LMFDB - Embedded newform 416.2.i.a.289.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [416,2,Mod(289,416)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(416, 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("416.289");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 416 = 2^{5} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	416.i (of order $3$, degree $2$, 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:	$3.32177672409$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{9}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			289.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	416.289
Dual form			416.2.i.a.321.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.00000 - 1.73205i) q^{3} +1.00000 q^{5} +(-0.500000 + 0.866025i) q^{9} +O(q^{10})$	\(q+(-1.00000 - 1.73205i) q^{3} +1.00000 q^{5} +(-0.500000 + 0.866025i) q^{9} +(-2.00000 - 3.46410i) q^{11} +(2.50000 - 2.59808i) q^{13} +(-1.00000 - 1.73205i) q^{15} +(-1.50000 + 2.59808i) q^{17} +(1.00000 - 1.73205i) q^{19} +(-1.00000 - 1.73205i) q^{23} -4.00000 q^{25} -4.00000 q^{27} +(-2.50000 - 4.33013i) q^{29} +2.00000 q^{31} +(-4.00000 + 6.92820i) q^{33} +(-2.50000 - 4.33013i) q^{37} +(-7.00000 - 1.73205i) q^{39} +(-1.50000 - 2.59808i) q^{41} +(2.00000 - 3.46410i) q^{43} +(-0.500000 + 0.866025i) q^{45} +6.00000 q^{47} +(3.50000 + 6.06218i) q^{49} +6.00000 q^{51} +13.0000 q^{53} +(-2.00000 - 3.46410i) q^{55} -4.00000 q^{57} +(-6.00000 + 10.3923i) q^{59} +(3.50000 - 6.06218i) q^{61} +(2.50000 - 2.59808i) q^{65} +(7.00000 + 12.1244i) q^{67} +(-2.00000 + 3.46410i) q^{69} +(-3.00000 + 5.19615i) q^{71} +7.00000 q^{73} +(4.00000 + 6.92820i) q^{75} +8.00000 q^{79} +(5.50000 + 9.52628i) q^{81} +4.00000 q^{83} +(-1.50000 + 2.59808i) q^{85} +(-5.00000 + 8.66025i) q^{87} +(-7.00000 - 12.1244i) q^{89} +(-2.00000 - 3.46410i) q^{93} +(1.00000 - 1.73205i) q^{95} +(1.00000 - 1.73205i) q^{97} +4.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} + 2 q^{5} - q^{9}+O(q^{10}) $ 2 * q - 2 * q^3 + 2 * q^5 - q^9	$ 2 q - 2 q^{3} + 2 q^{5} - q^{9} - 4 q^{11} + 5 q^{13} - 2 q^{15} - 3 q^{17} + 2 q^{19} - 2 q^{23} - 8 q^{25} - 8 q^{27} - 5 q^{29} + 4 q^{31} - 8 q^{33} - 5 q^{37} - 14 q^{39} - 3 q^{41} + 4 q^{43} - q^{45} + 12 q^{47} + 7 q^{49} + 12 q^{51} + 26 q^{53} - 4 q^{55} - 8 q^{57} - 12 q^{59} + 7 q^{61} + 5 q^{65} + 14 q^{67} - 4 q^{69} - 6 q^{71} + 14 q^{73} + 8 q^{75} + 16 q^{79} + 11 q^{81} + 8 q^{83} - 3 q^{85} - 10 q^{87} - 14 q^{89} - 4 q^{93} + 2 q^{95} + 2 q^{97} + 8 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 + 2 * q^5 - q^9 - 4 * q^11 + 5 * q^13 - 2 * q^15 - 3 * q^17 + 2 * q^19 - 2 * q^23 - 8 * q^25 - 8 * q^27 - 5 * q^29 + 4 * q^31 - 8 * q^33 - 5 * q^37 - 14 * q^39 - 3 * q^41 + 4 * q^43 - q^45 + 12 * q^47 + 7 * q^49 + 12 * q^51 + 26 * q^53 - 4 * q^55 - 8 * q^57 - 12 * q^59 + 7 * q^61 + 5 * q^65 + 14 * q^67 - 4 * q^69 - 6 * q^71 + 14 * q^73 + 8 * q^75 + 16 * q^79 + 11 * q^81 + 8 * q^83 - 3 * q^85 - 10 * q^87 - 14 * q^89 - 4 * q^93 + 2 * q^95 + 2 * q^97 + 8 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/416\mathbb{Z}\right)^\times$.

$n$	$261$	$287$	$353$
$\chi(n)$	$1$	$1$	$e\left(\frac{1}{3}\right)$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$	−1.00000	−	1.73205i	−0.577350	−	1.00000i	−0.995782	−	0.0917517i	$-0.970753\pi$
$3$	−1.00000	−	1.73205i	−0.577350	−	1.00000i	0.418432	−	0.908248i	$-0.362580\pi$
$4$		0			0
$5$	1.00000			0.447214			0.223607	−	0.974679i	$-0.428217\pi$
$5$	1.00000			0.447214			0.223607	+	0.974679i	$0.428217\pi$
$6$		0			0
$7$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$7$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$8$		0			0
$9$	−0.500000	+	0.866025i	−0.166667	+	0.288675i
$10$		0			0
$11$	−2.00000	−	3.46410i	−0.603023	−	1.04447i	−0.992361	−	0.123371i	$-0.960630\pi$
$11$	−2.00000	−	3.46410i	−0.603023	−	1.04447i	0.389338	−	0.921095i	$-0.372704\pi$
$12$		0			0
$13$	2.50000	−	2.59808i	0.693375	−	0.720577i
$13$	2.50000	−	2.59808i	0.693375	−	0.720577i
$14$		0			0
$15$	−1.00000	−	1.73205i	−0.258199	−	0.447214i
$16$		0			0
$17$	−1.50000	+	2.59808i	−0.363803	+	0.630126i	−0.988583	−	0.150675i	$-0.951855\pi$
$17$	−1.50000	+	2.59808i	−0.363803	+	0.630126i	0.624780	+	0.780801i	$0.285189\pi$
$18$		0			0
$19$	1.00000	−	1.73205i	0.229416	−	0.397360i	−0.728219	−	0.685344i	$-0.759652\pi$
$19$	1.00000	−	1.73205i	0.229416	−	0.397360i	0.957635	+	0.287984i	$0.0929851\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−1.00000	−	1.73205i	−0.208514	−	0.361158i	0.742732	−	0.669588i	$-0.233529\pi$
$23$	−1.00000	−	1.73205i	−0.208514	−	0.361158i	−0.951247	+	0.308431i	$0.900196\pi$
$24$		0			0
$25$	−4.00000			−0.800000
$26$		0			0
$27$	−4.00000			−0.769800
$28$		0			0
$29$	−2.50000	−	4.33013i	−0.464238	−	0.804084i	0.534928	−	0.844897i	$-0.320339\pi$
$29$	−2.50000	−	4.33013i	−0.464238	−	0.804084i	−0.999167	+	0.0408130i	$0.987005\pi$
$30$		0			0
$31$	2.00000			0.359211			0.179605	−	0.983739i	$-0.442518\pi$
$31$	2.00000			0.359211			0.179605	+	0.983739i	$0.442518\pi$
$32$		0			0
$33$	−4.00000	+	6.92820i	−0.696311	+	1.20605i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−2.50000	−	4.33013i	−0.410997	−	0.711868i	0.584002	−	0.811752i	$-0.301486\pi$
