# Properties

 Label 1170.2.i.k Level $1170$ Weight $2$ Character orbit 1170.i Analytic conductor $9.342$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1170 = 2 \cdot 3^{2} \cdot 5 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1170.i (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$9.34249703649$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 390) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 1 - \zeta_{6} ) q^{2} -\zeta_{6} q^{4} + q^{5} -2 \zeta_{6} q^{7} - q^{8} +O(q^{10})$$ $$q + ( 1 - \zeta_{6} ) q^{2} -\zeta_{6} q^{4} + q^{5} -2 \zeta_{6} q^{7} - q^{8} + ( 1 - \zeta_{6} ) q^{10} + ( 1 - \zeta_{6} ) q^{11} + ( 1 - 4 \zeta_{6} ) q^{13} -2 q^{14} + ( -1 + \zeta_{6} ) q^{16} + 2 \zeta_{6} q^{17} -6 \zeta_{6} q^{19} -\zeta_{6} q^{20} -\zeta_{6} q^{22} + ( -3 + 3 \zeta_{6} ) q^{23} + q^{25} + ( -3 - \zeta_{6} ) q^{26} + ( -2 + 2 \zeta_{6} ) q^{28} + ( -1 + \zeta_{6} ) q^{29} -3 q^{31} + \zeta_{6} q^{32} + 2 q^{34} -2 \zeta_{6} q^{35} + ( 5 - 5 \zeta_{6} ) q^{37} -6 q^{38} - q^{40} + ( 10 - 10 \zeta_{6} ) q^{41} -5 \zeta_{6} q^{43} - q^{44} + 3 \zeta_{6} q^{46} -3 q^{47} + ( 3 - 3 \zeta_{6} ) q^{49} + ( 1 - \zeta_{6} ) q^{50} + ( -4 + 3 \zeta_{6} ) q^{52} -14 q^{53} + ( 1 - \zeta_{6} ) q^{55} + 2 \zeta_{6} q^{56} + \zeta_{6} q^{58} -5 \zeta_{6} q^{59} + 10 \zeta_{6} q^{61} + ( -3 + 3 \zeta_{6} ) q^{62} + q^{64} + ( 1 - 4 \zeta_{6} ) q^{65} + ( 2 - 2 \zeta_{6} ) q^{68} -2 q^{70} + 4 \zeta_{6} q^{71} -2 q^{73} -5 \zeta_{6} q^{74} + ( -6 + 6 \zeta_{6} ) q^{76} -2 q^{77} + 5 q^{79} + ( -1 + \zeta_{6} ) q^{80} -10 \zeta_{6} q^{82} + 6 q^{83} + 2 \zeta_{6} q^{85} -5 q^{86} + ( -1 + \zeta_{6} ) q^{88} + ( 10 - 10 \zeta_{6} ) q^{89} + ( -8 + 6 \zeta_{6} ) q^{91} + 3 q^{92} + ( -3 + 3 \zeta_{6} ) q^{94} -6 \zeta_{6} q^{95} + 10 \zeta_{6} q^{97} -3 \zeta_{6} q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + q^{2} - q^{4} + 2q^{5} - 2q^{7} - 2q^{8} + O(q^{10})$$ $$2q + q^{2} - q^{4} + 2q^{5} - 2q^{7} - 2q^{8} + q^{10} + q^{11} - 2q^{13} - 4q^{14} - q^{16} + 2q^{17} - 6q^{19} - q^{20} - q^{22} - 3q^{23} + 2q^{25} - 7q^{26} - 2q^{28} - q^{29} - 6q^{31} + q^{32} + 4q^{34} - 2q^{35} + 5q^{37} - 12q^{38} - 2q^{40} + 10q^{41} - 5q^{43} - 2q^{44} + 3q^{46} - 6q^{47} + 3q^{49} + q^{50} - 5q^{52} - 28q^{53} + q^{55} + 2q^{56} + q^{58} - 5q^{59} + 10q^{61} - 3q^{62} + 2q^{64} - 2q^{65} + 2q^{68} - 4q^{70} + 4q^{71} - 4q^{73} - 5q^{74} - 6q^{76} - 4q^{77} + 10q^{79} - q^{80} - 10q^{82} + 12q^{83} + 2q^{85} - 10q^{86} - q^{88} + 10q^{89} - 10q^{91} + 6q^{92} - 3q^{94} - 6q^{95} + 10q^{97} - 3q^{98} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1170\mathbb{Z}\right)^\times$$.

 $$n$$ $$911$$ $$937$$ $$1081$$ $$\chi(n)$$ $$1$$ $$1$$ $$-\zeta_{6}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
451.1
 0.5 + 0.866025i 0.5 − 0.866025i
0.500000 0.866025i 0 −0.500000 0.866025i 1.00000 0 −1.00000 1.73205i −1.00000 0 0.500000 0.866025i
991.1 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 1.00000 0 −1.00000 + 1.73205i −1.00000 0 0.500000 + 0.866025i
 $$n$$: 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
13.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1170.2.i.k 2
3.b odd 2 1 390.2.i.a 2
13.c even 3 1 inner 1170.2.i.k 2
15.d odd 2 1 1950.2.i.s 2
15.e even 4 2 1950.2.z.h 4
39.h odd 6 1 5070.2.a.f 1
39.i odd 6 1 390.2.i.a 2
39.i odd 6 1 5070.2.a.o 1
39.k even 12 2 5070.2.b.g 2
195.x odd 6 1 1950.2.i.s 2
195.bl even 12 2 1950.2.z.h 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.2.i.a 2 3.b odd 2 1
390.2.i.a 2 39.i odd 6 1
1170.2.i.k 2 1.a even 1 1 trivial
1170.2.i.k 2 13.c even 3 1 inner
1950.2.i.s 2 15.d odd 2 1
1950.2.i.s 2 195.x odd 6 1
1950.2.z.h 4 15.e even 4 2
1950.2.z.h 4 195.bl even 12 2
5070.2.a.f 1 39.h odd 6 1
5070.2.a.o 1 39.i odd 6 1
5070.2.b.g 2 39.k even 12 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1170, [\chi])$$:

 $$T_{7}^{2} + 2 T_{7} + 4$$ $$T_{11}^{2} - T_{11} + 1$$ $$T_{29}^{2} + T_{29} + 1$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T + T^{2}$$
$3$ $$T^{2}$$
$5$ $$( -1 + T )^{2}$$
$7$ $$4 + 2 T + T^{2}$$
$11$ $$1 - T + T^{2}$$
$13$ $$13 + 2 T + T^{2}$$
$17$ $$4 - 2 T + T^{2}$$
$19$ $$36 + 6 T + T^{2}$$
$23$ $$9 + 3 T + T^{2}$$
$29$ $$1 + T + T^{2}$$
$31$ $$( 3 + T )^{2}$$
$37$ $$25 - 5 T + T^{2}$$
$41$ $$100 - 10 T + T^{2}$$
$43$ $$25 + 5 T + T^{2}$$
$47$ $$( 3 + T )^{2}$$
$53$ $$( 14 + T )^{2}$$
$59$ $$25 + 5 T + T^{2}$$
$61$ $$100 - 10 T + T^{2}$$
$67$ $$T^{2}$$
$71$ $$16 - 4 T + T^{2}$$
$73$ $$( 2 + T )^{2}$$
$79$ $$( -5 + T )^{2}$$
$83$ $$( -6 + T )^{2}$$
$89$ $$100 - 10 T + T^{2}$$
$97$ $$100 - 10 T + T^{2}$$