Properties

Label 3024.2.b.e
Level 3024
Weight 2
Character orbit 3024.b
Analytic conductor 24.147
Analytic rank 0
Dimension 2
CM no
Inner twists 2

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(24.1467615712\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 1 - 2 \zeta_{6} ) q^{5} + ( -1 - 2 \zeta_{6} ) q^{7} +O(q^{10})\) \( q + ( 1 - 2 \zeta_{6} ) q^{5} + ( -1 - 2 \zeta_{6} ) q^{7} + ( 2 - 4 \zeta_{6} ) q^{11} + ( 2 - 4 \zeta_{6} ) q^{13} + ( -3 + 6 \zeta_{6} ) q^{17} + 2 q^{19} + ( 4 - 8 \zeta_{6} ) q^{23} + 2 q^{25} + 6 q^{29} + 10 q^{31} + ( -5 + 4 \zeta_{6} ) q^{35} -11 q^{37} + ( -1 + 2 \zeta_{6} ) q^{41} + ( 3 - 6 \zeta_{6} ) q^{43} -9 q^{47} + ( -3 + 8 \zeta_{6} ) q^{49} -12 q^{53} -6 q^{55} -3 q^{59} + ( 2 - 4 \zeta_{6} ) q^{61} -6 q^{65} + ( 2 - 4 \zeta_{6} ) q^{67} + ( 8 - 16 \zeta_{6} ) q^{71} + ( 4 - 8 \zeta_{6} ) q^{73} + ( -10 + 8 \zeta_{6} ) q^{77} + ( -7 + 14 \zeta_{6} ) q^{79} -15 q^{83} + 9 q^{85} + ( -4 + 8 \zeta_{6} ) q^{89} + ( -10 + 8 \zeta_{6} ) q^{91} + ( 2 - 4 \zeta_{6} ) q^{95} + ( -10 + 20 \zeta_{6} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 4q^{7} + O(q^{10}) \) \( 2q - 4q^{7} + 4q^{19} + 4q^{25} + 12q^{29} + 20q^{31} - 6q^{35} - 22q^{37} - 18q^{47} + 2q^{49} - 24q^{53} - 12q^{55} - 6q^{59} - 12q^{65} - 12q^{77} - 30q^{83} + 18q^{85} - 12q^{91} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1567.1
0.500000 + 0.866025i
0.500000 0.866025i
0 0 0 1.73205i 0 −2.00000 1.73205i 0 0 0
1567.2 0 0 0 1.73205i 0 −2.00000 + 1.73205i 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
28.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3024.2.b.e yes 2
3.b odd 2 1 3024.2.b.c 2
4.b odd 2 1 3024.2.b.n yes 2
7.b odd 2 1 3024.2.b.n yes 2
12.b even 2 1 3024.2.b.l yes 2
21.c even 2 1 3024.2.b.l yes 2
28.d even 2 1 inner 3024.2.b.e yes 2
84.h odd 2 1 3024.2.b.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3024.2.b.c 2 3.b odd 2 1
3024.2.b.c 2 84.h odd 2 1
3024.2.b.e yes 2 1.a even 1 1 trivial
3024.2.b.e yes 2 28.d even 2 1 inner
3024.2.b.l yes 2 12.b even 2 1
3024.2.b.l yes 2 21.c even 2 1
3024.2.b.n yes 2 4.b odd 2 1
3024.2.b.n yes 2 7.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(3024, [\chi])\):

\( T_{5}^{2} + 3 \)
\( T_{11}^{2} + 12 \)
\( T_{13}^{2} + 12 \)
\( T_{17}^{2} + 27 \)
\( T_{19} - 2 \)
\( T_{29} - 6 \)
\( T_{47} + 9 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ \( 1 - 7 T^{2} + 25 T^{4} \)
$7$ \( 1 + 4 T + 7 T^{2} \)
$11$ \( 1 - 10 T^{2} + 121 T^{4} \)
$13$ \( 1 - 14 T^{2} + 169 T^{4} \)
$17$ \( 1 - 7 T^{2} + 289 T^{4} \)
$19$ \( ( 1 - 2 T + 19 T^{2} )^{2} \)
$23$ \( 1 + 2 T^{2} + 529 T^{4} \)
$29$ \( ( 1 - 6 T + 29 T^{2} )^{2} \)
$31$ \( ( 1 - 10 T + 31 T^{2} )^{2} \)
$37$ \( ( 1 + 11 T + 37 T^{2} )^{2} \)
$41$ \( 1 - 79 T^{2} + 1681 T^{4} \)
$43$ \( 1 - 59 T^{2} + 1849 T^{4} \)
$47$ \( ( 1 + 9 T + 47 T^{2} )^{2} \)
$53$ \( ( 1 + 12 T + 53 T^{2} )^{2} \)
$59$ \( ( 1 + 3 T + 59 T^{2} )^{2} \)
$61$ \( 1 - 110 T^{2} + 3721 T^{4} \)
$67$ \( ( 1 - 16 T + 67 T^{2} )( 1 + 16 T + 67 T^{2} ) \)
$71$ \( 1 + 50 T^{2} + 5041 T^{4} \)
$73$ \( 1 - 98 T^{2} + 5329 T^{4} \)
$79$ \( ( 1 - 13 T + 79 T^{2} )( 1 + 13 T + 79 T^{2} ) \)
$83$ \( ( 1 + 15 T + 83 T^{2} )^{2} \)
$89$ \( 1 - 130 T^{2} + 7921 T^{4} \)
$97$ \( 1 + 106 T^{2} + 9409 T^{4} \)
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