Properties

Label 1960.2.a.j
Level $1960$
Weight $2$
Character orbit 1960.a
Self dual yes
Analytic conductor $15.651$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1960 = 2^{3} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1960.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(15.6506787962\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + q^{3} - q^{5} - 2 q^{9} + O(q^{10}) \) \( q + q^{3} - q^{5} - 2 q^{9} + 3 q^{11} - q^{13} - q^{15} + 5 q^{17} + 6 q^{19} + q^{25} - 5 q^{27} - 5 q^{29} - 2 q^{31} + 3 q^{33} - 4 q^{37} - q^{39} + 2 q^{41} + 10 q^{43} + 2 q^{45} + 9 q^{47} + 5 q^{51} + 6 q^{53} - 3 q^{55} + 6 q^{57} + 6 q^{59} + 12 q^{61} + q^{65} - 2 q^{67} + 14 q^{73} + q^{75} + q^{79} + q^{81} - 12 q^{83} - 5 q^{85} - 5 q^{87} - 2 q^{93} - 6 q^{95} + 9 q^{97} - 6 q^{99} + O(q^{100}) \)

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 1.00000 0 −1.00000 0 0 0 −2.00000 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 1960.2.a.j yes 1
4.b odd 2 1 3920.2.a.l 1
5.b even 2 1 9800.2.a.t 1
7.b odd 2 1 1960.2.a.f 1
7.c even 3 2 1960.2.q.f 2
7.d odd 6 2 1960.2.q.j 2
28.d even 2 1 3920.2.a.x 1
35.c odd 2 1 9800.2.a.bd 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1960.2.a.f 1 7.b odd 2 1
1960.2.a.j yes 1 1.a even 1 1 trivial
1960.2.q.f 2 7.c even 3 2
1960.2.q.j 2 7.d odd 6 2
3920.2.a.l 1 4.b odd 2 1
3920.2.a.x 1 28.d even 2 1
9800.2.a.t 1 5.b even 2 1
9800.2.a.bd 1 35.c odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1960))\):

\( T_{3} - 1 \)
\( T_{11} - 3 \)
\( T_{13} + 1 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( -1 + T \)
$5$ \( 1 + T \)
$7$ \( T \)
$11$ \( -3 + T \)
$13$ \( 1 + T \)
$17$ \( -5 + T \)
$19$ \( -6 + T \)
$23$ \( T \)
$29$ \( 5 + T \)
$31$ \( 2 + T \)
$37$ \( 4 + T \)
$41$ \( -2 + T \)
$43$ \( -10 + T \)
$47$ \( -9 + T \)
$53$ \( -6 + T \)
$59$ \( -6 + T \)
$61$ \( -12 + T \)
$67$ \( 2 + T \)
$71$ \( T \)
$73$ \( -14 + T \)
$79$ \( -1 + T \)
$83$ \( 12 + T \)
$89$ \( T \)
$97$ \( -9 + T \)
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