LMFDB - Embedded newform 1600.2.o.c.607.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1600,2,Mod(543,1600)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1600.543"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1600, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 2, 3])) N = Newforms(chi, 2, names="a")

Level:	$N$	$=$	$1600 = 2^{6} \cdot 5^{2}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	1600.o (of order $4$ , degree $2$ , minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,2,0,0,0,-2,0,0,0,-8,0,-6,0,0,0,6,0,0,0,0,0,-6,0,0,0,8,0, -4,0,0,0,-8,0,0,0,-6,0,-12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(39)] == traces)

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 320)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			607.1
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	1600.607
Dual form			1600.2.o.c.543.1

$q$ -expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.00000 - 1.00000i) q^{3} +(-1.00000 + 1.00000i) q^{7} +1.00000i q^{9} -4.00000 q^{11} +(-3.00000 - 3.00000i) q^{13} +(3.00000 + 3.00000i) q^{17} +6.00000i q^{19} +2.00000i q^{21} +(-3.00000 - 3.00000i) q^{23} +(4.00000 + 4.00000i) q^{27} -2.00000 q^{29} +6.00000i q^{31} +(-4.00000 + 4.00000i) q^{33} +(-3.00000 + 3.00000i) q^{37} -6.00000 q^{39} +6.00000 q^{41} +(-3.00000 + 3.00000i) q^{43} +(-9.00000 + 9.00000i) q^{47} +5.00000i q^{49} +6.00000 q^{51} +(5.00000 + 5.00000i) q^{53} +(6.00000 + 6.00000i) q^{57} -10.0000i q^{59} -12.0000i q^{61} +(-1.00000 - 1.00000i) q^{63} +(-9.00000 - 9.00000i) q^{67} -6.00000 q^{69} +6.00000i q^{71} +(-5.00000 + 5.00000i) q^{73} +(4.00000 - 4.00000i) q^{77} +5.00000 q^{81} +(-3.00000 + 3.00000i) q^{83} +(-2.00000 + 2.00000i) q^{87} +6.00000 q^{91} +(6.00000 + 6.00000i) q^{93} +(7.00000 + 7.00000i) q^{97} -4.00000i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$2 q + 2 q^{3} - 2 q^{7} - 8 q^{11} - 6 q^{13} + 6 q^{17} - 6 q^{23} + 8 q^{27} - 4 q^{29} - 8 q^{33} - 6 q^{37} - 12 q^{39} + 12 q^{41} - 6 q^{43} - 18 q^{47} + 12 q^{51} + 10 q^{53} + 12 q^{57} - 2 q^{63}+ \cdots + 14 q^{97}+O(q^{100})$ 2 * q + 2 * q^3 - 2 * q^7 - 8 * q^11 - 6 * q^13 + 6 * q^17 - 6 * q^23 + 8 * q^27 - 4 * q^29 - 8 * q^33 - 6 * q^37 - 12 * q^39 + 12 * q^41 - 6 * q^43 - 18 * q^47 + 12 * q^51 + 10 * q^53 + 12 * q^57 - 2 * q^63 - 18 * q^67 - 12 * q^69 - 10 * q^73 + 8 * q^77 + 10 * q^81 - 6 * q^83 - 4 * q^87 + 12 * q^91 + 12 * q^93 + 14 * q^97

Character values

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

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

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
$3$	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	−0.870367	−	0.492403i	$-0.836119\pi$
$7$	−1.00000	+	1.00000i	−0.377964	+	0.377964i	0.492403	+	0.870367i	$0.336119\pi$
$8$		0			0
$9$			1.00000i			0.333333i
$10$		0			0
$11$	−4.00000			−1.20605			−0.603023	−	0.797724i	$-0.706037\pi$
$11$	−4.00000			−1.20605			−0.603023	+	0.797724i	$0.706037\pi$
$12$		0			0
$13$	−3.00000	−	3.00000i	−0.832050	−	0.832050i	0.155747	−	0.987797i	$-0.450222\pi$
$13$	−3.00000	−	3.00000i	−0.832050	−	0.832050i	−0.987797	+	0.155747i	$0.950222\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	3.00000	+	3.00000i	0.727607	+	0.727607i	0.970143	−	0.242536i	$-0.0779791\pi$
$17$	3.00000	+	3.00000i	0.727607	+	0.727607i	−0.242536	+	0.970143i	$0.577979\pi$
$18$		0			0
$19$			6.00000i			1.37649i	0.725476	+	0.688247i	$0.241620\pi$
$19$			6.00000i			1.37649i	−0.725476	+	0.688247i	$0.758380\pi$
$20$		0			0
$21$			2.00000i			0.436436i
$22$		0			0
$23$	−3.00000	−	3.00000i	−0.625543	−	0.625543i	0.321400	−	0.946943i	$-0.395847\pi$
$23$	−3.00000	−	3.00000i	−0.625543	−	0.625543i	−0.946943	+	0.321400i	$0.895847\pi$
$24$		0			0
$25$		0			0
$26$		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
$31$			6.00000i			1.07763i	0.842424	+	0.538816i	$0.181128\pi$
$31$			6.00000i			1.07763i	−0.842424	+	0.538816i	$0.818872\pi$
$32$		0			0
$33$	−4.00000	+	4.00000i	−0.696311	+	0.696311i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−3.00000	+	3.00000i	−0.493197	+	0.493197i	−0.909312	−	0.416115i	$-0.863391\pi$
$37$	−3.00000	+	3.00000i	−0.493197	+	0.493197i	0.416115	+	0.909312i	$0.363391\pi$
$38$		0			0
$39$	−6.00000			−0.960769
$40$		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
$47$	−9.00000	+	9.00000i	−1.31278	+	1.31278i	−0.393431	+	0.919354i	$0.628712\pi$
$47$	−9.00000	+	9.00000i	−1.31278	+	1.31278i	−0.919354	+	0.393431i	$0.871288\pi$
$48$		0			0
$49$			5.00000i			0.714286i
$50$		0			0
$51$	6.00000			0.840168
$52$		0			0
$53$	5.00000	+	5.00000i	0.686803	+	0.686803i	0.961524	−	0.274721i	$-0.0885855\pi$
$53$	5.00000	+	5.00000i	0.686803	+	0.686803i	−0.274721	+	0.961524i	$0.588586\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	6.00000	+	6.00000i	0.794719	+	0.794719i
$58$		0			0
$59$		−	10.0000i		−	1.30189i	−0.759125	−	0.650945i	$-0.774373\pi$
$59$		−	10.0000i		−	1.30189i	0.759125	−	0.650945i	$-0.225627\pi$
$60$		0			0
$61$		−	12.0000i		−	1.53644i	−0.640184	−	0.768221i	$-0.721142\pi$
$61$		−	12.0000i		−	1.53644i	0.640184	−	0.768221i	$-0.278858\pi$
$62$		0			0
$63$	−1.00000	−	1.00000i	−0.125988	−	0.125988i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−9.00000	−	9.00000i	−1.09952	−	1.09952i	−0.994466	−	0.105059i	$-0.966497\pi$
$67$	−9.00000	−	9.00000i	−1.09952	−	1.09952i	−0.105059	−	0.994466i	$-0.533503\pi$
$68$		0			0
$69$	−6.00000			−0.722315
