LMFDB - Embedded newform 390.2.e.d.79.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [390,2,Mod(79,390)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(390, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0]))

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

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

chi := DirichletCharacter("390.79");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 390 = 2 \cdot 3 \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	390.e (of order $2$, degree $1$, 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:	$3.11416567883$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			79.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	390.79
Dual form			390.2.e.d.79.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +(2.00000 - 1.00000i) q^{5} +1.00000 q^{6} -4.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$	\(q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +(2.00000 - 1.00000i) q^{5} +1.00000 q^{6} -4.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} +(1.00000 + 2.00000i) q^{10} -6.00000 q^{11} +1.00000i q^{12} -1.00000i q^{13} +4.00000 q^{14} +(-1.00000 - 2.00000i) q^{15} +1.00000 q^{16} -4.00000i q^{17} -1.00000i q^{18} -2.00000 q^{19} +(-2.00000 + 1.00000i) q^{20} -4.00000 q^{21} -6.00000i q^{22} +6.00000i q^{23} -1.00000 q^{24} +(3.00000 - 4.00000i) q^{25} +1.00000 q^{26} +1.00000i q^{27} +4.00000i q^{28} +10.0000 q^{29} +(2.00000 - 1.00000i) q^{30} +4.00000 q^{31} +1.00000i q^{32} +6.00000i q^{33} +4.00000 q^{34} +(-4.00000 - 8.00000i) q^{35} +1.00000 q^{36} -6.00000i q^{37} -2.00000i q^{38} -1.00000 q^{39} +(-1.00000 - 2.00000i) q^{40} +10.0000 q^{41} -4.00000i q^{42} +6.00000 q^{44} +(-2.00000 + 1.00000i) q^{45} -6.00000 q^{46} +8.00000i q^{47} -1.00000i q^{48} -9.00000 q^{49} +(4.00000 + 3.00000i) q^{50} -4.00000 q^{51} +1.00000i q^{52} +6.00000i q^{53} -1.00000 q^{54} +(-12.0000 + 6.00000i) q^{55} -4.00000 q^{56} +2.00000i q^{57} +10.0000i q^{58} +6.00000 q^{59} +(1.00000 + 2.00000i) q^{60} -6.00000 q^{61} +4.00000i q^{62} +4.00000i q^{63} -1.00000 q^{64} +(-1.00000 - 2.00000i) q^{65} -6.00000 q^{66} -12.0000i q^{67} +4.00000i q^{68} +6.00000 q^{69} +(8.00000 - 4.00000i) q^{70} +1.00000i q^{72} -2.00000i q^{73} +6.00000 q^{74} +(-4.00000 - 3.00000i) q^{75} +2.00000 q^{76} +24.0000i q^{77} -1.00000i q^{78} +8.00000 q^{79} +(2.00000 - 1.00000i) q^{80} +1.00000 q^{81} +10.0000i q^{82} +4.00000i q^{83} +4.00000 q^{84} +(-4.00000 - 8.00000i) q^{85} -10.0000i q^{87} +6.00000i q^{88} -14.0000 q^{89} +(-1.00000 - 2.00000i) q^{90} -4.00000 q^{91} -6.00000i q^{92} -4.00000i q^{93} -8.00000 q^{94} +(-4.00000 + 2.00000i) q^{95} +1.00000 q^{96} +14.0000i q^{97} -9.00000i q^{98} +6.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{4} + 4 q^{5} + 2 q^{6} - 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^4 + 4 * q^5 + 2 * q^6 - 2 * q^9	$ 2 q - 2 q^{4} + 4 q^{5} + 2 q^{6} - 2 q^{9} + 2 q^{10} - 12 q^{11} + 8 q^{14} - 2 q^{15} + 2 q^{16} - 4 q^{19} - 4 q^{20} - 8 q^{21} - 2 q^{24} + 6 q^{25} + 2 q^{26} + 20 q^{29} + 4 q^{30} + 8 q^{31} + 8 q^{34} - 8 q^{35} + 2 q^{36} - 2 q^{39} - 2 q^{40} + 20 q^{41} + 12 q^{44} - 4 q^{45} - 12 q^{46} - 18 q^{49} + 8 q^{50} - 8 q^{51} - 2 q^{54} - 24 q^{55} - 8 q^{56} + 12 q^{59} + 2 q^{60} - 12 q^{61} - 2 q^{64} - 2 q^{65} - 12 q^{66} + 12 q^{69} + 16 q^{70} + 12 q^{74} - 8 q^{75} + 4 q^{76} + 16 q^{79} + 4 q^{80} + 2 q^{81} + 8 q^{84} - 8 q^{85} - 28 q^{89} - 2 q^{90} - 8 q^{91} - 16 q^{94} - 8 q^{95} + 2 q^{96} + 12 q^{99}+O(q^{100}) $ 2 * q - 2 * q^4 + 4 * q^5 + 2 * q^6 - 2 * q^9 + 2 * q^10 - 12 * q^11 + 8 * q^14 - 2 * q^15 + 2 * q^16 - 4 * q^19 - 4 * q^20 - 8 * q^21 - 2 * q^24 + 6 * q^25 + 2 * q^26 + 20 * q^29 + 4 * q^30 + 8 * q^31 + 8 * q^34 - 8 * q^35 + 2 * q^36 - 2 * q^39 - 2 * q^40 + 20 * q^41 + 12 * q^44 - 4 * q^45 - 12 * q^46 - 18 * q^49 + 8 * q^50 - 8 * q^51 - 2 * q^54 - 24 * q^55 - 8 * q^56 + 12 * q^59 + 2 * q^60 - 12 * q^61 - 2 * q^64 - 2 * q^65 - 12 * q^66 + 12 * q^69 + 16 * q^70 + 12 * q^74 - 8 * q^75 + 4 * q^76 + 16 * q^79 + 4 * q^80 + 2 * q^81 + 8 * q^84 - 8 * q^85 - 28 * q^89 - 2 * q^90 - 8 * q^91 - 16 * q^94 - 8 * q^95 + 2 * q^96 + 12 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/390\mathbb{Z}\right)^\times$.

$n$	$131$	$157$	$301$
$\chi(n)$	$1$	$-1$	$1$

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 $

$2$			1.00000i			0.707107i
$2$			1.00000i			0.707107i
$3$		−	1.00000i		−	0.577350i
$3$		−	1.00000i		−	0.577350i
$4$	−1.00000			−0.500000
$5$	2.00000	−	1.00000i	0.894427	−	0.447214i
$5$	2.00000	−	1.00000i	0.894427	−	0.447214i
$6$	1.00000			0.408248
$7$		−	4.00000i		−	1.51186i	−0.654654	−	0.755929i	$-0.727186\pi$
$7$		−	4.00000i		−	1.51186i	0.654654	−	0.755929i	$-0.272814\pi$
$8$		−	1.00000i		−	0.353553i
$9$	−1.00000			−0.333333
$10$	1.00000	+	2.00000i	0.316228	+	0.632456i
$11$	−6.00000			−1.80907			−0.904534	−	0.426401i	$-0.859781\pi$
$11$	−6.00000			−1.80907			−0.904534	+	0.426401i	$0.859781\pi$
$12$			1.00000i			0.288675i
$13$		−	1.00000i		−	0.277350i
$13$		−	1.00000i		−	0.277350i
$14$	4.00000			1.06904
$15$	−1.00000	−	2.00000i	−0.258199	−	0.516398i
$16$	1.00000			0.250000
$17$		−	4.00000i		−	0.970143i	−0.874475	−	0.485071i	$-0.838794\pi$
$17$		−	4.00000i		−	0.970143i	0.874475	−	0.485071i	$-0.161206\pi$
$18$		−	1.00000i		−	0.235702i
$19$	−2.00000			−0.458831			−0.229416	−	0.973329i	$-0.573682\pi$
$19$	−2.00000			−0.458831			−0.229416	+	0.973329i	$0.573682\pi$
$20$	−2.00000	+	1.00000i	−0.447214	+	0.223607i
$21$	−4.00000			−0.872872
$22$		−	6.00000i		−	1.27920i
$23$			6.00000i			1.25109i	0.780189	+	0.625543i	$0.215123\pi$
$23$			6.00000i			1.25109i	−0.780189	+	0.625543i	$0.784877\pi$
