Properties

Label 3024.1.eg.b
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

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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 4.2.2709504.10

$q$-expansion

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

\(f(q)\) \(=\) \( q -\zeta_{24}^{7} q^{2} -\zeta_{24}^{2} q^{4} + \zeta_{24}^{10} q^{7} + \zeta_{24}^{9} q^{8} +O(q^{10})\) \( q -\zeta_{24}^{7} q^{2} -\zeta_{24}^{2} q^{4} + \zeta_{24}^{10} q^{7} + \zeta_{24}^{9} q^{8} -\zeta_{24}^{11} q^{11} + ( -1 - \zeta_{24}^{6} ) q^{13} + \zeta_{24}^{5} q^{14} + \zeta_{24}^{4} q^{16} + ( -\zeta_{24}^{4} - \zeta_{24}^{10} ) q^{19} -\zeta_{24}^{6} q^{22} + ( -\zeta_{24} - \zeta_{24}^{7} ) q^{23} + \zeta_{24}^{2} q^{25} + ( -\zeta_{24} + \zeta_{24}^{7} ) q^{26} + q^{28} + \zeta_{24}^{9} q^{29} -\zeta_{24}^{8} q^{31} -\zeta_{24}^{11} q^{32} + ( -\zeta_{24}^{4} + \zeta_{24}^{10} ) q^{37} + ( -\zeta_{24}^{5} + \zeta_{24}^{11} ) q^{38} + ( -\zeta_{24}^{3} + \zeta_{24}^{9} ) q^{41} -\zeta_{24} q^{44} + ( -\zeta_{24}^{2} + \zeta_{24}^{8} ) q^{46} -\zeta_{24}^{8} q^{49} -\zeta_{24}^{9} q^{50} + ( \zeta_{24}^{2} + \zeta_{24}^{8} ) q^{52} + 2 \zeta_{24}^{11} q^{53} -\zeta_{24}^{7} q^{56} + \zeta_{24}^{4} q^{58} + \zeta_{24}^{11} q^{59} + ( -\zeta_{24}^{4} - \zeta_{24}^{10} ) q^{61} -\zeta_{24}^{3} q^{62} -\zeta_{24}^{6} q^{64} + ( \zeta_{24}^{3} - \zeta_{24}^{9} ) q^{71} -\zeta_{24}^{2} q^{73} + ( \zeta_{24}^{5} + \zeta_{24}^{11} ) q^{74} + ( -1 + \zeta_{24}^{6} ) q^{76} + \zeta_{24}^{9} q^{77} -\zeta_{24}^{4} q^{79} + ( \zeta_{24}^{4} + \zeta_{24}^{10} ) q^{82} -\zeta_{24}^{9} q^{83} + \zeta_{24}^{8} q^{88} + ( \zeta_{24}^{4} - \zeta_{24}^{10} ) q^{91} + ( \zeta_{24}^{3} + \zeta_{24}^{9} ) q^{92} + q^{97} -\zeta_{24}^{3} q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + O(q^{10}) \) \( 8q - 8q^{13} + 4q^{16} - 4q^{19} + 8q^{28} + 4q^{31} - 4q^{37} - 4q^{46} + 4q^{49} - 4q^{52} + 4q^{58} - 4q^{61} - 8q^{76} - 4q^{79} + 4q^{82} - 4q^{88} + 4q^{91} + 8q^{97} + O(q^{100}) \)

Character values

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

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

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
−0.965926 0.258819i 0 0.866025 + 0.500000i 0 0 0.866025 0.500000i −0.707107 0.707107i 0 0
53.2 0.965926 + 0.258819i 0 0.866025 + 0.500000i 0 0 0.866025 0.500000i 0.707107 + 0.707107i 0 0
485.1 −0.258819 0.965926i 0 −0.866025 + 0.500000i 0 0 −0.866025 0.500000i 0.707107 + 0.707107i 0 0
485.2 0.258819 + 0.965926i 0 −0.866025 + 0.500000i 0 0 −0.866025 0.500000i −0.707107 0.707107i 0 0
1565.1 −0.258819 + 0.965926i 0 −0.866025 0.500000i 0 0 −0.866025 + 0.500000i 0.707107 0.707107i 0 0
1565.2 0.258819 0.965926i 0 −0.866025 0.500000i 0 0 −0.866025 + 0.500000i −0.707107 + 0.707107i 0 0
1997.1 −0.965926 + 0.258819i 0 0.866025 0.500000i 0 0 0.866025 + 0.500000i −0.707107 + 0.707107i 0 0
1997.2 0.965926 0.258819i 0 0.866025 0.500000i 0 0 0.866025 + 0.500000i 0.707107 0.707107i 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1997.2
Significant digits:
Format:

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

Hecke kernels

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

Hecke characteristic polynomials

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