2		900.2.r.c.851.4		8
120.77	even	4		900.2.o.a.599.2		16
120.83	odd	4		900.2.o.a.599.5		16
120.107	odd	4		900.2.o.a.599.4		16
360.43	even	12		900.2.o.a.299.2		16
360.133	odd	12		900.2.o.a.299.4		16
360.187	even	12		900.2.o.a.299.7		16
360.259	odd	6		900.2.r.c.551.3		8
360.277	odd	12		900.2.o.a.299.5		16
360.349	even	6		900.2.r.c.551.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
36.2.h.a.11.1	✓	8	72.61	even	6
36.2.h.a.11.2	yes	8	72.43	odd	6
36.2.h.a.23.1	yes	8	24.11	even	2
36.2.h.a.23.2	yes	8	24.5	odd	2
108.2.h.a.35.3		8	72.11	even	6
108.2.h.a.35.4		8	72.29	odd	6
108.2.h.a.71.3		8	8.5	even	2
108.2.h.a.71.4		8	8.3	odd	2
324.2.b.b.323.3		8	72.5	odd	6
324.2.b.b.323.4		8	72.67	odd	6
324.2.b.b.323.5		8	72.59	even	6
324.2.b.b.323.6		8	72.13	even	6
576.2.s.f.191.2		8	9.7	even	3
576.2.s.f.191.3		8	36.7	odd	6
576.2.s.f.383.2		8	12.11	even	2
576.2.s.f.383.3		8	3.2	odd	2
900.2.o.a.299.2		16	360.43	even	12
900.2.o.a.299.4		16	360.133	odd	12
900.2.o.a.299.5		16	360.277	odd	12
900.2.o.a.299.7		16	360.187	even	12
900.2.o.a.599.2		16	120.77	even	4
900.2.o.a.599.4		16	120.107	odd	4
900.2.o.a.599.5		16	120.83	odd	4
900.2.o.a.599.7		16	120.53	even	4
900.2.r.c.551.3		8	360.259	odd	6
900.2.r.c.551.4		8	360.349	even	6
900.2.r.c.851.3		8	120.29	odd	2
900.2.r.c.851.4		8	120.59	even	2
1728.2.s.f.575.3		8	9.2	odd	6	inner
1728.2.s.f.575.4		8	36.11	even	6	inner
1728.2.s.f.1151.3		8	4.3	odd	2	inner
1728.2.s.f.1151.4		8	1.1	even	1	trivial
5184.2.c.j.5183.3		8	36.31	odd	6
5184.2.c.j.5183.4		8	9.4	even	3
5184.2.c.j.5183.5		8	36.23	even	6
5184.2.c.j.5183.6		8	9.5	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:	\( N \)	\(=\)	\( 1728 = 2^{6} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1728.s (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.686141 + 0.396143i) q^{5} +(2.35143 - 1.35760i) q^{7} +O(q^{10})\)	\(q+(0.686141 + 0.396143i) q^{5} +(2.35143 - 1.35760i) q^{7} +(-1.71352 - 2.96790i) q^{11} +(1.68614 - 2.92048i) q^{13} +2.52434i q^{17} -2.20979i q^{19} +(1.07561 - 1.86301i) q^{23} +(-2.18614 - 3.78651i) q^{25} +(-0.686141 + 0.396143i) q^{29} +(-1.47603 - 0.852189i) q^{31} +2.15121 q^{35} -4.74456 q^{37} +(-0.127719 - 0.0737384i) q^{41} +(6.01594 - 3.47331i) q^{43} +(-5.77846 - 10.0086i) q^{47} +(0.186141 - 0.322405i) q^{49} +8.51278i q^{53} -2.71519i q^{55} +(2.58891 - 4.48412i) q^{59} +(1.68614 + 2.92048i) q^{61} +(2.31386 - 1.33591i) q^{65} +(6.01594 + 3.47331i) q^{67} +1.75079 q^{71} -2.37228 q^{73} +(-8.05842 - 4.65253i) q^{77} +(8.80507 - 5.08361i) q^{79} +(-3.62725 - 6.28258i) q^{83} +(-1.00000 + 1.73205i) q^{85} +5.34363i q^{89} -9.15640i q^{91} +(0.875393 - 1.51622i) q^{95} +(6.24456 + 10.8159i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 6 q^{5}+O(q^{10}) \) 8 * q - 6 * q^5	\( 8 q - 6 q^{5} + 2 q^{13} - 6 q^{25} + 6 q^{29} + 8 q^{37} - 24 q^{41} - 10 q^{49} + 2 q^{61} + 30 q^{65} + 4 q^{73} - 30 q^{77} - 8 q^{85} + 4 q^{97}+O(q^{100}) \) 8 * q - 6 * q^5 + 2 * q^13 - 6 * q^25 + 6 * q^29 + 8 * q^37 - 24 * q^41 - 10 * q^49 + 2 * q^61 + 30 * q^65 + 4 * q^73 - 30 * q^77 - 8 * q^85 + 4 * q^97

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