Properties

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

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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

\(f(q)\) \(=\) \( q + q^{2} + q^{4} - 3 q^{5} + q^{8} + O(q^{10}) \) \( q + q^{2} + q^{4} - 3 q^{5} + q^{8} - 3 q^{10} - q^{11} - 2 q^{13} + q^{16} + 3 q^{17} - 2 q^{19} - 3 q^{20} - q^{22} + 3 q^{23} + 4 q^{25} - 2 q^{26} - 2 q^{31} + q^{32} + 3 q^{34} + 8 q^{37} - 2 q^{38} - 3 q^{40} + 9 q^{41} - 4 q^{43} - q^{44} + 3 q^{46} - 3 q^{47} + 4 q^{50} - 2 q^{52} + 6 q^{53} + 3 q^{55} - 6 q^{59} - 5 q^{61} - 2 q^{62} + q^{64} + 6 q^{65} + 11 q^{67} + 3 q^{68} - 2 q^{73} + 8 q^{74} - 2 q^{76} - 13 q^{79} - 3 q^{80} + 9 q^{82} - 9 q^{83} - 9 q^{85} - 4 q^{86} - q^{88} - 12 q^{89} + 3 q^{92} - 3 q^{94} + 6 q^{95} - 5 q^{97} + O(q^{100}) \)

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 −3.00000 0 0 1.00000 0 −3.00000
\(n\): 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.bc 1
3.b odd 2 1 9702.2.a.z 1
7.b odd 2 1 9702.2.a.cd 1
7.d odd 6 2 1386.2.k.b 2
21.c even 2 1 9702.2.a.c 1
21.g even 6 2 1386.2.k.p yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1386.2.k.b 2 7.d odd 6 2
1386.2.k.p yes 2 21.g even 6 2
9702.2.a.c 1 21.c even 2 1
9702.2.a.z 1 3.b odd 2 1
9702.2.a.bc 1 1.a even 1 1 trivial
9702.2.a.cd 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9702))\):

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

Hecke characteristic polynomials

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