Properties

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

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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: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 144)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{-3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta q^{5} + \beta q^{7} +O(q^{10}) \) Copy content Toggle raw display \( q - \beta q^{5} + \beta q^{7} + 3 q^{11} + 5 q^{13} - 4 \beta q^{17} - 2 \beta q^{19} - 9 q^{23} + 2 q^{25} + \beta q^{29} - 3 \beta q^{31} + 3 q^{35} - 2 q^{37} - 3 \beta q^{41} + 3 \beta q^{43} - 3 q^{47} + 4 q^{49} - 3 \beta q^{55} - 3 q^{59} + q^{61} - 5 \beta q^{65} - 5 \beta q^{67} - 12 q^{71} - 2 q^{73} + 3 \beta q^{77} + 5 \beta q^{79} + 15 q^{83} - 12 q^{85} + 4 \beta q^{89} + 5 \beta q^{91} - 6 q^{95} - 5 q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 6 q^{11} + 10 q^{13} - 18 q^{23} + 4 q^{25} + 6 q^{35} - 4 q^{37} - 6 q^{47} + 8 q^{49} - 6 q^{59} + 2 q^{61} - 24 q^{71} - 4 q^{73} + 30 q^{83} - 24 q^{85} - 12 q^{95} - 10 q^{97}+O(q^{100}) \) 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
0.500000 + 0.866025i
0.500000 0.866025i
0 0 0 1.73205i 0 1.73205i 0 0 0
5183.2 0 0 0 1.73205i 0 1.73205i 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
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.d 2
3.b odd 2 1 5184.2.c.b 2
4.b odd 2 1 5184.2.c.b 2
8.b even 2 1 1296.2.c.a 2
8.d odd 2 1 1296.2.c.c 2
9.c even 3 1 576.2.s.b 2
9.c even 3 1 1728.2.s.d 2
9.d odd 6 1 576.2.s.c 2
9.d odd 6 1 1728.2.s.c 2
12.b even 2 1 inner 5184.2.c.d 2
24.f even 2 1 1296.2.c.a 2
24.h odd 2 1 1296.2.c.c 2
36.f odd 6 1 576.2.s.c 2
36.f odd 6 1 1728.2.s.c 2
36.h even 6 1 576.2.s.b 2
36.h even 6 1 1728.2.s.d 2
72.j odd 6 1 144.2.s.c yes 2
72.j odd 6 1 432.2.s.a 2
72.l even 6 1 144.2.s.b 2
72.l even 6 1 432.2.s.b 2
72.n even 6 1 144.2.s.b 2
72.n even 6 1 432.2.s.b 2
72.p odd 6 1 144.2.s.c yes 2
72.p odd 6 1 432.2.s.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
144.2.s.b 2 72.l even 6 1
144.2.s.b 2 72.n even 6 1
144.2.s.c yes 2 72.j odd 6 1
144.2.s.c yes 2 72.p odd 6 1
432.2.s.a 2 72.j odd 6 1
432.2.s.a 2 72.p odd 6 1
432.2.s.b 2 72.l even 6 1
432.2.s.b 2 72.n even 6 1
576.2.s.b 2 9.c even 3 1
576.2.s.b 2 36.h even 6 1
576.2.s.c 2 9.d odd 6 1
576.2.s.c 2 36.f odd 6 1
1296.2.c.a 2 8.b even 2 1
1296.2.c.a 2 24.f even 2 1
1296.2.c.c 2 8.d odd 2 1
1296.2.c.c 2 24.h odd 2 1
1728.2.s.c 2 9.d odd 6 1
1728.2.s.c 2 36.f odd 6 1
1728.2.s.d 2 9.c even 3 1
1728.2.s.d 2 36.h even 6 1
5184.2.c.b 2 3.b odd 2 1
5184.2.c.b 2 4.b odd 2 1
5184.2.c.d 2 1.a even 1 1 trivial
5184.2.c.d 2 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 \(S_{2}^{\mathrm{new}}(5184, [\chi])\):

\( T_{5}^{2} + 3 \) Copy content Toggle raw display
\( T_{7}^{2} + 3 \) Copy content Toggle raw display
\( T_{11} - 3 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 3 \) Copy content Toggle raw display
$7$ \( T^{2} + 3 \) Copy content Toggle raw display
$11$ \( (T - 3)^{2} \) Copy content Toggle raw display
$13$ \( (T - 5)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 48 \) Copy content Toggle raw display
$19$ \( T^{2} + 12 \) Copy content Toggle raw display
$23$ \( (T + 9)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} + 3 \) Copy content Toggle raw display
$31$ \( T^{2} + 27 \) Copy content Toggle raw display
$37$ \( (T + 2)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 27 \) Copy content Toggle raw display
$43$ \( T^{2} + 27 \) Copy content Toggle raw display
$47$ \( (T + 3)^{2} \) Copy content Toggle raw display
$53$ \( T^{2} \) Copy content Toggle raw display
$59$ \( (T + 3)^{2} \) Copy content Toggle raw display
$61$ \( (T - 1)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 75 \) Copy content Toggle raw display
$71$ \( (T + 12)^{2} \) Copy content Toggle raw display
$73$ \( (T + 2)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} + 75 \) Copy content Toggle raw display
$83$ \( (T - 15)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 48 \) Copy content Toggle raw display
$97$ \( (T + 5)^{2} \) Copy content Toggle raw display
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