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.

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

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
−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 \(T_{2}^{2} \) \(\mathstrut -\mathstrut T_{2} \) \(\mathstrut -\mathstrut 1 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8047))\).