$24$	−1.00000			−0.204124
$25$	3.00000	−	4.00000i	0.600000	−	0.800000i
$26$	1.00000			0.196116
$27$			1.00000i			0.192450i
$28$			4.00000i			0.755929i
$29$	10.0000			1.85695			0.928477	−	0.371391i	$-0.121119\pi$
$29$	10.0000			1.85695			0.928477	+	0.371391i	$0.121119\pi$
$30$	2.00000	−	1.00000i	0.365148	−	0.182574i
$31$	4.00000			0.718421			0.359211	−	0.933257i	$-0.383046\pi$
$31$	4.00000			0.718421			0.359211	+	0.933257i	$0.383046\pi$
$32$			1.00000i			0.176777i
$33$			6.00000i			1.04447i
$34$	4.00000			0.685994
$35$	−4.00000	−	8.00000i	−0.676123	−	1.35225i
$36$	1.00000			0.166667
$37$		−	6.00000i		−	0.986394i	−0.869918	−	0.493197i	$-0.835828\pi$
$37$		−	6.00000i		−	0.986394i	0.869918	−	0.493197i	$-0.164172\pi$
$38$		−	2.00000i		−	0.324443i
$39$	−1.00000			−0.160128
$40$	−1.00000	−	2.00000i	−0.158114	−	0.316228i
$41$	10.0000			1.56174			0.780869	−	0.624695i	$-0.214777\pi$
$41$	10.0000			1.56174			0.780869	+	0.624695i	$0.214777\pi$
$42$		−	4.00000i		−	0.617213i
$43$		0			0		1.00000			$0$
$43$		0			0		−1.00000			$\pi$
$44$	6.00000			0.904534
$45$	−2.00000	+	1.00000i	−0.298142	+	0.149071i
$46$	−6.00000			−0.884652
$47$			8.00000i			1.16692i	0.812142	+	0.583460i	$0.198301\pi$
$47$			8.00000i			1.16692i	−0.812142	+	0.583460i	$0.801699\pi$
$48$		−	1.00000i		−	0.144338i
$49$	−9.00000			−1.28571
$50$	4.00000	+	3.00000i	0.565685	+	0.424264i
$51$	−4.00000			−0.560112
$52$			1.00000i			0.138675i
$53$			6.00000i			0.824163i	0.911147	+	0.412082i	$0.135198\pi$
$53$			6.00000i			0.824163i	−0.911147	+	0.412082i	$0.864802\pi$
$54$	−1.00000			−0.136083
$55$	−12.0000	+	6.00000i	−1.61808	+	0.809040i
$56$	−4.00000			−0.534522
$57$			2.00000i			0.264906i
$58$			10.0000i			1.31306i
$59$	6.00000			0.781133			0.390567	−	0.920575i	$-0.372279\pi$
$59$	6.00000			0.781133			0.390567	+	0.920575i	$0.372279\pi$
$60$	1.00000	+	2.00000i	0.129099	+	0.258199i
$61$	−6.00000			−0.768221			−0.384111	−	0.923287i	$-0.625492\pi$
$61$	−6.00000			−0.768221			−0.384111	+	0.923287i	$0.625492\pi$
$62$			4.00000i			0.508001i
$63$			4.00000i			0.503953i
$64$	−1.00000			−0.125000
$65$	−1.00000	−	2.00000i	−0.124035	−	0.248069i
$66$	−6.00000			−0.738549
$67$		−	12.0000i		−	1.46603i	−0.680211	−	0.733017i	$-0.738112\pi$
$67$		−	12.0000i		−	1.46603i	0.680211	−	0.733017i	$-0.261888\pi$
$68$			4.00000i			0.485071i
$69$	6.00000			0.722315
$70$	8.00000	−	4.00000i	0.956183	−	0.478091i
$71$		0			0			−	1.00000i	$-0.5\pi$
$71$		0			0				1.00000i	$0.5\pi$
$72$			1.00000i			0.117851i
$73$		−	2.00000i		−	0.234082i	−0.993127	−	0.117041i	$-0.962659\pi$
$73$		−	2.00000i		−	0.234082i	0.993127	−	0.117041i	$-0.0373409\pi$
$74$	6.00000			0.697486
$75$	−4.00000	−	3.00000i	−0.461880	−	0.346410i
$76$	2.00000			0.229416
$77$			24.0000i			2.73505i
$78$		−	1.00000i		−	0.113228i
$79$	8.00000			0.900070			0.450035	−	0.893011i	$-0.351411\pi$
$79$	8.00000			0.900070			0.450035	+	0.893011i	$0.351411\pi$
$80$	2.00000	−	1.00000i	0.223607	−	0.111803i
$81$	1.00000			0.111111
$82$			10.0000i			1.10432i
$83$			4.00000i			0.439057i	0.975606	+	0.219529i	$0.0704519\pi$
$83$			4.00000i			0.439057i	−0.975606	+	0.219529i	$0.929548\pi$
$84$	4.00000			0.436436
$85$	−4.00000	−	8.00000i	−0.433861	−	0.867722i
$86$		0			0
$87$		−	10.0000i		−	1.07211i
$88$			6.00000i			0.639602i
$89$	−14.0000			−1.48400			−0.741999	−	0.670402i	$-0.766122\pi$
$89$	−14.0000			−1.48400			−0.741999	+	0.670402i	$0.766122\pi$
$90$	−1.00000	−	2.00000i	−0.105409	−	0.210819i
$91$	−4.00000			−0.419314
$92$		−	6.00000i		−	0.625543i
$93$		−	4.00000i		−	0.414781i
$94$	−8.00000			−0.825137
$95$	−4.00000	+	2.00000i	−0.410391	+	0.205196i
$96$	1.00000			0.102062
$97$			14.0000i			1.42148i	0.703452	+	0.710742i	$0.251641\pi$
$97$			14.0000i			1.42148i	−0.703452	+	0.710742i	$0.748359\pi$
$98$		−	9.00000i		−	0.909137i
$99$	6.00000			0.603023
$100$	−3.00000	+	4.00000i	−0.300000	+	0.400000i
$101$	10.0000			0.995037			0.497519	−	0.867453i	$-0.334245\pi$
$101$	10.0000			0.995037			0.497519	+	0.867453i	$0.334245\pi$
$102$		−	4.00000i		−	0.396059i
$103$			2.00000i			0.197066i	0.995134	+	0.0985329i	$0.0314150\pi$
$103$			2.00000i			0.197066i	−0.995134	+	0.0985329i	$0.968585\pi$
$104$	−1.00000			−0.0980581
$105$	−8.00000	+	4.00000i	−0.780720	+	0.390360i
$106$	−6.00000			−0.582772
$107$		0			0		1.00000			$0$
$107$		0			0		−1.00000			$\pi$
$108$		−	1.00000i		−	0.0962250i
$109$	−4.00000			−0.383131			−0.191565	−	0.981480i	$-0.561356\pi$
$109$	−4.00000			−0.383131			−0.191565	+	0.981480i	$0.561356\pi$
$110$	−6.00000	−	12.0000i	−0.572078	−	1.14416i
$111$	−6.00000			−0.569495
$112$		−	4.00000i		−	0.377964i
$113$			4.00000i			0.376288i	0.982141	+	0.188144i	$0.0602472\pi$
$113$			4.00000i			0.376288i	−0.982141	+	0.188144i	$0.939753\pi$
$114$	−2.00000			−0.187317
$115$	6.00000	+	12.0000i	0.559503	+	1.11901i
$116$	−10.0000			−0.928477
$117$			1.00000i			0.0924500i
$118$			6.00000i			0.552345i
$119$	−16.0000			−1.46672
$120$	−2.00000	+	1.00000i	−0.182574	+	0.0912871i
$121$	25.0000			2.27273
$122$		−	6.00000i		−	0.543214i
$123$		−	10.0000i		−	0.901670i
$124$	−4.00000			−0.359211
$125$	2.00000	−	11.0000i	0.178885	−	0.983870i
$126$	−4.00000			−0.356348
$127$		−	10.0000i		−	0.887357i	−0.896186	−	0.443678i	$-0.853673\pi$
$127$		−	10.0000i		−	0.887357i	0.896186	−	0.443678i	$-0.146327\pi$
$128$		−	1.00000i		−	0.0883883i
$129$		0			0
$130$	2.00000	−	1.00000i	0.175412	−	0.0877058i
$131$	−8.00000			−0.698963			−0.349482	−	0.936943i	$-0.613642\pi$
$131$	−8.00000			−0.698963			−0.349482	+	0.936943i	$0.613642\pi$
$132$		−	6.00000i		−	0.522233i
$133$			8.00000i			0.693688i
$134$	12.0000			1.03664
$135$	1.00000	+	2.00000i	0.0860663	+	0.172133i
$136$	−4.00000			−0.342997
$137$		−	14.0000i		−	1.19610i	−0.801459	−	0.598050i	$-0.795942\pi$
