Properties

Label 8006.2.a.b
Level 8006
Weight 2
Character orbit 8006.a
Self dual Yes
Analytic conductor 63.928
Analytic rank 1
Dimension 75
CM No

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Newspace parameters

Level: \( N \) = \( 8006 = 2 \cdot 4003 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 8006.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(63.9282318582\)
Analytic rank: \(1\)
Dimension: \(75\)
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.

\(\operatorname{Tr}(f)(q) = \) \(75q \) \(\mathstrut -\mathstrut 75q^{2} \) \(\mathstrut +\mathstrut q^{3} \) \(\mathstrut +\mathstrut 75q^{4} \) \(\mathstrut -\mathstrut 9q^{5} \) \(\mathstrut -\mathstrut q^{6} \) \(\mathstrut -\mathstrut 8q^{7} \) \(\mathstrut -\mathstrut 75q^{8} \) \(\mathstrut +\mathstrut 66q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \(75q \) \(\mathstrut -\mathstrut 75q^{2} \) \(\mathstrut +\mathstrut q^{3} \) \(\mathstrut +\mathstrut 75q^{4} \) \(\mathstrut -\mathstrut 9q^{5} \) \(\mathstrut -\mathstrut q^{6} \) \(\mathstrut -\mathstrut 8q^{7} \) \(\mathstrut -\mathstrut 75q^{8} \) \(\mathstrut +\mathstrut 66q^{9} \) \(\mathstrut +\mathstrut 9q^{10} \) \(\mathstrut -\mathstrut 5q^{11} \) \(\mathstrut +\mathstrut q^{12} \) \(\mathstrut -\mathstrut 35q^{13} \) \(\mathstrut +\mathstrut 8q^{14} \) \(\mathstrut -\mathstrut 21q^{15} \) \(\mathstrut +\mathstrut 75q^{16} \) \(\mathstrut +\mathstrut 4q^{17} \) \(\mathstrut -\mathstrut 66q^{18} \) \(\mathstrut -\mathstrut 59q^{19} \) \(\mathstrut -\mathstrut 9q^{20} \) \(\mathstrut -\mathstrut 62q^{21} \) \(\mathstrut +\mathstrut 5q^{22} \) \(\mathstrut +\mathstrut 43q^{23} \) \(\mathstrut -\mathstrut q^{24} \) \(\mathstrut +\mathstrut 44q^{25} \) \(\mathstrut +\mathstrut 35q^{26} \) \(\mathstrut +\mathstrut 4q^{27} \) \(\mathstrut -\mathstrut 8q^{28} \) \(\mathstrut -\mathstrut 38q^{29} \) \(\mathstrut +\mathstrut 21q^{30} \) \(\mathstrut -\mathstrut 51q^{31} \) \(\mathstrut -\mathstrut 75q^{32} \) \(\mathstrut -\mathstrut 19q^{33} \) \(\mathstrut -\mathstrut 4q^{34} \) \(\mathstrut +\mathstrut 14q^{35} \) \(\mathstrut +\mathstrut 66q^{36} \) \(\mathstrut -\mathstrut 63q^{37} \) \(\mathstrut +\mathstrut 59q^{38} \) \(\mathstrut -\mathstrut 34q^{39} \) \(\mathstrut +\mathstrut 9q^{40} \) \(\mathstrut -\mathstrut 27q^{41} \) \(\mathstrut +\mathstrut 62q^{42} \) \(\mathstrut -\mathstrut 39q^{43} \) \(\mathstrut -\mathstrut 5q^{44} \) \(\mathstrut -\mathstrut 52q^{45} \) \(\mathstrut -\mathstrut 43q^{46} \) \(\mathstrut +\mathstrut 40q^{47} \) \(\mathstrut +\mathstrut q^{48} \) \(\mathstrut +\mathstrut 29q^{49} \) \(\mathstrut -\mathstrut 44q^{50} \) \(\mathstrut -\mathstrut 34q^{51} \) \(\mathstrut -\mathstrut 35q^{52} \) \(\mathstrut -\mathstrut 39q^{53} \) \(\mathstrut -\mathstrut 4q^{54} \) \(\mathstrut -\mathstrut 48q^{55} \) \(\mathstrut +\mathstrut 8q^{56} \) \(\mathstrut -\mathstrut 28q^{57} \) \(\mathstrut +\mathstrut 38q^{58} \) \(\mathstrut +\mathstrut 5q^{59} \) \(\mathstrut -\mathstrut 21q^{60} \) \(\mathstrut -\mathstrut 98q^{61} \) \(\mathstrut +\mathstrut 51q^{62} \) \(\mathstrut +\mathstrut 2q^{63} \) \(\mathstrut +\mathstrut 75q^{64} \) \(\mathstrut -\mathstrut q^{65} \) \(\mathstrut +\mathstrut 19q^{66} \) \(\mathstrut -\mathstrut 59q^{67} \) \(\mathstrut +\mathstrut 4q^{68} \) \(\mathstrut -\mathstrut 69q^{69} \) \(\mathstrut -\mathstrut 14q^{70} \) \(\mathstrut -\mathstrut 9q^{71} \) \(\mathstrut -\mathstrut 66q^{72} \) \(\mathstrut -\mathstrut 51q^{73} \) \(\mathstrut +\mathstrut 63q^{74} \) \(\mathstrut -\mathstrut q^{75} \) \(\mathstrut -\mathstrut 59q^{76} \) \(\mathstrut -\mathstrut 25q^{77} \) \(\mathstrut +\mathstrut 34q^{78} \) \(\mathstrut -\mathstrut 139q^{79} \) \(\mathstrut -\mathstrut 9q^{80} \) \(\mathstrut +\mathstrut 23q^{81} \) \(\mathstrut +\mathstrut 27q^{82} \) \(\mathstrut +\mathstrut 31q^{83} \) \(\mathstrut -\mathstrut 62q^{84} \) \(\mathstrut -\mathstrut 149q^{85} \) \(\mathstrut +\mathstrut 39q^{86} \) \(\mathstrut +\mathstrut q^{87} \) \(\mathstrut +\mathstrut 5q^{88} \) \(\mathstrut -\mathstrut 39q^{89} \) \(\mathstrut +\mathstrut 52q^{90} \) \(\mathstrut -\mathstrut 93q^{91} \) \(\mathstrut +\mathstrut 43q^{92} \) \(\mathstrut -\mathstrut 83q^{93} \) \(\mathstrut -\mathstrut 40q^{94} \) \(\mathstrut +\mathstrut 2q^{95} \) \(\mathstrut -\mathstrut q^{96} \) \(\mathstrut -\mathstrut 70q^{97} \) \(\mathstrut -\mathstrut 29q^{98} \) \(\mathstrut -\mathstrut 61q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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 −3.43129 1.00000 0.837927 3.43129 3.88626 −1.00000 8.77378 −0.837927
1.2 −1.00000 −3.12981 1.00000 −3.93566 3.12981 0.0773402 −1.00000 6.79570 3.93566
1.3 −1.00000 −2.99270 1.00000 −0.0677361 2.99270 −2.03477 −1.00000 5.95625 0.0677361
1.4 −1.00000 −2.91158 1.00000 1.02232 2.91158 −2.06566 −1.00000 5.47731 −1.02232
1.5 −1.00000 −2.90089 1.00000 −2.73153 2.90089 4.86656 −1.00000 5.41514 2.73153
1.6 −1.00000 −2.82604 1.00000 3.06326 2.82604 0.692372 −1.00000 4.98649 −3.06326
1.7 −1.00000 −2.82362 1.00000 −2.07564 2.82362 0.568941 −1.00000 4.97283 2.07564
1.8 −1.00000 −2.77545 1.00000 4.25856 2.77545 2.96891 −1.00000 4.70314 −4.25856
1.9 −1.00000 −2.68989 1.00000 −1.33052 2.68989 1.26652 −1.00000 4.23551 1.33052
1.10 −1.00000 −2.63254 1.00000 −2.25670 2.63254 −0.475543 −1.00000 3.93027 2.25670
1.11 −1.00000 −2.43887 1.00000 2.67337 2.43887 3.98930 −1.00000 2.94808 −2.67337
1.12 −1.00000 −2.34883 1.00000 2.23352 2.34883 0.995037 −1.00000 2.51700 −2.23352
1.13 −1.00000 −2.29820 1.00000 −2.18058 2.29820 −2.19565 −1.00000 2.28173 2.18058
1.14 −1.00000 −2.22415 1.00000 2.52556 2.22415 1.21093 −1.00000 1.94686 −2.52556
1.15 −1.00000 −2.19035 1.00000 2.34635 2.19035 −2.32589 −1.00000 1.79762 −2.34635
1.16 −1.00000 −2.05716 1.00000 −1.30027 2.05716 −1.55677 −1.00000 1.23189 1.30027
1.17 −1.00000 −1.79620 1.00000 −0.413445 1.79620 −3.61034 −1.00000 0.226347 0.413445
1.18 −1.00000 −1.74787 1.00000 −1.73571 1.74787 2.21934 −1.00000 0.0550411 1.73571
1.19 −1.00000 −1.72100 1.00000 1.49175 1.72100 −1.07018 −1.00000 −0.0381718 −1.49175
1.20 −1.00000 −1.69760 1.00000 2.70478 1.69760 −3.93964 −1.00000 −0.118138 −2.70478
See all 75 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.75
Significant digits:
Format:

Inner twists

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

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(4003\) \(1\)