Properties

Label 6031.2.a.a
Level 6031
Weight 2
Character orbit 6031.a
Self dual yes
Analytic conductor 48.158
Analytic rank 0
Dimension 1
CM no
Inner twists 1

Related objects

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Newspace parameters

Level: \( N \) = \( 6031 = 37 \cdot 163 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 6031.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(48.1577774590\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 3q^{3} - 2q^{4} + 4q^{5} + q^{7} + 6q^{9} + O(q^{10}) \) \( q + 3q^{3} - 2q^{4} + 4q^{5} + q^{7} + 6q^{9} + q^{11} - 6q^{12} + 4q^{13} + 12q^{15} + 4q^{16} - 2q^{17} - 8q^{20} + 3q^{21} - 8q^{23} + 11q^{25} + 9q^{27} - 2q^{28} + 2q^{31} + 3q^{33} + 4q^{35} - 12q^{36} + q^{37} + 12q^{39} + 3q^{41} + 4q^{43} - 2q^{44} + 24q^{45} - 9q^{47} + 12q^{48} - 6q^{49} - 6q^{51} - 8q^{52} + 9q^{53} + 4q^{55} - 6q^{59} - 24q^{60} - 6q^{61} + 6q^{63} - 8q^{64} + 16q^{65} + 8q^{67} + 4q^{68} - 24q^{69} + 13q^{71} + 5q^{73} + 33q^{75} + q^{77} - 14q^{79} + 16q^{80} + 9q^{81} + 9q^{83} - 6q^{84} - 8q^{85} - 12q^{89} + 4q^{91} + 16q^{92} + 6q^{93} + 12q^{97} + 6q^{99} + O(q^{100}) \)

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 3.00000 −2.00000 4.00000 0 1.00000 0 6.00000 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 6031.2.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6031.2.a.a 1 1.a even 1 1 trivial

Atkin-Lehner signs

\( p \) Sign
\(37\) \(-1\)
\(163\) \(1\)

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6031))\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + 2 T^{2} \)
$3$ \( 1 - 3 T + 3 T^{2} \)
$5$ \( 1 - 4 T + 5 T^{2} \)
$7$ \( 1 - T + 7 T^{2} \)
$11$ \( 1 - T + 11 T^{2} \)
$13$ \( 1 - 4 T + 13 T^{2} \)
$17$ \( 1 + 2 T + 17 T^{2} \)
$19$ \( 1 + 19 T^{2} \)
$23$ \( 1 + 8 T + 23 T^{2} \)
$29$ \( 1 + 29 T^{2} \)
$31$ \( 1 - 2 T + 31 T^{2} \)
$37$ \( 1 - T \)
$41$ \( 1 - 3 T + 41 T^{2} \)
$43$ \( 1 - 4 T + 43 T^{2} \)
$47$ \( 1 + 9 T + 47 T^{2} \)
$53$ \( 1 - 9 T + 53 T^{2} \)
$59$ \( 1 + 6 T + 59 T^{2} \)
$61$ \( 1 + 6 T + 61 T^{2} \)
$67$ \( 1 - 8 T + 67 T^{2} \)
$71$ \( 1 - 13 T + 71 T^{2} \)
$73$ \( 1 - 5 T + 73 T^{2} \)
$79$ \( 1 + 14 T + 79 T^{2} \)
$83$ \( 1 - 9 T + 83 T^{2} \)
$89$ \( 1 + 12 T + 89 T^{2} \)
$97$ \( 1 - 12 T + 97 T^{2} \)
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