Properties

Label 6038.2.a.b
Level 6038
Weight 2
Character orbit 6038.a
Self dual Yes
Analytic conductor 48.214
Analytic rank 1
Dimension 54
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: \(1\)
Dimension: \(54\)
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.

\(\operatorname{Tr}(f)(q) = \) \(54q \) \(\mathstrut +\mathstrut 54q^{2} \) \(\mathstrut -\mathstrut 21q^{3} \) \(\mathstrut +\mathstrut 54q^{4} \) \(\mathstrut -\mathstrut 14q^{5} \) \(\mathstrut -\mathstrut 21q^{6} \) \(\mathstrut -\mathstrut 44q^{7} \) \(\mathstrut +\mathstrut 54q^{8} \) \(\mathstrut +\mathstrut 39q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \(54q \) \(\mathstrut +\mathstrut 54q^{2} \) \(\mathstrut -\mathstrut 21q^{3} \) \(\mathstrut +\mathstrut 54q^{4} \) \(\mathstrut -\mathstrut 14q^{5} \) \(\mathstrut -\mathstrut 21q^{6} \) \(\mathstrut -\mathstrut 44q^{7} \) \(\mathstrut +\mathstrut 54q^{8} \) \(\mathstrut +\mathstrut 39q^{9} \) \(\mathstrut -\mathstrut 14q^{10} \) \(\mathstrut -\mathstrut 31q^{11} \) \(\mathstrut -\mathstrut 21q^{12} \) \(\mathstrut -\mathstrut 34q^{13} \) \(\mathstrut -\mathstrut 44q^{14} \) \(\mathstrut -\mathstrut 22q^{15} \) \(\mathstrut +\mathstrut 54q^{16} \) \(\mathstrut -\mathstrut 40q^{17} \) \(\mathstrut +\mathstrut 39q^{18} \) \(\mathstrut -\mathstrut 44q^{19} \) \(\mathstrut -\mathstrut 14q^{20} \) \(\mathstrut -\mathstrut 3q^{21} \) \(\mathstrut -\mathstrut 31q^{22} \) \(\mathstrut -\mathstrut 33q^{23} \) \(\mathstrut -\mathstrut 21q^{24} \) \(\mathstrut +\mathstrut 14q^{25} \) \(\mathstrut -\mathstrut 34q^{26} \) \(\mathstrut -\mathstrut 66q^{27} \) \(\mathstrut -\mathstrut 44q^{28} \) \(\mathstrut -\mathstrut 22q^{30} \) \(\mathstrut -\mathstrut 65q^{31} \) \(\mathstrut +\mathstrut 54q^{32} \) \(\mathstrut -\mathstrut 43q^{33} \) \(\mathstrut -\mathstrut 40q^{34} \) \(\mathstrut -\mathstrut 46q^{35} \) \(\mathstrut +\mathstrut 39q^{36} \) \(\mathstrut -\mathstrut 58q^{37} \) \(\mathstrut -\mathstrut 44q^{38} \) \(\mathstrut -\mathstrut 36q^{39} \) \(\mathstrut -\mathstrut 14q^{40} \) \(\mathstrut -\mathstrut 49q^{41} \) \(\mathstrut -\mathstrut 3q^{42} \) \(\mathstrut -\mathstrut 47q^{43} \) \(\mathstrut -\mathstrut 31q^{44} \) \(\mathstrut -\mathstrut 45q^{45} \) \(\mathstrut -\mathstrut 33q^{46} \) \(\mathstrut -\mathstrut 66q^{47} \) \(\mathstrut -\mathstrut 21q^{48} \) \(\mathstrut +\mathstrut 16q^{49} \) \(\mathstrut +\mathstrut 14q^{50} \) \(\mathstrut -\mathstrut 33q^{51} \) \(\mathstrut -\mathstrut 34q^{52} \) \(\mathstrut -\mathstrut 16q^{53} \) \(\mathstrut -\mathstrut 66q^{54} \) \(\mathstrut -\mathstrut 50q^{55} \) \(\mathstrut -\mathstrut 44q^{56} \) \(\mathstrut -\mathstrut 33q^{57} \) \(\mathstrut -\mathstrut 70q^{59} \) \(\mathstrut -\mathstrut 22q^{60} \) \(\mathstrut -\mathstrut 40q^{61} \) \(\mathstrut -\mathstrut 65q^{62} \) \(\mathstrut -\mathstrut 117q^{63} \) \(\mathstrut +\mathstrut 54q^{64} \) \(\mathstrut -\mathstrut 33q^{65} \) \(\mathstrut -\mathstrut 43q^{66} \) \(\mathstrut -\mathstrut 82q^{67} \) \(\mathstrut -\mathstrut 40q^{68} \) \(\mathstrut -\mathstrut q^{69} \) \(\mathstrut -\mathstrut 46q^{70} \) \(\mathstrut -\mathstrut 60q^{71} \) \(\mathstrut +\mathstrut 39q^{72} \) \(\mathstrut -\mathstrut 92q^{73} \) \(\mathstrut -\mathstrut 58q^{74} \) \(\mathstrut -\mathstrut 68q^{75} \) \(\mathstrut -\mathstrut 44q^{76} \) \(\mathstrut +\mathstrut 13q^{77} \) \(\mathstrut -\mathstrut 36q^{78} \) \(\mathstrut -\mathstrut 57q^{79} \) \(\mathstrut -\mathstrut 14q^{80} \) \(\mathstrut +\mathstrut 26q^{81} \) \(\mathstrut -\mathstrut 49q^{82} \) \(\mathstrut -\mathstrut 77q^{83} \) \(\mathstrut -\mathstrut 3q^{84} \) \(\mathstrut -\mathstrut 24q^{85} \) \(\mathstrut -\mathstrut 47q^{86} \) \(\mathstrut -\mathstrut 61q^{87} \) \(\mathstrut -\mathstrut 31q^{88} \) \(\mathstrut -\mathstrut 54q^{89} \) \(\mathstrut -\mathstrut 45q^{90} \) \(\mathstrut -\mathstrut 46q^{91} \) \(\mathstrut -\mathstrut 33q^{92} \) \(\mathstrut -\mathstrut 24q^{93} \) \(\mathstrut -\mathstrut 66q^{94} \) \(\mathstrut -\mathstrut 66q^{95} \) \(\mathstrut -\mathstrut 21q^{96} \) \(\mathstrut -\mathstrut 137q^{97} \) \(\mathstrut +\mathstrut 16q^{98} \) \(\mathstrut -\mathstrut 71q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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 −3.41141 1.00000 0.211160 −3.41141 −3.35690 1.00000 8.63772 0.211160
1.2 1.00000 −3.37741 1.00000 −2.92787 −3.37741 −0.403159 1.00000 8.40689 −2.92787
1.3 1.00000 −3.34616 1.00000 0.269513 −3.34616 1.67569 1.00000 8.19680 0.269513
1.4 1.00000 −3.17799 1.00000 −0.0583780 −3.17799 −5.13519 1.00000 7.09963 −0.0583780
1.5 1.00000 −2.98721 1.00000 2.38189 −2.98721 −0.220104 1.00000 5.92341 2.38189
1.6 1.00000 −2.82052 1.00000 −0.526511 −2.82052 2.78989 1.00000 4.95531 −0.526511
1.7 1.00000 −2.77973 1.00000 3.72096 −2.77973 −2.52302 1.00000 4.72690 3.72096
1.8 1.00000 −2.56435 1.00000 −4.01554 −2.56435 −3.36655 1.00000 3.57587 −4.01554
1.9 1.00000 −2.48349 1.00000 3.70736 −2.48349 −1.05138 1.00000 3.16770 3.70736
1.10 1.00000 −2.40402 1.00000 −1.73865 −2.40402 4.25060 1.00000 2.77932 −1.73865
1.11 1.00000 −2.32590 1.00000 3.29375 −2.32590 −5.00561 1.00000 2.40980 3.29375
1.12 1.00000 −2.27315 1.00000 −2.64091 −2.27315 0.657447 1.00000 2.16722 −2.64091
1.13 1.00000 −2.11795 1.00000 1.57843 −2.11795 1.37381 1.00000 1.48570 1.57843
1.14 1.00000 −2.02410 1.00000 −4.35432 −2.02410 −1.05145 1.00000 1.09700 −4.35432
1.15 1.00000 −1.99905 1.00000 −3.02278 −1.99905 −5.14057 1.00000 0.996217 −3.02278
1.16 1.00000 −1.93892 1.00000 0.524165 −1.93892 −1.38269 1.00000 0.759396 0.524165
1.17 1.00000 −1.77991 1.00000 −0.815447 −1.77991 −3.69164 1.00000 0.168091 −0.815447
1.18 1.00000 −1.61049 1.00000 1.72129 −1.61049 1.31284 1.00000 −0.406311 1.72129
1.19 1.00000 −1.56837 1.00000 −1.56733 −1.56837 2.76703 1.00000 −0.540224 −1.56733
1.20 1.00000 −1.51709 1.00000 2.00493 −1.51709 2.93208 1.00000 −0.698426 2.00493
See all 54 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.54
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 \(T_{3}^{54} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6038))\).