LMFDB - Embedded newform 1600.2.n.j.1343.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1600,2,Mod(1343,1600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1600, 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("1600.1343");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1600 = 2^{6} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1600.n (of order $4$, degree $2$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$12.7760643234$
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:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 160)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			1343.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	1600.1343
Dual form			1600.2.n.j.1407.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.00000 - 1.00000i) q^{3} +(-1.00000 - 1.00000i) q^{7} +1.00000i q^{9} +O(q^{10})$	$q+(1.00000 - 1.00000i) q^{3} +(-1.00000 - 1.00000i) q^{7} +1.00000i q^{9} +6.00000i q^{11} +(-1.00000 - 1.00000i) q^{13} +(-1.00000 + 1.00000i) q^{17} +4.00000 q^{19} -2.00000 q^{21} +(5.00000 - 5.00000i) q^{23} +(4.00000 + 4.00000i) q^{27} +8.00000i q^{29} +2.00000i q^{31} +(6.00000 + 6.00000i) q^{33} +(-5.00000 + 5.00000i) q^{37} -2.00000 q^{39} +6.00000 q^{41} +(-3.00000 + 3.00000i) q^{43} +(7.00000 + 7.00000i) q^{47} -5.00000i q^{49} +2.00000i q^{51} +(-1.00000 - 1.00000i) q^{53} +(4.00000 - 4.00000i) q^{57} +4.00000 q^{59} -2.00000 q^{61} +(1.00000 - 1.00000i) q^{63} +(7.00000 + 7.00000i) q^{67} -10.0000i q^{69} -6.00000i q^{71} +(-9.00000 - 9.00000i) q^{73} +(6.00000 - 6.00000i) q^{77} +8.00000 q^{79} +5.00000 q^{81} +(5.00000 - 5.00000i) q^{83} +(8.00000 + 8.00000i) q^{87} +2.00000i q^{91} +(2.00000 + 2.00000i) q^{93} +(3.00000 - 3.00000i) q^{97} -6.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} - 2 q^{7}+O(q^{10}) $ 2 * q + 2 * q^3 - 2 * q^7	$ 2 q + 2 q^{3} - 2 q^{7} - 2 q^{13} - 2 q^{17} + 8 q^{19} - 4 q^{21} + 10 q^{23} + 8 q^{27} + 12 q^{33} - 10 q^{37} - 4 q^{39} + 12 q^{41} - 6 q^{43} + 14 q^{47} - 2 q^{53} + 8 q^{57} + 8 q^{59} - 4 q^{61} + 2 q^{63} + 14 q^{67} - 18 q^{73} + 12 q^{77} + 16 q^{79} + 10 q^{81} + 10 q^{83} + 16 q^{87} + 4 q^{93} + 6 q^{97} - 12 q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 - 2 * q^7 - 2 * q^13 - 2 * q^17 + 8 * q^19 - 4 * q^21 + 10 * q^23 + 8 * q^27 + 12 * q^33 - 10 * q^37 - 4 * q^39 + 12 * q^41 - 6 * q^43 + 14 * q^47 - 2 * q^53 + 8 * q^57 + 8 * q^59 - 4 * q^61 + 2 * q^63 + 14 * q^67 - 18 * q^73 + 12 * q^77 + 16 * q^79 + 10 * q^81 + 10 * q^83 + 16 * q^87 + 4 * q^93 + 6 * q^97 - 12 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1600\mathbb{Z}\right)^\times$.

$n$	$577$	$901$	$1151$
$\chi(n)$	$e\left(\frac{3}{4}\right)$	$1$	$-1$

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
$3$	1.00000	−	1.00000i	0.577350	−	0.577350i	−0.356822	−	0.934172i	$-0.616140\pi$
$3$	1.00000	−	1.00000i	0.577350	−	0.577350i	0.934172	+	0.356822i	$0.116140\pi$
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	−1.00000	−	1.00000i	−0.377964	−	0.377964i	0.492403	−	0.870367i	$-0.336119\pi$
$7$	−1.00000	−	1.00000i	−0.377964	−	0.377964i	−0.870367	+	0.492403i	$0.836119\pi$
$8$		0			0
$9$			1.00000i			0.333333i
$10$		0			0
$11$			6.00000i			1.80907i	0.426401	+	0.904534i	$0.359781\pi$
$11$			6.00000i			1.80907i	−0.426401	+	0.904534i	$0.640219\pi$
$12$		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
$15$		0			0
$16$		0			0
$17$	−1.00000	+	1.00000i	−0.242536	+	0.242536i	−0.817898	−	0.575363i	$-0.804861\pi$
$17$	−1.00000	+	1.00000i	−0.242536	+	0.242536i	0.575363	+	0.817898i	$0.304861\pi$
$18$		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
$21$	−2.00000			−0.436436
$22$		0			0
$23$	5.00000	−	5.00000i	1.04257	−	1.04257i	0.0435195	−	0.999053i	$-0.486143\pi$
$23$	5.00000	−	5.00000i	1.04257	−	1.04257i	0.999053	−	0.0435195i	$-0.0138571\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	4.00000	+	4.00000i	0.769800	+	0.769800i
$28$		0			0
$29$			8.00000i			1.48556i	0.669534	+	0.742781i	$0.266494\pi$
$29$			8.00000i			1.48556i	−0.669534	+	0.742781i	$0.733506\pi$
$30$		0			0
$31$			2.00000i			0.359211i	0.983739	+	0.179605i	$0.0574821\pi$
$31$			2.00000i			0.359211i	−0.983739	+	0.179605i	$0.942518\pi$
$32$		0			0
$33$	6.00000	+	6.00000i	1.04447	+	1.04447i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−5.00000	+	5.00000i	−0.821995	+	0.821995i	−0.986394	−	0.164399i	$-0.947432\pi$
$37$	−5.00000	+	5.00000i	−0.821995	+	0.821995i	0.164399	+	0.986394i	$0.447432\pi$
$38$		0			0
$39$	−2.00000			−0.320256
$40$		0			0
$41$	6.00000			0.937043			0.468521	−	0.883452i	$-0.344787\pi$
