LMFDB - Embedded newform 1368.2.s.a.577.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1368,2,Mod(505,1368)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1368.505"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1368, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 0, 4])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1368 = 2^{3} \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1368.s (of order $3$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,-4,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	no
Analytic conductor:	$10.9235349965$
Analytic rank:	$1$
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:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 152)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			577.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	1368.577
Dual form			1368.2.s.a.505.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.00000 - 3.46410i) q^{5} -3.00000 q^{11} +(-1.00000 + 1.73205i) q^{13} +(1.00000 + 1.73205i) q^{17} +(0.500000 - 4.33013i) q^{19} +(3.00000 - 5.19615i) q^{23} +(-5.50000 + 9.52628i) q^{25} +(-2.00000 + 3.46410i) q^{29} -10.0000 q^{31} +2.00000 q^{37} +(4.50000 + 7.79423i) q^{41} +(2.00000 + 3.46410i) q^{43} +(-6.00000 + 10.3923i) q^{47} -7.00000 q^{49} +(-1.00000 + 1.73205i) q^{53} +(6.00000 + 10.3923i) q^{55} +(-0.500000 - 0.866025i) q^{59} +(4.00000 - 6.92820i) q^{61} +8.00000 q^{65} +(-4.50000 + 7.79423i) q^{67} +(-3.00000 - 5.19615i) q^{71} +(4.50000 + 7.79423i) q^{73} +(2.00000 + 3.46410i) q^{79} +5.00000 q^{83} +(4.00000 - 6.92820i) q^{85} +(-9.00000 + 15.5885i) q^{89} +(-16.0000 + 6.92820i) q^{95} +(-0.500000 - 0.866025i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{5} - 6 q^{11} - 2 q^{13} + 2 q^{17} + q^{19} + 6 q^{23} - 11 q^{25} - 4 q^{29} - 20 q^{31} + 4 q^{37} + 9 q^{41} + 4 q^{43} - 12 q^{47} - 14 q^{49} - 2 q^{53} + 12 q^{55} - q^{59} + 8 q^{61}+ \cdots - q^{97}+O(q^{100}) $ 2 * q - 4 * q^5 - 6 * q^11 - 2 * q^13 + 2 * q^17 + q^19 + 6 * q^23 - 11 * q^25 - 4 * q^29 - 20 * q^31 + 4 * q^37 + 9 * q^41 + 4 * q^43 - 12 * q^47 - 14 * q^49 - 2 * q^53 + 12 * q^55 - q^59 + 8 * q^61 + 16 * q^65 - 9 * q^67 - 6 * q^71 + 9 * q^73 + 4 * q^79 + 10 * q^83 + 8 * q^85 - 18 * q^89 - 32 * q^95 - q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1368\mathbb{Z}\right)^\times$.

$n$	$343$	$685$	$1009$	$1217$
$\chi(n)$	$1$	$1$	$e\left(\frac{1}{3}\right)$	$1$

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
$5$	−2.00000	−	3.46410i	−0.894427	−	1.54919i	−0.834512	−	0.550990i	$-0.814250\pi$
$5$	−2.00000	−	3.46410i	−0.894427	−	1.54919i	−0.0599153	−	0.998203i	$-0.519083\pi$
$6$		0			0
$7$		0			0			−	1.00000i	$-0.5\pi$
$7$		0			0				1.00000i	$0.5\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−3.00000			−0.904534			−0.452267	−	0.891883i	$-0.649385\pi$
$11$	−3.00000			−0.904534			−0.452267	+	0.891883i	$0.649385\pi$
$12$		0			0
$13$	−1.00000	+	1.73205i	−0.277350	+	0.480384i	−0.970725	−	0.240192i	$-0.922790\pi$
$13$	−1.00000	+	1.73205i	−0.277350	+	0.480384i	0.693375	+	0.720577i	$0.256123\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	1.00000	+	1.73205i	0.242536	+	0.420084i	0.961436	−	0.275029i	$-0.0886875\pi$
$17$	1.00000	+	1.73205i	0.242536	+	0.420084i	−0.718900	+	0.695113i	$0.755354\pi$
$18$		0			0
$19$	0.500000	−	4.33013i	0.114708	−	0.993399i
$19$	0.500000	−	4.33013i	0.114708	−	0.993399i
$20$		0			0
$21$		0			0
$22$		0			0
$23$	3.00000	−	5.19615i	0.625543	−	1.08347i	−0.362892	−	0.931831i	$-0.618211\pi$
$23$	3.00000	−	5.19615i	0.625543	−	1.08347i	0.988436	−	0.151642i	$-0.0484560\pi$
$24$		0			0
$25$	−5.50000	+	9.52628i	−1.10000	+	1.90526i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−2.00000	+	3.46410i	−0.371391	+	0.643268i	−0.989780	−	0.142605i	$-0.954452\pi$
$29$	−2.00000	+	3.46410i	−0.371391	+	0.643268i	0.618389	+	0.785872i	$0.287786\pi$
$30$		0			0
$31$	−10.0000			−1.79605			−0.898027	−	0.439941i	$-0.854999\pi$
$31$	−10.0000			−1.79605			−0.898027	+	0.439941i	$0.854999\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	2.00000			0.328798			0.164399	−	0.986394i	$-0.447432\pi$
$37$	2.00000			0.328798			0.164399	+	0.986394i	$0.447432\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	4.50000	+	7.79423i	0.702782	+	1.21725i	0.967486	+	0.252924i	$0.0813924\pi$
$41$	4.50000	+	7.79423i	0.702782	+	1.21725i	−0.264704	+	0.964330i	$0.585274\pi$
$42$		0			0
$43$	2.00000	+	3.46410i	0.304997	+	0.528271i	0.977261	−	0.212041i	$-0.0680112\pi$
$43$	2.00000	+	3.46410i	0.304997	+	0.528271i	−0.672264	+	0.740312i	$0.734678\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−6.00000	+	10.3923i	−0.875190	+	1.51587i	−0.0186297	+	0.999826i	$0.505930\pi$
$47$	−6.00000	+	10.3923i	−0.875190	+	1.51587i	−0.856560	+	0.516047i	$0.827403\pi$
$48$		0			0
$49$	−7.00000			−1.00000
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−1.00000	+	1.73205i	−0.137361	+	0.237915i	−0.926497	−	0.376303i	$-0.877195\pi$
$53$	−1.00000	+	1.73205i	−0.137361	+	0.237915i	0.789136	+	0.614218i	$0.210529\pi$
$54$		0			0
$55$	6.00000	+	10.3923i	0.809040	+	1.40130i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−0.500000	−	0.866025i	−0.0650945	−	0.112747i	0.831641	−	0.555313i	$-0.187402\pi$
$59$	−0.500000	−	0.866025i	−0.0650945	−	0.112747i	−0.896736	+	0.442566i	$0.854068\pi$
$60$		0			0
$61$	4.00000	−	6.92820i	0.512148	−	0.887066i	−0.487753	−	0.872982i	$-0.662183\pi$
$61$	4.00000	−	6.92820i	0.512148	−	0.887066i	0.999901	−	0.0140840i	$-0.00448323\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	8.00000			0.992278
$66$		0			0