$37$	−2.50000	−	4.33013i	−0.410997	−	0.711868i	−0.994999	+	0.0998840i	$0.968153\pi$
$38$		0			0
$39$	−7.00000	−	1.73205i	−1.12090	−	0.277350i
$40$		0			0
$41$	−1.50000	−	2.59808i	−0.234261	−	0.405751i	0.724797	−	0.688963i	$-0.241934\pi$
$41$	−1.50000	−	2.59808i	−0.234261	−	0.405751i	−0.959058	+	0.283211i	$0.908600\pi$
$42$		0			0
$43$	2.00000	−	3.46410i	0.304997	−	0.528271i	−0.672264	−	0.740312i	$-0.734678\pi$
$43$	2.00000	−	3.46410i	0.304997	−	0.528271i	0.977261	+	0.212041i	$0.0680112\pi$
$44$		0			0
$45$	−0.500000	+	0.866025i	−0.0745356	+	0.129099i
$46$		0			0
$47$	6.00000			0.875190			0.437595	−	0.899172i	$-0.355830\pi$
$47$	6.00000			0.875190			0.437595	+	0.899172i	$0.355830\pi$
$48$		0			0
$49$	3.50000	+	6.06218i	0.500000	+	0.866025i
$50$		0			0
$51$	6.00000			0.840168
$52$		0			0
$53$	13.0000			1.78569			0.892844	−	0.450367i	$-0.148707\pi$
$53$	13.0000			1.78569			0.892844	+	0.450367i	$0.148707\pi$
$54$		0			0
$55$	−2.00000	−	3.46410i	−0.269680	−	0.467099i
$56$		0			0
$57$	−4.00000			−0.529813
$58$		0			0
$59$	−6.00000	+	10.3923i	−0.781133	+	1.35296i	0.150148	+	0.988663i	$0.452025\pi$
$59$	−6.00000	+	10.3923i	−0.781133	+	1.35296i	−0.931282	+	0.364299i	$0.881308\pi$
$60$		0			0
$61$	3.50000	−	6.06218i	0.448129	−	0.776182i	−0.550135	−	0.835076i	$-0.685424\pi$
$61$	3.50000	−	6.06218i	0.448129	−	0.776182i	0.998264	+	0.0588933i	$0.0187572\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	2.50000	−	2.59808i	0.310087	−	0.322252i
$66$		0			0
$67$	7.00000	+	12.1244i	0.855186	+	1.48123i	0.876472	+	0.481452i	$0.159891\pi$
$67$	7.00000	+	12.1244i	0.855186	+	1.48123i	−0.0212861	+	0.999773i	$0.506776\pi$
$68$		0			0
$69$	−2.00000	+	3.46410i	−0.240772	+	0.417029i
$70$		0			0
$71$	−3.00000	+	5.19615i	−0.356034	+	0.616670i	−0.987294	−	0.158901i	$-0.949205\pi$
$71$	−3.00000	+	5.19615i	−0.356034	+	0.616670i	0.631260	+	0.775571i	$0.282538\pi$
$72$		0			0
$73$	7.00000			0.819288			0.409644	−	0.912245i	$-0.365653\pi$
$73$	7.00000			0.819288			0.409644	+	0.912245i	$0.365653\pi$
$74$		0			0
$75$	4.00000	+	6.92820i	0.461880	+	0.800000i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	8.00000			0.900070			0.450035	−	0.893011i	$-0.351411\pi$
$79$	8.00000			0.900070			0.450035	+	0.893011i	$0.351411\pi$
$80$		0			0
$81$	5.50000	+	9.52628i	0.611111	+	1.05848i
$82$		0			0
$83$	4.00000			0.439057			0.219529	−	0.975606i	$-0.429548\pi$
$83$	4.00000			0.439057			0.219529	+	0.975606i	$0.429548\pi$
$84$		0			0
$85$	−1.50000	+	2.59808i	−0.162698	+	0.281801i
$86$		0			0
$87$	−5.00000	+	8.66025i	−0.536056	+	0.928477i
$88$		0			0
$89$	−7.00000	−	12.1244i	−0.741999	−	1.28518i	−0.951584	−	0.307389i	$-0.900545\pi$
$89$	−7.00000	−	12.1244i	−0.741999	−	1.28518i	0.209585	−	0.977790i	$-0.432789\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	−2.00000	−	3.46410i	−0.207390	−	0.359211i
$94$		0			0
$95$	1.00000	−	1.73205i	0.102598	−	0.177705i
$96$		0			0
$97$	1.00000	−	1.73205i	0.101535	−	0.175863i	−0.810782	−	0.585348i	$-0.800958\pi$
$97$	1.00000	−	1.73205i	0.101535	−	0.175863i	0.912317	+	0.409484i	$0.134291\pi$
$98$		0			0
$99$	4.00000			0.402015
$100$		0			0
$101$	5.50000	+	9.52628i	0.547270	+	0.947900i	0.998460	+	0.0554722i	$0.0176664\pi$
$101$	5.50000	+	9.52628i	0.547270	+	0.947900i	−0.451190	+	0.892428i	$0.649000\pi$
$102$		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
$107$	−3.00000	−	5.19615i	−0.290021	−	0.502331i	0.683793	−	0.729676i	$-0.260329\pi$
$107$	−3.00000	−	5.19615i	−0.290021	−	0.502331i	−0.973814	+	0.227345i	$0.926996\pi$
$108$		0			0
$109$	−10.0000			−0.957826			−0.478913	−	0.877862i	$-0.658969\pi$
$109$	−10.0000			−0.957826			−0.478913	+	0.877862i	$0.658969\pi$
$110$		0			0
$111$	−5.00000	+	8.66025i	−0.474579	+	0.821995i
$112$		0			0
$113$	4.50000	−	7.79423i	0.423324	−	0.733219i	−0.572938	−	0.819599i	$-0.694196\pi$
$113$	4.50000	−	7.79423i	0.423324	−	0.733219i	0.996262	+	0.0863794i	$0.0275297\pi$
$114$		0			0
$115$	−1.00000	−	1.73205i	−0.0932505	−	0.161515i
$116$		0			0
$117$	1.00000	+	3.46410i	0.0924500	+	0.320256i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−2.50000	+	4.33013i	−0.227273	+	0.393648i
$122$		0			0
$123$	−3.00000	+	5.19615i	−0.270501	+	0.468521i
$124$		0			0
$125$	−9.00000			−0.804984
$126$		0			0
$127$	11.0000	+	19.0526i	0.976092	+	1.69064i	0.676283	+	0.736642i	$0.263590\pi$
$127$	11.0000	+	19.0526i	0.976092	+	1.69064i	0.299809	+	0.953999i	$0.403077\pi$
$128$		0			0
$129$	−8.00000			−0.704361
$130$		0			0
$131$	−18.0000			−1.57267			−0.786334	−	0.617802i	$-0.788023\pi$
$131$	−18.0000			−1.57267			−0.786334	+	0.617802i	$0.788023\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	−4.00000			−0.344265
$136$		0			0
$137$	0.500000	−	0.866025i	0.0427179	−	0.0739895i	−0.843876	−	0.536538i	$-0.819732\pi$
$137$	0.500000	−	0.866025i	0.0427179	−	0.0739895i	0.886594	+	0.462549i	$0.153065\pi$
$138$		0			0
$139$	8.00000	−	13.8564i	0.678551	−	1.17529i	−0.296866	−	0.954919i	$-0.595942\pi$
$139$	8.00000	−	13.8564i	0.678551	−	1.17529i	0.975417	−	0.220366i	$-0.0707252\pi$
$140$		0			0
$141$	−6.00000	−	10.3923i	−0.505291	−	0.875190i
$142$		0			0