$70$		0			0
$71$			6.00000i			0.712069i	0.934473	+	0.356034i	$0.115871\pi$
$71$			6.00000i			0.712069i	−0.934473	+	0.356034i	$0.884129\pi$
$72$		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
$75$		0			0
$76$		0			0
$77$	4.00000	−	4.00000i	0.455842	−	0.455842i
$78$		0			0
$79$		0			0			−	1.00000i	$-0.5\pi$
$79$		0			0				1.00000i	$0.5\pi$
$80$		0			0
$81$	5.00000			0.555556
$82$		0			0
$83$	−3.00000	+	3.00000i	−0.329293	+	0.329293i	−0.852318	−	0.523025i	$-0.824804\pi$
$83$	−3.00000	+	3.00000i	−0.329293	+	0.329293i	0.523025	+	0.852318i	$0.324804\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	−2.00000	+	2.00000i	−0.214423	+	0.214423i
$88$		0			0
$89$		0			0		1.00000			$0$
$89$		0			0		−1.00000			$\pi$
$90$		0			0
$91$	6.00000			0.628971
$92$		0			0
$93$	6.00000	+	6.00000i	0.622171	+	0.622171i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	7.00000	+	7.00000i	0.710742	+	0.710742i	0.966691	−	0.255948i	$-0.0823876\pi$
$97$	7.00000	+	7.00000i	0.710742	+	0.710742i	−0.255948	+	0.966691i	$0.582388\pi$
$98$		0			0
$99$		−	4.00000i		−	0.402015i
$100$		0			0
$101$			8.00000i			0.796030i	0.917379	+	0.398015i	$0.130301\pi$
$101$			8.00000i			0.796030i	−0.917379	+	0.398015i	$0.869699\pi$
$102$		0			0
$103$	−11.0000	−	11.0000i	−1.08386	−	1.08386i	−0.996145	−	0.0877167i	$-0.972043\pi$
$103$	−11.0000	−	11.0000i	−1.08386	−	1.08386i	−0.0877167	−	0.996145i	$-0.527957\pi$
$104$		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
$109$	18.0000			1.72409			0.862044	−	0.506834i	$-0.169184\pi$
$109$	18.0000			1.72409			0.862044	+	0.506834i	$0.169184\pi$
$110$		0			0
$111$			6.00000i			0.569495i
$112$		0			0
$113$	3.00000	−	3.00000i	0.282216	−	0.282216i	−0.551776	−	0.833992i	$-0.686050\pi$
$113$	3.00000	−	3.00000i	0.282216	−	0.282216i	0.833992	+	0.551776i	$0.186050\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	3.00000	−	3.00000i	0.277350	−	0.277350i
$118$		0			0
$119$	−6.00000			−0.550019
$120$		0			0
$121$	5.00000			0.454545
$122$		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	−0.893246	−	0.449568i	$-0.851578\pi$
$127$	−5.00000	+	5.00000i	−0.443678	+	0.443678i	0.449568	+	0.893246i	$0.351578\pi$
$128$		0			0
$129$			6.00000i			0.528271i
$130$		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
$133$	−6.00000	−	6.00000i	−0.520266	−	0.520266i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	3.00000	+	3.00000i	0.256307	+	0.256307i	0.823550	−	0.567243i	$-0.191990\pi$
$137$	3.00000	+	3.00000i	0.256307	+	0.256307i	−0.567243	+	0.823550i	$0.691990\pi$
$138$		0			0
$139$			6.00000i			0.508913i	0.967084	+	0.254457i	$0.0818966\pi$
$139$			6.00000i			0.508913i	−0.967084	+	0.254457i	$0.918103\pi$
$140$		0			0
$141$			18.0000i			1.51587i
$142$		0			0
$143$	12.0000	+	12.0000i	1.00349	+	1.00349i
$144$		0			0
$145$		0			0
$146$		0			0
$147$	5.00000	+	5.00000i	0.412393	+	0.412393i
$148$		0			0
$149$	2.00000			0.163846			0.0819232	−	0.996639i	$-0.473894\pi$
$149$	2.00000			0.163846			0.0819232	+	0.996639i	$0.473894\pi$
$150$		0			0
$151$		−	10.0000i		−	0.813788i	−0.913475	−	0.406894i	$-0.866612\pi$
$151$		−	10.0000i		−	0.813788i	0.913475	−	0.406894i	$-0.133388\pi$
$152$		0			0
$153$	−3.00000	+	3.00000i	−0.242536	+	0.242536i
$154$		0			0
$155$		0			0
$156$		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
$159$	10.0000			0.793052
$160$		0			0
$161$	6.00000			0.472866
$162$		0			0
$163$	9.00000	−	9.00000i	0.704934	−	0.704934i	−0.260531	−	0.965465i	$-0.583898\pi$
$163$	9.00000	−	9.00000i	0.704934	−	0.704934i	0.965465	+	0.260531i	$0.0838976\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	3.00000	−	3.00000i	0.232147	−	0.232147i	−0.581441	−	0.813588i	$-0.697511\pi$
$167$	3.00000	−	3.00000i	0.232147	−	0.232147i	0.813588	+	0.581441i	$0.197511\pi$
$168$		0			0
$169$			5.00000i			0.384615i
$170$		0			0
$171$	−6.00000			−0.458831
$172$		0			0
$173$	−15.0000	−	15.0000i	−1.14043	−	1.14043i	−0.988372	−	0.152057i	$-0.951410\pi$
$173$	−15.0000	−	15.0000i	−1.14043	−	1.14043i	−0.152057	−	0.988372i	$-0.548590\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	−10.0000	−	10.0000i	−0.751646	−	0.751646i
$178$		0			0
$179$		−	2.00000i		−	0.149487i	−0.997203	−	0.0747435i	$-0.976186\pi$
$179$		−	2.00000i		−	0.149487i	0.997203	−	0.0747435i	$-0.0238138\pi$
$180$		0			0
$181$		0			0		1.00000			$0$
$181$		0			0		−1.00000			$\pi$
$182$		0			0
$183$	−12.0000	−	12.0000i	−0.887066	−	0.887066i
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−12.0000	−	12.0000i	−0.877527	−	0.877527i
$188$		0			0
$189$	−8.00000			−0.581914
$190$		0			0
$191$			6.00000i			0.434145i	0.976156	+	0.217072i	$0.0696508\pi$
$191$			6.00000i			0.434145i	−0.976156	+	0.217072i	$0.930349\pi$
$192$		0			0
$193$	−5.00000	+	5.00000i	−0.359908	+	0.359908i	−0.863779	−	0.503871i	$-0.831909\pi$
$193$	−5.00000	+	5.00000i	−0.359908	+	0.359908i	0.503871	+	0.863779i	$0.331909\pi$
$194$		0			0
$195$		0			0
$196$		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
$199$	−16.0000			−1.13421			−0.567105	−	0.823646i	$-0.691937\pi$
$199$	−16.0000			−1.13421			−0.567105	+	0.823646i	$0.691937\pi$
$200$		0			0
$201$	−18.0000			−1.26962
$202$		0			0
$203$	2.00000	−	2.00000i	0.140372	−	0.140372i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	3.00000	−	3.00000i	0.208514	−	0.208514i
$208$		0			0
$209$		−	24.0000i		−	1.66011i
$210$		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
