Properties

Label 6048.2.h.f
Level 6048
Weight 2
Character orbit 6048.h
Analytic conductor 48.294
Analytic rank 0
Dimension 8
CM no
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 6048 = 2^{5} \cdot 3^{3} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 6048.h (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(48.2935231425\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\zeta_{24})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{4} \)
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 \(\zeta_{24}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 2 \zeta_{24} - 2 \zeta_{24}^{3} - \zeta_{24}^{5} + \zeta_{24}^{7} ) q^{5} + \zeta_{24}^{6} q^{7} +O(q^{10})\) \( q + ( 2 \zeta_{24} - 2 \zeta_{24}^{3} - \zeta_{24}^{5} + \zeta_{24}^{7} ) q^{5} + \zeta_{24}^{6} q^{7} + ( 2 \zeta_{24} + 2 \zeta_{24}^{3} - \zeta_{24}^{5} - \zeta_{24}^{7} ) q^{11} + 4 q^{13} + ( -3 \zeta_{24} + 3 \zeta_{24}^{3} - \zeta_{24}^{5} - 4 \zeta_{24}^{7} ) q^{17} -\zeta_{24}^{6} q^{19} + ( -\zeta_{24} - \zeta_{24}^{3} + 4 \zeta_{24}^{5} - 3 \zeta_{24}^{7} ) q^{23} + ( -1 + 6 \zeta_{24}^{2} - 3 \zeta_{24}^{6} ) q^{25} + ( -4 \zeta_{24} + 4 \zeta_{24}^{3} - 4 \zeta_{24}^{7} ) q^{29} + ( -1 + 2 \zeta_{24}^{4} - 2 \zeta_{24}^{6} ) q^{31} + ( \zeta_{24} + \zeta_{24}^{3} - 2 \zeta_{24}^{5} + \zeta_{24}^{7} ) q^{35} + ( -10 + 2 \zeta_{24}^{2} - \zeta_{24}^{6} ) q^{37} + ( -3 \zeta_{24} + 3 \zeta_{24}^{3} - 4 \zeta_{24}^{5} - 7 \zeta_{24}^{7} ) q^{41} + ( -2 + 4 \zeta_{24}^{4} - 4 \zeta_{24}^{6} ) q^{43} + ( 2 \zeta_{24} + 2 \zeta_{24}^{3} + 4 \zeta_{24}^{5} - 6 \zeta_{24}^{7} ) q^{47} - q^{49} + ( -2 \zeta_{24} + 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} ) q^{53} -3 \zeta_{24}^{6} q^{55} + ( -2 \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{59} + ( -2 + 12 \zeta_{24}^{2} - 6 \zeta_{24}^{6} ) q^{61} + ( 8 \zeta_{24} - 8 \zeta_{24}^{3} - 4 \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{65} + ( -4 + 8 \zeta_{24}^{4} - 4 \zeta_{24}^{6} ) q^{67} + ( 3 \zeta_{24}^{5} - 3 \zeta_{24}^{7} ) q^{71} + ( -4 + 12 \zeta_{24}^{2} - 6 \zeta_{24}^{6} ) q^{73} + ( -\zeta_{24} + \zeta_{24}^{3} + 2 \zeta_{24}^{5} + \zeta_{24}^{7} ) q^{77} + ( 2 - 4 \zeta_{24}^{4} + 4 \zeta_{24}^{6} ) q^{79} + ( -6 \zeta_{24} - 6 \zeta_{24}^{3} - 2 \zeta_{24}^{5} + 8 \zeta_{24}^{7} ) q^{83} + ( 9 - 14 \zeta_{24}^{2} + 7 \zeta_{24}^{6} ) q^{85} + ( -9 \zeta_{24} + 9 \zeta_{24}^{3} + 8 \zeta_{24}^{5} - \zeta_{24}^{7} ) q^{89} + 4 \zeta_{24}^{6} q^{91} + ( -\zeta_{24} - \zeta_{24}^{3} + 2 \zeta_{24}^{5} - \zeta_{24}^{7} ) q^{95} + ( -4 - 8 \zeta_{24}^{2} + 4 \zeta_{24}^{6} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + O(q^{10}) \) \( 8q + 32q^{13} - 8q^{25} - 80q^{37} - 8q^{49} - 16q^{61} - 32q^{73} + 72q^{85} - 32q^{97} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2591.1
