# Properties

 Label 8047.2.a.b Level 8047 Weight 2 Character orbit 8047.a Self dual Yes Analytic conductor 64.256 Analytic rank 1 Dimension 142 CM No

# Related objects

## Newspace parameters

 Level: $$N$$ = $$8047 = 13 \cdot 619$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 8047.a (trivial)

## Newform invariants

 Self dual: Yes Analytic conductor: $$64.2556185065$$ Analytic rank: $$1$$ Dimension: $$142$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$142q - 13q^{2} - 26q^{3} + 129q^{4} - 37q^{5} - 15q^{6} - 14q^{7} - 39q^{8} + 98q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$142q - 13q^{2} - 26q^{3} + 129q^{4} - 37q^{5} - 15q^{6} - 14q^{7} - 39q^{8} + 98q^{9} - 25q^{10} - 25q^{11} - 62q^{12} + 142q^{13} - 57q^{14} - 14q^{15} + 111q^{16} - 141q^{17} - 29q^{18} - 3q^{19} - 87q^{20} - 19q^{21} - 24q^{22} - 69q^{23} - 40q^{24} + 87q^{25} - 13q^{26} - 95q^{27} - 34q^{28} - 147q^{29} - 2q^{30} - 21q^{31} - 66q^{32} - 62q^{33} - 6q^{34} - 59q^{35} + 74q^{36} - 37q^{37} - 76q^{38} - 26q^{39} - 61q^{40} - 97q^{41} - 29q^{42} - 33q^{43} - 57q^{44} - 86q^{45} - q^{46} - 102q^{47} - 141q^{48} + 70q^{49} - 28q^{50} - 13q^{51} + 129q^{52} - 137q^{53} - 29q^{54} - 24q^{55} - 130q^{56} - 65q^{57} - 15q^{58} - 56q^{59} + 11q^{60} - 77q^{61} - 150q^{62} - 32q^{63} + 73q^{64} - 37q^{65} - 32q^{66} - 9q^{67} - 226q^{68} - 113q^{69} + 6q^{70} - 18q^{71} - 82q^{72} - 117q^{73} - 70q^{74} - 83q^{75} + 40q^{76} - 214q^{77} - 15q^{78} - 52q^{79} - 161q^{80} - 10q^{81} - 36q^{82} - 74q^{83} + 53q^{84} + 2q^{85} + 17q^{86} - 49q^{87} - 29q^{88} - 171q^{89} - 57q^{90} - 14q^{91} - 187q^{92} - 39q^{93} + 13q^{94} - 150q^{95} - 47q^{96} - 126q^{97} - 85q^{98} + O(q^{100})$$

## 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 −2.78532 −0.218444 5.75799 3.30077 0.608437 −0.196953 −10.4672 −2.95228 −9.19368
1.2 −2.78511 −1.73890 5.75683 −2.18885 4.84303 2.36535 −10.4632 0.0237834 6.09618
1.3 −2.74745 −2.94528 5.54847 1.95253 8.09201 3.66595 −9.74923 5.67469 −5.36448
1.4 −2.73051 1.50429 5.45570 −3.60561 −4.10747 2.19221 −9.43585 −0.737125 9.84515
1.5 −2.66381 1.33027 5.09589 −0.440221 −3.54360 2.49453 −8.24687 −1.23037 1.17267
1.6 −2.65986 −2.58294 5.07483 0.599360 6.87024 −3.90786 −8.17861 3.67156 −1.59421
1.7 −2.65232 2.65488 5.03481 −0.675897 −7.04159 −2.02772 −8.04928 4.04839 1.79269
1.8 −2.56842 0.368799 4.59680 −0.479572 −0.947232 4.01594 −6.66967 −2.86399 1.23174
1.9 −2.55403 1.62979 4.52309 1.02400 −4.16254 −2.90866 −6.44404 −0.343777 −2.61533
1.10 −2.52587 −3.25300 4.38004 −2.46689 8.21667 −3.38874 −6.01169 7.58202 6.23106
1.11 −2.52091 −0.134089 4.35498 2.66858 0.338025 −2.31114 −5.93668 −2.98202 −6.72724
1.12 −2.51095 −2.93259 4.30485 −3.06715 7.36357 3.82085 −5.78735 5.60007 7.70146
1.13 −2.50524 −0.199696 4.27622 −1.81098 0.500286 −1.67033 −5.70247 −2.96012 4.53694
1.14 −2.45771 2.98686 4.04034 −1.52748 −7.34083 5.18217 −5.01456 5.92132 3.75410
1.15 −2.43267 −0.839960 3.91786 −3.64522 2.04334 −2.93716 −4.66552 −2.29447 8.86761
1.16 −2.41564 2.60884 3.83533 1.90553 −6.30203 −0.0638029 −4.43349 3.80607 −4.60307
1.17 −2.27884 −1.20174 3.19313 −1.58295 2.73859 1.25281 −2.71895 −1.55581 3.60730
1.18 −2.23848 1.32587 3.01078 3.90699 −2.96792 −1.03210 −2.26261 −1.24208 −8.74571
1.19 −2.23735 −0.938044 3.00574 3.28954 2.09873 1.35040 −2.25019 −2.12007 −7.35985
1.20 −2.15511 1.98414 2.64449 −3.20612 −4.27605 1.44494 −1.38895 0.936828 6.90953
See next 80 embeddings (of 142 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.142 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
$$13$$ $$-1$$
$$619$$ $$-1$$

## Hecke kernels

This newform can be constructed as the kernel of the linear operator $$T_{2}^{142} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(8047))$$.