LMFDB - Embedded newform 1216.2.i.e.961.1

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			961.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	1216.961
Dual form			1216.2.i.e.577.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.500000 + 0.866025i) q^{3} +(1.50000 - 2.59808i) q^{5} +(1.00000 + 1.73205i) q^{9} +O(q^{10})$	\(q+(-0.500000 + 0.866025i) q^{3} +(1.50000 - 2.59808i) q^{5} +(1.00000 + 1.73205i) q^{9} -4.00000 q^{11} +(-2.50000 - 4.33013i) q^{13} +(1.50000 + 2.59808i) q^{15} +(2.50000 - 4.33013i) q^{17} +(4.00000 - 1.73205i) q^{19} +(-0.500000 - 0.866025i) q^{23} +(-2.00000 - 3.46410i) q^{25} -5.00000 q^{27} +(1.50000 + 2.59808i) q^{29} -4.00000 q^{31} +(2.00000 - 3.46410i) q^{33} -2.00000 q^{37} +5.00000 q^{39} +(2.50000 - 4.33013i) q^{41} +(5.50000 - 9.52628i) q^{43} +6.00000 q^{45} +(-2.50000 - 4.33013i) q^{47} -7.00000 q^{49} +(2.50000 + 4.33013i) q^{51} +(-4.50000 - 7.79423i) q^{53} +(-6.00000 + 10.3923i) q^{55} +(-0.500000 + 4.33013i) q^{57} +(-6.50000 + 11.2583i) q^{59} +(-0.500000 - 0.866025i) q^{61} -15.0000 q^{65} +(2.50000 + 4.33013i) q^{67} +1.00000 q^{69} +(0.500000 - 0.866025i) q^{71} +(4.50000 - 7.79423i) q^{73} +4.00000 q^{75} +(8.50000 - 14.7224i) q^{79} +(-0.500000 + 0.866025i) q^{81} +16.0000 q^{83} +(-7.50000 - 12.9904i) q^{85} -3.00000 q^{87} +(-1.50000 - 2.59808i) q^{89} +(2.00000 - 3.46410i) q^{93} +(1.50000 - 12.9904i) q^{95} +(6.50000 - 11.2583i) q^{97} +(-4.00000 - 6.92820i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{3} + 3 q^{5} + 2 q^{9}+O(q^{10}) $ 2 * q - q^3 + 3 * q^5 + 2 * q^9	$ 2 q - q^{3} + 3 q^{5} + 2 q^{9} - 8 q^{11} - 5 q^{13} + 3 q^{15} + 5 q^{17} + 8 q^{19} - q^{23} - 4 q^{25} - 10 q^{27} + 3 q^{29} - 8 q^{31} + 4 q^{33} - 4 q^{37} + 10 q^{39} + 5 q^{41} + 11 q^{43} + 12 q^{45} - 5 q^{47} - 14 q^{49} + 5 q^{51} - 9 q^{53} - 12 q^{55} - q^{57} - 13 q^{59} - q^{61} - 30 q^{65} + 5 q^{67} + 2 q^{69} + q^{71} + 9 q^{73} + 8 q^{75} + 17 q^{79} - q^{81} + 32 q^{83} - 15 q^{85} - 6 q^{87} - 3 q^{89} + 4 q^{93} + 3 q^{95} + 13 q^{97} - 8 q^{99}+O(q^{100}) $ 2 * q - q^3 + 3 * q^5 + 2 * q^9 - 8 * q^11 - 5 * q^13 + 3 * q^15 + 5 * q^17 + 8 * q^19 - q^23 - 4 * q^25 - 10 * q^27 + 3 * q^29 - 8 * q^31 + 4 * q^33 - 4 * q^37 + 10 * q^39 + 5 * q^41 + 11 * q^43 + 12 * q^45 - 5 * q^47 - 14 * q^49 + 5 * q^51 - 9 * q^53 - 12 * q^55 - q^57 - 13 * q^59 - q^61 - 30 * q^65 + 5 * q^67 + 2 * q^69 + q^71 + 9 * q^73 + 8 * q^75 + 17 * q^79 - q^81 + 32 * q^83 - 15 * q^85 - 6 * q^87 - 3 * q^89 + 4 * q^93 + 3 * q^95 + 13 * q^97 - 8 * q^99

Display 100 coefficients 10 coefficients

Character values

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

$n$	$191$	$705$	$837$
$\chi(n)$	$1$	$e\left(\frac{2}{3}\right)$	$1$

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
$3$	−0.500000	+	0.866025i	−0.288675	+	0.500000i	−0.973494	−	0.228714i	$-0.926548\pi$
$3$	−0.500000	+	0.866025i	−0.288675	+	0.500000i	0.684819	+	0.728714i	$0.259881\pi$
$4$		0			0
$5$	1.50000	−	2.59808i	0.670820	−	1.16190i	−0.306851	−	0.951757i	$-0.599275\pi$
$5$	1.50000	−	2.59808i	0.670820	−	1.16190i	0.977672	−	0.210138i	$-0.0673912\pi$
$6$		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
$11$	−4.00000			−1.20605			−0.603023	−	0.797724i	$-0.706037\pi$
$11$	−4.00000			−1.20605			−0.603023	+	0.797724i	$0.706037\pi$
$12$		0			0
$13$	−2.50000	−	4.33013i	−0.693375	−	1.20096i	−0.970725	−	0.240192i	$-0.922790\pi$
$13$	−2.50000	−	4.33013i	−0.693375	−	1.20096i	0.277350	−	0.960769i	$-0.410544\pi$
$14$		0			0
$15$	1.50000	+	2.59808i	0.387298	+	0.670820i
$16$		0			0
$17$	2.50000	−	4.33013i	0.606339	−	1.05021i	−0.385499	−	0.922708i	$-0.625971\pi$
$17$	2.50000	−	4.33013i	0.606339	−	1.05021i	0.991838	−	0.127502i	$-0.0406959\pi$
$18$		0			0
$19$	4.00000	−	1.73205i	0.917663	−	0.397360i
$19$	4.00000	−	1.73205i	0.917663	−	0.397360i
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−0.500000	−	0.866025i	−0.104257	−	0.180579i	0.809177	−	0.587565i	$-0.199913\pi$
$23$	−0.500000	−	0.866025i	−0.104257	−	0.180579i	−0.913434	+	0.406986i	$0.866580\pi$
$24$		0			0
$25$	−2.00000	−	3.46410i	−0.400000	−	0.692820i
$26$		0			0
$27$	−5.00000			−0.962250
$28$		0			0
$29$	1.50000	+	2.59808i	0.278543	+	0.482451i	0.971023	−	0.238987i	$-0.0768152\pi$
$29$	1.50000	+	2.59808i	0.278543	+	0.482451i	−0.692480	+	0.721437i	$0.743482\pi$
$30$		0			0
$31$	−4.00000			−0.718421			−0.359211	−	0.933257i	$-0.616954\pi$
$31$	−4.00000			−0.718421			−0.359211	+	0.933257i	$0.616954\pi$
$32$		0			0
$33$	2.00000	−	3.46410i	0.348155	−	0.603023i
$34$		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
$39$	5.00000			0.800641
$40$		0			0
$41$	2.50000	−	4.33013i	0.390434	−	0.676252i	−0.602072	−	0.798441i	$-0.705658\pi$
$41$	2.50000	−	4.33013i	0.390434	−	0.676252i	0.992507	+	0.122189i	$0.0389915\pi$
$42$		0			0
$43$	5.50000	−	9.52628i	0.838742	−	1.45274i	−0.0522047	−	0.998636i	$-0.516625\pi$
$43$	5.50000	−	9.52628i	0.838742	−	1.45274i	0.890947	−	0.454108i	$-0.150042\pi$
