LMFDB - Embedded newform 160.2.n.f.63.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [160,2,Mod(63,160)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(160, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 0, 3]))

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

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

chi := DirichletCharacter("160.63");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 160 = 2^{5} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	160.n (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:	$1.27760643234$
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			63.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	160.63
Dual form			160.2.n.f.127.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+(2.00000 - 2.00000i) q^{3} +(-2.00000 - 1.00000i) q^{5} +(2.00000 + 2.00000i) q^{7} -5.00000i q^{9} +(-1.00000 - 1.00000i) q^{13} +(-6.00000 + 2.00000i) q^{15} +(-5.00000 + 5.00000i) q^{17} +4.00000 q^{19} +8.00000 q^{21} +(2.00000 - 2.00000i) q^{23} +(3.00000 + 4.00000i) q^{25} +(-4.00000 - 4.00000i) q^{27} +4.00000i q^{29} +4.00000i q^{31} +(-2.00000 - 6.00000i) q^{35} +(1.00000 - 1.00000i) q^{37} -4.00000 q^{39} +(-6.00000 + 6.00000i) q^{43} +(-5.00000 + 10.0000i) q^{45} +(-2.00000 - 2.00000i) q^{47} +1.00000i q^{49} +20.0000i q^{51} +(-7.00000 - 7.00000i) q^{53} +(8.00000 - 8.00000i) q^{57} +4.00000 q^{59} -4.00000 q^{61} +(10.0000 - 10.0000i) q^{63} +(1.00000 + 3.00000i) q^{65} +(-10.0000 - 10.0000i) q^{67} -8.00000i q^{69} -12.0000i q^{71} +(-3.00000 - 3.00000i) q^{73} +(14.0000 + 2.00000i) q^{75} +16.0000 q^{79} -1.00000 q^{81} +(-2.00000 + 2.00000i) q^{83} +(15.0000 - 5.00000i) q^{85} +(8.00000 + 8.00000i) q^{87} -4.00000i q^{91} +(8.00000 + 8.00000i) q^{93} +(-8.00000 - 4.00000i) q^{95} +(-3.00000 + 3.00000i) q^{97} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{3} - 4 q^{5} + 4 q^{7} - 2 q^{13} - 12 q^{15} - 10 q^{17} + 8 q^{19} + 16 q^{21} + 4 q^{23} + 6 q^{25} - 8 q^{27} - 4 q^{35} + 2 q^{37} - 8 q^{39} - 12 q^{43} - 10 q^{45} - 4 q^{47} - 14 q^{53}+ \cdots - 6 q^{97}+O(q^{100}) $ 2 * q + 4 * q^3 - 4 * q^5 + 4 * q^7 - 2 * q^13 - 12 * q^15 - 10 * q^17 + 8 * q^19 + 16 * q^21 + 4 * q^23 + 6 * q^25 - 8 * q^27 - 4 * q^35 + 2 * q^37 - 8 * q^39 - 12 * q^43 - 10 * q^45 - 4 * q^47 - 14 * q^53 + 16 * q^57 + 8 * q^59 - 8 * q^61 + 20 * q^63 + 2 * q^65 - 20 * q^67 - 6 * q^73 + 28 * q^75 + 32 * q^79 - 2 * q^81 - 4 * q^83 + 30 * q^85 + 16 * q^87 + 16 * q^93 - 16 * q^95 - 6 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/160\mathbb{Z}\right)^\times$.

$n$	$31$	$97$	$101$
$\chi(n)$	$-1$	$e\left(\frac{3}{4}\right)$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	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	0.975579	−	0.219650i	$-0.0704915\pi$
$7$	2.00000	+	2.00000i	0.755929	+	0.755929i	−0.219650	+	0.975579i	$0.570491\pi$
$8$		0			0
$9$		−	5.00000i		−	1.66667i
$10$		0			0
$11$		0			0		1.00000			$0$
$11$		0			0		−1.00000			$\pi$
$12$		0			0
$13$	−1.00000	−	1.00000i	−0.277350	−	0.277350i	0.554700	−	0.832050i	$-0.312833\pi$
$13$	−1.00000	−	1.00000i	−0.277350	−	0.277350i	−0.832050	+	0.554700i	$0.812833\pi$
$14$		0			0
$15$	−6.00000	+	2.00000i	−1.54919	+	0.516398i
$16$		0			0
$17$	−5.00000	+	5.00000i	−1.21268	+	1.21268i	−0.242536	+	0.970143i	$0.577979\pi$
$17$	−5.00000	+	5.00000i	−1.21268	+	1.21268i	−0.970143	+	0.242536i	$0.922021\pi$
$18$		0			0
$19$	4.00000			0.917663			0.458831	−	0.888523i	$-0.348268\pi$
$19$	4.00000			0.917663			0.458831	+	0.888523i	$0.348268\pi$
$20$		0			0
$21$	8.00000			1.74574
$22$		0			0
$23$	2.00000	−	2.00000i	0.417029	−	0.417029i	−0.467150	−	0.884178i	$-0.654719\pi$
$23$	2.00000	−	2.00000i	0.417029	−	0.417029i	0.884178	+	0.467150i	$0.154719\pi$
$24$		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
$29$			4.00000i			0.742781i	0.928477	+	0.371391i	$0.121119\pi$
$29$			4.00000i			0.742781i	−0.928477	+	0.371391i	$0.878881\pi$
$30$		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
$33$		0			0
$34$		0			0
$35$	−2.00000	−	6.00000i	−0.338062	−	1.01419i
$36$		0			0
$37$	1.00000	−	1.00000i	0.164399	−	0.164399i	−0.620113	−	0.784512i	$-0.712913\pi$
$37$	1.00000	−	1.00000i	0.164399	−	0.164399i	0.784512	+	0.620113i	$0.212913\pi$
$38$		0			0
$39$	−4.00000			−0.640513
$40$		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
$47$	−2.00000	−	2.00000i	−0.291730	−	0.291730i	0.546033	−	0.837763i	$-0.316137\pi$
$47$	−2.00000	−	2.00000i	−0.291730	−	0.291730i	−0.837763	+	0.546033i	$0.816137\pi$
$48$		0			0
$49$			1.00000i			0.142857i
$50$		0			0
$51$			20.0000i			2.80056i
$52$		0			0
$53$	−7.00000	−	7.00000i	−0.961524	−	0.961524i	0.0377628	−	0.999287i	$-0.487977\pi$
