Properties

Label 288.2.f.a
Level $288$
Weight $2$
Character orbit 288.f
Analytic conductor $2.300$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 288 = 2^{5} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 288.f (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.29969157821\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-3})\)
Defining polynomial: \(x^{4} - 2 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 72)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q -\beta_{2} q^{5} -\beta_{3} q^{7} +O(q^{10})\) \( q -\beta_{2} q^{5} -\beta_{3} q^{7} -2 \beta_{1} q^{11} -\beta_{3} q^{13} -\beta_{1} q^{17} + 4 q^{19} -2 \beta_{2} q^{23} + q^{25} + \beta_{2} q^{29} + \beta_{3} q^{31} + 6 \beta_{1} q^{35} -\beta_{1} q^{41} -8 q^{43} + 2 \beta_{2} q^{47} -5 q^{49} + 3 \beta_{2} q^{53} + 2 \beta_{3} q^{55} -8 \beta_{1} q^{59} + 4 \beta_{3} q^{61} + 6 \beta_{1} q^{65} + 4 q^{67} + 6 \beta_{2} q^{71} -4 q^{73} -4 \beta_{2} q^{77} -\beta_{3} q^{79} + 10 \beta_{1} q^{83} + \beta_{3} q^{85} + 5 \beta_{1} q^{89} -12 q^{91} -4 \beta_{2} q^{95} + 8 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + O(q^{10}) \) \( 4q + 16q^{19} + 4q^{25} - 32q^{43} - 20q^{49} + 16q^{67} - 16q^{73} - 48q^{91} + 32q^{97} + O(q^{100}) \)

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{3} \)\(/2\)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{3} + 4 \nu \)\()/2\)
\(\beta_{3}\)\(=\)\( 2 \nu^{2} - 2 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2} + \beta_{1}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3} + 2\)\()/2\)
\(\nu^{3}\)\(=\)\(2 \beta_{1}\)

