Properties

Label 8013.2.a.c
Level 8013
Weight 2
Character orbit 8013.a
Self dual Yes
Analytic conductor 63.984
Analytic rank 1
Dimension 116
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: \(1\)
Dimension: \(116\)
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.

\(\operatorname{Tr}(f)(q) = \) \(116q \) \(\mathstrut -\mathstrut 16q^{2} \) \(\mathstrut -\mathstrut 116q^{3} \) \(\mathstrut +\mathstrut 116q^{4} \) \(\mathstrut -\mathstrut 20q^{5} \) \(\mathstrut +\mathstrut 16q^{6} \) \(\mathstrut -\mathstrut 33q^{7} \) \(\mathstrut -\mathstrut 45q^{8} \) \(\mathstrut +\mathstrut 116q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \(116q \) \(\mathstrut -\mathstrut 16q^{2} \) \(\mathstrut -\mathstrut 116q^{3} \) \(\mathstrut +\mathstrut 116q^{4} \) \(\mathstrut -\mathstrut 20q^{5} \) \(\mathstrut +\mathstrut 16q^{6} \) \(\mathstrut -\mathstrut 33q^{7} \) \(\mathstrut -\mathstrut 45q^{8} \) \(\mathstrut +\mathstrut 116q^{9} \) \(\mathstrut +\mathstrut 3q^{10} \) \(\mathstrut -\mathstrut 57q^{11} \) \(\mathstrut -\mathstrut 116q^{12} \) \(\mathstrut +\mathstrut 6q^{13} \) \(\mathstrut -\mathstrut 9q^{14} \) \(\mathstrut +\mathstrut 20q^{15} \) \(\mathstrut +\mathstrut 112q^{16} \) \(\mathstrut -\mathstrut 30q^{17} \) \(\mathstrut -\mathstrut 16q^{18} \) \(\mathstrut +\mathstrut 3q^{19} \) \(\mathstrut -\mathstrut 54q^{20} \) \(\mathstrut +\mathstrut 33q^{21} \) \(\mathstrut -\mathstrut 22q^{22} \) \(\mathstrut -\mathstrut 58q^{23} \) \(\mathstrut +\mathstrut 45q^{24} \) \(\mathstrut +\mathstrut 126q^{25} \) \(\mathstrut -\mathstrut 21q^{26} \) \(\mathstrut -\mathstrut 116q^{27} \) \(\mathstrut -\mathstrut 77q^{28} \) \(\mathstrut -\mathstrut 38q^{29} \) \(\mathstrut -\mathstrut 3q^{30} \) \(\mathstrut +\mathstrut 17q^{31} \) \(\mathstrut -\mathstrut 106q^{32} \) \(\mathstrut +\mathstrut 57q^{33} \) \(\mathstrut +\mathstrut 35q^{34} \) \(\mathstrut -\mathstrut 72q^{35} \) \(\mathstrut +\mathstrut 116q^{36} \) \(\mathstrut -\mathstrut 41q^{37} \) \(\mathstrut -\mathstrut 45q^{38} \) \(\mathstrut -\mathstrut 6q^{39} \) \(\mathstrut +\mathstrut 5q^{40} \) \(\mathstrut -\mathstrut 39q^{41} \) \(\mathstrut +\mathstrut 9q^{42} \) \(\mathstrut -\mathstrut 118q^{43} \) \(\mathstrut -\mathstrut 103q^{44} \) \(\mathstrut -\mathstrut 20q^{45} \) \(\mathstrut -\mathstrut 8q^{46} \) \(\mathstrut -\mathstrut 65q^{47} \) \(\mathstrut -\mathstrut 112q^{48} \) \(\mathstrut +\mathstrut 165q^{49} \) \(\mathstrut -\mathstrut 72q^{50} \) \(\mathstrut +\mathstrut 30q^{51} \) \(\mathstrut -\mathstrut 10q^{52} \) \(\mathstrut -\mathstrut 58q^{53} \) \(\mathstrut +\mathstrut 16q^{54} \) \(\mathstrut +\mathstrut 14q^{55} \) \(\mathstrut -\mathstrut 23q^{56} \) \(\mathstrut -\mathstrut 3q^{57} \) \(\mathstrut -\mathstrut 27q^{58} \) \(\mathstrut -\mathstrut 75q^{59} \) \(\mathstrut +\mathstrut 54q^{60} \) \(\mathstrut +\mathstrut 45q^{61} \) \(\mathstrut -\mathstrut 73q^{62} \) \(\mathstrut -\mathstrut 33q^{63} \) \(\mathstrut +\mathstrut 111q^{64} \) \(\mathstrut -\mathstrut 86q^{65} \) \(\mathstrut +\mathstrut 22q^{66} \) \(\mathstrut -\mathstrut 127q^{67} \) \(\mathstrut -\mathstrut 