$53$	−7.00000	−	7.00000i	−0.961524	−	0.961524i	−0.999287	+	0.0377628i	$0.987977\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	8.00000	−	8.00000i	1.05963	−	1.05963i
$58$		0			0
$59$	4.00000			0.520756			0.260378	−	0.965507i	$-0.416153\pi$
$59$	4.00000			0.520756			0.260378	+	0.965507i	$0.416153\pi$
$60$		0			0
$61$	−4.00000			−0.512148			−0.256074	−	0.966657i	$-0.582429\pi$
$61$	−4.00000			−0.512148			−0.256074	+	0.966657i	$0.582429\pi$
$62$		0			0
$63$	10.0000	−	10.0000i	1.25988	−	1.25988i
$64$		0			0
$65$	1.00000	+	3.00000i	0.124035	+	0.372104i
$66$		0			0
$67$	−10.0000	−	10.0000i	−1.22169	−	1.22169i	−0.967029	−	0.254665i	$-0.918035\pi$
$67$	−10.0000	−	10.0000i	−1.22169	−	1.22169i	−0.254665	−	0.967029i	$-0.581965\pi$
$68$		0			0
$69$		−	8.00000i		−	0.963087i
$70$		0			0
$71$		−	12.0000i		−	1.42414i	−0.702109	−	0.712069i	$-0.747758\pi$
$71$		−	12.0000i		−	1.42414i	0.702109	−	0.712069i	$-0.252242\pi$
$72$		0			0
$73$	−3.00000	−	3.00000i	−0.351123	−	0.351123i	0.509404	−	0.860527i	$-0.329866\pi$
$73$	−3.00000	−	3.00000i	−0.351123	−	0.351123i	−0.860527	+	0.509404i	$0.829866\pi$
$74$		0			0
$75$	14.0000	+	2.00000i	1.61658	+	0.230940i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	16.0000			1.80014			0.900070	−	0.435745i	$-0.143515\pi$
$79$	16.0000			1.80014			0.900070	+	0.435745i	$0.143515\pi$
$80$		0			0
$81$	−1.00000			−0.111111
$82$		0			0
$83$	−2.00000	+	2.00000i	−0.219529	+	0.219529i	−0.808300	−	0.588771i	$-0.799612\pi$
$83$	−2.00000	+	2.00000i	−0.219529	+	0.219529i	0.588771	+	0.808300i	$0.299612\pi$
$84$		0			0
$85$	15.0000	−	5.00000i	1.62698	−	0.542326i
$86$		0			0
$87$	8.00000	+	8.00000i	0.857690	+	0.857690i
$88$		0			0
$89$		0			0		1.00000			$0$
$89$		0			0		−1.00000			$\pi$
$90$		0			0
$91$		−	4.00000i		−	0.419314i
$92$		0			0
$93$	8.00000	+	8.00000i	0.829561	+	0.829561i
$94$		0			0
$95$	−8.00000	−	4.00000i	−0.820783	−	0.410391i
$96$		0			0
$97$	−3.00000	+	3.00000i	−0.304604	+	0.304604i	−0.842812	−	0.538208i	$-0.819101\pi$
$97$	−3.00000	+	3.00000i	−0.304604	+	0.304604i	0.538208	+	0.842812i	$0.319101\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−6.00000			−0.597022			−0.298511	−	0.954406i	$-0.596490\pi$
$101$	−6.00000			−0.597022			−0.298511	+	0.954406i	$0.596490\pi$
$102$		0			0
$103$	−6.00000	+	6.00000i	−0.591198	+	0.591198i	−0.937955	−	0.346757i	$-0.887283\pi$
$103$	−6.00000	+	6.00000i	−0.591198	+	0.591198i	0.346757	+	0.937955i	$0.387283\pi$
$104$		0			0
$105$	−16.0000	−	8.00000i	−1.56144	−	0.780720i
$106$		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
$109$		−	10.0000i		−	0.957826i	−0.877862	−	0.478913i	$-0.841031\pi$
$109$		−	10.0000i		−	0.957826i	0.877862	−	0.478913i	$-0.158969\pi$
$110$		0			0
$111$		−	4.00000i		−	0.379663i
$112$		0			0
$113$	−9.00000	−	9.00000i	−0.846649	−	0.846649i	0.143065	−	0.989713i	$-0.454304\pi$
$113$	−9.00000	−	9.00000i	−0.846649	−	0.846649i	−0.989713	+	0.143065i	$0.954304\pi$
$114$		0			0
$115$	−6.00000	+	2.00000i	−0.559503	+	0.186501i
$116$		0			0
$117$	−5.00000	+	5.00000i	−0.462250	+	0.462250i
$118$		0			0
$119$	−20.0000			−1.83340
$120$		0			0
$121$	11.0000			1.00000
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−2.00000	−	11.0000i	−0.178885	−	0.983870i
$126$		0			0
$127$	10.0000	+	10.0000i	0.887357	+	0.887357i	0.994268	−	0.106912i	$-0.0340963\pi$
$127$	10.0000	+	10.0000i	0.887357	+	0.887357i	−0.106912	+	0.994268i	$0.534096\pi$
$128$		0			0
$129$			24.0000i			2.11308i
$130$		0			0
$131$		−	8.00000i		−	0.698963i	−0.936943	−	0.349482i	$-0.886358\pi$
$131$		−	8.00000i		−	0.698963i	0.936943	−	0.349482i	$-0.113642\pi$
$132$		0			0
$133$	8.00000	+	8.00000i	0.693688	+	0.693688i
$134$		0			0
$135$	4.00000	+	12.0000i	0.344265	+	1.03280i
$136$		0			0
$137$	1.00000	−	1.00000i	0.0854358	−	0.0854358i	−0.663097	−	0.748533i	$-0.730758\pi$
$137$	1.00000	−	1.00000i	0.0854358	−	0.0854358i	0.748533	+	0.663097i	$0.230758\pi$
$138$		0			0
$139$	−12.0000			−1.01783			−0.508913	−	0.860818i	$-0.669953\pi$
$139$	−12.0000			−1.01783			−0.508913	+	0.860818i	$0.669953\pi$
$140$		0			0
$141$	−8.00000			−0.673722
$142$		0			0
$143$		0			0
$144$		0			0
$145$	4.00000	−	8.00000i	0.332182	−	0.664364i
$146$		0			0
$147$	2.00000	+	2.00000i	0.164957	+	0.164957i
$148$		0			0
$149$			18.0000i			1.47462i	0.675556	+	0.737309i	$0.263904\pi$
$149$			18.0000i			1.47462i	−0.675556	+	0.737309i	$0.736096\pi$
$150$		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
$153$	25.0000	+	25.0000i	2.02113	+	2.02113i
$154$		0			0
$155$	4.00000	−	8.00000i	0.321288	−	0.642575i
$156$		0			0
$157$	−9.00000	+	9.00000i	−0.718278	+	0.718278i	−0.968252	−	0.249974i	$-0.919578\pi$
$157$	−9.00000	+	9.00000i	−0.718278	+	0.718278i	0.249974	+	0.968252i	$0.419578\pi$
$158$		0			0
$159$	−28.0000			−2.22054
