# Properties

 Label 8041.2.a.e 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.

 $$\operatorname{Tr}(f)(q) =$$ $$66q + 7q^{2} + 3q^{3} + 61q^{4} + 4q^{5} + 10q^{6} + 14q^{7} + 21q^{8} + 65q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$66q + 7q^{2} + 3q^{3} + 61q^{4} + 4q^{5} + 10q^{6} + 14q^{7} + 21q^{8} + 65q^{9} + 13q^{10} - 66q^{11} + 9q^{12} - 12q^{13} + 25q^{14} + 13q^{15} + 47q^{16} - 66q^{17} + 37q^{18} + 19q^{20} + 26q^{21} - 7q^{22} + 47q^{23} + 15q^{24} + 52q^{25} + 16q^{26} + 9q^{27} + 3q^{28} + 57q^{29} + 2q^{30} + 31q^{31} + 39q^{32} - 3q^{33} - 7q^{34} + 36q^{35} + 39q^{36} - 14q^{37} + 18q^{38} + 71q^{39} + 29q^{40} + 62q^{41} - 3q^{42} + 66q^{43} - 61q^{44} - 2q^{45} + 19q^{46} + 32q^{47} + 26q^{48} + 42q^{49} + 10q^{50} - 3q^{51} - 7q^{52} + 33q^{53} + 100q^{54} - 4q^{55} + 61q^{56} + 35q^{57} - 16q^{58} + 59q^{59} + 50q^{60} + 26q^{61} + 29q^{62} + 62q^{63} + 29q^{64} + 55q^{65} - 10q^{66} + 5q^{67} - 61q^{68} - 36q^{69} - 35q^{70} + 128q^{71} + 87q^{72} + 23q^{73} + 64q^{74} - 11q^{75} + 74q^{76} - 14q^{77} + 45q^{78} + 39q^{79} + 95q^{80} + 54q^{81} - 6q^{82} + 48q^{83} + 38q^{84} - 4q^{85} + 7q^{86} + 14q^{87} - 21q^{88} + 28q^{89} + 135q^{90} - 18q^{91} + 108q^{92} - 9q^{93} + 37q^{94} + 149q^{95} + 104q^{96} + 19q^{97} + 30q^{98} - 65q^{99} + O(q^{100})$$

## 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 −2.67289 −2.02637 5.14435 2.69749 5.41626 1.58680 −8.40451 1.10617 −7.21011
1.2 −2.55220 1.68778 4.51375 −1.92240 −4.30756 −1.71256 −6.41560 −0.151402 4.90636
1.3 −2.49759 2.42195 4.23793 −1.17913 −6.04903 1.05331 −5.58943 2.86584 2.94498
1.4 −2.44538 −0.190207 3.97990 3.44061 0.465129 3.54065 −4.84161 −2.96382 −8.41362
1.5 −2.44026 −2.32024 3.95484 −3.72812 5.66197 −3.99727 −4.77032 2.38350 9.09757
1.6 −2.41404 0.194701 3.82757 −3.15933 −0.470017 −0.226574 −4.41183 −2.96209 7.62675
1.7 −2.39048 1.48105 3.71439 1.30814 −3.54043 −2.95859 −4.09821 −0.806479 −3.12708
1.8 −2.38493 −1.31374 3.68787 1.21176 3.13318 −4.42296 −4.02545 −1.27408 −2.88995
1.9 −1.96819 1.90311 1.87379 3.64418 −3.74569 3.36449 0.248405 0.621828 −7.17246
1.10 −1.96076 −1.83181 1.84459 −1.45283 3.59175 0.993575 0.304726 0.355533 2.84866
1.11 −1.92404 0.667839 1.70195 1.74197 −1.28495 −1.15955 0.573464 −2.55399 −3.35162
1.12 −1.88145 −3.19345 1.53984 2.05646 6.00831 −0.224588 0.865759 7.19812 −3.86912
1.13 −1.79269 1.29476 1.21374 −0.440748 −2.32111 4.25701 1.40952 −1.32359 0.790124
1.14 −1.70598 0.804154 0.910358 −1.55515 −1.37187 1.24699 1.85890 −2.35334 2.65305
1.15 −1.64902 −1.36167 0.719266 −2.44108 2.24541 −1.20008 2.11196 −1.14587 4.02539
1.16 −1.40918 0.980513 −0.0142062 1.84997 −1.38172 −3.81355 2.83838 −2.03859 −2.60695
1.17 −1.35222 2.90367 −0.171490 −2.71204 −3.92642 −3.82442 2.93634 5.43132 3.66729
1.18 −1.32441 −2.65196 −0.245926 −0.221912 3.51229 0.607593 2.97454 4.03289 0.293903
1.19 −1.31054 3.05736 −0.282496 3.37199 −4.00679 1.80971 2.99129 6.34748 −4.41911
1.20 −1.19284 −3.20142 −0.577127 −3.80611 3.81880 4.55130 3.07411 7.24912 4.54009
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 $$S_{2}^{\mathrm{new}}(\Gamma_0(8041))$$.