LMFDB - Embedded newform 1296.2.s.g.863.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1296,2,Mod(431,1296)]

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([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("1296.431");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1296 = 2^{4} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1296.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:	$10.3486121020$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{6})$
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^{2} $
Twist minimal:	no (minimal twist has level 432)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			863.1
Root			$-0.866025 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	1296.863
Dual form			1296.2.s.g.431.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.59808 - 1.50000i) q^{5} +(-1.50000 + 0.866025i) q^{7} +O(q^{10})$	$q+(-2.59808 - 1.50000i) q^{5} +(-1.50000 + 0.866025i) q^{7} +(-2.59808 - 4.50000i) q^{11} +(1.00000 - 1.73205i) q^{13} +6.00000i q^{17} +6.92820i q^{19} +(2.00000 + 3.46410i) q^{25} +(5.19615 - 3.00000i) q^{29} +(4.50000 + 2.59808i) q^{31} +5.19615 q^{35} +8.00000 q^{37} +(-9.00000 + 5.19615i) q^{43} +(5.19615 + 9.00000i) q^{47} +(-2.00000 + 3.46410i) q^{49} -9.00000i q^{53} +15.5885i q^{55} +(-5.19615 + 9.00000i) q^{59} +(2.00000 + 3.46410i) q^{61} +(-5.19615 + 3.00000i) q^{65} +(3.00000 + 1.73205i) q^{67} -10.3923 q^{71} +1.00000 q^{73} +(7.79423 + 4.50000i) q^{77} +(-3.00000 + 1.73205i) q^{79} +(2.59808 + 4.50000i) q^{83} +(9.00000 - 15.5885i) q^{85} -6.00000i q^{89} +3.46410i q^{91} +(10.3923 - 18.0000i) q^{95} +(2.50000 + 4.33013i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 6 q^{7}+O(q^{10}) $ 4 * q - 6 * q^7	$ 4 q - 6 q^{7} + 4 q^{13} + 8 q^{25} + 18 q^{31} + 32 q^{37} - 36 q^{43} - 8 q^{49} + 8 q^{61} + 12 q^{67} + 4 q^{73} - 12 q^{79} + 36 q^{85} + 10 q^{97}+O(q^{100}) $ 4 * q - 6 * q^7 + 4 * q^13 + 8 * q^25 + 18 * q^31 + 32 * q^37 - 36 * q^43 - 8 * q^49 + 8 * q^61 + 12 * q^67 + 4 * q^73 - 12 * q^79 + 36 * q^85 + 10 * q^97

Display 100 coefficients 10 coefficients

Character values

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

$n$	$325$	$1135$	$1217$
$\chi(n)$	$1$	$-1$	$e\left(\frac{5}{6}\right)$

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
$5$	−2.59808	−	1.50000i	−1.16190	−	0.670820i	−0.210138	−	0.977672i	$-0.567391\pi$
$5$	−2.59808	−	1.50000i	−1.16190	−	0.670820i	−0.951757	+	0.306851i	$0.900725\pi$
$6$		0			0
$7$	−1.50000	+	0.866025i	−0.566947	+	0.327327i	−0.755929	−	0.654654i	$-0.772814\pi$
$7$	−1.50000	+	0.866025i	−0.566947	+	0.327327i	0.188982	+	0.981981i	$0.439481\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−2.59808	−	4.50000i	−0.783349	−	1.35680i	−0.929980	−	0.367610i	$-0.880176\pi$
$11$	−2.59808	−	4.50000i	−0.783349	−	1.35680i	0.146631	−	0.989191i	$-0.453157\pi$
$12$		0			0
$13$	1.00000	−	1.73205i	0.277350	−	0.480384i	−0.693375	−	0.720577i	$-0.743877\pi$
$13$	1.00000	−	1.73205i	0.277350	−	0.480384i	0.970725	+	0.240192i	$0.0772105\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			6.00000i			1.45521i	0.685994	+	0.727607i	$0.259367\pi$
$17$			6.00000i			1.45521i	−0.685994	+	0.727607i	$0.740633\pi$
$18$		0			0
$19$			6.92820i			1.58944i	0.606977	+	0.794719i	$0.292382\pi$
$19$			6.92820i			1.58944i	−0.606977	+	0.794719i	$0.707618\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$23$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$24$		0			0
$25$	2.00000	+	3.46410i	0.400000	+	0.692820i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	5.19615	−	3.00000i	0.964901	−	0.557086i	0.0672232	−	0.997738i	$-0.478586\pi$
$29$	5.19615	−	3.00000i	0.964901	−	0.557086i	0.897678	+	0.440652i	$0.145253\pi$
$30$		0			0
$31$	4.50000	+	2.59808i	0.808224	+	0.466628i	0.846339	−	0.532645i	$-0.178802\pi$
$31$	4.50000	+	2.59808i	0.808224	+	0.466628i	−0.0381148	+	0.999273i	$0.512135\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	5.19615			0.878310
$36$		0			0
$37$	8.00000			1.31519			0.657596	−	0.753371i	$-0.271573\pi$
$37$	8.00000			1.31519			0.657596	+	0.753371i	$0.271573\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$41$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$42$		0			0
$43$	−9.00000	+	5.19615i	−1.37249	+	0.792406i	−0.991241	−	0.132068i	$-0.957838\pi$
$43$	−9.00000	+	5.19615i	−1.37249	+	0.792406i	−0.381246	+	0.924473i	$0.624505\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	5.19615	+	9.00000i	0.757937	+	1.31278i	0.943901	+	0.330228i	$0.107126\pi$
$47$	5.19615	+	9.00000i	0.757937	+	1.31278i	−0.185964	+	0.982556i	$0.559541\pi$
$48$		0			0
$49$	−2.00000	+	3.46410i	−0.285714	+	0.494872i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	9.00000i		−	1.23625i	−0.786082	−	0.618123i	$-0.787894\pi$
$53$		−	9.00000i		−	1.23625i	0.786082	−	0.618123i	$-0.212106\pi$
$54$		0			0
$55$			15.5885i			2.10195i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−5.19615	+	9.00000i	−0.676481	+	1.17170i	0.299552	+	0.954080i	$0.403163\pi$
$59$	−5.19615	+	9.00000i	−0.676481	+	1.17170i	−0.976034	+	0.217620i	$0.930171\pi$
$60$		0			0
$61$	2.00000	+	3.46410i	0.256074	+	0.443533i	0.965187	−	0.261562i	$-0.0842377\pi$
$61$	2.00000	+	3.46410i	0.256074	+	0.443533i	−0.709113	+	0.705095i	$0.750904\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−5.19615	+	3.00000i	−0.644503	+	0.372104i
$66$		0			0
$67$	3.00000	+	1.73205i	0.366508	+	0.211604i	0.671932	−	0.740613i	$-0.265465\pi$
$67$	3.00000	+	1.73205i	0.366508	+	0.211604i	−0.305424	+	0.952217i	$0.598798\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−10.3923			−1.23334			−0.616670	−	0.787222i	$-0.711519\pi$
$71$	−10.3923			−1.23334			−0.616670	+	0.787222i	$0.711519\pi$