$160$		0			0
$161$	8.00000			0.630488
$162$		0			0
$163$	2.00000	−	2.00000i	0.156652	−	0.156652i	−0.624429	−	0.781081i	$-0.714668\pi$
$163$	2.00000	−	2.00000i	0.156652	−	0.156652i	0.781081	+	0.624429i	$0.214668\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−2.00000	−	2.00000i	−0.154765	−	0.154765i	0.625478	−	0.780242i	$-0.284904\pi$
$167$	−2.00000	−	2.00000i	−0.154765	−	0.154765i	−0.780242	+	0.625478i	$0.784904\pi$
$168$		0			0
$169$		−	11.0000i		−	0.846154i
$170$		0			0
$171$		−	20.0000i		−	1.52944i
$172$		0			0
$173$	13.0000	+	13.0000i	0.988372	+	0.988372i	0.999933	−	0.0115615i	$-0.00368021\pi$
$173$	13.0000	+	13.0000i	0.988372	+	0.988372i	−0.0115615	+	0.999933i	$0.503680\pi$
$174$		0			0
$175$	−2.00000	+	14.0000i	−0.151186	+	1.05830i
$176$		0			0
$177$	8.00000	−	8.00000i	0.601317	−	0.601317i
$178$		0			0
$179$	12.0000			0.896922			0.448461	−	0.893802i	$-0.351972\pi$
$179$	12.0000			0.896922			0.448461	+	0.893802i	$0.351972\pi$
$180$		0			0
$181$	−10.0000			−0.743294			−0.371647	−	0.928374i	$-0.621207\pi$
$181$	−10.0000			−0.743294			−0.371647	+	0.928374i	$0.621207\pi$
$182$		0			0
$183$	−8.00000	+	8.00000i	−0.591377	+	0.591377i
$184$		0			0
$185$	−3.00000	+	1.00000i	−0.220564	+	0.0735215i
$186$		0			0
$187$		0			0
$188$		0			0
$189$		−	16.0000i		−	1.16383i
$190$		0			0
$191$			20.0000i			1.44715i	0.690246	+	0.723575i	$0.257502\pi$
$191$			20.0000i			1.44715i	−0.690246	+	0.723575i	$0.742498\pi$
$192$		0			0
$193$	−5.00000	−	5.00000i	−0.359908	−	0.359908i	0.503871	−	0.863779i	$-0.331909\pi$
$193$	−5.00000	−	5.00000i	−0.359908	−	0.359908i	−0.863779	+	0.503871i	$0.831909\pi$
$194$		0			0
$195$	8.00000	+	4.00000i	0.572892	+	0.286446i
$196$		0			0
$197$	5.00000	−	5.00000i	0.356235	−	0.356235i	−0.506188	−	0.862423i	$-0.668946\pi$
$197$	5.00000	−	5.00000i	0.356235	−	0.356235i	0.862423	+	0.506188i	$0.168946\pi$
$198$		0			0
$199$	24.0000			1.70131			0.850657	−	0.525720i	$-0.176204\pi$
$199$	24.0000			1.70131			0.850657	+	0.525720i	$0.176204\pi$
$200$		0			0
$201$	−40.0000			−2.82138
$202$		0			0
$203$	−8.00000	+	8.00000i	−0.561490	+	0.561490i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−10.0000	−	10.0000i	−0.695048	−	0.695048i
$208$		0			0
$209$		0			0
$210$		0			0
$211$		−	16.0000i		−	1.10149i	−0.834675	−	0.550743i	$-0.814345\pi$
$211$		−	16.0000i		−	1.10149i	0.834675	−	0.550743i	$-0.185655\pi$
$212$		0			0
$213$	−24.0000	−	24.0000i	−1.64445	−	1.64445i
$214$		0			0
$215$	18.0000	−	6.00000i	1.22759	−	0.409197i
$216$		0			0
$217$	−8.00000	+	8.00000i	−0.543075	+	0.543075i
$218$		0			0
$219$	−12.0000			−0.810885
$220$		0			0
$221$	10.0000			0.672673
$222$		0			0
$223$	−10.0000	+	10.0000i	−0.669650	+	0.669650i	−0.957635	−	0.287985i	$-0.907015\pi$
$223$	−10.0000	+	10.0000i	−0.669650	+	0.669650i	0.287985	+	0.957635i	$0.407015\pi$
$224$		0			0
$225$	20.0000	−	15.0000i	1.33333	−	1.00000i
$226$		0			0
$227$	10.0000	+	10.0000i	0.663723	+	0.663723i	0.956256	−	0.292532i	$-0.0944979\pi$
$227$	10.0000	+	10.0000i	0.663723	+	0.663723i	−0.292532	+	0.956256i	$0.594498\pi$
$228$		0			0
$229$			20.0000i			1.32164i	0.750546	+	0.660819i	$0.229791\pi$
$229$			20.0000i			1.32164i	−0.750546	+	0.660819i	$0.770209\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−5.00000	−	5.00000i	−0.327561	−	0.327561i	0.524097	−	0.851658i	$-0.324403\pi$
$233$	−5.00000	−	5.00000i	−0.327561	−	0.327561i	−0.851658	+	0.524097i	$0.824403\pi$
$234$		0			0
$235$	2.00000	+	6.00000i	0.130466	+	0.391397i
$236$		0			0
$237$	32.0000	−	32.0000i	2.07862	−	2.07862i
$238$		0			0
$239$	8.00000			0.517477			0.258738	−	0.965947i	$-0.416693\pi$
$239$	8.00000			0.517477			0.258738	+	0.965947i	$0.416693\pi$
$240$		0			0
$241$	−16.0000			−1.03065			−0.515325	−	0.856995i	$-0.672329\pi$
$241$	−16.0000			−1.03065			−0.515325	+	0.856995i	$0.672329\pi$
$242$		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
$247$	−4.00000	−	4.00000i	−0.254514	−	0.254514i
$248$		0			0
$249$			8.00000i			0.506979i
$250$		0			0
$251$			24.0000i			1.51487i	0.652913	+	0.757433i	$0.273547\pi$
$251$			24.0000i			1.51487i	−0.652913	+	0.757433i	$0.726453\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$	20.0000	−	40.0000i	1.25245	−	2.50490i
$256$		0			0
$257$	7.00000	−	7.00000i	0.436648	−	0.436648i	−0.454234	−	0.890882i	$-0.650087\pi$
$257$	7.00000	−	7.00000i	0.436648	−	0.436648i	0.890882	+	0.454234i	$0.150087\pi$
$258$		0			0
$259$	4.00000			0.248548
$260$		0			0
$261$	20.0000			1.23797
$262$		0			0
$263$	6.00000	−	6.00000i	0.369976	−	0.369976i	−0.497492	−	0.867468i	$-0.665746\pi$
$263$	6.00000	−	6.00000i	0.369976	−	0.369976i	0.867468	+	0.497492i	$0.165746\pi$
$264$		0			0
$265$	7.00000	+	21.0000i	0.430007	+	1.29002i
$266$		0			0
$267$		0			0
