Properties

Label 8013.2.a.d
Level 8013
Weight 2
Character orbit 8013.a
Self dual Yes
Analytic conductor 63.984
Analytic rank 0
Dimension 129
CM No

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Newspace parameters

Level: \( N \) = \( 8013 = 3 \cdot 2671 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 8013.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(63.9841271397\)
Analytic rank: \(0\)
Dimension: \(129\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 129q + 15q^{2} + 129q^{3} + 151q^{4} + 16q^{5} + 15q^{6} + 61q^{7} + 42q^{8} + 129q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 129q + 15q^{2} + 129q^{3} + 151q^{4} + 16q^{5} + 15q^{6} + 61q^{7} + 42q^{8} + 129q^{9} + 41q^{10} + 51q^{11} + 151q^{12} + 56q^{13} + 5q^{14} + 16q^{15} + 195q^{16} + 18q^{17} + 15q^{18} + 93q^{19} + 44q^{20} + 61q^{21} + 46q^{22} + 50q^{23} + 42q^{24} + 193q^{25} + q^{26} + 129q^{27} + 145q^{28} + 24q^{29} + 41q^{30} + 67q^{31} + 89q^{32} + 51q^{33} + 73q^{34} + 56q^{35} + 151q^{36} + 95q^{37} + 9q^{38} + 56q^{39} + 103q^{40} + 7q^{41} + 5q^{42} + 150q^{43} + 69q^{44} + 16q^{45} + 72q^{46} + 53q^{47} + 195q^{48} + 240q^{49} + 17q^{50} + 18q^{51} + 124q^{52} + 34q^{53} + 15q^{54} + 66q^{55} - 17q^{56} + 93q^{57} + 57q^{58} + 49q^{59} + 44q^{60} + 113q^{61} + 27q^{62} + 61q^{63} + 262q^{64} + 22q^{65} + 46q^{66} + 185q^{67} + 2q^{68} + 50q^{69} + 25q^{70} + 41q^{71} + 42q^{72} + 153q^{73} - q^{74} + 193q^{75} + 190q^{76} + 39q^{77} + q^{78} + 101q^{79} + 48q^{80} + 129q^{81} + 15q^{82} + 162q^{83} + 145q^{84} + 99q^{85} + 13q^{86} + 24q^{87} + 86q^{88} - 4q^{89} + 41q^{90} + 117q^{91} + 56q^{92} + 67q^{93} + 49q^{94} + 71q^{95} + 89q^{96} + 159q^{97} + 40q^{98} + 51q^{99} + O(q^{100}) \)

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 −2.80990 1.00000 5.89553 3.04419 −2.80990 3.35815 −10.9460 1.00000 −8.55386
1.2 −2.75607 1.00000 5.59595 0.110089 −2.75607 1.43985 −9.91070 1.00000 −0.303413
1.3 −2.69445 1.00000 5.26005 −1.49008 −2.69445 1.50692 −8.78405 1.00000 4.01496
1.4 −2.69326 1.00000 5.25363 −3.14797 −2.69326 0.0430508 −8.76286 1.00000 8.47829
1.5 −2.62323 1.00000 4.88134 −3.97507 −2.62323 3.25996 −7.55842 1.00000 10.4275
1.6 −2.59970 1.00000 4.75846 1.02648 −2.59970 −4.07018 −7.17119 1.00000 −2.66854
1.7 −2.59380 1.00000 4.72779 3.19035 −2.59380 −1.98667 −7.07533 1.00000 −8.27513
1.8 −2.58900 1.00000 4.70290 −3.13677 −2.58900 5.18243 −6.99780 1.00000 8.12109
1.9 −2.52915 1.00000 4.39659 0.815749 −2.52915 5.25378 −6.06134 1.00000 −2.06315
1.10 −2.52474 1.00000 4.37429 −3.85140 −2.52474 −2.38848 −5.99445 1.00000 9.72375
1.11 −2.51884 1.00000 4.34456 3.26876 −2.51884 4.07622 −5.90558 1.00000 −8.23349
1.12 −2.41705 1.00000 3.84214 −2.77464 −2.41705 0.900586 −4.45255 1.00000 6.70646
1.13 −2.41059 1.00000 3.81094 −0.0291843 −2.41059 −4.28110 −4.36543 1.00000 0.0703514
1.14 −2.34305 1.00000 3.48991 4.17579 −2.34305 2.00162 −3.49093 1.00000 −9.78411
1.15 −2.27654 1.00000 3.18262 −2.00184 −2.27654 2.08885 −2.69227 1.00000 4.55727
1.16 −2.21092 1.00000 2.88816 1.51753 −2.21092 0.0486600 −1.96364 1.00000 −3.35514
1.17 −2.18651 1.00000 2.78083 −0.652555 −2.18651 2.68769 −1.70730 1.00000 1.42682
1.18 −2.17979 1.00000 2.75150 1.63083 −2.17979 4.53329 −1.63812 1.00000 −3.55488
1.19 −2.08790 1.00000 2.35934 0.999516 −2.08790 −2.01365 −0.750272 1.00000 −2.08689
1.20 −2.06267 1.00000 2.25459 −1.75345 −2.06267 −2.34820 −0.525142 1.00000 3.61679
See next 80 embeddings (of 129 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.129
Significant digits:
Format:

Inner twists

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

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(2671\) \(1\)