LMFDB - Embedded newform 592.2.i.c.433.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [592,2,Mod(417,592)] 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("592.417"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(592, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 4])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 592 = 2^{4} \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	592.i (of order $3$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	no
Analytic conductor:	$4.72714379966$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 37)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			433.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	592.433
Dual form			592.2.i.c.417.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.500000 + 0.866025i) q^{5} +(1.00000 - 1.73205i) q^{7} +(1.50000 + 2.59808i) q^{9} +2.00000 q^{11} +(1.00000 - 1.73205i) q^{13} +(-1.50000 - 2.59808i) q^{17} +(-3.00000 + 5.19615i) q^{19} +4.00000 q^{23} +(2.00000 + 3.46410i) q^{25} +9.00000 q^{29} +10.0000 q^{31} +(1.00000 + 1.73205i) q^{35} +(-5.50000 + 2.59808i) q^{37} +(4.50000 - 7.79423i) q^{41} -2.00000 q^{43} -3.00000 q^{45} -6.00000 q^{47} +(1.50000 + 2.59808i) q^{49} +(1.00000 + 1.73205i) q^{53} +(-1.00000 + 1.73205i) q^{55} +(-2.00000 - 3.46410i) q^{59} +(-0.500000 + 0.866025i) q^{61} +6.00000 q^{63} +(1.00000 + 1.73205i) q^{65} +(-5.00000 + 8.66025i) q^{67} +(3.00000 - 5.19615i) q^{71} -10.0000 q^{73} +(2.00000 - 3.46410i) q^{77} +(5.00000 - 8.66025i) q^{79} +(-4.50000 + 7.79423i) q^{81} +(-6.00000 - 10.3923i) q^{83} +3.00000 q^{85} +(-3.50000 - 6.06218i) q^{89} +(-2.00000 - 3.46410i) q^{91} +(-3.00000 - 5.19615i) q^{95} +7.00000 q^{97} +(3.00000 + 5.19615i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{5} + 2 q^{7} + 3 q^{9} + 4 q^{11} + 2 q^{13} - 3 q^{17} - 6 q^{19} + 8 q^{23} + 4 q^{25} + 18 q^{29} + 20 q^{31} + 2 q^{35} - 11 q^{37} + 9 q^{41} - 4 q^{43} - 6 q^{45} - 12 q^{47} + 3 q^{49}+ \cdots + 6 q^{99}+O(q^{100}) $ 2 * q - q^5 + 2 * q^7 + 3 * q^9 + 4 * q^11 + 2 * q^13 - 3 * q^17 - 6 * q^19 + 8 * q^23 + 4 * q^25 + 18 * q^29 + 20 * q^31 + 2 * q^35 - 11 * q^37 + 9 * q^41 - 4 * q^43 - 6 * q^45 - 12 * q^47 + 3 * q^49 + 2 * q^53 - 2 * q^55 - 4 * q^59 - q^61 + 12 * q^63 + 2 * q^65 - 10 * q^67 + 6 * q^71 - 20 * q^73 + 4 * q^77 + 10 * q^79 - 9 * q^81 - 12 * q^83 + 6 * q^85 - 7 * q^89 - 4 * q^91 - 6 * q^95 + 14 * q^97 + 6 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/592\mathbb{Z}\right)^\times$.

