Newform orbit 59.7.b.a

• Label: 59.7.b.a
• Level: $59$
• Weight: $7$
• Character orbit: 59.b
• Self dual: yes
• Analytic conductor: $13.573$
• Analytic rank: $0$
• Dimension: $1$
• CM discriminant: -59
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 59 \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	59.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	yes
Analytic conductor:	\(13.5731909336\)
Analytic rank:	\(0\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

$q$-expansion

\(f(q)\)

\(=\)

\( q - 5 q^{3} + 64 q^{4} + 191 q^{5} + 155 q^{7} - 704 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/59\mathbb{Z}\right)^\times\).

\(n\)	\(2\)
\(\chi(n)\)	\(1\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

58.1

0

0

−5.00000

64.0000

191.000

0

155.000

0

−704.000

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
59.b	odd	2	1	CM by \(\Q(\sqrt{-59}) \)

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	59.7.b.a	✓	1
59.b	odd	2	1	CM	59.7.b.a	✓	1

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
59.7.b.a	✓	1	1.a	even	1	1	trivial
59.7.b.a	✓	1	59.b	odd	2	1	CM

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{7}^{\mathrm{new}}(59, [\chi])\):

\( T_{2} \)

\( T_{3} + 5 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T \)
$3$	\( T + 5 \)
$5$	\( T - 191 \)
$7$	\( T - 155 \)
$11$	\( T \)