LMFDB - Embedded newform 640.2.o.g.447.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [640,2,Mod(63,640)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(640, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 2, 3]))

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

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

chi := DirichletCharacter("640.63");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 640 = 2^{7} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	640.o (of order $4$, 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:	$5.11042572936$
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:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			447.1
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	640.447
Dual form			640.2.o.g.63.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(2.00000 - 2.00000i) q^{3} +(-2.00000 + 1.00000i) q^{5} +(2.00000 - 2.00000i) q^{7} -5.00000i q^{9} +O(q^{10})$	\(q+(2.00000 - 2.00000i) q^{3} +(-2.00000 + 1.00000i) q^{5} +(2.00000 - 2.00000i) q^{7} -5.00000i q^{9} +4.00000 q^{11} +(3.00000 + 3.00000i) q^{13} +(-2.00000 + 6.00000i) q^{15} +(-3.00000 - 3.00000i) q^{17} -8.00000i q^{21} +(-6.00000 - 6.00000i) q^{23} +(3.00000 - 4.00000i) q^{25} +(-4.00000 - 4.00000i) q^{27} -2.00000 q^{29} +4.00000i q^{31} +(8.00000 - 8.00000i) q^{33} +(-2.00000 + 6.00000i) q^{35} +(3.00000 - 3.00000i) q^{37} +12.0000 q^{39} +(-6.00000 + 6.00000i) q^{43} +(5.00000 + 10.0000i) q^{45} +(6.00000 - 6.00000i) q^{47} -1.00000i q^{49} -12.0000 q^{51} +(-3.00000 - 3.00000i) q^{53} +(-8.00000 + 4.00000i) q^{55} +8.00000i q^{59} +6.00000i q^{61} +(-10.0000 - 10.0000i) q^{63} +(-9.00000 - 3.00000i) q^{65} +(6.00000 + 6.00000i) q^{67} -24.0000 q^{69} +12.0000i q^{71} +(-5.00000 + 5.00000i) q^{73} +(-2.00000 - 14.0000i) q^{75} +(8.00000 - 8.00000i) q^{77} +8.00000 q^{79} -1.00000 q^{81} +(6.00000 - 6.00000i) q^{83} +(9.00000 + 3.00000i) q^{85} +(-4.00000 + 4.00000i) q^{87} +12.0000 q^{91} +(8.00000 + 8.00000i) q^{93} +(11.0000 + 11.0000i) q^{97} -20.0000i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{3} - 4 q^{5} + 4 q^{7}+O(q^{10}) $ 2 * q + 4 * q^3 - 4 * q^5 + 4 * q^7	$ 2 q + 4 q^{3} - 4 q^{5} + 4 q^{7} + 8 q^{11} + 6 q^{13} - 4 q^{15} - 6 q^{17} - 12 q^{23} + 6 q^{25} - 8 q^{27} - 4 q^{29} + 16 q^{33} - 4 q^{35} + 6 q^{37} + 24 q^{39} - 12 q^{43} + 10 q^{45} + 12 q^{47} - 24 q^{51} - 6 q^{53} - 16 q^{55} - 20 q^{63} - 18 q^{65} + 12 q^{67} - 48 q^{69} - 10 q^{73} - 4 q^{75} + 16 q^{77} + 16 q^{79} - 2 q^{81} + 12 q^{83} + 18 q^{85} - 8 q^{87} + 24 q^{91} + 16 q^{93} + 22 q^{97}+O(q^{100}) $ 2 * q + 4 * q^3 - 4 * q^5 + 4 * q^7 + 8 * q^11 + 6 * q^13 - 4 * q^15 - 6 * q^17 - 12 * q^23 + 6 * q^25 - 8 * q^27 - 4 * q^29 + 16 * q^33 - 4 * q^35 + 6 * q^37 + 24 * q^39 - 12 * q^43 + 10 * q^45 + 12 * q^47 - 24 * q^51 - 6 * q^53 - 16 * q^55 - 20 * q^63 - 18 * q^65 + 12 * q^67 - 48 * q^69 - 10 * q^73 - 4 * q^75 + 16 * q^77 + 16 * q^79 - 2 * q^81 + 12 * q^83 + 18 * q^85 - 8 * q^87 + 24 * q^91 + 16 * q^93 + 22 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/640\mathbb{Z}\right)^\times$.

$n$	$257$	$261$	$511$
$\chi(n)$	$e\left(\frac{1}{4}\right)$	$-1$	$-1$

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 $

$2$		0			0
$2$		0			0
$3$	2.00000	−	2.00000i	1.15470	−	1.15470i	0.169102	−	0.985599i	$-0.445913\pi$
$3$	2.00000	−	2.00000i	1.15470	−	1.15470i	0.985599	−	0.169102i	$-0.0540867\pi$
$4$		0			0
$5$	−2.00000	+	1.00000i	−0.894427	+	0.447214i
$5$	−2.00000	+	1.00000i	−0.894427	+	0.447214i
$6$		0			0
$7$	2.00000	−	2.00000i	0.755929	−	0.755929i	−0.219650	−	0.975579i	$-0.570491\pi$
$7$	2.00000	−	2.00000i	0.755929	−	0.755929i	0.975579	+	0.219650i	$0.0704915\pi$
$8$		0			0
$9$		−	5.00000i		−	1.66667i
$10$		0			0
$11$	4.00000			1.20605			0.603023	−	0.797724i	$-0.293963\pi$
$11$	4.00000			1.20605			0.603023	+	0.797724i	$0.293963\pi$
$12$		0			0
$13$	3.00000	+	3.00000i	0.832050	+	0.832050i	0.987797	−	0.155747i	$-0.0497784\pi$
$13$	3.00000	+	3.00000i	0.832050	+	0.832050i	−0.155747	+	0.987797i	$0.549778\pi$
$14$		0			0
$15$	−2.00000	+	6.00000i	−0.516398	+	1.54919i
$16$		0			0
$17$	−3.00000	−	3.00000i	−0.727607	−	0.727607i	0.242536	−	0.970143i	$-0.422021\pi$
$17$	−3.00000	−	3.00000i	−0.727607	−	0.727607i	−0.970143	+	0.242536i	$0.922021\pi$
$18$		0			0
$19$		0			0		1.00000			$0$
$19$		0			0		−1.00000			$\pi$
$20$		0			0
$21$		−	8.00000i		−	1.74574i
$22$		0			0
$23$	−6.00000	−	6.00000i	−1.25109	−	1.25109i	−0.955233	−	0.295853i	$-0.904396\pi$
$23$	−6.00000	−	6.00000i	−1.25109	−	1.25109i	−0.295853	−	0.955233i	$-0.595604\pi$
$24$		0			0
$25$	3.00000	−	4.00000i	0.600000	−	0.800000i
$26$		0			0
$27$	−4.00000	−	4.00000i	−0.769800	−	0.769800i
$28$		0			0
$29$	−2.00000			−0.371391			−0.185695	−	0.982607i	$-0.559454\pi$
$29$	−2.00000			−0.371391			−0.185695	+	0.982607i	$0.559454\pi$
$30$		0			0
$31$			4.00000i			0.718421i	0.933257	+	0.359211i	$0.116954\pi$
$31$			4.00000i			0.718421i	−0.933257	+	0.359211i	$0.883046\pi$
$32$		0			0