$67$	−4.50000	+	7.79423i	−0.549762	+	0.952217i	0.448528	+	0.893769i	$0.351948\pi$
$67$	−4.50000	+	7.79423i	−0.549762	+	0.952217i	−0.998290	+	0.0584478i	$0.981385\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−3.00000	−	5.19615i	−0.356034	−	0.616670i	0.631260	−	0.775571i	$-0.282538\pi$
$71$	−3.00000	−	5.19615i	−0.356034	−	0.616670i	−0.987294	+	0.158901i	$0.949205\pi$
$72$		0			0
$73$	4.50000	+	7.79423i	0.526685	+	0.912245i	0.999517	+	0.0310925i	$0.00989865\pi$
$73$	4.50000	+	7.79423i	0.526685	+	0.912245i	−0.472831	+	0.881153i	$0.656768\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	2.00000	+	3.46410i	0.225018	+	0.389742i	0.956325	−	0.292306i	$-0.0944227\pi$
$79$	2.00000	+	3.46410i	0.225018	+	0.389742i	−0.731307	+	0.682048i	$0.761089\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	5.00000			0.548821			0.274411	−	0.961613i	$-0.411517\pi$
$83$	5.00000			0.548821			0.274411	+	0.961613i	$0.411517\pi$
$84$		0			0
$85$	4.00000	−	6.92820i	0.433861	−	0.751469i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−9.00000	+	15.5885i	−0.953998	+	1.65237i	−0.217354	+	0.976093i	$0.569742\pi$
$89$	−9.00000	+	15.5885i	−0.953998	+	1.65237i	−0.736644	+	0.676280i	$0.763591\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−16.0000	+	6.92820i	−1.64157	+	0.710819i
$96$		0			0
$97$	−0.500000	−	0.866025i	−0.0507673	−	0.0879316i	0.839525	−	0.543321i	$-0.182833\pi$
$97$	−0.500000	−	0.866025i	−0.0507673	−	0.0879316i	−0.890292	+	0.455389i	$0.849500\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	6.00000	−	10.3923i	0.597022	−	1.03407i	−0.396236	−	0.918149i	$-0.629684\pi$
$101$	6.00000	−	10.3923i	0.597022	−	1.03407i	0.993258	−	0.115924i	$-0.0369830\pi$
$102$		0			0
$103$	−14.0000			−1.37946			−0.689730	−	0.724066i	$-0.742271\pi$
$103$	−14.0000			−1.37946			−0.689730	+	0.724066i	$0.742271\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−16.0000			−1.54678			−0.773389	−	0.633932i	$-0.781440\pi$
$107$	−16.0000			−1.54678			−0.773389	+	0.633932i	$0.781440\pi$
$108$		0			0
$109$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$109$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	1.00000			0.0940721			0.0470360	−	0.998893i	$-0.485022\pi$
$113$	1.00000			0.0940721			0.0470360	+	0.998893i	$0.485022\pi$
$114$		0			0
$115$	−24.0000			−2.23801
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−2.00000			−0.181818
$122$		0			0
$123$		0			0
$124$		0			0
$125$	24.0000			2.14663
$126$		0			0
$127$	3.00000	−	5.19615i	0.266207	−	0.461084i	−0.701672	−	0.712500i	$-0.747563\pi$
$127$	3.00000	−	5.19615i	0.266207	−	0.461084i	0.967879	+	0.251416i	$0.0808962\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−7.50000	−	12.9904i	−0.655278	−	1.13497i	−0.981824	−	0.189794i	$-0.939218\pi$
$131$	−7.50000	−	12.9904i	−0.655278	−	1.13497i	0.326546	−	0.945181i	$-0.394115\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	4.50000	−	7.79423i	0.384461	−	0.665906i	−0.607233	−	0.794524i	$-0.707721\pi$
$137$	4.50000	−	7.79423i	0.384461	−	0.665906i	0.991694	+	0.128618i	$0.0410540\pi$
$138$		0			0
$139$	6.50000	−	11.2583i	0.551323	−	0.954919i	−0.446857	−	0.894606i	$-0.647457\pi$
$139$	6.50000	−	11.2583i	0.551323	−	0.954919i	0.998179	−	0.0603135i	$-0.0192101\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	3.00000	−	5.19615i	0.250873	−	0.434524i
$144$		0			0
$145$	16.0000			1.32873
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−5.00000	−	8.66025i	−0.409616	−	0.709476i	0.585231	−	0.810867i	$-0.301004\pi$
$149$	−5.00000	−	8.66025i	−0.409616	−	0.709476i	−0.994847	+	0.101391i	$0.967671\pi$
$150$		0			0
$151$	2.00000			0.162758			0.0813788	−	0.996683i	$-0.474068\pi$
$151$	2.00000			0.162758			0.0813788	+	0.996683i	$0.474068\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	20.0000	+	34.6410i	1.60644	+	2.78243i
$156$		0			0
$157$	−4.00000	−	6.92820i	−0.319235	−	0.552931i	0.661094	−	0.750303i	$-0.270093\pi$
$157$	−4.00000	−	6.92820i	−0.319235	−	0.552931i	−0.980329	+	0.197372i	$0.936759\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−11.0000			−0.861586			−0.430793	−	0.902451i	$-0.641766\pi$
$163$	−11.0000			−0.861586			−0.430793	+	0.902451i	$0.641766\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	8.00000	−	13.8564i	0.619059	−	1.07224i	−0.370599	−	0.928793i	$-0.620848\pi$
$167$	8.00000	−	13.8564i	0.619059	−	1.07224i	0.989658	−	0.143448i	$-0.0458190\pi$
$168$		0			0
$169$	4.50000	+	7.79423i	0.346154	+	0.599556i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−13.0000	−	22.5167i	−0.988372	−	1.71191i	−0.625871	−	0.779926i	$-0.715256\pi$
$173$	−13.0000	−	22.5167i	−0.988372	−	1.71191i	−0.362500	−	0.931984i	$-0.618077\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−9.00000			−0.672692			−0.336346	−	0.941739i	$-0.609191\pi$
$179$	−9.00000			−0.672692			−0.336346	+	0.941739i	$0.609191\pi$
$180$		0			0
$181$	−7.00000	+	12.1244i	−0.520306	+	0.901196i	0.479415	+	0.877588i	$0.340849\pi$
$181$	−7.00000	+	12.1244i	−0.520306	+	0.901196i	−0.999721	+	0.0236082i	$0.992485\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−4.00000	−	6.92820i	−0.294086	−	0.509372i
$186$		0			0
$187$	−3.00000	−	5.19615i	−0.219382	−	0.379980i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−12.0000			−0.868290			−0.434145	−	0.900843i	$-0.642949\pi$
$191$	−12.0000			−0.868290			−0.434145	+	0.900843i	$0.642949\pi$
$192$		0			0
$193$	3.00000	+	5.19615i	0.215945	+	0.374027i	0.953564	−	0.301189i	$-0.0973836\pi$
$193$	3.00000	+	5.19615i	0.215945	+	0.374027i	−0.737620	+	0.675216i	$0.764050\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	18.0000			1.28245			0.641223	−	0.767354i	$-0.278427\pi$
