LMFDB - Embedded newform 1216.2.i.b.577.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1216,2,Mod(577,1216)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1216, base_ring=CyclotomicField(6))

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

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

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

chi := DirichletCharacter("1216.577");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1216 = 2^{6} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1216.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:	$9.70980888579$
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:	$\Z[a_1, a_2, a_3]$
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	$\chi$	$=$	1216.577
Dual form			1216.2.i.b.961.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.500000 - 0.866025i) q^{3} +(-2.00000 - 3.46410i) q^{5} +(1.00000 - 1.73205i) q^{9} +O(q^{10})$	\(q+(-0.500000 - 0.866025i) q^{3} +(-2.00000 - 3.46410i) q^{5} +(1.00000 - 1.73205i) q^{9} +3.00000 q^{11} +(1.00000 - 1.73205i) q^{13} +(-2.00000 + 3.46410i) q^{15} +(-1.00000 - 1.73205i) q^{17} +(0.500000 - 4.33013i) q^{19} +(3.00000 - 5.19615i) q^{23} +(-5.50000 + 9.52628i) q^{25} -5.00000 q^{27} +(-2.00000 + 3.46410i) q^{29} +10.0000 q^{31} +(-1.50000 - 2.59808i) q^{33} -2.00000 q^{37} -2.00000 q^{39} +(-4.50000 - 7.79423i) q^{41} +(2.00000 + 3.46410i) q^{43} -8.00000 q^{45} +(-6.00000 + 10.3923i) q^{47} -7.00000 q^{49} +(-1.00000 + 1.73205i) q^{51} +(-1.00000 + 1.73205i) q^{53} +(-6.00000 - 10.3923i) q^{55} +(-4.00000 + 1.73205i) q^{57} +(0.500000 + 0.866025i) q^{59} +(-4.00000 + 6.92820i) q^{61} -8.00000 q^{65} +(-4.50000 + 7.79423i) q^{67} -6.00000 q^{69} +(-3.00000 - 5.19615i) q^{71} +(4.50000 + 7.79423i) q^{73} +11.0000 q^{75} +(-2.00000 - 3.46410i) q^{79} +(-0.500000 - 0.866025i) q^{81} -5.00000 q^{83} +(-4.00000 + 6.92820i) q^{85} +4.00000 q^{87} +(9.00000 - 15.5885i) q^{89} +(-5.00000 - 8.66025i) q^{93} +(-16.0000 + 6.92820i) q^{95} +(-0.500000 - 0.866025i) q^{97} +(3.00000 - 5.19615i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{3} - 4 q^{5} + 2 q^{9}+O(q^{10}) $ 2 * q - q^3 - 4 * q^5 + 2 * q^9	$ 2 q - q^{3} - 4 q^{5} + 2 q^{9} + 6 q^{11} + 2 q^{13} - 4 q^{15} - 2 q^{17} + q^{19} + 6 q^{23} - 11 q^{25} - 10 q^{27} - 4 q^{29} + 20 q^{31} - 3 q^{33} - 4 q^{37} - 4 q^{39} - 9 q^{41} + 4 q^{43} - 16 q^{45} - 12 q^{47} - 14 q^{49} - 2 q^{51} - 2 q^{53} - 12 q^{55} - 8 q^{57} + q^{59} - 8 q^{61} - 16 q^{65} - 9 q^{67} - 12 q^{69} - 6 q^{71} + 9 q^{73} + 22 q^{75} - 4 q^{79} - q^{81} - 10 q^{83} - 8 q^{85} + 8 q^{87} + 18 q^{89} - 10 q^{93} - 32 q^{95} - q^{97} + 6 q^{99}+O(q^{100}) $ 2 * q - q^3 - 4 * q^5 + 2 * q^9 + 6 * q^11 + 2 * q^13 - 4 * q^15 - 2 * q^17 + q^19 + 6 * q^23 - 11 * q^25 - 10 * q^27 - 4 * q^29 + 20 * q^31 - 3 * q^33 - 4 * q^37 - 4 * q^39 - 9 * q^41 + 4 * q^43 - 16 * q^45 - 12 * q^47 - 14 * q^49 - 2 * q^51 - 2 * q^53 - 12 * q^55 - 8 * q^57 + q^59 - 8 * q^61 - 16 * q^65 - 9 * q^67 - 12 * q^69 - 6 * q^71 + 9 * q^73 + 22 * q^75 - 4 * q^79 - q^81 - 10 * q^83 - 8 * q^85 + 8 * q^87 + 18 * q^89 - 10 * q^93 - 32 * q^95 - q^97 + 6 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1216\mathbb{Z}\right)^\times$.