$33$	8.00000	−	8.00000i	1.39262	−	1.39262i
$34$		0			0
$35$	−2.00000	+	6.00000i	−0.338062	+	1.01419i
$36$		0			0
$37$	3.00000	−	3.00000i	0.493197	−	0.493197i	−0.416115	−	0.909312i	$-0.636609\pi$
$37$	3.00000	−	3.00000i	0.493197	−	0.493197i	0.909312	+	0.416115i	$0.136609\pi$
$38$		0			0
$39$	12.0000			1.92154
$40$		0			0
$41$		0			0			−	1.00000i	$-0.5\pi$
$41$		0			0				1.00000i	$0.5\pi$
$42$		0			0
$43$	−6.00000	+	6.00000i	−0.914991	+	0.914991i	−0.996660	−	0.0816682i	$-0.973975\pi$
$43$	−6.00000	+	6.00000i	−0.914991	+	0.914991i	0.0816682	+	0.996660i	$0.473975\pi$
$44$		0			0
$45$	5.00000	+	10.0000i	0.745356	+	1.49071i
$46$		0			0
$47$	6.00000	−	6.00000i	0.875190	−	0.875190i	−0.117842	−	0.993032i	$-0.537598\pi$
$47$	6.00000	−	6.00000i	0.875190	−	0.875190i	0.993032	+	0.117842i	$0.0375978\pi$
$48$		0			0
$49$		−	1.00000i		−	0.142857i
$50$		0			0
$51$	−12.0000			−1.68034
$52$		0			0
$53$	−3.00000	−	3.00000i	−0.412082	−	0.412082i	0.470381	−	0.882463i	$-0.344116\pi$
$53$	−3.00000	−	3.00000i	−0.412082	−	0.412082i	−0.882463	+	0.470381i	$0.844116\pi$
$54$		0			0
$55$	−8.00000	+	4.00000i	−1.07872	+	0.539360i
$56$		0			0
$57$		0			0
$58$		0			0
$59$			8.00000i			1.04151i	0.853706	+	0.520756i	$0.174350\pi$
$59$			8.00000i			1.04151i	−0.853706	+	0.520756i	$0.825650\pi$
$60$		0			0
$61$			6.00000i			0.768221i	0.923287	+	0.384111i	$0.125492\pi$
$61$			6.00000i			0.768221i	−0.923287	+	0.384111i	$0.874508\pi$
$62$		0			0
$63$	−10.0000	−	10.0000i	−1.25988	−	1.25988i
$64$		0			0
$65$	−9.00000	−	3.00000i	−1.11631	−	0.372104i
$66$		0			0
$67$	6.00000	+	6.00000i	0.733017	+	0.733017i	0.971216	−	0.238200i	$-0.0765572\pi$
$67$	6.00000	+	6.00000i	0.733017	+	0.733017i	−0.238200	+	0.971216i	$0.576557\pi$
$68$		0			0
$69$	−24.0000			−2.88926
$70$		0			0
$71$			12.0000i			1.42414i	0.702109	+	0.712069i	$0.252242\pi$
$71$			12.0000i			1.42414i	−0.702109	+	0.712069i	$0.747758\pi$
$72$		0			0
$73$	−5.00000	+	5.00000i	−0.585206	+	0.585206i	−0.936329	−	0.351123i	$-0.885800\pi$
$73$	−5.00000	+	5.00000i	−0.585206	+	0.585206i	0.351123	+	0.936329i	$0.385800\pi$
$74$		0			0
$75$	−2.00000	−	14.0000i	−0.230940	−	1.61658i
$76$		0			0
$77$	8.00000	−	8.00000i	0.911685	−	0.911685i
$78$		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
$81$	−1.00000			−0.111111
$82$		0			0
$83$	6.00000	−	6.00000i	0.658586	−	0.658586i	−0.296460	−	0.955045i	$-0.595806\pi$
$83$	6.00000	−	6.00000i	0.658586	−	0.658586i	0.955045	+	0.296460i	$0.0958061\pi$
$84$		0			0
$85$	9.00000	+	3.00000i	0.976187	+	0.325396i
$86$		0			0
$87$	−4.00000	+	4.00000i	−0.428845	+	0.428845i
$88$		0			0
$89$		0			0		1.00000			$0$
$89$		0			0		−1.00000			$\pi$
$90$		0			0
$91$	12.0000			1.25794
$92$		0			0
$93$	8.00000	+	8.00000i	0.829561	+	0.829561i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	11.0000	+	11.0000i	1.11688	+	1.11688i	0.992196	+	0.124684i	$0.0397918\pi$
$97$	11.0000	+	11.0000i	1.11688	+	1.11688i	0.124684	+	0.992196i	$0.460208\pi$
$98$		0			0
$99$		−	20.0000i		−	2.01008i
$100$		0			0
$101$		−	4.00000i		−	0.398015i	−0.979998	−	0.199007i	$-0.936228\pi$
$101$		−	4.00000i		−	0.398015i	0.979998	−	0.199007i	$-0.0637718\pi$
$102$		0			0
$103$	2.00000	+	2.00000i	0.197066	+	0.197066i	0.798741	−	0.601675i	$-0.205500\pi$
$103$	2.00000	+	2.00000i	0.197066	+	0.197066i	−0.601675	+	0.798741i	$0.705500\pi$
$104$		0			0
$105$	8.00000	+	16.0000i	0.780720	+	1.56144i
$106$		0			0
$107$	6.00000	+	6.00000i	0.580042	+	0.580042i	0.934915	−	0.354873i	$-0.115476\pi$
$107$	6.00000	+	6.00000i	0.580042	+	0.580042i	−0.354873	+	0.934915i	$0.615476\pi$
$108$		0			0
$109$	−12.0000			−1.14939			−0.574696	−	0.818367i	$-0.694880\pi$
$109$	−12.0000			−1.14939			−0.574696	+	0.818367i	$0.694880\pi$
$110$		0			0
$111$		−	12.0000i		−	1.13899i
$112$		0			0
$113$	−9.00000	+	9.00000i	−0.846649	+	0.846649i	−0.989713	−	0.143065i	$-0.954304\pi$
$113$	−9.00000	+	9.00000i	−0.846649	+	0.846649i	0.143065	+	0.989713i	$0.454304\pi$
$114$		0			0
$115$	18.0000	+	6.00000i	1.67851	+	0.559503i
$116$		0			0
$117$	15.0000	−	15.0000i	1.38675	−	1.38675i
$118$		0			0
$119$	−12.0000			−1.10004
$120$		0			0
$121$	5.00000			0.454545
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−2.00000	+	11.0000i	−0.178885	+	0.983870i
$126$		0			0
$127$	2.00000	−	2.00000i	0.177471	−	0.177471i	−0.612781	−	0.790253i	$-0.709949\pi$
$127$	2.00000	−	2.00000i	0.177471	−	0.177471i	0.790253	+	0.612781i	$0.209949\pi$
$128$		0			0
$129$			24.0000i			2.11308i
$130$		0			0
$131$	−4.00000			−0.349482			−0.174741	−	0.984614i	$-0.555909\pi$
$131$	−4.00000			−0.349482			−0.174741	+	0.984614i	$0.555909\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	12.0000	+	4.00000i	1.03280	+	0.344265i
$136$		0			0
$137$	−15.0000	−	15.0000i	−1.28154	−	1.28154i	−0.939793	−	0.341743i	$-0.888983\pi$
$137$	−15.0000	−	15.0000i	−1.28154	−	1.28154i	−0.341743	−	0.939793i	$-0.611017\pi$
$138$		0			0
$139$		0			0		1.00000			$0$
$139$		0			0		−1.00000			$\pi$
$140$		0			0
$141$		−	24.0000i		−	2.02116i
$142$		0			0
$143$	12.0000	+	12.0000i	1.00349	+	1.00349i
