Properties

Label 6048.2.a.bt
Level 6048
Weight 2
Character orbit 6048.a
Self dual yes
Analytic conductor 48.294
Analytic rank 0
Dimension 4
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 6048 = 2^{5} \cdot 3^{3} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 6048.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(48.2935231425\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.39528.1
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 9 x^{2} - 2 x + 12\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 2 \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 5 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - \nu^{2} - 7 \nu + 3 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)\(/2\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 5\)
\(\nu^{3}\)\(=\)\((\)\(2 \beta_{3} + 2 \beta_{2} + 7 \beta_{1} + 4\)\()/2\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
1.12265
−1.48380
−2.50948
2.87063
0 0 0 −2.73965 0 1.00000 0 0 0
1.2 0 0 0 −1.79835 0 1.00000 0 0 0
1.3 0 0 0 2.29751 0 1.00000 0 0 0
1.4 0 0 0 4.24049 0 1.00000 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 6048.2.a.bt yes 4
3.b odd 2 1 6048.2.a.bp yes 4
4.b odd 2 1 6048.2.a.bs yes 4
12.b even 2 1 6048.2.a.bk 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6048.2.a.bk 4 12.b even 2 1
6048.2.a.bp yes 4 3.b odd 2 1
6048.2.a.bs yes 4 4.b odd 2 1
6048.2.a.bt yes 4 1.a even 1 1 trivial

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(7\) \(-1\)

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6048))\):

\( T_{5}^{4} - 2 T_{5}^{3} - 15 T_{5}^{2} + 12 T_{5} + 48 \)
\( T_{11}^{4} - 36 T_{11}^{2} + 16 T_{11} + 192 \)
\( T_{13}^{4} - 8 T_{13}^{3} - 21 T_{13}^{2} + 184 T_{13} + 88 \)
\( T_{17}^{4} - 2 T_{17}^{3} - 48 T_{17}^{2} + 154 T_{17} - 89 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( \)
$3$ \( \)
$5$ \( 1 - 2 T + 5 T^{2} - 18 T^{3} + 48 T^{4} - 90 T^{5} + 125 T^{6} - 250 T^{7} + 625 T^{8} \)
$7$ \( ( 1 - T )^{4} \)
$11$ \( 1 + 8 T^{2} + 16 T^{3} + 126 T^{4} + 176 T^{5} + 968 T^{6} + 14641 T^{8} \)
$13$ \( 1 - 8 T + 31 T^{2} - 128 T^{3} + 556 T^{4} - 1664 T^{5} + 5239 T^{6} - 17576 T^{7} + 28561 T^{8} \)
$17$ \( 1 - 2 T + 20 T^{2} + 52 T^{3} + 13 T^{4} + 884 T^{5} + 5780 T^{6} - 9826 T^{7} + 83521 T^{8} \)
$19$ \( 1 - 8 T + 64 T^{2} - 360 T^{3} + 1806 T^{4} - 6840 T^{5} + 23104 T^{6} - 54872 T^{7} + 130321 T^{8} \)
$23$ \( 1 + 47 T^{2} - 128 T^{3} + 1008 T^{4} - 2944 T^{5} + 24863 T^{6} + 279841 T^{8} \)
$29$ \( 1 - 6 T + 113 T^{2} - 490 T^{3} + 4896 T^{4} - 14210 T^{5} + 95033 T^{6} - 146334 T^{7} + 707281 T^{8} \)
$31$ \( 1 - 6 T + 121 T^{2} - 526 T^{3} + 5604 T^{4} - 16306 T^{5} + 116281 T^{6} - 178746 T^{7} + 923521 T^{8} \)
$37$ \( 1 - 6 T + 97 T^{2} - 262 T^{3} + 3804 T^{4} - 9694 T^{5} + 132793 T^{6} - 303918 T^{7} + 1874161 T^{8} \)
$41$ \( 1 + 6 T + 161 T^{2} + 698 T^{3} + 9852 T^{4} + 28618 T^{5} + 270641 T^{6} + 413526 T^{7} + 2825761 T^{8} \)
$43$ \( 1 + 2 T + 100 T^{2} - 104 T^{3} + 4469 T^{4} - 4472 T^{5} + 184900 T^{6} + 159014 T^{7} + 3418801 T^{8} \)
$47$ \( 1 + 2 T + 41 T^{2} + 146 T^{3} + 3640 T^{4} + 6862 T^{5} + 90569 T^{6} + 207646 T^{7} + 4879681 T^{8} \)
$53$ \( 1 - 6 T + 53 T^{2} + 370 T^{3} - 2412 T^{4} + 19610 T^{5} + 148877 T^{6} - 893262 T^{7} + 7890481 T^{8} \)
$59$ \( 1 + 170 T^{2} + 32 T^{3} + 13899 T^{4} + 1888 T^{5} + 591770 T^{6} + 12117361 T^{8} \)
$61$ \( 1 - 4 T + 124 T^{2} + 116 T^{3} + 6358 T^{4} + 7076 T^{5} + 461404 T^{6} - 907924 T^{7} + 13845841 T^{8} \)
$67$ \( 1 + 4 T + 139 T^{2} + 892 T^{3} + 10912 T^{4} + 59764 T^{5} + 623971 T^{6} + 1203052 T^{7} + 20151121 T^{8} \)
$71$ \( 1 + 16 T + 335 T^{2} + 3432 T^{3} + 37440 T^{4} + 243672 T^{5} + 1688735 T^{6} + 5726576 T^{7} + 25411681 T^{8} \)
$73$ \( 1 - 12 T + 184 T^{2} - 2148 T^{3} + 16782 T^{4} - 156804 T^{5} + 980536 T^{6} - 4668204 T^{7} + 28398241 T^{8} \)
$79$ \( 1 + 2 T + 97 T^{2} - 1498 T^{3} - 1952 T^{4} - 118342 T^{5} + 605377 T^{6} + 986078 T^{7} + 38950081 T^{8} \)
$83$ \( 1 - 18 T + 389 T^{2} - 4490 T^{3} + 50748 T^{4} - 372670 T^{5} + 2679821 T^{6} - 10292166 T^{7} + 47458321 T^{8} \)
$89$ \( 1 + 8 T + 263 T^{2} + 1500 T^{3} + 30168 T^{4} + 133500 T^{5} + 2083223 T^{6} + 5639752 T^{7} + 62742241 T^{8} \)
$97$ \( 1 - 24 T + 460 T^{2} - 5704 T^{3} + 65814 T^{4} - 553288 T^{5} + 4328140 T^{6} - 21904152 T^{7} + 88529281 T^{8} \)
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