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.

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

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 −3.30817 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4010))\):

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