Properties

Label 4010.2.a.n
Level 4010
Weight 2
Character orbit 4010.a
Self dual Yes
Analytic conductor 32.020
Analytic rank 0
Dimension 22
CM No

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Newspace parameters

Level: \( N \) = \( 4010 = 2 \cdot 5 \cdot 401 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4010.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(32.0200112105\)
Analytic rank: \(0\)
Dimension: \(22\)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

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

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1 1.00000 −3.32876 1.00000 1.00000 −3.32876 2.40644 1.00000 8.08065 1.00000
1.2 1.00000 −3.24170 1.00000 1.00000 −3.24170 1.32182 1.00000 7.50859 1.00000
1.3 1.00000 −3.04466 1.00000 1.00000 −3.04466 −0.894428 1.00000 6.26997 1.00000
1.4 1.00000 −2.94924 1.00000 1.00000 −2.94924 −5.23818 1.00000 5.69803 1.00000
1.5 1.00000 −1.93387 1.00000 1.00000 −1.93387 3.26149 1.00000 0.739836 1.00000
1.6 1.00000 −1.81588 1.00000 1.00000 −1.81588 −3.01167 1.00000 0.297426 1.00000
1.7 1.00000 −1.81499 1.00000 1.00000 −1.81499 −3.30313 1.00000 0.294193 1.00000
1.8 1.00000 −1.38202 1.00000 1.00000 −1.38202 3.56623 1.00000 −1.09003 1.00000
1.9 1.00000 −1.18240 1.00000 1.00000 −1.18240 −2.63299 1.00000 −1.60192 1.00000
1.10 1.00000 −0.591795 1.00000 1.00000 −0.591795 −0.164076 1.00000 −2.64978 1.00000
1.11 1.00000 −0.103543 1.00000 1.00000 −0.103543 2.39084 1.00000 −2.98928 1.00000
1.12 1.00000 0.470815 1.00000 1.00000 0.470815 2.26316 1.00000 −2.77833 1.00000
1.13 1.00000 0.822612 1.00000 1.00000 0.822612 0.710831 1.00000 −2.32331 1.00000
1.14 1.00000 0.825899 1.00000 1.00000 0.825899 −3.88199 1.00000 −2.31789 1.00000
1.15 1.00000 1.38303 1.00000 1.00000 1.38303 3.80913 1.00000 −1.08722 1.00000
1.16 1.00000 2.15640 1.00000 1.00000 2.15640 −4.76824 1.00000 1.65006 1.00000
1.17 1.00000 2.40993 1.00000 1.00000 2.40993 4.92260 1.00000 2.80778 1.00000
1.18 1.00000 2.44141 1.00000 1.00000 2.44141 −0.373089 1.00000 2.96048 1.00000
1.19 1.00000 2.70959 1.00000 1.00000 2.70959 0.508884 1.00000 4.34189 1.00000
1.20 1.00000 2.88697 1.00000 1.00000 2.88697 −3.00491 1.00000 5.33461 1.00000
See all 22 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.22
Significant digits:
Format:

Inner twists

This newform does not have CM; other inner twists have not been computed.

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(5\) \(-1\)
\(401\) \(-1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4010))\):

\(T_{3}^{22} - \cdots\)
\(T_{7}^{22} - \cdots\)
\(T_{11}^{22} - \cdots\)