LMFDB - Embedded newform 2592.2.i.q.865.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2592,2,Mod(865,2592)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2592, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 4]))

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

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

chi := DirichletCharacter("2592.865");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2592 = 2^{5} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2592.i (of order $3$, degree $2$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$20.6972242039$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{25}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 96)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			865.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	2592.865
Dual form			2592.2.i.q.1729.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.00000 + 1.73205i) q^{5} +(-2.00000 + 3.46410i) q^{7} +O(q^{10})$	$q+(1.00000 + 1.73205i) q^{5} +(-2.00000 + 3.46410i) q^{7} +(-2.00000 + 3.46410i) q^{11} +(1.00000 + 1.73205i) q^{13} +6.00000 q^{17} +4.00000 q^{19} +(0.500000 - 0.866025i) q^{25} +(1.00000 - 1.73205i) q^{29} +(2.00000 + 3.46410i) q^{31} -8.00000 q^{35} -2.00000 q^{37} +(1.00000 + 1.73205i) q^{41} +(2.00000 - 3.46410i) q^{43} +(-4.00000 + 6.92820i) q^{47} +(-4.50000 - 7.79423i) q^{49} -10.0000 q^{53} -8.00000 q^{55} +(2.00000 + 3.46410i) q^{59} +(-3.00000 + 5.19615i) q^{61} +(-2.00000 + 3.46410i) q^{65} +(2.00000 + 3.46410i) q^{67} -16.0000 q^{71} -6.00000 q^{73} +(-8.00000 - 13.8564i) q^{77} +(2.00000 - 3.46410i) q^{79} +(-6.00000 + 10.3923i) q^{83} +(6.00000 + 10.3923i) q^{85} -10.0000 q^{89} -8.00000 q^{91} +(4.00000 + 6.92820i) q^{95} +(7.00000 - 12.1244i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{5} - 4 q^{7}+O(q^{10}) $ 2 * q + 2 * q^5 - 4 * q^7	$ 2 q + 2 q^{5} - 4 q^{7} - 4 q^{11} + 2 q^{13} + 12 q^{17} + 8 q^{19} + q^{25} + 2 q^{29} + 4 q^{31} - 16 q^{35} - 4 q^{37} + 2 q^{41} + 4 q^{43} - 8 q^{47} - 9 q^{49} - 20 q^{53} - 16 q^{55} + 4 q^{59} - 6 q^{61} - 4 q^{65} + 4 q^{67} - 32 q^{71} - 12 q^{73} - 16 q^{77} + 4 q^{79} - 12 q^{83} + 12 q^{85} - 20 q^{89} - 16 q^{91} + 8 q^{95} + 14 q^{97}+O(q^{100}) $ 2 * q + 2 * q^5 - 4 * q^7 - 4 * q^11 + 2 * q^13 + 12 * q^17 + 8 * q^19 + q^25 + 2 * q^29 + 4 * q^31 - 16 * q^35 - 4 * q^37 + 2 * q^41 + 4 * q^43 - 8 * q^47 - 9 * q^49 - 20 * q^53 - 16 * q^55 + 4 * q^59 - 6 * q^61 - 4 * q^65 + 4 * q^67 - 32 * q^71 - 12 * q^73 - 16 * q^77 + 4 * q^79 - 12 * q^83 + 12 * q^85 - 20 * q^89 - 16 * q^91 + 8 * q^95 + 14 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2592\mathbb{Z}\right)^\times$.

$n$	$325$	$1217$	$2431$
$\chi(n)$	$1$	$e\left(\frac{2}{3}\right)$	$1$

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
$5$	1.00000	+	1.73205i	0.447214	+	0.774597i	0.998203	−	0.0599153i	$-0.0190830\pi$
$5$	1.00000	+	1.73205i	0.447214	+	0.774597i	−0.550990	+	0.834512i	$0.685750\pi$
$6$		0			0
$7$	−2.00000	+	3.46410i	−0.755929	+	1.30931i	0.188982	+	0.981981i	$0.439481\pi$
$7$	−2.00000	+	3.46410i	−0.755929	+	1.30931i	−0.944911	+	0.327327i	$0.893852\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−2.00000	+	3.46410i	−0.603023	+	1.04447i	0.389338	+	0.921095i	$0.372704\pi$
$11$	−2.00000	+	3.46410i	−0.603023	+	1.04447i	−0.992361	+	0.123371i	$0.960630\pi$
$12$		0			0
$13$	1.00000	+	1.73205i	0.277350	+	0.480384i	0.970725	−	0.240192i	$-0.0772105\pi$
$13$	1.00000	+	1.73205i	0.277350	+	0.480384i	−0.693375	+	0.720577i	$0.743877\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	6.00000			1.45521			0.727607	−	0.685994i	$-0.240633\pi$
$17$	6.00000			1.45521			0.727607	+	0.685994i	$0.240633\pi$
$18$		0			0
$19$	4.00000			0.917663			0.458831	−	0.888523i	$-0.348268\pi$
$19$	4.00000			0.917663			0.458831	+	0.888523i	$0.348268\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$23$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$24$		0			0
$25$	0.500000	−	0.866025i	0.100000	−	0.173205i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	1.00000	−	1.73205i	0.185695	−	0.321634i	−0.758115	−	0.652121i	$-0.773880\pi$
$29$	1.00000	−	1.73205i	0.185695	−	0.321634i	0.943811	+	0.330487i	$0.107213\pi$
$30$		0			0
$31$	2.00000	+	3.46410i	0.359211	+	0.622171i	0.987829	−	0.155543i	$-0.0497126\pi$
$31$	2.00000	+	3.46410i	0.359211	+	0.622171i	−0.628619	+	0.777714i	$0.716379\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−8.00000			−1.35225
$36$		0			0
$37$	−2.00000			−0.328798			−0.164399	−	0.986394i	$-0.552568\pi$
$37$	−2.00000			−0.328798			−0.164399	+	0.986394i	$0.552568\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	1.00000	+	1.73205i	0.156174	+	0.270501i	0.933486	−	0.358614i	$-0.116751\pi$
$41$	1.00000	+	1.73205i	0.156174	+	0.270501i	−0.777312	+	0.629115i	$0.783417\pi$
$42$		0			0
$43$	2.00000	−	3.46410i	0.304997	−	0.528271i	−0.672264	−	0.740312i	$-0.734678\pi$
$43$	2.00000	−	3.46410i	0.304997	−	0.528271i	0.977261	+	0.212041i	$0.0680112\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−4.00000	+	6.92820i	−0.583460	+	1.01058i	0.411606	+	0.911362i	$0.364968\pi$
$47$	−4.00000	+	6.92820i	−0.583460	+	1.01058i	−0.995066	+	0.0992202i	$0.968365\pi$
$48$		0			0
$49$	−4.50000	−	7.79423i	−0.642857	−	1.11346i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−10.0000			−1.37361			−0.686803	−	0.726844i	$-0.740986\pi$
$53$	−10.0000			−1.37361			−0.686803	+	0.726844i	$0.740986\pi$
$54$		0			0
$55$	−8.00000			−1.07872
$56$		0			0
$57$		0			0
$58$		0			0
$59$	2.00000	+	3.46410i	0.260378	+	0.450988i	0.966342	−	0.257260i	$-0.0828195\pi$
$59$	2.00000	+	3.46410i	0.260378	+	0.450988i	−0.705965	+	0.708247i	$0.749486\pi$
$60$		0			0
$61$	−3.00000	+	5.19615i	−0.384111	+	0.665299i	−0.991645	−	0.128994i	$-0.958825\pi$
$61$	−3.00000	+	5.19615i	−0.384111	+	0.665299i	0.607535	+	0.794293i	$0.292159\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−2.00000	+	3.46410i	−0.248069	+	0.429669i
$66$		0			0
$67$	2.00000	+	3.46410i	0.244339	+	0.423207i	0.961946	−	0.273241i	$-0.0880957\pi$
