# Properties

 Label 170.2.d.c Level $170$ Weight $2$ Character orbit 170.d Analytic conductor $1.357$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$1.35745683436$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{8})$$ Defining polynomial: $$x^{4} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ 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 a primitive root of unity $$\zeta_{8}$$. We also show the integral $$q$$-expansion of the trace form.

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
169.1
 0.707107 − 0.707107i −0.707107 + 0.707107i 0.707107 + 0.707107i −0.707107 − 0.707107i
1.00000i −2.82843 −1.00000 −0.707107 + 2.12132i 2.82843i 4.24264 1.00000i 5.00000 2.12132 + 0.707107i
169.2 1.00000i 2.82843 −1.00000 0.707107 2.12132i 2.82843i −4.24264 1.00000i 5.00000 −2.12132 0.707107i
169.3 1.00000i −2.82843 −1.00000 −0.707107 2.12132i 2.82843i 4.24264 1.00000i 5.00000 2.12132 0.707107i
169.4 1.00000i 2.82843 −1.00000 0.707107 + 2.12132i 2.82843i −4.24264 1.00000i 5.00000 −2.12132 + 0.707107i
 $$n$$: 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
17.b even 2 1 inner
85.c 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.d.c 4
3.b odd 2 1 1530.2.f.h 4
4.b odd 2 1 1360.2.o.d 4
5.b even 2 1 inner 170.2.d.c 4
5.c odd 4 1 850.2.b.b 2
5.c odd 4 1 850.2.b.g 2
15.d odd 2 1 1530.2.f.h 4
17.b even 2 1 inner 170.2.d.c 4
20.d odd 2 1 1360.2.o.d 4
51.c odd 2 1 1530.2.f.h 4
68.d odd 2 1 1360.2.o.d 4
85.c even 2 1 inner 170.2.d.c 4
85.g odd 4 1 850.2.b.b 2
85.g odd 4 1 850.2.b.g 2
255.h odd 2 1 1530.2.f.h 4
340.d odd 2 1 1360.2.o.d 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
170.2.d.c 4 1.a even 1 1 trivial
170.2.d.c 4 5.b even 2 1 inner
170.2.d.c 4 17.b even 2 1 inner
170.2.d.c 4 85.c even 2 1 inner
850.2.b.b 2 5.c odd 4 1
850.2.b.b 2 85.g odd 4 1
850.2.b.g 2 5.c odd 4 1
850.2.b.g 2 85.g odd 4 1
1360.2.o.d 4 4.b odd 2 1
1360.2.o.d 4 20.d odd 2 1
1360.2.o.d 4 68.d odd 2 1
1360.2.o.d 4 340.d odd 2 1
1530.2.f.h 4 3.b odd 2 1
1530.2.f.h 4 15.d odd 2 1
1530.2.f.h 4 51.c odd 2 1
1530.2.f.h 4 255.h odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} - 8$$ acting on $$S_{2}^{\mathrm{new}}(170, [\chi])$$.