$144$		0			0
$145$	4.00000	−	2.00000i	0.332182	−	0.166091i
$146$		0			0
$147$	−2.00000	−	2.00000i	−0.164957	−	0.164957i
$148$		0			0
$149$	4.00000			0.327693			0.163846	−	0.986486i	$-0.447610\pi$
$149$	4.00000			0.327693			0.163846	+	0.986486i	$0.447610\pi$
$150$		0			0
$151$		−	12.0000i		−	0.976546i	−0.872691	−	0.488273i	$-0.837627\pi$
$151$		−	12.0000i		−	0.976546i	0.872691	−	0.488273i	$-0.162373\pi$
$152$		0			0
$153$	−15.0000	+	15.0000i	−1.21268	+	1.21268i
$154$		0			0
$155$	−4.00000	−	8.00000i	−0.321288	−	0.642575i
$156$		0			0
$157$	−3.00000	+	3.00000i	−0.239426	+	0.239426i	−0.816612	−	0.577186i	$-0.804151\pi$
$157$	−3.00000	+	3.00000i	−0.239426	+	0.239426i	0.577186	+	0.816612i	$0.304151\pi$
$158$		0			0
$159$	−12.0000			−0.951662
$160$		0			0
$161$	−24.0000			−1.89146
$162$		0			0
$163$	−6.00000	+	6.00000i	−0.469956	+	0.469956i	−0.901900	−	0.431944i	$-0.857828\pi$
$163$	−6.00000	+	6.00000i	−0.469956	+	0.469956i	0.431944	+	0.901900i	$0.357828\pi$
$164$		0			0
$165$	−8.00000	+	24.0000i	−0.622799	+	1.86840i
$166$		0			0
$167$	6.00000	−	6.00000i	0.464294	−	0.464294i	−0.435766	−	0.900060i	$-0.643522\pi$
$167$	6.00000	−	6.00000i	0.464294	−	0.464294i	0.900060	+	0.435766i	$0.143522\pi$
$168$		0			0
$169$			5.00000i			0.384615i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	7.00000	+	7.00000i	0.532200	+	0.532200i	0.921227	−	0.389026i	$-0.127189\pi$
$173$	7.00000	+	7.00000i	0.532200	+	0.532200i	−0.389026	+	0.921227i	$0.627189\pi$
$174$		0			0
$175$	−2.00000	−	14.0000i	−0.151186	−	1.05830i
$176$		0			0
$177$	16.0000	+	16.0000i	1.20263	+	1.20263i
$178$		0			0
$179$			16.0000i			1.19590i	0.801535	+	0.597948i	$0.204017\pi$
$179$			16.0000i			1.19590i	−0.801535	+	0.597948i	$0.795983\pi$
$180$		0			0
$181$		−	12.0000i		−	0.891953i	−0.895045	−	0.445976i	$-0.852856\pi$
$181$		−	12.0000i		−	0.891953i	0.895045	−	0.445976i	$-0.147144\pi$
$182$		0			0
$183$	12.0000	+	12.0000i	0.887066	+	0.887066i
$184$		0			0
$185$	−3.00000	+	9.00000i	−0.220564	+	0.661693i
$186$		0			0
$187$	−12.0000	−	12.0000i	−0.877527	−	0.877527i
$188$		0			0
$189$	−16.0000			−1.16383
$190$		0			0
$191$			12.0000i			0.868290i	0.900843	+	0.434145i	$0.142949\pi$
$191$			12.0000i			0.868290i	−0.900843	+	0.434145i	$0.857051\pi$
$192$		0			0
$193$	13.0000	−	13.0000i	0.935760	−	0.935760i	−0.0622972	−	0.998058i	$-0.519843\pi$
$193$	13.0000	−	13.0000i	0.935760	−	0.935760i	0.998058	+	0.0622972i	$0.0198427\pi$
$194$		0			0
$195$	−24.0000	+	12.0000i	−1.71868	+	0.859338i
$196$		0			0
$197$	−15.0000	+	15.0000i	−1.06871	+	1.06871i	−0.0712470	+	0.997459i	$0.522698\pi$
$197$	−15.0000	+	15.0000i	−1.06871	+	1.06871i	−0.997459	+	0.0712470i	$0.977302\pi$
$198$		0			0
$199$	−24.0000			−1.70131			−0.850657	−	0.525720i	$-0.823796\pi$
$199$	−24.0000			−1.70131			−0.850657	+	0.525720i	$0.823796\pi$
$200$		0			0
$201$	24.0000			1.69283
$202$		0			0
$203$	−4.00000	+	4.00000i	−0.280745	+	0.280745i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−30.0000	+	30.0000i	−2.08514	+	2.08514i
$208$		0			0
$209$		0			0
$210$		0			0
$211$	−12.0000			−0.826114			−0.413057	−	0.910705i	$-0.635539\pi$
$211$	−12.0000			−0.826114			−0.413057	+	0.910705i	$0.635539\pi$
$212$		0			0
$213$	24.0000	+	24.0000i	1.64445	+	1.64445i
$214$		0			0
$215$	6.00000	−	18.0000i	0.409197	−	1.22759i
$216$		0			0
$217$	8.00000	+	8.00000i	0.543075	+	0.543075i
$218$		0			0
$219$			20.0000i			1.35147i
$220$		0			0
$221$		−	18.0000i		−	1.21081i
$222$		0			0
$223$	14.0000	+	14.0000i	0.937509	+	0.937509i	0.998159	−	0.0606498i	$-0.0193173\pi$
$223$	14.0000	+	14.0000i	0.937509	+	0.937509i	−0.0606498	+	0.998159i	$0.519317\pi$
$224$		0			0
$225$	−20.0000	−	15.0000i	−1.33333	−	1.00000i
$226$		0			0
$227$	2.00000	+	2.00000i	0.132745	+	0.132745i	0.770357	−	0.637613i	$-0.220078\pi$
$227$	2.00000	+	2.00000i	0.132745	+	0.132745i	−0.637613	+	0.770357i	$0.720078\pi$
$228$		0			0
$229$	−6.00000			−0.396491			−0.198246	−	0.980152i	$-0.563524\pi$
$229$	−6.00000			−0.396491			−0.198246	+	0.980152i	$0.563524\pi$
$230$		0			0
$231$		−	32.0000i		−	2.10545i
$232$		0			0
$233$	−3.00000	+	3.00000i	−0.196537	+	0.196537i	−0.798513	−	0.601977i	$-0.794380\pi$
$233$	−3.00000	+	3.00000i	−0.196537	+	0.196537i	0.601977	+	0.798513i	$0.294380\pi$
$234$		0			0
$235$	−6.00000	+	18.0000i	−0.391397	+	1.17419i
$236$		0			0
$237$	16.0000	−	16.0000i	1.03931	−	1.03931i
$238$		0			0
$239$		0			0			−	1.00000i	$-0.5\pi$
$239$		0			0				1.00000i	$0.5\pi$
$240$		0			0
$241$		0			0			−	1.00000i	$-0.5\pi$
$241$		0			0				1.00000i	$0.5\pi$
$242$		0			0
$243$	10.0000	−	10.0000i	0.641500	−	0.641500i
$244$		0			0
$245$	1.00000	+	2.00000i	0.0638877	+	0.127775i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		−	24.0000i		−	1.52094i
$250$		0			0
$251$	−4.00000			−0.252478			−0.126239	−	0.992000i	$-0.540291\pi$
$251$	−4.00000			−0.252478			−0.126239	+	0.992000i	$0.540291\pi$
$252$		0			0
$253$	−24.0000	−	24.0000i	−1.50887	−	1.50887i
$254$		0			0
$255$	24.0000	−	12.0000i	1.50294	−	0.751469i
$256$		0			0
