Properties

Label 8007.2.a.e
Level 8007
Weight 2
Character orbit 8007.a
Self dual Yes
Analytic conductor 63.936
Analytic rank 1
Dimension 46
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: \(46\)
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.

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

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 −2.80793 1.00000 5.88448 −2.90815 −2.80793 −1.28847 −10.9074 1.00000 8.16589
1.2 −2.76060 1.00000 5.62090 −0.677149 −2.76060 3.14387 −9.99585 1.00000 1.86934
1.3 −2.60999 1.00000 4.81203 −2.83562 −2.60999 1.53827 −7.33937 1.00000 7.40094
1.4 −2.58291 1.00000 4.67145 2.37262 −2.58291 −1.51271 −6.90012 1.00000 −6.12828
1.5 −2.45922 1.00000 4.04774 2.82362 −2.45922 4.27749 −5.03584 1.00000 −6.94390
1.6 −2.38281 1.00000 3.67776 −2.44898 −2.38281 −2.76989 −3.99778 1.00000 5.83545
1.7 −2.18789 1.00000 2.78687 1.88275 −2.18789 −4.23490 −1.72159 1.00000 −4.11926
1.8 −2.12891 1.00000 2.53226 −4.00748 −2.12891 2.85209 −1.13314 1.00000 8.53157
1.9 −2.00485 1.00000 2.01944 2.86654 −2.00485 −1.37721 −0.0389655 1.00000 −5.74699
1.10 −1.83962 1.00000 1.38420 0.446319 −1.83962 3.21050 1.13283 1.00000 −0.821058
1.11 −1.59838 1.00000 0.554812 1.16995 −1.59838 1.82514 2.30996 1.00000 −1.87002
1.12 −1.58086 1.00000 0.499118 −3.53760 −1.58086 −3.85230 2.37268 1.00000 5.59245
1.13 −1.50188 1.00000 0.255640 −0.915140 −1.50188 −3.11181 2.61982 1.00000 1.37443
1.14 −1.42820 1.00000 0.0397473 −3.43580 −1.42820 0.0177517 2.79963 1.00000 4.90700
1.15 −1.42497 1.00000 0.0305258 0.348852 −1.42497 −0.489415 2.80643 1.00000 −0.497102
1.16 −1.41678 1.00000 0.00726459 2.31073 −1.41678 2.67795 2.82327 1.00000 −3.27380
1.17 −0.943890 1.00000 −1.10907 3.50883 −0.943890 −2.32814 2.93462 1.00000 −3.31195
1.18 −0.938469 1.00000 −1.11928 −3.16176 −0.938469 3.19339 2.92734 1.00000 2.96721
1.19 −0.756159 1.00000 −1.42822 −3.23827 −0.756159 4.69700 2.59228 1.00000 2.44865
1.20 −0.576474 1.00000 −1.66768 0.828352 −0.576474 −0.479856 2.11432 1.00000 −0.477524
See all 46 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.46
Significant digits:
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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 \(T_{2}^{46} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8007))\).