Properties

Label 8007.2.a.d
Level 8007
Weight 2
Character orbit 8007.a
Self dual Yes
Analytic conductor 63.936
Analytic rank 1
Dimension 40
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: \(1\)
Dimension: \(40\)
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.

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

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 −2.69841 1.00000 5.28144 2.43860 −2.69841 −1.23437 −8.85467 1.00000 −6.58035
1.2 −2.61866 1.00000 4.85739 0.391872 −2.61866 2.70632 −7.48254 1.00000 −1.02618
1.3 −2.59865 1.00000 4.75296 −3.62998 −2.59865 4.35344 −7.15398 1.00000 9.43304
1.4 −2.49435 1.00000 4.22179 −2.73091 −2.49435 0.916542 −5.54193 1.00000 6.81185
1.5 −2.23744 1.00000 3.00614 1.45299 −2.23744 −4.68307 −2.25118 1.00000 −3.25099
1.6 −2.17139 1.00000 2.71494 −0.590074 −2.17139 −3.96431 −1.55242 1.00000 1.28128
1.7 −2.12487 1.00000 2.51508 2.60620 −2.12487 −0.251909 −1.09447 1.00000 −5.53783
1.8 −1.86400 1.00000 1.47451 −2.73802 −1.86400 2.48904 0.979516 1.00000 5.10367
1.9 −1.85562 1.00000 1.44333 −0.427574 −1.85562 2.91879 1.03298 1.00000 0.793415
1.10 −1.83549 1.00000 1.36903 −0.437870 −1.83549 1.24463 1.15814 1.00000 0.803707
1.11 −1.72962 1.00000 0.991576 0.136397 −1.72962 −2.36016 1.74419 1.00000 −0.235915
1.12 −1.55034 1.00000 0.403543 −3.10689 −1.55034 −2.24451 2.47505 1.00000 4.81672
1.13 −1.32783 1.00000 −0.236868 2.41386 −1.32783 0.553664 2.97018 1.00000 −3.20519
1.14 −1.21037 1.00000 −0.535004 3.43189 −1.21037 −1.42621 3.06829 1.00000 −4.15386
1.15 −1.15648 1.00000 −0.662543 −2.13083 −1.15648 1.76837 3.07919 1.00000 2.46428
1.16 −0.802287 1.00000 −1.35634 0.612480 −0.802287 −2.13461 2.69274 1.00000 −0.491385
1.17 −0.656002 1.00000 −1.56966 0.401662 −0.656002 −2.40151 2.34170 1.00000 −0.263491
1.18 −0.524942 1.00000 −1.72444 −0.0309405 −0.524942 4.53114 1.95511 1.00000 0.0162420
1.19 −0.481635 1.00000 −1.76803 −2.37410 −0.481635 −1.68399 1.81481 1.00000 1.14345
1.20 −0.422786 1.00000 −1.82125 −3.08259 −0.422786 −4.30268 1.61557 1.00000 1.30328
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.40
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 \(T_{2}^{40} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8007))\).