Newform orbit 72.4.d.c

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

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:	\(4\)
Coefficient field:	\(\Q(\sqrt{-10}, \sqrt{22})\)
Defining polynomial:	\( x^{4} - 6x^{2} + 64 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 2^{2} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\)	\(=\)	\( q + \beta_1 q^{2} + (\beta_{3} + 3) q^{4} + (\beta_{2} + \beta_1) q^{5} + 10 q^{7} + (4 \beta_{2} + 2 \beta_1) q^{8}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 12 q^{4} + 40 q^{7}+O(q^{10}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 6x^{2} + 64 \) :

\(\beta_{1}\)	\(=\)	\( \nu \)
\(\beta_{2}\)	\(=\)	\( ( \nu^{3} - 2\nu ) / 4 \)
\(\beta_{3}\)	\(=\)	\( \nu^{2} - 3 \)

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

−2.34521	−	1.58114i
−2.34521	+	1.58114i
2.34521	−	1.58114i
2.34521	+	1.58114i

−2.34521

−

1.58114i

0

3.00000

+

7.41620i

−

6.32456i

0

10.0000

4.69042

−

22.1359i

0

−10.0000

+

14.8324i

37.2

−2.34521

+

1.58114i

0

3.00000

−

7.41620i

6.32456i

0

10.0000

4.69042

+

22.1359i

0

−10.0000

−

14.8324i

37.3

2.34521

−

1.58114i

0

3.00000

−

7.41620i

−

6.32456i

0

10.0000

−4.69042

−

22.1359i

0

−10.0000

−

14.8324i

37.4

2.34521

+

1.58114i

0

3.00000

+

7.41620i

6.32456i

0

10.0000

−4.69042

+

22.1359i

0

−10.0000

+

14.8324i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
8.b	even	2	1	inner
24.h	odd	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.c	✓	4
3.b	odd	2	1	inner	72.4.d.c	✓	4
4.b	odd	2	1		288.4.d.c		4
8.b	even	2	1	inner	72.4.d.c	✓	4
8.d	odd	2	1		288.4.d.c		4
12.b	even	2	1		288.4.d.c		4
16.e	even	4	2		2304.4.a.bx		4
16.f	odd	4	2		2304.4.a.cc		4
24.f	even	2	1		288.4.d.c		4
24.h	odd	2	1	inner	72.4.d.c	✓	4
48.i	odd	4	2		2304.4.a.bx		4
48.k	even	4	2		2304.4.a.cc		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
72.4.d.c	✓	4	1.a	even	1	1	trivial
72.4.d.c	✓	4	3.b	odd	2	1	inner
72.4.d.c	✓	4	8.b	even	2	1	inner
72.4.d.c	✓	4	24.h	odd	2	1	inner
288.4.d.c		4	4.b	odd	2	1
288.4.d.c		4	8.d	odd	2	1
288.4.d.c		4	12.b	even	2	1
288.4.d.c		4	24.f	even	2	1
2304.4.a.bx		4	16.e	even	4	2
2304.4.a.bx		4	48.i	odd	4	2
2304.4.a.cc		4	16.f	odd	4	2
2304.4.a.cc		4	48.k	even	4	2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} + 40 \) acting on \(S_{4}^{\mathrm{new}}(72, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{4} - 6T^{2} + 64 \)
$3$	\( T^{4} \)
$5$	\( (T^{2} + 40)^{2} \)
$7$	\( (T - 10)^{4} \)
$11$	\( (T^{2} + 1440)^{2} \)