LMFDB - Embedded newform 1248.1.cm.a.1013.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1248,1,Mod(77,1248)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1248, base_ring=CyclotomicField(8)) chi = DirichletCharacter(H, H._module([0, 7, 4, 4])) N = Newforms(chi, 1, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1248.77"); S:= CuspForms(chi, 1); N := Newforms(S);

Level:	$ N $	$=$	$ 1248 = 2^{5} \cdot 3 \cdot 13 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	1248.cm (of order $8$, degree $4$, minimal)

Newform invariants

comment:select newform

sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)

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

Self dual:	no
Analytic conductor:	$0.622833135766$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$4$ over $\Q(\zeta_{8})$
Coefficient field:	$\Q(\zeta_{32})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 1 $ x^16 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{16}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{16} - \cdots)$

Embedding invariants

Embedding label			1013.4
Root			$0.555570 - 0.831470i$ of defining polynomial
Character	$\chi$	$=$	1248.1013
Dual form			1248.1.cm.a.701.4

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(0.980785 + 0.195090i) q^{2} +(-0.923880 + 0.382683i) q^{3} +(0.923880 + 0.382683i) q^{4} +(-0.750661 + 1.81225i) q^{5} +(-0.980785 + 0.195090i) q^{6} +(0.831470 + 0.555570i) q^{8} +(0.707107 - 0.707107i) q^{9} +(-1.08979 + 1.63099i) q^{10} +(-1.53636 - 0.636379i) q^{11} -1.00000 q^{12} +(0.382683 + 0.923880i) q^{13} -1.96157i q^{15} +(0.707107 + 0.707107i) q^{16} +(0.831470 - 0.555570i) q^{18} +(-1.38704 + 1.38704i) q^{20} +(-1.38268 - 0.923880i) q^{22} +(-0.980785 - 0.195090i) q^{24} +(-2.01367 - 2.01367i) q^{25} +(0.195090 + 0.980785i) q^{26} +(-0.382683 + 0.923880i) q^{27} +(0.382683 - 1.92388i) q^{30} +(0.555570 + 0.831470i) q^{32} +1.66294 q^{33} +(0.923880 - 0.382683i) q^{36} +(-0.707107 - 0.707107i) q^{39} +(-1.63099 + 1.08979i) q^{40} +(-0.275899 + 0.275899i) q^{41} +(0.707107 + 0.292893i) q^{43} +(-1.17588 - 1.17588i) q^{44} +(0.750661 + 1.81225i) q^{45} +1.11114i q^{47} +(-0.923880 - 0.382683i) q^{48} +1.00000i q^{49} +(-1.58213 - 2.36783i) q^{50} +1.00000i q^{52} +(-0.555570 + 0.831470i) q^{54} +(2.30656 - 2.30656i) q^{55} +(0.750661 - 1.81225i) q^{59} +(0.750661 - 1.81225i) q^{60} +(1.30656 - 0.541196i) q^{61} +(0.382683 + 0.923880i) q^{64} -1.96157 q^{65} +(1.63099 + 0.324423i) q^{66} +(1.17588 + 1.17588i) q^{71} +(0.980785 - 0.195090i) q^{72} +(2.63099 + 1.08979i) q^{75} +(-0.555570 - 0.831470i) q^{78} -0.765367i q^{79} +(-1.81225 + 0.750661i) q^{80} -1.00000i q^{81} +(-0.324423 + 0.216773i) q^{82} +(0.149316 + 0.360480i) q^{83} +(0.636379 + 0.425215i) q^{86} +(-0.923880 - 1.38268i) q^{88} +(0.785695 + 0.785695i) q^{89} +(0.382683 + 1.92388i) q^{90} +(-0.216773 + 1.08979i) q^{94} +(-0.831470 - 0.555570i) q^{96} +(-0.195090 + 0.980785i) q^{98} +(-1.53636 + 0.636379i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 16 q^{12} - 16 q^{22} + 16 q^{55} + 16 q^{75}+O(q^{100}) $ 16 * q - 16 * q^12 - 16 * q^22 + 16 * q^55 + 16 * q^75

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1248\mathbb{Z}\right)^\times$.

