Properties

Label 6013.2.a.a
Level $6013$
Weight $2$
Character orbit 6013.a
Self dual yes
Analytic conductor $48.014$
Analytic rank $2$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 6013 = 7 \cdot 859 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6013.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(48.0140467354\)
Analytic rank: \(2\)
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

\(f(q)\) \(=\) \( q - 2q^{2} - 2q^{3} + 2q^{4} - q^{5} + 4q^{6} + q^{7} + q^{9} + O(q^{10}) \) \( q - 2q^{2} - 2q^{3} + 2q^{4} - q^{5} + 4q^{6} + q^{7} + q^{9} + 2q^{10} - 4q^{12} - 2q^{13} - 2q^{14} + 2q^{15} - 4q^{16} - 3q^{17} - 2q^{18} - 4q^{19} - 2q^{20} - 2q^{21} - 6q^{23} - 4q^{25} + 4q^{26} + 4q^{27} + 2q^{28} - 4q^{30} - 9q^{31} + 8q^{32} + 6q^{34} - q^{35} + 2q^{36} + 2q^{37} + 8q^{38} + 4q^{39} - 7q^{41} + 4q^{42} - 11q^{43} - q^{45} + 12q^{46} + 4q^{47} + 8q^{48} + q^{49} + 8q^{50} + 6q^{51} - 4q^{52} - 13q^{53} - 8q^{54} + 8q^{57} - 9q^{59} + 4q^{60} + 11q^{61} + 18q^{62} + q^{63} - 8q^{64} + 2q^{65} - 2q^{67} - 6q^{68} + 12q^{69} + 2q^{70} - 6q^{73} - 4q^{74} + 8q^{75} - 8q^{76} - 8q^{78} + 13q^{79} + 4q^{80} - 11q^{81} + 14q^{82} + 6q^{83} - 4q^{84} + 3q^{85} + 22q^{86} + 6q^{89} + 2q^{90} - 2q^{91} - 12q^{92} + 18q^{93} - 8q^{94} + 4q^{95} - 16q^{96} - 6q^{97} - 2q^{98} + O(q^{100}) \)

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

Atkin-Lehner signs

\( p \) Sign
\(7\) \(-1\)
\(859\) \(1\)

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 6013.2.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6013.2.a.a 1 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} + 2 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6013))\).

Hecke characteristic polynomials

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