$41$	6.00000			0.937043			0.468521	+	0.883452i	$0.344787\pi$
$42$		0			0
$43$	−3.00000	+	3.00000i	−0.457496	+	0.457496i	−0.897833	−	0.440337i	$-0.854859\pi$
$43$	−3.00000	+	3.00000i	−0.457496	+	0.457496i	0.440337	+	0.897833i	$0.354859\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	7.00000	+	7.00000i	1.02105	+	1.02105i	0.999774	+	0.0212814i	$0.00677460\pi$
$47$	7.00000	+	7.00000i	1.02105	+	1.02105i	0.0212814	+	0.999774i	$0.493225\pi$
$48$		0			0
$49$		−	5.00000i		−	0.714286i
$50$		0			0
$51$			2.00000i			0.280056i
$52$		0			0
$53$	−1.00000	−	1.00000i	−0.137361	−	0.137361i	0.635083	−	0.772444i	$-0.280966\pi$
$53$	−1.00000	−	1.00000i	−0.137361	−	0.137361i	−0.772444	+	0.635083i	$0.780966\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	4.00000	−	4.00000i	0.529813	−	0.529813i
$58$		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
$61$	−2.00000			−0.256074			−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000			−0.256074			−0.128037	+	0.991769i	$0.540868\pi$
$62$		0			0
$63$	1.00000	−	1.00000i	0.125988	−	0.125988i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	7.00000	+	7.00000i	0.855186	+	0.855186i	0.990766	−	0.135580i	$-0.0432899\pi$
$67$	7.00000	+	7.00000i	0.855186	+	0.855186i	−0.135580	+	0.990766i	$0.543290\pi$
$68$		0			0
$69$		−	10.0000i		−	1.20386i
$70$		0			0
$71$		−	6.00000i		−	0.712069i	−0.934473	−	0.356034i	$-0.884129\pi$
$71$		−	6.00000i		−	0.712069i	0.934473	−	0.356034i	$-0.115871\pi$
$72$		0			0
$73$	−9.00000	−	9.00000i	−1.05337	−	1.05337i	−0.998493	−	0.0548772i	$-0.982523\pi$
$73$	−9.00000	−	9.00000i	−1.05337	−	1.05337i	−0.0548772	−	0.998493i	$-0.517477\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	6.00000	−	6.00000i	0.683763	−	0.683763i
$78$		0			0
$79$	8.00000			0.900070			0.450035	−	0.893011i	$-0.351411\pi$
$79$	8.00000			0.900070			0.450035	+	0.893011i	$0.351411\pi$
$80$		0			0
$81$	5.00000			0.555556
$82$		0			0
$83$	5.00000	−	5.00000i	0.548821	−	0.548821i	−0.377279	−	0.926100i	$-0.623140\pi$
$83$	5.00000	−	5.00000i	0.548821	−	0.548821i	0.926100	+	0.377279i	$0.123140\pi$
$84$		0			0
$85$		0			0
$86$		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
$91$			2.00000i			0.209657i
$92$		0			0
$93$	2.00000	+	2.00000i	0.207390	+	0.207390i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	3.00000	−	3.00000i	0.304604	−	0.304604i	−0.538208	−	0.842812i	$-0.680899\pi$
$97$	3.00000	−	3.00000i	0.304604	−	0.304604i	0.842812	+	0.538208i	$0.180899\pi$
$98$		0			0
$99$	−6.00000			−0.603023
$100$		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
$103$	−3.00000	+	3.00000i	−0.295599	+	0.295599i	−0.839287	−	0.543688i	$-0.817027\pi$
$103$	−3.00000	+	3.00000i	−0.295599	+	0.295599i	0.543688	+	0.839287i	$0.317027\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	3.00000	+	3.00000i	0.290021	+	0.290021i	0.837088	−	0.547068i	$-0.184256\pi$
$107$	3.00000	+	3.00000i	0.290021	+	0.290021i	−0.547068	+	0.837088i	$0.684256\pi$
$108$		0			0
$109$			4.00000i			0.383131i	0.981480	+	0.191565i	$0.0613564\pi$
$109$			4.00000i			0.383131i	−0.981480	+	0.191565i	$0.938644\pi$
$110$		0			0
$111$			10.0000i			0.949158i
$112$		0			0
$113$	3.00000	+	3.00000i	0.282216	+	0.282216i	0.833992	−	0.551776i	$-0.186050\pi$
$113$	3.00000	+	3.00000i	0.282216	+	0.282216i	−0.551776	+	0.833992i	$0.686050\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	1.00000	−	1.00000i	0.0924500	−	0.0924500i
$118$		0			0
$119$	2.00000			0.183340
$120$		0			0
$121$	−25.0000			−2.27273
$122$		0			0
$123$	6.00000	−	6.00000i	0.541002	−	0.541002i
$124$		0			0
$125$		0			0
$126$		0			0
$127$	−5.00000	−	5.00000i	−0.443678	−	0.443678i	0.449568	−	0.893246i	$-0.351578\pi$
$127$	−5.00000	−	5.00000i	−0.443678	−	0.443678i	−0.893246	+	0.449568i	$0.851578\pi$
$128$		0			0
$129$			6.00000i			0.528271i
$130$		0			0
$131$		−	2.00000i		−	0.174741i	−0.996176	−	0.0873704i	$-0.972154\pi$
$131$		−	2.00000i		−	0.174741i	0.996176	−	0.0873704i	$-0.0278464\pi$
$132$		0			0
$133$	−4.00000	−	4.00000i	−0.346844	−	0.346844i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−13.0000	+	13.0000i	−1.11066	+	1.11066i	−0.117604	+	0.993061i	$0.537521\pi$
$137$	−13.0000	+	13.0000i	−1.11066	+	1.11066i	−0.993061	+	0.117604i	$0.962479\pi$
$138$		0			0
$139$	12.0000			1.01783			0.508913	−	0.860818i	$-0.330047\pi$
$139$	12.0000			1.01783			0.508913	+	0.860818i	$0.330047\pi$
$140$		0			0
$141$	14.0000			1.17901
$142$		0			0
$143$	6.00000	−	6.00000i	0.501745	−	0.501745i
$144$		0			0
$145$		0			0
$146$		0			0
$147$	−5.00000	−	5.00000i	−0.412393	−	0.412393i
$148$		0			0