$n$	$191$	$705$	$837$
$\chi(n)$	$1$	$e\left(\frac{1}{3}\right)$	$1$

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
$3$	−0.500000	−	0.866025i	−0.288675	−	0.500000i	0.684819	−	0.728714i	$-0.259881\pi$
$3$	−0.500000	−	0.866025i	−0.288675	−	0.500000i	−0.973494	+	0.228714i	$0.926548\pi$
$4$		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
$9$	1.00000	−	1.73205i	0.333333	−	0.577350i
$10$		0			0
$11$	3.00000			0.904534			0.452267	−	0.891883i	$-0.350615\pi$
$11$	3.00000			0.904534			0.452267	+	0.891883i	$0.350615\pi$
$12$		0			0
$13$	1.00000	−	1.73205i	0.277350	−	0.480384i	−0.693375	−	0.720577i	$-0.743877\pi$
$13$	1.00000	−	1.73205i	0.277350	−	0.480384i	0.970725	+	0.240192i	$0.0772105\pi$
$14$		0			0
$15$	−2.00000	+	3.46410i	−0.516398	+	0.894427i
$16$		0			0
$17$	−1.00000	−	1.73205i	−0.242536	−	0.420084i	0.718900	−	0.695113i	$-0.244646\pi$
$17$	−1.00000	−	1.73205i	−0.242536	−	0.420084i	−0.961436	+	0.275029i	$0.911312\pi$
$18$		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
$27$	−5.00000			−0.962250
$28$		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	$-0.145001\pi$
$31$	10.0000			1.79605			0.898027	+	0.439941i	$0.145001\pi$
$32$		0			0
$33$	−1.50000	−	2.59808i	−0.261116	−	0.452267i
$34$		0			0
$35$		0			0
$36$		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
$39$	−2.00000			−0.320256
$40$		0			0
$41$	−4.50000	−	7.79423i	−0.702782	−	1.21725i	−0.967486	−	0.252924i	$-0.918608\pi$
$41$	−4.50000	−	7.79423i	−0.702782	−	1.21725i	0.264704	−	0.964330i	$-0.414726\pi$
$42$		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
$45$	−8.00000			−1.19257
$46$		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
$51$	−1.00000	+	1.73205i	−0.140028	+	0.242536i
$52$		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
$57$	−4.00000	+	1.73205i	−0.529813	+	0.229416i
$58$		0			0
$59$	0.500000	+	0.866025i	0.0650945	+	0.112747i	0.896736	−	0.442566i	$-0.145932\pi$
$59$	0.500000	+	0.866025i	0.0650945	+	0.112747i	−0.831641	+	0.555313i	$0.812598\pi$
$60$		0			0
$61$	−4.00000	+	6.92820i	−0.512148	+	0.887066i	0.487753	+	0.872982i	$0.337817\pi$
$61$	−4.00000	+	6.92820i	−0.512148	+	0.887066i	−0.999901	+	0.0140840i	$0.995517\pi$
$62$		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
$69$	−6.00000			−0.722315
$70$		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
$75$	11.0000			1.27017
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−2.00000	−	3.46410i	−0.225018	−	0.389742i	0.731307	−	0.682048i	$-0.238911\pi$
$79$	−2.00000	−	3.46410i	−0.225018	−	0.389742i	−0.956325	+	0.292306i	$0.905577\pi$
$80$		0			0
$81$	−0.500000	−	0.866025i	−0.0555556	−	0.0962250i
$82$		0			0
$83$	−5.00000			−0.548821			−0.274411	−	0.961613i	$-0.588483\pi$
$83$	−5.00000			−0.548821			−0.274411	+	0.961613i	$0.588483\pi$
$84$		0			0
$85$	−4.00000	+	6.92820i	−0.433861	+	0.751469i
$86$		0			0
$87$	4.00000			0.428845
$88$		0			0
$89$	9.00000	−	15.5885i	0.953998	−	1.65237i	0.217354	−	0.976093i	$-0.430258\pi$
$89$	9.00000	−	15.5885i	0.953998	−	1.65237i	0.736644	−	0.676280i	$-0.236409\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	−5.00000	−	8.66025i	−0.518476	−	0.898027i
$94$		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
$99$	3.00000	−	5.19615i	0.301511	−	0.522233i
$100$		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	$-0.257729\pi$
$103$	14.0000			1.37946			0.689730	+	0.724066i	$0.257729\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	16.0000			1.54678			0.773389	−	0.633932i	$-0.218560\pi$
$107$	16.0000			1.54678			0.773389	+	0.633932i	$0.218560\pi$
$108$		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
