Properties

Label 288.2.p.a
Level $288$
Weight $2$
Character orbit 288.p
Analytic conductor $2.300$
Analytic rank $0$
Dimension $4$
CM discriminant -8
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 288 = 2^{5} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 288.p (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.29969157821\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-3})\)
Defining polynomial: \(x^{4} - 2 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 72)
Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( \beta_{1} - \beta_{2} - \beta_{3} ) q^{3} + ( 1 - 2 \beta_{1} - \beta_{2} ) q^{9} +O(q^{10})\) \( q + ( \beta_{1} - \beta_{2} - \beta_{3} ) q^{3} + ( 1 - 2 \beta_{1} - \beta_{2} ) q^{9} + ( 6 + \beta_{1} - 3 \beta_{2} - \beta_{3} ) q^{11} + ( 3 - 6 \beta_{2} + 2 \beta_{3} ) q^{17} + ( 1 - 6 \beta_{1} + 3 \beta_{3} ) q^{19} + 5 \beta_{2} q^{25} + ( -5 + \beta_{3} ) q^{27} + ( -1 + 2 \beta_{1} - 5 \beta_{2} - 6 \beta_{3} ) q^{33} + ( 3 + 4 \beta_{1} + 3 \beta_{2} ) q^{41} + ( 3 \beta_{1} + 5 \beta_{2} + 3 \beta_{3} ) q^{43} + ( -7 + 7 \beta_{2} ) q^{49} + ( -6 - \beta_{1} + 7 \beta_{2} - 5 \beta_{3} ) q^{51} + ( -12 + 4 \beta_{1} + 5 \beta_{2} + 2 \beta_{3} ) q^{57} + ( -3 + 5 \beta_{1} - 3 \beta_{2} ) q^{59} + ( -7 + 3 \beta_{1} + 7 \beta_{2} - 6 \beta_{3} ) q^{67} + ( -1 - 12 \beta_{1} + 6 \beta_{3} ) q^{73} + ( 5 + 5 \beta_{1} - 5 \beta_{2} ) q^{75} + ( -4 \beta_{1} + 7 \beta_{2} + 4 \beta_{3} ) q^{81} + ( -2 \beta_{1} + 2 \beta_{3} ) q^{83} -4 \beta_{3} q^{89} + ( 6 \beta_{1} - 5 \beta_{2} + 6 \beta_{3} ) q^{97} + ( -1 - 12 \beta_{1} - 6 \beta_{2} + 5 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 2q^{3} + 2q^{9} + O(q^{10}) \) \( 4q - 2q^{3} + 2q^{9} + 18q^{11} + 4q^{19} + 10q^{25} - 20q^{27} - 14q^{33} + 18q^{41} + 10q^{43} - 14q^{49} - 10q^{51} - 38q^{57} - 18q^{59} - 14q^{67} - 4q^{73} + 10q^{75} + 14q^{81} - 10q^{97} - 16q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 2 x^{2} + 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} \)\(/2\)
\(\beta_{3}\)\(=\)\( \nu^{3} \)\(/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(2 \beta_{2}\)
\(\nu^{3}\)\(=\)\(2 \beta_{3}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/288\mathbb{Z}\right)^\times\).

