Properties

Label 8041.2.a.d
Level 8041
Weight 2
Character orbit 8041.a
Self dual Yes
Analytic conductor 64.208
Analytic rank 1
Dimension 62
CM No

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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: \(62\)
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.

\(\operatorname{Tr}(f)(q) = \) \(62q \) \(\mathstrut -\mathstrut 7q^{2} \) \(\mathstrut -\mathstrut 8q^{3} \) \(\mathstrut +\mathstrut 49q^{4} \) \(\mathstrut -\mathstrut 13q^{5} \) \(\mathstrut -\mathstrut 2q^{6} \) \(\mathstrut -\mathstrut 11q^{7} \) \(\mathstrut -\mathstrut 9q^{8} \) \(\mathstrut +\mathstrut 40q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \(62q \) \(\mathstrut -\mathstrut 7q^{2} \) \(\mathstrut -\mathstrut 8q^{3} \) \(\mathstrut +\mathstrut 49q^{4} \) \(\mathstrut -\mathstrut 13q^{5} \) \(\mathstrut -\mathstrut 2q^{6} \) \(\mathstrut -\mathstrut 11q^{7} \) \(\mathstrut -\mathstrut 9q^{8} \) \(\mathstrut +\mathstrut 40q^{9} \) \(\mathstrut -\mathstrut 7q^{10} \) \(\mathstrut +\mathstrut 62q^{11} \) \(\mathstrut -\mathstrut 17q^{12} \) \(\mathstrut -\mathstrut 31q^{14} \) \(\mathstrut -\mathstrut 20q^{15} \) \(\mathstrut +\mathstrut 27q^{16} \) \(\mathstrut -\mathstrut 62q^{17} \) \(\mathstrut +\mathstrut 3q^{18} \) \(\mathstrut -\mathstrut 29q^{20} \) \(\mathstrut -\mathstrut 18q^{21} \) \(\mathstrut -\mathstrut 7q^{22} \) \(\mathstrut -\mathstrut 50q^{23} \) \(\mathstrut -\mathstrut 31q^{24} \) \(\mathstrut +\mathstrut 35q^{25} \) \(\mathstrut -\mathstrut 32q^{26} \) \(\mathstrut -\mathstrut 14q^{27} \) \(\mathstrut -\mathstrut 13q^{28} \) \(\mathstrut -\mathstrut 26q^{29} \) \(\mathstrut -\mathstrut 10q^{30} \) \(\mathstrut -\mathstrut 58q^{31} \) \(\mathstrut -\mathstrut 5q^{32} \) \(\mathstrut -\mathstrut 8q^{33} \) \(\mathstrut +\mathstrut 7q^{34} \) \(\mathstrut -\mathstrut 32q^{35} \) \(\mathstrut -\mathstrut 29q^{36} \) \(\mathstrut -\mathstrut 41q^{37} \) \(\mathstrut -\mathstrut 10q^{38} \) \(\mathstrut -\mathstrut 53q^{39} \) \(\mathstrut -\mathstrut 31q^{40} \) \(\mathstrut -\mathstrut 55q^{41} \) \(\mathstrut -\mathstrut 7q^{42} \) \(\mathstrut +\mathstrut 62q^{43} \) \(\mathstrut +\mathstrut 49q^{44} \) \(\mathstrut -\mathstrut 34q^{45} \) \(\mathstrut -\mathstrut 39q^{46} \) \(\mathstrut -\mathstrut 31q^{47} \) \(\mathstrut -\mathstrut 30q^{48} \) \(\mathstrut +\mathstrut 35q^{49} \) \(\mathstrut -\mathstrut 40q^{50} \) \(\mathstrut +\mathstrut 8q^{51} \) \(\mathstrut +\mathstrut 13q^{52} \) \(\mathstrut -\mathstrut 74q^{53} \) \(\mathstrut +\mathstrut 48q^{54} \) \(\mathstrut -\mathstrut 13q^{55} \) \(\mathstrut -\mathstrut 75q^{56} \) \(\mathstrut -\mathstrut 43q^{57} \) \(\mathstrut -\mathstrut 46q^{58} \) \(\mathstrut -\mathstrut 65q^{59} \) \(\mathstrut -\mathstrut 8q^{60} \) \(\mathstrut -\mathstrut 14q^{61} \) \(\mathstrut -\mathstrut 29q^{62} \) \(\mathstrut -\mathstrut 23q^{63} \) \(\mathstrut -\mathstrut 15q^{64} \) \(\mathstrut -\mathstrut 9q^{65} \) \(\mathstrut -\mathstrut 2q^{66} \) \(\mathstrut -\mathstrut q^{67} \) \(\mathstrut -\mathstrut 