Properties

Label 576.2.i.h
Level $576$
Weight $2$
Character orbit 576.i
Analytic conductor $4.599$
Analytic rank $0$
Dimension $2$
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.i (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.59938315643\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 288)
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.

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

Character values

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

\(n\) \(65\) \(127\) \(325\)
\(\chi(n)\) \(-\zeta_{6}\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
193.1
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
385.1 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 576.2.i.h 2
3.b odd 2 1 1728.2.i.a 2
4.b odd 2 1 576.2.i.b 2
8.b even 2 1 288.2.i.a 2
8.d odd 2 1 288.2.i.b yes 2
9.c even 3 1 inner 576.2.i.h 2
9.c even 3 1 5184.2.a.b 1
9.d odd 6 1 1728.2.i.a 2
9.d odd 6 1 5184.2.a.bf 1
12.b even 2 1 1728.2.i.b 2
24.f even 2 1 864.2.i.b 2
24.h odd 2 1 864.2.i.a 2
36.f odd 6 1 576.2.i.b 2
36.f odd 6 1 5184.2.a.a 1
36.h even 6 1 1728.2.i.b 2
36.h even 6 1 5184.2.a.be 1
72.j odd 6 1 864.2.i.a 2
72.j odd 6 1 2592.2.a.b 1
72.l even 6 1 864.2.i.b 2
72.l even 6 1 2592.2.a.a 1
72.n even 6 1 288.2.i.a 2
72.n even 6 1 2592.2.a.h 1
72.p odd 6 1 288.2.i.b yes 2
72.p odd 6 1 2592.2.a.g 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
288.2.i.a 2 8.b even 2 1
288.2.i.a 2 72.n even 6 1
288.2.i.b yes 2 8.d odd 2 1
288.2.i.b yes 2 72.p odd 6 1
576.2.i.b 2 4.b odd 2 1
576.2.i.b 2 36.f odd 6 1
576.2.i.h 2 1.a even 1 1 trivial
576.2.i.h 2 9.c even 3 1 inner
864.2.i.a 2 24.h odd 2 1
864.2.i.a 2 72.j odd 6 1
864.2.i.b 2 24.f even 2 1
864.2.i.b 2 72.l even 6 1
1728.2.i.a 2 3.b odd 2 1
1728.2.i.a 2 9.d odd 6 1
1728.2.i.b 2 12.b even 2 1
1728.2.i.b 2 36.h even 6 1
2592.2.a.a 1 72.l even 6 1
2592.2.a.b 1 72.j odd 6 1
2592.2.a.g 1 72.p odd 6 1
2592.2.a.h 1 72.n even 6 1
5184.2.a.a 1 36.f odd 6 1
5184.2.a.b 1 9.c even 3 1
5184.2.a.be 1 36.h even 6 1
5184.2.a.bf 1 9.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(576, [\chi])\):

\( T_{5}^{2} - 4T_{5} + 16 \) Copy content Toggle raw display
\( T_{7}^{2} + 2T_{7} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

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