$143$	−14.0000	−	3.46410i	−1.17074	−	0.289683i
$144$		0			0
$145$	−2.50000	−	4.33013i	−0.207614	−	0.359597i
$146$		0			0
$147$	7.00000	−	12.1244i	0.577350	−	1.00000i
$148$		0			0
$149$	−10.5000	+	18.1865i	−0.860194	+	1.48990i	0.0115483	+	0.999933i	$0.496324\pi$
$149$	−10.5000	+	18.1865i	−0.860194	+	1.48990i	−0.871742	+	0.489966i	$0.837009\pi$
$150$		0			0
$151$	22.0000			1.79033			0.895167	−	0.445730i	$-0.147056\pi$
$151$	22.0000			1.79033			0.895167	+	0.445730i	$0.147056\pi$
$152$		0			0
$153$	−1.50000	−	2.59808i	−0.121268	−	0.210042i
$154$		0			0
$155$	2.00000			0.160644
$156$		0			0
$157$	5.00000			0.399043			0.199522	−	0.979893i	$-0.436061\pi$
$157$	5.00000			0.399043			0.199522	+	0.979893i	$0.436061\pi$
$158$		0			0
$159$	−13.0000	−	22.5167i	−1.03097	−	1.78569i
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−8.00000	+	13.8564i	−0.626608	+	1.08532i	0.361619	+	0.932326i	$0.382224\pi$
$163$	−8.00000	+	13.8564i	−0.626608	+	1.08532i	−0.988227	+	0.152992i	$0.951109\pi$
$164$		0			0
$165$	−4.00000	+	6.92820i	−0.311400	+	0.539360i
$166$		0			0
$167$	2.00000	+	3.46410i	0.154765	+	0.268060i	0.932973	−	0.359946i	$-0.117205\pi$
$167$	2.00000	+	3.46410i	0.154765	+	0.268060i	−0.778209	+	0.628006i	$0.783871\pi$
$168$		0			0
$169$	−0.500000	−	12.9904i	−0.0384615	−	0.999260i
$170$		0			0
$171$	1.00000	+	1.73205i	0.0764719	+	0.132453i
$172$		0			0
$173$	1.00000	−	1.73205i	0.0760286	−	0.131685i	−0.825505	−	0.564396i	$-0.809109\pi$
$173$	1.00000	−	1.73205i	0.0760286	−	0.131685i	0.901533	+	0.432710i	$0.142443\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	24.0000			1.80395
$178$		0			0
$179$	−8.00000	−	13.8564i	−0.597948	−	1.03568i	−0.993124	−	0.117071i	$-0.962650\pi$
$179$	−8.00000	−	13.8564i	−0.597948	−	1.03568i	0.395175	−	0.918606i	$-0.370684\pi$
$180$		0			0
$181$	5.00000			0.371647			0.185824	−	0.982583i	$-0.440505\pi$
$181$	5.00000			0.371647			0.185824	+	0.982583i	$0.440505\pi$
$182$		0			0
$183$	−14.0000			−1.03491
$184$		0			0
$185$	−2.50000	−	4.33013i	−0.183804	−	0.318357i
$186$		0			0
$187$	12.0000			0.877527
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−3.00000	+	5.19615i	−0.217072	+	0.375980i	−0.953912	−	0.300088i	$-0.902984\pi$
$191$	−3.00000	+	5.19615i	−0.217072	+	0.375980i	0.736839	+	0.676068i	$0.236317\pi$
$192$		0			0
$193$	10.5000	+	18.1865i	0.755807	+	1.30910i	0.944972	+	0.327150i	$0.106088\pi$
$193$	10.5000	+	18.1865i	0.755807	+	1.30910i	−0.189166	+	0.981945i	$0.560578\pi$
$194$		0			0
$195$	−7.00000	−	1.73205i	−0.501280	−	0.124035i
$196$		0			0
$197$	9.00000	+	15.5885i	0.641223	+	1.11063i	0.985160	+	0.171639i	$0.0549062\pi$
$197$	9.00000	+	15.5885i	0.641223	+	1.11063i	−0.343937	+	0.938993i	$0.611761\pi$
$198$		0			0
$199$	7.00000	−	12.1244i	0.496217	−	0.859473i	−0.503774	−	0.863836i	$-0.668055\pi$
$199$	7.00000	−	12.1244i	0.496217	−	0.859473i	0.999990	+	0.00436292i	$0.00138876\pi$
$200$		0			0
$201$	14.0000	−	24.2487i	0.987484	−	1.71037i
$202$		0			0
$203$		0			0
$204$		0			0
$205$	−1.50000	−	2.59808i	−0.104765	−	0.181458i
$206$		0			0
$207$	2.00000			0.139010
$208$		0			0
$209$	−8.00000			−0.553372
$210$		0			0
$211$	−9.00000	−	15.5885i	−0.619586	−	1.07315i	−0.989561	−	0.144112i	$-0.953967\pi$
$211$	−9.00000	−	15.5885i	−0.619586	−	1.07315i	0.369976	−	0.929041i	$-0.379366\pi$
$212$		0			0
$213$	12.0000			0.822226
$214$		0			0
$215$	2.00000	−	3.46410i	0.136399	−	0.236250i
$216$		0			0
$217$		0			0
$218$		0			0
$219$	−7.00000	−	12.1244i	−0.473016	−	0.819288i
$220$		0			0
$221$	3.00000	+	10.3923i	0.201802	+	0.699062i
$222$		0			0
$223$	−9.00000	−	15.5885i	−0.602685	−	1.04388i	−0.992413	−	0.122950i	$-0.960764\pi$
$223$	−9.00000	−	15.5885i	−0.602685	−	1.04388i	0.389728	−	0.920930i	$-0.372569\pi$
$224$		0			0
$225$	2.00000	−	3.46410i	0.133333	−	0.230940i
$226$		0			0
$227$	−9.00000	+	15.5885i	−0.597351	+	1.03464i	0.395860	+	0.918311i	$0.370447\pi$
$227$	−9.00000	+	15.5885i	−0.597351	+	1.03464i	−0.993210	+	0.116331i	$0.962887\pi$
$228$		0			0
$229$	−18.0000			−1.18947			−0.594737	−	0.803921i	$-0.702744\pi$
$229$	−18.0000			−1.18947			−0.594737	+	0.803921i	$0.702744\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	6.00000			0.393073			0.196537	−	0.980497i	$-0.437031\pi$
$233$	6.00000			0.393073			0.196537	+	0.980497i	$0.437031\pi$
$234$		0			0
$235$	6.00000			0.391397
$236$		0			0
$237$	−8.00000	−	13.8564i	−0.519656	−	0.900070i
$238$		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
$241$	−3.50000	+	6.06218i	−0.225455	+	0.390499i	−0.956456	−	0.291877i	$-0.905720\pi$
$241$	−3.50000	+	6.06218i	−0.225455	+	0.390499i	0.731001	+	0.682376i	$0.239053\pi$
$242$		0			0
$243$	5.00000	−	8.66025i	0.320750	−	0.555556i
$244$		0			0
$245$	3.50000	+	6.06218i	0.223607	+	0.387298i
$246$		0			0
$247$	−2.00000	−	6.92820i	−0.127257	−	0.440831i
$248$		0			0
$249$	−4.00000	−	6.92820i	−0.253490	−	0.439057i
$250$		0			0
$251$	11.0000	−	19.0526i	0.694314	−	1.20259i	−0.276098	−	0.961130i	$-0.589041\pi$
$251$	11.0000	−	19.0526i	0.694314	−	1.20259i	0.970411	−	0.241457i	$-0.0776254\pi$
$252$		0			0
$253$	−4.00000	+	6.92820i	−0.251478	+	0.435572i
$254$		0			0
$255$	6.00000			0.375735
$256$		0			0
