Properties

Label 3024.1.bw.a
Level $3024$
Weight $1$
Character orbit 3024.bw
Analytic conductor $1.509$
Analytic rank $0$
Dimension $4$
Projective image $A_{4}$
CM/RM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 3024 = 2^{4} \cdot 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3024.bw (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.50917259820\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1008)
Projective image \(A_{4}\)
Projective field Galois closure of 4.0.63504.1

$q$-expansion

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q -\zeta_{12}^{2} q^{5} -\zeta_{12}^{3} q^{7} +O(q^{10})\) \( q -\zeta_{12}^{2} q^{5} -\zeta_{12}^{3} q^{7} + \zeta_{12} q^{11} -\zeta_{12}^{4} q^{13} + \zeta_{12}^{2} q^{17} + \zeta_{12} q^{19} + \zeta_{12}^{5} q^{23} -\zeta_{12}^{2} q^{29} + \zeta_{12}^{5} q^{35} -\zeta_{12}^{4} q^{37} -\zeta_{12}^{4} q^{41} + \zeta_{12}^{5} q^{43} - q^{49} -\zeta_{12}^{2} q^{53} -\zeta_{12}^{3} q^{55} -2 \zeta_{12}^{3} q^{59} - q^{65} + 2 \zeta_{12}^{3} q^{71} -\zeta_{12}^{2} q^{73} -\zeta_{12}^{4} q^{77} -\zeta_{12}^{5} q^{83} -\zeta_{12}^{4} q^{85} -\zeta_{12}^{4} q^{89} -\zeta_{12} q^{91} -\zeta_{12}^{3} q^{95} + \zeta_{12}^{2} q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 2q^{5} + O(q^{10}) \) \( 4q - 2q^{5} + 2q^{13} + 2q^{17} - 2q^{29} + 2q^{37} + 2q^{41} - 4q^{49} - 2q^{53} - 4q^{65} - 2q^{73} + 2q^{77} + 2q^{85} + 2q^{89} + 2q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3024\mathbb{Z}\right)^\times\).

\(n\) \(757\) \(785\) \(1135\) \(2593\)
\(\chi(n)\) \(1\) \(\zeta_{12}^{4}\) \(-1\) \(\zeta_{12}^{4}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
415.1
0.866025 + 0.500000i
−0.866025 0.500000i
−0.866025 + 0.500000i
0.866025 0.500000i
0 0 0 −0.500000 0.866025i 0 1.00000i 0 0 0
415.2 0 0 0 −0.500000 0.866025i 0 1.00000i 0 0 0
991.1 0 0 0 −0.500000 + 0.866025i 0 1.00000i 0 0 0
991.2 0 0 0 −0.500000 + 0.866025i 0 1.00000i 0 0 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
63.h even 3 1 inner
252.u odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3024.1.bw.a 4
3.b odd 2 1 1008.1.bw.a 4
4.b odd 2 1 inner 3024.1.bw.a 4
7.c even 3 1 3024.1.dd.a 4
9.c even 3 1 3024.1.dd.a 4
9.d odd 6 1 1008.1.dd.a yes 4
12.b even 2 1 1008.1.bw.a 4
21.h odd 6 1 1008.1.dd.a yes 4
28.g odd 6 1 3024.1.dd.a 4
36.f odd 6 1 3024.1.dd.a 4
36.h even 6 1 1008.1.dd.a yes 4
63.h even 3 1 inner 3024.1.bw.a 4
63.j odd 6 1 1008.1.bw.a 4
84.n even 6 1 1008.1.dd.a yes 4
252.u odd 6 1 inner 3024.1.bw.a 4
252.bb even 6 1 1008.1.bw.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1008.1.bw.a 4 3.b odd 2 1
1008.1.bw.a 4 12.b even 2 1
1008.1.bw.a 4 63.j odd 6 1
1008.1.bw.a 4 252.bb even 6 1
1008.1.dd.a yes 4 9.d odd 6 1
1008.1.dd.a yes 4 21.h odd 6 1
1008.1.dd.a yes 4 36.h even 6 1
1008.1.dd.a yes 4 84.n even 6 1
3024.1.bw.a 4 1.a even 1 1 trivial
3024.1.bw.a 4 4.b odd 2 1 inner
3024.1.bw.a 4 63.h even 3 1 inner
3024.1.bw.a 4 252.u odd 6 1 inner
3024.1.dd.a 4 7.c even 3 1
3024.1.dd.a 4 9.c even 3 1
3024.1.dd.a 4 28.g odd 6 1
3024.1.dd.a 4 36.f odd 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(3024, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \)
$7$ \( ( 1 + T^{2} )^{2} \)
$11$ \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \)
$13$ \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \)
$17$ \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \)
$19$ \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \)
$23$ \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \)
$29$ \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \)
$31$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
$37$ \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \)
$41$ \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \)
$43$ \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \)
$47$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
$53$ \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \)
$59$ \( ( 1 + T^{2} )^{4} \)
$61$ \( ( 1 + T^{2} )^{4} \)
$67$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
$71$ \( ( 1 + T^{2} )^{4} \)
$73$ \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \)
$79$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
$83$ \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \)
$89$ \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \)
$97$ \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \)
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