Properties

Label 6022.2.a.d
Level 6022
Weight 2
Character orbit 6022.a
Self dual Yes
Analytic conductor 48.086
Analytic rank 1
Dimension 64
CM No

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Newspace parameters

Level: \( N \) = \( 6022 = 2 \cdot 3011 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 6022.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(48.0859120972\)
Analytic rank: \(1\)
Dimension: \(64\)
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.

\(\operatorname{Tr}(f)(q) = \) \(64q \) \(\mathstrut -\mathstrut 64q^{2} \) \(\mathstrut -\mathstrut 9q^{3} \) \(\mathstrut +\mathstrut 64q^{4} \) \(\mathstrut -\mathstrut 17q^{5} \) \(\mathstrut +\mathstrut 9q^{6} \) \(\mathstrut -\mathstrut 2q^{7} \) \(\mathstrut -\mathstrut 64q^{8} \) \(\mathstrut +\mathstrut 61q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \(64q \) \(\mathstrut -\mathstrut 64q^{2} \) \(\mathstrut -\mathstrut 9q^{3} \) \(\mathstrut +\mathstrut 64q^{4} \) \(\mathstrut -\mathstrut 17q^{5} \) \(\mathstrut +\mathstrut 9q^{6} \) \(\mathstrut -\mathstrut 2q^{7} \) \(\mathstrut -\mathstrut 64q^{8} \) \(\mathstrut +\mathstrut 61q^{9} \) \(\mathstrut +\mathstrut 17q^{10} \) \(\mathstrut -\mathstrut 15q^{11} \) \(\mathstrut -\mathstrut 9q^{12} \) \(\mathstrut -\mathstrut 28q^{13} \) \(\mathstrut +\mathstrut 2q^{14} \) \(\mathstrut +\mathstrut 64q^{16} \) \(\mathstrut -\mathstrut 62q^{17} \) \(\mathstrut -\mathstrut 61q^{18} \) \(\mathstrut +\mathstrut 24q^{19} \) \(\mathstrut -\mathstrut 17q^{20} \) \(\mathstrut -\mathstrut 20q^{21} \) \(\mathstrut +\mathstrut 15q^{22} \) \(\mathstrut -\mathstrut 41q^{23} \) \(\mathstrut +\mathstrut 9q^{24} \) \(\mathstrut +\mathstrut 61q^{25} \) \(\mathstrut +\mathstrut 28q^{26} \) \(\mathstrut -\mathstrut 36q^{27} \) \(\mathstrut -\mathstrut 2q^{28} \) \(\mathstrut -\mathstrut 45q^{29} \) \(\mathstrut +\mathstrut 40q^{31} \) \(\mathstrut -\mathstrut 64q^{32} \) \(\mathstrut -\mathstrut 36q^{33} \) \(\mathstrut +\mathstrut 62q^{34} \) \(\mathstrut -\mathstrut 59q^{35} \) \(\mathstrut +\mathstrut 61q^{36} \) \(\mathstrut -\mathstrut 27q^{37} \) \(\mathstrut -\mathstrut 24q^{38} \) \(\mathstrut +\mathstrut 5q^{39} \) \(\mathstrut +\mathstrut 17q^{40} \) \(\mathstrut -\mathstrut 42q^{41} \) \(\mathstrut +\mathstrut 20q^{42} \) \(\mathstrut -\mathstrut 25q^{43} \) \(\mathstrut -\mathstrut 15q^{44} \) \(\mathstrut -\mathstrut 47q^{45} \) \(\mathstrut +\mathstrut 41q^{46} \) \(\mathstrut -\mathstrut 64q^{47} \) \(\mathstrut -\mathstrut 9q^{48} \) \(\mathstrut +\mathstrut 76q^{49} \) \(\mathstrut -\mathstrut 61q^{50} \) \(\mathstrut +\mathstrut 5q^{51} \) \(\mathstrut -\mathstrut 28q^{52} \) \(\mathstrut -\mathstrut 70q^{53} \) \(\mathstrut +\mathstrut 36q^{54} \) \(\mathstrut +\mathstrut 9q^{55} \) \(\mathstrut +\mathstrut 2q^{56} \) \(\mathstrut -\mathstrut 47q^{57} \) \(\mathstrut +\mathstrut 45q^{58} \) \(\mathstrut -\mathstrut 17q^{59} \) \(\mathstrut -\mathstrut 52q^{61} \) \(\mathstrut -\mathstrut 40q^{62} \) \(\mathstrut -\mathstrut 36q^{63} \) \(\mathstrut +\mathstrut 64q^{64} \) \(\mathstrut -\mathstrut 49q^{65} \) \(\mathstrut +\mathstrut 36q^{66} \) \(\mathstrut +\mathstrut 5q^{67} \) \(\mathstrut -\mathstrut 62q^{68} \) \(\mathstrut -\mathstrut 69q^{69} \) \(\mathstrut +\mathstrut 59q^{70} \) \(\mathstrut -\mathstrut 9q^{71} \) \(\mathstrut -\mathstrut 61q^{72} \) \(\mathstrut -\mathstrut 39q^{73} \) \(\mathstrut +\mathstrut 27q^{74} \) \(\mathstrut -\mathstrut 28q^{75} \) \(\mathstrut +\mathstrut 24q^{76} \) \(\mathstrut -\mathstrut 149q^{77} \) \(\mathstrut -\mathstrut 5q^{78} \) \(\mathstrut +\mathstrut 31q^{79} \) \(\mathstrut -\mathstrut 17q^{80} \) \(\mathstrut +\mathstrut 52q^{81} \) \(\mathstrut +\mathstrut 42q^{82} \) \(\mathstrut -\mathstrut 121q^{83} \) \(\mathstrut -\mathstrut 20q^{84} \) \(\mathstrut -\mathstrut 54q^{85} \) \(\mathstrut +\mathstrut 25q^{86} \) \(\mathstrut -\mathstrut 78q^{87} \) \(\mathstrut +\mathstrut 15q^{88} \) \(\mathstrut -\mathstrut 24q^{89} \) \(\mathstrut +\mathstrut 47q^{90} \) \(\mathstrut +\mathstrut 74q^{91} \) \(\mathstrut -\mathstrut 41q^{92} \) \(\mathstrut -\mathstrut 74q^{93} \) \(\mathstrut +\mathstrut 64q^{94} \) \(\mathstrut -\mathstrut 74q^{95} \) \(\mathstrut +\mathstrut 9q^{96} \) \(\mathstrut -\mathstrut 5q^{97} \) \(\mathstrut -\mathstrut 76q^{98} \) \(\mathstrut -\mathstrut 34q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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 −3.41719 1.00000 0.0151085 3.41719 −2.13229 −1.00000 8.67716 −0.0151085
1.2 −1.00000 −3.28572 1.00000 −3.45147 3.28572 −1.04970 −1.00000 7.79593 3.45147
1.3 −1.00000 −3.15898 1.00000 −0.473594 3.15898 −3.23851 −1.00000 6.97917 0.473594
1.4 −1.00000 −3.12891 1.00000 3.39803 3.12891 2.97458 −1.00000 6.79008 −3.39803
1.5 −1.00000 −3.12816 1.00000 −3.78389 3.12816 3.19899 −1.00000 6.78536 3.78389
1.6 −1.00000 −3.07409 1.00000 0.701582 3.07409 1.60448 −1.00000 6.45002 −0.701582
1.7 −1.00000 −2.84165 1.00000 −2.41894 2.84165 4.66695 −1.00000 5.07496 2.41894
1.8 −1.00000 −2.65762 1.00000 −0.0671999 2.65762 1.45309 −1.00000 4.06295 0.0671999
1.9 −1.00000 −2.46766 1.00000 0.852564 2.46766 −2.40231 −1.00000 3.08936 −0.852564
1.10 −1.00000 −2.38767 1.00000 3.65260 2.38767 −5.18773 −1.00000 2.70099 −3.65260
1.11 −1.00000 −2.35063 1.00000 −2.93009 2.35063 −2.60832 −1.00000 2.52545 2.93009
1.12 −1.00000 −2.23814 1.00000 3.11624 2.23814 2.78730 −1.00000 2.00926 −3.11624
1.13 −1.00000 −2.21674 1.00000 1.41824 2.21674 −0.343724 −1.00000 1.91395 −1.41824
1.14 −1.00000 −2.19043 1.00000 −0.477734 2.19043 −4.48401 −1.00000 1.79797 0.477734
1.15 −1.00000 −2.16331 1.00000 −3.33265 2.16331 3.92579 −1.00000 1.67989 3.33265
1.16 −1.00000 −2.12241 1.00000 −2.62309 2.12241 −2.27034 −1.00000 1.50461 2.62309
1.17 −1.00000 −2.10760 1.00000 3.51120 2.10760 0.432948 −1.00000 1.44198 −3.51120
1.18 −1.00000 −1.92893 1.00000 1.84484 1.92893 2.06972 −1.00000 0.720763 −1.84484
1.19 −1.00000 −1.85756 1.00000 −0.233219 1.85756 0.276932 −1.00000 0.450535 0.233219
1.20 −1.00000 −1.43853 1.00000 −4.29281 1.43853 1.20415 −1.00000 −0.930640 4.29281
See all 64 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.64
Significant digits:
Format:

Inner twists

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

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3011\) \(1\)

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{3}^{64} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6022))\).