Properties

Label 8047.2.a.c
Level 8047
Weight 2
Character orbit 8047.a
Self dual Yes
Analytic conductor 64.256
Analytic rank 1
Dimension 151
CM No

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Newspace parameters

Level: \( N \) = \( 8047 = 13 \cdot 619 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 8047.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(64.2556185065\)
Analytic rank: \(1\)
Dimension: \(151\)
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.

\(\operatorname{Tr}(f)(q) = \) \(151q \) \(\mathstrut -\mathstrut 13q^{2} \) \(\mathstrut -\mathstrut 16q^{3} \) \(\mathstrut +\mathstrut 151q^{4} \) \(\mathstrut -\mathstrut 43q^{5} \) \(\mathstrut -\mathstrut 17q^{6} \) \(\mathstrut -\mathstrut 18q^{7} \) \(\mathstrut -\mathstrut 39q^{8} \) \(\mathstrut +\mathstrut 135q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \(151q \) \(\mathstrut -\mathstrut 13q^{2} \) \(\mathstrut -\mathstrut 16q^{3} \) \(\mathstrut +\mathstrut 151q^{4} \) \(\mathstrut -\mathstrut 43q^{5} \) \(\mathstrut -\mathstrut 17q^{6} \) \(\mathstrut -\mathstrut 18q^{7} \) \(\mathstrut -\mathstrut 39q^{8} \) \(\mathstrut +\mathstrut 135q^{9} \) \(\mathstrut -\mathstrut 3q^{10} \) \(\mathstrut -\mathstrut 27q^{11} \) \(\mathstrut -\mathstrut 52q^{12} \) \(\mathstrut -\mathstrut 151q^{13} \) \(\mathstrut -\mathstrut 9q^{14} \) \(\mathstrut -\mathstrut 14q^{15} \) \(\mathstrut +\mathstrut 143q^{16} \) \(\mathstrut -\mathstrut 111q^{17} \) \(\mathstrut -\mathstrut 37q^{18} \) \(\mathstrut -\mathstrut 17q^{19} \) \(\mathstrut -\mathstrut 107q^{20} \) \(\mathstrut -\mathstrut 29q^{21} \) \(\mathstrut -\mathstrut 16q^{22} \) \(\mathstrut -\mathstrut 47q^{23} \) \(\mathstrut -\mathstrut 46q^{24} \) \(\mathstrut +\mathstrut 122q^{25} \) \(\mathstrut +\mathstrut 13q^{26} \) \(\mathstrut -\mathstrut 55q^{27} \) \(\mathstrut -\mathstrut 44q^{28} \) \(\mathstrut +\mathstrut 37q^{29} \) \(\mathstrut -\mathstrut 14q^{30} \) \(\mathstrut -\mathstrut 27q^{31} \) \(\mathstrut -\mathstrut 86q^{32} \) \(\mathstrut -\mathstrut 94q^{33} \) \(\mathstrut -\mathstrut 10q^{34} \) \(\mathstrut -\mathstrut 47q^{35} \) \(\mathstrut +\mathstrut 124q^{36} \) \(\mathstrut -\mathstrut 59q^{37} \) \(\mathstrut -\mathstrut 80q^{38} \) \(\mathstrut +\mathstrut 16q^{39} \) \(\mathstrut +\mathstrut 5q^{40} \) \(\mathstrut -\mathstrut 129q^{41} \) \(\mathstrut -\mathstrut 77q^{42} \) \(\mathstrut -\mathstrut 11q^{43} \) \(\mathstrut -\mathstrut 99q^{44} \) \(\mathstrut -\mathstrut 122q^{45} \) \(\mathstrut -\mathstrut 17q^{46} \) \(\mathstrut -\mathstrut 130q^{47} \) \(\mathstrut -\mathstrut 111q^{48} \) \(\mathstrut +\mathstrut 99q^{49} \) \(\mathstrut -\mathstrut 72q^{50} \) \(\mathstrut +\mathstrut 15q^{51} \) \(\mathstrut -\mathstrut 151q^{52} \) \(\mathstrut -\mathstrut 43q^{53} \) \(\mathstrut -\mathstrut 49q^{54} \) \(\mathstrut -\mathstrut 40q^{55} \) \(\mathstrut -\mathstrut 50q^{56} \) \(\mathstrut -\mathstrut 85q^{57} \) \(\mathstrut -\mathstrut 73q^{58} \) \(\mathstrut -\mathstrut 74q^{59} \) \(\mathstrut -\mathstrut 43q^{60} \) \(\mathstrut -\mathstrut 7q^{61} \) \(\mathstrut -\mathstrut 110q^{62} \) \(\mathstrut -\mathstrut 70q^{63} \) \(\mathstrut +\mathstrut 141q^{64} \) \(\mathstrut +\mathstrut 43q^{65} \) \(\mathstrut -\mathstrut 16q^{66} \) \(\mathstrut -\mathstrut 