$44$		0			0
$45$	6.00000			0.894427
$46$		0			0
$47$	−2.50000	−	4.33013i	−0.364662	−	0.631614i	0.624059	−	0.781377i	$-0.285482\pi$
$47$	−2.50000	−	4.33013i	−0.364662	−	0.631614i	−0.988722	+	0.149763i	$0.952149\pi$
$48$		0			0
$49$	−7.00000			−1.00000
$50$		0			0
$51$	2.50000	+	4.33013i	0.350070	+	0.606339i
$52$		0			0
$53$	−4.50000	−	7.79423i	−0.618123	−	1.07062i	−0.989828	−	0.142269i	$-0.954560\pi$
$53$	−4.50000	−	7.79423i	−0.618123	−	1.07062i	0.371706	−	0.928351i	$-0.378773\pi$
$54$		0			0
$55$	−6.00000	+	10.3923i	−0.809040	+	1.40130i
$56$		0			0
$57$	−0.500000	+	4.33013i	−0.0662266	+	0.573539i
$58$		0			0
$59$	−6.50000	+	11.2583i	−0.846228	+	1.46571i	0.0383226	+	0.999265i	$0.487799\pi$
$59$	−6.50000	+	11.2583i	−0.846228	+	1.46571i	−0.884551	+	0.466444i	$0.845535\pi$
$60$		0			0
$61$	−0.500000	−	0.866025i	−0.0640184	−	0.110883i	0.832240	−	0.554416i	$-0.187058\pi$
$61$	−0.500000	−	0.866025i	−0.0640184	−	0.110883i	−0.896258	+	0.443533i	$0.853725\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−15.0000			−1.86052
$66$		0			0
$67$	2.50000	+	4.33013i	0.305424	+	0.529009i	0.977356	−	0.211604i	$-0.0678686\pi$
$67$	2.50000	+	4.33013i	0.305424	+	0.529009i	−0.671932	+	0.740613i	$0.734535\pi$
$68$		0			0
$69$	1.00000			0.120386
$70$		0			0
$71$	0.500000	−	0.866025i	0.0593391	−	0.102778i	−0.834830	−	0.550508i	$-0.814434\pi$
$71$	0.500000	−	0.866025i	0.0593391	−	0.102778i	0.894169	+	0.447730i	$0.147767\pi$
$72$		0			0
$73$	4.50000	−	7.79423i	0.526685	−	0.912245i	−0.472831	−	0.881153i	$-0.656768\pi$
$73$	4.50000	−	7.79423i	0.526685	−	0.912245i	0.999517	−	0.0310925i	$-0.00989865\pi$
$74$		0			0
$75$	4.00000			0.461880
$76$		0			0
$77$		0			0
$78$		0			0
$79$	8.50000	−	14.7224i	0.956325	−	1.65640i	0.225018	−	0.974355i	$-0.427756\pi$
$79$	8.50000	−	14.7224i	0.956325	−	1.65640i	0.731307	−	0.682048i	$-0.238911\pi$
$80$		0			0
$81$	−0.500000	+	0.866025i	−0.0555556	+	0.0962250i
$82$		0			0
$83$	16.0000			1.75623			0.878114	−	0.478451i	$-0.158802\pi$
$83$	16.0000			1.75623			0.878114	+	0.478451i	$0.158802\pi$
$84$		0			0
$85$	−7.50000	−	12.9904i	−0.813489	−	1.40900i
$86$		0			0
$87$	−3.00000			−0.321634
$88$		0			0
$89$	−1.50000	−	2.59808i	−0.159000	−	0.275396i	0.775509	−	0.631337i	$-0.217494\pi$
$89$	−1.50000	−	2.59808i	−0.159000	−	0.275396i	−0.934508	+	0.355942i	$0.884160\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	2.00000	−	3.46410i	0.207390	−	0.359211i
$94$		0			0
$95$	1.50000	−	12.9904i	0.153897	−	1.33278i
$96$		0			0
$97$	6.50000	−	11.2583i	0.659975	−	1.14311i	−0.320647	−	0.947199i	$-0.603900\pi$
$97$	6.50000	−	11.2583i	0.659975	−	1.14311i	0.980622	−	0.195911i	$-0.0627665\pi$
$98$		0			0
$99$	−4.00000	−	6.92820i	−0.402015	−	0.696311i
$100$		0			0
$101$	9.50000	+	16.4545i	0.945285	+	1.63728i	0.755179	+	0.655519i	$0.227550\pi$
$101$	9.50000	+	16.4545i	0.945285	+	1.63728i	0.190106	+	0.981763i	$0.439117\pi$
$102$		0			0
$103$		0			0			−	1.00000i	$-0.5\pi$
$103$		0			0				1.00000i	$0.5\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−12.0000			−1.16008			−0.580042	−	0.814587i	$-0.696964\pi$
$107$	−12.0000			−1.16008			−0.580042	+	0.814587i	$0.696964\pi$
$108$		0			0
$109$	3.50000	−	6.06218i	0.335239	−	0.580651i	−0.648292	−	0.761392i	$-0.724516\pi$
$109$	3.50000	−	6.06218i	0.335239	−	0.580651i	0.983531	+	0.180741i	$0.0578495\pi$
$110$		0			0
$111$	1.00000	−	1.73205i	0.0949158	−	0.164399i
$112$		0			0
$113$	6.00000			0.564433			0.282216	−	0.959351i	$-0.408930\pi$
$113$	6.00000			0.564433			0.282216	+	0.959351i	$0.408930\pi$
$114$		0			0
$115$	−3.00000			−0.279751
$116$		0			0
$117$	5.00000	−	8.66025i	0.462250	−	0.800641i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	5.00000			0.454545
$122$		0			0
$123$	2.50000	+	4.33013i	0.225417	+	0.390434i
$124$		0			0
$125$	3.00000			0.268328
$126$		0			0
$127$	7.50000	+	12.9904i	0.665517	+	1.15271i	0.979145	+	0.203164i	$0.0651224\pi$
$127$	7.50000	+	12.9904i	0.665517	+	1.15271i	−0.313627	+	0.949546i	$0.601544\pi$
$128$		0			0
$129$	5.50000	+	9.52628i	0.484248	+	0.838742i
$130$		0			0
$131$	7.50000	−	12.9904i	0.655278	−	1.13497i	−0.326546	−	0.945181i	$-0.605885\pi$
$131$	7.50000	−	12.9904i	0.655278	−	1.13497i	0.981824	−	0.189794i	$-0.0607819\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	−7.50000	+	12.9904i	−0.645497	+	1.11803i
$136$		0			0
$137$	2.50000	+	4.33013i	0.213589	+	0.369948i	0.952835	−	0.303488i	$-0.0981512\pi$
$137$	2.50000	+	4.33013i	0.213589	+	0.369948i	−0.739246	+	0.673436i	$0.764818\pi$
$138$		0			0
$139$	−7.50000	−	12.9904i	−0.636142	−	1.10183i	−0.986272	−	0.165129i	$-0.947196\pi$
$139$	−7.50000	−	12.9904i	−0.636142	−	1.10183i	0.350130	−	0.936701i	$-0.386137\pi$
$140$		0			0
$141$	5.00000			0.421076
$142$		0			0
$143$	10.0000	+	17.3205i	0.836242	+	1.44841i
$144$		0			0
$145$	9.00000			0.747409
$146$		0			0
$147$	3.50000	−	6.06218i	0.288675	−	0.500000i
$148$		0			0
$149$	−8.50000	+	14.7224i	−0.696347	+	1.20611i	0.273377	+	0.961907i	$0.411859\pi$
