Properties

Label 9072.2.a.bp
Level $9072$
Weight $2$
Character orbit 9072.a
Self dual yes
Analytic conductor $72.440$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 9072 = 2^{4} \cdot 3^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9072.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(72.4402847137\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{3}) \)
Defining polynomial: \(x^{2} - 3\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1134)
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.

\(f(q)\) \(=\) \( q + ( 2 + \beta ) q^{5} + q^{7} +O(q^{10})\) \( q + ( 2 + \beta ) q^{5} + q^{7} + ( -1 + 3 \beta ) q^{11} + ( -3 + 2 \beta ) q^{13} + 7 q^{17} + ( 1 + \beta ) q^{19} + ( -1 - 3 \beta ) q^{23} + ( 2 + 4 \beta ) q^{25} + ( 5 + 2 \beta ) q^{29} + ( -3 + 3 \beta ) q^{31} + ( 2 + \beta ) q^{35} + ( 2 - 5 \beta ) q^{37} + ( 6 + 2 \beta ) q^{41} + ( -2 - 2 \beta ) q^{43} + ( -3 + \beta ) q^{47} + q^{49} + ( -6 + 2 \beta ) q^{53} + ( 7 + 5 \beta ) q^{55} + ( 1 + 3 \beta ) q^{59} + ( -3 - 4 \beta ) q^{61} + \beta q^{65} + ( 5 - \beta ) q^{67} + ( -10 - 2 \beta ) q^{71} + ( 10 + \beta ) q^{73} + ( -1 + 3 \beta ) q^{77} + ( -3 - 7 \beta ) q^{79} + ( -1 + 9 \beta ) q^{83} + ( 14 + 7 \beta ) q^{85} + ( 3 - 4 \beta ) q^{89} + ( -3 + 2 \beta ) q^{91} + ( 5 + 3 \beta ) q^{95} + ( 4 - 4 \beta ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 4q^{5} + 2q^{7} + O(q^{10}) \) \( 2q + 4q^{5} + 2q^{7} - 2q^{11} - 6q^{13} + 14q^{17} + 2q^{19} - 2q^{23} + 4q^{25} + 10q^{29} - 6q^{31} + 4q^{35} + 4q^{37} + 12q^{41} - 4q^{43} - 6q^{47} + 2q^{49} - 12q^{53} + 14q^{55} + 2q^{59} - 6q^{61} + 10q^{67} - 20q^{71} + 20q^{73} - 2q^{77} - 6q^{79} - 2q^{83} + 28q^{85} + 6q^{89} - 6q^{91} + 10q^{95} + 8q^{97} + O(q^{100}) \)

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 0 0 0.267949 0 1.00000 0 0 0
1.2 0 0 0 3.73205 0 1.00000 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(7\) \(-1\)

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 9072.2.a.bp 2
3.b odd 2 1 9072.2.a.y 2
4.b odd 2 1 1134.2.a.l 2
12.b even 2 1 1134.2.a.m yes 2
28.d even 2 1 7938.2.a.bg 2
36.f odd 6 2 1134.2.f.s 4
36.h even 6 2 1134.2.f.r 4
84.h odd 2 1 7938.2.a.bt 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1134.2.a.l 2 4.b odd 2 1
1134.2.a.m yes 2 12.b even 2 1
1134.2.f.r 4 36.h even 6 2
1134.2.f.s 4 36.f odd 6 2
7938.2.a.bg 2 28.d even 2 1
7938.2.a.bt 2 84.h odd 2 1
9072.2.a.y 2 3.b odd 2 1
9072.2.a.bp 2 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9072))\):

\( T_{5}^{2} - 4 T_{5} + 1 \)
\( T_{11}^{2} + 2 T_{11} - 26 \)
\( T_{13}^{2} + 6 T_{13} - 3 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( 1 - 4 T + T^{2} \)
$7$ \( ( -1 + T )^{2} \)
$11$ \( -26 + 2 T + T^{2} \)
$13$ \( -3 + 6 T + T^{2} \)
$17$ \( ( -7 + T )^{2} \)
$19$ \( -2 - 2 T + T^{2} \)
$23$ \( -26 + 2 T + T^{2} \)
$29$ \( 13 - 10 T + T^{2} \)
$31$ \( -18 + 6 T + T^{2} \)
$37$ \( -71 - 4 T + T^{2} \)
$41$ \( 24 - 12 T + T^{2} \)
$43$ \( -8 + 4 T + T^{2} \)
$47$ \( 6 + 6 T + T^{2} \)
$53$ \( 24 + 12 T + T^{2} \)
$59$ \( -26 - 2 T + T^{2} \)
$61$ \( -39 + 6 T + T^{2} \)
$67$ \( 22 - 10 T + T^{2} \)
$71$ \( 88 + 20 T + T^{2} \)
$73$ \( 97 - 20 T + T^{2} \)
$79$ \( -138 + 6 T + T^{2} \)
$83$ \( -242 + 2 T + T^{2} \)
$89$ \( -39 - 6 T + T^{2} \)
$97$ \( -32 - 8 T + T^{2} \)
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