$197$	18.0000			1.28245			0.641223	+	0.767354i	$0.278427\pi$
$198$		0			0
$199$	9.00000	−	15.5885i	0.637993	−	1.10504i	−0.347879	−	0.937539i	$-0.613098\pi$
$199$	9.00000	−	15.5885i	0.637993	−	1.10504i	0.985873	−	0.167497i	$-0.0535685\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$	18.0000	−	31.1769i	1.25717	−	2.17749i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−1.50000	+	12.9904i	−0.103757	+	0.898563i
$210$		0			0
$211$	6.00000	+	10.3923i	0.413057	+	0.715436i	0.995222	−	0.0976347i	$-0.0311277\pi$
$211$	6.00000	+	10.3923i	0.413057	+	0.715436i	−0.582165	+	0.813070i	$0.697794\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	8.00000	−	13.8564i	0.545595	−	0.944999i
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−4.00000			−0.269069
$222$		0			0
$223$	−5.00000	−	8.66025i	−0.334825	−	0.579934i	0.648626	−	0.761107i	$-0.275344\pi$
$223$	−5.00000	−	8.66025i	−0.334825	−	0.579934i	−0.983451	+	0.181173i	$0.942010\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−19.0000			−1.26107			−0.630537	−	0.776159i	$-0.717165\pi$
$227$	−19.0000			−1.26107			−0.630537	+	0.776159i	$0.717165\pi$
$228$		0			0
$229$	−8.00000			−0.528655			−0.264327	−	0.964433i	$-0.585150\pi$
$229$	−8.00000			−0.528655			−0.264327	+	0.964433i	$0.585150\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	5.50000	+	9.52628i	0.360317	+	0.624087i	0.988013	−	0.154371i	$-0.0493352\pi$
$233$	5.50000	+	9.52628i	0.360317	+	0.624087i	−0.627696	+	0.778459i	$0.716002\pi$
$234$		0			0
$235$	48.0000			3.13117
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−12.0000			−0.776215			−0.388108	−	0.921614i	$-0.626871\pi$
$239$	−12.0000			−0.776215			−0.388108	+	0.921614i	$0.626871\pi$
$240$		0			0
$241$	−10.5000	+	18.1865i	−0.676364	+	1.17150i	0.299704	+	0.954032i	$0.403112\pi$
$241$	−10.5000	+	18.1865i	−0.676364	+	1.17150i	−0.976068	+	0.217465i	$0.930221\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	14.0000	+	24.2487i	0.894427	+	1.54919i
$246$		0			0
$247$	7.00000	+	5.19615i	0.445399	+	0.330623i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	2.50000	−	4.33013i	0.157799	−	0.273315i	−0.776276	−	0.630393i	$-0.782894\pi$
$251$	2.50000	−	4.33013i	0.157799	−	0.273315i	0.934075	+	0.357078i	$0.116227\pi$
$252$		0			0
$253$	−9.00000	+	15.5885i	−0.565825	+	0.980038i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	1.50000	−	2.59808i	0.0935674	−	0.162064i	−0.815442	−	0.578838i	$-0.803506\pi$
$257$	1.50000	−	2.59808i	0.0935674	−	0.162064i	0.909010	+	0.416775i	$0.136840\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	8.00000	+	13.8564i	0.493301	+	0.854423i	0.999970	−	0.00771799i	$-0.00245674\pi$
$263$	8.00000	+	13.8564i	0.493301	+	0.854423i	−0.506669	+	0.862141i	$0.669123\pi$
$264$		0			0
$265$	8.00000			0.491436
$266$		0			0
$267$		0			0
$268$		0			0
$269$	2.00000	+	3.46410i	0.121942	+	0.211210i	0.920534	−	0.390664i	$-0.127754\pi$
$269$	2.00000	+	3.46410i	0.121942	+	0.211210i	−0.798591	+	0.601874i	$0.794421\pi$
$270$		0			0
$271$	10.0000	+	17.3205i	0.607457	+	1.05215i	0.991658	+	0.128897i	$0.0411435\pi$
$271$	10.0000	+	17.3205i	0.607457	+	1.05215i	−0.384201	+	0.923249i	$0.625523\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	16.5000	−	28.5788i	0.994987	−	1.72337i
$276$		0			0
$277$	12.0000			0.721010			0.360505	−	0.932757i	$-0.382604\pi$
$277$	12.0000			0.721010			0.360505	+	0.932757i	$0.382604\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−6.50000	+	11.2583i	−0.387757	+	0.671616i	−0.992148	−	0.125073i	$-0.960084\pi$
$281$	−6.50000	+	11.2583i	−0.387757	+	0.671616i	0.604390	+	0.796689i	$0.293417\pi$
$282$		0			0
$283$	−6.50000	−	11.2583i	−0.386385	−	0.669238i	0.605575	−	0.795788i	$-0.292943\pi$
$283$	−6.50000	−	11.2583i	−0.386385	−	0.669238i	−0.991960	+	0.126550i	$0.959610\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	6.50000	−	11.2583i	0.382353	−	0.662255i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	4.00000			0.233682			0.116841	−	0.993151i	$-0.462723\pi$
$293$	4.00000			0.233682			0.116841	+	0.993151i	$0.462723\pi$
$294$		0			0
$295$	−2.00000	+	3.46410i	−0.116445	+	0.201688i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	6.00000	+	10.3923i	0.346989	+	0.601003i
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−32.0000			−1.83231
$306$		0			0
$307$	−12.5000	−	21.6506i	−0.713413	−	1.23567i	−0.963569	−	0.267461i	$-0.913815\pi$
$307$	−12.5000	−	21.6506i	−0.713413	−	1.23567i	0.250156	−	0.968206i	$-0.419518\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	2.00000			0.113410			0.0567048	−	0.998391i	$-0.481941\pi$
$311$	2.00000			0.113410			0.0567048	+	0.998391i	$0.481941\pi$
$312$		0			0
$313$	9.50000	−	16.4545i	0.536972	−	0.930062i	−0.462093	−	0.886831i	$-0.652902\pi$
$313$	9.50000	−	16.4545i	0.536972	−	0.930062i	0.999065	−	0.0432311i	$-0.0137652\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	3.00000	−	5.19615i	0.168497	−	0.291845i	−0.769395	−	0.638774i	$-0.779442\pi$
$317$	3.00000	−	5.19615i	0.168497	−	0.291845i	0.937892	+	0.346929i	$0.112775\pi$
$318$		0			0
$319$	6.00000	−	10.3923i	0.335936	−	0.581857i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	8.00000	−	3.46410i	0.445132	−	0.192748i
$324$		0			0
$325$	−11.0000	−	19.0526i	−0.610170	−	1.05685i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	13.0000			0.714545			0.357272	−	0.934000i	$-0.383707\pi$
$331$	13.0000			0.714545			0.357272	+	0.934000i	$0.383707\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	36.0000			1.96689
$336$		0			0
