Properties

Label 6034.2.a.i
Level 6034
Weight 2
Character orbit 6034.a
Self dual Yes
Analytic conductor 48.182
Analytic rank 0
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: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{3}) \)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
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 \(\beta = \sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} + 2 q^{3} + q^{4} + 2 q^{6} + q^{7} + q^{8} + q^{9} +O(q^{10})\) \( q + q^{2} + 2 q^{3} + q^{4} + 2 q^{6} + q^{7} + q^{8} + q^{9} + ( 2 - 2 \beta ) q^{11} + 2 q^{12} + ( -3 + \beta ) q^{13} + q^{14} + q^{16} -2 \beta q^{17} + q^{18} + ( 4 + 2 \beta ) q^{19} + 2 q^{21} + ( 2 - 2 \beta ) q^{22} + 6 q^{23} + 2 q^{24} -5 q^{25} + ( -3 + \beta ) q^{26} -4 q^{27} + q^{28} + ( 4 - 2 \beta ) q^{29} + ( 2 + 2 \beta ) q^{31} + q^{32} + ( 4 - 4 \beta ) q^{33} -2 \beta q^{34} + q^{36} + ( 7 + \beta ) q^{37} + ( 4 + 2 \beta ) q^{38} + ( -6 + 2 \beta ) q^{39} + 4 \beta q^{41} + 2 q^{42} + ( 3 - 5 \beta ) q^{43} + ( 2 - 2 \beta ) q^{44} + 6 q^{46} + ( 2 + 6 \beta ) q^{47} + 2 q^{48} + q^{49} -5 q^{50} -4 \beta q^{51} + ( -3 + \beta ) q^{52} + 2 q^{53} -4 q^{54} + q^{56} + ( 8 + 4 \beta ) q^{57} + ( 4 - 2 \beta ) q^{58} + 2 \beta q^{59} + ( 2 + 2 \beta ) q^{61} + ( 2 + 2 \beta ) q^{62} + q^{63} + q^{64} + ( 4 - 4 \beta ) q^{66} + ( -1 - \beta ) q^{67} -2 \beta q^{68} + 12 q^{69} + ( 4 + 4 \beta ) q^{71} + q^{72} + 2 q^{73} + ( 7 + \beta ) q^{74} -10 q^{75} + ( 4 + 2 \beta ) q^{76} + ( 2 - 2 \beta ) q^{77} + ( -6 + 2 \beta ) q^{78} + 12 q^{79} -11 q^{81} + 4 \beta q^{82} + ( -3 - 5 \beta ) q^{83} + 2 q^{84} + ( 3 - 5 \beta ) q^{86} + ( 8 - 4 \beta ) q^{87} + ( 2 - 2 \beta ) q^{88} -2 q^{89} + ( -3 + \beta ) q^{91} + 6 q^{92} + ( 4 + 4 \beta ) q^{93} + ( 2 + 6 \beta ) q^{94} + 2 q^{96} -8 q^{97} + q^{98} + ( 2 - 2 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{2} + 4q^{3} + 2q^{4} + 4q^{6} + 2q^{7} + 2q^{8} + 2q^{9} + O(q^{10}) \) \( 2q + 2q^{2} + 4q^{3} + 2q^{4} + 4q^{6} + 2q^{7} + 2q^{8} + 2q^{9} + 4q^{11} + 4q^{12} - 6q^{13} + 2q^{14} + 2q^{16} + 2q^{18} + 8q^{19} + 4q^{21} + 4q^{22} + 12q^{23} + 4q^{24} - 10q^{25} - 6q^{26} - 8q^{27} + 2q^{28} + 8q^{29} + 4q^{31} + 2q^{32} + 8q^{33} + 2q^{36} + 14q^{37} + 8q^{38} - 12q^{39} + 4q^{42} + 6q^{43} + 4q^{44} + 12q^{46} + 4q^{47} + 4q^{48} + 2q^{49} - 10q^{50} - 6q^{52} + 4q^{53} - 8q^{54} + 2q^{56} + 16q^{57} + 8q^{58} + 4q^{61} + 4q^{62} + 2q^{63} + 2q^{64} + 8q^{66} - 2q^{67} + 24q^{69} + 8q^{71} + 2q^{72} + 4q^{73} + 14q^{74} - 20q^{75} + 8q^{76} + 4q^{77} - 12q^{78} + 24q^{79} - 22q^{81} - 6q^{83} + 4q^{84} + 6q^{86} + 16q^{87} + 4q^{88} - 4q^{89} - 6q^{91} + 12q^{92} + 8q^{93} + 4q^{94} + 4q^{96} - 16q^{97} + 2q^{98} + 4q^{99} + O(q^{100}) \)

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
1.73205
−1.73205
1.00000 2.00000 1.00000 0 2.00000 1.00000 1.00000 1.00000 0
1.2 1.00000 2.00000 1.00000 0 2.00000 1.00000 1.00000 1.00000 0
\(n\): 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6034))\):

\( T_{3} - 2 \)
\( T_{5} \)
\( T_{11}^{2} - 4 T_{11} - 8 \)