Properties

Label 1960.2.a.h
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

\(f(q)\) \(=\) \( q + q^{3} - q^{5} - 2 q^{9} + O(q^{10}) \) \( q + q^{3} - q^{5} - 2 q^{9} - 5 q^{11} + 7 q^{13} - q^{15} - 3 q^{17} - 2 q^{19} + 8 q^{23} + q^{25} - 5 q^{27} - 5 q^{29} - 10 q^{31} - 5 q^{33} + 4 q^{37} + 7 q^{39} - 6 q^{41} + 2 q^{43} + 2 q^{45} - 7 q^{47} - 3 q^{51} - 10 q^{53} + 5 q^{55} - 2 q^{57} - 10 q^{59} - 12 q^{61} - 7 q^{65} - 2 q^{67} + 8 q^{69} - 2 q^{73} + q^{75} - 7 q^{79} + q^{81} + 4 q^{83} + 3 q^{85} - 5 q^{87} - 8 q^{89} - 10 q^{93} + 2 q^{95} + 17 q^{97} + 10 q^{99} + O(q^{100}) \)

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.h yes 1
4.b odd 2 1 3920.2.a.o 1
5.b even 2 1 9800.2.a.m 1
7.b odd 2 1 1960.2.a.d 1
7.c even 3 2 1960.2.q.g 2
7.d odd 6 2 1960.2.q.l 2
28.d even 2 1 3920.2.a.bb 1
35.c odd 2 1 9800.2.a.y 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1960.2.a.d 1 7.b odd 2 1
1960.2.a.h yes 1 1.a even 1 1 trivial
1960.2.q.g 2 7.c even 3 2
1960.2.q.l 2 7.d odd 6 2
3920.2.a.o 1 4.b odd 2 1
3920.2.a.bb 1 28.d even 2 1
9800.2.a.m 1 5.b even 2 1
9800.2.a.y 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(1960))\):

\( T_{3} - 1 \)
\( T_{11} + 5 \)
\( T_{13} - 7 \)

Hecke characteristic polynomials

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