$n$	$113$	$149$	$223$
$\chi(n)$	$e\left(\frac{1}{3}\right)$	$1$	$1$

Coefficient data

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

Currently showing only $a_p$; display all $a_n$ Currently showing all $a_n$; display 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
$3$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$3$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$4$		0			0
$5$	−0.500000	+	0.866025i	−0.223607	+	0.387298i	−0.955901	−	0.293691i	$-0.905116\pi$
$5$	−0.500000	+	0.866025i	−0.223607	+	0.387298i	0.732294	+	0.680989i	$0.238450\pi$
$6$		0			0
$7$	1.00000	−	1.73205i	0.377964	−	0.654654i	−0.612801	−	0.790237i	$-0.709957\pi$
$7$	1.00000	−	1.73205i	0.377964	−	0.654654i	0.990766	+	0.135583i	$0.0432908\pi$
$8$		0			0
$9$	1.50000	+	2.59808i	0.500000	+	0.866025i
$10$		0			0
$11$	2.00000			0.603023			0.301511	−	0.953463i	$-0.402509\pi$
$11$	2.00000			0.603023			0.301511	+	0.953463i	$0.402509\pi$
$12$		0			0
$13$	1.00000	−	1.73205i	0.277350	−	0.480384i	−0.693375	−	0.720577i	$-0.743877\pi$
$13$	1.00000	−	1.73205i	0.277350	−	0.480384i	0.970725	+	0.240192i	$0.0772105\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−1.50000	−	2.59808i	−0.363803	−	0.630126i	0.624780	−	0.780801i	$-0.285189\pi$
$17$	−1.50000	−	2.59808i	−0.363803	−	0.630126i	−0.988583	+	0.150675i	$0.951855\pi$
$18$		0			0
$19$	−3.00000	+	5.19615i	−0.688247	+	1.19208i	0.284157	+	0.958778i	$0.408286\pi$
$19$	−3.00000	+	5.19615i	−0.688247	+	1.19208i	−0.972404	+	0.233301i	$0.925047\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	4.00000			0.834058			0.417029	−	0.908893i	$-0.363071\pi$
$23$	4.00000			0.834058			0.417029	+	0.908893i	$0.363071\pi$
$24$		0			0
$25$	2.00000	+	3.46410i	0.400000	+	0.692820i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	9.00000			1.67126			0.835629	−	0.549294i	$-0.185103\pi$
$29$	9.00000			1.67126			0.835629	+	0.549294i	$0.185103\pi$
$30$		0			0
$31$	10.0000			1.79605			0.898027	−	0.439941i	$-0.145001\pi$
$31$	10.0000			1.79605			0.898027	+	0.439941i	$0.145001\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	1.00000	+	1.73205i	0.169031	+	0.292770i
$36$		0			0
$37$	−5.50000	+	2.59808i	−0.904194	+	0.427121i
$37$	−5.50000	+	2.59808i	−0.904194	+	0.427121i
$38$		0			0
$39$		0			0
$40$		0			0
$41$	4.50000	−	7.79423i	0.702782	−	1.21725i	−0.264704	−	0.964330i	$-0.585274\pi$
$41$	4.50000	−	7.79423i	0.702782	−	1.21725i	0.967486	−	0.252924i	$-0.0813924\pi$
$42$		0			0
$43$	−2.00000			−0.304997			−0.152499	−	0.988304i	$-0.548732\pi$
$43$	−2.00000			−0.304997			−0.152499	+	0.988304i	$0.548732\pi$
$44$		0			0
$45$	−3.00000			−0.447214
$46$		0			0
$47$	−6.00000			−0.875190			−0.437595	−	0.899172i	$-0.644170\pi$
$47$	−6.00000			−0.875190			−0.437595	+	0.899172i	$0.644170\pi$
$48$		0			0
$49$	1.50000	+	2.59808i	0.214286	+	0.371154i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	1.00000	+	1.73205i	0.137361	+	0.237915i	0.926497	−	0.376303i	$-0.122805\pi$
$53$	1.00000	+	1.73205i	0.137361	+	0.237915i	−0.789136	+	0.614218i	$0.789471\pi$
$54$		0			0
$55$	−1.00000	+	1.73205i	−0.134840	+	0.233550i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−2.00000	−	3.46410i	−0.260378	−	0.450988i	0.705965	−	0.708247i	$-0.250514\pi$
$59$	−2.00000	−	3.46410i	−0.260378	−	0.450988i	−0.966342	+	0.257260i	$0.917180\pi$
$60$		0			0
$61$	−0.500000	+	0.866025i	−0.0640184	+	0.110883i	−0.896258	−	0.443533i	$-0.853725\pi$
$61$	−0.500000	+	0.866025i	−0.0640184	+	0.110883i	0.832240	+	0.554416i	$0.187058\pi$
$62$		0			0
$63$	6.00000			0.755929
$64$		0			0
$65$	1.00000	+	1.73205i	0.124035	+	0.214834i
$66$		0			0
$67$	−5.00000	+	8.66025i	−0.610847	+	1.05802i	0.380251	+	0.924883i	$0.375838\pi$
$67$	−5.00000	+	8.66025i	−0.610847	+	1.05802i	−0.991098	+	0.133135i	$0.957496\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	3.00000	−	5.19615i	0.356034	−	0.616670i	−0.631260	−	0.775571i	$-0.717462\pi$
$71$	3.00000	−	5.19615i	0.356034	−	0.616670i	0.987294	+	0.158901i	$0.0507952\pi$
$72$		0			0
$73$	−10.0000			−1.17041			−0.585206	−	0.810885i	$-0.698986\pi$
$73$	−10.0000			−1.17041			−0.585206	+	0.810885i	$0.698986\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	2.00000	−	3.46410i	0.227921	−	0.394771i
$78$		0			0
$79$	5.00000	−	8.66025i	0.562544	−	0.974355i	−0.434730	−	0.900561i	$-0.643156\pi$
$79$	5.00000	−	8.66025i	0.562544	−	0.974355i	0.997274	−	0.0737937i	$-0.0235106\pi$
$80$		0			0
$81$	−4.50000	+	7.79423i	−0.500000	+	0.866025i
$82$		0			0
$83$	−6.00000	−	10.3923i	−0.658586	−	1.14070i	−0.980982	−	0.194099i	$-0.937822\pi$
$83$	−6.00000	−	10.3923i	−0.658586	−	1.14070i	0.322396	−	0.946605i	$-0.395512\pi$
$84$		0			0
$85$	3.00000			0.325396
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−3.50000	−	6.06218i	−0.370999	−	0.642590i	0.618720	−	0.785611i	$-0.287651\pi$
$89$	−3.50000	−	6.06218i	−0.370999	−	0.642590i	−0.989720	+	0.143022i	$0.954318\pi$
$90$		0			0
$91$	−2.00000	−	3.46410i	−0.209657	−	0.363137i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−3.00000	−	5.19615i	−0.307794	−	0.533114i
$96$		0			0
$97$	7.00000			0.710742			0.355371	−	0.934725i	$-0.384354\pi$
$97$	7.00000			0.710742			0.355371	+	0.934725i	$0.384354\pi$
$98$		0			0
$99$	3.00000	+	5.19615i	0.301511	+	0.522233i

