# Properties

 Label 8041.2.a.f Level 8041 Weight 2 Character orbit 8041.a Self dual Yes Analytic conductor 64.208 Analytic rank 0 Dimension 66 CM No

# Related objects

## 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: $$0$$ Dimension: $$66$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$66q + 12q^{2} + 66q^{4} + 6q^{5} + 7q^{6} + 13q^{7} + 30q^{8} + 58q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$66q + 12q^{2} + 66q^{4} + 6q^{5} + 7q^{6} + 13q^{7} + 30q^{8} + 58q^{9} + 7q^{10} - 66q^{11} + 12q^{12} + 12q^{13} + 13q^{14} + 35q^{15} + 58q^{16} + 66q^{17} + 37q^{18} + 24q^{19} + 17q^{20} + 16q^{21} - 12q^{22} + 25q^{23} + 22q^{24} + 56q^{25} + 36q^{26} + 17q^{28} + 29q^{29} + 28q^{30} + 37q^{31} + 62q^{32} + 12q^{34} + 40q^{35} + 107q^{36} - 34q^{37} + 22q^{38} + 61q^{39} + 37q^{40} + 41q^{41} + 19q^{42} - 66q^{43} - 66q^{44} + 10q^{45} + 43q^{46} + 61q^{47} + 29q^{48} + 33q^{49} + 59q^{50} + 51q^{52} - 35q^{53} - 37q^{54} - 6q^{55} + 37q^{56} - 7q^{57} + 17q^{58} + 48q^{59} - 56q^{60} + q^{61} + 37q^{62} + 43q^{63} + 68q^{64} + 41q^{65} - 7q^{66} + 10q^{67} + 66q^{68} + 18q^{69} + 77q^{70} + 84q^{71} + 83q^{72} + 5q^{73} + 36q^{74} + 14q^{75} + 14q^{76} - 13q^{77} + 41q^{78} + 58q^{79} + 25q^{80} + 78q^{81} - 28q^{82} + 47q^{83} + 44q^{84} + 6q^{85} - 12q^{86} + 101q^{87} - 30q^{88} + 53q^{89} + q^{90} + 2q^{91} + 34q^{92} - 3q^{93} + 17q^{94} + 91q^{95} + 27q^{96} - 28q^{97} + 87q^{98} - 58q^{99} + O(q^{100})$$

## 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 −2.69318 −2.72821 5.25323 0.571799 7.34757 0.634478 −8.76155 4.44313 −1.53996
1.2 −2.64866 1.22889 5.01539 −0.305577 −3.25491 −2.34072 −7.98675 −1.48983 0.809369
1.3 −2.63329 −0.751907 4.93424 0.880982 1.97999 −0.474275 −7.72671 −2.43464 −2.31989
1.4 −2.41177 3.28542 3.81663 −1.23247 −7.92367 1.42521 −4.38130 7.79396 2.97244
1.5 −2.28377 1.73431 3.21562 −2.87299 −3.96077 2.79436 −2.77620 0.00783032 6.56126
1.6 −2.25118 2.59454 3.06780 2.27169 −5.84076 2.89150 −2.40380 3.73163 −5.11397
1.7 −2.23777 −0.164695 3.00762 1.12990 0.368549 0.279747 −2.25482 −2.97288 −2.52847
1.8 −2.16346 −1.64599 2.68057 −3.89271 3.56104 −1.32552 −1.47238 −0.290720 8.42173
1.9 −2.13623 −2.11060 2.56346 3.94938 4.50873 −1.45786 −1.20369 1.45464 −8.43678
1.10 −1.99436 −0.822682 1.97746 −2.77484 1.64072 2.78471 0.0449503 −2.32319 5.53402
1.11 −1.92381 −0.292020 1.70106 −2.65493 0.561792 −3.53150 0.575100 −2.91472 5.10760
1.12 −1.88001 −1.81223 1.53444 2.13867 3.40702 1.99941 0.875255 0.284183 −4.02072
1.13 −1.74475 2.06451 1.04415 3.04620 −3.60205 −4.86539 1.66772 1.26220 −5.31485
1.14 −1.74128 −2.85862 1.03206 −1.07282 4.97766 2.85165 1.68545 5.17170 1.86807
1.15 −1.63713 2.43295 0.680210 −0.845755 −3.98306 −3.87420 2.16067 2.91923 1.38461
1.16 −1.36974 0.455204 −0.123817 −1.28468 −0.623510 0.875632 2.90907 −2.79279 1.75968
1.17 −1.30842 −0.505193 −0.288029 1.89375 0.661007 −3.03323 2.99371 −2.74478 −2.47782
1.18 −1.25548 −0.219421 −0.423759 4.17472 0.275480 −0.198048 3.04299 −2.95185 −5.24130
1.19 −1.23666 0.919561 −0.470672 0.683092 −1.13718 3.69740 3.05538 −2.15441 −0.844753
1.20 −1.16270 −2.59083 −0.648140 −0.596152 3.01234 −2.62369 3.07898 3.71239 0.693143
See all 66 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.66 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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 $$T_{2}^{66} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(8041))$$.