# Properties

 Label 3024.1.eg.a Level $3024$ Weight $1$ Character orbit 3024.eg Analytic conductor $1.509$ Analytic rank $0$ Dimension $8$ Projective image $S_{4}$ CM/RM no Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3024 = 2^{4} \cdot 3^{3} \cdot 7$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 3024.eg (of order $$12$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.50917259820$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$2$$ over $$\Q(\zeta_{12})$$ Coefficient field: $$\Q(\zeta_{24})$$ Defining polynomial: $$x^{8} - x^{4} + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image $$S_{4}$$ Projective field Galois closure of 4.2.2709504.12

## $q$-expansion

The $$q$$-expansion and trace form are shown below.

 $$f(q)$$ $$=$$ $$q -\zeta_{24} q^{2} + \zeta_{24}^{2} q^{4} -\zeta_{24}^{7} q^{5} -\zeta_{24}^{6} q^{7} -\zeta_{24}^{3} q^{8} +O(q^{10})$$ $$q -\zeta_{24} q^{2} + \zeta_{24}^{2} q^{4} -\zeta_{24}^{7} q^{5} -\zeta_{24}^{6} q^{7} -\zeta_{24}^{3} q^{8} + \zeta_{24}^{8} q^{10} + \zeta_{24}^{7} q^{14} + \zeta_{24}^{4} q^{16} + ( \zeta_{24}^{4} + \zeta_{24}^{10} ) q^{19} -\zeta_{24}^{9} q^{20} + ( -\zeta_{24} - \zeta_{24}^{7} ) q^{23} -\zeta_{24}^{8} q^{28} -\zeta_{24}^{9} q^{29} -2 \zeta_{24}^{8} q^{31} -\zeta_{24}^{5} q^{32} -\zeta_{24} q^{35} + ( -\zeta_{24}^{5} - \zeta_{24}^{11} ) q^{38} + \zeta_{24}^{10} q^{40} + ( \zeta_{24}^{3} - \zeta_{24}^{9} ) q^{41} + ( 1 - \zeta_{24}^{6} ) q^{43} + ( \zeta_{24}^{2} + \zeta_{24}^{8} ) q^{46} + ( -\zeta_{24} + \zeta_{24}^{7} ) q^{47} - q^{49} + \zeta_{24}^{11} q^{53} + \zeta_{24}^{9} q^{56} + \zeta_{24}^{10} q^{58} -\zeta_{24}^{11} q^{59} + 2 \zeta_{24}^{9} q^{62} + \zeta_{24}^{6} q^{64} + ( -\zeta_{24}^{2} + \zeta_{24}^{8} ) q^{67} + \zeta_{24}^{2} q^{70} + ( -\zeta_{24}^{3} + \zeta_{24}^{9} ) q^{71} -\zeta_{24}^{2} q^{73} + ( -1 + \zeta_{24}^{6} ) q^{76} -\zeta_{24}^{4} q^{79} -\zeta_{24}^{11} q^{80} + ( -\zeta_{24}^{4} + \zeta_{24}^{10} ) q^{82} + ( -\zeta_{24} + \zeta_{24}^{7} ) q^{86} + ( -\zeta_{24}^{3} - \zeta_{24}^{9} ) q^{92} + ( \zeta_{24}^{2} - \zeta_{24}^{8} ) q^{94} + ( \zeta_{24}^{5} - \zeta_{24}^{11} ) q^{95} + \zeta_{24} q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + O(q^{10})$$ $$8q - 4q^{10} + 4q^{16} + 4q^{19} + 4q^{28} + 8q^{31} + 8q^{43} - 4q^{46} - 8q^{49} - 4q^{67} - 8q^{76} - 4q^{79} - 4q^{82} + 4q^{94} + O(q^{100})$$

