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 $4$

Related objects

Downloads

Learn more

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: \(\Q(\zeta_{12})\)
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
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 basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{3} q^{3} + ( - \beta_1 + 1) q^{5} + \beta_{2} q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{3} q^{3} + ( - \beta_1 + 1) q^{5} + \beta_{2} q^{7} + 3 q^{9} - \beta_{2} q^{11} + ( - 3 \beta_1 + 3) q^{13} + ( - \beta_{3} + \beta_{2}) q^{15} + 4 q^{17} + 4 \beta_{3} q^{19} - 3 \beta_1 q^{21} + (5 \beta_{3} - 5 \beta_{2}) q^{23} + 4 \beta_1 q^{25} - 3 \beta_{3} q^{27} - \beta_1 q^{29} + ( - 3 \beta_{3} + 3 \beta_{2}) q^{31} + 3 \beta_1 q^{33} + \beta_{3} q^{35} - 8 q^{37} + ( - 3 \beta_{3} + 3 \beta_{2}) q^{39} + (5 \beta_1 - 5) q^{41} + 5 \beta_{2} q^{43} + ( - 3 \beta_1 + 3) q^{45} - 7 \beta_{2} q^{47} + ( - 4 \beta_1 + 4) q^{49} - 4 \beta_{3} q^{51} - 8 q^{53} - \beta_{3} q^{55} - 12 q^{57} + ( - \beta_{3} + \beta_{2}) q^{59} + 7 \beta_1 q^{61} + 3 \beta_{2} q^{63} - 3 \beta_1 q^{65} + (5 \beta_{3} - 5 \beta_{2}) q^{67} + (15 \beta_1 - 15) q^{69} + 2 \beta_{3} q^{71} - 12 q^{73} - 4 \beta_{2} q^{75} + ( - 3 \beta_1 + 3) q^{77} + 3 \beta_{2} q^{79} + 9 q^{81} + 5 \beta_{2} q^{83} + ( - 4 \beta_1 + 4) q^{85} + \beta_{2} q^{87} - 4 q^{89} + 3 \beta_{3} q^{91} + ( - 9 \beta_1 + 9) q^{93} + (4 \beta_{3} - 4 \beta_{2}) q^{95} + 3 \beta_1 q^{97} - 3 \beta_{2} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{5} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{5} + 12 q^{9} + 6 q^{13} + 16 q^{17} - 6 q^{21} + 8 q^{25} - 2 q^{29} + 6 q^{33} - 32 q^{37} - 10 q^{41} + 6 q^{45} + 8 q^{49} - 32 q^{53} - 48 q^{57} + 14 q^{61} - 6 q^{65} - 30 q^{69} - 48 q^{73} + 6 q^{77} + 36 q^{81} + 8 q^{85} - 16 q^{89} + 18 q^{93} + 6 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{12}^{2} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{12}^{3} + \zeta_{12} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\zeta_{12}^{3} + 2\zeta_{12} \) Copy content Toggle raw display
\(\zeta_{12}\)\(=\) \( ( \beta_{3} + \beta_{2} ) / 3 \) Copy content Toggle raw display
\(\zeta_{12}^{2}\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\zeta_{12}^{3}\)\(=\) \( ( -\beta_{3} + 2\beta_{2} ) / 3 \) Copy content Toggle raw display

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
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 \(S_{2}^{\mathrm{new}}(288, [\chi])\):

\( T_{5}^{2} - T_{5} + 1 \) Copy content Toggle raw display
\( T_{7}^{4} + 3T_{7}^{2} + 9 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} - 3)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} + 3T^{2} + 9 \) Copy content Toggle raw display
$11$ \( T^{4} + 3T^{2} + 9 \) Copy content Toggle raw display
$13$ \( (T^{2} - 3 T + 9)^{2} \) Copy content Toggle raw display
$17$ \( (T - 4)^{4} \) Copy content Toggle raw display
$19$ \( (T^{2} - 48)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 75T^{2} + 5625 \) Copy content Toggle raw display
$29$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} + 27T^{2} + 729 \) Copy content Toggle raw display
$37$ \( (T + 8)^{4} \) Copy content Toggle raw display
$41$ \( (T^{2} + 5 T + 25)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + 75T^{2} + 5625 \) Copy content Toggle raw display
$47$ \( T^{4} + 147 T^{2} + 21609 \) Copy content Toggle raw display
$53$ \( (T + 8)^{4} \) Copy content Toggle raw display
$59$ \( T^{4} + 3T^{2} + 9 \) Copy content Toggle raw display
$61$ \( (T^{2} - 7 T + 49)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 75T^{2} + 5625 \) Copy content Toggle raw display
$71$ \( (T^{2} - 12)^{2} \) Copy content Toggle raw display
$73$ \( (T + 12)^{4} \) Copy content Toggle raw display
$79$ \( T^{4} + 27T^{2} + 729 \) Copy content Toggle raw display
$83$ \( T^{4} + 75T^{2} + 5625 \) Copy content Toggle raw display
$89$ \( (T + 4)^{4} \) Copy content Toggle raw display
$97$ \( (T^{2} - 3 T + 9)^{2} \) Copy content Toggle raw display
show more
show less