# Properties

 Label 85.2.a.b Level 85 Weight 2 Character orbit 85.a Self dual Yes Analytic conductor 0.679 Analytic rank 1 Dimension 2 CM No Inner twists 1

# Related objects

## Newspace parameters

 Level: $$N$$ = $$85 = 5 \cdot 17$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 85.a (trivial)

## Newform invariants

 Self dual: Yes Analytic conductor: $$0.678728417181$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{2})$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -1 + \beta ) q^{2} + ( -2 - \beta ) q^{3} + ( 1 - 2 \beta ) q^{4} - q^{5} -\beta q^{6} + ( -2 + \beta ) q^{7} + ( -3 + \beta ) q^{8} + ( 3 + 4 \beta ) q^{9} +O(q^{10})$$ $$q + ( -1 + \beta ) q^{2} + ( -2 - \beta ) q^{3} + ( 1 - 2 \beta ) q^{4} - q^{5} -\beta q^{6} + ( -2 + \beta ) q^{7} + ( -3 + \beta ) q^{8} + ( 3 + 4 \beta ) q^{9} + ( 1 - \beta ) q^{10} + ( -4 + \beta ) q^{11} + ( 2 + 3 \beta ) q^{12} -2 \beta q^{13} + ( 4 - 3 \beta ) q^{14} + ( 2 + \beta ) q^{15} + 3 q^{16} - q^{17} + ( 5 - \beta ) q^{18} -2 \beta q^{19} + ( -1 + 2 \beta ) q^{20} + 2 q^{21} + ( 6 - 5 \beta ) q^{22} + ( -2 - \beta ) q^{23} + ( 4 + \beta ) q^{24} + q^{25} + ( -4 + 2 \beta ) q^{26} + ( -8 - 8 \beta ) q^{27} + ( -6 + 5 \beta ) q^{28} + ( -2 - 2 \beta ) q^{29} + \beta q^{30} + 3 \beta q^{31} + ( 3 + \beta ) q^{32} + ( 6 + 2 \beta ) q^{33} + ( 1 - \beta ) q^{34} + ( 2 - \beta ) q^{35} + ( -13 - 2 \beta ) q^{36} + ( -2 + 6 \beta ) q^{37} + ( -4 + 2 \beta ) q^{38} + ( 4 + 4 \beta ) q^{39} + ( 3 - \beta ) q^{40} + ( 2 - 6 \beta ) q^{41} + ( -2 + 2 \beta ) q^{42} + ( 2 + 4 \beta ) q^{43} + ( -8 + 9 \beta ) q^{44} + ( -3 - 4 \beta ) q^{45} -\beta q^{46} + ( -2 - 2 \beta ) q^{47} + ( -6 - 3 \beta ) q^{48} + ( -1 - 4 \beta ) q^{49} + ( -1 + \beta ) q^{50} + ( 2 + \beta ) q^{51} + ( 8 - 2 \beta ) q^{52} + ( 6 - 4 \beta ) q^{53} -8 q^{54} + ( 4 - \beta ) q^{55} + ( 8 - 5 \beta ) q^{56} + ( 4 + 4 \beta ) q^{57} -2 q^{58} + ( -12 + 2 \beta ) q^{59} + ( -2 - 3 \beta ) q^{60} + ( 2 + 4 \beta ) q^{61} + ( 6 - 3 \beta ) q^{62} + ( 2 - 5 \beta ) q^{63} + ( -7 + 2 \beta ) q^{64} + 2 \beta q^{65} + ( -2 + 4 \beta ) q^{66} + ( -6 + 2 \beta ) q^{67} + ( -1 + 2 \beta ) q^{68} + ( 6 + 4 \beta ) q^{69} + ( -4 + 3 \beta ) q^{70} -3 \beta q^{71} + ( -1 - 9 \beta ) q^{72} + ( -2 - 2 \beta ) q^{73} + ( 14 - 8 \beta ) q^{74} + ( -2 - \beta ) q^{75} + ( 8 - 2 \beta ) q^{76} + ( 10 - 6 \beta ) q^{77} + 4 q^{78} + ( 4 + \beta ) q^{79} -3 q^{80} + ( 23 + 12 \beta ) q^{81} + ( -14 + 8 \beta ) q^{82} + ( -2 + 8 \beta ) q^{83} + ( 2 - 4 \beta ) q^{84} + q^{85} + ( 6 - 2 \beta ) q^{86} + ( 8 + 6 \beta ) q^{87} + ( 14 - 7 \beta ) q^{88} + ( -8 + 4 \beta ) q^{89} + ( -5 + \beta ) q^{90} + ( -4 + 4 \beta ) q^{91} + ( 2 + 3 \beta ) q^{92} + ( -6 - 6 \beta ) q^{93} -2 q^{94} + 2 \beta q^{95} + ( -8 - 5 \beta ) q^{96} + ( -2 + 4 \beta ) q^{97} + ( -7 + 3 \beta ) q^{98} + ( -4 - 13 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{2} - 4q^{3} + 2q^{4} - 2q^{5} - 4q^{7} - 6q^{8} + 6q^{9} + O(q^{10})$$ $$2q - 2q^{2} - 4q^{3} + 2q^{4} - 2q^{5} - 4q^{7} - 6q^{8} + 6q^{9} + 2q^{10} - 8q^{11} + 4q^{12} + 8q^{14} + 4q^{15} + 6q^{16} - 2q^{17} + 10q^{18} - 2q^{20} + 4q^{21} + 12q^{22} - 4q^{23} + 8q^{24} + 2q^{25} - 8q^{26} - 16q^{27} - 12q^{28} - 4q^{29} + 6q^{32} + 12q^{33} + 2q^{34} + 4q^{35} - 26q^{36} - 4q^{37} - 8q^{38} + 8q^{39} + 6q^{40} + 4q^{41} - 4q^{42} + 4q^{43} - 16q^{44} - 6q^{45} - 4q^{47} - 12q^{48} - 2q^{49} - 2q^{50} + 4q^{51} + 16q^{52} + 12q^{53} - 16q^{54} + 8q^{55} + 16q^{56} + 8q^{57} - 4q^{58} - 24q^{59} - 4q^{60} + 4q^{61} + 12q^{62} + 4q^{63} - 14q^{64} - 4q^{66} - 12q^{67} - 2q^{68} + 12q^{69} - 8q^{70} - 2q^{72} - 4q^{73} + 28q^{74} - 4q^{75} + 16q^{76} + 20q^{77} + 8q^{78} + 8q^{79} - 6q^{80} + 46q^{81} - 28q^{82} - 4q^{83} + 4q^{84} + 2q^{85} + 12q^{86} + 16q^{87} + 28q^{88} - 16q^{89} - 10q^{90} - 8q^{91} + 4q^{92} - 12q^{93} - 4q^{94} - 16q^{96} - 4q^{97} - 14q^{98} - 8q^{99} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −1.41421 1.41421
−2.41421 −0.585786 3.82843 −1.00000 1.41421 −3.41421 −4.41421 −2.65685 2.41421
1.2 0.414214 −3.41421 −1.82843 −1.00000 −1.41421 −0.585786 −1.58579 8.65685 −0.414214
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

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

## Atkin-Lehner signs

$$p$$ Sign
$$5$$ $$1$$
$$17$$ $$1$$

## Hecke kernels

This newform can be constructed as the kernel of the linear operator $$T_{2}^{2} + 2 T_{2} - 1$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(85))$$.