# Properties

 Label 170.2.c.b Level $170$ Weight $2$ Character orbit 170.c Analytic conductor $1.357$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$1.35745683436$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: 6.0.5161984.1 Defining polynomial: $$x^{6} - 4 x^{3} + 25 x^{2} - 20 x + 8$$ 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 basis $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 4 x^{3} + 25 x^{2} - 20 x + 8$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-5 \nu^{5} - 2 \nu^{4} - 25 \nu^{3} + 10 \nu^{2} - 121 \nu + 100$$$$)/121$$ $$\beta_{3}$$ $$=$$ $$($$$$7 \nu^{5} + 27 \nu^{4} + 35 \nu^{3} - 14 \nu^{2} + 223$$$$)/121$$ $$\beta_{4}$$ $$=$$ $$($$$$-25 \nu^{5} - 10 \nu^{4} - 4 \nu^{3} + 50 \nu^{2} - 605 \nu + 258$$$$)/242$$ $$\beta_{5}$$ $$=$$ $$($$$$-65 \nu^{5} - 26 \nu^{4} + 38 \nu^{3} + 372 \nu^{2} - 1331 \nu + 574$$$$)/242$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{5} - 3 \beta_{4} + \beta_{2} - \beta_{1}$$ $$\nu^{3}$$ $$=$$ $$2 \beta_{4} - 5 \beta_{2} + 2$$ $$\nu^{4}$$ $$=$$ $$5 \beta_{3} + 7 \beta_{2} + 7 \beta_{1} - 15$$ $$\nu^{5}$$ $$=$$ $$2 \beta_{5} - 16 \beta_{4} - 2 \beta_{3} - 29 \beta_{1} + 16$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
69.1
 0.432320 + 0.432320i 1.32001 + 1.32001i −1.75233 − 1.75233i −1.75233 + 1.75233i 1.32001 − 1.32001i 0.432320 − 0.432320i
1.00000i 2.62620i −1.00000 −2.19388 + 0.432320i −2.62620 0.864641i 1.00000i −3.89692 0.432320 + 2.19388i
69.2 1.00000i 0.484862i −1.00000 1.80487 + 1.32001i 0.484862 2.64002i 1.00000i 2.76491 1.32001 1.80487i
69.3 1.00000i 3.14134i −1.00000 1.38900 1.75233i 3.14134 3.50466i 1.00000i −6.86799 −1.75233 1.38900i
69.4 1.00000i 3.14134i −1.00000 1.38900 + 1.75233i 3.14134 3.50466i 1.00000i −6.86799 −1.75233 + 1.38900i
69.5 1.00000i 0.484862i −1.00000 1.80487 1.32001i 0.484862 2.64002i 1.00000i 2.76491 1.32001 + 1.80487i
69.6 1.00000i 2.62620i −1.00000 −2.19388 0.432320i −2.62620 0.864641i 1.00000i −3.89692 0.432320 2.19388i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 69.6 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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 170.2.c.b 6
3.b odd 2 1 1530.2.d.g 6
4.b odd 2 1 1360.2.e.c 6
5.b even 2 1 inner 170.2.c.b 6
5.c odd 4 1 850.2.a.p 3
5.c odd 4 1 850.2.a.q 3
15.d odd 2 1 1530.2.d.g 6
15.e even 4 1 7650.2.a.dj 3
15.e even 4 1 7650.2.a.do 3
20.d odd 2 1 1360.2.e.c 6
20.e even 4 1 6800.2.a.bk 3
20.e even 4 1 6800.2.a.bp 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
170.2.c.b 6 1.a even 1 1 trivial
170.2.c.b 6 5.b even 2 1 inner
850.2.a.p 3 5.c odd 4 1
850.2.a.q 3 5.c odd 4 1
1360.2.e.c 6 4.b odd 2 1