$149$		−	12.0000i		−	0.983078i	−0.870855	−	0.491539i	$-0.836434\pi$
$149$		−	12.0000i		−	0.983078i	0.870855	−	0.491539i	$-0.163566\pi$
$150$		0			0
$151$			18.0000i			1.46482i	0.680864	+	0.732410i	$0.261604\pi$
$151$			18.0000i			1.46482i	−0.680864	+	0.732410i	$0.738396\pi$
$152$		0			0
$153$	−1.00000	−	1.00000i	−0.0808452	−	0.0808452i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	3.00000	−	3.00000i	0.239426	−	0.239426i	−0.577186	−	0.816612i	$-0.695849\pi$
$157$	3.00000	−	3.00000i	0.239426	−	0.239426i	0.816612	+	0.577186i	$0.195849\pi$
$158$		0			0
$159$	−2.00000			−0.158610
$160$		0			0
$161$	−10.0000			−0.788110
$162$		0			0
$163$	1.00000	−	1.00000i	0.0783260	−	0.0783260i	−0.666858	−	0.745184i	$-0.732361\pi$
$163$	1.00000	−	1.00000i	0.0783260	−	0.0783260i	0.745184	+	0.666858i	$0.232361\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−5.00000	−	5.00000i	−0.386912	−	0.386912i	0.486673	−	0.873584i	$-0.338210\pi$
$167$	−5.00000	−	5.00000i	−0.386912	−	0.386912i	−0.873584	+	0.486673i	$0.838210\pi$
$168$		0			0
$169$		−	11.0000i		−	0.846154i
$170$		0			0
$171$			4.00000i			0.305888i
$172$		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
$175$		0			0
$176$		0			0
$177$	4.00000	−	4.00000i	0.300658	−	0.300658i
$178$		0			0
$179$	−12.0000			−0.896922			−0.448461	−	0.893802i	$-0.648028\pi$
$179$	−12.0000			−0.896922			−0.448461	+	0.893802i	$0.648028\pi$
$180$		0			0
$181$	10.0000			0.743294			0.371647	−	0.928374i	$-0.378793\pi$
$181$	10.0000			0.743294			0.371647	+	0.928374i	$0.378793\pi$
$182$		0			0
$183$	−2.00000	+	2.00000i	−0.147844	+	0.147844i
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−6.00000	−	6.00000i	−0.438763	−	0.438763i
$188$		0			0
$189$		−	8.00000i		−	0.581914i
$190$		0			0
$191$			10.0000i			0.723575i	0.932261	+	0.361787i	$0.117833\pi$
$191$			10.0000i			0.723575i	−0.932261	+	0.361787i	$0.882167\pi$
$192$		0			0
$193$	−1.00000	−	1.00000i	−0.0719816	−	0.0719816i	0.670199	−	0.742181i	$-0.266209\pi$
$193$	−1.00000	−	1.00000i	−0.0719816	−	0.0719816i	−0.742181	+	0.670199i	$0.766209\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−1.00000	+	1.00000i	−0.0712470	+	0.0712470i	−0.741832	−	0.670585i	$-0.766043\pi$
$197$	−1.00000	+	1.00000i	−0.0712470	+	0.0712470i	0.670585	+	0.741832i	$0.266043\pi$
$198$		0			0
$199$		0			0			−	1.00000i	$-0.5\pi$
$199$		0			0				1.00000i	$0.5\pi$
$200$		0			0
$201$	14.0000			0.987484
$202$		0			0
$203$	8.00000	−	8.00000i	0.561490	−	0.561490i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	5.00000	+	5.00000i	0.347524	+	0.347524i
$208$		0			0
$209$			24.0000i			1.66011i
$210$		0			0
$211$		−	10.0000i		−	0.688428i	−0.938891	−	0.344214i	$-0.888145\pi$
$211$		−	10.0000i		−	0.688428i	0.938891	−	0.344214i	$-0.111855\pi$
$212$		0			0
$213$	−6.00000	−	6.00000i	−0.411113	−	0.411113i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	2.00000	−	2.00000i	0.135769	−	0.135769i
$218$		0			0
$219$	−18.0000			−1.21633
$220$		0			0
$221$	2.00000			0.134535
$222$		0			0
$223$	−19.0000	+	19.0000i	−1.27233	+	1.27233i	−0.327474	+	0.944860i	$0.606197\pi$
$223$	−19.0000	+	19.0000i	−1.27233	+	1.27233i	−0.944860	+	0.327474i	$0.893803\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−13.0000	−	13.0000i	−0.862840	−	0.862840i	0.128827	−	0.991667i	$-0.458879\pi$
$227$	−13.0000	−	13.0000i	−0.862840	−	0.862840i	−0.991667	+	0.128827i	$0.958879\pi$
$228$		0			0
$229$			16.0000i			1.05731i	0.848837	+	0.528655i	$0.177303\pi$
$229$			16.0000i			1.05731i	−0.848837	+	0.528655i	$0.822697\pi$
$230$		0			0
$231$		−	12.0000i		−	0.789542i
$232$		0			0
$233$	−13.0000	−	13.0000i	−0.851658	−	0.851658i	0.138679	−	0.990337i	$-0.455714\pi$
$233$	−13.0000	−	13.0000i	−0.851658	−	0.851658i	−0.990337	+	0.138679i	$0.955714\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	8.00000	−	8.00000i	0.519656	−	0.519656i
$238$		0			0
$239$	16.0000			1.03495			0.517477	−	0.855697i	$-0.326871\pi$
$239$	16.0000			1.03495			0.517477	+	0.855697i	$0.326871\pi$
$240$		0			0
$241$	14.0000			0.901819			0.450910	−	0.892570i	$-0.351100\pi$
$241$	14.0000			0.901819			0.450910	+	0.892570i	$0.351100\pi$
$242$		0			0
$243$	−7.00000	+	7.00000i	−0.449050	+	0.449050i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−4.00000	−	4.00000i	−0.254514	−	0.254514i
$248$		0			0
$249$		−	10.0000i		−	0.633724i
$250$		0			0
$251$		−	18.0000i		−	1.13615i	−0.822977	−	0.568075i	$-0.807688\pi$
$251$		−	18.0000i		−	1.13615i	0.822977	−	0.568075i	$-0.192312\pi$
$252$		0			0
$253$	30.0000	+	30.0000i	1.88608	+	1.88608i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	11.0000	−	11.0000i	0.686161	−	0.686161i	−0.275220	−	0.961381i	$-0.588751\pi$
