LMFDB - Embedded newform 1600.2.o.c.543.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1600,2,Mod(543,1600)]

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, 2, 3]))

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

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

chi := DirichletCharacter("1600.543");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Embedding invariants

Embedding label			543.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	1600.543
Dual form			1600.2.o.c.607.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

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

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	0.934172	−	0.356822i	$-0.116140\pi$
$3$	1.00000	+	1.00000i	0.577350	+	0.577350i	−0.356822	+	0.934172i	$0.616140\pi$
$4$		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
$11$	−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	−0.987797	−	0.155747i	$-0.950222\pi$
$13$	−3.00000	+	3.00000i	−0.832050	+	0.832050i	0.155747	+	0.987797i	$0.450222\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	3.00000	−	3.00000i	0.727607	−	0.727607i	−0.242536	−	0.970143i	$-0.577979\pi$
$17$	3.00000	−	3.00000i	0.727607	−	0.727607i	0.970143	+	0.242536i	$0.0779791\pi$
$18$		0			0
$19$		−	6.00000i		−	1.37649i	−0.725476	−	0.688247i	$-0.758380\pi$
$19$		−	6.00000i		−	1.37649i	0.725476	−	0.688247i	$-0.241620\pi$
$20$		0			0
$21$		−	2.00000i		−	0.436436i
$22$		0			0
$23$	−3.00000	+	3.00000i	−0.625543	+	0.625543i	−0.946943	−	0.321400i	$-0.895847\pi$
$23$	−3.00000	+	3.00000i	−0.625543	+	0.625543i	0.321400	+	0.946943i	$0.395847\pi$
$24$		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	$-0.818872\pi$
$31$		−	6.00000i		−	1.07763i	0.842424	−	0.538816i	$-0.181128\pi$
$32$		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	0.416115	−	0.909312i	$-0.363391\pi$
$37$	−3.00000	−	3.00000i	−0.493197	−	0.493197i	−0.909312	+	0.416115i	$0.863391\pi$
$38$		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	0.440337	−	0.897833i	$-0.354859\pi$
$43$	−3.00000	−	3.00000i	−0.457496	−	0.457496i	−0.897833	+	0.440337i	$0.854859\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−9.00000	−	9.00000i	−1.31278	−	1.31278i	−0.919354	−	0.393431i	$-0.871288\pi$
$47$	−9.00000	−	9.00000i	−1.31278	−	1.31278i	−0.393431	−	0.919354i	$-0.628712\pi$
$48$		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	−0.274721	−	0.961524i	$-0.588586\pi$
$53$	5.00000	−	5.00000i	0.686803	−	0.686803i	0.961524	+	0.274721i	$0.0885855\pi$
$54$		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	$0.225627\pi$
$59$			10.0000i			1.30189i	−0.759125	+	0.650945i	$0.774373\pi$
$60$		0			0
$61$			12.0000i			1.53644i	0.640184	+	0.768221i	$0.278858\pi$
$61$			12.0000i			1.53644i	−0.640184	+	0.768221i	$0.721142\pi$
$62$		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	−0.105059	+	0.994466i	$0.533503\pi$
$67$	−9.00000	+	9.00000i	−1.09952	+	1.09952i	−0.994466	+	0.105059i	$0.966497\pi$
$68$		0			0
$69$	−6.00000			−0.722315
$70$		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
$73$	−5.00000	−	5.00000i	−0.585206	−	0.585206i	0.351123	−	0.936329i	$-0.385800\pi$
$73$	−5.00000	−	5.00000i	−0.585206	−	0.585206i	−0.936329	+	0.351123i	$0.885800\pi$
$74$		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	0.523025	−	0.852318i	$-0.324804\pi$
$83$	−3.00000	−	3.00000i	−0.329293	−	0.329293i	−0.852318	+	0.523025i	$0.824804\pi$
$84$		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	−0.255948	−	0.966691i	$-0.582388\pi$
$97$	7.00000	−	7.00000i	0.710742	−	0.710742i	0.966691	+	0.255948i	$0.0823876\pi$
$98$		0			0
$99$			4.00000i			0.402015i
$100$		0			0
