Properties

Label 6038.2.a.a
Level 6038
Weight 2
Character orbit 6038.a
Self dual Yes
Analytic conductor 48.214
Analytic rank 2
Dimension 2
CM No
Inner twists 1

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) = \( 6038 = 2 \cdot 3019 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 6038.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(48.2136727404\)
Analytic rank: \(2\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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.

\(f(q)\) \(=\) \( q\) \(+ q^{2}\) \( + ( -1 - \beta ) q^{3} \) \(+ q^{4}\) \( + ( -3 + \beta ) q^{5} \) \( + ( -1 - \beta ) q^{6} \) \( -3 q^{7} \) \(+ q^{8}\) \( + ( -1 + 3 \beta ) q^{9} \) \(+O(q^{10})\) \( q\) \(+ q^{2}\) \( + ( -1 - \beta ) q^{3} \) \(+ q^{4}\) \( + ( -3 + \beta ) q^{5} \) \( + ( -1 - \beta ) q^{6} \) \( -3 q^{7} \) \(+ q^{8}\) \( + ( -1 + 3 \beta ) q^{9} \) \( + ( -3 + \beta ) q^{10} \) \( + ( -4 + \beta ) q^{11} \) \( + ( -1 - \beta ) q^{12} \) \( -3 q^{13} \) \( -3 q^{14} \) \( + ( 2 + \beta ) q^{15} \) \(+ q^{16}\) \( + ( -1 - 4 \beta ) q^{17} \) \( + ( -1 + 3 \beta ) q^{18} \) \( -3 q^{19} \) \( + ( -3 + \beta ) q^{20} \) \( + ( 3 + 3 \beta ) q^{21} \) \( + ( -4 + \beta ) q^{22} \) \( + ( -5 + 4 \beta ) q^{23} \) \( + ( -1 - \beta ) q^{24} \) \( + ( 5 - 5 \beta ) q^{25} \) \( -3 q^{26} \) \( + ( 1 - 2 \beta ) q^{27} \) \( -3 q^{28} \) \( + ( -6 - 2 \beta ) q^{29} \) \( + ( 2 + \beta ) q^{30} \) \( + ( 2 - 5 \beta ) q^{31} \) \(+ q^{32}\) \( + ( 3 + 2 \beta ) q^{33} \) \( + ( -1 - 4 \beta ) q^{34} \) \( + ( 9 - 3 \beta ) q^{35} \) \( + ( -1 + 3 \beta ) q^{36} \) \( + ( 3 - 3 \beta ) q^{37} \) \( -3 q^{38} \) \( + ( 3 + 3 \beta ) q^{39} \) \( + ( -3 + \beta ) q^{40} \) \( + ( -5 + 10 \beta ) q^{41} \) \( + ( 3 + 3 \beta ) q^{42} \) \( + ( -6 - 2 \beta ) q^{43} \) \( + ( -4 + \beta ) q^{44} \) \( + ( 6 - 7 \beta ) q^{45} \) \( + ( -5 + 4 \beta ) q^{46} \) \( + ( -3 + 9 \beta ) q^{47} \) \( + ( -1 - \beta ) q^{48} \) \( + 2 q^{49} \) \( + ( 5 - 5 \beta ) q^{50} \) \( + ( 5 + 9 \beta ) q^{51} \) \( -3 q^{52} \) \( -9 q^{53} \) \( + ( 1 - 2 \beta ) q^{54} \) \( + ( 13 - 6 \beta ) q^{55} \) \( -3 q^{56} \) \( + ( 3 + 3 \beta ) q^{57} \) \( + ( -6 - 2 \beta ) q^{58} \) \( + ( -13 + 2 \beta ) q^{59} \) \( + ( 2 + \beta ) q^{60} \) \( + ( -4 + \beta ) q^{61} \) \( + ( 2 - 5 \beta ) q^{62} \) \( + ( 3 - 9 \beta ) q^{63} \) \(+ q^{64}\) \( + ( 9 - 3 \beta ) q^{65} \) \( + ( 3 + 2 \beta ) q^{66} \) \( + ( -3 + 3 \beta ) q^{67} \) \( + ( -1 - 4 \beta ) q^{68} \) \( + ( 1 - 3 \beta ) q^{69} \) \( + ( 9 - 3 \beta ) q^{70} \) \( + ( 