$72$		0			0
$73$	1.00000			0.117041			0.0585206	−	0.998286i	$-0.481362\pi$
$73$	1.00000			0.117041			0.0585206	+	0.998286i	$0.481362\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	7.79423	+	4.50000i	0.888235	+	0.512823i
$78$		0			0
$79$	−3.00000	+	1.73205i	−0.337526	+	0.194871i	−0.659178	−	0.751987i	$-0.729095\pi$
$79$	−3.00000	+	1.73205i	−0.337526	+	0.194871i	0.321651	+	0.946858i	$0.395762\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	2.59808	+	4.50000i	0.285176	+	0.493939i	0.972652	−	0.232268i	$-0.0746146\pi$
$83$	2.59808	+	4.50000i	0.285176	+	0.493939i	−0.687476	+	0.726207i	$0.741281\pi$
$84$		0			0
$85$	9.00000	−	15.5885i	0.976187	−	1.69081i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	6.00000i		−	0.635999i	−0.948091	−	0.317999i	$-0.896989\pi$
$89$		−	6.00000i		−	0.635999i	0.948091	−	0.317999i	$-0.103011\pi$
$90$		0			0
$91$			3.46410i			0.363137i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	10.3923	−	18.0000i	1.06623	−	1.84676i
$96$		0			0
$97$	2.50000	+	4.33013i	0.253837	+	0.439658i	0.964579	−	0.263795i	$-0.0849741\pi$
$97$	2.50000	+	4.33013i	0.253837	+	0.439658i	−0.710742	+	0.703452i	$0.751641\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−7.79423	+	4.50000i	−0.775555	+	0.447767i	−0.834853	−	0.550474i	$-0.814447\pi$
$101$	−7.79423	+	4.50000i	−0.775555	+	0.447767i	0.0592978	+	0.998240i	$0.481114\pi$
$102$		0			0
$103$	−3.00000	−	1.73205i	−0.295599	−	0.170664i	0.344865	−	0.938652i	$-0.387925\pi$
$103$	−3.00000	−	1.73205i	−0.295599	−	0.170664i	−0.640464	+	0.767988i	$0.721258\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	5.19615			0.502331			0.251166	−	0.967944i	$-0.419186\pi$
$107$	5.19615			0.502331			0.251166	+	0.967944i	$0.419186\pi$
$108$		0			0
$109$	16.0000			1.53252			0.766261	−	0.642529i	$-0.222115\pi$
$109$	16.0000			1.53252			0.766261	+	0.642529i	$0.222115\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	10.3923	+	6.00000i	0.977626	+	0.564433i	0.901553	−	0.432670i	$-0.142428\pi$
$113$	10.3923	+	6.00000i	0.977626	+	0.564433i	0.0760733	+	0.997102i	$0.475762\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−5.19615	−	9.00000i	−0.476331	−	0.825029i
$120$		0			0
$121$	−8.00000	+	13.8564i	−0.727273	+	1.25967i
$122$		0			0
$123$		0			0
$124$		0			0
$125$			3.00000i			0.268328i
$126$		0			0
$127$		−	5.19615i		−	0.461084i	−0.973062	−	0.230542i	$-0.925950\pi$
$127$		−	5.19615i		−	0.461084i	0.973062	−	0.230542i	$-0.0740499\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	2.59808	−	4.50000i	0.226995	−	0.393167i	−0.729921	−	0.683531i	$-0.760443\pi$
$131$	2.59808	−	4.50000i	0.226995	−	0.393167i	0.956916	+	0.290365i	$0.0937766\pi$
$132$		0			0
$133$	−6.00000	−	10.3923i	−0.520266	−	0.901127i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−5.19615	+	3.00000i	−0.443937	+	0.256307i	−0.705266	−	0.708942i	$-0.749173\pi$
$137$	−5.19615	+	3.00000i	−0.443937	+	0.256307i	0.261329	+	0.965250i	$0.415839\pi$
$138$		0			0
$139$	6.00000	+	3.46410i	0.508913	+	0.293821i	0.732387	−	0.680889i	$-0.238406\pi$
$139$	6.00000	+	3.46410i	0.508913	+	0.293821i	−0.223474	+	0.974710i	$0.571740\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−10.3923			−0.869048
$144$		0			0
$145$	−18.0000			−1.49482
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−12.9904	−	7.50000i	−1.06421	−	0.614424i	−0.137619	−	0.990485i	$-0.543945\pi$
$149$	−12.9904	−	7.50000i	−1.06421	−	0.614424i	−0.926595	+	0.376061i	$0.877278\pi$
$150$		0			0
$151$	−7.50000	+	4.33013i	−0.610341	+	0.352381i	−0.773099	−	0.634285i	$-0.781294\pi$
$151$	−7.50000	+	4.33013i	−0.610341	+	0.352381i	0.162758	+	0.986666i	$0.447961\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−7.79423	−	13.5000i	−0.626048	−	1.08435i
$156$		0			0
$157$	2.00000	−	3.46410i	0.159617	−	0.276465i	−0.775113	−	0.631822i	$-0.782307\pi$
$157$	2.00000	−	3.46410i	0.159617	−	0.276465i	0.934731	+	0.355357i	$0.115641\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$			3.46410i			0.271329i	0.990755	+	0.135665i	$0.0433170\pi$
$163$			3.46410i			0.271329i	−0.990755	+	0.135665i	$0.956683\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−5.19615	+	9.00000i	−0.402090	+	0.696441i	−0.993978	−	0.109580i	$-0.965050\pi$
$167$	−5.19615	+	9.00000i	−0.402090	+	0.696441i	0.591888	+	0.806020i	$0.298383\pi$
$168$		0			0
$169$	4.50000	+	7.79423i	0.346154	+	0.599556i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−2.59808	+	1.50000i	−0.197528	+	0.114043i	−0.595502	−	0.803354i	$-0.703047\pi$
$173$	−2.59808	+	1.50000i	−0.197528	+	0.114043i	0.397974	+	0.917397i	$0.369713\pi$
$174$		0			0
$175$	−6.00000	−	3.46410i	−0.453557	−	0.261861i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−5.19615			−0.388379			−0.194189	−	0.980964i	$-0.562208\pi$
$179$	−5.19615			−0.388379			−0.194189	+	0.980964i	$0.562208\pi$
$180$		0			0
$181$	−8.00000			−0.594635			−0.297318	−	0.954779i	$-0.596092\pi$
$181$	−8.00000			−0.594635			−0.297318	+	0.954779i	$0.596092\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−20.7846	−	12.0000i	−1.52811	−	0.882258i
$186$		0			0
$187$	27.0000	−	15.5885i	1.97444	−	1.13994i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	5.19615	+	9.00000i	0.375980	+	0.651217i	0.990473	−	0.137705i	$-0.0439727\pi$
$191$	5.19615	+	9.00000i	0.375980	+	0.651217i	−0.614493	+	0.788922i	$0.710639\pi$
$192$		0			0
$193$	−5.50000	+	9.52628i	−0.395899	+	0.685717i	−0.993215	−	0.116289i	$-0.962900\pi$
$193$	−5.50000	+	9.52628i	−0.395899	+	0.685717i	0.597317	+	0.802005i	$0.296234\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			15.0000i			1.06871i	0.845262	+	0.534353i	$0.179445\pi$
$197$			15.0000i			1.06871i	−0.845262	+	0.534353i	$0.820555\pi$
