# Properties

 Label 170.2.g.a Level $170$ Weight $2$ Character orbit 170.g 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.g (of order $$4$$, degree $$2$$, 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: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-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.

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

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
89.1
 − 1.00000i 1.00000i
−1.00000 1.00000 + 1.00000i 1.00000 1.00000 + 2.00000i −1.00000 1.00000i −3.00000 + 3.00000i −1.00000 1.00000i −1.00000 2.00000i
149.1 −1.00000 1.00000 1.00000i 1.00000 1.00000 2.00000i −1.00000 + 1.00000i −3.00000 3.00000i −1.00000 1.00000i −1.00000 + 2.00000i
 $$n$$: 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
85.j even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 170.2.g.a 2
3.b odd 2 1 1530.2.n.e 2
5.b even 2 1 170.2.g.c yes 2
5.c odd 4 1 850.2.h.b 2
5.c odd 4 1 850.2.h.d 2
15.d odd 2 1 1530.2.n.d 2
17.c even 4 1 170.2.g.c yes 2
51.f odd 4 1 1530.2.n.d 2
85.f odd 4 1 850.2.h.b 2
85.i odd 4 1 850.2.h.d 2
85.j even 4 1 inner 170.2.g.a 2
255.i odd 4 1 1530.2.n.e 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
170.2.g.a 2 1.a even 1 1 trivial
170.2.g.a 2 85.j even 4 1 inner
170.2.g.c yes 2 5.b even 2 1
170.2.g.c yes 2 17.c even 4 1
850.2.h.b 2 5.c odd 4 1
850.2.h.b 2 85.f odd 4 1
850.2.h.d 2 5.c odd 4 1
850.2.h.d 2 85.i odd 4 1
1530.2.n.d 2 15.d odd 2 1
1530.2.n.d 2 51.f odd 4 1
1530.2.n.e 2 3.b odd 2 1
1530.2.n.e 2 255.i odd 4 1

## Hecke kernels

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

 $$T_{3}^{2} - 2 T_{3} + 2$$ $$T_{7}^{2} + 6 T_{7} + 18$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T )^{2}$$
$3$ $$1 - 2 T + 2 T^{2} - 6 T^{3} + 9 T^{4}$$
$5$ $$1 - 2 T + 5 T^{2}$$
$7$ $$1 + 6 T + 18 T^{2} + 42 T^{3} + 49 T^{4}$$
$11$ $$1 - 2 T + 2 T^{2} - 22 T^{3} + 121 T^{4}$$
$13$ $$( 1 - 6 T + 13 T^{2} )( 1 + 6 T + 13 T^{2} )$$
$17$ $$1 - 2 T + 17 T^{2}$$
$19$ $$1 - 2 T^{2} + 361 T^{4}$$
$23$ $$1 - 10 T + 50 T^{2} - 230 T^{3} + 529 T^{4}$$
$29$ $$( 1 - 10 T + 29 T^{2} )( 1 - 4 T + 29 T^{2} )$$
$31$ $$1 - 2 T + 2 T^{2} - 62 T^{3} + 961 T^{4}$$
$37$ $$( 1 - 2 T + 37 T^{2} )( 1 + 12 T + 37 T^{2} )$$
$41$ $$( 1 - 10 T + 41 T^{2} )( 1 + 8 T + 41 T^{2} )$$
$43$ $$( 1 + 4 T + 43 T^{2} )^{2}$$
$47$ $$1 - 90 T^{2} + 2209 T^{4}$$
$53$ $$( 1 - 6 T + 53 T^{2} )^{2}$$
$59$ $$1 - 82 T^{2} + 3481 T^{4}$$
$61$ $$1 + 18 T + 162 T^{2} + 1098 T^{3} + 3721 T^{4}$$
$67$ $$1 - 130 T^{2} + 4489 T^{4}$$
$71$ $$1 - 2 T + 2 T^{2} - 142 T^{3} + 5041 T^{4}$$
$73$ $$1 - 2 T + 2 T^{2} - 146 T^{3} + 5329 T^{4}$$
$79$ $$1 - 6 T + 18 T^{2} - 474 T^{3} + 6241 T^{4}$$
$83$ $$( 1 + 4 T + 83 T^{2} )^{2}$$
$89$ $$( 1 + 10 T + 89 T^{2} )^{2}$$
$97$ $$( 1 - 18 T + 97 T^{2} )( 1 + 8 T + 97 T^{2} )$$