Newform orbit 72.4.d.b

• Label: 72.4.d.b
• Level: $72$
• Weight: $4$
• Character orbit: 72.d
• Analytic conductor: $4.248$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 72 = 2^{3} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	72.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.24813752041\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{-7}) \)
Defining polynomial:	\( x^{2} - x + 2 \)
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 2 \)
Twist minimal:	no (minimal twist has level 8)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{-7}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q + (\beta + 1) q^{2} + (2 \beta - 6) q^{4} + 4 \beta q^{5} - 8 q^{7} + ( - 4 \beta - 20) q^{8}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} - 12 q^{4} - 16 q^{7} - 40 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(37\)	\(55\)	\(65\)
\(\chi(n)\)	\(-1\)	\(1\)	\(1\)

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

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

37.1

0.500000	−	1.32288i
0.500000	+	1.32288i

1.00000

−

2.64575i

0

−6.00000

−

5.29150i

−

10.5830i

0

−8.00000

−20.0000

+

10.5830i

0

−28.0000

−

10.5830i

37.2

1.00000

+

2.64575i

0

−6.00000

+

5.29150i

10.5830i

0

−8.00000

−20.0000

−

10.5830i

0

−28.0000

+

10.5830i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
8.b	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	72.4.d.b		2
3.b	odd	2	1		8.4.b.a	✓	2
4.b	odd	2	1		288.4.d.a		2
8.b	even	2	1	inner	72.4.d.b		2
8.d	odd	2	1		288.4.d.a		2
12.b	even	2	1		32.4.b.a		2
15.d	odd	2	1		200.4.d.a		2
15.e	even	4	2		200.4.f.a		4
16.e	even	4	2		2304.4.a.bn		2
16.f	odd	4	2		2304.4.a.v		2
24.f	even	2	1		32.4.b.a		2
24.h	odd	2	1		8.4.b.a	✓	2
48.i	odd	4	2		256.4.a.l		2
48.k	even	4	2		256.4.a.j		2
60.h	even	2	1		800.4.d.a		2
60.l	odd	4	2		800.4.f.a		4
120.i	odd	2	1		200.4.d.a		2
120.m	even	2	1		800.4.d.a		2
120.q	odd	4	2		800.4.f.a		4
120.w	even	4	2		200.4.f.a		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
8.4.b.a	✓	2	3.b	odd	2	1
8.4.b.a	✓	2	24.h	odd	2	1
32.4.b.a		2	12.b	even	2	1
32.4.b.a		2	24.f	even	2	1
72.4.d.b		2	1.a	even	1	1	trivial
72.4.d.b		2	8.b	even	2	1	inner
200.4.d.a		2	15.d	odd	2	1
200.4.d.a		2	120.i	odd	2	1
200.4.f.a		4	15.e	even	4	2
200.4.f.a		4	120.w	even	4	2
256.4.a.j		2	48.k	even	4	2
256.4.a.l		2	48.i	odd	4	2
288.4.d.a		2	4.b	odd	2	1
288.4.d.a		2	8.d	odd	2	1
800.4.d.a		2	60.h	even	2	1
800.4.d.a		2	120.m	even	2	1
800.4.f.a		4	60.l	odd	4	2
800.4.f.a		4	120.q	odd	4	2
2304.4.a.v		2	16.f	odd	4	2
2304.4.a.bn		2	16.e	even	4	2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} + 112 \) acting on \(S_{4}^{\mathrm{new}}(72, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} - 2T + 8 \)
$3$	\( T^{2} \)
$5$	\( T^{2} + 112 \)
$7$	\( (T + 8)^{2} \)
$11$	\( T^{2} + 252 \)