$67$	2.00000	+	3.46410i	0.244339	+	0.423207i	−0.717607	+	0.696449i	$0.754762\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−16.0000			−1.89885			−0.949425	−	0.313993i	$-0.898333\pi$
$71$	−16.0000			−1.89885			−0.949425	+	0.313993i	$0.898333\pi$
$72$		0			0
$73$	−6.00000			−0.702247			−0.351123	−	0.936329i	$-0.614200\pi$
$73$	−6.00000			−0.702247			−0.351123	+	0.936329i	$0.614200\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−8.00000	−	13.8564i	−0.911685	−	1.57908i
$78$		0			0
$79$	2.00000	−	3.46410i	0.225018	−	0.389742i	−0.731307	−	0.682048i	$-0.761089\pi$
$79$	2.00000	−	3.46410i	0.225018	−	0.389742i	0.956325	+	0.292306i	$0.0944227\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−6.00000	+	10.3923i	−0.658586	+	1.14070i	0.322396	+	0.946605i	$0.395512\pi$
$83$	−6.00000	+	10.3923i	−0.658586	+	1.14070i	−0.980982	+	0.194099i	$0.937822\pi$
$84$		0			0
$85$	6.00000	+	10.3923i	0.650791	+	1.12720i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−10.0000			−1.06000			−0.529999	−	0.847998i	$-0.677808\pi$
$89$	−10.0000			−1.06000			−0.529999	+	0.847998i	$0.677808\pi$
$90$		0			0
$91$	−8.00000			−0.838628
$92$		0			0
$93$		0			0
$94$		0			0
$95$	4.00000	+	6.92820i	0.410391	+	0.710819i
$96$		0			0
$97$	7.00000	−	12.1244i	0.710742	−	1.23104i	−0.253837	−	0.967247i	$-0.581693\pi$
$97$	7.00000	−	12.1244i	0.710742	−	1.23104i	0.964579	−	0.263795i	$-0.0849741\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−3.00000	+	5.19615i	−0.298511	+	0.517036i	−0.975796	−	0.218685i	$-0.929823\pi$
$101$	−3.00000	+	5.19615i	−0.298511	+	0.517036i	0.677284	+	0.735721i	$0.263157\pi$
$102$		0			0
$103$	−6.00000	−	10.3923i	−0.591198	−	1.02398i	−0.994071	−	0.108729i	$-0.965322\pi$
$103$	−6.00000	−	10.3923i	−0.591198	−	1.02398i	0.402874	−	0.915255i	$-0.368011\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−4.00000			−0.386695			−0.193347	−	0.981130i	$-0.561934\pi$
$107$	−4.00000			−0.386695			−0.193347	+	0.981130i	$0.561934\pi$
$108$		0			0
$109$	14.0000			1.34096			0.670478	−	0.741929i	$-0.266089\pi$
$109$	14.0000			1.34096			0.670478	+	0.741929i	$0.266089\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	1.00000	+	1.73205i	0.0940721	+	0.162938i	0.909221	−	0.416314i	$-0.136678\pi$
$113$	1.00000	+	1.73205i	0.0940721	+	0.162938i	−0.815149	+	0.579252i	$0.803345\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−12.0000	+	20.7846i	−1.10004	+	1.90532i
$120$		0			0
$121$	−2.50000	−	4.33013i	−0.227273	−	0.393648i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	12.0000			1.07331
$126$		0			0
$127$	20.0000			1.77471			0.887357	−	0.461084i	$-0.152539\pi$
$127$	20.0000			1.77471			0.887357	+	0.461084i	$0.152539\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−2.00000	−	3.46410i	−0.174741	−	0.302660i	0.765331	−	0.643637i	$-0.222575\pi$
$131$	−2.00000	−	3.46410i	−0.174741	−	0.302660i	−0.940072	+	0.340977i	$0.889242\pi$
$132$		0			0
$133$	−8.00000	+	13.8564i	−0.693688	+	1.20150i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	9.00000	−	15.5885i	0.768922	−	1.33181i	−0.169226	−	0.985577i	$-0.554127\pi$
$137$	9.00000	−	15.5885i	0.768922	−	1.33181i	0.938148	−	0.346235i	$-0.112540\pi$
$138$		0			0
$139$	−10.0000	−	17.3205i	−0.848189	−	1.46911i	−0.882823	−	0.469706i	$-0.844360\pi$
$139$	−10.0000	−	17.3205i	−0.848189	−	1.46911i	0.0346338	−	0.999400i	$-0.488974\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−8.00000			−0.668994
$144$		0			0
$145$	4.00000			0.332182
$146$		0			0
$147$		0			0
$148$		0			0
$149$	9.00000	+	15.5885i	0.737309	+	1.27706i	0.953703	+	0.300750i	$0.0972370\pi$
$149$	9.00000	+	15.5885i	0.737309	+	1.27706i	−0.216394	+	0.976306i	$0.569430\pi$
$150$		0			0
$151$	6.00000	−	10.3923i	0.488273	−	0.845714i	−0.511636	−	0.859202i	$-0.670960\pi$
$151$	6.00000	−	10.3923i	0.488273	−	0.845714i	0.999909	+	0.0134886i	$0.00429367\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−4.00000	+	6.92820i	−0.321288	+	0.556487i
$156$		0			0
$157$	5.00000	+	8.66025i	0.399043	+	0.691164i	0.993608	−	0.112884i	$-0.0360089\pi$
$157$	5.00000	+	8.66025i	0.399043	+	0.691164i	−0.594565	+	0.804048i	$0.702676\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	4.00000			0.313304			0.156652	−	0.987654i	$-0.449930\pi$
$163$	4.00000			0.313304			0.156652	+	0.987654i	$0.449930\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−4.00000	−	6.92820i	−0.309529	−	0.536120i	0.668730	−	0.743505i	$-0.266838\pi$
$167$	−4.00000	−	6.92820i	−0.309529	−	0.536120i	−0.978259	+	0.207385i	$0.933505\pi$
$168$		0			0
$169$	4.50000	−	7.79423i	0.346154	−	0.599556i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−3.00000	+	5.19615i	−0.228086	+	0.395056i	−0.957241	−	0.289292i	$-0.906580\pi$
$173$	−3.00000	+	5.19615i	−0.228086	+	0.395056i	0.729155	+	0.684349i	$0.239913\pi$
$174$		0			0
$175$	2.00000	+	3.46410i	0.151186	+	0.261861i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−12.0000			−0.896922			−0.448461	−	0.893802i	$-0.648028\pi$
$179$	−12.0000			−0.896922			−0.448461	+	0.893802i	$0.648028\pi$
$180$		0			0
$181$	−10.0000			−0.743294			−0.371647	−	0.928374i	$-0.621207\pi$
$181$	−10.0000			−0.743294			−0.371647	+	0.928374i	$0.621207\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−2.00000	−	3.46410i	−0.147043	−	0.254686i
$186$		0			0
$187$	−12.0000	+	20.7846i	−0.877527	+	1.51992i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$191$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$192$		0			0
$193$	−9.00000	−	15.5885i	−0.647834	−	1.12208i	−0.983639	−	0.180150i	$-0.942342\pi$
$193$	−9.00000	−	15.5885i	−0.647834	−	1.12208i	0.335805	−	0.941932i	$-0.390992\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	22.0000			1.56744			0.783718	−	0.621117i	$-0.213321\pi$
$197$	22.0000			1.56744			0.783718	+	0.621117i	$0.213321\pi$
$198$		0			0
$199$	−4.00000			−0.283552			−0.141776	−	0.989899i	$-0.545281\pi$
$199$	−4.00000			−0.283552			−0.141776	+	0.989899i	$0.545281\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	4.00000	+	6.92820i	0.280745	+	0.486265i
$204$		0			0
$205$	−2.00000	+	3.46410i	−0.139686	+	0.241943i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−8.00000	+	13.8564i	−0.553372	+	0.958468i
