Properties

Label 9702.2.a.y
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 \( q - q^{2} + q^{4} + 2 q^{5} - q^{8} - 2 q^{10} + q^{11} + 7 q^{13} + q^{16} + 2 q^{17} + 2 q^{20} - q^{22} + 8 q^{23} - q^{25} - 7 q^{26} + 5 q^{29} - 4 q^{31} - q^{32} - 2 q^{34} + 4 q^{37} - 2 q^{40} + 4 q^{41} - 8 q^{43} + q^{44} - 8 q^{46} + 2 q^{47} + q^{50} + 7 q^{52} + 6 q^{53} + 2 q^{55} - 5 q^{58} + 3 q^{59} - q^{61} + 4 q^{62} + q^{64} + 14 q^{65} + 9 q^{67} + 2 q^{68} + 2 q^{71} - 4 q^{73} - 4 q^{74} + 9 q^{79} + 2 q^{80} - 4 q^{82} + 6 q^{83} + 4 q^{85} + 8 q^{86} - q^{88} + 6 q^{89} + 8 q^{92} - 2 q^{94} - 7 q^{97}+O(q^{100}) \) 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.y 1
3.b odd 2 1 1078.2.a.g 1
7.b odd 2 1 9702.2.a.i 1
7.d odd 6 2 1386.2.k.o 2
12.b even 2 1 8624.2.a.be 1
21.c even 2 1 1078.2.a.m 1
21.g even 6 2 154.2.e.a 2
21.h odd 6 2 1078.2.e.f 2
84.h odd 2 1 8624.2.a.b 1
84.j odd 6 2 1232.2.q.e 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
154.2.e.a 2 21.g even 6 2
1078.2.a.g 1 3.b odd 2 1
1078.2.a.m 1 21.c even 2 1
1078.2.e.f 2 21.h odd 6 2
1232.2.q.e 2 84.j odd 6 2
1386.2.k.o 2 7.d odd 6 2
8624.2.a.b 1 84.h odd 2 1
8624.2.a.be 1 12.b even 2 1
9702.2.a.i 1 7.b odd 2 1
9702.2.a.y 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
\( T_{13} - 7 \) Copy content Toggle raw display
\( T_{17} - 2 \) Copy content Toggle raw display
\( T_{19} \) Copy content Toggle raw display
\( T_{23} - 8 \) Copy content Toggle raw display
\( T_{29} - 5 \) 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
$13$ \( T - 7 \) Copy content Toggle raw display
$17$ \( T - 2 \) Copy content Toggle raw display
$19$ \( T \) Copy content Toggle raw display
$23$ \( T - 8 \) Copy content Toggle raw display
$29$ \( T - 5 \) Copy content Toggle raw display
$31$ \( T + 4 \) Copy content Toggle raw display
$37$ \( T - 4 \) Copy content Toggle raw display
$41$ \( T - 4 \) Copy content Toggle raw display
$43$ \( T + 8 \) Copy content Toggle raw display
$47$ \( T - 2 \) Copy content Toggle raw display
$53$ \( T - 6 \) Copy content Toggle raw display
$59$ \( T - 3 \) Copy content Toggle raw display
$61$ \( T + 1 \) Copy content Toggle raw display
$67$ \( T - 9 \) Copy content Toggle raw display
$71$ \( T - 2 \) Copy content Toggle raw display
$73$ \( T + 4 \) Copy content Toggle raw display
$79$ \( T - 9 \) Copy content Toggle raw display
$83$ \( T - 6 \) Copy content Toggle raw display
$89$ \( T - 6 \) Copy content Toggle raw display
$97$ \( T + 7 \) Copy content Toggle raw display
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