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: \(x^{4} - x^{2} + 1\)
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 primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -1 + 2 \zeta_{12}^{2} ) q^{5} + ( -4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{7} +O(q^{10})\) \( q + ( -1 + 2 \zeta_{12}^{2} ) q^{5} + ( -4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{7} + ( -1 + 2 \zeta_{12}^{2} ) q^{13} + 3 q^{17} -2 \zeta_{12}^{3} q^{19} + ( -8 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{23} + 2 q^{25} + ( 5 - 10 \zeta_{12}^{2} ) q^{29} + ( -4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{31} -6 \zeta_{12}^{3} q^{35} + ( -5 + 10 \zeta_{12}^{2} ) q^{37} -6 q^{41} -4 \zeta_{12}^{3} q^{43} + ( 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{47} + 5 q^{49} + ( 4 - 8 \zeta_{12}^{2} ) q^{53} + 12 \zeta_{12}^{3} q^{59} + ( 3 - 6 \zeta_{12}^{2} ) q^{61} -3 q^{65} -10 \zeta_{12}^{3} q^{67} + ( -4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{71} + 5 q^{73} + ( -8 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{79} + 6 \zeta_{12}^{3} q^{83} + ( -3 + 6 \zeta_{12}^{2} ) q^{85} + 15 q^{89} -6 \zeta_{12}^{3} q^{91} + ( 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{95} + 10 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + O(q^{10}) \) \( 4 q + 12 q^{17} + 8 q^{25} - 24 q^{41} + 20 q^{49} - 12 q^{65} + 20 q^{73} + 60 q^{89} + 40 q^{97} + O(q^{100}) \)

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 \(S_{2}^{\mathrm{new}}(5184, [\chi])\):

\( T_{5}^{2} + 3 \)
\( T_{7}^{2} - 12 \)
\( T_{17} - 3 \)
\( T_{23}^{2} - 48 \)
\( T_{41} + 6 \)

Hecke characteristic polynomials

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