Properties

Label 8015.2.a.h
Level 8015
Weight 2
Character orbit 8015.a
Self dual Yes
Analytic conductor 64.000
Analytic rank 1
Dimension 38
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: \(38\)
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.

\(\operatorname{Tr}(f)(q) = \) \(38q \) \(\mathstrut -\mathstrut 6q^{2} \) \(\mathstrut -\mathstrut 9q^{3} \) \(\mathstrut +\mathstrut 24q^{4} \) \(\mathstrut +\mathstrut 38q^{5} \) \(\mathstrut -\mathstrut 10q^{6} \) \(\mathstrut +\mathstrut 38q^{7} \) \(\mathstrut -\mathstrut 21q^{8} \) \(\mathstrut +\mathstrut 13q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \(38q \) \(\mathstrut -\mathstrut 6q^{2} \) \(\mathstrut -\mathstrut 9q^{3} \) \(\mathstrut +\mathstrut 24q^{4} \) \(\mathstrut +\mathstrut 38q^{5} \) \(\mathstrut -\mathstrut 10q^{6} \) \(\mathstrut +\mathstrut 38q^{7} \) \(\mathstrut -\mathstrut 21q^{8} \) \(\mathstrut +\mathstrut 13q^{9} \) \(\mathstrut -\mathstrut 6q^{10} \) \(\mathstrut -\mathstrut 18q^{11} \) \(\mathstrut -\mathstrut 20q^{12} \) \(\mathstrut -\mathstrut 25q^{13} \) \(\mathstrut -\mathstrut 6q^{14} \) \(\mathstrut -\mathstrut 9q^{15} \) \(\mathstrut -\mathstrut 21q^{17} \) \(\mathstrut -\mathstrut 7q^{18} \) \(\mathstrut -\mathstrut 14q^{19} \) \(\mathstrut +\mathstrut 24q^{20} \) \(\mathstrut -\mathstrut 9q^{21} \) \(\mathstrut -\mathstrut 11q^{22} \) \(\mathstrut -\mathstrut 16q^{23} \) \(\mathstrut -\mathstrut 10q^{24} \) \(\mathstrut +\mathstrut 38q^{25} \) \(\mathstrut -\mathstrut 21q^{26} \) \(\mathstrut -\mathstrut 15q^{27} \) \(\mathstrut +\mathstrut 24q^{28} \) \(\mathstrut -\mathstrut 52q^{29} \) \(\mathstrut -\mathstrut 10q^{30} \) \(\mathstrut -\mathstrut 30q^{31} \) \(\mathstrut -\mathstrut 8q^{32} \) \(\mathstrut -\mathstrut 13q^{33} \) \(\mathstrut -\mathstrut 9q^{34} \) \(\mathstrut +\mathstrut 38q^{35} \) \(\mathstrut -\mathstrut 16q^{36} \) \(\mathstrut -\mathstrut 47q^{37} \) \(\mathstrut -\mathstrut 10q^{38} \) \(\mathstrut -\mathstrut 50q^{39} \) \(\mathstrut -\mathstrut 21q^{40} \) \(\mathstrut -\mathstrut 35q^{41} \) \(\mathstrut -\mathstrut 10q^{42} \) \(\mathstrut -\mathstrut 18q^{43} \) \(\mathstrut -\mathstrut 22q^{44} \) \(\mathstrut +\mathstrut 13q^{45} \) \(\mathstrut -\mathstrut 17q^{46} \) \(\mathstrut -\mathstrut 23q^{47} \) \(\mathstrut -\mathstrut 13q^{48} \) \(\mathstrut +\mathstrut 38q^{49} \) \(\mathstrut -\mathstrut 6q^{50} \) \(\mathstrut -\mathstrut 5q^{51} \) \(\mathstrut -\mathstrut 41q^{52} \) \(\mathstrut -\mathstrut 12q^{53} \) \(\mathstrut +\mathstrut 35q^{54} \) \(\mathstrut -\mathstrut 18q^{55} \) \(\mathstrut -\mathstrut 21q^{56} \) \(\mathstrut -\mathstrut 3q^{57} \) \(\mathstrut -\mathstrut 16q^{58} \) \(\mathstrut -\mathstrut 16q^{59} \) \(\mathstrut -\mathstrut 20q^{60} \) \(\mathstrut -\mathstrut 47q^{61} \) \(\mathstrut +\mathstrut 14q^{62} \) \(\mathstrut +\mathstrut 13q^{63} \) \(\mathstrut -\mathstrut 55q^{64} \) \(\mathstrut -\mathstrut 25q^{65} \) \(\mathstrut -\mathstrut 6q^{66} \) \(\mathstrut -\mathstrut 54q^{67} \) \(\mathstrut +\mathstrut 