$337$	−1.50000	−	2.59808i	−0.0817102	−	0.141526i	0.822274	−	0.569091i	$-0.192705\pi$
$337$	−1.50000	−	2.59808i	−0.0817102	−	0.141526i	−0.903985	+	0.427565i	$0.859372\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	30.0000			1.62459
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−4.50000	−	7.79423i	−0.241573	−	0.418416i	0.719590	−	0.694399i	$-0.244330\pi$
$347$	−4.50000	−	7.79423i	−0.241573	−	0.418416i	−0.961162	+	0.275983i	$0.910997\pi$
$348$		0			0
$349$	−28.0000			−1.49881			−0.749403	−	0.662114i	$-0.769659\pi$
$349$	−28.0000			−1.49881			−0.749403	+	0.662114i	$0.769659\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−11.0000			−0.585471			−0.292735	−	0.956193i	$-0.594566\pi$
$353$	−11.0000			−0.585471			−0.292735	+	0.956193i	$0.594566\pi$
$354$		0			0
$355$	−12.0000	+	20.7846i	−0.636894	+	1.10313i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−1.00000	−	1.73205i	−0.0527780	−	0.0914141i	0.838429	−	0.545010i	$-0.183474\pi$
$359$	−1.00000	−	1.73205i	−0.0527780	−	0.0914141i	−0.891207	+	0.453596i	$0.850141\pi$
$360$		0			0
$361$	−18.5000	−	4.33013i	−0.973684	−	0.227901i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	18.0000	−	31.1769i	0.942163	−	1.63187i
$366$		0			0
$367$	13.0000	−	22.5167i	0.678594	−	1.17536i	−0.296810	−	0.954937i	$-0.595923\pi$
$367$	13.0000	−	22.5167i	0.678594	−	1.17536i	0.975404	−	0.220423i	$-0.0707439\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	4.00000			0.207112			0.103556	−	0.994624i	$-0.466978\pi$
$373$	4.00000			0.207112			0.103556	+	0.994624i	$0.466978\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−4.00000	−	6.92820i	−0.206010	−	0.356821i
$378$		0			0
$379$	−12.0000			−0.616399			−0.308199	−	0.951322i	$-0.599726\pi$
$379$	−12.0000			−0.616399			−0.308199	+	0.951322i	$0.599726\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−4.00000	−	6.92820i	−0.204390	−	0.354015i	0.745548	−	0.666452i	$-0.232188\pi$
$383$	−4.00000	−	6.92820i	−0.204390	−	0.354015i	−0.949938	+	0.312437i	$0.898855\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	2.00000	−	3.46410i	0.101404	−	0.175637i	−0.810859	−	0.585241i	$-0.801000\pi$
$389$	2.00000	−	3.46410i	0.101404	−	0.175637i	0.912263	+	0.409604i	$0.134333\pi$
$390$		0			0
$391$	12.0000			0.606866
$392$		0			0
$393$		0			0
$394$		0			0
$395$	8.00000	−	13.8564i	0.402524	−	0.697191i
$396$		0			0
$397$	7.00000	+	12.1244i	0.351320	+	0.608504i	0.986481	−	0.163876i	$-0.0523996\pi$
$397$	7.00000	+	12.1244i	0.351320	+	0.608504i	−0.635161	+	0.772380i	$0.719066\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−1.50000	−	2.59808i	−0.0749064	−	0.129742i	0.826139	−	0.563466i	$-0.190532\pi$
$401$	−1.50000	−	2.59808i	−0.0749064	−	0.129742i	−0.901046	+	0.433724i	$0.857199\pi$
$402$		0			0
$403$	10.0000	−	17.3205i	0.498135	−	0.862796i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−6.00000			−0.297409
$408$		0			0
$409$	17.5000	−	30.3109i	0.865319	−	1.49878i	−0.00141047	−	0.999999i	$-0.500449\pi$
$409$	17.5000	−	30.3109i	0.865319	−	1.49878i	0.866730	−	0.498778i	$-0.166218\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$	−10.0000	−	17.3205i	−0.490881	−	0.850230i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−12.0000			−0.586238			−0.293119	−	0.956076i	$-0.594693\pi$
$419$	−12.0000			−0.586238			−0.293119	+	0.956076i	$0.594693\pi$
$420$		0			0
$421$	11.0000	+	19.0526i	0.536107	+	0.928565i	0.999109	+	0.0422075i	$0.0134391\pi$
$421$	11.0000	+	19.0526i	0.536107	+	0.928565i	−0.463002	+	0.886357i	$0.653228\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−22.0000			−1.06716
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−15.0000	+	25.9808i	−0.722525	+	1.25145i	0.237460	+	0.971397i	$0.423685\pi$
$431$	−15.0000	+	25.9808i	−0.722525	+	1.25145i	−0.959985	+	0.280052i	$0.909648\pi$
$432$		0			0
$433$	−5.00000	+	8.66025i	−0.240285	+	0.416185i	−0.960795	−	0.277259i	$-0.910574\pi$
$433$	−5.00000	+	8.66025i	−0.240285	+	0.416185i	0.720511	+	0.693444i	$0.243907\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−21.0000	−	15.5885i	−1.00457	−	0.745697i
$438$		0			0
$439$	7.00000	+	12.1244i	0.334092	+	0.578664i	0.983310	−	0.181938i	$-0.0582371\pi$
$439$	7.00000	+	12.1244i	0.334092	+	0.578664i	−0.649218	+	0.760602i	$0.724904\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	0.500000	−	0.866025i	0.0237557	−	0.0411461i	−0.853903	−	0.520432i	$-0.825771\pi$
$443$	0.500000	−	0.866025i	0.0237557	−	0.0411461i	0.877659	+	0.479286i	$0.159104\pi$
$444$		0			0
$445$	72.0000			3.41313
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−33.0000			−1.55737			−0.778683	−	0.627417i	$-0.784112\pi$
$449$	−33.0000			−1.55737			−0.778683	+	0.627417i	$0.784112\pi$
$450$		0			0
$451$	−13.5000	−	23.3827i	−0.635690	−	1.10105i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−11.0000			−0.514558			−0.257279	−	0.966337i	$-0.582826\pi$
$457$	−11.0000			−0.514558			−0.257279	+	0.966337i	$0.582826\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	3.00000	+	5.19615i	0.139724	+	0.242009i	0.927392	−	0.374091i	$-0.122045\pi$
$461$	3.00000	+	5.19615i	0.139724	+	0.242009i	−0.787668	+	0.616100i	$0.788712\pi$
$462$		0			0
$463$	−34.0000			−1.58011			−0.790057	−	0.613033i	$-0.789949\pi$
$463$	−34.0000			−1.58011			−0.790057	+	0.613033i	$0.789949\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−5.00000			−0.231372			−0.115686	−	0.993286i	$-0.536907\pi$
$467$	−5.00000			−0.231372			−0.115686	+	0.993286i	$0.536907\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−6.00000	−	10.3923i	−0.275880	−	0.477839i
$474$		0			0
