Properties

Label 6048.2.h.d
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: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{6} \)
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.

\(f(q)\) \(=\) \( q + ( 1 - \zeta_{24} + \zeta_{24}^{3} - 2 \zeta_{24}^{4} + \zeta_{24}^{5} ) q^{5} -\zeta_{24}^{6} q^{7} +O(q^{10})\) \( q + ( 1 - \zeta_{24} + \zeta_{24}^{3} - 2 \zeta_{24}^{4} + \zeta_{24}^{5} ) q^{5} -\zeta_{24}^{6} q^{7} + ( \zeta_{24} + 4 \zeta_{24}^{2} + \zeta_{24}^{3} - \zeta_{24}^{5} - 2 \zeta_{24}^{6} ) q^{11} + ( -\zeta_{24} - \zeta_{24}^{3} - \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{13} + ( 1 - \zeta_{24} + \zeta_{24}^{3} - 2 \zeta_{24}^{4} + \zeta_{24}^{5} ) q^{17} + ( \zeta_{24} - \zeta_{24}^{3} + \zeta_{24}^{5} - 2 \zeta_{24}^{6} + 2 \zeta_{24}^{7} ) q^{19} + ( -2 \zeta_{24} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} ) q^{23} + ( 2 \zeta_{24} + 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} - 4 \zeta_{24}^{7} ) q^{25} + ( -2 - 3 \zeta_{24} + 3 \zeta_{24}^{3} + 4 \zeta_{24}^{4} + 3 \zeta_{24}^{5} ) q^{29} + ( \zeta_{24} - \zeta_{24}^{3} + \zeta_{24}^{5} + 2 \zeta_{24}^{6} + 2 \zeta_{24}^{7} ) q^{31} + ( \zeta_{24} - 2 \zeta_{24}^{2} + \zeta_{24}^{3} - \zeta_{24}^{5} + \zeta_{24}^{6} ) q^{35} -7 q^{37} + ( -1 - 3 \zeta_{24} + 3 \zeta_{24}^{3} + 2 \zeta_{24}^{4} + 3 \zeta_{24}^{5} ) q^{41} + ( 3 \zeta_{24} - 3 \zeta_{24}^{3} + 3 \zeta_{24}^{5} + 3 \zeta_{24}^{6} + 6 \zeta_{24}^{7} ) q^{43} + ( -4 \zeta_{24} + 2 \zeta_{24}^{2} - 4 \zeta_{24}^{3} + 4 \zeta_{24}^{5} - \zeta_{24}^{6} ) q^{47} - q^{49} + ( 2 - 2 \zeta_{24} + 2 \zeta_{24}^{3} - 4 \zeta_{24}^{4} + 2 \zeta_{24}^{5} ) q^{53} + ( \zeta_{24} - \zeta_{24}^{3} + \zeta_{24}^{5} - 4 \zeta_{24}^{6} + 2 \zeta_{24}^{7} ) q^{55} + ( 2 \zeta_{24} + 10 \zeta_{24}^{2} + 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} - 5 \zeta_{24}^{6} ) q^{59} + ( 2 + 3 \zeta_{24} + 3 \zeta_{24}^{3} + 3 \zeta_{24}^{5} - 6 \zeta_{24}^{7} ) q^{61} + ( 2 - 3 \zeta_{24} + 3 \zeta_{24}^{3} - 4 \zeta_{24}^{4} + 3 \zeta_{24}^{5} ) q^{65} -8 \zeta_{24}^{6} q^{67} + ( 8 - 2 \zeta_{24} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{73} + ( 2 + \zeta_{24} - \zeta_{24}^{3} - 4 \zeta_{24}^{4} - \zeta_{24}^{5} ) q^{77} + ( -\zeta_{24} + \zeta_{24}^{3} - \zeta_{24}^{5} + 11 \zeta_{24}^{6} - 2 \zeta_{24}^{7} ) q^{79} + ( -6 \zeta_{24} - 2 \zeta_{24}^{2} - 6 \zeta_{24}^{3} + 6 \zeta_{24}^{5} + \zeta_{24}^{6} ) q^{83} + ( -5 + 2 \zeta_{24} + 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} - 4 \zeta_{24}^{7} ) q^{85} + ( 10 \zeta_{24} - 10 \zeta_{24}^{3} - 10 \zeta_{24}^{5} ) q^{89} + ( \zeta_{24} - \zeta_{24}^{3} + \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{91} + ( 5 \zeta_{24} - 8 \zeta_{24}^{2} + 5 \zeta_{24}^{3} - 5 \zeta_{24}^{5} + 4 \zeta_{24}^{6} ) q^{95} + ( 5 \zeta_{24} + 5 \zeta_{24}^{3} + 5 \zeta_{24}^{5} - 10 \zeta_{24}^{7} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + O(q^{10}) \) \( 8q - 56q^{37} - 8q^{49} + 16q^{61} + 64q^{73} - 40q^{85} + O(q^{100}) \)

