Properties

Label 8043.2.a.l
Level 8043
Weight 2
Character orbit 8043.a
Self dual Yes
Analytic conductor 64.224
Analytic rank 1
Dimension 2
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 8043 = 3 \cdot 7 \cdot 383 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 8043.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(64.2236783457\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
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 \(\beta = \frac{1}{2}(1 + \sqrt{17})\). We also show the integral \(q\)-expansion of the trace form.

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

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
2.56155
−1.56155
−2.56155 1.00000 4.56155 −1.56155 −2.56155 1.00000 −6.56155 1.00000 4.00000
1.2 1.56155 1.00000 0.438447 2.56155 1.56155 1.00000 −2.43845 1.00000 4.00000
\(n\): 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
\(3\) \(-1\)
\(7\) \(-1\)
\(383\) \(1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8043))\):

\(T_{2}^{2} \) \(\mathstrut +\mathstrut T_{2} \) \(\mathstrut -\mathstrut 4 \)
\(T_{5}^{2} \) \(\mathstrut -\mathstrut T_{5} \) \(\mathstrut -\mathstrut 4 \)
\(T_{11}^{2} \) \(\mathstrut +\mathstrut 9 T_{11} \) \(\mathstrut +\mathstrut 16 \)