Newform 116.2.c.a

• Label: 116.2.c.a
• Level: 116
• Weight: 2
• Character orbit: 116.c
• Analytic conductor: 0.926
• Analytic rank: 0
• Dimension: 2
• CM: No
• Inner twists: 2

Newspace parameters

Level:	\( N \)	=	\( 116 = 2^{2} \cdot 29 \)
Weight:	\( k \)	=	\( 2 \)
Character orbit:	\([\chi]\)	=	116.c (of order \(2\) and degree \(1\))

Newform invariants

Self dual:	No
Analytic conductor:	\(0.926264663447\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{-7}) \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 2 \)
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.

\(f(q)\)	\(=\)	\( q -\beta q^{3} + q^{5} -2 q^{7} -4 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2q + 2q^{5} - 4q^{7} - 8q^{9} + O(q^{10}) \)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/116\mathbb{Z}\right)^\times\).

\(n\)	\(59\)	\(89\)
\(\chi(n)\)	\(1\)	\(-1\)

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

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

57.1

0.500000	+	1.32288i
0.500000	−	1.32288i

0

−

2.64575i

0

1.00000

0

−2.00000

0

−4.00000

0

57.2

0

2.64575i

0

1.00000

0

−2.00000

0

−4.00000

0

Inner twists

Char. orbit	Parity	Mult.	Self Twist	Proved
1.a	Even	1	trivial	yes
29.b	Even	1		yes

Hecke kernels

There are no other newforms in \(S_{2}^{\mathrm{new}}(116, [\chi])\).