Properties

Label 9408.2.a.ei
Level $9408$
Weight $2$
Character orbit 9408.a
Self dual yes
Analytic conductor $75.123$
Analytic rank $1$
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: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.621.1
Defining polynomial: \(x^{3} - 6 x - 3\)
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 a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 2 \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 4 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)\(/2\)
\(\nu^{2}\)\(=\)\((\)\(2 \beta_{2} + \beta_{1} + 8\)\()/2\)

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

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( ( 1 - T )^{3} \)
$5$ \( 1 + 6 T^{2} + 4 T^{3} + 30 T^{4} + 125 T^{6} \)
$7$ 1
$11$ \( 1 + 6 T^{2} - 38 T^{3} + 66 T^{4} + 1331 T^{6} \)
$13$ \( 1 + 3 T + 3 T^{2} - 34 T^{3} + 39 T^{4} + 507 T^{5} + 2197 T^{6} \)
$17$ \( 1 + 6 T + 27 T^{2} + 108 T^{3} + 459 T^{4} + 1734 T^{5} + 4913 T^{6} \)
$19$ \( 1 - 3 T + 21 T^{2} - 2 T^{3} + 399 T^{4} - 1083 T^{5} + 6859 T^{6} \)
$23$ \( 1 + 6 T + 45 T^{2} + 180 T^{3} + 1035 T^{4} + 3174 T^{5} + 12167 T^{6} \)
$29$ \( 1 + 12 T + 126 T^{2} + 728 T^{3} + 3654 T^{4} + 10092 T^{5} + 24389 T^{6} \)
$31$ \( 1 + 3 T + 72 T^{2} + 139 T^{3} + 2232 T^{4} + 2883 T^{5} + 29791 T^{6} \)
$37$ \( 1 + 3 T + 27 T^{2} - 146 T^{3} + 999 T^{4} + 4107 T^{5} + 50653 T^{6} \)
$41$ \( 1 - 6 T + 27 T^{2} + 20 T^{3} + 1107 T^{4} - 10086 T^{5} + 68921 T^{6} \)
$43$ \( 1 + 15 T + 177 T^{2} + 1318 T^{3} + 7611 T^{4} + 27735 T^{5} + 79507 T^{6} \)
$47$ \( 1 + 12 T + 153 T^{2} + 1016 T^{3} + 7191 T^{4} + 26508 T^{5} + 103823 T^{6} \)
$53$ \( 1 + 6 T + 78 T^{2} + 802 T^{3} + 4134 T^{4} + 16854 T^{5} + 148877 T^{6} \)
$59$ \( 1 - 12 T + 198 T^{2} - 1334 T^{3} + 11682 T^{4} - 41772 T^{5} + 205379 T^{6} \)
$61$ \( 1 + 18 T + 195 T^{2} + 1644 T^{3} + 11895 T^{4} + 66978 T^{5} + 226981 T^{6} \)
$67$ \( 1 + 9 T + 189 T^{2} + 1042 T^{3} + 12663 T^{4} + 40401 T^{5} + 300763 T^{6} \)
$71$ \( 1 + 177 T^{2} - 32 T^{3} + 12567 T^{4} + 357911 T^{6} \)
$73$ \( 1 + 33 T + 555 T^{2} + 5814 T^{3} + 40515 T^{4} + 175857 T^{5} + 389017 T^{6} \)
$79$ \( 1 + 27 T + 432 T^{2} + 4619 T^{3} + 34128 T^{4} + 168507 T^{5} + 493039 T^{6} \)
$83$ \( 1 - 18 T + 234 T^{2} - 2040 T^{3} + 19422 T^{4} - 124002 T^{5} + 571787 T^{6} \)
$89$ \( 1 - 12 T + 279 T^{2} - 2024 T^{3} + 24831 T^{4} - 95052 T^{5} + 704969 T^{6} \)
$97$ \( 1 + 264 T^{2} - 38 T^{3} + 25608 T^{4} + 912673 T^{6} \)
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