Properties

Label 8047.2.a.a
Level 8047
Weight 2
Character orbit 8047.a
Self dual Yes
Analytic conductor 64.256
Analytic rank 0
Dimension 2
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 8047 = 13 \cdot 619 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 8047.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(64.2556185065\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
Coefficient ring: \(\Z[a_1, a_2]\)
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.

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

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
−0.618034
1.61803
−0.618034 −0.381966 −1.61803 3.23607 0.236068 −2.61803 2.23607 −2.85410 −2.00000
1.2 1.61803 −2.61803 0.618034 −1.23607 −4.23607 −0.381966 −2.23607 3.85410 −2.00000
\(n\): 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
\(13\) \(1\)
\(619\) \(-1\)

Hecke kernels

This newform can be constructed as the kernel of the linear operator \( T_{2}^{2} - T_{2} - 1 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8047))\).