$257$	−5.50000	−	9.52628i	−0.343081	−	0.594233i	0.641923	−	0.766769i	$-0.278137\pi$
$257$	−5.50000	−	9.52628i	−0.343081	−	0.594233i	−0.985003	+	0.172536i	$0.944804\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	5.00000			0.309492
$262$		0			0
$263$	−12.0000	−	20.7846i	−0.739952	−	1.28163i	−0.952517	−	0.304487i	$-0.901515\pi$
$263$	−12.0000	−	20.7846i	−0.739952	−	1.28163i	0.212565	−	0.977147i	$-0.431818\pi$
$264$		0			0
$265$	13.0000			0.798584
$266$		0			0
$267$	−14.0000	+	24.2487i	−0.856786	+	1.48400i
$268$		0			0
$269$	−7.00000	+	12.1244i	−0.426798	+	0.739235i	−0.996586	−	0.0825561i	$-0.973692\pi$
$269$	−7.00000	+	12.1244i	−0.426798	+	0.739235i	0.569789	+	0.821791i	$0.307025\pi$
$270$		0			0
$271$	−6.00000	−	10.3923i	−0.364474	−	0.631288i	0.624218	−	0.781251i	$-0.285418\pi$
$271$	−6.00000	−	10.3923i	−0.364474	−	0.631288i	−0.988692	+	0.149963i	$0.952085\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	8.00000	+	13.8564i	0.482418	+	0.835573i
$276$		0			0
$277$	−0.500000	+	0.866025i	−0.0300421	+	0.0520344i	−0.880656	−	0.473757i	$-0.842897\pi$
$277$	−0.500000	+	0.866025i	−0.0300421	+	0.0520344i	0.850613	+	0.525792i	$0.176231\pi$
$278$		0			0
$279$	−1.00000	+	1.73205i	−0.0598684	+	0.103695i
$280$		0			0
$281$	−21.0000			−1.25275			−0.626377	−	0.779520i	$-0.715463\pi$
$281$	−21.0000			−1.25275			−0.626377	+	0.779520i	$0.715463\pi$
$282$		0			0
$283$	10.0000	+	17.3205i	0.594438	+	1.02960i	0.993626	+	0.112728i	$0.0359589\pi$
$283$	10.0000	+	17.3205i	0.594438	+	1.02960i	−0.399188	+	0.916869i	$0.630708\pi$
$284$		0			0
$285$	−4.00000			−0.236940
$286$		0			0
$287$		0			0
$288$		0			0
$289$	4.00000	+	6.92820i	0.235294	+	0.407541i
$290$		0			0
$291$	−4.00000			−0.234484
$292$		0			0
$293$	9.50000	−	16.4545i	0.554996	−	0.961281i	−0.442908	−	0.896567i	$-0.646053\pi$
$293$	9.50000	−	16.4545i	0.554996	−	0.961281i	0.997904	−	0.0647140i	$-0.0206135\pi$
$294$		0			0
$295$	−6.00000	+	10.3923i	−0.349334	+	0.605063i
$296$		0			0
$297$	8.00000	+	13.8564i	0.464207	+	0.804030i
$298$		0			0
$299$	−7.00000	−	1.73205i	−0.404820	−	0.100167i
$300$		0			0
$301$		0			0
$302$		0			0
$303$	11.0000	−	19.0526i	0.631933	−	1.09454i
$304$		0			0
$305$	3.50000	−	6.06218i	0.200409	−	0.347119i
$306$		0			0
$307$	26.0000			1.48390			0.741949	−	0.670456i	$-0.233902\pi$
$307$	26.0000			1.48390			0.741949	+	0.670456i	$0.233902\pi$
$308$		0			0
$309$	−14.0000	−	24.2487i	−0.796432	−	1.37946i
$310$		0			0
$311$	10.0000			0.567048			0.283524	−	0.958965i	$-0.408496\pi$
$311$	10.0000			0.567048			0.283524	+	0.958965i	$0.408496\pi$
$312$		0			0
$313$	−34.0000			−1.92179			−0.960897	−	0.276907i	$-0.910691\pi$
$313$	−34.0000			−1.92179			−0.960897	+	0.276907i	$0.910691\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−3.00000			−0.168497			−0.0842484	−	0.996445i	$-0.526849\pi$
$317$	−3.00000			−0.168497			−0.0842484	+	0.996445i	$0.526849\pi$
$318$		0			0
$319$	−10.0000	+	17.3205i	−0.559893	+	0.969762i
$320$		0			0
$321$	−6.00000	+	10.3923i	−0.334887	+	0.580042i
$322$		0			0
$323$	3.00000	+	5.19615i	0.166924	+	0.289122i
$324$		0			0
$325$	−10.0000	+	10.3923i	−0.554700	+	0.576461i
$326$		0			0
$327$	10.0000	+	17.3205i	0.553001	+	0.957826i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	4.00000	−	6.92820i	0.219860	−	0.380808i	−0.734905	−	0.678170i	$-0.762773\pi$
$331$	4.00000	−	6.92820i	0.219860	−	0.380808i	0.954765	+	0.297361i	$0.0961066\pi$
$332$		0			0
$333$	5.00000			0.273998
$334$		0			0
$335$	7.00000	+	12.1244i	0.382451	+	0.662424i
$336$		0			0
$337$	−1.00000			−0.0544735			−0.0272367	−	0.999629i	$-0.508671\pi$
$337$	−1.00000			−0.0544735			−0.0272367	+	0.999629i	$0.508671\pi$
$338$		0			0
$339$	−18.0000			−0.977626
$340$		0			0
$341$	−4.00000	−	6.92820i	−0.216612	−	0.375183i
$342$		0			0
$343$		0			0
$344$		0			0
$345$	−2.00000	+	3.46410i	−0.107676	+	0.186501i
$346$		0			0
$347$	−5.00000	+	8.66025i	−0.268414	+	0.464907i	−0.968452	−	0.249198i	$-0.919833\pi$
$347$	−5.00000	+	8.66025i	−0.268414	+	0.464907i	0.700038	+	0.714105i	$0.253166\pi$
$348$		0			0
$349$	5.00000	+	8.66025i	0.267644	+	0.463573i	0.968253	−	0.249973i	$-0.0804216\pi$
$349$	5.00000	+	8.66025i	0.267644	+	0.463573i	−0.700609	+	0.713545i	$0.747088\pi$
$350$		0			0
$351$	−10.0000	+	10.3923i	−0.533761	+	0.554700i
$352$		0			0
$353$	−1.50000	−	2.59808i	−0.0798369	−	0.138282i	0.823343	−	0.567545i	$-0.192107\pi$
$353$	−1.50000	−	2.59808i	−0.0798369	−	0.138282i	−0.903179	+	0.429263i	$0.858773\pi$
$354$		0			0
$355$	−3.00000	+	5.19615i	−0.159223	+	0.275783i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	20.0000			1.05556			0.527780	−	0.849381i	$-0.323025\pi$
$359$	20.0000			1.05556			0.527780	+	0.849381i	$0.323025\pi$
$360$		0			0
$361$	7.50000	+	12.9904i	0.394737	+	0.683704i
$362$		0			0
$363$	10.0000			0.524864
$364$		0			0
$365$	7.00000			0.366397
$366$		0			0
$367$	11.0000	+	19.0526i	0.574195	+	0.994535i	0.996129	+	0.0879086i	$0.0280183\pi$
$367$	11.0000	+	19.0526i	0.574195	+	0.994535i	−0.421933	+	0.906627i	$0.638648\pi$
$368$		0			0
$369$	3.00000			0.156174
$370$		0			0
$371$		0			0
$372$		0			0
$373$	1.50000	−	2.59808i	0.0776671	−	0.134523i	−0.824576	−	0.565751i	$-0.808586\pi$
