Properties

Label 1960.2.a.d
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

\(f(q)\) \(=\) \( q - q^{3} + q^{5} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{3} + q^{5} - 2 q^{9} - 5 q^{11} - 7 q^{13} - q^{15} + 3 q^{17} + 2 q^{19} + 8 q^{23} + q^{25} + 5 q^{27} - 5 q^{29} + 10 q^{31} + 5 q^{33} + 4 q^{37} + 7 q^{39} + 6 q^{41} + 2 q^{43} - 2 q^{45} + 7 q^{47} - 3 q^{51} - 10 q^{53} - 5 q^{55} - 2 q^{57} + 10 q^{59} + 12 q^{61} - 7 q^{65} - 2 q^{67} - 8 q^{69} + 2 q^{73} - q^{75} - 7 q^{79} + q^{81} - 4 q^{83} + 3 q^{85} + 5 q^{87} + 8 q^{89} - 10 q^{93} + 2 q^{95} - 17 q^{97} + 10 q^{99}+O(q^{100}) \) 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.d 1
4.b odd 2 1 3920.2.a.bb 1
5.b even 2 1 9800.2.a.y 1
7.b odd 2 1 1960.2.a.h yes 1
7.c even 3 2 1960.2.q.l 2
7.d odd 6 2 1960.2.q.g 2
28.d even 2 1 3920.2.a.o 1
35.c odd 2 1 9800.2.a.m 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1960.2.a.d 1 1.a even 1 1 trivial
1960.2.a.h yes 1 7.b odd 2 1
1960.2.q.g 2 7.d odd 6 2
1960.2.q.l 2 7.c even 3 2
3920.2.a.o 1 28.d even 2 1
3920.2.a.bb 1 4.b odd 2 1
9800.2.a.m 1 35.c odd 2 1
9800.2.a.y 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
\( T_{11} + 5 \) Copy content Toggle raw display
\( T_{13} + 7 \) 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
$11$ \( T + 5 \) Copy content Toggle raw display
$13$ \( T + 7 \) Copy content Toggle raw display
$17$ \( T - 3 \) Copy content Toggle raw display
$19$ \( T - 2 \) Copy content Toggle raw display
$23$ \( T - 8 \) Copy content Toggle raw display
$29$ \( T + 5 \) Copy content Toggle raw display
$31$ \( T - 10 \) Copy content Toggle raw display
$37$ \( T - 4 \) Copy content Toggle raw display
$41$ \( T - 6 \) Copy content Toggle raw display
$43$ \( T - 2 \) Copy content Toggle raw display
$47$ \( T - 7 \) Copy content Toggle raw display
$53$ \( T + 10 \) Copy content Toggle raw display
$59$ \( T - 10 \) 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
$73$ \( T - 2 \) Copy content Toggle raw display
$79$ \( T + 7 \) Copy content Toggle raw display
$83$ \( T + 4 \) Copy content Toggle raw display
$89$ \( T - 8 \) Copy content Toggle raw display
$97$ \( T + 17 \) Copy content Toggle raw display
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