Properties

Label 4002.2.a.u
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.

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

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
−1.56155
2.56155
−1.00000 1.00000 1.00000 −1.56155 −1.00000 −1.56155 −1.00000 1.00000 1.56155
1.2 −1.00000 1.00000 1.00000 2.56155 −1.00000 2.56155 −1.00000 1.00000 −2.56155
\(n\): 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4002))\):

\(T_{5}^{2} \) \(\mathstrut -\mathstrut T_{5} \) \(\mathstrut -\mathstrut 4 \)
\(T_{7}^{2} \) \(\mathstrut -\mathstrut T_{7} \) \(\mathstrut -\mathstrut 4 \)
\(T_{11}^{2} \) \(\mathstrut -\mathstrut 2 T_{11} \) \(\mathstrut -\mathstrut 16 \)