Properties

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

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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: \(2\)
Coefficient field: \(\Q(\sqrt{3}) \)
Defining polynomial: \(x^{2} - 3\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1815)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{3} -2 q^{4} + \beta q^{7} + q^{9} +O(q^{10})\) \( q + q^{3} -2 q^{4} + \beta q^{7} + q^{9} -2 q^{12} + 4 q^{16} -2 \beta q^{17} + 3 \beta q^{19} + \beta q^{21} -6 q^{23} + q^{27} -2 \beta q^{28} -4 \beta q^{29} + q^{31} -2 q^{36} + 5 q^{37} -2 \beta q^{41} + 6 \beta q^{43} + 12 q^{47} + 4 q^{48} -4 q^{49} -2 \beta q^{51} -6 q^{53} + 3 \beta q^{57} + 7 \beta q^{61} + \beta q^{63} -8 q^{64} + 5 q^{67} + 4 \beta q^{68} -6 q^{69} -6 q^{71} -\beta q^{73} -6 \beta q^{76} + 9 \beta q^{79} + q^{81} -4 \beta q^{83} -2 \beta q^{84} -4 \beta q^{87} -6 q^{89} + 12 q^{92} + q^{93} + 13 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{3} - 4q^{4} + 2q^{9} + O(q^{10}) \) \( 2q + 2q^{3} - 4q^{4} + 2q^{9} - 4q^{12} + 8q^{16} - 12q^{23} + 2q^{27} + 2q^{31} - 4q^{36} + 10q^{37} + 24q^{47} + 8q^{48} - 8q^{49} - 12q^{53} - 16q^{64} + 10q^{67} - 12q^{69} - 12q^{71} + 2q^{81} - 12q^{89} + 24q^{92} + 2q^{93} + 26q^{97} + O(q^{100}) \)

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
−1.73205
1.73205
0 1.00000 −2.00000 0 0 −1.73205 0 1.00000 0
1.2 0 1.00000 −2.00000 0 0 1.73205 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

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 9075.2.a.bl 2
5.b even 2 1 1815.2.a.g 2
11.b odd 2 1 inner 9075.2.a.bl 2
15.d odd 2 1 5445.2.a.q 2
55.d odd 2 1 1815.2.a.g 2
165.d even 2 1 5445.2.a.q 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1815.2.a.g 2 5.b even 2 1
1815.2.a.g 2 55.d odd 2 1
5445.2.a.q 2 15.d odd 2 1
5445.2.a.q 2 165.d even 2 1
9075.2.a.bl 2 1.a even 1 1 trivial
9075.2.a.bl 2 11.b odd 2 1 inner

Hecke kernels

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

\( T_{2} \)
\( T_{7}^{2} - 3 \)
\( T_{13} \)
\( T_{17}^{2} - 12 \)
\( T_{19}^{2} - 27 \)
\( T_{23} + 6 \)
\( T_{37} - 5 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( ( -1 + T )^{2} \)
$5$ \( T^{2} \)
$7$ \( -3 + T^{2} \)
$11$ \( T^{2} \)
$13$ \( T^{2} \)
$17$ \( -12 + T^{2} \)
$19$ \( -27 + T^{2} \)
$23$ \( ( 6 + T )^{2} \)
$29$ \( -48 + T^{2} \)
$31$ \( ( -1 + T )^{2} \)
$37$ \( ( -5 + T )^{2} \)
$41$ \( -12 + T^{2} \)
$43$ \( -108 + T^{2} \)
$47$ \( ( -12 + T )^{2} \)
$53$ \( ( 6 + T )^{2} \)
$59$ \( T^{2} \)
$61$ \( -147 + T^{2} \)
$67$ \( ( -5 + T )^{2} \)
$71$ \( ( 6 + T )^{2} \)
$73$ \( -3 + T^{2} \)
$79$ \( -243 + T^{2} \)
$83$ \( -48 + T^{2} \)
$89$ \( ( 6 + T )^{2} \)
$97$ \( ( -13 + T )^{2} \)
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