Properties

Label 9072.2.a.by
Level $9072$
Weight $2$
Character orbit 9072.a
Self dual yes
Analytic conductor $72.440$
Analytic rank $1$
Dimension $3$
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: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.321.1
Defining polynomial: \(x^{3} - x^{2} - 4 x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 252)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{5} + q^{7} +O(q^{10})\) \( q + \beta_{1} q^{5} + q^{7} + ( -1 - \beta_{1} - 2 \beta_{2} ) q^{11} + ( 1 + 2 \beta_{1} + 2 \beta_{2} ) q^{13} + ( 1 - 2 \beta_{1} - \beta_{2} ) q^{17} + ( -2 \beta_{1} + \beta_{2} ) q^{19} + ( -4 - \beta_{1} + \beta_{2} ) q^{23} + ( -2 + \beta_{1} + \beta_{2} ) q^{25} + ( 1 - 3 \beta_{1} - \beta_{2} ) q^{29} + 3 \beta_{1} q^{31} + \beta_{1} q^{35} + ( -2 + \beta_{1} - 2 \beta_{2} ) q^{37} + ( -1 + \beta_{1} - 2 \beta_{2} ) q^{41} + ( \beta_{1} + 4 \beta_{2} ) q^{43} + ( -8 + 2 \beta_{1} - \beta_{2} ) q^{47} + q^{49} + ( 2 + \beta_{1} + \beta_{2} ) q^{53} + ( -1 - 4 \beta_{1} - \beta_{2} ) q^{55} + ( -11 + \beta_{1} - \beta_{2} ) q^{59} + ( 4 - 7 \beta_{1} - \beta_{2} ) q^{61} + ( 4 + 5 \beta_{1} + 2 \beta_{2} ) q^{65} + ( -3 + 2 \beta_{1} - \beta_{2} ) q^{67} + ( -3 - 5 \beta_{1} + 3 \beta_{2} ) q^{71} + ( -2 \beta_{1} + \beta_{2} ) q^{73} + ( -1 - \beta_{1} - 2 \beta_{2} ) q^{77} + ( 4 - 4 \beta_{1} - \beta_{2} ) q^{79} + ( -7 + 2 \beta_{1} + \beta_{2} ) q^{83} + ( -5 - 2 \beta_{1} - 2 \beta_{2} ) q^{85} + ( 5 + \beta_{1} + 4 \beta_{2} ) q^{89} + ( 1 + 2 \beta_{1} + 2 \beta_{2} ) q^{91} + ( -7 - \beta_{1} - 2 \beta_{2} ) q^{95} + ( -1 - \beta_{1} + 5 \beta_{2} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + q^{5} + 3q^{7} + O(q^{10}) \) \( 3q + q^{5} + 3q^{7} - 2q^{11} + 3q^{13} + 2q^{17} - 3q^{19} - 14q^{23} - 6q^{25} + q^{29} + 3q^{31} + q^{35} - 3q^{37} - 3q^{43} - 21q^{47} + 3q^{49} + 6q^{53} - 6q^{55} - 31q^{59} + 6q^{61} + 15q^{65} - 6q^{67} - 17q^{71} - 3q^{73} - 2q^{77} + 9q^{79} - 20q^{83} - 15q^{85} + 12q^{89} + 3q^{91} - 20q^{95} - 9q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - x^{2} - 4 x + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 3 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + \beta_{1} + 3\)

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
−1.69963
0.239123
2.46050
0 0 0 −1.69963 0 1.00000 0 0 0
1.2 0 0 0 0.239123 0 1.00000 0 0 0
1.3 0 0 0 2.46050 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.by 3
3.b odd 2 1 9072.2.a.bv 3
4.b odd 2 1 2268.2.a.i 3
9.c even 3 2 1008.2.r.j 6
9.d odd 6 2 3024.2.r.j 6
12.b even 2 1 2268.2.a.h 3
36.f odd 6 2 252.2.j.a 6
36.h even 6 2 756.2.j.b 6
252.n even 6 2 1764.2.l.f 6
252.o even 6 2 5292.2.l.e 6
252.r odd 6 2 5292.2.i.e 6
252.s odd 6 2 5292.2.j.d 6
252.u odd 6 2 1764.2.i.g 6
252.bb even 6 2 5292.2.i.f 6
252.bi even 6 2 1764.2.j.e 6
252.bj even 6 2 1764.2.i.d 6
252.bl odd 6 2 1764.2.l.e 6
252.bn odd 6 2 5292.2.l.f 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
252.2.j.a 6 36.f odd 6 2
756.2.j.b 6 36.h even 6 2
1008.2.r.j 6 9.c even 3 2
1764.2.i.d 6 252.bj even 6 2
1764.2.i.g 6 252.u odd 6 2
1764.2.j.e 6 252.bi even 6 2
1764.2.l.e 6 252.bl odd 6 2
1764.2.l.f 6 252.n even 6 2
2268.2.a.h 3 12.b even 2 1
2268.2.a.i 3 4.b odd 2 1
3024.2.r.j 6 9.d odd 6 2
5292.2.i.e 6 252.r odd 6 2
5292.2.i.f 6 252.bb even 6 2
5292.2.j.d 6 252.s odd 6 2
5292.2.l.e 6 252.o even 6 2
5292.2.l.f 6 252.bn odd 6 2
9072.2.a.bv 3 3.b odd 2 1
9072.2.a.by 3 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9072))\):