49q^{68} \) \(\mathstrut -\mathstrut 59q^{69} \) \(\mathstrut -\mathstrut 31q^{70} \) \(\mathstrut -\mathstrut 141q^{71} \) \(\mathstrut +\mathstrut 9q^{72} \) \(\mathstrut -\mathstrut 4q^{73} \) \(\mathstrut -\mathstrut 94q^{74} \) \(\mathstrut -\mathstrut 43q^{75} \) \(\mathstrut +\mathstrut 34q^{76} \) \(\mathstrut -\mathstrut 11q^{77} \) \(\mathstrut -\mathstrut 11q^{78} \) \(\mathstrut -\mathstrut 63q^{79} \) \(\mathstrut -\mathstrut 41q^{80} \) \(\mathstrut -\mathstrut 30q^{81} \) \(\mathstrut +\mathstrut 38q^{82} \) \(\mathstrut -\mathstrut 44q^{83} \) \(\mathstrut -\mathstrut 16q^{84} \) \(\mathstrut +\mathstrut 13q^{85} \) \(\mathstrut -\mathstrut 7q^{86} \) \(\mathstrut -\mathstrut 8q^{87} \) \(\mathstrut -\mathstrut 9q^{88} \) \(\mathstrut -\mathstrut 58q^{89} \) \(\mathstrut -\mathstrut 55q^{90} \) \(\mathstrut -\mathstrut 78q^{91} \) \(\mathstrut -\mathstrut 104q^{92} \) \(\mathstrut -\mathstrut 5q^{94} \) \(\mathstrut -\mathstrut 99q^{95} \) \(\mathstrut -\mathstrut 148q^{96} \) \(\mathstrut -\mathstrut 26q^{97} \) \(\mathstrut +\mathstrut 16q^{98} \) \(\mathstrut +\mathstrut 40q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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 −2.76025 2.03734 5.61901 −1.74676 −5.62359 0.752232 −9.98938 1.15077 4.82150
1.2 −2.70684 −0.791084 5.32698 2.18902 2.14134 2.25606 −9.00562 −2.37419 −5.92532
1.3 −2.51837 −0.407906 4.34217 −4.11514 1.02726 −1.84586 −5.89846 −2.83361 10.3634
1.4 −2.46781 0.627778 4.09007 2.55941 −1.54924 −1.57206 −5.15790 −2.60589 −6.31614
1.5 −2.41829 −2.95622 3.84815 1.73328 7.14900 3.91504 −4.46936 5.73922 −4.19159
1.6 −2.40656 −2.51592 3.79153 0.539420 6.05472 −2.08610 −4.31144 3.32987 −1.29815
1.7 −2.36585 2.45510 3.59727 1.09940 −5.80841 2.69770 −3.77890 3.02752 −2.60102
1.8 −2.25455 2.07382 3.08298 1.31329 −4.67552 −0.325302 −2.44162 1.30073 −2.96087
1.9 −2.21985 0.267340 2.92774 −2.37360 −0.593455 −4.48408 −2.05944 −2.92853 5.26904
1.10 −2.09677 0.607527 2.39646 −1.49692 −1.27385 −0.898210 −0.831287 −2.63091 3.13870
1.11 −1.99324 1.03243 1.97299 −3.57518 −2.05788 3.51904 0.0538292 −1.93409 7.12618
1.12 −1.97447 −2.45298 1.89852 −4.33084 4.84333 2.34067 0.200361 3.01712 8.55111
1.13 −1.89473 −0.654571 1.59002 2.95213 1.24024 −2.97686 0.776803 −2.57154 −5.59350
1.14 −1.81914 −2.58004 1.30927 −0.309884 4.69345 2.19724 1.25653 3.65659 0.563723
1.15 −1.79824 −0.403049 1.23368 −0.0361831 0.724781 4.76370 1.37802 −2.83755 0.0650661
1.16 −1.70186 −1.87988 0.896329 −0.570066 3.19929 −0.204447 1.87829 0.533941 0.970172
1.17 −1.47469 −1.86783 0.174714 4.42666 2.75448 0.892669 2.69173 0.488806 −6.52796
1.18 −1.44193 2.68271 0.0791741 −0.0677310 −3.86829 1.81686 2.76970 4.19693 0.0976637
1.19 −1.34818 −2.19319 −0.182400 −0.782983 2.95682 −4.69818 2.94228 1.81007 1.05561
1.20 −1.29109 −0.909633 −0.333090 −2.82704 1.17442 −1.79091 3.01223 −2.17257 3.64996
See all 62 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.62
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 \(T_{2}^{62} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8041))\).