Properties

Label 8007.2.a.b
Level 8007
Weight 2
Character orbit 8007.a
Self dual Yes
Analytic conductor 63.936
Analytic rank 1
Dimension 2
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 8007 = 3 \cdot 17 \cdot 157 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 8007.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(63.9362168984\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
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 \(\beta = \sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

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

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
−1.41421
1.41421
−2.41421 1.00000 3.82843 −0.585786 −2.41421 −0.585786 −4.41421 1.00000 1.41421
1.2 0.414214 1.00000 −1.82843 −3.41421 0.414214 −3.41421 −1.58579 1.00000 −1.41421
\(n\): 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
\(3\) \(-1\)
\(17\) \(1\)
\(157\) \(-1\)

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{2}^{2} \) \(\mathstrut +\mathstrut 2 T_{2} \) \(\mathstrut -\mathstrut 1 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8007))\).