Properties

Label 9408.2.a.ej
Level $9408$
Weight $2$
Character orbit 9408.a
Self dual yes
Analytic conductor $75.123$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 9408 = 2^{6} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9408.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(75.1232582216\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.621.1
Defining polynomial: \( x^{3} - 6x - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 672)
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.

\(f(q)\) \(=\) \( q + q^{3} - \beta_{2} q^{5} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{3} - \beta_{2} q^{5} + q^{9} + ( - \beta_{2} - \beta_1) q^{11} + ( - \beta_{2} + \beta_1 + 1) q^{13} - \beta_{2} q^{15} + (2 \beta_{2} + 2) q^{17} + ( - \beta_{2} + \beta_1 + 1) q^{19} + (2 \beta_{2} + 2) q^{23} + ( - \beta_{2} - \beta_1 + 1) q^{25} + q^{27} + (\beta_{2} - 4) q^{29} + (\beta_1 - 1) q^{31} + ( - \beta_{2} - \beta_1) q^{33} + ( - 3 \beta_{2} - \beta_1 - 1) q^{37} + ( - \beta_{2} + \beta_1 + 1) q^{39} + (2 \beta_{2} + 2 \beta_1 - 2) q^{41} + (\beta_{2} + \beta_1 + 5) q^{43} - \beta_{2} q^{45} + ( - 2 \beta_{2} - 4) q^{47} + (2 \beta_{2} + 2) q^{51} + ( - \beta_{2} - 2 \beta_1 - 2) q^{53} + ( - 3 \beta_{2} + 4) q^{55} + ( - \beta_{2} + \beta_1 + 1) q^{57} + ( - \beta_{2} - \beta_1 + 4) q^{59} + ( - 2 \beta_1 + 6) q^{61} + ( - 2 \beta_1 + 8) q^{65} + ( - \beta_{2} + \beta_1 + 3) q^{67} + (2 \beta_{2} + 2) q^{69} + 2 \beta_{2} q^{71} + ( - \beta_{2} - \beta_1 + 11) q^{73} + ( - \beta_{2} - \beta_1 + 1) q^{75} + ( - 2 \beta_{2} - \beta_1 + 9) q^{79} + q^{81} + ( - 3 \beta_{2} + \beta_1 + 6) q^{83} + (2 \beta_1 - 12) q^{85} + (\beta_{2} - 4) q^{87} + ( - 2 \beta_{2} - 4) q^{89} + (\beta_1 - 1) q^{93} + ( - 2 \beta_1 + 8) q^{95} + ( - \beta_{2} - \beta_1) q^{97} + ( - \beta_{2} - \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{3} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 3 q^{3} + 3 q^{9} + 3 q^{13} + 6 q^{17} + 3 q^{19} + 6 q^{23} + 3 q^{25} + 3 q^{27} - 12 q^{29} - 3 q^{31} - 3 q^{37} + 3 q^{39} - 6 q^{41} + 15 q^{43} - 12 q^{47} + 6 q^{51} - 6 q^{53} + 12 q^{55} + 3 q^{57} + 12 q^{59} + 18 q^{61} + 24 q^{65} + 9 q^{67} + 6 q^{69} + 33 q^{73} + 3 q^{75} + 27 q^{79} + 3 q^{81} + 18 q^{83} - 36 q^{85} - 12 q^{87} - 12 q^{89} - 3 q^{93} + 24 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - 6x - 3 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 4 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{2} + \beta _1 + 8 ) / 2 \) Copy content Toggle raw display

