Properties

Label 9196.2.a.f
Level $9196$
Weight $2$
Character orbit 9196.a
Self dual yes
Analytic conductor $73.430$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 9196 = 2^{2} \cdot 11^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9196.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(73.4304296988\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 76)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 2 q^{3} - q^{5} + 3 q^{7} + q^{9} + O(q^{10}) \) \( q + 2 q^{3} - q^{5} + 3 q^{7} + q^{9} + 4 q^{13} - 2 q^{15} + 3 q^{17} + q^{19} + 6 q^{21} + 8 q^{23} - 4 q^{25} - 4 q^{27} + 2 q^{29} + 4 q^{31} - 3 q^{35} + 10 q^{37} + 8 q^{39} - 10 q^{41} - q^{43} - q^{45} - q^{47} + 2 q^{49} + 6 q^{51} - 4 q^{53} + 2 q^{57} + 6 q^{59} + 13 q^{61} + 3 q^{63} - 4 q^{65} - 12 q^{67} + 16 q^{69} + 2 q^{71} - 9 q^{73} - 8 q^{75} - 8 q^{79} - 11 q^{81} + 12 q^{83} - 3 q^{85} + 4 q^{87} + 12 q^{89} + 12 q^{91} + 8 q^{93} - q^{95} - 8 q^{97} + O(q^{100}) \)

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
0 2.00000 0 −1.00000 0 3.00000 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(11\) \(-1\)
\(19\) \(-1\)

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 9196.2.a.f 1
11.b odd 2 1 76.2.a.a 1
33.d even 2 1 684.2.a.b 1
44.c even 2 1 304.2.a.a 1
55.d odd 2 1 1900.2.a.b 1
55.e even 4 2 1900.2.c.b 2
77.b even 2 1 3724.2.a.a 1
88.b odd 2 1 1216.2.a.c 1
88.g even 2 1 1216.2.a.q 1
132.d odd 2 1 2736.2.a.q 1
209.d even 2 1 1444.2.a.a 1
209.g even 6 2 1444.2.e.c 2
209.h odd 6 2 1444.2.e.a 2
220.g even 2 1 7600.2.a.p 1
836.h odd 2 1 5776.2.a.p 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
76.2.a.a 1 11.b odd 2 1
304.2.a.a 1 44.c even 2 1
684.2.a.b 1 33.d even 2 1
1216.2.a.c 1 88.b odd 2 1
1216.2.a.q 1 88.g even 2 1
1444.2.a.a 1 209.d even 2 1
1444.2.e.a 2 209.h odd 6 2
1444.2.e.c 2 209.g even 6 2
1900.2.a.b 1 55.d odd 2 1
1900.2.c.b 2 55.e even 4 2
2736.2.a.q 1 132.d odd 2 1
3724.2.a.a 1 77.b even 2 1
5776.2.a.p 1 836.h odd 2 1
7600.2.a.p 1 220.g even 2 1
9196.2.a.f 1 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(9196))\):

\( T_{3} - 2 \)
\( T_{5} + 1 \)
\( T_{7} - 3 \)
\( T_{13} - 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( -2 + T \)
$5$ \( 1 + T \)
$7$ \( -3 + T \)
$11$ \( T \)
$13$ \( -4 + T \)
$17$ \( -3 + T \)
$19$ \( -1 + T \)
$23$ \( -8 + T \)
$29$ \( -2 + T \)
$31$ \( -4 + T \)
$37$ \( -10 + T \)
$41$ \( 10 + T \)
$43$ \( 1 + T \)
$47$ \( 1 + T \)
$53$ \( 4 + T \)
$59$ \( -6 + T \)
$61$ \( -13 + T \)
$67$ \( 12 + T \)
$71$ \( -2 + T \)
$73$ \( 9 + T \)
$79$ \( 8 + T \)
$83$ \( -12 + T \)
$89$ \( -12 + T \)
$97$ \( 8 + T \)
show more
show less