Properties

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

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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 75)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 2q^{2} - q^{3} + 2q^{4} + 2q^{6} + 3q^{7} + q^{9} + O(q^{10}) \) \( q - 2q^{2} - q^{3} + 2q^{4} + 2q^{6} + 3q^{7} + q^{9} - 2q^{12} - q^{13} - 6q^{14} - 4q^{16} - 2q^{17} - 2q^{18} + 5q^{19} - 3q^{21} + 6q^{23} + 2q^{26} - q^{27} + 6q^{28} - 10q^{29} - 3q^{31} + 8q^{32} + 4q^{34} + 2q^{36} + 2q^{37} - 10q^{38} + q^{39} + 8q^{41} + 6q^{42} - q^{43} - 12q^{46} + 2q^{47} + 4q^{48} + 2q^{49} + 2q^{51} - 2q^{52} - 4q^{53} + 2q^{54} - 5q^{57} + 20q^{58} - 10q^{59} - 7q^{61} + 6q^{62} + 3q^{63} - 8q^{64} - 3q^{67} - 4q^{68} - 6q^{69} - 8q^{71} + 14q^{73} - 4q^{74} + 10q^{76} - 2q^{78} + q^{81} - 16q^{82} - 6q^{83} - 6q^{84} + 2q^{86} + 10q^{87} - 3q^{91} + 12q^{92} + 3q^{93} - 4q^{94} - 8q^{96} + 17q^{97} - 4q^{98} + O(q^{100}) \)

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 3.00000 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.a 1
5.b even 2 1 9075.2.a.s 1
11.b odd 2 1 75.2.a.c yes 1
33.d even 2 1 225.2.a.a 1
44.c even 2 1 1200.2.a.p 1
55.d odd 2 1 75.2.a.a 1
55.e even 4 2 75.2.b.a 2
77.b even 2 1 3675.2.a.q 1
88.b odd 2 1 4800.2.a.bq 1
88.g even 2 1 4800.2.a.be 1
132.d odd 2 1 3600.2.a.bk 1
165.d even 2 1 225.2.a.e 1
165.l odd 4 2 225.2.b.a 2
220.g even 2 1 1200.2.a.c 1
220.i odd 4 2 1200.2.f.d 2
385.h even 2 1 3675.2.a.b 1
440.c even 2 1 4800.2.a.br 1
440.o odd 2 1 4800.2.a.bb 1
440.t even 4 2 4800.2.f.l 2
440.w odd 4 2 4800.2.f.y 2
660.g odd 2 1 3600.2.a.j 1
660.q even 4 2 3600.2.f.p 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
75.2.a.a 1 55.d odd 2 1
75.2.a.c yes 1 11.b odd 2 1
75.2.b.a 2 55.e even 4 2
225.2.a.a 1 33.d even 2 1
225.2.a.e 1 165.d even 2 1
225.2.b.a 2 165.l odd 4 2
1200.2.a.c 1 220.g even 2 1
1200.2.a.p 1 44.c even 2 1
1200.2.f.d 2 220.i odd 4 2
3600.2.a.j 1 660.g odd 2 1
3600.2.a.bk 1 132.d odd 2 1
3600.2.f.p 2 660.q even 4 2
3675.2.a.b 1 385.h even 2 1
3675.2.a.q 1 77.b even 2 1
4800.2.a.bb 1 440.o odd 2 1
4800.2.a.be 1 88.g even 2 1
4800.2.a.bq 1 88.b odd 2 1
4800.2.a.br 1 440.c even 2 1
4800.2.f.l 2 440.t even 4 2
4800.2.f.y 2 440.w odd 4 2
9075.2.a.a 1 1.a even 1 1 trivial
9075.2.a.s 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9075))\):

\( T_{2} + 2 \)
\( T_{7} - 3 \)
\( T_{13} + 1 \)
\( T_{17} + 2 \)
\( T_{19} - 5 \)
\( T_{23} - 6 \)
\( T_{37} - 2 \)

Hecke characteristic polynomials

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