Properties

Label 6048.2.c.b
Level 6048
Weight 2
Character orbit 6048.c
Analytic conductor 48.294
Analytic rank 0
Dimension 2
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 6048 = 2^{5} \cdot 3^{3} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 6048.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(48.2935231425\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1512)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 i q^{5} + q^{7} +O(q^{10})\) \( q + 2 i q^{5} + q^{7} + 5 i q^{13} + q^{17} -4 i q^{19} + 5 q^{23} + q^{25} -9 i q^{29} + 7 q^{31} + 2 i q^{35} + 2 i q^{37} + 2 q^{41} -5 i q^{43} + q^{49} -9 i q^{53} -i q^{59} + 6 i q^{61} -10 q^{65} + 9 i q^{67} + 15 q^{71} + 14 q^{79} -4 i q^{83} + 2 i q^{85} -13 q^{89} + 5 i q^{91} + 8 q^{95} -14 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{7} + O(q^{10}) \) \( 2q + 2q^{7} + 2q^{17} + 10q^{23} + 2q^{25} + 14q^{31} + 4q^{41} + 2q^{49} - 20q^{65} + 30q^{71} + 28q^{79} - 26q^{89} + 16q^{95} - 28q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/6048\mathbb{Z}\right)^\times\).

\(n\) \(2593\) \(3781\) \(3809\) \(4159\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3025.1
1.00000i
1.00000i
0 0 0 2.00000i 0 1.00000 0 0 0
3025.2 0 0 0 2.00000i 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
8.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 6048.2.c.b 2
3.b odd 2 1 6048.2.c.a 2
4.b odd 2 1 1512.2.c.a 2
8.b even 2 1 inner 6048.2.c.b 2
8.d odd 2 1 1512.2.c.a 2
12.b even 2 1 1512.2.c.b yes 2
24.f even 2 1 1512.2.c.b yes 2
24.h odd 2 1 6048.2.c.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1512.2.c.a 2 4.b odd 2 1
1512.2.c.a 2 8.d odd 2 1
1512.2.c.b yes 2 12.b even 2 1
1512.2.c.b yes 2 24.f even 2 1
6048.2.c.a 2 3.b odd 2 1
6048.2.c.a 2 24.h odd 2 1
6048.2.c.b 2 1.a even 1 1 trivial
6048.2.c.b 2 8.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(6048, [\chi])\):

\( T_{5}^{2} + 4 \)
\( T_{17} - 1 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( \)
$3$ \( \)
$5$ \( ( 1 - 4 T + 5 T^{2} )( 1 + 4 T + 5 T^{2} ) \)
$7$ \( ( 1 - T )^{2} \)
$11$ \( ( 1 - 11 T^{2} )^{2} \)
$13$ \( 1 - T^{2} + 169 T^{4} \)
$17$ \( ( 1 - T + 17 T^{2} )^{2} \)
$19$ \( 1 - 22 T^{2} + 361 T^{4} \)
$23$ \( ( 1 - 5 T + 23 T^{2} )^{2} \)
$29$ \( 1 + 23 T^{2} + 841 T^{4} \)
$31$ \( ( 1 - 7 T + 31 T^{2} )^{2} \)
$37$ \( ( 1 - 12 T + 37 T^{2} )( 1 + 12 T + 37 T^{2} ) \)
$41$ \( ( 1 - 2 T + 41 T^{2} )^{2} \)
$43$ \( 1 - 61 T^{2} + 1849 T^{4} \)
$47$ \( ( 1 + 47 T^{2} )^{2} \)
$53$ \( 1 - 25 T^{2} + 2809 T^{4} \)
$59$ \( 1 - 117 T^{2} + 3481 T^{4} \)
$61$ \( 1 - 86 T^{2} + 3721 T^{4} \)
$67$ \( 1 - 53 T^{2} + 4489 T^{4} \)
$71$ \( ( 1 - 15 T + 71 T^{2} )^{2} \)
$73$ \( ( 1 + 73 T^{2} )^{2} \)
$79$ \( ( 1 - 14 T + 79 T^{2} )^{2} \)
$83$ \( 1 - 150 T^{2} + 6889 T^{4} \)
$89$ \( ( 1 + 13 T + 89 T^{2} )^{2} \)
$97$ \( ( 1 + 14 T + 97 T^{2} )^{2} \)
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