Properties

Label 9800.2.a.v
Level $9800$
Weight $2$
Character orbit 9800.a
Self dual yes
Analytic conductor $78.253$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 9800 = 2^{3} \cdot 5^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9800.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(78.2533939809\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1400)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 3q^{9} + O(q^{10}) \) \( q - 3q^{9} + q^{11} - 2q^{13} - 4q^{17} + 2q^{19} + 5q^{23} + q^{29} + 2q^{31} + 3q^{37} - 12q^{41} + 11q^{43} - 2q^{47} + 6q^{53} + 10q^{59} - 4q^{61} + q^{67} - 3q^{71} - 9q^{79} + 9q^{81} + 2q^{83} + 6q^{89} - 14q^{97} - 3q^{99} + O(q^{100}) \)

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
0 0 0 0 0 0 0 −3.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(5\) \(1\)
\(7\) \(-1\)

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 9800.2.a.v 1
5.b even 2 1 9800.2.a.w 1
7.b odd 2 1 1400.2.a.f 1
28.d even 2 1 2800.2.a.r 1
35.c odd 2 1 1400.2.a.h yes 1
35.f even 4 2 1400.2.g.h 2
140.c even 2 1 2800.2.a.n 1
140.j odd 4 2 2800.2.g.o 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1400.2.a.f 1 7.b odd 2 1
1400.2.a.h yes 1 35.c odd 2 1
1400.2.g.h 2 35.f even 4 2
2800.2.a.n 1 140.c even 2 1
2800.2.a.r 1 28.d even 2 1
2800.2.g.o 2 140.j odd 4 2
9800.2.a.v 1 1.a even 1 1 trivial
9800.2.a.w 1 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(9800))\):

\( T_{3} \)
\( T_{11} - 1 \)
\( T_{13} + 2 \)
\( T_{19} - 2 \)
\( T_{23} - 5 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( T \)
$5$ \( T \)
$7$ \( T \)
$11$ \( -1 + T \)
$13$ \( 2 + T \)
$17$ \( 4 + T \)
$19$ \( -2 + T \)
$23$ \( -5 + T \)
$29$ \( -1 + T \)
$31$ \( -2 + T \)
$37$ \( -3 + T \)
$41$ \( 12 + T \)
$43$ \( -11 + T \)
$47$ \( 2 + T \)
$53$ \( -6 + T \)
$59$ \( -10 + T \)
$61$ \( 4 + T \)
$67$ \( -1 + T \)
$71$ \( 3 + T \)
$73$ \( T \)
$79$ \( 9 + T \)
$83$ \( -2 + T \)
$89$ \( -6 + T \)
$97$ \( 14 + T \)
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