$268$		0			0
$269$			10.0000i			0.609711i	0.952399	+	0.304855i	$0.0986081\pi$
$269$			10.0000i			0.609711i	−0.952399	+	0.304855i	$0.901392\pi$
$270$		0			0
$271$		−	20.0000i		−	1.21491i	−0.794353	−	0.607457i	$-0.792190\pi$
$271$		−	20.0000i		−	1.21491i	0.794353	−	0.607457i	$-0.207810\pi$
$272$		0			0
$273$	−8.00000	−	8.00000i	−0.484182	−	0.484182i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−9.00000	+	9.00000i	−0.540758	+	0.540758i	−0.923751	−	0.382993i	$-0.874893\pi$
$277$	−9.00000	+	9.00000i	−0.540758	+	0.540758i	0.382993	+	0.923751i	$0.374893\pi$
$278$		0			0
$279$	20.0000			1.19737
$280$		0			0
$281$	8.00000			0.477240			0.238620	−	0.971113i	$-0.423305\pi$
$281$	8.00000			0.477240			0.238620	+	0.971113i	$0.423305\pi$
$282$		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
$285$	−24.0000	+	8.00000i	−1.42164	+	0.473879i
$286$		0			0
$287$		0			0
$288$		0			0
$289$		−	33.0000i		−	1.94118i
$290$		0			0
$291$			12.0000i			0.703452i
$292$		0			0
$293$	−5.00000	−	5.00000i	−0.292103	−	0.292103i	0.545807	−	0.837911i	$-0.316223\pi$
$293$	−5.00000	−	5.00000i	−0.292103	−	0.292103i	−0.837911	+	0.545807i	$0.816223\pi$
$294$		0			0
$295$	−8.00000	−	4.00000i	−0.465778	−	0.232889i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−4.00000			−0.231326
$300$		0			0
$301$	−24.0000			−1.38334
$302$		0			0
$303$	−12.0000	+	12.0000i	−0.689382	+	0.689382i
$304$		0			0
$305$	8.00000	+	4.00000i	0.458079	+	0.229039i
$306$		0			0
$307$	−10.0000	−	10.0000i	−0.570730	−	0.570730i	0.361602	−	0.932332i	$-0.382230\pi$
$307$	−10.0000	−	10.0000i	−0.570730	−	0.570730i	−0.932332	+	0.361602i	$0.882230\pi$
$308$		0			0
$309$			24.0000i			1.36531i
$310$		0			0
$311$			28.0000i			1.58773i	0.608091	+	0.793867i	$0.291935\pi$
$311$			28.0000i			1.58773i	−0.608091	+	0.793867i	$0.708065\pi$
$312$		0			0
$313$	15.0000	+	15.0000i	0.847850	+	0.847850i	0.989865	−	0.142014i	$-0.0453579\pi$
$313$	15.0000	+	15.0000i	0.847850	+	0.847850i	−0.142014	+	0.989865i	$0.545358\pi$
$314$		0			0
$315$	−30.0000	+	10.0000i	−1.69031	+	0.563436i
$316$		0			0
$317$	11.0000	−	11.0000i	0.617822	−	0.617822i	−0.327151	−	0.944972i	$-0.606088\pi$
$317$	11.0000	−	11.0000i	0.617822	−	0.617822i	0.944972	+	0.327151i	$0.106088\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$	24.0000			1.33955
$322$		0			0
$323$	−20.0000	+	20.0000i	−1.11283	+	1.11283i
$324$		0			0
$325$	1.00000	−	7.00000i	0.0554700	−	0.388290i
$326$		0			0
$327$	−20.0000	−	20.0000i	−1.10600	−	1.10600i
$328$		0			0
$329$		−	8.00000i		−	0.441054i
$330$		0			0
$331$		0			0		1.00000			$0$
$331$		0			0		−1.00000			$\pi$
$332$		0			0
$333$	−5.00000	−	5.00000i	−0.273998	−	0.273998i
$334$		0			0
$335$	10.0000	+	30.0000i	0.546358	+	1.63908i
$336$		0			0
$337$	−23.0000	+	23.0000i	−1.25289	+	1.25289i	−0.298471	+	0.954419i	$0.596477\pi$
$337$	−23.0000	+	23.0000i	−1.25289	+	1.25289i	−0.954419	+	0.298471i	$0.903523\pi$
$338$		0			0
$339$	−36.0000			−1.95525
$340$		0			0
$341$		0			0
$342$		0			0
$343$	12.0000	−	12.0000i	0.647939	−	0.647939i
$344$		0			0
$345$	−8.00000	+	16.0000i	−0.430706	+	0.861411i
$346$		0			0
$347$	18.0000	+	18.0000i	0.966291	+	0.966291i	0.999450	−	0.0331594i	$-0.0105569\pi$
$347$	18.0000	+	18.0000i	0.966291	+	0.966291i	−0.0331594	+	0.999450i	$0.510557\pi$
$348$		0			0
$349$		−	20.0000i		−	1.07058i	−0.844670	−	0.535288i	$-0.820203\pi$
$349$		−	20.0000i		−	1.07058i	0.844670	−	0.535288i	$-0.179797\pi$
$350$		0			0
$351$			8.00000i			0.427008i
$352$		0			0
$353$	9.00000	+	9.00000i	0.479022	+	0.479022i	0.904819	−	0.425797i	$-0.140006\pi$
$353$	9.00000	+	9.00000i	0.479022	+	0.479022i	−0.425797	+	0.904819i	$0.640006\pi$
$354$		0			0
$355$	−12.0000	+	24.0000i	−0.636894	+	1.27379i
$356$		0			0
$357$	−40.0000	+	40.0000i	−2.11702	+	2.11702i
$358$		0			0
$359$	−16.0000			−0.844448			−0.422224	−	0.906492i	$-0.638750\pi$
$359$	−16.0000			−0.844448			−0.422224	+	0.906492i	$0.638750\pi$
$360$		0			0
$361$	−3.00000			−0.157895
$362$		0			0
$363$	22.0000	−	22.0000i	1.15470	−	1.15470i
$364$		0			0
$365$	3.00000	+	9.00000i	0.157027	+	0.471082i
$366$		0			0
$367$	−22.0000	−	22.0000i	−1.14839	−	1.14839i	−0.986869	−	0.161521i	$-0.948360\pi$
$367$	−22.0000	−	22.0000i	−1.14839	−	1.14839i	−0.161521	−	0.986869i	$-0.551640\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		−	28.0000i		−	1.45369i
$372$		0			0
$373$	21.0000	+	21.0000i	1.08734	+	1.08734i	0.995802	+	0.0915371i	$0.0291780\pi$
$373$	21.0000	+	21.0000i	1.08734	+	1.08734i	0.0915371	+	0.995802i	$0.470822\pi$
$374$		0			0
$375$	−26.0000	−	18.0000i	−1.34263	−	0.929516i
$376$		0			0
$377$	4.00000	−	4.00000i	0.206010	−	0.206010i
$378$		0			0
$379$	28.0000			1.43826			0.719132	−	0.694874i	$-0.244540\pi$
