# Properties

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

# 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: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ 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 $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

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

## 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
 − 1.00000i 1.00000i
1.00000i 1.00000 −1.00000 2.00000 + 1.00000i 1.00000i 2.00000 1.00000i −2.00000 1.00000 2.00000i
169.2 1.00000i 1.00000 −1.00000 2.00000 1.00000i 1.00000i 2.00000 1.00000i −2.00000 1.00000 + 2.00000i
 $$n$$: 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.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.b yes 2
3.b odd 2 1 1530.2.f.b 2
4.b odd 2 1 1360.2.o.a 2
5.b even 2 1 170.2.d.a 2
5.c odd 4 1 850.2.b.c 2
5.c odd 4 1 850.2.b.i 2
15.d odd 2 1 1530.2.f.e 2
17.b even 2 1 170.2.d.a 2
20.d odd 2 1 1360.2.o.b 2
51.c odd 2 1 1530.2.f.e 2
68.d odd 2 1 1360.2.o.b 2
85.c even 2 1 inner 170.2.d.b yes 2
85.g odd 4 1 850.2.b.c 2
85.g odd 4 1 850.2.b.i 2
255.h odd 2 1 1530.2.f.b 2
340.d odd 2 1 1360.2.o.a 2

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

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T^{2}$$
$3$ $$( 1 - T + 3 T^{2} )^{2}$$
$5$ $$1 - 4 T + 5 T^{2}$$
$7$ $$( 1 - 2 T + 7 T^{2} )^{2}$$
$11$ $$( 1 - 11 T^{2} )^{2}$$
$13$ $$1 - 25 T^{2} + 169 T^{4}$$
$17$ $$1 - 2 T + 17 T^{2}$$
$19$ $$( 1 + 5 T + 19 T^{2} )^{2}$$
$23$ $$( 1 + 4 T + 23 T^{2} )^{2}$$
$29$ $$1 + 23 T^{2} + 841 T^{4}$$
$31$ $$1 - 37 T^{2} + 961 T^{4}$$
$37$ $$( 1 - 2 T + 37 T^{2} )^{2}$$
$41$ $$( 1 - 8 T + 41 T^{2} )( 1 + 8 T + 41 T^{2} )$$
$43$ $$1 - 50 T^{2} + 1849 T^{4}$$
$47$ $$1 - 45 T^{2} + 2209 T^{4}$$
$53$ $$1 - 105 T^{2} + 2809 T^{4}$$
$59$ $$( 1 + 5 T + 59 T^{2} )^{2}$$
$61$ $$1 - 97 T^{2} + 3721 T^{4}$$
$67$ $$1 - 130 T^{2} + 4489 T^{4}$$
$71$ $$1 - 117 T^{2} + 5041 T^{4}$$
$73$ $$( 1 - 11 T + 73 T^{2} )^{2}$$
$79$ $$1 + 98 T^{2} + 6241 T^{4}$$
$83$ $$1 - 130 T^{2} + 6889 T^{4}$$
$89$ $$( 1 - 5 T + 89 T^{2} )^{2}$$
$97$ $$( 1 - 7 T + 97 T^{2} )^{2}$$