Properties

Label 6038.2.a.d
Level 6038
Weight 2
Character orbit 6038.a
Self dual Yes
Analytic conductor 48.214
Analytic rank 0
Dimension 69
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: \(69\)
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.

\(\operatorname{Tr}(f)(q) = \) \(69q \) \(\mathstrut -\mathstrut 69q^{2} \) \(\mathstrut +\mathstrut 8q^{3} \) \(\mathstrut +\mathstrut 69q^{4} \) \(\mathstrut +\mathstrut 18q^{5} \) \(\mathstrut -\mathstrut 8q^{6} \) \(\mathstrut +\mathstrut 32q^{7} \) \(\mathstrut -\mathstrut 69q^{8} \) \(\mathstrut +\mathstrut 79q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \(69q \) \(\mathstrut -\mathstrut 69q^{2} \) \(\mathstrut +\mathstrut 8q^{3} \) \(\mathstrut +\mathstrut 69q^{4} \) \(\mathstrut +\mathstrut 18q^{5} \) \(\mathstrut -\mathstrut 8q^{6} \) \(\mathstrut +\mathstrut 32q^{7} \) \(\mathstrut -\mathstrut 69q^{8} \) \(\mathstrut +\mathstrut 79q^{9} \) \(\mathstrut -\mathstrut 18q^{10} \) \(\mathstrut -\mathstrut 4q^{11} \) \(\mathstrut +\mathstrut 8q^{12} \) \(\mathstrut +\mathstrut 40q^{13} \) \(\mathstrut -\mathstrut 32q^{14} \) \(\mathstrut +\mathstrut 6q^{15} \) \(\mathstrut +\mathstrut 69q^{16} \) \(\mathstrut +\mathstrut 6q^{17} \) \(\mathstrut -\mathstrut 79q^{18} \) \(\mathstrut +\mathstrut 13q^{19} \) \(\mathstrut +\mathstrut 18q^{20} \) \(\mathstrut +\mathstrut 23q^{21} \) \(\mathstrut +\mathstrut 4q^{22} \) \(\mathstrut +\mathstrut 11q^{23} \) \(\mathstrut -\mathstrut 8q^{24} \) \(\mathstrut +\mathstrut 111q^{25} \) \(\mathstrut -\mathstrut 40q^{26} \) \(\mathstrut +\mathstrut 32q^{27} \) \(\mathstrut +\mathstrut 32q^{28} \) \(\mathstrut +\mathstrut 23q^{29} \) \(\mathstrut -\mathstrut 6q^{30} \) \(\mathstrut +\mathstrut 30q^{31} \) \(\mathstrut -\mathstrut 69q^{32} \) \(\mathstrut +\mathstrut 37q^{33} \) \(\mathstrut -\mathstrut 6q^{34} \) \(\mathstrut -\mathstrut 12q^{35} \) \(\mathstrut +\mathstrut 79q^{36} \) \(\mathstrut +\mathstrut 81q^{37} \) \(\mathstrut -\mathstrut 13q^{38} \) \(\mathstrut -\mathstrut 8q^{39} \) \(\mathstrut -\mathstrut 18q^{40} \) \(\mathstrut -\mathstrut 13q^{41} \) \(\mathstrut -\mathstrut 23q^{42} \) \(\mathstrut +\mathstrut 42q^{43} \) \(\mathstrut -\mathstrut 4q^{44} \) \(\mathstrut +\mathstrut 89q^{45} \) \(\mathstrut -\mathstrut 11q^{46} \) \(\mathstrut +\mathstrut 35q^{47} \) \(\mathstrut +\mathstrut 8q^{48} \) \(\mathstrut +\mathstrut 115q^{49} \) \(\mathstrut -\mathstrut 111q^{50} \) \(\mathstrut -\mathstrut 21q^{51} \) \(\mathstrut +\mathstrut 40q^{52} \) \(\mathstrut +\mathstrut 41q^{53} \) \(\mathstrut -\mathstrut 32q^{54} \) \(\mathstrut +\mathstrut 28q^{55} \) \(\mathstrut -\mathstrut 32q^{56} \) \(\mathstrut +\mathstrut 39q^{57} \) \(\mathstrut -\mathstrut 23q^{58} \) \(\mathstrut -\mathstrut 24q^{59} \) \(\mathstrut +\mathstrut 6q^{60} \) \(\mathstrut +\mathstrut 69q^{61} \) \(\mathstrut -\mathstrut 30q^{62} \) \(\mathstrut +\mathstrut 79q^{63} \) \(\mathstrut +\mathstrut 69q^{64} \) \(\mathstrut +\mathstrut 15q^{65} \) \(\mathstrut -\mathstrut 37q^{66} \) \(\mathstrut +\mathstrut 87q^{67} \) \(\mathstrut +\mathstrut 