$475$	38.5000	+	28.5788i	1.76650	+	1.31129i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	10.0000	−	17.3205i	0.456912	−	0.791394i	−0.541884	−	0.840453i	$-0.682289\pi$
$479$	10.0000	−	17.3205i	0.456912	−	0.791394i	0.998796	+	0.0490589i	$0.0156222\pi$
$480$		0			0
$481$	−2.00000	+	3.46410i	−0.0911922	+	0.157949i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−2.00000	+	3.46410i	−0.0908153	+	0.157297i
$486$		0			0
$487$	−26.0000			−1.17817			−0.589086	−	0.808070i	$-0.700512\pi$
$487$	−26.0000			−1.17817			−0.589086	+	0.808070i	$0.700512\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	8.00000	+	13.8564i	0.361035	+	0.625331i	0.988131	−	0.153611i	$-0.0490902\pi$
$491$	8.00000	+	13.8564i	0.361035	+	0.625331i	−0.627096	+	0.778942i	$0.715757\pi$
$492$		0			0
$493$	−8.00000			−0.360302
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−3.50000	−	6.06218i	−0.156682	−	0.271380i	0.776989	−	0.629515i	$-0.216746\pi$
$499$	−3.50000	−	6.06218i	−0.156682	−	0.271380i	−0.933670	+	0.358134i	$0.883413\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−3.00000	+	5.19615i	−0.133763	+	0.231685i	−0.925124	−	0.379664i	$-0.876040\pi$
$503$	−3.00000	+	5.19615i	−0.133763	+	0.231685i	0.791361	+	0.611349i	$0.209373\pi$
$504$		0			0
$505$	−48.0000			−2.13597
$506$		0			0
$507$		0			0
$508$		0			0
$509$	8.00000	−	13.8564i	0.354594	−	0.614174i	−0.632455	−	0.774597i	$-0.717953\pi$
$509$	8.00000	−	13.8564i	0.354594	−	0.614174i	0.987048	+	0.160423i	$0.0512858\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$	28.0000	+	48.4974i	1.23383	+	2.13705i
$516$		0			0
$517$	18.0000	−	31.1769i	0.791639	−	1.37116i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−1.00000			−0.0438108			−0.0219054	−	0.999760i	$-0.506973\pi$
$521$	−1.00000			−0.0438108			−0.0219054	+	0.999760i	$0.506973\pi$
$522$		0			0
$523$	−10.0000	+	17.3205i	−0.437269	+	0.757373i	−0.997478	−	0.0709788i	$-0.977388\pi$
$523$	−10.0000	+	17.3205i	−0.437269	+	0.757373i	0.560208	+	0.828352i	$0.310721\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−10.0000	−	17.3205i	−0.435607	−	0.754493i
$528$		0			0
$529$	−6.50000	−	11.2583i	−0.282609	−	0.489493i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−18.0000			−0.779667
$534$		0			0
$535$	32.0000	+	55.4256i	1.38348	+	2.39626i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	21.0000			0.904534
$540$		0			0
$541$	−2.00000	+	3.46410i	−0.0859867	+	0.148933i	−0.905811	−	0.423681i	$-0.860738\pi$
$541$	−2.00000	+	3.46410i	−0.0859867	+	0.148933i	0.819825	+	0.572615i	$0.194071\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−14.0000	+	24.2487i	−0.598597	+	1.03680i	0.394432	+	0.918925i	$0.370941\pi$
$547$	−14.0000	+	24.2487i	−0.598597	+	1.03680i	−0.993028	+	0.117875i	$0.962392\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	14.0000	+	10.3923i	0.596420	+	0.442727i
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−20.0000	+	34.6410i	−0.847427	+	1.46779i	0.0360693	+	0.999349i	$0.488516\pi$
$557$	−20.0000	+	34.6410i	−0.847427	+	1.46779i	−0.883497	+	0.468438i	$0.844817\pi$
$558$		0			0
$559$	−8.00000			−0.338364
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−11.0000			−0.463595			−0.231797	−	0.972764i	$-0.574461\pi$
$563$	−11.0000			−0.463595			−0.231797	+	0.972764i	$0.574461\pi$
$564$		0			0
$565$	−2.00000	−	3.46410i	−0.0841406	−	0.145736i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	46.0000			1.92842			0.964210	−	0.265139i	$-0.0854179\pi$
$569$	46.0000			1.92842			0.964210	+	0.265139i	$0.0854179\pi$
$570$		0			0
$571$	17.0000			0.711428			0.355714	−	0.934595i	$-0.384238\pi$
$571$	17.0000			0.711428			0.355714	+	0.934595i	$0.384238\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	33.0000	+	57.1577i	1.37620	+	2.38364i
$576$		0			0
$577$	−37.0000			−1.54033			−0.770165	−	0.637845i	$-0.779826\pi$
$577$	−37.0000			−1.54033			−0.770165	+	0.637845i	$0.779826\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	3.00000	−	5.19615i	0.124247	−	0.215203i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	2.00000	+	3.46410i	0.0825488	+	0.142979i	0.904344	−	0.426804i	$-0.140361\pi$
$587$	2.00000	+	3.46410i	0.0825488	+	0.142979i	−0.821795	+	0.569783i	$0.807027\pi$
$588$		0			0
$589$	−5.00000	+	43.3013i	−0.206021	+	1.78420i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−10.5000	+	18.1865i	−0.431183	+	0.746831i	−0.996976	−	0.0777165i	$-0.975237\pi$
$593$	−10.5000	+	18.1865i	−0.431183	+	0.746831i	0.565792	+	0.824548i	$0.308570\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	19.0000	−	32.9090i	0.776319	−	1.34462i	−0.157731	−	0.987482i	$-0.550418\pi$
$599$	19.0000	−	32.9090i	0.776319	−	1.34462i	0.934050	−	0.357142i	$-0.116249\pi$
$600$		0			0
$601$	−13.0000			−0.530281			−0.265141	−	0.964210i	$-0.585418\pi$
$601$	−13.0000			−0.530281			−0.265141	+	0.964210i	$0.585418\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	4.00000	+	6.92820i	0.162623	+	0.281672i
$606$		0			0
$607$		0			0			−	1.00000i	$-0.5\pi$
$607$		0			0				1.00000i	$0.5\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−12.0000	−	20.7846i	−0.485468	−	0.840855i
$612$		0			0
$613$	11.0000	+	19.0526i	0.444286	+	0.769526i	0.998002	−	0.0631797i	$-0.0201241\pi$
$613$	11.0000	+	19.0526i	0.444286	+	0.769526i	−0.553716	+	0.832705i	$0.686791\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	1.50000	−	2.59808i	0.0603877	−	0.104595i	−0.834251	−	0.551385i	$-0.814100\pi$
$617$	1.50000	−	2.59808i	0.0603877	−	0.104595i	0.894639	+	0.446790i	$0.147433\pi$
$618$		0			0
$619$	−20.0000			−0.803868			−0.401934	−	0.915669i	$-0.631662\pi$
