Properties

Label 288.2.a.d
Level 288
Weight 2
Character orbit 288.a
Self dual yes
Analytic conductor 2.300
Analytic rank 0
Dimension 1
CM discriminant -4
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 288 = 2^{5} \cdot 3^{2} \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 288.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(2.29969157821\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 32)
Fricke sign: \(-1\)
Sato-Tate group: $N(\mathrm{U}(1))$

$q$-expansion

\(f(q)\) \(=\) \( q + 2q^{5} + O(q^{10}) \) \( q + 2q^{5} + 6q^{13} - 2q^{17} - q^{25} + 10q^{29} - 2q^{37} - 10q^{41} - 7q^{49} - 14q^{53} - 10q^{61} + 12q^{65} - 6q^{73} - 4q^{85} - 10q^{89} + 18q^{97} + O(q^{100}) \)

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
0 0 0 2.00000 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
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 288.2.a.d 1
3.b odd 2 1 32.2.a.a 1
4.b odd 2 1 CM 288.2.a.d 1
5.b even 2 1 7200.2.a.v 1
5.c odd 4 2 7200.2.f.m 2
8.b even 2 1 576.2.a.c 1
8.d odd 2 1 576.2.a.c 1
9.c even 3 2 2592.2.i.e 2
9.d odd 6 2 2592.2.i.t 2
12.b even 2 1 32.2.a.a 1
15.d odd 2 1 800.2.a.d 1
15.e even 4 2 800.2.c.e 2
16.e even 4 2 2304.2.d.j 2
16.f odd 4 2 2304.2.d.j 2
20.d odd 2 1 7200.2.a.v 1
20.e even 4 2 7200.2.f.m 2
21.c even 2 1 1568.2.a.e 1
21.g even 6 2 1568.2.i.f 2
21.h odd 6 2 1568.2.i.g 2
24.f even 2 1 64.2.a.a 1
24.h odd 2 1 64.2.a.a 1
33.d even 2 1 3872.2.a.f 1
36.f odd 6 2 2592.2.i.e 2
36.h even 6 2 2592.2.i.t 2
39.d odd 2 1 5408.2.a.g 1
48.i odd 4 2 256.2.b.b 2
48.k even 4 2 256.2.b.b 2
51.c odd 2 1 9248.2.a.f 1
60.h even 2 1 800.2.a.d 1
60.l odd 4 2 800.2.c.e 2
84.h odd 2 1 1568.2.a.e 1
84.j odd 6 2 1568.2.i.f 2
84.n even 6 2 1568.2.i.g 2
96.o even 8 4 1024.2.e.j 4
96.p odd 8 4 1024.2.e.j 4
120.i odd 2 1 1600.2.a.n 1
120.m even 2 1 1600.2.a.n 1
120.q odd 4 2 1600.2.c.l 2
120.w even 4 2 1600.2.c.l 2
132.d odd 2 1 3872.2.a.f 1
156.h even 2 1 5408.2.a.g 1
168.e odd 2 1 3136.2.a.m 1
168.i even 2 1 3136.2.a.m 1
204.h even 2 1 9248.2.a.f 1
264.m even 2 1 7744.2.a.v 1
264.p odd 2 1 7744.2.a.v 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
32.2.a.a 1 3.b odd 2 1
32.2.a.a 1 12.b even 2 1
64.2.a.a 1 24.f even 2 1
64.2.a.a 1 24.h odd 2 1
256.2.b.b 2 48.i odd 4 2
256.2.b.b 2 48.k even 4 2
288.2.a.d 1 1.a even 1 1 trivial
288.2.a.d 1 4.b odd 2 1 CM
576.2.a.c 1 8.b even 2 1
576.2.a.c 1 8.d odd 2 1
800.2.a.d 1 15.d odd 2 1
800.2.a.d 1 60.h even 2 1
800.2.c.e 2 15.e even 4 2
800.2.c.e 2 60.l odd 4 2
1024.2.e.j 4 96.o even 8 4
1024.2.e.j 4 96.p odd 8 4
1568.2.a.e 1 21.c even 2 1
1568.2.a.e 1 84.h odd 2 1
1568.2.i.f 2 21.g even 6 2
1568.2.i.f 2 84.j odd 6 2
1568.2.i.g 2 21.h odd 6 2
1568.2.i.g 2 84.n even 6 2
1600.2.a.n 1 120.i odd 2 1
1600.2.a.n 1 120.m even 2 1
1600.2.c.l 2 120.q odd 4 2
1600.2.c.l 2 120.w even 4 2
2304.2.d.j 2 16.e even 4 2
2304.2.d.j 2 16.f odd 4 2
2592.2.i.e 2 9.c even 3 2
2592.2.i.e 2 36.f odd 6 2
2592.2.i.t 2 9.d odd 6 2
2592.2.i.t 2 36.h even 6 2
3136.2.a.m 1 168.e odd 2 1
3136.2.a.m 1 168.i even 2 1
3872.2.a.f 1 33.d even 2 1
3872.2.a.f 1 132.d odd 2 1
5408.2.a.g 1 39.d odd 2 1
5408.2.a.g 1 156.h even 2 1
7200.2.a.v 1 5.b even 2 1
7200.2.a.v 1 20.d odd 2 1
7200.2.f.m 2 5.c odd 4 2
7200.2.f.m 2 20.e even 4 2
7744.2.a.v 1 264.m even 2 1
7744.2.a.v 1 264.p odd 2 1
9248.2.a.f 1 51.c odd 2 1
9248.2.a.f 1 204.h even 2 1

Atkin-Lehner signs

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

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(288))\):

\( T_{5} - 2 \)
\( T_{7} \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( \)
$3$ \( \)
$5$ \( 1 - 2 T + 5 T^{2} \)
$7$ \( 1 + 7 T^{2} \)
$11$ \( 1 + 11 T^{2} \)
$13$ \( 1 - 6 T + 13 T^{2} \)
$17$ \( 1 + 2 T + 17 T^{2} \)
$19$ \( 1 + 19 T^{2} \)
$23$ \( 1 + 23 T^{2} \)
$29$ \( 1 - 10 T + 29 T^{2} \)
$31$ \( 1 + 31 T^{2} \)
$37$ \( 1 + 2 T + 37 T^{2} \)
$41$ \( 1 + 10 T + 41 T^{2} \)
$43$ \( 1 + 43 T^{2} \)
$47$ \( 1 + 47 T^{2} \)
$53$ \( 1 + 14 T + 53 T^{2} \)
$59$ \( 1 + 59 T^{2} \)
$61$ \( 1 + 10 T + 61 T^{2} \)
$67$ \( 1 + 67 T^{2} \)
$71$ \( 1 + 71 T^{2} \)
$73$ \( 1 + 6 T + 73 T^{2} \)
$79$ \( 1 + 79 T^{2} \)
$83$ \( 1 + 83 T^{2} \)
$89$ \( 1 + 10 T + 89 T^{2} \)
$97$ \( 1 - 18 T + 97 T^{2} \)
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