$210$		0			0
$211$	10.0000	+	17.3205i	0.688428	+	1.19239i	0.972346	+	0.233544i	$0.0750324\pi$
$211$	10.0000	+	17.3205i	0.688428	+	1.19239i	−0.283918	+	0.958849i	$0.591634\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	8.00000			0.545595
$216$		0			0
$217$	−16.0000			−1.08615
$218$		0			0
$219$		0			0
$220$		0			0
$221$	6.00000	+	10.3923i	0.403604	+	0.699062i
$222$		0			0
$223$	−2.00000	+	3.46410i	−0.133930	+	0.231973i	−0.925188	−	0.379509i	$-0.876093\pi$
$223$	−2.00000	+	3.46410i	−0.133930	+	0.231973i	0.791258	+	0.611482i	$0.209426\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−6.00000	+	10.3923i	−0.398234	+	0.689761i	−0.993508	−	0.113761i	$-0.963710\pi$
$227$	−6.00000	+	10.3923i	−0.398234	+	0.689761i	0.595274	+	0.803523i	$0.297043\pi$
$228$		0			0
$229$	5.00000	+	8.66025i	0.330409	+	0.572286i	0.982592	−	0.185776i	$-0.0594799\pi$
$229$	5.00000	+	8.66025i	0.330409	+	0.572286i	−0.652183	+	0.758062i	$0.726147\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	6.00000			0.393073			0.196537	−	0.980497i	$-0.437031\pi$
$233$	6.00000			0.393073			0.196537	+	0.980497i	$0.437031\pi$
$234$		0			0
$235$	−16.0000			−1.04372
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$239$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$240$		0			0
$241$	7.00000	−	12.1244i	0.450910	−	0.780998i	−0.547533	−	0.836784i	$-0.684433\pi$
$241$	7.00000	−	12.1244i	0.450910	−	0.780998i	0.998443	+	0.0557856i	$0.0177663\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	9.00000	−	15.5885i	0.574989	−	0.995910i
$246$		0			0
$247$	4.00000	+	6.92820i	0.254514	+	0.440831i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−12.0000			−0.757433			−0.378717	−	0.925513i	$-0.623635\pi$
$251$	−12.0000			−0.757433			−0.378717	+	0.925513i	$0.623635\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$	1.00000	+	1.73205i	0.0623783	+	0.108042i	0.895528	−	0.445005i	$-0.146798\pi$
$257$	1.00000	+	1.73205i	0.0623783	+	0.108042i	−0.833150	+	0.553047i	$0.813465\pi$
$258$		0			0
$259$	4.00000	−	6.92820i	0.248548	−	0.430498i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	12.0000	−	20.7846i	0.739952	−	1.28163i	−0.212565	−	0.977147i	$-0.568182\pi$
$263$	12.0000	−	20.7846i	0.739952	−	1.28163i	0.952517	−	0.304487i	$-0.0984850\pi$
$264$		0			0
$265$	−10.0000	−	17.3205i	−0.614295	−	1.06399i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−18.0000			−1.09748			−0.548740	−	0.835993i	$-0.684892\pi$
$269$	−18.0000			−1.09748			−0.548740	+	0.835993i	$0.684892\pi$
$270$		0			0
$271$	−12.0000			−0.728948			−0.364474	−	0.931214i	$-0.618751\pi$
$271$	−12.0000			−0.728948			−0.364474	+	0.931214i	$0.618751\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	2.00000	+	3.46410i	0.120605	+	0.208893i
$276$		0			0
$277$	−11.0000	+	19.0526i	−0.660926	+	1.14476i	0.319447	+	0.947604i	$0.396503\pi$
$277$	−11.0000	+	19.0526i	−0.660926	+	1.14476i	−0.980373	+	0.197153i	$0.936830\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−3.00000	+	5.19615i	−0.178965	+	0.309976i	−0.941526	−	0.336939i	$-0.890608\pi$
$281$	−3.00000	+	5.19615i	−0.178965	+	0.309976i	0.762561	+	0.646916i	$0.223942\pi$
$282$		0			0
$283$	14.0000	+	24.2487i	0.832214	+	1.44144i	0.896279	+	0.443491i	$0.146260\pi$
$283$	14.0000	+	24.2487i	0.832214	+	1.44144i	−0.0640654	+	0.997946i	$0.520407\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−8.00000			−0.472225
$288$		0			0
$289$	19.0000			1.11765
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−3.00000	−	5.19615i	−0.175262	−	0.303562i	0.764990	−	0.644042i	$-0.222744\pi$
$293$	−3.00000	−	5.19615i	−0.175262	−	0.303562i	−0.940252	+	0.340480i	$0.889411\pi$
$294$		0			0
$295$	−4.00000	+	6.92820i	−0.232889	+	0.403376i
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$	8.00000	+	13.8564i	0.461112	+	0.798670i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−12.0000			−0.687118
$306$		0			0
$307$	12.0000			0.684876			0.342438	−	0.939540i	$-0.388747\pi$
$307$	12.0000			0.684876			0.342438	+	0.939540i	$0.388747\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	12.0000	+	20.7846i	0.680458	+	1.17859i	0.974841	+	0.222900i	$0.0715523\pi$
$311$	12.0000	+	20.7846i	0.680458	+	1.17859i	−0.294384	+	0.955687i	$0.595114\pi$
$312$		0			0
$313$	−5.00000	+	8.66025i	−0.282617	+	0.489506i	−0.972028	−	0.234863i	$-0.924536\pi$
$313$	−5.00000	+	8.66025i	−0.282617	+	0.489506i	0.689412	+	0.724370i	$0.257869\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	9.00000	−	15.5885i	0.505490	−	0.875535i	−0.494489	−	0.869184i	$-0.664645\pi$
$317$	9.00000	−	15.5885i	0.505490	−	0.875535i	0.999980	−	0.00635137i	$-0.00202172\pi$
$318$		0			0
$319$	4.00000	+	6.92820i	0.223957	+	0.387905i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	24.0000			1.33540
$324$		0			0
$325$	2.00000			0.110940
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−16.0000	−	27.7128i	−0.882109	−	1.52786i
$330$		0			0
$331$	14.0000	−	24.2487i	0.769510	−	1.33283i	−0.168320	−	0.985732i	$-0.553834\pi$
$331$	14.0000	−	24.2487i	0.769510	−	1.33283i	0.937829	−	0.347097i	$-0.112833\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−4.00000	+	6.92820i	−0.218543	+	0.378528i
$336$		0			0
$337$	−1.00000	−	1.73205i	−0.0544735	−	0.0943508i	0.837503	−	0.546433i	$-0.184015\pi$
$337$	−1.00000	−	1.73205i	−0.0544735	−	0.0943508i	−0.891976	+	0.452082i	$0.850681\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−16.0000			−0.866449
$342$		0			0
$343$	8.00000			0.431959
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−10.0000	−	17.3205i	−0.536828	−	0.929814i	−0.999072	−	0.0430610i	$-0.986289\pi$
$347$	−10.0000	−	17.3205i	−0.536828	−	0.929814i	0.462244	−	0.886753i	$-0.347044\pi$
$348$		0			0
$349$	13.0000	−	22.5167i	0.695874	−	1.20529i	−0.274011	−	0.961727i	$-0.588351\pi$
$349$	13.0000	−	22.5167i	0.695874	−	1.20529i	0.969885	−	0.243563i	$-0.0783162\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−7.00000	+	12.1244i	−0.372572	+	0.645314i	−0.989960	−	0.141344i	$-0.954858\pi$
$353$	−7.00000	+	12.1244i	−0.372572	+	0.645314i	0.617388	+	0.786659i	$0.288191\pi$
