Properties

Label 9408.2.a.en
Level $9408$
Weight $2$
Character orbit 9408.a
Self dual yes
Analytic conductor $75.123$
Analytic rank $0$
Dimension $4$
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: \(4\)
Coefficient field: 4.4.2624.1
Defining polynomial: \( x^{4} - 2x^{3} - 3x^{2} + 2x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 4704)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{3} + (\beta_{3} + 1) q^{5} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{3} + (\beta_{3} + 1) q^{5} + q^{9} + ( - \beta_{2} - 1) q^{11} + ( - \beta_{3} - \beta_{2} + 2) q^{13} + (\beta_{3} + 1) q^{15} + ( - \beta_{2} - \beta_1 - 1) q^{17} + (\beta_{3} + \beta_{2} + \beta_1 + 2) q^{19} + ( - \beta_{2} + 3) q^{23} + (2 \beta_{3} + 2 \beta_1 + 3) q^{25} + q^{27} + ( - \beta_{3} + \beta_{2} - \beta_1) q^{29} + ( - \beta_{3} - \beta_{2} + 3 \beta_1 + 2) q^{31} + ( - \beta_{2} - 1) q^{33} + 4 \beta_1 q^{37} + ( - \beta_{3} - \beta_{2} + 2) q^{39} + (2 \beta_{3} + \beta_{2} + 3 \beta_1 - 5) q^{41} + (2 \beta_{3} - 2 \beta_1 + 2) q^{43} + (\beta_{3} + 1) q^{45} + (3 \beta_{3} + \beta_{2} - \beta_1 + 4) q^{47} + ( - \beta_{2} - \beta_1 - 1) q^{51} + ( - 2 \beta_{3} - 2 \beta_1) q^{53} + ( - \beta_{3} - \beta_{2} - 5 \beta_1 + 2) q^{55} + (\beta_{3} + \beta_{2} + \beta_1 + 2) q^{57} + ( - 3 \beta_{3} - \beta_{2} - 3 \beta_1) q^{59} + ( - \beta_{3} + \beta_{2} + 4) q^{61} + (\beta_{3} - \beta_{2} - 7 \beta_1 - 2) q^{65} + (2 \beta_{2} + 4 \beta_1 + 2) q^{67} + ( - \beta_{2} + 3) q^{69} + ( - \beta_{2} - 4 \beta_1 + 3) q^{71} + ( - 2 \beta_{3} - \beta_1 - 2) q^{73} + (2 \beta_{3} + 2 \beta_1 + 3) q^{75} + (2 \beta_{3} + 2 \beta_{2} - 2 \beta_1 + 4) q^{79} + q^{81} + ( - 2 \beta_{3} - 2 \beta_{2} - 2 \beta_1) q^{83} + ( - 2 \beta_{3} - 2 \beta_{2} - 6 \beta_1 + 2) q^{85} + ( - \beta_{3} + \beta_{2} - \beta_1) q^{87} + (\beta_{2} - 5 \beta_1 - 7) q^{89} + ( - \beta_{3} - \beta_{2} + 3 \beta_1 + 2) q^{93} + (4 \beta_{3} + 2 \beta_{2} + 8 \beta_1 + 6) q^{95} + ( - 2 \beta_{3} + 3 \beta_1 - 10) q^{97} + ( - \beta_{2} - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{3} + 4 q^{5} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{3} + 4 q^{5} + 4 q^{9} - 4 q^{11} + 8 q^{13} + 4 q^{15} - 4 q^{17} + 8 q^{19} + 12 q^{23} + 12 q^{25} + 4 q^{27} + 8 q^{31} - 4 q^{33} + 8 q^{39} - 20 q^{41} + 8 q^{43} + 4 q^{45} + 16 q^{47} - 4 q^{51} + 8 q^{55} + 8 q^{57} + 16 q^{61} - 8 q^{65} + 8 q^{67} + 12 q^{69} + 12 q^{71} - 8 q^{73} + 12 q^{75} + 16 q^{79} + 4 q^{81} + 8 q^{85} - 28 q^{89} + 8 q^{93} + 24 q^{95} - 40 q^{97} - 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( \nu^{3} - 2\nu^{2} - 2\nu + 1 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 2\nu^{2} - 4\nu - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\nu^{3} + 2\nu^{2} + 4\nu - 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{3} + \beta_{2} + 2\beta _1 + 5 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 3\beta_{3} + \beta_{2} + 4\beta _1 + 5 \) 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
−0.360409
−1.22833
0.814115
2.77462
0 1.00000 0 −2.13503 0 0 0 1.00000 0
1.2 0 1.00000 0 −1.04244 0 0 0 1.00000 0
