Properties

Label 8041.2.a.c
Level 8041
Weight 2
Character orbit 8041.a
Self dual Yes
Analytic conductor 64.208
Analytic rank 1
Dimension 60
CM No

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) = \( 8041 = 11 \cdot 17 \cdot 43 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 8041.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(64.2077082653\)
Analytic rank: \(1\)
Dimension: \(60\)
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.

\(\operatorname{Tr}(f)(q) = \) \(60q \) \(\mathstrut -\mathstrut 9q^{2} \) \(\mathstrut -\mathstrut 6q^{3} \) \(\mathstrut +\mathstrut 43q^{4} \) \(\mathstrut -\mathstrut 15q^{5} \) \(\mathstrut -\mathstrut 4q^{6} \) \(\mathstrut -\mathstrut 17q^{7} \) \(\mathstrut -\mathstrut 21q^{8} \) \(\mathstrut +\mathstrut 40q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \(60q \) \(\mathstrut -\mathstrut 9q^{2} \) \(\mathstrut -\mathstrut 6q^{3} \) \(\mathstrut +\mathstrut 43q^{4} \) \(\mathstrut -\mathstrut 15q^{5} \) \(\mathstrut -\mathstrut 4q^{6} \) \(\mathstrut -\mathstrut 17q^{7} \) \(\mathstrut -\mathstrut 21q^{8} \) \(\mathstrut +\mathstrut 40q^{9} \) \(\mathstrut +\mathstrut q^{10} \) \(\mathstrut +\mathstrut 60q^{11} \) \(\mathstrut -\mathstrut 13q^{12} \) \(\mathstrut -\mathstrut 2q^{13} \) \(\mathstrut -\mathstrut 15q^{14} \) \(\mathstrut -\mathstrut 32q^{15} \) \(\mathstrut +\mathstrut 21q^{16} \) \(\mathstrut +\mathstrut 60q^{17} \) \(\mathstrut -\mathstrut 41q^{18} \) \(\mathstrut -\mathstrut 24q^{19} \) \(\mathstrut -\mathstrut 33q^{20} \) \(\mathstrut -\mathstrut 12q^{21} \) \(\mathstrut -\mathstrut 9q^{22} \) \(\mathstrut -\mathstrut 20q^{23} \) \(\mathstrut +\mathstrut 9q^{24} \) \(\mathstrut +\mathstrut 25q^{25} \) \(\mathstrut -\mathstrut 40q^{26} \) \(\mathstrut -\mathstrut 18q^{27} \) \(\mathstrut -\mathstrut 3q^{28} \) \(\mathstrut -\mathstrut 54q^{29} \) \(\mathstrut -\mathstrut 18q^{30} \) \(\mathstrut -\mathstrut 50q^{31} \) \(\mathstrut -\mathstrut 51q^{32} \) \(\mathstrut -\mathstrut 6q^{33} \) \(\mathstrut -\mathstrut 9q^{34} \) \(\mathstrut -\mathstrut 30q^{35} \) \(\mathstrut +\mathstrut 27q^{36} \) \(\mathstrut -\mathstrut 63q^{37} \) \(\mathstrut -\mathstrut 38q^{38} \) \(\mathstrut -\mathstrut 55q^{39} \) \(\mathstrut -\mathstrut 5q^{40} \) \(\mathstrut -\mathstrut 53q^{41} \) \(\mathstrut -\mathstrut 47q^{42} \) \(\mathstrut -\mathstrut 60q^{43} \) \(\mathstrut +\mathstrut 43q^{44} \) \(\mathstrut -\mathstrut 36q^{45} \) \(\mathstrut -\mathstrut 27q^{46} \) \(\mathstrut -\mathstrut 47q^{47} \) \(\mathstrut -\mathstrut 38q^{48} \) \(\mathstrut +\mathstrut 5q^{49} \) \(\mathstrut -\mathstrut 4q^{50} \) \(\mathstrut -\mathstrut 6q^{51} \) \(\mathstrut -\mathstrut 13q^{52} \) \(\mathstrut -\mathstrut 24q^{53} \) \(\mathstrut -\mathstrut 58q^{54} \) \(\mathstrut -\mathstrut 15q^{55} \) \(\mathstrut -\mathstrut 45q^{56} \) \(\mathstrut +\mathstrut 23q^{57} \) \(\mathstrut +\mathstrut 16q^{58} \) \(\mathstrut -\mathstrut 57q^{59} \) \(\mathstrut -\mathstrut 4q^{60} \) \(\mathstrut -\mathstrut 8q^{61} \) \(\mathstrut -\mathstrut 3q^{62} \) \(\mathstrut -\mathstrut 57q^{63} \) \(\mathstrut -\mathstrut 5q^{64} \) \(\mathstrut -\mathstrut 27q^{65} \) \(\mathstrut -\mathstrut 4q^{66} \) \(\mathstrut -\mathstrut 49q^{67} \) \(\mathstrut +\mathstrut 