1 - 8 \beta ) q^{71} \) \( + ( -1 + 3 \beta ) q^{72} \) \( + ( -9 + 6 \beta ) q^{73} \) \( + ( 3 - 3 \beta ) q^{74} \) \( + 5 \beta q^{75} \) \( -3 q^{76} \) \( + ( 12 - 3 \beta ) q^{77} \) \( + ( 3 + 3 \beta ) q^{78} \) \( + ( 3 - 3 \beta ) q^{79} \) \( + ( -3 + \beta ) q^{80} \) \( + ( 4 - 6 \beta ) q^{81} \) \( + ( -5 + 10 \beta ) q^{82} \) \( + ( 11 - 8 \beta ) q^{83} \) \( + ( 3 + 3 \beta ) q^{84} \) \( + ( -1 + 7 \beta ) q^{85} \) \( + ( -6 - 2 \beta ) q^{86} \) \( + ( 8 + 10 \beta ) q^{87} \) \( + ( -4 + \beta ) q^{88} \) \( + ( 1 - 11 \beta ) q^{89} \) \( + ( 6 - 7 \beta ) q^{90} \) \( + 9 q^{91} \) \( + ( -5 + 4 \beta ) q^{92} \) \( + ( 3 + 8 \beta ) q^{93} \) \( + ( -3 + 9 \beta ) q^{94} \) \( + ( 9 - 3 \beta ) q^{95} \) \( + ( -1 - \beta ) q^{96} \) \( + ( 17 - 2 \beta ) q^{97} \) \( + 2 q^{98} \) \( + ( 7 - 10 \beta ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(2q \) \(\mathstrut +\mathstrut 2q^{2} \) \(\mathstrut -\mathstrut 3q^{3} \) \(\mathstrut +\mathstrut 2q^{4} \) \(\mathstrut -\mathstrut 5q^{5} \) \(\mathstrut -\mathstrut 3q^{6} \) \(\mathstrut -\mathstrut 6q^{7} \) \(\mathstrut +\mathstrut 2q^{8} \) \(\mathstrut +\mathstrut q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut +\mathstrut 2q^{2} \) \(\mathstrut -\mathstrut 3q^{3} \) \(\mathstrut +\mathstrut 2q^{4} \) \(\mathstrut -\mathstrut 5q^{5} \) \(\mathstrut -\mathstrut 3q^{6} \) \(\mathstrut -\mathstrut 6q^{7} \) \(\mathstrut +\mathstrut 2q^{8} \) \(\mathstrut +\mathstrut q^{9} \) \(\mathstrut -\mathstrut 5q^{10} \) \(\mathstrut -\mathstrut 7q^{11} \) \(\mathstrut -\mathstrut 3q^{12} \) \(\mathstrut -\mathstrut 6q^{13} \) \(\mathstrut -\mathstrut 6q^{14} \) \(\mathstrut +\mathstrut 5q^{15} \) \(\mathstrut +\mathstrut 2q^{16} \) \(\mathstrut -\mathstrut 6q^{17} \) \(\mathstrut +\mathstrut q^{18} \) \(\mathstrut -\mathstrut 6q^{19} \) \(\mathstrut -\mathstrut 5q^{20} \) \(\mathstrut +\mathstrut 9q^{21} \) \(\mathstrut -\mathstrut 7q^{22} \) \(\mathstrut -\mathstrut 6q^{23} \) \(\mathstrut -\mathstrut 3q^{24} \) \(\mathstrut +\mathstrut 5q^{25} \) \(\mathstrut -\mathstrut 6q^{26} \) \(\mathstrut -\mathstrut 6q^{28} \) \(\mathstrut -\mathstrut 14q^{29} \) \(\mathstrut +\mathstrut 5q^{30} \) \(\mathstrut -\mathstrut q^{31} \) \(\mathstrut +\mathstrut 2q^{32} \) \(\mathstrut +\mathstrut 8q^{33} \) \(\mathstrut -\mathstrut 6q^{34} \) \(\mathstrut +\mathstrut 15q^{35} \) \(\mathstrut +\mathstrut q^{36} \) \(\mathstrut +\mathstrut 3q^{37} \) \(\mathstrut -\mathstrut 6q^{38} \) \(\mathstrut +\mathstrut 9q^{39} \) \(\mathstrut -\mathstrut 