# Properties

 Label 170.2.a.d 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

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

## 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
 0
−1.00000 3.00000 1.00000 −1.00000 −3.00000 2.00000 −1.00000 6.00000 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.d 1
3.b odd 2 1 1530.2.a.o 1
4.b odd 2 1 1360.2.a.a 1
5.b even 2 1 850.2.a.f 1
5.c odd 4 2 850.2.c.a 2
7.b odd 2 1 8330.2.a.a 1
8.b even 2 1 5440.2.a.b 1
8.d odd 2 1 5440.2.a.y 1
15.d odd 2 1 7650.2.a.l 1
17.b even 2 1 2890.2.a.b 1
17.c even 4 2 2890.2.b.d 2
20.d odd 2 1 6800.2.a.z 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
170.2.a.d 1 1.a even 1 1 trivial
850.2.a.f 1 5.b even 2 1
850.2.c.a 2 5.c odd 4 2
1360.2.a.a 1 4.b odd 2 1
1530.2.a.o 1 3.b odd 2 1
2890.2.a.b 1 17.b even 2 1
2890.2.b.d 2 17.c even 4 2
5440.2.a.b 1 8.b even 2 1
5440.2.a.y 1 8.d odd 2 1
6800.2.a.z 1 20.d odd 2 1
7650.2.a.l 1 15.d odd 2 1
8330.2.a.a 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(170))$$:

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