Currently showing only $a_p$; display all $a_n$ Currently showing all $a_n$; display only $a_p$

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	592.2.i.c.433.1		2
4.3	odd	2		37.2.c.a.26.1	yes	2
12.11	even	2		333.2.f.a.100.1		2
20.3	even	4		925.2.o.a.174.1		4
20.7	even	4		925.2.o.a.174.2		4
20.19	odd	2		925.2.e.a.26.1		2
37.10	even	3	inner	592.2.i.c.417.1		2
148.11	odd	6		1369.2.a.b.1.1		1
148.47	odd	6		37.2.c.a.10.1	✓	2
148.63	odd	6		1369.2.a.d.1.1		1
148.103	even	12		1369.2.b.b.1368.2		2
148.119	even	12		1369.2.b.b.1368.1		2
444.47	even	6		333.2.f.a.10.1		2
740.47	even	12		925.2.o.a.824.1		4
740.343	even	12		925.2.o.a.824.2		4
740.639	odd	6		925.2.e.a.676.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
37.2.c.a.10.1	✓	2	148.47	odd	6
37.2.c.a.26.1	yes	2	4.3	odd	2
333.2.f.a.10.1		2	444.47	even	6
333.2.f.a.100.1		2	12.11	even	2
592.2.i.c.417.1		2	37.10	even	3	inner
592.2.i.c.433.1		2	1.1	even	1	trivial
925.2.e.a.26.1		2	20.19	odd	2
925.2.e.a.676.1		2	740.639	odd	6
925.2.o.a.174.1		4	20.3	even	4
925.2.o.a.174.2		4	20.7	even	4
925.2.o.a.824.1		4	740.47	even	12
925.2.o.a.824.2		4	740.343	even	12
1369.2.a.b.1.1		1	148.11	odd	6
1369.2.a.d.1.1		1	148.63	odd	6
1369.2.b.b.1368.1		2	148.119	even	12
1369.2.b.b.1368.2		2	148.103	even	12