$149$	−8.50000	+	14.7224i	−0.696347	+	1.20611i	−0.969724	+	0.244202i	$0.921474\pi$
$150$		0			0
$151$	−16.0000			−1.30206			−0.651031	−	0.759051i	$-0.725663\pi$
$151$	−16.0000			−1.30206			−0.651031	+	0.759051i	$0.725663\pi$
$152$		0			0
$153$	10.0000			0.808452
$154$		0			0
$155$	−6.00000	+	10.3923i	−0.481932	+	0.834730i
$156$		0			0
$157$	−6.50000	+	11.2583i	−0.518756	+	0.898513i	0.481006	+	0.876717i	$0.340272\pi$
$157$	−6.50000	+	11.2583i	−0.518756	+	0.898513i	−0.999762	+	0.0217953i	$0.993062\pi$
$158$		0			0
$159$	9.00000			0.713746
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−4.00000			−0.313304			−0.156652	−	0.987654i	$-0.550070\pi$
$163$	−4.00000			−0.313304			−0.156652	+	0.987654i	$0.550070\pi$
$164$		0			0
$165$	−6.00000	−	10.3923i	−0.467099	−	0.809040i
$166$		0			0
$167$	−2.50000	−	4.33013i	−0.193456	−	0.335075i	0.752937	−	0.658092i	$-0.228636\pi$
$167$	−2.50000	−	4.33013i	−0.193456	−	0.335075i	−0.946393	+	0.323017i	$0.895303\pi$
$168$		0			0
$169$	−6.00000	+	10.3923i	−0.461538	+	0.799408i
$170$		0			0
$171$	7.00000	+	5.19615i	0.535303	+	0.397360i
$172$		0			0
$173$	−2.50000	+	4.33013i	−0.190071	+	0.329213i	−0.945274	−	0.326278i	$-0.894205\pi$
$173$	−2.50000	+	4.33013i	−0.190071	+	0.329213i	0.755202	+	0.655492i	$0.227539\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	−6.50000	−	11.2583i	−0.488570	−	0.846228i
$178$		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
$181$	3.50000	+	6.06218i	0.260153	+	0.450598i	0.966282	−	0.257485i	$-0.0828937\pi$
$181$	3.50000	+	6.06218i	0.260153	+	0.450598i	−0.706129	+	0.708083i	$0.749560\pi$
$182$		0			0
$183$	1.00000			0.0739221
$184$		0			0
$185$	−3.00000	+	5.19615i	−0.220564	+	0.382029i
$186$		0			0
$187$	−10.0000	+	17.3205i	−0.731272	+	1.26660i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	16.0000			1.15772			0.578860	−	0.815427i	$-0.303498\pi$
$191$	16.0000			1.15772			0.578860	+	0.815427i	$0.303498\pi$
$192$		0			0
$193$	−7.50000	+	12.9904i	−0.539862	+	0.935068i	0.459049	+	0.888411i	$0.348190\pi$
$193$	−7.50000	+	12.9904i	−0.539862	+	0.935068i	−0.998911	+	0.0466572i	$0.985143\pi$
$194$		0			0
$195$	7.50000	−	12.9904i	0.537086	−	0.930261i
$196$		0			0
$197$	−10.0000			−0.712470			−0.356235	−	0.934396i	$-0.615940\pi$
$197$	−10.0000			−0.712470			−0.356235	+	0.934396i	$0.615940\pi$
$198$		0			0
$199$	1.50000	+	2.59808i	0.106332	+	0.184173i	0.914282	−	0.405079i	$-0.132756\pi$
$199$	1.50000	+	2.59808i	0.106332	+	0.184173i	−0.807950	+	0.589252i	$0.799423\pi$
$200$		0			0
$201$	−5.00000			−0.352673
$202$		0			0
$203$		0			0
$204$		0			0
$205$	−7.50000	−	12.9904i	−0.523823	−	0.907288i
$206$		0			0
$207$	1.00000	−	1.73205i	0.0695048	−	0.120386i
$208$		0			0
$209$	−16.0000	+	6.92820i	−1.10674	+	0.479234i
$210$		0			0
$211$	−4.50000	+	7.79423i	−0.309793	+	0.536577i	−0.978317	−	0.207114i	$-0.933593\pi$
$211$	−4.50000	+	7.79423i	−0.309793	+	0.536577i	0.668524	+	0.743690i	$0.266926\pi$
$212$		0			0
$213$	0.500000	+	0.866025i	0.0342594	+	0.0593391i
$214$		0			0
$215$	−16.5000	−	28.5788i	−1.12529	−	1.94906i
$216$		0			0
$217$		0			0
$218$		0			0
$219$	4.50000	+	7.79423i	0.304082	+	0.526685i
$220$		0			0
$221$	−25.0000			−1.68168
$222$		0			0
$223$	−5.50000	+	9.52628i	−0.368307	+	0.637927i	−0.989301	−	0.145889i	$-0.953396\pi$
$223$	−5.50000	+	9.52628i	−0.368307	+	0.637927i	0.620994	+	0.783815i	$0.286729\pi$
$224$		0			0
$225$	4.00000	−	6.92820i	0.266667	−	0.461880i
$226$		0			0
$227$	12.0000			0.796468			0.398234	−	0.917284i	$-0.369623\pi$
$227$	12.0000			0.796468			0.398234	+	0.917284i	$0.369623\pi$
$228$		0			0
$229$	22.0000			1.45380			0.726900	−	0.686743i	$-0.240960\pi$
$229$	22.0000			1.45380			0.726900	+	0.686743i	$0.240960\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−5.50000	+	9.52628i	−0.360317	+	0.624087i	−0.988013	−	0.154371i	$-0.950665\pi$
$233$	−5.50000	+	9.52628i	−0.360317	+	0.624087i	0.627696	+	0.778459i	$0.283998\pi$
$234$		0			0
$235$	−15.0000			−0.978492
$236$		0			0
$237$	8.50000	+	14.7224i	0.552134	+	0.956325i
$238$		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	0.976068	+	0.217465i	$0.0697789\pi$
$241$	10.5000	+	18.1865i	0.676364	+	1.17150i	−0.299704	+	0.954032i	$0.596888\pi$
$242$		0			0
$243$	−8.00000	−	13.8564i	−0.513200	−	0.888889i
$244$		0			0
$245$	−10.5000	+	18.1865i	−0.670820	+	1.16190i
$246$		0			0
$247$	−17.5000	−	12.9904i	−1.11350	−	0.826558i
$248$		0			0
$249$	−8.00000	+	13.8564i	−0.506979	+	0.878114i
$250$		0			0
$251$	4.50000	+	7.79423i	0.284037	+	0.491967i	0.972375	−	0.233423i	$-0.0749927\pi$
$251$	4.50000	+	7.79423i	0.284037	+	0.491967i	−0.688338	+	0.725390i	$0.741659\pi$
$252$		0			0
$253$	2.00000	+	3.46410i	0.125739	+	0.217786i
$254$		0			0
$255$	15.0000			0.939336
$256$		0			0
$257$	12.5000	+	21.6506i	0.779729	+	1.35053i	0.932098	+	0.362206i	$0.117976\pi$
$257$	12.5000	+	21.6506i	0.779729	+	1.35053i	−0.152370	+	0.988324i	$0.548690\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	−3.00000	+	5.19615i	−0.185695	+	0.321634i