$257$	11.0000	−	11.0000i	0.686161	−	0.686161i	0.961381	+	0.275220i	$0.0887507\pi$
$258$		0			0
$259$	10.0000			0.621370
$260$		0			0
$261$	−8.00000			−0.495188
$262$		0			0
$263$	9.00000	−	9.00000i	0.554964	−	0.554964i	−0.372906	−	0.927869i	$-0.621638\pi$
$263$	9.00000	−	9.00000i	0.554964	−	0.554964i	0.927869	+	0.372906i	$0.121638\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$		−	4.00000i		−	0.243884i	−0.992537	−	0.121942i	$-0.961088\pi$
$269$		−	4.00000i		−	0.243884i	0.992537	−	0.121942i	$-0.0389122\pi$
$270$		0			0
$271$		−	22.0000i		−	1.33640i	−0.743980	−	0.668202i	$-0.767064\pi$
$271$		−	22.0000i		−	1.33640i	0.743980	−	0.668202i	$-0.232936\pi$
$272$		0			0
$273$	2.00000	+	2.00000i	0.121046	+	0.121046i
$274$		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
$279$	−2.00000			−0.119737
$280$		0			0
$281$	−10.0000			−0.596550			−0.298275	−	0.954480i	$-0.596411\pi$
$281$	−10.0000			−0.596550			−0.298275	+	0.954480i	$0.596411\pi$
$282$		0			0
$283$	−15.0000	+	15.0000i	−0.891657	+	0.891657i	−0.994679	−	0.103022i	$-0.967149\pi$
$283$	−15.0000	+	15.0000i	−0.891657	+	0.891657i	0.103022	+	0.994679i	$0.467149\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−6.00000	−	6.00000i	−0.354169	−	0.354169i
$288$		0			0
$289$			15.0000i			0.882353i
$290$		0			0
$291$		−	6.00000i		−	0.351726i
$292$		0			0
$293$	−17.0000	−	17.0000i	−0.993151	−	0.993151i	0.00682610	−	0.999977i	$-0.497827\pi$
$293$	−17.0000	−	17.0000i	−0.993151	−	0.993151i	−0.999977	+	0.00682610i	$0.997827\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	−24.0000	+	24.0000i	−1.39262	+	1.39262i
$298$		0			0
$299$	−10.0000			−0.578315
$300$		0			0
$301$	6.00000			0.345834
$302$		0			0
$303$	−6.00000	+	6.00000i	−0.344691	+	0.344691i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	7.00000	+	7.00000i	0.399511	+	0.399511i	0.878061	−	0.478549i	$-0.158837\pi$
$307$	7.00000	+	7.00000i	0.399511	+	0.399511i	−0.478549	+	0.878061i	$0.658837\pi$
$308$		0			0
$309$			6.00000i			0.341328i
$310$		0			0
$311$		−	22.0000i		−	1.24751i	−0.781622	−	0.623753i	$-0.785607\pi$
$311$		−	22.0000i		−	1.24751i	0.781622	−	0.623753i	$-0.214393\pi$
$312$		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
$315$		0			0
$316$		0			0
$317$	−25.0000	+	25.0000i	−1.40414	+	1.40414i	−0.617822	+	0.786318i	$0.711985\pi$
$317$	−25.0000	+	25.0000i	−1.40414	+	1.40414i	−0.786318	+	0.617822i	$0.788015\pi$
$318$		0			0
$319$	−48.0000			−2.68748
$320$		0			0
$321$	6.00000			0.334887
$322$		0			0
$323$	−4.00000	+	4.00000i	−0.222566	+	0.222566i
$324$		0			0
$325$		0			0
$326$		0			0
$327$	4.00000	+	4.00000i	0.221201	+	0.221201i
$328$		0			0
$329$		−	14.0000i		−	0.771845i
$330$		0			0
$331$		−	18.0000i		−	0.989369i	−0.869072	−	0.494685i	$-0.835284\pi$
$331$		−	18.0000i		−	0.989369i	0.869072	−	0.494685i	$-0.164716\pi$
$332$		0			0
$333$	−5.00000	−	5.00000i	−0.273998	−	0.273998i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−1.00000	+	1.00000i	−0.0544735	+	0.0544735i	−0.733819	−	0.679345i	$-0.762264\pi$
$337$	−1.00000	+	1.00000i	−0.0544735	+	0.0544735i	0.679345	+	0.733819i	$0.262264\pi$
$338$		0			0
$339$	6.00000			0.325875
$340$		0			0
$341$	−12.0000			−0.649836
$342$		0			0
$343$	−12.0000	+	12.0000i	−0.647939	+	0.647939i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−9.00000	−	9.00000i	−0.483145	−	0.483145i	0.422989	−	0.906135i	$-0.360981\pi$
$347$	−9.00000	−	9.00000i	−0.483145	−	0.483145i	−0.906135	+	0.422989i	$0.860981\pi$
$348$		0			0
$349$			32.0000i			1.71292i	0.516213	+	0.856460i	$0.327341\pi$
$349$			32.0000i			1.71292i	−0.516213	+	0.856460i	$0.672659\pi$
$350$		0			0
$351$		−	8.00000i		−	0.427008i
$352$		0			0
$353$	15.0000	+	15.0000i	0.798369	+	0.798369i	0.982838	−	0.184469i	$-0.0590565\pi$
$353$	15.0000	+	15.0000i	0.798369	+	0.798369i	−0.184469	+	0.982838i	$0.559057\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	2.00000	−	2.00000i	0.105851	−	0.105851i
$358$		0			0
$359$	16.0000			0.844448			0.422224	−	0.906492i	$-0.361250\pi$
$359$	16.0000			0.844448			0.422224	+	0.906492i	$0.361250\pi$
$360$		0			0
$361$	−3.00000			−0.157895
$362$		0			0
$363$	−25.0000	+	25.0000i	−1.31216	+	1.31216i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−13.0000	−	13.0000i	−0.678594	−	0.678594i	0.281088	−	0.959682i	$-0.409305\pi$
$367$	−13.0000	−	13.0000i	−0.678594	−	0.678594i	−0.959682	+	0.281088i	$0.909305\pi$
$368$		0			0
$369$			6.00000i			0.312348i
$370$		0			0
$371$			2.00000i			0.103835i
$372$		0			0