$101$		−	8.00000i		−	0.796030i	−0.917379	−	0.398015i	$-0.869699\pi$
$101$		−	8.00000i		−	0.796030i	0.917379	−	0.398015i	$-0.130301\pi$
$102$		0			0
$103$	−11.0000	+	11.0000i	−1.08386	+	1.08386i	−0.0877167	+	0.996145i	$0.527957\pi$
$103$	−11.0000	+	11.0000i	−1.08386	+	1.08386i	−0.996145	+	0.0877167i	$0.972043\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	3.00000	−	3.00000i	0.290021	−	0.290021i	−0.547068	−	0.837088i	$-0.684256\pi$
$107$	3.00000	−	3.00000i	0.290021	−	0.290021i	0.837088	+	0.547068i	$0.184256\pi$
$108$		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	0.833992	−	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
$117$	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	0.449568	−	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
$131$	−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	−0.567243	−	0.823550i	$-0.691990\pi$
$137$	3.00000	−	3.00000i	0.256307	−	0.256307i	0.823550	+	0.567243i	$0.191990\pi$
$138$		0			0
$139$		−	6.00000i		−	0.508913i	−0.967084	−	0.254457i	$-0.918103\pi$
$139$		−	6.00000i		−	0.508913i	0.967084	−	0.254457i	$-0.0818966\pi$
$140$		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	$0.133388\pi$
$151$			10.0000i			0.813788i	−0.913475	+	0.406894i	$0.866612\pi$
$152$		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	0.577186	−	0.816612i	$-0.304151\pi$
$157$	−3.00000	−	3.00000i	−0.239426	−	0.239426i	−0.816612	+	0.577186i	$0.804151\pi$
$158$		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	0.965465	−	0.260531i	$-0.0838976\pi$
$163$	9.00000	+	9.00000i	0.704934	+	0.704934i	−0.260531	+	0.965465i	$0.583898\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	3.00000	+	3.00000i	0.232147	+	0.232147i	0.813588	−	0.581441i	$-0.197511\pi$
$167$	3.00000	+	3.00000i	0.232147	+	0.232147i	−0.581441	+	0.813588i	$0.697511\pi$
$168$		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	−0.152057	+	0.988372i	$0.548590\pi$
$173$	−15.0000	+	15.0000i	−1.14043	+	1.14043i	−0.988372	+	0.152057i	$0.951410\pi$
$174$		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	$0.0238138\pi$
$179$			2.00000i			0.149487i	−0.997203	+	0.0747435i	$0.976186\pi$
$180$		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	$-0.930349\pi$
$191$		−	6.00000i		−	0.434145i	0.976156	−	0.217072i	$-0.0696508\pi$
$192$		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
$195$		0			0
$196$		0			0
$197$	5.00000	+	5.00000i	0.356235	+	0.356235i	0.862423	−	0.506188i	$-0.168946\pi$
$197$	5.00000	+	5.00000i	0.356235	+	0.356235i	−0.506188	+	0.862423i	$0.668946\pi$
$198$		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	−0.121987	−	0.992532i	$-0.538927\pi$
$223$	13.0000	−	13.0000i	0.870544	−	0.870544i	0.992532	+	0.121987i	$0.0389267\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	11.0000	−	11.0000i	0.730096	−	0.730096i	−0.240543	−	0.970639i	$-0.577325\pi$
$227$	11.0000	−	11.0000i	0.730096	−	0.730096i	0.970639	+	0.240543i	$0.0773255\pi$
$228$		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	0.999853	−	0.0171699i	$-0.00546562\pi$
$233$	15.0000	+	15.0000i	0.982683	+	0.982683i	−0.0171699	+	0.999853i	$0.505466\pi$
$234$		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	−0.607321	−	0.794456i	$-0.707756\pi$
$257$	3.00000	−	3.00000i	0.187135	−	0.187135i	0.794456	+	0.607321i	$0.207756\pi$
$258$		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	−0.372906	−	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
$269$	−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	$0.0193479\pi$
$271$			2.00000i			0.121491i	−0.998153	+	0.0607457i	$0.980652\pi$
$272$		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	0.0942828	−	0.995545i	$-0.469944\pi$