$262$		0			0
$263$	−9.50000	+	16.4545i	−0.585795	+	1.01463i	0.408981	+	0.912543i	$0.365884\pi$
$263$	−9.50000	+	16.4545i	−0.585795	+	1.01463i	−0.994776	+	0.102084i	$0.967449\pi$
$264$		0			0
$265$	−27.0000			−1.65860
$266$		0			0
$267$	3.00000			0.183597
$268$		0			0
$269$	−8.50000	+	14.7224i	−0.518254	+	0.897643i	0.481521	+	0.876435i	$0.340085\pi$
$269$	−8.50000	+	14.7224i	−0.518254	+	0.897643i	−0.999775	+	0.0212079i	$0.993249\pi$
$270$		0			0
$271$	14.5000	−	25.1147i	0.880812	−	1.52561i	0.0303728	−	0.999539i	$-0.490331\pi$
$271$	14.5000	−	25.1147i	0.880812	−	1.52561i	0.850439	−	0.526073i	$-0.176336\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	8.00000	+	13.8564i	0.482418	+	0.835573i
$276$		0			0
$277$	30.0000			1.80253			0.901263	−	0.433273i	$-0.142641\pi$
$277$	30.0000			1.80253			0.901263	+	0.433273i	$0.142641\pi$
$278$		0			0
$279$	−4.00000	−	6.92820i	−0.239474	−	0.414781i
$280$		0			0
$281$	−7.50000	−	12.9904i	−0.447412	−	0.774941i	0.550804	−	0.834634i	$-0.314321\pi$
$281$	−7.50000	−	12.9904i	−0.447412	−	0.774941i	−0.998217	+	0.0596933i	$0.980988\pi$
$282$		0			0
$283$	7.50000	−	12.9904i	0.445829	−	0.772198i	−0.552281	−	0.833658i	$-0.686242\pi$
$283$	7.50000	−	12.9904i	0.445829	−	0.772198i	0.998110	+	0.0614601i	$0.0195757\pi$
$284$		0			0
$285$	10.5000	+	7.79423i	0.621966	+	0.461690i
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−4.00000	−	6.92820i	−0.235294	−	0.407541i
$290$		0			0
$291$	6.50000	+	11.2583i	0.381037	+	0.659975i
$292$		0			0
$293$	−10.0000			−0.584206			−0.292103	−	0.956387i	$-0.594355\pi$
$293$	−10.0000			−0.584206			−0.292103	+	0.956387i	$0.594355\pi$
$294$		0			0
$295$	19.5000	+	33.7750i	1.13533	+	1.96646i
$296$		0			0
$297$	20.0000			1.16052
$298$		0			0
$299$	−2.50000	+	4.33013i	−0.144579	+	0.250418i
$300$		0			0
$301$		0			0
$302$		0			0
$303$	−19.0000			−1.09152
$304$		0			0
$305$	−3.00000			−0.171780
$306$		0			0
$307$	1.50000	−	2.59808i	0.0856095	−	0.148280i	−0.820041	−	0.572304i	$-0.806050\pi$
$307$	1.50000	−	2.59808i	0.0856095	−	0.148280i	0.905651	+	0.424024i	$0.139383\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−12.0000			−0.680458			−0.340229	−	0.940343i	$-0.610505\pi$
$311$	−12.0000			−0.680458			−0.340229	+	0.940343i	$0.610505\pi$
$312$		0			0
$313$	2.50000	+	4.33013i	0.141308	+	0.244753i	0.927990	−	0.372606i	$-0.121536\pi$
$313$	2.50000	+	4.33013i	0.141308	+	0.244753i	−0.786681	+	0.617359i	$0.788202\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	13.5000	+	23.3827i	0.758236	+	1.31330i	0.943750	+	0.330661i	$0.107272\pi$
$317$	13.5000	+	23.3827i	0.758236	+	1.31330i	−0.185514	+	0.982642i	$0.559395\pi$
$318$		0			0
$319$	−6.00000	−	10.3923i	−0.335936	−	0.581857i
$320$		0			0
$321$	6.00000	−	10.3923i	0.334887	−	0.580042i
$322$		0			0
$323$	2.50000	−	21.6506i	0.139104	−	1.20467i
$324$		0			0
$325$	−10.0000	+	17.3205i	−0.554700	+	0.960769i
$326$		0			0
$327$	3.50000	+	6.06218i	0.193550	+	0.335239i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	20.0000			1.09930			0.549650	−	0.835395i	$-0.314761\pi$
$331$	20.0000			1.09930			0.549650	+	0.835395i	$0.314761\pi$
$332$		0			0
$333$	−2.00000	−	3.46410i	−0.109599	−	0.189832i
$334$		0			0
$335$	15.0000			0.819538
$336$		0			0
$337$	−1.50000	+	2.59808i	−0.0817102	+	0.141526i	−0.903985	−	0.427565i	$-0.859372\pi$
$337$	−1.50000	+	2.59808i	−0.0817102	+	0.141526i	0.822274	+	0.569091i	$0.192705\pi$
$338$		0			0
$339$	−3.00000	+	5.19615i	−0.162938	+	0.282216i
$340$		0			0
$341$	16.0000			0.866449
$342$		0			0
$343$		0			0
$344$		0			0
$345$	1.50000	−	2.59808i	0.0807573	−	0.139876i
$346$		0			0
$347$	−2.50000	+	4.33013i	−0.134207	+	0.232453i	−0.925294	−	0.379250i	$-0.876182\pi$
$347$	−2.50000	+	4.33013i	−0.134207	+	0.232453i	0.791087	+	0.611703i	$0.209515\pi$
$348$		0			0
$349$	14.0000			0.749403			0.374701	−	0.927146i	$-0.377745\pi$
$349$	14.0000			0.749403			0.374701	+	0.927146i	$0.377745\pi$
$350$		0			0
$351$	12.5000	+	21.6506i	0.667201	+	1.15563i
$352$		0			0
$353$	−10.0000			−0.532246			−0.266123	−	0.963939i	$-0.585743\pi$
$353$	−10.0000			−0.532246			−0.266123	+	0.963939i	$0.585743\pi$
$354$		0			0
$355$	−1.50000	−	2.59808i	−0.0796117	−	0.137892i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	2.50000	−	4.33013i	0.131945	−	0.228535i	−0.792481	−	0.609896i	$-0.791211\pi$
$359$	2.50000	−	4.33013i	0.131945	−	0.228535i	0.924426	+	0.381361i	$0.124544\pi$
$360$		0			0
$361$	13.0000	−	13.8564i	0.684211	−	0.729285i
$362$		0			0
$363$	−2.50000	+	4.33013i	−0.131216	+	0.227273i
$364$		0			0
$365$	−13.5000	−	23.3827i	−0.706622	−	1.22391i
$366$		0			0
$367$	−2.50000	−	4.33013i	−0.130499	−	0.226031i	0.793370	−	0.608740i	$-0.208325\pi$
$367$	−2.50000	−	4.33013i	−0.130499	−	0.226031i	−0.923869	+	0.382709i	$0.874991\pi$
$368$		0			0
$369$	10.0000			0.520579
$370$		0			0
$371$		0			0
$372$		0			0
$373$	10.0000			0.517780			0.258890	−	0.965907i	$-0.416643\pi$
