Properties

Label 9408.2.a.em
Level $9408$
Weight $2$
Character orbit 9408.a
Self dual yes
Analytic conductor $75.123$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

Related objects

Downloads

Learn more about

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: \(x^{4} - 2 x^{3} - 3 x^{2} + 2 x + 1\)
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.

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

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{3} - 2 \nu^{2} - 2 \nu + 1 \)
\(\beta_{2}\)\(=\)\( 2 \nu^{2} - 4 \nu - 3 \)
\(\beta_{3}\)\(=\)\( -\nu^{3} + 2 \nu^{2} + 4 \nu - 2 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} + \beta_{1} + 1\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(2 \beta_{3} + \beta_{2} + 2 \beta_{1} + 5\)\()/2\)
\(\nu^{3}\)\(=\)\(3 \beta_{3} + \beta_{2} + 4 \beta_{1} + 5\)

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 −4.13503 0 0 0 1.00000 0
1.2 0 1.00000 0 −3.04244 0 0 0 1.00000 0
1.3 0 1.00000 0 1.04244 0 0 0 1.00000 0
1.4 0 1.00000 0 2.13503 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.em 4
4.b odd 2 1 9408.2.a.ek 4
7.b odd 2 1 9408.2.a.el 4
8.b even 2 1 4704.2.a.bx yes 4
8.d odd 2 1 4704.2.a.bz yes 4
28.d even 2 1 9408.2.a.en 4
56.e even 2 1 4704.2.a.bw 4
56.h odd 2 1 4704.2.a.by yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4704.2.a.bw 4 56.e even 2 1
4704.2.a.bx yes 4 8.b even 2 1
4704.2.a.by yes 4 56.h odd 2 1
4704.2.a.bz yes 4 8.d odd 2 1
9408.2.a.ek 4 4.b odd 2 1
9408.2.a.el 4 7.b odd 2 1
9408.2.a.em 4 1.a even 1 1 trivial
9408.2.a.en 4 28.d even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(9408))\):

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