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
−0.965926 0.258819i
−0.258819 + 0.965926i
0.965926 + 0.258819i
0.258819 0.965926i
0.258819 + 0.965926i
0.965926 0.258819i
−0.258819 0.965926i
−0.965926 + 0.258819i
0 0 0 3.14626i 0 1.00000i 0 0 0
2591.2 0 0 0 3.14626i 0 1.00000i 0 0 0
2591.3 0 0 0 0.317837i 0 1.00000i 0 0 0
2591.4 0 0 0 0.317837i 0 1.00000i 0 0 0
2591.5 0 0 0 0.317837i 0 1.00000i 0 0 0
2591.6 0 0 0 0.317837i 0 1.00000i 0 0 0
2591.7 0 0 0 3.14626i 0 1.00000i 0 0 0
2591.8 0 0 0 3.14626i 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.d 8
3.b odd 2 1 inner 6048.2.h.d 8
4.b odd 2 1 inner 6048.2.h.d 8
12.b even 2 1 inner 6048.2.h.d 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6048.2.h.d 8 1.a even 1 1 trivial
6048.2.h.d 8 3.b odd 2 1 inner
6048.2.h.d 8 4.b odd 2 1 inner
6048.2.h.d 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 \(S_{2}^{\mathrm{new}}(6048, [\chi])\):

\( T_{5}^{4} + 10 T_{5}^{2} + 1 \)
\( T_{11}^{4} - 28 T_{11}^{2} + 100 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ \( ( 1 - 10 T^{2} + 51 T^{4} - 250 T^{6} + 625 T^{8} )^{2} \)
$7$ \( ( 1 + T^{2} )^{4} \)
$11$ \( ( 1 + 16 T^{2} + 210 T^{4} + 1936 T^{6} + 14641 T^{8} )^{2} \)
$13$ \( ( 1 + 20 T^{2} + 169 T^{4} )^{4} \)
$17$ \( ( 1 - 58 T^{2} + 1395 T^{4} - 16762 T^{6} + 83521 T^{8} )^{2} \)
$19$ \( ( 1 - 56 T^{2} + 1410 T^{4} - 20216 T^{6} + 130321 T^{8} )^{2} \)
$23$ \( ( 1 + 38 T^{2} + 529 T^{4} )^{4} \)
$29$ \( ( 1 - 56 T^{2} + 1602 T^{4} - 47096 T^{6} + 707281 T^{8} )^{2} \)
$31$ \( ( 1 - 104 T^{2} + 4530 T^{4} - 99944 T^{6} + 923521 T^{8} )^{2} \)
$37$ \( ( 1 + 7 T + 37 T^{2} )^{8} \)
$41$ \( ( 1 - 122 T^{2} + 6867 T^{4} - 205082 T^{6} + 2825761 T^{8} )^{2} \)
$43$ \( ( 1 - 46 T^{2} + 2283 T^{4} - 85054 T^{6} + 3418801 T^{8} )^{2} \)
$47$ \( ( 1 + 118 T^{2} + 7515 T^{4} + 260662 T^{6} + 4879681 T^{8} )^{2} \)
$53$ \( ( 1 - 172 T^{2} + 12630 T^{4} - 483148 T^{6} + 7890481 T^{8} )^{2} \)
$59$ \( ( 1 + 70 T^{2} + 5787 T^{4} + 243670 T^{6} + 12117361 T^{8} )^{2} \)
$61$ \( ( 1 - 4 T + 72 T^{2} - 244 T^{3} + 3721 T^{4} )^{4} \)
$67$ \( ( 1 - 70 T^{2} + 4489 T^{4} )^{4} \)
$71$ \( ( 1 + 71 T^{2} )^{8} \)
$73$ \( ( 1 - 16 T + 186 T^{2} - 1168 T^{3} + 5329 T^{4} )^{4} \)
$79$ \( ( 1 - 62 T^{2} + 10539 T^{4} - 386942 T^{6} + 38950081 T^{8} )^{2} \)
$83$ \( ( 1 + 182 T^{2} + 21195 T^{4} + 1253798 T^{6} + 47458321 T^{8} )^{2} \)
$89$ \( ( 1 + 22 T^{2} + 7921 T^{4} )^{4} \)
$97$ \( ( 1 + 44 T^{2} + 9409 T^{4} )^{4} \)
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