5q^{40} \) \(\mathstrut +\mathstrut 9q^{42} \) \(\mathstrut -\mathstrut 14q^{43} \) \(\mathstrut -\mathstrut 7q^{44} \) \(\mathstrut +\mathstrut 5q^{45} \) \(\mathstrut -\mathstrut 6q^{46} \) \(\mathstrut +\mathstrut 3q^{47} \) \(\mathstrut -\mathstrut 3q^{48} \) \(\mathstrut +\mathstrut 4q^{49} \) \(\mathstrut +\mathstrut 5q^{50} \) \(\mathstrut +\mathstrut 19q^{51} \) \(\mathstrut -\mathstrut 6q^{52} \) \(\mathstrut -\mathstrut 18q^{53} \) \(\mathstrut +\mathstrut 20q^{55} \) \(\mathstrut -\mathstrut 6q^{56} \) \(\mathstrut +\mathstrut 9q^{57} \) \(\mathstrut -\mathstrut 14q^{58} \) \(\mathstrut -\mathstrut 24q^{59} \) \(\mathstrut +\mathstrut 5q^{60} \) \(\mathstrut -\mathstrut 7q^{61} \) \(\mathstrut -\mathstrut q^{62} \) \(\mathstrut -\mathstrut 3q^{63} \) \(\mathstrut +\mathstrut 2q^{64} \) \(\mathstrut +\mathstrut 15q^{65} \) \(\mathstrut +\mathstrut 8q^{66} \) \(\mathstrut -\mathstrut 3q^{67} \) \(\mathstrut -\mathstrut 6q^{68} \) \(\mathstrut -\mathstrut q^{69} \) \(\mathstrut +\mathstrut 15q^{70} \) \(\mathstrut -\mathstrut 6q^{71} \) \(\mathstrut +\mathstrut q^{72} \) \(\mathstrut -\mathstrut 12q^{73} \) \(\mathstrut +\mathstrut 3q^{74} \) \(\mathstrut +\mathstrut 5q^{75} \) \(\mathstrut -\mathstrut 6q^{76} \) \(\mathstrut +\mathstrut 21q^{77} \) \(\mathstrut +\mathstrut 9q^{78} \) \(\mathstrut +\mathstrut 3q^{79} \) \(\mathstrut -\mathstrut 5q^{80} \) \(\mathstrut +\mathstrut 2q^{81} \) \(\mathstrut +\mathstrut 14q^{83} \) \(\mathstrut +\mathstrut 9q^{84} \) \(\mathstrut +\mathstrut 5q^{85} \) \(\mathstrut -\mathstrut 14q^{86} \) \(\mathstrut +\mathstrut 26q^{87} \) \(\mathstrut -\mathstrut 7q^{88} \) \(\mathstrut -\mathstrut 9q^{89} \) \(\mathstrut +\mathstrut 5q^{90} \) \(\mathstrut +\mathstrut 18q^{91} \) \(\mathstrut -\mathstrut 6q^{92} \) \(\mathstrut +\mathstrut 14q^{93} \) \(\mathstrut +\mathstrut 3q^{94} \) \(\mathstrut +\mathstrut 15q^{95} \) \(\mathstrut -\mathstrut 3q^{96} \) \(\mathstrut +\mathstrut 32q^{97} \) \(\mathstrut +\mathstrut 4q^{98} \) \(\mathstrut +\mathstrut 4q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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 −2.61803 1.00000 −1.38197 −2.61803 −3.00000 1.00000 3.85410 −1.38197
1.2 1.00000 −0.381966 1.00000 −3.61803 −0.381966 −3.00000 1.00000 −2.85410 −3.61803
\(n\): 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
\(2\) \(-1\)
\(3019\) \(1\)

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{3}^{2} \) \(\mathstrut +\mathstrut 3 T_{3} \) \(\mathstrut +\mathstrut 1 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6038))\).