Properties

Label 9800.2.a.bj
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 56)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 2q^{3} + q^{9} + O(q^{10}) \) \( q + 2q^{3} + q^{9} - 2q^{17} + 2q^{19} - 8q^{23} - 4q^{27} + 2q^{29} - 4q^{31} + 6q^{37} + 2q^{41} - 8q^{43} - 4q^{47} - 4q^{51} + 10q^{53} + 4q^{57} - 6q^{59} - 4q^{61} + 12q^{67} - 16q^{69} - 14q^{73} - 8q^{79} - 11q^{81} + 6q^{83} + 4q^{87} - 10q^{89} - 8q^{93} - 2q^{97} + O(q^{100}) \)

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 2.00000 0 0 0 0 0 1.00000 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.bj 1
5.b even 2 1 392.2.a.b 1
7.b odd 2 1 1400.2.a.a 1
15.d odd 2 1 3528.2.a.b 1
20.d odd 2 1 784.2.a.i 1
28.d even 2 1 2800.2.a.bd 1
35.c odd 2 1 56.2.a.b 1
35.f even 4 2 1400.2.g.b 2
35.i odd 6 2 392.2.i.a 2
35.j even 6 2 392.2.i.e 2
40.e odd 2 1 3136.2.a.c 1
40.f even 2 1 3136.2.a.w 1
60.h even 2 1 7056.2.a.c 1
105.g even 2 1 504.2.a.h 1
105.o odd 6 2 3528.2.s.ba 2
105.p even 6 2 3528.2.s.a 2
140.c even 2 1 112.2.a.a 1
140.j odd 4 2 2800.2.g.g 2
140.p odd 6 2 784.2.i.b 2
140.s even 6 2 784.2.i.j 2
280.c odd 2 1 448.2.a.c 1
280.n even 2 1 448.2.a.h 1
385.h even 2 1 6776.2.a.h 1
420.o odd 2 1 1008.2.a.m 1
455.h odd 2 1 9464.2.a.h 1
560.be even 4 2 1792.2.b.h 2
560.bf odd 4 2 1792.2.b.a 2
840.b odd 2 1 4032.2.a.a 1
840.u even 2 1 4032.2.a.d 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.2.a.b 1 35.c odd 2 1
112.2.a.a 1 140.c even 2 1
392.2.a.b 1 5.b even 2 1
392.2.i.a 2 35.i odd 6 2
392.2.i.e 2 35.j even 6 2
448.2.a.c 1 280.c odd 2 1
448.2.a.h 1 280.n even 2 1
504.2.a.h 1 105.g even 2 1
784.2.a.i 1 20.d odd 2 1
784.2.i.b 2 140.p odd 6 2
784.2.i.j 2 140.s even 6 2
1008.2.a.m 1 420.o odd 2 1
1400.2.a.a 1 7.b odd 2 1
1400.2.g.b 2 35.f even 4 2
1792.2.b.a 2 560.bf odd 4 2
1792.2.b.h 2 560.be even 4 2
2800.2.a.bd 1 28.d even 2 1
2800.2.g.g 2 140.j odd 4 2
3136.2.a.c 1 40.e odd 2 1
3136.2.a.w 1 40.f even 2 1
3528.2.a.b 1 15.d odd 2 1
3528.2.s.a 2 105.p even 6 2
3528.2.s.ba 2 105.o odd 6 2
4032.2.a.a 1 840.b odd 2 1
4032.2.a.d 1 840.u even 2 1
6776.2.a.h 1 385.h even 2 1
7056.2.a.c 1 60.h even 2 1
9464.2.a.h 1 455.h odd 2 1
9800.2.a.bj 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9800))\):

\( T_{3} - 2 \)
\( T_{11} \)
\( T_{13} \)
\( T_{19} - 2 \)
\( T_{23} + 8 \)

Hecke characteristic polynomials

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