Properties

Label 288.2.i.a
Level 288
Weight 2
Character orbit 288.i
Analytic conductor 2.300
Analytic rank 1
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.i (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.29969157821\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -1 - \zeta_{6} ) q^{3} -4 \zeta_{6} q^{5} + ( -2 + 2 \zeta_{6} ) q^{7} + 3 \zeta_{6} q^{9} +O(q^{10})\) \( q + ( -1 - \zeta_{6} ) q^{3} -4 \zeta_{6} q^{5} + ( -2 + 2 \zeta_{6} ) q^{7} + 3 \zeta_{6} q^{9} + ( -5 + 5 \zeta_{6} ) q^{11} + 2 \zeta_{6} q^{13} + ( -4 + 8 \zeta_{6} ) q^{15} -3 q^{17} - q^{19} + ( 4 - 2 \zeta_{6} ) q^{21} -6 \zeta_{6} q^{23} + ( -11 + 11 \zeta_{6} ) q^{25} + ( 3 - 6 \zeta_{6} ) q^{27} + ( 2 - 2 \zeta_{6} ) q^{29} -4 \zeta_{6} q^{31} + ( 10 - 5 \zeta_{6} ) q^{33} + 8 q^{35} -8 q^{37} + ( 2 - 4 \zeta_{6} ) q^{39} -\zeta_{6} q^{41} + ( -7 + 7 \zeta_{6} ) q^{43} + ( 12 - 12 \zeta_{6} ) q^{45} + ( 2 - 2 \zeta_{6} ) q^{47} + 3 \zeta_{6} q^{49} + ( 3 + 3 \zeta_{6} ) q^{51} -4 q^{53} + 20 q^{55} + ( 1 + \zeta_{6} ) q^{57} + 5 \zeta_{6} q^{59} -6 q^{63} + ( 8 - 8 \zeta_{6} ) q^{65} -13 \zeta_{6} q^{67} + ( -6 + 12 \zeta_{6} ) q^{69} -8 q^{71} + 3 q^{73} + ( 22 - 11 \zeta_{6} ) q^{75} -10 \zeta_{6} q^{77} + ( 8 - 8 \zeta_{6} ) q^{79} + ( -9 + 9 \zeta_{6} ) q^{81} + ( 12 - 12 \zeta_{6} ) q^{83} + 12 \zeta_{6} q^{85} + ( -4 + 2 \zeta_{6} ) q^{87} -10 q^{89} -4 q^{91} + ( -4 + 8 \zeta_{6} ) q^{93} + 4 \zeta_{6} q^{95} + ( 11 - 11 \zeta_{6} ) q^{97} -15 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 3q^{3} - 4q^{5} - 2q^{7} + 3q^{9} + O(q^{10}) \) \( 2q - 3q^{3} - 4q^{5} - 2q^{7} + 3q^{9} - 5q^{11} + 2q^{13} - 6q^{17} - 2q^{19} + 6q^{21} - 6q^{23} - 11q^{25} + 2q^{29} - 4q^{31} + 15q^{33} + 16q^{35} - 16q^{37} - q^{41} - 7q^{43} + 12q^{45} + 2q^{47} + 3q^{49} + 9q^{51} - 8q^{53} + 40q^{55} + 3q^{57} + 5q^{59} - 12q^{63} + 8q^{65} - 13q^{67} - 16q^{71} + 6q^{73} + 33q^{75} - 10q^{77} + 8q^{79} - 9q^{81} + 12q^{83} + 12q^{85} - 6q^{87} - 20q^{89} - 8q^{91} + 4q^{95} + 11q^{97} - 30q^{99} + O(q^{100}) \)

