Properties

Label 4009.2.a.e
Level 4009
Weight 2
Character orbit 4009.a
Self dual Yes
Analytic conductor 32.012
Analytic rank 0
Dimension 82
CM No

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Newspace parameters

Level: \( N \) = \( 4009 = 19 \cdot 211 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4009.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(32.0120261703\)
Analytic rank: \(0\)
Dimension: \(82\)
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.

\(\operatorname{Tr}(f)(q) = \) \(82q \) \(\mathstrut +\mathstrut 15q^{2} \) \(\mathstrut +\mathstrut 12q^{3} \) \(\mathstrut +\mathstrut 89q^{4} \) \(\mathstrut +\mathstrut 9q^{5} \) \(\mathstrut +\mathstrut 9q^{6} \) \(\mathstrut +\mathstrut 14q^{7} \) \(\mathstrut +\mathstrut 42q^{8} \) \(\mathstrut +\mathstrut 92q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \(82q \) \(\mathstrut +\mathstrut 15q^{2} \) \(\mathstrut +\mathstrut 12q^{3} \) \(\mathstrut +\mathstrut 89q^{4} \) \(\mathstrut +\mathstrut 9q^{5} \) \(\mathstrut +\mathstrut 9q^{6} \) \(\mathstrut +\mathstrut 14q^{7} \) \(\mathstrut +\mathstrut 42q^{8} \) \(\mathstrut +\mathstrut 92q^{9} \) \(\mathstrut +\mathstrut 4q^{10} \) \(\mathstrut +\mathstrut 41q^{11} \) \(\mathstrut +\mathstrut 26q^{12} \) \(\mathstrut +\mathstrut 13q^{13} \) \(\mathstrut +\mathstrut 22q^{14} \) \(\mathstrut +\mathstrut 41q^{15} \) \(\mathstrut +\mathstrut 87q^{16} \) \(\mathstrut +\mathstrut 12q^{17} \) \(\mathstrut +\mathstrut 24q^{18} \) \(\mathstrut +\mathstrut 82q^{19} \) \(\mathstrut +\mathstrut 26q^{20} \) \(\mathstrut +\mathstrut 29q^{21} \) \(\mathstrut +\mathstrut 2q^{22} \) \(\mathstrut +\mathstrut 59q^{23} \) \(\mathstrut +\mathstrut 16q^{24} \) \(\mathstrut +\mathstrut 67q^{25} \) \(\mathstrut +\mathstrut 24q^{26} \) \(\mathstrut +\mathstrut 42q^{27} \) \(\mathstrut -\mathstrut 2q^{28} \) \(\mathstrut +\mathstrut 101q^{29} \) \(\mathstrut -\mathstrut 22q^{30} \) \(\mathstrut +\mathstrut 48q^{31} \) \(\mathstrut +\mathstrut 69q^{32} \) \(\mathstrut +\mathstrut 3q^{33} \) \(\mathstrut +\mathstrut q^{34} \) \(\mathstrut +\mathstrut 38q^{35} \) \(\mathstrut +\mathstrut 82q^{36} \) \(\mathstrut +\mathstrut 16q^{37} \) \(\mathstrut +\mathstrut 15q^{38} \) \(\mathstrut +\mathstrut 82q^{39} \) \(\mathstrut +\mathstrut 20q^{40} \) \(\mathstrut +\mathstrut 86q^{41} \) \(\mathstrut -\mathstrut q^{42} \) \(\mathstrut +\mathstrut 9q^{43} \) \(\mathstrut +\mathstrut 82q^{44} \) \(\mathstrut -\mathstrut 8q^{45} \) \(\mathstrut +\mathstrut 43q^{46} \) \(\mathstrut +\mathstrut 24q^{47} \) \(\mathstrut +\mathstrut 34q^{48} \) \(\mathstrut +\mathstrut 76q^{49} \) \(\mathstrut +\mathstrut 82q^{50} \) \(\mathstrut +\mathstrut 57q^{51} \) \(\mathstrut -\mathstrut 22q^{52} \) \(\mathstrut +\mathstrut 39q^{53} \) \(\mathstrut +\mathstrut 17q^{54} \) \(\mathstrut -\mathstrut 21q^{55} \) \(\mathstrut +\mathstrut 50q^{56} \) \(\mathstrut +\mathstrut 12q^{57} \) \(\mathstrut +\mathstrut 33q^{58} \) \(\mathstrut +\mathstrut 79q^{59} \) \(\mathstrut +\mathstrut 87q^{60} \) \(\mathstrut +\mathstrut 4q^{61} \) \(\mathstrut +\mathstrut 40q^{62} \) \(\mathstrut +\mathstrut 44q^{63} \) \(\mathstrut +\mathstrut 90q^{64} \) \(\mathstrut +\mathstrut 66q^{65} \) \(\mathstrut -\mathstrut 39q^{66} \) \(\mathstrut +\mathstrut 33q^{67} \) \(\mathstrut -\mathstrut 9q^{68} \) \(\mathstrut +\mathstrut 60q^{69} \) \(\mathstrut +\mathstrut 30q^{70} \) \(\mathstrut +\mathstrut 168q^{71} \) \(\mathstrut +\mathstrut 15q^{72} \) \(\mathstrut -\mathstrut 28q^{73} \) \(\mathstrut +\mathstrut 35q^{74} \) \(\mathstrut +\mathstrut 55q^{75} \) \(\mathstrut +\mathstrut 89q^{76} \) \(\mathstrut +\mathstrut 19q^{77} \) \(\mathstrut -\mathstrut 41q^{78} \) \(\mathstrut +\mathstrut 121q^{79} \) \(\mathstrut +\mathstrut 64q^{80} \) \(\mathstrut +\mathstrut 110q^{81} \) \(\mathstrut +\mathstrut 41q^{82} \) \(\mathstrut +\mathstrut 28q^{84} \) \(\mathstrut +\mathstrut 17q^{85} \) \(\mathstrut +\mathstrut 80q^{86} \) \(\mathstrut +\mathstrut 29q^{87} \) \(\mathstrut +\mathstrut 49q^{88} \) \(\mathstrut +\mathstrut 83q^{89} \) \(\mathstrut -\mathstrut 42q^{90} \) \(\mathstrut +\mathstrut 38q^{91} \) \(\mathstrut +\mathstrut 71q^{92} \) \(\mathstrut -\mathstrut q^{93} \) \(\mathstrut +\mathstrut 89q^{94} \) \(\mathstrut +\mathstrut 9q^{95} \) \(\mathstrut +\mathstrut 35q^{96} \) \(\mathstrut -\mathstrut 23q^{97} \) \(\mathstrut +\mathstrut 135q^{98} \) \(\mathstrut +\mathstrut 93q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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 −2.71228 −0.799829 5.35647 2.45685 2.16936 −1.47623 −9.10370 −2.36027 −6.66368
1.2 −2.64355 −0.244924 4.98837 −2.38993 0.647469 −3.32754 −7.89992 −2.94001 6.31792
1.3 −2.62936 3.14041 4.91352 2.46626 −8.25726 1.15889 −7.66068 6.86218 −6.48468
1.4 −2.62201 −2.46509 4.87496 −1.74208 6.46350 −2.61324 −7.53817 3.07667 4.56776
1.5 −2.40525 0.482964 3.78524 1.80217 −1.16165 −1.90696 −4.29395 −2.76675 −4.33466
1.6 −2.39399 1.52449 3.73121 −2.51919 −3.64962 −1.00997 −4.14450 −0.675931 6.03093
1.7 −2.39206 −0.112128 3.72195 0.840987 0.268217 2.66760 −4.11902 −2.98743 −2.01169
1.8 −2.27080 −1.94834 3.15654 0.0772315 4.42430 2.91283 −2.62627 0.796044 −0.175377
1.9 −2.22938 2.96636 2.97012 −0.893171 −6.61314 4.38195 −2.16277 5.79930 1.99121
1.10 −2.21501 1.93892 2.90626 3.78369 −4.29473 0.133386 −2.00737 0.759427 −8.38090
1.11 −2.16486 3.17577 2.68660 0.570979 −6.87507 −2.83752 −1.48639 7.08548 −1.23609
1.12 −2.09995 −2.73690 2.40979 2.29135 5.74735 −1.74327 −0.860544 4.49062 −4.81171
1.13 −1.93317 −1.52394 1.73714 −2.99074 2.94604 −0.667472 0.508147 −0.677595 5.78161
1.14 −1.90886 1.35744 1.64375 −1.48428 −2.59115 2.77982 0.680038 −1.15737 2.83328
1.15 −1.89977 −2.93667 1.60912 −0.296578 5.57898 2.59029 0.742589 5.62400 0.563430
1.16 −1.69115 −1.97765 0.859975 −4.28235 3.34450 4.00531 1.92795 0.911117 7.24208
1.17 −1.66851 −1.79206 0.783934 2.47432 2.99007 0.566931 2.02902 0.211475 −4.12843
1.18 −1.53574 2.00602 0.358512 1.47205 −3.08073 −4.88163 2.52091 1.02410 −2.26069
1.19 −1.53259 1.29314 0.348820 3.28950 −1.98185 4.48074 2.53058 −1.32779 −5.04144
1.20 −1.51298 0.189655 0.289107 −0.503823 −0.286944 −1.74123 2.58855 −2.96403 0.762274
See all 82 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.82
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
\(19\) \(-1\)
\(211\) \(1\)

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{2}^{82} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4009))\).