$257$	−9.00000	−	9.00000i	−0.561405	−	0.561405i	0.368302	−	0.929706i	$-0.379939\pi$
$257$	−9.00000	−	9.00000i	−0.561405	−	0.561405i	−0.929706	+	0.368302i	$0.879939\pi$
$258$		0			0
$259$		−	12.0000i		−	0.745644i
$260$		0			0
$261$			10.0000i			0.618984i
$262$		0			0
$263$	6.00000	+	6.00000i	0.369976	+	0.369976i	0.867468	−	0.497492i	$-0.165746\pi$
$263$	6.00000	+	6.00000i	0.369976	+	0.369976i	−0.497492	+	0.867468i	$0.665746\pi$
$264$		0			0
$265$	9.00000	+	3.00000i	0.552866	+	0.184289i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−4.00000			−0.243884			−0.121942	−	0.992537i	$-0.538912\pi$
$269$	−4.00000			−0.243884			−0.121942	+	0.992537i	$0.538912\pi$
$270$		0			0
$271$			12.0000i			0.728948i	0.931214	+	0.364474i	$0.118751\pi$
$271$			12.0000i			0.728948i	−0.931214	+	0.364474i	$0.881249\pi$
$272$		0			0
$273$	24.0000	−	24.0000i	1.45255	−	1.45255i
$274$		0			0
$275$	12.0000	−	16.0000i	0.723627	−	0.964836i
$276$		0			0
$277$	21.0000	−	21.0000i	1.26177	−	1.26177i	0.311532	−	0.950236i	$-0.399158\pi$
$277$	21.0000	−	21.0000i	1.26177	−	1.26177i	0.950236	−	0.311532i	$-0.100842\pi$
$278$		0			0
$279$	20.0000			1.19737
$280$		0			0
$281$	−24.0000			−1.43172			−0.715860	−	0.698244i	$-0.753965\pi$
$281$	−24.0000			−1.43172			−0.715860	+	0.698244i	$0.753965\pi$
$282$		0			0
$283$	6.00000	−	6.00000i	0.356663	−	0.356663i	−0.505918	−	0.862581i	$-0.668846\pi$
$283$	6.00000	−	6.00000i	0.356663	−	0.356663i	0.862581	+	0.505918i	$0.168846\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$			1.00000i			0.0588235i
$290$		0			0
$291$	44.0000			2.57933
$292$		0			0
$293$	9.00000	+	9.00000i	0.525786	+	0.525786i	0.919313	−	0.393527i	$-0.128745\pi$
$293$	9.00000	+	9.00000i	0.525786	+	0.525786i	−0.393527	+	0.919313i	$0.628745\pi$
$294$		0			0
$295$	−8.00000	−	16.0000i	−0.465778	−	0.931556i
$296$		0			0
$297$	−16.0000	−	16.0000i	−0.928414	−	0.928414i
$298$		0			0
$299$		−	36.0000i		−	2.08193i
$300$		0			0
$301$			24.0000i			1.38334i
$302$		0			0
$303$	−8.00000	−	8.00000i	−0.459588	−	0.459588i
$304$		0			0
$305$	−6.00000	−	12.0000i	−0.343559	−	0.687118i
$306$		0			0
$307$	−18.0000	−	18.0000i	−1.02731	−	1.02731i	−0.999616	−	0.0276979i	$-0.991182\pi$
$307$	−18.0000	−	18.0000i	−1.02731	−	1.02731i	−0.0276979	−	0.999616i	$-0.508818\pi$
$308$		0			0
$309$	8.00000			0.455104
$310$		0			0
$311$			12.0000i			0.680458i	0.940343	+	0.340229i	$0.110505\pi$
$311$			12.0000i			0.680458i	−0.940343	+	0.340229i	$0.889495\pi$
$312$		0			0
$313$	−1.00000	+	1.00000i	−0.0565233	+	0.0565233i	−0.734803	−	0.678280i	$-0.762726\pi$
$313$	−1.00000	+	1.00000i	−0.0565233	+	0.0565233i	0.678280	+	0.734803i	$0.262726\pi$
$314$		0			0
$315$	30.0000	+	10.0000i	1.69031	+	0.563436i
$316$		0			0
$317$	7.00000	−	7.00000i	0.393159	−	0.393159i	−0.482653	−	0.875812i	$-0.660327\pi$
$317$	7.00000	−	7.00000i	0.393159	−	0.393159i	0.875812	+	0.482653i	$0.160327\pi$
$318$		0			0
$319$	−8.00000			−0.447914
$320$		0			0
$321$	24.0000			1.33955
$322$		0			0
$323$		0			0
$324$		0			0
$325$	21.0000	−	3.00000i	1.16487	−	0.166410i
$326$		0			0
$327$	−24.0000	+	24.0000i	−1.32720	+	1.32720i
$328$		0			0
$329$		−	24.0000i		−	1.32316i
$330$		0			0
$331$	12.0000			0.659580			0.329790	−	0.944054i	$-0.393022\pi$
$331$	12.0000			0.659580			0.329790	+	0.944054i	$0.393022\pi$
$332$		0			0
$333$	−15.0000	−	15.0000i	−0.821995	−	0.821995i
$334$		0			0
$335$	−18.0000	−	6.00000i	−0.983445	−	0.327815i
$336$		0			0
$337$	−7.00000	−	7.00000i	−0.381314	−	0.381314i	0.490261	−	0.871576i	$-0.336901\pi$
$337$	−7.00000	−	7.00000i	−0.381314	−	0.381314i	−0.871576	+	0.490261i	$0.836901\pi$
$338$		0			0
$339$			36.0000i			1.95525i
$340$		0			0
$341$			16.0000i			0.866449i
$342$		0			0
$343$	12.0000	+	12.0000i	0.647939	+	0.647939i
$344$		0			0
$345$	48.0000	−	24.0000i	2.58423	−	1.29212i
$346$		0			0
$347$	10.0000	+	10.0000i	0.536828	+	0.536828i	0.922596	−	0.385768i	$-0.126063\pi$
$347$	10.0000	+	10.0000i	0.536828	+	0.536828i	−0.385768	+	0.922596i	$0.626063\pi$
$348$		0			0
$349$	−30.0000			−1.60586			−0.802932	−	0.596071i	$-0.796728\pi$
$349$	−30.0000			−1.60586			−0.802932	+	0.596071i	$0.796728\pi$
$350$		0			0
$351$		−	24.0000i		−	1.28103i
$352$		0			0
$353$	9.00000	−	9.00000i	0.479022	−	0.479022i	−0.425797	−	0.904819i	$-0.640006\pi$
$353$	9.00000	−	9.00000i	0.479022	−	0.479022i	0.904819	+	0.425797i	$0.140006\pi$
$354$		0			0
$355$	−12.0000	−	24.0000i	−0.636894	−	1.27379i
$356$		0			0
$357$	−24.0000	+	24.0000i	−1.27021	+	1.27021i
$358$		0			0
$359$	−24.0000			−1.26667			−0.633336	−	0.773877i	$-0.718315\pi$
$359$	−24.0000			−1.26667			−0.633336	+	0.773877i	$0.718315\pi$
$360$		0			0
$361$	19.0000			1.00000
$362$		0			0
$363$	10.0000	−	10.0000i	0.524864	−	0.524864i
$364$		0			0
$365$	5.00000	−	15.0000i	0.261712	−	0.785136i
$366$		0			0
$367$	26.0000	−	26.0000i	1.35719	−	1.35719i	0.479824	−	0.877365i	$-0.340700\pi$
$367$	26.0000	−	26.0000i	1.35719	−	1.35719i	0.877365	−	0.479824i	$-0.159300\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−12.0000			−0.623009
$372$		0			0