$198$		0			0
$199$		−	22.5167i		−	1.59616i	−0.602549	−	0.798082i	$-0.705848\pi$
$199$		−	22.5167i		−	1.59616i	0.602549	−	0.798082i	$-0.294152\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−5.19615	+	9.00000i	−0.364698	+	0.631676i
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	31.1769	−	18.0000i	2.15655	−	1.24509i
$210$		0			0
$211$	−12.0000	−	6.92820i	−0.826114	−	0.476957i	0.0264062	−	0.999651i	$-0.491594\pi$
$211$	−12.0000	−	6.92820i	−0.826114	−	0.476957i	−0.852520	+	0.522694i	$0.824927\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	31.1769			2.12625
$216$		0			0
$217$	−9.00000			−0.610960
$218$		0			0
$219$		0			0
$220$		0			0
$221$	10.3923	+	6.00000i	0.699062	+	0.403604i
$222$		0			0
$223$	9.00000	−	5.19615i	0.602685	−	0.347960i	−0.167412	−	0.985887i	$-0.553541\pi$
$223$	9.00000	−	5.19615i	0.602685	−	0.347960i	0.770097	+	0.637927i	$0.220208\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	5.19615	+	9.00000i	0.344881	+	0.597351i	0.985332	−	0.170648i	$-0.0545860\pi$
$227$	5.19615	+	9.00000i	0.344881	+	0.597351i	−0.640451	+	0.767999i	$0.721253\pi$
$228$		0			0
$229$	7.00000	−	12.1244i	0.462573	−	0.801200i	−0.536515	−	0.843891i	$-0.680260\pi$
$229$	7.00000	−	12.1244i	0.462573	−	0.801200i	0.999088	+	0.0426906i	$0.0135930\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	6.00000i		−	0.393073i	−0.980497	−	0.196537i	$-0.937031\pi$
$233$		−	6.00000i		−	0.393073i	0.980497	−	0.196537i	$-0.0629694\pi$
$234$		0			0
$235$		−	31.1769i		−	2.03376i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−10.3923	+	18.0000i	−0.672222	+	1.16432i	0.305050	+	0.952336i	$0.401327\pi$
$239$	−10.3923	+	18.0000i	−0.672222	+	1.16432i	−0.977273	+	0.211987i	$0.932007\pi$
$240$		0			0
$241$	−11.0000	−	19.0526i	−0.708572	−	1.22728i	−0.965387	−	0.260822i	$-0.916006\pi$
$241$	−11.0000	−	19.0526i	−0.708572	−	1.22728i	0.256814	−	0.966461i	$-0.417327\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	10.3923	−	6.00000i	0.663940	−	0.383326i
$246$		0			0
$247$	12.0000	+	6.92820i	0.763542	+	0.440831i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	10.3923			0.655956			0.327978	−	0.944685i	$-0.393633\pi$
$251$	10.3923			0.655956			0.327978	+	0.944685i	$0.393633\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$	5.19615	+	3.00000i	0.324127	+	0.187135i	0.653231	−	0.757159i	$-0.273413\pi$
$257$	5.19615	+	3.00000i	0.324127	+	0.187135i	−0.329104	+	0.944294i	$0.606747\pi$
$258$		0			0
$259$	−12.0000	+	6.92820i	−0.745644	+	0.430498i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$263$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$264$		0			0
$265$	−13.5000	+	23.3827i	−0.829298	+	1.43639i
$266$		0			0
$267$		0			0
$268$		0			0
$269$			30.0000i			1.82913i	0.404436	+	0.914566i	$0.367468\pi$
$269$			30.0000i			1.82913i	−0.404436	+	0.914566i	$0.632532\pi$
$270$		0			0
$271$			1.73205i			0.105215i	0.998615	+	0.0526073i	$0.0167532\pi$
$271$			1.73205i			0.105215i	−0.998615	+	0.0526073i	$0.983247\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	10.3923	−	18.0000i	0.626680	−	1.08544i
$276$		0			0
$277$	−14.0000	−	24.2487i	−0.841178	−	1.45696i	−0.888899	−	0.458103i	$-0.848529\pi$
$277$	−14.0000	−	24.2487i	−0.841178	−	1.45696i	0.0477206	−	0.998861i	$-0.484804\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	5.19615	−	3.00000i	0.309976	−	0.178965i	−0.336939	−	0.941526i	$-0.609392\pi$
$281$	5.19615	−	3.00000i	0.309976	−	0.178965i	0.646916	+	0.762561i	$0.276058\pi$
$282$		0			0
$283$	−18.0000	−	10.3923i	−1.06999	−	0.617758i	−0.141810	−	0.989894i	$-0.545292\pi$
$283$	−18.0000	−	10.3923i	−1.06999	−	0.617758i	−0.928178	+	0.372135i	$0.878626\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−19.0000			−1.11765
$290$		0			0
$291$		0			0
$292$		0			0
$293$	5.19615	+	3.00000i	0.303562	+	0.175262i	0.644042	−	0.764990i	$-0.277256\pi$
$293$	5.19615	+	3.00000i	0.303562	+	0.175262i	−0.340480	+	0.940252i	$0.610589\pi$
$294$		0			0
$295$	27.0000	−	15.5885i	1.57200	−	0.907595i
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$	9.00000	−	15.5885i	0.518751	−	0.898504i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		−	12.0000i		−	0.687118i
$306$		0			0
$307$		−	31.1769i		−	1.77936i	−0.456584	−	0.889680i	$-0.650927\pi$
$307$		−	31.1769i		−	1.77936i	0.456584	−	0.889680i	$-0.349073\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−10.3923	+	18.0000i	−0.589294	+	1.02069i	0.405032	+	0.914303i	$0.367261\pi$
$311$	−10.3923	+	18.0000i	−0.589294	+	1.02069i	−0.994325	+	0.106384i	$0.966073\pi$
$312$		0			0
$313$	15.5000	+	26.8468i	0.876112	+	1.51747i	0.855574	+	0.517681i	$0.173205\pi$
$313$	15.5000	+	26.8468i	0.876112	+	1.51747i	0.0205381	+	0.999789i	$0.493462\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−12.9904	+	7.50000i	−0.729612	+	0.421242i	−0.818280	−	0.574819i	$-0.805072\pi$
$317$	−12.9904	+	7.50000i	−0.729612	+	0.421242i	0.0886679	+	0.996061i	$0.471739\pi$
$318$		0			0
$319$	−27.0000	−	15.5885i	−1.51171	−	0.872786i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−41.5692			−2.31297
$324$		0			0
$325$	8.00000			0.443760
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−15.5885	−	9.00000i	−0.859419	−	0.496186i
$330$		0			0
$331$	3.00000	−	1.73205i	0.164895	−	0.0952021i	−0.415282	−	0.909693i	$-0.636317\pi$
$331$	3.00000	−	1.73205i	0.164895	−	0.0952021i	0.580176	+	0.814491i	$0.302984\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−5.19615	−	9.00000i	−0.283896	−	0.491723i
$336$		0			0
$337$	1.00000	−	1.73205i	0.0544735	−	0.0943508i	−0.837503	−	0.546433i	$-0.815985\pi$
$337$	1.00000	−	1.73205i	0.0544735	−	0.0943508i	0.891976	+	0.452082i	$0.149319\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		−	27.0000i		−	1.46213i