1360.2.e.c 6 20.d odd 2 1
1530.2.d.g 6 3.b odd 2 1
1530.2.d.g 6 15.d odd 2 1
6800.2.a.bk 3 20.e even 4 1
6800.2.a.bp 3 20.e even 4 1
7650.2.a.dj 3 15.e even 4 1
7650.2.a.do 3 15.e even 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{6} + 17 T_{3}^{4} + 72 T_{3}^{2} + 16$$ acting on $$S_{2}^{\mathrm{new}}(170, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T^{2} )^{3}$$
$3$ $$1 - T^{2} + 3 T^{4} - 38 T^{6} + 27 T^{8} - 81 T^{10} + 729 T^{12}$$
$5$ $$1 - 2 T - 3 T^{2} + 24 T^{3} - 15 T^{4} - 50 T^{5} + 125 T^{6}$$
$7$ $$1 - 22 T^{2} + 275 T^{4} - 2316 T^{6} + 13475 T^{8} - 52822 T^{10} + 117649 T^{12}$$
$11$ $$( 1 - 2 T + 11 T^{2} )^{6}$$
$13$ $$1 - 61 T^{2} + 1723 T^{4} - 28558 T^{6} + 291187 T^{8} - 1742221 T^{10} + 4826809 T^{12}$$
$17$ $$( 1 + T^{2} )^{3}$$
$19$ $$( 1 + T + 33 T^{2} + 58 T^{3} + 627 T^{4} + 361 T^{5} + 6859 T^{6} )^{2}$$
$23$ $$1 - 70 T^{2} + 2355 T^{4} - 57580 T^{6} + 1245795 T^{8} - 19588870 T^{10} + 148035889 T^{12}$$
$29$ $$( 1 + 17 T + 153 T^{2} + 936 T^{3} + 4437 T^{4} + 14297 T^{5} + 24389 T^{6} )^{2}$$
$31$ $$( 1 - 3 T + 47 T^{2} - 260 T^{3} + 1457 T^{4} - 2883 T^{5} + 29791 T^{6} )^{2}$$
$37$ $$( 1 - 10 T + 93 T^{2} - 472 T^{3} + 3441 T^{4} - 13690 T^{5} + 50653 T^{6} )( 1 + 10 T + 93 T^{2} + 472 T^{3} + 3441 T^{4} + 13690 T^{5} + 50653 T^{6} )$$
$41$ $$( 1 - 16 T + 183 T^{2} - 1296 T^{3} + 7503 T^{4} - 26896 T^{5} + 68921 T^{6} )^{2}$$
$43$ $$1 - 186 T^{2} + 16391 T^{4} - 879788 T^{6} + 30306959 T^{8} - 635896986 T^{10} + 6321363049 T^{12}$$
$47$ $$1 - 25 T^{2} + 4995 T^{4} - 125830 T^{6} + 11033955 T^{8} - 121992025 T^{10} + 10779215329 T^{12}$$
$53$ $$1 - 53 T^{2} + 7995 T^{4} - 263774 T^{6} + 22457955 T^{8} - 418195493 T^{10} + 22164361129 T^{12}$$
$59$ $$( 1 + 9 T + 161 T^{2} + 902 T^{3} + 9499 T^{4} + 31329 T^{5} + 205379 T^{6} )^{2}$$
$61$ $$( 1 + 9 T + 153 T^{2} + 1064 T^{3} + 9333 T^{4} + 33489 T^{5} + 226981 T^{6} )^{2}$$
$67$ $$1 - 270 T^{2} + 34775 T^{4} - 2833220 T^{6} + 156104975 T^{8} - 5440802670 T^{10} + 90458382169 T^{12}$$
$71$ $$( 1 + T + 95 T^{2} + 476 T^{3} + 6745 T^{4} + 5041 T^{5} + 357911 T^{6} )^{2}$$
$73$ $$1 - 421 T^{2} + 75043 T^{4} - 7247278 T^{6} + 399904147 T^{8} - 11955659461 T^{10} + 151334226289 T^{12}$$
$79$ $$( 1 + 4 T + 163 T^{2} + 472 T^{3} + 12877 T^{4} + 24964 T^{5} + 493039 T^{6} )^{2}$$
$83$ $$1 - 170 T^{2} + 855 T^{4} + 1073140 T^{6} + 5890095 T^{8} - 8067914570 T^{10} + 326940373369 T^{12}$$
$89$ $$( 1 - 9 T + 251 T^{2} - 1442 T^{3} + 22339 T^{4} - 71289 T^{5} + 704969 T^{6} )^{2}$$
$97$ $$1 - 493 T^{2} + 108803 T^{4} - 13645854 T^{6} + 1023727427 T^{8} - 43644935533 T^{10} + 832972004929 T^{12}$$