Properties

Label 5184.2.d.k
Level $5184$
Weight $2$
Character orbit 5184.d
Analytic conductor $41.394$
Analytic rank $0$
Dimension $4$
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.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(41.3944484078\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(f(q)\) \(=\) \( q + \beta_{2} q^{5} - \beta_{3} q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{5} - \beta_{3} q^{7} + \beta_{2} q^{13} + 3 q^{17} - \beta_1 q^{19} - 2 \beta_{3} q^{23} + 2 q^{25} - 5 \beta_{2} q^{29} - \beta_{3} q^{31} - 3 \beta_1 q^{35} + 5 \beta_{2} q^{37} - 6 q^{41} - 2 \beta_1 q^{43} + \beta_{3} q^{47} + 5 q^{49} - 4 \beta_{2} q^{53} + 6 \beta_1 q^{59} - 3 \beta_{2} q^{61} - 3 q^{65} - 5 \beta_1 q^{67} - \beta_{3} q^{71} + 5 q^{73} - 2 \beta_{3} q^{79} + 3 \beta_1 q^{83} + 3 \beta_{2} q^{85} + 15 q^{89} - 3 \beta_1 q^{91} + \beta_{3} q^{95} + 10 q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 12 q^{17} + 8 q^{25} - 24 q^{41} + 20 q^{49} - 12 q^{65} + 20 q^{73} + 60 q^{89} + 40 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( 2\zeta_{12}^{3} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 2\zeta_{12}^{2} - 1 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -2\zeta_{12}^{3} + 4\zeta_{12} \) Copy content Toggle raw display
\(\zeta_{12}\)\(=\) \( ( \beta_{3} + \beta_1 ) / 4 \) Copy content Toggle raw display
\(\zeta_{12}^{2}\)\(=\) \( ( \beta_{2} + 1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{12}^{3}\)\(=\) \( ( \beta_1 ) / 2 \) 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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2593.1
0.866025 0.500000i
−0.866025 + 0.500000i
0.866025 + 0.500000i
−0.866025 0.500000i
0 0 0 1.73205i 0 −3.46410 0 0 0
2593.2 0 0 0 1.73205i 0 3.46410 0 0 0
2593.3 0 0 0 1.73205i 0 −3.46410 0 0 0
2593.4 0 0 0 1.73205i 0 3.46410 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
8.b even 2 1 inner
8.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 5184.2.d.k yes 4
3.b odd 2 1 5184.2.d.d 4
4.b odd 2 1 inner 5184.2.d.k yes 4
8.b even 2 1 inner 5184.2.d.k yes 4
8.d odd 2 1 inner 5184.2.d.k yes 4
12.b even 2 1 5184.2.d.d 4
24.f even 2 1 5184.2.d.d 4
24.h odd 2 1 5184.2.d.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5184.2.d.d 4 3.b odd 2 1
5184.2.d.d 4 12.b even 2 1
5184.2.d.d 4 24.f even 2 1
5184.2.d.d 4 24.h odd 2 1
5184.2.d.k yes 4 1.a even 1 1 trivial
5184.2.d.k yes 4 4.b odd 2 1 inner
5184.2.d.k yes 4 8.b even 2 1 inner
5184.2.d.k yes 4 8.d odd 2 1 inner

Hecke kernels

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

\( T_{5}^{2} + 3 \) Copy content Toggle raw display
\( T_{7}^{2} - 12 \) Copy content Toggle raw display
\( T_{17} - 3 \) Copy content Toggle raw display
\( T_{23}^{2} - 48 \) Copy content Toggle raw display
\( T_{41} + 6 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( (T^{2} + 3)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} - 12)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} \) Copy content Toggle raw display
$13$ \( (T^{2} + 3)^{2} \) Copy content Toggle raw display
$17$ \( (T - 3)^{4} \) Copy content Toggle raw display
$19$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} - 48)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 75)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} - 12)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 75)^{2} \) Copy content Toggle raw display
$41$ \( (T + 6)^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} + 16)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} - 12)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 48)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} + 144)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 27)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 100)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} - 12)^{2} \) Copy content Toggle raw display
$73$ \( (T - 5)^{4} \) Copy content Toggle raw display
$79$ \( (T^{2} - 48)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} + 36)^{2} \) Copy content Toggle raw display
$89$ \( (T - 15)^{4} \) Copy content Toggle raw display
$97$ \( (T - 10)^{4} \) Copy content Toggle raw display
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