$373$	1.50000	−	2.59808i	0.0776671	−	0.134523i	0.902243	+	0.431228i	$0.141920\pi$
$374$		0			0
$375$	9.00000	+	15.5885i	0.464758	+	0.804984i
$376$		0			0
$377$	−17.5000	−	4.33013i	−0.901296	−	0.223013i
$378$		0			0
$379$	−1.00000	−	1.73205i	−0.0513665	−	0.0889695i	0.839199	−	0.543825i	$-0.183024\pi$
$379$	−1.00000	−	1.73205i	−0.0513665	−	0.0889695i	−0.890565	+	0.454855i	$0.849691\pi$
$380$		0			0
$381$	22.0000	−	38.1051i	1.12709	−	1.95218i
$382$		0			0
$383$	−2.00000	+	3.46410i	−0.102195	+	0.177007i	−0.912589	−	0.408879i	$-0.865920\pi$
$383$	−2.00000	+	3.46410i	−0.102195	+	0.177007i	0.810394	+	0.585886i	$0.199253\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	2.00000	+	3.46410i	0.101666	+	0.176090i
$388$		0			0
$389$	33.0000			1.67317			0.836583	−	0.547840i	$-0.184550\pi$
$389$	33.0000			1.67317			0.836583	+	0.547840i	$0.184550\pi$
$390$		0			0
$391$	6.00000			0.303433
$392$		0			0
$393$	18.0000	+	31.1769i	0.907980	+	1.57267i
$394$		0			0
$395$	8.00000			0.402524
$396$		0			0
$397$	17.0000	−	29.4449i	0.853206	−	1.47780i	−0.0250943	−	0.999685i	$-0.507989\pi$
$397$	17.0000	−	29.4449i	0.853206	−	1.47780i	0.878300	−	0.478110i	$-0.158678\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	16.5000	+	28.5788i	0.823971	+	1.42716i	0.902703	+	0.430263i	$0.141579\pi$
$401$	16.5000	+	28.5788i	0.823971	+	1.42716i	−0.0787327	+	0.996896i	$0.525087\pi$
$402$		0			0
$403$	5.00000	−	5.19615i	0.249068	−	0.258839i
$404$		0			0
$405$	5.50000	+	9.52628i	0.273297	+	0.473365i
$406$		0			0
$407$	−10.0000	+	17.3205i	−0.495682	+	0.858546i
$408$		0			0
$409$	4.50000	−	7.79423i	0.222511	−	0.385400i	−0.733059	−	0.680165i	$-0.761908\pi$
$409$	4.50000	−	7.79423i	0.222511	−	0.385400i	0.955570	+	0.294765i	$0.0952414\pi$
$410$		0			0
$411$	−2.00000			−0.0986527
$412$		0			0
$413$		0			0
$414$		0			0
$415$	4.00000			0.196352
$416$		0			0
$417$	−32.0000			−1.56705
$418$		0			0
$419$	−7.00000	−	12.1244i	−0.341972	−	0.592314i	0.642827	−	0.766012i	$-0.277762\pi$
$419$	−7.00000	−	12.1244i	−0.341972	−	0.592314i	−0.984799	+	0.173698i	$0.944428\pi$
$420$		0			0
$421$	9.00000			0.438633			0.219317	−	0.975654i	$-0.429617\pi$
$421$	9.00000			0.438633			0.219317	+	0.975654i	$0.429617\pi$
$422$		0			0
$423$	−3.00000	+	5.19615i	−0.145865	+	0.252646i
$424$		0			0
$425$	6.00000	−	10.3923i	0.291043	−	0.504101i
$426$		0			0
$427$		0			0
$428$		0			0
$429$	8.00000	+	27.7128i	0.386244	+	1.33799i
$430$		0			0
$431$	6.00000	+	10.3923i	0.289010	+	0.500580i	0.973574	−	0.228373i	$-0.0733406\pi$
$431$	6.00000	+	10.3923i	0.289010	+	0.500580i	−0.684564	+	0.728953i	$0.740007\pi$
$432$		0			0
$433$	0.500000	−	0.866025i	0.0240285	−	0.0416185i	−0.853761	−	0.520665i	$-0.825684\pi$
$433$	0.500000	−	0.866025i	0.0240285	−	0.0416185i	0.877790	+	0.479046i	$0.159017\pi$
$434$		0			0
$435$	−5.00000	+	8.66025i	−0.239732	+	0.415227i
$436$		0			0
$437$	−4.00000			−0.191346
$438$		0			0
$439$	−4.00000	−	6.92820i	−0.190910	−	0.330665i	0.754642	−	0.656136i	$-0.227810\pi$
$439$	−4.00000	−	6.92820i	−0.190910	−	0.330665i	−0.945552	+	0.325471i	$0.894477\pi$
$440$		0			0
$441$	−7.00000			−0.333333
$442$		0			0
$443$	−4.00000			−0.190046			−0.0950229	−	0.995475i	$-0.530292\pi$
$443$	−4.00000			−0.190046			−0.0950229	+	0.995475i	$0.530292\pi$
$444$		0			0
$445$	−7.00000	−	12.1244i	−0.331832	−	0.574750i
$446$		0			0
$447$	42.0000			1.98653
$448$		0			0
$449$	5.00000	−	8.66025i	0.235965	−	0.408703i	−0.723588	−	0.690232i	$-0.757508\pi$
$449$	5.00000	−	8.66025i	0.235965	−	0.408703i	0.959553	+	0.281529i	$0.0908417\pi$
$450$		0			0
$451$	−6.00000	+	10.3923i	−0.282529	+	0.489355i
$452$		0			0
$453$	−22.0000	−	38.1051i	−1.03365	−	1.79033i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	12.5000	+	21.6506i	0.584725	+	1.01277i	0.994910	+	0.100771i	$0.0321310\pi$
$457$	12.5000	+	21.6506i	0.584725	+	1.01277i	−0.410184	+	0.912003i	$0.634536\pi$
$458$		0			0
$459$	6.00000	−	10.3923i	0.280056	−	0.485071i
$460$		0			0
$461$	−10.5000	+	18.1865i	−0.489034	+	0.847031i	−0.999920	−	0.0126168i	$-0.995984\pi$
$461$	−10.5000	+	18.1865i	−0.489034	+	0.847031i	0.510887	+	0.859648i	$0.329317\pi$
$462$		0			0
$463$	−20.0000			−0.929479			−0.464739	−	0.885448i	$-0.653852\pi$
$463$	−20.0000			−0.929479			−0.464739	+	0.885448i	$0.653852\pi$
$464$		0			0
$465$	−2.00000	−	3.46410i	−0.0927478	−	0.160644i
$466$		0			0
$467$	24.0000			1.11059			0.555294	−	0.831654i	$-0.312606\pi$
$467$	24.0000			1.11059			0.555294	+	0.831654i	$0.312606\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	−5.00000	−	8.66025i	−0.230388	−	0.399043i
$472$		0			0
$473$	−16.0000			−0.735681
$474$		0			0
$475$	−4.00000	+	6.92820i	−0.183533	+	0.317888i
$476$		0			0
$477$	−6.50000	+	11.2583i	−0.297615	+	0.515484i
$478$		0			0
$479$	−3.00000	−	5.19615i	−0.137073	−	0.237418i	0.789314	−	0.613990i	$-0.210436\pi$
$479$	−3.00000	−	5.19615i	−0.137073	−	0.237418i	−0.926388	+	0.376571i	$0.877103\pi$
$480$		0			0
$481$	−17.5000	−	4.33013i	−0.797931	−	0.197437i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	1.00000	−	1.73205i	0.0454077	−	0.0786484i
$486$		0			0
$487$	−16.0000	+	27.7128i	−0.725029	+	1.25579i	0.233933	+	0.972253i	$0.424840\pi$
$487$	−16.0000	+	27.7128i	−0.725029	+	1.25579i	−0.958962	+	0.283535i	$0.908493\pi$
