# Properties

 Label 170.2.a.e Level $170$ Weight $2$ Character orbit 170.a Self dual yes Analytic conductor $1.357$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$170 = 2 \cdot 5 \cdot 17$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 170.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$1.35745683436$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0
1.00000 1.00000 1.00000 −1.00000 1.00000 2.00000 1.00000 −2.00000 −1.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$5$$ $$1$$
$$17$$ $$1$$

## 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 170.2.a.e 1
3.b odd 2 1 1530.2.a.g 1
4.b odd 2 1 1360.2.a.d 1
5.b even 2 1 850.2.a.b 1
5.c odd 4 2 850.2.c.e 2
7.b odd 2 1 8330.2.a.q 1
8.b even 2 1 5440.2.a.k 1
8.d odd 2 1 5440.2.a.r 1
15.d odd 2 1 7650.2.a.bo 1
17.b even 2 1 2890.2.a.n 1
17.c even 4 2 2890.2.b.b 2
20.d odd 2 1 6800.2.a.t 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
170.2.a.e 1 1.a even 1 1 trivial
850.2.a.b 1 5.b even 2 1
850.2.c.e 2 5.c odd 4 2
1360.2.a.d 1 4.b odd 2 1
1530.2.a.g 1 3.b odd 2 1
2890.2.a.n 1 17.b even 2 1
2890.2.b.b 2 17.c even 4 2
5440.2.a.k 1 8.b even 2 1
5440.2.a.r 1 8.d odd 2 1
6800.2.a.t 1 20.d odd 2 1
7650.2.a.bo 1 15.d odd 2 1
8330.2.a.q 1 7.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(170))$$:

 $$T_{3} - 1$$ $$T_{7} - 2$$ $$T_{13} + 1$$