Newform orbit 99.8.a.a

• Label: 99.8.a.a
• Level: $99$
• Weight: $8$
• Character orbit: 99.a
• Self dual: yes
• Analytic conductor: $30.926$
• Analytic rank: $0$
• Dimension: $1$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 99 = 3^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	99.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(30.9261175229\)
Analytic rank:	\(0\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 33)
Fricke sign:	\(1\)
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

\(f(q)\)

\(=\)

\( q - 10 q^{2} - 28 q^{4} + 410 q^{5} - 1028 q^{7} + 1560 q^{8}+O(q^{10}) \)

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

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

1.1

0

−10.0000

0

−28.0000

410.000

0

−1028.00

1560.00

0

−4100.00

Atkin-Lehner signs

\( p \)	Sign
\(3\)	\(-1\)
\(11\)	\(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	99.8.a.a		1
3.b	odd	2	1		33.8.a.a	✓	1
12.b	even	2	1		528.8.a.a		1
33.d	even	2	1		363.8.a.a		1

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
33.8.a.a	✓	1	3.b	odd	2	1
99.8.a.a		1	1.a	even	1	1	trivial
363.8.a.a		1	33.d	even	2	1
528.8.a.a		1	12.b	even	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} + 10 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(99))\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T + 10 \)
$3$	\( T \)
$5$	\( T - 410 \)
$7$	\( T + 1028 \)
$11$	\( T - 1331 \)