Properties

Label 576.2.bb.a
Level $576$
Weight $2$
Character orbit 576.bb
Analytic conductor $4.599$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 576 = 2^{6} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 576.bb (of order \(12\), degree \(4\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.59938315643\)
Analytic rank: \(0\)
Dimension: \(4\)
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: \( 1 \)
Twist minimal: no (minimal twist has level 144)
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-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.

\(f(q)\) \(=\) \( q + (\zeta_{12}^{3} - 2 \zeta_{12}) q^{3} + (\zeta_{12}^{3} - \zeta_{12} - 1) q^{5} + (\zeta_{12}^{3} + 2 \zeta_{12}^{2} - \zeta_{12} - 4) q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\zeta_{12}^{3} - 2 \zeta_{12}) q^{3} + (\zeta_{12}^{3} - \zeta_{12} - 1) q^{5} + (\zeta_{12}^{3} + 2 \zeta_{12}^{2} - \zeta_{12} - 4) q^{7} + 3 q^{9} + ( - \zeta_{12}^{3} - \zeta_{12}^{2} + 1) q^{11} + (\zeta_{12}^{3} - 2 \zeta_{12}^{2} - 3 \zeta_{12} + 3) q^{13} + ( - \zeta_{12}^{3} - \zeta_{12}^{2} + 2 \zeta_{12} + 2) q^{15} + 4 q^{17} + ( - 3 \zeta_{12}^{3} + 3) q^{19} + ( - 6 \zeta_{12}^{3} - \zeta_{12}^{2} + 6 \zeta_{12} + 2) q^{21} + (2 \zeta_{12}^{2} + 3 \zeta_{12} + 2) q^{23} + (3 \zeta_{12}^{3} - \zeta_{12}^{2} - 3 \zeta_{12} + 2) q^{25} + (3 \zeta_{12}^{3} - 6 \zeta_{12}) q^{27} + (2 \zeta_{12}^{3} - 2 \zeta_{12}^{2} - \zeta_{12} + 1) q^{29} + ( - 6 \zeta_{12}^{3} - 4 \zeta_{12}^{2} + 3 \zeta_{12} + 4) q^{31} + (2 \zeta_{12}^{3} + 2 \zeta_{12}^{2} - \zeta_{12} - 1) q^{33} + ( - 5 \zeta_{12}^{3} - 3 \zeta_{12}^{2} + 3 \zeta_{12} + 5) q^{35} + ( - 7 \zeta_{12}^{3} + 2 \zeta_{12}^{2} + 2 \zeta_{12} - 7) q^{37} + (5 \zeta_{12}^{3} + \zeta_{12}^{2} - 4 \zeta_{12} + 4) q^{39} + ( - 3 \zeta_{12}^{2} + 6 \zeta_{12} - 3) q^{41} + ( - 5 \zeta_{12}^{3} - 5 \zeta_{12}^{2} + 6 \zeta_{12} - 1) q^{43} + (3 \zeta_{12}^{3} - 3 \zeta_{12} - 3) q^{45} + (3 \zeta_{12}^{3} - 4 \zeta_{12}^{2} + 3 \zeta_{12}) q^{47} + ( - 8 \zeta_{12}^{3} - 