Properties

Label 6034.2.a.e
Level 6034
Weight 2
Character orbit 6034.a
Self dual yes
Analytic conductor 48.182
Analytic rank 1
Dimension 1
CM no
Inner twists 1

Related objects

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Newspace parameters

Level: \( N \) = \( 6034 = 2 \cdot 7 \cdot 431 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 6034.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(48.1817325796\)
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

\(f(q)\) \(=\) \( q + q^{2} - q^{3} + q^{4} - 3q^{5} - q^{6} - q^{7} + q^{8} - 2q^{9} + O(q^{10}) \) \( q + q^{2} - q^{3} + q^{4} - 3q^{5} - q^{6} - q^{7} + q^{8} - 2q^{9} - 3q^{10} + 3q^{11} - q^{12} - 2q^{13} - q^{14} + 3q^{15} + q^{16} + 8q^{17} - 2q^{18} - 5q^{19} - 3q^{20} + q^{21} + 3q^{22} + 5q^{23} - q^{24} + 4q^{25} - 2q^{26} + 5q^{27} - q^{28} + 3q^{29} + 3q^{30} - 6q^{31} + q^{32} - 3q^{33} + 8q^{34} + 3q^{35} - 2q^{36} - 8q^{37} - 5q^{38} + 2q^{39} - 3q^{40} - 2q^{41} + q^{42} + 4q^{43} + 3q^{44} + 6q^{45} + 5q^{46} - 2q^{47} - q^{48} + q^{49} + 4q^{50} - 8q^{51} - 2q^{52} + 9q^{53} + 5q^{54} - 9q^{55} - q^{56} + 5q^{57} + 3q^{58} + 9q^{59} + 3q^{60} - 6q^{61} - 6q^{62} + 2q^{63} + q^{64} + 6q^{65} - 3q^{66} - 8q^{67} + 8q^{68} - 5q^{69} + 3q^{70} + 10q^{71} - 2q^{72} - 4q^{73} - 8q^{74} - 4q^{75} - 5q^{76} - 3q^{77} + 2q^{78} + 4q^{79} - 3q^{80} + q^{81} - 2q^{82} - 6q^{83} + q^{84} - 24q^{85} + 4q^{86} - 3q^{87} + 3q^{88} - 6q^{89} + 6q^{90} + 2q^{91} + 5q^{92} + 6q^{93} - 2q^{94} + 15q^{95} - q^{96} - 7q^{97} + q^{98} - 6q^{99} + O(q^{100}) \)

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
1.00000 −1.00000 1.00000 −3.00000 −1.00000 −1.00000 1.00000 −2.00000 −3.00000
\(n\): 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 6034.2.a.e 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6034.2.a.e 1 1.a even 1 1 trivial

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(7\) \(1\)
\(431\) \(-1\)

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6034))\):

\( T_{3} + 1 \)
\( T_{5} + 3 \)
\( T_{11} - 3 \)

Hecke Characteristic Polynomials

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