$277$	−15.0000	−	15.0000i	−0.901263	−	0.901263i	−0.995545	+	0.0942828i	$0.969944\pi$
$278$		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	0.103022	−	0.994679i	$-0.467149\pi$
$283$	−15.0000	−	15.0000i	−0.891657	−	0.891657i	−0.994679	+	0.103022i	$0.967149\pi$
$284$		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	−0.677293	−	0.735714i	$-0.736847\pi$
$293$	1.00000	−	1.00000i	0.0584206	−	0.0584206i	0.735714	+	0.677293i	$0.236847\pi$
$294$		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	−0.134780	−	0.990876i	$-0.543033\pi$
$307$	15.0000	−	15.0000i	0.856095	−	0.856095i	0.990876	+	0.134780i	$0.0430329\pi$
$308$		0			0
$309$	−22.0000			−1.25154
$310$		0			0
$311$			18.0000i			1.02069i	0.859971	+	0.510343i	$0.170482\pi$
$311$			18.0000i			1.02069i	−0.859971	+	0.510343i	$0.829518\pi$
$312$		0			0
$313$	−1.00000	−	1.00000i	−0.0565233	−	0.0565233i	0.678280	−	0.734803i	$-0.262726\pi$
$313$	−1.00000	−	1.00000i	−0.0565233	−	0.0565233i	−0.734803	+	0.678280i	$0.762726\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−3.00000	−	3.00000i	−0.168497	−	0.168497i	0.617822	−	0.786318i	$-0.288015\pi$
$317$	−3.00000	−	3.00000i	−0.168497	−	0.168497i	−0.786318	+	0.617822i	$0.788015\pi$
$318$		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	−0.490261	+	0.871576i	$0.663099\pi$
$337$	−25.0000	+	25.0000i	−1.36184	+	1.36184i	−0.871576	+	0.490261i	$0.836901\pi$
$338$		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	−0.996477	−	0.0838690i	$-0.973272\pi$
$347$	−17.0000	+	17.0000i	−0.912608	+	0.912608i	0.0838690	+	0.996477i	$0.473272\pi$
$348$		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	0.982838	−	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
$357$	−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	0.933298	−	0.359103i	$-0.116917\pi$
$367$	11.0000	+	11.0000i	0.574195	+	0.574195i	−0.359103	+	0.933298i	$0.616917\pi$
$368$		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	−0.780496	−	0.625161i	$-0.785033\pi$
$373$	−3.00000	+	3.00000i	−0.155334	+	0.155334i	0.625161	+	0.780496i	$0.285033\pi$
$374$		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	$0.152975\pi$
$379$			18.0000i			0.924598i	−0.886724	+	0.462299i	$0.847025\pi$
$380$		0			0
$381$		−	10.0000i		−	0.512316i
$382$		0			0
$383$	21.0000	−	21.0000i	1.07305	−	1.07305i	0.0759373	−	0.997113i	$-0.475805\pi$
$383$	21.0000	−	21.0000i	1.07305	−	1.07305i	0.997113	−	0.0759373i	$-0.0241949\pi$
$384$		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	0.895918	−	0.444220i	$-0.146519\pi$
$397$	9.00000	+	9.00000i	0.451697	+	0.451697i	−0.444220	+	0.895918i	$0.646519\pi$
$398$		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	$0.0958798\pi$
$409$			12.0000i			0.593362i	−0.954977	+	0.296681i	$0.904120\pi$
$410$		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	$0.0785470\pi$
$419$			10.0000i			0.488532i	−0.969708	+	0.244266i	$0.921453\pi$
$420$		0			0
$421$		−	12.0000i		−	0.584844i	−0.956289	−	0.292422i	$-0.905539\pi$
$421$		−	12.0000i		−	0.584844i	0.956289	−	0.292422i	$-0.0944612\pi$
$422$		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	$-0.742982\pi$
$431$		−	30.0000i		−	1.44505i	0.691345	−	0.722525i	$-0.257018\pi$
$432$		0			0
$433$	7.00000	+	7.00000i	0.336399	+	0.336399i	0.855010	−	0.518611i	$-0.173551\pi$
$433$	7.00000	+	7.00000i	0.336399	+	0.336399i	−0.518611	+	0.855010i	$0.673551\pi$
$434$		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	0.977678	+	0.210108i	$0.0673814\pi$
$443$	25.0000	+	25.0000i	1.18779	+	1.18779i	0.210108	+	0.977678i	$0.432619\pi$
$444$		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	$-0.908618\pi$