$342$		0			0
$343$		−	19.0526i		−	1.02874i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	7.79423	−	13.5000i	0.418416	−	0.724718i	−0.577364	−	0.816487i	$-0.695919\pi$
$347$	7.79423	−	13.5000i	0.418416	−	0.724718i	0.995780	+	0.0917687i	$0.0292521\pi$
$348$		0			0
$349$	2.00000	+	3.46410i	0.107058	+	0.185429i	0.914577	−	0.404412i	$-0.132524\pi$
$349$	2.00000	+	3.46410i	0.107058	+	0.185429i	−0.807519	+	0.589841i	$0.799190\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−31.1769	+	18.0000i	−1.65938	+	0.958043i	−0.686378	+	0.727245i	$0.740800\pi$
$353$	−31.1769	+	18.0000i	−1.65938	+	0.958043i	−0.973002	+	0.230799i	$0.925866\pi$
$354$		0			0
$355$	27.0000	+	15.5885i	1.43301	+	0.827349i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	10.3923			0.548485			0.274242	−	0.961661i	$-0.411573\pi$
$359$	10.3923			0.548485			0.274242	+	0.961661i	$0.411573\pi$
$360$		0			0
$361$	−29.0000			−1.52632
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−2.59808	−	1.50000i	−0.135990	−	0.0785136i
$366$		0			0
$367$	7.50000	−	4.33013i	0.391497	−	0.226031i	−0.291312	−	0.956628i	$-0.594092\pi$
$367$	7.50000	−	4.33013i	0.391497	−	0.226031i	0.682808	+	0.730597i	$0.260758\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	7.79423	+	13.5000i	0.404656	+	0.700885i
$372$		0			0
$373$	−1.00000	+	1.73205i	−0.0517780	+	0.0896822i	−0.890753	−	0.454488i	$-0.849822\pi$
$373$	−1.00000	+	1.73205i	−0.0517780	+	0.0896822i	0.838975	+	0.544170i	$0.183156\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	12.0000i		−	0.618031i
$378$		0			0
$379$			27.7128i			1.42351i	0.702427	+	0.711756i	$0.252100\pi$
$379$			27.7128i			1.42351i	−0.702427	+	0.711756i	$0.747900\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	15.5885	−	27.0000i	0.796533	−	1.37964i	−0.125328	−	0.992115i	$-0.539998\pi$
$383$	15.5885	−	27.0000i	0.796533	−	1.37964i	0.921861	−	0.387520i	$-0.126668\pi$
$384$		0			0
$385$	−13.5000	−	23.3827i	−0.688024	−	1.19169i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−7.79423	+	4.50000i	−0.395183	+	0.228159i	−0.684403	−	0.729103i	$-0.739937\pi$
$389$	−7.79423	+	4.50000i	−0.395183	+	0.228159i	0.289220	+	0.957263i	$0.406604\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$	10.3923			0.522894
$396$		0			0
$397$	2.00000			0.100377			0.0501886	−	0.998740i	$-0.484018\pi$
$397$	2.00000			0.100377			0.0501886	+	0.998740i	$0.484018\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	20.7846	+	12.0000i	1.03793	+	0.599251i	0.919247	−	0.393680i	$-0.128798\pi$
$401$	20.7846	+	12.0000i	1.03793	+	0.599251i	0.118686	+	0.992932i	$0.462132\pi$
$402$		0			0
$403$	9.00000	−	5.19615i	0.448322	−	0.258839i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−20.7846	−	36.0000i	−1.03025	−	1.78445i
$408$		0			0
$409$	−12.5000	+	21.6506i	−0.618085	+	1.07056i	0.371750	+	0.928333i	$0.378758\pi$
$409$	−12.5000	+	21.6506i	−0.618085	+	1.07056i	−0.989835	+	0.142222i	$0.954575\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		−	18.0000i		−	0.885722i
$414$		0			0
$415$		−	15.5885i		−	0.765207i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	5.19615	−	9.00000i	0.253849	−	0.439679i	−0.710734	−	0.703461i	$-0.751637\pi$
$419$	5.19615	−	9.00000i	0.253849	−	0.439679i	0.964582	+	0.263783i	$0.0849701\pi$
$420$		0			0
$421$	−2.00000	−	3.46410i	−0.0974740	−	0.168830i	0.813164	−	0.582034i	$-0.197743\pi$
$421$	−2.00000	−	3.46410i	−0.0974740	−	0.168830i	−0.910638	+	0.413204i	$0.864410\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−20.7846	+	12.0000i	−1.00820	+	0.582086i
$426$		0			0
$427$	−6.00000	−	3.46410i	−0.290360	−	0.167640i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−31.1769			−1.50174			−0.750870	−	0.660451i	$-0.770365\pi$
$431$	−31.1769			−1.50174			−0.750870	+	0.660451i	$0.770365\pi$
$432$		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
$437$		0			0
$438$		0			0
$439$	−13.5000	+	7.79423i	−0.644320	+	0.371998i	−0.786277	−	0.617875i	$-0.787994\pi$
$439$	−13.5000	+	7.79423i	−0.644320	+	0.371998i	0.141957	+	0.989873i	$0.454661\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	15.5885	+	27.0000i	0.740630	+	1.28281i	0.952209	+	0.305448i	$0.0988061\pi$
$443$	15.5885	+	27.0000i	0.740630	+	1.28281i	−0.211579	+	0.977361i	$0.567861\pi$
$444$		0			0
$445$	−9.00000	+	15.5885i	−0.426641	+	0.738964i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			36.0000i			1.69895i	0.527633	+	0.849473i	$0.323080\pi$
$449$			36.0000i			1.69895i	−0.527633	+	0.849473i	$0.676920\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$	5.19615	−	9.00000i	0.243599	−	0.421927i
$456$		0			0
$457$	−0.500000	−	0.866025i	−0.0233890	−	0.0405110i	0.854094	−	0.520119i	$-0.174112\pi$
$457$	−0.500000	−	0.866025i	−0.0233890	−	0.0405110i	−0.877483	+	0.479608i	$0.840779\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	7.79423	−	4.50000i	0.363013	−	0.209586i	−0.307388	−	0.951584i	$-0.599455\pi$
$461$	7.79423	−	4.50000i	0.363013	−	0.209586i	0.670402	+	0.741998i	$0.266122\pi$
$462$		0			0
$463$	28.5000	+	16.4545i	1.32451	+	0.764705i	0.984444	−	0.175698i	$-0.0562182\pi$
$463$	28.5000	+	16.4545i	1.32451	+	0.764705i	0.340063	+	0.940403i	$0.389552\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−25.9808			−1.20225			−0.601123	−	0.799156i	$-0.705280\pi$
$467$	−25.9808			−1.20225			−0.601123	+	0.799156i	$0.705280\pi$
$468$		0			0
$469$	−6.00000			−0.277054
$470$		0			0
$471$		0			0
$472$		0			0
$473$	46.7654	+	27.0000i	2.15027	+	1.24146i
$474$		0			0
$475$	−24.0000	+	13.8564i	−1.10120	+	0.635776i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−5.19615	−	9.00000i	−0.237418	−	0.411220i	0.722554	−	0.691314i	$-0.242968\pi$
$479$	−5.19615	−	9.00000i	−0.237418	−	0.411220i	−0.959973	+	0.280094i	$0.909635\pi$
