Properties

Label 6048.2.a.bn
Level 6048
Weight 2
Character orbit 6048.a
Self dual yes
Analytic conductor 48.294
Analytic rank 1
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: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.25808.1
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
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.

\(f(q)\) \(=\) \( q + ( -1 - \beta_{2} ) q^{5} + q^{7} +O(q^{10})\) \( q + ( -1 - \beta_{2} ) q^{5} + q^{7} + ( -1 - \beta_{1} ) q^{11} + ( -1 + \beta_{1} - \beta_{3} ) q^{13} + ( -1 + \beta_{2} - \beta_{3} ) q^{17} + ( \beta_{2} + \beta_{3} ) q^{19} + ( -2 + \beta_{3} ) q^{23} + ( 3 + \beta_{2} + \beta_{3} ) q^{25} + ( 1 - \beta_{1} + \beta_{3} ) q^{29} + ( 3 + \beta_{1} + \beta_{2} ) q^{31} + ( -1 - \beta_{2} ) q^{35} + ( -2 + \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{37} + ( \beta_{1} + \beta_{2} - \beta_{3} ) q^{41} + ( 1 - \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{43} + ( -1 + \beta_{1} + \beta_{3} ) q^{47} + q^{49} + ( -4 + 2 \beta_{1} - 2 \beta_{2} ) q^{53} + ( 2 - \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{55} + ( -5 - \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{59} + ( -2 + 2 \beta_{2} - 2 \beta_{3} ) q^{61} + ( 3 - \beta_{1} + 2 \beta_{2} + 3 \beta_{3} ) q^{65} + ( 1 + \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{67} + ( -3 + \beta_{2} - 2 \beta_{3} ) q^{71} + ( -1 - \beta_{1} - 4 \beta_{2} + \beta_{3} ) q^{73} + ( -1 - \beta_{1} ) q^{77} + ( -1 - 3 \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{79} + ( -5 + \beta_{1} - \beta_{3} ) q^{83} + ( -3 - 2 \beta_{1} + 3 \beta_{2} + \beta_{3} ) q^{85} + ( -6 - \beta_{1} + \beta_{2} + \beta_{3} ) q^{89} + ( -1 + \beta_{1} - \beta_{3} ) q^{91} + ( -10 + 2 \beta_{1} - 2 \beta_{2} - 3 \beta_{3} ) q^{95} + ( 4 + 4 \beta_{2} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 2q^{5} + 4q^{7} + O(q^{10}) \) \( 4q - 2q^{5} + 4q^{7} - 2q^{11} - 4q^{13} - 4q^{17} - 4q^{19} - 10q^{23} + 8q^{25} + 4q^{29} + 8q^{31} - 2q^{35} - 8q^{37} - 2q^{41} + 4q^{43} - 8q^{47} + 4q^{49} - 16q^{53} + 8q^{55} - 24q^{59} - 8q^{61} + 4q^{65} - 4q^{67} - 10q^{71} + 4q^{73} - 2q^{77} - 4q^{79} - 20q^{83} - 16q^{85} - 26q^{89} - 4q^{91} - 34q^{95} + 8q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 10 x^{2} - 6 x + 9\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{3} - 3 \nu^{2} - \nu + 9 \)\()/3\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{3} - 3 \nu^{2} - 7 \nu + 9 \)\()/3\)
\(\beta_{3}\)\(=\)\( \nu^{3} - \nu^{2} - 7 \nu \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{2} + \beta_{1}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3} - 3 \beta_{2} + 9\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(3 \beta_{3} - 10 \beta_{2} + 7 \beta_{1} + 9\)\()/2\)

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
−1.55466
0.707500
−2.46810
3.31526
0 0 0 −3.95805 0 1.00000 0 0 0
1.2 0 0 0 −1.96666 0 1.00000 0 0 0
1.3 0 0 0 1.34410 0 1.00000 0 0 0
1.4 0 0 0 2.58060 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.bn yes 4
3.b odd 2 1 6048.2.a.bu yes 4
4.b odd 2 1 6048.2.a.bl 4
12.b even 2 1 6048.2.a.bq yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6048.2.a.bl 4 4.b odd 2 1
6048.2.a.bn yes 4 1.a even 1 1 trivial
6048.2.a.bq yes 4 12.b even 2 1
6048.2.a.bu yes 4 3.b odd 2 1

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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6048))\):