\( T_{5}^{3} - T_{5}^{2} - 4 T_{5} + 1 \)
\( T_{11}^{3} + 2 T_{11}^{2} - 25 T_{11} - 59 \)
\( T_{13}^{3} - 3 T_{13}^{2} - 33 T_{13} + 27 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ \( 1 - T + 11 T^{2} - 9 T^{3} + 55 T^{4} - 25 T^{5} + 125 T^{6} \)
$7$ \( ( 1 - T )^{3} \)
$11$ \( 1 + 2 T + 8 T^{2} - 15 T^{3} + 88 T^{4} + 242 T^{5} + 1331 T^{6} \)
$13$ \( 1 - 3 T + 6 T^{2} - 51 T^{3} + 78 T^{4} - 507 T^{5} + 2197 T^{6} \)
$17$ \( 1 - 2 T + 32 T^{2} - 21 T^{3} + 544 T^{4} - 578 T^{5} + 4913 T^{6} \)
$19$ \( 1 + 3 T + 33 T^{2} + 35 T^{3} + 627 T^{4} + 1083 T^{5} + 6859 T^{6} \)
$23$ \( 1 + 14 T + 122 T^{2} + 675 T^{3} + 2806 T^{4} + 7406 T^{5} + 12167 T^{6} \)
$29$ \( 1 - T + 47 T^{2} + 51 T^{3} + 1363 T^{4} - 841 T^{5} + 24389 T^{6} \)
$31$ \( 1 - 3 T + 57 T^{2} - 159 T^{3} + 1767 T^{4} - 2883 T^{5} + 29791 T^{6} \)
$37$ \( 1 + 3 T + 81 T^{2} + 199 T^{3} + 2997 T^{4} + 4107 T^{5} + 50653 T^{6} \)
$41$ \( 1 + 90 T^{2} + 9 T^{3} + 3690 T^{4} + 68921 T^{6} \)
$43$ \( 1 + 3 T + 33 T^{2} + 539 T^{3} + 1419 T^{4} + 5547 T^{5} + 79507 T^{6} \)
$47$ \( 1 + 21 T + 261 T^{2} + 2181 T^{3} + 12267 T^{4} + 46389 T^{5} + 103823 T^{6} \)
$53$ \( 1 - 6 T + 162 T^{2} - 627 T^{3} + 8586 T^{4} - 16854 T^{5} + 148877 T^{6} \)
$59$ \( 1 + 31 T + 485 T^{2} + 4647 T^{3} + 28615 T^{4} + 107911 T^{5} + 205379 T^{6} \)
$61$ \( 1 - 6 T - 12 T^{2} + 357 T^{3} - 732 T^{4} - 22326 T^{5} + 226981 T^{6} \)
$67$ \( 1 + 6 T + 186 T^{2} + 811 T^{3} + 12462 T^{4} + 26934 T^{5} + 300763 T^{6} \)
$71$ \( 1 + 17 T + 119 T^{2} + 507 T^{3} + 8449 T^{4} + 85697 T^{5} + 357911 T^{6} \)
$73$ \( 1 + 3 T + 195 T^{2} + 359 T^{3} + 14235 T^{4} + 15987 T^{5} + 389017 T^{6} \)
$79$ \( 1 - 9 T + 195 T^{2} - 1053 T^{3} + 15405 T^{4} - 56169 T^{5} + 493039 T^{6} \)
$83$ \( 1 + 20 T + 362 T^{2} + 3447 T^{3} + 30046 T^{4} + 137780 T^{5} + 571787 T^{6} \)
$89$ \( 1 - 12 T + 216 T^{2} - 1425 T^{3} + 19224 T^{4} - 95052 T^{5} + 704969 T^{6} \)
$97$ \( 1 + 9 T + 147 T^{2} + 1673 T^{3} + 14259 T^{4} + 84681 T^{5} + 912673 T^{6} \)
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