6 \zeta_{12}^{2} + 4 \zeta_{12} + 6) q^{49} + (4 \zeta_{12}^{3} - 8 \zeta_{12}) q^{51} + (3 \zeta_{12}^{3} + 2 \zeta_{12}^{2} + 2 \zeta_{12} + 3) q^{53} + (2 \zeta_{12}^{3} + 2 \zeta_{12}^{2} - 1) q^{55} + (3 \zeta_{12}^{3} + 6 \zeta_{12}^{2} - 6 \zeta_{12} - 3) q^{57} + ( - 3 \zeta_{12}^{3} + 3 \zeta_{12}^{2}) q^{59} + ( - 8 \zeta_{12}^{3} + 8 \zeta_{12}^{2} + \zeta_{12} - 7) q^{61} + (3 \zeta_{12}^{3} + 6 \zeta_{12}^{2} - 3 \zeta_{12} - 12) q^{63} + (2 \zeta_{12}^{3} + \zeta_{12}^{2} + 2 \zeta_{12}) q^{65} + (3 \zeta_{12}^{3} + 5 \zeta_{12}^{2} + 2 \zeta_{12} - 2) q^{67} + ( - 3 \zeta_{12}^{2} - 6 \zeta_{12} - 3) q^{69} + (4 \zeta_{12}^{3} - 8 \zeta_{12}^{2} + 4) q^{71} + ( - 4 \zeta_{12}^{3} - 4 \zeta_{12}^{2} + 2) q^{73} + (3 \zeta_{12}^{3} - 3 \zeta_{12}^{2} - 3 \zeta_{12} + 6) q^{75} + (3 \zeta_{12}^{3} + 5 \zeta_{12}^{2} + 2 \zeta_{12} - 2) q^{77} + (\zeta_{12}^{3} + \zeta_{12}) q^{79} + 9 q^{81} + ( - 9 \zeta_{12}^{3} + 9 \zeta_{12}^{2} + 10 \zeta_{12} + 1) q^{83} + (4 \zeta_{12}^{3} - 4 \zeta_{12} - 4) q^{85} + (3 \zeta_{12}^{3} - 3 \zeta_{12}^{2} + 3) q^{87} + ( - 2 \zeta_{12}^{3} - 16 \zeta_{12}^{2} + 8) q^{89} + ( - 5 \zeta_{12}^{3} + 9 \zeta_{12}^{2} + 9 \zeta_{12} - 5) q^{91} + (8 \zeta_{12}^{3} + 9 \zeta_{12}^{2} - 4 \zeta_{12} - 9) q^{93} + (6 \zeta_{12}^{3} + 3 \zeta_{12}^{2} - 3 \zeta_{12} - 3) q^{95} - \zeta_{12}^{2} q^{97} + ( - 3 \zeta_{12}^{3} - 3 \zeta_{12}^{2} + 3) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{5} - 12 q^{7} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{5} - 12 q^{7} + 12 q^{9} + 2 q^{11} + 8 q^{13} + 6 q^{15} + 16 q^{17} + 12 q^{19} + 6 q^{21} + 12 q^{23} + 6 q^{25} + 8 q^{31} + 14 q^{35} - 24 q^{37} + 18 q^{39} - 18 q^{41} - 14 q^{43} - 12 q^{45} - 8 q^{47} + 12 q^{49} + 16 q^{53} + 6 q^{59} - 12 q^{61} - 36 q^{63} + 2 q^{65} + 2 q^{67} - 18 q^{69} + 18 q^{75} + 2 q^{77} + 36 q^{81} + 22 q^{83} - 16 q^{85} + 6 q^{87} - 2 q^{91} - 18 q^{93} - 6 q^{95} - 2 q^{97} + 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/576\mathbb{Z}\right)^\times\).