$379$	28.0000			1.43826			0.719132	+	0.694874i	$0.244540\pi$
$380$		0			0
$381$	40.0000			2.04926
$382$		0			0
$383$	22.0000	−	22.0000i	1.12415	−	1.12415i	0.133036	−	0.991111i	$-0.457527\pi$
$383$	22.0000	−	22.0000i	1.12415	−	1.12415i	0.991111	−	0.133036i	$-0.0424727\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	30.0000	+	30.0000i	1.52499	+	1.52499i
$388$		0			0
$389$		−	18.0000i		−	0.912636i	−0.889817	−	0.456318i	$-0.849168\pi$
$389$		−	18.0000i		−	0.912636i	0.889817	−	0.456318i	$-0.150832\pi$
$390$		0			0
$391$			20.0000i			1.01144i
$392$		0			0
$393$	−16.0000	−	16.0000i	−0.807093	−	0.807093i
$394$		0			0
$395$	−32.0000	−	16.0000i	−1.61009	−	0.805047i
$396$		0			0
$397$	13.0000	−	13.0000i	0.652451	−	0.652451i	−0.301131	−	0.953583i	$-0.597364\pi$
$397$	13.0000	−	13.0000i	0.652451	−	0.652451i	0.953583	+	0.301131i	$0.0973643\pi$
$398$		0			0
$399$	32.0000			1.60200
$400$		0			0
$401$	34.0000			1.69788			0.848939	−	0.528490i	$-0.177242\pi$
$401$	34.0000			1.69788			0.848939	+	0.528490i	$0.177242\pi$
$402$		0			0
$403$	4.00000	−	4.00000i	0.199254	−	0.199254i
$404$		0			0
$405$	2.00000	+	1.00000i	0.0993808	+	0.0496904i
$406$		0			0
$407$		0			0
$408$		0			0
$409$		−	2.00000i		−	0.0988936i	−0.998777	−	0.0494468i	$-0.984254\pi$
$409$		−	2.00000i		−	0.0988936i	0.998777	−	0.0494468i	$-0.0157458\pi$
$410$		0			0
$411$		−	4.00000i		−	0.197305i
$412$		0			0
$413$	8.00000	+	8.00000i	0.393654	+	0.393654i
$414$		0			0
$415$	6.00000	−	2.00000i	0.294528	−	0.0981761i
$416$		0			0
$417$	−24.0000	+	24.0000i	−1.17529	+	1.17529i
$418$		0			0
$419$	−12.0000			−0.586238			−0.293119	−	0.956076i	$-0.594693\pi$
$419$	−12.0000			−0.586238			−0.293119	+	0.956076i	$0.594693\pi$
$420$		0			0
$421$	20.0000			0.974740			0.487370	−	0.873195i	$-0.337956\pi$
$421$	20.0000			0.974740			0.487370	+	0.873195i	$0.337956\pi$
$422$		0			0
$423$	−10.0000	+	10.0000i	−0.486217	+	0.486217i
$424$		0			0
$425$	−35.0000	−	5.00000i	−1.69775	−	0.242536i
$426$		0			0
$427$	−8.00000	−	8.00000i	−0.387147	−	0.387147i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	4.00000i		−	0.192673i	−0.995349	−	0.0963366i	$-0.969287\pi$
$431$		−	4.00000i		−	0.192673i	0.995349	−	0.0963366i	$-0.0307125\pi$
$432$		0			0
$433$	−19.0000	−	19.0000i	−0.913082	−	0.913082i	0.0834318	−	0.996513i	$-0.473412\pi$
$433$	−19.0000	−	19.0000i	−0.913082	−	0.913082i	−0.996513	+	0.0834318i	$0.973412\pi$
$434$		0			0
$435$	−8.00000	−	24.0000i	−0.383571	−	1.15071i
$436$		0			0
$437$	8.00000	−	8.00000i	0.382692	−	0.382692i
$438$		0			0
$439$	−16.0000			−0.763638			−0.381819	−	0.924237i	$-0.624702\pi$
$439$	−16.0000			−0.763638			−0.381819	+	0.924237i	$0.624702\pi$
$440$		0			0
$441$	5.00000			0.238095
$442$		0			0
$443$	22.0000	−	22.0000i	1.04525	−	1.04525i	0.0463251	−	0.998926i	$-0.485249\pi$
$443$	22.0000	−	22.0000i	1.04525	−	1.04525i	0.998926	−	0.0463251i	$-0.0147510\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	36.0000	+	36.0000i	1.70274	+	1.70274i
$448$		0			0
$449$			26.0000i			1.22702i	0.789689	+	0.613508i	$0.210242\pi$
$449$			26.0000i			1.22702i	−0.789689	+	0.613508i	$0.789758\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	−24.0000	−	24.0000i	−1.12762	−	1.12762i
$454$		0			0
$455$	−4.00000	+	8.00000i	−0.187523	+	0.375046i
$456$		0			0
$457$	15.0000	−	15.0000i	0.701670	−	0.701670i	−0.263099	−	0.964769i	$-0.584744\pi$
$457$	15.0000	−	15.0000i	0.701670	−	0.701670i	0.964769	+	0.263099i	$0.0847444\pi$
$458$		0			0
$459$	40.0000			1.86704
$460$		0			0
$461$	−14.0000			−0.652045			−0.326023	−	0.945362i	$-0.605709\pi$
$461$	−14.0000			−0.652045			−0.326023	+	0.945362i	$0.605709\pi$
$462$		0			0
$463$	−22.0000	+	22.0000i	−1.02243	+	1.02243i	−0.0226840	+	0.999743i	$0.507221\pi$
$463$	−22.0000	+	22.0000i	−1.02243	+	1.02243i	−0.999743	+	0.0226840i	$0.992779\pi$
$464$		0			0
$465$	−8.00000	−	24.0000i	−0.370991	−	1.11297i
$466$		0			0
$467$	2.00000	+	2.00000i	0.0925490	+	0.0925490i	0.751865	−	0.659317i	$-0.229154\pi$
$467$	2.00000	+	2.00000i	0.0925490	+	0.0925490i	−0.659317	+	0.751865i	$0.729154\pi$
$468$		0			0
$469$		−	40.0000i		−	1.84703i
$470$		0			0
$471$			36.0000i			1.65879i
$472$		0			0
$473$		0			0
$474$		0			0
$475$	12.0000	+	16.0000i	0.550598	+	0.734130i
$476$		0			0
$477$	−35.0000	+	35.0000i	−1.60254	+	1.60254i
$478$		0			0
$479$	−24.0000			−1.09659			−0.548294	−	0.836286i	$-0.684723\pi$
$479$	−24.0000			−1.09659			−0.548294	+	0.836286i	$0.684723\pi$
$480$		0			0
$481$	−2.00000			−0.0911922
$482$		0			0
$483$	16.0000	−	16.0000i	0.728025	−	0.728025i
$484$		0			0
$485$	9.00000	−	3.00000i	0.408669	−	0.136223i
$486$		0			0
$487$	−6.00000	−	6.00000i	−0.271886	−	0.271886i	0.557973	−	0.829859i	$-0.311579\pi$
$487$	−6.00000	−	6.00000i	−0.271886	−	0.271886i	−0.829859	+	0.557973i	$0.811579\pi$