$213$	6.00000	+	6.00000i	0.411113	+	0.411113i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−6.00000	−	6.00000i	−0.407307	−	0.407307i
$218$		0			0
$219$			10.0000i			0.675737i
$220$		0			0
$221$		−	18.0000i		−	1.21081i
$222$		0			0
$223$	13.0000	+	13.0000i	0.870544	+	0.870544i	0.992532	−	0.121987i	$-0.0389267\pi$
$223$	13.0000	+	13.0000i	0.870544	+	0.870544i	−0.121987	+	0.992532i	$0.538927\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	11.0000	+	11.0000i	0.730096	+	0.730096i	0.970639	−	0.240543i	$-0.0773255\pi$
$227$	11.0000	+	11.0000i	0.730096	+	0.730096i	−0.240543	+	0.970639i	$0.577325\pi$
$228$		0			0
$229$	6.00000			0.396491			0.198246	−	0.980152i	$-0.436476\pi$
$229$	6.00000			0.396491			0.198246	+	0.980152i	$0.436476\pi$
$230$		0			0
$231$		−	8.00000i		−	0.526361i
$232$		0			0
$233$	15.0000	−	15.0000i	0.982683	−	0.982683i	−0.0171699	−	0.999853i	$-0.505466\pi$
$233$	15.0000	−	15.0000i	0.982683	−	0.982683i	0.999853	+	0.0171699i	$0.00546562\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$	24.0000			1.55243			0.776215	−	0.630468i	$-0.217137\pi$
$239$	24.0000			1.55243			0.776215	+	0.630468i	$0.217137\pi$
$240$		0			0
$241$	−18.0000			−1.15948			−0.579741	−	0.814801i	$-0.696846\pi$
$241$	−18.0000			−1.15948			−0.579741	+	0.814801i	$0.696846\pi$
$242$		0			0
$243$	−7.00000	+	7.00000i	−0.449050	+	0.449050i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	18.0000	−	18.0000i	1.14531	−	1.14531i
$248$		0			0
$249$			6.00000i			0.380235i
$250$		0			0
$251$	4.00000			0.252478			0.126239	−	0.992000i	$-0.459709\pi$
$251$	4.00000			0.252478			0.126239	+	0.992000i	$0.459709\pi$
$252$		0			0
$253$	12.0000	+	12.0000i	0.754434	+	0.754434i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	3.00000	+	3.00000i	0.187135	+	0.187135i	0.794456	−	0.607321i	$-0.207756\pi$
$257$	3.00000	+	3.00000i	0.187135	+	0.187135i	−0.607321	+	0.794456i	$0.707756\pi$
$258$		0			0
$259$		−	6.00000i		−	0.372822i
$260$		0			0
$261$		−	2.00000i		−	0.123797i
$262$		0			0
$263$	9.00000	+	9.00000i	0.554964	+	0.554964i	0.927869	−	0.372906i	$-0.121638\pi$
$263$	9.00000	+	9.00000i	0.554964	+	0.554964i	−0.372906	+	0.927869i	$0.621638\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−14.0000			−0.853595			−0.426798	−	0.904347i	$-0.640358\pi$
$269$	−14.0000			−0.853595			−0.426798	+	0.904347i	$0.640358\pi$
$270$		0			0
$271$		−	2.00000i		−	0.121491i	−0.998153	−	0.0607457i	$-0.980652\pi$
$271$		−	2.00000i		−	0.121491i	0.998153	−	0.0607457i	$-0.0193479\pi$
$272$		0			0
$273$	6.00000	−	6.00000i	0.363137	−	0.363137i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−15.0000	+	15.0000i	−0.901263	+	0.901263i	−0.995545	−	0.0942828i	$-0.969944\pi$
$277$	−15.0000	+	15.0000i	−0.901263	+	0.901263i	0.0942828	+	0.995545i	$0.469944\pi$
$278$		0			0
$279$	−6.00000			−0.359211
$280$		0			0
$281$	6.00000			0.357930			0.178965	−	0.983855i	$-0.442725\pi$
$281$	6.00000			0.357930			0.178965	+	0.983855i	$0.442725\pi$
$282$		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
$289$			1.00000i			0.0588235i
$290$		0			0
$291$	14.0000			0.820695
$292$		0			0
$293$	1.00000	+	1.00000i	0.0584206	+	0.0584206i	0.735714	−	0.677293i	$-0.236847\pi$
$293$	1.00000	+	1.00000i	0.0584206	+	0.0584206i	−0.677293	+	0.735714i	$0.736847\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	−16.0000	−	16.0000i	−0.928414	−	0.928414i
$298$		0			0
$299$			18.0000i			1.04097i
$300$		0			0
$301$		−	6.00000i		−	0.345834i
$302$		0			0
$303$	8.00000	+	8.00000i	0.459588	+	0.459588i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	15.0000	+	15.0000i	0.856095	+	0.856095i	0.990876	−	0.134780i	$-0.0430329\pi$
$307$	15.0000	+	15.0000i	0.856095	+	0.856095i	−0.134780	+	0.990876i	$0.543033\pi$
$308$		0			0
$309$	−22.0000			−1.25154
$310$		0			0
$311$		−	18.0000i		−	1.02069i	−0.859971	−	0.510343i	$-0.829518\pi$
$311$		−	18.0000i		−	1.02069i	0.859971	−	0.510343i	$-0.170482\pi$
$312$		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
$315$		0			0
$316$		0			0
$317$	−3.00000	+	3.00000i	−0.168497	+	0.168497i	−0.786318	−	0.617822i	$-0.788015\pi$
$317$	−3.00000	+	3.00000i	−0.168497	+	0.168497i	0.617822	+	0.786318i	$0.288015\pi$
$318$		0			0
$319$	8.00000			0.447914
$320$		0			0
$321$	6.00000			0.334887
$322$		0			0
$323$	−18.0000	+	18.0000i	−1.00155	+	1.00155i
$324$		0			0
$325$		0			0
$326$		0			0
$327$	18.0000	−	18.0000i	0.995402	−	0.995402i
$328$		0			0
$329$		−	18.0000i		−	0.992372i
$330$		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
$333$	−3.00000	−	3.00000i	−0.164399	−	0.164399i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−25.0000	−	25.0000i	−1.36184	−	1.36184i	−0.871576	−	0.490261i	$-0.836901\pi$
$337$	−25.0000	−	25.0000i	−1.36184	−	1.36184i	−0.490261	−	0.871576i	$-0.663099\pi$
$338$		0			0
$339$		−	6.00000i		−	0.325875i
$340$		0			0
$341$		−	24.0000i		−	1.29967i
$342$		0			0
$343$	−12.0000	−	12.0000i	−0.647939	−	0.647939i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−17.0000	−	17.0000i	−0.912608	−	0.912608i	0.0838690	−	0.996477i	$-0.473272\pi$
$347$	−17.0000	−	17.0000i	−0.912608	−	0.912608i	−0.996477	+	0.0838690i	$0.973272\pi$
$348$		0			0
$349$	−18.0000			−0.963518			−0.481759	−	0.876304i	$-0.660002\pi$
$349$	−18.0000			−0.963518			−0.481759	+	0.876304i	$0.660002\pi$
$350$		0			0
$351$		−	24.0000i		−	1.28103i
$352$		0			0
$353$	15.0000	−	15.0000i	0.798369	−	0.798369i	−0.184469	−	0.982838i	$-0.559057\pi$
