Properties

Label 3024.1.f.b
Level 3024
Weight 1
Character orbit 3024.f
Analytic conductor 1.509
Analytic rank 0
Dimension 2
Projective image \(D_{6}\)
RM discriminant 21
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.f (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.50917259820\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 756)
Projective image \(D_{6}\)
Projective field Galois closure of 6.0.4000752.1

$q$-expansion

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

\(f(q)\) \(=\) \( q + ( -\zeta_{6} - \zeta_{6}^{2} ) q^{5} + q^{7} +O(q^{10})\) \( q + ( -\zeta_{6} - \zeta_{6}^{2} ) q^{5} + q^{7} + ( -\zeta_{6} - \zeta_{6}^{2} ) q^{17} + ( -1 - \zeta_{6} + \zeta_{6}^{2} ) q^{25} + ( -\zeta_{6} - \zeta_{6}^{2} ) q^{35} + q^{37} + ( \zeta_{6} + \zeta_{6}^{2} ) q^{41} - q^{43} + ( \zeta_{6} + \zeta_{6}^{2} ) q^{47} + q^{49} + ( -\zeta_{6} - \zeta_{6}^{2} ) q^{59} -2 q^{67} + q^{79} + ( -\zeta_{6} - \zeta_{6}^{2} ) q^{83} + ( -2 - \zeta_{6} + \zeta_{6}^{2} ) q^{85} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{7} + O(q^{10}) \) \( 2q + 2q^{7} - 4q^{25} + 2q^{37} - 2q^{43} + 2q^{49} - 4q^{67} + 2q^{79} - 6q^{85} + O(q^{100}) \)

Character values

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

\(n\) \(757\) \(785\) \(1135\) \(2593\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
433.1
0.500000 + 0.866025i
0.500000 0.866025i
0 0 0 1.73205i 0 1.00000 0 0 0
433.2 0 0 0 1.73205i 0 1.00000 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
21.c even 2 1 RM by \(\Q(\sqrt{21}) \)
3.b odd 2 1 inner
7.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3024.1.f.b 2
3.b odd 2 1 inner 3024.1.f.b 2
4.b odd 2 1 756.1.d.a 2
7.b odd 2 1 inner 3024.1.f.b 2
12.b even 2 1 756.1.d.a 2
21.c even 2 1 RM 3024.1.f.b 2
28.d even 2 1 756.1.d.a 2
36.f odd 6 1 2268.1.bc.a 2
36.f odd 6 1 2268.1.bc.d 2
36.h even 6 1 2268.1.bc.a 2
36.h even 6 1 2268.1.bc.d 2
84.h odd 2 1 756.1.d.a 2
252.s odd 6 1 2268.1.bc.a 2
252.s odd 6 1 2268.1.bc.d 2
252.bi even 6 1 2268.1.bc.a 2
252.bi even 6 1 2268.1.bc.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
756.1.d.a 2 4.b odd 2 1
756.1.d.a 2 12.b even 2 1
756.1.d.a 2 28.d even 2 1
756.1.d.a 2 84.h odd 2 1
2268.1.bc.a 2 36.f odd 6 1
2268.1.bc.a 2 36.h even 6 1
2268.1.bc.a 2 252.s odd 6 1
2268.1.bc.a 2 252.bi even 6 1
2268.1.bc.d 2 36.f odd 6 1
2268.1.bc.d 2 36.h even 6 1
2268.1.bc.d 2 252.s odd 6 1
2268.1.bc.d 2 252.bi even 6 1
3024.1.f.b 2 1.a even 1 1 trivial
3024.1.f.b 2 3.b odd 2 1 inner
3024.1.f.b 2 7.b odd 2 1 inner
3024.1.f.b 2 21.c even 2 1 RM

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} + 3 \) acting on \(S_{1}^{\mathrm{new}}(3024, [\chi])\).

Hecke characteristic polynomials

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