Properties

Label 288.2.d.b
Level 288
Weight 2
Character orbit 288.d
Analytic conductor 2.300
Analytic rank 0
Dimension 2
CM no
Inner twists 2

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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: \(\Q(\sqrt{-1}) \)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 24)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 i q^{5} + 2 q^{7} +O(q^{10})\) \( q + 2 i q^{5} + 2 q^{7} + 4 i q^{13} + 2 q^{17} + 4 i q^{19} + 4 q^{23} + q^{25} -6 i q^{29} -2 q^{31} + 4 i q^{35} -8 i q^{37} -2 q^{41} -4 i q^{43} -12 q^{47} -3 q^{49} + 6 i q^{53} -4 i q^{59} -8 q^{65} -12 i q^{67} + 12 q^{71} -6 q^{73} -10 q^{79} -16 i q^{83} + 4 i q^{85} + 10 q^{89} + 8 i q^{91} -8 q^{95} -2 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 4q^{7} + O(q^{10}) \) \( 2q + 4q^{7} + 4q^{17} + 8q^{23} + 2q^{25} - 4q^{31} - 4q^{41} - 24q^{47} - 6q^{49} - 16q^{65} + 24q^{71} - 12q^{73} - 20q^{79} + 20q^{89} - 16q^{95} - 4q^{97} + O(q^{100}) \)