\( T_{5}^{4} + 2 T_{5}^{3} - 12 T_{5}^{2} - 10 T_{5} + 27 \)
\( T_{11}^{4} + 2 T_{11}^{3} - 32 T_{11}^{2} - 114 T_{11} - 73 \)
\( T_{13}^{4} + 4 T_{13}^{3} - 40 T_{13}^{2} - 176 T_{13} - 32 \)
\( T_{17}^{4} + 4 T_{17}^{3} - 32 T_{17}^{2} - 80 T_{17} + 48 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( \)
$3$ \( \)
$5$ \( 1 + 2 T + 8 T^{2} + 20 T^{3} + 57 T^{4} + 100 T^{5} + 200 T^{6} + 250 T^{7} + 625 T^{8} \)
$7$ \( ( 1 - T )^{4} \)
$11$ \( 1 + 2 T + 12 T^{2} - 48 T^{3} - 51 T^{4} - 528 T^{5} + 1452 T^{6} + 2662 T^{7} + 14641 T^{8} \)
$13$ \( 1 + 4 T + 12 T^{2} - 20 T^{3} - 58 T^{4} - 260 T^{5} + 2028 T^{6} + 8788 T^{7} + 28561 T^{8} \)
$17$ \( 1 + 4 T + 36 T^{2} + 124 T^{3} + 694 T^{4} + 2108 T^{5} + 10404 T^{6} + 19652 T^{7} + 83521 T^{8} \)
$19$ \( 1 + 4 T + 26 T^{2} - 40 T^{3} + 3 T^{4} - 760 T^{5} + 9386 T^{6} + 27436 T^{7} + 130321 T^{8} \)
$23$ \( 1 + 10 T + 96 T^{2} + 576 T^{3} + 3385 T^{4} + 13248 T^{5} + 50784 T^{6} + 121670 T^{7} + 279841 T^{8} \)
$29$ \( 1 - 4 T + 76 T^{2} - 172 T^{3} + 2694 T^{4} - 4988 T^{5} + 63916 T^{6} - 97556 T^{7} + 707281 T^{8} \)
$31$ \( 1 - 8 T + 94 T^{2} - 392 T^{3} + 3303 T^{4} - 12152 T^{5} + 90334 T^{6} - 238328 T^{7} + 923521 T^{8} \)
$37$ \( 1 + 8 T + 62 T^{2} - 40 T^{3} - 121 T^{4} - 1480 T^{5} + 84878 T^{6} + 405224 T^{7} + 1874161 T^{8} \)
$41$ \( 1 + 2 T + 108 T^{2} + 4 T^{3} + 5237 T^{4} + 164 T^{5} + 181548 T^{6} + 137842 T^{7} + 2825761 T^{8} \)
$43$ \( 1 - 4 T + 68 T^{2} + 220 T^{3} + 1014 T^{4} + 9460 T^{5} + 125732 T^{6} - 318028 T^{7} + 3418801 T^{8} \)
$47$ \( 1 + 8 T + 124 T^{2} + 1016 T^{3} + 7926 T^{4} + 47752 T^{5} + 273916 T^{6} + 830584 T^{7} + 4879681 T^{8} \)
$53$ \( 1 + 16 T + 148 T^{2} + 1136 T^{3} + 8534 T^{4} + 60208 T^{5} + 415732 T^{6} + 2382032 T^{7} + 7890481 T^{8} \)
$59$ \( 1 + 24 T + 348 T^{2} + 3464 T^{3} + 29158 T^{4} + 204376 T^{5} + 1211388 T^{6} + 4929096 T^{7} + 12117361 T^{8} \)
$61$ \( 1 + 8 T + 116 T^{2} + 824 T^{3} + 7478 T^{4} + 50264 T^{5} + 431636 T^{6} + 1815848 T^{7} + 13845841 T^{8} \)
$67$ \( 1 + 4 T + 100 T^{2} + 692 T^{3} + 4454 T^{4} + 46364 T^{5} + 448900 T^{6} + 1203052 T^{7} + 20151121 T^{8} \)
$71$ \( 1 + 10 T + 192 T^{2} + 1424 T^{3} + 19193 T^{4} + 101104 T^{5} + 967872 T^{6} + 3579110 T^{7} + 25411681 T^{8} \)
$73$ \( 1 - 4 T + 44 T^{2} - 268 T^{3} + 10310 T^{4} - 19564 T^{5} + 234476 T^{6} - 1556068 T^{7} + 28398241 T^{8} \)
$79$ \( 1 + 4 T + 20 T^{2} + 404 T^{3} + 11814 T^{4} + 31916 T^{5} + 124820 T^{6} + 1972156 T^{7} + 38950081 T^{8} \)
$83$ \( 1 + 20 T + 436 T^{2} + 4932 T^{3} + 57734 T^{4} + 409356 T^{5} + 3003604 T^{6} + 11435740 T^{7} + 47458321 T^{8} \)
$89$ \( 1 + 26 T + 548 T^{2} + 7140 T^{3} + 80541 T^{4} + 635460 T^{5} + 4340708 T^{6} + 18329194 T^{7} + 62742241 T^{8} \)
$97$ \( 1 - 8 T + 196 T^{2} - 1688 T^{3} + 26118 T^{4} - 163736 T^{5} + 1844164 T^{6} - 7301384 T^{7} + 88529281 T^{8} \)
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