Properties

Label 9075.2.a.k
Level $9075$
Weight $2$
Character orbit 9075.a
Self dual yes
Analytic conductor $72.464$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 9075 = 3 \cdot 5^{2} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9075.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(72.4642398343\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1815)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - q^{3} - 2q^{4} + q^{7} + q^{9} + O(q^{10}) \) \( q - q^{3} - 2q^{4} + q^{7} + q^{9} + 2q^{12} - 2q^{13} + 4q^{16} - 6q^{17} - 7q^{19} - q^{21} + 6q^{23} - q^{27} - 2q^{28} - q^{31} - 2q^{36} + 7q^{37} + 2q^{39} - 6q^{41} - 8q^{43} - 4q^{48} - 6q^{49} + 6q^{51} + 4q^{52} + 6q^{53} + 7q^{57} - 12q^{59} - q^{61} + q^{63} - 8q^{64} + 7q^{67} + 12q^{68} - 6q^{69} + 6q^{71} + 13q^{73} + 14q^{76} + 11q^{79} + q^{81} + 2q^{84} - 18q^{89} - 2q^{91} - 12q^{92} + q^{93} + q^{97} + O(q^{100}) \)

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 −1.00000 −2.00000 0 0 1.00000 0 1.00000 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 9075.2.a.k 1
5.b even 2 1 1815.2.a.b 1
11.b odd 2 1 9075.2.a.j 1
15.d odd 2 1 5445.2.a.f 1
55.d odd 2 1 1815.2.a.c yes 1
165.d even 2 1 5445.2.a.g 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1815.2.a.b 1 5.b even 2 1
1815.2.a.c yes 1 55.d odd 2 1
5445.2.a.f 1 15.d odd 2 1
5445.2.a.g 1 165.d even 2 1
9075.2.a.j 1 11.b odd 2 1
9075.2.a.k 1 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(9075))\):

\( T_{2} \)
\( T_{7} - 1 \)
\( T_{13} + 2 \)
\( T_{17} + 6 \)
\( T_{19} + 7 \)
\( T_{23} - 6 \)
\( T_{37} - 7 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( 1 + T \)
$5$ \( T \)
$7$ \( -1 + T \)
$11$ \( T \)
$13$ \( 2 + T \)
$17$ \( 6 + T \)
$19$ \( 7 + T \)
$23$ \( -6 + T \)
$29$ \( T \)
$31$ \( 1 + T \)
$37$ \( -7 + T \)
$41$ \( 6 + T \)
$43$ \( 8 + T \)
$47$ \( T \)
$53$ \( -6 + T \)
$59$ \( 12 + T \)
$61$ \( 1 + T \)
$67$ \( -7 + T \)
$71$ \( -6 + T \)
$73$ \( -13 + T \)
$79$ \( -11 + T \)
$83$ \( T \)
$89$ \( 18 + T \)
$97$ \( -1 + T \)
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