$137$		−	14.0000i		−	1.19610i	0.801459	−	0.598050i	$-0.204058\pi$
$138$			6.00000i			0.510754i
$139$	−16.0000			−1.35710			−0.678551	−	0.734553i	$-0.737392\pi$
$139$	−16.0000			−1.35710			−0.678551	+	0.734553i	$0.737392\pi$
$140$	4.00000	+	8.00000i	0.338062	+	0.676123i
$141$	8.00000			0.673722
$142$		0			0
$143$			6.00000i			0.501745i
$144$	−1.00000			−0.0833333
$145$	20.0000	−	10.0000i	1.66091	−	0.830455i
$146$	2.00000			0.165521
$147$			9.00000i			0.742307i
$148$			6.00000i			0.493197i
$149$	−16.0000			−1.31077			−0.655386	−	0.755295i	$-0.727494\pi$
$149$	−16.0000			−1.31077			−0.655386	+	0.755295i	$0.727494\pi$
$150$	3.00000	−	4.00000i	0.244949	−	0.326599i
$151$	16.0000			1.30206			0.651031	−	0.759051i	$-0.274337\pi$
$151$	16.0000			1.30206			0.651031	+	0.759051i	$0.274337\pi$
$152$			2.00000i			0.162221i
$153$			4.00000i			0.323381i
$154$	−24.0000			−1.93398
$155$	8.00000	−	4.00000i	0.642575	−	0.321288i
$156$	1.00000			0.0800641
$157$			6.00000i			0.478852i	0.970915	+	0.239426i	$0.0769593\pi$
$157$			6.00000i			0.478852i	−0.970915	+	0.239426i	$0.923041\pi$
$158$			8.00000i			0.636446i
$159$	6.00000			0.475831
$160$	1.00000	+	2.00000i	0.0790569	+	0.158114i
$161$	24.0000			1.89146
$162$			1.00000i			0.0785674i
$163$			4.00000i			0.313304i	0.987654	+	0.156652i	$0.0500701\pi$
$163$			4.00000i			0.313304i	−0.987654	+	0.156652i	$0.949930\pi$
$164$	−10.0000			−0.780869
$165$	6.00000	+	12.0000i	0.467099	+	0.934199i
$166$	−4.00000			−0.310460
$167$		−	16.0000i		−	1.23812i	−0.785345	−	0.619059i	$-0.787514\pi$
$167$		−	16.0000i		−	1.23812i	0.785345	−	0.619059i	$-0.212486\pi$
$168$			4.00000i			0.308607i
$169$	−1.00000			−0.0769231
$170$	8.00000	−	4.00000i	0.613572	−	0.306786i
$171$	2.00000			0.152944
$172$		0			0
$173$			22.0000i			1.67263i	0.548250	+	0.836315i	$0.315294\pi$
$173$			22.0000i			1.67263i	−0.548250	+	0.836315i	$0.684706\pi$
$174$	10.0000			0.758098
$175$	−16.0000	−	12.0000i	−1.20949	−	0.907115i
$176$	−6.00000			−0.452267
$177$		−	6.00000i		−	0.450988i
$178$		−	14.0000i		−	1.04934i
$179$	24.0000			1.79384			0.896922	−	0.442189i	$-0.145798\pi$
$179$	24.0000			1.79384			0.896922	+	0.442189i	$0.145798\pi$
$180$	2.00000	−	1.00000i	0.149071	−	0.0745356i
$181$	6.00000			0.445976			0.222988	−	0.974821i	$-0.428419\pi$
$181$	6.00000			0.445976			0.222988	+	0.974821i	$0.428419\pi$
$182$		−	4.00000i		−	0.296500i
$183$			6.00000i			0.443533i
$184$	6.00000			0.442326
$185$	−6.00000	−	12.0000i	−0.441129	−	0.882258i
$186$	4.00000			0.293294
$187$			24.0000i			1.75505i
$188$		−	8.00000i		−	0.583460i
$189$	4.00000			0.290957
$190$	−2.00000	−	4.00000i	−0.145095	−	0.290191i
$191$		0			0			−	1.00000i	$-0.5\pi$
$191$		0			0				1.00000i	$0.5\pi$
$192$			1.00000i			0.0721688i
$193$			2.00000i			0.143963i	0.997406	+	0.0719816i	$0.0229323\pi$
$193$			2.00000i			0.143963i	−0.997406	+	0.0719816i	$0.977068\pi$
$194$	−14.0000			−1.00514
$195$	−2.00000	+	1.00000i	−0.143223	+	0.0716115i
$196$	9.00000			0.642857
$197$		−	6.00000i		−	0.427482i	−0.976890	−	0.213741i	$-0.931435\pi$
$197$		−	6.00000i		−	0.427482i	0.976890	−	0.213741i	$-0.0685649\pi$
$198$			6.00000i			0.426401i
$199$	8.00000			0.567105			0.283552	−	0.958957i	$-0.408487\pi$
$199$	8.00000			0.567105			0.283552	+	0.958957i	$0.408487\pi$
$200$	−4.00000	−	3.00000i	−0.282843	−	0.212132i
$201$	−12.0000			−0.846415
$202$			10.0000i			0.703598i
$203$		−	40.0000i		−	2.80745i
$204$	4.00000			0.280056
$205$	20.0000	−	10.0000i	1.39686	−	0.698430i
$206$	−2.00000			−0.139347
$207$		−	6.00000i		−	0.417029i
$208$		−	1.00000i		−	0.0693375i
$209$	12.0000			0.830057
$210$	−4.00000	−	8.00000i	−0.276026	−	0.552052i
$211$	4.00000			0.275371			0.137686	−	0.990476i	$-0.456034\pi$
$211$	4.00000			0.275371			0.137686	+	0.990476i	$0.456034\pi$
$212$		−	6.00000i		−	0.412082i
$213$		0			0
$214$		0			0
$215$		0			0
$216$	1.00000			0.0680414
$217$		−	16.0000i		−	1.08615i
$218$		−	4.00000i		−	0.270914i
$219$	−2.00000			−0.135147
$220$	12.0000	−	6.00000i	0.809040	−	0.404520i
$221$	−4.00000			−0.269069
$222$		−	6.00000i		−	0.402694i
$223$		−	8.00000i		−	0.535720i	−0.963458	−	0.267860i	$-0.913684\pi$
$223$		−	8.00000i		−	0.535720i	0.963458	−	0.267860i	$-0.0863164\pi$
$224$	4.00000			0.267261
$225$	−3.00000	+	4.00000i	−0.200000	+	0.266667i
$226$	−4.00000			−0.266076
$227$		−	4.00000i		−	0.265489i	−0.991150	−	0.132745i	$-0.957621\pi$
$227$		−	4.00000i		−	0.265489i	0.991150	−	0.132745i	$-0.0423790\pi$
$228$		−	2.00000i		−	0.132453i
$229$	−8.00000			−0.528655			−0.264327	−	0.964433i	$-0.585150\pi$
$229$	−8.00000			−0.528655			−0.264327	+	0.964433i	$0.585150\pi$
$230$	−12.0000	+	6.00000i	−0.791257	+	0.395628i
$231$	24.0000			1.57908
$232$		−	10.0000i		−	0.656532i
$233$			12.0000i			0.786146i	0.919507	+	0.393073i	$0.128588\pi$
$233$			12.0000i			0.786146i	−0.919507	+	0.393073i	$0.871412\pi$
$234$	−1.00000			−0.0653720
$235$	8.00000	+	16.0000i	0.521862	+	1.04372i
$236$	−6.00000			−0.390567
$237$		−	8.00000i		−	0.519656i
$238$		−	16.0000i		−	1.03713i
$239$	4.00000			0.258738			0.129369	−	0.991596i	$-0.458705\pi$
$239$	4.00000			0.258738			0.129369	+	0.991596i	$0.458705\pi$
$240$	−1.00000	−	2.00000i	−0.0645497	−	0.129099i
$241$	18.0000			1.15948			0.579741	−	0.814801i	$-0.303154\pi$
$241$	18.0000			1.15948			0.579741	+	0.814801i	$0.303154\pi$
$242$			25.0000i			1.60706i
$243$		−	1.00000i		−	0.0641500i
$244$	6.00000			0.384111
$245$	−18.0000	+	9.00000i	−1.14998	+	0.574989i
$246$	10.0000			0.637577
$247$			2.00000i			0.127257i
$248$		−	4.00000i		−	0.254000i
$249$	4.00000			0.253490
$250$	11.0000	+	2.00000i	0.695701	+	0.126491i
$251$	−20.0000			−1.26239			−0.631194	−	0.775625i	$-0.717435\pi$
$251$	−20.0000			−1.26239			−0.631194	+	0.775625i	$0.717435\pi$
