Properties

Label 288.2.i.d
Level 288
Weight 2
Character orbit 288.i
Analytic conductor 2.300
Analytic rank 0
Dimension 4
CM no
Inner twists 4

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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: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\zeta_{12})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3 \)
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 \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -2 \zeta_{12} + \zeta_{12}^{3} ) q^{3} + ( 1 - \zeta_{12}^{2} ) q^{5} + ( \zeta_{12} + \zeta_{12}^{3} ) q^{7} + 3 q^{9} +O(q^{10})\) \( q + ( -2 \zeta_{12} + \zeta_{12}^{3} ) q^{3} + ( 1 - \zeta_{12}^{2} ) q^{5} + ( \zeta_{12} + \zeta_{12}^{3} ) q^{7} + 3 q^{9} + ( -\zeta_{12} - \zeta_{12}^{3} ) q^{11} + ( 3 - 3 \zeta_{12}^{2} ) q^{13} + ( -\zeta_{12} + 2 \zeta_{12}^{3} ) q^{15} + 4 q^{17} + ( 8 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{19} -3 \zeta_{12}^{2} q^{21} + ( 5 \zeta_{12} - 10 \zeta_{12}^{3} ) q^{23} + 4 \zeta_{12}^{2} q^{25} + ( -6 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{27} -\zeta_{12}^{2} q^{29} + ( -3 \zeta_{12} + 6 \zeta_{12}^{3} ) q^{31} + 3 \zeta_{12}^{2} q^{33} + ( 2 \zeta_{12} - \zeta_{12}^{3} ) q^{35} -8 q^{37} + ( -3 \zeta_{12} + 6 \zeta_{12}^{3} ) q^{39} + ( -5 + 5 \zeta_{12}^{2} ) q^{41} + ( 5 \zeta_{12} + 5 \zeta_{12}^{3} ) q^{43} + ( 3 - 3 \zeta_{12}^{2} ) q^{45} + ( -7 \zeta_{12} - 7 \zeta_{12}^{3} ) q^{47} + ( 4 - 4 \zeta_{12}^{2} ) q^{49} + ( -8 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{51} -8 q^{53} + ( -2 \zeta_{12} + \zeta_{12}^{3} ) q^{55} -12 q^{57} + ( -\zeta_{12} + 2 \zeta_{12}^{3} ) q^{59} + 7 \zeta_{12}^{2} q^{61} + ( 3 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{63} -3 \zeta_{12}^{2} q^{65} + ( 5 \zeta_{12} - 10 \zeta_{12}^{3} ) q^{67} + ( -15 + 15 \zeta_{12}^{2} ) q^{69} + ( 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{71} -12 q^{73} + ( -4 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{75} + ( 3 - 3 \zeta_{12}^{2} ) q^{77} + ( 3 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{79} + 9 q^{81} + ( 5 \zeta_{12} + 5 \zeta_{12}^{3} ) q^{83} + ( 4 - 4 \zeta_{12}^{2} ) q^{85} + ( \zeta_{12} + \zeta_{12}^{3} ) q^{87} -4 q^{89} + ( 6 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{91} + ( 9 - 9 \zeta_{12}^{2} ) q^{93} + ( 4 \zeta_{12} - 8 \zeta_{12}^{3} ) q^{95} + 3 \zeta_{12}^{2} q^{97} + ( -3 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 2q^{5} + 12q^{9} + O(q^{10}) \) \( 4q + 2q^{5} + 12q^{9} + 6q^{13} + 16q^{17} - 6q^{21} + 8q^{25} - 2q^{29} + 6q^{33} - 32q^{37} - 10q^{41} + 6q^{45} + 8q^{49} - 32q^{53} - 48q^{57} + 14q^{61} - 6q^{65} - 30q^{69} - 48q^{73} + 6q^{77} + 36q^{81} + 8q^{85} - 16q^{89} + 18q^{93} + 6q^{97} + O(q^{100}) \)

Character values

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

\(n\) \(37\) \(65\) \(127\)
\(\chi(n)\) \(1\) \(-1 + \zeta_{12}\) \(1\)

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
0.866025 0.500000i
−0.866025 + 0.500000i
0.866025 + 0.500000i
−0.866025 0.500000i
0 −1.73205 0 0.500000 + 0.866025i 0 0.866025 1.50000i 0 3.00000 0
97.2 0 1.73205 0 0.500000 + 0.866025i 0 −0.866025 + 1.50000i 0 3.00000 0
193.1 0 −1.73205 0 0.500000 0.866025i 0 0.866025 + 1.50000i 0 3.00000 0
193.2 0 1.73205 0 0.500000 0.866025i 0 −0.866025 1.50000i 0 3.00000 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 inner
9.c even 3 1 inner
36.f odd 6 1 inner

Twists

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

Hecke kernels

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

\( T_{5}^{2} - T_{5} + 1 \)
\( T_{7}^{4} + 3 T_{7}^{2} + 9 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( ( 1 - 3 T^{2} )^{2} \)
$5$ \( ( 1 - T - 4 T^{2} - 5 T^{3} + 25 T^{4} )^{2} \)
$7$ \( ( 1 - 13 T^{2} + 49 T^{4} )( 1 + 2 T^{2} + 49 T^{4} ) \)
$11$ \( 1 - 19 T^{2} + 240 T^{4} - 2299 T^{6} + 14641 T^{8} \)
$13$ \( ( 1 - 3 T - 4 T^{2} - 39 T^{3} + 169 T^{4} )^{2} \)
$17$ \( ( 1 - 4 T + 17 T^{2} )^{4} \)
$19$ \( ( 1 - 10 T^{2} + 361 T^{4} )^{2} \)
$23$ \( 1 + 29 T^{2} + 312 T^{4} + 15341 T^{6} + 279841 T^{8} \)
$29$ \( ( 1 + T - 28 T^{2} + 29 T^{3} + 841 T^{4} )^{2} \)
$31$ \( 1 - 35 T^{2} + 264 T^{4} - 33635 T^{6} + 923521 T^{8} \)
$37$ \( ( 1 + 8 T + 37 T^{2} )^{4} \)
$41$ \( ( 1 + 5 T - 16 T^{2} + 205 T^{3} + 1681 T^{4} )^{2} \)
$43$ \( 1 - 11 T^{2} - 1728 T^{4} - 20339 T^{6} + 3418801 T^{8} \)
$47$ \( 1 + 53 T^{2} + 600 T^{4} + 117077 T^{6} + 4879681 T^{8} \)
$53$ \( ( 1 + 8 T + 53 T^{2} )^{4} \)
$59$ \( 1 - 115 T^{2} + 9744 T^{4} - 400315 T^{6} + 12117361 T^{8} \)
$61$ \( ( 1 - 7 T - 12 T^{2} - 427 T^{3} + 3721 T^{4} )^{2} \)
$67$ \( 1 - 59 T^{2} - 1008 T^{4} - 264851 T^{6} + 20151121 T^{8} \)
$71$ \( ( 1 + 130 T^{2} + 5041 T^{4} )^{2} \)
$73$ \( ( 1 + 12 T + 73 T^{2} )^{4} \)
$79$ \( ( 1 - 142 T^{2} + 6241 T^{4} )( 1 + 11 T^{2} + 6241 T^{4} ) \)
$83$ \( 1 - 91 T^{2} + 1392 T^{4} - 626899 T^{6} + 47458321 T^{8} \)
$89$ \( ( 1 + 4 T + 89 T^{2} )^{4} \)
$97$ \( ( 1 - 3 T - 88 T^{2} - 291 T^{3} + 9409 T^{4} )^{2} \)
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