Properties

Label 8048.2.a.c
Level $8048$
Weight $2$
Character orbit 8048.a
Self dual yes
Analytic conductor $64.264$
Analytic rank $2$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 8048 = 2^{4} \cdot 503 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8048.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(64.2636035467\)
Analytic rank: \(2\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 503)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - q^{3} - 4q^{5} + 3q^{7} - 2q^{9} + O(q^{10}) \) \( q - q^{3} - 4q^{5} + 3q^{7} - 2q^{9} - 5q^{11} + q^{13} + 4q^{15} - 8q^{19} - 3q^{21} - 9q^{23} + 11q^{25} + 5q^{27} - 6q^{29} + 2q^{31} + 5q^{33} - 12q^{35} + 2q^{37} - q^{39} - 10q^{41} - 5q^{43} + 8q^{45} + q^{47} + 2q^{49} - 6q^{53} + 20q^{55} + 8q^{57} - 4q^{59} + 5q^{61} - 6q^{63} - 4q^{65} - 13q^{67} + 9q^{69} - 6q^{71} + 6q^{73} - 11q^{75} - 15q^{77} - 16q^{79} + q^{81} - 9q^{83} + 6q^{87} - 6q^{89} + 3q^{91} - 2q^{93} + 32q^{95} + 10q^{97} + 10q^{99} + O(q^{100}) \)

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
0
0 −1.00000 0 −4.00000 0 3.00000 0 −2.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(503\) \(1\)

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 8048.2.a.c 1
4.b odd 2 1 503.2.a.a 1
12.b even 2 1 4527.2.a.g 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
503.2.a.a 1 4.b odd 2 1
4527.2.a.g 1 12.b even 2 1
8048.2.a.c 1 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8048))\):

\( T_{3} + 1 \)
\( T_{5} + 4 \)
\( T_{7} - 3 \)
\( T_{13} - 1 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 + T + 3 T^{2} \)
$5$ \( 1 + 4 T + 5 T^{2} \)
$7$ \( 1 - 3 T + 7 T^{2} \)
$11$ \( 1 + 5 T + 11 T^{2} \)
$13$ \( 1 - T + 13 T^{2} \)
$17$ \( 1 + 17 T^{2} \)
$19$ \( 1 + 8 T + 19 T^{2} \)
$23$ \( 1 + 9 T + 23 T^{2} \)
$29$ \( 1 + 6 T + 29 T^{2} \)
$31$ \( 1 - 2 T + 31 T^{2} \)
$37$ \( 1 - 2 T + 37 T^{2} \)
$41$ \( 1 + 10 T + 41 T^{2} \)
$43$ \( 1 + 5 T + 43 T^{2} \)
$47$ \( 1 - T + 47 T^{2} \)
$53$ \( 1 + 6 T + 53 T^{2} \)
$59$ \( 1 + 4 T + 59 T^{2} \)
$61$ \( 1 - 5 T + 61 T^{2} \)
$67$ \( 1 + 13 T + 67 T^{2} \)
$71$ \( 1 + 6 T + 71 T^{2} \)
$73$ \( 1 - 6 T + 73 T^{2} \)
$79$ \( 1 + 16 T + 79 T^{2} \)
$83$ \( 1 + 9 T + 83 T^{2} \)
$89$ \( 1 + 6 T + 89 T^{2} \)
$97$ \( 1 - 10 T + 97 T^{2} \)
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