$449$		−	12.0000i		−	0.566315i	0.959073	−	0.283158i	$-0.0913819\pi$
$450$		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	−0.982339	−	0.187112i	$-0.940087\pi$
$457$	−17.0000	+	17.0000i	−0.795226	+	0.795226i	0.187112	+	0.982339i	$0.440087\pi$
$458$		0			0
$459$		−	24.0000i		−	1.12022i
$460$		0			0
$461$		−	40.0000i		−	1.86299i	−0.363760	−	0.931493i	$-0.618507\pi$
$461$		−	40.0000i		−	1.86299i	0.363760	−	0.931493i	$-0.381493\pi$
$462$		0			0
$463$	1.00000	−	1.00000i	0.0464739	−	0.0464739i	−0.683488	−	0.729962i	$-0.739538\pi$
$463$	1.00000	−	1.00000i	0.0464739	−	0.0464739i	0.729962	+	0.683488i	$0.239538\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−21.0000	+	21.0000i	−0.971764	+	0.971764i	−0.999612	−	0.0278481i	$-0.991135\pi$
$467$	−21.0000	+	21.0000i	−0.971764	+	0.971764i	0.0278481	+	0.999612i	$0.491135\pi$
$468$		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	0.684087	−	0.729401i	$-0.260201\pi$
$487$	−1.00000	−	1.00000i	−0.0453143	−	0.0453143i	−0.729401	+	0.684087i	$0.760201\pi$
$488$		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	$-0.957122\pi$
$499$		−	6.00000i		−	0.268597i	0.990941	−	0.134298i	$-0.0428781\pi$
$500$		0			0
$501$			6.00000i			0.268060i
$502$		0			0
$503$	21.0000	−	21.0000i	0.936344	−	0.936344i	−0.0617480	−	0.998092i	$-0.519668\pi$
$503$	21.0000	−	21.0000i	0.936344	−	0.936344i	0.998092	+	0.0617480i	$0.0196675\pi$
$504$		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	0.298504	−	0.954408i	$-0.403512\pi$
$523$	−15.0000	−	15.0000i	−0.655904	−	0.655904i	−0.954408	+	0.298504i	$0.903512\pi$
$524$		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	$-0.827450\pi$
$541$		−	24.0000i		−	1.03184i	0.856637	−	0.515920i	$-0.172550\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	3.00000	−	3.00000i	0.128271	−	0.128271i	−0.640057	−	0.768328i	$-0.721089\pi$
$547$	3.00000	−	3.00000i	0.128271	−	0.128271i	0.768328	+	0.640057i	$0.221089\pi$
$548$		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	−0.918808	−	0.394704i	$-0.870847\pi$
$557$	−31.0000	−	31.0000i	−1.31351	−	1.31351i	−0.394704	−	0.918808i	$-0.629153\pi$
$558$		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	0.727865	−	0.685720i	$-0.240513\pi$
$563$	1.00000	+	1.00000i	0.0421450	+	0.0421450i	−0.685720	+	0.727865i	$0.740513\pi$
$564$		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	$-0.919065\pi$
$569$		−	12.0000i		−	0.503066i	0.967849	−	0.251533i	$-0.0809347\pi$
$570$		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	−0.190673	−	0.981654i	$-0.561067\pi$
$577$	19.0000	−	19.0000i	0.790980	−	0.790980i	0.981654	+	0.190673i	$0.0610671\pi$
$578$		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	−0.326188	−	0.945305i	$-0.605764\pi$
$587$	15.0000	−	15.0000i	0.619116	−	0.619116i	0.945305	+	0.326188i	$0.105764\pi$
$588$		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	0.944497	−	0.328521i	$-0.106550\pi$
$593$	15.0000	+	15.0000i	0.615976	+	0.615976i	−0.328521	+	0.944497i	$0.606550\pi$
$594$		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	0.834750	−	0.550629i	$-0.185612\pi$
$607$	7.00000	+	7.00000i	0.284121	+	0.284121i	−0.550629	+	0.834750i	$0.685612\pi$
$608$		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	−0.501596	−	0.865102i	$-0.667253\pi$
$613$	9.00000	−	9.00000i	0.363507	−	0.363507i	0.865102	+	0.501596i	$0.167253\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	15.0000	−	15.0000i	0.603877	−	0.603877i	−0.337462	−	0.941339i	$-0.609568\pi$
$617$	15.0000	−	15.0000i	0.603877	−	0.603877i	0.941339	+	0.337462i	$0.109568\pi$
$618$		0			0
$619$			42.0000i			1.68812i	0.536247	+	0.844061i	$0.319842\pi$