39q^{67} \) \(\mathstrut -\mathstrut 222q^{68} \) \(\mathstrut +\mathstrut 19q^{69} \) \(\mathstrut -\mathstrut 52q^{70} \) \(\mathstrut -\mathstrut 72q^{71} \) \(\mathstrut -\mathstrut 106q^{72} \) \(\mathstrut -\mathstrut 143q^{73} \) \(\mathstrut +\mathstrut 20q^{74} \) \(\mathstrut -\mathstrut 73q^{75} \) \(\mathstrut -\mathstrut 88q^{76} \) \(\mathstrut -\mathstrut 86q^{77} \) \(\mathstrut +\mathstrut 17q^{78} \) \(\mathstrut +\mathstrut 10q^{79} \) \(\mathstrut -\mathstrut 239q^{80} \) \(\mathstrut +\mathstrut 103q^{81} \) \(\mathstrut -\mathstrut 96q^{82} \) \(\mathstrut -\mathstrut 96q^{83} \) \(\mathstrut -\mathstrut 75q^{84} \) \(\mathstrut -\mathstrut 24q^{85} \) \(\mathstrut -\mathstrut 109q^{86} \) \(\mathstrut -\mathstrut 65q^{87} \) \(\mathstrut -\mathstrut 45q^{88} \) \(\mathstrut -\mathstrut 237q^{89} \) \(\mathstrut -\mathstrut 79q^{90} \) \(\mathstrut +\mathstrut 18q^{91} \) \(\mathstrut -\mathstrut 153q^{92} \) \(\mathstrut -\mathstrut 137q^{93} \) \(\mathstrut -\mathstrut 23q^{94} \) \(\mathstrut +\mathstrut 10q^{95} \) \(\mathstrut -\mathstrut 109q^{96} \) \(\mathstrut -\mathstrut 160q^{97} \) \(\mathstrut -\mathstrut 119q^{98} \) \(\mathstrut -\mathstrut 86q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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 −2.80247 −1.33128 5.85385 −2.29720 3.73087 −2.08383 −10.8003 −1.22769 6.43783
1.2 −2.76685 2.61084 5.65545 1.02194 −7.22381 1.46189 −10.1141 3.81650 −2.82754
1.3 −2.75813 −3.14715 5.60727 −3.12480 8.68025 −2.02299 −9.94932 6.90456 8.61861
1.4 −2.73460 0.234090 5.47804 −4.22814 −0.640142 4.15879 −9.51105 −2.94520 11.5623
1.5 −2.70117 −0.798295 5.29633 1.50406 2.15633 −4.14726 −8.90395 −2.36273 −4.06273
1.6 −2.69703 2.83475 5.27397 −3.14903 −7.64541 1.25916 −8.82999 5.03583 8.49303
1.7 −2.66878 1.69053 5.12237 −3.40805 −4.51165 −4.86127 −8.33291 −0.142111 9.09532
1.8 −2.66015 −2.71534 5.07642 1.80330 7.22323 −0.0430519 −8.18376 4.37308 −4.79706
1.9 −2.65702 1.31176 5.05973 3.40250 −3.48538 2.98608 −8.12975 −1.27927 −9.04051
1.10 −2.62707 −2.08646 4.90149 1.65471 5.48127 −3.37709 −7.62241 1.35330 −4.34704
1.11 −2.59872 0.688709 4.75335 −1.04737 −1.78976 2.76372 −7.15519 −2.52568 2.72181
1.12 −2.58760 −0.469727 4.69569 −2.90773 1.21547 0.483013 −6.97536 −2.77936 7.52406
1.13 −2.53026 −1.57653 4.40222 −1.20027 3.98902 4.10699 −6.07823 −0.514564 3.03700
1.14 −2.42822 2.05739 3.89625 1.53452 −4.99579 1.53522 −4.60451 1.23285 −3.72615
1.15 −2.37031 2.63830 3.61835 3.06059 −6.25357 −2.14204 −3.83599 3.96060 −7.25455
1.16 −2.36725 0.254210 3.60387 1.17067 −0.601779 −2.57756 −3.79676 −2.93538 −2.77126
1.17 −2.34671 2.23237 3.50704 −1.50112 −5.23873 −4.09520 −3.53658 1.98350 3.52269
1.18 −2.30742 −2.51530 3.32419 −4.19325 5.80386 2.00418 −3.05546 3.32673 9.67559
1.19 −2.28288 −3.31283 3.21152 2.87679 7.56278 −0.777401 −2.76575 7.97486 −6.56735
1.20 −2.25853 −0.662024 3.10095 2.15399 1.49520 0.474761 −2.48652 −2.56172 −4.86485
See next 80 embeddings (of 151 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.151
Significant digits:
Format:

Inner twists

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

Atkin-Lehner signs

\( p \) Sign
\(13\) \(1\)
\(619\) \(1\)

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{2}^{151} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8047))\).