Properties

Label 9800.2.a.bc
Level $9800$
Weight $2$
Character orbit 9800.a
Self dual yes
Analytic conductor $78.253$
Analytic rank $0$
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: \(0\)
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

\(f(q)\) \(=\) \( q + q^{3} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{3} - 2 q^{9} + 2 q^{11} + 4 q^{17} + 2 q^{19} - q^{23} - 5 q^{27} + 9 q^{29} - 4 q^{31} + 2 q^{33} - 4 q^{37} - q^{41} - 9 q^{43} + 4 q^{51} + 10 q^{53} + 2 q^{57} + 10 q^{59} - 9 q^{61} - 5 q^{67} - q^{69} + 14 q^{71} + 12 q^{73} + 14 q^{79} + q^{81} + 11 q^{83} + 9 q^{87} + 15 q^{89} - 4 q^{93} - 18 q^{97} - 4 q^{99}+O(q^{100}) \) 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
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.bc 1
5.b even 2 1 1960.2.a.e 1
7.b odd 2 1 9800.2.a.r 1
7.d odd 6 2 1400.2.q.e 2
20.d odd 2 1 3920.2.a.y 1
35.c odd 2 1 1960.2.a.i 1
35.i odd 6 2 280.2.q.a 2
35.j even 6 2 1960.2.q.k 2
35.k even 12 4 1400.2.bh.b 4
105.p even 6 2 2520.2.bi.a 2
140.c even 2 1 3920.2.a.m 1
140.s even 6 2 560.2.q.h 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.2.q.a 2 35.i odd 6 2
560.2.q.h 2 140.s even 6 2
1400.2.q.e 2 7.d odd 6 2
1400.2.bh.b 4 35.k even 12 4
1960.2.a.e 1 5.b even 2 1
1960.2.a.i 1 35.c odd 2 1
1960.2.q.k 2 35.j even 6 2
2520.2.bi.a 2 105.p even 6 2
3920.2.a.m 1 140.c even 2 1
3920.2.a.y 1 20.d odd 2 1
9800.2.a.r 1 7.b odd 2 1
9800.2.a.bc 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9800))\):

\( T_{3} - 1 \) Copy content Toggle raw display
\( T_{11} - 2 \) Copy content Toggle raw display
\( T_{13} \) Copy content Toggle raw display
\( T_{19} - 2 \) Copy content Toggle raw display
\( T_{23} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T - 1 \) Copy content Toggle raw display
$5$ \( T \) Copy content Toggle raw display
$7$ \( T \) Copy content Toggle raw display
$11$ \( T - 2 \) Copy content Toggle raw display
$13$ \( T \) Copy content Toggle raw display
$17$ \( T - 4 \) Copy content Toggle raw display
$19$ \( T - 2 \) Copy content Toggle raw display
$23$ \( T + 1 \) Copy content Toggle raw display
$29$ \( T - 9 \) Copy content Toggle raw display
$31$ \( T + 4 \) Copy content Toggle raw display
$37$ \( T + 4 \) Copy content Toggle raw display
$41$ \( T + 1 \) Copy content Toggle raw display
$43$ \( T + 9 \) Copy content Toggle raw display
$47$ \( T \) Copy content Toggle raw display
$53$ \( T - 10 \) Copy content Toggle raw display
$59$ \( T - 10 \) Copy content Toggle raw display
$61$ \( T + 9 \) Copy content Toggle raw display
$67$ \( T + 5 \) Copy content Toggle raw display
$71$ \( T - 14 \) Copy content Toggle raw display
$73$ \( T - 12 \) Copy content Toggle raw display
$79$ \( T - 14 \) Copy content Toggle raw display
$83$ \( T - 11 \) Copy content Toggle raw display
$89$ \( T - 15 \) Copy content Toggle raw display
$97$ \( T + 18 \) Copy content Toggle raw display
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