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 0 0 3.34607i 0 1.00000i 0 0 0
2591.2 0 0 0 3.34607i 0 1.00000i 0 0 0
2591.3 0 0 0 0.896575i 0 1.00000i 0 0 0
2591.4 0 0 0 0.896575i 0 1.00000i 0 0 0
2591.5 0 0 0 0.896575i 0 1.00000i 0 0 0
2591.6 0 0 0 0.896575i 0 1.00000i 0 0 0
2591.7 0 0 0 3.34607i 0 1.00000i 0 0 0
2591.8 0 0 0 3.34607i 0 1.00000i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2591.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
12.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.h.f 8
3.b odd 2 1 inner 6048.2.h.f 8
4.b odd 2 1 inner 6048.2.h.f 8
12.b even 2 1 inner 6048.2.h.f 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6048.2.h.f 8 1.a even 1 1 trivial
6048.2.h.f 8 3.b odd 2 1 inner
6048.2.h.f 8 4.b odd 2 1 inner
6048.2.h.f 8 12.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 \(S_{2}^{\mathrm{new}}(6048, [\chi])\):

\( T_{5}^{4} + 12 T_{5}^{2} + 9 \)
\( T_{11}^{4} - 12 T_{11}^{2} + 9 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ \( ( 1 - 8 T^{2} + 39 T^{4} - 200 T^{6} + 625 T^{8} )^{2} \)
$7$ \( ( 1 + T^{2} )^{4} \)
$11$ \( ( 1 + 32 T^{2} + 471 T^{4} + 3872 T^{6} + 14641 T^{8} )^{2} \)
$13$ \( ( 1 - 4 T + 13 T^{2} )^{8} \)
$17$ \( ( 1 - 16 T^{2} + 450 T^{4} - 4624 T^{6} + 83521 T^{8} )^{2} \)
$19$ \( ( 1 - 37 T^{2} + 361 T^{4} )^{4} \)
$23$ \( ( 1 + 40 T^{2} + 783 T^{4} + 21160 T^{6} + 279841 T^{8} )^{2} \)
$29$ \( ( 1 - 52 T^{2} + 1590 T^{4} - 43732 T^{6} + 707281 T^{8} )^{2} \)
$31$ \( ( 1 - 110 T^{2} + 4899 T^{4} - 105710 T^{6} + 923521 T^{8} )^{2} \)
$37$ \( ( 1 + 20 T + 171 T^{2} + 740 T^{3} + 1369 T^{4} )^{4} \)
$41$ \( ( 1 - 16 T^{2} + 3279 T^{4} - 26896 T^{6} + 2825761 T^{8} )^{2} \)
$43$ \( ( 1 - 116 T^{2} + 6294 T^{4} - 214484 T^{6} + 3418801 T^{8} )^{2} \)
$47$ \( ( 1 + 76 T^{2} + 5430 T^{4} + 167884 T^{6} + 4879681 T^{8} )^{2} \)
$53$ \( ( 1 - 98 T^{2} + 2809 T^{4} )^{4} \)
$59$ \( ( 1 + 220 T^{2} + 19014 T^{4} + 765820 T^{6} + 12117361 T^{8} )^{2} \)
$61$ \( ( 1 + 4 T + 18 T^{2} + 244 T^{3} + 3721 T^{4} )^{4} \)
$67$ \( ( 1 - 140 T^{2} + 10806 T^{4} - 628460 T^{6} + 20151121 T^{8} )^{2} \)
$71$ \( ( 1 + 248 T^{2} + 25215 T^{4} + 1250168 T^{6} + 25411681 T^{8} )^{2} \)
$73$ \( ( 1 + 8 T + 54 T^{2} + 584 T^{3} + 5329 T^{4} )^{4} \)
$79$ \( ( 1 - 260 T^{2} + 28614 T^{4} - 1622660 T^{6} + 38950081 T^{8} )^{2} \)
$83$ \( ( 1 + 124 T^{2} + 14550 T^{4} + 854236 T^{6} + 47458321 T^{8} )^{2} \)
$89$ \( ( 1 - 64 T^{2} + 15999 T^{4} - 506944 T^{6} + 62742241 T^{8} )^{2} \)
$97$ \( ( 1 + 8 T + 162 T^{2} + 776 T^{3} + 9409 T^{4} )^{4} \)
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