$488$		0			0
$489$		−	8.00000i		−	0.361773i
$490$		0			0
$491$			16.0000i			0.722070i	0.932552	+	0.361035i	$0.117576\pi$
$491$			16.0000i			0.722070i	−0.932552	+	0.361035i	$0.882424\pi$
$492$		0			0
$493$	−20.0000	−	20.0000i	−0.900755	−	0.900755i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	24.0000	−	24.0000i	1.07655	−	1.07655i
$498$		0			0
$499$	4.00000			0.179065			0.0895323	−	0.995984i	$-0.471463\pi$
$499$	4.00000			0.179065			0.0895323	+	0.995984i	$0.471463\pi$
$500$		0			0
$501$	−8.00000			−0.357414
$502$		0			0
$503$	10.0000	−	10.0000i	0.445878	−	0.445878i	−0.448104	−	0.893982i	$-0.647900\pi$
$503$	10.0000	−	10.0000i	0.445878	−	0.445878i	0.893982	+	0.448104i	$0.147900\pi$
$504$		0			0
$505$	12.0000	+	6.00000i	0.533993	+	0.266996i
$506$		0			0
$507$	−22.0000	−	22.0000i	−0.977054	−	0.977054i
$508$		0			0
$509$		−	36.0000i		−	1.59567i	−0.602875	−	0.797836i	$-0.705978\pi$
$509$		−	36.0000i		−	1.59567i	0.602875	−	0.797836i	$-0.294022\pi$
$510$		0			0
$511$		−	12.0000i		−	0.530849i
$512$		0			0
$513$	−16.0000	−	16.0000i	−0.706417	−	0.706417i
$514$		0			0
$515$	18.0000	−	6.00000i	0.793175	−	0.264392i
$516$		0			0
$517$		0			0
$518$		0			0
$519$	52.0000			2.28255
$520$		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
$523$	14.0000	−	14.0000i	0.612177	−	0.612177i	−0.331336	−	0.943513i	$-0.607499\pi$
$523$	14.0000	−	14.0000i	0.612177	−	0.612177i	0.943513	+	0.331336i	$0.107499\pi$
$524$		0			0
$525$	24.0000	+	32.0000i	1.04745	+	1.39659i
$526$		0			0
$527$	−20.0000	−	20.0000i	−0.871214	−	0.871214i
$528$		0			0
$529$			15.0000i			0.652174i
$530$		0			0
$531$		−	20.0000i		−	0.867926i
$532$		0			0
$533$		0			0
$534$		0			0
$535$	−6.00000	−	18.0000i	−0.259403	−	0.778208i
$536$		0			0
$537$	24.0000	−	24.0000i	1.03568	−	1.03568i
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−30.0000			−1.28980			−0.644900	−	0.764267i	$-0.723101\pi$
$541$	−30.0000			−1.28980			−0.644900	+	0.764267i	$0.723101\pi$
$542$		0			0
$543$	−20.0000	+	20.0000i	−0.858282	+	0.858282i
$544$		0			0
$545$	−10.0000	+	20.0000i	−0.428353	+	0.856706i
$546$		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
$549$			20.0000i			0.853579i
$550$		0			0
$551$			16.0000i			0.681623i
$552$		0			0
$553$	32.0000	+	32.0000i	1.36078	+	1.36078i
$554$		0			0
$555$	−4.00000	+	8.00000i	−0.169791	+	0.339581i
$556$		0			0
$557$	−15.0000	+	15.0000i	−0.635570	+	0.635570i	−0.949460	−	0.313889i	$-0.898368\pi$
$557$	−15.0000	+	15.0000i	−0.635570	+	0.635570i	0.313889	+	0.949460i	$0.398368\pi$
$558$		0			0
$559$	12.0000			0.507546
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−6.00000	+	6.00000i	−0.252870	+	0.252870i	−0.822146	−	0.569276i	$-0.807223\pi$
$563$	−6.00000	+	6.00000i	−0.252870	+	0.252870i	0.569276	+	0.822146i	$0.307223\pi$
$564$		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
$569$			2.00000i			0.0838444i	0.999121	+	0.0419222i	$0.0133482\pi$
$569$			2.00000i			0.0838444i	−0.999121	+	0.0419222i	$0.986652\pi$
$570$		0			0
$571$			16.0000i			0.669579i	0.942293	+	0.334790i	$0.108665\pi$
$571$			16.0000i			0.669579i	−0.942293	+	0.334790i	$0.891335\pi$
$572$		0			0
$573$	40.0000	+	40.0000i	1.67102	+	1.67102i
$574$		0			0
$575$	14.0000	+	2.00000i	0.583840	+	0.0834058i
$576$		0			0
$577$	15.0000	−	15.0000i	0.624458	−	0.624458i	−0.322210	−	0.946668i	$-0.604426\pi$
$577$	15.0000	−	15.0000i	0.624458	−	0.624458i	0.946668	+	0.322210i	$0.104426\pi$
$578$		0			0
$579$	−20.0000			−0.831172
$580$		0			0
$581$	−8.00000			−0.331896
$582$		0			0
$583$		0			0
$584$		0			0
$585$	15.0000	−	5.00000i	0.620174	−	0.206725i
$586$		0			0
$587$	−14.0000	−	14.0000i	−0.577842	−	0.577842i	0.356466	−	0.934308i	$-0.383981\pi$
$587$	−14.0000	−	14.0000i	−0.577842	−	0.577842i	−0.934308	+	0.356466i	$0.883981\pi$
$588$		0			0
$589$			16.0000i			0.659269i
$590$		0			0
$591$		−	20.0000i		−	0.822690i
$592$		0			0
$593$	1.00000	+	1.00000i	0.0410651	+	0.0410651i	0.727341	−	0.686276i	$-0.240756\pi$
$593$	1.00000	+	1.00000i	0.0410651	+	0.0410651i	−0.686276	+	0.727341i	$0.740756\pi$
$594$		0			0
$595$	40.0000	+	20.0000i	1.63984	+	0.819920i
$596$		0			0
$597$	48.0000	−	48.0000i	1.96451	−	1.96451i
$598$		0			0
$599$	−8.00000			−0.326871			−0.163436	−	0.986554i	$-0.552258\pi$
$599$	−8.00000			−0.326871			−0.163436	+	0.986554i	$0.552258\pi$
$600$		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
$603$	−50.0000	+	50.0000i	−2.03616	+	2.03616i
$604$		0			0
$605$	−22.0000	−	11.0000i	−0.894427	−	0.447214i
$606$		0			0
$607$	−18.0000	−	18.0000i	−0.730597	−	0.730597i	0.240141	−	0.970738i	$-0.422806\pi$
$607$	−18.0000	−	18.0000i	−0.730597	−	0.730597i	−0.970738	+	0.240141i	$0.922806\pi$
$608$		0			0
$609$			32.0000i			1.29671i
