Properties

Label 8038.2.a.a
Level 8038
Weight 2
Character orbit 8038.a
Self dual Yes
Analytic conductor 64.184
Analytic rank 1
Dimension 75
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: \(75\)
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.

\(\operatorname{Tr}(f)(q) = \) \(75q \) \(\mathstrut +\mathstrut 75q^{2} \) \(\mathstrut -\mathstrut 30q^{3} \) \(\mathstrut +\mathstrut 75q^{4} \) \(\mathstrut -\mathstrut 29q^{5} \) \(\mathstrut -\mathstrut 30q^{6} \) \(\mathstrut -\mathstrut 31q^{7} \) \(\mathstrut +\mathstrut 75q^{8} \) \(\mathstrut +\mathstrut 55q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \(75q \) \(\mathstrut +\mathstrut 75q^{2} \) \(\mathstrut -\mathstrut 30q^{3} \) \(\mathstrut +\mathstrut 75q^{4} \) \(\mathstrut -\mathstrut 29q^{5} \) \(\mathstrut -\mathstrut 30q^{6} \) \(\mathstrut -\mathstrut 31q^{7} \) \(\mathstrut +\mathstrut 75q^{8} \) \(\mathstrut +\mathstrut 55q^{9} \) \(\mathstrut -\mathstrut 29q^{10} \) \(\mathstrut -\mathstrut 30q^{11} \) \(\mathstrut -\mathstrut 30q^{12} \) \(\mathstrut -\mathstrut 23q^{13} \) \(\mathstrut -\mathstrut 31q^{14} \) \(\mathstrut -\mathstrut 22q^{15} \) \(\mathstrut +\mathstrut 75q^{16} \) \(\mathstrut -\mathstrut 48q^{17} \) \(\mathstrut +\mathstrut 55q^{18} \) \(\mathstrut -\mathstrut 58q^{19} \) \(\mathstrut -\mathstrut 29q^{20} \) \(\mathstrut -\mathstrut 5q^{21} \) \(\mathstrut -\mathstrut 30q^{22} \) \(\mathstrut -\mathstrut 79q^{23} \) \(\mathstrut -\mathstrut 30q^{24} \) \(\mathstrut +\mathstrut 42q^{25} \) \(\mathstrut -\mathstrut 23q^{26} \) \(\mathstrut -\mathstrut 108q^{27} \) \(\mathstrut -\mathstrut 31q^{28} \) \(\mathstrut -\mathstrut 39q^{29} \) \(\mathstrut -\mathstrut 22q^{30} \) \(\mathstrut -\mathstrut 95q^{31} \) \(\mathstrut +\mathstrut 75q^{32} \) \(\mathstrut -\mathstrut 44q^{33} \) \(\mathstrut -\mathstrut 48q^{34} \) \(\mathstrut -\mathstrut 60q^{35} \) \(\mathstrut +\mathstrut 55q^{36} \) \(\mathstrut -\mathstrut 16q^{37} \) \(\mathstrut -\mathstrut 58q^{38} \) \(\mathstrut -\mathstrut 57q^{39} \) \(\mathstrut -\mathstrut 29q^{40} \) \(\mathstrut -\mathstrut 85q^{41} \) \(\mathstrut -\mathstrut 5q^{42} \) \(\mathstrut -\mathstrut 55q^{43} \) \(\mathstrut -\mathstrut 30q^{44} \) \(\mathstrut -\mathstrut 48q^{45} \) \(\mathstrut -\mathstrut 79q^{46} \) \(\mathstrut -\mathstrut 88q^{47} \) \(\mathstrut -\mathstrut 30q^{48} \) \(\mathstrut +\mathstrut 26q^{49} \) \(\mathstrut +\mathstrut 42q^{50} \) \(\mathstrut -\mathstrut 39q^{51} \) \(\mathstrut -\mathstrut 23q^{52} \) \(\mathstrut -\mathstrut 79q^{53} \) \(\mathstrut -\mathstrut 108q^{54} \) \(\mathstrut -\mathstrut 78q^{55} \) \(\mathstrut -\mathstrut 31q^{56} \) \(\mathstrut -\mathstrut 40q^{57} \) \(\mathstrut -\mathstrut 39q^{58} \) \(\mathstrut -\mathstrut 74q^{59} \) \(\mathstrut -\mathstrut 22q^{60} \) \(\mathstrut -\mathstrut 21q^{61} \) \(\mathstrut -\mathstrut 95q^{62} \) \(\mathstrut -\mathstrut 97q^{63} \) \(\mathstrut +\mathstrut 75q^{64} \) \(\mathstrut -\mathstrut 63q^{65} \) \(\mathstrut -\mathstrut 44q^{66} \) \(\mathstrut -\mathstrut 59q^{67} \) \(\mathstrut -\mathstrut 