$353$	15.0000	−	15.0000i	0.798369	−	0.798369i	0.982838	+	0.184469i	$0.0590565\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	−6.00000	+	6.00000i	−0.317554	+	0.317554i
$358$		0			0
$359$	24.0000			1.26667			0.633336	−	0.773877i	$-0.281685\pi$
$359$	24.0000			1.26667			0.633336	+	0.773877i	$0.281685\pi$
$360$		0			0
$361$	−17.0000			−0.894737
$362$		0			0
$363$	5.00000	−	5.00000i	0.262432	−	0.262432i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	11.0000	−	11.0000i	0.574195	−	0.574195i	−0.359103	−	0.933298i	$-0.616917\pi$
$367$	11.0000	−	11.0000i	0.574195	−	0.574195i	0.933298	+	0.359103i	$0.116917\pi$
$368$		0			0
$369$			6.00000i			0.312348i
$370$		0			0
$371$	−10.0000			−0.519174
$372$		0			0
$373$	−3.00000	−	3.00000i	−0.155334	−	0.155334i	0.625161	−	0.780496i	$-0.285033\pi$
$373$	−3.00000	−	3.00000i	−0.155334	−	0.155334i	−0.780496	+	0.625161i	$0.785033\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	6.00000	+	6.00000i	0.309016	+	0.309016i
$378$		0			0
$379$		−	18.0000i		−	0.924598i	−0.886724	−	0.462299i	$-0.847025\pi$
$379$		−	18.0000i		−	0.924598i	0.886724	−	0.462299i	$-0.152975\pi$
$380$		0			0
$381$			10.0000i			0.512316i
$382$		0			0
$383$	21.0000	+	21.0000i	1.07305	+	1.07305i	0.997113	+	0.0759373i	$0.0241949\pi$
$383$	21.0000	+	21.0000i	1.07305	+	1.07305i	0.0759373	+	0.997113i	$0.475805\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−3.00000	−	3.00000i	−0.152499	−	0.152499i
$388$		0			0
$389$	26.0000			1.31825			0.659126	−	0.752032i	$-0.270926\pi$
$389$	26.0000			1.31825			0.659126	+	0.752032i	$0.270926\pi$
$390$		0			0
$391$		−	18.0000i		−	0.910299i
$392$		0			0
$393$	−4.00000	+	4.00000i	−0.201773	+	0.201773i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	9.00000	−	9.00000i	0.451697	−	0.451697i	−0.444220	−	0.895918i	$-0.646519\pi$
$397$	9.00000	−	9.00000i	0.451697	−	0.451697i	0.895918	+	0.444220i	$0.146519\pi$
$398$		0			0
$399$	−12.0000			−0.600751
$400$		0			0
$401$	−30.0000			−1.49813			−0.749064	−	0.662497i	$-0.769497\pi$
$401$	−30.0000			−1.49813			−0.749064	+	0.662497i	$0.769497\pi$
$402$		0			0
$403$	18.0000	−	18.0000i	0.896644	−	0.896644i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	12.0000	−	12.0000i	0.594818	−	0.594818i
$408$		0			0
$409$		−	12.0000i		−	0.593362i	−0.954977	−	0.296681i	$-0.904120\pi$
$409$		−	12.0000i		−	0.593362i	0.954977	−	0.296681i	$-0.0958798\pi$
$410$		0			0
$411$	6.00000			0.295958
$412$		0			0
$413$	10.0000	+	10.0000i	0.492068	+	0.492068i
$414$		0			0
$415$		0			0
$416$		0			0
$417$	6.00000	+	6.00000i	0.293821	+	0.293821i
$418$		0			0
$419$		−	10.0000i		−	0.488532i	−0.969708	−	0.244266i	$-0.921453\pi$
$419$		−	10.0000i		−	0.488532i	0.969708	−	0.244266i	$-0.0785470\pi$
$420$		0			0
$421$			12.0000i			0.584844i	0.956289	+	0.292422i	$0.0944612\pi$
$421$			12.0000i			0.584844i	−0.956289	+	0.292422i	$0.905539\pi$
$422$		0			0
$423$	−9.00000	−	9.00000i	−0.437595	−	0.437595i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	12.0000	+	12.0000i	0.580721	+	0.580721i
$428$		0			0
$429$	24.0000			1.15873
$430$		0			0
$431$			30.0000i			1.44505i	0.691345	+	0.722525i	$0.257018\pi$
$431$			30.0000i			1.44505i	−0.691345	+	0.722525i	$0.742982\pi$
$432$		0			0
$433$	7.00000	−	7.00000i	0.336399	−	0.336399i	−0.518611	−	0.855010i	$-0.673551\pi$
$433$	7.00000	−	7.00000i	0.336399	−	0.336399i	0.855010	+	0.518611i	$0.173551\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	18.0000	−	18.0000i	0.861057	−	0.861057i
$438$		0			0
$439$	−8.00000			−0.381819			−0.190910	−	0.981608i	$-0.561144\pi$
$439$	−8.00000			−0.381819			−0.190910	+	0.981608i	$0.561144\pi$
$440$		0			0
$441$	−5.00000			−0.238095
$442$		0			0
$443$	25.0000	−	25.0000i	1.18779	−	1.18779i	0.210108	−	0.977678i	$-0.432619\pi$
$443$	25.0000	−	25.0000i	1.18779	−	1.18779i	0.977678	−	0.210108i	$-0.0673814\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	2.00000	−	2.00000i	0.0945968	−	0.0945968i
$448$		0			0
$449$			12.0000i			0.566315i	0.959073	+	0.283158i	$0.0913819\pi$
$449$			12.0000i			0.566315i	−0.959073	+	0.283158i	$0.908618\pi$
$450$		0			0
$451$	−24.0000			−1.13012
$452$		0			0
$453$	−10.0000	−	10.0000i	−0.469841	−	0.469841i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−17.0000	−	17.0000i	−0.795226	−	0.795226i	0.187112	−	0.982339i	$-0.440087\pi$
$457$	−17.0000	−	17.0000i	−0.795226	−	0.795226i	−0.982339	+	0.187112i	$0.940087\pi$
$458$		0			0
$459$			24.0000i			1.12022i
$460$		0			0
$461$			40.0000i			1.86299i	0.363760	+	0.931493i	$0.381493\pi$
$461$			40.0000i			1.86299i	−0.363760	+	0.931493i	$0.618507\pi$
$462$		0			0
$463$	1.00000	+	1.00000i	0.0464739	+	0.0464739i	0.729962	−	0.683488i	$-0.239538\pi$
$463$	1.00000	+	1.00000i	0.0464739	+	0.0464739i	−0.683488	+	0.729962i	$0.739538\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−21.0000	−	21.0000i	−0.971764	−	0.971764i	0.0278481	−	0.999612i	$-0.491135\pi$
$467$	−21.0000	−	21.0000i	−0.971764	−	0.971764i	−0.999612	+	0.0278481i	$0.991135\pi$
$468$		0			0
$469$	18.0000			0.831163
$470$		0			0
$471$			6.00000i			0.276465i
$472$		0			0
$473$	12.0000	−	12.0000i	0.551761	−	0.551761i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−5.00000	+	5.00000i	−0.228934	+	0.228934i
$478$		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
$483$	6.00000	−	6.00000i	0.273009	−	0.273009i
$484$		0			0
$485$		0			0
$486$		0			0
$487$	−1.00000	+	1.00000i	−0.0453143	+	0.0453143i	−0.729401	−	0.684087i	$-0.760201\pi$
$487$	−1.00000	+	1.00000i	−0.0453143	+	0.0453143i	0.684087	+	0.729401i	$0.260201\pi$