$619$	−20.0000			−0.803868			−0.401934	+	0.915669i	$0.631662\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−20.5000	−	35.5070i	−0.820000	−	1.42028i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	2.00000	+	3.46410i	0.0797452	+	0.138123i
$630$		0			0
$631$	4.00000	−	6.92820i	0.159237	−	0.275807i	−0.775356	−	0.631524i	$-0.782430\pi$
$631$	4.00000	−	6.92820i	0.159237	−	0.275807i	0.934594	+	0.355716i	$0.115763\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−24.0000			−0.952411
$636$		0			0
$637$	7.00000	−	12.1244i	0.277350	−	0.480384i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	16.5000	+	28.5788i	0.651711	+	1.12880i	0.982708	+	0.185164i	$0.0592817\pi$
$641$	16.5000	+	28.5788i	0.651711	+	1.12880i	−0.330997	+	0.943632i	$0.607385\pi$
$642$		0			0
$643$	−2.50000	−	4.33013i	−0.0985904	−	0.170764i	0.812511	−	0.582946i	$-0.198100\pi$
$643$	−2.50000	−	4.33013i	−0.0985904	−	0.170764i	−0.911101	+	0.412182i	$0.864767\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	14.0000			0.550397			0.275198	−	0.961387i	$-0.411256\pi$
$647$	14.0000			0.550397			0.275198	+	0.961387i	$0.411256\pi$
$648$		0			0
$649$	1.50000	+	2.59808i	0.0588802	+	0.101983i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−36.0000			−1.40879			−0.704394	−	0.709809i	$-0.748781\pi$
$653$	−36.0000			−1.40879			−0.704394	+	0.709809i	$0.748781\pi$
$654$		0			0
$655$	−30.0000	+	51.9615i	−1.17220	+	2.03030i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	6.00000	−	10.3923i	0.233727	−	0.404827i	−0.725175	−	0.688565i	$-0.758241\pi$
$659$	6.00000	−	10.3923i	0.233727	−	0.404827i	0.958902	+	0.283738i	$0.0915745\pi$
$660$		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
$665$		0			0
$666$		0			0
$667$	12.0000	+	20.7846i	0.464642	+	0.804783i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−12.0000	+	20.7846i	−0.463255	+	0.802381i
$672$		0			0
$673$	30.0000			1.15642			0.578208	−	0.815890i	$-0.303752\pi$
$673$	30.0000			1.15642			0.578208	+	0.815890i	$0.303752\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	2.00000			0.0768662			0.0384331	−	0.999261i	$-0.487763\pi$
$677$	2.00000			0.0768662			0.0384331	+	0.999261i	$0.487763\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	28.0000			1.07139			0.535695	−	0.844411i	$-0.320050\pi$
$683$	28.0000			1.07139			0.535695	+	0.844411i	$0.320050\pi$
$684$		0			0
$685$	−36.0000			−1.37549
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−2.00000	−	3.46410i	−0.0761939	−	0.131972i
$690$		0			0
$691$	36.0000			1.36950			0.684752	−	0.728776i	$-0.259910\pi$
$691$	36.0000			1.36950			0.684752	+	0.728776i	$0.259910\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−52.0000			−1.97247
$696$		0			0
$697$	−9.00000	+	15.5885i	−0.340899	+	0.590455i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−6.00000	−	10.3923i	−0.226617	−	0.392512i	0.730186	−	0.683248i	$-0.239433\pi$
$701$	−6.00000	−	10.3923i	−0.226617	−	0.392512i	−0.956803	+	0.290736i	$0.906100\pi$
$702$		0			0
$703$	1.00000	−	8.66025i	0.0377157	−	0.326628i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−9.00000	+	15.5885i	−0.338002	+	0.585437i	−0.984057	−	0.177854i	$-0.943084\pi$
$709$	−9.00000	+	15.5885i	−0.338002	+	0.585437i	0.646055	+	0.763291i	$0.276418\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−30.0000	+	51.9615i	−1.12351	+	1.94597i
$714$		0			0
$715$	−24.0000			−0.897549
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−17.0000	−	29.4449i	−0.633993	−	1.09811i	−0.986728	−	0.162385i	$-0.948081\pi$
$719$	−17.0000	−	29.4449i	−0.633993	−	1.09811i	0.352735	−	0.935723i	$-0.385252\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−22.0000	−	38.1051i	−0.817059	−	1.41519i
$726$		0			0
$727$	4.00000	+	6.92820i	0.148352	+	0.256953i	0.930618	−	0.365991i	$-0.119270\pi$
$727$	4.00000	+	6.92820i	0.148352	+	0.256953i	−0.782267	+	0.622944i	$0.785937\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−4.00000	+	6.92820i	−0.147945	+	0.256249i
$732$		0			0
$733$	22.0000			0.812589			0.406294	−	0.913742i	$-0.366821\pi$
$733$	22.0000			0.812589			0.406294	+	0.913742i	$0.366821\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	13.5000	−	23.3827i	0.497279	−	0.861312i
$738$		0			0
$739$	2.50000	+	4.33013i	0.0919640	+	0.159286i	0.908337	−	0.418238i	$-0.137352\pi$
$739$	2.50000	+	4.33013i	0.0919640	+	0.159286i	−0.816373	+	0.577524i	$0.804019\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−3.00000	−	5.19615i	−0.110059	−	0.190628i	0.805735	−	0.592277i	$-0.201771\pi$
$743$	−3.00000	−	5.19615i	−0.110059	−	0.190628i	−0.915794	+	0.401648i	$0.868437\pi$
$744$		0			0
$745$	−20.0000	+	34.6410i	−0.732743	+	1.26915i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−19.0000	+	32.9090i	−0.693320	+	1.20087i	0.277424	+	0.960748i	$0.410519\pi$
$751$	−19.0000	+	32.9090i	−0.693320	+	1.20087i	−0.970744	+	0.240118i	$0.922814\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−4.00000	−	6.92820i	−0.145575	−	0.252143i
$756$		0			0
$757$	−7.00000	−	12.1244i	−0.254419	−	0.440667i	0.710318	−	0.703881i	$-0.248551\pi$
$757$	−7.00000	−	12.1244i	−0.254419	−	0.440667i	−0.964738	+	0.263213i	$0.915218\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−17.0000			−0.616250			−0.308125	−	0.951346i	$-0.599701\pi$
$761$	−17.0000			−0.616250			−0.308125	+	0.951346i	$0.599701\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$	2.00000			0.0722158
$768$		0			0
$769$	−1.00000	+	1.73205i	−0.0360609	+	0.0624593i	−0.883493	−	0.468445i	$-0.844814\pi$
