Properties

Label 6038.2.a.e
Level 6038
Weight 2
Character orbit 6038.a
Self dual Yes
Analytic conductor 48.214
Analytic rank 0
Dimension 70
CM No

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Newspace parameters

Level: \( N \) = \( 6038 = 2 \cdot 3019 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 6038.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(48.2136727404\)
Analytic rank: \(0\)
Dimension: \(70\)
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.

\(\operatorname{Tr}(f)(q) = \) \(70q \) \(\mathstrut +\mathstrut 70q^{2} \) \(\mathstrut +\mathstrut 25q^{3} \) \(\mathstrut +\mathstrut 70q^{4} \) \(\mathstrut +\mathstrut 18q^{5} \) \(\mathstrut +\mathstrut 25q^{6} \) \(\mathstrut +\mathstrut 50q^{7} \) \(\mathstrut +\mathstrut 70q^{8} \) \(\mathstrut +\mathstrut 89q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \(70q \) \(\mathstrut +\mathstrut 70q^{2} \) \(\mathstrut +\mathstrut 25q^{3} \) \(\mathstrut +\mathstrut 70q^{4} \) \(\mathstrut +\mathstrut 18q^{5} \) \(\mathstrut +\mathstrut 25q^{6} \) \(\mathstrut +\mathstrut 50q^{7} \) \(\mathstrut +\mathstrut 70q^{8} \) \(\mathstrut +\mathstrut 89q^{9} \) \(\mathstrut +\mathstrut 18q^{10} \) \(\mathstrut +\mathstrut 41q^{11} \) \(\mathstrut +\mathstrut 25q^{12} \) \(\mathstrut +\mathstrut 41q^{13} \) \(\mathstrut +\mathstrut 50q^{14} \) \(\mathstrut +\mathstrut 13q^{15} \) \(\mathstrut +\mathstrut 70q^{16} \) \(\mathstrut +\mathstrut 40q^{17} \) \(\mathstrut +\mathstrut 89q^{18} \) \(\mathstrut +\mathstrut 55q^{19} \) \(\mathstrut +\mathstrut 18q^{20} \) \(\mathstrut +\mathstrut 2q^{21} \) \(\mathstrut +\mathstrut 41q^{22} \) \(\mathstrut +\mathstrut 41q^{23} \) \(\mathstrut +\mathstrut 25q^{24} \) \(\mathstrut +\mathstrut 104q^{25} \) \(\mathstrut +\mathstrut 41q^{26} \) \(\mathstrut +\mathstrut 82q^{27} \) \(\mathstrut +\mathstrut 50q^{28} \) \(\mathstrut +\mathstrut 11q^{29} \) \(\mathstrut +\mathstrut 13q^{30} \) \(\mathstrut +\mathstrut 78q^{31} \) \(\mathstrut +\mathstrut 70q^{32} \) \(\mathstrut +\mathstrut 45q^{33} \) \(\mathstrut +\mathstrut 40q^{34} \) \(\mathstrut +\mathstrut 25q^{35} \) \(\mathstrut +\mathstrut 89q^{36} \) \(\mathstrut +\mathstrut 46q^{37} \) \(\mathstrut +\mathstrut 55q^{38} \) \(\mathstrut +\mathstrut 19q^{39} \) \(\mathstrut +\mathstrut 18q^{40} \) \(\mathstrut +\mathstrut 51q^{41} \) \(\mathstrut +\mathstrut 2q^{42} \) \(\mathstrut +\mathstrut 68q^{43} \) \(\mathstrut +\mathstrut 41q^{44} \) \(\mathstrut +\mathstrut 37q^{45} \) \(\mathstrut +\mathstrut 41q^{46} \) \(\mathstrut +\mathstrut 69q^{47} \) \(\mathstrut +\mathstrut 25q^{48} \) \(\mathstrut +\mathstrut 126q^{49} \) \(\mathstrut +\mathstrut 104q^{50} \) \(\mathstrut +\mathstrut 36q^{51} \) \(\mathstrut +\mathstrut 41q^{52} \) \(\mathstrut +\mathstrut 23q^{53} \) \(\mathstrut +\mathstrut 82q^{54} \) \(\mathstrut +\mathstrut 42q^{55} \) \(\mathstrut +\mathstrut 50q^{56} \) \(\mathstrut +\mathstrut 14q^{57} \) \(\mathstrut +\mathstrut 11q^{58} \) \(\mathstrut +\mathstrut 89q^{59} \) \(\mathstrut +\mathstrut 13q^{60} \) \(\mathstrut +\mathstrut 32q^{61} \) \(\mathstrut +\mathstrut 78q^{62} \) \(\mathstrut +\mathstrut 106q^{63} \) \(\mathstrut +\mathstrut 70q^{64} \) \(\mathstrut +\mathstrut 18q^{65} \) \(\mathstrut +\mathstrut 45q^{66} \) \(\mathstrut +\mathstrut 90q^{67} \) \(\mathstrut +\mathstrut 40q^{68} \) \(\mathstrut -\mathstrut 12q^{69} \) \(\mathstrut +\mathstrut 25q^{70} \) \(\mathstrut +\mathstrut 54q^{71} \) \(\mathstrut +\mathstrut 89q^{72} \) \(\mathstrut +\mathstrut 94q^{73} \) \(\mathstrut +\mathstrut 46q^{74} \) \(\mathstrut +\mathstrut 72q^{75} \) \(\mathstrut +\mathstrut 55q^{76} \) \(\mathstrut -\mathstrut 16q^{77} \) \(\mathstrut +\mathstrut 19q^{78} \) \(\mathstrut +\mathstrut 54q^{79} \) \(\mathstrut +\mathstrut 18q^{80} \) \(\mathstrut +\mathstrut 102q^{81} \) \(\mathstrut +\mathstrut 51q^{82} \) \(\mathstrut +\mathstrut 60q^{83} \) \(\mathstrut +\mathstrut 2q^{84} \) \(\mathstrut -\mathstrut 5q^{85} \) \(\mathstrut +\mathstrut 68q^{86} \) \(\mathstrut +\mathstrut 9q^{87} \) \(\mathstrut +\mathstrut 41q^{88} \) \(\mathstrut +\mathstrut 77q^{89} \) \(\mathstrut +\mathstrut 37q^{90} \) \(\mathstrut +\mathstrut 54q^{91} \) \(\mathstrut +\mathstrut 41q^{92} \) \(\mathstrut -\mathstrut 2q^{93} \) \(\mathstrut +\mathstrut 69q^{94} \) \(\mathstrut +\mathstrut 39q^{95} \) \(\mathstrut +\mathstrut 25q^{96} \) \(\mathstrut +\mathstrut 139q^{97} \) \(\mathstrut +\mathstrut 126q^{98} \) \(\mathstrut +\mathstrut 60q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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 −3.07733 1.00000 0.607795 −3.07733 4.61887 1.00000 6.46998 0.607795
1.2 1.00000 −3.00085 1.00000 −0.777425 −3.00085 3.31557 1.00000 6.00511 −0.777425
1.3 1.00000 −2.94455 1.00000 3.92217 −2.94455 4.11176 1.00000 5.67038 3.92217
1.4 1.00000 −2.93737 1.00000 −3.45627 −2.93737 2.34218 1.00000 5.62815 −3.45627
1.5 1.00000 −2.81205 1.00000 2.74866 −2.81205 −2.36582 1.00000 4.90764 2.74866
1.6 1.00000 −2.80845 1.00000 1.85496 −2.80845 −0.963784 1.00000 4.88737 1.85496
1.7 1.00000 −2.79976 1.00000 −2.91282 −2.79976 −0.947335 1.00000 4.83868 −2.91282
1.8 1.00000 −2.73176 1.00000 1.22157 −2.73176 −0.448839 1.00000 4.46253 1.22157
1.9 1.00000 −2.44970 1.00000 −2.67633 −2.44970 2.27602 1.00000 3.00102 −2.67633
1.10 1.00000 −2.34935 1.00000 −0.768821 −2.34935 −1.35127 1.00000 2.51947 −0.768821
1.11 1.00000 −2.23514 1.00000 3.00438 −2.23514 1.56958 1.00000 1.99587 3.00438
1.12 1.00000 −2.08835 1.00000 −0.884909 −2.08835 −2.93795 1.00000 1.36120 −0.884909
1.13 1.00000 −2.05123 1.00000 3.94879 −2.05123 5.22685 1.00000 1.20754 3.94879
1.14 1.00000 −1.76800 1.00000 1.61299 −1.76800 3.26382 1.00000 0.125835 1.61299
1.15 1.00000 −1.60124 1.00000 −1.76472 −1.60124 −0.621570 1.00000 −0.436015 −1.76472
1.16 1.00000 −1.50199 1.00000 −3.07076 −1.50199 4.02705 1.00000 −0.744026 −3.07076
1.17 1.00000 −1.49174 1.00000 1.96904 −1.49174 1.25123 1.00000 −0.774726 1.96904
1.18 1.00000 −1.19115 1.00000 −1.23162 −1.19115 −2.44516 1.00000 −1.58116 −1.23162
1.19 1.00000 −1.12952 1.00000 −0.938247 −1.12952 0.310012 1.00000 −1.72418 −0.938247
1.20 1.00000 −1.00890 1.00000 1.23789 −1.00890 −4.49878 1.00000 −1.98213 1.23789
See all 70 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.70
Significant digits:
Format:

Inner twists

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

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3019\) \(1\)

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{3}^{70} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6038))\).