Properties

Label 5184.2.a.cf
Level $5184$
Weight $2$
Character orbit 5184.a
Self dual yes
Analytic conductor $41.394$
Analytic rank $0$
Dimension $4$
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.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(41.3944484078\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.13068.1
Defining polynomial: \( x^{4} - x^{3} - 6x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2\cdot 3 \)
Twist minimal: no (minimal twist has level 288)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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.

\(f(q)\) \(=\) \( q + (\beta_1 + 1) q^{5} + \beta_{3} q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 + 1) q^{5} + \beta_{3} q^{7} + \beta_{2} q^{11} + (\beta_1 - 1) q^{13} - \beta_1 q^{17} + ( - \beta_{3} + \beta_{2}) q^{19} + \beta_{3} q^{23} + (\beta_1 + 4) q^{25} + (\beta_1 + 3) q^{29} + ( - \beta_{3} + 2 \beta_{2}) q^{31} + (3 \beta_{3} - 2 \beta_{2}) q^{35} - 4 q^{37} + q^{41} - \beta_{2} q^{43} + \beta_{3} q^{47} + (3 \beta_1 + 8) q^{49} + 4 q^{53} + ( - \beta_{3} - 2 \beta_{2}) q^{55} + ( - 2 \beta_{3} - \beta_{2}) q^{59} + (3 \beta_1 - 5) q^{61} + ( - \beta_1 + 7) q^{65} + (2 \beta_{3} + \beta_{2}) q^{67} + 2 \beta_{3} q^{71} + (\beta_1 + 8) q^{73} + ( - 3 \beta_1 + 3) q^{77} + (\beta_{3} - 2 \beta_{2}) q^{79} + (\beta_{3} + 2 \beta_{2}) q^{83} - 8 q^{85} + (2 \beta_1 + 8) q^{89} + (\beta_{3} - 2 \beta_{2}) q^{91} - 4 \beta_{3} q^{95} + 9 q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{5} - 6 q^{13} + 2 q^{17} + 14 q^{25} + 10 q^{29} - 16 q^{37} + 4 q^{41} + 26 q^{49} + 16 q^{53} - 26 q^{61} + 30 q^{65} + 30 q^{73} + 18 q^{77} - 32 q^{85} + 28 q^{89} + 36 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - 6x^{2} - x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{3} - \nu^{2} - 7\nu - 1 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{3} - 6\nu - 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 2\nu^{3} - 3\nu^{2} - 9\nu + 1 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + \beta_{2} - 3\beta_1 ) / 6 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{3} + 5\beta_{2} - 3\beta _1 + 18 ) / 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 2\beta_{2} - 3\beta _1 + 4 \) Copy content Toggle raw display

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0.328543
3.04374
−1.82405
−0.548230
0 0 0 −2.37228 0 −2.20979 0 0 0
1.2 0 0 0 −2.37228 0 2.20979 0 0 0
1.3 0 0 0 3.37228 0 −4.70285 0 0 0
1.4 0 0 0 3.37228 0 4.70285 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b 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.a.cf 4
3.b odd 2 1 5184.2.a.cc 4
4.b odd 2 1 inner 5184.2.a.cf 4
8.b even 2 1 2592.2.a.u 4
8.d odd 2 1 2592.2.a.u 4
9.c even 3 2 576.2.i.n 8
9.d odd 6 2 1728.2.i.n 8
12.b even 2 1 5184.2.a.cc 4
24.f even 2 1 2592.2.a.x 4
24.h odd 2 1 2592.2.a.x 4
36.f odd 6 2 576.2.i.n 8
36.h even 6 2 1728.2.i.n 8
72.j odd 6 2 864.2.i.f 8
72.l even 6 2 864.2.i.f 8
72.n even 6 2 288.2.i.f 8
72.p odd 6 2 288.2.i.f 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
288.2.i.f 8 72.n even 6 2
288.2.i.f 8 72.p odd 6 2
576.2.i.n 8 9.c even 3 2
576.2.i.n 8 36.f odd 6 2
864.2.i.f 8 72.j odd 6 2
864.2.i.f 8 72.l even 6 2
1728.2.i.n 8 9.d odd 6 2
1728.2.i.n 8 36.h even 6 2
2592.2.a.u 4 8.b even 2 1
2592.2.a.u 4 8.d odd 2 1
2592.2.a.x 4 24.f even 2 1
2592.2.a.x 4 24.h odd 2 1
5184.2.a.cc 4 3.b odd 2 1
5184.2.a.cc 4 12.b even 2 1
5184.2.a.cf 4 1.a even 1 1 trivial
5184.2.a.cf 4 4.b 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(5184))\):

\( T_{5}^{2} - T_{5} - 8 \) Copy content Toggle raw display
\( T_{7}^{4} - 27T_{7}^{2} + 108 \) Copy content Toggle raw display
\( T_{11}^{4} - 36T_{11}^{2} + 27 \) 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
$5$ \( (T^{2} - T - 8)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} - 27T^{2} + 108 \) Copy content Toggle raw display
$11$ \( T^{4} - 36T^{2} + 27 \) Copy content Toggle raw display
$13$ \( (T^{2} + 3 T - 6)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} - T - 8)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} - 45T^{2} + 432 \) Copy content Toggle raw display
$23$ \( T^{4} - 27T^{2} + 108 \) Copy content Toggle raw display
$29$ \( (T^{2} - 5 T - 2)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} - 135T^{2} + 3888 \) Copy content Toggle raw display
$37$ \( (T + 4)^{4} \) Copy content Toggle raw display
$41$ \( (T - 1)^{4} \) Copy content Toggle raw display
$43$ \( T^{4} - 36T^{2} + 27 \) Copy content Toggle raw display
$47$ \( T^{4} - 27T^{2} + 108 \) Copy content Toggle raw display
$53$ \( (T - 4)^{4} \) Copy content Toggle raw display
$59$ \( T^{4} - 180T^{2} + 7803 \) Copy content Toggle raw display
$61$ \( (T^{2} + 13 T - 32)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} - 180T^{2} + 7803 \) Copy content Toggle raw display
$71$ \( T^{4} - 108T^{2} + 1728 \) Copy content Toggle raw display
$73$ \( (T^{2} - 15 T + 48)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} - 135T^{2} + 3888 \) Copy content Toggle raw display
$83$ \( T^{4} - 207T^{2} + 1728 \) Copy content Toggle raw display
$89$ \( (T^{2} - 14 T + 16)^{2} \) Copy content Toggle raw display
$97$ \( (T - 9)^{4} \) Copy content Toggle raw display
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