$373$	−21.0000	−	21.0000i	−1.08734	−	1.08734i	−0.995802	−	0.0915371i	$-0.970822\pi$
$373$	−21.0000	−	21.0000i	−1.08734	−	1.08734i	−0.0915371	−	0.995802i	$-0.529178\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	8.00000	−	8.00000i	0.412021	−	0.412021i
$378$		0			0
$379$	4.00000			0.205466			0.102733	−	0.994709i	$-0.467241\pi$
$379$	4.00000			0.205466			0.102733	+	0.994709i	$0.467241\pi$
$380$		0			0
$381$	−10.0000			−0.512316
$382$		0			0
$383$	13.0000	−	13.0000i	0.664269	−	0.664269i	−0.292114	−	0.956383i	$-0.594359\pi$
$383$	13.0000	−	13.0000i	0.664269	−	0.664269i	0.956383	+	0.292114i	$0.0943588\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−3.00000	−	3.00000i	−0.152499	−	0.152499i
$388$		0			0
$389$		−	12.0000i		−	0.608424i	−0.952604	−	0.304212i	$-0.901607\pi$
$389$		−	12.0000i		−	0.608424i	0.952604	−	0.304212i	$-0.0983931\pi$
$390$		0			0
$391$			10.0000i			0.505722i
$392$		0			0
$393$	−2.00000	−	2.00000i	−0.100887	−	0.100887i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	19.0000	−	19.0000i	0.953583	−	0.953583i	−0.0453868	−	0.998969i	$-0.514452\pi$
$397$	19.0000	−	19.0000i	0.953583	−	0.953583i	0.998969	+	0.0453868i	$0.0144520\pi$
$398$		0			0
$399$	−8.00000			−0.400501
$400$		0			0
$401$	10.0000			0.499376			0.249688	−	0.968326i	$-0.419672\pi$
$401$	10.0000			0.499376			0.249688	+	0.968326i	$0.419672\pi$
$402$		0			0
$403$	2.00000	−	2.00000i	0.0996271	−	0.0996271i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−30.0000	−	30.0000i	−1.48704	−	1.48704i
$408$		0			0
$409$		−	20.0000i		−	0.988936i	−0.869196	−	0.494468i	$-0.835363\pi$
$409$		−	20.0000i		−	0.988936i	0.869196	−	0.494468i	$-0.164637\pi$
$410$		0			0
$411$			26.0000i			1.28249i
$412$		0			0
$413$	−4.00000	−	4.00000i	−0.196827	−	0.196827i
$414$		0			0
$415$		0			0
$416$		0			0
$417$	12.0000	−	12.0000i	0.587643	−	0.587643i
$418$		0			0
$419$	12.0000			0.586238			0.293119	−	0.956076i	$-0.405307\pi$
$419$	12.0000			0.586238			0.293119	+	0.956076i	$0.405307\pi$
$420$		0			0
$421$	−2.00000			−0.0974740			−0.0487370	−	0.998812i	$-0.515520\pi$
$421$	−2.00000			−0.0974740			−0.0487370	+	0.998812i	$0.515520\pi$
$422$		0			0
$423$	−7.00000	+	7.00000i	−0.340352	+	0.340352i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	2.00000	+	2.00000i	0.0967868	+	0.0967868i
$428$		0			0
$429$		−	12.0000i		−	0.579365i
$430$		0			0
$431$		−	38.0000i		−	1.83040i	−0.403005	−	0.915198i	$-0.632034\pi$
$431$		−	38.0000i		−	1.83040i	0.403005	−	0.915198i	$-0.367966\pi$
$432$		0			0
$433$	−5.00000	−	5.00000i	−0.240285	−	0.240285i	0.576683	−	0.816968i	$-0.304347\pi$
$433$	−5.00000	−	5.00000i	−0.240285	−	0.240285i	−0.816968	+	0.576683i	$0.804347\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	20.0000	−	20.0000i	0.956730	−	0.956730i
$438$		0			0
$439$	−32.0000			−1.52728			−0.763638	−	0.645644i	$-0.776589\pi$
$439$	−32.0000			−1.52728			−0.763638	+	0.645644i	$0.776589\pi$
$440$		0			0
$441$	5.00000			0.238095
$442$		0			0
$443$	17.0000	−	17.0000i	0.807694	−	0.807694i	−0.176590	−	0.984284i	$-0.556507\pi$
$443$	17.0000	−	17.0000i	0.807694	−	0.807694i	0.984284	+	0.176590i	$0.0565067\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	−12.0000	−	12.0000i	−0.567581	−	0.567581i
$448$		0			0
$449$		−	4.00000i		−	0.188772i	−0.995536	−	0.0943858i	$-0.969911\pi$
$449$		−	4.00000i		−	0.188772i	0.995536	−	0.0943858i	$-0.0300887\pi$
$450$		0			0
$451$			36.0000i			1.69517i
$452$		0			0
$453$	18.0000	+	18.0000i	0.845714	+	0.845714i
$454$		0			0
$455$		0			0
$456$		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
$459$	−8.00000			−0.373408
$460$		0			0
$461$	2.00000			0.0931493			0.0465746	−	0.998915i	$-0.485169\pi$
$461$	2.00000			0.0931493			0.0465746	+	0.998915i	$0.485169\pi$
$462$		0			0
$463$	17.0000	−	17.0000i	0.790057	−	0.790057i	−0.191446	−	0.981503i	$-0.561318\pi$
$463$	17.0000	−	17.0000i	0.790057	−	0.790057i	0.981503	+	0.191446i	$0.0613177\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−5.00000	−	5.00000i	−0.231372	−	0.231372i	0.581893	−	0.813265i	$-0.302312\pi$
$467$	−5.00000	−	5.00000i	−0.231372	−	0.231372i	−0.813265	+	0.581893i	$0.802312\pi$
$468$		0			0
$469$		−	14.0000i		−	0.646460i
$470$		0			0
$471$		−	6.00000i		−	0.276465i
$472$		0			0
$473$	−18.0000	−	18.0000i	−0.827641	−	0.827641i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	1.00000	−	1.00000i	0.0457869	−	0.0457869i
$478$		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
$481$	10.0000			0.455961
$482$		0			0
$483$	−10.0000	+	10.0000i	−0.455016	+	0.455016i
$484$		0			0
$485$		0			0
$486$		0			0