$619$			42.0000i			1.68812i	−0.536247	+	0.844061i	$0.680158\pi$
$620$		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	$-0.796303\pi$
$631$		−	30.0000i		−	1.19428i	0.802137	−	0.597141i	$-0.203697\pi$
$632$		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	0.645474	−	0.763782i	$-0.276660\pi$
$643$	−3.00000	−	3.00000i	−0.118308	−	0.118308i	−0.763782	+	0.645474i	$0.776660\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−9.00000	−	9.00000i	−0.353827	−	0.353827i	0.507705	−	0.861531i	$-0.330494\pi$
$647$	−9.00000	−	9.00000i	−0.353827	−	0.353827i	−0.861531	+	0.507705i	$0.830494\pi$
$648$		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	0.357462	−	0.933928i	$-0.383642\pi$
$653$	33.0000	−	33.0000i	1.29139	−	1.29139i	0.933928	−	0.357462i	$-0.116358\pi$
$654$		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	$-0.859042\pi$
$659$		−	22.0000i		−	0.856998i	0.903542	−	0.428499i	$-0.140958\pi$
$660$		0			0
$661$			12.0000i			0.466746i	0.972387	+	0.233373i	$0.0749763\pi$
$661$			12.0000i			0.466746i	−0.972387	+	0.233373i	$0.925024\pi$
$662$		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	0.0356839	−	0.999363i	$-0.488639\pi$
$673$	−25.0000	−	25.0000i	−0.963679	−	0.963679i	−0.999363	+	0.0356839i	$0.988639\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−27.0000	−	27.0000i	−1.03769	−	1.03769i	−0.999261	−	0.0384331i	$-0.987763\pi$
$677$	−27.0000	−	27.0000i	−1.03769	−	1.03769i	−0.0384331	−	0.999261i	$-0.512237\pi$
$678$		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	−0.999432	−	0.0336941i	$-0.989273\pi$
$683$	−27.0000	−	27.0000i	−1.03313	−	1.03313i	−0.0336941	−	0.999432i	$-0.510727\pi$
$684$		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	$0.123283\pi$
$701$			20.0000i			0.755390i	−0.925930	+	0.377695i	$0.876717\pi$
$702$		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	0.0703254	−	0.997524i	$-0.477596\pi$
$727$	−25.0000	−	25.0000i	−0.927199	−	0.927199i	−0.997524	+	0.0703254i	$0.977596\pi$
$728$		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	−0.927604	−	0.373566i	$-0.878135\pi$
$733$	−15.0000	+	15.0000i	−0.554038	+	0.554038i	0.373566	+	0.927604i	$0.378135\pi$
$734$		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	$-0.813947\pi$
$739$		−	30.0000i		−	1.10357i	0.833987	−	0.551784i	$-0.186053\pi$
$740$		0			0
$741$			36.0000i			1.32249i
$742$		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
$747$	−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	$0.277905\pi$
$751$			42.0000i			1.53260i	−0.642482	+	0.766301i	$0.722095\pi$
$752$		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	0.851487	−	0.524376i	$-0.175701\pi$
$757$	9.00000	+	9.00000i	0.327111	+	0.327111i	−0.524376	+	0.851487i	$0.675701\pi$
$758$		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	$-0.743582\pi$
$769$		−	40.0000i		−	1.44244i	0.692708	−	0.721218i	$-0.256418\pi$
$770$		0			0
$771$	6.00000			0.216085
$772$		0			0
$773$	9.00000	−	9.00000i	0.323708	−	0.323708i	−0.526480	−	0.850188i	$-0.676489\pi$
$773$	9.00000	−	9.00000i	0.323708	−	0.323708i	0.850188	+	0.526480i	$0.176489\pi$
$774$		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	−0.387273	−	0.921965i	$-0.626583\pi$
$787$	15.0000	−	15.0000i	0.534692	−	0.534692i	0.921965	+	0.387273i	$0.126583\pi$
$788$		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
$796$

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.o (of order \(4\), degree \(2\), minimal)

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

Introduction

L-functions

Modular forms

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Database

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