Properties

Label 1989.2.a.e
Level $1989$
Weight $2$
Character orbit 1989.a
Self dual yes
Analytic conductor $15.882$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1989 = 3^{2} \cdot 13 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1989.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(15.8822449620\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 663)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + q^{2} - q^{4} + 2q^{5} - 3q^{8} + O(q^{10}) \) \( q + q^{2} - q^{4} + 2q^{5} - 3q^{8} + 2q^{10} - 4q^{11} + q^{13} - q^{16} - q^{17} - 4q^{19} - 2q^{20} - 4q^{22} - q^{25} + q^{26} + 2q^{29} - 8q^{31} + 5q^{32} - q^{34} - 2q^{37} - 4q^{38} - 6q^{40} - 2q^{41} - 4q^{43} + 4q^{44} - 8q^{47} - 7q^{49} - q^{50} - q^{52} + 10q^{53} - 8q^{55} + 2q^{58} - 4q^{59} + 14q^{61} - 8q^{62} + 7q^{64} + 2q^{65} - 4q^{67} + q^{68} - 14q^{73} - 2q^{74} + 4q^{76} - 8q^{79} - 2q^{80} - 2q^{82} + 4q^{83} - 2q^{85} - 4q^{86} + 12q^{88} + 6q^{89} - 8q^{94} - 8q^{95} - 6q^{97} - 7q^{98} + O(q^{100}) \)

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 2.00000 0 0 −3.00000 0 2.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(13\) \(-1\)
\(17\) \(1\)

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 1989.2.a.e 1
3.b odd 2 1 663.2.a.a 1
39.d odd 2 1 8619.2.a.i 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
663.2.a.a 1 3.b odd 2 1
1989.2.a.e 1 1.a even 1 1 trivial
8619.2.a.i 1 39.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1989))\):

\( T_{2} - 1 \)
\( T_{5} - 2 \)

Hecke characteristic polynomials

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