$480$		0			0
$481$	8.00000	−	13.8564i	0.364769	−	0.631798i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		−	15.0000i		−	0.681115i
$486$		0			0
$487$			24.2487i			1.09881i	0.835555	+	0.549407i	$0.185146\pi$
$487$			24.2487i			1.09881i	−0.835555	+	0.549407i	$0.814854\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−2.59808	+	4.50000i	−0.117250	+	0.203082i	−0.918677	−	0.395010i	$-0.870741\pi$
$491$	−2.59808	+	4.50000i	−0.117250	+	0.203082i	0.801427	+	0.598092i	$0.204074\pi$
$492$		0			0
$493$	18.0000	+	31.1769i	0.810679	+	1.40414i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	15.5885	−	9.00000i	0.699238	−	0.403705i
$498$		0			0
$499$	−27.0000	−	15.5885i	−1.20869	−	0.697835i	−0.246214	−	0.969216i	$-0.579187\pi$
$499$	−27.0000	−	15.5885i	−1.20869	−	0.697835i	−0.962472	+	0.271380i	$0.912520\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	41.5692			1.85348			0.926740	−	0.375703i	$-0.122599\pi$
$503$	41.5692			1.85348			0.926740	+	0.375703i	$0.122599\pi$
$504$		0			0
$505$	27.0000			1.20148
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−7.79423	−	4.50000i	−0.345473	−	0.199459i	0.317217	−	0.948353i	$-0.397252\pi$
$509$	−7.79423	−	4.50000i	−0.345473	−	0.199459i	−0.662690	+	0.748894i	$0.730585\pi$
$510$		0			0
$511$	−1.50000	+	0.866025i	−0.0663561	+	0.0383107i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	5.19615	+	9.00000i	0.228970	+	0.396587i
$516$		0			0
$517$	27.0000	−	46.7654i	1.18746	−	2.05674i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			18.0000i			0.788594i	0.918983	+	0.394297i	$0.129012\pi$
$521$			18.0000i			0.788594i	−0.918983	+	0.394297i	$0.870988\pi$
$522$		0			0
$523$			17.3205i			0.757373i	0.925525	+	0.378686i	$0.123624\pi$
$523$			17.3205i			0.757373i	−0.925525	+	0.378686i	$0.876376\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−15.5885	+	27.0000i	−0.679044	+	1.17614i
$528$		0			0
$529$	11.5000	+	19.9186i	0.500000	+	0.866025i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$	−13.5000	−	7.79423i	−0.583656	−	0.336974i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	20.7846			0.895257
$540$		0			0
$541$	28.0000			1.20381			0.601907	−	0.798566i	$-0.294408\pi$
$541$	28.0000			1.20381			0.601907	+	0.798566i	$0.294408\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−41.5692	−	24.0000i	−1.78063	−	1.02805i
$546$		0			0
$547$	21.0000	−	12.1244i	0.897895	−	0.518400i	0.0213785	−	0.999771i	$-0.493195\pi$
$547$	21.0000	−	12.1244i	0.897895	−	0.518400i	0.876517	+	0.481371i	$0.159861\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	20.7846	+	36.0000i	0.885454	+	1.53365i
$552$		0			0
$553$	3.00000	−	5.19615i	0.127573	−	0.220963i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	39.0000i		−	1.65248i	−0.563316	−	0.826242i	$-0.690475\pi$
$557$		−	39.0000i		−	1.65248i	0.563316	−	0.826242i	$-0.309525\pi$
$558$		0			0
$559$			20.7846i			0.879095i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−7.79423	+	13.5000i	−0.328488	+	0.568957i	−0.982212	−	0.187776i	$-0.939872\pi$
$563$	−7.79423	+	13.5000i	−0.328488	+	0.568957i	0.653724	+	0.756733i	$0.273206\pi$
$564$		0			0
$565$	−18.0000	−	31.1769i	−0.757266	−	1.31162i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	20.7846	−	12.0000i	0.871336	−	0.503066i	0.00354413	−	0.999994i	$-0.498872\pi$
$569$	20.7846	−	12.0000i	0.871336	−	0.503066i	0.867792	+	0.496928i	$0.165539\pi$
$570$		0			0
$571$	21.0000	+	12.1244i	0.878823	+	0.507388i	0.870270	−	0.492575i	$-0.163944\pi$
$571$	21.0000	+	12.1244i	0.878823	+	0.507388i	0.00855261	+	0.999963i	$0.497278\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−10.0000			−0.416305			−0.208153	−	0.978096i	$-0.566745\pi$
$577$	−10.0000			−0.416305			−0.208153	+	0.978096i	$0.566745\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−7.79423	−	4.50000i	−0.323359	−	0.186691i
$582$		0			0
$583$	−40.5000	+	23.3827i	−1.67734	+	0.968412i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−2.59808	−	4.50000i	−0.107234	−	0.185735i	0.807415	−	0.589984i	$-0.200866\pi$
$587$	−2.59808	−	4.50000i	−0.107234	−	0.185735i	−0.914649	+	0.404249i	$0.867533\pi$
$588$		0			0
$589$	−18.0000	+	31.1769i	−0.741677	+	1.28462i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	6.00000i		−	0.246390i	−0.992382	−	0.123195i	$-0.960686\pi$
$593$		−	6.00000i		−	0.246390i	0.992382	−	0.123195i	$-0.0393141\pi$
$594$		0			0
$595$			31.1769i			1.27813i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−15.5885	+	27.0000i	−0.636927	+	1.10319i	0.349176	+	0.937057i	$0.386461\pi$
$599$	−15.5885	+	27.0000i	−0.636927	+	1.10319i	−0.986103	+	0.166133i	$0.946872\pi$
$600$		0			0
$601$	9.50000	+	16.4545i	0.387513	+	0.671192i	0.992114	−	0.125336i	$-0.0400009\pi$
$601$	9.50000	+	16.4545i	0.387513	+	0.671192i	−0.604601	+	0.796528i	$0.706668\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	41.5692	−	24.0000i	1.69003	−	0.975739i
$606$		0			0
$607$	33.0000	+	19.0526i	1.33943	+	0.773320i	0.986723	−	0.162415i	$-0.0519282\pi$
$607$	33.0000	+	19.0526i	1.33943	+	0.773320i	0.352706	+	0.935734i	$0.385262\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	20.7846			0.840855
$612$		0			0
$613$	−10.0000			−0.403896			−0.201948	−	0.979396i	$-0.564727\pi$
$613$	−10.0000			−0.403896			−0.201948	+	0.979396i	$0.564727\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−10.3923	−	6.00000i	−0.418378	−	0.241551i	0.276005	−	0.961156i	$-0.410989\pi$
$617$	−10.3923	−	6.00000i	−0.418378	−	0.241551i	−0.694383	+	0.719605i	$0.744323\pi$
$618$		0			0
$619$	−21.0000	+	12.1244i	−0.844061	+	0.487319i	−0.858643	−	0.512575i	$-0.828692\pi$