$488$		0			0
$489$	32.0000			1.44709
$490$		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
$493$	15.0000			0.675566
$494$		0			0
$495$	4.00000			0.179787
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−14.0000			−0.626726			−0.313363	−	0.949633i	$-0.601456\pi$
$499$	−14.0000			−0.626726			−0.313363	+	0.949633i	$0.601456\pi$
$500$		0			0
$501$	4.00000	−	6.92820i	0.178707	−	0.309529i
$502$		0			0
$503$	14.0000	−	24.2487i	0.624229	−	1.08120i	−0.364460	−	0.931219i	$-0.618746\pi$
$503$	14.0000	−	24.2487i	0.624229	−	1.08120i	0.988689	−	0.149978i	$-0.0479202\pi$
$504$		0			0
$505$	5.50000	+	9.52628i	0.244747	+	0.423914i
$506$		0			0
$507$	−22.0000	+	13.8564i	−0.977054	+	0.615385i
$508$		0			0
$509$	1.50000	+	2.59808i	0.0664863	+	0.115158i	0.897352	−	0.441315i	$-0.145488\pi$
$509$	1.50000	+	2.59808i	0.0664863	+	0.115158i	−0.830866	+	0.556473i	$0.812154\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$	−4.00000	+	6.92820i	−0.176604	+	0.305888i
$514$		0			0
$515$	14.0000			0.616914
$516$		0			0
$517$	−12.0000	−	20.7846i	−0.527759	−	0.914106i
$518$		0			0
$519$	−4.00000			−0.175581
$520$		0			0
$521$	−25.0000			−1.09527			−0.547635	−	0.836717i	$-0.684472\pi$
$521$	−25.0000			−1.09527			−0.547635	+	0.836717i	$0.684472\pi$
$522$		0			0
$523$	−8.00000	−	13.8564i	−0.349816	−	0.605898i	0.636401	−	0.771358i	$-0.280422\pi$
$523$	−8.00000	−	13.8564i	−0.349816	−	0.605898i	−0.986216	+	0.165460i	$0.947089\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−3.00000	+	5.19615i	−0.130682	+	0.226348i
$528$		0			0
$529$	9.50000	−	16.4545i	0.413043	−	0.715412i
$530$		0			0
$531$	−6.00000	−	10.3923i	−0.260378	−	0.450988i
$532$		0			0
$533$	−10.5000	−	2.59808i	−0.454805	−	0.112535i
$534$		0			0
$535$	−3.00000	−	5.19615i	−0.129701	−	0.224649i
$536$		0			0
$537$	−16.0000	+	27.7128i	−0.690451	+	1.19590i
$538$		0			0
$539$	14.0000	−	24.2487i	0.603023	−	1.04447i
$540$		0			0
$541$	−23.0000			−0.988847			−0.494424	−	0.869221i	$-0.664621\pi$
$541$	−23.0000			−0.988847			−0.494424	+	0.869221i	$0.664621\pi$
$542$		0			0
$543$	−5.00000	−	8.66025i	−0.214571	−	0.371647i
$544$		0			0
$545$	−10.0000			−0.428353
$546$		0			0
$547$	22.0000			0.940652			0.470326	−	0.882493i	$-0.344136\pi$
$547$	22.0000			0.940652			0.470326	+	0.882493i	$0.344136\pi$
$548$		0			0
$549$	3.50000	+	6.06218i	0.149376	+	0.258727i
$550$		0			0
$551$	−10.0000			−0.426014
$552$		0			0
$553$		0			0
$554$		0			0
$555$	−5.00000	+	8.66025i	−0.212238	+	0.367607i
$556$		0			0
$557$	11.5000	+	19.9186i	0.487271	+	0.843978i	0.999893	−	0.0146368i	$-0.00465919\pi$
$557$	11.5000	+	19.9186i	0.487271	+	0.843978i	−0.512622	+	0.858614i	$0.671326\pi$
$558$		0			0
$559$	−4.00000	−	13.8564i	−0.169182	−	0.586064i
$560$		0			0
$561$	−12.0000	−	20.7846i	−0.506640	−	0.877527i
$562$		0			0
$563$	−12.0000	+	20.7846i	−0.505740	+	0.875967i	0.494238	+	0.869326i	$0.335447\pi$
$563$	−12.0000	+	20.7846i	−0.505740	+	0.875967i	−0.999978	+	0.00664037i	$0.997886\pi$
$564$		0			0
$565$	4.50000	−	7.79423i	0.189316	−	0.327906i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−3.00000	−	5.19615i	−0.125767	−	0.217834i	0.796266	−	0.604947i	$-0.206806\pi$
$569$	−3.00000	−	5.19615i	−0.125767	−	0.217834i	−0.922032	+	0.387113i	$0.873472\pi$
$570$		0			0
$571$	−20.0000			−0.836974			−0.418487	−	0.908223i	$-0.637439\pi$
$571$	−20.0000			−0.836974			−0.418487	+	0.908223i	$0.637439\pi$
$572$		0			0
$573$	12.0000			0.501307
$574$		0			0
$575$	4.00000	+	6.92820i	0.166812	+	0.288926i
$576$		0			0
$577$	−29.0000			−1.20729			−0.603643	−	0.797255i	$-0.706285\pi$
$577$	−29.0000			−1.20729			−0.603643	+	0.797255i	$0.706285\pi$
$578$		0			0
$579$	21.0000	−	36.3731i	0.872730	−	1.51161i
$580$		0			0
$581$		0			0
$582$		0			0
$583$	−26.0000	−	45.0333i	−1.07681	−	1.86509i
$584$		0			0
$585$	1.00000	+	3.46410i	0.0413449	+	0.143223i
$586$		0			0
$587$	−8.00000	−	13.8564i	−0.330195	−	0.571915i	0.652355	−	0.757914i	$-0.273781\pi$
$587$	−8.00000	−	13.8564i	−0.330195	−	0.571915i	−0.982550	+	0.185999i	$0.940448\pi$
$588$		0			0
$589$	2.00000	−	3.46410i	0.0824086	−	0.142736i
$590$		0			0
$591$	18.0000	−	31.1769i	0.740421	−	1.28245i
$592$		0			0
$593$	−1.00000			−0.0410651			−0.0205325	−	0.999789i	$-0.506536\pi$
$593$	−1.00000			−0.0410651			−0.0205325	+	0.999789i	$0.506536\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−28.0000			−1.14596
$598$		0			0
$599$	−30.0000			−1.22577			−0.612883	−	0.790173i	$-0.709990\pi$
$599$	−30.0000			−1.22577			−0.612883	+	0.790173i	$0.709990\pi$
$600$		0			0
$601$	4.50000	+	7.79423i	0.183559	+	0.317933i	0.943090	−	0.332538i	$-0.107905\pi$
$601$	4.50000	+	7.79423i	0.183559	+	0.317933i	−0.759531	+	0.650471i	$0.774572\pi$
$602$		0			0
$603$	−14.0000			−0.570124
$604$		0			0
$605$	−2.50000	+	4.33013i	−0.101639	+	0.176045i
$606$		0			0
$607$	17.0000	−	29.4449i	0.690009	−	1.19513i	−0.281826	−	0.959466i	$-0.590940\pi$
$607$	17.0000	−	29.4449i	0.690009	−	1.19513i	0.971834	−	0.235665i	$-0.0757267\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	15.0000	−	15.5885i	0.606835	−	0.630641i
$612$		0			0
$613$	−0.500000	−	0.866025i	−0.0201948	−	0.0349784i	0.855751	−	0.517387i	$-0.173095\pi$
