# Properties

 Label 170.2.h.a Level $170$ Weight $2$ Character orbit 170.h Analytic conductor $1.357$ Analytic rank $0$ Dimension $4$ 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: $$4$$ Relative dimension: $$2$$ over $$\Q(i)$$ Coefficient field: $$\Q(\zeta_{8})$$ Defining polynomial: $$x^{4} + 1$$ 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 a primitive root of unity $$\zeta_{8}$$. We also show the integral $$q$$-expansion of the trace form.

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

## Character values

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

 $$n$$ $$71$$ $$137$$ $$\chi(n)$$ $$\zeta_{8}^{2}$$ $$1$$

## 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
 −0.707107 + 0.707107i 0.707107 − 0.707107i −0.707107 − 0.707107i 0.707107 + 0.707107i
1.00000i 0.292893 0.292893i −1.00000 −0.707107 + 0.707107i 0.292893 + 0.292893i 2.41421 + 2.41421i 1.00000i 2.82843i −0.707107 0.707107i
21.2 1.00000i 1.70711 1.70711i −1.00000 0.707107 0.707107i 1.70711 + 1.70711i −0.414214 0.414214i 1.00000i 2.82843i 0.707107 + 0.707107i
81.1 1.00000i 0.292893 + 0.292893i −1.00000 −0.707107 0.707107i 0.292893 0.292893i 2.41421 2.41421i 1.00000i 2.82843i −0.707107 + 0.707107i
81.2 1.00000i 1.70711 + 1.70711i −1.00000 0.707107 + 0.707107i 1.70711 1.70711i −0.414214 + 0.414214i 1.00000i 2.82843i 0.707107 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
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.a 4
3.b odd 2 1 1530.2.q.c 4
4.b odd 2 1 1360.2.bt.a 4
5.b even 2 1 850.2.h.g 4
5.c odd 4 1 850.2.g.e 4
5.c odd 4 1 850.2.g.h 4
17.c even 4 1 inner 170.2.h.a 4
17.d even 8 1 2890.2.a.t 2
17.d even 8 1 2890.2.a.v 2
17.d even 8 2 2890.2.b.j 4
51.f odd 4 1 1530.2.q.c 4
68.f odd 4 1 1360.2.bt.a 4
85.f odd 4 1 850.2.g.e 4
85.i odd 4 1 850.2.g.h 4
85.j even 4 1 850.2.h.g 4

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{4} - 4 T_{3}^{3} + 8 T_{3}^{2} - 4 T_{3} + 1$$ acting on $$S_{2}^{\mathrm{new}}(170, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T^{2} )^{2}$$
$3$ $$1 - 4 T + 8 T^{2} - 16 T^{3} + 31 T^{4} - 48 T^{5} + 72 T^{6} - 108 T^{7} + 81 T^{8}$$
$5$ $$1 + T^{4}$$
$7$ $$1 - 4 T + 8 T^{2} - 20 T^{3} + 46 T^{4} - 140 T^{5} + 392 T^{6} - 1372 T^{7} + 2401 T^{8}$$
$11$ $$( 1 - 2 T + 2 T^{2} - 22 T^{3} + 121 T^{4} )^{2}$$
$13$ $$( 1 - T + 13 T^{2} )^{4}$$
$17$ $$1 + 8 T + 32 T^{2} + 136 T^{3} + 289 T^{4}$$
$19$ $$1 - 70 T^{2} + 1939 T^{4} - 25270 T^{6} + 130321 T^{8}$$
$23$ $$1 + 8 T + 32 T^{2} - 40 T^{3} - 734 T^{4} - 920 T^{5} + 16928 T^{6} + 97336 T^{7} + 279841 T^{8}$$
$29$ $$1 + 8 T + 32 T^{2} - 96 T^{3} - 1393 T^{4} - 2784 T^{5} + 26912 T^{6} + 195112 T^{7} + 707281 T^{8}$$
$31$ $$1 + 12 T + 72 T^{2} + 576 T^{3} + 4319 T^{4} + 17856 T^{5} + 69192 T^{6} + 357492 T^{7} + 923521 T^{8}$$
$37$ $$1 - 12 T + 72 T^{2} - 612 T^{3} + 5006 T^{4} - 22644 T^{5} + 98568 T^{6} - 607836 T^{7} + 1874161 T^{8}$$
$41$ $$1 + 4 T + 8 T^{2} - 84 T^{3} - 2962 T^{4} - 3444 T^{5} + 13448 T^{6} + 275684 T^{7} + 2825761 T^{8}$$
$43$ $$1 - 64 T^{2} + 2130 T^{4} - 118336 T^{6} + 3418801 T^{8}$$
$47$ $$( 1 + 2 T + 77 T^{2} + 94 T^{3} + 2209 T^{4} )^{2}$$
$53$ $$1 - 18 T^{2} - 1501 T^{4} - 50562 T^{6} + 7890481 T^{8}$$
$59$ $$1 - 38 T^{2} + 6931 T^{4} - 132278 T^{6} + 12117361 T^{8}$$
$61$ $$1 - 8 T + 32 T^{2} + 416 T^{3} - 7361 T^{4} + 25376 T^{5} + 119072 T^{6} - 1815848 T^{7} + 13845841 T^{8}$$
$67$ $$( 1 - 12 T + 162 T^{2} - 804 T^{3} + 4489 T^{4} )^{2}$$
$71$ $$1 + 36 T + 648 T^{2} + 8352 T^{3} + 81727 T^{4} + 592992 T^{5} + 3266568 T^{6} + 12884796 T^{7} + 25411681 T^{8}$$
$73$ $$1 - 8 T + 32 T^{2} - 448 T^{3} + 5983 T^{4} - 32704 T^{5} + 170528 T^{6} - 3112136 T^{7} + 28398241 T^{8}$$
$79$ $$1 - 16 T + 128 T^{2} - 1200 T^{3} + 11234 T^{4} - 94800 T^{5} + 798848 T^{6} - 7888624 T^{7} + 38950081 T^{8}$$
$83$ $$( 1 - 68 T^{2} + 6889 T^{4} )^{2}$$
$89$ $$( 1 - 18 T + 227 T^{2} - 1602 T^{3} + 7921 T^{4} )^{2}$$
$97$ $$1 - 32 T + 512 T^{2} - 6912 T^{3} + 79151 T^{4} - 670464 T^{5} + 4817408 T^{6} - 29205536 T^{7} + 88529281 T^{8}$$