$373$	−9.00000	−	9.00000i	−0.466002	−	0.466002i	0.434614	−	0.900617i	$-0.356885\pi$
$373$	−9.00000	−	9.00000i	−0.466002	−	0.466002i	−0.900617	+	0.434614i	$0.856885\pi$
$374$		0			0
$375$	18.0000	+	26.0000i	0.929516	+	1.34263i
$376$		0			0
$377$	−6.00000	−	6.00000i	−0.309016	−	0.309016i
$378$		0			0
$379$		−	24.0000i		−	1.23280i	−0.787434	−	0.616399i	$-0.788591\pi$
$379$		−	24.0000i		−	1.23280i	0.787434	−	0.616399i	$-0.211409\pi$
$380$		0			0
$381$		−	8.00000i		−	0.409852i
$382$		0			0
$383$	6.00000	+	6.00000i	0.306586	+	0.306586i	0.843584	−	0.536998i	$-0.180442\pi$
$383$	6.00000	+	6.00000i	0.306586	+	0.306586i	−0.536998	+	0.843584i	$0.680442\pi$
$384$		0			0
$385$	−8.00000	+	24.0000i	−0.407718	+	1.22315i
$386$		0			0
$387$	30.0000	+	30.0000i	1.52499	+	1.52499i
$388$		0			0
$389$	−4.00000			−0.202808			−0.101404	−	0.994845i	$-0.532333\pi$
$389$	−4.00000			−0.202808			−0.101404	+	0.994845i	$0.532333\pi$
$390$		0			0
$391$			36.0000i			1.82060i
$392$		0			0
$393$	−8.00000	+	8.00000i	−0.403547	+	0.403547i
$394$		0			0
$395$	−16.0000	+	8.00000i	−0.805047	+	0.402524i
$396$		0			0
$397$	−15.0000	+	15.0000i	−0.752828	+	0.752828i	−0.975006	−	0.222178i	$-0.928683\pi$
$397$	−15.0000	+	15.0000i	−0.752828	+	0.752828i	0.222178	+	0.975006i	$0.428683\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−18.0000			−0.898877			−0.449439	−	0.893311i	$-0.648376\pi$
$401$	−18.0000			−0.898877			−0.449439	+	0.893311i	$0.648376\pi$
$402$		0			0
$403$	−12.0000	+	12.0000i	−0.597763	+	0.597763i
$404$		0			0
$405$	2.00000	−	1.00000i	0.0993808	−	0.0496904i
$406$		0			0
$407$	12.0000	−	12.0000i	0.594818	−	0.594818i
$408$		0			0
$409$		−	30.0000i		−	1.48340i	−0.670729	−	0.741702i	$-0.734019\pi$
$409$		−	30.0000i		−	1.48340i	0.670729	−	0.741702i	$-0.265981\pi$
$410$		0			0
$411$	−60.0000			−2.95958
$412$		0			0
$413$	16.0000	+	16.0000i	0.787309	+	0.787309i
$414$		0			0
$415$	−6.00000	+	18.0000i	−0.294528	+	0.883585i
$416$		0			0
$417$		0			0
$418$		0			0
$419$		−	8.00000i		−	0.390826i	−0.980721	−	0.195413i	$-0.937395\pi$
$419$		−	8.00000i		−	0.390826i	0.980721	−	0.195413i	$-0.0626047\pi$
$420$		0			0
$421$			30.0000i			1.46211i	0.682318	+	0.731055i	$0.260972\pi$
$421$			30.0000i			1.46211i	−0.682318	+	0.731055i	$0.739028\pi$
$422$		0			0
$423$	−30.0000	−	30.0000i	−1.45865	−	1.45865i
$424$		0			0
$425$	−21.0000	+	3.00000i	−1.01865	+	0.145521i
$426$		0			0
$427$	12.0000	+	12.0000i	0.580721	+	0.580721i
$428$		0			0
$429$	48.0000			2.31746
$430$		0			0
$431$		−	36.0000i		−	1.73406i	−0.498257	−	0.867029i	$-0.666026\pi$
$431$		−	36.0000i		−	1.73406i	0.498257	−	0.867029i	$-0.333974\pi$
$432$		0			0
$433$	11.0000	−	11.0000i	0.528626	−	0.528626i	−0.391536	−	0.920163i	$-0.628056\pi$
$433$	11.0000	−	11.0000i	0.528626	−	0.528626i	0.920163	+	0.391536i	$0.128056\pi$
$434$		0			0
$435$	4.00000	−	12.0000i	0.191785	−	0.575356i
$436$		0			0
$437$		0			0
$438$		0			0
$439$	24.0000			1.14546			0.572729	−	0.819745i	$-0.305885\pi$
$439$	24.0000			1.14546			0.572729	+	0.819745i	$0.305885\pi$
$440$		0			0
$441$	−5.00000			−0.238095
$442$		0			0
$443$	−2.00000	+	2.00000i	−0.0950229	+	0.0950229i	−0.753020	−	0.657997i	$-0.771404\pi$
$443$	−2.00000	+	2.00000i	−0.0950229	+	0.0950229i	0.657997	+	0.753020i	$0.271404\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	8.00000	−	8.00000i	0.378387	−	0.378387i
$448$		0			0
$449$		−	6.00000i		−	0.283158i	−0.989927	−	0.141579i	$-0.954782\pi$
$449$		−	6.00000i		−	0.283158i	0.989927	−	0.141579i	$-0.0452178\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	−24.0000	−	24.0000i	−1.12762	−	1.12762i
$454$		0			0
$455$	−24.0000	+	12.0000i	−1.12514	+	0.562569i
$456$		0			0
$457$	−1.00000	−	1.00000i	−0.0467780	−	0.0467780i	0.683331	−	0.730109i	$-0.260531\pi$
$457$	−1.00000	−	1.00000i	−0.0467780	−	0.0467780i	−0.730109	+	0.683331i	$0.760531\pi$
$458$		0			0
$459$			24.0000i			1.12022i
$460$		0			0
$461$		−	28.0000i		−	1.30409i	−0.758180	−	0.652045i	$-0.773911\pi$
$461$		−	28.0000i		−	1.30409i	0.758180	−	0.652045i	$-0.226089\pi$
$462$		0			0
$463$	−14.0000	−	14.0000i	−0.650635	−	0.650635i	0.302511	−	0.953146i	$-0.402175\pi$
$463$	−14.0000	−	14.0000i	−0.650635	−	0.650635i	−0.953146	+	0.302511i	$0.902175\pi$
$464$		0			0
$465$	−24.0000	−	8.00000i	−1.11297	−	0.370991i
$466$		0			0
$467$	−30.0000	−	30.0000i	−1.38823	−	1.38823i	−0.829029	−	0.559205i	$-0.811106\pi$
$467$	−30.0000	−	30.0000i	−1.38823	−	1.38823i	−0.559205	−	0.829029i	$-0.688894\pi$
$468$		0			0
$469$	24.0000			1.10822
$470$		0			0
$471$			12.0000i			0.552931i
$472$		0			0
$473$	−24.0000	+	24.0000i	−1.10352	+	1.10352i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−15.0000	+	15.0000i	−0.686803	+	0.686803i
$478$		0			0
$479$	24.0000			1.09659			0.548294	−	0.836286i	$-0.315277\pi$
$479$	24.0000			1.09659			0.548294	+	0.836286i	$0.315277\pi$
$480$		0			0
$481$	18.0000			0.820729
$482$		0			0
$483$	−48.0000	+	48.0000i	−2.18408	+	2.18408i
$484$		0			0
$485$	−33.0000	−	11.0000i	−1.49845	−	0.499484i
$486$		0			0
$487$	10.0000	−	10.0000i	0.453143	−	0.453143i	−0.443253	−	0.896396i	$-0.646176\pi$
