Properties

Label 6048.2.a.bm
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: 4.4.22896.1
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 9 x^{2} - 4 x + 3\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{3} + \nu^{2} - 5 \nu - 9 \)\()/3\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{3} + \nu^{2} - 11 \nu - 9 \)\()/3\)
\(\beta_{3}\)\(=\)\( -\nu^{3} + \nu^{2} + 7 \nu - 2 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{2} + \beta_{1}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3} + \beta_{2} + 2 \beta_{1} + 11\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(-\beta_{3} - 6 \beta_{2} + 9 \beta_{1} + 7\)\()/2\)

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
0.398683
3.15690
−0.896343
−2.65924
0 0 0 −4.38773 0 −1.00000 0 0 0
1.2 0 0 0 −0.766021 0 −1.00000 0 0 0
1.3 0 0 0 0.314350 0 −1.00000 0 0 0
1.4 0 0 0 2.83940 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.bm 4
3.b odd 2 1 6048.2.a.br yes 4
4.b odd 2 1 6048.2.a.bo yes 4
12.b even 2 1 6048.2.a.bv yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6048.2.a.bm 4 1.a even 1 1 trivial
6048.2.a.bo yes 4 4.b odd 2 1
6048.2.a.br yes 4 3.b odd 2 1
6048.2.a.bv yes 4 12.b even 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6048))\):

\( T_{5}^{4} + 2 T_{5}^{3} - 12 T_{5}^{2} - 6 T_{5} + 3 \)
\( T_{11}^{4} - 2 T_{11}^{3} - 24 T_{11}^{2} + 86 T_{11} - 73 \)
\( T_{13}^{4} - 36 T_{13}^{2} - 32 T_{13} + 48 \)
\( T_{17}^{4} + 8 T_{17}^{3} - 24 T_{17}^{2} - 224 T_{17} - 64 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ \( 1 + 2 T + 8 T^{2} + 24 T^{3} + 33 T^{4} + 120 T^{5} + 200 T^{6} + 250 T^{7} + 625 T^{8} \)
$7$ \( ( 1 + T )^{4} \)
$11$ \( 1 - 2 T + 20 T^{2} + 20 T^{3} + 125 T^{4} + 220 T^{5} + 2420 T^{6} - 2662 T^{7} + 14641 T^{8} \)
$13$ \( 1 + 16 T^{2} - 32 T^{3} + 126 T^{4} - 416 T^{5} + 2704 T^{6} + 28561 T^{8} \)
$17$ \( 1 + 8 T + 44 T^{2} + 184 T^{3} + 854 T^{4} + 3128 T^{5} + 12716 T^{6} + 39304 T^{7} + 83521 T^{8} \)
$19$ \( 1 - 4 T + 34 T^{2} - 72 T^{3} + 699 T^{4} - 1368 T^{5} + 12274 T^{6} - 27436 T^{7} + 130321 T^{8} \)
$23$ \( 1 - 2 T + 8 T^{2} - 4 T^{3} + 761 T^{4} - 92 T^{5} + 4232 T^{6} - 24334 T^{7} + 279841 T^{8} \)
$29$ \( 1 + 8 T + 80 T^{2} + 296 T^{3} + 2206 T^{4} + 8584 T^{5} + 67280 T^{6} + 195112 T^{7} + 707281 T^{8} \)
$31$ \( 1 + 4 T + 70 T^{2} + 448 T^{3} + 2407 T^{4} + 13888 T^{5} + 67270 T^{6} + 119164 T^{7} + 923521 T^{8} \)
$37$ \( 1 + 4 T + 118 T^{2} + 344 T^{3} + 5975 T^{4} + 12728 T^{5} + 161542 T^{6} + 202612 T^{7} + 1874161 T^{8} \)
$41$ \( 1 + 2 T + 92 T^{2} + 80 T^{3} + 4093 T^{4} + 3280 T^{5} + 154652 T^{6} + 137842 T^{7} + 2825761 T^{8} \)
$43$ \( 1 - 8 T + 160 T^{2} - 952 T^{3} + 10046 T^{4} - 40936 T^{5} + 295840 T^{6} - 636056 T^{7} + 3418801 T^{8} \)
$47$ \( 1 - 12 T + 200 T^{2} - 1484 T^{3} + 13950 T^{4} - 69748 T^{5} + 441800 T^{6} - 1245876 T^{7} + 4879681 T^{8} \)
$53$ \( ( 1 + 4 T + 53 T^{2} )^{4} \)
$59$ \( 1 - 12 T + 248 T^{2} - 1916 T^{3} + 21870 T^{4} - 113044 T^{5} + 863288 T^{6} - 2464548 T^{7} + 12117361 T^{8} \)
$61$ \( 1 + 8 T + 100 T^{2} + 440 T^{3} + 3478 T^{4} + 26840 T^{5} + 372100 T^{6} + 1815848 T^{7} + 13845841 T^{8} \)
$67$ \( 1 + 8 T + 88 T^{2} + 8 T^{3} + 814 T^{4} + 536 T^{5} + 395032 T^{6} + 2406104 T^{7} + 20151121 T^{8} \)
$71$ \( 1 + 18 T + 296 T^{2} + 2724 T^{3} + 27369 T^{4} + 193404 T^{5} + 1492136 T^{6} + 6442398 T^{7} + 25411681 T^{8} \)
$73$ \( 1 - 8 T + 136 T^{2} - 1032 T^{3} + 11166 T^{4} - 75336 T^{5} + 724744 T^{6} - 3112136 T^{7} + 28398241 T^{8} \)
$79$ \( 1 + 160 T^{2} - 848 T^{3} + 11886 T^{4} - 66992 T^{5} + 998560 T^{6} + 38950081 T^{8} \)
$83$ \( 1 - 40 T^{2} - 16 T^{3} + 13134 T^{4} - 1328 T^{5} - 275560 T^{6} + 47458321 T^{8} \)
$89$ \( 1 + 18 T + 404 T^{2} + 4416 T^{3} + 54549 T^{4} + 393024 T^{5} + 3200084 T^{6} + 12689442 T^{7} + 62742241 T^{8} \)
$97$ \( 1 - 8 T + 244 T^{2} - 2200 T^{3} + 29798 T^{4} - 213400 T^{5} + 2295796 T^{6} - 7301384 T^{7} + 88529281 T^{8} \)
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