$487$	15.0000	+	15.0000i	0.679715	+	0.679715i	0.959936	−	0.280221i	$-0.0904077\pi$
$487$	15.0000	+	15.0000i	0.679715	+	0.679715i	−0.280221	+	0.959936i	$0.590408\pi$
$488$		0			0
$489$		−	2.00000i		−	0.0904431i
$490$		0			0
$491$		−	2.00000i		−	0.0902587i	−0.998981	−	0.0451294i	$-0.985630\pi$
$491$		−	2.00000i		−	0.0902587i	0.998981	−	0.0451294i	$-0.0143700\pi$
$492$		0			0
$493$	−8.00000	−	8.00000i	−0.360302	−	0.360302i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−6.00000	+	6.00000i	−0.269137	+	0.269137i
$498$		0			0
$499$	−20.0000			−0.895323			−0.447661	−	0.894203i	$-0.647743\pi$
$499$	−20.0000			−0.895323			−0.447661	+	0.894203i	$0.647743\pi$
$500$		0			0
$501$	−10.0000			−0.446767
$502$		0			0
$503$	−11.0000	+	11.0000i	−0.490466	+	0.490466i	−0.908453	−	0.417987i	$-0.862736\pi$
$503$	−11.0000	+	11.0000i	−0.490466	+	0.490466i	0.417987	+	0.908453i	$0.362736\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−11.0000	−	11.0000i	−0.488527	−	0.488527i
$508$		0			0
$509$		0			0		1.00000			$0$
$509$		0			0		−1.00000			$\pi$
$510$		0			0
$511$			18.0000i			0.796273i
$512$		0			0
$513$	16.0000	+	16.0000i	0.706417	+	0.706417i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	−42.0000	+	42.0000i	−1.84716	+	1.84716i
$518$		0			0
$519$	14.0000			0.614532
$520$		0			0
$521$	18.0000			0.788594			0.394297	−	0.918983i	$-0.370988\pi$
$521$	18.0000			0.788594			0.394297	+	0.918983i	$0.370988\pi$
$522$		0			0
$523$	25.0000	−	25.0000i	1.09317	−	1.09317i	0.0979859	−	0.995188i	$-0.468760\pi$
$523$	25.0000	−	25.0000i	1.09317	−	1.09317i	0.995188	−	0.0979859i	$-0.0312400\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−2.00000	−	2.00000i	−0.0871214	−	0.0871214i
$528$		0			0
$529$		−	27.0000i		−	1.17391i
$530$		0			0
$531$			4.00000i			0.173585i
$532$		0			0
$533$	−6.00000	−	6.00000i	−0.259889	−	0.259889i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	−12.0000	+	12.0000i	−0.517838	+	0.517838i
$538$		0			0
$539$	30.0000			1.29219
$540$		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
$543$	10.0000	−	10.0000i	0.429141	−	0.429141i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	3.00000	+	3.00000i	0.128271	+	0.128271i	0.768328	−	0.640057i	$-0.221089\pi$
$547$	3.00000	+	3.00000i	0.128271	+	0.128271i	−0.640057	+	0.768328i	$0.721089\pi$
$548$		0			0
$549$		−	2.00000i		−	0.0853579i
$550$		0			0
$551$			32.0000i			1.36325i
$552$		0			0
$553$	−8.00000	−	8.00000i	−0.340195	−	0.340195i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	15.0000	−	15.0000i	0.635570	−	0.635570i	−0.313889	−	0.949460i	$-0.601632\pi$
$557$	15.0000	−	15.0000i	0.635570	−	0.635570i	0.949460	+	0.313889i	$0.101632\pi$
$558$		0			0
$559$	6.00000			0.253773
$560$		0			0
$561$	−12.0000			−0.506640
$562$		0			0
$563$	−15.0000	+	15.0000i	−0.632175	+	0.632175i	−0.948613	−	0.316438i	$-0.897513\pi$
$563$	−15.0000	+	15.0000i	−0.632175	+	0.632175i	0.316438	+	0.948613i	$0.397513\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−5.00000	−	5.00000i	−0.209980	−	0.209980i
$568$		0			0
$569$			20.0000i			0.838444i	0.907884	+	0.419222i	$0.137697\pi$
$569$			20.0000i			0.838444i	−0.907884	+	0.419222i	$0.862303\pi$
$570$		0			0
$571$		−	26.0000i		−	1.08807i	−0.839064	−	0.544033i	$-0.816897\pi$
$571$		−	26.0000i		−	1.08807i	0.839064	−	0.544033i	$-0.183103\pi$
$572$		0			0
$573$	10.0000	+	10.0000i	0.417756	+	0.417756i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	3.00000	−	3.00000i	0.124892	−	0.124892i	−0.641898	−	0.766790i	$-0.721853\pi$
$577$	3.00000	−	3.00000i	0.124892	−	0.124892i	0.766790	+	0.641898i	$0.221853\pi$
$578$		0			0
$579$	−2.00000			−0.0831172
$580$		0			0
$581$	−10.0000			−0.414870
$582$		0			0
$583$	6.00000	−	6.00000i	0.248495	−	0.248495i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−25.0000	−	25.0000i	−1.03186	−	1.03186i	−0.999475	−	0.0323850i	$-0.989690\pi$
$587$	−25.0000	−	25.0000i	−1.03186	−	1.03186i	−0.0323850	−	0.999475i	$-0.510310\pi$
$588$		0			0
$589$			8.00000i			0.329634i
$590$		0			0
$591$			2.00000i			0.0822690i
$592$		0			0
$593$	23.0000	+	23.0000i	0.944497	+	0.944497i	0.998539	−	0.0540419i	$-0.0172104\pi$
$593$	23.0000	+	23.0000i	0.944497	+	0.944497i	−0.0540419	+	0.998539i	$0.517210\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−16.0000			−0.653742			−0.326871	−	0.945069i	$-0.605994\pi$
$599$	−16.0000			−0.653742			−0.326871	+	0.945069i	$0.605994\pi$
$600$		0			0
$601$	−26.0000			−1.06056			−0.530281	−	0.847822i	$-0.677914\pi$
$601$	−26.0000			−1.06056			−0.530281	+	0.847822i	$0.677914\pi$
$602$		0			0
