Properties

Label 4002.2.a.t
Level 4002
Weight 2
Character orbit 4002.a
Self dual Yes
Analytic conductor 31.956
Analytic rank 0
Dimension 2
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 4002 = 2 \cdot 3 \cdot 23 \cdot 29 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4002.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(31.9561308889\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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.

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

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
−1.41421
1.41421
−1.00000 −1.00000 1.00000 −2.82843 1.00000 −1.41421 −1.00000 1.00000 2.82843
1.2 −1.00000 −1.00000 1.00000 2.82843 1.00000 1.41421 −1.00000 1.00000 −2.82843
\(n\): 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
\(2\) \(1\)
\(3\) \(1\)
\(23\) \(1\)
\(29\) \(-1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4002))\):

\(T_{5}^{2} \) \(\mathstrut -\mathstrut 8 \)
\(T_{7}^{2} \) \(\mathstrut -\mathstrut 2 \)
\(T_{11} \) \(\mathstrut -\mathstrut 4 \)