94q^{68} \) \(\mathstrut +\mathstrut 58q^{69} \) \(\mathstrut -\mathstrut 7q^{70} \) \(\mathstrut -\mathstrut 61q^{71} \) \(\mathstrut -\mathstrut 45q^{72} \) \(\mathstrut +\mathstrut 15q^{73} \) \(\mathstrut -\mathstrut 51q^{74} \) \(\mathstrut -\mathstrut 126q^{75} \) \(\mathstrut +\mathstrut 96q^{76} \) \(\mathstrut -\mathstrut 57q^{77} \) \(\mathstrut +\mathstrut 21q^{78} \) \(\mathstrut +\mathstrut 7q^{79} \) \(\mathstrut -\mathstrut 144q^{80} \) \(\mathstrut +\mathstrut 116q^{81} \) \(\mathstrut -\mathstrut 37q^{82} \) \(\mathstrut -\mathstrut 194q^{83} \) \(\mathstrut +\mathstrut 77q^{84} \) \(\mathstrut +\mathstrut 3q^{85} \) \(\mathstrut -\mathstrut 57q^{86} \) \(\mathstrut +\mathstrut 38q^{87} \) \(\mathstrut -\mathstrut 42q^{88} \) \(\mathstrut -\mathstrut 56q^{89} \) \(\mathstrut +\mathstrut 3q^{90} \) \(\mathstrut -\mathstrut 39q^{91} \) \(\mathstrut -\mathstrut 138q^{92} \) \(\mathstrut -\mathstrut 17q^{93} \) \(\mathstrut +\mathstrut 51q^{94} \) \(\mathstrut -\mathstrut 127q^{95} \) \(\mathstrut +\mathstrut 106q^{96} \) \(\mathstrut +\mathstrut 57q^{97} \) \(\mathstrut -\mathstrut 105q^{98} \) \(\mathstrut -\mathstrut 57q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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 −2.78241 −1.00000 5.74180 −1.84035 2.78241 −4.26418 −10.4112 1.00000 5.12061
1.2 −2.77241 −1.00000 5.68625 1.80856 2.77241 −1.74273 −10.2198 1.00000 −5.01408
1.3 −2.74646 −1.00000 5.54305 −2.85945 2.74646 1.69114 −9.73084 1.00000 7.85338
1.4 −2.74034 −1.00000 5.50946 3.71802 2.74034 −2.61622 −9.61710 1.00000 −10.1886
1.5 −2.72179 −1.00000 5.40814 2.52515 2.72179 2.50598 −9.27623 1.00000 −6.87293
1.6 −2.71385 −1.00000 5.36498 −0.752198 2.71385 4.51557 −9.13205 1.00000 2.04135
1.7 −2.66795 −1.00000 5.11795 −3.13798 2.66795 0.0813556 −8.31854 1.00000 8.37196
1.8 −2.59210 −1.00000 4.71896 −1.33515 2.59210 −1.92470 −7.04781 1.00000 3.46084
1.9 −2.58282 −1.00000 4.67097 −1.48132 2.58282 0.817005 −6.89864 1.00000 3.82598
1.10 −2.57433 −1.00000 4.62715 −4.27877 2.57433 −4.90523 −6.76315 1.00000 11.0149
1.11 −2.49870 −1.00000 4.24352 −4.37212 2.49870 1.33064 −5.60588 1.00000 10.9246
1.12 −2.45463 −1.00000 4.02523 1.82739 2.45463 −2.98046 −4.97119 1.00000 −4.48558
1.13 −2.42944 −1.00000 3.90217 −3.31824 2.42944 4.69634 −4.62122 1.00000 8.06146
1.14 −2.39685 −1.00000 3.74487 3.43258 2.39685 −5.25948 −4.18219 1.00000 −8.22737
1.15 −2.30332 −1.00000 3.30530 0.401271 2.30332 −3.38368 −3.00653 1.00000 −0.924257
1.16 −2.24523 −1.00000 3.04104 −1.84042 2.24523 1.63491 −2.33738 1.00000 4.13216
1.17 −2.23895 −1.00000 3.01291 1.55531 2.23895 1.34297 −2.26786 1.00000 −3.48228
1.18 −2.20341 −1.00000 2.85502 −0.504029 2.20341 −3.35275 −1.88396 1.00000 1.11058
1.19 −2.18030 −1.00000 2.75372 1.60452 2.18030 4.21617 −1.64333 1.00000 −3.49835
1.20 −2.15908 −1.00000 2.66162 −0.295164 2.15908 3.97405 −1.42849 1.00000 0.637282
See next 80 embeddings (of 116 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.116
Significant digits:
Format:

Inner twists

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

Atkin-Lehner signs

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