Properties

Label 9702.2.a.bg
Level $9702$
Weight $2$
Character orbit 9702.a
Self dual yes
Analytic conductor $77.471$
Analytic rank $0$
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: \(0\)
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

\(f(q)\) \(=\) \( q + q^{2} + q^{4} - 2q^{5} + q^{8} + O(q^{10}) \) \( q + q^{2} + q^{4} - 2q^{5} + q^{8} - 2q^{10} - q^{11} + 2q^{13} + q^{16} - 3q^{17} + 7q^{19} - 2q^{20} - q^{22} - 7q^{23} - q^{25} + 2q^{26} - 5q^{29} + 2q^{31} + q^{32} - 3q^{34} + 3q^{37} + 7q^{38} - 2q^{40} + 6q^{41} + 11q^{43} - q^{44} - 7q^{46} - 7q^{47} - q^{50} + 2q^{52} - 4q^{53} + 2q^{55} - 5q^{58} + 11q^{59} + 10q^{61} + 2q^{62} + q^{64} - 4q^{65} - 4q^{67} - 3q^{68} - 5q^{71} - 8q^{73} + 3q^{74} + 7q^{76} - 8q^{79} - 2q^{80} + 6q^{82} + 14q^{83} + 6q^{85} + 11q^{86} - q^{88} + 2q^{89} - 7q^{92} - 7q^{94} - 14q^{95} + 15q^{97} + O(q^{100}) \)

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.bg 1
3.b odd 2 1 9702.2.a.w 1
7.b odd 2 1 9702.2.a.bx 1
7.c even 3 2 1386.2.k.i 2
21.c even 2 1 9702.2.a.j 1
21.h odd 6 2 1386.2.k.k yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1386.2.k.i 2 7.c even 3 2
1386.2.k.k yes 2 21.h odd 6 2
9702.2.a.j 1 21.c even 2 1
9702.2.a.w 1 3.b odd 2 1
9702.2.a.bg 1 1.a even 1 1 trivial
9702.2.a.bx 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9702))\):

\( T_{5} + 2 \)
\( T_{13} - 2 \)
\( T_{17} + 3 \)
\( T_{19} - 7 \)
\( T_{23} + 7 \)
\( T_{29} + 5 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -1 + T \)
$3$ \( T \)
$5$ \( 2 + T \)
$7$ \( T \)
$11$ \( 1 + T \)
$13$ \( -2 + T \)
$17$ \( 3 + T \)
$19$ \( -7 + T \)
$23$ \( 7 + T \)
$29$ \( 5 + T \)
$31$ \( -2 + T \)
$37$ \( -3 + T \)
$41$ \( -6 + T \)
$43$ \( -11 + T \)
$47$ \( 7 + T \)
$53$ \( 4 + T \)
$59$ \( -11 + T \)
$61$ \( -10 + T \)
$67$ \( 4 + T \)
$71$ \( 5 + T \)
$73$ \( 8 + T \)
$79$ \( 8 + T \)
$83$ \( -14 + T \)
$89$ \( -2 + T \)
$97$ \( -15 + T \)
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