Character values

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

\(n\) \(37\) \(65\) \(127\)
\(\chi(n)\) \(1\) \(-\zeta_{6}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
97.1
0.500000 + 0.866025i
0.500000 0.866025i
0 −1.50000 0.866025i 0 −2.00000 3.46410i 0 −1.00000 + 1.73205i 0 1.50000 + 2.59808i 0
193.1 0 −1.50000 + 0.866025i 0 −2.00000 + 3.46410i 0 −1.00000 1.73205i 0 1.50000 2.59808i 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
9.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 288.2.i.a 2
3.b odd 2 1 864.2.i.a 2
4.b odd 2 1 288.2.i.b yes 2
8.b even 2 1 576.2.i.h 2
8.d odd 2 1 576.2.i.b 2
9.c even 3 1 inner 288.2.i.a 2
9.c even 3 1 2592.2.a.h 1
9.d odd 6 1 864.2.i.a 2
9.d odd 6 1 2592.2.a.b 1
12.b even 2 1 864.2.i.b 2
24.f even 2 1 1728.2.i.b 2
24.h odd 2 1 1728.2.i.a 2
36.f odd 6 1 288.2.i.b yes 2
36.f odd 6 1 2592.2.a.g 1
36.h even 6 1 864.2.i.b 2
36.h even 6 1 2592.2.a.a 1
72.j odd 6 1 1728.2.i.a 2
72.j odd 6 1 5184.2.a.bf 1
72.l even 6 1 1728.2.i.b 2
72.l even 6 1 5184.2.a.be 1
72.n even 6 1 576.2.i.h 2
72.n even 6 1 5184.2.a.b 1
72.p odd 6 1 576.2.i.b 2
72.p odd 6 1 5184.2.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
288.2.i.a 2 1.a even 1 1 trivial
288.2.i.a 2 9.c even 3 1 inner
288.2.i.b yes 2 4.b odd 2 1
288.2.i.b yes 2 36.f odd 6 1
576.2.i.b 2 8.d odd 2 1
576.2.i.b 2 72.p odd 6 1
576.2.i.h 2 8.b even 2 1
576.2.i.h 2 72.n even 6 1
864.2.i.a 2 3.b odd 2 1
864.2.i.a 2 9.d odd 6 1
864.2.i.b 2 12.b even 2 1
864.2.i.b 2 36.h even 6 1
1728.2.i.a 2 24.h odd 2 1
1728.2.i.a 2 72.j odd 6 1
1728.2.i.b 2 24.f even 2 1
1728.2.i.b 2 72.l even 6 1
2592.2.a.a 1 36.h even 6 1
2592.2.a.b 1 9.d odd 6 1
2592.2.a.g 1 36.f odd 6 1
2592.2.a.h 1 9.c even 3 1
5184.2.a.a 1 72.p odd 6 1
5184.2.a.b 1 72.n even 6 1
5184.2.a.be 1 72.l even 6 1
5184.2.a.bf 1 72.j odd 6 1

Hecke kernels

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

\( T_{5}^{2} + 4 T_{5} + 16 \)
\( T_{7}^{2} + 2 T_{7} + 4 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( \)
$3$ \( 1 + 3 T + 3 T^{2} \)
$5$ \( 1 + 4 T + 11 T^{2} + 20 T^{3} + 25 T^{4} \)
$7$ \( 1 + 2 T - 3 T^{2} + 14 T^{3} + 49 T^{4} \)
$11$ \( 1 + 5 T + 14 T^{2} + 55 T^{3} + 121 T^{4} \)
$13$ \( ( 1 - 7 T + 13 T^{2} )( 1 + 5 T + 13 T^{2} ) \)
$17$ \( ( 1 + 3 T + 17 T^{2} )^{2} \)
$19$ \( ( 1 + T + 19 T^{2} )^{2} \)
$23$ \( 1 + 6 T + 13 T^{2} + 138 T^{3} + 529 T^{4} \)
$29$ \( 1 - 2 T - 25 T^{2} - 58 T^{3} + 841 T^{4} \)
$31$ \( ( 1 - 7 T + 31 T^{2} )( 1 + 11 T + 31 T^{2} ) \)
$37$ \( ( 1 + 8 T + 37 T^{2} )^{2} \)
$41$ \( 1 + T - 40 T^{2} + 41 T^{3} + 1681 T^{4} \)
$43$ \( 1 + 7 T + 6 T^{2} + 301 T^{3} + 1849 T^{4} \)
$47$ \( 1 - 2 T - 43 T^{2} - 94 T^{3} + 2209 T^{4} \)
$53$ \( ( 1 + 4 T + 53 T^{2} )^{2} \)
$59$ \( 1 - 5 T - 34 T^{2} - 295 T^{3} + 3481 T^{4} \)
$61$ \( 1 - 61 T^{2} + 3721 T^{4} \)
$67$ \( 1 + 13 T + 102 T^{2} + 871 T^{3} + 4489 T^{4} \)
$71$ \( ( 1 + 8 T + 71 T^{2} )^{2} \)
$73$ \( ( 1 - 3 T + 73 T^{2} )^{2} \)
$79$ \( 1 - 8 T - 15 T^{2} - 632 T^{3} + 6241 T^{4} \)
$83$ \( 1 - 12 T + 61 T^{2} - 996 T^{3} + 6889 T^{4} \)
$89$ \( ( 1 + 10 T + 89 T^{2} )^{2} \)
$97$ \( 1 - 11 T + 24 T^{2} - 1067 T^{3} + 9409 T^{4} \)
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