Properties

Label 8038.2.a.c
Level 8038
Weight 2
Character orbit 8038.a
Self dual Yes
Analytic conductor 64.184
Analytic rank 1
Dimension 84
CM No

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Newspace parameters

Level: \( N \) = \( 8038 = 2 \cdot 4019 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 8038.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(64.1837531447\)
Analytic rank: \(1\)
Dimension: \(84\)
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.

\(\operatorname{Tr}(f)(q) = \) \(84q \) \(\mathstrut -\mathstrut 84q^{2} \) \(\mathstrut -\mathstrut 19q^{3} \) \(\mathstrut +\mathstrut 84q^{4} \) \(\mathstrut -\mathstrut 32q^{5} \) \(\mathstrut +\mathstrut 19q^{6} \) \(\mathstrut +\mathstrut q^{7} \) \(\mathstrut -\mathstrut 84q^{8} \) \(\mathstrut +\mathstrut 77q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \(84q \) \(\mathstrut -\mathstrut 84q^{2} \) \(\mathstrut -\mathstrut 19q^{3} \) \(\mathstrut +\mathstrut 84q^{4} \) \(\mathstrut -\mathstrut 32q^{5} \) \(\mathstrut +\mathstrut 19q^{6} \) \(\mathstrut +\mathstrut q^{7} \) \(\mathstrut -\mathstrut 84q^{8} \) \(\mathstrut +\mathstrut 77q^{9} \) \(\mathstrut +\mathstrut 32q^{10} \) \(\mathstrut -\mathstrut 6q^{11} \) \(\mathstrut -\mathstrut 19q^{12} \) \(\mathstrut -\mathstrut 29q^{13} \) \(\mathstrut -\mathstrut q^{14} \) \(\mathstrut -\mathstrut 4q^{15} \) \(\mathstrut +\mathstrut 84q^{16} \) \(\mathstrut -\mathstrut 36q^{17} \) \(\mathstrut -\mathstrut 77q^{18} \) \(\mathstrut +\mathstrut 33q^{19} \) \(\mathstrut -\mathstrut 32q^{20} \) \(\mathstrut -\mathstrut 21q^{21} \) \(\mathstrut +\mathstrut 6q^{22} \) \(\mathstrut -\mathstrut 62q^{23} \) \(\mathstrut +\mathstrut 19q^{24} \) \(\mathstrut +\mathstrut 82q^{25} \) \(\mathstrut +\mathstrut 29q^{26} \) \(\mathstrut -\mathstrut 82q^{27} \) \(\mathstrut +\mathstrut q^{28} \) \(\mathstrut -\mathstrut 51q^{29} \) \(\mathstrut +\mathstrut 4q^{30} \) \(\mathstrut +\mathstrut 39q^{31} \) \(\mathstrut -\mathstrut 84q^{32} \) \(\mathstrut -\mathstrut 32q^{33} \) \(\mathstrut +\mathstrut 36q^{34} \) \(\mathstrut -\mathstrut 34q^{35} \) \(\mathstrut +\mathstrut 77q^{36} \) \(\mathstrut -\mathstrut 32q^{37} \) \(\mathstrut -\mathstrut 33q^{38} \) \(\mathstrut +\mathstrut 29q^{39} \) \(\mathstrut +\mathstrut 32q^{40} \) \(\mathstrut -\mathstrut 38q^{41} \) \(\mathstrut +\mathstrut 21q^{42} \) \(\mathstrut -\mathstrut 6q^{44} \) \(\mathstrut -\mathstrut 91q^{45} \) \(\mathstrut +\mathstrut 62q^{46} \) \(\mathstrut -\mathstrut 58q^{47} \) \(\mathstrut -\mathstrut 19q^{48} \) \(\mathstrut +\mathstrut 83q^{49} \) \(\mathstrut -\mathstrut 82q^{50} \) \(\mathstrut -\mathstrut q^{51} \) \(\mathstrut -\mathstrut 29q^{52} \) \(\mathstrut -\mathstrut 106q^{53} \) \(\mathstrut +\mathstrut 82q^{54} \) \(\mathstrut +\mathstrut 32q^{55} \) \(\mathstrut -\mathstrut q^{56} \) \(\mathstrut -\mathstrut 44q^{57} \) \(\mathstrut +\mathstrut 51q^{58} \) \(\mathstrut -\mathstrut 42q^{59} \) \(\mathstrut -\mathstrut 4q^{60} \) \(\mathstrut -\mathstrut 41q^{61} \) \(\mathstrut -\mathstrut 39q^{62} \) \(\mathstrut -\mathstrut 9q^{63} \) \(\mathstrut +\mathstrut 84q^{64} \) \(\mathstrut -\mathstrut 49q^{65} \) \(\mathstrut +\mathstrut 32q^{66} \) \(\mathstrut -\mathstrut 16q^{67} \) \(\mathstrut -\mathstrut 36q^{68} \) \(\mathstrut -\mathstrut 45q^{69} \) \(\mathstrut +\mathstrut 34q^{70} \) \(\mathstrut -\mathstrut 62q^{71} \) \(\mathstrut -\mathstrut 77q^{72} \) \(\mathstrut -\mathstrut 16q^{73} \) \(\mathstrut +\mathstrut 32q^{74} \) \(\mathstrut -\mathstrut 80q^{75} \) \(\mathstrut +\mathstrut 33q^{76} \) \(\mathstrut -\mathstrut 134q^{77} \) \(\mathstrut -\mathstrut 29q^{78} \) \(\mathstrut +\mathstrut 53q^{79} \) \(\mathstrut -\mathstrut 32q^{80} \) \(\mathstrut +\mathstrut 56q^{81} \) \(\mathstrut +\mathstrut 38q^{82} \) \(\mathstrut -\mathstrut 90q^{83} \) \(\mathstrut -\mathstrut 21q^{84} \) \(\mathstrut -\mathstrut 60q^{85} \) \(\mathstrut -\mathstrut 3q^{87} \) \(\mathstrut +\mathstrut 6q^{88} \) \(\mathstrut -\mathstrut 54q^{89} \) \(\mathstrut +\mathstrut 91q^{90} \) \(\mathstrut +\mathstrut 33q^{91} \) \(\mathstrut -\mathstrut 62q^{92} \) \(\mathstrut -\mathstrut 69q^{93} \) \(\mathstrut +\mathstrut 58q^{94} \) \(\mathstrut -\mathstrut 47q^{95} \) \(\mathstrut +\mathstrut 19q^{96} \) \(\mathstrut -\mathstrut 31q^{97} \) \(\mathstrut -\mathstrut 83q^{98} \) \(\mathstrut +\mathstrut 23q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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 −1.00000 −3.36115 1.00000 −3.51543 3.36115 −3.19269 −1.00000 8.29735 3.51543
1.2 −1.00000 −3.35998 1.00000 2.98972 3.35998 3.64714 −1.00000 8.28947 −2.98972
1.3 −1.00000 −3.33190 1.00000 −1.90041 3.33190 −1.90946 −1.00000 8.10156 1.90041
1.4 −1.00000 −3.26564 1.00000 1.02149 3.26564 −2.73396 −1.00000 7.66439 −1.02149
1.5 −1.00000 −3.22723 1.00000 −4.01107 3.22723 2.38054 −1.00000 7.41504 4.01107
1.6 −1.00000 −3.18728 1.00000 −3.96813 3.18728 3.34102 −1.00000 7.15875 3.96813
1.7 −1.00000 −2.97502 1.00000 1.80363 2.97502 −0.809668 −1.00000 5.85072 −1.80363
1.8 −1.00000 −2.90498 1.00000 −2.22164 2.90498 −2.98182 −1.00000 5.43889 2.22164
1.9 −1.00000 −2.89385 1.00000 0.388076 2.89385 −0.0680600 −1.00000 5.37435 −0.388076
1.10 −1.00000 −2.71506 1.00000 −2.98033 2.71506 3.18004 −1.00000 4.37153 2.98033
1.11 −1.00000 −2.71043 1.00000 1.06338 2.71043 1.86356 −1.00000 4.34646 −1.06338
1.12 −1.00000 −2.66947 1.00000 0.308435 2.66947 −4.20139 −1.00000 4.12607 −0.308435
1.13 −1.00000 −2.63531 1.00000 2.31488 2.63531 −0.771560 −1.00000 3.94487 −2.31488
1.14 −1.00000 −2.62072 1.00000 −0.614579 2.62072 2.08468 −1.00000 3.86818 0.614579
1.15 −1.00000 −2.53354 1.00000 1.39163 2.53354 4.71879 −1.00000 3.41884 −1.39163
1.16 −1.00000 −2.39740 1.00000 −1.80214 2.39740 −2.60681 −1.00000 2.74755 1.80214
1.17 −1.00000 −2.39489 1.00000 4.12569 2.39489 3.01891 −1.00000 2.73549 −4.12569
1.18 −1.00000 −2.25732 1.00000 −3.39963 2.25732 3.01257 −1.00000 2.09550 3.39963
1.19 −1.00000 −2.22339 1.00000 −0.973252 2.22339 −3.76604 −1.00000 1.94347 0.973252
1.20 −1.00000 −1.99033 1.00000 −1.73876 1.99033 5.09180 −1.00000 0.961408 1.73876
See all 84 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.84
Significant digits:
Format:

Inner twists

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

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(4019\) \(1\)