Properties

Label 288.2.i.b
Level $288$
Weight $2$
Character orbit 288.i
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.i (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.29969157821\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \(x^{2} - x + 1\)
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.b yes 2
3.b odd 2 1 864.2.i.b 2
4.b odd 2 1 288.2.i.a 2
8.b even 2 1 576.2.i.b 2
8.d odd 2 1 576.2.i.h 2
9.c even 3 1 inner 288.2.i.b yes 2
9.c even 3 1 2592.2.a.g 1
9.d odd 6 1 864.2.i.b 2
9.d odd 6 1 2592.2.a.a 1
12.b even 2 1 864.2.i.a 2
24.f even 2 1 1728.2.i.a 2
24.h odd 2 1 1728.2.i.b 2
36.f odd 6 1 288.2.i.a 2
36.f odd 6 1 2592.2.a.h 1
36.h even 6 1 864.2.i.a 2
36.h even 6 1 2592.2.a.b 1
72.j odd 6 1 1728.2.i.b 2
72.j odd 6 1 5184.2.a.be 1
72.l even 6 1 1728.2.i.a 2
72.l even 6 1 5184.2.a.bf 1
72.n even 6 1 576.2.i.b 2
72.n even 6 1 5184.2.a.a 1
72.p odd 6 1 576.2.i.h 2
72.p odd 6 1 5184.2.a.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
288.2.i.a 2 4.b odd 2 1
288.2.i.a 2 36.f odd 6 1
288.2.i.b yes 2 1.a even 1 1 trivial
288.2.i.b yes 2 9.c even 3 1 inner
576.2.i.b 2 8.b even 2 1
576.2.i.b 2 72.n even 6 1
576.2.i.h 2 8.d odd 2 1
576.2.i.h 2 72.p odd 6 1
864.2.i.a 2 12.b even 2 1
864.2.i.a 2 36.h even 6 1
864.2.i.b 2 3.b odd 2 1
864.2.i.b 2 9.d odd 6 1
1728.2.i.a 2 24.f even 2 1
1728.2.i.a 2 72.l even 6 1
1728.2.i.b 2 24.h odd 2 1
1728.2.i.b 2 72.j odd 6 1
2592.2.a.a 1 9.d odd 6 1
2592.2.a.b 1 36.h even 6 1
2592.2.a.g 1 9.c even 3 1
2592.2.a.h 1 36.f odd 6 1
5184.2.a.a 1 72.n even 6 1
5184.2.a.b 1 72.p odd 6 1
5184.2.a.be 1 72.j odd 6 1
5184.2.a.bf 1 72.l even 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

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 3 - 3 T + T^{2} \)
$5$ \( 16 + 4 T + T^{2} \)
$7$ \( 4 - 2 T + T^{2} \)
$11$ \( 25 - 5 T + T^{2} \)
$13$ \( 4 - 2 T + T^{2} \)
$17$ \( ( 3 + T )^{2} \)
$19$ \( ( -1 + T )^{2} \)
$23$ \( 36 - 6 T + T^{2} \)
$29$ \( 4 - 2 T + T^{2} \)
$31$ \( 16 - 4 T + T^{2} \)
$37$ \( ( 8 + T )^{2} \)
$41$ \( 1 + T + T^{2} \)
$43$ \( 49 - 7 T + T^{2} \)
$47$ \( 4 + 2 T + T^{2} \)
$53$ \( ( 4 + T )^{2} \)
$59$ \( 25 + 5 T + T^{2} \)
$61$ \( T^{2} \)
$67$ \( 169 - 13 T + T^{2} \)
$71$ \( ( -8 + T )^{2} \)
$73$ \( ( -3 + T )^{2} \)
$79$ \( 64 + 8 T + T^{2} \)
$83$ \( 144 + 12 T + T^{2} \)
$89$ \( ( 10 + T )^{2} \)
$97$ \( 121 - 11 T + T^{2} \)
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