Properties

Label 6048.2.a.bg
Level 6048
Weight 2
Character orbit 6048.a
Self dual yes
Analytic conductor 48.294
Analytic rank 1
Dimension 2
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: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

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

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
−1.41421
1.41421
0 0 0 −0.414214 0 −1.00000 0 0 0
1.2 0 0 0 2.41421 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.bg yes 2
3.b odd 2 1 6048.2.a.z 2
4.b odd 2 1 6048.2.a.bj yes 2
12.b even 2 1 6048.2.a.ba yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6048.2.a.z 2 3.b odd 2 1
6048.2.a.ba yes 2 12.b even 2 1
6048.2.a.bg yes 2 1.a even 1 1 trivial
6048.2.a.bj yes 2 4.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6048))\):

\( T_{5}^{2} - 2 T_{5} - 1 \)
\( T_{11}^{2} + 2 T_{11} - 1 \)
\( T_{13}^{2} - 8 \)
\( T_{17}^{2} - 8 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( \)
$3$ \( \)
$5$ \( 1 - 2 T + 9 T^{2} - 10 T^{3} + 25 T^{4} \)
$7$ \( ( 1 + T )^{2} \)
$11$ \( 1 + 2 T + 21 T^{2} + 22 T^{3} + 121 T^{4} \)
$13$ \( 1 + 18 T^{2} + 169 T^{4} \)
$17$ \( 1 + 26 T^{2} + 289 T^{4} \)
$19$ \( 1 + 2 T + 31 T^{2} + 38 T^{3} + 361 T^{4} \)
$23$ \( 1 + 2 T + 29 T^{2} + 46 T^{3} + 529 T^{4} \)
$29$ \( 1 + 50 T^{2} + 841 T^{4} \)
$31$ \( 1 - 2 T + 31 T^{2} - 62 T^{3} + 961 T^{4} \)
$37$ \( 1 - 6 T + 75 T^{2} - 222 T^{3} + 1369 T^{4} \)
$41$ \( 1 - 10 T + 57 T^{2} - 410 T^{3} + 1681 T^{4} \)
$43$ \( 1 + 4 T + 82 T^{2} + 172 T^{3} + 1849 T^{4} \)
$47$ \( 1 + 12 T + 98 T^{2} + 564 T^{3} + 2209 T^{4} \)
$53$ \( ( 1 + 4 T + 53 T^{2} )^{2} \)
$59$ \( 1 + 4 T + 90 T^{2} + 236 T^{3} + 3481 T^{4} \)
$61$ \( 1 - 6 T^{2} + 3721 T^{4} \)
$67$ \( 1 + 8 T + 78 T^{2} + 536 T^{3} + 4489 T^{4} \)
$71$ \( 1 + 14 T + 189 T^{2} + 994 T^{3} + 5041 T^{4} \)
$73$ \( 1 + 12 T + 174 T^{2} + 876 T^{3} + 5329 T^{4} \)
$79$ \( 1 - 4 T + 154 T^{2} - 316 T^{3} + 6241 T^{4} \)
$83$ \( 1 + 16 T + 222 T^{2} + 1328 T^{3} + 6889 T^{4} \)
$89$ \( 1 + 6 T + 169 T^{2} + 534 T^{3} + 7921 T^{4} \)
$97$ \( 1 + 8 T + 178 T^{2} + 776 T^{3} + 9409 T^{4} \)
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