$488$		0			0
$489$		−	18.0000i		−	0.813988i
$490$		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
$495$		0			0
$496$		0			0
$497$	−6.00000	−	6.00000i	−0.269137	−	0.269137i
$498$		0			0
$499$			6.00000i			0.268597i	0.990941	+	0.134298i	$0.0428781\pi$
$499$			6.00000i			0.268597i	−0.990941	+	0.134298i	$0.957122\pi$
$500$		0			0
$501$		−	6.00000i		−	0.268060i
$502$		0			0
$503$	21.0000	+	21.0000i	0.936344	+	0.936344i	0.998092	−	0.0617480i	$-0.0196675\pi$
$503$	21.0000	+	21.0000i	0.936344	+	0.936344i	−0.0617480	+	0.998092i	$0.519668\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	5.00000	+	5.00000i	0.222058	+	0.222058i
$508$		0			0
$509$	14.0000			0.620539			0.310270	−	0.950649i	$-0.399581\pi$
$509$	14.0000			0.620539			0.310270	+	0.950649i	$0.399581\pi$
$510$		0			0
$511$		−	10.0000i		−	0.442374i
$512$		0			0
$513$	−24.0000	+	24.0000i	−1.05963	+	1.05963i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	36.0000	−	36.0000i	1.58328	−	1.58328i
$518$		0			0
$519$	−30.0000			−1.31685
$520$		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
$523$	−15.0000	+	15.0000i	−0.655904	+	0.655904i	−0.954408	−	0.298504i	$-0.903512\pi$
$523$	−15.0000	+	15.0000i	−0.655904	+	0.655904i	0.298504	+	0.954408i	$0.403512\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−18.0000	+	18.0000i	−0.784092	+	0.784092i
$528$		0			0
$529$		−	5.00000i		−	0.217391i
$530$		0			0
$531$	10.0000			0.433963
$532$		0			0
$533$	−18.0000	−	18.0000i	−0.779667	−	0.779667i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	−2.00000	−	2.00000i	−0.0863064	−	0.0863064i
$538$		0			0
$539$		−	20.0000i		−	0.861461i
$540$		0			0
$541$			24.0000i			1.03184i	0.856637	+	0.515920i	$0.172550\pi$
$541$			24.0000i			1.03184i	−0.856637	+	0.515920i	$0.827450\pi$
$542$		0			0
$543$		0			0
$544$		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
$549$	12.0000			0.512148
$550$		0			0
$551$		−	12.0000i		−	0.511217i
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−31.0000	+	31.0000i	−1.31351	+	1.31351i	−0.394704	+	0.918808i	$0.629153\pi$
$557$	−31.0000	+	31.0000i	−1.31351	+	1.31351i	−0.918808	+	0.394704i	$0.870847\pi$
$558$		0			0
$559$	18.0000			0.761319
$560$		0			0
$561$	−24.0000			−1.01328
$562$		0			0
$563$	1.00000	−	1.00000i	0.0421450	−	0.0421450i	−0.685720	−	0.727865i	$-0.740513\pi$
$563$	1.00000	−	1.00000i	0.0421450	−	0.0421450i	0.727865	+	0.685720i	$0.240513\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−5.00000	+	5.00000i	−0.209980	+	0.209980i
$568$		0			0
$569$			12.0000i			0.503066i	0.967849	+	0.251533i	$0.0809347\pi$
$569$			12.0000i			0.503066i	−0.967849	+	0.251533i	$0.919065\pi$
$570$		0			0
$571$	12.0000			0.502184			0.251092	−	0.967963i	$-0.419210\pi$
$571$	12.0000			0.502184			0.251092	+	0.967963i	$0.419210\pi$
$572$		0			0
$573$	6.00000	+	6.00000i	0.250654	+	0.250654i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	19.0000	+	19.0000i	0.790980	+	0.790980i	0.981654	−	0.190673i	$-0.0610671\pi$
$577$	19.0000	+	19.0000i	0.790980	+	0.790980i	−0.190673	+	0.981654i	$0.561067\pi$
$578$		0			0
$579$			10.0000i			0.415586i
$580$		0			0
$581$		−	6.00000i		−	0.248922i
$582$		0			0
$583$	−20.0000	−	20.0000i	−0.828315	−	0.828315i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	15.0000	+	15.0000i	0.619116	+	0.619116i	0.945305	−	0.326188i	$-0.105764\pi$
$587$	15.0000	+	15.0000i	0.619116	+	0.619116i	−0.326188	+	0.945305i	$0.605764\pi$
$588$		0			0
$589$	−36.0000			−1.48335
$590$		0			0
$591$		−	10.0000i		−	0.411345i
$592$		0			0
$593$	15.0000	−	15.0000i	0.615976	−	0.615976i	−0.328521	−	0.944497i	$-0.606550\pi$
$593$	15.0000	−	15.0000i	0.615976	−	0.615976i	0.944497	+	0.328521i	$0.106550\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−16.0000	+	16.0000i	−0.654836	+	0.654836i
$598$		0			0
$599$		0			0			−	1.00000i	$-0.5\pi$
$599$		0			0				1.00000i	$0.5\pi$
$600$		0			0
$601$	−10.0000			−0.407909			−0.203954	−	0.978980i	$-0.565379\pi$
$601$	−10.0000			−0.407909			−0.203954	+	0.978980i	$0.565379\pi$
$602$		0			0
$603$	9.00000	−	9.00000i	0.366508	−	0.366508i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	7.00000	−	7.00000i	0.284121	−	0.284121i	−0.550629	−	0.834750i	$-0.685612\pi$
$607$	7.00000	−	7.00000i	0.284121	−	0.284121i	0.834750	+	0.550629i	$0.185612\pi$
$608$		0			0
$609$		−	4.00000i		−	0.162088i
$610$		0			0
$611$	54.0000			2.18461
$612$		0			0
$613$	9.00000	+	9.00000i	0.363507	+	0.363507i	0.865102	−	0.501596i	$-0.167253\pi$
$613$	9.00000	+	9.00000i	0.363507	+	0.363507i	−0.501596	+	0.865102i	$0.667253\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	15.0000	+	15.0000i	0.603877	+	0.603877i	0.941339	−	0.337462i	$-0.109568\pi$
$617$	15.0000	+	15.0000i	0.603877	+	0.603877i	−0.337462	+	0.941339i	$0.609568\pi$
$618$		0			0
$619$		−	42.0000i		−	1.68812i	−0.536247	−	0.844061i	$-0.680158\pi$
$619$		−	42.0000i		−	1.68812i	0.536247	−	0.844061i	$-0.319842\pi$
$620$		0			0
$621$		−	24.0000i		−	0.963087i
$622$		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
$629$	−18.0000			−0.717707
$630$		0			0
$631$			30.0000i			1.19428i	0.802137	+	0.597141i	$0.203697\pi$
$631$			30.0000i			1.19428i	−0.802137	+	0.597141i	$0.796303\pi$
$632$		0			0
$633$	−12.0000	+	12.0000i	−0.476957	+	0.476957i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	15.0000	−	15.0000i	0.594322	−	0.594322i
$638$		0			0
$639$	−6.00000			−0.237356
$640$		0			0
$641$	30.0000			1.18493			0.592464	−	0.805597i	$-0.298155\pi$