$354$		0			0
$355$	−16.0000	−	27.7128i	−0.849192	−	1.47084i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	16.0000			0.844448			0.422224	−	0.906492i	$-0.361250\pi$
$359$	16.0000			0.844448			0.422224	+	0.906492i	$0.361250\pi$
$360$		0			0
$361$	−3.00000			−0.157895
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−6.00000	−	10.3923i	−0.314054	−	0.543958i
$366$		0			0
$367$	−6.00000	+	10.3923i	−0.313197	+	0.542474i	−0.979053	−	0.203607i	$-0.934733\pi$
$367$	−6.00000	+	10.3923i	−0.313197	+	0.542474i	0.665855	+	0.746081i	$0.268067\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	20.0000	−	34.6410i	1.03835	−	1.79847i
$372$		0			0
$373$	17.0000	+	29.4449i	0.880227	+	1.52460i	0.851089	+	0.525022i	$0.175943\pi$
$373$	17.0000	+	29.4449i	0.880227	+	1.52460i	0.0291379	+	0.999575i	$0.490724\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	4.00000			0.206010
$378$		0			0
$379$	28.0000			1.43826			0.719132	−	0.694874i	$-0.244540\pi$
$379$	28.0000			1.43826			0.719132	+	0.694874i	$0.244540\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−16.0000	−	27.7128i	−0.817562	−	1.41606i	−0.907474	−	0.420109i	$-0.861992\pi$
$383$	−16.0000	−	27.7128i	−0.817562	−	1.41606i	0.0899119	−	0.995950i	$-0.471341\pi$
$384$		0			0
$385$	16.0000	−	27.7128i	0.815436	−	1.41238i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−15.0000	+	25.9808i	−0.760530	+	1.31728i	0.182047	+	0.983290i	$0.441728\pi$
$389$	−15.0000	+	25.9808i	−0.760530	+	1.31728i	−0.942578	+	0.333987i	$0.891606\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$	8.00000			0.402524
$396$		0			0
$397$	22.0000			1.10415			0.552074	−	0.833795i	$-0.313837\pi$
$397$	22.0000			1.10415			0.552074	+	0.833795i	$0.313837\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	5.00000	+	8.66025i	0.249688	+	0.432472i	0.963439	−	0.267927i	$-0.0863386\pi$
$401$	5.00000	+	8.66025i	0.249688	+	0.432472i	−0.713751	+	0.700399i	$0.753005\pi$
$402$		0			0
$403$	−4.00000	+	6.92820i	−0.199254	+	0.345118i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	4.00000	−	6.92820i	0.198273	−	0.343418i
$408$		0			0
$409$	−5.00000	−	8.66025i	−0.247234	−	0.428222i	0.715523	−	0.698589i	$-0.246188\pi$
$409$	−5.00000	−	8.66025i	−0.247234	−	0.428222i	−0.962757	+	0.270367i	$0.912855\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−16.0000			−0.787309
$414$		0			0
$415$	−24.0000			−1.17811
$416$		0			0
$417$		0			0
$418$		0			0
$419$	10.0000	+	17.3205i	0.488532	+	0.846162i	0.999913	−	0.0131919i	$-0.00419923\pi$
$419$	10.0000	+	17.3205i	0.488532	+	0.846162i	−0.511381	+	0.859354i	$0.670866\pi$
$420$		0			0
$421$	−19.0000	+	32.9090i	−0.926003	+	1.60388i	−0.136064	+	0.990700i	$0.543445\pi$
$421$	−19.0000	+	32.9090i	−0.926003	+	1.60388i	−0.789940	+	0.613185i	$0.789888\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	3.00000	−	5.19615i	0.145521	−	0.252050i
$426$		0			0
$427$	−12.0000	−	20.7846i	−0.580721	−	1.00584i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	24.0000			1.15604			0.578020	−	0.816023i	$-0.303826\pi$
$431$	24.0000			1.15604			0.578020	+	0.816023i	$0.303826\pi$
$432$		0			0
$433$	−14.0000			−0.672797			−0.336399	−	0.941720i	$-0.609209\pi$
$433$	−14.0000			−0.672797			−0.336399	+	0.941720i	$0.609209\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$	−14.0000	+	24.2487i	−0.668184	+	1.15733i	0.310228	+	0.950662i	$0.399595\pi$
$439$	−14.0000	+	24.2487i	−0.668184	+	1.15733i	−0.978412	+	0.206666i	$0.933739\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	6.00000	−	10.3923i	0.285069	−	0.493753i	−0.687557	−	0.726130i	$-0.741317\pi$
$443$	6.00000	−	10.3923i	0.285069	−	0.493753i	0.972626	+	0.232377i	$0.0746503\pi$
$444$		0			0
$445$	−10.0000	−	17.3205i	−0.474045	−	0.821071i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−10.0000			−0.471929			−0.235965	−	0.971762i	$-0.575825\pi$
$449$	−10.0000			−0.471929			−0.235965	+	0.971762i	$0.575825\pi$
$450$		0			0
$451$	−8.00000			−0.376705
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−8.00000	−	13.8564i	−0.375046	−	0.649598i
$456$		0			0
$457$	−13.0000	+	22.5167i	−0.608114	+	1.05328i	0.383437	+	0.923567i	$0.374740\pi$
$457$	−13.0000	+	22.5167i	−0.608114	+	1.05328i	−0.991551	+	0.129718i	$0.958593\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−3.00000	+	5.19615i	−0.139724	+	0.242009i	−0.927392	−	0.374091i	$-0.877955\pi$
$461$	−3.00000	+	5.19615i	−0.139724	+	0.242009i	0.787668	+	0.616100i	$0.211288\pi$
$462$		0			0
$463$	−10.0000	−	17.3205i	−0.464739	−	0.804952i	0.534450	−	0.845200i	$-0.320519\pi$
$463$	−10.0000	−	17.3205i	−0.464739	−	0.804952i	−0.999190	+	0.0402476i	$0.987185\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	28.0000			1.29569			0.647843	−	0.761774i	$-0.275671\pi$
$467$	28.0000			1.29569			0.647843	+	0.761774i	$0.275671\pi$
$468$		0			0
$469$	−16.0000			−0.738811
$470$		0			0
$471$		0			0
$472$		0			0
$473$	8.00000	+	13.8564i	0.367840	+	0.637118i
$474$		0			0
$475$	2.00000	−	3.46410i	0.0917663	−	0.158944i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	12.0000	−	20.7846i	0.548294	−	0.949673i	−0.450098	−	0.892979i	$-0.648611\pi$
$479$	12.0000	−	20.7846i	0.548294	−	0.949673i	0.998392	−	0.0566937i	$-0.0180558\pi$
$480$		0			0
$481$	−2.00000	−	3.46410i	−0.0911922	−	0.157949i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	28.0000			1.27141
$486$		0			0
$487$	4.00000			0.181257			0.0906287	−	0.995885i	$-0.471112\pi$
$487$	4.00000			0.181257			0.0906287	+	0.995885i	$0.471112\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−14.0000	−	24.2487i	−0.631811	−	1.09433i	−0.987181	−	0.159603i	$-0.948978\pi$
$491$	−14.0000	−	24.2487i	−0.631811	−	1.09433i	0.355370	−	0.934726i	$-0.384355\pi$
$492$		0			0
$493$	6.00000	−	10.3923i	0.270226	−	0.468046i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	32.0000	−	55.4256i	1.43540	−	2.48618i
$498$		0			0
$499$	−6.00000	−	10.3923i	−0.268597	−	0.465223i	0.699903	−	0.714238i	$-0.253227\pi$
$499$	−6.00000	−	10.3923i	−0.268597	−	0.465223i	−0.968500	+	0.249015i	$0.919893\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0			−	1.00000i	$-0.5\pi$