$252$		−	4.00000i		−	0.251976i
$253$		−	36.0000i		−	2.26330i
$254$	10.0000			0.627456
$255$	−8.00000	+	4.00000i	−0.500979	+	0.250490i
$256$	1.00000			0.0625000
$257$			28.0000i			1.74659i	0.487190	+	0.873296i	$0.338022\pi$
$257$			28.0000i			1.74659i	−0.487190	+	0.873296i	$0.661978\pi$
$258$		0			0
$259$	−24.0000			−1.49129
$260$	1.00000	+	2.00000i	0.0620174	+	0.124035i
$261$	−10.0000			−0.618984
$262$		−	8.00000i		−	0.494242i
$263$			10.0000i			0.616626i	0.951285	+	0.308313i	$0.0997645\pi$
$263$			10.0000i			0.616626i	−0.951285	+	0.308313i	$0.900236\pi$
$264$	6.00000			0.369274
$265$	6.00000	+	12.0000i	0.368577	+	0.737154i
$266$	−8.00000			−0.490511
$267$			14.0000i			0.856786i
$268$			12.0000i			0.733017i
$269$	−10.0000			−0.609711			−0.304855	−	0.952399i	$-0.598608\pi$
$269$	−10.0000			−0.609711			−0.304855	+	0.952399i	$0.598608\pi$
$270$	−2.00000	+	1.00000i	−0.121716	+	0.0608581i
$271$	−8.00000			−0.485965			−0.242983	−	0.970031i	$-0.578126\pi$
$271$	−8.00000			−0.485965			−0.242983	+	0.970031i	$0.578126\pi$
$272$		−	4.00000i		−	0.242536i
$273$			4.00000i			0.242091i
$274$	14.0000			0.845771
$275$	−18.0000	+	24.0000i	−1.08544	+	1.44725i
$276$	−6.00000			−0.361158
$277$		−	14.0000i		−	0.841178i	−0.907251	−	0.420589i	$-0.861823\pi$
$277$		−	14.0000i		−	0.841178i	0.907251	−	0.420589i	$-0.138177\pi$
$278$		−	16.0000i		−	0.959616i
$279$	−4.00000			−0.239474
$280$	−8.00000	+	4.00000i	−0.478091	+	0.239046i
$281$	−18.0000			−1.07379			−0.536895	−	0.843649i	$-0.680403\pi$
$281$	−18.0000			−1.07379			−0.536895	+	0.843649i	$0.680403\pi$
$282$			8.00000i			0.476393i
$283$			12.0000i			0.713326i	0.934233	+	0.356663i	$0.116086\pi$
$283$			12.0000i			0.713326i	−0.934233	+	0.356663i	$0.883914\pi$
$284$		0			0
$285$	2.00000	+	4.00000i	0.118470	+	0.236940i
$286$	−6.00000			−0.354787
$287$		−	40.0000i		−	2.36113i
$288$		−	1.00000i		−	0.0589256i
$289$	1.00000			0.0588235
$290$	10.0000	+	20.0000i	0.587220	+	1.17444i
$291$	14.0000			0.820695
$292$			2.00000i			0.117041i
$293$		−	18.0000i		−	1.05157i	−0.850617	−	0.525786i	$-0.823771\pi$
$293$		−	18.0000i		−	1.05157i	0.850617	−	0.525786i	$-0.176229\pi$
$294$	−9.00000			−0.524891
$295$	12.0000	−	6.00000i	0.698667	−	0.349334i
$296$	−6.00000			−0.348743
$297$		−	6.00000i		−	0.348155i
$298$		−	16.0000i		−	0.926855i
$299$	6.00000			0.346989
$300$	4.00000	+	3.00000i	0.230940	+	0.173205i
$301$		0			0
$302$			16.0000i			0.920697i
$303$		−	10.0000i		−	0.574485i
$304$	−2.00000			−0.114708
$305$	−12.0000	+	6.00000i	−0.687118	+	0.343559i
$306$	−4.00000			−0.228665
$307$		−	4.00000i		−	0.228292i	−0.993464	−	0.114146i	$-0.963587\pi$
$307$		−	4.00000i		−	0.228292i	0.993464	−	0.114146i	$-0.0364132\pi$
$308$		−	24.0000i		−	1.36753i
$309$	2.00000			0.113776
$310$	4.00000	+	8.00000i	0.227185	+	0.454369i
$311$	−8.00000			−0.453638			−0.226819	−	0.973937i	$-0.572833\pi$
$311$	−8.00000			−0.453638			−0.226819	+	0.973937i	$0.572833\pi$
$312$			1.00000i			0.0566139i
$313$			8.00000i			0.452187i	0.974106	+	0.226093i	$0.0725954\pi$
$313$			8.00000i			0.452187i	−0.974106	+	0.226093i	$0.927405\pi$
$314$	−6.00000			−0.338600
$315$	4.00000	+	8.00000i	0.225374	+	0.450749i
$316$	−8.00000			−0.450035
$317$			2.00000i			0.112331i	0.998421	+	0.0561656i	$0.0178875\pi$
$317$			2.00000i			0.112331i	−0.998421	+	0.0561656i	$0.982113\pi$
$318$			6.00000i			0.336463i
$319$	−60.0000			−3.35936
$320$	−2.00000	+	1.00000i	−0.111803	+	0.0559017i
$321$		0			0
$322$			24.0000i			1.33747i
$323$			8.00000i			0.445132i
$324$	−1.00000			−0.0555556
$325$	−4.00000	−	3.00000i	−0.221880	−	0.166410i
$326$	−4.00000			−0.221540
$327$			4.00000i			0.221201i
$328$		−	10.0000i		−	0.552158i
$329$	32.0000			1.76422
$330$	−12.0000	+	6.00000i	−0.660578	+	0.330289i
$331$	2.00000			0.109930			0.0549650	−	0.998488i	$-0.482495\pi$
$331$	2.00000			0.109930			0.0549650	+	0.998488i	$0.482495\pi$
$332$		−	4.00000i		−	0.219529i
$333$			6.00000i			0.328798i
$334$	16.0000			0.875481
$335$	−12.0000	−	24.0000i	−0.655630	−	1.31126i
$336$	−4.00000			−0.218218
$337$			16.0000i			0.871576i	0.900049	+	0.435788i	$0.143530\pi$
$337$			16.0000i			0.871576i	−0.900049	+	0.435788i	$0.856470\pi$
$338$		−	1.00000i		−	0.0543928i
$339$	4.00000			0.217250
$340$	4.00000	+	8.00000i	0.216930	+	0.433861i
$341$	−24.0000			−1.29967
$342$			2.00000i			0.108148i
$343$			8.00000i			0.431959i
$344$		0			0
$345$	12.0000	−	6.00000i	0.646058	−	0.323029i
$346$	−22.0000			−1.18273
$347$			16.0000i			0.858925i	0.903085	+	0.429463i	$0.141297\pi$
$347$			16.0000i			0.858925i	−0.903085	+	0.429463i	$0.858703\pi$
$348$			10.0000i			0.536056i
$349$	4.00000			0.214115			0.107058	−	0.994253i	$-0.465857\pi$
$349$	4.00000			0.214115			0.107058	+	0.994253i	$0.465857\pi$
$350$	12.0000	−	16.0000i	0.641427	−	0.855236i
$351$	1.00000			0.0533761
$352$		−	6.00000i		−	0.319801i
$353$			26.0000i			1.38384i	0.721974	+	0.691920i	$0.243235\pi$
$353$			26.0000i			1.38384i	−0.721974	+	0.691920i	$0.756765\pi$
$354$	6.00000			0.318896
$355$		0			0
$356$	14.0000			0.741999
$357$			16.0000i			0.846810i
$358$			24.0000i			1.26844i
$359$	20.0000			1.05556			0.527780	−	0.849381i	$-0.323025\pi$
$359$	20.0000			1.05556			0.527780	+	0.849381i	$0.323025\pi$
$360$	1.00000	+	2.00000i	0.0527046	+	0.105409i
$361$	−15.0000			−0.789474
$362$			6.00000i			0.315353i
$363$		−	25.0000i		−	1.31216i
$364$	4.00000			0.209657
$365$	−2.00000	−	4.00000i	−0.104685	−	0.209370i
$366$	−6.00000			−0.313625
$367$		−	18.0000i		−	0.939592i	−0.882775	−	0.469796i	$-0.844327\pi$
$367$		−	18.0000i		−	0.939592i	0.882775	−	0.469796i	$-0.155673\pi$
$368$			6.00000i			0.312772i
$369$	−10.0000			−0.520579
$370$	12.0000	−	6.00000i	0.623850	−	0.311925i
$371$	24.0000			1.24602