$111$	1.00000	+	1.73205i	0.0949158	+	0.164399i
$112$		0			0
$113$	−1.00000			−0.0940721			−0.0470360	−	0.998893i	$-0.514978\pi$
$113$	−1.00000			−0.0940721			−0.0470360	+	0.998893i	$0.514978\pi$
$114$		0			0
$115$	−24.0000			−2.23801
$116$		0			0
$117$	−2.00000	−	3.46410i	−0.184900	−	0.320256i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−2.00000			−0.181818
$122$		0			0
$123$	−4.50000	+	7.79423i	−0.405751	+	0.702782i
$124$		0			0
$125$	24.0000			2.14663
$126$		0			0
$127$	−3.00000	+	5.19615i	−0.266207	+	0.461084i	−0.967879	−	0.251416i	$-0.919104\pi$
$127$	−3.00000	+	5.19615i	−0.266207	+	0.461084i	0.701672	+	0.712500i	$0.252437\pi$
$128$		0			0
$129$	2.00000	−	3.46410i	0.176090	−	0.304997i
$130$		0			0
$131$	7.50000	+	12.9904i	0.655278	+	1.13497i	0.981824	+	0.189794i	$0.0607819\pi$
$131$	7.50000	+	12.9904i	0.655278	+	1.13497i	−0.326546	+	0.945181i	$0.605885\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	10.0000	+	17.3205i	0.860663	+	1.49071i
$136$		0			0
$137$	−4.50000	+	7.79423i	−0.384461	+	0.665906i	−0.991694	−	0.128618i	$-0.958946\pi$
$137$	−4.50000	+	7.79423i	−0.384461	+	0.665906i	0.607233	+	0.794524i	$0.292279\pi$
$138$		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
$141$	12.0000			1.01058
$142$		0			0
$143$	3.00000	−	5.19615i	0.250873	−	0.434524i
$144$		0			0
$145$	16.0000			1.32873
$146$		0			0
$147$	3.50000	+	6.06218i	0.288675	+	0.500000i
$148$		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	$-0.525932\pi$
$151$	−2.00000			−0.162758			−0.0813788	+	0.996683i	$0.525932\pi$
$152$		0			0
$153$	−4.00000			−0.323381
$154$		0			0
$155$	−20.0000	−	34.6410i	−1.60644	−	2.78243i
$156$		0			0
$157$	4.00000	+	6.92820i	0.319235	+	0.552931i	0.980329	−	0.197372i	$-0.0632408\pi$
$157$	4.00000	+	6.92820i	0.319235	+	0.552931i	−0.661094	+	0.750303i	$0.729907\pi$
$158$		0			0
$159$	2.00000			0.158610
$160$		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
$165$	−6.00000	+	10.3923i	−0.467099	+	0.809040i
$166$		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
$171$	−7.00000	−	5.19615i	−0.535303	−	0.397360i
$172$		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
$177$	0.500000	−	0.866025i	0.0375823	−	0.0650945i
$178$		0			0
$179$	9.00000			0.672692			0.336346	−	0.941739i	$-0.390809\pi$
$179$	9.00000			0.672692			0.336346	+	0.941739i	$0.390809\pi$
$180$		0			0
$181$	7.00000	−	12.1244i	0.520306	−	0.901196i	−0.479415	−	0.877588i	$-0.659151\pi$
$181$	7.00000	−	12.1244i	0.520306	−	0.901196i	0.999721	−	0.0236082i	$-0.00751541\pi$
$182$		0			0
$183$	8.00000			0.591377
$184$		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
$195$	4.00000	+	6.92820i	0.286446	+	0.496139i
$196$		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	$0.386902\pi$
$199$	−9.00000	+	15.5885i	−0.637993	+	1.10504i	−0.985873	+	0.167497i	$0.946431\pi$
$200$		0			0
$201$	9.00000			0.634811
$202$		0			0
$203$		0			0
$204$		0			0
$205$	−18.0000	+	31.1769i	−1.25717	+	2.17749i
$206$		0			0
$207$	−6.00000	−	10.3923i	−0.417029	−	0.722315i
$208$		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
$213$	−3.00000	+	5.19615i	−0.205557	+	0.356034i
$214$		0			0
$215$	8.00000	−	13.8564i	0.545595	−	0.944999i
$216$		0			0
$217$		0			0
$218$		0			0
$219$	4.50000	−	7.79423i	0.304082	−	0.526685i
$220$		0			0
$221$	−4.00000			−0.269069
$222$		0			0
$223$	5.00000	+	8.66025i	0.334825	+	0.579934i	0.983451	−	0.181173i	$-0.0579895\pi$
$223$	5.00000	+	8.66025i	0.334825	+	0.579934i	−0.648626	+	0.761107i	$0.724656\pi$
$224$		0			0
$225$	11.0000	+	19.0526i	0.733333	+	1.27017i
$226$		0			0
$227$	19.0000			1.26107			0.630537	−	0.776159i	$-0.282835\pi$