$610$		0			0
$611$			4.00000i			0.161823i
$612$		0			0
$613$	−9.00000	−	9.00000i	−0.363507	−	0.363507i	0.501596	−	0.865102i	$-0.332747\pi$
$613$	−9.00000	−	9.00000i	−0.363507	−	0.363507i	−0.865102	+	0.501596i	$0.832747\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−29.0000	+	29.0000i	−1.16750	+	1.16750i	−0.184701	+	0.982795i	$0.559132\pi$
$617$	−29.0000	+	29.0000i	−1.16750	+	1.16750i	−0.982795	+	0.184701i	$0.940868\pi$
$618$		0			0
$619$	−28.0000			−1.12542			−0.562708	−	0.826656i	$-0.690240\pi$
$619$	−28.0000			−1.12542			−0.562708	+	0.826656i	$0.690240\pi$
$620$		0			0
$621$	−16.0000			−0.642058
$622$		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
$629$			10.0000i			0.398726i
$630$		0			0
$631$		−	4.00000i		−	0.159237i	−0.996825	−	0.0796187i	$-0.974630\pi$
$631$		−	4.00000i		−	0.159237i	0.996825	−	0.0796187i	$-0.0253703\pi$
$632$		0			0
$633$	−32.0000	−	32.0000i	−1.27189	−	1.27189i
$634$		0			0
$635$	−10.0000	−	30.0000i	−0.396838	−	1.19051i
$636$		0			0
$637$	1.00000	−	1.00000i	0.0396214	−	0.0396214i
$638$		0			0
$639$	−60.0000			−2.37356
$640$		0			0
$641$	−48.0000			−1.89589			−0.947943	−	0.318440i	$-0.896841\pi$
$641$	−48.0000			−1.89589			−0.947943	+	0.318440i	$0.896841\pi$
$642$		0			0
$643$	−10.0000	+	10.0000i	−0.394362	+	0.394362i	−0.876239	−	0.481877i	$-0.839955\pi$
$643$	−10.0000	+	10.0000i	−0.394362	+	0.394362i	0.481877	+	0.876239i	$0.339955\pi$
$644$		0			0
$645$	24.0000	−	48.0000i	0.944999	−	1.89000i
$646$		0			0
$647$	10.0000	+	10.0000i	0.393141	+	0.393141i	0.875805	−	0.482665i	$-0.160331\pi$
$647$	10.0000	+	10.0000i	0.393141	+	0.393141i	−0.482665	+	0.875805i	$0.660331\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$			32.0000i			1.25418i
$652$		0			0
$653$	1.00000	+	1.00000i	0.0391330	+	0.0391330i	0.726403	−	0.687270i	$-0.241191\pi$
$653$	1.00000	+	1.00000i	0.0391330	+	0.0391330i	−0.687270	+	0.726403i	$0.741191\pi$
$654$		0			0
$655$	−8.00000	+	16.0000i	−0.312586	+	0.625172i
$656$		0			0
$657$	−15.0000	+	15.0000i	−0.585206	+	0.585206i
$658$		0			0
$659$	20.0000			0.779089			0.389545	−	0.921008i	$-0.372632\pi$
$659$	20.0000			0.779089			0.389545	+	0.921008i	$0.372632\pi$
$660$		0			0
$661$	12.0000			0.466746			0.233373	−	0.972387i	$-0.425024\pi$
$661$	12.0000			0.466746			0.233373	+	0.972387i	$0.425024\pi$
$662$		0			0
$663$	20.0000	−	20.0000i	0.776736	−	0.776736i
$664$		0			0
$665$	−8.00000	−	24.0000i	−0.310227	−	0.930680i
$666$		0			0
$667$	8.00000	+	8.00000i	0.309761	+	0.309761i
$668$		0			0
$669$			40.0000i			1.54649i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	5.00000	+	5.00000i	0.192736	+	0.192736i	0.796877	−	0.604141i	$-0.206484\pi$
$673$	5.00000	+	5.00000i	0.192736	+	0.192736i	−0.604141	+	0.796877i	$0.706484\pi$
$674$		0			0
$675$	4.00000	−	28.0000i	0.153960	−	1.07772i
$676$		0			0
$677$	−3.00000	+	3.00000i	−0.115299	+	0.115299i	−0.762402	−	0.647103i	$-0.775980\pi$
$677$	−3.00000	+	3.00000i	−0.115299	+	0.115299i	0.647103	+	0.762402i	$0.275980\pi$
$678$		0			0
$679$	−12.0000			−0.460518
$680$		0			0
$681$	40.0000			1.53280
$682$		0			0
$683$	−22.0000	+	22.0000i	−0.841807	+	0.841807i	−0.989094	−	0.147287i	$-0.952946\pi$
$683$	−22.0000	+	22.0000i	−0.841807	+	0.841807i	0.147287	+	0.989094i	$0.452946\pi$
$684$		0			0
$685$	−3.00000	+	1.00000i	−0.114624	+	0.0382080i
$686$		0			0
$687$	40.0000	+	40.0000i	1.52610	+	1.52610i
$688$		0			0
$689$			14.0000i			0.533358i
$690$		0			0
$691$			32.0000i			1.21734i	0.793424	+	0.608669i	$0.208296\pi$
$691$			32.0000i			1.21734i	−0.793424	+	0.608669i	$0.791704\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	24.0000	+	12.0000i	0.910372	+	0.455186i
$696$		0			0
$697$		0			0
$698$		0			0
$699$	−20.0000			−0.756469
$700$		0			0
$701$	20.0000			0.755390			0.377695	−	0.925930i	$-0.376717\pi$
$701$	20.0000			0.755390			0.377695	+	0.925930i	$0.376717\pi$
$702$		0			0
$703$	4.00000	−	4.00000i	0.150863	−	0.150863i
$704$		0			0
$705$	16.0000	+	8.00000i	0.602595	+	0.301297i
$706$		0			0
$707$	−12.0000	−	12.0000i	−0.451306	−	0.451306i
$708$		0			0
$709$			12.0000i			0.450669i	0.974281	+	0.225335i	$0.0723476\pi$
$709$			12.0000i			0.450669i	−0.974281	+	0.225335i	$0.927652\pi$
$710$		0			0
$711$		−	80.0000i		−	3.00023i
$712$		0			0
$713$	8.00000	+	8.00000i	0.299602	+	0.299602i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	16.0000	−	16.0000i	0.597531	−	0.597531i
$718$		0			0
$719$	8.00000			0.298350			0.149175	−	0.988811i	$-0.452338\pi$
$719$	8.00000			0.298350			0.149175	+	0.988811i	$0.452338\pi$
$720$		0			0
$721$	−24.0000			−0.893807
$722$		0			0
$723$	−32.0000	+	32.0000i	−1.19009	+	1.19009i
$724$		0			0
$725$	−16.0000	+	12.0000i	−0.594225	+	0.445669i
$726$		0			0
