Properties

Label 6038.2.a.c
Level 6038
Weight 2
Character orbit 6038.a
Self dual Yes
Analytic conductor 48.214
Analytic rank 1
Dimension 57
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: \(57\)
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.

\(\operatorname{Tr}(f)(q) = \) \(57q \) \(\mathstrut -\mathstrut 57q^{2} \) \(\mathstrut -\mathstrut 5q^{3} \) \(\mathstrut +\mathstrut 57q^{4} \) \(\mathstrut -\mathstrut 15q^{5} \) \(\mathstrut +\mathstrut 5q^{6} \) \(\mathstrut -\mathstrut 28q^{7} \) \(\mathstrut -\mathstrut 57q^{8} \) \(\mathstrut +\mathstrut 50q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \(57q \) \(\mathstrut -\mathstrut 57q^{2} \) \(\mathstrut -\mathstrut 5q^{3} \) \(\mathstrut +\mathstrut 57q^{4} \) \(\mathstrut -\mathstrut 15q^{5} \) \(\mathstrut +\mathstrut 5q^{6} \) \(\mathstrut -\mathstrut 28q^{7} \) \(\mathstrut -\mathstrut 57q^{8} \) \(\mathstrut +\mathstrut 50q^{9} \) \(\mathstrut +\mathstrut 15q^{10} \) \(\mathstrut +\mathstrut 13q^{11} \) \(\mathstrut -\mathstrut 5q^{12} \) \(\mathstrut -\mathstrut 43q^{13} \) \(\mathstrut +\mathstrut 28q^{14} \) \(\mathstrut -\mathstrut 10q^{15} \) \(\mathstrut +\mathstrut 57q^{16} \) \(\mathstrut -\mathstrut 50q^{18} \) \(\mathstrut -\mathstrut 6q^{19} \) \(\mathstrut -\mathstrut 15q^{20} \) \(\mathstrut -\mathstrut 23q^{21} \) \(\mathstrut -\mathstrut 13q^{22} \) \(\mathstrut -\mathstrut q^{23} \) \(\mathstrut +\mathstrut 5q^{24} \) \(\mathstrut +\mathstrut 20q^{25} \) \(\mathstrut +\mathstrut 43q^{26} \) \(\mathstrut -\mathstrut 20q^{27} \) \(\mathstrut -\mathstrut 28q^{28} \) \(\mathstrut -\mathstrut 4q^{29} \) \(\mathstrut +\mathstrut 10q^{30} \) \(\mathstrut -\mathstrut 34q^{31} \) \(\mathstrut -\mathstrut 57q^{32} \) \(\mathstrut -\mathstrut 43q^{33} \) \(\mathstrut +\mathstrut 26q^{35} \) \(\mathstrut +\mathstrut 50q^{36} \) \(\mathstrut -\mathstrut 64q^{37} \) \(\mathstrut +\mathstrut 6q^{38} \) \(\mathstrut +\mathstrut 8q^{39} \) \(\mathstrut +\mathstrut 15q^{40} \) \(\mathstrut +\mathstrut 27q^{41} \) \(\mathstrut +\mathstrut 23q^{42} \) \(\mathstrut -\mathstrut 29q^{43} \) \(\mathstrut +\mathstrut 13q^{44} \) \(\mathstrut -\mathstrut 76q^{45} \) \(\mathstrut +\mathstrut q^{46} \) \(\mathstrut -\mathstrut 25q^{47} \) \(\mathstrut -\mathstrut 5q^{48} \) \(\mathstrut +\mathstrut 7q^{49} \) \(\mathstrut -\mathstrut 20q^{50} \) \(\mathstrut +\mathstrut 27q^{51} \) \(\mathstrut -\mathstrut 43q^{52} \) \(\mathstrut -\mathstrut 34q^{53} \) \(\mathstrut +\mathstrut 20q^{54} \) \(\mathstrut -\mathstrut 36q^{55} \) \(\mathstrut +\mathstrut 28q^{56} \) \(\mathstrut -\mathstrut 33q^{57} \) \(\mathstrut +\mathstrut 4q^{58} \) \(\mathstrut +\mathstrut 19q^{59} \) \(\mathstrut -\mathstrut 10q^{60} \) \(\mathstrut -\mathstrut 58q^{61} \) \(\mathstrut +\mathstrut 34q^{62} \) \(\mathstrut -\mathstrut 65q^{63} \) \(\mathstrut +\mathstrut 57q^{64} \) \(\mathstrut +\mathstrut 17q^{65} \) \(\mathstrut +\mathstrut 43q^{66} \) \(\mathstrut -\mathstrut 84q^{67} \) \(\mathstrut -\mathstrut 33q^{69} \) \(\mathstrut -\mathstrut 26q^{70} \) \(\mathstrut +\mathstrut 22q^{71} \) \(\mathstrut -\mathstrut 50q^{72} \) \(\mathstrut -\mathstrut 82q^{73} \) \(\mathstrut +\mathstrut 64q^{74} \) \(\mathstrut +\mathstrut 8q^{75} \) \(\mathstrut -\mathstrut 6q^{76} \) \(\mathstrut -\mathstrut 41q^{77} \) \(\mathstrut -\mathstrut 8q^{78} \) \(\mathstrut +\mathstrut 8q^{79} \) \(\mathstrut -\mathstrut 15q^{80} \) \(\mathstrut +\mathstrut 25q^{81} \) \(\mathstrut -\mathstrut 27q^{82} \) \(\mathstrut -\mathstrut 23q^{83} \) \(\mathstrut -\mathstrut 23q^{84} \) \(\mathstrut -\mathstrut 58q^{85} \) \(\mathstrut +\mathstrut 29q^{86} \) \(\mathstrut -\mathstrut 17q^{87} \) \(\mathstrut -\mathstrut 13q^{88} \) \(\mathstrut +\mathstrut 18q^{89} \) \(\mathstrut +\mathstrut 76q^{90} \) \(\mathstrut -\mathstrut 4q^{91} \) \(\mathstrut -\mathstrut q^{92} \) \(\mathstrut -\mathstrut 60q^{93} \) \(\mathstrut +\mathstrut 25q^{94} \) \(\mathstrut +\mathstrut 36q^{95} \) \(\mathstrut +\mathstrut 5q^{96} \) \(\mathstrut -\mathstrut 156q^{97} \) \(\mathstrut -\mathstrut 7q^{98} \) \(\mathstrut +\mathstrut 25q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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 −3.40132 1.00000 −0.259244 3.40132 0.796720 −1.00000 8.56895 0.259244
1.2 −1.00000 −3.09911 1.00000 −3.06696 3.09911 −3.37062 −1.00000 6.60450 3.06696
1.3 −1.00000 −3.09194 1.00000 −2.28798 3.09194 0.749696 −1.00000 6.56008 2.28798
1.4 −1.00000 −2.99307 1.00000 2.62099 2.99307 −1.28357 −1.00000 5.95844 −2.62099
1.5 −1.00000 −2.94445 1.00000 −1.42608 2.94445 1.57265 −1.00000 5.66979 1.42608
1.6 −1.00000 −2.85836 1.00000 −3.68447 2.85836 2.06392 −1.00000 5.17024 3.68447
1.7 −1.00000 −2.75036 1.00000 0.139827 2.75036 0.205180 −1.00000 4.56449 −0.139827
1.8 −1.00000 −2.68115 1.00000 1.78050 2.68115 −4.24119 −1.00000 4.18856 −1.78050
1.9 −1.00000 −2.46647 1.00000 3.27492 2.46647 3.14216 −1.00000 3.08346 −3.27492
1.10 −1.00000 −2.39662 1.00000 −1.22776 2.39662 −4.40908 −1.00000 2.74377 1.22776
1.11 −1.00000 −2.25784 1.00000 0.481380 2.25784 −0.819021 −1.00000 2.09784 −0.481380
1.12 −1.00000 −2.15416 1.00000 1.67498 2.15416 2.16308 −1.00000 1.64039 −1.67498
1.13 −1.00000 −2.12089 1.00000 −1.22072 2.12089 −2.75912 −1.00000 1.49816 1.22072
1.14 −1.00000 −1.79555 1.00000 3.22751 1.79555 −1.51507 −1.00000 0.223993 −3.22751
1.15 −1.00000 −1.71541 1.00000 1.06645 1.71541 4.85447 −1.00000 −0.0573586 −1.06645
1.16 −1.00000 −1.68023 1.00000 −2.51181 1.68023 3.66839 −1.00000 −0.176828 2.51181
1.17 −1.00000 −1.57568 1.00000 1.43358 1.57568 2.19919 −1.00000 −0.517233 −1.43358
1.18 −1.00000 −1.49224 1.00000 −4.20825 1.49224 −4.82054 −1.00000 −0.773233 4.20825
1.19 −1.00000 −1.42937 1.00000 −1.03254 1.42937 −0.605000 −1.00000 −0.956891 1.03254
1.20 −1.00000 −1.21264 1.00000 1.98344 1.21264 −0.999136 −1.00000 −1.52951 −1.98344
See all 57 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.57
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 \(T_{3}^{57} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6038))\).