$227$	19.0000			1.26107			0.630537	+	0.776159i	$0.282835\pi$
$228$		0			0
$229$	8.00000			0.528655			0.264327	−	0.964433i	$-0.414850\pi$
$229$	8.00000			0.528655			0.264327	+	0.964433i	$0.414850\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−5.50000	−	9.52628i	−0.360317	−	0.624087i	0.627696	−	0.778459i	$-0.283998\pi$
$233$	−5.50000	−	9.52628i	−0.360317	−	0.624087i	−0.988013	+	0.154371i	$0.950665\pi$
$234$		0			0
$235$	48.0000			3.13117
$236$		0			0
$237$	−2.00000	+	3.46410i	−0.129914	+	0.225018i
$238$		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
$243$	−8.00000	+	13.8564i	−0.513200	+	0.888889i
$244$		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
$249$	2.50000	+	4.33013i	0.158431	+	0.274411i
$250$		0			0
$251$	−2.50000	+	4.33013i	−0.157799	+	0.273315i	−0.934075	−	0.357078i	$-0.883773\pi$
$251$	−2.50000	+	4.33013i	−0.157799	+	0.273315i	0.776276	+	0.630393i	$0.217106\pi$
$252$		0			0
$253$	9.00000	−	15.5885i	0.565825	−	0.980038i
$254$		0			0
$255$	8.00000			0.500979
$256$		0			0
$257$	−1.50000	+	2.59808i	−0.0935674	+	0.162064i	−0.909010	−	0.416775i	$-0.863160\pi$
$257$	−1.50000	+	2.59808i	−0.0935674	+	0.162064i	0.815442	+	0.578838i	$0.196494\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	4.00000	+	6.92820i	0.247594	+	0.428845i
$262$		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
$267$	−18.0000			−1.10158
$268$		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	$-0.958856\pi$
$271$	−10.0000	−	17.3205i	−0.607457	−	1.05215i	0.384201	−	0.923249i	$-0.374477\pi$
$272$		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	$-0.617396\pi$
$277$	−12.0000			−0.721010			−0.360505	+	0.932757i	$0.617396\pi$
$278$		0			0
$279$	10.0000	−	17.3205i	0.598684	−	1.03695i
$280$		0			0
$281$	6.50000	−	11.2583i	0.387757	−	0.671616i	−0.604390	−	0.796689i	$-0.706583\pi$
$281$	6.50000	−	11.2583i	0.387757	−	0.671616i	0.992148	+	0.125073i	$0.0399165\pi$
$282$		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
$285$	14.0000	+	10.3923i	0.829288	+	0.615587i
$286$		0			0
$287$		0			0
$288$		0			0
$289$	6.50000	−	11.2583i	0.382353	−	0.662255i
$290$		0			0
$291$	−0.500000	+	0.866025i	−0.0293105	+	0.0507673i
$292$		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
$297$	−15.0000			−0.870388
$298$		0			0
$299$	−6.00000	−	10.3923i	−0.346989	−	0.601003i
$300$		0			0
$301$		0			0
$302$		0			0
$303$	−12.0000			−0.689382
$304$		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
$309$	−7.00000	−	12.1244i	−0.398216	−	0.689730i
$310$		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
$321$	−8.00000	−	13.8564i	−0.446516	−	0.773389i
$322$		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
$333$	−2.00000	+	3.46410i	−0.109599	+	0.189832i
$334$		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
$339$	0.500000	+	0.866025i	0.0271563	+	0.0470360i
$340$		0			0
$341$	30.0000			1.62459
$342$		0			0
$343$		0			0
$344$		0			0
$345$	12.0000	+	20.7846i	0.646058	+	1.11901i
$346$		0			0
$347$	4.50000	+	7.79423i	0.241573	+	0.418416i	0.961162	−	0.275983i	$-0.0890035\pi$
$347$	4.50000	+	7.79423i	0.241573	+	0.418416i	−0.719590	+	0.694399i	$0.755670\pi$
$348$		0			0
$349$	28.0000			1.49881			0.749403	−	0.662114i	$-0.230341\pi$
$349$	28.0000			1.49881			0.749403	+	0.662114i	$0.230341\pi$
$350$		0			0
$351$	−5.00000	+	8.66025i	−0.266880	+	0.462250i
$352$		0			0
$353$	11.0000			0.585471			0.292735	−	0.956193i	$-0.405434\pi$
$353$	11.0000			0.585471			0.292735	+	0.956193i	$0.405434\pi$
$354$		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
$363$	1.00000	+	1.73205i	0.0524864	+	0.0909091i
