Properties

Label 6034.2.a.g
Level 6034
Weight 2
Character orbit 6034.a
Self dual Yes
Analytic conductor 48.182
Analytic rank 1
Dimension 2
CM No
Inner twists 1

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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: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{5})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} + 2 q^{3} + q^{4} + ( -2 - \beta ) q^{5} -2 q^{6} - q^{7} - q^{8} + q^{9} +O(q^{10})\) \( q - q^{2} + 2 q^{3} + q^{4} + ( -2 - \beta ) q^{5} -2 q^{6} - q^{7} - q^{8} + q^{9} + ( 2 + \beta ) q^{10} + ( 2 - 5 \beta ) q^{11} + 2 q^{12} + ( -2 + 4 \beta ) q^{13} + q^{14} + ( -4 - 2 \beta ) q^{15} + q^{16} + ( 1 - \beta ) q^{17} - q^{18} + 4 \beta q^{19} + ( -2 - \beta ) q^{20} -2 q^{21} + ( -2 + 5 \beta ) q^{22} + ( 6 + 2 \beta ) q^{23} -2 q^{24} + 5 \beta q^{25} + ( 2 - 4 \beta ) q^{26} -4 q^{27} - q^{28} + ( -2 + 2 \beta ) q^{29} + ( 4 + 2 \beta ) q^{30} + ( 2 + 2 \beta ) q^{31} - q^{32} + ( 4 - 10 \beta ) q^{33} + ( -1 + \beta ) q^{34} + ( 2 + \beta ) q^{35} + q^{36} + ( -3 - 3 \beta ) q^{37} -4 \beta q^{38} + ( -4 + 8 \beta ) q^{39} + ( 2 + \beta ) q^{40} + ( 2 - 8 \beta ) q^{41} + 2 q^{42} + ( 2 - 5 \beta ) q^{44} + ( -2 - \beta ) q^{45} + ( -6 - 2 \beta ) q^{46} + 2 q^{47} + 2 q^{48} + q^{49} -5 \beta q^{50} + ( 2 - 2 \beta ) q^{51} + ( -2 + 4 \beta ) q^{52} + ( -8 - 2 \beta ) q^{53} + 4 q^{54} + ( 1 + 13 \beta ) q^{55} + q^{56} + 8 \beta q^{57} + ( 2 - 2 \beta ) q^{58} + ( -4 + 10 \beta ) q^{59} + ( -4 - 2 \beta ) q^{60} + ( -3 - 3 \beta ) q^{61} + ( -2 - 2 \beta ) q^{62} - q^{63} + q^{64} -10 \beta q^{65} + ( -4 + 10 \beta ) q^{66} + ( -2 + 4 \beta ) q^{67} + ( 1 - \beta ) q^{68} + ( 12 + 4 \beta ) q^{69} + ( -2 - \beta ) q^{70} + ( 8 + \beta ) q^{71} - q^{72} + ( -2 + 4 \beta ) q^{73} + ( 3 + 3 \beta ) q^{74} + 10 \beta q^{75} + 4 \beta q^{76} + ( -2 + 5 \beta ) q^{77} + ( 4 - 8 \beta ) q^{78} + ( -12 + 3 \beta ) q^{79} + ( -2 - \beta ) q^{80} -11 q^{81} + ( -2 + 8 \beta ) q^{82} + ( 4 - 4 \beta ) q^{83} -2 q^{84} + ( -1 + 2 \beta ) q^{85} + ( -4 + 4 \beta ) q^{87} + ( -2 + 5 \beta ) q^{88} + ( -12 - \beta ) q^{89} + ( 2 + \beta ) q^{90} + ( 2 - 4 \beta ) q^{91} + ( 6 + 2 \beta ) q^{92} + ( 4 + 4 \beta ) q^{93} -2 q^{94} + ( -4 - 12 \beta ) q^{95} -2 q^{96} + ( 2 - 10 \beta ) q^{97} - q^{98} + ( 2 - 5 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{2} + 4q^{3} + 2q^{4} - 5q^{5} - 4q^{6} - 2q^{7} - 2q^{8} + 2q^{9} + O(q^{10}) \) \( 2q - 2q^{2} + 4q^{3} + 2q^{4} - 5q^{5} - 4q^{6} - 2q^{7} - 2q^{8} + 2q^{9} + 5q^{10} - q^{11} + 4q^{12} + 2q^{14} - 10q^{15} + 2q^{16} + q^{17} - 2q^{18} + 4q^{19} - 5q^{20} - 4q^{21} + q^{22} + 14q^{23} - 4q^{24} + 5q^{25} - 8q^{27} - 2q^{28} - 2q^{29} + 10q^{30} + 6q^{31} - 2q^{32} - 2q^{33} - q^{34} + 5q^{35} + 2q^{36} - 9q^{37} - 4q^{38} + 5q^{40} - 4q^{41} + 4q^{42} - q^{44} - 5q^{45} - 14q^{46} + 4q^{47} + 4q^{48} + 2q^{49} - 5q^{50} + 2q^{51} - 18q^{53} + 8q^{54} + 15q^{55} + 2q^{56} + 8q^{57} + 2q^{58} + 2q^{59} - 10q^{60} - 9q^{61} - 6q^{62} - 2q^{63} + 2q^{64} - 10q^{65} + 2q^{66} + q^{68} + 28q^{69} - 5q^{70} + 17q^{71} - 2q^{72} + 9q^{74} + 10q^{75} + 4q^{76} + q^{77} - 21q^{79} - 5q^{80} - 22q^{81} + 4q^{82} + 4q^{83} - 4q^{84} - 4q^{87} + q^{88} - 25q^{89} + 5q^{90} + 14q^{92} + 12q^{93} - 4q^{94} - 20q^{95} - 4q^{96} - 6q^{97} - 2q^{98} - q^{99} + O(q^{100}) \)

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
1.61803
−0.618034
−1.00000 2.00000 1.00000 −3.61803 −2.00000 −1.00000 −1.00000 1.00000 3.61803
1.2 −1.00000 2.00000 1.00000 −1.38197 −2.00000 −1.00000 −1.00000 1.00000 1.38197
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

This newform does not admit any (nontrivial) inner twists.

Atkin-Lehner signs

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

Hecke kernels

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

\( T_{3} - 2 \)
\( T_{5}^{2} + 5 T_{5} + 5 \)
\( T_{11}^{2} + T_{11} - 31 \)