48q^{68} \) \(\mathstrut +\mathstrut 3q^{69} \) \(\mathstrut -\mathstrut 60q^{70} \) \(\mathstrut -\mathstrut 72q^{71} \) \(\mathstrut +\mathstrut 55q^{72} \) \(\mathstrut -\mathstrut 91q^{73} \) \(\mathstrut -\mathstrut 16q^{74} \) \(\mathstrut -\mathstrut 95q^{75} \) \(\mathstrut -\mathstrut 58q^{76} \) \(\mathstrut -\mathstrut 60q^{77} \) \(\mathstrut -\mathstrut 57q^{78} \) \(\mathstrut -\mathstrut 64q^{79} \) \(\mathstrut -\mathstrut 29q^{80} \) \(\mathstrut +\mathstrut 47q^{81} \) \(\mathstrut -\mathstrut 85q^{82} \) \(\mathstrut -\mathstrut 105q^{83} \) \(\mathstrut -\mathstrut 5q^{84} \) \(\mathstrut +\mathstrut 14q^{85} \) \(\mathstrut -\mathstrut 55q^{86} \) \(\mathstrut -\mathstrut 75q^{87} \) \(\mathstrut -\mathstrut 30q^{88} \) \(\mathstrut -\mathstrut 78q^{89} \) \(\mathstrut -\mathstrut 48q^{90} \) \(\mathstrut -\mathstrut 89q^{91} \) \(\mathstrut -\mathstrut 79q^{92} \) \(\mathstrut +\mathstrut 27q^{93} \) \(\mathstrut -\mathstrut 88q^{94} \) \(\mathstrut -\mathstrut 53q^{95} \) \(\mathstrut -\mathstrut 30q^{96} \) \(\mathstrut -\mathstrut 79q^{97} \) \(\mathstrut +\mathstrut 26q^{98} \) \(\mathstrut -\mathstrut 57q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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 −3.39091 1.00000 3.77047 −3.39091 −1.27110 1.00000 8.49828 3.77047
1.2 1.00000 −3.38305 1.00000 1.91300 −3.38305 −4.34480 1.00000 8.44501 1.91300
1.3 1.00000 −3.32095 1.00000 −3.93433 −3.32095 −4.89708 1.00000 8.02870 −3.93433
1.4 1.00000 −3.27949 1.00000 −1.91910 −3.27949 3.63095 1.00000 7.75505 −1.91910
1.5 1.00000 −3.25539 1.00000 −2.85398 −3.25539 2.81527 1.00000 7.59759 −2.85398
1.6 1.00000 −3.20148 1.00000 0.754953 −3.20148 1.04626 1.00000 7.24947 0.754953
1.7 1.00000 −3.17284 1.00000 −1.87632 −3.17284 −0.162262 1.00000 7.06690 −1.87632
1.8 1.00000 −2.93359 1.00000 2.13721 −2.93359 0.512181 1.00000 5.60596 2.13721
1.9 1.00000 −2.85626 1.00000 0.772797 −2.85626 −4.64843 1.00000 5.15824 0.772797
1.10 1.00000 −2.79660 1.00000 −1.17558 −2.79660 1.04192 1.00000 4.82097 −1.17558
1.11 1.00000 −2.59529 1.00000 3.17530 −2.59529 −0.203775 1.00000 3.73551 3.17530
1.12 1.00000 −2.54835 1.00000 −2.91194 −2.54835 −2.80949 1.00000 3.49408 −2.91194
1.13 1.00000 −2.53842 1.00000 2.78579 −2.53842 −2.33236 1.00000 3.44358 2.78579
1.14 1.00000 −2.46931 1.00000 0.696459 −2.46931 −2.94189 1.00000 3.09749 0.696459
1.15 1.00000 −2.32361 1.00000 0.467112 −2.32361 3.12912 1.00000 2.39917 0.467112
1.16 1.00000 −2.28450 1.00000 −2.56431 −2.28450 0.255204 1.00000 2.21893 −2.56431
1.17 1.00000 −2.22920 1.00000 −4.17184 −2.22920 −0.836281 1.00000 1.96935 −4.17184
1.18 1.00000 −2.12788 1.00000 3.19034 −2.12788 2.73143 1.00000 1.52787 3.19034
1.19 1.00000 −2.10614 1.00000 −0.0733427 −2.10614 4.17239 1.00000 1.43582 −0.0733427
1.20 1.00000 −1.99460 1.00000 1.10571 −1.99460 2.73220 1.00000 0.978446 1.10571
See all 75 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.75
Significant digits:
Format:

Inner twists

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

Atkin-Lehner signs

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