Properties

Label 5184.2.d.l
Level $5184$
Weight $2$
Character orbit 5184.d
Analytic conductor $41.394$
Analytic rank $0$
Dimension $4$
CM discriminant -8
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: \(\Q(i, \sqrt{6})\)
Defining polynomial: \( x^{4} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 576)
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.

\(f(q)\) \(=\) \( q+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{2} + \beta_1) q^{11} + ( - \beta_{3} + 3) q^{17} + ( - 3 \beta_{2} + \beta_1) q^{19} + 5 q^{25} + (2 \beta_{3} + 3) q^{41} + (3 \beta_{2} - 4 \beta_1) q^{43} - 7 q^{49} + ( - 5 \beta_{2} + 4 \beta_1) q^{59} + (3 \beta_{2} + 2 \beta_1) q^{67} + (3 \beta_{3} - 1) q^{73} + 9 \beta_1 q^{83} + 18 q^{89} + ( - 3 \beta_{3} - 5) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 12 q^{17} + 20 q^{25} + 12 q^{41} - 28 q^{49} - 4 q^{73} + 72 q^{89} - 20 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 2\nu^{2} ) / 3 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + \nu^{2} + 3\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -2\nu^{3} + 6\nu ) / 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + 2\beta_{2} - \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 3\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -3\beta_{3} + 6\beta_{2} - 3\beta_1 ) / 4 \) 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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2593.1
1.22474 1.22474i
−1.22474 + 1.22474i
−1.22474 1.22474i
1.22474 + 1.22474i
0 0 0 0 0 0 0 0 0
2593.2 0 0 0 0 0 0 0 0 0
2593.3 0 0 0 0 0 0 0 0 0
2593.4 0 0 0 0 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
8.d odd 2 1 CM by \(\Q(\sqrt{-2}) \)
4.b odd 2 1 inner
8.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.d.l 4
3.b odd 2 1 5184.2.d.e 4
4.b odd 2 1 inner 5184.2.d.l 4
8.b even 2 1 inner 5184.2.d.l 4
8.d odd 2 1 CM 5184.2.d.l 4
9.c even 3 2 576.2.r.c 8
9.d odd 6 2 1728.2.r.c 8
12.b even 2 1 5184.2.d.e 4
24.f even 2 1 5184.2.d.e 4
24.h odd 2 1 5184.2.d.e 4
36.f odd 6 2 576.2.r.c 8
36.h even 6 2 1728.2.r.c 8
72.j odd 6 2 1728.2.r.c 8
72.l even 6 2 1728.2.r.c 8
72.n even 6 2 576.2.r.c 8
72.p odd 6 2 576.2.r.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
576.2.r.c 8 9.c even 3 2
576.2.r.c 8 36.f odd 6 2
576.2.r.c 8 72.n even 6 2
576.2.r.c 8 72.p odd 6 2
1728.2.r.c 8 9.d odd 6 2
1728.2.r.c 8 36.h even 6 2
1728.2.r.c 8 72.j odd 6 2
1728.2.r.c 8 72.l even 6 2
5184.2.d.e 4 3.b odd 2 1
5184.2.d.e 4 12.b even 2 1
5184.2.d.e 4 24.f even 2 1
5184.2.d.e 4 24.h odd 2 1
5184.2.d.l 4 1.a even 1 1 trivial
5184.2.d.l 4 4.b odd 2 1 inner
5184.2.d.l 4 8.b even 2 1 inner
5184.2.d.l 4 8.d odd 2 1 CM

Hecke kernels

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

\( T_{5} \) Copy content Toggle raw display
\( T_{7} \) Copy content Toggle raw display
\( T_{17}^{2} - 6T_{17} - 15 \) Copy content Toggle raw display
\( T_{23} \) Copy content Toggle raw display
\( T_{41}^{2} - 6T_{41} - 87 \) 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
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} + 30T^{2} + 9 \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( (T^{2} - 6 T - 15)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} + 110T^{2} + 2809 \) Copy content Toggle raw display
$23$ \( T^{4} \) Copy content Toggle raw display
$29$ \( T^{4} \) Copy content Toggle raw display
$31$ \( T^{4} \) Copy content Toggle raw display
$37$ \( T^{4} \) Copy content Toggle raw display
$41$ \( (T^{2} - 6 T - 87)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + 158T^{2} + 841 \) Copy content Toggle raw display
$47$ \( T^{4} \) Copy content Toggle raw display
$53$ \( T^{4} \) Copy content Toggle raw display
$59$ \( T^{4} + 318 T^{2} + 19881 \) Copy content Toggle raw display
$61$ \( T^{4} \) Copy content Toggle raw display
$67$ \( T^{4} + 206T^{2} + 25 \) Copy content Toggle raw display
$71$ \( T^{4} \) Copy content Toggle raw display
$73$ \( (T^{2} + 2 T - 215)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} + 324)^{2} \) Copy content Toggle raw display
$89$ \( (T - 18)^{4} \) Copy content Toggle raw display
$97$ \( (T^{2} + 10 T - 191)^{2} \) Copy content Toggle raw display
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