$n$	$703$	$769$	$833$	$1093$
$\chi(n)$	$1$	$-1$	$-1$	$e\left(\frac{5}{8}\right)$

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 $

$2$	0.980785	+	0.195090i	0.980785	+	0.195090i
$2$	0.980785	+	0.195090i	0.980785	+	0.195090i
$3$	−0.923880	+	0.382683i	−0.923880	+	0.382683i
$3$	−0.923880	+	0.382683i	−0.923880	+	0.382683i
$4$	0.923880	+	0.382683i	0.923880	+	0.382683i
$5$	−0.750661	+	1.81225i	−0.750661	+	1.81225i	−0.195090	+	0.980785i	$0.562500\pi$
$5$	−0.750661	+	1.81225i	−0.750661	+	1.81225i	−0.555570	+	0.831470i	$0.687500\pi$
$6$	−0.980785	+	0.195090i	−0.980785	+	0.195090i
$7$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$7$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$8$	0.831470	+	0.555570i	0.831470	+	0.555570i
$9$	0.707107	−	0.707107i	0.707107	−	0.707107i
$10$	−1.08979	+	1.63099i	−1.08979	+	1.63099i
$11$	−1.53636	−	0.636379i	−1.53636	−	0.636379i	−0.555570	−	0.831470i	$-0.687500\pi$
$11$	−1.53636	−	0.636379i	−1.53636	−	0.636379i	−0.980785	+	0.195090i	$0.937500\pi$
$12$	−1.00000			−1.00000
$13$	0.382683	+	0.923880i	0.382683	+	0.923880i
$13$	0.382683	+	0.923880i	0.382683	+	0.923880i
$14$		0			0
$15$		−	1.96157i		−	1.96157i
$16$	0.707107	+	0.707107i	0.707107	+	0.707107i
$17$		0			0			−	1.00000i	$-0.5\pi$
$17$		0			0				1.00000i	$0.5\pi$
$18$	0.831470	−	0.555570i	0.831470	−	0.555570i
$19$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$19$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$20$	−1.38704	+	1.38704i	−1.38704	+	1.38704i
$21$		0			0
$22$	−1.38268	−	0.923880i	−1.38268	−	0.923880i
$23$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$23$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$24$	−0.980785	−	0.195090i	−0.980785	−	0.195090i
$25$	−2.01367	−	2.01367i	−2.01367	−	2.01367i
$26$	0.195090	+	0.980785i	0.195090	+	0.980785i
$27$	−0.382683	+	0.923880i	−0.382683	+	0.923880i
$28$		0			0
$29$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$29$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$30$	0.382683	−	1.92388i	0.382683	−	1.92388i
$31$		0			0		1.00000			$0$
$31$		0			0		−1.00000			$\pi$
$32$	0.555570	+	0.831470i	0.555570	+	0.831470i
$33$	1.66294			1.66294
$34$		0			0
$35$		0			0
$36$	0.923880	−	0.382683i	0.923880	−	0.382683i
$37$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$37$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$38$		0			0
$39$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$40$	−1.63099	+	1.08979i	−1.63099	+	1.08979i
$41$	−0.275899	+	0.275899i	−0.275899	+	0.275899i	−0.831470	−	0.555570i	$-0.812500\pi$
$41$	−0.275899	+	0.275899i	−0.275899	+	0.275899i	0.555570	+	0.831470i	$0.312500\pi$
$42$		0			0
$43$	0.707107	+	0.292893i	0.707107	+	0.292893i	0.707107	−	0.707107i	$-0.250000\pi$
$43$	0.707107	+	0.292893i	0.707107	+	0.292893i			1.00000i	$0.5\pi$
$44$	−1.17588	−	1.17588i	−1.17588	−	1.17588i
$45$	0.750661	+	1.81225i	0.750661	+	1.81225i
$46$		0			0
$47$			1.11114i			1.11114i	0.831470	+	0.555570i	$0.187500\pi$
$47$			1.11114i			1.11114i	−0.831470	+	0.555570i	$0.812500\pi$
$48$	−0.923880	−	0.382683i	−0.923880	−	0.382683i
$49$			1.00000i			1.00000i
$50$	−1.58213	−	2.36783i	−1.58213	−	2.36783i
$51$		0			0
$52$			1.00000i			1.00000i
$53$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$53$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$54$	−0.555570	+	0.831470i	−0.555570	+	0.831470i
$55$	2.30656	−	2.30656i	2.30656	−	2.30656i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	0.750661	−	1.81225i	0.750661	−	1.81225i	0.195090	−	0.980785i	$-0.437500\pi$
$59$	0.750661	−	1.81225i	0.750661	−	1.81225i	0.555570	−	0.831470i	$-0.312500\pi$
$60$	0.750661	−	1.81225i	0.750661	−	1.81225i
$61$	1.30656	−	0.541196i	1.30656	−	0.541196i	0.382683	−	0.923880i	$-0.375000\pi$
$61$	1.30656	−	0.541196i	1.30656	−	0.541196i	0.923880	+	0.382683i	$0.125000\pi$
$62$		0			0
$63$		0			0
$64$	0.382683	+	0.923880i	0.382683	+	0.923880i
$65$	−1.96157			−1.96157
$66$	1.63099	+	0.324423i	1.63099	+	0.324423i
$67$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$67$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	1.17588	+	1.17588i	1.17588	+	1.17588i	0.980785	+	0.195090i	$0.0625000\pi$
$71$	1.17588	+	1.17588i	1.17588	+	1.17588i	0.195090	+	0.980785i	$0.437500\pi$
$72$	0.980785	−	0.195090i	0.980785	−	0.195090i
$73$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$73$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$74$		0			0
$75$	2.63099	+	1.08979i	2.63099	+	1.08979i
$76$		0			0
$77$		0			0
$78$	−0.555570	−	0.831470i	−0.555570	−	0.831470i
$79$		−	0.765367i		−	0.765367i	−0.923880	−	0.382683i	$-0.875000\pi$
$79$		−	0.765367i		−	0.765367i	0.923880	−	0.382683i	$-0.125000\pi$
$80$	−1.81225	+	0.750661i	−1.81225	+	0.750661i
$81$		−	1.00000i		−	1.00000i
$82$	−0.324423	+	0.216773i	−0.324423	+	0.216773i
$83$	0.149316	+	0.360480i	0.149316	+	0.360480i	0.980785	−	0.195090i	$-0.0625000\pi$
$83$	0.149316	+	0.360480i	0.149316	+	0.360480i	−0.831470	+	0.555570i	$0.812500\pi$
$84$		0			0
$85$		0			0
$86$	0.636379	+	0.425215i	0.636379	+	0.425215i
$87$		0			0
$88$	−0.923880	−	1.38268i	−0.923880	−	1.38268i
$89$	0.785695	+	0.785695i	0.785695	+	0.785695i	0.980785	−	0.195090i	$-0.0625000\pi$
$89$	0.785695	+	0.785695i	0.785695	+	0.785695i	−0.195090	+	0.980785i	$0.562500\pi$
$90$	0.382683	+	1.92388i	0.382683	+	1.92388i
$91$		0			0
$92$		0			0
$93$		0			0
$94$	−0.216773	+	1.08979i	−0.216773	+	1.08979i
$95$		0			0
$96$	−0.831470	−	0.555570i	−0.831470	−	0.555570i
$97$		0			0		1.00000			$0$
$97$		0			0		−1.00000			$\pi$
$98$	−0.195090	+	0.980785i	−0.195090	+	0.980785i
$99$	−1.53636	+	0.636379i	−1.53636	+	0.636379i
$100$	−1.08979	−	2.63099i	−1.08979	−	2.63099i
$101$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$101$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$102$		0			0
$103$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$103$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$104$	−0.195090	+	0.980785i	−0.195090	+	0.980785i
$105$		0			0
$106$		0			0
$107$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$107$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$108$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$109$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$109$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$110$	2.71223	−	1.81225i	2.71223	−	1.81225i
$111$		0			0
$112$		0			0
$113$		0			0			−	1.00000i	$-0.5\pi$
$113$		0			0				1.00000i	$0.5\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	0.923880	+	0.382683i	0.923880	+	0.382683i
$118$	1.08979	−	1.63099i	1.08979	−	1.63099i
$119$		0			0
$120$	1.08979	−	1.63099i	1.08979	−	1.63099i
$121$	1.24830	+	1.24830i	1.24830	+	1.24830i
$122$	1.38704	−	0.275899i	1.38704	−	0.275899i
$123$	0.149316	−	0.360480i	0.149316	−	0.360480i
$124$		0			0
$125$	3.34861	−	1.38704i	3.34861	−	1.38704i
$126$		0			0
$127$	0.765367			0.765367			0.382683	−	0.923880i	$-0.375000\pi$
$127$	0.765367			0.765367			0.382683	+	0.923880i	$0.375000\pi$
$128$	0.195090	+	0.980785i	0.195090	+	0.980785i
$129$	−0.765367			−0.765367
$130$	−1.92388	−	0.382683i	−1.92388	−	0.382683i
$131$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$131$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$132$	1.53636	+	0.636379i	1.53636	+	0.636379i
$133$		0			0
$134$		0			0
