Properties

Label 6041.2.a.b
Level 6041
Weight 2
Character orbit 6041.a
Self dual Yes
Analytic conductor 48.238
Analytic rank 1
Dimension 2
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 6041 = 7 \cdot 863 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 6041.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(48.2376278611\)
Analytic rank: \(1\)
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.

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

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
−0.618034
1.61803
−1.00000 0.381966 −1.00000 1.61803 −0.381966 −1.00000 3.00000 −2.85410 −1.61803
1.2 −1.00000 2.61803 −1.00000 −0.618034 −2.61803 −1.00000 3.00000 3.85410 0.618034
\(n\): 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
\(7\) \(1\)
\(863\) \(1\)

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{2} \) \(\mathstrut +\mathstrut 1 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6041))\).