Properties

Label 9702.2.a.bz
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 154)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

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.bz 1
3.b odd 2 1 1078.2.a.b 1
7.b odd 2 1 1386.2.a.f 1
12.b even 2 1 8624.2.a.z 1
21.c even 2 1 154.2.a.b 1
21.g even 6 2 1078.2.e.h 2
21.h odd 6 2 1078.2.e.l 2
84.h odd 2 1 1232.2.a.c 1
105.g even 2 1 3850.2.a.o 1
105.k odd 4 2 3850.2.c.d 2
168.e odd 2 1 4928.2.a.bf 1
168.i even 2 1 4928.2.a.d 1
231.h odd 2 1 1694.2.a.i 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
154.2.a.b 1 21.c even 2 1
1078.2.a.b 1 3.b odd 2 1
1078.2.e.h 2 21.g even 6 2
1078.2.e.l 2 21.h odd 6 2
1232.2.a.c 1 84.h odd 2 1
1386.2.a.f 1 7.b odd 2 1
1694.2.a.i 1 231.h odd 2 1
3850.2.a.o 1 105.g even 2 1
3850.2.c.d 2 105.k odd 4 2
4928.2.a.d 1 168.i even 2 1
4928.2.a.bf 1 168.e odd 2 1
8624.2.a.z 1 12.b even 2 1
9702.2.a.bz 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9702))\):

\( T_{5} - 2 \) Copy content Toggle raw display
\( T_{13} - 4 \) Copy content Toggle raw display
\( T_{17} \) Copy content Toggle raw display
\( T_{19} + 4 \) Copy content Toggle raw display
\( T_{23} + 4 \) Copy content Toggle raw display
\( T_{29} + 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 1 \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T - 2 \) Copy content Toggle raw display
$7$ \( T \) Copy content Toggle raw display
$11$ \( T + 1 \) Copy content Toggle raw display
$13$ \( T - 4 \) Copy content Toggle raw display
$17$ \( T \) Copy content Toggle raw display
$19$ \( T + 4 \) Copy content Toggle raw display
$23$ \( T + 4 \) Copy content Toggle raw display
$29$ \( T + 2 \) Copy content Toggle raw display
$31$ \( T - 10 \) Copy content Toggle raw display
$37$ \( T + 6 \) Copy content Toggle raw display
$41$ \( T \) Copy content Toggle raw display
$43$ \( T + 4 \) Copy content Toggle raw display
$47$ \( T - 10 \) Copy content Toggle raw display
$53$ \( T - 14 \) Copy content Toggle raw display
$59$ \( T - 10 \) Copy content Toggle raw display
$61$ \( T - 8 \) Copy content Toggle raw display
$67$ \( T - 8 \) Copy content Toggle raw display
$71$ \( T - 4 \) Copy content Toggle raw display
$73$ \( T + 4 \) Copy content Toggle raw display
$79$ \( T - 16 \) Copy content Toggle raw display
$83$ \( T - 4 \) Copy content Toggle raw display
$89$ \( T - 10 \) Copy content Toggle raw display
$97$ \( T + 6 \) Copy content Toggle raw display
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