$487$	10.0000	−	10.0000i	0.453143	−	0.453143i	0.896396	+	0.443253i	$0.146176\pi$
$488$		0			0
$489$			24.0000i			1.08532i
$490$		0			0
$491$	−20.0000			−0.902587			−0.451294	−	0.892375i	$-0.649037\pi$
$491$	−20.0000			−0.902587			−0.451294	+	0.892375i	$0.649037\pi$
$492$		0			0
$493$	6.00000	+	6.00000i	0.270226	+	0.270226i
$494$		0			0
$495$	20.0000	+	40.0000i	0.898933	+	1.79787i
$496$		0			0
$497$	24.0000	+	24.0000i	1.07655	+	1.07655i
$498$		0			0
$499$			24.0000i			1.07439i	0.843459	+	0.537194i	$0.180516\pi$
$499$			24.0000i			1.07439i	−0.843459	+	0.537194i	$0.819484\pi$
$500$		0			0
$501$		−	24.0000i		−	1.07224i
$502$		0			0
$503$	−6.00000	−	6.00000i	−0.267527	−	0.267527i	0.560576	−	0.828103i	$-0.310580\pi$
$503$	−6.00000	−	6.00000i	−0.267527	−	0.267527i	−0.828103	+	0.560576i	$0.810580\pi$
$504$		0			0
$505$	4.00000	+	8.00000i	0.177998	+	0.355995i
$506$		0			0
$507$	10.0000	+	10.0000i	0.444116	+	0.444116i
$508$		0			0
$509$	34.0000			1.50702			0.753512	−	0.657434i	$-0.228358\pi$
$509$	34.0000			1.50702			0.753512	+	0.657434i	$0.228358\pi$
$510$		0			0
$511$			20.0000i			0.884748i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−6.00000	−	2.00000i	−0.264392	−	0.0881305i
$516$		0			0
$517$	24.0000	−	24.0000i	1.05552	−	1.05552i
$518$		0			0
$519$	28.0000			1.22906
$520$		0			0
$521$	6.00000			0.262865			0.131432	−	0.991325i	$-0.458042\pi$
$521$	6.00000			0.262865			0.131432	+	0.991325i	$0.458042\pi$
$522$		0			0
$523$	6.00000	−	6.00000i	0.262362	−	0.262362i	−0.563651	−	0.826013i	$-0.690604\pi$
$523$	6.00000	−	6.00000i	0.262362	−	0.262362i	0.826013	+	0.563651i	$0.190604\pi$
$524$		0			0
$525$	−32.0000	−	24.0000i	−1.39659	−	1.04745i
$526$		0			0
$527$	12.0000	−	12.0000i	0.522728	−	0.522728i
$528$		0			0
$529$			49.0000i			2.13043i
$530$		0			0
$531$	40.0000			1.73585
$532$		0			0
$533$		0			0
$534$		0			0
$535$	−18.0000	−	6.00000i	−0.778208	−	0.259403i
$536$		0			0
$537$	32.0000	+	32.0000i	1.38090	+	1.38090i
$538$		0			0
$539$		−	4.00000i		−	0.172292i
$540$		0			0
$541$			36.0000i			1.54776i	0.633332	+	0.773880i	$0.281687\pi$
$541$			36.0000i			1.54776i	−0.633332	+	0.773880i	$0.718313\pi$
$542$		0			0
$543$	−24.0000	−	24.0000i	−1.02994	−	1.02994i
$544$		0			0
$545$	24.0000	−	12.0000i	1.02805	−	0.514024i
$546$		0			0
$547$	−6.00000	−	6.00000i	−0.256541	−	0.256541i	0.567104	−	0.823646i	$-0.308064\pi$
$547$	−6.00000	−	6.00000i	−0.256541	−	0.256541i	−0.823646	+	0.567104i	$0.808064\pi$
$548$		0			0
$549$	30.0000			1.28037
$550$		0			0
$551$		0			0
$552$		0			0
$553$	16.0000	−	16.0000i	0.680389	−	0.680389i
$554$		0			0
$555$	12.0000	+	24.0000i	0.509372	+	1.01874i
$556$		0			0
$557$	−21.0000	+	21.0000i	−0.889799	+	0.889799i	−0.994503	−	0.104705i	$-0.966610\pi$
$557$	−21.0000	+	21.0000i	−0.889799	+	0.889799i	0.104705	+	0.994503i	$0.466610\pi$
$558$		0			0
$559$	−36.0000			−1.52264
$560$		0			0
$561$	−48.0000			−2.02656
$562$		0			0
$563$	10.0000	−	10.0000i	0.421450	−	0.421450i	−0.464253	−	0.885703i	$-0.653677\pi$
$563$	10.0000	−	10.0000i	0.421450	−	0.421450i	0.885703	+	0.464253i	$0.153677\pi$
$564$		0			0
$565$	9.00000	−	27.0000i	0.378633	−	1.13590i
$566$		0			0
$567$	−2.00000	+	2.00000i	−0.0839921	+	0.0839921i
$568$		0			0
$569$			30.0000i			1.25767i	0.777541	+	0.628833i	$0.216467\pi$
$569$			30.0000i			1.25767i	−0.777541	+	0.628833i	$0.783533\pi$
$570$		0			0
$571$	−12.0000			−0.502184			−0.251092	−	0.967963i	$-0.580790\pi$
$571$	−12.0000			−0.502184			−0.251092	+	0.967963i	$0.580790\pi$
$572$		0			0
$573$	24.0000	+	24.0000i	1.00261	+	1.00261i
$574$		0			0
$575$	−42.0000	+	6.00000i	−1.75152	+	0.250217i
$576$		0			0
$577$	−17.0000	−	17.0000i	−0.707719	−	0.707719i	0.258336	−	0.966055i	$-0.416826\pi$
$577$	−17.0000	−	17.0000i	−0.707719	−	0.707719i	−0.966055	+	0.258336i	$0.916826\pi$
$578$		0			0
$579$		−	52.0000i		−	2.16105i
$580$		0			0
$581$		−	24.0000i		−	0.995688i
$582$		0			0
$583$	−12.0000	−	12.0000i	−0.496989	−	0.496989i
$584$		0			0
$585$	−15.0000	+	45.0000i	−0.620174	+	1.86052i
$586$		0			0
$587$	−6.00000	−	6.00000i	−0.247647	−	0.247647i	0.572358	−	0.820004i	$-0.306029\pi$
$587$	−6.00000	−	6.00000i	−0.247647	−	0.247647i	−0.820004	+	0.572358i	$0.806029\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$			60.0000i			2.46807i
$592$		0			0
$593$	−15.0000	+	15.0000i	−0.615976	+	0.615976i	−0.944497	−	0.328521i	$-0.893450\pi$
$593$	−15.0000	+	15.0000i	−0.615976	+	0.615976i	0.328521	+	0.944497i	$0.393450\pi$
$594$		0			0
$595$	24.0000	−	12.0000i	0.983904	−	0.491952i
$596$		0			0
$597$	−48.0000	+	48.0000i	−1.96451	+	1.96451i
$598$		0			0
$599$	24.0000			0.980613			0.490307	−	0.871550i	$-0.336885\pi$
$599$	24.0000			0.980613			0.490307	+	0.871550i	$0.336885\pi$
$600$		0			0
$601$	−8.00000			−0.326327			−0.163163	−	0.986599i	$-0.552170\pi$
$601$	−8.00000			−0.326327			−0.163163	+	0.986599i	$0.552170\pi$
$602$		0			0
$603$	30.0000	−	30.0000i	1.22169	−	1.22169i
$604$		0			0
$605$	−10.0000	+	5.00000i	−0.406558	+	0.203279i
$606$		0			0
