Properties

Label 288.2.d.a
Level $288$
Weight $2$
Character orbit 288.d
Analytic conductor $2.300$
Analytic rank $0$
Dimension $2$
CM discriminant -24
Inner twists $4$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(2.29969157821\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-2}) \)
Defining polynomial: \(x^{2} + 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 72)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 2\sqrt{-2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{5} -2 q^{7} +O(q^{10})\) \( q + \beta q^{5} -2 q^{7} + 2 \beta q^{11} -3 q^{25} + \beta q^{29} + 10 q^{31} -2 \beta q^{35} -3 q^{49} -5 \beta q^{53} -16 q^{55} -4 \beta q^{59} + 14 q^{73} -4 \beta q^{77} + 10 q^{79} + 2 \beta q^{83} + 2 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 4q^{7} + O(q^{10}) \) \( 2q - 4q^{7} - 6q^{25} + 20q^{31} - 6q^{49} - 32q^{55} + 28q^{73} + 20q^{79} + 4q^{97} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
145.1
1.41421i
1.41421i
0 0 0 2.82843i 0 −2.00000 0 0 0
145.2 0 0 0 2.82843i 0 −2.00000 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
24.h odd 2 1 CM by \(\Q(\sqrt{-6}) \)
3.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 288.2.d.a 2
3.b odd 2 1 inner 288.2.d.a 2
4.b odd 2 1 72.2.d.a 2
5.b even 2 1 7200.2.k.h 2
5.c odd 4 2 7200.2.d.p 4
8.b even 2 1 inner 288.2.d.a 2
8.d odd 2 1 72.2.d.a 2
9.c even 3 2 2592.2.r.i 4
9.d odd 6 2 2592.2.r.i 4
12.b even 2 1 72.2.d.a 2
15.d odd 2 1 7200.2.k.h 2
15.e even 4 2 7200.2.d.p 4
16.e even 4 2 2304.2.a.y 2
16.f odd 4 2 2304.2.a.q 2
20.d odd 2 1 1800.2.k.e 2
20.e even 4 2 1800.2.d.n 4
24.f even 2 1 72.2.d.a 2
24.h odd 2 1 CM 288.2.d.a 2
36.f odd 6 2 648.2.n.h 4
36.h even 6 2 648.2.n.h 4
40.e odd 2 1 1800.2.k.e 2
40.f even 2 1 7200.2.k.h 2
40.i odd 4 2 7200.2.d.p 4
40.k even 4 2 1800.2.d.n 4
48.i odd 4 2 2304.2.a.y 2
48.k even 4 2 2304.2.a.q 2
60.h even 2 1 1800.2.k.e 2
60.l odd 4 2 1800.2.d.n 4
72.j odd 6 2 2592.2.r.i 4
72.l even 6 2 648.2.n.h 4
72.n even 6 2 2592.2.r.i 4
72.p odd 6 2 648.2.n.h 4
120.i odd 2 1 7200.2.k.h 2
120.m even 2 1 1800.2.k.e 2
120.q odd 4 2 1800.2.d.n 4
120.w even 4 2 7200.2.d.p 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
72.2.d.a 2 4.b odd 2 1
72.2.d.a 2 8.d odd 2 1
72.2.d.a 2 12.b even 2 1
72.2.d.a 2 24.f even 2 1
288.2.d.a 2 1.a even 1 1 trivial
288.2.d.a 2 3.b odd 2 1 inner
288.2.d.a 2 8.b even 2 1 inner
288.2.d.a 2 24.h odd 2 1 CM
648.2.n.h 4 36.f odd 6 2
648.2.n.h 4 36.h even 6 2
648.2.n.h 4 72.l even 6 2
648.2.n.h 4 72.p odd 6 2
1800.2.d.n 4 20.e even 4 2
1800.2.d.n 4 40.k even 4 2
1800.2.d.n 4 60.l odd 4 2
1800.2.d.n 4 120.q odd 4 2
1800.2.k.e 2 20.d odd 2 1
1800.2.k.e 2 40.e odd 2 1
1800.2.k.e 2 60.h even 2 1
1800.2.k.e 2 120.m even 2 1
2304.2.a.q 2 16.f odd 4 2
2304.2.a.q 2 48.k even 4 2
2304.2.a.y 2 16.e even 4 2
2304.2.a.y 2 48.i odd 4 2
2592.2.r.i 4 9.c even 3 2
2592.2.r.i 4 9.d odd 6 2
2592.2.r.i 4 72.j odd 6 2
2592.2.r.i 4 72.n even 6 2
7200.2.d.p 4 5.c odd 4 2
7200.2.d.p 4 15.e even 4 2
7200.2.d.p 4 40.i odd 4 2
7200.2.d.p 4 120.w even 4 2
7200.2.k.h 2 5.b even 2 1
7200.2.k.h 2 15.d odd 2 1
7200.2.k.h 2 40.f even 2 1
7200.2.k.h 2 120.i odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} + 8 \) acting on \(S_{2}^{\mathrm{new}}(288, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( 8 + T^{2} \)
$7$ \( ( 2 + T )^{2} \)
$11$ \( 32 + T^{2} \)
$13$ \( T^{2} \)
$17$ \( T^{2} \)
$19$ \( T^{2} \)
$23$ \( T^{2} \)
$29$ \( 8 + T^{2} \)
$31$ \( ( -10 + T )^{2} \)
$37$ \( T^{2} \)
$41$ \( T^{2} \)
$43$ \( T^{2} \)
$47$ \( T^{2} \)
$53$ \( 200 + T^{2} \)
$59$ \( 128 + T^{2} \)
$61$ \( T^{2} \)
$67$ \( T^{2} \)
$71$ \( T^{2} \)
$73$ \( ( -14 + T )^{2} \)
$79$ \( ( -10 + T )^{2} \)
$83$ \( 32 + T^{2} \)
$89$ \( T^{2} \)
$97$ \( ( -2 + T )^{2} \)
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