$613$	−0.500000	−	0.866025i	−0.0201948	−	0.0349784i	−0.875946	+	0.482409i	$0.839762\pi$
$614$		0			0
$615$	−3.00000	+	5.19615i	−0.120972	+	0.209529i
$616$		0			0
$617$	18.5000	−	32.0429i	0.744782	−	1.29000i	−0.205515	−	0.978654i	$-0.565887\pi$
$617$	18.5000	−	32.0429i	0.744782	−	1.29000i	0.950297	−	0.311346i	$-0.100780\pi$
$618$		0			0
$619$	−36.0000			−1.44696			−0.723481	−	0.690344i	$-0.757459\pi$
$619$	−36.0000			−1.44696			−0.723481	+	0.690344i	$0.757459\pi$
$620$		0			0
$621$	4.00000	+	6.92820i	0.160514	+	0.278019i
$622$		0			0
$623$		0			0
$624$		0			0
$625$	11.0000			0.440000
$626$		0			0
$627$	8.00000	+	13.8564i	0.319489	+	0.553372i
$628$		0			0
$629$	15.0000			0.598089
$630$		0			0
$631$	−10.0000	+	17.3205i	−0.398094	+	0.689519i	−0.993491	−	0.113913i	$-0.963661\pi$
$631$	−10.0000	+	17.3205i	−0.398094	+	0.689519i	0.595397	+	0.803432i	$0.296995\pi$
$632$		0			0
$633$	−18.0000	+	31.1769i	−0.715436	+	1.23917i
$634$		0			0
$635$	11.0000	+	19.0526i	0.436522	+	0.756078i
$636$		0			0
$637$	24.5000	+	6.06218i	0.970725	+	0.240192i
$638$		0			0
$639$	−3.00000	−	5.19615i	−0.118678	−	0.205557i
$640$		0			0
$641$	8.50000	−	14.7224i	0.335730	−	0.581501i	−0.647895	−	0.761730i	$-0.724350\pi$
$641$	8.50000	−	14.7224i	0.335730	−	0.581501i	0.983625	+	0.180229i	$0.0576838\pi$
$642$		0			0
$643$	2.00000	−	3.46410i	0.0788723	−	0.136611i	−0.823891	−	0.566748i	$-0.808201\pi$
$643$	2.00000	−	3.46410i	0.0788723	−	0.136611i	0.902764	+	0.430137i	$0.141535\pi$
$644$		0			0
$645$	−8.00000			−0.315000
$646$		0			0
$647$	−19.0000	−	32.9090i	−0.746967	−	1.29378i	−0.949270	−	0.314462i	$-0.898176\pi$
$647$	−19.0000	−	32.9090i	−0.746967	−	1.29378i	0.202303	−	0.979323i	$-0.435157\pi$
$648$		0			0
$649$	48.0000			1.88416
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−7.00000	−	12.1244i	−0.273931	−	0.474463i	0.695934	−	0.718106i	$-0.254991\pi$
$653$	−7.00000	−	12.1244i	−0.273931	−	0.474463i	−0.969865	+	0.243643i	$0.921657\pi$
$654$		0			0
$655$	−18.0000			−0.703318
$656$		0			0
$657$	−3.50000	+	6.06218i	−0.136548	+	0.236508i
$658$		0			0
$659$	−14.0000	+	24.2487i	−0.545363	+	0.944596i	0.453221	+	0.891398i	$0.350275\pi$
$659$	−14.0000	+	24.2487i	−0.545363	+	0.944596i	−0.998584	+	0.0531977i	$0.983059\pi$
$660$		0			0
$661$	−6.50000	−	11.2583i	−0.252821	−	0.437898i	0.711481	−	0.702706i	$-0.248025\pi$
$661$	−6.50000	−	11.2583i	−0.252821	−	0.437898i	−0.964301	+	0.264807i	$0.914692\pi$
$662$		0			0
$663$	15.0000	−	15.5885i	0.582552	−	0.605406i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−5.00000	+	8.66025i	−0.193601	+	0.335326i
$668$		0			0
$669$	−18.0000	+	31.1769i	−0.695920	+	1.20537i
$670$		0			0
$671$	−28.0000			−1.08093
$672$		0			0
$673$	2.50000	+	4.33013i	0.0963679	+	0.166914i	0.910179	−	0.414216i	$-0.135944\pi$
$673$	2.50000	+	4.33013i	0.0963679	+	0.166914i	−0.813811	+	0.581130i	$0.802611\pi$
$674$		0			0
$675$	16.0000			0.615840
$676$		0			0
$677$	26.0000			0.999261			0.499631	−	0.866239i	$-0.333469\pi$
$677$	26.0000			0.999261			0.499631	+	0.866239i	$0.333469\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	36.0000			1.37952
$682$		0			0
$683$	19.0000	−	32.9090i	0.727015	−	1.25923i	−0.231125	−	0.972924i	$-0.574240\pi$
$683$	19.0000	−	32.9090i	0.727015	−	1.25923i	0.958139	−	0.286302i	$-0.0924262\pi$
$684$		0			0
$685$	0.500000	−	0.866025i	0.0191040	−	0.0330891i
$686$		0			0
$687$	18.0000	+	31.1769i	0.686743	+	1.18947i
$688$		0			0
$689$	32.5000	−	33.7750i	1.23815	−	1.28672i
$690$		0			0
$691$	−10.0000	−	17.3205i	−0.380418	−	0.658903i	0.610704	−	0.791859i	$-0.290887\pi$
$691$	−10.0000	−	17.3205i	−0.380418	−	0.658903i	−0.991122	+	0.132956i	$0.957553\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	8.00000	−	13.8564i	0.303457	−	0.525603i
$696$		0			0
$697$	9.00000			0.340899
$698$		0			0
$699$	−6.00000	−	10.3923i	−0.226941	−	0.393073i
$700$		0			0
$701$	−10.0000			−0.377695			−0.188847	−	0.982006i	$-0.560475\pi$
$701$	−10.0000			−0.377695			−0.188847	+	0.982006i	$0.560475\pi$
$702$		0			0
$703$	−10.0000			−0.377157
$704$		0			0
$705$	−6.00000	−	10.3923i	−0.225973	−	0.391397i
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−10.5000	+	18.1865i	−0.394336	+	0.683010i	−0.993016	−	0.117978i	$-0.962359\pi$
$709$	−10.5000	+	18.1865i	−0.394336	+	0.683010i	0.598680	+	0.800988i	$0.295692\pi$
$710$		0			0
$711$	−4.00000	+	6.92820i	−0.150012	+	0.259828i
$712$		0			0
$713$	−2.00000	−	3.46410i	−0.0749006	−	0.129732i
$714$		0			0
$715$	−14.0000	−	3.46410i	−0.523570	−	0.129550i
$716$		0			0
$717$	12.0000	+	20.7846i	0.448148	+	0.776215i
$718$		0			0
$719$	24.0000	−	41.5692i	0.895049	−	1.55027i	0.0613050	−	0.998119i	$-0.480474\pi$
$719$	24.0000	−	41.5692i	0.895049	−	1.55027i	0.833744	−	0.552151i	$-0.186193\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$	14.0000			0.520666
$724$		0			0
$725$	10.0000	+	17.3205i	0.371391	+	0.643268i
$726$		0			0
$727$	−28.0000			−1.03846			−0.519231	−	0.854634i	$-0.673782\pi$
$727$	−28.0000			−1.03846			−0.519231	+	0.854634i	$0.673782\pi$
$728$		0			0
$729$	13.0000			0.481481
$730$		0			0
$731$	6.00000	+	10.3923i	0.221918	+	0.384373i
$732$		0			0
$733$	1.00000			0.0369358			0.0184679	−	0.999829i	$-0.494121\pi$
