Properties

Label 8013.2.a.b
Level 8013
Weight 2
Character orbit 8013.a
Self dual Yes
Analytic conductor 63.984
Analytic rank 0
Dimension 106
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: \(106\)
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.

\(\operatorname{Tr}(f)(q) = \) \(106q \) \(\mathstrut +\mathstrut 15q^{2} \) \(\mathstrut -\mathstrut 106q^{3} \) \(\mathstrut +\mathstrut 109q^{4} \) \(\mathstrut +\mathstrut 16q^{5} \) \(\mathstrut -\mathstrut 15q^{6} \) \(\mathstrut +\mathstrut 35q^{7} \) \(\mathstrut +\mathstrut 48q^{8} \) \(\mathstrut +\mathstrut 106q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \(106q \) \(\mathstrut +\mathstrut 15q^{2} \) \(\mathstrut -\mathstrut 106q^{3} \) \(\mathstrut +\mathstrut 109q^{4} \) \(\mathstrut +\mathstrut 16q^{5} \) \(\mathstrut -\mathstrut 15q^{6} \) \(\mathstrut +\mathstrut 35q^{7} \) \(\mathstrut +\mathstrut 48q^{8} \) \(\mathstrut +\mathstrut 106q^{9} \) \(\mathstrut -\mathstrut 3q^{10} \) \(\mathstrut +\mathstrut 55q^{11} \) \(\mathstrut -\mathstrut 109q^{12} \) \(\mathstrut -\mathstrut 8q^{13} \) \(\mathstrut +\mathstrut 27q^{14} \) \(\mathstrut -\mathstrut 16q^{15} \) \(\mathstrut +\mathstrut 111q^{16} \) \(\mathstrut +\mathstrut 28q^{17} \) \(\mathstrut +\mathstrut 15q^{18} \) \(\mathstrut +\mathstrut q^{19} \) \(\mathstrut +\mathstrut 54q^{20} \) \(\mathstrut -\mathstrut 35q^{21} \) \(\mathstrut +\mathstrut 20q^{22} \) \(\mathstrut +\mathstrut 62q^{23} \) \(\mathstrut -\mathstrut 48q^{24} \) \(\mathstrut +\mathstrut 102q^{25} \) \(\mathstrut +\mathstrut 21q^{26} \) \(\mathstrut -\mathstrut 106q^{27} \) \(\mathstrut +\mathstrut 79q^{28} \) \(\mathstrut +\mathstrut 36q^{29} \) \(\mathstrut +\mathstrut 3q^{30} \) \(\mathstrut +\mathstrut q^{31} \) \(\mathstrut +\mathstrut 111q^{32} \) \(\mathstrut -\mathstrut 55q^{33} \) \(\mathstrut -\mathstrut 27q^{34} \) \(\mathstrut +\mathstrut 72q^{35} \) \(\mathstrut +\mathstrut 109q^{36} \) \(\mathstrut +\mathstrut 31q^{37} \) \(\mathstrut +\mathstrut 43q^{38} \) \(\mathstrut +\mathstrut 8q^{39} \) \(\mathstrut -\mathstrut 13q^{40} \) \(\mathstrut +\mathstrut 35q^{41} \) \(\mathstrut -\mathstrut 27q^{42} \) \(\mathstrut +\mathstrut 98q^{43} \) \(\mathstrut +\mathstrut 121q^{44} \) \(\mathstrut +\mathstrut 16q^{45} \) \(\mathstrut +\mathstrut 8q^{46} \) \(\mathstrut +\mathstrut 75q^{47} \) \(\mathstrut -\mathstrut 111q^{48} \) \(\mathstrut +\mathstrut 49q^{49} \) \(\mathstrut +\mathstrut 83q^{50} \) \(\mathstrut -\mathstrut 28q^{51} \) \(\mathstrut -\mathstrut 18q^{52} \) \(\mathstrut +\mathstrut 60q^{53} \) \(\mathstrut -\mathstrut 15q^{54} \) \(\mathstrut +\mathstrut 14q^{55} \) \(\mathstrut +\mathstrut 85q^{56} \) \(\mathstrut -\mathstrut q^{57} \) \(\mathstrut +\mathstrut 65q^{58} \) \(\mathstrut +\mathstrut 77q^{59} \) \(\mathstrut -\mathstrut 54q^{60} \) \(\mathstrut -\mathstrut 55q^{61} \) \(\mathstrut +\mathstrut 83q^{62} \) \(\mathstrut +\mathstrut 35q^{63} \) \(\mathstrut +\mathstrut 122q^{64} \) \(\mathstrut +\mathstrut 86q^{65} \) \(\mathstrut -\mathstrut 20q^{66} \) \(\mathstrut +\mathstrut 121q^{67} \) \(\mathstrut +\mathstrut 80q^{68} \) \(\mathstrut -\mathstrut 62q^{69} \) \(\mathstrut -\mathstrut 11q^{70} \) \(\mathstrut +\mathstrut 79q^{71} \) \(\mathstrut +\mathstrut 48q^{72} \) \(\mathstrut -\mathstrut 29q^{73} \) \(\mathstrut +\mathstrut 91q^{74} \) \(\mathstrut -\mathstrut 102q^{75} \) \(\mathstrut -\mathstrut 10q^{76} \) \(\mathstrut +\mathstrut 87q^{77} \) \(\mathstrut -\mathstrut 21q^{78} \) \(\mathstrut +\mathstrut 15q^{79} \) \(\mathstrut +\mathstrut 108q^{80} \) \(\mathstrut +\mathstrut 106q^{81} \) \(\mathstrut +\mathstrut 21q^{82} \) \(\mathstrut +\mathstrut 196q^{83} \) \(\mathstrut -\mathstrut 79q^{84} \) \(\mathstrut -\mathstrut 5q^{85} \) \(\mathstrut +\mathstrut 65q^{86} \) \(\mathstrut -\mathstrut 36q^{87} \) \(\mathstrut +\mathstrut 84q^{88} \) \(\mathstrut +\mathstrut 34q^{89} \) \(\mathstrut -\mathstrut 3q^{90} \) \(\mathstrut +\mathstrut 17q^{91} \) \(\mathstrut +\mathstrut 162q^{92} \) \(\mathstrut -\mathstrut q^{93} \) \(\mathstrut -\mathstrut 35q^{94} \) \(\mathstrut +\mathstrut 113q^{95} \) \(\mathstrut -\mathstrut 111q^{96} \) \(\mathstrut -\mathstrut 63q^{97} \) \(\mathstrut +\mathstrut 112q^{98} \) \(\mathstrut +\mathstrut 55q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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 −2.68955 −1.00000 5.23367 −0.143035 2.68955 0.169841 −8.69711 1.00000 0.384699
1.2 −2.68470 −1.00000 5.20764 3.29964 2.68470 2.53834 −8.61155 1.00000 −8.85857
1.3 −2.57806 −1.00000 4.64638 −0.00185853 2.57806 4.00944 −6.82252 1.00000 0.00479139
1.4 −2.57103 −1.00000 4.61019 2.04961 2.57103 −2.14469 −6.71087 1.00000 −5.26961
1.5 −2.57051 −1.00000 4.60751 −0.585495 2.57051 −2.49504 −6.70262 1.00000 1.50502
1.6 −2.45069 −1.00000 4.00587 0.123141 2.45069 −0.667497 −4.91577 1.00000 −0.301780
1.7 −2.44992 −1.00000 4.00213 3.65851 2.44992 −1.10640 −4.90506 1.00000 −8.96306
1.8 −2.43314 −1.00000 3.92015 4.32607 2.43314 4.27561 −4.67199 1.00000 −10.5259
1.9 −2.35891 −1.00000 3.56446 −3.22536 2.35891 −2.62511 −3.69041 1.00000 7.60832
1.10 −2.32341 −1.00000 3.39824 0.581135 2.32341 1.71987 −3.24868 1.00000 −1.35022
1.11 −2.30949 −1.00000 3.33375 0.518255 2.30949 1.93365 −3.08028 1.00000 −1.19691
1.12 −2.30696 −1.00000 3.32206 −3.03521 2.30696 1.29495 −3.04993 1.00000 7.00211
1.13 −2.24374 −1.00000 3.03438 2.67702 2.24374 −0.787815 −2.32089 1.00000 −6.00654
1.14 −2.16031 −1.00000 2.66695 −0.855629 2.16031 2.67269 −1.44083 1.00000 1.84843
1.15 −2.12375 −1.00000 2.51030 −3.37551 2.12375 −2.22714 −1.08375 1.00000 7.16873
1.16 −2.09214 −1.00000 2.37707 −2.25066 2.09214 −3.42799 −0.788875 1.00000 4.70871
1.17 −1.85225 −1.00000 1.43083 2.49048 1.85225 −3.91286 1.05424 1.00000 −4.61299
1.18 −1.84122 −1.00000 1.39011 −1.17426 1.84122 1.69712 1.12295 1.00000 2.16208
1.19 −1.83416 −1.00000 1.36415 −3.29261 1.83416 −0.883876 1.16624 1.00000 6.03918
1.20 −1.79051 −1.00000 1.20593 −0.580006 1.79051 0.129222 1.42179 1.00000 1.03851
See next 80 embeddings (of 106 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.106
Significant digits:
Format:

Inner twists

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

Atkin-Lehner signs

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