$769$	−1.00000	+	1.73205i	−0.0360609	+	0.0624593i	0.847432	+	0.530904i	$0.178148\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	26.0000	−	45.0333i	0.935155	−	1.61974i	0.160798	−	0.986987i	$-0.448593\pi$
$773$	26.0000	−	45.0333i	0.935155	−	1.61974i	0.774357	−	0.632749i	$-0.218073\pi$
$774$		0			0
$775$	55.0000	−	95.2628i	1.97566	−	3.42194i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	36.0000	−	15.5885i	1.28983	−	0.558514i
$780$		0			0
$781$	9.00000	+	15.5885i	0.322045	+	0.557799i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−16.0000	+	27.7128i	−0.571064	+	0.989113i
$786$		0			0
$787$	41.0000			1.46149			0.730746	−	0.682649i	$-0.239172\pi$
$787$	41.0000			1.46149			0.730746	+	0.682649i	$0.239172\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	8.00000	+	13.8564i	0.284088	+	0.492055i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−42.0000			−1.48772			−0.743858	−	0.668338i	$-0.767006\pi$
$797$	−42.0000			−1.48772			−0.743858	+	0.668338i	$0.767006\pi$
$798$		0			0
$799$	−24.0000			−0.849059
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−13.5000	−	23.3827i	−0.476405	−	0.825157i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−21.0000			−0.738321			−0.369160	−	0.929366i	$-0.620355\pi$
$809$	−21.0000			−0.738321			−0.369160	+	0.929366i	$0.620355\pi$
$810$		0			0
$811$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$811$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	22.0000	+	38.1051i	0.770626	+	1.33476i
$816$		0			0
$817$	16.0000	−	6.92820i	0.559769	−	0.242387i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−19.0000	+	32.9090i	−0.663105	+	1.14853i	0.316691	+	0.948529i	$0.397428\pi$
$821$	−19.0000	+	32.9090i	−0.663105	+	1.14853i	−0.979795	+	0.200002i	$0.935905\pi$
$822$		0			0
$823$	13.0000	−	22.5167i	0.453152	−	0.784881i	−0.545428	−	0.838157i	$-0.683633\pi$
$823$	13.0000	−	22.5167i	0.453152	−	0.784881i	0.998580	+	0.0532760i	$0.0169663\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−27.5000	+	47.6314i	−0.956269	+	1.65631i	−0.224831	+	0.974398i	$0.572183\pi$
$827$	−27.5000	+	47.6314i	−0.956269	+	1.65631i	−0.731438	+	0.681908i	$0.761150\pi$
$828$		0			0
$829$	−12.0000			−0.416777			−0.208389	−	0.978046i	$-0.566822\pi$
$829$	−12.0000			−0.416777			−0.208389	+	0.978046i	$0.566822\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−7.00000	−	12.1244i	−0.242536	−	0.420084i
$834$		0			0
$835$	−64.0000			−2.21481
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$839$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$840$		0			0
$841$	6.50000	+	11.2583i	0.224138	+	0.388218i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	18.0000	−	31.1769i	0.619219	−	1.07252i
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$	6.00000	−	10.3923i	0.205677	−	0.356244i
$852$		0			0
$853$	−21.0000	−	36.3731i	−0.719026	−	1.24539i	−0.961386	−	0.275204i	$-0.911255\pi$
$853$	−21.0000	−	36.3731i	−0.719026	−	1.24539i	0.242360	−	0.970186i	$-0.422079\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−17.5000	−	30.3109i	−0.597789	−	1.03540i	−0.993147	−	0.116873i	$-0.962713\pi$
$857$	−17.5000	−	30.3109i	−0.597789	−	1.03540i	0.395358	−	0.918527i	$-0.370620\pi$
$858$		0			0
$859$	−14.5000	+	25.1147i	−0.494734	+	0.856904i	−0.999982	−	0.00607046i	$-0.998068\pi$
$859$	−14.5000	+	25.1147i	−0.494734	+	0.856904i	0.505248	+	0.862974i	$0.331401\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−6.00000			−0.204242			−0.102121	−	0.994772i	$-0.532563\pi$
$863$	−6.00000			−0.204242			−0.102121	+	0.994772i	$0.532563\pi$
$864$		0			0
$865$	−52.0000	+	90.0666i	−1.76805	+	3.06236i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−6.00000	−	10.3923i	−0.203536	−	0.352535i
$870$		0			0
$871$	−9.00000	−	15.5885i	−0.304953	−	0.528195i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−20.0000	−	34.6410i	−0.675352	−	1.16974i	−0.976366	−	0.216124i	$-0.930658\pi$
$877$	−20.0000	−	34.6410i	−0.675352	−	1.16974i	0.301014	−	0.953620i	$-0.402675\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−7.00000			−0.235836			−0.117918	−	0.993023i	$-0.537622\pi$
$881$	−7.00000			−0.235836			−0.117918	+	0.993023i	$0.537622\pi$
$882$		0			0
$883$	21.5000	−	37.2391i	0.723533	−	1.25320i	−0.236043	−	0.971743i	$-0.575850\pi$
$883$	21.5000	−	37.2391i	0.723533	−	1.25320i	0.959575	−	0.281453i	$-0.0908162\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	16.0000	−	27.7128i	0.537227	−	0.930505i	−0.461825	−	0.886971i	$-0.652805\pi$
$887$	16.0000	−	27.7128i	0.537227	−	0.930505i	0.999052	−	0.0435339i	$-0.0138616\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$	42.0000	+	31.1769i	1.40548	+	1.04330i
$894$		0			0
$895$	18.0000	+	31.1769i	0.601674	+	1.04213i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	20.0000	−	34.6410i	0.667037	−	1.15534i
$900$		0			0
$901$	−4.00000			−0.133259
$902$		0			0
$903$		0			0
$904$		0			0
$905$	56.0000			1.86150
$906$		0			0
$907$	−24.5000	−	42.4352i	−0.813509	−	1.40904i	−0.910393	−	0.413744i	$-0.864221\pi$
$907$	−24.5000	−	42.4352i	−0.813509	−	1.40904i	0.0968843	−	0.995296i	$-0.469112\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	14.0000			0.463841			0.231920	−	0.972735i	$-0.425499\pi$
$911$	14.0000			0.463841			0.231920	+	0.972735i	$0.425499\pi$
$912$		0			0
$913$	−15.0000			−0.496428
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−26.0000			−0.857661			−0.428830	−	0.903385i	$-0.641074\pi$
$919$	−26.0000			−0.857661			−0.428830	+	0.903385i	$0.641074\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	12.0000			0.394985
$924$		0			0
$925$	−11.0000	+	19.0526i	−0.361678	+	0.626444i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	10.5000	+	18.1865i	0.344494	+	0.596681i	0.985262	−	0.171054i	$-0.0547172\pi$