$373$	10.0000			0.517780			0.258890	+	0.965907i	$0.416643\pi$
$374$		0			0
$375$	−1.50000	+	2.59808i	−0.0774597	+	0.134164i
$376$		0			0
$377$	7.50000	−	12.9904i	0.386270	−	0.669039i
$378$		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
$381$	−15.0000			−0.768473
$382$		0			0
$383$	−7.50000	+	12.9904i	−0.383232	+	0.663777i	−0.991522	−	0.129937i	$-0.958522\pi$
$383$	−7.50000	+	12.9904i	−0.383232	+	0.663777i	0.608290	+	0.793715i	$0.291856\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	22.0000			1.11832
$388$		0			0
$389$	−8.50000	−	14.7224i	−0.430967	−	0.746457i	0.565990	−	0.824412i	$-0.308494\pi$
$389$	−8.50000	−	14.7224i	−0.430967	−	0.746457i	−0.996957	+	0.0779554i	$0.975161\pi$
$390$		0			0
$391$	−5.00000			−0.252861
$392$		0			0
$393$	7.50000	+	12.9904i	0.378325	+	0.655278i
$394$		0			0
$395$	−25.5000	−	44.1673i	−1.28304	−	2.22230i
$396$		0			0
$397$	17.5000	−	30.3109i	0.878300	−	1.52126i	0.0250943	−	0.999685i	$-0.492011\pi$
$397$	17.5000	−	30.3109i	0.878300	−	1.52126i	0.853206	−	0.521575i	$-0.174655\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−5.50000	+	9.52628i	−0.274657	+	0.475720i	−0.970049	−	0.242911i	$-0.921898\pi$
$401$	−5.50000	+	9.52628i	−0.274657	+	0.475720i	0.695392	+	0.718631i	$0.255231\pi$
$402$		0			0
$403$	10.0000	+	17.3205i	0.498135	+	0.862796i
$404$		0			0
$405$	1.50000	+	2.59808i	0.0745356	+	0.129099i
$406$		0			0
$407$	8.00000			0.396545
$408$		0			0
$409$	−3.50000	−	6.06218i	−0.173064	−	0.299755i	0.766426	−	0.642333i	$-0.222033\pi$
$409$	−3.50000	−	6.06218i	−0.173064	−	0.299755i	−0.939490	+	0.342578i	$0.888700\pi$
$410$		0			0
$411$	−5.00000			−0.246632
$412$		0			0
$413$		0			0
$414$		0			0
$415$	24.0000	−	41.5692i	1.17811	−	2.04055i
$416$		0			0
$417$	15.0000			0.734553
$418$		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
$421$	−14.5000	+	25.1147i	−0.706687	+	1.22402i	0.259393	+	0.965772i	$0.416478\pi$
$421$	−14.5000	+	25.1147i	−0.706687	+	1.22402i	−0.966079	+	0.258245i	$0.916856\pi$
$422$		0			0
$423$	5.00000	−	8.66025i	0.243108	−	0.421076i
$424$		0			0
$425$	−20.0000			−0.970143
$426$		0			0
$427$		0			0
$428$		0			0
$429$	−20.0000			−0.965609
$430$		0			0
$431$	−4.50000	−	7.79423i	−0.216757	−	0.375435i	0.737057	−	0.675830i	$-0.236215\pi$
$431$	−4.50000	−	7.79423i	−0.216757	−	0.375435i	−0.953815	+	0.300395i	$0.902881\pi$
$432$		0			0
$433$	12.5000	+	21.6506i	0.600712	+	1.04046i	0.992713	+	0.120499i	$0.0384494\pi$
$433$	12.5000	+	21.6506i	0.600712	+	1.04046i	−0.392002	+	0.919964i	$0.628217\pi$
$434$		0			0
$435$	−4.50000	+	7.79423i	−0.215758	+	0.373705i
$436$		0			0
$437$	−3.50000	−	2.59808i	−0.167428	−	0.124283i
$438$		0			0
$439$	−3.50000	+	6.06218i	−0.167046	+	0.289332i	−0.937380	−	0.348309i	$-0.886756\pi$
$439$	−3.50000	+	6.06218i	−0.167046	+	0.289332i	0.770334	+	0.637641i	$0.220089\pi$
$440$		0			0
$441$	−7.00000	−	12.1244i	−0.333333	−	0.577350i
$442$		0			0
$443$	−7.50000	−	12.9904i	−0.356336	−	0.617192i	0.631010	−	0.775775i	$-0.282641\pi$
$443$	−7.50000	−	12.9904i	−0.356336	−	0.617192i	−0.987346	+	0.158583i	$0.949307\pi$
$444$		0			0
$445$	−9.00000			−0.426641
$446$		0			0
$447$	−8.50000	−	14.7224i	−0.402036	−	0.696347i
$448$		0			0
$449$	−2.00000			−0.0943858			−0.0471929	−	0.998886i	$-0.515028\pi$
$449$	−2.00000			−0.0943858			−0.0471929	+	0.998886i	$0.515028\pi$
$450$		0			0
$451$	−10.0000	+	17.3205i	−0.470882	+	0.815591i
$452$		0			0
$453$	8.00000	−	13.8564i	0.375873	−	0.651031i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	10.0000			0.467780			0.233890	−	0.972263i	$-0.424854\pi$
$457$	10.0000			0.467780			0.233890	+	0.972263i	$0.424854\pi$
$458$		0			0
$459$	−12.5000	+	21.6506i	−0.583450	+	1.01057i
$460$		0			0
$461$	−0.500000	+	0.866025i	−0.0232873	+	0.0403348i	−0.877434	−	0.479697i	$-0.840747\pi$
$461$	−0.500000	+	0.866025i	−0.0232873	+	0.0403348i	0.854147	+	0.520032i	$0.174080\pi$
$462$		0			0
$463$	20.0000			0.929479			0.464739	−	0.885448i	$-0.346148\pi$
$463$	20.0000			0.929479			0.464739	+	0.885448i	$0.346148\pi$
$464$		0			0
$465$	−6.00000	−	10.3923i	−0.278243	−	0.481932i
$466$		0			0
$467$	12.0000			0.555294			0.277647	−	0.960683i	$-0.410445\pi$
$467$	12.0000			0.555294			0.277647	+	0.960683i	$0.410445\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	−6.50000	−	11.2583i	−0.299504	−	0.518756i
$472$		0			0
$473$	−22.0000	+	38.1051i	−1.01156	+	1.75208i
$474$		0			0
$475$	−14.0000	−	10.3923i	−0.642364	−	0.476832i
$476$		0			0
$477$	9.00000	−	15.5885i	0.412082	−	0.713746i
$478$		0			0
$479$	13.5000	+	23.3827i	0.616831	+	1.06838i	0.990060	+	0.140643i	$0.0449170\pi$
$479$	13.5000	+	23.3827i	0.616831	+	1.06838i	−0.373230	+	0.927739i	$0.621750\pi$
$480$		0			0
$481$	5.00000	+	8.66025i	0.227980	+	0.394874i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−19.5000	−	33.7750i	−0.885449	−	1.53364i
$486$		0			0
$487$	40.0000			1.81257			0.906287	−	0.422664i	$-0.138905\pi$
