Properties

Label 9072.2.a.cd
Level $9072$
Weight $2$
Character orbit 9072.a
Self dual yes
Analytic conductor $72.440$
Analytic rank $0$
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: \(0\)
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 63)
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.

\(f(q)\) \(=\) \( q + ( 2 + \beta_{2} ) q^{5} + q^{7} +O(q^{10})\) \( q + ( 2 + \beta_{2} ) q^{5} + q^{7} + ( -1 + 2 \beta_{1} + \beta_{2} ) q^{11} + q^{13} + ( 4 - \beta_{1} - \beta_{2} ) q^{17} + ( 1 + \beta_{1} + \beta_{2} ) q^{19} + ( 1 - \beta_{1} + 2 \beta_{2} ) q^{23} + ( 3 - \beta_{1} + 2 \beta_{2} ) q^{25} -\beta_{1} q^{29} + ( 2 - 2 \beta_{1} + \beta_{2} ) q^{31} + ( 2 + \beta_{2} ) q^{35} + 3 \beta_{2} q^{37} + ( 7 - \beta_{2} ) q^{41} + ( 4 \beta_{1} + \beta_{2} ) q^{43} + ( -3 - 3 \beta_{1} - 3 \beta_{2} ) q^{47} + q^{49} + ( 5 + \beta_{1} - 2 \beta_{2} ) q^{53} + ( 5 \beta_{1} - \beta_{2} ) q^{55} + ( -4 + \beta_{1} - 2 \beta_{2} ) q^{59} + ( -1 - \beta_{1} + 2 \beta_{2} ) q^{61} + ( 2 + \beta_{2} ) q^{65} + ( -2 + 7 \beta_{1} + \beta_{2} ) q^{67} + ( 2 + \beta_{1} - 2 \beta_{2} ) q^{71} + ( -1 + \beta_{1} - 5 \beta_{2} ) q^{73} + ( -1 + 2 \beta_{1} + \beta_{2} ) q^{77} + ( -3 - 5 \beta_{1} + \beta_{2} ) q^{79} + ( -4 + \beta_{1} + \beta_{2} ) q^{83} + ( 5 - 2 \beta_{1} + 4 \beta_{2} ) q^{85} + ( -1 + 6 \beta_{1} + \beta_{2} ) q^{89} + q^{91} + ( 5 + 2 \beta_{1} + \beta_{2} ) q^{95} + ( 2 - 5 \beta_{1} - 2 \beta_{2} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + 5q^{5} + 3q^{7} + O(q^{10}) \) \( 3q + 5q^{5} + 3q^{7} - 2q^{11} + 3q^{13} + 12q^{17} + 3q^{19} + 6q^{25} - q^{29} + 3q^{31} + 5q^{35} - 3q^{37} + 22q^{41} + 3q^{43} - 9q^{47} + 3q^{49} + 18q^{53} + 6q^{55} - 9q^{59} - 6q^{61} + 5q^{65} + 9q^{71} + 3q^{73} - 2q^{77} - 15q^{79} - 12q^{83} + 9q^{85} + 2q^{89} + 3q^{91} + 16q^{95} + 3q^{97} + O(q^{100}) \)

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
0.239123
2.46050
−1.69963
0 0 0 −1.18194 0 1.00000 0 0 0
1.2 0 0 0 2.59358 0 1.00000 0 0 0
1.3 0 0 0 3.58836 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.cd 3
3.b odd 2 1 9072.2.a.bq 3
4.b odd 2 1 567.2.a.g 3
9.c even 3 2 3024.2.r.g 6
9.d odd 6 2 1008.2.r.k 6
12.b even 2 1 567.2.a.d 3
28.d even 2 1 3969.2.a.p 3
36.f odd 6 2 189.2.f.a 6
36.h even 6 2 63.2.f.b 6
84.h odd 2 1 3969.2.a.m 3
252.n even 6 2 1323.2.g.b 6
252.o even 6 2 441.2.g.e 6
252.r odd 6 2 441.2.h.b 6
252.s odd 6 2 441.2.f.d 6
252.u odd 6 2 1323.2.h.d 6
252.bb even 6 2 441.2.h.c 6
252.bi even 6 2 1323.2.f.c 6
252.bj even 6 2 1323.2.h.e 6
252.bl odd 6 2 1323.2.g.c 6
252.bn odd 6 2 441.2.g.d 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.2.f.b 6 36.h even 6 2
189.2.f.a 6 36.f odd 6 2
441.2.f.d 6 252.s odd 6 2
441.2.g.d 6 252.bn odd 6 2
441.2.g.e 6 252.o even 6 2
441.2.h.b 6 252.r odd 6 2
441.2.h.c 6 252.bb even 6 2
567.2.a.d 3 12.b even 2 1
567.2.a.g 3 4.b odd 2 1
1008.2.r.k 6 9.d odd 6 2
1323.2.f.c 6 252.bi even 6 2
1323.2.g.b 6 252.n even 6 2
1323.2.g.c 6 252.bl odd 6 2
1323.2.h.d 6 252.u odd 6 2
1323.2.h.e 6 252.bj even 6 2
3024.2.r.g 6 9.c even 3 2
3969.2.a.m 3 84.h odd 2 1
3969.2.a.p 3 28.d even 2 1
9072.2.a.bq 3 3.b odd 2 1
9072.2.a.cd 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9072))\):

