Properties

Label 9800.2.a.bq
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 200)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 3q^{3} + 6q^{9} + O(q^{10}) \) \( q + 3q^{3} + 6q^{9} + q^{11} - 4q^{13} - 5q^{17} - q^{19} - 2q^{23} + 9q^{27} - 8q^{29} - 10q^{31} + 3q^{33} - 6q^{37} - 12q^{39} + 3q^{41} + 4q^{43} - 4q^{47} - 15q^{51} + 6q^{53} - 3q^{57} - 8q^{59} - 10q^{61} - q^{67} - 6q^{69} - 12q^{71} - 3q^{73} + 6q^{79} + 9q^{81} + 13q^{83} - 24q^{87} + 9q^{89} - 30q^{93} + 14q^{97} + 6q^{99} + O(q^{100}) \)

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.bq 1
5.b even 2 1 9800.2.a.c 1
7.b odd 2 1 200.2.a.a 1
21.c even 2 1 1800.2.a.r 1
28.d even 2 1 400.2.a.h 1
35.c odd 2 1 200.2.a.e yes 1
35.f even 4 2 200.2.c.a 2
56.e even 2 1 1600.2.a.b 1
56.h odd 2 1 1600.2.a.x 1
84.h odd 2 1 3600.2.a.m 1
105.g even 2 1 1800.2.a.h 1
105.k odd 4 2 1800.2.f.f 2
140.c even 2 1 400.2.a.a 1
140.j odd 4 2 400.2.c.a 2
280.c odd 2 1 1600.2.a.a 1
280.n even 2 1 1600.2.a.y 1
280.s even 4 2 1600.2.c.a 2
280.y odd 4 2 1600.2.c.b 2
420.o odd 2 1 3600.2.a.bf 1
420.w even 4 2 3600.2.f.n 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
200.2.a.a 1 7.b odd 2 1
200.2.a.e yes 1 35.c odd 2 1
200.2.c.a 2 35.f even 4 2
400.2.a.a 1 140.c even 2 1
400.2.a.h 1 28.d even 2 1
400.2.c.a 2 140.j odd 4 2
1600.2.a.a 1 280.c odd 2 1
1600.2.a.b 1 56.e even 2 1
1600.2.a.x 1 56.h odd 2 1
1600.2.a.y 1 280.n even 2 1
1600.2.c.a 2 280.s even 4 2
1600.2.c.b 2 280.y odd 4 2
1800.2.a.h 1 105.g even 2 1
1800.2.a.r 1 21.c even 2 1
1800.2.f.f 2 105.k odd 4 2
3600.2.a.m 1 84.h odd 2 1
3600.2.a.bf 1 420.o odd 2 1
3600.2.f.n 2 420.w even 4 2
9800.2.a.c 1 5.b even 2 1
9800.2.a.bq 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9800))\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( -3 + T \)
$5$ \( T \)
$7$ \( T \)
$11$ \( -1 + T \)
$13$ \( 4 + T \)
$17$ \( 5 + T \)
$19$ \( 1 + T \)
$23$ \( 2 + T \)
$29$ \( 8 + T \)
$31$ \( 10 + T \)
$37$ \( 6 + T \)
$41$ \( -3 + T \)
$43$ \( -4 + T \)
$47$ \( 4 + T \)
$53$ \( -6 + T \)
$59$ \( 8 + T \)
$61$ \( 10 + T \)
$67$ \( 1 + T \)
$71$ \( 12 + T \)
$73$ \( 3 + T \)
$79$ \( -6 + T \)
$83$ \( -13 + T \)
$89$ \( -9 + T \)
$97$ \( -14 + T \)
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