$727$	18.0000	+	18.0000i	0.667583	+	0.667583i	0.957156	−	0.289573i	$-0.0935133\pi$
$727$	18.0000	+	18.0000i	0.667583	+	0.667583i	−0.289573	+	0.957156i	$0.593513\pi$
$728$		0			0
$729$		−	43.0000i		−	1.59259i
$730$		0			0
$731$		−	60.0000i		−	2.21918i
$732$		0			0
$733$	21.0000	+	21.0000i	0.775653	+	0.775653i	0.979088	−	0.203436i	$-0.0652108\pi$
$733$	21.0000	+	21.0000i	0.775653	+	0.775653i	−0.203436	+	0.979088i	$0.565211\pi$
$734$		0			0
$735$	−2.00000	−	6.00000i	−0.0737711	−	0.221313i
$736$		0			0
$737$		0			0
$738$		0			0
$739$	44.0000			1.61857			0.809283	−	0.587419i	$-0.199856\pi$
$739$	44.0000			1.61857			0.809283	+	0.587419i	$0.199856\pi$
$740$		0			0
$741$	−16.0000			−0.587775
$742$		0			0
$743$	30.0000	−	30.0000i	1.10059	−	1.10059i	0.106254	−	0.994339i	$-0.466114\pi$
$743$	30.0000	−	30.0000i	1.10059	−	1.10059i	0.994339	−	0.106254i	$-0.0338857\pi$
$744$		0			0
$745$	18.0000	−	36.0000i	0.659469	−	1.31894i
$746$		0			0
$747$	10.0000	+	10.0000i	0.365881	+	0.365881i
$748$		0			0
$749$			24.0000i			0.876941i
$750$		0			0
$751$		−	44.0000i		−	1.60558i	−0.596260	−	0.802791i	$-0.703347\pi$
$751$		−	44.0000i		−	1.60558i	0.596260	−	0.802791i	$-0.296653\pi$
$752$		0			0
$753$	48.0000	+	48.0000i	1.74922	+	1.74922i
$754$		0			0
$755$	−12.0000	+	24.0000i	−0.436725	+	0.873449i
$756$		0			0
$757$	−1.00000	+	1.00000i	−0.0363456	+	0.0363456i	−0.725046	−	0.688700i	$-0.758182\pi$
$757$	−1.00000	+	1.00000i	−0.0363456	+	0.0363456i	0.688700	+	0.725046i	$0.258182\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−22.0000			−0.797499			−0.398750	−	0.917060i	$-0.630556\pi$
$761$	−22.0000			−0.797499			−0.398750	+	0.917060i	$0.630556\pi$
$762$		0			0
$763$	20.0000	−	20.0000i	0.724049	−	0.724049i
$764$		0			0
$765$	−25.0000	−	75.0000i	−0.903877	−	2.71163i
$766$		0			0
$767$	−4.00000	−	4.00000i	−0.144432	−	0.144432i
$768$		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
$771$		−	28.0000i		−	1.00840i
$772$		0			0
$773$	−1.00000	−	1.00000i	−0.0359675	−	0.0359675i	0.688894	−	0.724862i	$-0.258096\pi$
$773$	−1.00000	−	1.00000i	−0.0359675	−	0.0359675i	−0.724862	+	0.688894i	$0.758096\pi$
$774$		0			0
$775$	−16.0000	+	12.0000i	−0.574737	+	0.431053i
$776$		0			0
$777$	8.00000	−	8.00000i	0.286998	−	0.286998i
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$	16.0000	−	16.0000i	0.571793	−	0.571793i
$784$		0			0
$785$	27.0000	−	9.00000i	0.963671	−	0.321224i
$786$		0			0
$787$	30.0000	+	30.0000i	1.06938	+	1.06938i	0.997406	+	0.0719783i	$0.0229312\pi$
$787$	30.0000	+	30.0000i	1.06938	+	1.06938i	0.0719783	+	0.997406i	$0.477069\pi$
$788$		0			0
$789$		−	24.0000i		−	0.854423i
$790$		0			0
$791$		−	36.0000i		−	1.28001i
$792$		0			0
$793$	4.00000	+	4.00000i	0.142044	+	0.142044i

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:	\( N \)	\(=\)	\( 160 = 2^{5} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	160.n (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(2.00000 - 2.00000i) q^{3} +(-2.00000 - 1.00000i) q^{5} +(2.00000 + 2.00000i) q^{7} -5.00000i q^{9} +(-1.00000 - 1.00000i) q^{13} +(-6.00000 + 2.00000i) q^{15} +(-5.00000 + 5.00000i) q^{17} +4.00000 q^{19} +8.00000 q^{21} +(2.00000 - 2.00000i) q^{23} +(3.00000 + 4.00000i) q^{25} +(-4.00000 - 4.00000i) q^{27} +4.00000i q^{29} +4.00000i q^{31} +(-2.00000 - 6.00000i) q^{35} +(1.00000 - 1.00000i) q^{37} -4.00000 q^{39} +(-6.00000 + 6.00000i) q^{43} +(-5.00000 + 10.0000i) q^{45} +(-2.00000 - 2.00000i) q^{47} +1.00000i q^{49} +20.0000i q^{51} +(-7.00000 - 7.00000i) q^{53} +(8.00000 - 8.00000i) q^{57} +4.00000 q^{59} -4.00000 q^{61} +(10.0000 - 10.0000i) q^{63} +(1.00000 + 3.00000i) q^{65} +(-10.0000 - 10.0000i) q^{67} -8.00000i q^{69} -12.0000i q^{71} +(-3.00000 - 3.00000i) q^{73} +(14.0000 + 2.00000i) q^{75} +16.0000 q^{79} -1.00000 q^{81} +(-2.00000 + 2.00000i) q^{83} +(15.0000 - 5.00000i) q^{85} +(8.00000 + 8.00000i) q^{87} -4.00000i q^{91} +(8.00000 + 8.00000i) q^{93} +(-8.00000 - 4.00000i) q^{95} +(-3.00000 + 3.00000i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{3} - 4 q^{5} + 4 q^{7} - 2 q^{13} - 12 q^{15} - 10 q^{17} + 8 q^{19} + 16 q^{21} + 4 q^{23} + 6 q^{25} - 8 q^{27} - 4 q^{35} + 2 q^{37} - 8 q^{39} - 12 q^{43} - 10 q^{45} - 4 q^{47} - 14 q^{53}+ \cdots - 6 q^{97}+O(q^{100}) \) 2 * q + 4 * q^3 - 4 * q^5 + 4 * q^7 - 2 * q^13 - 12 * q^15 - 10 * q^17 + 8 * q^19 + 16 * q^21 + 4 * q^23 + 6 * q^25 - 8 * q^27 - 4 * q^35 + 2 * q^37 - 8 * q^39 - 12 * q^43 - 10 * q^45 - 4 * q^47 - 14 * q^53 + 16 * q^57 + 8 * q^59 - 8 * q^61 + 20 * q^63 + 2 * q^65 - 20 * q^67 - 6 * q^73 + 28 * q^75 + 32 * q^79 - 2 * q^81 - 4 * q^83 + 30 * q^85 + 16 * q^87 + 16 * q^93 - 16 * q^95 - 6 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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