43q^{68} \) \(\mathstrut -\mathstrut 57q^{69} \) \(\mathstrut -\mathstrut 7q^{70} \) \(\mathstrut -\mathstrut 151q^{71} \) \(\mathstrut -\mathstrut 29q^{72} \) \(\mathstrut -\mathstrut 12q^{73} \) \(\mathstrut +\mathstrut 18q^{74} \) \(\mathstrut -\mathstrut 23q^{75} \) \(\mathstrut -\mathstrut 38q^{76} \) \(\mathstrut -\mathstrut 17q^{77} \) \(\mathstrut +\mathstrut 37q^{78} \) \(\mathstrut -\mathstrut 11q^{79} \) \(\mathstrut -\mathstrut 21q^{80} \) \(\mathstrut +\mathstrut 28q^{81} \) \(\mathstrut +\mathstrut 44q^{82} \) \(\mathstrut -\mathstrut 42q^{83} \) \(\mathstrut +\mathstrut 16q^{84} \) \(\mathstrut -\mathstrut 15q^{85} \) \(\mathstrut +\mathstrut 9q^{86} \) \(\mathstrut -\mathstrut 56q^{87} \) \(\mathstrut -\mathstrut 21q^{88} \) \(\mathstrut -\mathstrut 88q^{89} \) \(\mathstrut +\mathstrut 47q^{90} \) \(\mathstrut -\mathstrut 60q^{91} \) \(\mathstrut +\mathstrut 26q^{92} \) \(\mathstrut -\mathstrut 16q^{93} \) \(\mathstrut +\mathstrut 37q^{94} \) \(\mathstrut -\mathstrut 57q^{95} \) \(\mathstrut +\mathstrut 108q^{96} \) \(\mathstrut -\mathstrut 22q^{97} \) \(\mathstrut +\mathstrut 8q^{98} \) \(\mathstrut +\mathstrut 40q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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 −2.78000 −1.81181 5.72839 1.28422 5.03683 3.89203 −10.3649 0.282655 −3.57012
1.2 −2.66007 −2.34749 5.07599 −2.77927 6.24449 0.942946 −8.18236 2.51070 7.39305
1.3 −2.54628 1.17861 4.48352 1.63315 −3.00107 −0.708864 −6.32373 −1.61088 −4.15845
1.4 −2.53062 2.95421 4.40406 −1.92556 −7.47600 2.94104 −6.08377 5.72738 4.87287
1.5 −2.51152 −1.72847 4.30774 −3.07277 4.34109 −4.07006 −5.79593 −0.0123937 7.71734
1.6 −2.42660 2.76366 3.88839 −0.843592 −6.70630 −2.86536 −4.58236 4.63781 2.04706
1.7 −2.38636 −2.80769 3.69472 2.77009 6.70017 −2.49366 −4.04422 4.88313 −6.61043
1.8 −2.36766 0.323518 3.60581 −3.56257 −0.765980 3.82937 −3.80202 −2.89534 8.43496
1.9 −2.17121 0.274303 2.71415 −0.625848 −0.595570 −3.36796 −1.55058 −2.92476 1.35885
1.10 −2.12431 0.433471 2.51268 3.47058 −0.920826 2.89316 −1.08909 −2.81210 −7.37258
1.11 −2.07976 −1.74571 2.32541 −0.111413 3.63066 −0.659970 −0.676772 0.0475049 0.231713
1.12 −1.94215 1.51854 1.77194 3.53502 −2.94923 −1.41025 0.442922 −0.694041 −6.86554
1.13 −1.83493 1.92990 1.36698 0.407070 −3.54123 0.437202 1.16155 0.724496 −0.746946
1.14 −1.57436 0.0662834 0.478605 −1.53153 −0.104354 0.748849 2.39522 −2.99561 2.41117
1.15 −1.55010 2.25408 0.402812 −2.41752 −3.49405 2.04645 2.47580 2.08087 3.74740
1.16 −1.50893 −3.04602 0.276872 −1.87360 4.59624 −2.31151 2.60008 6.27825 2.82713
1.17 −1.49915 −0.148971 0.247445 −2.56328 0.223329 2.63900 2.62734 −2.97781 3.84274
1.18 −1.33974 3.18054 −0.205105 0.454728 −4.26109 −3.38234 2.95426 7.11585 −0.609216
1.19 −1.32373 −3.06819 −0.247749 3.29067 4.06144 −2.61144 2.97540 6.41379 −4.35594
1.20 −1.23062 −0.724033 −0.485567 2.55129 0.891011 −2.06260 3.05880 −2.47578 −3.13968
See all 60 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.60
Significant digits:
Format:

Inner twists

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

Atkin-Lehner signs

\( p \) Sign
\(11\) \(-1\)
\(17\) \(-1\)
\(43\) \(1\)

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{2}^{60} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8041))\).