$135$	−1.38704	−	1.38704i	−1.38704	−	1.38704i
$136$		0			0
$137$	0.785695	−	0.785695i	0.785695	−	0.785695i	−0.195090	−	0.980785i	$-0.562500\pi$
$137$	0.785695	−	0.785695i	0.785695	−	0.785695i	0.980785	+	0.195090i	$0.0625000\pi$
$138$		0			0
$139$	−1.70711	−	0.707107i	−1.70711	−	0.707107i	−0.707107	−	0.707107i	$-0.750000\pi$
$139$	−1.70711	−	0.707107i	−1.70711	−	0.707107i	−1.00000			$\pi$
$140$		0			0
$141$	−0.425215	−	1.02656i	−0.425215	−	1.02656i
$142$	0.923880	+	1.38268i	0.923880	+	1.38268i
$143$		−	1.66294i		−	1.66294i
$144$	1.00000			1.00000
$145$		0			0
$146$		0			0
$147$	−0.382683	−	0.923880i	−0.382683	−	0.923880i
$148$		0			0
$149$	−1.02656	−	0.425215i	−1.02656	−	0.425215i	−0.195090	−	0.980785i	$-0.562500\pi$
$149$	−1.02656	−	0.425215i	−1.02656	−	0.425215i	−0.831470	+	0.555570i	$0.812500\pi$
$150$	2.36783	+	1.58213i	2.36783	+	1.58213i
$151$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$151$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$	−0.382683	−	0.923880i	−0.382683	−	0.923880i
$157$	0.707107	−	0.292893i	0.707107	−	0.292893i		−	1.00000i	$-0.5\pi$
$157$	0.707107	−	0.292893i	0.707107	−	0.292893i	0.707107	+	0.707107i	$0.250000\pi$
$158$	0.149316	−	0.750661i	0.149316	−	0.750661i
$159$		0			0
$160$	−1.92388	+	0.382683i	−1.92388	+	0.382683i
$161$		0			0
$162$	0.195090	−	0.980785i	0.195090	−	0.980785i
$163$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$163$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$164$	−0.360480	+	0.149316i	−0.360480	+	0.149316i
$165$	−1.24830	+	3.01367i	−1.24830	+	3.01367i
$166$	0.0761205	+	0.382683i	0.0761205	+	0.382683i
$167$	0.275899	+	0.275899i	0.275899	+	0.275899i	0.831470	−	0.555570i	$-0.187500\pi$
$167$	0.275899	+	0.275899i	0.275899	+	0.275899i	−0.555570	+	0.831470i	$0.687500\pi$
$168$		0			0
$169$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$170$		0			0
$171$		0			0
$172$	0.541196	+	0.541196i	0.541196	+	0.541196i
$173$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$173$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$174$		0			0
$175$		0			0
$176$	−0.636379	−	1.53636i	−0.636379	−	1.53636i
$177$			1.96157i			1.96157i
$178$	0.617317	+	0.923880i	0.617317	+	0.923880i
$179$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$179$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$180$			1.96157i			1.96157i
$181$	−1.70711	−	0.707107i	−1.70711	−	0.707107i	−0.707107	−	0.707107i	$-0.750000\pi$
$181$	−1.70711	−	0.707107i	−1.70711	−	0.707107i	−1.00000			$\pi$
$182$		0			0
$183$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$	−0.425215	+	1.02656i	−0.425215	+	1.02656i
$189$		0			0
$190$		0			0
$191$		0			0		1.00000			$0$
$191$		0			0		−1.00000			$\pi$
$192$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$193$		0			0		1.00000			$0$
$193$		0			0		−1.00000			$\pi$
$194$		0			0
$195$	1.81225	−	0.750661i	1.81225	−	0.750661i
$196$	−0.382683	+	0.923880i	−0.382683	+	0.923880i
$197$	0.636379	−	1.53636i	0.636379	−	1.53636i	−0.195090	−	0.980785i	$-0.562500\pi$
$197$	0.636379	−	1.53636i	0.636379	−	1.53636i	0.831470	−	0.555570i	$-0.187500\pi$
$198$	−1.63099	+	0.324423i	−1.63099	+	0.324423i
$199$	−1.30656	−	1.30656i	−1.30656	−	1.30656i	−0.923880	−	0.382683i	$-0.875000\pi$
$199$	−1.30656	−	1.30656i	−1.30656	−	1.30656i	−0.382683	−	0.923880i	$-0.625000\pi$
$200$	−0.555570	−	2.79304i	−0.555570	−	2.79304i
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$	−0.292893	−	0.707107i	−0.292893	−	0.707107i
$206$		0			0
$207$		0			0
$208$	−0.382683	+	0.923880i	−0.382683	+	0.923880i
$209$		0			0
$210$		0			0
$211$	0.541196	+	1.30656i	0.541196	+	1.30656i	0.923880	+	0.382683i	$0.125000\pi$
$211$	0.541196	+	1.30656i	0.541196	+	1.30656i	−0.382683	+	0.923880i	$0.625000\pi$
$212$		0			0
$213$	−1.53636	−	0.636379i	−1.53636	−	0.636379i
$214$		0			0
$215$	−1.06159	+	1.06159i	−1.06159	+	1.06159i
$216$	−0.831470	+	0.555570i	−0.831470	+	0.555570i
$217$		0			0
$218$		0			0
$219$		0			0
$220$	3.01367	−	1.24830i	3.01367	−	1.24830i
$221$		0			0
$222$		0			0
$223$		0			0		1.00000			$0$
$223$		0			0		−1.00000			$\pi$
$224$		0			0
$225$	−2.84776			−2.84776
$226$		0			0
$227$	1.02656	−	0.425215i	1.02656	−	0.425215i	0.195090	−	0.980785i	$-0.437500\pi$
$227$	1.02656	−	0.425215i	1.02656	−	0.425215i	0.831470	+	0.555570i	$0.187500\pi$
$228$		0			0
$229$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$229$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$233$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$234$	0.831470	+	0.555570i	0.831470	+	0.555570i
$235$	−2.01367	−	0.834089i	−2.01367	−	0.834089i
$236$	1.38704	−	1.38704i	1.38704	−	1.38704i
$237$	0.292893	+	0.707107i	0.292893	+	0.707107i
$238$		0			0
$239$		−	0.390181i		−	0.390181i	−0.980785	−	0.195090i	$-0.937500\pi$
$239$		−	0.390181i		−	0.390181i	0.980785	−	0.195090i	$-0.0625000\pi$
$240$	1.38704	−	1.38704i	1.38704	−	1.38704i
$241$		0			0			−	1.00000i	$-0.5\pi$
$241$		0			0				1.00000i	$0.5\pi$
$242$	0.980785	+	1.46785i	0.980785	+	1.46785i
$243$	0.382683	+	0.923880i	0.382683	+	0.923880i
$244$	1.41421			1.41421
$245$	−1.81225	−	0.750661i	−1.81225	−	0.750661i
$246$	0.216773	−	0.324423i	0.216773	−	0.324423i
$247$		0			0
$248$		0			0
$249$	−0.275899	−	0.275899i	−0.275899	−	0.275899i
$250$	3.55487	−	0.707107i	3.55487	−	0.707107i
$251$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$251$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$252$		0			0
$253$		0			0
$254$	0.750661	+	0.149316i	0.750661	+	0.149316i
$255$		0			0
$256$			1.00000i			1.00000i
$257$		0			0		1.00000			$0$
$257$		0			0		−1.00000			$\pi$
$258$	−0.750661	−	0.149316i	−0.750661	−	0.149316i
$259$		0			0
$260$	−1.81225	−	0.750661i	−1.81225	−	0.750661i
$261$		0			0
$262$		0			0
$263$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$263$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$264$	1.38268	+	0.923880i	1.38268	+	0.923880i
$265$		0			0
$266$		0			0
$267$	−1.02656	−	0.425215i	−1.02656	−	0.425215i
$268$		0			0
$269$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$269$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$270$	−1.08979	−	1.63099i	−1.08979	−	1.63099i
$271$		0			0			−	1.00000i	$-0.5\pi$
$271$		0			0				1.00000i	$0.5\pi$
$272$		0			0
$273$		0			0
$274$	0.923880	−	0.617317i	0.923880	−	0.617317i
$275$	1.81225	+	4.37517i	1.81225	+	4.37517i
$276$		0			0
$277$	1.70711	+	0.707107i	1.70711	+	0.707107i	1.00000			$0$
$277$	1.70711	+	0.707107i	1.70711	+	0.707107i	0.707107	+	0.707107i	$0.250000\pi$
$278$	−1.53636	−	1.02656i	−1.53636	−	1.02656i
$279$		0			0
$280$		0			0
$281$	−1.38704	−	1.38704i	−1.38704	−	1.38704i	−0.831470	−	0.555570i	$-0.812500\pi$
$281$	−1.38704	−	1.38704i	−1.38704	−	1.38704i	−0.555570	−	0.831470i	$-0.687500\pi$
$282$	−0.216773	−	1.08979i	−0.216773	−	1.08979i
$283$	−0.541196	+	1.30656i	−0.541196	+	1.30656i	0.382683	+	0.923880i	$0.375000\pi$
$283$	−0.541196	+	1.30656i	−0.541196	+	1.30656i	−0.923880	+	0.382683i	$0.875000\pi$
$284$	0.636379	+	1.53636i	0.636379	+	1.53636i
$285$		0			0
$286$	0.324423	−	1.63099i	0.324423	−	1.63099i
$287$		0			0
$288$	0.980785	+	0.195090i	0.980785	+	0.195090i
$289$	−1.00000			−1.00000
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−0.425215	+	1.02656i	−0.425215	+	1.02656i	0.555570	+	0.831470i	$0.312500\pi$