$487$	40.0000			1.81257			0.906287	+	0.422664i	$0.138905\pi$
$488$		0			0
$489$	2.00000	−	3.46410i	0.0904431	−	0.156652i
$490$		0			0
$491$	−4.50000	+	7.79423i	−0.203082	+	0.351749i	−0.949520	−	0.313707i	$-0.898429\pi$
$491$	−4.50000	+	7.79423i	−0.203082	+	0.351749i	0.746438	+	0.665455i	$0.231763\pi$
$492$		0			0
$493$	15.0000			0.675566
$494$		0			0
$495$	−24.0000			−1.07872
$496$		0			0
$497$		0			0
$498$		0			0
$499$	17.5000	−	30.3109i	0.783408	−	1.35690i	−0.146538	−	0.989205i	$-0.546813\pi$
$499$	17.5000	−	30.3109i	0.783408	−	1.35690i	0.929946	−	0.367697i	$-0.119854\pi$
$500$		0			0
$501$	5.00000			0.223384
$502$		0			0
$503$	7.50000	+	12.9904i	0.334408	+	0.579212i	0.983371	−	0.181608i	$-0.0581302\pi$
$503$	7.50000	+	12.9904i	0.334408	+	0.579212i	−0.648963	+	0.760820i	$0.724797\pi$
$504$		0			0
$505$	57.0000			2.53647
$506$		0			0
$507$	−6.00000	−	10.3923i	−0.266469	−	0.461538i
$508$		0			0
$509$	−2.50000	−	4.33013i	−0.110811	−	0.191930i	0.805287	−	0.592886i	$-0.202011\pi$
$509$	−2.50000	−	4.33013i	−0.110811	−	0.191930i	−0.916097	+	0.400956i	$0.868678\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$	−20.0000	+	8.66025i	−0.883022	+	0.382360i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	10.0000	+	17.3205i	0.439799	+	0.761755i
$518$		0			0
$519$	−2.50000	−	4.33013i	−0.109738	−	0.190071i
$520$		0			0
$521$	−6.00000			−0.262865			−0.131432	−	0.991325i	$-0.541958\pi$
$521$	−6.00000			−0.262865			−0.131432	+	0.991325i	$0.541958\pi$
$522$		0			0
$523$	14.5000	+	25.1147i	0.634041	+	1.09819i	0.986718	+	0.162446i	$0.0519382\pi$
$523$	14.5000	+	25.1147i	0.634041	+	1.09819i	−0.352677	+	0.935745i	$0.614728\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−10.0000	+	17.3205i	−0.435607	+	0.754493i
$528$		0			0
$529$	11.0000	−	19.0526i	0.478261	−	0.828372i
$530$		0			0
$531$	−26.0000			−1.12830
$532$		0			0
$533$	−25.0000			−1.08287
$534$		0			0
$535$	−18.0000	+	31.1769i	−0.778208	+	1.34790i
$536$		0			0
$537$	6.00000	−	10.3923i	0.258919	−	0.448461i
$538$		0			0
$539$	28.0000			1.20605
$540$		0			0
$541$	5.50000	+	9.52628i	0.236463	+	0.409567i	0.959697	−	0.281037i	$-0.0906783\pi$
$541$	5.50000	+	9.52628i	0.236463	+	0.409567i	−0.723234	+	0.690604i	$0.757345\pi$
$542$		0			0
$543$	−7.00000			−0.300399
$544$		0			0
$545$	−10.5000	−	18.1865i	−0.449771	−	0.779026i
$546$		0			0
$547$	−3.50000	−	6.06218i	−0.149649	−	0.259200i	0.781449	−	0.623970i	$-0.214481\pi$
$547$	−3.50000	−	6.06218i	−0.149649	−	0.259200i	−0.931098	+	0.364770i	$0.881148\pi$
$548$		0			0
$549$	1.00000	−	1.73205i	0.0426790	−	0.0739221i
$550$		0			0
$551$	10.5000	+	7.79423i	0.447315	+	0.332045i
$552$		0			0
$553$		0			0
$554$		0			0
$555$	−3.00000	−	5.19615i	−0.127343	−	0.220564i
$556$		0			0
$557$	−16.5000	−	28.5788i	−0.699127	−	1.21092i	−0.968769	−	0.247964i	$-0.920239\pi$
$557$	−16.5000	−	28.5788i	−0.699127	−	1.21092i	0.269642	−	0.962961i	$-0.413095\pi$
$558$		0			0
$559$	−55.0000			−2.32625
$560$		0			0
$561$	−10.0000	−	17.3205i	−0.422200	−	0.731272i
$562$		0			0
$563$	4.00000			0.168580			0.0842900	−	0.996441i	$-0.473138\pi$
$563$	4.00000			0.168580			0.0842900	+	0.996441i	$0.473138\pi$
$564$		0			0
$565$	9.00000	−	15.5885i	0.378633	−	0.655811i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−18.0000			−0.754599			−0.377300	−	0.926091i	$-0.623147\pi$
$569$	−18.0000			−0.754599			−0.377300	+	0.926091i	$0.623147\pi$
$570$		0			0
$571$	−4.00000			−0.167395			−0.0836974	−	0.996491i	$-0.526673\pi$
$571$	−4.00000			−0.167395			−0.0836974	+	0.996491i	$0.526673\pi$
$572$		0			0
$573$	−8.00000	+	13.8564i	−0.334205	+	0.578860i
$574$		0			0
$575$	−2.00000	+	3.46410i	−0.0834058	+	0.144463i
$576$		0			0
$577$	−2.00000			−0.0832611			−0.0416305	−	0.999133i	$-0.513255\pi$
$577$	−2.00000			−0.0832611			−0.0416305	+	0.999133i	$0.513255\pi$
$578$		0			0
$579$	−7.50000	−	12.9904i	−0.311689	−	0.539862i
$580$		0			0
$581$		0			0
$582$		0			0
$583$	18.0000	+	31.1769i	0.745484	+	1.29122i
$584$		0			0
$585$	−15.0000	−	25.9808i	−0.620174	−	1.07417i
$586$		0			0
$587$	−12.5000	+	21.6506i	−0.515930	+	0.893617i	0.483899	+	0.875124i	$0.339220\pi$
$587$	−12.5000	+	21.6506i	−0.515930	+	0.893617i	−0.999829	+	0.0184934i	$0.994113\pi$
$588$		0			0
$589$	−16.0000	+	6.92820i	−0.659269	+	0.285472i
$590$		0			0
$591$	5.00000	−	8.66025i	0.205673	−	0.356235i
$592$		0			0
$593$	−17.5000	−	30.3109i	−0.718639	−	1.24472i	−0.961539	−	0.274668i	$-0.911432\pi$
$593$	−17.5000	−	30.3109i	−0.718639	−	1.24472i	0.242900	−	0.970051i	$-0.421901\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−3.00000			−0.122782
$598$		0			0
$599$	−12.5000	−	21.6506i	−0.510736	−	0.884621i	−0.999923	−	0.0124417i	$-0.996040\pi$
$599$	−12.5000	−	21.6506i	−0.510736	−	0.884621i	0.489186	−	0.872179i	$-0.337294\pi$
$600$		0			0
$601$	−34.0000			−1.38689			−0.693444	−	0.720510i	$-0.743908\pi$
$601$	−34.0000			−1.38689			−0.693444	+	0.720510i	$0.743908\pi$
$602$		0			0
$603$	−5.00000	+	8.66025i	−0.203616	+	0.352673i
$604$		0			0