$364$		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	$0.404077\pi$
$367$	−13.0000	+	22.5167i	−0.678594	+	1.17536i	−0.975404	+	0.220423i	$0.929256\pi$
$368$		0			0
$369$	−18.0000			−0.937043
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−4.00000			−0.207112			−0.103556	−	0.994624i	$-0.533022\pi$
$373$	−4.00000			−0.207112			−0.103556	+	0.994624i	$0.533022\pi$
$374$		0			0
$375$	−12.0000	−	20.7846i	−0.619677	−	1.07331i
$376$		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
$381$	6.00000			0.307389
$382$		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
$387$	8.00000			0.406663
$388$		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
$393$	7.50000	−	12.9904i	0.378325	−	0.655278i
$394$		0			0
$395$	−8.00000	+	13.8564i	−0.402524	+	0.697191i
$396$		0			0
$397$	−7.00000	−	12.1244i	−0.351320	−	0.608504i	0.635161	−	0.772380i	$-0.280934\pi$
$397$	−7.00000	−	12.1244i	−0.351320	−	0.608504i	−0.986481	+	0.163876i	$0.947600\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	1.50000	+	2.59808i	0.0749064	+	0.129742i	0.901046	−	0.433724i	$-0.142801\pi$
$401$	1.50000	+	2.59808i	0.0749064	+	0.129742i	−0.826139	+	0.563466i	$0.809468\pi$
$402$		0			0
$403$	10.0000	−	17.3205i	0.498135	−	0.862796i
$404$		0			0
$405$	−2.00000	+	3.46410i	−0.0993808	+	0.172133i
$406$		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
$411$	9.00000			0.443937
$412$		0			0
$413$		0			0
$414$		0			0
$415$	10.0000	+	17.3205i	0.490881	+	0.850230i
$416$		0			0
$417$	−13.0000			−0.636613
$418$		0			0
$419$	12.0000			0.586238			0.293119	−	0.956076i	$-0.405307\pi$
$419$	12.0000			0.586238			0.293119	+	0.956076i	$0.405307\pi$
$420$		0			0
$421$	−11.0000	−	19.0526i	−0.536107	−	0.928565i	−0.999109	−	0.0422075i	$-0.986561\pi$
$421$	−11.0000	−	19.0526i	−0.536107	−	0.928565i	0.463002	−	0.886357i	$-0.346772\pi$
$422$		0			0
$423$	12.0000	+	20.7846i	0.583460	+	1.01058i
$424$		0			0
$425$	22.0000			1.06716
$426$		0			0
$427$		0			0
$428$		0			0
$429$	−6.00000			−0.289683
$430$		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
$435$	−8.00000	−	13.8564i	−0.383571	−	0.664364i
$436$		0			0
$437$	−21.0000	−	15.5885i	−1.00457	−	0.745697i
$438$		0			0
$439$	−7.00000	−	12.1244i	−0.334092	−	0.578664i	0.649218	−	0.760602i	$-0.275096\pi$
$439$	−7.00000	−	12.1244i	−0.334092	−	0.578664i	−0.983310	+	0.181938i	$0.941763\pi$
$440$		0			0
$441$	−7.00000	+	12.1244i	−0.333333	+	0.577350i
$442$		0			0
$443$	−0.500000	+	0.866025i	−0.0237557	+	0.0411461i	−0.877659	−	0.479286i	$-0.840896\pi$
$443$	−0.500000	+	0.866025i	−0.0237557	+	0.0411461i	0.853903	+	0.520432i	$0.174229\pi$
$444$		0			0
$445$	−72.0000			−3.41313
$446$		0			0
$447$	−5.00000	+	8.66025i	−0.236492	+	0.409616i
$448$		0			0
$449$	33.0000			1.55737			0.778683	−	0.627417i	$-0.215888\pi$
$449$	33.0000			1.55737			0.778683	+	0.627417i	$0.215888\pi$
$450$		0			0
$451$	−13.5000	−	23.3827i	−0.635690	−	1.10105i
$452$		0			0
$453$	1.00000	+	1.73205i	0.0469841	+	0.0813788i
$454$		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
$459$	5.00000	+	8.66025i	0.233380	+	0.404226i
$460$		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	$-0.210051\pi$
$463$	34.0000			1.58011			0.790057	+	0.613033i	$0.210051\pi$
$464$		0			0
$465$	−20.0000	+	34.6410i	−0.927478	+	1.60644i
$466$		0			0
$467$	5.00000			0.231372			0.115686	−	0.993286i	$-0.463093\pi$
$467$	5.00000			0.231372			0.115686	+	0.993286i	$0.463093\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	4.00000	−	6.92820i	0.184310	−	0.319235i
$472$		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
$477$	2.00000	+	3.46410i	0.0915737	+	0.158610i