$503$		0			0				1.00000i	$0.5\pi$
$504$		0			0
$505$	−12.0000			−0.533993
$506$		0			0
$507$		0			0
$508$		0			0
$509$	9.00000	+	15.5885i	0.398918	+	0.690946i	0.993593	−	0.113020i	$-0.0360525\pi$
$509$	9.00000	+	15.5885i	0.398918	+	0.690946i	−0.594675	+	0.803966i	$0.702719\pi$
$510$		0			0
$511$	12.0000	−	20.7846i	0.530849	−	0.919457i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	12.0000	−	20.7846i	0.528783	−	0.915879i
$516$		0			0
$517$	−16.0000	−	27.7128i	−0.703679	−	1.21881i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	30.0000			1.31432			0.657162	−	0.753749i	$-0.271757\pi$
$521$	30.0000			1.31432			0.657162	+	0.753749i	$0.271757\pi$
$522$		0			0
$523$	12.0000			0.524723			0.262362	−	0.964970i	$-0.415499\pi$
$523$	12.0000			0.524723			0.262362	+	0.964970i	$0.415499\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	12.0000	+	20.7846i	0.522728	+	0.905392i
$528$		0			0
$529$	11.5000	−	19.9186i	0.500000	−	0.866025i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−2.00000	+	3.46410i	−0.0866296	+	0.150047i
$534$		0			0
$535$	−4.00000	−	6.92820i	−0.172935	−	0.299532i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	36.0000			1.55063
$540$		0			0
$541$	−2.00000			−0.0859867			−0.0429934	−	0.999075i	$-0.513689\pi$
$541$	−2.00000			−0.0859867			−0.0429934	+	0.999075i	$0.513689\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	14.0000	+	24.2487i	0.599694	+	1.03870i
$546$		0			0
$547$	6.00000	−	10.3923i	0.256541	−	0.444343i	−0.708772	−	0.705438i	$-0.750750\pi$
$547$	6.00000	−	10.3923i	0.256541	−	0.444343i	0.965313	+	0.261095i	$0.0840836\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	4.00000	−	6.92820i	0.170406	−	0.295151i
$552$		0			0
$553$	8.00000	+	13.8564i	0.340195	+	0.589234i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	22.0000			0.932170			0.466085	−	0.884740i	$-0.345664\pi$
$557$	22.0000			0.932170			0.466085	+	0.884740i	$0.345664\pi$
$558$		0			0
$559$	8.00000			0.338364
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−6.00000	−	10.3923i	−0.252870	−	0.437983i	0.711445	−	0.702742i	$-0.248041\pi$
$563$	−6.00000	−	10.3923i	−0.252870	−	0.437983i	−0.964315	+	0.264758i	$0.914708\pi$
$564$		0			0
$565$	−2.00000	+	3.46410i	−0.0841406	+	0.145736i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−23.0000	+	39.8372i	−0.964210	+	1.67006i	−0.252488	+	0.967600i	$0.581249\pi$
$569$	−23.0000	+	39.8372i	−0.964210	+	1.67006i	−0.711722	+	0.702461i	$0.752085\pi$
$570$		0			0
$571$	22.0000	+	38.1051i	0.920671	+	1.59465i	0.798379	+	0.602155i	$0.205691\pi$
$571$	22.0000	+	38.1051i	0.920671	+	1.59465i	0.122292	+	0.992494i	$0.460975\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	18.0000			0.749350			0.374675	−	0.927156i	$-0.377754\pi$
$577$	18.0000			0.749350			0.374675	+	0.927156i	$0.377754\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−24.0000	−	41.5692i	−0.995688	−	1.72458i
$582$		0			0
$583$	20.0000	−	34.6410i	0.828315	−	1.43468i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	18.0000	−	31.1769i	0.742940	−	1.28681i	−0.208212	−	0.978084i	$-0.566764\pi$
$587$	18.0000	−	31.1769i	0.742940	−	1.28681i	0.951151	−	0.308725i	$-0.0999023\pi$
$588$		0			0
$589$	8.00000	+	13.8564i	0.329634	+	0.570943i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	30.0000			1.23195			0.615976	−	0.787765i	$-0.288762\pi$
$593$	30.0000			1.23195			0.615976	+	0.787765i	$0.288762\pi$
$594$		0			0
$595$	−48.0000			−1.96781
$596$		0			0
$597$		0			0
$598$		0			0
$599$	8.00000	+	13.8564i	0.326871	+	0.566157i	0.981889	−	0.189456i	$-0.0606724\pi$
$599$	8.00000	+	13.8564i	0.326871	+	0.566157i	−0.655018	+	0.755613i	$0.727339\pi$
$600$		0			0
$601$	−5.00000	+	8.66025i	−0.203954	+	0.353259i	−0.949799	−	0.312861i	$-0.898713\pi$
$601$	−5.00000	+	8.66025i	−0.203954	+	0.353259i	0.745845	+	0.666120i	$0.232046\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	5.00000	−	8.66025i	0.203279	−	0.352089i
$606$		0			0
$607$	−10.0000	−	17.3205i	−0.405887	−	0.703018i	0.588537	−	0.808470i	$-0.299704\pi$
$607$	−10.0000	−	17.3205i	−0.405887	−	0.703018i	−0.994424	+	0.105453i	$0.966371\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−16.0000			−0.647291
$612$		0			0
$613$	−18.0000			−0.727013			−0.363507	−	0.931592i	$-0.618421\pi$
$613$	−18.0000			−0.727013			−0.363507	+	0.931592i	$0.618421\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	21.0000	+	36.3731i	0.845428	+	1.46432i	0.885249	+	0.465118i	$0.153988\pi$
$617$	21.0000	+	36.3731i	0.845428	+	1.46432i	−0.0398207	+	0.999207i	$0.512679\pi$
$618$		0			0
$619$	6.00000	−	10.3923i	0.241160	−	0.417702i	−0.719885	−	0.694094i	$-0.755805\pi$
$619$	6.00000	−	10.3923i	0.241160	−	0.417702i	0.961045	+	0.276392i	$0.0891387\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	20.0000	−	34.6410i	0.801283	−	1.38786i
$624$		0			0
$625$	9.50000	+	16.4545i	0.380000	+	0.658179i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−12.0000			−0.478471
$630$		0			0
$631$	20.0000			0.796187			0.398094	−	0.917345i	$-0.369672\pi$
$631$	20.0000			0.796187			0.398094	+	0.917345i	$0.369672\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	20.0000	+	34.6410i	0.793676	+	1.37469i
$636$		0			0
$637$	9.00000	−	15.5885i	0.356593	−	0.617637i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	5.00000	−	8.66025i	0.197488	−	0.342059i	−0.750225	−	0.661182i	$-0.770055\pi$
$641$	5.00000	−	8.66025i	0.197488	−	0.342059i	0.947713	+	0.319123i	$0.103388\pi$
$642$		0			0
$643$	6.00000	+	10.3923i	0.236617	+	0.409832i	0.959741	−	0.280885i	$-0.0906280\pi$
$643$	6.00000	+	10.3923i	0.236617	+	0.409832i	−0.723124	+	0.690718i	$0.757295\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	48.0000			1.88707			0.943537	−	0.331266i	$-0.107476\pi$
$647$	48.0000			1.88707			0.943537	+	0.331266i	$0.107476\pi$
$648$		0			0
$649$	−16.0000			−0.628055
$650$		0			0
$651$		0			0
$652$		0			0
$653$	5.00000	+	8.66025i	0.195665	+	0.338902i	0.947118	−	0.320884i	$-0.103980\pi$
$653$	5.00000	+	8.66025i	0.195665	+	0.338902i	−0.751453	+	0.659786i	$0.770647\pi$
$654$		0			0
