Properties

Label 6045.2.a.b
Level 6045
Weight 2
Character orbit 6045.a
Self dual yes
Analytic conductor 48.270
Analytic rank 1
Dimension 1
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 6045 = 3 \cdot 5 \cdot 13 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6045.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(48.2695680219\)
Analytic rank: \(1\)
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

\(f(q)\) \(=\) \( q - 2q^{2} - q^{3} + 2q^{4} + q^{5} + 2q^{6} + 2q^{7} + q^{9} + O(q^{10}) \) \( q - 2q^{2} - q^{3} + 2q^{4} + q^{5} + 2q^{6} + 2q^{7} + q^{9} - 2q^{10} - 5q^{11} - 2q^{12} - q^{13} - 4q^{14} - q^{15} - 4q^{16} + 4q^{17} - 2q^{18} - 2q^{19} + 2q^{20} - 2q^{21} + 10q^{22} - 2q^{23} + q^{25} + 2q^{26} - q^{27} + 4q^{28} + 2q^{30} + q^{31} + 8q^{32} + 5q^{33} - 8q^{34} + 2q^{35} + 2q^{36} + q^{37} + 4q^{38} + q^{39} + 10q^{41} + 4q^{42} - 10q^{43} - 10q^{44} + q^{45} + 4q^{46} + 4q^{47} + 4q^{48} - 3q^{49} - 2q^{50} - 4q^{51} - 2q^{52} + 12q^{53} + 2q^{54} - 5q^{55} + 2q^{57} - 10q^{59} - 2q^{60} + 14q^{61} - 2q^{62} + 2q^{63} - 8q^{64} - q^{65} - 10q^{66} - 12q^{67} + 8q^{68} + 2q^{69} - 4q^{70} - 4q^{71} - 7q^{73} - 2q^{74} - q^{75} - 4q^{76} - 10q^{77} - 2q^{78} - 8q^{79} - 4q^{80} + q^{81} - 20q^{82} - 4q^{83} - 4q^{84} + 4q^{85} + 20q^{86} + 15q^{89} - 2q^{90} - 2q^{91} - 4q^{92} - q^{93} - 8q^{94} - 2q^{95} - 8q^{96} + 10q^{97} + 6q^{98} - 5q^{99} + O(q^{100}) \)

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
−2.00000 −1.00000 2.00000 1.00000 2.00000 2.00000 0 1.00000 −2.00000
\(n\): 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 6045.2.a.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6045.2.a.b 1 1.a even 1 1 trivial

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(5\) \(-1\)
\(13\) \(1\)
\(31\) \(-1\)

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6045))\):

\( T_{2} + 2 \)
\( T_{7} - 2 \)
\( T_{11} + 5 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 2 T + 2 T^{2} \)
$3$ \( 1 + T \)
$5$ \( 1 - T \)
$7$ \( 1 - 2 T + 7 T^{2} \)
$11$ \( 1 + 5 T + 11 T^{2} \)
$13$ \( 1 + T \)
$17$ \( 1 - 4 T + 17 T^{2} \)
$19$ \( 1 + 2 T + 19 T^{2} \)
$23$ \( 1 + 2 T + 23 T^{2} \)
$29$ \( 1 + 29 T^{2} \)
$31$ \( 1 - T \)
$37$ \( 1 - T + 37 T^{2} \)
$41$ \( 1 - 10 T + 41 T^{2} \)
$43$ \( 1 + 10 T + 43 T^{2} \)
$47$ \( 1 - 4 T + 47 T^{2} \)
$53$ \( 1 - 12 T + 53 T^{2} \)
$59$ \( 1 + 10 T + 59 T^{2} \)
$61$ \( 1 - 14 T + 61 T^{2} \)
$67$ \( 1 + 12 T + 67 T^{2} \)
$71$ \( 1 + 4 T + 71 T^{2} \)
$73$ \( 1 + 7 T + 73 T^{2} \)
$79$ \( 1 + 8 T + 79 T^{2} \)
$83$ \( 1 + 4 T + 83 T^{2} \)
$89$ \( 1 - 15 T + 89 T^{2} \)
$97$ \( 1 - 10 T + 97 T^{2} \)
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