$293$	−0.425215	+	1.02656i	−0.425215	+	1.02656i	−0.980785	+	0.195090i	$0.937500\pi$
$294$	−0.195090	−	0.980785i	−0.195090	−	0.980785i
$295$	2.72078	+	2.72078i	2.72078	+	2.72078i
$296$		0			0
$297$	1.17588	−	1.17588i	1.17588	−	1.17588i
$298$	−0.923880	−	0.617317i	−0.923880	−	0.617317i
$299$		0			0
$300$	2.01367	+	2.01367i	2.01367	+	2.01367i
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$			2.77408i			2.77408i
$306$		0			0
$307$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$307$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$311$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$312$	−0.195090	−	0.980785i	−0.195090	−	0.980785i
$313$	−0.541196	−	0.541196i	−0.541196	−	0.541196i	0.382683	−	0.923880i	$-0.375000\pi$
$313$	−0.541196	−	0.541196i	−0.541196	−	0.541196i	−0.923880	+	0.382683i	$0.875000\pi$
$314$	0.750661	−	0.149316i	0.750661	−	0.149316i
$315$		0			0
$316$	0.292893	−	0.707107i	0.292893	−	0.707107i
$317$	1.02656	−	0.425215i	1.02656	−	0.425215i	0.195090	−	0.980785i	$-0.437500\pi$
$317$	1.02656	−	0.425215i	1.02656	−	0.425215i	0.831470	+	0.555570i	$0.187500\pi$
$318$		0			0
$319$		0			0
$320$	−1.96157			−1.96157
$321$		0			0
$322$		0			0
$323$		0			0
$324$	0.382683	−	0.923880i	0.382683	−	0.923880i
$325$	1.08979	−	2.63099i	1.08979	−	2.63099i
$326$		0			0
$327$		0			0
$328$	−0.382683	+	0.0761205i	−0.382683	+	0.0761205i
$329$		0			0
$330$	−1.81225	+	2.71223i	−1.81225	+	2.71223i
$331$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$331$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$332$			0.390181i			0.390181i
$333$		0			0
$334$	0.216773	+	0.324423i	0.216773	+	0.324423i
$335$		0			0
$336$		0			0
$337$		−	0.765367i		−	0.765367i	−0.923880	−	0.382683i	$-0.875000\pi$
$337$		−	0.765367i		−	0.765367i	0.923880	−	0.382683i	$-0.125000\pi$
$338$	−0.831470	+	0.555570i	−0.831470	+	0.555570i
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$	0.425215	+	0.636379i	0.425215	+	0.636379i
$345$		0			0
$346$		0			0
$347$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$347$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$348$		0			0
$349$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$349$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$350$		0			0
$351$	−1.00000			−1.00000
$352$	−0.324423	−	1.63099i	−0.324423	−	1.63099i
$353$	0.390181			0.390181			0.195090	−	0.980785i	$-0.437500\pi$
$353$	0.390181			0.390181			0.195090	+	0.980785i	$0.437500\pi$
$354$	−0.382683	+	1.92388i	−0.382683	+	1.92388i
$355$	−3.01367	+	1.24830i	−3.01367	+	1.24830i
$356$	0.425215	+	1.02656i	0.425215	+	1.02656i
$357$		0			0
$358$		0			0
$359$	−0.785695	−	0.785695i	−0.785695	−	0.785695i	0.195090	−	0.980785i	$-0.437500\pi$
$359$	−0.785695	−	0.785695i	−0.785695	−	0.785695i	−0.980785	+	0.195090i	$0.937500\pi$
$360$	−0.382683	+	1.92388i	−0.382683	+	1.92388i
$361$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$362$	−1.53636	−	1.02656i	−1.53636	−	1.02656i
$363$	−1.63099	−	0.675577i	−1.63099	−	0.675577i
$364$		0			0
$365$		0			0
$366$	−1.17588	+	0.785695i	−1.17588	+	0.785695i
$367$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$367$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$368$		0			0
$369$			0.390181i			0.390181i
$370$		0			0
$371$		0			0
$372$		0			0
$373$	1.30656	+	0.541196i	1.30656	+	0.541196i	0.923880	−	0.382683i	$-0.125000\pi$
$373$	1.30656	+	0.541196i	1.30656	+	0.541196i	0.382683	+	0.923880i	$0.375000\pi$
$374$		0			0
$375$	−2.56292	+	2.56292i	−2.56292	+	2.56292i
$376$	−0.617317	+	0.923880i	−0.617317	+	0.923880i
$377$		0			0
$378$		0			0
$379$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$379$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$380$		0			0
$381$	−0.707107	+	0.292893i	−0.707107	+	0.292893i
$382$		0			0
$383$	−0.390181			−0.390181			−0.195090	−	0.980785i	$-0.562500\pi$
$383$	−0.390181			−0.390181			−0.195090	+	0.980785i	$0.562500\pi$
$384$	−0.555570	−	0.831470i	−0.555570	−	0.831470i
$385$		0			0
$386$		0			0
$387$	0.707107	−	0.292893i	0.707107	−	0.292893i
$388$		0			0
$389$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$389$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$390$	1.92388	−	0.382683i	1.92388	−	0.382683i
$391$		0			0
$392$	−0.555570	+	0.831470i	−0.555570	+	0.831470i
$393$		0			0
$394$	0.923880	−	1.38268i	0.923880	−	1.38268i
$395$	1.38704	+	0.574531i	1.38704	+	0.574531i
$396$	−1.66294			−1.66294
$397$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$397$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$398$	−1.02656	−	1.53636i	−1.02656	−	1.53636i
$399$		0			0
$400$		−	2.84776i		−	2.84776i
$401$		−	1.66294i		−	1.66294i	−0.555570	−	0.831470i	$-0.687500\pi$
$401$		−	1.66294i		−	1.66294i	0.555570	−	0.831470i	$-0.312500\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	1.81225	+	0.750661i	1.81225	+	0.750661i
$406$		0			0
$407$		0			0
$408$		0			0
$409$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$409$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$410$	−0.149316	−	0.750661i	−0.149316	−	0.750661i
$411$	−0.425215	+	1.02656i	−0.425215	+	1.02656i
$412$		0			0
$413$		0			0
$414$		0			0
$415$	−0.765367			−0.765367
$416$	−0.555570	+	0.831470i	−0.555570	+	0.831470i
$417$	1.84776			1.84776
$418$		0			0
$419$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$419$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$420$		0			0
$421$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$421$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$422$	0.275899	+	1.38704i	0.275899	+	1.38704i
$423$	0.785695	+	0.785695i	0.785695	+	0.785695i
$424$		0			0
$425$		0			0
$426$	−1.38268	−	0.923880i	−1.38268	−	0.923880i
$427$		0			0
$428$		0			0
$429$	0.636379	+	1.53636i	0.636379	+	1.53636i
$430$	−1.24830	+	0.834089i	−1.24830	+	0.834089i
$431$			0.390181i			0.390181i	0.980785	+	0.195090i	$0.0625000\pi$
$431$			0.390181i			0.390181i	−0.980785	+	0.195090i	$0.937500\pi$
$432$	−0.923880	+	0.382683i	−0.923880	+	0.382683i
$433$		−	1.84776i		−	1.84776i	−0.382683	−	0.923880i	$-0.625000\pi$
$433$		−	1.84776i		−	1.84776i	0.382683	−	0.923880i	$-0.375000\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$	−1.00000	+	1.00000i	−1.00000	+	1.00000i			1.00000i	$0.5\pi$
$439$	−1.00000	+	1.00000i	−1.00000	+	1.00000i	−1.00000			$\pi$
$440$	3.19929	−	0.636379i	3.19929	−	0.636379i
$441$	0.707107	+	0.707107i	0.707107	+	0.707107i
$442$		0			0
$443$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$443$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$444$		0			0
$445$	−2.01367	+	0.834089i	−2.01367	+	0.834089i
$446$		0			0
$447$	1.11114			1.11114
$448$		0			0
$449$	−1.11114			−1.11114			−0.555570	−	0.831470i	$-0.687500\pi$
$449$	−1.11114			−1.11114			−0.555570	+	0.831470i	$0.687500\pi$
$450$	−2.79304	−	0.555570i	−2.79304	−	0.555570i
$451$	0.599456	−	0.248303i	0.599456	−	0.248303i
$452$		0			0
$453$		0			0
$454$	1.08979	−	0.216773i	1.08979	−	0.216773i
$455$		0			0
$456$		0			0
$457$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$457$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−0.636379	−	1.53636i	−0.636379	−	1.53636i	−0.831470	−	0.555570i	$-0.812500\pi$
$461$	−0.636379	−	1.53636i	−0.636379	−	1.53636i	0.195090	−	0.980785i	$-0.437500\pi$
$462$		0			0
$463$		0			0			−	1.00000i	$-0.5\pi$
