Properties

Label 1960.2.a.f
Level $1960$
Weight $2$
Character orbit 1960.a
Self dual yes
Analytic conductor $15.651$
Analytic rank $1$
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: \(1\)
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

\(f(q)\) \(=\) \( q - q^{3} + q^{5} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{3} + q^{5} - 2 q^{9} + 3 q^{11} + q^{13} - q^{15} - 5 q^{17} - 6 q^{19} + q^{25} + 5 q^{27} - 5 q^{29} + 2 q^{31} - 3 q^{33} - 4 q^{37} - q^{39} - 2 q^{41} + 10 q^{43} - 2 q^{45} - 9 q^{47} + 5 q^{51} + 6 q^{53} + 3 q^{55} + 6 q^{57} - 6 q^{59} - 12 q^{61} + q^{65} - 2 q^{67} - 14 q^{73} - q^{75} + q^{79} + q^{81} + 12 q^{83} - 5 q^{85} + 5 q^{87} - 2 q^{93} - 6 q^{95} - 9 q^{97} - 6 q^{99}+O(q^{100}) \) 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 −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.f 1
4.b odd 2 1 3920.2.a.x 1
5.b even 2 1 9800.2.a.bd 1
7.b odd 2 1 1960.2.a.j yes 1
7.c even 3 2 1960.2.q.j 2
7.d odd 6 2 1960.2.q.f 2
28.d even 2 1 3920.2.a.l 1
35.c odd 2 1 9800.2.a.t 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1960.2.a.f 1 1.a even 1 1 trivial
1960.2.a.j yes 1 7.b odd 2 1
1960.2.q.f 2 7.d odd 6 2
1960.2.q.j 2 7.c even 3 2
3920.2.a.l 1 28.d even 2 1
3920.2.a.x 1 4.b odd 2 1
9800.2.a.t 1 35.c odd 2 1
9800.2.a.bd 1 5.b even 2 1

Hecke kernels

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

\( T_{3} + 1 \) Copy content Toggle raw display
\( T_{11} - 3 \) Copy content Toggle raw display
\( T_{13} - 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

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