# Properties

 Label 8002.2.a.d Level $8002$ Weight $2$ Character orbit 8002.a Self dual yes Analytic conductor $63.896$ Analytic rank $1$ Dimension $69$ CM no Inner twists $1$

# Learn more about

## Newspace parameters

 Level: $$N$$ $$=$$ $$8002 = 2 \cdot 4001$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 8002.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$63.8962916974$$ Analytic rank: $$1$$ Dimension: $$69$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$69q + 69q^{2} - 25q^{3} + 69q^{4} - 33q^{5} - 25q^{6} - 19q^{7} + 69q^{8} + 54q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$69q + 69q^{2} - 25q^{3} + 69q^{4} - 33q^{5} - 25q^{6} - 19q^{7} + 69q^{8} + 54q^{9} - 33q^{10} - 30q^{11} - 25q^{12} - 58q^{13} - 19q^{14} + 2q^{15} + 69q^{16} - 80q^{17} + 54q^{18} - 40q^{19} - 33q^{20} - 32q^{21} - 30q^{22} - 45q^{23} - 25q^{24} + 42q^{25} - 58q^{26} - 76q^{27} - 19q^{28} - 44q^{29} + 2q^{30} - 12q^{31} + 69q^{32} - 41q^{33} - 80q^{34} - 49q^{35} + 54q^{36} - 47q^{37} - 40q^{38} - 14q^{39} - 33q^{40} - 94q^{41} - 32q^{42} - 10q^{43} - 30q^{44} - 89q^{45} - 45q^{46} - 85q^{47} - 25q^{48} + 32q^{49} + 42q^{50} - 10q^{51} - 58q^{52} - 41q^{53} - 76q^{54} - 27q^{55} - 19q^{56} - 72q^{57} - 44q^{58} - 75q^{59} + 2q^{60} - 98q^{61} - 12q^{62} - 61q^{63} + 69q^{64} - 47q^{65} - 41q^{66} - 22q^{67} - 80q^{68} - 74q^{69} - 49q^{70} - 22q^{71} + 54q^{72} - 129q^{73} - 47q^{74} - 106q^{75} - 40q^{76} - 108q^{77} - 14q^{78} + 21q^{79} - 33q^{80} + 13q^{81} - 94q^{82} - 111q^{83} - 32q^{84} - 67q^{85} - 10q^{86} - 38q^{87} - 30q^{88} - 112q^{89} - 89q^{90} - 55q^{91} - 45q^{92} - 90q^{93} - 85q^{94} - 38q^{95} - 25q^{96} - 98q^{97} + 32q^{98} - 51q^{99} + O(q^{100})$$

## 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 1.00000 −3.43715 1.00000 −2.28285 −3.43715 −3.85963 1.00000 8.81400 −2.28285
1.2 1.00000 −3.40667 1.00000 2.18421 −3.40667 −1.24057 1.00000 8.60538 2.18421
1.3 1.00000 −3.22228 1.00000 −3.24951 −3.22228 4.15498 1.00000 7.38311 −3.24951
1.4 1.00000 −3.09800 1.00000 −2.64538 −3.09800 3.07801 1.00000 6.59762 −2.64538
1.5 1.00000 −3.04857 1.00000 −0.106661 −3.04857 −0.613952 1.00000 6.29376 −0.106661
1.6 1.00000 −3.03319 1.00000 −3.55972 −3.03319 −4.22109 1.00000 6.20023 −3.55972
1.7 1.00000 −2.87355 1.00000 2.00628 −2.87355 0.474155 1.00000 5.25731 2.00628
1.8 1.00000 −2.80104 1.00000 0.676703 −2.80104 3.76033 1.00000 4.84585 0.676703
1.9 1.00000 −2.76808 1.00000 4.07666 −2.76808 2.61903 1.00000 4.66227 4.07666
1.10 1.00000 −2.73719 1.00000 −4.28393 −2.73719 2.06732 1.00000 4.49223 −4.28393
1.11 1.00000 −2.64782 1.00000 −0.675974 −2.64782 −0.999289 1.00000 4.01094 −0.675974
1.12 1.00000 −2.62997 1.00000 −0.133179 −2.62997 −2.98290 1.00000 3.91674 −0.133179
1.13 1.00000 −2.42898 1.00000 −1.63165 −2.42898 3.93412 1.00000 2.89993 −1.63165
1.14 1.00000 −2.38948 1.00000 1.42184 −2.38948 −5.14183 1.00000 2.70964 1.42184
1.15 1.00000 −2.15867 1.00000 0.496482 −2.15867 −0.313465 1.00000 1.65985 0.496482
1.16 1.00000 −2.04234 1.00000 −4.37946 −2.04234 −3.34038 1.00000 1.17114 −4.37946
1.17 1.00000 −2.01476 1.00000 2.07868 −2.01476 −1.64070 1.00000 1.05926 2.07868
1.18 1.00000 −1.99879 1.00000 2.05838 −1.99879 2.31041 1.00000 0.995149 2.05838
1.19 1.00000 −1.96903 1.00000 3.30115 −1.96903 1.85739 1.00000 0.877064 3.30115
1.20 1.00000 −1.96349 1.00000 0.136598 −1.96349 −0.645165 1.00000 0.855285 0.136598
See all 69 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.69 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$4001$$ $$-1$$

## 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 8002.2.a.d 69

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8002.2.a.d 69 1.a even 1 1 trivial

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{69} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(8002))$$.