Properties

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

Related objects

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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 280)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 3q^{3} + 6q^{9} + O(q^{10}) \) \( q - 3q^{3} + 6q^{9} - 5q^{11} - 5q^{13} - 7q^{17} + 2q^{19} + 2q^{23} - 9q^{27} + 7q^{29} - 4q^{31} + 15q^{33} + 6q^{37} + 15q^{39} + 12q^{41} + 2q^{43} + q^{47} + 21q^{51} - 6q^{57} + 4q^{59} - 4q^{61} - 8q^{67} - 6q^{69} + 6q^{73} - 3q^{79} + 9q^{81} - 4q^{83} - 21q^{87} + 12q^{93} + 13q^{97} - 30q^{99} + O(q^{100}) \)

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 −3.00000 0 0 0 0 0 6.00000 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.a 1
5.b even 2 1 1960.2.a.o 1
7.b odd 2 1 1400.2.a.n 1
20.d odd 2 1 3920.2.a.c 1
28.d even 2 1 2800.2.a.c 1
35.c odd 2 1 280.2.a.a 1
35.f even 4 2 1400.2.g.a 2
35.i odd 6 2 1960.2.q.o 2
35.j even 6 2 1960.2.q.a 2
105.g even 2 1 2520.2.a.i 1
140.c even 2 1 560.2.a.f 1
140.j odd 4 2 2800.2.g.b 2
280.c odd 2 1 2240.2.a.z 1
280.n even 2 1 2240.2.a.a 1
420.o odd 2 1 5040.2.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.2.a.a 1 35.c odd 2 1
560.2.a.f 1 140.c even 2 1
1400.2.a.n 1 7.b odd 2 1
1400.2.g.a 2 35.f even 4 2
1960.2.a.o 1 5.b even 2 1
1960.2.q.a 2 35.j even 6 2
1960.2.q.o 2 35.i odd 6 2
2240.2.a.a 1 280.n even 2 1
2240.2.a.z 1 280.c odd 2 1
2520.2.a.i 1 105.g even 2 1
2800.2.a.c 1 28.d even 2 1
2800.2.g.b 2 140.j odd 4 2
3920.2.a.c 1 20.d odd 2 1
5040.2.a.a 1 420.o odd 2 1
9800.2.a.a 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9800))\):

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

Hecke characteristic polynomials

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