# Properties

 Label 170.2.h.b Level $170$ Weight $2$ Character orbit 170.h Analytic conductor $1.357$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$170 = 2 \cdot 5 \cdot 17$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 170.h (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.35745683436$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(i)$$ Coefficient field: 8.0.23045668864.1 Defining polynomial: $$x^{8} + 26 x^{6} + 237 x^{4} + 892 x^{2} + 1156$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ 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 a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 26 x^{6} + 237 x^{4} + 892 x^{2} + 1156$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-5 \nu^{7} - 96 \nu^{5} - 505 \nu^{3} - 618 \nu$$$$)/272$$ $$\beta_{3}$$ $$=$$ $$($$$$-2 \nu^{7} - 17 \nu^{6} - 52 \nu^{5} - 340 \nu^{4} - 406 \nu^{3} - 2057 \nu^{2} - 900 \nu - 3706$$$$)/272$$ $$\beta_{4}$$ $$=$$ $$($$$$2 \nu^{7} - 17 \nu^{6} + 52 \nu^{5} - 340 \nu^{4} + 406 \nu^{3} - 2057 \nu^{2} + 900 \nu - 3706$$$$)/272$$ $$\beta_{5}$$ $$=$$ $$($$$$-\nu^{6} - 20 \nu^{4} - 113 \nu^{2} - 162$$$$)/8$$ $$\beta_{6}$$ $$=$$ $$($$$$-\nu^{7} - 20 \nu^{5} - 4 \nu^{4} - 121 \nu^{3} - 52 \nu^{2} - 218 \nu - 136$$$$)/16$$ $$\beta_{7}$$ $$=$$ $$($$$$-\nu^{7} - 20 \nu^{5} + 4 \nu^{4} - 121 \nu^{3} + 52 \nu^{2} - 218 \nu + 136$$$$)/16$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{5} - \beta_{4} - \beta_{3} - 7$$ $$\nu^{3}$$ $$=$$ $$-\beta_{7} - \beta_{6} - \beta_{4} + \beta_{3} + 6 \beta_{2} - 7 \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$2 \beta_{7} - 2 \beta_{6} - 13 \beta_{5} + 13 \beta_{4} + 13 \beta_{3} + 57$$ $$\nu^{5}$$ $$=$$ $$15 \beta_{7} + 15 \beta_{6} + 25 \beta_{4} - 25 \beta_{3} - 82 \beta_{2} + 57 \beta_{1}$$ $$\nu^{6}$$ $$=$$ $$-40 \beta_{7} + 40 \beta_{6} + 139 \beta_{5} - 147 \beta_{4} - 147 \beta_{3} - 511$$ $$\nu^{7}$$ $$=$$ $$-187 \beta_{7} - 187 \beta_{6} - 379 \beta_{4} + 379 \beta_{3} + 914 \beta_{2} - 511 \beta_{1}$$

## Character values

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

 $$n$$ $$71$$ $$137$$ $$\chi(n)$$ $$-\beta_{2}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
21.1
 − 3.26843i 1.69904i 2.26843i − 2.69904i 3.26843i − 1.69904i − 2.26843i 2.69904i
1.00000i −2.31113 + 2.31113i −1.00000 0.707107 0.707107i 2.31113 + 2.31113i −3.26843 3.26843i 1.00000i 7.68265i −0.707107 0.707107i
21.2 1.00000i −1.20140 + 1.20140i −1.00000 −0.707107 + 0.707107i 1.20140 + 1.20140i 1.69904 + 1.69904i 1.00000i 0.113256i 0.707107 + 0.707107i
21.3 1.00000i 1.60402 1.60402i −1.00000 0.707107 0.707107i −1.60402 1.60402i 2.26843 + 2.26843i 1.00000i 2.14578i −0.707107 0.707107i
21.4 1.00000i 1.90851 1.90851i −1.00000 −0.707107 + 0.707107i −1.90851 1.90851i −2.69904 2.69904i 1.00000i 4.28483i 0.707107 + 0.707107i
81.1 1.00000i −2.31113 2.31113i −1.00000 0.707107 + 0.707107i 2.31113 2.31113i −3.26843 + 3.26843i 1.00000i 7.68265i −0.707107 + 0.707107i
81.2 1.00000i −1.20140 1.20140i −1.00000 −0.707107 0.707107i 1.20140 1.20140i 1.69904 1.69904i 1.00000i 0.113256i 0.707107 0.707107i
81.3 1.00000i 1.60402 + 1.60402i −1.00000 0.707107 + 0.707107i −1.60402 + 1.60402i 2.26843 2.26843i 1.00000i 2.14578i −0.707107 + 0.707107i
81.4 1.00000i 1.90851 + 1.90851i −1.00000 −0.707107 0.707107i −1.90851 + 1.90851i −2.69904 + 2.69904i 1.00000i 4.28483i 0.707107 0.707107i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 81.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
17.c 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.h.b 8
3.b odd 2 1 1530.2.q.g 8
4.b odd 2 1 1360.2.bt.b 8
5.b even 2 1 850.2.h.n 8
5.c odd 4 1 850.2.g.i 8
5.c odd 4 1 850.2.g.l 8
17.c even 4 1 inner 170.2.h.b 8
17.d even 8 1 2890.2.a.bd 4
17.d even 8 1 2890.2.a.be 4
17.d even 8 2 2890.2.b.o 8
51.f odd 4 1 1530.2.q.g 8
68.f odd 4 1 1360.2.bt.b 8
85.f odd 4 1 850.2.g.l 8
85.i odd 4 1 850.2.g.i 8
85.j even 4 1 850.2.h.n 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
170.2.h.b 8 1.a even 1 1 trivial
170.2.h.b 8 17.c even 4 1 inner
850.2.g.i 8 5.c odd 4 1
850.2.g.i 8 85.i odd 4 1
850.2.g.l 8 5.c odd 4 1
850.2.g.l 8 85.f odd 4 1
850.2.h.n 8 5.b even 2 1
850.2.h.n 8 85.j even 4 1
1360.2.bt.b 8 4.b odd 2 1
1360.2.bt.b 8 68.f odd 4 1
1530.2.q.g 8 3.b odd 2 1
1530.2.q.g 8 51.f odd 4 1
2890.2.a.bd 4 17.d even 8 1
2890.2.a.be 4 17.d even 8 1
2890.2.b.o 8 17.d even 8 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{8} - 4 T_{3}^{5} + 101 T_{3}^{4} - 52 T_{3}^{3} + 8 T_{3}^{2} + 136 T_{3} + 1156$$ acting on $$S_{2}^{\mathrm{new}}(170, [\chi])$$.