$655$	4.00000	−	6.92820i	0.156293	−	0.270707i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−18.0000	+	31.1769i	−0.701180	+	1.21448i	0.266872	+	0.963732i	$0.414010\pi$
$659$	−18.0000	+	31.1769i	−0.701180	+	1.21448i	−0.968052	+	0.250748i	$0.919323\pi$
$660$		0			0
$661$	−7.00000	−	12.1244i	−0.272268	−	0.471583i	0.697174	−	0.716902i	$-0.254441\pi$
$661$	−7.00000	−	12.1244i	−0.272268	−	0.471583i	−0.969442	+	0.245319i	$0.921107\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−32.0000			−1.24091
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−12.0000	−	20.7846i	−0.463255	−	0.802381i
$672$		0			0
$673$	−1.00000	+	1.73205i	−0.0385472	+	0.0667657i	−0.884655	−	0.466246i	$-0.845606\pi$
$673$	−1.00000	+	1.73205i	−0.0385472	+	0.0667657i	0.846108	+	0.533011i	$0.178940\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−7.00000	+	12.1244i	−0.269032	+	0.465977i	−0.968612	−	0.248577i	$-0.920037\pi$
$677$	−7.00000	+	12.1244i	−0.269032	+	0.465977i	0.699580	+	0.714554i	$0.253370\pi$
$678$		0			0
$679$	28.0000	+	48.4974i	1.07454	+	1.86116i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−12.0000			−0.459167			−0.229584	−	0.973289i	$-0.573736\pi$
$683$	−12.0000			−0.459167			−0.229584	+	0.973289i	$0.573736\pi$
$684$		0			0
$685$	36.0000			1.37549
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−10.0000	−	17.3205i	−0.380970	−	0.659859i
$690$		0			0
$691$	−26.0000	+	45.0333i	−0.989087	+	1.71315i	−0.366947	+	0.930242i	$0.619597\pi$
$691$	−26.0000	+	45.0333i	−0.989087	+	1.71315i	−0.622139	+	0.782907i	$0.713736\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	20.0000	−	34.6410i	0.758643	−	1.31401i
$696$		0			0
$697$	6.00000	+	10.3923i	0.227266	+	0.393637i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−18.0000			−0.679851			−0.339925	−	0.940452i	$-0.610402\pi$
$701$	−18.0000			−0.679851			−0.339925	+	0.940452i	$0.610402\pi$
$702$		0			0
$703$	−8.00000			−0.301726
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−12.0000	−	20.7846i	−0.451306	−	0.781686i
$708$		0			0
$709$	−11.0000	+	19.0526i	−0.413114	+	0.715534i	−0.995228	−	0.0975728i	$-0.968892\pi$
$709$	−11.0000	+	19.0526i	−0.413114	+	0.715534i	0.582115	+	0.813107i	$0.302225\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$	−8.00000	−	13.8564i	−0.299183	−	0.518200i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	8.00000			0.298350			0.149175	−	0.988811i	$-0.452338\pi$
$719$	8.00000			0.298350			0.149175	+	0.988811i	$0.452338\pi$
$720$		0			0
$721$	48.0000			1.78761
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−1.00000	−	1.73205i	−0.0371391	−	0.0643268i
$726$		0			0
$727$	6.00000	−	10.3923i	0.222528	−	0.385429i	−0.733047	−	0.680178i	$-0.761903\pi$
$727$	6.00000	−	10.3923i	0.222528	−	0.385429i	0.955575	+	0.294749i	$0.0952359\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	12.0000	−	20.7846i	0.443836	−	0.768747i
$732$		0			0
$733$	−7.00000	−	12.1244i	−0.258551	−	0.447823i	0.707303	−	0.706910i	$-0.249912\pi$
$733$	−7.00000	−	12.1244i	−0.258551	−	0.447823i	−0.965854	+	0.259087i	$0.916578\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−16.0000			−0.589368
$738$		0			0
$739$	−20.0000			−0.735712			−0.367856	−	0.929883i	$-0.619908\pi$
$739$	−20.0000			−0.735712			−0.367856	+	0.929883i	$0.619908\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−12.0000	−	20.7846i	−0.440237	−	0.762513i	0.557470	−	0.830197i	$-0.311772\pi$
$743$	−12.0000	−	20.7846i	−0.440237	−	0.762513i	−0.997707	+	0.0676840i	$0.978439\pi$
$744$		0			0
$745$	−18.0000	+	31.1769i	−0.659469	+	1.14223i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	8.00000	−	13.8564i	0.292314	−	0.506302i
$750$		0			0
$751$	2.00000	+	3.46410i	0.0729810	+	0.126407i	0.900207	−	0.435463i	$-0.143415\pi$
$751$	2.00000	+	3.46410i	0.0729810	+	0.126407i	−0.827225	+	0.561870i	$0.810082\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	24.0000			0.873449
$756$		0			0
$757$	6.00000			0.218074			0.109037	−	0.994038i	$-0.465223\pi$
$757$	6.00000			0.218074			0.109037	+	0.994038i	$0.465223\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−15.0000	−	25.9808i	−0.543750	−	0.941802i	−0.998684	−	0.0512772i	$-0.983671\pi$
$761$	−15.0000	−	25.9808i	−0.543750	−	0.941802i	0.454935	−	0.890525i	$-0.349663\pi$
$762$		0			0
$763$	−28.0000	+	48.4974i	−1.01367	+	1.75572i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−4.00000	+	6.92820i	−0.144432	+	0.250163i
$768$		0			0
$769$	7.00000	+	12.1244i	0.252426	+	0.437215i	0.964193	−	0.265200i	$-0.0854381\pi$
$769$	7.00000	+	12.1244i	0.252426	+	0.437215i	−0.711767	+	0.702416i	$0.752105\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−42.0000			−1.51064			−0.755318	−	0.655359i	$-0.772517\pi$
$773$	−42.0000			−1.51064			−0.755318	+	0.655359i	$0.772517\pi$
$774$		0			0
$775$	4.00000			0.143684
$776$		0			0
$777$		0			0
$778$		0			0
$779$	4.00000	+	6.92820i	0.143315	+	0.248229i
$780$		0			0
$781$	32.0000	−	55.4256i	1.14505	−	1.98328i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−10.0000	+	17.3205i	−0.356915	+	0.618195i
$786$		0			0
$787$	6.00000	+	10.3923i	0.213877	+	0.370446i	0.952925	−	0.303207i	$-0.0980575\pi$
$787$	6.00000	+	10.3923i	0.213877	+	0.370446i	−0.739048	+	0.673653i	$0.764724\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−8.00000			−0.284447
$792$		0			0
$793$	−12.0000			−0.426132
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−3.00000	−	5.19615i	−0.106265	−	0.184057i	0.807989	−	0.589197i	$-0.200556\pi$
$797$	−3.00000	−	5.19615i	−0.106265	−	0.184057i	−0.914255	+	0.405140i	$0.867223\pi$
$798$		0			0
$799$	−24.0000	+	41.5692i	−0.849059	+	1.47061i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	12.0000	−	20.7846i	0.423471	−	0.733473i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	14.0000			0.492214			0.246107	−	0.969243i	$-0.420849\pi$
$809$	14.0000			0.492214			0.246107	+	0.969243i	$0.420849\pi$
$810$		0			0
$811$	28.0000			0.983213			0.491606	−	0.870817i	$-0.336410\pi$
$811$	28.0000			0.983213			0.491606	+	0.870817i	$0.336410\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	4.00000	+	6.92820i	0.140114	+	0.242684i