6q^{68} \) \(\mathstrut +\mathstrut 51q^{69} \) \(\mathstrut +\mathstrut 12q^{70} \) \(\mathstrut -\mathstrut 22q^{71} \) \(\mathstrut -\mathstrut 79q^{72} \) \(\mathstrut +\mathstrut 104q^{73} \) \(\mathstrut -\mathstrut 81q^{74} \) \(\mathstrut +\mathstrut 39q^{75} \) \(\mathstrut +\mathstrut 13q^{76} \) \(\mathstrut +\mathstrut 47q^{77} \) \(\mathstrut +\mathstrut 8q^{78} \) \(\mathstrut +\mathstrut 16q^{79} \) \(\mathstrut +\mathstrut 18q^{80} \) \(\mathstrut +\mathstrut 105q^{81} \) \(\mathstrut +\mathstrut 13q^{82} \) \(\mathstrut +\mathstrut 30q^{83} \) \(\mathstrut +\mathstrut 23q^{84} \) \(\mathstrut +\mathstrut 74q^{85} \) \(\mathstrut -\mathstrut 42q^{86} \) \(\mathstrut +\mathstrut 47q^{87} \) \(\mathstrut +\mathstrut 4q^{88} \) \(\mathstrut -\mathstrut 4q^{89} \) \(\mathstrut -\mathstrut 89q^{90} \) \(\mathstrut +\mathstrut 70q^{91} \) \(\mathstrut +\mathstrut 11q^{92} \) \(\mathstrut +\mathstrut 104q^{93} \) \(\mathstrut -\mathstrut 35q^{94} \) \(\mathstrut -\mathstrut 36q^{95} \) \(\mathstrut -\mathstrut 8q^{96} \) \(\mathstrut +\mathstrut 158q^{97} \) \(\mathstrut -\mathstrut 115q^{98} \) \(\mathstrut +\mathstrut 34q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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 −3.38264 1.00000 3.55013 3.38264 −0.929098 −1.00000 8.44223 −3.55013
1.2 −1.00000 −3.22022 1.00000 2.70863 3.22022 4.47236 −1.00000 7.36980 −2.70863
1.3 −1.00000 −3.17982 1.00000 −0.0288660 3.17982 0.302400 −1.00000 7.11126 0.0288660
1.4 −1.00000 −2.96065 1.00000 0.357077 2.96065 −2.81647 −1.00000 5.76543 −0.357077
1.5 −1.00000 −2.91924 1.00000 −2.24533 2.91924 −3.88903 −1.00000 5.52197 2.24533
1.6 −1.00000 −2.88111 1.00000 2.68970 2.88111 4.10918 −1.00000 5.30078 −2.68970
1.7 −1.00000 −2.79695 1.00000 −3.22250 2.79695 3.03239 −1.00000 4.82293 3.22250
1.8 −1.00000 −2.63337 1.00000 −0.783693 2.63337 4.82371 −1.00000 3.93463 0.783693
1.9 −1.00000 −2.40625 1.00000 −0.733460 2.40625 −2.00296 −1.00000 2.79002 0.733460
1.10 −1.00000 −2.29973 1.00000 4.30527 2.29973 −2.14109 −1.00000 2.28878 −4.30527
1.11 −1.00000 −2.23277 1.00000 1.71292 2.23277 0.0975688 −1.00000 1.98527 −1.71292
1.12 −1.00000 −2.22571 1.00000 1.16705 2.22571 1.13637 −1.00000 1.95376 −1.16705
1.13 −1.00000 −2.18407 1.00000 2.82880 2.18407 −4.34635 −1.00000 1.77018 −2.82880
1.14 −1.00000 −2.15476 1.00000 −2.42408 2.15476 −0.124964 −1.00000 1.64299 2.42408
1.15 −1.00000 −2.10359 1.00000 2.00437 2.10359 2.39594 −1.00000 1.42511 −2.00437
1.16 −1.00000 −2.10090 1.00000 −3.50554 2.10090 3.00259 −1.00000 1.41376 3.50554
1.17 −1.00000 −1.78840 1.00000 −4.39275 1.78840 −0.240516 −1.00000 0.198361 4.39275
1.18 −1.00000 −1.59881 1.00000 4.26429 1.59881 −0.810088 −1.00000 −0.443805 −4.26429
1.19 −1.00000 −1.58255 1.00000 −3.72234 1.58255 −1.18949 −1.00000 −0.495522 3.72234
1.20 −1.00000 −1.49405 1.00000 −0.329476 1.49405 1.19581 −1.00000 −0.767828 0.329476
See all 69 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.69
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 \(T_{3}^{69} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6038))\).