\(n\) \(65\) \(127\) \(325\)
\(\chi(n)\) \(-1 + \zeta_{12}^{2}\) \(1\) \(\zeta_{12}^{3}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1
0.866025 + 0.500000i
−0.866025 + 0.500000i
−0.866025 0.500000i
0.866025 0.500000i
0 −1.73205 0 −1.86603 + 0.500000i 0 −3.86603 + 2.23205i 0 3.00000 0
241.1 0 1.73205 0 −0.133975 + 0.500000i 0 −2.13397 1.23205i 0 3.00000 0
337.1 0 1.73205 0 −0.133975 0.500000i 0 −2.13397 + 1.23205i 0 3.00000 0
529.1 0 −1.73205 0 −1.86603 0.500000i 0 −3.86603 2.23205i 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
144.x even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 576.2.bb.a 4
3.b odd 2 1 1728.2.bc.c 4
4.b odd 2 1 144.2.x.a 4
9.c even 3 1 576.2.bb.b 4
9.d odd 6 1 1728.2.bc.b 4
12.b even 2 1 432.2.y.d 4
16.e even 4 1 576.2.bb.b 4
16.f odd 4 1 144.2.x.d yes 4
36.f odd 6 1 144.2.x.d yes 4
36.h even 6 1 432.2.y.a 4
48.i odd 4 1 1728.2.bc.b 4
48.k even 4 1 432.2.y.a 4
144.u even 12 1 432.2.y.d 4
144.v odd 12 1 144.2.x.a 4
144.w odd 12 1 1728.2.bc.c 4
144.x even 12 1 inner 576.2.bb.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
144.2.x.a 4 4.b odd 2 1
144.2.x.a 4 144.v odd 12 1
144.2.x.d yes 4 16.f odd 4 1
144.2.x.d yes 4 36.f odd 6 1
432.2.y.a 4 36.h even 6 1
432.2.y.a 4 48.k even 4 1
432.2.y.d 4 12.b even 2 1
432.2.y.d 4 144.u even 12 1
576.2.bb.a 4 1.a even 1 1 trivial
576.2.bb.a 4 144.x even 12 1 inner
576.2.bb.b 4 9.c even 3 1
576.2.bb.b 4 16.e even 4 1
1728.2.bc.b 4 9.d odd 6 1
1728.2.bc.b 4 48.i odd 4 1
1728.2.bc.c 4 3.b odd 2 1
1728.2.bc.c 4 144.w odd 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} + 4T_{5}^{3} + 5T_{5}^{2} + 2T_{5} + 1 \) acting on \(S_{2}^{\mathrm{new}}(576, [\chi])\). 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
$5$ \( T^{4} + 4 T^{3} + 5 T^{2} + 2 T + 1 \) Copy content Toggle raw display
$7$ \( T^{4} + 12 T^{3} + 59 T^{2} + \cdots + 121 \) Copy content Toggle raw display
$11$ \( T^{4} - 2 T^{3} + 5 T^{2} - 4 T + 1 \) Copy content Toggle raw display
$13$ \( T^{4} - 8 T^{3} + 17 T^{2} - 22 T + 121 \) Copy content Toggle raw display
$17$ \( (T - 4)^{4} \) Copy content Toggle raw display
$19$ \( (T^{2} - 6 T + 18)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} - 12 T^{3} + 51 T^{2} - 36 T + 9 \) Copy content Toggle raw display
$29$ \( T^{4} + 9 T^{2} + 18 T + 9 \) Copy content Toggle raw display
$31$ \( T^{4} - 8 T^{3} + 75 T^{2} + 88 T + 121 \) Copy content Toggle raw display
$37$ \( T^{4} + 24 T^{3} + 288 T^{2} + \cdots + 4356 \) Copy content Toggle raw display
$41$ \( T^{4} + 18 T^{3} + 99 T^{2} - 162 T + 81 \) Copy content Toggle raw display
$43$ \( T^{4} + 14 T^{3} + 65 T^{2} + \cdots + 3481 \) Copy content Toggle raw display
$47$ \( T^{4} + 8 T^{3} + 75 T^{2} - 88 T + 121 \) Copy content Toggle raw display
$53$ \( T^{4} - 16 T^{3} + 128 T^{2} + \cdots + 676 \) Copy content Toggle raw display
$59$ \( T^{4} - 6 T^{3} + 45 T^{2} - 108 T + 81 \) Copy content Toggle raw display
$61$ \( T^{4} + 12 T^{3} + 261 T^{2} + \cdots + 1089 \) Copy content Toggle raw display
$67$ \( T^{4} - 2 T^{3} + 65 T^{2} + 176 T + 121 \) Copy content Toggle raw display
$71$ \( T^{4} + 128T^{2} + 1024 \) Copy content Toggle raw display
$73$ \( T^{4} + 56T^{2} + 16 \) Copy content Toggle raw display
$79$ \( T^{4} + 3T^{2} + 9 \) Copy content Toggle raw display
$83$ \( T^{4} - 22 T^{3} + 185 T^{2} + \cdots + 32041 \) Copy content Toggle raw display
$89$ \( T^{4} + 392 T^{2} + 35344 \) Copy content Toggle raw display
$97$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
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