Overview	Random
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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:	\( N \)	\(=\)	\( 592 = 2^{4} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	592.i (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.500000 + 0.866025i) q^{5} +(1.00000 - 1.73205i) q^{7} +(1.50000 + 2.59808i) q^{9} +2.00000 q^{11} +(1.00000 - 1.73205i) q^{13} +(-1.50000 - 2.59808i) q^{17} +(-3.00000 + 5.19615i) q^{19} +4.00000 q^{23} +(2.00000 + 3.46410i) q^{25} +9.00000 q^{29} +10.0000 q^{31} +(1.00000 + 1.73205i) q^{35} +(-5.50000 + 2.59808i) q^{37} +(4.50000 - 7.79423i) q^{41} -2.00000 q^{43} -3.00000 q^{45} -6.00000 q^{47} +(1.50000 + 2.59808i) q^{49} +(1.00000 + 1.73205i) q^{53} +(-1.00000 + 1.73205i) q^{55} +(-2.00000 - 3.46410i) q^{59} +(-0.500000 + 0.866025i) q^{61} +6.00000 q^{63} +(1.00000 + 1.73205i) q^{65} +(-5.00000 + 8.66025i) q^{67} +(3.00000 - 5.19615i) q^{71} -10.0000 q^{73} +(2.00000 - 3.46410i) q^{77} +(5.00000 - 8.66025i) q^{79} +(-4.50000 + 7.79423i) q^{81} +(-6.00000 - 10.3923i) q^{83} +3.00000 q^{85} +(-3.50000 - 6.06218i) q^{89} +(-2.00000 - 3.46410i) q^{91} +(-3.00000 - 5.19615i) q^{95} +7.00000 q^{97} +(3.00000 + 5.19615i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{5} + 2 q^{7} + 3 q^{9} + 4 q^{11} + 2 q^{13} - 3 q^{17} - 6 q^{19} + 8 q^{23} + 4 q^{25} + 18 q^{29} + 20 q^{31} + 2 q^{35} - 11 q^{37} + 9 q^{41} - 4 q^{43} - 6 q^{45} - 12 q^{47} + 3 q^{49}+ \cdots + 6 q^{99}+O(q^{100}) \) 2 * q - q^5 + 2 * q^7 + 3 * q^9 + 4 * q^11 + 2 * q^13 - 3 * q^17 - 6 * q^19 + 8 * q^23 + 4 * q^25 + 18 * q^29 + 20 * q^31 + 2 * q^35 - 11 * q^37 + 9 * q^41 - 4 * q^43 - 6 * q^45 - 12 * q^47 + 3 * q^49 + 2 * q^53 - 2 * q^55 - 4 * q^59 - q^61 + 12 * q^63 + 2 * q^65 - 10 * q^67 + 6 * q^71 - 20 * q^73 + 4 * q^77 + 10 * q^79 - 9 * q^81 - 12 * q^83 + 6 * q^85 - 7 * q^89 - 4 * q^91 - 6 * q^95 + 14 * q^97 + 6 * q^99

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