$372$			4.00000i			0.207390i
$373$			18.0000i			0.932005i	0.884783	+	0.466002i	$0.154306\pi$
$373$			18.0000i			0.932005i	−0.884783	+	0.466002i	$0.845694\pi$
$374$	−24.0000			−1.24101
$375$	−11.0000	−	2.00000i	−0.568038	−	0.103280i
$376$	8.00000			0.412568
$377$		−	10.0000i		−	0.515026i
$378$			4.00000i			0.205738i
$379$	−10.0000			−0.513665			−0.256833	−	0.966456i	$-0.582679\pi$
$379$	−10.0000			−0.513665			−0.256833	+	0.966456i	$0.582679\pi$
$380$	4.00000	−	2.00000i	0.205196	−	0.102598i
$381$	−10.0000			−0.512316
$382$		0			0
$383$		−	20.0000i		−	1.02195i	−0.859595	−	0.510976i	$-0.829284\pi$
$383$		−	20.0000i		−	1.02195i	0.859595	−	0.510976i	$-0.170716\pi$
$384$	−1.00000			−0.0510310
$385$	24.0000	+	48.0000i	1.22315	+	2.44631i
$386$	−2.00000			−0.101797
$387$		0			0
$388$		−	14.0000i		−	0.710742i
$389$	−30.0000			−1.52106			−0.760530	−	0.649303i	$-0.775061\pi$
$389$	−30.0000			−1.52106			−0.760530	+	0.649303i	$0.775061\pi$
$390$	−1.00000	−	2.00000i	−0.0506370	−	0.101274i
$391$	24.0000			1.21373
$392$			9.00000i			0.454569i
$393$			8.00000i			0.403547i
$394$	6.00000			0.302276
$395$	16.0000	−	8.00000i	0.805047	−	0.402524i
$396$	−6.00000			−0.301511
$397$		−	10.0000i		−	0.501886i	−0.968002	−	0.250943i	$-0.919259\pi$
$397$		−	10.0000i		−	0.501886i	0.968002	−	0.250943i	$-0.0807406\pi$
$398$			8.00000i			0.401004i
$399$	8.00000			0.400501
$400$	3.00000	−	4.00000i	0.150000	−	0.200000i
$401$	30.0000			1.49813			0.749064	−	0.662497i	$-0.230503\pi$
$401$	30.0000			1.49813			0.749064	+	0.662497i	$0.230503\pi$
$402$		−	12.0000i		−	0.598506i
$403$		−	4.00000i		−	0.199254i
$404$	−10.0000			−0.497519
$405$	2.00000	−	1.00000i	0.0993808	−	0.0496904i
$406$	40.0000			1.98517
$407$			36.0000i			1.78445i
$408$			4.00000i			0.198030i
$409$	34.0000			1.68119			0.840596	−	0.541663i	$-0.182205\pi$
$409$	34.0000			1.68119			0.840596	+	0.541663i	$0.182205\pi$
$410$	10.0000	+	20.0000i	0.493865	+	0.987730i
$411$	−14.0000			−0.690569
$412$		−	2.00000i		−	0.0985329i
$413$		−	24.0000i		−	1.18096i
$414$	6.00000			0.294884
$415$	4.00000	+	8.00000i	0.196352	+	0.392705i
$416$	1.00000			0.0490290
$417$			16.0000i			0.783523i
$418$			12.0000i			0.586939i
$419$	−8.00000			−0.390826			−0.195413	−	0.980721i	$-0.562605\pi$
$419$	−8.00000			−0.390826			−0.195413	+	0.980721i	$0.562605\pi$
$420$	8.00000	−	4.00000i	0.390360	−	0.195180i
$421$	4.00000			0.194948			0.0974740	−	0.995238i	$-0.468924\pi$
$421$	4.00000			0.194948			0.0974740	+	0.995238i	$0.468924\pi$
$422$			4.00000i			0.194717i
$423$		−	8.00000i		−	0.388973i
$424$	6.00000			0.291386
$425$	−16.0000	−	12.0000i	−0.776114	−	0.582086i
$426$		0			0
$427$			24.0000i			1.16144i
$428$		0			0
$429$	6.00000			0.289683
$430$		0			0
$431$	−16.0000			−0.770693			−0.385346	−	0.922772i	$-0.625918\pi$
$431$	−16.0000			−0.770693			−0.385346	+	0.922772i	$0.625918\pi$
$432$			1.00000i			0.0481125i
$433$		−	16.0000i		−	0.768911i	−0.923144	−	0.384455i	$-0.874389\pi$
$433$		−	16.0000i		−	0.768911i	0.923144	−	0.384455i	$-0.125611\pi$
$434$	16.0000			0.768025
$435$	−10.0000	−	20.0000i	−0.479463	−	0.958927i
$436$	4.00000			0.191565
$437$		−	12.0000i		−	0.574038i
$438$		−	2.00000i		−	0.0955637i
$439$	32.0000			1.52728			0.763638	−	0.645644i	$-0.223411\pi$
$439$	32.0000			1.52728			0.763638	+	0.645644i	$0.223411\pi$
$440$	6.00000	+	12.0000i	0.286039	+	0.572078i
$441$	9.00000			0.428571
$442$		−	4.00000i		−	0.190261i
$443$		0			0		1.00000			$0$
$443$		0			0		−1.00000			$\pi$
$444$	6.00000			0.284747
$445$	−28.0000	+	14.0000i	−1.32733	+	0.663664i
$446$	8.00000			0.378811
$447$			16.0000i			0.756774i
$448$			4.00000i			0.188982i
$449$	−38.0000			−1.79333			−0.896665	−	0.442709i	$-0.854018\pi$
$449$	−38.0000			−1.79333			−0.896665	+	0.442709i	$0.854018\pi$
$450$	−4.00000	−	3.00000i	−0.188562	−	0.141421i
$451$	−60.0000			−2.82529
$452$		−	4.00000i		−	0.188144i
$453$		−	16.0000i		−	0.751746i
$454$	4.00000			0.187729
$455$	−8.00000	+	4.00000i	−0.375046	+	0.187523i
$456$	2.00000			0.0936586
$457$			14.0000i			0.654892i	0.944870	+	0.327446i	$0.106188\pi$
$457$			14.0000i			0.654892i	−0.944870	+	0.327446i	$0.893812\pi$
$458$		−	8.00000i		−	0.373815i
$459$	4.00000			0.186704
$460$	−6.00000	−	12.0000i	−0.279751	−	0.559503i
$461$	−4.00000			−0.186299			−0.0931493	−	0.995652i	$-0.529693\pi$
$461$	−4.00000			−0.186299			−0.0931493	+	0.995652i	$0.529693\pi$
$462$			24.0000i			1.11658i
$463$		−	4.00000i		−	0.185896i	−0.995671	−	0.0929479i	$-0.970371\pi$
$463$		−	4.00000i		−	0.185896i	0.995671	−	0.0929479i	$-0.0296290\pi$
$464$	10.0000			0.464238
$465$	−4.00000	−	8.00000i	−0.185496	−	0.370991i
$466$	−12.0000			−0.555889
$467$			8.00000i			0.370196i	0.982720	+	0.185098i	$0.0592602\pi$
$467$			8.00000i			0.370196i	−0.982720	+	0.185098i	$0.940740\pi$
$468$		−	1.00000i		−	0.0462250i
$469$	−48.0000			−2.21643
$470$	−16.0000	+	8.00000i	−0.738025	+	0.369012i
$471$	6.00000			0.276465
$472$		−	6.00000i		−	0.276172i
$473$		0			0
$474$	8.00000			0.367452
$475$	−6.00000	+	8.00000i	−0.275299	+	0.367065i
$476$	16.0000			0.733359
$477$		−	6.00000i		−	0.274721i
$478$			4.00000i			0.182956i
$479$	20.0000			0.913823			0.456912	−	0.889512i	$-0.348956\pi$
$479$	20.0000			0.913823			0.456912	+	0.889512i	$0.348956\pi$
$480$	2.00000	−	1.00000i	0.0912871	−	0.0456435i
$481$	−6.00000			−0.273576
$482$			18.0000i			0.819878i
$483$		−	24.0000i		−	1.09204i
$484$	−25.0000			−1.13636
$485$	14.0000	+	28.0000i	0.635707	+	1.27141i
$486$	1.00000			0.0453609
$487$			32.0000i			1.45006i	0.688718	+	0.725029i	$0.258174\pi$
$487$			32.0000i			1.45006i	−0.688718	+	0.725029i	$0.741826\pi$
$488$			6.00000i			0.271607i
$489$	4.00000			0.180886
$490$	−9.00000	−	18.0000i	−0.406579	−	0.813157i
$491$		0			0			−	1.00000i	$-0.5\pi$
