Properties

Label 9075.2.a.i
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 825)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - q^{3} - 2 q^{4} - q^{7} + q^{9} + O(q^{10}) \) \( q - q^{3} - 2 q^{4} - q^{7} + q^{9} + 2 q^{12} - q^{13} + 4 q^{16} - 6 q^{17} + 7 q^{19} + q^{21} - 6 q^{23} - q^{27} + 2 q^{28} + 6 q^{29} - 7 q^{31} - 2 q^{36} - 2 q^{37} + q^{39} + 6 q^{41} - q^{43} - 4 q^{48} - 6 q^{49} + 6 q^{51} + 2 q^{52} + 6 q^{53} - 7 q^{57} - 5 q^{61} - q^{63} - 8 q^{64} - 5 q^{67} + 12 q^{68} + 6 q^{69} - 12 q^{71} + 14 q^{73} - 14 q^{76} + 4 q^{79} + q^{81} - 6 q^{83} - 2 q^{84} - 6 q^{87} + 6 q^{89} + q^{91} + 12 q^{92} + 7 q^{93} - 17 q^{97} + O(q^{100}) \)

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.i 1
5.b even 2 1 9075.2.a.l 1
11.b odd 2 1 825.2.a.b 1
33.d even 2 1 2475.2.a.f 1
55.d odd 2 1 825.2.a.c yes 1
55.e even 4 2 825.2.c.b 2
165.d even 2 1 2475.2.a.e 1
165.l odd 4 2 2475.2.c.h 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
825.2.a.b 1 11.b odd 2 1
825.2.a.c yes 1 55.d odd 2 1
825.2.c.b 2 55.e even 4 2
2475.2.a.e 1 165.d even 2 1
2475.2.a.f 1 33.d even 2 1
2475.2.c.h 2 165.l odd 4 2
9075.2.a.i 1 1.a even 1 1 trivial
9075.2.a.l 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9075))\):

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

Hecke characteristic polynomials

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