$641$	30.0000			1.18493			0.592464	+	0.805597i	$0.298155\pi$
$642$		0			0
$643$	−3.00000	+	3.00000i	−0.118308	+	0.118308i	−0.763782	−	0.645474i	$-0.776660\pi$
$643$	−3.00000	+	3.00000i	−0.118308	+	0.118308i	0.645474	+	0.763782i	$0.276660\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−9.00000	+	9.00000i	−0.353827	+	0.353827i	−0.861531	−	0.507705i	$-0.830494\pi$
$647$	−9.00000	+	9.00000i	−0.353827	+	0.353827i	0.507705	+	0.861531i	$0.330494\pi$
$648$		0			0
$649$			40.0000i			1.57014i
$650$		0			0
$651$	−12.0000			−0.470317
$652$		0			0
$653$	33.0000	+	33.0000i	1.29139	+	1.29139i	0.933928	+	0.357462i	$0.116358\pi$
$653$	33.0000	+	33.0000i	1.29139	+	1.29139i	0.357462	+	0.933928i	$0.383642\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−5.00000	−	5.00000i	−0.195069	−	0.195069i
$658$		0			0
$659$			22.0000i			0.856998i	0.903542	+	0.428499i	$0.140958\pi$
$659$			22.0000i			0.856998i	−0.903542	+	0.428499i	$0.859042\pi$
$660$		0			0
$661$		−	12.0000i		−	0.466746i	−0.972387	−	0.233373i	$-0.925024\pi$
$661$		−	12.0000i		−	0.466746i	0.972387	−	0.233373i	$-0.0749763\pi$
$662$		0			0
$663$	−18.0000	−	18.0000i	−0.699062	−	0.699062i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	6.00000	+	6.00000i	0.232321	+	0.232321i
$668$		0			0
$669$	26.0000			1.00522
$670$		0			0
$671$			48.0000i			1.85302i
$672$		0			0
$673$	−25.0000	+	25.0000i	−0.963679	+	0.963679i	−0.999363	−	0.0356839i	$-0.988639\pi$
$673$	−25.0000	+	25.0000i	−0.963679	+	0.963679i	0.0356839	+	0.999363i	$0.488639\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−27.0000	+	27.0000i	−1.03769	+	1.03769i	−0.0384331	+	0.999261i	$0.512237\pi$
$677$	−27.0000	+	27.0000i	−1.03769	+	1.03769i	−0.999261	+	0.0384331i	$0.987763\pi$
$678$		0			0
$679$	−14.0000			−0.537271
$680$		0			0
$681$	22.0000			0.843042
$682$		0			0
$683$	−27.0000	+	27.0000i	−1.03313	+	1.03313i	−0.0336941	+	0.999432i	$0.510727\pi$
$683$	−27.0000	+	27.0000i	−1.03313	+	1.03313i	−0.999432	+	0.0336941i	$0.989273\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	6.00000	−	6.00000i	0.228914	−	0.228914i
$688$		0			0
$689$		−	30.0000i		−	1.14291i
$690$		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
$693$	4.00000	+	4.00000i	0.151947	+	0.151947i
$694$		0			0
$695$		0			0
$696$		0			0
$697$	18.0000	+	18.0000i	0.681799	+	0.681799i
$698$		0			0
$699$		−	30.0000i		−	1.13470i
$700$		0			0
$701$		−	20.0000i		−	0.755390i	−0.925930	−	0.377695i	$-0.876717\pi$
$701$		−	20.0000i		−	0.755390i	0.925930	−	0.377695i	$-0.123283\pi$
$702$		0			0
$703$	−18.0000	−	18.0000i	−0.678883	−	0.678883i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−8.00000	−	8.00000i	−0.300871	−	0.300871i
$708$		0			0
$709$	−42.0000			−1.57734			−0.788672	−	0.614815i	$-0.789231\pi$
$709$	−42.0000			−1.57734			−0.788672	+	0.614815i	$0.789231\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	18.0000	−	18.0000i	0.674105	−	0.674105i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	24.0000	−	24.0000i	0.896296	−	0.896296i
$718$		0			0
$719$	−24.0000			−0.895049			−0.447524	−	0.894272i	$-0.647694\pi$
$719$	−24.0000			−0.895049			−0.447524	+	0.894272i	$0.647694\pi$
$720$		0			0
$721$	22.0000			0.819323
$722$		0			0
$723$	−18.0000	+	18.0000i	−0.669427	+	0.669427i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−25.0000	+	25.0000i	−0.927199	+	0.927199i	−0.997524	−	0.0703254i	$-0.977596\pi$
$727$	−25.0000	+	25.0000i	−0.927199	+	0.927199i	0.0703254	+	0.997524i	$0.477596\pi$
$728$		0			0
$729$			29.0000i			1.07407i
$730$		0			0
$731$	−18.0000			−0.665754
$732$		0			0
$733$	−15.0000	−	15.0000i	−0.554038	−	0.554038i	0.373566	−	0.927604i	$-0.378135\pi$
$733$	−15.0000	−	15.0000i	−0.554038	−	0.554038i	−0.927604	+	0.373566i	$0.878135\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	36.0000	+	36.0000i	1.32608	+	1.32608i
$738$		0			0
$739$			30.0000i			1.10357i	0.833987	+	0.551784i	$0.186053\pi$
$739$			30.0000i			1.10357i	−0.833987	+	0.551784i	$0.813947\pi$
$740$		0			0
$741$		−	36.0000i		−	1.32249i
$742$		0			0
$743$	9.00000	+	9.00000i	0.330178	+	0.330178i	0.852654	−	0.522476i	$-0.174992\pi$
$743$	9.00000	+	9.00000i	0.330178	+	0.330178i	−0.522476	+	0.852654i	$0.674992\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	−3.00000	−	3.00000i	−0.109764	−	0.109764i
$748$		0			0
$749$	−6.00000			−0.219235
$750$		0			0
$751$		−	42.0000i		−	1.53260i	−0.642482	−	0.766301i	$-0.722095\pi$
$751$		−	42.0000i		−	1.53260i	0.642482	−	0.766301i	$-0.277905\pi$
$752$		0			0
$753$	4.00000	−	4.00000i	0.145768	−	0.145768i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	9.00000	−	9.00000i	0.327111	−	0.327111i	−0.524376	−	0.851487i	$-0.675701\pi$
$757$	9.00000	−	9.00000i	0.327111	−	0.327111i	0.851487	+	0.524376i	$0.175701\pi$
$758$		0			0
$759$	24.0000			0.871145
$760$		0			0
$761$	18.0000			0.652499			0.326250	−	0.945284i	$-0.394215\pi$
$761$	18.0000			0.652499			0.326250	+	0.945284i	$0.394215\pi$
$762$		0			0
$763$	−18.0000	+	18.0000i	−0.651644	+	0.651644i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−30.0000	+	30.0000i	−1.08324	+	1.08324i
$768$		0			0
$769$			40.0000i			1.44244i	0.692708	+	0.721218i	$0.256418\pi$
$769$			40.0000i			1.44244i	−0.692708	+	0.721218i	$0.743582\pi$
$770$		0			0
$771$	6.00000			0.216085
$772$		0			0
$773$	9.00000	+	9.00000i	0.323708	+	0.323708i	0.850188	−	0.526480i	$-0.176489\pi$