$491$		0			0				1.00000i	$0.5\pi$
$492$			10.0000i			0.450835i
$493$		−	40.0000i		−	1.80151i
$494$	−2.00000			−0.0899843
$495$	12.0000	−	6.00000i	0.539360	−	0.269680i
$496$	4.00000			0.179605
$497$		0			0
$498$			4.00000i			0.179244i
$499$	18.0000			0.805791			0.402895	−	0.915246i	$-0.368004\pi$
$499$	18.0000			0.805791			0.402895	+	0.915246i	$0.368004\pi$
$500$	−2.00000	+	11.0000i	−0.0894427	+	0.491935i
$501$	−16.0000			−0.714827
$502$		−	20.0000i		−	0.892644i
$503$			14.0000i			0.624229i	0.950044	+	0.312115i	$0.101037\pi$
$503$			14.0000i			0.624229i	−0.950044	+	0.312115i	$0.898963\pi$
$504$	4.00000			0.178174
$505$	20.0000	−	10.0000i	0.889988	−	0.444994i
$506$	36.0000			1.60040
$507$			1.00000i			0.0444116i
$508$			10.0000i			0.443678i
$509$	−16.0000			−0.709188			−0.354594	−	0.935020i	$-0.615381\pi$
$509$	−16.0000			−0.709188			−0.354594	+	0.935020i	$0.615381\pi$
$510$	−4.00000	−	8.00000i	−0.177123	−	0.354246i
$511$	−8.00000			−0.353899
$512$			1.00000i			0.0441942i
$513$		−	2.00000i		−	0.0883022i
$514$	−28.0000			−1.23503
$515$	2.00000	+	4.00000i	0.0881305	+	0.176261i
$516$		0			0
$517$		−	48.0000i		−	2.11104i
$518$		−	24.0000i		−	1.05450i
$519$	22.0000			0.965693
$520$	−2.00000	+	1.00000i	−0.0877058	+	0.0438529i
$521$	6.00000			0.262865			0.131432	−	0.991325i	$-0.458042\pi$
$521$	6.00000			0.262865			0.131432	+	0.991325i	$0.458042\pi$
$522$		−	10.0000i		−	0.437688i
$523$		−	16.0000i		−	0.699631i	−0.936819	−	0.349816i	$-0.886244\pi$
$523$		−	16.0000i		−	0.699631i	0.936819	−	0.349816i	$-0.113756\pi$
$524$	8.00000			0.349482
$525$	−12.0000	+	16.0000i	−0.523723	+	0.698297i
$526$	−10.0000			−0.436021
$527$		−	16.0000i		−	0.696971i
$528$			6.00000i			0.261116i
$529$	−13.0000			−0.565217
$530$	−12.0000	+	6.00000i	−0.521247	+	0.260623i
$531$	−6.00000			−0.260378
$532$		−	8.00000i		−	0.346844i
$533$		−	10.0000i		−	0.433148i
$534$	−14.0000			−0.605839
$535$		0			0
$536$	−12.0000			−0.518321
$537$		−	24.0000i		−	1.03568i
$538$		−	10.0000i		−	0.431131i
$539$	54.0000			2.32594
$540$	−1.00000	−	2.00000i	−0.0430331	−	0.0860663i
$541$	−32.0000			−1.37579			−0.687894	−	0.725811i	$-0.741464\pi$
$541$	−32.0000			−1.37579			−0.687894	+	0.725811i	$0.741464\pi$
$542$		−	8.00000i		−	0.343629i
$543$		−	6.00000i		−	0.257485i
$544$	4.00000			0.171499
$545$	−8.00000	+	4.00000i	−0.342682	+	0.171341i
$546$	−4.00000			−0.171184
$547$			36.0000i			1.53925i	0.638497	+	0.769624i	$0.279557\pi$
$547$			36.0000i			1.53925i	−0.638497	+	0.769624i	$0.720443\pi$
$548$			14.0000i			0.598050i
$549$	6.00000			0.256074
$550$	−24.0000	−	18.0000i	−1.02336	−	0.767523i
$551$	−20.0000			−0.852029
$552$		−	6.00000i		−	0.255377i
$553$		−	32.0000i		−	1.36078i
$554$	14.0000			0.594803
$555$	−12.0000	+	6.00000i	−0.509372	+	0.254686i
$556$	16.0000			0.678551
$557$		−	18.0000i		−	0.762684i	−0.924434	−	0.381342i	$-0.875462\pi$
$557$		−	18.0000i		−	0.762684i	0.924434	−	0.381342i	$-0.124538\pi$
$558$		−	4.00000i		−	0.169334i
$559$		0			0
$560$	−4.00000	−	8.00000i	−0.169031	−	0.338062i
$561$	24.0000			1.01328
$562$		−	18.0000i		−	0.759284i
$563$		−	24.0000i		−	1.01148i	−0.862686	−	0.505740i	$-0.831220\pi$
$563$		−	24.0000i		−	1.01148i	0.862686	−	0.505740i	$-0.168780\pi$
$564$	−8.00000			−0.336861
$565$	4.00000	+	8.00000i	0.168281	+	0.336563i
$566$	−12.0000			−0.504398
$567$		−	4.00000i		−	0.167984i
$568$		0			0
$569$	42.0000			1.76073			0.880366	−	0.474295i	$-0.157297\pi$
$569$	42.0000			1.76073			0.880366	+	0.474295i	$0.157297\pi$
$570$	−4.00000	+	2.00000i	−0.167542	+	0.0837708i
$571$	12.0000			0.502184			0.251092	−	0.967963i	$-0.419210\pi$
$571$	12.0000			0.502184			0.251092	+	0.967963i	$0.419210\pi$
$572$		−	6.00000i		−	0.250873i
$573$		0			0
$574$	40.0000			1.66957
$575$	24.0000	+	18.0000i	1.00087	+	0.750652i
$576$	1.00000			0.0416667
$577$			2.00000i			0.0832611i	0.999133	+	0.0416305i	$0.0132552\pi$
$577$			2.00000i			0.0832611i	−0.999133	+	0.0416305i	$0.986745\pi$
$578$			1.00000i			0.0415945i
$579$	2.00000			0.0831172
$580$	−20.0000	+	10.0000i	−0.830455	+	0.415227i
$581$	16.0000			0.663792
$582$			14.0000i			0.580319i
$583$		−	36.0000i		−	1.49097i
$584$	−2.00000			−0.0827606
$585$	1.00000	+	2.00000i	0.0413449	+	0.0826898i
$586$	18.0000			0.743573
$587$		−	28.0000i		−	1.15568i	−0.816149	−	0.577842i	$-0.803895\pi$
$587$		−	28.0000i		−	1.15568i	0.816149	−	0.577842i	$-0.196105\pi$
$588$		−	9.00000i		−	0.371154i
$589$	−8.00000			−0.329634
$590$	6.00000	+	12.0000i	0.247016	+	0.494032i
$591$	−6.00000			−0.246807
$592$		−	6.00000i		−	0.246598i
$593$		−	14.0000i		−	0.574911i	−0.957794	−	0.287456i	$-0.907191\pi$
$593$		−	14.0000i		−	0.574911i	0.957794	−	0.287456i	$-0.0928094\pi$
$594$	6.00000			0.246183
$595$	−32.0000	+	16.0000i	−1.31187	+	0.655936i
$596$	16.0000			0.655386
$597$		−	8.00000i		−	0.327418i
$598$			6.00000i			0.245358i
$599$	32.0000			1.30748			0.653742	−	0.756717i	$-0.273198\pi$
$599$	32.0000			1.30748			0.653742	+	0.756717i	$0.273198\pi$
$600$	−3.00000	+	4.00000i	−0.122474	+	0.163299i
$601$	22.0000			0.897399			0.448699	−	0.893683i	$-0.351887\pi$
$601$	22.0000			0.897399			0.448699	+	0.893683i	$0.351887\pi$
$602$		0			0
$603$			12.0000i			0.488678i
$604$	−16.0000			−0.651031
$605$	50.0000	−	25.0000i	2.03279	−	1.01639i
$606$	10.0000			0.406222
$607$			42.0000i			1.70473i	0.522949	+	0.852364i	$0.324832\pi$
$607$			42.0000i			1.70473i	−0.522949	+	0.852364i	$0.675168\pi$
$608$		−	2.00000i		−	0.0811107i
$609$	−40.0000			−1.62088
$610$	−6.00000	−	12.0000i	−0.242933	−	0.485866i
$611$	8.00000			0.323645
$612$		−	4.00000i		−	0.161690i
$613$			38.0000i			1.53481i	0.641165	+	0.767403i	$0.278451\pi$
$613$			38.0000i			1.53481i	−0.641165	+	0.767403i	$0.721549\pi$
$614$	4.00000			0.161427
