Properties

Label 4002.2.a.x
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: \(\Q(\sqrt{6}) \)
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 \(\beta = \sqrt{6}\). We also show the integral \(q\)-expansion of the trace form.

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

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
2.44949
−2.44949
1.00000 1.00000 1.00000 4.00000 1.00000 −0.449490 1.00000 1.00000 4.00000
1.2 1.00000 1.00000 1.00000 4.00000 1.00000 4.44949 1.00000 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
\(2\) \(-1\)
\(3\) \(-1\)
\(23\) \(1\)
\(29\) \(-1\)

Hecke kernels

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

\(T_{5} \) \(\mathstrut -\mathstrut 4 \)
\(T_{7}^{2} \) \(\mathstrut -\mathstrut 4 T_{7} \) \(\mathstrut -\mathstrut 2 \)
\(T_{11}^{2} \) \(\mathstrut -\mathstrut 24 \)