Properties

Label 2475.2.a.f
Level $2475$
Weight $2$
Character orbit 2475.a
Self dual yes
Analytic conductor $19.763$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2475 = 3^{2} \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2475.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(19.7629745003\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 825)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 2q^{4} + q^{7} + O(q^{10}) \) \( q - 2q^{4} + q^{7} + q^{11} + q^{13} + 4q^{16} - 6q^{17} - 7q^{19} + 6q^{23} - 2q^{28} + 6q^{29} - 7q^{31} - 2q^{37} + 6q^{41} + q^{43} - 2q^{44} - 6q^{49} - 2q^{52} - 6q^{53} + 5q^{61} - 8q^{64} - 5q^{67} + 12q^{68} + 12q^{71} - 14q^{73} + 14q^{76} + q^{77} - 4q^{79} - 6q^{83} - 6q^{89} + q^{91} - 12q^{92} - 17q^{97} + O(q^{100}) \)

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 0 −2.00000 0 0 1.00000 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(5\) \(1\)
\(11\) \(-1\)

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 2475.2.a.f 1
3.b odd 2 1 825.2.a.b 1
5.b even 2 1 2475.2.a.e 1
5.c odd 4 2 2475.2.c.h 2
15.d odd 2 1 825.2.a.c yes 1
15.e even 4 2 825.2.c.b 2
33.d even 2 1 9075.2.a.i 1
165.d even 2 1 9075.2.a.l 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
825.2.a.b 1 3.b odd 2 1
825.2.a.c yes 1 15.d odd 2 1
825.2.c.b 2 15.e even 4 2
2475.2.a.e 1 5.b even 2 1
2475.2.a.f 1 1.a even 1 1 trivial
2475.2.c.h 2 5.c odd 4 2
9075.2.a.i 1 33.d even 2 1
9075.2.a.l 1 165.d even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2475))\):

\( T_{2} \)
\( T_{7} - 1 \)
\( T_{29} - 6 \)

Hecke characteristic polynomials

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