Properties

Label 9702.2.a.bu
Level $9702$
Weight $2$
Character orbit 9702.a
Self dual yes
Analytic conductor $77.471$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 9702 = 2 \cdot 3^{2} \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9702.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(77.4708600410\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 66)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + q^{2} + q^{4} + q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + q^{4} + q^{8} + q^{11} + 4 q^{13} + q^{16} - 6 q^{17} + 4 q^{19} + q^{22} - 6 q^{23} - 5 q^{25} + 4 q^{26} - 6 q^{29} - 8 q^{31} + q^{32} - 6 q^{34} - 10 q^{37} + 4 q^{38} + 6 q^{41} + 8 q^{43} + q^{44} - 6 q^{46} - 6 q^{47} - 5 q^{50} + 4 q^{52} - 6 q^{58} - 8 q^{61} - 8 q^{62} + q^{64} - 4 q^{67} - 6 q^{68} - 6 q^{71} - 2 q^{73} - 10 q^{74} + 4 q^{76} + 14 q^{79} + 6 q^{82} - 12 q^{83} + 8 q^{86} + q^{88} - 6 q^{89} - 6 q^{92} - 6 q^{94} - 14 q^{97}+O(q^{100}) \) 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
1.00000 0 1.00000 0 0 0 1.00000 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(7\) \(-1\)
\(11\) \(-1\)

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 9702.2.a.bu 1
3.b odd 2 1 3234.2.a.d 1
7.b odd 2 1 198.2.a.e 1
21.c even 2 1 66.2.a.a 1
28.d even 2 1 1584.2.a.h 1
35.c odd 2 1 4950.2.a.g 1
35.f even 4 2 4950.2.c.r 2
56.e even 2 1 6336.2.a.bf 1
56.h odd 2 1 6336.2.a.bj 1
63.l odd 6 2 1782.2.e.f 2
63.o even 6 2 1782.2.e.s 2
77.b even 2 1 2178.2.a.b 1
84.h odd 2 1 528.2.a.d 1
105.g even 2 1 1650.2.a.m 1
105.k odd 4 2 1650.2.c.d 2
168.e odd 2 1 2112.2.a.v 1
168.i even 2 1 2112.2.a.i 1
231.h odd 2 1 726.2.a.i 1
231.r odd 10 4 726.2.e.b 4
231.u even 10 4 726.2.e.k 4
924.n even 2 1 5808.2.a.l 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
66.2.a.a 1 21.c even 2 1
198.2.a.e 1 7.b odd 2 1
528.2.a.d 1 84.h odd 2 1
726.2.a.i 1 231.h odd 2 1
726.2.e.b 4 231.r odd 10 4
726.2.e.k 4 231.u even 10 4
1584.2.a.h 1 28.d even 2 1
1650.2.a.m 1 105.g even 2 1
1650.2.c.d 2 105.k odd 4 2
1782.2.e.f 2 63.l odd 6 2
1782.2.e.s 2 63.o even 6 2
2112.2.a.i 1 168.i even 2 1
2112.2.a.v 1 168.e odd 2 1
2178.2.a.b 1 77.b even 2 1
3234.2.a.d 1 3.b odd 2 1
4950.2.a.g 1 35.c odd 2 1
4950.2.c.r 2 35.f even 4 2
5808.2.a.l 1 924.n even 2 1
6336.2.a.bf 1 56.e even 2 1
6336.2.a.bj 1 56.h odd 2 1
9702.2.a.bu 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9702))\):

\( T_{5} \) Copy content Toggle raw display
\( T_{13} - 4 \) Copy content Toggle raw display
\( T_{17} + 6 \) Copy content Toggle raw display
\( T_{19} - 4 \) Copy content Toggle raw display
\( T_{23} + 6 \) Copy content Toggle raw display
\( T_{29} + 6 \) Copy content Toggle raw display

Hecke characteristic polynomials

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