Properties

Label 8007.2.a.h
Level 8007
Weight 2
Character orbit 8007.a
Self dual Yes
Analytic conductor 63.936
Analytic rank 0
Dimension 56
CM No

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Newspace parameters

Level: \( N \) = \( 8007 = 3 \cdot 17 \cdot 157 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 8007.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(63.9362168984\)
Analytic rank: \(0\)
Dimension: \(56\)
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.

\(\operatorname{Tr}(f)(q) = \) \(56q \) \(\mathstrut +\mathstrut 7q^{2} \) \(\mathstrut +\mathstrut 56q^{3} \) \(\mathstrut +\mathstrut 61q^{4} \) \(\mathstrut +\mathstrut 17q^{5} \) \(\mathstrut +\mathstrut 7q^{6} \) \(\mathstrut +\mathstrut 5q^{7} \) \(\mathstrut +\mathstrut 18q^{8} \) \(\mathstrut +\mathstrut 56q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \(56q \) \(\mathstrut +\mathstrut 7q^{2} \) \(\mathstrut +\mathstrut 56q^{3} \) \(\mathstrut +\mathstrut 61q^{4} \) \(\mathstrut +\mathstrut 17q^{5} \) \(\mathstrut +\mathstrut 7q^{6} \) \(\mathstrut +\mathstrut 5q^{7} \) \(\mathstrut +\mathstrut 18q^{8} \) \(\mathstrut +\mathstrut 56q^{9} \) \(\mathstrut -\mathstrut 2q^{10} \) \(\mathstrut +\mathstrut 35q^{11} \) \(\mathstrut +\mathstrut 61q^{12} \) \(\mathstrut +\mathstrut 8q^{13} \) \(\mathstrut +\mathstrut 36q^{14} \) \(\mathstrut +\mathstrut 17q^{15} \) \(\mathstrut +\mathstrut 71q^{16} \) \(\mathstrut -\mathstrut 56q^{17} \) \(\mathstrut +\mathstrut 7q^{18} \) \(\mathstrut -\mathstrut 2q^{19} \) \(\mathstrut +\mathstrut 58q^{20} \) \(\mathstrut +\mathstrut 5q^{21} \) \(\mathstrut +\mathstrut 27q^{22} \) \(\mathstrut +\mathstrut 40q^{23} \) \(\mathstrut +\mathstrut 18q^{24} \) \(\mathstrut +\mathstrut 85q^{25} \) \(\mathstrut +\mathstrut 15q^{26} \) \(\mathstrut +\mathstrut 56q^{27} \) \(\mathstrut -\mathstrut 4q^{28} \) \(\mathstrut +\mathstrut 41q^{29} \) \(\mathstrut -\mathstrut 2q^{30} \) \(\mathstrut +\mathstrut q^{31} \) \(\mathstrut +\mathstrut 43q^{32} \) \(\mathstrut +\mathstrut 35q^{33} \) \(\mathstrut -\mathstrut 7q^{34} \) \(\mathstrut +\mathstrut 57q^{35} \) \(\mathstrut +\mathstrut 61q^{36} \) \(\mathstrut +\mathstrut 34q^{37} \) \(\mathstrut +\mathstrut 52q^{38} \) \(\mathstrut +\mathstrut 8q^{39} \) \(\mathstrut +\mathstrut 14q^{40} \) \(\mathstrut +\mathstrut 49q^{41} \) \(\mathstrut +\mathstrut 36q^{42} \) \(\mathstrut +\mathstrut 27q^{43} \) \(\mathstrut +\mathstrut 66q^{44} \) \(\mathstrut +\mathstrut 17q^{45} \) \(\mathstrut +\mathstrut 10q^{46} \) \(\mathstrut +\mathstrut 43q^{47} \) \(\mathstrut +\mathstrut 71q^{48} \) \(\mathstrut +\mathstrut 51q^{49} \) \(\mathstrut +\mathstrut 30q^{50} \) \(\mathstrut -\mathstrut 56q^{51} \) \(\mathstrut -\mathstrut 7q^{52} \) \(\mathstrut +\mathstrut 73q^{53} \) \(\mathstrut +\mathstrut 7q^{54} \) \(\mathstrut +\mathstrut 15q^{55} \) \(\mathstrut +\mathstrut 118q^{56} \) \(\mathstrut -\mathstrut 2q^{57} \) \(\mathstrut -\mathstrut q^{58} \) \(\mathstrut +\mathstrut 53q^{59} \) \(\mathstrut +\mathstrut 58q^{60} \) \(\mathstrut +\mathstrut 15q^{61} \) \(\mathstrut +\mathstrut 16q^{62} \) \(\mathstrut +\mathstrut 5q^{63} \) \(\mathstrut +\mathstrut 124q^{64} \) \(\mathstrut +\mathstrut 107q^{65} \) \(\mathstrut +\mathstrut 27q^{66} \) \(\mathstrut +\mathstrut 