Properties

Label 3024.2.t.a
Level 3024
Weight 2
Character orbit 3024.t
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.t (of order \(3\), degree \(2\), not 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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 126)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q -3 q^{5} + ( -3 + \zeta_{6} ) q^{7} +O(q^{10})\) \( q -3 q^{5} + ( -3 + \zeta_{6} ) q^{7} -3 q^{11} + ( 1 - \zeta_{6} ) q^{13} + ( 3 - 3 \zeta_{6} ) q^{17} -7 \zeta_{6} q^{19} -9 q^{23} + 4 q^{25} + 3 \zeta_{6} q^{29} + 8 \zeta_{6} q^{31} + ( 9 - 3 \zeta_{6} ) q^{35} + \zeta_{6} q^{37} + ( 3 - 3 \zeta_{6} ) q^{41} -\zeta_{6} q^{43} + ( 8 - 5 \zeta_{6} ) q^{49} + ( 3 - 3 \zeta_{6} ) q^{53} + 9 q^{55} + ( -2 + 2 \zeta_{6} ) q^{61} + ( -3 + 3 \zeta_{6} ) q^{65} -4 \zeta_{6} q^{67} + 12 q^{71} + ( -11 + 11 \zeta_{6} ) q^{73} + ( 9 - 3 \zeta_{6} ) q^{77} + ( -16 + 16 \zeta_{6} ) q^{79} + 9 \zeta_{6} q^{83} + ( -9 + 9 \zeta_{6} ) q^{85} + 3 \zeta_{6} q^{89} + ( -2 + 3 \zeta_{6} ) q^{91} + 21 \zeta_{6} q^{95} + \zeta_{6} q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 6q^{5} - 5q^{7} + O(q^{10}) \) \( 2q - 6q^{5} - 5q^{7} - 6q^{11} + q^{13} + 3q^{17} - 7q^{19} - 18q^{23} + 8q^{25} + 3q^{29} + 8q^{31} + 15q^{35} + q^{37} + 3q^{41} - q^{43} + 11q^{49} + 3q^{53} + 18q^{55} - 2q^{61} - 3q^{65} - 4q^{67} + 24q^{71} - 11q^{73} + 15q^{77} - 16q^{79} + 9q^{83} - 9q^{85} + 3q^{89} - q^{91} + 21q^{95} + q^{97} + O(q^{100}) \)

Character values

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

\(n\) \(757\) \(785\) \(1135\) \(2593\)
\(\chi(n)\) \(1\) \(-1 + \zeta_{6}\) \(1\) \(-\zeta_{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
289.1
0.500000 0.866025i
0.500000 + 0.866025i
0 0 0 −3.00000 0 −2.50000 0.866025i 0 0 0
1873.1 0 0 0 −3.00000 0 −2.50000 + 0.866025i 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
63.g even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3024.2.t.a 2
3.b odd 2 1 1008.2.t.f 2
4.b odd 2 1 378.2.h.a 2
7.c even 3 1 3024.2.q.f 2
9.c even 3 1 3024.2.q.f 2
9.d odd 6 1 1008.2.q.a 2
12.b even 2 1 126.2.h.b yes 2
21.h odd 6 1 1008.2.q.a 2
28.d even 2 1 2646.2.h.d 2
28.f even 6 1 2646.2.e.g 2
28.f even 6 1 2646.2.f.a 2
28.g odd 6 1 378.2.e.b 2
28.g odd 6 1 2646.2.f.d 2
36.f odd 6 1 378.2.e.b 2
36.f odd 6 1 1134.2.g.c 2
36.h even 6 1 126.2.e.a 2
36.h even 6 1 1134.2.g.e 2
63.g even 3 1 inner 3024.2.t.a 2
63.n odd 6 1 1008.2.t.f 2
84.h odd 2 1 882.2.h.i 2
84.j odd 6 1 882.2.e.c 2
84.j odd 6 1 882.2.f.g 2
84.n even 6 1 126.2.e.a 2
84.n even 6 1 882.2.f.i 2
252.n even 6 1 2646.2.h.d 2
252.n even 6 1 7938.2.a.be 1
252.o even 6 1 126.2.h.b yes 2
252.o even 6 1 7938.2.a.m 1
252.r odd 6 1 882.2.f.g 2
252.s odd 6 1 882.2.e.c 2
252.u odd 6 1 1134.2.g.c 2
252.u odd 6 1 2646.2.f.d 2
252.bb even 6 1 882.2.f.i 2
252.bb even 6 1 1134.2.g.e 2
252.bi even 6 1 2646.2.e.g 2
252.bj even 6 1 2646.2.f.a 2
252.bl odd 6 1 378.2.h.a 2
252.bl odd 6 1 7938.2.a.t 1
252.bn odd 6 1 882.2.h.i 2
252.bn odd 6 1 7938.2.a.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.2.e.a 2 36.h even 6 1
126.2.e.a 2 84.n even 6 1
126.2.h.b yes 2 12.b even 2 1
126.2.h.b yes 2 252.o even 6 1
378.2.e.b 2 28.g odd 6 1
378.2.e.b 2 36.f odd 6 1
378.2.h.a 2 4.b odd 2 1
378.2.h.a 2 252.bl odd 6 1
882.2.e.c 2 84.j odd 6 1
882.2.e.c 2 252.s odd 6 1
882.2.f.g 2 84.j odd 6 1
882.2.f.g 2 252.r odd 6 1
882.2.f.i 2 84.n even 6 1
882.2.f.i 2 252.bb even 6 1
882.2.h.i 2 84.h odd 2 1
882.2.h.i 2 252.bn odd 6 1
1008.2.q.a 2 9.d odd 6 1
1008.2.q.a 2 21.h odd 6 1
1008.2.t.f 2 3.b odd 2 1
1008.2.t.f 2 63.n odd 6 1
1134.2.g.c 2 36.f odd 6 1
1134.2.g.c 2 252.u odd 6 1
1134.2.g.e 2 36.h even 6 1
1134.2.g.e 2 252.bb even 6 1
2646.2.e.g 2 28.f even 6 1
2646.2.e.g 2 252.bi even 6 1
2646.2.f.a 2 28.f even 6 1
2646.2.f.a 2 252.bj even 6 1
2646.2.f.d 2 28.g odd 6 1
2646.2.f.d 2 252.u odd 6 1
2646.2.h.d 2 28.d even 2 1
2646.2.h.d 2 252.n even 6 1
3024.2.q.f 2 7.c even 3 1
3024.2.q.f 2 9.c even 3 1
3024.2.t.a 2 1.a even 1 1 trivial
3024.2.t.a 2 63.g even 3 1 inner
7938.2.a.b 1 252.bn odd 6 1
7938.2.a.m 1 252.o even 6 1
7938.2.a.t 1 252.bl odd 6 1
7938.2.a.be 1 252.n even 6 1