$478$		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	$-0.299488\pi$
$487$	26.0000			1.17817			0.589086	+	0.808070i	$0.299488\pi$
$488$		0			0
$489$	5.50000	+	9.52628i	0.248719	+	0.430793i
$490$		0			0
$491$	−8.00000	−	13.8564i	−0.361035	−	0.625331i	0.627096	−	0.778942i	$-0.284243\pi$
$491$	−8.00000	−	13.8564i	−0.361035	−	0.625331i	−0.988131	+	0.153611i	$0.950910\pi$
$492$		0			0
$493$	8.00000			0.360302
$494$		0			0
$495$	−24.0000			−1.07872
$496$		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
$501$	−16.0000			−0.714827
$502$		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
$507$	4.50000	−	7.79423i	0.199852	−	0.346154i
$508$		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
$513$	−2.50000	+	21.6506i	−0.110378	+	0.955899i
$514$		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
$519$	−13.0000	+	22.5167i	−0.570637	+	0.988372i
$520$		0			0
$521$	1.00000			0.0438108			0.0219054	−	0.999760i	$-0.493027\pi$
$521$	1.00000			0.0438108			0.0219054	+	0.999760i	$0.493027\pi$
$522$		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
$531$	2.00000			0.0867926
$532$		0			0
$533$	−18.0000			−0.779667
$534$		0			0
$535$	−32.0000	−	55.4256i	−1.38348	−	2.39626i
$536$		0			0
$537$	−4.50000	−	7.79423i	−0.194189	−	0.336346i
$538$		0			0
$539$	−21.0000			−0.904534
$540$		0			0
$541$	2.00000	−	3.46410i	0.0859867	−	0.148933i	−0.819825	−	0.572615i	$-0.805929\pi$
$541$	2.00000	−	3.46410i	0.0859867	−	0.148933i	0.905811	+	0.423681i	$0.139262\pi$
$542$		0			0
$543$	−14.0000			−0.600798
$544$		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
$549$	8.00000	+	13.8564i	0.341432	+	0.591377i
$550$		0			0
$551$	14.0000	+	10.3923i	0.596420	+	0.442727i
$552$		0			0
$553$		0			0
$554$		0			0
$555$	4.00000	−	6.92820i	0.169791	−	0.294086i
$556$		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
$561$	−3.00000	+	5.19615i	−0.126660	+	0.219382i
$562$		0			0
$563$	11.0000			0.463595			0.231797	−	0.972764i	$-0.425539\pi$
$563$	11.0000			0.463595			0.231797	+	0.972764i	$0.425539\pi$
$564$		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	$-0.914582\pi$
$569$	−46.0000			−1.92842			−0.964210	+	0.265139i	$0.914582\pi$
$570$		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
$573$	6.00000	+	10.3923i	0.250654	+	0.434145i
$574$		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
$579$	3.00000	−	5.19615i	0.124676	−	0.215945i
$580$		0			0
$581$		0			0
$582$		0			0
$583$	−3.00000	+	5.19615i	−0.124247	+	0.215203i
$584$		0			0
$585$	−8.00000	+	13.8564i	−0.330759	+	0.572892i
$586$		0			0
$587$	−2.00000	−	3.46410i	−0.0825488	−	0.142979i	0.821795	−	0.569783i	$-0.192973\pi$
$587$	−2.00000	−	3.46410i	−0.0825488	−	0.142979i	−0.904344	+	0.426804i	$0.859639\pi$
$588$		0			0
$589$	5.00000	−	43.3013i	0.206021	−	1.78420i
$590$		0			0
$591$	−9.00000	−	15.5885i	−0.370211	−	0.641223i
$592$		0			0
$593$	10.5000	−	18.1865i	0.431183	−	0.746831i	−0.565792	−	0.824548i	$-0.691430\pi$
$593$	10.5000	−	18.1865i	0.431183	−	0.746831i	0.996976	+	0.0777165i	$0.0247629\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	18.0000			0.736691
$598$		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
$603$	9.00000	+	15.5885i	0.366508	+	0.634811i
$604$		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	0.553716	−	0.832705i	$-0.313209\pi$
$613$	−11.0000	−	19.0526i	−0.444286	−	0.769526i	−0.998002	+	0.0631797i	$0.979876\pi$
$614$		0			0
$615$	36.0000			1.45166
$616$		0			0
$617$	−1.50000	+	2.59808i	−0.0603877	+	0.104595i	−0.894639	−	0.446790i	$-0.852567\pi$
$617$	−1.50000	+	2.59808i	−0.0603877	+	0.104595i	0.834251	+	0.551385i	$0.185900\pi$
$618$		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
