Newform orbit 72.2.i.b

• Label: 72.2.i.b
• Level: $72$
• Weight: $2$
• Character orbit: 72.i
• Analytic conductor: $0.575$
• Analytic rank: $0$
• Dimension: $4$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 72 = 2^{3} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	72.i (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.574922894553\)
Analytic rank:	\(0\)
Dimension:	\(4\)
Relative dimension:	\(2\) over \(\Q(\zeta_{3})\)
Coefficient field:	\(\Q(\sqrt{-3}, \sqrt{-11})\)
Defining polynomial:	\( x^{4} - x^{3} - 2x^{2} - 3x + 9 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q + \beta_1 q^{3} + ( - 2 \beta_{3} - \beta_{2} + \beta_1) q^{5} + (\beta_{3} - 2 \beta_{2} - 2 \beta_1 + 1) q^{7} + (\beta_{3} + 3 \beta_{2}) q^{9}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + q^{3} + q^{5} - 3 q^{7} + 5 q^{9}+O(q^{10}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - 2x^{2} - 3x + 9 \) :

\(\beta_{1}\)	\(=\)	\( \nu \)
\(\beta_{2}\)	\(=\)	\( ( \nu^{3} + 2\nu^{2} - 2\nu - 3 ) / 6 \)
\(\beta_{3}\)	\(=\)	\( ( -\nu^{3} + 2\nu + 3 ) / 2 \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/72\mathbb{Z}\right)^\times\).

\(n\)	\(37\)	\(55\)	\(65\)
\(\chi(n)\)	\(1\)	\(1\)	\(-1 + \beta_{2}\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

25.1

−1.18614	+	1.26217i
1.68614	−	0.396143i
−1.18614	−	1.26217i
1.68614	+	0.396143i

0

−1.18614

+

1.26217i

0

1.68614

+

2.92048i

0

0.686141

−

1.18843i

0

−0.186141

−

2.99422i

0

25.2

0

1.68614

−

0.396143i

0

−1.18614

−

2.05446i

0

−2.18614

+

3.78651i

0

2.68614

−

1.33591i

0

49.1

0

−1.18614

−

1.26217i

0

1.68614

−

2.92048i

0

0.686141

+

1.18843i

0

−0.186141

+

2.99422i

0

49.2

0

1.68614

+

0.396143i

0

−1.18614

+

2.05446i

0

−2.18614

−

3.78651i

0

2.68614

+

1.33591i

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
9.c	even	3	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	72.2.i.b	✓	4
3.b	odd	2	1		216.2.i.b		4
4.b	odd	2	1		144.2.i.d		4
8.b	even	2	1		576.2.i.j		4
8.d	odd	2	1		576.2.i.l		4
9.c	even	3	1	inner	72.2.i.b	✓	4
9.c	even	3	1		648.2.a.f		2
9.d	odd	6	1		216.2.i.b		4
9.d	odd	6	1		648.2.a.g		2
12.b	even	2	1		432.2.i.d		4
24.f	even	2	1		1728.2.i.j		4
24.h	odd	2	1		1728.2.i.i		4
36.f	odd	6	1		144.2.i.d		4
36.f	odd	6	1		1296.2.a.n		2
36.h	even	6	1		432.2.i.d		4
36.h	even	6	1		1296.2.a.p		2
72.j	odd	6	1		1728.2.i.i		4
72.j	odd	6	1		5184.2.a.bp		2
72.l	even	6	1		1728.2.i.j		4
72.l	even	6	1		5184.2.a.bo		2
72.n	even	6	1		576.2.i.j		4
72.n	even	6	1		5184.2.a.bt		2
72.p	odd	6	1		576.2.i.l		4
72.p	odd	6	1		5184.2.a.bs		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
72.2.i.b	✓	4	1.a	even	1	1	trivial
72.2.i.b	✓	4	9.c	even	3	1	inner
144.2.i.d		4	4.b	odd	2	1
144.2.i.d		4	36.f	odd	6	1
216.2.i.b		4	3.b	odd	2	1
216.2.i.b		4	9.d	odd	6	1
432.2.i.d		4	12.b	even	2	1
432.2.i.d		4	36.h	even	6	1
576.2.i.j		4	8.b	even	2	1
576.2.i.j		4	72.n	even	6	1
576.2.i.l		4	8.d	odd	2	1
576.2.i.l		4	72.p	odd	6	1
648.2.a.f		2	9.c	even	3	1
648.2.a.g		2	9.d	odd	6	1
1296.2.a.n		2	36.f	odd	6	1
1296.2.a.p		2	36.h	even	6	1
1728.2.i.i		4	24.h	odd	2	1
1728.2.i.i		4	72.j	odd	6	1
1728.2.i.j		4	24.f	even	2	1
1728.2.i.j		4	72.l	even	6	1
5184.2.a.bo		2	72.l	even	6	1
5184.2.a.bp		2	72.j	odd	6	1
5184.2.a.bs		2	72.p	odd	6	1
5184.2.a.bt		2	72.n	even	6	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} - T_{5}^{3} + 9T_{5}^{2} + 8T_{5} + 64 \) acting on \(S_{2}^{\mathrm{new}}(72, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{4} \)
$3$	T^4 - T^3 - 2*T^2 - 3*T + 9
$5$	T^4 - T^3 + 9*T^2 + 8*T + 64
$7$	T^4 + 3*T^3 + 15*T^2 - 18*T + 36
$11$	\( (T^{2} + T + 1)^{2} \)