Newform orbit 72.2.i.a

• Label: 72.2.i.a
• Level: $72$
• Weight: $2$
• Character orbit: 72.i
• Analytic conductor: $0.575$
• Analytic rank: $0$
• Dimension: $2$
• 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:	\(2\)
Coefficient field:	\(\Q(\sqrt{-3}) \)
Defining polynomial:	\( x^{2} - x + 1 \)
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 primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q + (2 \zeta_{6} - 1) q^{3} + ( - \zeta_{6} + 1) q^{5} + 3 \zeta_{6} q^{7} - 3 q^{9}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{5} + 3 q^{7} - 6 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(37\)	\(55\)	\(65\)
\(\chi(n)\)	\(1\)	\(1\)	\(-1 + \zeta_{6}\)

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

0.500000	−	0.866025i
0.500000	+	0.866025i

0

−

1.73205i

0

0.500000

+

0.866025i

0

1.50000

−

2.59808i

0

−3.00000

0

49.1

0

1.73205i

0

0.500000

−

0.866025i

0

1.50000

+

2.59808i

0

−3.00000

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.a	✓	2
3.b	odd	2	1		216.2.i.a		2
4.b	odd	2	1		144.2.i.b		2
8.b	even	2	1		576.2.i.d		2
8.d	odd	2	1		576.2.i.c		2
9.c	even	3	1	inner	72.2.i.a	✓	2
9.c	even	3	1		648.2.a.a		1
9.d	odd	6	1		216.2.i.a		2
9.d	odd	6	1		648.2.a.c		1
12.b	even	2	1		432.2.i.a		2
24.f	even	2	1		1728.2.i.g		2
24.h	odd	2	1		1728.2.i.h		2
36.f	odd	6	1		144.2.i.b		2
36.f	odd	6	1		1296.2.a.e		1
36.h	even	6	1		432.2.i.a		2
36.h	even	6	1		1296.2.a.i		1
72.j	odd	6	1		1728.2.i.h		2
72.j	odd	6	1		5184.2.a.i		1
72.l	even	6	1		1728.2.i.g		2
72.l	even	6	1		5184.2.a.n		1
72.n	even	6	1		576.2.i.d		2
72.n	even	6	1		5184.2.a.s		1
72.p	odd	6	1		576.2.i.c		2
72.p	odd	6	1		5184.2.a.x		1

    

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} - T_{5} + 1 \) acting on \(S_{2}^{\mathrm{new}}(72, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} \)
$3$	\( T^{2} + 3 \)
$5$	\( T^{2} - T + 1 \)
$7$	\( T^{2} - 3T + 9 \)
$11$	\( T^{2} + 5T + 25 \)