Properties

Label 8007.2.a.g
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.

\(\operatorname{Tr}(f)(q) = \) \(56q \) \(\mathstrut +\mathstrut q^{2} \) \(\mathstrut -\mathstrut 56q^{3} \) \(\mathstrut +\mathstrut 61q^{4} \) \(\mathstrut +\mathstrut q^{5} \) \(\mathstrut -\mathstrut q^{6} \) \(\mathstrut +\mathstrut 19q^{7} \) \(\mathstrut +\mathstrut 56q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \(56q \) \(\mathstrut +\mathstrut q^{2} \) \(\mathstrut -\mathstrut 56q^{3} \) \(\mathstrut +\mathstrut 61q^{4} \) \(\mathstrut +\mathstrut q^{5} \) \(\mathstrut -\mathstrut q^{6} \) \(\mathstrut +\mathstrut 19q^{7} \) \(\mathstrut +\mathstrut 56q^{9} \) \(\mathstrut +\mathstrut 8q^{10} \) \(\mathstrut -\mathstrut 7q^{11} \) \(\mathstrut -\mathstrut 61q^{12} \) \(\mathstrut +\mathstrut 8q^{13} \) \(\mathstrut -\mathstrut 8q^{14} \) \(\mathstrut -\mathstrut q^{15} \) \(\mathstrut +\mathstrut 71q^{16} \) \(\mathstrut -\mathstrut 56q^{17} \) \(\mathstrut +\mathstrut q^{18} \) \(\mathstrut -\mathstrut 2q^{19} \) \(\mathstrut -\mathstrut 4q^{20} \) \(\mathstrut -\mathstrut 19q^{21} \) \(\mathstrut +\mathstrut 47q^{22} \) \(\mathstrut +\mathstrut 16q^{23} \) \(\mathstrut +\mathstrut 85q^{25} \) \(\mathstrut -\mathstrut 11q^{26} \) \(\mathstrut -\mathstrut 56q^{27} \) \(\mathstrut +\mathstrut 52q^{28} \) \(\mathstrut +\mathstrut 17q^{29} \) \(\mathstrut -\mathstrut 8q^{30} \) \(\mathstrut +\mathstrut 23q^{31} \) \(\mathstrut +\mathstrut 11q^{32} \) \(\mathstrut +\mathstrut 7q^{33} \) \(\mathstrut -\mathstrut q^{34} \) \(\mathstrut -\mathstrut 41q^{35} \) \(\mathstrut +\mathstrut 61q^{36} \) \(\mathstrut +\mathstrut 58q^{37} \) \(\mathstrut -\mathstrut 22q^{38} \) \(\mathstrut -\mathstrut 8q^{39} \) \(\mathstrut +\mathstrut 38q^{40} \) \(\mathstrut -\mathstrut q^{41} \) \(\mathstrut +\mathstrut 8q^{42} \) \(\mathstrut +\mathstrut 27q^{43} \) \(\mathstrut +\mathstrut 2q^{44} \) \(\mathstrut +\mathstrut q^{45} \) \(\mathstrut +\mathstrut 46q^{46} \) \(\mathstrut +\mathstrut 5q^{47} \) \(\mathstrut -\mathstrut 71q^{48} \) \(\mathstrut +\mathstrut 59q^{49} \) \(\mathstrut -\mathstrut 4q^{50} \) \(\mathstrut +\mathstrut 56q^{51} \) \(\mathstrut +\mathstrut 25q^{52} \) \(\mathstrut +\mathstrut 15q^{53} \) \(\mathstrut -\mathstrut q^{54} \) \(\mathstrut +\mathstrut 9q^{55} \) \(\mathstrut -\mathstrut 36q^{56} \) \(\mathstrut +\mathstrut 2q^{57} \) \(\mathstrut +\mathstrut 89q^{58} \) \(\mathstrut -\mathstrut 61q^{59} \) \(\mathstrut +\mathstrut 4q^{60} \) \(\mathstrut +\mathstrut 47q^{61} \) \(\mathstrut +\mathstrut 8q^{62} \) \(\mathstrut +\mathstrut 19q^{63} \) \(\mathstrut +\mathstrut 88q^{64} \) \(\mathstrut +\mathstrut 39q^{65} \) \(\mathstrut -\mathstrut 47q^{66} \) \(\mathstrut +\mathstrut 20q^{67} \) \(\mathstrut -\mathstrut 61q^{68} \) \(\mathstrut -\mathstrut 16q^{69} \) \(\mathstrut +\mathstrut 36q^{70} \) \(\mathstrut -\mathstrut 2q^{71} \) \(\mathstrut +\mathstrut 93q^{73} \) \(\mathstrut +\mathstrut 48q^{74} \) \(\mathstrut -\mathstrut 85q^{75} \) \(\mathstrut +\mathstrut 38q^{76} \) \(\mathstrut +\mathstrut 26q^{77} \) \(\mathstrut +\mathstrut 11q^{78} \) \(\mathstrut +\mathstrut 72q^{79} \) \(\mathstrut +\mathstrut 42q^{80} \) \(\mathstrut +\mathstrut 56q^{81} \) \(\mathstrut +\mathstrut 33q^{82} \) \(\mathstrut -\mathstrut 11q^{83} \) \(\mathstrut -\mathstrut 52q^{84} \) \(\mathstrut -\mathstrut q^{85} \) \(\mathstrut -\mathstrut 4q^{86} \) \(\mathstrut -\mathstrut 17q^{87} \) \(\mathstrut +\mathstrut 130q^{88} \) \(\mathstrut -\mathstrut 6q^{89} \) \(\mathstrut +\mathstrut 8q^{90} \) \(\mathstrut +\mathstrut 37q^{91} \) \(\mathstrut +\mathstrut 132q^{92} \) \(\mathstrut -\mathstrut 23q^{93} \) \(\mathstrut -\mathstrut 32q^{94} \) \(\mathstrut +\mathstrut 12q^{95} \) \(\mathstrut -\mathstrut 11q^{96} \) \(\mathstrut +\mathstrut 100q^{97} \) \(\mathstrut +\mathstrut 42q^{98} \) \(\mathstrut -\mathstrut 7q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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 −2.76474 −1.00000 5.64378 −1.85589 2.76474 1.33970 −10.0741 1.00000 5.13105
1.2 −2.71794 −1.00000 5.38720 2.54834 2.71794 4.22064 −9.20622 1.00000 −6.92625
1.3 −2.65517 −1.00000 5.04993 −2.53504 2.65517 2.76662 −8.09809 1.00000 6.73097
1.4 −2.51234 −1.00000 4.31187 −0.437106 2.51234 0.108032 −5.80821 1.00000 1.09816
1.5 −2.49549 −1.00000 4.22749 2.28895 2.49549 −3.97589 −5.55870 1.00000 −5.71206
1.6 −2.45801 −1.00000 4.04182 −3.57546 2.45801 3.37294 −5.01881 1.00000 8.78852
1.7 −2.36108 −1.00000 3.57471 4.34691 2.36108 −0.239132 −3.71802 1.00000 −10.2634
1.8 −2.27923 −1.00000 3.19488 1.61810 2.27923 2.49401 −2.72340 1.00000 −3.68801
1.9 −2.11176 −1.00000 2.45954 −3.52613 2.11176 −3.47105 −0.970439 1.00000 7.44635
1.10 −1.95968 −1.00000 1.84033 −3.13690 1.95968 −0.0823024 0.312893 1.00000 6.14731
1.11 −1.93055 −1.00000 1.72702 −2.51216 1.93055 −1.67367 0.526999 1.00000 4.84984
1.12 −1.84970 −1.00000 1.42137 0.505373 1.84970 −1.72444 1.07028 1.00000 −0.934786
1.13 −1.84393 −1.00000 1.40007 2.87244 1.84393 −1.74244 1.10623 1.00000 −5.29656
1.14 −1.75940 −1.00000 1.09547 −0.617389 1.75940 3.20947 1.59142 1.00000 1.08623
1.15 −1.59327 −1.00000 0.538495 2.65587 1.59327 −0.355214 2.32857 1.00000 −4.23151
1.16 −1.50676 −1.00000 0.270331 −1.85890 1.50676 4.43102 2.60620 1.00000 2.80092
1.17 −1.26654 −1.00000 −0.395877 2.14407 1.26654 1.86831 3.03447 1.00000 −2.71555
1.18 −1.23565 −1.00000 −0.473178 −0.721310 1.23565 −3.93183 3.05597 1.00000 0.891284
1.19 −1.15978 −1.00000 −0.654916 −0.00396327 1.15978 3.25506 3.07911 1.00000 0.00459651
1.20 −0.882919 −1.00000 −1.22045 −0.454468 0.882919 1.65699 2.84340 1.00000 0.401259
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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8007))\).