$607$	−2.00000	+	2.00000i	−0.0811775	+	0.0811775i	−0.746530	−	0.665352i	$-0.768281\pi$
$607$	−2.00000	+	2.00000i	−0.0811775	+	0.0811775i	0.665352	+	0.746530i	$0.268281\pi$
$608$		0			0
$609$			16.0000i			0.648353i
$610$		0			0
$611$	36.0000			1.45640
$612$		0			0
$613$	3.00000	+	3.00000i	0.121169	+	0.121169i	0.765091	−	0.643922i	$-0.222694\pi$
$613$	3.00000	+	3.00000i	0.121169	+	0.121169i	−0.643922	+	0.765091i	$0.722694\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−27.0000	−	27.0000i	−1.08698	−	1.08698i	−0.995838	−	0.0911411i	$-0.970949\pi$
$617$	−27.0000	−	27.0000i	−1.08698	−	1.08698i	−0.0911411	−	0.995838i	$-0.529051\pi$
$618$		0			0
$619$		−	24.0000i		−	0.964641i	−0.875995	−	0.482321i	$-0.839794\pi$
$619$		−	24.0000i		−	0.964641i	0.875995	−	0.482321i	$-0.160206\pi$
$620$		0			0
$621$			48.0000i			1.92617i
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−7.00000	−	24.0000i	−0.280000	−	0.960000i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−18.0000			−0.717707
$630$		0			0
$631$		−	20.0000i		−	0.796187i	−0.917345	−	0.398094i	$-0.869672\pi$
$631$		−	20.0000i		−	0.796187i	0.917345	−	0.398094i	$-0.130328\pi$
$632$		0			0
$633$	−24.0000	+	24.0000i	−0.953914	+	0.953914i
$634$		0			0
$635$	−2.00000	+	6.00000i	−0.0793676	+	0.238103i
$636$		0			0
$637$	3.00000	−	3.00000i	0.118864	−	0.118864i
$638$		0			0
$639$	60.0000			2.37356
$640$		0			0
$641$		0			0			−	1.00000i	$-0.5\pi$
$641$		0			0				1.00000i	$0.5\pi$
$642$		0			0
$643$	30.0000	−	30.0000i	1.18308	−	1.18308i	0.204144	−	0.978941i	$-0.434559\pi$
$643$	30.0000	−	30.0000i	1.18308	−	1.18308i	0.978941	−	0.204144i	$-0.0654409\pi$
$644$		0			0
$645$	−24.0000	−	48.0000i	−0.944999	−	1.89000i
$646$		0			0
$647$	18.0000	−	18.0000i	0.707653	−	0.707653i	−0.258388	−	0.966041i	$-0.583191\pi$
$647$	18.0000	−	18.0000i	0.707653	−	0.707653i	0.966041	+	0.258388i	$0.0831913\pi$
$648$		0			0
$649$			32.0000i			1.25611i
$650$		0			0
$651$	32.0000			1.25418
$652$		0			0
$653$	−19.0000	−	19.0000i	−0.743527	−	0.743527i	0.229728	−	0.973255i	$-0.426216\pi$
$653$	−19.0000	−	19.0000i	−0.743527	−	0.743527i	−0.973255	+	0.229728i	$0.926216\pi$
$654$		0			0
$655$	8.00000	−	4.00000i	0.312586	−	0.156293i
$656$		0			0
$657$	25.0000	+	25.0000i	0.975343	+	0.975343i
$658$		0			0
$659$			40.0000i			1.55818i	0.626913	+	0.779089i	$0.284318\pi$
$659$			40.0000i			1.55818i	−0.626913	+	0.779089i	$0.715682\pi$
$660$		0			0
$661$			18.0000i			0.700119i	0.936727	+	0.350059i	$0.113839\pi$
$661$			18.0000i			0.700119i	−0.936727	+	0.350059i	$0.886161\pi$
$662$		0			0
$663$	−36.0000	−	36.0000i	−1.39812	−	1.39812i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	12.0000	+	12.0000i	0.464642	+	0.464642i
$668$		0			0
$669$	56.0000			2.16509
$670$		0			0
$671$			24.0000i			0.926510i
$672$		0			0
$673$	−29.0000	+	29.0000i	−1.11787	+	1.11787i	−0.125814	+	0.992054i	$0.540154\pi$
$673$	−29.0000	+	29.0000i	−1.11787	+	1.11787i	−0.992054	+	0.125814i	$0.959846\pi$
$674$		0			0
$675$	−28.0000	+	4.00000i	−1.07772	+	0.153960i
$676$		0			0
$677$	−7.00000	+	7.00000i	−0.269032	+	0.269032i	−0.828710	−	0.559678i	$-0.810925\pi$
$677$	−7.00000	+	7.00000i	−0.269032	+	0.269032i	0.559678	+	0.828710i	$0.310925\pi$
$678$		0			0
$679$	44.0000			1.68857
$680$		0			0
$681$	8.00000			0.306561
$682$		0			0
$683$	18.0000	−	18.0000i	0.688751	−	0.688751i	−0.273205	−	0.961956i	$-0.588084\pi$
$683$	18.0000	−	18.0000i	0.688751	−	0.688751i	0.961956	+	0.273205i	$0.0880837\pi$
$684$		0			0
$685$	45.0000	+	15.0000i	1.71936	+	0.573121i
$686$		0			0
$687$	−12.0000	+	12.0000i	−0.457829	+	0.457829i
$688$		0			0
$689$		−	18.0000i		−	0.685745i
$690$		0			0
$691$	−12.0000			−0.456502			−0.228251	−	0.973602i	$-0.573301\pi$
$691$	−12.0000			−0.456502			−0.228251	+	0.973602i	$0.573301\pi$
$692$		0			0
$693$	−40.0000	−	40.0000i	−1.51947	−	1.51947i
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$			12.0000i			0.453882i
$700$		0			0
$701$			26.0000i			0.982006i	0.871158	+	0.491003i	$0.163370\pi$
$701$			26.0000i			0.982006i	−0.871158	+	0.491003i	$0.836630\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$	24.0000	+	48.0000i	0.903892	+	1.80778i
$706$		0			0
$707$	−8.00000	−	8.00000i	−0.300871	−	0.300871i
$708$		0			0
$709$	6.00000			0.225335			0.112667	−	0.993633i	$-0.464061\pi$
$709$	6.00000			0.225335			0.112667	+	0.993633i	$0.464061\pi$
$710$		0			0
$711$		−	40.0000i		−	1.50012i
$712$		0			0
$713$	24.0000	−	24.0000i	0.898807	−	0.898807i
$714$		0			0
$715$	−36.0000	−	12.0000i	−1.34632	−	0.448775i
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0			−	1.00000i	$-0.5\pi$
$719$		0			0				1.00000i	$0.5\pi$
$720$		0			0
$721$	8.00000			0.297936
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−6.00000	+	8.00000i	−0.222834	+	0.297113i
$726$		0			0
$727$	10.0000	−	10.0000i	0.370879	−	0.370879i	−0.496918	−	0.867797i	$-0.665535\pi$
$727$	10.0000	−	10.0000i	0.370879	−	0.370879i	0.867797	+	0.496918i	$0.165535\pi$
$728$		0			0
$729$		−	43.0000i		−	1.59259i
$730$		0			0
$731$	36.0000			1.33151
$732$		0			0