Character values

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

\(n\) \(37\) \(65\) \(127\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
143.1
1.22474 + 0.707107i
1.22474 0.707107i
−1.22474 0.707107i
−1.22474 + 0.707107i
0 0 0 −2.44949 0 3.46410i 0 0 0
143.2 0 0 0 −2.44949 0 3.46410i 0 0 0
143.3 0 0 0 2.44949 0 3.46410i 0 0 0
143.4 0 0 0 2.44949 0 3.46410i 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
3.b odd 2 1 inner
8.d odd 2 1 inner
24.f even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 288.2.f.a 4
3.b odd 2 1 inner 288.2.f.a 4
4.b odd 2 1 72.2.f.a 4
5.b even 2 1 7200.2.b.c 4
5.c odd 4 2 7200.2.m.c 8
8.b even 2 1 72.2.f.a 4
8.d odd 2 1 inner 288.2.f.a 4
9.c even 3 1 2592.2.p.a 4
9.c even 3 1 2592.2.p.c 4
9.d odd 6 1 2592.2.p.a 4
9.d odd 6 1 2592.2.p.c 4
12.b even 2 1 72.2.f.a 4
15.d odd 2 1 7200.2.b.c 4
15.e even 4 2 7200.2.m.c 8
16.e even 4 2 2304.2.c.i 8
16.f odd 4 2 2304.2.c.i 8
20.d odd 2 1 1800.2.b.c 4
20.e even 4 2 1800.2.m.c 8
24.f even 2 1 inner 288.2.f.a 4
24.h odd 2 1 72.2.f.a 4
36.f odd 6 1 648.2.l.a 4
36.f odd 6 1 648.2.l.c 4
36.h even 6 1 648.2.l.a 4
36.h even 6 1 648.2.l.c 4
40.e odd 2 1 7200.2.b.c 4
40.f even 2 1 1800.2.b.c 4
40.i odd 4 2 1800.2.m.c 8
40.k even 4 2 7200.2.m.c 8
48.i odd 4 2 2304.2.c.i 8
48.k even 4 2 2304.2.c.i 8
60.h even 2 1 1800.2.b.c 4
60.l odd 4 2 1800.2.m.c 8
72.j odd 6 1 648.2.l.a 4
72.j odd 6 1 648.2.l.c 4
72.l even 6 1 2592.2.p.a 4
72.l even 6 1 2592.2.p.c 4
72.n even 6 1 648.2.l.a 4
72.n even 6 1 648.2.l.c 4
72.p odd 6 1 2592.2.p.a 4
72.p odd 6 1 2592.2.p.c 4
120.i odd 2 1 1800.2.b.c 4
120.m even 2 1 7200.2.b.c 4
120.q odd 4 2 7200.2.m.c 8
120.w even 4 2 1800.2.m.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
72.2.f.a 4 4.b odd 2 1
72.2.f.a 4 8.b even 2 1
72.2.f.a 4 12.b even 2 1
72.2.f.a 4 24.h odd 2 1
288.2.f.a 4 1.a even 1 1 trivial
288.2.f.a 4 3.b odd 2 1 inner
288.2.f.a 4 8.d odd 2 1 inner
288.2.f.a 4 24.f even 2 1 inner
648.2.l.a 4 36.f odd 6 1
648.2.l.a 4 36.h even 6 1
648.2.l.a 4 72.j odd 6 1
648.2.l.a 4 72.n even 6 1
648.2.l.c 4 36.f odd 6 1
648.2.l.c 4 36.h even 6 1
648.2.l.c 4 72.j odd 6 1
648.2.l.c 4 72.n even 6 1
1800.2.b.c 4 20.d odd 2 1
1800.2.b.c 4 40.f even 2 1
1800.2.b.c 4 60.h even 2 1
1800.2.b.c 4 120.i odd 2 1
1800.2.m.c 8 20.e even 4 2
1800.2.m.c 8 40.i odd 4 2
1800.2.m.c 8 60.l odd 4 2
1800.2.m.c 8 120.w even 4 2
2304.2.c.i 8 16.e even 4 2
2304.2.c.i 8 16.f odd 4 2
2304.2.c.i 8 48.i odd 4 2
2304.2.c.i 8 48.k even 4 2
2592.2.p.a 4 9.c even 3 1
2592.2.p.a 4 9.d odd 6 1
2592.2.p.a 4 72.l even 6 1
2592.2.p.a 4 72.p odd 6 1
2592.2.p.c 4 9.c even 3 1
2592.2.p.c 4 9.d odd 6 1
2592.2.p.c 4 72.l even 6 1
2592.2.p.c 4 72.p odd 6 1
7200.2.b.c 4 5.b even 2 1
7200.2.b.c 4 15.d odd 2 1
7200.2.b.c 4 40.e odd 2 1
7200.2.b.c 4 120.m even 2 1
7200.2.m.c 8 5.c odd 4 2
7200.2.m.c 8 15.e even 4 2
7200.2.m.c 8 40.k even 4 2
7200.2.m.c 8 120.q odd 4 2

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(288, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( T^{4} \)
$5$ \( ( -6 + T^{2} )^{2} \)
$7$ \( ( 12 + T^{2} )^{2} \)
$11$ \( ( 8 + T^{2} )^{2} \)
$13$ \( ( 12 + T^{2} )^{2} \)
$17$ \( ( 2 + T^{2} )^{2} \)
$19$ \( ( -4 + T )^{4} \)
$23$ \( ( -24 + T^{2} )^{2} \)
$29$ \( ( -6 + T^{2} )^{2} \)
$31$ \( ( 12 + T^{2} )^{2} \)
$37$ \( T^{4} \)
$41$ \( ( 2 + T^{2} )^{2} \)
$43$ \( ( 8 + T )^{4} \)
$47$ \( ( -24 + T^{2} )^{2} \)
$53$ \( ( -54 + T^{2} )^{2} \)
$59$ \( ( 128 + T^{2} )^{2} \)
$61$ \( ( 192 + T^{2} )^{2} \)
$67$ \( ( -4 + T )^{4} \)
$71$ \( ( -216 + T^{2} )^{2} \)
$73$ \( ( 4 + T )^{4} \)
$79$ \( ( 12 + T^{2} )^{2} \)
$83$ \( ( 200 + T^{2} )^{2} \)
$89$ \( ( 50 + T^{2} )^{2} \)
$97$ \( ( -8 + T )^{4} \)
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