$603$	−7.00000	+	7.00000i	−0.285062	+	0.285062i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	15.0000	+	15.0000i	0.608831	+	0.608831i	0.942641	−	0.333809i	$-0.108334\pi$
$607$	15.0000	+	15.0000i	0.608831	+	0.608831i	−0.333809	+	0.942641i	$0.608334\pi$
$608$		0			0
$609$		−	16.0000i		−	0.648353i
$610$		0			0
$611$		−	14.0000i		−	0.566379i
$612$		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
$617$	11.0000	−	11.0000i	0.442843	−	0.442843i	−0.450123	−	0.892966i	$-0.648620\pi$
$617$	11.0000	−	11.0000i	0.442843	−	0.442843i	0.892966	+	0.450123i	$0.148620\pi$
$618$		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
$621$	40.0000			1.60514
$622$		0			0
$623$		0			0
$624$		0			0
$625$		0			0
$626$		0			0
$627$	24.0000	+	24.0000i	0.958468	+	0.958468i
$628$		0			0
$629$		−	10.0000i		−	0.398726i
$630$		0			0
$631$			10.0000i			0.398094i	0.979990	+	0.199047i	$0.0637846\pi$
$631$			10.0000i			0.398094i	−0.979990	+	0.199047i	$0.936215\pi$
$632$		0			0
$633$	−10.0000	−	10.0000i	−0.397464	−	0.397464i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−5.00000	+	5.00000i	−0.198107	+	0.198107i
$638$		0			0
$639$	6.00000			0.237356
$640$		0			0
$641$	−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
$643$	−11.0000	+	11.0000i	−0.433798	+	0.433798i	−0.889918	−	0.456120i	$-0.849239\pi$
$643$	−11.0000	+	11.0000i	−0.433798	+	0.433798i	0.456120	+	0.889918i	$0.349239\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	31.0000	+	31.0000i	1.21874	+	1.21874i	0.968075	+	0.250661i	$0.0806479\pi$
$647$	31.0000	+	31.0000i	1.21874	+	1.21874i	0.250661	+	0.968075i	$0.419352\pi$
$648$		0			0
$649$			24.0000i			0.942082i
$650$		0			0
$651$		−	4.00000i		−	0.156772i
$652$		0			0
$653$	−5.00000	−	5.00000i	−0.195665	−	0.195665i	0.602474	−	0.798139i	$-0.294182\pi$
$653$	−5.00000	−	5.00000i	−0.195665	−	0.195665i	−0.798139	+	0.602474i	$0.794182\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	9.00000	−	9.00000i	0.351123	−	0.351123i
$658$		0			0
$659$	44.0000			1.71400			0.856998	−	0.515319i	$-0.172327\pi$
$659$	44.0000			1.71400			0.856998	+	0.515319i	$0.172327\pi$
$660$		0			0
$661$	30.0000			1.16686			0.583432	−	0.812162i	$-0.301709\pi$
$661$	30.0000			1.16686			0.583432	+	0.812162i	$0.301709\pi$
$662$		0			0
$663$	2.00000	−	2.00000i	0.0776736	−	0.0776736i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	40.0000	+	40.0000i	1.54881	+	1.54881i
$668$		0			0
$669$			38.0000i			1.46916i
$670$		0			0
$671$		−	12.0000i		−	0.463255i
$672$		0			0
$673$	−29.0000	−	29.0000i	−1.11787	−	1.11787i	−0.992054	−	0.125814i	$-0.959846\pi$
$673$	−29.0000	−	29.0000i	−1.11787	−	1.11787i	−0.125814	−	0.992054i	$-0.540154\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	15.0000	−	15.0000i	0.576497	−	0.576497i	−0.357439	−	0.933936i	$-0.616350\pi$
$677$	15.0000	−	15.0000i	0.576497	−	0.576497i	0.933936	+	0.357439i	$0.116350\pi$
$678$		0			0
$679$	−6.00000			−0.230259
$680$		0			0
$681$	−26.0000			−0.996322
$682$		0			0
$683$	13.0000	−	13.0000i	0.497431	−	0.497431i	−0.413206	−	0.910637i	$-0.635591\pi$
$683$	13.0000	−	13.0000i	0.497431	−	0.497431i	0.910637	+	0.413206i	$0.135591\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	16.0000	+	16.0000i	0.610438	+	0.610438i
$688$		0			0
$689$			2.00000i			0.0761939i
$690$		0			0
$691$		−	10.0000i		−	0.380418i	−0.981744	−	0.190209i	$-0.939083\pi$
$691$		−	10.0000i		−	0.380418i	0.981744	−	0.190209i	$-0.0609166\pi$
$692$		0			0
$693$	6.00000	+	6.00000i	0.227921	+	0.227921i
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−6.00000	+	6.00000i	−0.227266	+	0.227266i
$698$		0			0
$699$	−26.0000			−0.983410
$700$		0			0
$701$	−2.00000			−0.0755390			−0.0377695	−	0.999286i	$-0.512025\pi$
$701$	−2.00000			−0.0755390			−0.0377695	+	0.999286i	$0.512025\pi$
$702$		0			0
$703$	−20.0000	+	20.0000i	−0.754314	+	0.754314i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	6.00000	+	6.00000i	0.225653	+	0.225653i
$708$		0			0
$709$		−	24.0000i		−	0.901339i	−0.892691	−	0.450669i	$-0.851185\pi$
$709$		−	24.0000i		−	0.901339i	0.892691	−	0.450669i	$-0.148815\pi$
$710$		0			0
$711$			8.00000i			0.300023i
$712$		0			0
$713$	10.0000	+	10.0000i	0.374503	+	0.374503i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	16.0000	−	16.0000i	0.597531	−	0.597531i
$718$		0			0
$719$	16.0000			0.596699			0.298350	−	0.954457i	$-0.403564\pi$
$719$	16.0000			0.596699			0.298350	+	0.954457i	$0.403564\pi$
$720$		0			0
$721$	6.00000			0.223452
$722$		0			0
$723$	14.0000	−	14.0000i	0.520666	−	0.520666i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	15.0000	+	15.0000i	0.556319	+	0.556319i	0.928257	−	0.371938i	$-0.121307\pi$