$619$	−21.0000	+	12.1244i	−0.844061	+	0.487319i	0.0145814	+	0.999894i	$0.495358\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	5.19615	+	9.00000i	0.208179	+	0.360577i
$624$		0			0
$625$	14.5000	−	25.1147i	0.580000	−	1.00459i
$626$		0			0
$627$		0			0
$628$		0			0
$629$			48.0000i			1.91389i
$630$		0			0
$631$			12.1244i			0.482663i	0.970443	+	0.241331i	$0.0775841\pi$
$631$			12.1244i			0.482663i	−0.970443	+	0.241331i	$0.922416\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−7.79423	+	13.5000i	−0.309305	+	0.535731i
$636$		0			0
$637$	4.00000	+	6.92820i	0.158486	+	0.274505i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−20.7846	+	12.0000i	−0.820943	+	0.473972i	−0.850741	−	0.525584i	$-0.823847\pi$
$641$	−20.7846	+	12.0000i	−0.820943	+	0.473972i	0.0297987	+	0.999556i	$0.490513\pi$
$642$		0			0
$643$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$643$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−20.7846			−0.817127			−0.408564	−	0.912730i	$-0.633970\pi$
$647$	−20.7846			−0.817127			−0.408564	+	0.912730i	$0.633970\pi$
$648$		0			0
$649$	54.0000			2.11969
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−7.79423	−	4.50000i	−0.305012	−	0.176099i	0.339680	−	0.940541i	$-0.389681\pi$
$653$	−7.79423	−	4.50000i	−0.305012	−	0.176099i	−0.644692	+	0.764442i	$0.723014\pi$
$654$		0			0
$655$	−13.5000	+	7.79423i	−0.527489	+	0.304546i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−18.1865	−	31.5000i	−0.708447	−	1.22707i	−0.965433	−	0.260651i	$-0.916063\pi$
$659$	−18.1865	−	31.5000i	−0.708447	−	1.22707i	0.256986	−	0.966415i	$-0.417270\pi$
$660$		0			0
$661$	−10.0000	+	17.3205i	−0.388955	+	0.673690i	−0.992309	−	0.123784i	$-0.960497\pi$
$661$	−10.0000	+	17.3205i	−0.388955	+	0.673690i	0.603354	+	0.797473i	$0.293830\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$			36.0000i			1.39602i
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$	10.3923	−	18.0000i	0.401190	−	0.694882i
$672$		0			0
$673$	−11.5000	−	19.9186i	−0.443292	−	0.767805i	0.554639	−	0.832091i	$-0.312856\pi$
$673$	−11.5000	−	19.9186i	−0.443292	−	0.767805i	−0.997932	+	0.0642860i	$0.979523\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	15.5885	−	9.00000i	0.599113	−	0.345898i	−0.169580	−	0.985517i	$-0.554241\pi$
$677$	15.5885	−	9.00000i	0.599113	−	0.345898i	0.768693	+	0.639618i	$0.220908\pi$
$678$		0			0
$679$	−7.50000	−	4.33013i	−0.287824	−	0.166175i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−10.3923			−0.397650			−0.198825	−	0.980035i	$-0.563713\pi$
$683$	−10.3923			−0.397650			−0.198825	+	0.980035i	$0.563713\pi$
$684$		0			0
$685$	18.0000			0.687745
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−15.5885	−	9.00000i	−0.593873	−	0.342873i
$690$		0			0
$691$	−18.0000	+	10.3923i	−0.684752	+	0.395342i	−0.801643	−	0.597803i	$-0.796041\pi$
$691$	−18.0000	+	10.3923i	−0.684752	+	0.395342i	0.116891	+	0.993145i	$0.462707\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−10.3923	−	18.0000i	−0.394203	−	0.682779i
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$			21.0000i			0.793159i	0.918000	+	0.396580i	$0.129803\pi$
$701$			21.0000i			0.793159i	−0.918000	+	0.396580i	$0.870197\pi$
$702$		0			0
$703$			55.4256i			2.09042i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	7.79423	−	13.5000i	0.293132	−	0.507720i
$708$		0			0
$709$	16.0000	+	27.7128i	0.600893	+	1.04078i	0.992686	+	0.120723i	$0.0385214\pi$
$709$	16.0000	+	27.7128i	0.600893	+	1.04078i	−0.391794	+	0.920053i	$0.628145\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$	27.0000	+	15.5885i	1.00974	+	0.582975i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	20.7846			0.775135			0.387568	−	0.921841i	$-0.373315\pi$
$719$	20.7846			0.775135			0.387568	+	0.921841i	$0.373315\pi$
$720$		0			0
$721$	6.00000			0.223452
$722$		0			0
$723$		0			0
$724$		0			0
$725$	20.7846	+	12.0000i	0.771921	+	0.445669i
$726$		0			0
$727$	40.5000	−	23.3827i	1.50206	−	0.867216i	0.502065	−	0.864830i	$-0.332574\pi$
$727$	40.5000	−	23.3827i	1.50206	−	0.867216i	0.999997	−	0.00238576i	$-0.000759411\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−31.1769	−	54.0000i	−1.15312	−	1.99726i
$732$		0			0
$733$	−2.00000	+	3.46410i	−0.0738717	+	0.127950i	−0.900595	−	0.434659i	$-0.856869\pi$
$733$	−2.00000	+	3.46410i	−0.0738717	+	0.127950i	0.826723	+	0.562609i	$0.190202\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	18.0000i		−	0.663039i
$738$		0			0
$739$			10.3923i			0.382287i	0.981562	+	0.191144i	$0.0612196\pi$
$739$			10.3923i			0.382287i	−0.981562	+	0.191144i	$0.938780\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	15.5885	−	27.0000i	0.571885	−	0.990534i	−0.424487	−	0.905434i	$-0.639546\pi$
$743$	15.5885	−	27.0000i	0.571885	−	0.990534i	0.996372	−	0.0851001i	$-0.0271210\pi$
$744$		0			0
$745$	22.5000	+	38.9711i	0.824336	+	1.42779i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−7.79423	+	4.50000i	−0.284795	+	0.164426i
$750$		0			0
$751$	−19.5000	−	11.2583i	−0.711565	−	0.410822i	0.100075	−	0.994980i	$-0.468092\pi$
$751$	−19.5000	−	11.2583i	−0.711565	−	0.410822i	−0.811640	+	0.584158i	$0.801425\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	25.9808			0.945537
$756$		0			0
$757$	−26.0000			−0.944986			−0.472493	−	0.881334i	$-0.656646\pi$
$757$	−26.0000			−0.944986			−0.472493	+	0.881334i	$0.656646\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	20.7846	+	12.0000i	0.753442	+	0.435000i	0.826936	−	0.562296i	$-0.190082\pi$
$761$	20.7846	+	12.0000i	0.753442	+	0.435000i	−0.0734946	+	0.997296i	$0.523415\pi$
$762$		0			0
$763$	−24.0000	+	13.8564i	−0.868858	+	0.501636i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	10.3923	+	18.0000i	0.375244	+	0.649942i
$768$		0			0
$769$	12.5000	−	21.6506i	0.450762	−	0.780742i	−0.547672	−	0.836693i	$-0.684486\pi$