$621$	−15.0000	+	25.9808i	−0.601929	+	1.04257i
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−20.5000	−	35.5070i	−0.820000	−	1.42028i
$626$		0			0
$627$	−12.0000	+	5.19615i	−0.479234	+	0.207514i
$628$		0			0
$629$	2.00000	+	3.46410i	0.0797452	+	0.138123i
$630$		0			0
$631$	−4.00000	+	6.92820i	−0.159237	+	0.275807i	−0.934594	−	0.355716i	$-0.884237\pi$
$631$	−4.00000	+	6.92820i	−0.159237	+	0.275807i	0.775356	+	0.631524i	$0.217570\pi$
$632$		0			0
$633$	6.00000	−	10.3923i	0.238479	−	0.413057i
$634$		0			0
$635$	24.0000			0.952411
$636$		0			0
$637$	−7.00000	+	12.1244i	−0.277350	+	0.480384i
$638$		0			0
$639$	−12.0000			−0.474713
$640$		0			0
$641$	−16.5000	−	28.5788i	−0.651711	−	1.12880i	−0.982708	−	0.185164i	$-0.940718\pi$
$641$	−16.5000	−	28.5788i	−0.651711	−	1.12880i	0.330997	−	0.943632i	$-0.392615\pi$
$642$		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
$645$	−16.0000			−0.629999
$646$		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
$657$	18.0000			0.702247
$658$		0			0
$659$	−6.00000	+	10.3923i	−0.233727	+	0.404827i	−0.958902	−	0.283738i	$-0.908425\pi$
$659$	−6.00000	+	10.3923i	−0.233727	+	0.404827i	0.725175	+	0.688565i	$0.241759\pi$
$660$		0			0
$661$	10.0000	−	17.3205i	0.388955	−	0.673690i	−0.603354	−	0.797473i	$-0.706170\pi$
$661$	10.0000	−	17.3205i	0.388955	−	0.673690i	0.992309	+	0.123784i	$0.0395028\pi$
$662$		0			0
$663$	2.00000	+	3.46410i	0.0776736	+	0.134535i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	12.0000	+	20.7846i	0.464642	+	0.804783i
$668$		0			0
$669$	5.00000	−	8.66025i	0.193311	−	0.334825i
$670$		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
$675$	27.5000	−	47.6314i	1.05848	−	1.83333i
$676$		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
$681$	−9.50000	−	16.4545i	−0.364041	−	0.630537i
$682$		0			0
$683$	−28.0000			−1.07139			−0.535695	−	0.844411i	$-0.679950\pi$
$683$	−28.0000			−1.07139			−0.535695	+	0.844411i	$0.679950\pi$
$684$		0			0
$685$	36.0000			1.37549
$686$		0			0
$687$	−4.00000	−	6.92820i	−0.152610	−	0.264327i
$688$		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
$699$	−5.50000	+	9.52628i	−0.208029	+	0.360317i
$700$		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
$705$	−24.0000	−	41.5692i	−0.903892	−	1.56559i
$706$		0			0
$707$		0			0
$708$		0			0
$709$	9.00000	−	15.5885i	0.338002	−	0.585437i	−0.646055	−	0.763291i	$-0.723582\pi$
$709$	9.00000	−	15.5885i	0.338002	−	0.585437i	0.984057	+	0.177854i	$0.0569156\pi$
$710$		0			0
$711$	−8.00000			−0.300023
$712$		0			0
$713$	30.0000	−	51.9615i	1.12351	−	1.94597i
$714$		0			0
$715$	−24.0000			−0.897549
$716$		0			0
$717$	6.00000	+	10.3923i	0.224074	+	0.388108i
$718$		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
$723$	21.0000			0.780998
$724$		0			0
$725$	−22.0000	−	38.1051i	−0.817059	−	1.41519i
$726$		0			0
$727$	−4.00000	−	6.92820i	−0.148352	−	0.256953i	0.782267	−	0.622944i	$-0.214063\pi$
$727$	−4.00000	−	6.92820i	−0.148352	−	0.256953i	−0.930618	+	0.365991i	$0.880730\pi$
$728$		0			0
$729$	13.0000			0.481481
$730$		0			0
$731$	4.00000	−	6.92820i	0.147945	−	0.256249i
$732$		0			0
$733$	−22.0000			−0.812589			−0.406294	−	0.913742i	$-0.633179\pi$
$733$	−22.0000			−0.812589			−0.406294	+	0.913742i	$0.633179\pi$
$734$		0			0
$735$	14.0000	−	24.2487i	0.516398	−	0.894427i
$736$		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
$741$	−1.00000	+	8.66025i	−0.0367359	+	0.318142i
$742$		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
$747$	−5.00000	+	8.66025i	−0.182940	+	0.316862i
$748$		0			0
$749$		0			0
$750$		0			0
$751$	19.0000	−	32.9090i	0.693320	−	1.20087i	−0.277424	−	0.960748i	$-0.589481\pi$
