# Properties

 Label 170.2.g.e Level $170$ Weight $2$ Character orbit 170.g 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.g (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: $$\Z[a_1, \ldots, a_{7}]$$ 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.

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

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
89.1
 −0.707107 + 0.707107i 0.707107 − 0.707107i −0.707107 − 0.707107i 0.707107 + 0.707107i
−1.00000 −1.70711 1.70711i 1.00000 2.12132 + 0.707107i 1.70711 + 1.70711i 1.00000 1.00000i −1.00000 2.82843i −2.12132 0.707107i
89.2 −1.00000 −0.292893 0.292893i 1.00000 −2.12132 0.707107i 0.292893 + 0.292893i 1.00000 1.00000i −1.00000 2.82843i 2.12132 + 0.707107i
149.1 −1.00000 −1.70711 + 1.70711i 1.00000 2.12132 0.707107i 1.70711 1.70711i 1.00000 + 1.00000i −1.00000 2.82843i −2.12132 + 0.707107i
149.2 −1.00000 −0.292893 + 0.292893i 1.00000 −2.12132 + 0.707107i 0.292893 0.292893i 1.00000 + 1.00000i −1.00000 2.82843i 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
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.e 4
3.b odd 2 1 1530.2.n.o 4
5.b even 2 1 170.2.g.f yes 4
5.c odd 4 1 850.2.h.h 4
5.c odd 4 1 850.2.h.k 4
15.d odd 2 1 1530.2.n.j 4
17.c even 4 1 170.2.g.f yes 4
51.f odd 4 1 1530.2.n.j 4
85.f odd 4 1 850.2.h.k 4
85.i odd 4 1 850.2.h.h 4
85.j even 4 1 inner 170.2.g.e 4
255.i odd 4 1 1530.2.n.o 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
170.2.g.e 4 1.a even 1 1 trivial
170.2.g.e 4 85.j even 4 1 inner
170.2.g.f yes 4 5.b even 2 1
170.2.g.f yes 4 17.c even 4 1
850.2.h.h 4 5.c odd 4 1
850.2.h.h 4 85.i odd 4 1
850.2.h.k 4 5.c odd 4 1
850.2.h.k 4 85.f odd 4 1
1530.2.n.j 4 15.d odd 2 1
1530.2.n.j 4 51.f odd 4 1
1530.2.n.o 4 3.b odd 2 1
1530.2.n.o 4 255.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 $$S_{2}^{\mathrm{new}}(170, [\chi])$$:

 $$T_{3}^{4} + 4 T_{3}^{3} + 8 T_{3}^{2} + 4 T_{3} + 1$$ $$T_{7}^{2} - 2 T_{7} + 2$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T )^{4}$$
$3$ $$1 + 4 T + 8 T^{2} + 16 T^{3} + 31 T^{4} + 48 T^{5} + 72 T^{6} + 108 T^{7} + 81 T^{8}$$
$5$ $$1 - 8 T^{2} + 25 T^{4}$$
$7$ $$( 1 - 2 T + 2 T^{2} - 14 T^{3} + 49 T^{4} )^{2}$$
$11$ $$( 1 + 6 T + 11 T^{2} )^{2}( 1 + 14 T^{2} + 121 T^{4} )$$
$13$ $$( 1 - 17 T^{2} + 169 T^{4} )^{2}$$
$17$ $$1 + 16 T^{2} + 289 T^{4}$$
$19$ $$1 - 22 T^{2} + 195 T^{4} - 7942 T^{6} + 130321 T^{8}$$
$23$ $$1 - 158 T^{4} + 279841 T^{8}$$
$29$ $$1 + 1567 T^{4} + 707281 T^{8}$$
$31$ $$1 - 4 T + 8 T^{2} + 192 T^{3} - 1633 T^{4} + 5952 T^{5} + 7688 T^{6} - 119164 T^{7} + 923521 T^{8}$$
$37$ $$1 - 4 T + 8 T^{2} - 12 T^{3} - 1138 T^{4} - 444 T^{5} + 10952 T^{6} - 202612 T^{7} + 1874161 T^{8}$$
$41$ $$1 - 12 T + 72 T^{2} - 660 T^{3} + 5854 T^{4} - 27060 T^{5} + 121032 T^{6} - 827052 T^{7} + 2825761 T^{8}$$
$43$ $$( 1 - 16 T + 132 T^{2} - 688 T^{3} + 1849 T^{4} )^{2}$$
$47$ $$1 - 166 T^{2} + 11235 T^{4} - 366694 T^{6} + 4879681 T^{8}$$
$53$ $$( 1 + 3 T + 53 T^{2} )^{4}$$
$59$ $$1 - 22 T^{2} + 3555 T^{4} - 76582 T^{6} + 12117361 T^{8}$$
$61$ $$1 - 16 T + 128 T^{2} - 1344 T^{3} + 13583 T^{4} - 81984 T^{5} + 476288 T^{6} - 3631696 T^{7} + 13845841 T^{8}$$
$67$ $$1 - 52 T^{2} - 714 T^{4} - 233428 T^{6} + 20151121 T^{8}$$
$71$ $$1 - 12 T + 72 T^{2} - 1056 T^{3} + 15199 T^{4} - 74976 T^{5} + 362952 T^{6} - 4294932 T^{7} + 25411681 T^{8}$$
$73$ $$1 + 8 T + 32 T^{2} - 5329 T^{4} + 170528 T^{6} + 3112136 T^{7} + 28398241 T^{8}$$
$79$ $$1 - 16 T + 128 T^{2} - 1200 T^{3} + 11234 T^{4} - 94800 T^{5} + 798848 T^{6} - 7888624 T^{7} + 38950081 T^{8}$$
$83$ $$( 1 + 148 T^{2} + 6889 T^{4} )^{2}$$
$89$ $$( 1 - 6 T + 115 T^{2} - 534 T^{3} + 7921 T^{4} )^{2}$$
$97$ $$1 + 8 T + 32 T^{2} + 768 T^{3} + 18431 T^{4} + 74496 T^{5} + 301088 T^{6} + 7301384 T^{7} + 88529281 T^{8}$$