$463$		0			0				1.00000i	$0.5\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$467$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$468$	0.707107	+	0.707107i	0.707107	+	0.707107i
$469$		0			0
$470$	−1.81225	−	1.21091i	−1.81225	−	1.21091i
$471$	−0.541196	+	0.541196i	−0.541196	+	0.541196i
$472$	1.63099	−	1.08979i	1.63099	−	1.08979i
$473$	−0.899976	−	0.899976i	−0.899976	−	0.899976i
$474$	0.149316	+	0.750661i	0.149316	+	0.750661i
$475$		0			0
$476$		0			0
$477$		0			0
$478$	0.0761205	−	0.382683i	0.0761205	−	0.382683i
$479$	1.66294			1.66294			0.831470	−	0.555570i	$-0.187500\pi$
$479$	1.66294			1.66294			0.831470	+	0.555570i	$0.187500\pi$
$480$	1.63099	−	1.08979i	1.63099	−	1.08979i
$481$		0			0
$482$		0			0
$483$		0			0
$484$	0.675577	+	1.63099i	0.675577	+	1.63099i
$485$		0			0
$486$	0.195090	+	0.980785i	0.195090	+	0.980785i
$487$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$487$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$488$	1.38704	+	0.275899i	1.38704	+	0.275899i
$489$		0			0
$490$	−1.63099	−	1.08979i	−1.63099	−	1.08979i
$491$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$491$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$492$	0.275899	−	0.275899i	0.275899	−	0.275899i
$493$		0			0
$494$		0			0
$495$		−	3.26197i		−	3.26197i
$496$		0			0
$497$		0			0
$498$	−0.216773	−	0.324423i	−0.216773	−	0.324423i
$499$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$499$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$500$	3.62451			3.62451
$501$	−0.360480	−	0.149316i	−0.360480	−	0.149316i
$502$		0			0
$503$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$503$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	0.382683	−	0.923880i	0.382683	−	0.923880i
$508$	0.707107	+	0.292893i	0.707107	+	0.292893i
$509$	−0.360480	+	0.149316i	−0.360480	+	0.149316i	−0.555570	−	0.831470i	$-0.687500\pi$
$509$	−0.360480	+	0.149316i	−0.360480	+	0.149316i	0.195090	+	0.980785i	$0.437500\pi$
$510$		0			0
$511$		0			0
$512$	−0.195090	+	0.980785i	−0.195090	+	0.980785i
$513$		0			0
$514$		0			0
$515$		0			0
$516$	−0.707107	−	0.292893i	−0.707107	−	0.292893i
$517$	0.707107	−	1.70711i	0.707107	−	1.70711i
$518$		0			0
$519$		0			0
$520$	−1.63099	−	1.08979i	−1.63099	−	1.08979i
$521$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$521$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$522$		0			0
$523$	−1.70711	−	0.707107i	−1.70711	−	0.707107i	−0.707107	−	0.707107i	$-0.750000\pi$
$523$	−1.70711	−	0.707107i	−1.70711	−	0.707107i	−1.00000			$\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$	1.17588	+	1.17588i	1.17588	+	1.17588i
$529$		−	1.00000i		−	1.00000i
$530$		0			0
$531$	−0.750661	−	1.81225i	−0.750661	−	1.81225i
$532$		0			0
$533$	−0.360480	−	0.149316i	−0.360480	−	0.149316i
$534$	−0.923880	−	0.617317i	−0.923880	−	0.617317i
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$	0.636379	−	1.53636i	0.636379	−	1.53636i
$540$	−0.750661	−	1.81225i	−0.750661	−	1.81225i
$541$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$541$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$542$		0			0
$543$	1.84776			1.84776
$544$		0			0
$545$		0			0
$546$		0			0
$547$	1.70711	−	0.707107i	1.70711	−	0.707107i	0.707107	−	0.707107i	$-0.250000\pi$
$547$	1.70711	−	0.707107i	1.70711	−	0.707107i	1.00000			$0$
$548$	1.02656	−	0.425215i	1.02656	−	0.425215i
$549$	0.541196	−	1.30656i	0.541196	−	1.30656i
$550$	0.923880	+	4.64466i	0.923880	+	4.64466i
$551$		0			0
$552$		0			0
$553$		0			0
$554$	1.53636	+	1.02656i	1.53636	+	1.02656i
$555$		0			0
$556$	−1.30656	−	1.30656i	−1.30656	−	1.30656i
$557$	0.149316	+	0.360480i	0.149316	+	0.360480i	0.980785	−	0.195090i	$-0.0625000\pi$
$557$	0.149316	+	0.360480i	0.149316	+	0.360480i	−0.831470	+	0.555570i	$0.812500\pi$
$558$		0			0
$559$			0.765367i			0.765367i
$560$		0			0
$561$		0			0
$562$	−1.08979	−	1.63099i	−1.08979	−	1.63099i
$563$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$563$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$564$		−	1.11114i		−	1.11114i
$565$		0			0
$566$	−0.785695	+	1.17588i	−0.785695	+	1.17588i
$567$		0			0
$568$	0.324423	+	1.63099i	0.324423	+	1.63099i
$569$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$569$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$570$		0			0
$571$	0.541196	−	1.30656i	0.541196	−	1.30656i	−0.382683	−	0.923880i	$-0.625000\pi$
$571$	0.541196	−	1.30656i	0.541196	−	1.30656i	0.923880	−	0.382683i	$-0.125000\pi$
$572$	0.636379	−	1.53636i	0.636379	−	1.53636i
$573$		0			0
$574$		0			0
$575$		0			0
$576$	0.923880	+	0.382683i	0.923880	+	0.382683i
$577$		0			0		1.00000			$0$
$577$		0			0		−1.00000			$\pi$
$578$	−0.980785	−	0.195090i	−0.980785	−	0.195090i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$	−1.38704	+	1.38704i	−1.38704	+	1.38704i
$586$	−0.617317	+	0.923880i	−0.617317	+	0.923880i
$587$	−1.81225	−	0.750661i	−1.81225	−	0.750661i	−0.980785	−	0.195090i	$-0.937500\pi$
$587$	−1.81225	−	0.750661i	−1.81225	−	0.750661i	−0.831470	−	0.555570i	$-0.812500\pi$
$588$		−	1.00000i		−	1.00000i
$589$		0			0
$590$	2.13770	+	3.19929i	2.13770	+	3.19929i
$591$			1.66294i			1.66294i
$592$		0			0
$593$		−	1.96157i		−	1.96157i	−0.195090	−	0.980785i	$-0.562500\pi$
$593$		−	1.96157i		−	1.96157i	0.195090	−	0.980785i	$-0.437500\pi$
$594$	1.38268	−	0.923880i	1.38268	−	0.923880i
$595$		0			0
$596$	−0.785695	−	0.785695i	−0.785695	−	0.785695i
$597$	1.70711	+	0.707107i	1.70711	+	0.707107i
$598$		0			0
$599$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$599$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$600$	1.58213	+	2.36783i	1.58213	+	2.36783i
$601$	1.30656	+	1.30656i	1.30656	+	1.30656i	0.923880	+	0.382683i	$0.125000\pi$
$601$	1.30656	+	1.30656i	1.30656	+	1.30656i	0.382683	+	0.923880i	$0.375000\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−3.19929	+	1.32519i	−3.19929	+	1.32519i
$606$		0			0
$607$	2.00000			2.00000			1.00000			$0$
$607$	2.00000			2.00000			1.00000			$0$
$608$		0			0
$609$		0			0
$610$	−0.541196	+	2.72078i	−0.541196	+	2.72078i
$611$	−1.02656	+	0.425215i	−1.02656	+	0.425215i
$612$		0			0
$613$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$613$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$614$		0			0
$615$	0.541196	+	0.541196i	0.541196	+	0.541196i
$616$		0			0
$617$	1.38704	−	1.38704i	1.38704	−	1.38704i	0.555570	−	0.831470i	$-0.312500\pi$
$617$	1.38704	−	1.38704i	1.38704	−	1.38704i	0.831470	−	0.555570i	$-0.187500\pi$
$618$		0			0
$619$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$619$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		−	1.00000i		−	1.00000i
$625$			4.26197i			4.26197i
$626$	−0.425215	−	0.636379i	−0.425215	−	0.636379i
$627$		0			0
$628$	0.765367			0.765367
$629$		0			0
$630$		0			0
$631$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$631$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$632$	0.425215	−	0.636379i	0.425215	−	0.636379i
$633$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$634$	1.08979	−	0.216773i	1.08979	−	0.216773i
$635$	−0.574531	+	1.38704i	−0.574531	+	1.38704i
$636$		0			0
$637$	−0.923880	+	0.382683i	−0.923880	+	0.382683i
$638$		0			0
$639$	1.66294			1.66294
$640$	−1.92388	−	0.382683i	−1.92388	−	0.382683i
$641$		0			0		1.00000			$0$
$641$		0			0		−1.00000			$\pi$
$642$		0			0