$605$	7.50000	−	12.9904i	0.304918	−	0.528134i
$606$		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
$611$	−12.5000	+	21.6506i	−0.505696	+	0.875891i
$612$		0			0
$613$	−0.500000	+	0.866025i	−0.0201948	+	0.0349784i	−0.875946	−	0.482409i	$-0.839762\pi$
$613$	−0.500000	+	0.866025i	−0.0201948	+	0.0349784i	0.855751	+	0.517387i	$0.173095\pi$
$614$		0			0
$615$	15.0000			0.604858
$616$		0			0
$617$	−1.50000	−	2.59808i	−0.0603877	−	0.104595i	0.834251	−	0.551385i	$-0.185900\pi$
$617$	−1.50000	−	2.59808i	−0.0603877	−	0.104595i	−0.894639	+	0.446790i	$0.852567\pi$
$618$		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
$621$	2.50000	+	4.33013i	0.100322	+	0.173762i
$622$		0			0
$623$		0			0
$624$		0			0
$625$	14.5000	−	25.1147i	0.580000	−	1.00459i
$626$		0			0
$627$	2.00000	−	17.3205i	0.0798723	−	0.691714i
$628$		0			0
$629$	−5.00000	+	8.66025i	−0.199363	+	0.345307i
$630$		0			0
$631$	−14.5000	−	25.1147i	−0.577236	−	0.999802i	−0.995795	−	0.0916122i	$-0.970798\pi$
$631$	−14.5000	−	25.1147i	−0.577236	−	0.999802i	0.418559	−	0.908190i	$-0.362535\pi$
$632$		0			0
$633$	−4.50000	−	7.79423i	−0.178859	−	0.309793i
$634$		0			0
$635$	45.0000			1.78577
$636$		0			0
$637$	17.5000	+	30.3109i	0.693375	+	1.20096i
$638$		0			0
$639$	2.00000			0.0791188
$640$		0			0
$641$	4.50000	−	7.79423i	0.177739	−	0.307854i	−0.763367	−	0.645966i	$-0.776455\pi$
$641$	4.50000	−	7.79423i	0.177739	−	0.307854i	0.941106	+	0.338112i	$0.109788\pi$
$642$		0			0
$643$	−2.50000	+	4.33013i	−0.0985904	+	0.170764i	−0.911101	−	0.412182i	$-0.864767\pi$
$643$	−2.50000	+	4.33013i	−0.0985904	+	0.170764i	0.812511	+	0.582946i	$0.198100\pi$
$644$		0			0
$645$	33.0000			1.29937
$646$		0			0
$647$		0			0			−	1.00000i	$-0.5\pi$
$647$		0			0				1.00000i	$0.5\pi$
$648$		0			0
$649$	26.0000	−	45.0333i	1.02059	−	1.76771i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	6.00000			0.234798			0.117399	−	0.993085i	$-0.462544\pi$
$653$	6.00000			0.234798			0.117399	+	0.993085i	$0.462544\pi$
$654$		0			0
$655$	−22.5000	−	38.9711i	−0.879148	−	1.52273i
$656$		0			0
$657$	18.0000			0.702247
$658$		0			0
$659$	−9.50000	−	16.4545i	−0.370067	−	0.640976i	0.619508	−	0.784990i	$-0.287332\pi$
$659$	−9.50000	−	16.4545i	−0.370067	−	0.640976i	−0.989576	+	0.144015i	$0.953999\pi$
$660$		0			0
$661$	−14.5000	−	25.1147i	−0.563985	−	0.976850i	−0.997143	−	0.0755324i	$-0.975934\pi$
$661$	−14.5000	−	25.1147i	−0.563985	−	0.976850i	0.433159	−	0.901318i	$-0.357399\pi$
$662$		0			0
$663$	12.5000	−	21.6506i	0.485460	−	0.840841i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	1.50000	−	2.59808i	0.0580802	−	0.100598i
$668$		0			0
$669$	−5.50000	−	9.52628i	−0.212642	−	0.368307i
$670$		0			0
$671$	2.00000	+	3.46410i	0.0772091	+	0.133730i
$672$		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
$675$	10.0000	+	17.3205i	0.384900	+	0.666667i
$676$		0			0
$677$	30.0000			1.15299			0.576497	−	0.817099i	$-0.304419\pi$
$677$	30.0000			1.15299			0.576497	+	0.817099i	$0.304419\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	−6.00000	+	10.3923i	−0.229920	+	0.398234i
$682$		0			0
$683$		0			0			−	1.00000i	$-0.5\pi$
$683$		0			0				1.00000i	$0.5\pi$
$684$		0			0
$685$	15.0000			0.573121
$686$		0			0
$687$	−11.0000	+	19.0526i	−0.419676	+	0.726900i
$688$		0			0
$689$	−22.5000	+	38.9711i	−0.857182	+	1.48468i
$690$		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
$695$	−45.0000			−1.70695
$696$		0			0
$697$	−12.5000	−	21.6506i	−0.473471	−	0.820076i
$698$		0			0
$699$	−5.50000	−	9.52628i	−0.208029	−	0.360317i
$700$		0			0
$701$	11.5000	−	19.9186i	0.434349	−	0.752315i	−0.562893	−	0.826530i	$-0.690312\pi$
$701$	11.5000	−	19.9186i	0.434349	−	0.752315i	0.997242	+	0.0742151i	$0.0236451\pi$
$702$		0			0
$703$	−8.00000	+	3.46410i	−0.301726	+	0.130651i
$704$		0			0
$705$	7.50000	−	12.9904i	0.282466	−	0.489246i
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−8.50000	−	14.7224i	−0.319224	−	0.552913i	0.661102	−	0.750296i	$-0.270089\pi$
$709$	−8.50000	−	14.7224i	−0.319224	−	0.552913i	−0.980326	+	0.197383i	$0.936756\pi$
$710$		0			0
$711$	34.0000			1.27510
$712$		0			0
$713$	2.00000	+	3.46410i	0.0749006	+	0.129732i
$714$		0			0
$715$	60.0000			2.24387
$716$		0			0
$717$	6.00000	−	10.3923i	0.224074	−	0.388108i
$718$		0			0
$719$	−13.5000	+	23.3827i	−0.503465	+	0.872027i	0.496527	+	0.868021i	$0.334608\pi$
$719$	−13.5000	+	23.3827i	−0.503465	+	0.872027i	−0.999992	+	0.00400572i	$0.998725\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$	−21.0000			−0.780998
$724$		0			0
$725$	6.00000	−	10.3923i	0.222834	−	0.385961i
$726$		0			0
$727$	6.50000	−	11.2583i	0.241072	−	0.417548i	−0.719948	−	0.694028i	$-0.755834\pi$
$727$	6.50000	−	11.2583i	0.241072	−	0.417548i	0.961020	+	0.276479i	$0.0891678\pi$
$728$		0			0
$729$	13.0000			0.481481
$730$		0			0
$731$	−27.5000	−	47.6314i	−1.01712	−	1.76171i
$732$		0			0
$733$	6.00000			0.221615			0.110808	−	0.993842i	$-0.464656\pi$
$733$	6.00000			0.221615			0.110808	+	0.993842i	$0.464656\pi$
