Properties

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

Related objects

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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: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 363)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 2 q^{2} + q^{3} + 2 q^{4} + 2 q^{6} - q^{7} + q^{9} + O(q^{10}) \) \( q + 2 q^{2} + q^{3} + 2 q^{4} + 2 q^{6} - q^{7} + q^{9} + 2 q^{12} + 2 q^{13} - 2 q^{14} - 4 q^{16} - 4 q^{17} + 2 q^{18} - 3 q^{19} - q^{21} - 2 q^{23} + 4 q^{26} + q^{27} - 2 q^{28} + 6 q^{29} - 5 q^{31} - 8 q^{32} - 8 q^{34} + 2 q^{36} - 3 q^{37} - 6 q^{38} + 2 q^{39} - 2 q^{41} - 2 q^{42} - 12 q^{43} - 4 q^{46} - 2 q^{47} - 4 q^{48} - 6 q^{49} - 4 q^{51} + 4 q^{52} - 6 q^{53} + 2 q^{54} - 3 q^{57} + 12 q^{58} - 10 q^{59} + 3 q^{61} - 10 q^{62} - q^{63} - 8 q^{64} + q^{67} - 8 q^{68} - 2 q^{69} + 11 q^{73} - 6 q^{74} - 6 q^{76} + 4 q^{78} + 11 q^{79} + q^{81} - 4 q^{82} - 6 q^{83} - 2 q^{84} - 24 q^{86} + 6 q^{87} + 12 q^{89} - 2 q^{91} - 4 q^{92} - 5 q^{93} - 4 q^{94} - 8 q^{96} - 5 q^{97} - 12 q^{98} + O(q^{100}) \)

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
2.00000 1.00000 2.00000 0 2.00000 −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.t 1
5.b even 2 1 363.2.a.a 1
11.b odd 2 1 9075.2.a.b 1
15.d odd 2 1 1089.2.a.k 1
20.d odd 2 1 5808.2.a.bh 1
55.d odd 2 1 363.2.a.c yes 1
55.h odd 10 4 363.2.e.d 4
55.j even 10 4 363.2.e.i 4
165.d even 2 1 1089.2.a.a 1
220.g even 2 1 5808.2.a.bi 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
363.2.a.a 1 5.b even 2 1
363.2.a.c yes 1 55.d odd 2 1
363.2.e.d 4 55.h odd 10 4
363.2.e.i 4 55.j even 10 4
1089.2.a.a 1 165.d even 2 1
1089.2.a.k 1 15.d odd 2 1
5808.2.a.bh 1 20.d odd 2 1
5808.2.a.bi 1 220.g even 2 1
9075.2.a.b 1 11.b odd 2 1
9075.2.a.t 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9075))\):

\( T_{2} - 2 \)
\( T_{7} + 1 \)
\( T_{13} - 2 \)
\( T_{17} + 4 \)
\( T_{19} + 3 \)
\( T_{23} + 2 \)
\( T_{37} + 3 \)

Hecke characteristic polynomials

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