Properties

Label 990.2.a.b
Level $990$
Weight $2$
Character orbit 990.a
Self dual yes
Analytic conductor $7.905$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 990 = 2 \cdot 3^{2} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 990.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(7.90518980011\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 330)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - q^{2} + q^{4} - q^{5} - q^{8} + O(q^{10}) \) \( q - q^{2} + q^{4} - q^{5} - q^{8} + q^{10} - q^{11} + 6q^{13} + q^{16} - 2q^{17} - 4q^{19} - q^{20} + q^{22} + q^{25} - 6q^{26} + 10q^{29} - q^{32} + 2q^{34} + 6q^{37} + 4q^{38} + q^{40} - 2q^{41} + 4q^{43} - q^{44} + 8q^{47} - 7q^{49} - q^{50} + 6q^{52} + 10q^{53} + q^{55} - 10q^{58} + 4q^{59} - 2q^{61} + q^{64} - 6q^{65} - 4q^{67} - 2q^{68} + 8q^{71} + 2q^{73} - 6q^{74} - 4q^{76} - 8q^{79} - q^{80} + 2q^{82} + 12q^{83} + 2q^{85} - 4q^{86} + q^{88} + 6q^{89} - 8q^{94} + 4q^{95} + 18q^{97} + 7q^{98} + O(q^{100}) \)

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
−1.00000 0 1.00000 −1.00000 0 0 −1.00000 0 1.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(-1\)
\(5\) \(1\)
\(11\) \(1\)

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 990.2.a.b 1
3.b odd 2 1 330.2.a.d 1
4.b odd 2 1 7920.2.a.m 1
5.b even 2 1 4950.2.a.bg 1
5.c odd 4 2 4950.2.c.j 2
12.b even 2 1 2640.2.a.t 1
15.d odd 2 1 1650.2.a.h 1
15.e even 4 2 1650.2.c.g 2
33.d even 2 1 3630.2.a.f 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
330.2.a.d 1 3.b odd 2 1
990.2.a.b 1 1.a even 1 1 trivial
1650.2.a.h 1 15.d odd 2 1
1650.2.c.g 2 15.e even 4 2
2640.2.a.t 1 12.b even 2 1
3630.2.a.f 1 33.d even 2 1
4950.2.a.bg 1 5.b even 2 1
4950.2.c.j 2 5.c odd 4 2
7920.2.a.m 1 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(990))\):

\( T_{7} \)
\( T_{13} - 6 \)
\( T_{17} + 2 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T \)
$3$ \( T \)
$5$ \( 1 + T \)
$7$ \( T \)
$11$ \( 1 + T \)
$13$ \( -6 + T \)
$17$ \( 2 + T \)
$19$ \( 4 + T \)
$23$ \( T \)
$29$ \( -10 + T \)
$31$ \( T \)
$37$ \( -6 + T \)
$41$ \( 2 + T \)
$43$ \( -4 + T \)
$47$ \( -8 + T \)
$53$ \( -10 + T \)
$59$ \( -4 + T \)
$61$ \( 2 + T \)
$67$ \( 4 + T \)
$71$ \( -8 + T \)
$73$ \( -2 + T \)
$79$ \( 8 + T \)
$83$ \( -12 + T \)
$89$ \( -6 + T \)
$97$ \( -18 + T \)
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