Properties

Label 4002.2.a.s
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: \(\Q(\sqrt{17}) \)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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.

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

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
−1.00000 −1.00000 1.00000 −3.56155 1.00000 3.56155 −1.00000 1.00000 3.56155
1.2 −1.00000 −1.00000 1.00000 0.561553 1.00000 −0.561553 −1.00000 1.00000 −0.561553
\(n\): 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4002))\):

\(T_{5}^{2} \) \(\mathstrut +\mathstrut 3 T_{5} \) \(\mathstrut -\mathstrut 2 \)
\(T_{7}^{2} \) \(\mathstrut -\mathstrut 3 T_{7} \) \(\mathstrut -\mathstrut 2 \)
\(T_{11} \) \(\mathstrut +\mathstrut 2 \)