$615$	−10.0000	−	20.0000i	−0.403239	−	0.806478i
$616$	24.0000			0.966988
$617$		−	18.0000i		−	0.724653i	−0.932051	−	0.362326i	$-0.881983\pi$
$617$		−	18.0000i		−	0.724653i	0.932051	−	0.362326i	$-0.118017\pi$
$618$			2.00000i			0.0804518i
$619$	22.0000			0.884255			0.442127	−	0.896952i	$-0.354224\pi$
$619$	22.0000			0.884255			0.442127	+	0.896952i	$0.354224\pi$
$620$	−8.00000	+	4.00000i	−0.321288	+	0.160644i
$621$	−6.00000			−0.240772
$622$		−	8.00000i		−	0.320771i
$623$			56.0000i			2.24359i
$624$	−1.00000			−0.0400320
$625$	−7.00000	−	24.0000i	−0.280000	−	0.960000i
$626$	−8.00000			−0.319744
$627$		−	12.0000i		−	0.479234i
$628$		−	6.00000i		−	0.239426i
$629$	−24.0000			−0.956943
$630$	−8.00000	+	4.00000i	−0.318728	+	0.159364i
$631$	40.0000			1.59237			0.796187	−	0.605050i	$-0.206847\pi$
$631$	40.0000			1.59237			0.796187	+	0.605050i	$0.206847\pi$
$632$		−	8.00000i		−	0.318223i
$633$		−	4.00000i		−	0.158986i
$634$	−2.00000			−0.0794301
$635$	−10.0000	−	20.0000i	−0.396838	−	0.793676i
$636$	−6.00000			−0.237915
$637$			9.00000i			0.356593i
$638$		−	60.0000i		−	2.37542i
$639$		0			0
$640$	−1.00000	−	2.00000i	−0.0395285	−	0.0790569i
$641$	−18.0000			−0.710957			−0.355479	−	0.934684i	$-0.615682\pi$
$641$	−18.0000			−0.710957			−0.355479	+	0.934684i	$0.615682\pi$
$642$		0			0
$643$			20.0000i			0.788723i	0.918955	+	0.394362i	$0.129034\pi$
$643$			20.0000i			0.788723i	−0.918955	+	0.394362i	$0.870966\pi$
$644$	−24.0000			−0.945732
$645$		0			0
$646$	−8.00000			−0.314756
$647$		−	6.00000i		−	0.235884i	−0.993020	−	0.117942i	$-0.962370\pi$
$647$		−	6.00000i		−	0.235884i	0.993020	−	0.117942i	$-0.0376297\pi$
$648$		−	1.00000i		−	0.0392837i
$649$	−36.0000			−1.41312
$650$	3.00000	−	4.00000i	0.117670	−	0.156893i
$651$	−16.0000			−0.627089
$652$		−	4.00000i		−	0.156652i
$653$			10.0000i			0.391330i	0.980671	+	0.195665i	$0.0626866\pi$
$653$			10.0000i			0.391330i	−0.980671	+	0.195665i	$0.937313\pi$
$654$	−4.00000			−0.156412
$655$	−16.0000	+	8.00000i	−0.625172	+	0.312586i
$656$	10.0000			0.390434
$657$			2.00000i			0.0780274i
$658$			32.0000i			1.24749i
$659$	32.0000			1.24654			0.623272	−	0.782006i	$-0.285803\pi$
$659$	32.0000			1.24654			0.623272	+	0.782006i	$0.285803\pi$
$660$	−6.00000	−	12.0000i	−0.233550	−	0.467099i
$661$	−32.0000			−1.24466			−0.622328	−	0.782757i	$-0.713813\pi$
$661$	−32.0000			−1.24466			−0.622328	+	0.782757i	$0.713813\pi$
$662$			2.00000i			0.0777322i
$663$			4.00000i			0.155347i
$664$	4.00000			0.155230
$665$	8.00000	+	16.0000i	0.310227	+	0.620453i
$666$	−6.00000			−0.232495
$667$			60.0000i			2.32321i
$668$			16.0000i			0.619059i
$669$	−8.00000			−0.309298
$670$	24.0000	−	12.0000i	0.927201	−	0.463600i
$671$	36.0000			1.38976
$672$		−	4.00000i		−	0.154303i
$673$		−	44.0000i		−	1.69608i	−0.529936	−	0.848038i	$-0.677784\pi$
$673$		−	44.0000i		−	1.69608i	0.529936	−	0.848038i	$-0.322216\pi$
$674$	−16.0000			−0.616297
$675$	4.00000	+	3.00000i	0.153960	+	0.115470i
$676$	1.00000			0.0384615
$677$			6.00000i			0.230599i	0.993331	+	0.115299i	$0.0367827\pi$
$677$			6.00000i			0.230599i	−0.993331	+	0.115299i	$0.963217\pi$
$678$			4.00000i			0.153619i
$679$	56.0000			2.14908
$680$	−8.00000	+	4.00000i	−0.306786	+	0.153393i
$681$	−4.00000			−0.153280
$682$		−	24.0000i		−	0.919007i
$683$			20.0000i			0.765279i	0.923898	+	0.382639i	$0.124985\pi$
$683$			20.0000i			0.765279i	−0.923898	+	0.382639i	$0.875015\pi$
$684$	−2.00000			−0.0764719
$685$	−14.0000	−	28.0000i	−0.534913	−	1.06983i
$686$	−8.00000			−0.305441
$687$			8.00000i			0.305219i
$688$		0			0
$689$	6.00000			0.228582
$690$	6.00000	+	12.0000i	0.228416	+	0.456832i
$691$	18.0000			0.684752			0.342376	−	0.939563i	$-0.388768\pi$
$691$	18.0000			0.684752			0.342376	+	0.939563i	$0.388768\pi$
$692$		−	22.0000i		−	0.836315i
$693$		−	24.0000i		−	0.911685i
$694$	−16.0000			−0.607352
$695$	−32.0000	+	16.0000i	−1.21383	+	0.606915i
$696$	−10.0000			−0.379049
$697$		−	40.0000i		−	1.51511i
$698$			4.00000i			0.151402i
$699$	12.0000			0.453882
$700$	16.0000	+	12.0000i	0.604743	+	0.453557i
$701$	−6.00000			−0.226617			−0.113308	−	0.993560i	$-0.536145\pi$
$701$	−6.00000			−0.226617			−0.113308	+	0.993560i	$0.536145\pi$
$702$			1.00000i			0.0377426i
$703$			12.0000i			0.452589i
$704$	6.00000			0.226134
$705$	16.0000	−	8.00000i	0.602595	−	0.301297i
$706$	−26.0000			−0.978523
$707$		−	40.0000i		−	1.50435i
$708$			6.00000i			0.225494i
$709$	8.00000			0.300446			0.150223	−	0.988652i	$-0.452001\pi$
$709$	8.00000			0.300446			0.150223	+	0.988652i	$0.452001\pi$
$710$		0			0
$711$	−8.00000			−0.300023
$712$			14.0000i			0.524672i
$713$			24.0000i			0.898807i
$714$	−16.0000			−0.598785
$715$	6.00000	+	12.0000i	0.224387	+	0.448775i
$716$	−24.0000			−0.896922
$717$		−	4.00000i		−	0.149383i
$718$			20.0000i			0.746393i
$719$	−8.00000			−0.298350			−0.149175	−	0.988811i	$-0.547662\pi$
$719$	−8.00000			−0.298350			−0.149175	+	0.988811i	$0.547662\pi$
$720$	−2.00000	+	1.00000i	−0.0745356	+	0.0372678i
$721$	8.00000			0.297936
$722$		−	15.0000i		−	0.558242i
$723$		−	18.0000i		−	0.669427i
$724$	−6.00000			−0.222988
$725$	30.0000	−	40.0000i	1.11417	−	1.48556i
$726$	25.0000			0.927837
$727$			38.0000i			1.40934i	0.709534	+	0.704671i	$0.248905\pi$
$727$			38.0000i			1.40934i	−0.709534	+	0.704671i	$0.751095\pi$
$728$			4.00000i			0.148250i
$729$	−1.00000			−0.0370370
$730$	4.00000	−	2.00000i	0.148047	−	0.0740233i
$731$		0			0
$732$		−	6.00000i		−	0.221766i
$733$			18.0000i			0.664845i	0.943131	+	0.332423i	$0.107866\pi$
$733$			18.0000i			0.664845i	−0.943131	+	0.332423i	$0.892134\pi$
$734$	18.0000			0.664392
$735$	9.00000	+	18.0000i	0.331970	+	0.663940i
$736$	−6.00000			−0.221163
$737$			72.0000i			2.65215i
$738$		−	10.0000i		−	0.368105i