20q^{67} \) \(\mathstrut -\mathstrut 61q^{68} \) \(\mathstrut +\mathstrut 40q^{69} \) \(\mathstrut +\mathstrut 16q^{70} \) \(\mathstrut +\mathstrut 56q^{71} \) \(\mathstrut +\mathstrut 18q^{72} \) \(\mathstrut +\mathstrut 49q^{73} \) \(\mathstrut +\mathstrut 28q^{74} \) \(\mathstrut +\mathstrut 85q^{75} \) \(\mathstrut -\mathstrut 38q^{76} \) \(\mathstrut +\mathstrut 50q^{77} \) \(\mathstrut +\mathstrut 15q^{78} \) \(\mathstrut -\mathstrut 4q^{79} \) \(\mathstrut +\mathstrut 74q^{80} \) \(\mathstrut +\mathstrut 56q^{81} \) \(\mathstrut +\mathstrut 59q^{82} \) \(\mathstrut +\mathstrut 35q^{83} \) \(\mathstrut -\mathstrut 4q^{84} \) \(\mathstrut -\mathstrut 17q^{85} \) \(\mathstrut +\mathstrut 38q^{86} \) \(\mathstrut +\mathstrut 41q^{87} \) \(\mathstrut +\mathstrut 64q^{88} \) \(\mathstrut +\mathstrut 66q^{89} \) \(\mathstrut -\mathstrut 2q^{90} \) \(\mathstrut +\mathstrut 5q^{91} \) \(\mathstrut +\mathstrut 96q^{92} \) \(\mathstrut +\mathstrut q^{93} \) \(\mathstrut -\mathstrut 12q^{94} \) \(\mathstrut +\mathstrut 70q^{95} \) \(\mathstrut +\mathstrut 43q^{96} \) \(\mathstrut +\mathstrut 60q^{97} \) \(\mathstrut +\mathstrut 26q^{98} \) \(\mathstrut +\mathstrut 35q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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 −2.74837 1.00000 5.55354 2.55432 −2.74837 −2.04110 −9.76645 1.00000 −7.02023
1.2 −2.65002 1.00000 5.02263 3.74088 −2.65002 −4.42567 −8.01004 1.00000 −9.91341
1.3 −2.59527 1.00000 4.73543 −1.29433 −2.59527 2.82321 −7.09919 1.00000 3.35914
1.4 −2.47538 1.00000 4.12750 2.16981 −2.47538 1.33731 −5.26636 1.00000 −5.37109
1.5 −2.46910 1.00000 4.09645 1.22884 −2.46910 −1.04815 −5.17633 1.00000 −3.03411
1.6 −2.42729 1.00000 3.89172 −3.95283 −2.42729 −4.02222 −4.59173 1.00000 9.59464
1.7 −2.23272 1.00000 2.98503 −1.75437 −2.23272 0.669385 −2.19928 1.00000 3.91702
1.8 −2.14010 1.00000 2.58002 −0.881570 −2.14010 −2.47197 −1.24130 1.00000 1.88665
1.9 −2.04447 1.00000 2.17985 4.17783 −2.04447 2.27767 −0.367694 1.00000 −8.54145
1.10 −1.87368 1.00000 1.51068 −1.39015 −1.87368 0.634123 0.916837 1.00000 2.60469
1.11 −1.81598 1.00000 1.29778 −1.85785 −1.81598 −3.71645 1.27521 1.00000 3.37381
1.12 −1.81301 1.00000 1.28701 1.68912 −1.81301 2.36168 1.29265 1.00000 −3.06240
1.13 −1.59198 1.00000 0.534400 −1.86457 −1.59198 2.61461 2.33321 1.00000 2.96836
1.14 −1.32549 1.00000 −0.243069 4.13081 −1.32549 1.12393 2.97317 1.00000 −5.47535
1.15 −1.30899 1.00000 −0.286535 2.04407 −1.30899 −3.68915 2.99306 1.00000 −2.67568
1.16 −1.30269 1.00000 −0.302986 1.77397 −1.30269 4.96759 3.00009 1.00000 −2.31094
1.17 −1.21361 1.00000 −0.527156 4.28768 −1.21361 1.72310 3.06698 1.00000 −5.20356
1.18 −1.19021 1.00000 −0.583408 2.13314 −1.19021 −4.67655 3.07479 1.00000 −2.53888
1.19 −1.18880 1.00000 −0.586753 −1.07516 −1.18880 −2.55094 3.07513 1.00000 1.27815
1.20 −0.892239 1.00000 −1.20391 −0.989381 −0.892239 3.60802 2.85865 1.00000 0.882764
See all 56 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.56
Significant digits:
Format:

Inner twists

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

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(17\) \(1\)
\(157\) \(1\)

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{2}^{56} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8007))\).