$751$	19.0000	−	32.9090i	0.693320	−	1.20087i	0.970744	−	0.240118i	$-0.0771860\pi$
$752$		0			0
$753$	5.00000			0.182210
$754$		0			0
$755$	4.00000	+	6.92820i	0.145575	+	0.252143i
$756$		0			0
$757$	7.00000	+	12.1244i	0.254419	+	0.440667i	0.964738	−	0.263213i	$-0.0847823\pi$
$757$	7.00000	+	12.1244i	0.254419	+	0.440667i	−0.710318	+	0.703881i	$0.751449\pi$
$758$		0			0
$759$	−18.0000			−0.653359
$760$		0			0
$761$	17.0000			0.616250			0.308125	−	0.951346i	$-0.400299\pi$
$761$	17.0000			0.616250			0.308125	+	0.951346i	$0.400299\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	8.00000	+	13.8564i	0.289241	+	0.500979i
$766$		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
$771$	3.00000			0.108042
$772$		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
$783$	10.0000	−	17.3205i	0.357371	−	0.618984i
$784$		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
$789$	8.00000	−	13.8564i	0.284808	−	0.493301i
$790$		0			0
$791$		0			0

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:	\( N \)	\(=\)	\( 1216 = 2^{6} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1216.i (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.500000 - 0.866025i) q^{3} +(-2.00000 - 3.46410i) q^{5} +(1.00000 - 1.73205i) q^{9} +O(q^{10})\)	\(q+(-0.500000 - 0.866025i) q^{3} +(-2.00000 - 3.46410i) q^{5} +(1.00000 - 1.73205i) q^{9} +3.00000 q^{11} +(1.00000 - 1.73205i) q^{13} +(-2.00000 + 3.46410i) q^{15} +(-1.00000 - 1.73205i) q^{17} +(0.500000 - 4.33013i) q^{19} +(3.00000 - 5.19615i) q^{23} +(-5.50000 + 9.52628i) q^{25} -5.00000 q^{27} +(-2.00000 + 3.46410i) q^{29} +10.0000 q^{31} +(-1.50000 - 2.59808i) q^{33} -2.00000 q^{37} -2.00000 q^{39} +(-4.50000 - 7.79423i) q^{41} +(2.00000 + 3.46410i) q^{43} -8.00000 q^{45} +(-6.00000 + 10.3923i) q^{47} -7.00000 q^{49} +(-1.00000 + 1.73205i) q^{51} +(-1.00000 + 1.73205i) q^{53} +(-6.00000 - 10.3923i) q^{55} +(-4.00000 + 1.73205i) q^{57} +(0.500000 + 0.866025i) q^{59} +(-4.00000 + 6.92820i) q^{61} -8.00000 q^{65} +(-4.50000 + 7.79423i) q^{67} -6.00000 q^{69} +(-3.00000 - 5.19615i) q^{71} +(4.50000 + 7.79423i) q^{73} +11.0000 q^{75} +(-2.00000 - 3.46410i) q^{79} +(-0.500000 - 0.866025i) q^{81} -5.00000 q^{83} +(-4.00000 + 6.92820i) q^{85} +4.00000 q^{87} +(9.00000 - 15.5885i) q^{89} +(-5.00000 - 8.66025i) q^{93} +(-16.0000 + 6.92820i) q^{95} +(-0.500000 - 0.866025i) q^{97} +(3.00000 - 5.19615i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{3} - 4 q^{5} + 2 q^{9}+O(q^{10}) \) 2 * q - q^3 - 4 * q^5 + 2 * q^9	\( 2 q - q^{3} - 4 q^{5} + 2 q^{9} + 6 q^{11} + 2 q^{13} - 4 q^{15} - 2 q^{17} + q^{19} + 6 q^{23} - 11 q^{25} - 10 q^{27} - 4 q^{29} + 20 q^{31} - 3 q^{33} - 4 q^{37} - 4 q^{39} - 9 q^{41} + 4 q^{43} - 16 q^{45} - 12 q^{47} - 14 q^{49} - 2 q^{51} - 2 q^{53} - 12 q^{55} - 8 q^{57} + q^{59} - 8 q^{61} - 16 q^{65} - 9 q^{67} - 12 q^{69} - 6 q^{71} + 9 q^{73} + 22 q^{75} - 4 q^{79} - q^{81} - 10 q^{83} - 8 q^{85} + 8 q^{87} + 18 q^{89} - 10 q^{93} - 32 q^{95} - q^{97} + 6 q^{99}+O(q^{100}) \) 2 * q - q^3 - 4 * q^5 + 2 * q^9 + 6 * q^11 + 2 * q^13 - 4 * q^15 - 2 * q^17 + q^19 + 6 * q^23 - 11 * q^25 - 10 * q^27 - 4 * q^29 + 20 * q^31 - 3 * q^33 - 4 * q^37 - 4 * q^39 - 9 * q^41 + 4 * q^43 - 16 * q^45 - 12 * q^47 - 14 * q^49 - 2 * q^51 - 2 * q^53 - 12 * q^55 - 8 * q^57 + q^59 - 8 * q^61 - 16 * q^65 - 9 * q^67 - 12 * q^69 - 6 * q^71 + 9 * q^73 + 22 * q^75 - 4 * q^79 - q^81 - 10 * q^83 - 8 * q^85 + 8 * q^87 + 18 * q^89 - 10 * q^93 - 32 * q^95 - q^97 + 6 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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