$773$	9.00000	+	9.00000i	0.323708	+	0.323708i	−0.526480	+	0.850188i	$0.676489\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	−6.00000	−	6.00000i	−0.215249	−	0.215249i
$778$		0			0
$779$			36.0000i			1.28983i
$780$		0			0
$781$		−	24.0000i		−	0.858788i
$782$		0			0
$783$	−8.00000	−	8.00000i	−0.285897	−	0.285897i
$784$		0			0
$785$		0			0
$786$		0			0
$787$	15.0000	+	15.0000i	0.534692	+	0.534692i	0.921965	−	0.387273i	$-0.126583\pi$
$787$	15.0000	+	15.0000i	0.534692	+	0.534692i	−0.387273	+	0.921965i	$0.626583\pi$
$788$		0			0
$789$	18.0000			0.640817
$790$		0			0
$791$			6.00000i			0.213335i
$792$		0			0
$793$	−36.0000	+	36.0000i	−1.27840	+	1.27840i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−3.00000	+	3.00000i	−0.106265	+	0.106265i	−0.758240	−	0.651975i	$-0.773941\pi$
$797$	−3.00000	+	3.00000i	−0.106265	+	0.106265i	0.651975	+	0.758240i	$0.273941\pi$
$798$		0			0
$799$	−54.0000			−1.91038
$800$		0			0
$801$		0			0
$802$		0			0
$803$	20.0000	−	20.0000i	0.705785	−	0.705785i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	−14.0000	+	14.0000i	−0.492823	+	0.492823i
$808$		0			0
$809$		−	48.0000i		−	1.68759i	−0.536666	−	0.843795i	$-0.680316\pi$
$809$		−	48.0000i		−	1.68759i	0.536666	−	0.843795i	$-0.319684\pi$
$810$		0			0
$811$	−12.0000			−0.421377			−0.210688	−	0.977553i	$-0.567571\pi$
$811$	−12.0000			−0.421377			−0.210688	+	0.977553i	$0.567571\pi$
$812$		0			0
$813$	−2.00000	−	2.00000i	−0.0701431	−	0.0701431i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−18.0000	−	18.0000i	−0.629740	−	0.629740i
$818$		0			0
$819$			6.00000i			0.209657i
$820$		0			0
$821$			52.0000i			1.81481i	0.420255	+	0.907406i	$0.361941\pi$
$821$			52.0000i			1.81481i	−0.420255	+	0.907406i	$0.638059\pi$
$822$		0			0
$823$	5.00000	+	5.00000i	0.174289	+	0.174289i	0.788861	−	0.614572i	$-0.210671\pi$
$823$	5.00000	+	5.00000i	0.174289	+	0.174289i	−0.614572	+	0.788861i	$0.710671\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	23.0000	+	23.0000i	0.799788	+	0.799788i	0.983062	−	0.183274i	$-0.0586694\pi$
$827$	23.0000	+	23.0000i	0.799788	+	0.799788i	−0.183274	+	0.983062i	$0.558669\pi$
$828$		0			0
$829$	−30.0000			−1.04194			−0.520972	−	0.853574i	$-0.674430\pi$
$829$	−30.0000			−1.04194			−0.520972	+	0.853574i	$0.674430\pi$
$830$		0			0
$831$			30.0000i			1.04069i
$832$		0			0
$833$	−15.0000	+	15.0000i	−0.519719	+	0.519719i
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−24.0000	+	24.0000i	−0.829561	+	0.829561i
$838$		0			0
$839$	48.0000			1.65714			0.828572	−	0.559883i	$-0.189154\pi$
$839$	48.0000			1.65714			0.828572	+	0.559883i	$0.189154\pi$
$840$		0			0
$841$	−25.0000			−0.862069
$842$		0			0
$843$	6.00000	−	6.00000i	0.206651	−	0.206651i
$844$		0			0
$845$		0			0
$846$		0			0
$847$	−5.00000	+	5.00000i	−0.171802	+	0.171802i
$848$		0			0
$849$			30.0000i			1.02960i
$850$		0			0
$851$	18.0000			0.617032
$852$		0			0
$853$	9.00000	+	9.00000i	0.308154	+	0.308154i	0.844193	−	0.536039i	$-0.180080\pi$
$853$	9.00000	+	9.00000i	0.308154	+	0.308154i	−0.536039	+	0.844193i	$0.680080\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	27.0000	+	27.0000i	0.922302	+	0.922302i	0.997192	−	0.0748894i	$-0.0238604\pi$
$857$	27.0000	+	27.0000i	0.922302	+	0.922302i	−0.0748894	+	0.997192i	$0.523860\pi$
$858$		0			0
$859$			6.00000i			0.204717i	0.994748	+	0.102359i	$0.0326389\pi$
$859$			6.00000i			0.204717i	−0.994748	+	0.102359i	$0.967361\pi$
$860$		0			0
$861$			12.0000i			0.408959i
$862$		0			0
$863$	−15.0000	−	15.0000i	−0.510606	−	0.510606i	0.404106	−	0.914712i	$-0.367583\pi$
$863$	−15.0000	−	15.0000i	−0.510606	−	0.510606i	−0.914712	+	0.404106i	$0.867583\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	1.00000	+	1.00000i	0.0339618	+	0.0339618i
$868$		0			0
$869$		0			0
$870$		0			0
$871$			54.0000i			1.82972i
$872$		0			0
$873$	−7.00000	+	7.00000i	−0.236914	+	0.236914i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−27.0000	+	27.0000i	−0.911725	+	0.911725i	−0.996408	−	0.0846827i	$-0.973012\pi$
$877$	−27.0000	+	27.0000i	−0.911725	+	0.911725i	0.0846827	+	0.996408i	$0.473012\pi$
$878$		0			0
$879$	2.00000			0.0674583
$880$		0			0
$881$	30.0000			1.01073			0.505363	−	0.862907i	$-0.331359\pi$
$881$	30.0000			1.01073			0.505363	+	0.862907i	$0.331359\pi$
$882$		0			0
$883$	9.00000	−	9.00000i	0.302874	−	0.302874i	−0.539263	−	0.842137i	$-0.681297\pi$
$883$	9.00000	−	9.00000i	0.302874	−	0.302874i	0.842137	+	0.539263i	$0.181297\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−9.00000	+	9.00000i	−0.302190	+	0.302190i	−0.841870	−	0.539680i	$-0.818545\pi$
$887$	−9.00000	+	9.00000i	−0.302190	+	0.302190i	0.539680	+	0.841870i	$0.318545\pi$
$888$		0			0
$889$		−	10.0000i		−	0.335389i
$890$		0			0
$891$	−20.0000			−0.670025
$892$		0			0
$893$	−54.0000	−	54.0000i	−1.80704	−	1.80704i
$894$		0			0
$895$		0			0
$896$		0			0
$897$	18.0000	+	18.0000i	0.601003	+	0.601003i
$898$		0			0
$899$		−	12.0000i		−	0.400222i
$900$		0			0
$901$			30.0000i			0.999445i
$902$		0			0
$903$	−6.00000	−	6.00000i	−0.199667	−	0.199667i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	3.00000	+	3.00000i	0.0996134	+	0.0996134i	0.755157	−	0.655544i	$-0.227561\pi$
$907$	3.00000	+	3.00000i	0.0996134	+	0.0996134i	−0.655544	+	0.755157i	$0.727561\pi$
$908$		0			0
$909$	−8.00000			−0.265343
$910$		0			0
$911$			6.00000i			0.198789i	0.995048	+	0.0993944i	$0.0316906\pi$
$911$			6.00000i			0.198789i	−0.995048	+	0.0993944i	$0.968309\pi$
$912$		0			0
