Properties

Label 4002.2.a.w
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: \(\Q(\sqrt{3}) \)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
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 \(\beta = \sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

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

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
1.73205
−1.73205
1.00000 1.00000 1.00000 3.00000 1.00000 3.00000 1.00000 1.00000 3.00000
1.2 1.00000 1.00000 1.00000 3.00000 1.00000 3.00000 1.00000 1.00000 3.00000
\(n\): 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4002))\):

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