# Properties

 Label 8007.2.a.j Level 8007 Weight 2 Character orbit 8007.a Self dual yes Analytic conductor 63.936 Analytic rank 0 Dimension 64 CM no Inner twists 1

# Related objects

## Newspace parameters

 Level: $$N$$ = $$8007 = 3 \cdot 17 \cdot 157$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 8007.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$63.9362168984$$ Analytic rank: $$0$$ Dimension: $$64$$ Coefficient ring index: multiple of None Twist minimal: yes 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.

 $$\operatorname{Tr}(f)(q) =$$ $$64q + 5q^{2} - 64q^{3} + 77q^{4} - 3q^{5} - 5q^{6} + 5q^{7} + 18q^{8} + 64q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$64q + 5q^{2} - 64q^{3} + 77q^{4} - 3q^{5} - 5q^{6} + 5q^{7} + 18q^{8} + 64q^{9} + 12q^{10} - 7q^{11} - 77q^{12} + 24q^{13} - 14q^{14} + 3q^{15} + 103q^{16} + 64q^{17} + 5q^{18} + 26q^{19} - 24q^{20} - 5q^{21} + 25q^{22} + 20q^{23} - 18q^{24} + 141q^{25} + 9q^{26} - 64q^{27} + 14q^{28} + 5q^{29} - 12q^{30} + 11q^{31} + 31q^{32} + 7q^{33} + 5q^{34} - 3q^{35} + 77q^{36} + 50q^{37} + 8q^{38} - 24q^{39} + 28q^{40} - 9q^{41} + 14q^{42} + 59q^{43} - 6q^{44} - 3q^{45} + 11q^{47} - 103q^{48} + 163q^{49} + 20q^{50} - 64q^{51} + 65q^{52} + 39q^{53} - 5q^{54} + 35q^{55} - 34q^{56} - 26q^{57} - 27q^{58} - 65q^{59} + 24q^{60} + 15q^{61} + 18q^{62} + 5q^{63} + 152q^{64} + 49q^{65} - 25q^{66} + 56q^{67} + 77q^{68} - 20q^{69} + 28q^{70} - 18q^{71} + 18q^{72} + 37q^{73} - 76q^{74} - 141q^{75} + 30q^{76} + 80q^{77} - 9q^{78} + 20q^{79} - 144q^{80} + 64q^{81} + 27q^{82} + 3q^{83} - 14q^{84} - 3q^{85} + 12q^{86} - 5q^{87} + 108q^{88} + 42q^{89} + 12q^{90} + 25q^{91} + 18q^{92} - 11q^{93} + 60q^{94} + 42q^{95} - 31q^{96} + 72q^{97} + 18q^{98} - 7q^{99} + O(q^{100})$$

## 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 −2.81598 −1.00000 5.92975 −4.24598 2.81598 0.0516498 −11.0661 1.00000 11.9566
1.2 −2.69942 −1.00000 5.28690 3.29410 2.69942 4.13717 −8.87273 1.00000 −8.89217
1.3 −2.69904 −1.00000 5.28480 −1.39519 2.69904 −4.02432 −8.86580 1.00000 3.76567
1.4 −2.54273 −1.00000 4.46545 −2.67798 2.54273 2.94599 −6.26898 1.00000 6.80937
1.5 −2.53596 −1.00000 4.43108 −0.923448 2.53596 −0.313569 −6.16510 1.00000 2.34182
1.6 −2.38697 −1.00000 3.69761 2.25593 2.38697 −4.86795 −4.05215 1.00000 −5.38482
1.7 −2.37978 −1.00000 3.66334 0.734525 2.37978 2.72642 −3.95838 1.00000 −1.74801
1.8 −2.37098 −1.00000 3.62156 −2.47104 2.37098 5.22837 −3.84468 1.00000 5.85879
1.9 −2.32739 −1.00000 3.41675 3.32274 2.32739 1.34387 −3.29733 1.00000 −7.73332
1.10 −2.16822 −1.00000 2.70116 −4.31017 2.16822 −1.36362 −1.52027 1.00000 9.34539
1.11 −2.11517 −1.00000 2.47395 2.21169 2.11517 −2.99289 −1.00248 1.00000 −4.67809
1.12 −1.82641 −1.00000 1.33579 0.0479756 1.82641 −0.416922 1.21313 1.00000 −0.0876232
1.13 −1.78864 −1.00000 1.19923 −3.68881 1.78864 4.99843 1.43229 1.00000 6.59796
1.14 −1.70394 −1.00000 0.903397 3.92360 1.70394 −2.84859 1.86854 1.00000 −6.68557
1.15 −1.65285 −1.00000 0.731922 2.84674 1.65285 −0.210210 2.09595 1.00000 −4.70524
1.16 −1.62615 −1.00000 0.644366 0.536374 1.62615 4.53678 2.20447 1.00000 −0.872225
1.17 −1.58531 −1.00000 0.513198 −0.0940015 1.58531 −1.19806 2.35704 1.00000 0.149021
1.18 −1.55552 −1.00000 0.419651 −2.03554 1.55552 −4.57317 2.45827 1.00000 3.16632
1.19 −1.43835 −1.00000 0.0688642 −3.82222 1.43835 1.52533 2.77766 1.00000 5.49771
1.20 −1.28460 −1.00000 −0.349799 −0.580730 1.28460 −1.93204 3.01856 1.00000 0.746007
See all 64 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.64 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 8007.2.a.j 64

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8007.2.a.j 64 1.a even 1 1 trivial

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$1$$
$$17$$ $$-1$$
$$157$$ $$1$$

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{64} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(8007))$$.

## Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database