$727$	15.0000	+	15.0000i	0.556319	+	0.556319i	−0.371938	+	0.928257i	$0.621307\pi$
$728$		0			0
$729$			29.0000i			1.07407i
$730$		0			0
$731$		−	6.00000i		−	0.221918i
$732$		0			0
$733$	−9.00000	−	9.00000i	−0.332423	−	0.332423i	0.521083	−	0.853506i	$-0.325528\pi$
$733$	−9.00000	−	9.00000i	−0.332423	−	0.332423i	−0.853506	+	0.521083i	$0.825528\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−42.0000	+	42.0000i	−1.54709	+	1.54709i
$738$		0			0
$739$	−28.0000			−1.03000			−0.514998	−	0.857191i	$-0.672207\pi$
$739$	−28.0000			−1.03000			−0.514998	+	0.857191i	$0.672207\pi$
$740$		0			0
$741$	−8.00000			−0.293887
$742$		0			0
$743$	9.00000	−	9.00000i	0.330178	−	0.330178i	−0.522476	−	0.852654i	$-0.674992\pi$
$743$	9.00000	−	9.00000i	0.330178	−	0.330178i	0.852654	+	0.522476i	$0.174992\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	5.00000	+	5.00000i	0.182940	+	0.182940i
$748$		0			0
$749$		−	6.00000i		−	0.219235i
$750$		0			0
$751$			26.0000i			0.948753i	0.880322	+	0.474377i	$0.157327\pi$
$751$			26.0000i			0.948753i	−0.880322	+	0.474377i	$0.842673\pi$
$752$		0			0
$753$	−18.0000	−	18.0000i	−0.655956	−	0.655956i
$754$		0			0
$755$		0			0
$756$		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
$759$	60.0000			2.17786
$760$		0			0
$761$	50.0000			1.81250			0.906249	−	0.422744i	$-0.138933\pi$
$761$	50.0000			1.81250			0.906249	+	0.422744i	$0.138933\pi$
$762$		0			0
$763$	4.00000	−	4.00000i	0.144810	−	0.144810i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−4.00000	−	4.00000i	−0.144432	−	0.144432i
$768$		0			0
$769$			8.00000i			0.288487i	0.989542	+	0.144244i	$0.0460749\pi$
$769$			8.00000i			0.288487i	−0.989542	+	0.144244i	$0.953925\pi$
$770$		0			0
$771$		−	22.0000i		−	0.792311i
$772$		0			0
$773$	35.0000	+	35.0000i	1.25886	+	1.25886i	0.951637	+	0.307226i	$0.0994007\pi$
$773$	35.0000	+	35.0000i	1.25886	+	1.25886i	0.307226	+	0.951637i	$0.400599\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	10.0000	−	10.0000i	0.358748	−	0.358748i
$778$		0			0
$779$	24.0000			0.859889
$780$		0			0
$781$	36.0000			1.28818
$782$		0			0
$783$	−32.0000	+	32.0000i	−1.14359	+	1.14359i
$784$		0			0
$785$		0			0
$786$		0			0
$787$	−9.00000	−	9.00000i	−0.320815	−	0.320815i	0.528265	−	0.849080i	$-0.322843\pi$
$787$	−9.00000	−	9.00000i	−0.320815	−	0.320815i	−0.849080	+	0.528265i	$0.822843\pi$
$788$		0			0
$789$		−	18.0000i		−	0.640817i
$790$		0			0
$791$		−	6.00000i		−	0.213335i
$792$		0			0
$793$	2.00000	+	2.00000i	0.0710221	+	0.0710221i
$794$		0			0
$795$		0			0
$796$		0			0

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1600 = 2^{6} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1600.n (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(1.00000 - 1.00000i) q^{3} +(-1.00000 - 1.00000i) q^{7} +1.00000i q^{9} +O(q^{10})\)	\(q+(1.00000 - 1.00000i) q^{3} +(-1.00000 - 1.00000i) q^{7} +1.00000i q^{9} +6.00000i q^{11} +(-1.00000 - 1.00000i) q^{13} +(-1.00000 + 1.00000i) q^{17} +4.00000 q^{19} -2.00000 q^{21} +(5.00000 - 5.00000i) q^{23} +(4.00000 + 4.00000i) q^{27} +8.00000i q^{29} +2.00000i q^{31} +(6.00000 + 6.00000i) q^{33} +(-5.00000 + 5.00000i) q^{37} -2.00000 q^{39} +6.00000 q^{41} +(-3.00000 + 3.00000i) q^{43} +(7.00000 + 7.00000i) q^{47} -5.00000i q^{49} +2.00000i q^{51} +(-1.00000 - 1.00000i) q^{53} +(4.00000 - 4.00000i) q^{57} +4.00000 q^{59} -2.00000 q^{61} +(1.00000 - 1.00000i) q^{63} +(7.00000 + 7.00000i) q^{67} -10.0000i q^{69} -6.00000i q^{71} +(-9.00000 - 9.00000i) q^{73} +(6.00000 - 6.00000i) q^{77} +8.00000 q^{79} +5.00000 q^{81} +(5.00000 - 5.00000i) q^{83} +(8.00000 + 8.00000i) q^{87} +2.00000i q^{91} +(2.00000 + 2.00000i) q^{93} +(3.00000 - 3.00000i) q^{97} -6.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} - 2 q^{7}+O(q^{10}) \) 2 * q + 2 * q^3 - 2 * q^7	\( 2 q + 2 q^{3} - 2 q^{7} - 2 q^{13} - 2 q^{17} + 8 q^{19} - 4 q^{21} + 10 q^{23} + 8 q^{27} + 12 q^{33} - 10 q^{37} - 4 q^{39} + 12 q^{41} - 6 q^{43} + 14 q^{47} - 2 q^{53} + 8 q^{57} + 8 q^{59} - 4 q^{61} + 2 q^{63} + 14 q^{67} - 18 q^{73} + 12 q^{77} + 16 q^{79} + 10 q^{81} + 10 q^{83} + 16 q^{87} + 4 q^{93} + 6 q^{97} - 12 q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 - 2 * q^7 - 2 * q^13 - 2 * q^17 + 8 * q^19 - 4 * q^21 + 10 * q^23 + 8 * q^27 + 12 * q^33 - 10 * q^37 - 4 * q^39 + 12 * q^41 - 6 * q^43 + 14 * q^47 - 2 * q^53 + 8 * q^57 + 8 * q^59 - 4 * q^61 + 2 * q^63 + 14 * q^67 - 18 * q^73 + 12 * q^77 + 16 * q^79 + 10 * q^81 + 10 * q^83 + 16 * q^87 + 4 * q^93 + 6 * q^97 - 12 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more