Properties

Label 9702.2.a.bi
Level $9702$
Weight $2$
Character orbit 9702.a
Self dual yes
Analytic conductor $77.471$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 9702 = 2 \cdot 3^{2} \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9702.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(77.4708600410\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1386)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + q^{2} + q^{4} - 2 q^{5} + q^{8} + O(q^{10}) \) \( q + q^{2} + q^{4} - 2 q^{5} + q^{8} - 2 q^{10} + q^{11} + 2 q^{13} + q^{16} + q^{17} - 3 q^{19} - 2 q^{20} + q^{22} - q^{23} - q^{25} + 2 q^{26} - q^{29} - 2 q^{31} + q^{32} + q^{34} - 5 q^{37} - 3 q^{38} - 2 q^{40} - 10 q^{41} + q^{43} + q^{44} - q^{46} + 7 q^{47} - q^{50} + 2 q^{52} + 12 q^{53} - 2 q^{55} - q^{58} - 3 q^{59} - 14 q^{61} - 2 q^{62} + q^{64} - 4 q^{65} + 12 q^{67} + q^{68} + 5 q^{71} - 8 q^{73} - 5 q^{74} - 3 q^{76} - 2 q^{80} - 10 q^{82} - 6 q^{83} - 2 q^{85} + q^{86} + q^{88} - 6 q^{89} - q^{92} + 7 q^{94} + 6 q^{95} + 7 q^{97} + O(q^{100}) \)

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

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(7\) \(1\)
\(11\) \(-1\)

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 9702.2.a.bi 1
3.b odd 2 1 9702.2.a.s 1
7.b odd 2 1 9702.2.a.ca 1
7.c even 3 2 1386.2.k.h 2
21.c even 2 1 9702.2.a.f 1
21.h odd 6 2 1386.2.k.l yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1386.2.k.h 2 7.c even 3 2
1386.2.k.l yes 2 21.h odd 6 2
9702.2.a.f 1 21.c even 2 1
9702.2.a.s 1 3.b odd 2 1
9702.2.a.bi 1 1.a even 1 1 trivial
9702.2.a.ca 1 7.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(9702))\):

\( T_{5} + 2 \)
\( T_{13} - 2 \)
\( T_{17} - 1 \)
\( T_{19} + 3 \)
\( T_{23} + 1 \)
\( T_{29} + 1 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -1 + T \)
$3$ \( T \)
$5$ \( 2 + T \)
$7$ \( T \)
$11$ \( -1 + T \)
$13$ \( -2 + T \)
$17$ \( -1 + T \)
$19$ \( 3 + T \)
$23$ \( 1 + T \)
$29$ \( 1 + T \)
$31$ \( 2 + T \)
$37$ \( 5 + T \)
$41$ \( 10 + T \)
$43$ \( -1 + T \)
$47$ \( -7 + T \)
$53$ \( -12 + T \)
$59$ \( 3 + T \)
$61$ \( 14 + T \)
$67$ \( -12 + T \)
$71$ \( -5 + T \)
$73$ \( 8 + T \)
$79$ \( T \)
$83$ \( 6 + T \)
$89$ \( 6 + T \)
$97$ \( -7 + T \)
show more
show less