$643$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$643$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$644$		0			0
$645$	0.574531	−	1.38704i	0.574531	−	1.38704i
$646$		0			0
$647$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$647$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$648$	0.555570	−	0.831470i	0.555570	−	0.831470i
$649$	−2.30656	+	2.30656i	−2.30656	+	2.30656i
$650$	1.58213	−	2.36783i	1.58213	−	2.36783i
$651$		0			0
$652$		0			0
$653$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$653$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$654$		0			0
$655$		0			0
$656$	−0.390181			−0.390181
$657$		0			0
$658$		0			0
$659$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$659$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$660$	−2.30656	+	2.30656i	−2.30656	+	2.30656i
$661$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$661$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$662$		0			0
$663$		0			0
$664$	−0.0761205	+	0.382683i	−0.0761205	+	0.382683i
$665$		0			0
$666$		0			0
$667$		0			0
$668$	0.149316	+	0.360480i	0.149316	+	0.360480i
$669$		0			0
$670$		0			0
$671$	−2.35175			−2.35175
$672$		0			0
$673$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$673$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$674$	0.149316	−	0.750661i	0.149316	−	0.750661i
$675$	2.63099	−	1.08979i	2.63099	−	1.08979i
$676$	−0.923880	+	0.382683i	−0.923880	+	0.382683i
$677$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$677$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	−0.785695	+	0.785695i	−0.785695	+	0.785695i
$682$		0			0
$683$	−1.81225	−	0.750661i	−1.81225	−	0.750661i	−0.980785	−	0.195090i	$-0.937500\pi$
$683$	−1.81225	−	0.750661i	−1.81225	−	0.750661i	−0.831470	−	0.555570i	$-0.812500\pi$
$684$		0			0
$685$	0.834089	+	2.01367i	0.834089	+	2.01367i
$686$		0			0
$687$		0			0
$688$	0.292893	+	0.707107i	0.292893	+	0.707107i
$689$		0			0
$690$		0			0
$691$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$691$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	2.56292	−	2.56292i	2.56292	−	2.56292i
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$701$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$702$	−0.980785	−	0.195090i	−0.980785	−	0.195090i
$703$		0			0
$704$		−	1.66294i		−	1.66294i
$705$	2.17958			2.17958
$706$	0.382683	+	0.0761205i	0.382683	+	0.0761205i
$707$		0			0
$708$	−0.750661	+	1.81225i	−0.750661	+	1.81225i
$709$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$709$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$710$	−3.19929	+	0.636379i	−3.19929	+	0.636379i
$711$	−0.541196	−	0.541196i	−0.541196	−	0.541196i
$712$	0.216773	+	1.08979i	0.216773	+	1.08979i
$713$		0			0
$714$		0			0
$715$	3.01367	+	1.24830i	3.01367	+	1.24830i
$716$		0			0
$717$	0.149316	+	0.360480i	0.149316	+	0.360480i
$718$	−0.617317	−	0.923880i	−0.617317	−	0.923880i
$719$		0			0			−	1.00000i	$-0.5\pi$
$719$		0			0				1.00000i	$0.5\pi$
$720$	−0.750661	+	1.81225i	−0.750661	+	1.81225i
$721$		0			0
$722$	−0.831470	+	0.555570i	−0.831470	+	0.555570i
$723$		0			0
$724$	−1.30656	−	1.30656i	−1.30656	−	1.30656i
$725$		0			0
$726$	−1.46785	−	0.980785i	−1.46785	−	0.980785i
$727$	−1.30656	+	1.30656i	−1.30656	+	1.30656i	−0.382683	+	0.923880i	$0.625000\pi$
$727$	−1.30656	+	1.30656i	−1.30656	+	1.30656i	−0.923880	+	0.382683i	$0.875000\pi$
$728$		0			0
$729$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$730$		0			0
$731$		0			0
$732$	−1.30656	+	0.541196i	−1.30656	+	0.541196i
$733$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$733$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$734$	−0.275899	+	1.38704i	−0.275899	+	1.38704i
$735$	1.96157			1.96157
$736$		0			0
$737$		0			0
$738$	−0.0761205	+	0.382683i	−0.0761205	+	0.382683i
$739$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$739$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−0.275899	−	0.275899i	−0.275899	−	0.275899i	0.555570	−	0.831470i	$-0.312500\pi$
$743$	−0.275899	−	0.275899i	−0.275899	−	0.275899i	−0.831470	+	0.555570i	$0.812500\pi$
$744$		0			0
$745$	1.54120	−	1.54120i	1.54120	−	1.54120i
$746$	1.17588	+	0.785695i	1.17588	+	0.785695i
$747$	0.360480	+	0.149316i	0.360480	+	0.149316i
$748$		0			0
$749$		0			0
$750$	−3.01367	+	2.01367i	−3.01367	+	2.01367i
$751$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$751$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$752$	−0.785695	+	0.785695i	−0.785695	+	0.785695i
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−1.30656	−	0.541196i	−1.30656	−	0.541196i	−0.382683	−	0.923880i	$-0.625000\pi$
$757$	−1.30656	−	0.541196i	−1.30656	−	0.541196i	−0.923880	+	0.382683i	$0.875000\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	1.17588	+	1.17588i	1.17588	+	1.17588i	0.980785	+	0.195090i	$0.0625000\pi$
$761$	1.17588	+	1.17588i	1.17588	+	1.17588i	0.195090	+	0.980785i	$0.437500\pi$
$762$	−0.750661	+	0.149316i	−0.750661	+	0.149316i
$763$		0			0
$764$		0			0
$765$		0			0
$766$	−0.382683	−	0.0761205i	−0.382683	−	0.0761205i
$767$	1.96157			1.96157
$768$	−0.382683	−	0.923880i	−0.382683	−	0.923880i
$769$		0			0		1.00000			$0$
$769$		0			0		−1.00000			$\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−0.149316	+	0.360480i	−0.149316	+	0.360480i	−0.980785	−	0.195090i	$-0.937500\pi$
$773$	−0.149316	+	0.360480i	−0.149316	+	0.360480i	0.831470	+	0.555570i	$0.187500\pi$
$774$	0.750661	−	0.149316i	0.750661	−	0.149316i
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$	1.96157			1.96157
$781$	−1.05826	−	2.55487i	−1.05826	−	2.55487i
$782$		0			0
$783$		0			0
$784$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$785$			1.50132i			1.50132i
$786$		0			0
$787$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$787$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$788$	1.17588	−	1.17588i	1.17588	−	1.17588i
$789$		0			0
$790$	1.24830	+	0.834089i	1.24830	+	0.834089i
$791$		0			0
$792$	−1.63099	−	0.324423i	−1.63099	−	0.324423i
$793$	1.00000	+	1.00000i	1.00000	+	1.00000i
$794$		0			0
$795$		0			0
$796$	−0.707107	−	1.70711i	−0.707107	−	1.70711i
$797$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$797$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$798$		0			0
$799$		0			0
$800$	0.555570	−	2.79304i	0.555570	−	2.79304i
$801$	1.11114			1.11114
$802$	0.324423	−	1.63099i	0.324423	−	1.63099i
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$809$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$810$	1.63099	+	1.08979i	1.63099	+	1.08979i
$811$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$811$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		−	0.765367i		−	0.765367i
$821$	1.53636	+	0.636379i	1.53636	+	0.636379i	0.980785	−	0.195090i	$-0.0625000\pi$
$821$	1.53636	+	0.636379i	1.53636	+	0.636379i	0.555570	+	0.831470i	$0.312500\pi$
$822$	−0.617317	+	0.923880i	−0.617317	+	0.923880i
$823$	1.00000	−	1.00000i	1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$823$	1.00000	−	1.00000i	1.00000	−	1.00000i	1.00000			$0$
$824$		0			0
$825$	−3.34861	−	3.34861i	−3.34861	−	3.34861i
$826$		0			0
$827$	−0.750661	+	1.81225i	−0.750661	+	1.81225i	−0.195090	+	0.980785i	$0.562500\pi$
$827$	−0.750661	+	1.81225i	−0.750661	+	1.81225i	−0.555570	+	0.831470i	$0.687500\pi$
$828$		0			0
