Properties

Label 5184.2.c.i
Level $5184$
Weight $2$
Character orbit 5184.c
Analytic conductor $41.394$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 5184 = 2^{6} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5184.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(41.3944484078\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\zeta_{24})\)
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 2592)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{3} - \beta_1) q^{5} + \beta_{5} q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{3} - \beta_1) q^{5} + \beta_{5} q^{7} + (\beta_{7} + 2 \beta_{6}) q^{11} + ( - 2 \beta_{2} - 1) q^{13} + (\beta_{3} - \beta_1) q^{17} + (\beta_{5} - \beta_{4}) q^{19} - \beta_{6} q^{23} + ( - \beta_{2} + 3) q^{25} + ( - 5 \beta_{3} - 2 \beta_1) q^{29} + ( - 2 \beta_{5} - 3 \beta_{4}) q^{31} - \beta_{6} q^{35} + (\beta_{2} + 6) q^{37} + (2 \beta_{3} - \beta_1) q^{41} + (\beta_{5} + 4 \beta_{4}) q^{43} + ( - \beta_{7} + 5 \beta_{6}) q^{47} + (2 \beta_{2} + 3) q^{49} + ( - 2 \beta_{3} - \beta_1) q^{53} + (3 \beta_{5} + 4 \beta_{4}) q^{55} + (5 \beta_{7} - \beta_{6}) q^{59} + (3 \beta_{2} - 4) q^{61} + (3 \beta_{3} + 5 \beta_1) q^{65} + (7 \beta_{5} + 4 \beta_{4}) q^{67} + ( - 6 \beta_{7} + \beta_{6}) q^{71} + ( - \beta_{2} - 12) q^{73} + ( - 2 \beta_{3} - 2 \beta_1) q^{77} + (9 \beta_{5} + 5 \beta_{4}) q^{79} + (2 \beta_{7} + 4 \beta_{6}) q^{83} - \beta_{2} q^{85} + (7 \beta_{3} + 8 \beta_1) q^{89} + (\beta_{5} - 2 \beta_{4}) q^{91} + \beta_{7} q^{95} + ( - 6 \beta_{2} + 4) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{13} + 24 q^{25} + 48 q^{37} + 24 q^{49} - 32 q^{61} - 96 q^{73} + 32 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( -\zeta_{24}^{5} - \zeta_{24}^{3} + \zeta_{24} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( -\zeta_{24}^{6} + 2\zeta_{24}^{2} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\zeta_{24}^{7} + \zeta_{24}^{3} - \zeta_{24} \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( 2\zeta_{24}^{6} \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( -\zeta_{24}^{6} + 2\zeta_{24}^{4} - 1 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( -\zeta_{24}^{5} + \zeta_{24}^{3} + \zeta_{24} \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( -2\zeta_{24}^{7} + \zeta_{24}^{5} + \zeta_{24}^{3} + \zeta_{24} \) Copy content Toggle raw display
\(\zeta_{24}\)\(=\) \( ( \beta_{7} + \beta_{6} - 2\beta_{3} ) / 4 \) Copy content Toggle raw display
\(\zeta_{24}^{2}\)\(=\) \( ( \beta_{4} + 2\beta_{2} ) / 4 \) Copy content Toggle raw display
\(\zeta_{24}^{3}\)\(=\) \( ( \beta_{6} - \beta_1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{24}^{4}\)\(=\) \( ( 2\beta_{5} + \beta_{4} + 2 ) / 4 \) Copy content Toggle raw display
\(\zeta_{24}^{5}\)\(=\) \( ( \beta_{7} - \beta_{6} - 2\beta_{3} - 2\beta_1 ) / 4 \) Copy content Toggle raw display
\(\zeta_{24}^{6}\)\(=\) \( ( \beta_{4} ) / 2 \) Copy content Toggle raw display
\(\zeta_{24}^{7}\)\(=\) \( ( -\beta_{7} + \beta_{6} - 2\beta_{3} - 2\beta_1 ) / 4 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/5184\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(1217\) \(2431\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5183.1
0.965926 0.258819i
−0.965926 0.258819i
−0.258819 0.965926i
0.258819 0.965926i
0.258819 + 0.965926i
−0.258819 + 0.965926i