$733$	−33.0000	−	33.0000i	−1.21888	−	1.21888i	−0.968024	−	0.250859i	$-0.919287\pi$
$733$	−33.0000	−	33.0000i	−1.21888	−	1.21888i	−0.250859	−	0.968024i	$-0.580713\pi$
$734$		0			0
$735$	6.00000	+	2.00000i	0.221313	+	0.0737711i
$736$		0			0
$737$	24.0000	+	24.0000i	0.884051	+	0.884051i
$738$		0			0
$739$			48.0000i			1.76571i	0.469647	+	0.882854i	$0.344381\pi$
$739$			48.0000i			1.76571i	−0.469647	+	0.882854i	$0.655619\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	30.0000	+	30.0000i	1.10059	+	1.10059i	0.994339	+	0.106254i	$0.0338857\pi$
$743$	30.0000	+	30.0000i	1.10059	+	1.10059i	0.106254	+	0.994339i	$0.466114\pi$
$744$		0			0
$745$	−8.00000	+	4.00000i	−0.293097	+	0.146549i
$746$		0			0
$747$	−30.0000	−	30.0000i	−1.09764	−	1.09764i
$748$		0			0
$749$	24.0000			0.876941
$750$		0			0
$751$		−	4.00000i		−	0.145962i	−0.997333	−	0.0729810i	$-0.976749\pi$
$751$		−	4.00000i		−	0.145962i	0.997333	−	0.0729810i	$-0.0232513\pi$
$752$		0			0
$753$	−8.00000	+	8.00000i	−0.291536	+	0.291536i
$754$		0			0
$755$	12.0000	+	24.0000i	0.436725	+	0.873449i
$756$		0			0
$757$	−3.00000	+	3.00000i	−0.109037	+	0.109037i	−0.759520	−	0.650484i	$-0.774566\pi$
$757$	−3.00000	+	3.00000i	−0.109037	+	0.109037i	0.650484	+	0.759520i	$0.274566\pi$
$758$		0			0
$759$	−96.0000			−3.48458
$760$		0			0
$761$	42.0000			1.52250			0.761249	−	0.648459i	$-0.224586\pi$
$761$	42.0000			1.52250			0.761249	+	0.648459i	$0.224586\pi$
$762$		0			0
$763$	−24.0000	+	24.0000i	−0.868858	+	0.868858i
$764$		0			0
$765$	15.0000	−	45.0000i	0.542326	−	1.62698i
$766$		0			0
$767$	−24.0000	+	24.0000i	−0.866590	+	0.866590i
$768$		0			0
$769$		−	40.0000i		−	1.44244i	−0.692708	−	0.721218i	$-0.743582\pi$
$769$		−	40.0000i		−	1.44244i	0.692708	−	0.721218i	$-0.256418\pi$
$770$		0			0
$771$	−36.0000			−1.29651
$772$		0			0
$773$	−5.00000	−	5.00000i	−0.179838	−	0.179838i	0.611448	−	0.791285i	$-0.290588\pi$
$773$	−5.00000	−	5.00000i	−0.179838	−	0.179838i	−0.791285	+	0.611448i	$0.790588\pi$
$774$		0			0
$775$	16.0000	+	12.0000i	0.574737	+	0.431053i
$776$		0			0
$777$	−24.0000	−	24.0000i	−0.860995	−	0.860995i
$778$		0			0
$779$		0			0
$780$		0			0
$781$			48.0000i			1.71758i
$782$		0			0
$783$	8.00000	+	8.00000i	0.285897	+	0.285897i
$784$		0			0
$785$	3.00000	−	9.00000i	0.107075	−	0.321224i
$786$		0			0
$787$	6.00000	+	6.00000i	0.213877	+	0.213877i	0.805912	−	0.592035i	$-0.201675\pi$
$787$	6.00000	+	6.00000i	0.213877	+	0.213877i	−0.592035	+	0.805912i	$0.701675\pi$
$788$		0			0
$789$	24.0000			0.854423
$790$		0			0
$791$			36.0000i			1.28001i
$792$		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:	\( N \)	\(=\)	\( 640 = 2^{7} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	640.o (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(2.00000 - 2.00000i) q^{3} +(-2.00000 + 1.00000i) q^{5} +(2.00000 - 2.00000i) q^{7} -5.00000i q^{9} +O(q^{10})\)	\(q+(2.00000 - 2.00000i) q^{3} +(-2.00000 + 1.00000i) q^{5} +(2.00000 - 2.00000i) q^{7} -5.00000i q^{9} +4.00000 q^{11} +(3.00000 + 3.00000i) q^{13} +(-2.00000 + 6.00000i) q^{15} +(-3.00000 - 3.00000i) q^{17} -8.00000i q^{21} +(-6.00000 - 6.00000i) q^{23} +(3.00000 - 4.00000i) q^{25} +(-4.00000 - 4.00000i) q^{27} -2.00000 q^{29} +4.00000i q^{31} +(8.00000 - 8.00000i) q^{33} +(-2.00000 + 6.00000i) q^{35} +(3.00000 - 3.00000i) q^{37} +12.0000 q^{39} +(-6.00000 + 6.00000i) q^{43} +(5.00000 + 10.0000i) q^{45} +(6.00000 - 6.00000i) q^{47} -1.00000i q^{49} -12.0000 q^{51} +(-3.00000 - 3.00000i) q^{53} +(-8.00000 + 4.00000i) q^{55} +8.00000i q^{59} +6.00000i q^{61} +(-10.0000 - 10.0000i) q^{63} +(-9.00000 - 3.00000i) q^{65} +(6.00000 + 6.00000i) q^{67} -24.0000 q^{69} +12.0000i q^{71} +(-5.00000 + 5.00000i) q^{73} +(-2.00000 - 14.0000i) q^{75} +(8.00000 - 8.00000i) q^{77} +8.00000 q^{79} -1.00000 q^{81} +(6.00000 - 6.00000i) q^{83} +(9.00000 + 3.00000i) q^{85} +(-4.00000 + 4.00000i) q^{87} +12.0000 q^{91} +(8.00000 + 8.00000i) q^{93} +(11.0000 + 11.0000i) q^{97} -20.0000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{3} - 4 q^{5} + 4 q^{7}+O(q^{10}) \) 2 * q + 4 * q^3 - 4 * q^5 + 4 * q^7	\( 2 q + 4 q^{3} - 4 q^{5} + 4 q^{7} + 8 q^{11} + 6 q^{13} - 4 q^{15} - 6 q^{17} - 12 q^{23} + 6 q^{25} - 8 q^{27} - 4 q^{29} + 16 q^{33} - 4 q^{35} + 6 q^{37} + 24 q^{39} - 12 q^{43} + 10 q^{45} + 12 q^{47} - 24 q^{51} - 6 q^{53} - 16 q^{55} - 20 q^{63} - 18 q^{65} + 12 q^{67} - 48 q^{69} - 10 q^{73} - 4 q^{75} + 16 q^{77} + 16 q^{79} - 2 q^{81} + 12 q^{83} + 18 q^{85} - 8 q^{87} + 24 q^{91} + 16 q^{93} + 22 q^{97}+O(q^{100}) \) 2 * q + 4 * q^3 - 4 * q^5 + 4 * q^7 + 8 * q^11 + 6 * q^13 - 4 * q^15 - 6 * q^17 - 12 * q^23 + 6 * q^25 - 8 * q^27 - 4 * q^29 + 16 * q^33 - 4 * q^35 + 6 * q^37 + 24 * q^39 - 12 * q^43 + 10 * q^45 + 12 * q^47 - 24 * q^51 - 6 * q^53 - 16 * q^55 - 20 * q^63 - 18 * q^65 + 12 * q^67 - 48 * q^69 - 10 * q^73 - 4 * q^75 + 16 * q^77 + 16 * q^79 - 2 * q^81 + 12 * q^83 + 18 * q^85 - 8 * q^87 + 24 * q^91 + 16 * q^93 + 22 * q^97

Introduction

L-functions

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