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
1.00000i
1.00000i
0 0 0 2.00000i 0 2.00000 0 0 0
145.2 0 0 0 2.00000i 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
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.b 2
3.b odd 2 1 96.2.d.a 2
4.b odd 2 1 72.2.d.b 2
5.b even 2 1 7200.2.k.d 2
5.c odd 4 1 7200.2.d.d 2
5.c odd 4 1 7200.2.d.g 2
8.b even 2 1 inner 288.2.d.b 2
8.d odd 2 1 72.2.d.b 2
9.c even 3 2 2592.2.r.g 4
9.d odd 6 2 2592.2.r.f 4
12.b even 2 1 24.2.d.a 2
15.d odd 2 1 2400.2.k.a 2
15.e even 4 1 2400.2.d.b 2
15.e even 4 1 2400.2.d.c 2
16.e even 4 1 2304.2.a.b 1
16.e even 4 1 2304.2.a.l 1
16.f odd 4 1 2304.2.a.e 1
16.f odd 4 1 2304.2.a.o 1
20.d odd 2 1 1800.2.k.a 2
20.e even 4 1 1800.2.d.b 2
20.e even 4 1 1800.2.d.i 2
21.c even 2 1 4704.2.c.a 2
24.f even 2 1 24.2.d.a 2
24.h odd 2 1 96.2.d.a 2
36.f odd 6 2 648.2.n.c 4
36.h even 6 2 648.2.n.k 4
40.e odd 2 1 1800.2.k.a 2
40.f even 2 1 7200.2.k.d 2
40.i odd 4 1 7200.2.d.d 2
40.i odd 4 1 7200.2.d.g 2
40.k even 4 1 1800.2.d.b 2
40.k even 4 1 1800.2.d.i 2
48.i odd 4 1 768.2.a.d 1
48.i odd 4 1 768.2.a.e 1
48.k even 4 1 768.2.a.a 1
48.k even 4 1 768.2.a.h 1
60.h even 2 1 600.2.k.b 2
60.l odd 4 1 600.2.d.b 2
60.l odd 4 1 600.2.d.c 2
72.j odd 6 2 2592.2.r.f 4
72.l even 6 2 648.2.n.k 4
72.n even 6 2 2592.2.r.g 4
72.p odd 6 2 648.2.n.c 4
84.h odd 2 1 1176.2.c.a 2
120.i odd 2 1 2400.2.k.a 2
120.m even 2 1 600.2.k.b 2
120.q odd 4 1 600.2.d.b 2
120.q odd 4 1 600.2.d.c 2
120.w even 4 1 2400.2.d.b 2
120.w even 4 1 2400.2.d.c 2
168.e odd 2 1 1176.2.c.a 2
168.i even 2 1 4704.2.c.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.2.d.a 2 12.b even 2 1
24.2.d.a 2 24.f even 2 1
72.2.d.b 2 4.b odd 2 1
72.2.d.b 2 8.d odd 2 1
96.2.d.a 2 3.b odd 2 1
96.2.d.a 2 24.h odd 2 1
288.2.d.b 2 1.a even 1 1 trivial
288.2.d.b 2 8.b even 2 1 inner
600.2.d.b 2 60.l odd 4 1
600.2.d.b 2 120.q odd 4 1
600.2.d.c 2 60.l odd 4 1
600.2.d.c 2 120.q odd 4 1
600.2.k.b 2 60.h even 2 1
600.2.k.b 2 120.m even 2 1
648.2.n.c 4 36.f odd 6 2
648.2.n.c 4 72.p odd 6 2
648.2.n.k 4 36.h even 6 2
648.2.n.k 4 72.l even 6 2
768.2.a.a 1 48.k even 4 1
768.2.a.d 1 48.i odd 4 1
768.2.a.e 1 48.i odd 4 1
768.2.a.h 1 48.k even 4 1
1176.2.c.a 2 84.h odd 2 1
1176.2.c.a 2 168.e odd 2 1
1800.2.d.b 2 20.e even 4 1
1800.2.d.b 2 40.k even 4 1
1800.2.d.i 2 20.e even 4 1
1800.2.d.i 2 40.k even 4 1
1800.2.k.a 2 20.d odd 2 1
1800.2.k.a 2 40.e odd 2 1
2304.2.a.b 1 16.e even 4 1
2304.2.a.e 1 16.f odd 4 1
2304.2.a.l 1 16.e even 4 1
2304.2.a.o 1 16.f odd 4 1
2400.2.d.b 2 15.e even 4 1
2400.2.d.b 2 120.w even 4 1
2400.2.d.c 2 15.e even 4 1
2400.2.d.c 2 120.w even 4 1
2400.2.k.a 2 15.d odd 2 1
2400.2.k.a 2 120.i odd 2 1
2592.2.r.f 4 9.d odd 6 2
2592.2.r.f 4 72.j odd 6 2
2592.2.r.g 4 9.c even 3 2
2592.2.r.g 4 72.n even 6 2
4704.2.c.a 2 21.c even 2 1
4704.2.c.a 2 168.i even 2 1
7200.2.d.d 2 5.c odd 4 1
7200.2.d.d 2 40.i odd 4 1
7200.2.d.g 2 5.c odd 4 1
7200.2.d.g 2 40.i odd 4 1
7200.2.k.d 2 5.b even 2 1
7200.2.k.d 2 40.f even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} + 4 \) acting on \(S_{2}^{\mathrm{new}}(288, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ \( ( 1 - 4 T + 5 T^{2} )( 1 + 4 T + 5 T^{2} ) \)
$7$ \( ( 1 - 2 T + 7 T^{2} )^{2} \)
$11$ \( ( 1 - 11 T^{2} )^{2} \)
$13$ \( ( 1 - 6 T + 13 T^{2} )( 1 + 6 T + 13 T^{2} ) \)
$17$ \( ( 1 - 2 T + 17 T^{2} )^{2} \)
$19$ \( 1 - 22 T^{2} + 361 T^{4} \)
$23$ \( ( 1 - 4 T + 23 T^{2} )^{2} \)
$29$ \( 1 - 22 T^{2} + 841 T^{4} \)
$31$ \( ( 1 + 2 T + 31 T^{2} )^{2} \)
$37$ \( 1 - 10 T^{2} + 1369 T^{4} \)
$41$ \( ( 1 + 2 T + 41 T^{2} )^{2} \)
$43$ \( 1 - 70 T^{2} + 1849 T^{4} \)
$47$ \( ( 1 + 12 T + 47 T^{2} )^{2} \)
$53$ \( 1 - 70 T^{2} + 2809 T^{4} \)
$59$ \( 1 - 102 T^{2} + 3481 T^{4} \)
$61$ \( ( 1 - 61 T^{2} )^{2} \)
$67$ \( 1 + 10 T^{2} + 4489 T^{4} \)
$71$ \( ( 1 - 12 T + 71 T^{2} )^{2} \)
$73$ \( ( 1 + 6 T + 73 T^{2} )^{2} \)
$79$ \( ( 1 + 10 T + 79 T^{2} )^{2} \)
$83$ \( 1 + 90 T^{2} + 6889 T^{4} \)
$89$ \( ( 1 - 10 T + 89 T^{2} )^{2} \)
$97$ \( ( 1 + 2 T + 97 T^{2} )^{2} \)
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