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
−2.14510
2.66908
−0.523976
0 1.00000 0 −2.74657 0 0 0 1.00000 0
1.2 0 1.00000 0 −0.454904 0 0 0 1.00000 0
1.3 0 1.00000 0 3.20147 0 0 0 1.00000 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 9408.2.a.ej 3
4.b odd 2 1 9408.2.a.eh 3
7.b odd 2 1 9408.2.a.eg 3
7.c even 3 2 1344.2.q.y 6
8.b even 2 1 4704.2.a.bs 3
8.d odd 2 1 4704.2.a.bu 3
28.d even 2 1 9408.2.a.ei 3
28.g odd 6 2 1344.2.q.z 6
56.e even 2 1 4704.2.a.bt 3
56.h odd 2 1 4704.2.a.bv 3
56.k odd 6 2 672.2.q.k 6
56.p even 6 2 672.2.q.l yes 6
168.s odd 6 2 2016.2.s.v 6
168.v even 6 2 2016.2.s.u 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
672.2.q.k 6 56.k odd 6 2
672.2.q.l yes 6 56.p even 6 2
1344.2.q.y 6 7.c even 3 2
1344.2.q.z 6 28.g odd 6 2
2016.2.s.u 6 168.v even 6 2
2016.2.s.v 6 168.s odd 6 2
4704.2.a.bs 3 8.b even 2 1
4704.2.a.bt 3 56.e even 2 1
4704.2.a.bu 3 8.d odd 2 1
4704.2.a.bv 3 56.h odd 2 1
9408.2.a.eg 3 7.b odd 2 1
9408.2.a.eh 3 4.b odd 2 1
9408.2.a.ei 3 28.d even 2 1
9408.2.a.ej 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9408))\):

\( T_{5}^{3} - 9T_{5} - 4 \) Copy content Toggle raw display
\( T_{11}^{3} - 27T_{11} + 38 \) Copy content Toggle raw display
\( T_{13}^{3} - 3T_{13}^{2} - 36T_{13} + 112 \) Copy content Toggle raw display
\( T_{17}^{3} - 6T_{17}^{2} - 24T_{17} + 96 \) Copy content Toggle raw display
\( T_{19}^{3} - 3T_{19}^{2} - 36T_{19} + 112 \) Copy content Toggle raw display
\( T_{31}^{3} + 3T_{31}^{2} - 21T_{31} - 47 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \) Copy content Toggle raw display
$3$ \( (T - 1)^{3} \) Copy content Toggle raw display
$5$ \( T^{3} - 9T - 4 \) Copy content Toggle raw display
$7$ \( T^{3} \) Copy content Toggle raw display
$11$ \( T^{3} - 27T + 38 \) Copy content Toggle raw display
$13$ \( T^{3} - 3 T^{2} - 36 T + 112 \) Copy content Toggle raw display
$17$ \( T^{3} - 6 T^{2} - 24 T + 96 \) Copy content Toggle raw display
$19$ \( T^{3} - 3 T^{2} - 36 T + 112 \) Copy content Toggle raw display
$23$ \( T^{3} - 6 T^{2} - 24 T + 96 \) Copy content Toggle raw display
$29$ \( T^{3} + 12 T^{2} + 39 T + 32 \) Copy content Toggle raw display
$31$ \( T^{3} + 3 T^{2} - 21 T - 47 \) Copy content Toggle raw display
$37$ \( T^{3} + 3 T^{2} - 84 T - 368 \) Copy content Toggle raw display
$41$ \( T^{3} + 6 T^{2} - 96 T - 512 \) Copy content Toggle raw display
$43$ \( T^{3} - 15 T^{2} + 48 T - 28 \) Copy content Toggle raw display
$47$ \( T^{3} + 12 T^{2} + 12 T - 112 \) Copy content Toggle raw display
$53$ \( T^{3} + 6 T^{2} - 81 T + 166 \) Copy content Toggle raw display
$59$ \( T^{3} - 12 T^{2} + 21 T + 82 \) Copy content Toggle raw display
$61$ \( T^{3} - 18 T^{2} + 12 T + 552 \) Copy content Toggle raw display
$67$ \( T^{3} - 9 T^{2} - 12 T + 164 \) Copy content Toggle raw display
$71$ \( T^{3} - 36T + 32 \) Copy content Toggle raw display
$73$ \( T^{3} - 33 T^{2} + 336 T - 996 \) Copy content Toggle raw display
$79$ \( T^{3} - 27 T^{2} + 195 T - 353 \) Copy content Toggle raw display
$83$ \( T^{3} - 18 T^{2} - 15 T + 948 \) Copy content Toggle raw display
$89$ \( T^{3} + 12 T^{2} + 12 T - 112 \) Copy content Toggle raw display
$97$ \( T^{3} - 27T + 38 \) Copy content Toggle raw display
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