Properties

Label 9072.2.a.bd
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: \(\Q(\sqrt{6}) \)
Defining polynomial: \(x^{2} - 6\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 126)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -1 + \beta ) q^{5} + q^{7} +O(q^{10})\) \( q + ( -1 + \beta ) q^{5} + q^{7} + 2 q^{11} + 2 \beta q^{13} -2 q^{17} + ( -5 + \beta ) q^{19} - q^{23} + ( 2 - 2 \beta ) q^{25} + ( 2 + 2 \beta ) q^{29} -6 q^{31} + ( -1 + \beta ) q^{35} + ( 2 + 4 \beta ) q^{37} -4 \beta q^{41} + ( -2 - 2 \beta ) q^{43} + 4 \beta q^{47} + q^{49} + ( 6 + 2 \beta ) q^{53} + ( -2 + 2 \beta ) q^{55} -2 q^{59} + ( 9 - \beta ) q^{61} + ( 12 - 2 \beta ) q^{65} + ( 8 + 2 \beta ) q^{67} + ( 5 - 2 \beta ) q^{71} + ( -2 - 2 \beta ) q^{73} + 2 q^{77} + ( -3 + 2 \beta ) q^{79} + 2 q^{83} + ( 2 - 2 \beta ) q^{85} + ( 12 + 2 \beta ) q^{89} + 2 \beta q^{91} + ( 11 - 6 \beta ) q^{95} + ( -2 + 2 \beta ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{5} + 2 q^{7} + O(q^{10}) \) \( 2 q - 2 q^{5} + 2 q^{7} + 4 q^{11} - 4 q^{17} - 10 q^{19} - 2 q^{23} + 4 q^{25} + 4 q^{29} - 12 q^{31} - 2 q^{35} + 4 q^{37} - 4 q^{43} + 2 q^{49} + 12 q^{53} - 4 q^{55} - 4 q^{59} + 18 q^{61} + 24 q^{65} + 16 q^{67} + 10 q^{71} - 4 q^{73} + 4 q^{77} - 6 q^{79} + 4 q^{83} + 4 q^{85} + 24 q^{89} + 22 q^{95} - 4 q^{97} + O(q^{100}) \)

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
−2.44949
2.44949
0 0 0 −3.44949 0 1.00000 0 0 0
1.2 0 0 0 1.44949 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.bd 2
3.b odd 2 1 9072.2.a.bk 2
4.b odd 2 1 1134.2.a.i 2
9.c even 3 2 3024.2.r.e 4
9.d odd 6 2 1008.2.r.e 4
12.b even 2 1 1134.2.a.p 2
28.d even 2 1 7938.2.a.bm 2
36.f odd 6 2 378.2.f.d 4
36.h even 6 2 126.2.f.c 4
84.h odd 2 1 7938.2.a.bn 2
252.n even 6 2 2646.2.h.n 4
252.o even 6 2 882.2.h.k 4
252.r odd 6 2 882.2.e.n 4
252.s odd 6 2 882.2.f.j 4
252.u odd 6 2 2646.2.e.l 4
252.bb even 6 2 882.2.e.m 4
252.bi even 6 2 2646.2.f.k 4
252.bj even 6 2 2646.2.e.k 4
252.bl odd 6 2 2646.2.h.m 4
252.bn odd 6 2 882.2.h.l 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.2.f.c 4 36.h even 6 2
378.2.f.d 4 36.f odd 6 2
882.2.e.m 4 252.bb even 6 2
882.2.e.n 4 252.r odd 6 2
882.2.f.j 4 252.s odd 6 2
882.2.h.k 4 252.o even 6 2
882.2.h.l 4 252.bn odd 6 2
1008.2.r.e 4 9.d odd 6 2
1134.2.a.i 2 4.b odd 2 1
1134.2.a.p 2 12.b even 2 1
2646.2.e.k 4 252.bj even 6 2
2646.2.e.l 4 252.u odd 6 2
2646.2.f.k 4 252.bi even 6 2
2646.2.h.m 4 252.bl odd 6 2
2646.2.h.n 4 252.n even 6 2
3024.2.r.e 4 9.c even 3 2
7938.2.a.bm 2 28.d even 2 1
7938.2.a.bn 2 84.h odd 2 1
9072.2.a.bd 2 1.a even 1 1 trivial
9072.2.a.bk 2 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(9072))\):

\( T_{5}^{2} + 2 T_{5} - 5 \)
\( T_{11} - 2 \)
\( T_{13}^{2} - 24 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( -5 + 2 T + T^{2} \)
$7$ \( ( -1 + T )^{2} \)
$11$ \( ( -2 + T )^{2} \)
$13$ \( -24 + T^{2} \)
$17$ \( ( 2 + T )^{2} \)
$19$ \( 19 + 10 T + T^{2} \)
$23$ \( ( 1 + T )^{2} \)
$29$ \( -20 - 4 T + T^{2} \)
$31$ \( ( 6 + T )^{2} \)
$37$ \( -92 - 4 T + T^{2} \)
$41$ \( -96 + T^{2} \)
$43$ \( -20 + 4 T + T^{2} \)
$47$ \( -96 + T^{2} \)
$53$ \( 12 - 12 T + T^{2} \)
$59$ \( ( 2 + T )^{2} \)
$61$ \( 75 - 18 T + T^{2} \)
$67$ \( 40 - 16 T + T^{2} \)
$71$ \( 1 - 10 T + T^{2} \)
$73$ \( -20 + 4 T + T^{2} \)
$79$ \( -15 + 6 T + T^{2} \)
$83$ \( ( -2 + T )^{2} \)
$89$ \( 120 - 24 T + T^{2} \)
$97$ \( -20 + 4 T + T^{2} \)
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