Properties

Label 2475.2.a.k
Level $2475$
Weight $2$
Character orbit 2475.a
Self dual yes
Analytic conductor $19.763$
Analytic rank $0$
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: \(0\)
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

\(f(q)\) \(=\) \( q + q^{2} - q^{4} + 3q^{7} - 3q^{8} + O(q^{10}) \) \( q + q^{2} - q^{4} + 3q^{7} - 3q^{8} + q^{11} - 2q^{13} + 3q^{14} - q^{16} + 3q^{17} - q^{19} + q^{22} + q^{23} - 2q^{26} - 3q^{28} + 6q^{29} + 4q^{31} + 5q^{32} + 3q^{34} - q^{37} - q^{38} - 5q^{41} - 4q^{43} - q^{44} + q^{46} + 3q^{47} + 2q^{49} + 2q^{52} + 10q^{53} - 9q^{56} + 6q^{58} + 11q^{59} + 14q^{61} + 4q^{62} + 7q^{64} - 2q^{67} - 3q^{68} - 5q^{71} - 2q^{73} - q^{74} + q^{76} + 3q^{77} + 5q^{79} - 5q^{82} + 8q^{83} - 4q^{86} - 3q^{88} - 10q^{89} - 6q^{91} - q^{92} + 3q^{94} + 17q^{97} + 2q^{98} + O(q^{100}) \)

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
1.00000 0 −1.00000 0 0 3.00000 −3.00000 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.k yes 1
3.b odd 2 1 2475.2.a.d yes 1
5.b even 2 1 2475.2.a.b 1
5.c odd 4 2 2475.2.c.e 2
15.d odd 2 1 2475.2.a.h yes 1
15.e even 4 2 2475.2.c.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2475.2.a.b 1 5.b even 2 1
2475.2.a.d yes 1 3.b odd 2 1
2475.2.a.h yes 1 15.d odd 2 1
2475.2.a.k yes 1 1.a even 1 1 trivial
2475.2.c.c 2 15.e even 4 2
2475.2.c.e 2 5.c odd 4 2

Hecke kernels

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

\( T_{2} - 1 \)
\( T_{7} - 3 \)
\( T_{29} - 6 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -1 + T \)
$3$ \( T \)
$5$ \( T \)
$7$ \( -3 + T \)
$11$ \( -1 + T \)
$13$ \( 2 + T \)
$17$ \( -3 + T \)
$19$ \( 1 + T \)
$23$ \( -1 + T \)
$29$ \( -6 + T \)
$31$ \( -4 + T \)
$37$ \( 1 + T \)
$41$ \( 5 + T \)
$43$ \( 4 + T \)
$47$ \( -3 + T \)
$53$ \( -10 + T \)
$59$ \( -11 + T \)
$61$ \( -14 + T \)
$67$ \( 2 + T \)
$71$ \( 5 + T \)
$73$ \( 2 + T \)
$79$ \( -5 + T \)
$83$ \( -8 + T \)
$89$ \( 10 + T \)
$97$ \( -17 + T \)
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