Properties

Label 6048.2.a.bd
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: \(\Q(\sqrt{17}) \)
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 \(\beta = \frac{1}{2}(1 + \sqrt{17})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta q^{5} + q^{7} +O(q^{10})\) \( q -\beta q^{5} + q^{7} + 2 \beta q^{11} + ( -5 + \beta ) q^{13} + ( 1 - 2 \beta ) q^{17} + ( 2 - 2 \beta ) q^{19} + ( 1 + 3 \beta ) q^{23} + ( -1 + \beta ) q^{25} + ( -7 - \beta ) q^{29} + ( 1 - \beta ) q^{31} -\beta q^{35} + ( 2 - \beta ) q^{37} + ( 10 - \beta ) q^{41} + ( -3 + 4 \beta ) q^{43} + ( -2 + \beta ) q^{47} + q^{49} + ( -1 + 3 \beta ) q^{53} + ( -8 - 2 \beta ) q^{55} + ( -3 - 2 \beta ) q^{59} + ( 2 - 4 \beta ) q^{61} + ( -4 + 4 \beta ) q^{65} + ( -11 + 3 \beta ) q^{67} + ( -1 + \beta ) q^{71} + ( -2 - 2 \beta ) q^{73} + 2 \beta q^{77} + ( -6 - \beta ) q^{79} + \beta q^{83} + ( 8 + \beta ) q^{85} + ( 9 - 3 \beta ) q^{89} + ( -5 + \beta ) q^{91} + 8 q^{95} + ( -12 + 4 \beta ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - q^{5} + 2q^{7} + O(q^{10}) \) \( 2q - q^{5} + 2q^{7} + 2q^{11} - 9q^{13} + 2q^{19} + 5q^{23} - q^{25} - 15q^{29} + q^{31} - q^{35} + 3q^{37} + 19q^{41} - 2q^{43} - 3q^{47} + 2q^{49} + q^{53} - 18q^{55} - 8q^{59} - 4q^{65} - 19q^{67} - q^{71} - 6q^{73} + 2q^{77} - 13q^{79} + q^{83} + 17q^{85} + 15q^{89} - 9q^{91} + 16q^{95} - 20q^{97} + O(q^{100}) \)

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
2.56155
−1.56155
0 0 0 −2.56155 0 1.00000 0 0 0
1.2 0 0 0 1.56155 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.bd yes 2
3.b odd 2 1 6048.2.a.bf yes 2
4.b odd 2 1 6048.2.a.bc 2
12.b even 2 1 6048.2.a.be yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6048.2.a.bc 2 4.b odd 2 1
6048.2.a.bd yes 2 1.a even 1 1 trivial
6048.2.a.be yes 2 12.b even 2 1
6048.2.a.bf yes 2 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6048))\):

\( T_{5}^{2} + T_{5} - 4 \)
\( T_{11}^{2} - 2 T_{11} - 16 \)
\( T_{13}^{2} + 9 T_{13} + 16 \)
\( T_{17}^{2} - 17 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( \)
$3$ \( \)
$5$ \( 1 + T + 6 T^{2} + 5 T^{3} + 25 T^{4} \)
$7$ \( ( 1 - T )^{2} \)
$11$ \( 1 - 2 T + 6 T^{2} - 22 T^{3} + 121 T^{4} \)
$13$ \( 1 + 9 T + 42 T^{2} + 117 T^{3} + 169 T^{4} \)
$17$ \( 1 + 17 T^{2} + 289 T^{4} \)
$19$ \( 1 - 2 T + 22 T^{2} - 38 T^{3} + 361 T^{4} \)
$23$ \( 1 - 5 T + 14 T^{2} - 115 T^{3} + 529 T^{4} \)
$29$ \( 1 + 15 T + 110 T^{2} + 435 T^{3} + 841 T^{4} \)
$31$ \( 1 - T + 58 T^{2} - 31 T^{3} + 961 T^{4} \)
$37$ \( 1 - 3 T + 72 T^{2} - 111 T^{3} + 1369 T^{4} \)
$41$ \( 1 - 19 T + 168 T^{2} - 779 T^{3} + 1681 T^{4} \)
$43$ \( 1 + 2 T + 19 T^{2} + 86 T^{3} + 1849 T^{4} \)
$47$ \( 1 + 3 T + 92 T^{2} + 141 T^{3} + 2209 T^{4} \)
$53$ \( 1 - T + 68 T^{2} - 53 T^{3} + 2809 T^{4} \)
$59$ \( 1 + 8 T + 117 T^{2} + 472 T^{3} + 3481 T^{4} \)
$61$ \( 1 + 54 T^{2} + 3721 T^{4} \)
$67$ \( 1 + 19 T + 186 T^{2} + 1273 T^{3} + 4489 T^{4} \)
$71$ \( 1 + T + 138 T^{2} + 71 T^{3} + 5041 T^{4} \)
$73$ \( 1 + 6 T + 138 T^{2} + 438 T^{3} + 5329 T^{4} \)
$79$ \( 1 + 13 T + 196 T^{2} + 1027 T^{3} + 6241 T^{4} \)
$83$ \( 1 - T + 162 T^{2} - 83 T^{3} + 6889 T^{4} \)
$89$ \( 1 - 15 T + 196 T^{2} - 1335 T^{3} + 7921 T^{4} \)
$97$ \( 1 + 20 T + 226 T^{2} + 1940 T^{3} + 9409 T^{4} \)
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