$769$	12.5000	−	21.6506i	0.450762	−	0.780742i	0.998434	+	0.0559513i	$0.0178191\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			6.00000i			0.215805i	0.994161	+	0.107903i	$0.0344134\pi$
$773$			6.00000i			0.215805i	−0.994161	+	0.107903i	$0.965587\pi$
$774$		0			0
$775$			20.7846i			0.746605i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$	27.0000	+	46.7654i	0.966136	+	1.67340i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−10.3923	+	6.00000i	−0.370917	+	0.214149i
$786$		0			0
$787$	30.0000	+	17.3205i	1.06938	+	0.617409i	0.928013	−	0.372547i	$-0.121516\pi$
$787$	30.0000	+	17.3205i	1.06938	+	0.617409i	0.141371	+	0.989957i	$0.454849\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−20.7846			−0.739016
$792$		0			0
$793$	8.00000			0.284088
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−28.5788	−	16.5000i	−1.01231	−	0.584460i	−0.100446	−	0.994943i	$-0.532027\pi$
$797$	−28.5788	−	16.5000i	−1.01231	−	0.584460i	−0.911868	+	0.410483i	$0.865360\pi$
$798$		0			0
$799$	−54.0000	+	31.1769i	−1.91038	+	1.10296i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−2.59808	−	4.50000i	−0.0916841	−	0.158802i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	6.00000i		−	0.210949i	−0.994422	−	0.105474i	$-0.966364\pi$
$809$		−	6.00000i		−	0.210949i	0.994422	−	0.105474i	$-0.0336361\pi$
$810$		0			0
$811$			20.7846i			0.729846i	0.931038	+	0.364923i	$0.118905\pi$
$811$			20.7846i			0.729846i	−0.931038	+	0.364923i	$0.881095\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	5.19615	−	9.00000i	0.182013	−	0.315256i
$816$		0			0
$817$	−36.0000	−	62.3538i	−1.25948	−	2.18148i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−36.3731	+	21.0000i	−1.26943	+	0.732905i	−0.974880	−	0.222731i	$-0.928503\pi$
$821$	−36.3731	+	21.0000i	−1.26943	+	0.732905i	−0.294549	+	0.955636i	$0.595169\pi$
$822$		0			0
$823$	−28.5000	−	16.4545i	−0.993448	−	0.573567i	−0.0871445	−	0.996196i	$-0.527774\pi$
$823$	−28.5000	−	16.4545i	−0.993448	−	0.573567i	−0.906303	+	0.422628i	$0.861108\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−31.1769			−1.08413			−0.542064	−	0.840337i	$-0.682357\pi$
$827$	−31.1769			−1.08413			−0.542064	+	0.840337i	$0.682357\pi$
$828$		0			0
$829$	−26.0000			−0.903017			−0.451509	−	0.892267i	$-0.649114\pi$
$829$	−26.0000			−0.903017			−0.451509	+	0.892267i	$0.649114\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−20.7846	−	12.0000i	−0.720144	−	0.415775i
$834$		0			0
$835$	27.0000	−	15.5885i	0.934374	−	0.539461i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	15.5885	+	27.0000i	0.538173	+	0.932144i	0.999002	+	0.0446547i	$0.0142187\pi$
$839$	15.5885	+	27.0000i	0.538173	+	0.932144i	−0.460829	+	0.887489i	$0.652448\pi$
$840$		0			0
$841$	3.50000	−	6.06218i	0.120690	−	0.209041i
$842$		0			0
$843$		0			0
$844$		0			0
$845$		−	27.0000i		−	0.928828i
$846$		0			0
$847$		−	27.7128i		−	0.952224i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$	−25.0000	−	43.3013i	−0.855984	−	1.48261i	−0.875729	−	0.482802i	$-0.839619\pi$
$853$	−25.0000	−	43.3013i	−0.855984	−	1.48261i	0.0197457	−	0.999805i	$-0.493714\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	25.9808	−	15.0000i	0.887486	−	0.512390i	0.0143666	−	0.999897i	$-0.495427\pi$
$857$	25.9808	−	15.0000i	0.887486	−	0.512390i	0.873119	+	0.487507i	$0.162093\pi$
$858$		0			0
$859$	6.00000	+	3.46410i	0.204717	+	0.118194i	0.598854	−	0.800858i	$-0.295623\pi$
$859$	6.00000	+	3.46410i	0.204717	+	0.118194i	−0.394137	+	0.919052i	$0.628956\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−20.7846			−0.707516			−0.353758	−	0.935337i	$-0.615096\pi$
$863$	−20.7846			−0.707516			−0.353758	+	0.935337i	$0.615096\pi$
$864$		0			0
$865$	9.00000			0.306009
$866$		0			0
$867$		0			0
$868$		0			0
$869$	15.5885	+	9.00000i	0.528802	+	0.305304i
$870$		0			0
$871$	6.00000	−	3.46410i	0.203302	−	0.117377i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−2.59808	−	4.50000i	−0.0878310	−	0.152128i
$876$		0			0
$877$	−19.0000	+	32.9090i	−0.641584	+	1.11126i	0.343495	+	0.939155i	$0.388389\pi$
$877$	−19.0000	+	32.9090i	−0.641584	+	1.11126i	−0.985079	+	0.172102i	$0.944944\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	54.0000i		−	1.81931i	−0.415369	−	0.909653i	$-0.636347\pi$
$881$		−	54.0000i		−	1.81931i	0.415369	−	0.909653i	$-0.363653\pi$
$882$		0			0
$883$		−	6.92820i		−	0.233153i	−0.993182	−	0.116576i	$-0.962808\pi$
$883$		−	6.92820i		−	0.233153i	0.993182	−	0.116576i	$-0.0371920\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	5.19615	−	9.00000i	0.174470	−	0.302190i	−0.765508	−	0.643426i	$-0.777512\pi$
$887$	5.19615	−	9.00000i	0.174470	−	0.302190i	0.939978	+	0.341236i	$0.110846\pi$
$888$		0			0
$889$	4.50000	+	7.79423i	0.150925	+	0.261410i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−62.3538	+	36.0000i	−2.08659	+	1.20469i
$894$		0			0
$895$	13.5000	+	7.79423i	0.451255	+	0.260532i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	31.1769			1.03981
$900$		0			0
$901$	54.0000			1.79900
$902$		0			0
$903$		0			0
$904$		0			0
$905$	20.7846	+	12.0000i	0.690904	+	0.398893i
$906$		0			0
$907$	24.0000	−	13.8564i	0.796907	−	0.460094i	−0.0454815	−	0.998965i	$-0.514482\pi$
$907$	24.0000	−	13.8564i	0.796907	−	0.460094i	0.842388	+	0.538871i	$0.181149\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	5.19615	+	9.00000i	0.172156	+	0.298183i	0.939173	−	0.343443i	$-0.111593\pi$
$911$	5.19615	+	9.00000i	0.172156	+	0.298183i	−0.767017	+	0.641626i	$0.778260\pi$
$912$		0			0
$913$	13.5000	−	23.3827i	0.446785	−	0.773854i
$914$		0			0
$915$		0			0
$916$		0			0
$917$			9.00000i			0.297206i
$918$		0			0
$919$		−	15.5885i		−	0.514216i	−0.966383	−	0.257108i	$-0.917230\pi$
