Properties

Label 9408.2.a.do
Level 9408
Weight 2
Character orbit 9408.a
Self dual yes
Analytic conductor 75.123
Analytic rank 1
Dimension 2
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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{3}) \)
Defining polynomial: \(x^{2} - 3\)
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 \(\beta = 2\sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{3} + 2 \beta q^{5} + q^{9} +O(q^{10})\) \( q - q^{3} + 2 \beta q^{5} + q^{9} + ( 2 - 2 \beta ) q^{11} + 2 q^{13} -2 \beta q^{15} + ( -4 + 2 \beta ) q^{17} -4 \beta q^{19} + ( -2 + 2 \beta ) q^{23} + 7 q^{25} - q^{27} + ( -2 + 4 \beta ) q^{29} + ( -4 - 4 \beta ) q^{31} + ( -2 + 2 \beta ) q^{33} + 2 q^{37} -2 q^{39} + ( -8 - 2 \beta ) q^{41} -8 q^{43} + 2 \beta q^{45} + ( 4 + 4 \beta ) q^{47} + ( 4 - 2 \beta ) q^{51} + 2 q^{53} + ( -12 + 4 \beta ) q^{55} + 4 \beta q^{57} + ( 8 - 4 \beta ) q^{59} + ( -2 - 4 \beta ) q^{61} + 4 \beta q^{65} + ( 4 - 4 \beta ) q^{67} + ( 2 - 2 \beta ) q^{69} + ( -6 - 2 \beta ) q^{71} + ( -6 - 4 \beta ) q^{73} -7 q^{75} + ( 4 + 4 \beta ) q^{79} + q^{81} + 4 q^{83} + ( 12 - 8 \beta ) q^{85} + ( 2 - 4 \beta ) q^{87} -2 \beta q^{89} + ( 4 + 4 \beta ) q^{93} -24 q^{95} + ( 2 + 4 \beta ) q^{97} + ( 2 - 2 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{3} + 2q^{9} + O(q^{10}) \) \( 2q - 2q^{3} + 2q^{9} + 4q^{11} + 4q^{13} - 8q^{17} - 4q^{23} + 14q^{25} - 2q^{27} - 4q^{29} - 8q^{31} - 4q^{33} + 4q^{37} - 4q^{39} - 16q^{41} - 16q^{43} + 8q^{47} + 8q^{51} + 4q^{53} - 24q^{55} + 16q^{59} - 4q^{61} + 8q^{67} + 4q^{69} - 12q^{71} - 12q^{73} - 14q^{75} + 8q^{79} + 2q^{81} + 8q^{83} + 24q^{85} + 4q^{87} + 8q^{93} - 48q^{95} + 4q^{97} + 4q^{99} + O(q^{100}) \)

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
−1.73205
1.73205
0 −1.00000 0 −3.46410 0 0 0 1.00000 0
1.2 0 −1.00000 0 3.46410 0 0 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

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.do 2
4.b odd 2 1 9408.2.a.dx 2
7.b odd 2 1 1344.2.a.v 2
8.b even 2 1 4704.2.a.bn 2
8.d odd 2 1 4704.2.a.bm 2
21.c even 2 1 4032.2.a.bs 2
28.d even 2 1 1344.2.a.u 2
56.e even 2 1 672.2.a.j yes 2
56.h odd 2 1 672.2.a.i 2
84.h odd 2 1 4032.2.a.br 2
112.j even 4 2 5376.2.c.bn 4
112.l odd 4 2 5376.2.c.bh 4
168.e odd 2 1 2016.2.a.s 2
168.i even 2 1 2016.2.a.t 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
672.2.a.i 2 56.h odd 2 1
672.2.a.j yes 2 56.e even 2 1
1344.2.a.u 2 28.d even 2 1
1344.2.a.v 2 7.b odd 2 1
2016.2.a.s 2 168.e odd 2 1
2016.2.a.t 2 168.i even 2 1
4032.2.a.br 2 84.h odd 2 1
4032.2.a.bs 2 21.c even 2 1
4704.2.a.bm 2 8.d odd 2 1
4704.2.a.bn 2 8.b even 2 1
5376.2.c.bh 4 112.l odd 4 2
5376.2.c.bn 4 112.j even 4 2
9408.2.a.do 2 1.a even 1 1 trivial
9408.2.a.dx 2 4.b odd 2 1

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(7\) \(-1\)

Hecke kernels

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

\( T_{5}^{2} - 12 \)
\( T_{11}^{2} - 4 T_{11} - 8 \)
\( T_{13} - 2 \)
\( T_{17}^{2} + 8 T_{17} + 4 \)
\( T_{19}^{2} - 48 \)
\( T_{31}^{2} + 8 T_{31} - 32 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( ( 1 + T )^{2} \)
$5$ \( 1 - 2 T^{2} + 25 T^{4} \)
$7$ 1
$11$ \( 1 - 4 T + 14 T^{2} - 44 T^{3} + 121 T^{4} \)
$13$ \( ( 1 - 2 T + 13 T^{2} )^{2} \)
$17$ \( 1 + 8 T + 38 T^{2} + 136 T^{3} + 289 T^{4} \)
$19$ \( 1 - 10 T^{2} + 361 T^{4} \)
$23$ \( 1 + 4 T + 38 T^{2} + 92 T^{3} + 529 T^{4} \)
$29$ \( 1 + 4 T + 14 T^{2} + 116 T^{3} + 841 T^{4} \)
$31$ \( 1 + 8 T + 30 T^{2} + 248 T^{3} + 961 T^{4} \)
$37$ \( ( 1 - 2 T + 37 T^{2} )^{2} \)
$41$ \( 1 + 16 T + 134 T^{2} + 656 T^{3} + 1681 T^{4} \)
$43$ \( ( 1 + 8 T + 43 T^{2} )^{2} \)
$47$ \( 1 - 8 T + 62 T^{2} - 376 T^{3} + 2209 T^{4} \)
$53$ \( ( 1 - 2 T + 53 T^{2} )^{2} \)
$59$ \( 1 - 16 T + 134 T^{2} - 944 T^{3} + 3481 T^{4} \)
$61$ \( 1 + 4 T + 78 T^{2} + 244 T^{3} + 3721 T^{4} \)
$67$ \( 1 - 8 T + 102 T^{2} - 536 T^{3} + 4489 T^{4} \)
$71$ \( 1 + 12 T + 166 T^{2} + 852 T^{3} + 5041 T^{4} \)
$73$ \( 1 + 12 T + 134 T^{2} + 876 T^{3} + 5329 T^{4} \)
$79$ \( 1 - 8 T + 126 T^{2} - 632 T^{3} + 6241 T^{4} \)
$83$ \( ( 1 - 4 T + 83 T^{2} )^{2} \)
$89$ \( 1 + 166 T^{2} + 7921 T^{4} \)
$97$ \( 1 - 4 T + 150 T^{2} - 388 T^{3} + 9409 T^{4} \)
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