$829$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$829$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$830$	−0.750661	−	0.149316i	−0.750661	−	0.149316i
$831$	−1.84776			−1.84776
$832$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$833$		0			0
$834$	1.81225	+	0.360480i	1.81225	+	0.360480i
$835$	−0.707107	+	0.292893i	−0.707107	+	0.292893i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	1.38704	+	1.38704i	1.38704	+	1.38704i	0.831470	+	0.555570i	$0.187500\pi$
$839$	1.38704	+	1.38704i	1.38704	+	1.38704i	0.555570	+	0.831470i	$0.312500\pi$
$840$		0			0
$841$	0.707107	−	0.707107i	0.707107	−	0.707107i
$842$		0			0
$843$	1.81225	+	0.750661i	1.81225	+	0.750661i
$844$			1.41421i			1.41421i
$845$	−0.750661	−	1.81225i	−0.750661	−	1.81225i
$846$	0.617317	+	0.923880i	0.617317	+	0.923880i
$847$		0			0
$848$		0			0
$849$		−	1.41421i		−	1.41421i
$850$		0			0
$851$		0			0
$852$	−1.17588	−	1.17588i	−1.17588	−	1.17588i
$853$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$853$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$857$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$858$	0.324423	+	1.63099i	0.324423	+	1.63099i
$859$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$859$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$860$	−1.38704	+	0.574531i	−1.38704	+	0.574531i
$861$		0			0
$862$	−0.0761205	+	0.382683i	−0.0761205	+	0.382683i
$863$	1.11114			1.11114			0.555570	−	0.831470i	$-0.312500\pi$
$863$	1.11114			1.11114			0.555570	+	0.831470i	$0.312500\pi$
$864$	−0.980785	+	0.195090i	−0.980785	+	0.195090i
$865$		0			0
$866$	0.360480	−	1.81225i	0.360480	−	1.81225i
$867$	0.923880	−	0.382683i	0.923880	−	0.382683i
$868$		0			0
$869$	−0.487064	+	1.17588i	−0.487064	+	1.17588i
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$877$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$878$	−1.17588	+	0.785695i	−1.17588	+	0.785695i
$879$		−	1.11114i		−	1.11114i
$880$	3.26197			3.26197
$881$		0			0			−	1.00000i	$-0.5\pi$
$881$		0			0				1.00000i	$0.5\pi$
$882$	0.555570	+	0.831470i	0.555570	+	0.831470i
$883$	−0.541196	−	1.30656i	−0.541196	−	1.30656i	−0.923880	−	0.382683i	$-0.875000\pi$
$883$	−0.541196	−	1.30656i	−0.541196	−	1.30656i	0.382683	−	0.923880i	$-0.375000\pi$
$884$		0			0
$885$	−3.55487	−	1.47247i	−3.55487	−	1.47247i
$886$		0			0
$887$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$887$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$888$		0			0
$889$		0			0
$890$	−2.13770	+	0.425215i	−2.13770	+	0.425215i
$891$	−0.636379	+	1.53636i	−0.636379	+	1.53636i
$892$		0			0
$893$		0			0
$894$	1.08979	+	0.216773i	1.08979	+	0.216773i
$895$		0			0
$896$		0			0
$897$		0			0
$898$	−1.08979	−	0.216773i	−1.08979	−	0.216773i
$899$		0			0
$900$	−2.63099	−	1.08979i	−2.63099	−	1.08979i
$901$		0			0
$902$	0.636379	−	0.126584i	0.636379	−	0.126584i
$903$		0			0
$904$		0			0
$905$	2.56292	−	2.56292i	2.56292	−	2.56292i
$906$		0			0
$907$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$907$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$908$	1.11114			1.11114
$909$		0			0
$910$		0			0
$911$		0			0			−	1.00000i	$-0.5\pi$
$911$		0			0				1.00000i	$0.5\pi$
$912$		0			0
$913$		−	0.648847i		−	0.648847i
$914$		0			0
$915$	−1.06159	−	2.56292i	−1.06159	−	2.56292i
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−1.30656	+	1.30656i	−1.30656	+	1.30656i	−0.382683	+	0.923880i	$0.625000\pi$
$919$	−1.30656	+	1.30656i	−1.30656	+	1.30656i	−0.923880	+	0.382683i	$0.875000\pi$
$920$		0			0
$921$		0			0
$922$	−0.324423	−	1.63099i	−0.324423	−	1.63099i
$923$	−0.636379	+	1.53636i	−0.636379	+	1.53636i
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$	1.96157			1.96157			0.980785	−	0.195090i	$-0.0625000\pi$
$929$	1.96157			1.96157			0.980785	+	0.195090i	$0.0625000\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$	0.555570	+	0.831470i	0.555570	+	0.831470i
$937$	−0.541196	+	0.541196i	−0.541196	+	0.541196i	−0.923880	−	0.382683i	$-0.875000\pi$
$937$	−0.541196	+	0.541196i	−0.541196	+	0.541196i	0.382683	+	0.923880i	$0.375000\pi$
$938$		0			0
$939$	0.707107	+	0.292893i	0.707107	+	0.292893i
$940$	−1.54120	−	1.54120i	−1.54120	−	1.54120i
$941$	−0.149316	−	0.360480i	−0.149316	−	0.360480i	0.831470	−	0.555570i	$-0.187500\pi$
$941$	−0.149316	−	0.360480i	−0.149316	−	0.360480i	−0.980785	+	0.195090i	$0.937500\pi$
$942$	−0.636379	+	0.425215i	−0.636379	+	0.425215i
$943$		0			0
$944$	1.81225	−	0.750661i	1.81225	−	0.750661i
$945$		0			0
$946$	−0.707107	−	1.05826i	−0.707107	−	1.05826i
$947$	0.425215	+	1.02656i	0.425215	+	1.02656i	0.980785	+	0.195090i	$0.0625000\pi$
$947$	0.425215	+	1.02656i	0.425215	+	1.02656i	−0.555570	+	0.831470i	$0.687500\pi$
$948$			0.765367i			0.765367i
$949$		0			0
$950$		0			0
$951$	−0.785695	+	0.785695i	−0.785695	+	0.785695i
$952$		0			0
$953$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$953$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$954$		0			0
$955$		0			0
$956$	0.149316	−	0.360480i	0.149316	−	0.360480i
$957$		0			0
$958$	1.63099	+	0.324423i	1.63099	+	0.324423i
$959$		0			0
$960$	1.81225	−	0.750661i	1.81225	−	0.750661i
$961$	1.00000			1.00000
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$967$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$968$	0.344406	+	1.73145i	0.344406	+	1.73145i
$969$		0			0
$970$		0			0
$971$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$971$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$972$			1.00000i			1.00000i
$973$		0			0
$974$		0			0
$975$			2.84776i			2.84776i
$976$	1.30656	+	0.541196i	1.30656	+	0.541196i
$977$		−	0.390181i		−	0.390181i	−0.980785	−	0.195090i	$-0.937500\pi$
$977$		−	0.390181i		−	0.390181i	0.980785	−	0.195090i	$-0.0625000\pi$
$978$		0			0
$979$	−0.707107	−	1.70711i	−0.707107	−	1.70711i
$980$	−1.38704	−	1.38704i	−1.38704	−	1.38704i
$981$		0			0
$982$		0			0
$983$	−1.38704	+	1.38704i	−1.38704	+	1.38704i	−0.555570	+	0.831470i	$0.687500\pi$
$983$	−1.38704	+	1.38704i	−1.38704	+	1.38704i	−0.831470	+	0.555570i	$0.812500\pi$
$984$	0.324423	−	0.216773i	0.324423	−	0.216773i
$985$	2.30656	+	2.30656i	2.30656	+	2.30656i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$	0.636379	−	3.19929i	0.636379	−	3.19929i
$991$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$991$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	3.34861	−	1.38704i	3.34861	−	1.38704i
$996$	−0.149316	−	0.360480i	−0.149316	−	0.360480i
$997$	−0.707107	+	1.70711i	−0.707107	+	1.70711i			1.00000i	$0.5\pi$
$997$	−0.707107	+	1.70711i	−0.707107	+	1.70711i	−0.707107	+	0.707107i	$0.750000\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1248.1.cm.a.1013.4	yes	16
3.2	odd	2	inner	1248.1.cm.a.1013.1	yes	16
13.12	even	2	inner	1248.1.cm.a.1013.1	yes	16
32.29	even	8	inner	1248.1.cm.a.701.4	yes	16
39.38	odd	2	CM	1248.1.cm.a.1013.4	yes	16
96.29	odd	8	inner	1248.1.cm.a.701.1	✓	16
416.285	even	8	inner	1248.1.cm.a.701.1	✓	16
1248.701	odd	8	inner	1248.1.cm.a.701.4	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1248.1.cm.a.701.1	✓	16	96.29	odd	8	inner
1248.1.cm.a.701.1	✓	16	416.285	even	8	inner
1248.1.cm.a.701.4	yes	16	32.29	even	8	inner
1248.1.cm.a.701.4	yes	16	1248.701	odd	8	inner
1248.1.cm.a.1013.1	yes	16	3.2	odd	2	inner
1248.1.cm.a.1013.1	yes	16	13.12	even	2	inner
1248.1.cm.a.1013.4	yes	16	1.1	even	1	trivial
1248.1.cm.a.1013.4	yes	16	39.38	odd	2	CM

