Properties

Label 8013.2.a.a
Level 8013
Weight 2
Character orbit 8013.a
Self dual Yes
Analytic conductor 63.984
Analytic rank 1
Dimension 94
CM No

Related objects

Downloads

Learn more about

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: \(94\)
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.

\(\operatorname{Tr}(f)(q) = \) \(94q \) \(\mathstrut -\mathstrut 13q^{2} \) \(\mathstrut +\mathstrut 94q^{3} \) \(\mathstrut +\mathstrut 73q^{4} \) \(\mathstrut -\mathstrut 14q^{5} \) \(\mathstrut -\mathstrut 13q^{6} \) \(\mathstrut -\mathstrut 55q^{7} \) \(\mathstrut -\mathstrut 36q^{8} \) \(\mathstrut +\mathstrut 94q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \(94q \) \(\mathstrut -\mathstrut 13q^{2} \) \(\mathstrut +\mathstrut 94q^{3} \) \(\mathstrut +\mathstrut 73q^{4} \) \(\mathstrut -\mathstrut 14q^{5} \) \(\mathstrut -\mathstrut 13q^{6} \) \(\mathstrut -\mathstrut 55q^{7} \) \(\mathstrut -\mathstrut 36q^{8} \) \(\mathstrut +\mathstrut 94q^{9} \) \(\mathstrut -\mathstrut 39q^{10} \) \(\mathstrut -\mathstrut 49q^{11} \) \(\mathstrut +\mathstrut 73q^{12} \) \(\mathstrut -\mathstrut 52q^{13} \) \(\mathstrut -\mathstrut 7q^{14} \) \(\mathstrut -\mathstrut 14q^{15} \) \(\mathstrut +\mathstrut 43q^{16} \) \(\mathstrut -\mathstrut 22q^{17} \) \(\mathstrut -\mathstrut 13q^{18} \) \(\mathstrut -\mathstrut 89q^{19} \) \(\mathstrut -\mathstrut 22q^{20} \) \(\mathstrut -\mathstrut 55q^{21} \) \(\mathstrut -\mathstrut 36q^{22} \) \(\mathstrut -\mathstrut 46q^{23} \) \(\mathstrut -\mathstrut 36q^{24} \) \(\mathstrut +\mathstrut 18q^{25} \) \(\mathstrut +\mathstrut q^{26} \) \(\mathstrut +\mathstrut 94q^{27} \) \(\mathstrut -\mathstrut 123q^{28} \) \(\mathstrut -\mathstrut 20q^{29} \) \(\mathstrut -\mathstrut 39q^{30} \) \(\mathstrut -\mathstrut 61q^{31} \) \(\mathstrut -\mathstrut 65q^{32} \) \(\mathstrut -\mathstrut 49q^{33} \) \(\mathstrut -\mathstrut 67q^{34} \) \(\mathstrut -\mathstrut 40q^{35} \) \(\mathstrut +\mathstrut 73q^{36} \) \(\mathstrut -\mathstrut 83q^{37} \) \(\mathstrut -\mathstrut 19q^{38} \) \(\mathstrut -\mathstrut 52q^{39} \) \(\mathstrut -\mathstrut 101q^{40} \) \(\mathstrut -\mathstrut 25q^{41} \) \(\mathstrut -\mathstrut 7q^{42} \) \(\mathstrut -\mathstrut 150q^{43} \) \(\mathstrut -\mathstrut 71q^{44} \) \(\mathstrut -\mathstrut 14q^{45} \) \(\mathstrut -\mathstrut 72q^{46} \) \(\mathstrut -\mathstrut 39q^{47} \) \(\mathstrut +\mathstrut 43q^{48} \) \(\mathstrut -\mathstrut q^{49} \) \(\mathstrut -\mathstrut 45q^{50} \) \(\mathstrut -\mathstrut 22q^{51} \) \(\mathstrut -\mathstrut 110q^{52} \) \(\mathstrut -\mathstrut 30q^{53} \) \(\mathstrut -\mathstrut 13q^{54} \) \(\mathstrut -\mathstrut 54q^{55} \) \(\mathstrut -\mathstrut 5q^{56} \) \(\mathstrut -\mathstrut 89q^{57} \) \(\mathstrut -\mathstrut 77q^{58} \) \(\mathstrut -\mathstrut 43q^{59} \) \(\mathstrut -\mathstrut 22q^{60} \) \(\mathstrut -\mathstrut 109q^{61} \) \(\mathstrut -\mathstrut 33q^{62} \) \(\mathstrut -\mathstrut 55q^{63} \) \(\mathstrut +\mathstrut 10q^{64} \) \(\mathstrut -\mathstrut 66q^{65} \) \(\mathstrut -\mathstrut 36q^{66} \) \(\mathstrut -\mathstrut 155q^{67} \) \(\mathstrut -\mathstrut 46q^{68} \) \(\mathstrut -\mathstrut 46q^{69} \) \(\mathstrut -\mathstrut 43q^{70} \) \(\mathstrut -\mathstrut 27q^{71} \) \(\mathstrut -\mathstrut 36q^{72} \) \(\mathstrut -\mathstrut 157q^{73} \) \(\mathstrut -\mathstrut 29q^{74} \) \(\mathstrut +\mathstrut 18q^{75} \) \(\mathstrut -\mathstrut 176q^{76} \) \(\mathstrut -\mathstrut 9q^{77} \) \(\mathstrut +\mathstrut q^{78} \) \(\mathstrut -\mathstrut 99q^{79} \) \(\mathstrut -\mathstrut 18q^{80} \) \(\mathstrut +\mathstrut 94q^{81} \) \(\mathstrut -\mathstrut 53q^{82} \) \(\mathstrut -\mathstrut 144q^{83} \) \(\mathstrut -\mathstrut 123q^{84} \) \(\mathstrut -\mathstrut 105q^{85} \) \(\mathstrut +\mathstrut 23q^{86} \) \(\mathstrut -\mathstrut 20q^{87} \) \(\mathstrut -\mathstrut 88q^{88} \) \(\mathstrut -\mathstrut 4q^{89} \) \(\mathstrut -\mathstrut 39q^{90} \) \(\mathstrut -\mathstrut 99q^{91} \) \(\mathstrut -\mathstrut 76q^{92} \) \(\mathstrut -\mathstrut 61q^{93} \) \(\mathstrut -\mathstrut 65q^{94} \) \(\mathstrut -\mathstrut 49q^{95} \) \(\mathstrut -\mathstrut 65q^{96} \) \(\mathstrut -\mathstrut 139q^{97} \) \(\mathstrut -\mathstrut 6q^{98} \) \(\mathstrut -\mathstrut 49q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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 −2.78585 1.00000 5.76095 3.26911 −2.78585 −3.26627 −10.4774 1.00000 −9.10723
1.2 −2.73477 1.00000 5.47898 −0.238588 −2.73477 1.07409 −9.51422 1.00000 0.652485
1.3 −2.64959 1.00000 5.02034 −2.74590 −2.64959 −5.01160 −8.00268 1.00000 7.27553
1.4 −2.64705 1.00000 5.00689 −1.00090 −2.64705 −3.26297 −7.95941 1.00000 2.64944
1.5 −2.57838 1.00000 4.64804 1.93561 −2.57838 2.52768 −6.82766 1.00000 −4.99075
1.6 −2.56682 1.00000 4.58855 3.72342 −2.56682 −2.49598 −6.64435 1.00000 −9.55734
1.7 −2.55292 1.00000 4.51739 3.29007 −2.55292 1.28784 −6.42669 1.00000 −8.39927
1.8 −2.51426 1.00000 4.32151 −2.69736 −2.51426 −1.28921 −5.83689 1.00000 6.78186
1.9 −2.46856 1.00000 4.09380 0.0867826 −2.46856 0.912467 −5.16868 1.00000 −0.214228
1.10 −2.45728 1.00000 4.03824 0.781238 −2.45728 −3.06583 −5.00853 1.00000 −1.91972
1.11 −2.33315 1.00000 3.44357 −2.85647 −2.33315 −3.40806 −3.36806 1.00000 6.66457
1.12 −2.27625 1.00000 3.18130 1.70680 −2.27625 −0.170344 −2.68893 1.00000 −3.88509
1.13 −2.15734 1.00000 2.65411 1.74315 −2.15734 2.90909 −1.41114 1.00000 −3.76056
1.14 −2.15634 1.00000 2.64980 −1.53192 −2.15634 −0.507117 −1.40120 1.00000 3.30334
1.15 −2.06033 1.00000 2.24497 1.16597 −2.06033 0.252514 −0.504726 1.00000 −2.40228
1.16 −2.05369 1.00000 2.21762 −2.14256 −2.05369 −0.368331 −0.446932 1.00000 4.40015
1.17 −2.05347 1.00000 2.21673 −2.14654 −2.05347 −0.521799 −0.445046 1.00000 4.40785
1.18 −2.03943 1.00000 2.15929 −1.46688 −2.03943 4.57052 −0.324864 1.00000 2.99161
1.19 −1.99976 1.00000 1.99902 −2.89200 −1.99976 2.76984 0.00195305 1.00000 5.78329
1.20 −1.92790 1.00000 1.71679 3.85366 −1.92790 −4.10922 0.545993 1.00000 −7.42947
See all 94 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.94
Significant digits:
Format:

Inner twists

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

Atkin-Lehner signs

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