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.

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

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4010))\):

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