Properties

Label 4010.2.a.o
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.

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

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 −3.30817 1.00000 −1.00000 −3.30817 −2.76226 1.00000 7.94397 −1.00000
1.2 1.00000 −2.90289 1.00000 −1.00000 −2.90289 4.43766 1.00000 5.42676 −1.00000
1.3 1.00000 −2.88307 1.00000 −1.00000 −2.88307 −0.867210 1.00000 5.31208 −1.00000
1.4 1.00000 −2.52966 1.00000 −1.00000 −2.52966 2.69506 1.00000 3.39917 −1.00000
1.5 1.00000 −1.85051 1.00000 −1.00000 −1.85051 −3.61831 1.00000 0.424381 −1.00000
1.6 1.00000 −1.60919 1.00000 −1.00000 −1.60919 3.90377 1.00000 −0.410498 −1.00000
1.7 1.00000 −1.34023 1.00000 −1.00000 −1.34023 0.629438 1.00000 −1.20379 −1.00000
1.8 1.00000 −1.31697 1.00000 −1.00000 −1.31697 −3.01403 1.00000 −1.26559 −1.00000
1.9 1.00000 −1.18460 1.00000 −1.00000 −1.18460 4.68706 1.00000 −1.59673 −1.00000
1.10 1.00000 −0.310341 1.00000 −1.00000 −0.310341 1.61370 1.00000 −2.90369 −1.00000
1.11 1.00000 −0.165649 1.00000 −1.00000 −0.165649 −1.98987 1.00000 −2.97256 −1.00000
1.12 1.00000 0.417458 1.00000 −1.00000 0.417458 −4.20952 1.00000 −2.82573 −1.00000
1.13 1.00000 0.529822 1.00000 −1.00000 0.529822 −4.65586 1.00000 −2.71929 −1.00000
1.14 1.00000 0.642301 1.00000 −1.00000 0.642301 4.88072 1.00000 −2.58745 −1.00000
1.15 1.00000 1.74418 1.00000 −1.00000 1.74418 3.19985 1.00000 0.0421511 −1.00000
1.16 1.00000 1.77730 1.00000 −1.00000 1.77730 1.31771 1.00000 0.158778 −1.00000
1.17 1.00000 1.82747 1.00000 −1.00000 1.82747 0.568203 1.00000 0.339635 −1.00000
1.18 1.00000 2.45072 1.00000 −1.00000 2.45072 4.38820 1.00000 3.00604 −1.00000
1.19 1.00000 2.51296 1.00000 −1.00000 2.51296 −2.03791 1.00000 3.31499 −1.00000
1.20 1.00000 3.00389 1.00000 −1.00000 3.00389 −1.21989 1.00000 6.02333 −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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4010))\):

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