\(n\) \(37\) \(65\) \(127\)
\(\chi(n)\) \(-1\) \(1 - \beta_{2}\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
47.1
−1.22474 + 0.707107i
1.22474 0.707107i
−1.22474 0.707107i
1.22474 + 0.707107i
0 −1.72474 + 0.158919i 0 0 0 0 0 2.94949 0.548188i 0
47.2 0 0.724745 + 1.57313i 0 0 0 0 0 −1.94949 + 2.28024i 0
239.1 0 −1.72474 0.158919i 0 0 0 0 0 2.94949 + 0.548188i 0
239.2 0 0.724745 1.57313i 0 0 0 0 0 −1.94949 2.28024i 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 \(\Q(\sqrt{-2}) \)
9.d odd 6 1 inner
72.l even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 288.2.p.a 4
3.b odd 2 1 864.2.p.a 4
4.b odd 2 1 72.2.l.a 4
8.b even 2 1 72.2.l.a 4
8.d odd 2 1 CM 288.2.p.a 4
9.c even 3 1 864.2.p.a 4
9.c even 3 1 2592.2.f.a 4
9.d odd 6 1 inner 288.2.p.a 4
9.d odd 6 1 2592.2.f.a 4
12.b even 2 1 216.2.l.a 4
24.f even 2 1 864.2.p.a 4
24.h odd 2 1 216.2.l.a 4
36.f odd 6 1 216.2.l.a 4
36.f odd 6 1 648.2.f.a 4
36.h even 6 1 72.2.l.a 4
36.h even 6 1 648.2.f.a 4
72.j odd 6 1 72.2.l.a 4
72.j odd 6 1 648.2.f.a 4
72.l even 6 1 inner 288.2.p.a 4
72.l even 6 1 2592.2.f.a 4
72.n even 6 1 216.2.l.a 4
72.n even 6 1 648.2.f.a 4
72.p odd 6 1 864.2.p.a 4
72.p odd 6 1 2592.2.f.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
72.2.l.a 4 4.b odd 2 1
72.2.l.a 4 8.b even 2 1
72.2.l.a 4 36.h even 6 1
72.2.l.a 4 72.j odd 6 1
216.2.l.a 4 12.b even 2 1
216.2.l.a 4 24.h odd 2 1
216.2.l.a 4 36.f odd 6 1
216.2.l.a 4 72.n even 6 1
288.2.p.a 4 1.a even 1 1 trivial
288.2.p.a 4 8.d odd 2 1 CM
288.2.p.a 4 9.d odd 6 1 inner
288.2.p.a 4 72.l even 6 1 inner
648.2.f.a 4 36.f odd 6 1
648.2.f.a 4 36.h even 6 1
648.2.f.a 4 72.j odd 6 1
648.2.f.a 4 72.n even 6 1
864.2.p.a 4 3.b odd 2 1
864.2.p.a 4 9.c even 3 1
864.2.p.a 4 24.f even 2 1
864.2.p.a 4 72.p odd 6 1
2592.2.f.a 4 9.c even 3 1
2592.2.f.a 4 9.d odd 6 1
2592.2.f.a 4 72.l even 6 1
2592.2.f.a 4 72.p odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} \) acting on \(S_{2}^{\mathrm{new}}(288, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( 9 + 6 T + T^{2} + 2 T^{3} + T^{4} \)
$5$ \( T^{4} \)
$7$ \( T^{4} \)
$11$ \( 625 - 450 T + 133 T^{2} - 18 T^{3} + T^{4} \)
$13$ \( T^{4} \)
$17$ \( 361 + 70 T^{2} + T^{4} \)
$19$ \( ( -53 - 2 T + T^{2} )^{2} \)
$23$ \( T^{4} \)
$29$ \( T^{4} \)
$31$ \( T^{4} \)
$37$ \( T^{4} \)
$41$ \( 25 + 90 T + 103 T^{2} - 18 T^{3} + T^{4} \)
$43$ \( 841 + 290 T + 129 T^{2} - 10 T^{3} + T^{4} \)
$47$ \( T^{4} \)
$53$ \( T^{4} \)
$59$ \( 529 - 414 T + 85 T^{2} + 18 T^{3} + T^{4} \)
$61$ \( T^{4} \)
$67$ \( 25 - 70 T + 201 T^{2} + 14 T^{3} + T^{4} \)
$71$ \( T^{4} \)
$73$ \( ( -215 + 2 T + T^{2} )^{2} \)
$79$ \( T^{4} \)
$83$ \( 64 - 8 T^{2} + T^{4} \)
$89$ \( ( 32 + T^{2} )^{2} \)
$97$ \( 36481 - 1910 T + 291 T^{2} + 10 T^{3} + T^{4} \)
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