Properties

Label 6010.2.a.f
Level 6010
Weight 2
Character orbit 6010.a
Self dual Yes
Analytic conductor 47.990
Analytic rank 1
Dimension 22
CM No

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Newspace parameters

Level: \( N \) = \( 6010 = 2 \cdot 5 \cdot 601 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 6010.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(47.9900916148\)
Analytic rank: \(1\)
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.

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

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 −3.26920 1.00000 −1.00000 −3.26920 0.898513 1.00000 7.68765 −1.00000
1.2 1.00000 −3.13304 1.00000 −1.00000 −3.13304 −3.32735 1.00000 6.81597 −1.00000
1.3 1.00000 −2.63651 1.00000 −1.00000 −2.63651 2.17132 1.00000 3.95120 −1.00000
1.4 1.00000 −2.29642 1.00000 −1.00000 −2.29642 −4.86493 1.00000 2.27356 −1.00000
1.5 1.00000 −1.98091 1.00000 −1.00000 −1.98091 −1.31383 1.00000 0.924017 −1.00000
1.6 1.00000 −1.88167 1.00000 −1.00000 −1.88167 −3.66727 1.00000 0.540685 −1.00000
1.7 1.00000 −1.55148 1.00000 −1.00000 −1.55148 2.51624 1.00000 −0.592901 −1.00000
1.8 1.00000 −1.51693 1.00000 −1.00000 −1.51693 −1.44279 1.00000 −0.698933 −1.00000
1.9 1.00000 −1.30771 1.00000 −1.00000 −1.30771 4.54753 1.00000 −1.28989 −1.00000
1.10 1.00000 −1.30184 1.00000 −1.00000 −1.30184 0.511613 1.00000 −1.30521 −1.00000
1.11 1.00000 −0.310973 1.00000 −1.00000 −0.310973 1.47909 1.00000 −2.90330 −1.00000
1.12 1.00000 0.0740024 1.00000 −1.00000 0.0740024 2.79514 1.00000 −2.99452 −1.00000
1.13 1.00000 0.214164 1.00000 −1.00000 0.214164 −0.767476 1.00000 −2.95413 −1.00000
1.14 1.00000 0.530792 1.00000 −1.00000 0.530792 −3.74336 1.00000 −2.71826 −1.00000
1.15 1.00000 0.724060 1.00000 −1.00000 0.724060 −1.42945 1.00000 −2.47574 −1.00000
1.16 1.00000 1.17759 1.00000 −1.00000 1.17759 1.86080 1.00000 −1.61328 −1.00000
1.17 1.00000 1.34202 1.00000 −1.00000 1.34202 2.38251 1.00000 −1.19898 −1.00000
1.18 1.00000 1.73340 1.00000 −1.00000 1.73340 −0.557678 1.00000 0.00467827 −1.00000
1.19 1.00000 2.04538 1.00000 −1.00000 2.04538 −3.30792 1.00000 1.18360 −1.00000
1.20 1.00000 2.14756 1.00000 −1.00000 2.14756 −1.75294 1.00000 1.61200 −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
\(2\) \(-1\)
\(5\) \(1\)
\(601\) \(-1\)

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{3}^{22} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6010))\).