# Properties

 Label 170.2.c.a Level $170$ Weight $2$ Character orbit 170.c Analytic conductor $1.357$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$170 = 2 \cdot 5 \cdot 17$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 170.c (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.35745683436$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

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

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/170\mathbb{Z}\right)^\times$$.

 $$n$$ $$71$$ $$137$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
69.1
 − 1.00000i 1.00000i
1.00000i 1.00000i −1.00000 1.00000 2.00000i −1.00000 0 1.00000i 2.00000 −2.00000 1.00000i
69.2 1.00000i 1.00000i −1.00000 1.00000 + 2.00000i −1.00000 0 1.00000i 2.00000 −2.00000 + 1.00000i
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 170.2.c.a 2
3.b odd 2 1 1530.2.d.b 2
4.b odd 2 1 1360.2.e.b 2
5.b even 2 1 inner 170.2.c.a 2
5.c odd 4 1 850.2.a.d 1
5.c odd 4 1 850.2.a.h 1
15.d odd 2 1 1530.2.d.b 2
15.e even 4 1 7650.2.a.s 1
15.e even 4 1 7650.2.a.cb 1
20.d odd 2 1 1360.2.e.b 2
20.e even 4 1 6800.2.a.g 1
20.e even 4 1 6800.2.a.r 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
170.2.c.a 2 1.a even 1 1 trivial
170.2.c.a 2 5.b even 2 1 inner
850.2.a.d 1 5.c odd 4 1
850.2.a.h 1 5.c odd 4 1
1360.2.e.b 2 4.b odd 2 1
1360.2.e.b 2 20.d odd 2 1
1530.2.d.b 2 3.b odd 2 1
1530.2.d.b 2 15.d odd 2 1
6800.2.a.g 1 20.e even 4 1
6800.2.a.r 1 20.e even 4 1
7650.2.a.s 1 15.e even 4 1
7650.2.a.cb 1 15.e even 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(170, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T^{2}$$
$3$ $$1 + T^{2}$$
$5$ $$5 - 2 T + T^{2}$$
$7$ $$T^{2}$$
$11$ $$( 6 + T )^{2}$$
$13$ $$9 + T^{2}$$
$17$ $$1 + T^{2}$$
$19$ $$( -7 + T )^{2}$$
$23$ $$64 + T^{2}$$
$29$ $$( -5 + T )^{2}$$
$31$ $$( -5 + T )^{2}$$
$37$ $$64 + T^{2}$$
$41$ $$T^{2}$$
$43$ $$16 + T^{2}$$
$47$ $$9 + T^{2}$$
$53$ $$81 + T^{2}$$
$59$ $$( 5 + T )^{2}$$
$61$ $$( 3 + T )^{2}$$
$67$ $$4 + T^{2}$$
$71$ $$( 15 + T )^{2}$$
$73$ $$121 + T^{2}$$
$79$ $$( 8 + T )^{2}$$
$83$ $$16 + T^{2}$$
$89$ $$( -1 + T )^{2}$$
$97$ $$81 + T^{2}$$