# Properties

 Label 170.2.a.a 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

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

## 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 −2.00000 1.00000 −1.00000 2.00000 2.00000 −1.00000 1.00000 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.a 1
3.b odd 2 1 1530.2.a.n 1
4.b odd 2 1 1360.2.a.i 1
5.b even 2 1 850.2.a.j 1
5.c odd 4 2 850.2.c.g 2
7.b odd 2 1 8330.2.a.m 1
8.b even 2 1 5440.2.a.w 1
8.d odd 2 1 5440.2.a.f 1
15.d odd 2 1 7650.2.a.g 1
17.b even 2 1 2890.2.a.i 1
17.c even 4 2 2890.2.b.g 2
20.d odd 2 1 6800.2.a.e 1

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

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T$$
$3$ $$1 + 2 T + 3 T^{2}$$
$5$ $$1 + T$$
$7$ $$1 - 2 T + 7 T^{2}$$
$11$ $$1 - 6 T + 11 T^{2}$$
$13$ $$1 - 2 T + 13 T^{2}$$
$17$ $$1 - T$$
$19$ $$1 - 8 T + 19 T^{2}$$
$23$ $$1 + 6 T + 23 T^{2}$$
$29$ $$1 + 6 T + 29 T^{2}$$
$31$ $$1 - 2 T + 31 T^{2}$$
$37$ $$1 - 2 T + 37 T^{2}$$
$41$ $$1 + 6 T + 41 T^{2}$$
$43$ $$1 + 4 T + 43 T^{2}$$
$47$ $$1 - 12 T + 47 T^{2}$$
$53$ $$1 - 6 T + 53 T^{2}$$
$59$ $$1 + 59 T^{2}$$
$61$ $$1 - 2 T + 61 T^{2}$$
$67$ $$1 - 8 T + 67 T^{2}$$
$71$ $$1 + 6 T + 71 T^{2}$$
$73$ $$1 - 2 T + 73 T^{2}$$
$79$ $$1 + 10 T + 79 T^{2}$$
$83$ $$1 - 12 T + 83 T^{2}$$
$89$ $$1 - 6 T + 89 T^{2}$$
$97$ $$1 - 2 T + 97 T^{2}$$