$913$	12.0000	−	12.0000i	0.397142	−	0.397142i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	4.00000	−	4.00000i	0.132092	−	0.132092i
$918$		0			0
$919$	−48.0000			−1.58337			−0.791687	−	0.610927i	$-0.790797\pi$
$919$	−48.0000			−1.58337			−0.791687	+	0.610927i	$0.790797\pi$
$920$		0			0
$921$	30.0000			0.988534
$922$		0			0
$923$	18.0000	−	18.0000i	0.592477	−	0.592477i
$924$		0			0
$925$		0			0
$926$		0			0
$927$	11.0000	−	11.0000i	0.361287	−	0.361287i
$928$		0			0
$929$		−	12.0000i		−	0.393707i	−0.980433	−	0.196854i	$-0.936928\pi$
$929$		−	12.0000i		−	0.393707i	0.980433	−	0.196854i	$-0.0630724\pi$
$930$		0			0
$931$	−30.0000			−0.983210
$932$		0			0
$933$	−18.0000	−	18.0000i	−0.589294	−	0.589294i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	23.0000	+	23.0000i	0.751377	+	0.751377i	0.974736	−	0.223359i	$-0.0717022\pi$
$937$	23.0000	+	23.0000i	0.751377	+	0.751377i	−0.223359	+	0.974736i	$0.571702\pi$
$938$		0			0
$939$			2.00000i			0.0652675i
$940$		0			0
$941$		−	32.0000i		−	1.04317i	−0.853199	−	0.521585i	$-0.825341\pi$
$941$		−	32.0000i		−	1.04317i	0.853199	−	0.521585i	$-0.174659\pi$
$942$		0			0
$943$	−18.0000	−	18.0000i	−0.586161	−	0.586161i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	7.00000	+	7.00000i	0.227469	+	0.227469i	0.811635	−	0.584165i	$-0.198578\pi$
$947$	7.00000	+	7.00000i	0.227469	+	0.227469i	−0.584165	+	0.811635i	$0.698578\pi$
$948$		0			0
$949$	30.0000			0.973841
$950$		0			0
$951$			6.00000i			0.194563i
$952$		0			0
$953$	−9.00000	+	9.00000i	−0.291539	+	0.291539i	−0.837688	−	0.546149i	$-0.816093\pi$
$953$	−9.00000	+	9.00000i	−0.291539	+	0.291539i	0.546149	+	0.837688i	$0.316093\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	8.00000	−	8.00000i	0.258603	−	0.258603i
$958$		0			0
$959$	−6.00000			−0.193750
$960$		0			0
$961$	−5.00000			−0.161290
$962$		0			0
$963$	−3.00000	+	3.00000i	−0.0966736	+	0.0966736i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−5.00000	+	5.00000i	−0.160789	+	0.160789i	−0.782916	−	0.622127i	$-0.786269\pi$
$967$	−5.00000	+	5.00000i	−0.160789	+	0.160789i	0.622127	+	0.782916i	$0.286269\pi$
$968$		0			0
$969$			36.0000i			1.15649i
$970$		0			0
$971$	20.0000			0.641831			0.320915	−	0.947108i	$-0.396010\pi$
$971$	20.0000			0.641831			0.320915	+	0.947108i	$0.396010\pi$
$972$		0			0
$973$	−6.00000	−	6.00000i	−0.192351	−	0.192351i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−9.00000	−	9.00000i	−0.287936	−	0.287936i	0.548328	−	0.836263i	$-0.315265\pi$
$977$	−9.00000	−	9.00000i	−0.287936	−	0.287936i	−0.836263	+	0.548328i	$0.815265\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$			18.0000i			0.574696i
$982$		0			0
$983$	33.0000	+	33.0000i	1.05254	+	1.05254i	0.998541	+	0.0539954i	$0.0171956\pi$
$983$	33.0000	+	33.0000i	1.05254	+	1.05254i	0.0539954	+	0.998541i	$0.482804\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	−18.0000	−	18.0000i	−0.572946	−	0.572946i
$988$		0			0
$989$	18.0000			0.572367
$990$		0			0
$991$		−	2.00000i		−	0.0635321i	−0.999495	−	0.0317660i	$-0.989887\pi$
$991$		−	2.00000i		−	0.0635321i	0.999495	−	0.0317660i	$-0.0101131\pi$
$992$		0			0
$993$	12.0000	−	12.0000i	0.380808	−	0.380808i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	9.00000	−	9.00000i	0.285033	−	0.285033i	−0.550079	−	0.835112i	$-0.685403\pi$
$997$	9.00000	−	9.00000i	0.285033	−	0.285033i	0.835112	+	0.550079i	$0.185403\pi$
$998$		0			0
$999$	−24.0000			−0.759326

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1600.2.o.c.607.1		2
4.3	odd	2		1600.2.o.b.607.1		2
5.2	odd	4		320.2.o.a.223.1	✓	2
5.3	odd	4		1600.2.o.d.543.1		2
5.4	even	2		320.2.o.b.287.1	yes	2
8.3	odd	2		1600.2.o.d.607.1		2
8.5	even	2		1600.2.o.a.607.1		2
20.3	even	4		1600.2.o.a.543.1		2
20.7	even	4		320.2.o.c.223.1	yes	2
20.19	odd	2		320.2.o.d.287.1	yes	2
40.3	even	4	inner	1600.2.o.c.543.1		2
40.13	odd	4		1600.2.o.b.543.1		2
40.19	odd	2		320.2.o.a.287.1	yes	2
40.27	even	4		320.2.o.b.223.1	yes	2
40.29	even	2		320.2.o.c.287.1	yes	2
40.37	odd	4		320.2.o.d.223.1	yes	2
80.19	odd	4		1280.2.n.j.767.1		2
80.27	even	4		1280.2.n.i.1023.1		2
80.29	even	4		1280.2.n.d.767.1		2
80.37	odd	4		1280.2.n.c.1023.1		2
80.59	odd	4		1280.2.n.c.767.1		2
80.67	even	4		1280.2.n.d.1023.1		2
80.69	even	4		1280.2.n.i.767.1		2
80.77	odd	4		1280.2.n.j.1023.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
320.2.o.a.223.1	✓	2	5.2	odd	4
320.2.o.a.287.1	yes	2	40.19	odd	2
320.2.o.b.223.1	yes	2	40.27	even	4
320.2.o.b.287.1	yes	2	5.4	even	2
320.2.o.c.223.1	yes	2	20.7	even	4
320.2.o.c.287.1	yes	2	40.29	even	2
320.2.o.d.223.1	yes	2	40.37	odd	4
320.2.o.d.287.1	yes	2	20.19	odd	2
1280.2.n.c.767.1		2	80.59	odd	4
1280.2.n.c.1023.1		2	80.37	odd	4
1280.2.n.d.767.1		2	80.29	even	4
1280.2.n.d.1023.1		2	80.67	even	4
1280.2.n.i.767.1		2	80.69	even	4
1280.2.n.i.1023.1		2	80.27	even	4
1280.2.n.j.767.1		2	80.19	odd	4
1280.2.n.j.1023.1		2	80.77	odd	4
1600.2.o.a.543.1		2	20.3	even	4
1600.2.o.a.607.1		2	8.5	even	2
1600.2.o.b.543.1		2	40.13	odd	4
1600.2.o.b.607.1		2	4.3	odd	2
1600.2.o.c.543.1		2	40.3	even	4	inner
1600.2.o.c.607.1		2	1.1	even	1	trivial
1600.2.o.d.543.1		2	5.3	odd	4
1600.2.o.d.607.1		2	8.3	odd	2

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 $\F_{q}$
Belyi maps

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Character values

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	1600.2.o.c.607.1

Level	$1600$
Weight	$2$
Character	1600.607
Analytic conductor	$12.776$
Analytic rank	$0$
Dimension	$2$
Inner twists	$2$