1.3 0 1.00000 0 3.04244 0 0 0 1.00000 0
1.4 0 1.00000 0 4.13503 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.en 4
4.b odd 2 1 9408.2.a.el 4
7.b odd 2 1 9408.2.a.ek 4
8.b even 2 1 4704.2.a.bw 4
8.d odd 2 1 4704.2.a.by yes 4
28.d even 2 1 9408.2.a.em 4
56.e even 2 1 4704.2.a.bx yes 4
56.h odd 2 1 4704.2.a.bz yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4704.2.a.bw 4 8.b even 2 1
4704.2.a.bx yes 4 56.e even 2 1
4704.2.a.by yes 4 8.d odd 2 1
4704.2.a.bz yes 4 56.h odd 2 1
9408.2.a.ek 4 7.b odd 2 1
9408.2.a.el 4 4.b odd 2 1
9408.2.a.em 4 28.d even 2 1
9408.2.a.en 4 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9408))\):

\( T_{5}^{4} - 4T_{5}^{3} - 8T_{5}^{2} + 24T_{5} + 28 \) Copy content Toggle raw display
\( T_{11}^{4} + 4T_{11}^{3} - 20T_{11}^{2} - 48T_{11} + 16 \) Copy content Toggle raw display
\( T_{13}^{4} - 8T_{13}^{3} - 4T_{13}^{2} + 80T_{13} + 68 \) Copy content Toggle raw display
\( T_{17}^{4} + 4T_{17}^{3} - 24T_{17}^{2} - 120T_{17} - 100 \) Copy content Toggle raw display
\( T_{19}^{4} - 8T_{19}^{3} - 8T_{19}^{2} + 64T_{19} + 64 \) Copy content Toggle raw display
\( T_{31}^{4} - 8T_{31}^{3} - 40T_{31}^{2} + 128T_{31} - 64 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T - 1)^{4} \) Copy content Toggle raw display
$5$ \( T^{4} - 4 T^{3} - 8 T^{2} + 24 T + 28 \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} + 4 T^{3} - 20 T^{2} - 48 T + 16 \) Copy content Toggle raw display
$13$ \( T^{4} - 8 T^{3} - 4 T^{2} + 80 T + 68 \) Copy content Toggle raw display
$17$ \( T^{4} + 4 T^{3} - 24 T^{2} - 120 T - 100 \) Copy content Toggle raw display
$19$ \( T^{4} - 8 T^{3} - 8 T^{2} + 64 T + 64 \) Copy content Toggle raw display
$23$ \( T^{4} - 12 T^{3} + 28 T^{2} + \cdots - 112 \) Copy content Toggle raw display
$29$ \( T^{4} - 56 T^{2} - 128 T + 64 \) Copy content Toggle raw display
$31$ \( T^{4} - 8 T^{3} - 40 T^{2} + 128 T - 64 \) Copy content Toggle raw display
$37$ \( (T^{2} - 32)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} + 20 T^{3} + 56 T^{2} + \cdots - 4804 \) Copy content Toggle raw display
$43$ \( T^{4} - 8 T^{3} - 48 T^{2} + 384 T - 256 \) Copy content Toggle raw display
$47$ \( T^{4} - 16 T^{3} - 24 T^{2} + \cdots - 3008 \) Copy content Toggle raw display
$53$ \( T^{4} - 72 T^{2} + 128 T + 272 \) Copy content Toggle raw display
$59$ \( T^{4} - 152 T^{2} + 960 T - 1600 \) Copy content Toggle raw display
$61$ \( T^{4} - 16 T^{3} + 44 T^{2} + \cdots - 412 \) Copy content Toggle raw display
$67$ \( T^{4} - 8 T^{3} - 144 T^{2} + \cdots - 4352 \) Copy content Toggle raw display
$71$ \( T^{4} - 12 T^{3} - 36 T^{2} + \cdots + 272 \) Copy content Toggle raw display
$73$ \( T^{4} + 8 T^{3} - 36 T^{2} - 144 T + 452 \) Copy content Toggle raw display
$79$ \( T^{4} - 16 T^{3} - 32 T^{2} + \cdots - 1024 \) Copy content Toggle raw display
$83$ \( T^{4} - 128 T^{2} + 256 T + 1792 \) Copy content Toggle raw display
$89$ \( T^{4} + 28 T^{3} + 168 T^{2} + \cdots - 4772 \) Copy content Toggle raw display
$97$ \( T^{4} + 40 T^{3} + 508 T^{2} + \cdots - 1148 \) Copy content Toggle raw display
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