Properties

Label 8007.2.a.f
Level 8007
Weight 2
Character orbit 8007.a
Self dual Yes
Analytic conductor 63.936
Analytic rank 1
Dimension 48
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: \(48\)
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.

\(\operatorname{Tr}(f)(q) = \) \(48q \) \(\mathstrut -\mathstrut q^{2} \) \(\mathstrut -\mathstrut 48q^{3} \) \(\mathstrut +\mathstrut 45q^{4} \) \(\mathstrut +\mathstrut q^{5} \) \(\mathstrut +\mathstrut q^{6} \) \(\mathstrut -\mathstrut 13q^{7} \) \(\mathstrut -\mathstrut 6q^{8} \) \(\mathstrut +\mathstrut 48q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \(48q \) \(\mathstrut -\mathstrut q^{2} \) \(\mathstrut -\mathstrut 48q^{3} \) \(\mathstrut +\mathstrut 45q^{4} \) \(\mathstrut +\mathstrut q^{5} \) \(\mathstrut +\mathstrut q^{6} \) \(\mathstrut -\mathstrut 13q^{7} \) \(\mathstrut -\mathstrut 6q^{8} \) \(\mathstrut +\mathstrut 48q^{9} \) \(\mathstrut -\mathstrut 20q^{10} \) \(\mathstrut +\mathstrut 5q^{11} \) \(\mathstrut -\mathstrut 45q^{12} \) \(\mathstrut -\mathstrut 8q^{13} \) \(\mathstrut +\mathstrut 4q^{14} \) \(\mathstrut -\mathstrut q^{15} \) \(\mathstrut +\mathstrut 39q^{16} \) \(\mathstrut -\mathstrut 48q^{17} \) \(\mathstrut -\mathstrut q^{18} \) \(\mathstrut -\mathstrut 6q^{19} \) \(\mathstrut +\mathstrut 6q^{20} \) \(\mathstrut +\mathstrut 13q^{21} \) \(\mathstrut -\mathstrut 35q^{22} \) \(\mathstrut -\mathstrut 8q^{23} \) \(\mathstrut +\mathstrut 6q^{24} \) \(\mathstrut +\mathstrut 13q^{25} \) \(\mathstrut +\mathstrut 17q^{26} \) \(\mathstrut -\mathstrut 48q^{27} \) \(\mathstrut -\mathstrut 38q^{28} \) \(\mathstrut +\mathstrut q^{29} \) \(\mathstrut +\mathstrut 20q^{30} \) \(\mathstrut -\mathstrut 21q^{31} \) \(\mathstrut -\mathstrut 3q^{32} \) \(\mathstrut -\mathstrut 5q^{33} \) \(\mathstrut +\mathstrut q^{34} \) \(\mathstrut +\mathstrut 19q^{35} \) \(\mathstrut +\mathstrut 45q^{36} \) \(\mathstrut -\mathstrut 58q^{37} \) \(\mathstrut -\mathstrut 14q^{38} \) \(\mathstrut +\mathstrut 8q^{39} \) \(\mathstrut -\mathstrut 54q^{40} \) \(\mathstrut -\mathstrut 3q^{41} \) \(\mathstrut -\mathstrut 4q^{42} \) \(\mathstrut -\mathstrut 33q^{43} \) \(\mathstrut +\mathstrut 2q^{44} \) \(\mathstrut +\mathstrut q^{45} \) \(\mathstrut -\mathstrut 26q^{46} \) \(\mathstrut +\mathstrut 9q^{47} \) \(\mathstrut -\mathstrut 39q^{48} \) \(\mathstrut +\mathstrut 11q^{49} \) \(\mathstrut +\mathstrut 4q^{50} \) \(\mathstrut +\mathstrut 48q^{51} \) \(\mathstrut -\mathstrut 31q^{52} \) \(\mathstrut -\mathstrut 33q^{53} \) \(\mathstrut +\mathstrut q^{54} \) \(\mathstrut -\mathstrut 21q^{55} \) \(\mathstrut +\mathstrut 6q^{57} \) \(\mathstrut -\mathstrut 55q^{58} \) \(\mathstrut +\mathstrut 77q^{59} \) \(\mathstrut -\mathstrut 6q^{60} \) \(\mathstrut -\mathstrut 29q^{61} \) \(\mathstrut -\mathstrut 46q^{62} \) \(\mathstrut -\mathstrut 13q^{63} \) \(\mathstrut +\mathstrut 24q^{64} \) \(\mathstrut -\mathstrut 49q^{65} \) \(\mathstrut +\mathstrut 35q^{66} \) \(\mathstrut -\mathstrut 44q^{67} \) \(\mathstrut -\mathstrut 45q^{68} \) \(\mathstrut +\mathstrut 8q^{69} \) \(\mathstrut +\mathstrut 4q^{70} \) \(\mathstrut +\mathstrut 22q^{71} \) \(\mathstrut -\mathstrut 6q^{72} \) \(\mathstrut -\mathstrut 63q^{73} \) \(\mathstrut -\mathstrut 16q^{74} \) \(\mathstrut -\mathstrut 13q^{75} \) \(\mathstrut -\mathstrut 46q^{76} \) \(\mathstrut -\mathstrut 30q^{77} \) \(\mathstrut -\mathstrut 17q^{78} \) \(\mathstrut -\mathstrut 46q^{79} \) \(\mathstrut -\mathstrut 14q^{80} \) \(\mathstrut +\mathstrut 48q^{81} \) \(\mathstrut -\mathstrut 75q^{82} \) \(\mathstrut +\mathstrut 11q^{83} \) \(\mathstrut +\mathstrut 38q^{84} \) \(\mathstrut -\mathstrut q^{85} \) \(\mathstrut +\mathstrut 8q^{86} \) \(\mathstrut -\mathstrut q^{87} \) \(\mathstrut -\mathstrut 116q^{88} \) \(\mathstrut +\mathstrut 10q^{89} \) \(\mathstrut -\mathstrut 20q^{90} \) \(\mathstrut -\mathstrut 67q^{91} \) \(\mathstrut -\mathstrut 64q^{92} \) \(\mathstrut +\mathstrut 21q^{93} \) \(\mathstrut -\mathstrut 16q^{94} \) \(\mathstrut -\mathstrut 8q^{95} \) \(\mathstrut +\mathstrut 3q^{96} \) \(\mathstrut -\mathstrut 96q^{97} \) \(\mathstrut -\mathstrut 46q^{98} \) \(\mathstrut +\mathstrut 5q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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 −2.78117 −1.00000 5.73492 2.04412 2.78117 −0.793043 −10.3875 1.00000 −5.68506
1.2 −2.66751 −1.00000 5.11563 −0.759881 2.66751 −3.14723 −8.31098 1.00000 2.02699
1.3 −2.52044 −1.00000 4.35263 −1.87454 2.52044 −3.72156 −5.92968 1.00000 4.72466
1.4 −2.46582 −1.00000 4.08028 3.93570 2.46582 −1.94268 −5.12960 1.00000 −9.70474
1.5 −2.39228 −1.00000 3.72299 0.325110 2.39228 −0.363079 −4.12187 1.00000 −0.777753
1.6 −2.28381 −1.00000 3.21580 1.64317 2.28381 3.93496 −2.77667 1.00000 −3.75270
1.7 −2.27518 −1.00000 3.17643 2.61308 2.27518 3.54577 −2.67658 1.00000 −5.94522
1.8 −2.17906 −1.00000 2.74829 −2.11197 2.17906 2.69680 −1.63058 1.00000 4.60211
1.9 −2.00652 −1.00000 2.02613 −1.83243 2.00652 −1.92586 −0.0524220 1.00000 3.67682
1.10 −1.95217 −1.00000 1.81095 −1.54543 1.95217 2.71405 0.369057 1.00000 3.01694
1.11 −1.70660 −1.00000 0.912495 1.58325 1.70660 −2.98844 1.85594 1.00000 −2.70198
1.12 −1.54663 −1.00000 0.392054 3.59105 1.54663 −3.93711 2.48689 1.00000 −5.55401
1.13 −1.50666 −1.00000 0.270014 −0.475693 1.50666 −0.891979 2.60649 1.00000 0.716706
1.14 −1.44064 −1.00000 0.0754557 −3.22452 1.44064 1.08354 2.77258 1.00000 4.64538
1.15 −1.38255 −1.00000 −0.0885651 2.12725 1.38255 −0.203962 2.88754 1.00000 −2.94103
1.16 −1.24727 −1.00000 −0.444326 −3.92446 1.24727 −1.90187 3.04873 1.00000 4.89484
1.17 −0.980818 −1.00000 −1.03800 1.00436 0.980818 −3.53145 2.97972 1.00000 −0.985094
1.18 −0.901374 −1.00000 −1.18752 1.97657 0.901374 4.00403 2.87315 1.00000 −1.78163
1.19 −0.857233 −1.00000 −1.26515 0.241794 0.857233 −3.40893 2.79900 1.00000 −0.207274
1.20 −0.557584 −1.00000 −1.68910 2.57031 0.557584 2.28117 2.05698 1.00000 −1.43316
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.48
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 \(T_{2}^{48} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8007))\).