\( T_{5}^{3} - 5 T_{5}^{2} + 2 T_{5} + 11 \)
\( T_{11}^{3} + 2 T_{11}^{2} - 19 T_{11} - 47 \)
\( T_{13} - 1 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ \( 1 - 5 T + 17 T^{2} - 39 T^{3} + 85 T^{4} - 125 T^{5} + 125 T^{6} \)
$7$ \( ( 1 - T )^{3} \)
$11$ \( 1 + 2 T + 14 T^{2} - 3 T^{3} + 154 T^{4} + 242 T^{5} + 1331 T^{6} \)
$13$ \( ( 1 - T + 13 T^{2} )^{3} \)
$17$ \( 1 - 12 T + 90 T^{2} - 435 T^{3} + 1530 T^{4} - 3468 T^{5} + 4913 T^{6} \)
$19$ \( 1 - 3 T + 51 T^{2} - 107 T^{3} + 969 T^{4} - 1083 T^{5} + 6859 T^{6} \)
$23$ \( 1 + 36 T^{2} - 9 T^{3} + 828 T^{4} + 12167 T^{6} \)
$29$ \( 1 + T + 83 T^{2} + 57 T^{3} + 2407 T^{4} + 841 T^{5} + 24389 T^{6} \)
$31$ \( 1 - 3 T + 69 T^{2} - 213 T^{3} + 2139 T^{4} - 2883 T^{5} + 29791 T^{6} \)
$37$ \( 1 + 3 T + 57 T^{2} + 303 T^{3} + 2109 T^{4} + 4107 T^{5} + 50653 T^{6} \)
$41$ \( 1 - 22 T + 278 T^{2} - 2157 T^{3} + 11398 T^{4} - 36982 T^{5} + 68921 T^{6} \)
$43$ \( 1 - 3 T + 63 T^{2} - 379 T^{3} + 2709 T^{4} - 5547 T^{5} + 79507 T^{6} \)
$47$ \( 1 + 9 T + 87 T^{2} + 657 T^{3} + 4089 T^{4} + 19881 T^{5} + 103823 T^{6} \)
$53$ \( 1 - 18 T + 234 T^{2} - 1917 T^{3} + 12402 T^{4} - 50562 T^{5} + 148877 T^{6} \)
$59$ \( 1 + 9 T + 171 T^{2} + 999 T^{3} + 10089 T^{4} + 31329 T^{5} + 205379 T^{6} \)
$61$ \( 1 + 6 T + 162 T^{2} + 665 T^{3} + 9882 T^{4} + 22326 T^{5} + 226981 T^{6} \)
$67$ \( 1 - 6 T^{2} - 683 T^{3} - 402 T^{4} + 300763 T^{6} \)
$71$ \( 1 - 9 T + 207 T^{2} - 1197 T^{3} + 14697 T^{4} - 45369 T^{5} + 357911 T^{6} \)
$73$ \( 1 - 3 T + 51 T^{2} - 681 T^{3} + 3723 T^{4} - 15987 T^{5} + 389017 T^{6} \)
$79$ \( 1 + 15 T + 189 T^{2} + 1601 T^{3} + 14931 T^{4} + 93615 T^{5} + 493039 T^{6} \)
$83$ \( 1 + 12 T + 288 T^{2} + 2019 T^{3} + 23904 T^{4} + 82668 T^{5} + 571787 T^{6} \)
$89$ \( 1 - 2 T + 116 T^{2} - 735 T^{3} + 10324 T^{4} - 15842 T^{5} + 704969 T^{6} \)
$97$ \( 1 - 3 T + 177 T^{2} + 21 T^{3} + 17169 T^{4} - 28227 T^{5} + 912673 T^{6} \)
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