## Character values

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

 $$n$$ $$757$$ $$785$$ $$1135$$ $$2593$$ $$\chi(n)$$ $$-\zeta_{24}^{6}$$ $$-1$$ $$1$$ $$-\zeta_{24}^{4}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
53.1
 0.258819 − 0.965926i −0.258819 + 0.965926i 0.965926 − 0.258819i −0.965926 + 0.258819i 0.965926 + 0.258819i −0.965926 − 0.258819i 0.258819 + 0.965926i −0.258819 − 0.965926i
−0.258819 + 0.965926i 0 −0.866025 0.500000i 0.965926 + 0.258819i 0 1.00000i 0.707107 0.707107i 0 −0.500000 + 0.866025i
53.2 0.258819 0.965926i 0 −0.866025 0.500000i −0.965926 0.258819i 0 1.00000i −0.707107 + 0.707107i 0 −0.500000 + 0.866025i
485.1 −0.965926 + 0.258819i 0 0.866025 0.500000i 0.258819 + 0.965926i 0 1.00000i −0.707107 + 0.707107i 0 −0.500000 0.866025i
485.2 0.965926 0.258819i 0 0.866025 0.500000i −0.258819 0.965926i 0 1.00000i 0.707107 0.707107i 0 −0.500000 0.866025i
1565.1 −0.965926 0.258819i 0 0.866025 + 0.500000i 0.258819 0.965926i 0 1.00000i −0.707107 0.707107i 0 −0.500000 + 0.866025i
1565.2 0.965926 + 0.258819i 0 0.866025 + 0.500000i −0.258819 + 0.965926i 0 1.00000i 0.707107 + 0.707107i 0 −0.500000 + 0.866025i
1997.1 −0.258819 0.965926i 0 −0.866025 + 0.500000i 0.965926 0.258819i 0 1.00000i 0.707107 + 0.707107i 0 −0.500000 0.866025i
1997.2 0.258819 + 0.965926i 0 −0.866025 + 0.500000i −0.965926 + 0.258819i 0 1.00000i −0.707107 0.707107i 0 −0.500000 0.866025i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1997.2 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.c even 3 1 inner
16.e even 4 1 inner
21.h odd 6 1 inner
48.i odd 4 1 inner
112.w even 12 1 inner
336.bt odd 12 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3024.1.eg.a 8
3.b odd 2 1 inner 3024.1.eg.a 8
7.c even 3 1 inner 3024.1.eg.a 8
16.e even 4 1 inner 3024.1.eg.a 8
21.h odd 6 1 inner 3024.1.eg.a 8
48.i odd 4 1 inner 3024.1.eg.a 8
112.w even 12 1 inner 3024.1.eg.a 8
336.bt odd 12 1 inner 3024.1.eg.a 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3024.1.eg.a 8 1.a even 1 1 trivial
3024.1.eg.a 8 3.b odd 2 1 inner
3024.1.eg.a 8 7.c even 3 1 inner
3024.1.eg.a 8 16.e even 4 1 inner
3024.1.eg.a 8 21.h odd 6 1 inner
3024.1.eg.a 8 48.i odd 4 1 inner
3024.1.eg.a 8 112.w even 12 1 inner
3024.1.eg.a 8 336.bt odd 12 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{8} - T_{5}^{4} + 1$$ acting on $$S_{1}^{\mathrm{new}}(3024, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T^{4} + T^{8}$$
$3$ 1
$5$ $$( 1 + T^{4} )^{2}( 1 - T^{4} + T^{8} )$$
$7$ $$( 1 + T^{2} )^{4}$$
$11$ $$( 1 - T^{4} + T^{8} )^{2}$$
$13$ $$( 1 + T^{4} )^{4}$$
$17$ $$( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4}$$
$19$ $$( 1 - T + T^{2} )^{4}( 1 - T^{2} + T^{4} )^{2}$$
$23$ $$( 1 - T^{4} + T^{8} )^{2}$$
$29$ $$( 1 - T^{4} + T^{8} )^{2}$$
$31$ $$( 1 - T + T^{2} )^{8}$$
$37$ $$( 1 - T^{4} + T^{8} )^{2}$$
$41$ $$( 1 + T^{4} )^{4}$$
$43$ $$( 1 - T )^{8}( 1 + T^{2} )^{4}$$
$47$ $$( 1 - T^{4} + T^{8} )^{2}$$
$53$ $$( 1 + T^{4} )^{2}( 1 - T^{4} + T^{8} )$$
$59$ $$( 1 + T^{4} )^{2}( 1 - T^{4} + T^{8} )$$
$61$ $$( 1 - T^{4} + T^{8} )^{2}$$
$67$ $$( 1 + T + T^{2} )^{4}( 1 - T^{2} + T^{4} )^{2}$$
$71$ $$( 1 + T^{4} )^{4}$$
$73$ $$( 1 + T^{2} )^{4}( 1 - T^{2} + T^{4} )^{2}$$
$79$ $$( 1 + T )^{8}( 1 - T + T^{2} )^{4}$$
$83$ $$( 1 + T^{4} )^{4}$$
$89$ $$( 1 - T^{2} + T^{4} )^{4}$$
$97$ $$( 1 + T^{2} )^{8}$$