Properties

Label 9747.2.a.f
Level $9747$
Weight $2$
Character orbit 9747.a
Self dual yes
Analytic conductor $77.830$
Analytic rank $0$
Dimension $1$
CM discriminant -3
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 9747 = 3^{3} \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9747.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(77.8301868501\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 27)
Fricke sign: \(-1\)
Sato-Tate group: $N(\mathrm{U}(1))$

$q$-expansion

\(f(q)\) \(=\) \( q - 2 q^{4} - q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - 2 q^{4} - q^{7} - 5 q^{13} + 4 q^{16} - 5 q^{25} + 2 q^{28} + 4 q^{31} - 11 q^{37} + 8 q^{43} - 6 q^{49} + 10 q^{52} - q^{61} - 8 q^{64} - 5 q^{67} - 7 q^{73} - 17 q^{79} + 5 q^{91} + 19 q^{97}+O(q^{100}) \) 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
0 0 −2.00000 0 0 −1.00000 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(19\) \(-1\)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 9747.2.a.f 1
3.b odd 2 1 CM 9747.2.a.f 1
19.b odd 2 1 27.2.a.a 1
57.d even 2 1 27.2.a.a 1
76.d even 2 1 432.2.a.e 1
95.d odd 2 1 675.2.a.e 1
95.g even 4 2 675.2.b.f 2
133.c even 2 1 1323.2.a.i 1
152.b even 2 1 1728.2.a.o 1
152.g odd 2 1 1728.2.a.n 1
171.l even 6 2 81.2.c.a 2
171.o odd 6 2 81.2.c.a 2
209.d even 2 1 3267.2.a.f 1
228.b odd 2 1 432.2.a.e 1
247.d odd 2 1 4563.2.a.e 1
285.b even 2 1 675.2.a.e 1
285.j odd 4 2 675.2.b.f 2
323.c odd 2 1 7803.2.a.k 1
399.h odd 2 1 1323.2.a.i 1
456.l odd 2 1 1728.2.a.o 1
456.p even 2 1 1728.2.a.n 1
513.bw even 18 6 729.2.e.f 6
513.ca odd 18 6 729.2.e.f 6
627.b odd 2 1 3267.2.a.f 1
684.w even 6 2 1296.2.i.i 2
684.bh odd 6 2 1296.2.i.i 2
741.d even 2 1 4563.2.a.e 1
969.h even 2 1 7803.2.a.k 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
27.2.a.a 1 19.b odd 2 1
27.2.a.a 1 57.d even 2 1
81.2.c.a 2 171.l even 6 2
81.2.c.a 2 171.o odd 6 2
432.2.a.e 1 76.d even 2 1
432.2.a.e 1 228.b odd 2 1
675.2.a.e 1 95.d odd 2 1
675.2.a.e 1 285.b even 2 1
675.2.b.f 2 95.g even 4 2
675.2.b.f 2 285.j odd 4 2
729.2.e.f 6 513.bw even 18 6
729.2.e.f 6 513.ca odd 18 6
1296.2.i.i 2 684.w even 6 2
1296.2.i.i 2 684.bh odd 6 2
1323.2.a.i 1 133.c even 2 1
1323.2.a.i 1 399.h odd 2 1
1728.2.a.n 1 152.g odd 2 1
1728.2.a.n 1 456.p even 2 1
1728.2.a.o 1 152.b even 2 1
1728.2.a.o 1 456.l odd 2 1
3267.2.a.f 1 209.d even 2 1
3267.2.a.f 1 627.b odd 2 1
4563.2.a.e 1 247.d odd 2 1
4563.2.a.e 1 741.d even 2 1
7803.2.a.k 1 323.c odd 2 1
7803.2.a.k 1 969.h even 2 1
9747.2.a.f 1 1.a even 1 1 trivial
9747.2.a.f 1 3.b odd 2 1 CM

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(9747))\):

\( T_{2} \) Copy content Toggle raw display
\( T_{5} \) Copy content Toggle raw display
\( T_{7} + 1 \) Copy content Toggle raw display
\( T_{13} + 5 \) Copy content Toggle raw display

Hecke characteristic polynomials

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