Properties

Label 288.2.a.b
Level 288
Weight 2
Character orbit 288.a
Self dual yes
Analytic conductor 2.300
Analytic rank 1
Dimension 1
CM no
Inner twists 1

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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: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 96)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 2q^{5} - 4q^{7} + O(q^{10}) \) \( q - 2q^{5} - 4q^{7} - 4q^{11} - 2q^{13} + 6q^{17} - 4q^{19} - q^{25} - 2q^{29} + 4q^{31} + 8q^{35} - 2q^{37} - 2q^{41} + 4q^{43} - 8q^{47} + 9q^{49} - 10q^{53} + 8q^{55} + 4q^{59} + 6q^{61} + 4q^{65} + 4q^{67} + 16q^{71} - 6q^{73} + 16q^{77} + 4q^{79} - 12q^{83} - 12q^{85} - 10q^{89} + 8q^{91} + 8q^{95} - 14q^{97} + O(q^{100}) \)

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 −4.00000 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 288.2.a.b 1
3.b odd 2 1 96.2.a.b yes 1
4.b odd 2 1 288.2.a.c 1
5.b even 2 1 7200.2.a.bx 1
5.c odd 4 2 7200.2.f.f 2
8.b even 2 1 576.2.a.g 1
8.d odd 2 1 576.2.a.h 1
9.c even 3 2 2592.2.i.w 2
9.d odd 6 2 2592.2.i.h 2
12.b even 2 1 96.2.a.a 1
15.d odd 2 1 2400.2.a.q 1
15.e even 4 2 2400.2.f.r 2
16.e even 4 2 2304.2.d.s 2
16.f odd 4 2 2304.2.d.c 2
20.d odd 2 1 7200.2.a.e 1
20.e even 4 2 7200.2.f.x 2
21.c even 2 1 4704.2.a.e 1
24.f even 2 1 192.2.a.c 1
24.h odd 2 1 192.2.a.a 1
36.f odd 6 2 2592.2.i.q 2
36.h even 6 2 2592.2.i.b 2
48.i odd 4 2 768.2.d.h 2
48.k even 4 2 768.2.d.a 2
60.h even 2 1 2400.2.a.r 1
60.l odd 4 2 2400.2.f.a 2
84.h odd 2 1 4704.2.a.t 1
120.i odd 2 1 4800.2.a.co 1
120.m even 2 1 4800.2.a.f 1
120.q odd 4 2 4800.2.f.bh 2
120.w even 4 2 4800.2.f.e 2
168.e odd 2 1 9408.2.a.bj 1
168.i even 2 1 9408.2.a.ct 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
96.2.a.a 1 12.b even 2 1
96.2.a.b yes 1 3.b odd 2 1
192.2.a.a 1 24.h odd 2 1
192.2.a.c 1 24.f even 2 1
288.2.a.b 1 1.a even 1 1 trivial
288.2.a.c 1 4.b odd 2 1
576.2.a.g 1 8.b even 2 1
576.2.a.h 1 8.d odd 2 1
768.2.d.a 2 48.k even 4 2
768.2.d.h 2 48.i odd 4 2
2304.2.d.c 2 16.f odd 4 2
2304.2.d.s 2 16.e even 4 2
2400.2.a.q 1 15.d odd 2 1
2400.2.a.r 1 60.h even 2 1
2400.2.f.a 2 60.l odd 4 2
2400.2.f.r 2 15.e even 4 2
2592.2.i.b 2 36.h even 6 2
2592.2.i.h 2 9.d odd 6 2
2592.2.i.q 2 36.f odd 6 2
2592.2.i.w 2 9.c even 3 2
4704.2.a.e 1 21.c even 2 1
4704.2.a.t 1 84.h odd 2 1
4800.2.a.f 1 120.m even 2 1
4800.2.a.co 1 120.i odd 2 1
4800.2.f.e 2 120.w even 4 2
4800.2.f.bh 2 120.q odd 4 2
7200.2.a.e 1 20.d odd 2 1
7200.2.a.bx 1 5.b even 2 1
7200.2.f.f 2 5.c odd 4 2
7200.2.f.x 2 20.e even 4 2
9408.2.a.bj 1 168.e odd 2 1
9408.2.a.ct 1 168.i 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(288))\):

\( T_{5} + 2 \)
\( T_{7} + 4 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ \( 1 + 2 T + 5 T^{2} \)
$7$ \( 1 + 4 T + 7 T^{2} \)
$11$ \( 1 + 4 T + 11 T^{2} \)
$13$ \( 1 + 2 T + 13 T^{2} \)
$17$ \( 1 - 6 T + 17 T^{2} \)
$19$ \( 1 + 4 T + 19 T^{2} \)
$23$ \( 1 + 23 T^{2} \)
$29$ \( 1 + 2 T + 29 T^{2} \)
$31$ \( 1 - 4 T + 31 T^{2} \)
$37$ \( 1 + 2 T + 37 T^{2} \)
$41$ \( 1 + 2 T + 41 T^{2} \)
$43$ \( 1 - 4 T + 43 T^{2} \)
$47$ \( 1 + 8 T + 47 T^{2} \)
$53$ \( 1 + 10 T + 53 T^{2} \)
$59$ \( 1 - 4 T + 59 T^{2} \)
$61$ \( 1 - 6 T + 61 T^{2} \)
$67$ \( 1 - 4 T + 67 T^{2} \)
$71$ \( 1 - 16 T + 71 T^{2} \)
$73$ \( 1 + 6 T + 73 T^{2} \)
$79$ \( 1 - 4 T + 79 T^{2} \)
$83$ \( 1 + 12 T + 83 T^{2} \)
$89$ \( 1 + 10 T + 89 T^{2} \)
$97$ \( 1 + 14 T + 97 T^{2} \)
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