$734$		0			0
$735$	−10.5000	−	18.1865i	−0.387298	−	0.670820i
$736$		0			0
$737$	−10.0000	−	17.3205i	−0.368355	−	0.638009i
$738$		0			0
$739$	−18.5000	+	32.0429i	−0.680534	+	1.17872i	0.294285	+	0.955718i	$0.404919\pi$
$739$	−18.5000	+	32.0429i	−0.680534	+	1.17872i	−0.974818	+	0.223001i	$0.928415\pi$
$740$		0			0
$741$	20.0000	−	8.66025i	0.734718	−	0.318142i
$742$		0			0
$743$	14.5000	−	25.1147i	0.531953	−	0.921370i	−0.467351	−	0.884072i	$-0.654791\pi$
$743$	14.5000	−	25.1147i	0.531953	−	0.921370i	0.999304	−	0.0372984i	$-0.0118752\pi$
$744$		0			0
$745$	25.5000	+	44.1673i	0.934248	+	1.61816i
$746$		0			0
$747$	16.0000	+	27.7128i	0.585409	+	1.01396i
$748$		0			0
$749$		0			0
$750$		0			0
$751$	15.5000	+	26.8468i	0.565603	+	0.979653i	0.996993	+	0.0774878i	$0.0246899\pi$
$751$	15.5000	+	26.8468i	0.565603	+	0.979653i	−0.431390	+	0.902165i	$0.641977\pi$
$752$		0			0
$753$	−9.00000			−0.327978
$754$		0			0
$755$	−24.0000	+	41.5692i	−0.873449	+	1.51286i
$756$		0			0
$757$	17.5000	−	30.3109i	0.636048	−	1.10167i	−0.350244	−	0.936659i	$-0.613901\pi$
$757$	17.5000	−	30.3109i	0.636048	−	1.10167i	0.986292	−	0.165009i	$-0.0527654\pi$
$758$		0			0
$759$	−4.00000			−0.145191
$760$		0			0
$761$	38.0000			1.37750			0.688749	−	0.724999i	$-0.258160\pi$
$761$	38.0000			1.37750			0.688749	+	0.724999i	$0.258160\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	15.0000	−	25.9808i	0.542326	−	0.939336i
$766$		0			0
$767$	65.0000			2.34701
$768$		0			0
$769$	−11.5000	−	19.9186i	−0.414701	−	0.718283i	0.580696	−	0.814120i	$-0.302780\pi$
$769$	−11.5000	−	19.9186i	−0.414701	−	0.718283i	−0.995397	+	0.0958377i	$0.969447\pi$
$770$		0			0
$771$	−25.0000			−0.900353
$772$		0			0
$773$	−12.5000	−	21.6506i	−0.449594	−	0.778719i	0.548766	−	0.835976i	$-0.315098\pi$
$773$	−12.5000	−	21.6506i	−0.449594	−	0.778719i	−0.998359	+	0.0572570i	$0.981765\pi$
$774$		0			0
$775$	8.00000	+	13.8564i	0.287368	+	0.497737i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	2.50000	−	21.6506i	0.0895718	−	0.775715i
$780$		0			0
$781$	−2.00000	+	3.46410i	−0.0715656	+	0.123955i
$782$		0			0
$783$	−7.50000	−	12.9904i	−0.268028	−	0.464238i
$784$		0			0
$785$	19.5000	+	33.7750i	0.695985	+	1.20548i
$786$		0			0
$787$	20.0000			0.712923			0.356462	−	0.934310i	$-0.383983\pi$
$787$	20.0000			0.712923			0.356462	+	0.934310i	$0.383983\pi$
$788$		0			0
$789$	−9.50000	−	16.4545i	−0.338209	−	0.585795i
$790$		0			0
$791$		0			0
$792$		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)

\(f(q)\)	\(=\)	\(q+(-0.500000 + 0.866025i) q^{3} +(1.50000 - 2.59808i) q^{5} +(1.00000 + 1.73205i) q^{9} +O(q^{10})\)	\(q+(-0.500000 + 0.866025i) q^{3} +(1.50000 - 2.59808i) q^{5} +(1.00000 + 1.73205i) q^{9} -4.00000 q^{11} +(-2.50000 - 4.33013i) q^{13} +(1.50000 + 2.59808i) q^{15} +(2.50000 - 4.33013i) q^{17} +(4.00000 - 1.73205i) q^{19} +(-0.500000 - 0.866025i) q^{23} +(-2.00000 - 3.46410i) q^{25} -5.00000 q^{27} +(1.50000 + 2.59808i) q^{29} -4.00000 q^{31} +(2.00000 - 3.46410i) q^{33} -2.00000 q^{37} +5.00000 q^{39} +(2.50000 - 4.33013i) q^{41} +(5.50000 - 9.52628i) q^{43} +6.00000 q^{45} +(-2.50000 - 4.33013i) q^{47} -7.00000 q^{49} +(2.50000 + 4.33013i) q^{51} +(-4.50000 - 7.79423i) q^{53} +(-6.00000 + 10.3923i) q^{55} +(-0.500000 + 4.33013i) q^{57} +(-6.50000 + 11.2583i) q^{59} +(-0.500000 - 0.866025i) q^{61} -15.0000 q^{65} +(2.50000 + 4.33013i) q^{67} +1.00000 q^{69} +(0.500000 - 0.866025i) q^{71} +(4.50000 - 7.79423i) q^{73} +4.00000 q^{75} +(8.50000 - 14.7224i) q^{79} +(-0.500000 + 0.866025i) q^{81} +16.0000 q^{83} +(-7.50000 - 12.9904i) q^{85} -3.00000 q^{87} +(-1.50000 - 2.59808i) q^{89} +(2.00000 - 3.46410i) q^{93} +(1.50000 - 12.9904i) q^{95} +(6.50000 - 11.2583i) q^{97} +(-4.00000 - 6.92820i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{3} + 3 q^{5} + 2 q^{9}+O(q^{10}) \) 2 * q - q^3 + 3 * q^5 + 2 * q^9	\( 2 q - q^{3} + 3 q^{5} + 2 q^{9} - 8 q^{11} - 5 q^{13} + 3 q^{15} + 5 q^{17} + 8 q^{19} - q^{23} - 4 q^{25} - 10 q^{27} + 3 q^{29} - 8 q^{31} + 4 q^{33} - 4 q^{37} + 10 q^{39} + 5 q^{41} + 11 q^{43} + 12 q^{45} - 5 q^{47} - 14 q^{49} + 5 q^{51} - 9 q^{53} - 12 q^{55} - q^{57} - 13 q^{59} - q^{61} - 30 q^{65} + 5 q^{67} + 2 q^{69} + q^{71} + 9 q^{73} + 8 q^{75} + 17 q^{79} - q^{81} + 32 q^{83} - 15 q^{85} - 6 q^{87} - 3 q^{89} + 4 q^{93} + 3 q^{95} + 13 q^{97} - 8 q^{99}+O(q^{100}) \) 2 * q - q^3 + 3 * q^5 + 2 * q^9 - 8 * q^11 - 5 * q^13 + 3 * q^15 + 5 * q^17 + 8 * q^19 - q^23 - 4 * q^25 - 10 * q^27 + 3 * q^29 - 8 * q^31 + 4 * q^33 - 4 * q^37 + 10 * q^39 + 5 * q^41 + 11 * q^43 + 12 * q^45 - 5 * q^47 - 14 * q^49 + 5 * q^51 - 9 * q^53 - 12 * q^55 - q^57 - 13 * q^59 - q^61 - 30 * q^65 + 5 * q^67 + 2 * q^69 + q^71 + 9 * q^73 + 8 * q^75 + 17 * q^79 - q^81 + 32 * q^83 - 15 * q^85 - 6 * q^87 - 3 * q^89 + 4 * q^93 + 3 * q^95 + 13 * q^97 - 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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