Properties

Label 8038.2.a.d
Level 8038
Weight 2
Character orbit 8038.a
Self dual Yes
Analytic conductor 64.184
Analytic rank 0
Dimension 92
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: \(0\)
Dimension: \(92\)
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.

\(\operatorname{Tr}(f)(q) = \) \(92q \) \(\mathstrut +\mathstrut 92q^{2} \) \(\mathstrut +\mathstrut 31q^{3} \) \(\mathstrut +\mathstrut 92q^{4} \) \(\mathstrut +\mathstrut 28q^{5} \) \(\mathstrut +\mathstrut 31q^{6} \) \(\mathstrut +\mathstrut 29q^{7} \) \(\mathstrut +\mathstrut 92q^{8} \) \(\mathstrut +\mathstrut 113q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \(92q \) \(\mathstrut +\mathstrut 92q^{2} \) \(\mathstrut +\mathstrut 31q^{3} \) \(\mathstrut +\mathstrut 92q^{4} \) \(\mathstrut +\mathstrut 28q^{5} \) \(\mathstrut +\mathstrut 31q^{6} \) \(\mathstrut +\mathstrut 29q^{7} \) \(\mathstrut +\mathstrut 92q^{8} \) \(\mathstrut +\mathstrut 113q^{9} \) \(\mathstrut +\mathstrut 28q^{10} \) \(\mathstrut +\mathstrut 37q^{11} \) \(\mathstrut +\mathstrut 31q^{12} \) \(\mathstrut +\mathstrut 20q^{13} \) \(\mathstrut +\mathstrut 29q^{14} \) \(\mathstrut +\mathstrut 30q^{15} \) \(\mathstrut +\mathstrut 92q^{16} \) \(\mathstrut +\mathstrut 52q^{17} \) \(\mathstrut +\mathstrut 113q^{18} \) \(\mathstrut +\mathstrut 61q^{19} \) \(\mathstrut +\mathstrut 28q^{20} \) \(\mathstrut +\mathstrut 5q^{21} \) \(\mathstrut +\mathstrut 37q^{22} \) \(\mathstrut +\mathstrut 71q^{23} \) \(\mathstrut +\mathstrut 31q^{24} \) \(\mathstrut +\mathstrut 118q^{25} \) \(\mathstrut +\mathstrut 20q^{26} \) \(\mathstrut +\mathstrut 112q^{27} \) \(\mathstrut +\mathstrut 29q^{28} \) \(\mathstrut +\mathstrut 30q^{29} \) \(\mathstrut +\mathstrut 30q^{30} \) \(\mathstrut +\mathstrut 89q^{31} \) \(\mathstrut +\mathstrut 92q^{32} \) \(\mathstrut +\mathstrut 52q^{33} \) \(\mathstrut +\mathstrut 52q^{34} \) \(\mathstrut +\mathstrut 58q^{35} \) \(\mathstrut +\mathstrut 113q^{36} \) \(\mathstrut +\mathstrut 15q^{37} \) \(\mathstrut +\mathstrut 61q^{38} \) \(\mathstrut +\mathstrut 43q^{39} \) \(\mathstrut +\mathstrut 28q^{40} \) \(\mathstrut +\mathstrut 75q^{41} \) \(\mathstrut +\mathstrut 5q^{42} \) \(\mathstrut +\mathstrut 46q^{43} \) \(\mathstrut +\mathstrut 37q^{44} \) \(\mathstrut +\mathstrut 63q^{45} \) \(\mathstrut +\mathstrut 71q^{46} \) \(\mathstrut +\mathstrut 92q^{47} \) \(\mathstrut +\mathstrut 31q^{48} \) \(\mathstrut +\mathstrut 131q^{49} \) \(\mathstrut +\mathstrut 118q^{50} \) \(\mathstrut +\mathstrut 45q^{51} \) \(\mathstrut +\mathstrut 20q^{52} \) \(\mathstrut +\mathstrut 72q^{53} \) \(\mathstrut +\mathstrut 112q^{54} \) \(\mathstrut +\mathstrut 86q^{55} \) \(\mathstrut +\mathstrut 29q^{56} \) \(\mathstrut +\mathstrut 44q^{57} \) \(\mathstrut +\mathstrut 30q^{58} \) \(\mathstrut +\mathstrut 95q^{59} \) \(\mathstrut +\mathstrut 30q^{60} \) \(\mathstrut -\mathstrut 4q^{61} \) \(\mathstrut +\mathstrut 89q^{62} \) \(\mathstrut +\mathstrut 67q^{63} \) \(\mathstrut +\mathstrut 92q^{64} \) \(\mathstrut +\mathstrut 55q^{65} \) \(\mathstrut +\mathstrut 52q^{66} \) \(\mathstrut +\mathstrut 40q^{67} \) \(\mathstrut +\mathstrut 52q^{68} \) \(\mathstrut +\mathstrut 25q^{69} \) \(\mathstrut +\mathstrut 58q^{70} \) \(\mathstrut +\mathstrut 84q^{71} \) \(\mathstrut +\mathstrut 113q^{72} \) \(\mathstrut +\mathstrut 87q^{73} \) \(\mathstrut +\mathstrut 15q^{74} \) \(\mathstrut +\mathstrut 132q^{75} \) \(\mathstrut +\mathstrut 61q^{76} \) \(\mathstrut +\mathstrut 96q^{77} \) \(\mathstrut +\mathstrut 43q^{78} \) \(\mathstrut +\mathstrut 68q^{79} \) \(\mathstrut +\mathstrut 28q^{80} \) \(\mathstrut +\mathstrut 156q^{81} \) \(\mathstrut +\mathstrut 75q^{82} \) \(\mathstrut +\mathstrut 120q^{83} \) \(\mathstrut +\mathstrut 5q^{84} \) \(\mathstrut -\mathstrut 14q^{85} \) \(\mathstrut +\mathstrut 46q^{86} \) \(\mathstrut +\mathstrut 73q^{87} \) \(\mathstrut +\mathstrut 37q^{88} \) \(\mathstrut +\mathstrut 86q^{89} \) \(\mathstrut +\mathstrut 63q^{90} \) \(\mathstrut +\mathstrut 93q^{91} \) \(\mathstrut +\mathstrut 71q^{92} \) \(\mathstrut +\mathstrut 29q^{93} \) \(\mathstrut +\mathstrut 92q^{94} \) \(\mathstrut +\mathstrut 67q^{95} \) \(\mathstrut +\mathstrut 31q^{96} \) \(\mathstrut +\mathstrut 65q^{97} \) \(\mathstrut +\mathstrut 131q^{98} \) \(\mathstrut +\mathstrut 94q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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 −3.24576 1.00000 −1.41185 −3.24576 −1.89864 1.00000 7.53496 −1.41185
1.2 1.00000 −3.23664 1.00000 3.20472 −3.23664 2.34255 1.00000 7.47583 3.20472
1.3 1.00000 −3.14562 1.00000 −0.303255 −3.14562 2.95430 1.00000 6.89495 −0.303255
1.4 1.00000 −2.82708 1.00000 −2.48763 −2.82708 −1.73566 1.00000 4.99240 −2.48763
1.5 1.00000 −2.74439 1.00000 2.72118 −2.74439 4.78598 1.00000 4.53169 2.72118
1.6 1.00000 −2.70157 1.00000 0.143003 −2.70157 −2.92792 1.00000 4.29850 0.143003
1.7 1.00000 −2.67211 1.00000 1.54258 −2.67211 1.53058 1.00000 4.14019 1.54258
1.8 1.00000 −2.65852 1.00000 2.22462 −2.65852 0.870094 1.00000 4.06774 2.22462
1.9 1.00000 −2.58627 1.00000 −0.666816 −2.58627 −1.51739 1.00000 3.68880 −0.666816
1.10 1.00000 −2.54065 1.00000 −2.68248 −2.54065 5.07030 1.00000 3.45490 −2.68248
1.11 1.00000 −2.45077 1.00000 −1.39013 −2.45077 4.27652 1.00000 3.00630 −1.39013
1.12 1.00000 −2.41917 1.00000 2.79360 −2.41917 0.842715 1.00000 2.85238 2.79360
1.13 1.00000 −2.40015 1.00000 −4.19507 −2.40015 1.33387 1.00000 2.76072 −4.19507
1.14 1.00000 −2.39141 1.00000 3.31334 −2.39141 −3.74056 1.00000 2.71882 3.31334
1.15 1.00000 −2.38521 1.00000 −2.77606 −2.38521 −1.22030 1.00000 2.68923 −2.77606
1.16 1.00000 −2.15753 1.00000 −0.384990 −2.15753 −2.87024 1.00000 1.65493 −0.384990
1.17 1.00000 −1.94937 1.00000 2.21034 −1.94937 3.83725 1.00000 0.800030 2.21034
1.18 1.00000 −1.90955 1.00000 4.18412 −1.90955 −2.40151 1.00000 0.646387 4.18412
1.19 1.00000 −1.78655 1.00000 0.381834 −1.78655 1.78538 1.00000 0.191778 0.381834
1.20 1.00000 −1.75958 1.00000 2.55886 −1.75958 −0.837744 1.00000 0.0961261 2.55886
See all 92 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.92
Significant digits:
Format:

Inner twists

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

Atkin-Lehner signs

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