Hecke kernels

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

\( T_{5} + 3 \)
\( T_{11} + 3 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ \( ( 1 + 3 T + 5 T^{2} )^{2} \)
$7$ \( 1 + 5 T + 7 T^{2} \)
$11$ \( ( 1 + 3 T + 11 T^{2} )^{2} \)
$13$ \( 1 - T - 12 T^{2} - 13 T^{3} + 169 T^{4} \)
$17$ \( 1 - 3 T - 8 T^{2} - 51 T^{3} + 289 T^{4} \)
$19$ \( ( 1 - T + 19 T^{2} )( 1 + 8 T + 19 T^{2} ) \)
$23$ \( ( 1 + 9 T + 23 T^{2} )^{2} \)
$29$ \( 1 - 3 T - 20 T^{2} - 87 T^{3} + 841 T^{4} \)
$31$ \( 1 - 8 T + 33 T^{2} - 248 T^{3} + 961 T^{4} \)
$37$ \( ( 1 - 11 T + 37 T^{2} )( 1 + 10 T + 37 T^{2} ) \)
$41$ \( 1 - 3 T - 32 T^{2} - 123 T^{3} + 1681 T^{4} \)
$43$ \( 1 + T - 42 T^{2} + 43 T^{3} + 1849 T^{4} \)
$47$ \( 1 - 47 T^{2} + 2209 T^{4} \)
$53$ \( 1 - 3 T - 44 T^{2} - 159 T^{3} + 2809 T^{4} \)
$59$ \( 1 - 59 T^{2} + 3481 T^{4} \)
$61$ \( 1 + 2 T - 57 T^{2} + 122 T^{3} + 3721 T^{4} \)
$67$ \( 1 + 4 T - 51 T^{2} + 268 T^{3} + 4489 T^{4} \)
$71$ \( ( 1 - 12 T + 71 T^{2} )^{2} \)
$73$ \( 1 + 11 T + 48 T^{2} + 803 T^{3} + 5329 T^{4} \)
$79$ \( 1 + 16 T + 177 T^{2} + 1264 T^{3} + 6241 T^{4} \)
$83$ \( 1 - 9 T - 2 T^{2} - 747 T^{3} + 6889 T^{4} \)
$89$ \( 1 - 3 T - 80 T^{2} - 267 T^{3} + 7921 T^{4} \)
$97$ \( 1 - T - 96 T^{2} - 97 T^{3} + 9409 T^{4} \)
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