Newform orbit 144.7.e.c

• Label: 144.7.e.c
• Level: $144$
• Weight: $7$
• Character orbit: 144.e
• Analytic conductor: $33.128$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 144 = 2^{4} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	144.e (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(33.1277880413\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{-2}) \)
Defining polynomial:	\( x^{2} + 2 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index:	\( 3^{2} \)
Twist minimal:	no (minimal twist has level 36)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 9\sqrt{-2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q + 5 \beta q^{5} + 244 q^{7}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 488 q^{7}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/144\mathbb{Z}\right)^\times\).

\(n\)	\(37\)	\(65\)	\(127\)
\(\chi(n)\)	\(1\)	\(-1\)	\(1\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

17.1

	−	1.41421i
		1.41421i

0

0

0

−

63.6396i

0

244.000

0

0

0

17.2

0

0

0

63.6396i

0

244.000

0

0

0

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	144.7.e.c		2
3.b	odd	2	1	inner	144.7.e.c		2
4.b	odd	2	1		36.7.c.a	✓	2
8.b	even	2	1		576.7.e.j		2
8.d	odd	2	1		576.7.e.c		2
12.b	even	2	1		36.7.c.a	✓	2
24.f	even	2	1		576.7.e.c		2
24.h	odd	2	1		576.7.e.j		2
36.f	odd	6	2		324.7.g.e		4
36.h	even	6	2		324.7.g.e		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
36.7.c.a	✓	2	4.b	odd	2	1
36.7.c.a	✓	2	12.b	even	2	1
144.7.e.c		2	1.a	even	1	1	trivial
144.7.e.c		2	3.b	odd	2	1	inner
324.7.g.e		4	36.f	odd	6	2
324.7.g.e		4	36.h	even	6	2
576.7.e.c		2	8.d	odd	2	1
576.7.e.c		2	24.f	even	2	1
576.7.e.j		2	8.b	even	2	1
576.7.e.j		2	24.h	odd	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{7}^{\mathrm{new}}(144, [\chi])\):

\( T_{5}^{2} + 4050 \)

\( T_{7} - 244 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} \)
$3$	\( T^{2} \)
$5$	\( T^{2} + 4050 \)
$7$	\( (T - 244)^{2} \)
$11$	\( T^{2} + 5725728 \)