−0.965926 + 0.258819i
0.965926 + 0.258819i
0 0 0 1.93185i 0 0.732051i 0 0 0
5183.2 0 0 0 1.93185i 0 0.732051i 0 0 0
5183.3 0 0 0 0.517638i 0 2.73205i 0 0 0
5183.4 0 0 0 0.517638i 0 2.73205i 0 0 0
5183.5 0 0 0 0.517638i 0 2.73205i 0 0 0
5183.6 0 0 0 0.517638i 0 2.73205i 0 0 0
5183.7 0 0 0 1.93185i 0 0.732051i 0 0 0
5183.8 0 0 0 1.93185i 0 0.732051i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 5183.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
12.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 5184.2.c.i 8
3.b odd 2 1 inner 5184.2.c.i 8
4.b odd 2 1 inner 5184.2.c.i 8
8.b even 2 1 2592.2.c.a 8
8.d odd 2 1 2592.2.c.a 8
12.b even 2 1 inner 5184.2.c.i 8
24.f even 2 1 2592.2.c.a 8
24.h odd 2 1 2592.2.c.a 8
72.j odd 6 1 2592.2.s.b 8
72.j odd 6 1 2592.2.s.f 8
72.l even 6 1 2592.2.s.b 8
72.l even 6 1 2592.2.s.f 8
72.n even 6 1 2592.2.s.b 8
72.n even 6 1 2592.2.s.f 8
72.p odd 6 1 2592.2.s.b 8
72.p odd 6 1 2592.2.s.f 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2592.2.c.a 8 8.b even 2 1
2592.2.c.a 8 8.d odd 2 1
2592.2.c.a 8 24.f even 2 1
2592.2.c.a 8 24.h odd 2 1
2592.2.s.b 8 72.j odd 6 1
2592.2.s.b 8 72.l even 6 1
2592.2.s.b 8 72.n even 6 1
2592.2.s.b 8 72.p odd 6 1
2592.2.s.f 8 72.j odd 6 1
2592.2.s.f 8 72.l even 6 1
2592.2.s.f 8 72.n even 6 1
2592.2.s.f 8 72.p odd 6 1
5184.2.c.i 8 1.a even 1 1 trivial
5184.2.c.i 8 3.b odd 2 1 inner
5184.2.c.i 8 4.b odd 2 1 inner
5184.2.c.i 8 12.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(5184, [\chi])\):

\( T_{5}^{4} + 4T_{5}^{2} + 1 \) Copy content Toggle raw display
\( T_{7}^{4} + 8T_{7}^{2} + 4 \) Copy content Toggle raw display
\( T_{11}^{4} - 28T_{11}^{2} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( (T^{4} + 4 T^{2} + 1)^{2} \) Copy content Toggle raw display
$7$ \( (T^{4} + 8 T^{2} + 4)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} - 28 T^{2} + 4)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 2 T - 11)^{4} \) Copy content Toggle raw display
$17$ \( (T^{4} + 12 T^{2} + 9)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + 24 T^{2} + 36)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} - 2)^{4} \) Copy content Toggle raw display
$29$ \( (T^{4} + 76 T^{2} + 1369)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + 56 T^{2} + 16)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} - 12 T + 33)^{4} \) Copy content Toggle raw display
$41$ \( (T^{4} + 28 T^{2} + 4)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 104 T^{2} + 2116)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} - 112 T^{2} + 1936)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 6)^{4} \) Copy content Toggle raw display
$59$ \( (T^{4} - 304 T^{2} + 21904)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 8 T - 11)^{4} \) Copy content Toggle raw display
$67$ \( (T^{4} + 296 T^{2} + 21316)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} - 436 T^{2} + 45796)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 24 T + 141)^{4} \) Copy content Toggle raw display
$79$ \( (T^{4} + 488 T^{2} + 58564)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} - 112 T^{2} + 64)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + 228 T^{2} + 1089)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} - 8 T - 92)^{4} \) Copy content Toggle raw display
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