$816$		0			0
$817$	8.00000	−	13.8564i	0.279885	−	0.484774i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	13.0000	−	22.5167i	0.453703	−	0.785837i	−0.544909	−	0.838495i	$-0.683436\pi$
$821$	13.0000	−	22.5167i	0.453703	−	0.785837i	0.998613	+	0.0526580i	$0.0167693\pi$
$822$		0			0
$823$	−2.00000	−	3.46410i	−0.0697156	−	0.120751i	0.829060	−	0.559159i	$-0.188876\pi$
$823$	−2.00000	−	3.46410i	−0.0697156	−	0.120751i	−0.898776	+	0.438408i	$0.855543\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−20.0000			−0.695468			−0.347734	−	0.937593i	$-0.613049\pi$
$827$	−20.0000			−0.695468			−0.347734	+	0.937593i	$0.613049\pi$
$828$		0			0
$829$	−34.0000			−1.18087			−0.590434	−	0.807086i	$-0.701044\pi$
$829$	−34.0000			−1.18087			−0.590434	+	0.807086i	$0.701044\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−27.0000	−	46.7654i	−0.935495	−	1.62032i
$834$		0			0
$835$	8.00000	−	13.8564i	0.276851	−	0.479521i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−16.0000	+	27.7128i	−0.552381	+	0.956753i	0.445721	+	0.895172i	$0.352947\pi$
$839$	−16.0000	+	27.7128i	−0.552381	+	0.956753i	−0.998102	+	0.0615805i	$0.980386\pi$
$840$		0			0
$841$	12.5000	+	21.6506i	0.431034	+	0.746574i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	18.0000			0.619219
$846$		0			0
$847$	20.0000			0.687208
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$	9.00000	−	15.5885i	0.308154	−	0.533739i	−0.669804	−	0.742538i	$-0.733622\pi$
$853$	9.00000	−	15.5885i	0.308154	−	0.533739i	0.977959	+	0.208799i	$0.0669554\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−7.00000	+	12.1244i	−0.239115	+	0.414160i	−0.960461	−	0.278416i	$-0.910191\pi$
$857$	−7.00000	+	12.1244i	−0.239115	+	0.414160i	0.721345	+	0.692576i	$0.243524\pi$
$858$		0			0
$859$	2.00000	+	3.46410i	0.0682391	+	0.118194i	0.898126	−	0.439738i	$-0.144929\pi$
$859$	2.00000	+	3.46410i	0.0682391	+	0.118194i	−0.829887	+	0.557931i	$0.811595\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	32.0000			1.08929			0.544646	−	0.838666i	$-0.316664\pi$
$863$	32.0000			1.08929			0.544646	+	0.838666i	$0.316664\pi$
$864$		0			0
$865$	−12.0000			−0.408012
$866$		0			0
$867$		0			0
$868$		0			0
$869$	8.00000	+	13.8564i	0.271381	+	0.470046i
$870$		0			0
$871$	−4.00000	+	6.92820i	−0.135535	+	0.234753i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−24.0000	+	41.5692i	−0.811348	+	1.40530i
$876$		0			0
$877$	−11.0000	−	19.0526i	−0.371444	−	0.643359i	0.618344	−	0.785907i	$-0.287804\pi$
$877$	−11.0000	−	19.0526i	−0.371444	−	0.643359i	−0.989788	+	0.142548i	$0.954470\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−18.0000			−0.606435			−0.303218	−	0.952921i	$-0.598061\pi$
$881$	−18.0000			−0.606435			−0.303218	+	0.952921i	$0.598061\pi$
$882$		0			0
$883$	20.0000			0.673054			0.336527	−	0.941674i	$-0.390748\pi$
$883$	20.0000			0.673054			0.336527	+	0.941674i	$0.390748\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−12.0000	−	20.7846i	−0.402921	−	0.697879i	0.591156	−	0.806557i	$-0.298672\pi$
$887$	−12.0000	−	20.7846i	−0.402921	−	0.697879i	−0.994077	+	0.108678i	$0.965338\pi$
$888$		0			0
$889$	−40.0000	+	69.2820i	−1.34156	+	2.32364i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−16.0000	+	27.7128i	−0.535420	+	0.927374i
$894$		0			0
$895$	−12.0000	−	20.7846i	−0.401116	−	0.694753i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	8.00000			0.266815
$900$		0			0
$901$	−60.0000			−1.99889
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−10.0000	−	17.3205i	−0.332411	−	0.575753i
$906$		0			0
$907$	10.0000	−	17.3205i	0.332045	−	0.575118i	−0.650868	−	0.759191i	$-0.725595\pi$
$907$	10.0000	−	17.3205i	0.332045	−	0.575118i	0.982913	+	0.184073i	$0.0589282\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$911$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$912$		0			0
$913$	−24.0000	−	41.5692i	−0.794284	−	1.37574i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	16.0000			0.528367
$918$		0			0
$919$	36.0000			1.18753			0.593765	−	0.804638i	$-0.297641\pi$
$919$	36.0000			1.18753			0.593765	+	0.804638i	$0.297641\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−16.0000	−	27.7128i	−0.526646	−	0.912178i
$924$		0			0
$925$	−1.00000	+	1.73205i	−0.0328798	+	0.0569495i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−3.00000	+	5.19615i	−0.0984268	+	0.170480i	−0.911034	−	0.412332i	$-0.864714\pi$
$929$	−3.00000	+	5.19615i	−0.0984268	+	0.170480i	0.812607	+	0.582812i	$0.198048\pi$
$930$		0			0
$931$	−18.0000	−	31.1769i	−0.589926	−	1.02178i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−48.0000			−1.56977
$936$		0			0
$937$	−22.0000			−0.718709			−0.359354	−	0.933201i	$-0.617003\pi$
$937$	−22.0000			−0.718709			−0.359354	+	0.933201i	$0.617003\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	29.0000	+	50.2295i	0.945373	+	1.63743i	0.755003	+	0.655722i	$0.227636\pi$
$941$	29.0000	+	50.2295i	0.945373	+	1.63743i	0.190370	+	0.981712i	$0.439031\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$	6.00000	−	10.3923i	0.194974	−	0.337705i	−0.751918	−	0.659256i	$-0.770871\pi$
$947$	6.00000	−	10.3923i	0.194974	−	0.337705i	0.946892	+	0.321552i	$0.104204\pi$
$948$		0			0
$949$	−6.00000	−	10.3923i	−0.194768	−	0.337348i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−18.0000			−0.583077			−0.291539	−	0.956559i	$-0.594167\pi$
$953$	−18.0000			−0.583077			−0.291539	+	0.956559i	$0.594167\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$	36.0000	+	62.3538i	1.16250	+	2.01351i
$960$		0			0
$961$	7.50000	−	12.9904i	0.241935	−	0.419045i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	18.0000	−	31.1769i	0.579441	−	1.00362i
$966$		0			0
$967$	−6.00000	−	10.3923i	−0.192947	−	0.334194i	0.753279	−	0.657702i	$-0.228471\pi$
$967$	−6.00000	−	10.3923i	−0.192947	−	0.334194i	−0.946226	+	0.323508i	$0.895138\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	52.0000			1.66876			0.834380	−	0.551190i	$-0.185826\pi$
$971$	52.0000			1.66876			0.834380	+	0.551190i	$0.185826\pi$
$972$		0			0
$973$	80.0000			2.56468
