Properties

Label 8015.2.a.k
Level 8015
Weight 2
Character orbit 8015.a
Self dual Yes
Analytic conductor 64.000
Analytic rank 1
Dimension 49
CM No

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Newspace parameters

Level: \( N \) = \( 8015 = 5 \cdot 7 \cdot 229 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 8015.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(64.0000972201\)
Analytic rank: \(1\)
Dimension: \(49\)
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.

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

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 −2.67776 −3.22300 5.17039 −1.00000 8.63041 −1.00000 −8.48955 7.38771 2.67776
1.2 −2.66490 2.35676 5.10167 −1.00000 −6.28053 −1.00000 −8.26563 2.55434 2.66490
1.3 −2.62540 −0.161903 4.89272 −1.00000 0.425059 −1.00000 −7.59456 −2.97379 2.62540
1.4 −2.56687 −1.38440 4.58882 −1.00000 3.55358 −1.00000 −6.64517 −1.08344 2.56687
1.5 −2.52338 −1.42453 4.36744 −1.00000 3.59462 −1.00000 −5.97394 −0.970719 2.52338
1.6 −2.33828 2.05266 3.46754 −1.00000 −4.79969 −1.00000 −3.43150 1.21342 2.33828
1.7 −2.29229 −2.84815 3.25461 −1.00000 6.52878 −1.00000 −2.87593 5.11193 2.29229
1.8 −2.08886 −0.324198 2.36335 −1.00000 0.677204 −1.00000 −0.758978 −2.89490 2.08886
1.9 −2.05694 0.523578 2.23099 −1.00000 −1.07697 −1.00000 −0.475131 −2.72587 2.05694
1.10 −1.92336 −1.33966 1.69931 −1.00000 2.57665 −1.00000 0.578330 −1.20531 1.92336
1.11 −1.91964 0.294978 1.68500 −1.00000 −0.566250 −1.00000 0.604679 −2.91299 1.91964
1.12 −1.80610 2.35882 1.26200 −1.00000 −4.26027 −1.00000 1.33289 2.56402 1.80610
1.13 −1.73153 −2.59444 0.998212 −1.00000 4.49236 −1.00000 1.73463 3.73110 1.73153
1.14 −1.56783 2.04149 0.458083 −1.00000 −3.20071 −1.00000 2.41746 1.16770 1.56783
1.15 −1.36146 −0.153831 −0.146418 −1.00000 0.209435 −1.00000 2.92227 −2.97634 1.36146
1.16 −1.23019 2.26268 −0.486637 −1.00000 −2.78352 −1.00000 3.05903 2.11970 1.23019
1.17 −1.12638 −2.96921 −0.731259 −1.00000 3.34447 −1.00000 3.07645 5.81619 1.12638
1.18 −0.978041 −3.27758 −1.04343 −1.00000 3.20561 −1.00000 2.97661 7.74254 0.978041
1.19 −0.788296 −1.23031 −1.37859 −1.00000 0.969852 −1.00000 2.66333 −1.48633 0.788296
1.20 −0.736107 −1.41103 −1.45815 −1.00000 1.03867 −1.00000 2.54557 −1.00899 0.736107
See all 49 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.49
Significant digits:
Format:

Inner twists

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

Atkin-Lehner signs

\( p \) Sign
\(5\) \(1\)
\(7\) \(1\)
\(229\) \(1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8015))\):

\(T_{2}^{49} + \cdots\)
\(T_{3}^{49} + \cdots\)