$919$		−	15.5885i		−	0.514216i	0.966383	−	0.257108i	$-0.0827696\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−10.3923	+	18.0000i	−0.342067	+	0.592477i
$924$		0			0
$925$	16.0000	+	27.7128i	0.526077	+	0.911192i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−10.3923	+	6.00000i	−0.340960	+	0.196854i	−0.660697	−	0.750653i	$-0.729739\pi$
$929$	−10.3923	+	6.00000i	−0.340960	+	0.196854i	0.319736	+	0.947507i	$0.396406\pi$
$930$		0			0
$931$	−24.0000	−	13.8564i	−0.786568	−	0.454125i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−93.5307			−3.05878
$936$		0			0
$937$	17.0000			0.555366			0.277683	−	0.960673i	$-0.410434\pi$
$937$	17.0000			0.555366			0.277683	+	0.960673i	$0.410434\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	2.59808	+	1.50000i	0.0846949	+	0.0488986i	0.541749	−	0.840540i	$-0.317762\pi$
$941$	2.59808	+	1.50000i	0.0846949	+	0.0488986i	−0.457054	+	0.889439i	$0.651096\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$	23.3827	+	40.5000i	0.759835	+	1.31607i	0.942934	+	0.332979i	$0.108054\pi$
$947$	23.3827	+	40.5000i	0.759835	+	1.31607i	−0.183099	+	0.983094i	$0.558613\pi$
$948$		0			0
$949$	1.00000	−	1.73205i	0.0324614	−	0.0562247i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	24.0000i		−	0.777436i	−0.921357	−	0.388718i	$-0.872918\pi$
$953$		−	24.0000i		−	0.777436i	0.921357	−	0.388718i	$-0.127082\pi$
$954$		0			0
$955$		−	31.1769i		−	1.00886i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	5.19615	−	9.00000i	0.167793	−	0.290625i
$960$		0			0
$961$	−2.00000	−	3.46410i	−0.0645161	−	0.111745i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	28.5788	−	16.5000i	0.919985	−	0.531154i
$966$		0			0
$967$	43.5000	+	25.1147i	1.39887	+	0.807635i	0.994274	−	0.106862i	$-0.0340803\pi$
$967$	43.5000	+	25.1147i	1.39887	+	0.807635i	0.404592	+	0.914497i	$0.367414\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	25.9808			0.833762			0.416881	−	0.908961i	$-0.363123\pi$
$971$	25.9808			0.833762			0.416881	+	0.908961i	$0.363123\pi$
$972$		0			0
$973$	−12.0000			−0.384702
$974$		0			0
$975$		0			0
$976$		0			0
$977$	46.7654	+	27.0000i	1.49616	+	0.863807i	0.999990	−	0.00442082i	$-0.00140720\pi$
$977$	46.7654	+	27.0000i	1.49616	+	0.863807i	0.496167	+	0.868227i	$0.334741\pi$
$978$		0			0
$979$	−27.0000	+	15.5885i	−0.862924	+	0.498209i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−5.19615	−	9.00000i	−0.165732	−	0.287055i	0.771183	−	0.636613i	$-0.219665\pi$
$983$	−5.19615	−	9.00000i	−0.165732	−	0.287055i	−0.936915	+	0.349558i	$0.886332\pi$
$984$		0			0
$985$	22.5000	−	38.9711i	0.716910	−	1.24172i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		−	32.9090i		−	1.04539i	−0.852520	−	0.522694i	$-0.824927\pi$
$991$		−	32.9090i		−	1.04539i	0.852520	−	0.522694i	$-0.175073\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−33.7750	+	58.5000i	−1.07074	+	1.85457i
$996$		0			0
$997$	−13.0000	−	22.5167i	−0.411714	−	0.713110i	0.583363	−	0.812211i	$-0.301736\pi$
$997$	−13.0000	−	22.5167i	−0.411714	−	0.713110i	−0.995077	+	0.0991016i	$0.968403\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1296.2.s.g.863.1		4
3.2	odd	2	inner	1296.2.s.g.863.2		4
4.3	odd	2		1296.2.s.i.863.1		4
9.2	odd	6		1296.2.s.i.431.1		4
9.4	even	3		432.2.c.c.431.3	yes	4
9.5	odd	6		432.2.c.c.431.1	✓	4
9.7	even	3		1296.2.s.i.431.2		4
12.11	even	2		1296.2.s.i.863.2		4
36.7	odd	6	inner	1296.2.s.g.431.2		4
36.11	even	6	inner	1296.2.s.g.431.1		4
36.23	even	6		432.2.c.c.431.2	yes	4
36.31	odd	6		432.2.c.c.431.4	yes	4
72.5	odd	6		1728.2.c.e.1727.3		4
72.13	even	6		1728.2.c.e.1727.1		4
72.59	even	6		1728.2.c.e.1727.4		4
72.67	odd	6		1728.2.c.e.1727.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
432.2.c.c.431.1	✓	4	9.5	odd	6
432.2.c.c.431.2	yes	4	36.23	even	6
432.2.c.c.431.3	yes	4	9.4	even	3
432.2.c.c.431.4	yes	4	36.31	odd	6
1296.2.s.g.431.1		4	36.11	even	6	inner
1296.2.s.g.431.2		4	36.7	odd	6	inner
1296.2.s.g.863.1		4	1.1	even	1	trivial
1296.2.s.g.863.2		4	3.2	odd	2	inner
1296.2.s.i.431.1		4	9.2	odd	6
1296.2.s.i.431.2		4	9.7	even	3
1296.2.s.i.863.1		4	4.3	odd	2
1296.2.s.i.863.2		4	12.11	even	2
1728.2.c.e.1727.1		4	72.13	even	6
1728.2.c.e.1727.2		4	72.67	odd	6
1728.2.c.e.1727.3		4	72.5	odd	6
1728.2.c.e.1727.4		4	72.59	even	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.s (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-2.59808 - 1.50000i) q^{5} +(-1.50000 + 0.866025i) q^{7} +O(q^{10})\)	\(q+(-2.59808 - 1.50000i) q^{5} +(-1.50000 + 0.866025i) q^{7} +(-2.59808 - 4.50000i) q^{11} +(1.00000 - 1.73205i) q^{13} +6.00000i q^{17} +6.92820i q^{19} +(2.00000 + 3.46410i) q^{25} +(5.19615 - 3.00000i) q^{29} +(4.50000 + 2.59808i) q^{31} +5.19615 q^{35} +8.00000 q^{37} +(-9.00000 + 5.19615i) q^{43} +(5.19615 + 9.00000i) q^{47} +(-2.00000 + 3.46410i) q^{49} -9.00000i q^{53} +15.5885i q^{55} +(-5.19615 + 9.00000i) q^{59} +(2.00000 + 3.46410i) q^{61} +(-5.19615 + 3.00000i) q^{65} +(3.00000 + 1.73205i) q^{67} -10.3923 q^{71} +1.00000 q^{73} +(7.79423 + 4.50000i) q^{77} +(-3.00000 + 1.73205i) q^{79} +(2.59808 + 4.50000i) q^{83} +(9.00000 - 15.5885i) q^{85} -6.00000i q^{89} +3.46410i q^{91} +(10.3923 - 18.0000i) q^{95} +(2.50000 + 4.33013i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 6 q^{7}+O(q^{10}) \) 4 * q - 6 * q^7	\( 4 q - 6 q^{7} + 4 q^{13} + 8 q^{25} + 18 q^{31} + 32 q^{37} - 36 q^{43} - 8 q^{49} + 8 q^{61} + 12 q^{67} + 4 q^{73} - 12 q^{79} + 36 q^{85} + 10 q^{97}+O(q^{100}) \) 4 * q - 6 * q^7 + 4 * q^13 + 8 * q^25 + 18 * q^31 + 32 * q^37 - 36 * q^43 - 8 * q^49 + 8 * q^61 + 12 * q^67 + 4 * q^73 - 12 * q^79 + 36 * q^85 + 10 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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