Properties

Label 9800.2.a.o
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 280)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - q^{3} - 2q^{9} + O(q^{10}) \) \( q - q^{3} - 2q^{9} - 2q^{11} + 4q^{13} - 6q^{19} - 3q^{23} + 5q^{27} - 3q^{29} + 2q^{33} + 12q^{37} - 4q^{39} + 7q^{41} + 9q^{43} + 6q^{53} + 6q^{57} + 10q^{59} - 5q^{61} - 11q^{67} + 3q^{69} - 10q^{71} - 8q^{73} + 6q^{79} + q^{81} - 3q^{83} + 3q^{87} - 17q^{89} - 2q^{97} + 4q^{99} + O(q^{100}) \)

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.o 1
5.b even 2 1 1960.2.a.l 1
7.b odd 2 1 9800.2.a.z 1
7.d odd 6 2 1400.2.q.c 2
20.d odd 2 1 3920.2.a.q 1
35.c odd 2 1 1960.2.a.c 1
35.i odd 6 2 280.2.q.b 2
35.j even 6 2 1960.2.q.d 2
35.k even 12 4 1400.2.bh.c 4
105.p even 6 2 2520.2.bi.d 2
140.c even 2 1 3920.2.a.v 1
140.s even 6 2 560.2.q.e 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.2.q.b 2 35.i odd 6 2
560.2.q.e 2 140.s even 6 2
1400.2.q.c 2 7.d odd 6 2
1400.2.bh.c 4 35.k even 12 4
1960.2.a.c 1 35.c odd 2 1
1960.2.a.l 1 5.b even 2 1
1960.2.q.d 2 35.j even 6 2
2520.2.bi.d 2 105.p even 6 2
3920.2.a.q 1 20.d odd 2 1
3920.2.a.v 1 140.c even 2 1
9800.2.a.o 1 1.a even 1 1 trivial
9800.2.a.z 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9800))\):

\( T_{3} + 1 \)
\( T_{11} + 2 \)
\( T_{13} - 4 \)
\( T_{19} + 6 \)
\( T_{23} + 3 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( 1 + T \)
$5$ \( T \)
$7$ \( T \)
$11$ \( 2 + T \)
$13$ \( -4 + T \)
$17$ \( T \)
$19$ \( 6 + T \)
$23$ \( 3 + T \)
$29$ \( 3 + T \)
$31$ \( T \)
$37$ \( -12 + T \)
$41$ \( -7 + T \)
$43$ \( -9 + T \)
$47$ \( T \)
$53$ \( -6 + T \)
$59$ \( -10 + T \)
$61$ \( 5 + T \)
$67$ \( 11 + T \)
$71$ \( 10 + T \)
$73$ \( 8 + T \)
$79$ \( -6 + T \)
$83$ \( 3 + T \)
$89$ \( 17 + T \)
$97$ \( 2 + T \)
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