Properties

Label 5184.2.c.g
Level $5184$
Weight $2$
Character orbit 5184.c
Analytic conductor $41.394$
Analytic rank $0$
Dimension $4$
CM discriminant -4
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: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{3})\)
Defining polynomial: \( x^{4} + 4x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3^{3} \)
Twist minimal: no (minimal twist has level 1296)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_1 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{5} + ( - \beta_{3} + 2) q^{13} + (\beta_{2} - \beta_1) q^{17} + (\beta_{3} - 1) q^{25} + (\beta_{2} - 3 \beta_1) q^{29} + ( - 2 \beta_{3} + 1) q^{37} + ( - \beta_{2} - 4 \beta_1) q^{41} + 7 q^{49} + ( - \beta_{2} - 4 \beta_1) q^{53} + ( - 2 \beta_{3} - 5) q^{61} + ( - \beta_{2} + 7 \beta_1) q^{65} + (\beta_{3} - 8) q^{73} + ( - 2 \beta_{3} + 3) q^{85} + (2 \beta_{2} - 5 \beta_1) q^{89} - 8 q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 8 q^{13} - 4 q^{25} + 4 q^{37} + 28 q^{49} - 20 q^{61} - 32 q^{73} + 12 q^{85} - 32 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 4x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{3} + 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 5\nu^{3} + 19\nu \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 3\nu^{2} + 6 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} - 5\beta_1 ) / 9 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} - 6 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -2\beta_{2} + 19\beta_1 ) / 9 \) 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
1.93185i
0.517638i
0.517638i
1.93185i
0 0 0 3.34607i 0 0 0 0 0
5183.2 0 0 0 0.896575i 0 0 0 0 0
5183.3 0 0 0 0.896575i 0 0 0 0 0
5183.4 0 0 0 3.34607i 0 0 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 CM by \(\Q(\sqrt{-1}) \)
3.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.g 4
3.b odd 2 1 inner 5184.2.c.g 4
4.b odd 2 1 CM 5184.2.c.g 4
8.b even 2 1 1296.2.c.e 4
8.d odd 2 1 1296.2.c.e 4
12.b even 2 1 inner 5184.2.c.g 4
24.f even 2 1 1296.2.c.e 4
24.h odd 2 1 1296.2.c.e 4
72.j odd 6 2 1296.2.s.k 8
72.l even 6 2 1296.2.s.k 8
72.n even 6 2 1296.2.s.k 8
72.p odd 6 2 1296.2.s.k 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1296.2.c.e 4 8.b even 2 1
1296.2.c.e 4 8.d odd 2 1
1296.2.c.e 4 24.f even 2 1
1296.2.c.e 4 24.h odd 2 1
1296.2.s.k 8 72.j odd 6 2
1296.2.s.k 8 72.l even 6 2
1296.2.s.k 8 72.n even 6 2
1296.2.s.k 8 72.p odd 6 2
5184.2.c.g 4 1.a even 1 1 trivial
5184.2.c.g 4 3.b odd 2 1 inner
5184.2.c.g 4 4.b odd 2 1 CM
5184.2.c.g 4 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 \(S_{2}^{\mathrm{new}}(5184, [\chi])\):

\( T_{5}^{4} + 12T_{5}^{2} + 9 \) Copy content Toggle raw display
\( T_{7} \) Copy content Toggle raw display
\( T_{11} \) 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
$5$ \( T^{4} + 12T^{2} + 9 \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} \) Copy content Toggle raw display
$13$ \( (T^{2} - 4 T - 23)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + 84T^{2} + 1089 \) Copy content Toggle raw display
$19$ \( T^{4} \) Copy content Toggle raw display
$23$ \( T^{4} \) Copy content Toggle raw display
$29$ \( T^{4} + 156T^{2} + 4761 \) Copy content Toggle raw display
$31$ \( T^{4} \) Copy content Toggle raw display
$37$ \( (T^{2} - 2 T - 107)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 162)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} \) Copy content Toggle raw display
$47$ \( T^{4} \) Copy content Toggle raw display
$53$ \( (T^{2} + 162)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} \) Copy content Toggle raw display
$61$ \( (T^{2} + 10 T - 83)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} \) Copy content Toggle raw display
$71$ \( T^{4} \) Copy content Toggle raw display
$73$ \( (T^{2} + 16 T + 37)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} \) Copy content Toggle raw display
$83$ \( T^{4} \) Copy content Toggle raw display
$89$ \( T^{4} + 516 T^{2} + 62001 \) Copy content Toggle raw display
$97$ \( (T + 8)^{4} \) Copy content Toggle raw display
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