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−11.0000	−	19.0526i	−0.351921	−	0.609545i	0.634665	−	0.772787i	$-0.281138\pi$
$977$	−11.0000	−	19.0526i	−0.351921	−	0.609545i	−0.986586	+	0.163242i	$0.947805\pi$
$978$		0			0
$979$	20.0000	−	34.6410i	0.639203	−	1.10713i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−12.0000	+	20.7846i	−0.382741	+	0.662926i	−0.991453	−	0.130465i	$-0.958353\pi$
$983$	−12.0000	+	20.7846i	−0.382741	+	0.662926i	0.608712	+	0.793391i	$0.291686\pi$
$984$		0			0
$985$	22.0000	+	38.1051i	0.700978	+	1.21413i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	−52.0000			−1.65183			−0.825917	−	0.563791i	$-0.809342\pi$
$991$	−52.0000			−1.65183			−0.825917	+	0.563791i	$0.809342\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−4.00000	−	6.92820i	−0.126809	−	0.219639i
$996$		0			0
$997$	9.00000	−	15.5885i	0.285033	−	0.493691i	−0.687584	−	0.726105i	$-0.741329\pi$
$997$	9.00000	−	15.5885i	0.285033	−	0.493691i	0.972617	+	0.232413i	$0.0746622\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2592.2.i.q.865.1		2
3.2	odd	2		2592.2.i.b.865.1		2
4.3	odd	2		2592.2.i.w.865.1		2
9.2	odd	6		96.2.a.a.1.1	✓	1
9.4	even	3	inner	2592.2.i.q.1729.1		2
9.5	odd	6		2592.2.i.b.1729.1		2
9.7	even	3		288.2.a.c.1.1		1
12.11	even	2		2592.2.i.h.865.1		2
36.7	odd	6		288.2.a.b.1.1		1
36.11	even	6		96.2.a.b.1.1	yes	1
36.23	even	6		2592.2.i.h.1729.1		2
36.31	odd	6		2592.2.i.w.1729.1		2
45.2	even	12		2400.2.f.a.1249.2		2
45.7	odd	12		7200.2.f.x.6049.2		2
45.29	odd	6		2400.2.a.r.1.1		1
45.34	even	6		7200.2.a.e.1.1		1
45.38	even	12		2400.2.f.a.1249.1		2
45.43	odd	12		7200.2.f.x.6049.1		2
63.20	even	6		4704.2.a.t.1.1		1
72.11	even	6		192.2.a.a.1.1		1
72.29	odd	6		192.2.a.c.1.1		1
72.43	odd	6		576.2.a.g.1.1		1
72.61	even	6		576.2.a.h.1.1		1
144.11	even	12		768.2.d.h.385.1		2
144.29	odd	12		768.2.d.a.385.1		2
144.43	odd	12		2304.2.d.s.1153.1		2
144.61	even	12		2304.2.d.c.1153.2		2
144.83	even	12		768.2.d.h.385.2		2
144.101	odd	12		768.2.d.a.385.2		2
144.115	odd	12		2304.2.d.s.1153.2		2
144.133	even	12		2304.2.d.c.1153.1		2
180.7	even	12		7200.2.f.f.6049.1		2
180.43	even	12		7200.2.f.f.6049.2		2
180.47	odd	12		2400.2.f.r.1249.1		2
180.79	odd	6		7200.2.a.bx.1.1		1
180.83	odd	12		2400.2.f.r.1249.2		2
180.119	even	6		2400.2.a.q.1.1		1
252.83	odd	6		4704.2.a.e.1.1		1
360.29	odd	6		4800.2.a.f.1.1		1
360.83	odd	12		4800.2.f.e.3649.1		2
360.173	even	12		4800.2.f.bh.3649.2		2
360.227	odd	12		4800.2.f.e.3649.2		2
360.299	even	6		4800.2.a.co.1.1		1
360.317	even	12		4800.2.f.bh.3649.1		2
504.83	odd	6		9408.2.a.ct.1.1		1
504.461	even	6		9408.2.a.bj.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
96.2.a.a.1.1	✓	1	9.2	odd	6
96.2.a.b.1.1	yes	1	36.11	even	6
192.2.a.a.1.1		1	72.11	even	6
192.2.a.c.1.1		1	72.29	odd	6
288.2.a.b.1.1		1	36.7	odd	6
288.2.a.c.1.1		1	9.7	even	3
576.2.a.g.1.1		1	72.43	odd	6
576.2.a.h.1.1		1	72.61	even	6
768.2.d.a.385.1		2	144.29	odd	12
768.2.d.a.385.2		2	144.101	odd	12
768.2.d.h.385.1		2	144.11	even	12
768.2.d.h.385.2		2	144.83	even	12
2304.2.d.c.1153.1		2	144.133	even	12
2304.2.d.c.1153.2		2	144.61	even	12
2304.2.d.s.1153.1		2	144.43	odd	12
2304.2.d.s.1153.2		2	144.115	odd	12
2400.2.a.q.1.1		1	180.119	even	6
2400.2.a.r.1.1		1	45.29	odd	6
2400.2.f.a.1249.1		2	45.38	even	12
2400.2.f.a.1249.2		2	45.2	even	12
2400.2.f.r.1249.1		2	180.47	odd	12
2400.2.f.r.1249.2		2	180.83	odd	12
2592.2.i.b.865.1		2	3.2	odd	2
2592.2.i.b.1729.1		2	9.5	odd	6
2592.2.i.h.865.1		2	12.11	even	2
2592.2.i.h.1729.1		2	36.23	even	6
2592.2.i.q.865.1		2	1.1	even	1	trivial
2592.2.i.q.1729.1		2	9.4	even	3	inner
2592.2.i.w.865.1		2	4.3	odd	2
2592.2.i.w.1729.1		2	36.31	odd	6
4704.2.a.e.1.1		1	252.83	odd	6
4704.2.a.t.1.1		1	63.20	even	6
4800.2.a.f.1.1		1	360.29	odd	6
4800.2.a.co.1.1		1	360.299	even	6
4800.2.f.e.3649.1		2	360.83	odd	12
4800.2.f.e.3649.2		2	360.227	odd	12
4800.2.f.bh.3649.1		2	360.317	even	12
4800.2.f.bh.3649.2		2	360.173	even	12
7200.2.a.e.1.1		1	45.34	even	6
7200.2.a.bx.1.1		1	180.79	odd	6
7200.2.f.f.6049.1		2	180.7	even	12
7200.2.f.f.6049.2		2	180.43	even	12
7200.2.f.x.6049.1		2	45.43	odd	12
7200.2.f.x.6049.2		2	45.7	odd	12
9408.2.a.bj.1.1		1	504.461	even	6
9408.2.a.ct.1.1		1	504.83	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:	\( N \)	\(=\)	\( 2592 = 2^{5} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2592.i (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(1.00000 + 1.73205i) q^{5} +(-2.00000 + 3.46410i) q^{7} +O(q^{10})\)	\(q+(1.00000 + 1.73205i) q^{5} +(-2.00000 + 3.46410i) q^{7} +(-2.00000 + 3.46410i) q^{11} +(1.00000 + 1.73205i) q^{13} +6.00000 q^{17} +4.00000 q^{19} +(0.500000 - 0.866025i) q^{25} +(1.00000 - 1.73205i) q^{29} +(2.00000 + 3.46410i) q^{31} -8.00000 q^{35} -2.00000 q^{37} +(1.00000 + 1.73205i) q^{41} +(2.00000 - 3.46410i) q^{43} +(-4.00000 + 6.92820i) q^{47} +(-4.50000 - 7.79423i) q^{49} -10.0000 q^{53} -8.00000 q^{55} +(2.00000 + 3.46410i) q^{59} +(-3.00000 + 5.19615i) q^{61} +(-2.00000 + 3.46410i) q^{65} +(2.00000 + 3.46410i) q^{67} -16.0000 q^{71} -6.00000 q^{73} +(-8.00000 - 13.8564i) q^{77} +(2.00000 - 3.46410i) q^{79} +(-6.00000 + 10.3923i) q^{83} +(6.00000 + 10.3923i) q^{85} -10.0000 q^{89} -8.00000 q^{91} +(4.00000 + 6.92820i) q^{95} +(7.00000 - 12.1244i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{5} - 4 q^{7}+O(q^{10}) \) 2 * q + 2 * q^5 - 4 * q^7	\( 2 q + 2 q^{5} - 4 q^{7} - 4 q^{11} + 2 q^{13} + 12 q^{17} + 8 q^{19} + q^{25} + 2 q^{29} + 4 q^{31} - 16 q^{35} - 4 q^{37} + 2 q^{41} + 4 q^{43} - 8 q^{47} - 9 q^{49} - 20 q^{53} - 16 q^{55} + 4 q^{59} - 6 q^{61} - 4 q^{65} + 4 q^{67} - 32 q^{71} - 12 q^{73} - 16 q^{77} + 4 q^{79} - 12 q^{83} + 12 q^{85} - 20 q^{89} - 16 q^{91} + 8 q^{95} + 14 q^{97}+O(q^{100}) \) 2 * q + 2 * q^5 - 4 * q^7 - 4 * q^11 + 2 * q^13 + 12 * q^17 + 8 * q^19 + q^25 + 2 * q^29 + 4 * q^31 - 16 * q^35 - 4 * q^37 + 2 * q^41 + 4 * q^43 - 8 * q^47 - 9 * q^49 - 20 * q^53 - 16 * q^55 + 4 * q^59 - 6 * q^61 - 4 * q^65 + 4 * q^67 - 32 * q^71 - 12 * q^73 - 16 * q^77 + 4 * q^79 - 12 * q^83 + 12 * q^85 - 20 * q^89 - 16 * q^91 + 8 * q^95 + 14 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more