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

Level:	\( N \)	\(=\)	\( 1248 = 2^{5} \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1248.cm (of order \(8\), degree \(4\), minimal)

\(f(q)\)	\(=\)	\(q+(0.980785 + 0.195090i) q^{2} +(-0.923880 + 0.382683i) q^{3} +(0.923880 + 0.382683i) q^{4} +(-0.750661 + 1.81225i) q^{5} +(-0.980785 + 0.195090i) q^{6} +(0.831470 + 0.555570i) q^{8} +(0.707107 - 0.707107i) q^{9} +(-1.08979 + 1.63099i) q^{10} +(-1.53636 - 0.636379i) q^{11} -1.00000 q^{12} +(0.382683 + 0.923880i) q^{13} -1.96157i q^{15} +(0.707107 + 0.707107i) q^{16} +(0.831470 - 0.555570i) q^{18} +(-1.38704 + 1.38704i) q^{20} +(-1.38268 - 0.923880i) q^{22} +(-0.980785 - 0.195090i) q^{24} +(-2.01367 - 2.01367i) q^{25} +(0.195090 + 0.980785i) q^{26} +(-0.382683 + 0.923880i) q^{27} +(0.382683 - 1.92388i) q^{30} +(0.555570 + 0.831470i) q^{32} +1.66294 q^{33} +(0.923880 - 0.382683i) q^{36} +(-0.707107 - 0.707107i) q^{39} +(-1.63099 + 1.08979i) q^{40} +(-0.275899 + 0.275899i) q^{41} +(0.707107 + 0.292893i) q^{43} +(-1.17588 - 1.17588i) q^{44} +(0.750661 + 1.81225i) q^{45} +1.11114i q^{47} +(-0.923880 - 0.382683i) q^{48} +1.00000i q^{49} +(-1.58213 - 2.36783i) q^{50} +1.00000i q^{52} +(-0.555570 + 0.831470i) q^{54} +(2.30656 - 2.30656i) q^{55} +(0.750661 - 1.81225i) q^{59} +(0.750661 - 1.81225i) q^{60} +(1.30656 - 0.541196i) q^{61} +(0.382683 + 0.923880i) q^{64} -1.96157 q^{65} +(1.63099 + 0.324423i) q^{66} +(1.17588 + 1.17588i) q^{71} +(0.980785 - 0.195090i) q^{72} +(2.63099 + 1.08979i) q^{75} +(-0.555570 - 0.831470i) q^{78} -0.765367i q^{79} +(-1.81225 + 0.750661i) q^{80} -1.00000i q^{81} +(-0.324423 + 0.216773i) q^{82} +(0.149316 + 0.360480i) q^{83} +(0.636379 + 0.425215i) q^{86} +(-0.923880 - 1.38268i) q^{88} +(0.785695 + 0.785695i) q^{89} +(0.382683 + 1.92388i) q^{90} +(-0.216773 + 1.08979i) q^{94} +(-0.831470 - 0.555570i) q^{96} +(-0.195090 + 0.980785i) q^{98} +(-1.53636 + 0.636379i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 16 q^{12} - 16 q^{22} + 16 q^{55} + 16 q^{75}+O(q^{100}) \) 16 * q - 16 * q^12 - 16 * q^22 + 16 * q^55 + 16 * q^75

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

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Database

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