$929$	10.5000	+	18.1865i	0.344494	+	0.596681i	−0.640768	+	0.767735i	$0.721384\pi$
$930$		0			0
$931$	−3.50000	+	30.3109i	−0.114708	+	0.993399i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−12.0000	+	20.7846i	−0.392442	+	0.679729i
$936$		0			0
$937$	−5.50000	+	9.52628i	−0.179677	+	0.311210i	−0.941770	−	0.336258i	$-0.890839\pi$
$937$	−5.50000	+	9.52628i	−0.179677	+	0.311210i	0.762093	+	0.647468i	$0.224172\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−5.00000	+	8.66025i	−0.162995	+	0.282316i	−0.935942	−	0.352155i	$-0.885449\pi$
$941$	−5.00000	+	8.66025i	−0.162995	+	0.282316i	0.772946	+	0.634472i	$0.218782\pi$
$942$		0			0
$943$	54.0000			1.75848
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−14.0000	−	24.2487i	−0.454939	−	0.787977i	0.543746	−	0.839250i	$-0.317006\pi$
$947$	−14.0000	−	24.2487i	−0.454939	−	0.787977i	−0.998685	+	0.0512727i	$0.983672\pi$
$948$		0			0
$949$	−18.0000			−0.584305
$950$		0			0
$951$		0			0
$952$		0			0
$953$	0.500000	+	0.866025i	0.0161966	+	0.0280533i	0.874010	−	0.485908i	$-0.161511\pi$
$953$	0.500000	+	0.866025i	0.0161966	+	0.0280533i	−0.857814	+	0.513961i	$0.828178\pi$
$954$		0			0
$955$	24.0000	+	41.5692i	0.776622	+	1.34515i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	69.0000			2.22581
$962$		0			0
$963$		0			0
$964$		0			0
$965$	12.0000	−	20.7846i	0.386294	−	0.669080i
$966$		0			0
$967$	−27.0000	−	46.7654i	−0.868261	−	1.50387i	−0.863772	−	0.503883i	$-0.831904\pi$
$967$	−27.0000	−	46.7654i	−0.868261	−	1.50387i	−0.00448958	−	0.999990i	$-0.501429\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	26.5000	+	45.8993i	0.850425	+	1.47298i	0.880825	+	0.473442i	$0.156989\pi$
$971$	26.5000	+	45.8993i	0.850425	+	1.47298i	−0.0303994	+	0.999538i	$0.509678\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	47.0000			1.50366			0.751832	−	0.659355i	$-0.229171\pi$
$977$	47.0000			1.50366			0.751832	+	0.659355i	$0.229171\pi$
$978$		0			0
$979$	27.0000	−	46.7654i	0.862924	−	1.49463i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	16.0000	+	27.7128i	0.510321	+	0.883901i	0.999928	+	0.0119587i	$0.00380665\pi$
$983$	16.0000	+	27.7128i	0.510321	+	0.883901i	−0.489608	+	0.871943i	$0.662860\pi$
$984$		0			0
$985$	−36.0000	−	62.3538i	−1.14706	−	1.98676i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	24.0000			0.763156
$990$		0			0
$991$	−4.00000	−	6.92820i	−0.127064	−	0.220082i	0.795474	−	0.605988i	$-0.207222\pi$
$991$	−4.00000	−	6.92820i	−0.127064	−	0.220082i	−0.922538	+	0.385906i	$0.873889\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−72.0000			−2.28255
$996$		0			0
$997$	16.0000	−	27.7128i	0.506725	−	0.877674i	−0.493245	−	0.869891i	$-0.664189\pi$
$997$	16.0000	−	27.7128i	0.506725	−	0.877674i	0.999970	−	0.00778294i	$-0.00247741\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1368.2.s.a.577.1		2
3.2	odd	2		152.2.i.b.121.1	yes	2
4.3	odd	2		2736.2.s.a.577.1		2
12.11	even	2		304.2.i.d.273.1		2
19.11	even	3	inner	1368.2.s.a.505.1		2
24.5	odd	2		1216.2.i.f.577.1		2
24.11	even	2		1216.2.i.b.577.1		2
57.11	odd	6		152.2.i.b.49.1	✓	2
57.26	odd	6		2888.2.a.d.1.1		1
57.50	even	6		2888.2.a.a.1.1		1
76.11	odd	6		2736.2.s.a.1873.1		2
228.11	even	6		304.2.i.d.49.1		2
228.83	even	6		5776.2.a.e.1.1		1
228.107	odd	6		5776.2.a.j.1.1		1
456.11	even	6		1216.2.i.b.961.1		2
456.125	odd	6		1216.2.i.f.961.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
152.2.i.b.49.1	✓	2	57.11	odd	6
152.2.i.b.121.1	yes	2	3.2	odd	2
304.2.i.d.49.1		2	228.11	even	6
304.2.i.d.273.1		2	12.11	even	2
1216.2.i.b.577.1		2	24.11	even	2
1216.2.i.b.961.1		2	456.11	even	6
1216.2.i.f.577.1		2	24.5	odd	2
1216.2.i.f.961.1		2	456.125	odd	6
1368.2.s.a.505.1		2	19.11	even	3	inner
1368.2.s.a.577.1		2	1.1	even	1	trivial
2736.2.s.a.577.1		2	4.3	odd	2
2736.2.s.a.1873.1		2	76.11	odd	6
2888.2.a.a.1.1		1	57.50	even	6
2888.2.a.d.1.1		1	57.26	odd	6
5776.2.a.e.1.1		1	228.83	even	6
5776.2.a.j.1.1		1	228.107	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1368 = 2^{3} \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1368.s (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-2.00000 - 3.46410i) q^{5} -3.00000 q^{11} +(-1.00000 + 1.73205i) q^{13} +(1.00000 + 1.73205i) q^{17} +(0.500000 - 4.33013i) q^{19} +(3.00000 - 5.19615i) q^{23} +(-5.50000 + 9.52628i) q^{25} +(-2.00000 + 3.46410i) q^{29} -10.0000 q^{31} +2.00000 q^{37} +(4.50000 + 7.79423i) q^{41} +(2.00000 + 3.46410i) q^{43} +(-6.00000 + 10.3923i) q^{47} -7.00000 q^{49} +(-1.00000 + 1.73205i) q^{53} +(6.00000 + 10.3923i) q^{55} +(-0.500000 - 0.866025i) q^{59} +(4.00000 - 6.92820i) q^{61} +8.00000 q^{65} +(-4.50000 + 7.79423i) q^{67} +(-3.00000 - 5.19615i) q^{71} +(4.50000 + 7.79423i) q^{73} +(2.00000 + 3.46410i) q^{79} +5.00000 q^{83} +(4.00000 - 6.92820i) q^{85} +(-9.00000 + 15.5885i) q^{89} +(-16.0000 + 6.92820i) q^{95} +(-0.500000 - 0.866025i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{5} - 6 q^{11} - 2 q^{13} + 2 q^{17} + q^{19} + 6 q^{23} - 11 q^{25} - 4 q^{29} - 20 q^{31} + 4 q^{37} + 9 q^{41} + 4 q^{43} - 12 q^{47} - 14 q^{49} - 2 q^{53} + 12 q^{55} - q^{59} + 8 q^{61}+ \cdots - q^{97}+O(q^{100}) \) 2 * q - 4 * q^5 - 6 * q^11 - 2 * q^13 + 2 * q^17 + q^19 + 6 * q^23 - 11 * q^25 - 4 * q^29 - 20 * q^31 + 4 * q^37 + 9 * q^41 + 4 * q^43 - 12 * q^47 - 14 * q^49 - 2 * q^53 + 12 * q^55 - q^59 + 8 * q^61 + 16 * q^65 - 9 * q^67 - 6 * q^71 + 9 * q^73 + 4 * q^79 + 10 * q^83 + 8 * q^85 - 18 * q^89 - 32 * q^95 - q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more