$739$	2.00000			0.0735712			0.0367856	−	0.999323i	$-0.488288\pi$
$739$	2.00000			0.0735712			0.0367856	+	0.999323i	$0.488288\pi$
$740$	6.00000	+	12.0000i	0.220564	+	0.441129i
$741$	2.00000			0.0734718
$742$			24.0000i			0.881068i
$743$			36.0000i			1.32071i	0.750953	+	0.660356i	$0.229595\pi$
$743$			36.0000i			1.32071i	−0.750953	+	0.660356i	$0.770405\pi$
$744$	−4.00000			−0.146647
$745$	−32.0000	+	16.0000i	−1.17239	+	0.586195i
$746$	−18.0000			−0.659027
$747$		−	4.00000i		−	0.146352i
$748$		−	24.0000i		−	0.877527i
$749$		0			0
$750$	2.00000	−	11.0000i	0.0730297	−	0.401663i
$751$	−48.0000			−1.75154			−0.875772	−	0.482724i	$-0.839647\pi$
$751$	−48.0000			−1.75154			−0.875772	+	0.482724i	$0.839647\pi$
$752$			8.00000i			0.291730i
$753$			20.0000i			0.728841i
$754$	10.0000			0.364179
$755$	32.0000	−	16.0000i	1.16460	−	0.582300i
$756$	−4.00000			−0.145479
$757$		−	22.0000i		−	0.799604i	−0.916602	−	0.399802i	$-0.869079\pi$
$757$		−	22.0000i		−	0.799604i	0.916602	−	0.399802i	$-0.130921\pi$
$758$		−	10.0000i		−	0.363216i
$759$	−36.0000			−1.30672
$760$	2.00000	+	4.00000i	0.0725476	+	0.145095i
$761$	30.0000			1.08750			0.543750	−	0.839248i	$-0.317004\pi$
$761$	30.0000			1.08750			0.543750	+	0.839248i	$0.317004\pi$
$762$		−	10.0000i		−	0.362262i
$763$			16.0000i			0.579239i
$764$		0			0
$765$	4.00000	+	8.00000i	0.144620	+	0.289241i
$766$	20.0000			0.722629
$767$		−	6.00000i		−	0.216647i
$768$		−	1.00000i		−	0.0360844i
$769$	−10.0000			−0.360609			−0.180305	−	0.983611i	$-0.557708\pi$
$769$	−10.0000			−0.360609			−0.180305	+	0.983611i	$0.557708\pi$
$770$	−48.0000	+	24.0000i	−1.72980	+	0.864900i
$771$	28.0000			1.00840
$772$		−	2.00000i		−	0.0719816i
$773$			18.0000i			0.647415i	0.946157	+	0.323708i	$0.104929\pi$
$773$			18.0000i			0.647415i	−0.946157	+	0.323708i	$0.895071\pi$
$774$		0			0
$775$	12.0000	−	16.0000i	0.431053	−	0.574737i
$776$	14.0000			0.502571
$777$			24.0000i			0.860995i
$778$		−	30.0000i		−	1.07555i
$779$	−20.0000			−0.716574
$780$	2.00000	−	1.00000i	0.0716115	−	0.0358057i
$781$		0			0
$782$			24.0000i			0.858238i
$783$			10.0000i			0.357371i

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:	\( N \)	\(=\)	\( 390 = 2 \cdot 3 \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	390.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +(2.00000 - 1.00000i) q^{5} +1.00000 q^{6} -4.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +(2.00000 - 1.00000i) q^{5} +1.00000 q^{6} -4.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} +(1.00000 + 2.00000i) q^{10} -6.00000 q^{11} +1.00000i q^{12} -1.00000i q^{13} +4.00000 q^{14} +(-1.00000 - 2.00000i) q^{15} +1.00000 q^{16} -4.00000i q^{17} -1.00000i q^{18} -2.00000 q^{19} +(-2.00000 + 1.00000i) q^{20} -4.00000 q^{21} -6.00000i q^{22} +6.00000i q^{23} -1.00000 q^{24} +(3.00000 - 4.00000i) q^{25} +1.00000 q^{26} +1.00000i q^{27} +4.00000i q^{28} +10.0000 q^{29} +(2.00000 - 1.00000i) q^{30} +4.00000 q^{31} +1.00000i q^{32} +6.00000i q^{33} +4.00000 q^{34} +(-4.00000 - 8.00000i) q^{35} +1.00000 q^{36} -6.00000i q^{37} -2.00000i q^{38} -1.00000 q^{39} +(-1.00000 - 2.00000i) q^{40} +10.0000 q^{41} -4.00000i q^{42} +6.00000 q^{44} +(-2.00000 + 1.00000i) q^{45} -6.00000 q^{46} +8.00000i q^{47} -1.00000i q^{48} -9.00000 q^{49} +(4.00000 + 3.00000i) q^{50} -4.00000 q^{51} +1.00000i q^{52} +6.00000i q^{53} -1.00000 q^{54} +(-12.0000 + 6.00000i) q^{55} -4.00000 q^{56} +2.00000i q^{57} +10.0000i q^{58} +6.00000 q^{59} +(1.00000 + 2.00000i) q^{60} -6.00000 q^{61} +4.00000i q^{62} +4.00000i q^{63} -1.00000 q^{64} +(-1.00000 - 2.00000i) q^{65} -6.00000 q^{66} -12.0000i q^{67} +4.00000i q^{68} +6.00000 q^{69} +(8.00000 - 4.00000i) q^{70} +1.00000i q^{72} -2.00000i q^{73} +6.00000 q^{74} +(-4.00000 - 3.00000i) q^{75} +2.00000 q^{76} +24.0000i q^{77} -1.00000i q^{78} +8.00000 q^{79} +(2.00000 - 1.00000i) q^{80} +1.00000 q^{81} +10.0000i q^{82} +4.00000i q^{83} +4.00000 q^{84} +(-4.00000 - 8.00000i) q^{85} -10.0000i q^{87} +6.00000i q^{88} -14.0000 q^{89} +(-1.00000 - 2.00000i) q^{90} -4.00000 q^{91} -6.00000i q^{92} -4.00000i q^{93} -8.00000 q^{94} +(-4.00000 + 2.00000i) q^{95} +1.00000 q^{96} +14.0000i q^{97} -9.00000i q^{98} +6.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} + 4 q^{5} + 2 q^{6} - 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^4 + 4 * q^5 + 2 * q^6 - 2 * q^9	\( 2 q - 2 q^{4} + 4 q^{5} + 2 q^{6} - 2 q^{9} + 2 q^{10} - 12 q^{11} + 8 q^{14} - 2 q^{15} + 2 q^{16} - 4 q^{19} - 4 q^{20} - 8 q^{21} - 2 q^{24} + 6 q^{25} + 2 q^{26} + 20 q^{29} + 4 q^{30} + 8 q^{31} + 8 q^{34} - 8 q^{35} + 2 q^{36} - 2 q^{39} - 2 q^{40} + 20 q^{41} + 12 q^{44} - 4 q^{45} - 12 q^{46} - 18 q^{49} + 8 q^{50} - 8 q^{51} - 2 q^{54} - 24 q^{55} - 8 q^{56} + 12 q^{59} + 2 q^{60} - 12 q^{61} - 2 q^{64} - 2 q^{65} - 12 q^{66} + 12 q^{69} + 16 q^{70} + 12 q^{74} - 8 q^{75} + 4 q^{76} + 16 q^{79} + 4 q^{80} + 2 q^{81} + 8 q^{84} - 8 q^{85} - 28 q^{89} - 2 q^{90} - 8 q^{91} - 16 q^{94} - 8 q^{95} + 2 q^{96} + 12 q^{99}+O(q^{100}) \) 2 * q - 2 * q^4 + 4 * q^5 + 2 * q^6 - 2 * q^9 + 2 * q^10 - 12 * q^11 + 8 * q^14 - 2 * q^15 + 2 * q^16 - 4 * q^19 - 4 * q^20 - 8 * q^21 - 2 * q^24 + 6 * q^25 + 2 * q^26 + 20 * q^29 + 4 * q^30 + 8 * q^31 + 8 * q^34 - 8 * q^35 + 2 * q^36 - 2 * q^39 - 2 * q^40 + 20 * q^41 + 12 * q^44 - 4 * q^45 - 12 * q^46 - 18 * q^49 + 8 * q^50 - 8 * q^51 - 2 * q^54 - 24 * q^55 - 8 * q^56 + 12 * q^59 + 2 * q^60 - 12 * q^61 - 2 * q^64 - 2 * q^65 - 12 * q^66 + 12 * q^69 + 16 * q^70 + 12 * q^74 - 8 * q^75 + 4 * q^76 + 16 * q^79 + 4 * q^80 + 2 * q^81 + 8 * q^84 - 8 * q^85 - 28 * q^89 - 2 * q^90 - 8 * q^91 - 16 * q^94 - 8 * q^95 + 2 * q^96 + 12 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more