# 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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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