31q^{68} \) \(\mathstrut -\mathstrut 95q^{69} \) \(\mathstrut -\mathstrut 6q^{70} \) \(\mathstrut -\mathstrut 41q^{71} \) \(\mathstrut -\mathstrut 59q^{73} \) \(\mathstrut -\mathstrut q^{74} \) \(\mathstrut -\mathstrut 9q^{75} \) \(\mathstrut -\mathstrut 23q^{76} \) \(\mathstrut -\mathstrut 18q^{77} \) \(\mathstrut +\mathstrut 19q^{78} \) \(\mathstrut -\mathstrut 117q^{79} \) \(\mathstrut -\mathstrut 38q^{81} \) \(\mathstrut +\mathstrut 19q^{82} \) \(\mathstrut -\mathstrut 21q^{83} \) \(\mathstrut -\mathstrut 20q^{84} \) \(\mathstrut -\mathstrut 21q^{85} \) \(\mathstrut -\mathstrut 12q^{86} \) \(\mathstrut -\mathstrut 14q^{87} \) \(\mathstrut -\mathstrut 40q^{88} \) \(\mathstrut -\mathstrut 96q^{89} \) \(\mathstrut -\mathstrut 7q^{90} \) \(\mathstrut -\mathstrut 25q^{91} \) \(\mathstrut +\mathstrut 53q^{92} \) \(\mathstrut -\mathstrut 53q^{93} \) \(\mathstrut -\mathstrut 28q^{94} \) \(\mathstrut -\mathstrut 14q^{95} \) \(\mathstrut +\mathstrut 40q^{96} \) \(\mathstrut -\mathstrut 70q^{97} \) \(\mathstrut -\mathstrut 6q^{98} \) \(\mathstrut -\mathstrut 52q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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 −2.60567 −1.96600 4.78952 1.00000 5.12274 1.00000 −7.26856 0.865151 −2.60567
1.2 −2.59504 0.990389 4.73422 1.00000 −2.57010 1.00000 −7.09539 −2.01913 −2.59504
1.3 −2.51011 −2.35176 4.30066 1.00000 5.90318 1.00000 −5.77493 2.53077 −2.51011
1.4 −2.46620 0.936734 4.08216 1.00000 −2.31018 1.00000 −5.13504 −2.12253 −2.46620
1.5 −2.38102 2.43622 3.66928 1.00000 −5.80069 1.00000 −3.97459 2.93515 −2.38102
1.6 −2.02615 0.680619 2.10529 1.00000 −1.37904 1.00000 −0.213342 −2.53676 −2.02615
1.7 −1.89678 −2.50257 1.59778 1.00000 4.74683 1.00000 0.762920 3.26285 −1.89678
1.8 −1.84626 −2.48565 1.40867 1.00000 4.58916 1.00000 1.09175 3.17847 −1.84626
1.9 −1.83849 1.40018 1.38005 1.00000 −2.57423 1.00000 1.13977 −1.03949 −1.83849
1.10 −1.74109 −0.00460406 1.03141 1.00000 0.00801611 1.00000 1.68640 −2.99998 −1.74109
1.11 −1.66719 2.43876 0.779527 1.00000 −4.06588 1.00000 2.03476 2.94756 −1.66719
1.12 −1.65687 −0.430057 0.745215 1.00000 0.712549 1.00000 2.07901 −2.81505 −1.65687
1.13 −1.23429 −1.22031 −0.476531 1.00000 1.50621 1.00000 3.05675 −1.51085 −1.23429
1.14 −1.08292 −0.496145 −0.827289 1.00000 0.537284 1.00000 3.06172 −2.75384 −1.08292
1.15 −0.697175 1.52901 −1.51395 1.00000 −1.06599 1.00000 2.44984 −0.662121 −0.697175
1.16 −0.656564 2.76581 −1.56892 1.00000 −1.81593 1.00000 2.34323 4.64973 −0.656564
1.17 −0.520713 −3.35310 −1.72886 1.00000 1.74600 1.00000 1.94166 8.24329 −0.520713
1.18 −0.217856 −2.35998 −1.95254 1.00000 0.514135 1.00000 0.861085 2.56949 −0.217856
1.19 −0.0912753 1.39308 −1.99167 1.00000 −0.127154 1.00000 0.364341 −1.05934 −0.0912753
1.20 −0.0830831 2.10839 −1.99310 1.00000 −0.175171 1.00000 0.331759 1.44530 −0.0830831
See all 38 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.38
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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8015))\):

\(T_{2}^{38} + \cdots\)
\(T_{3}^{38} + \cdots\)