Properties

Label 9800.2.a.m
Level $9800$
Weight $2$
Character orbit 9800.a
Self dual yes
Analytic conductor $78.253$
Analytic rank $2$
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: \(2\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1960)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - q^{3} - 2q^{9} + O(q^{10}) \) \( q - q^{3} - 2q^{9} - 5q^{11} - 7q^{13} + 3q^{17} - 2q^{19} - 8q^{23} + 5q^{27} - 5q^{29} - 10q^{31} + 5q^{33} - 4q^{37} + 7q^{39} - 6q^{41} - 2q^{43} + 7q^{47} - 3q^{51} + 10q^{53} + 2q^{57} - 10q^{59} - 12q^{61} + 2q^{67} + 8q^{69} + 2q^{73} - 7q^{79} + q^{81} - 4q^{83} + 5q^{87} - 8q^{89} + 10q^{93} - 17q^{97} + 10q^{99} + O(q^{100}) \)

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 −1.00000 0 0 0 0 0 −2.00000 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.m 1
5.b even 2 1 1960.2.a.h yes 1
7.b odd 2 1 9800.2.a.y 1
20.d odd 2 1 3920.2.a.o 1
35.c odd 2 1 1960.2.a.d 1
35.i odd 6 2 1960.2.q.l 2
35.j even 6 2 1960.2.q.g 2
140.c even 2 1 3920.2.a.bb 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1960.2.a.d 1 35.c odd 2 1
1960.2.a.h yes 1 5.b even 2 1
1960.2.q.g 2 35.j even 6 2
1960.2.q.l 2 35.i odd 6 2
3920.2.a.o 1 20.d odd 2 1
3920.2.a.bb 1 140.c even 2 1
9800.2.a.m 1 1.a even 1 1 trivial
9800.2.a.y 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9800))\):

\( T_{3} + 1 \)
\( T_{11} + 5 \)
\( T_{13} + 7 \)
\( T_{19} + 2 \)
\( T_{23} + 8 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( 1 + T \)
$5$ \( T \)
$7$ \( T \)
$11$ \( 5 + T \)
$13$ \( 7 + T \)
$17$ \( -3 + T \)
$19$ \( 2 + T \)
$23$ \( 8 + T \)
$29$ \( 5 + T \)
$31$ \( 10 + T \)
$37$ \( 4 + T \)
$41$ \( 6 + T \)
$43$ \( 2 + T \)
$47$ \( -7 + T \)
$53$ \( -10 + T \)
$59$ \( 10 + T \)
$61$ \( 12 + T \)
$67$ \( -2 + T \)
$71$ \( T \)
$73$ \( -2 + T \)
$79$ \( 7 + T \)
$83$ \( 4 + T \)
$89$ \( 8 + T \)
$97$ \( 17 + T \)
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