$733$	1.00000			0.0369358			0.0184679	+	0.999829i	$0.494121\pi$
$734$		0			0
$735$	7.00000	−	12.1244i	0.258199	−	0.447214i
$736$		0			0
$737$	28.0000	−	48.4974i	1.03139	−	1.78643i
$738$		0			0
$739$	−10.0000	−	17.3205i	−0.367856	−	0.637145i	0.621374	−	0.783514i	$-0.286575\pi$
$739$	−10.0000	−	17.3205i	−0.367856	−	0.637145i	−0.989230	+	0.146369i	$0.953241\pi$
$740$		0			0
$741$	−10.0000	+	10.3923i	−0.367359	+	0.381771i
$742$		0			0
$743$	24.0000	+	41.5692i	0.880475	+	1.52503i	0.850814	+	0.525467i	$0.176109\pi$
$743$	24.0000	+	41.5692i	0.880475	+	1.52503i	0.0296605	+	0.999560i	$0.490557\pi$
$744$		0			0
$745$	−10.5000	+	18.1865i	−0.384690	+	0.666303i
$746$		0			0
$747$	−2.00000	+	3.46410i	−0.0731762	+	0.126745i
$748$		0			0
$749$		0			0
$750$		0			0
$751$	10.0000	+	17.3205i	0.364905	+	0.632034i	0.988761	−	0.149505i	$-0.0477681\pi$
$751$	10.0000	+	17.3205i	0.364905	+	0.632034i	−0.623856	+	0.781540i	$0.714435\pi$
$752$		0			0
$753$	−44.0000			−1.60345
$754$		0			0
$755$	22.0000			0.800662
$756$		0			0
$757$	−19.0000	−	32.9090i	−0.690567	−	1.19610i	−0.971652	−	0.236414i	$-0.924028\pi$
$757$	−19.0000	−	32.9090i	−0.690567	−	1.19610i	0.281086	−	0.959683i	$-0.409305\pi$
$758$		0			0
$759$	16.0000			0.580763
$760$		0			0
$761$	−19.0000	+	32.9090i	−0.688749	+	1.19295i	0.283493	+	0.958974i	$0.408507\pi$
$761$	−19.0000	+	32.9090i	−0.688749	+	1.19295i	−0.972243	+	0.233975i	$0.924827\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	−1.50000	−	2.59808i	−0.0542326	−	0.0939336i
$766$		0			0
$767$	12.0000	+	41.5692i	0.433295	+	1.50098i
$768$		0			0
$769$	−19.0000	−	32.9090i	−0.685158	−	1.18673i	−0.973387	−	0.229166i	$-0.926400\pi$
$769$	−19.0000	−	32.9090i	−0.685158	−	1.18673i	0.288230	−	0.957561i	$-0.406933\pi$
$770$		0			0
$771$	−11.0000	+	19.0526i	−0.396155	+	0.686161i
$772$		0			0
$773$	3.00000	−	5.19615i	0.107903	−	0.186893i	−0.807018	−	0.590527i	$-0.798920\pi$
$773$	3.00000	−	5.19615i	0.107903	−	0.186893i	0.914920	+	0.403634i	$0.132253\pi$
$774$		0			0
$775$	−8.00000			−0.287368
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−6.00000			−0.214972
$780$		0			0
$781$	24.0000			0.858788
$782$		0			0
$783$	10.0000	+	17.3205i	0.357371	+	0.618984i
$784$		0			0
$785$	5.00000			0.178458
$786$		0			0
$787$	−11.0000	+	19.0526i	−0.392108	+	0.679150i	−0.992727	−	0.120384i	$-0.961587\pi$
$787$	−11.0000	+	19.0526i	−0.392108	+	0.679150i	0.600620	+	0.799535i	$0.294921\pi$
$788$		0			0
$789$	−24.0000	+	41.5692i	−0.854423	+	1.47990i
$790$		0			0
$791$		0			0
$792$		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:	\( N \)	\(=\)	\( 416 = 2^{5} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	416.i (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.00000 - 1.73205i) q^{3} +1.00000 q^{5} +(-0.500000 + 0.866025i) q^{9} +O(q^{10})\)	\(q+(-1.00000 - 1.73205i) q^{3} +1.00000 q^{5} +(-0.500000 + 0.866025i) q^{9} +(-2.00000 - 3.46410i) q^{11} +(2.50000 - 2.59808i) q^{13} +(-1.00000 - 1.73205i) q^{15} +(-1.50000 + 2.59808i) q^{17} +(1.00000 - 1.73205i) q^{19} +(-1.00000 - 1.73205i) q^{23} -4.00000 q^{25} -4.00000 q^{27} +(-2.50000 - 4.33013i) q^{29} +2.00000 q^{31} +(-4.00000 + 6.92820i) q^{33} +(-2.50000 - 4.33013i) q^{37} +(-7.00000 - 1.73205i) q^{39} +(-1.50000 - 2.59808i) q^{41} +(2.00000 - 3.46410i) q^{43} +(-0.500000 + 0.866025i) q^{45} +6.00000 q^{47} +(3.50000 + 6.06218i) q^{49} +6.00000 q^{51} +13.0000 q^{53} +(-2.00000 - 3.46410i) q^{55} -4.00000 q^{57} +(-6.00000 + 10.3923i) q^{59} +(3.50000 - 6.06218i) q^{61} +(2.50000 - 2.59808i) q^{65} +(7.00000 + 12.1244i) q^{67} +(-2.00000 + 3.46410i) q^{69} +(-3.00000 + 5.19615i) q^{71} +7.00000 q^{73} +(4.00000 + 6.92820i) q^{75} +8.00000 q^{79} +(5.50000 + 9.52628i) q^{81} +4.00000 q^{83} +(-1.50000 + 2.59808i) q^{85} +(-5.00000 + 8.66025i) q^{87} +(-7.00000 - 12.1244i) q^{89} +(-2.00000 - 3.46410i) q^{93} +(1.00000 - 1.73205i) q^{95} +(1.00000 - 1.73205i) q^{97} +4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} + 2 q^{5} - q^{9}+O(q^{10}) \) 2 * q - 2 * q^3 + 2 * q^5 - q^9	\( 2 q - 2 q^{3} + 2 q^{5} - q^{9} - 4 q^{11} + 5 q^{13} - 2 q^{15} - 3 q^{17} + 2 q^{19} - 2 q^{23} - 8 q^{25} - 8 q^{27} - 5 q^{29} + 4 q^{31} - 8 q^{33} - 5 q^{37} - 14 q^{39} - 3 q^{41} + 4 q^{43} - q^{45} + 12 q^{47} + 7 q^{49} + 12 q^{51} + 26 q^{53} - 4 q^{55} - 8 q^{57} - 12 q^{59} + 7 q^{61} + 5 q^{65} + 14 q^{67} - 4 q^{69} - 6 q^{71} + 14 q^{73} + 8 q^{75} + 16 q^{79} + 11 q^{81} + 8 q^{83} - 3 q^{85} - 10 q^{87} - 14 q^{89} - 4 q^{93} + 2 q^{95} + 2 q^{97} + 8 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 + 2 * q^5 - q^9 - 4 * q^11 + 5 * q^13 - 2 * q^15 - 3 * q^17 + 2 * q^19 - 2 * q^23 - 8 * q^25 - 8 * q^27 - 5 * q^29 + 4 * q^31 - 8 * q^33 - 5 * q^37 - 14 * q^39 - 3 * q^41 + 4 * q^43 - q^45 + 12 * q^47 + 7 * q^49 + 12 * q^51 + 26 * q^53 - 4 * q^55 - 8 * q^57 - 12 * q^59 + 7 * q^61 + 5 * q^65 + 14 * q^67 - 4 * q^69 - 6 * q^71 + 14 * q^73 + 8 * q^75 + 16 * q^79 + 11 * q^81 + 8 * q^83 - 3 * q^85 - 10 * q^87 - 14 * q^89 - 4 * q^93 + 2 * q^95 + 2 * q^97 + 8 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

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