# Properties

 Label 1170.2.bp.d Level $1170$ Weight $2$ Character orbit 1170.bp Analytic conductor $9.342$ Analytic rank $0$ Dimension $4$ 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.bp (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$9.34249703649$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 390) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( \zeta_{12} - \zeta_{12}^{3} ) q^{2} + ( 1 - \zeta_{12}^{2} ) q^{4} + ( -2 \zeta_{12} + \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{5} + ( -1 + \zeta_{12} - \zeta_{12}^{2} ) q^{7} -\zeta_{12}^{3} q^{8} +O(q^{10})$$ $$q + ( \zeta_{12} - \zeta_{12}^{3} ) q^{2} + ( 1 - \zeta_{12}^{2} ) q^{4} + ( -2 \zeta_{12} + \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{5} + ( -1 + \zeta_{12} - \zeta_{12}^{2} ) q^{7} -\zeta_{12}^{3} q^{8} + ( -2 + \zeta_{12} + 2 \zeta_{12}^{2} ) q^{10} + ( -\zeta_{12} - \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{11} + ( 3 \zeta_{12} - 2 \zeta_{12}^{2} - 3 \zeta_{12}^{3} ) q^{13} + ( 1 - 2 \zeta_{12} + \zeta_{12}^{3} ) q^{14} -\zeta_{12}^{2} q^{16} + ( 2 - \zeta_{12} + 2 \zeta_{12}^{2} ) q^{17} + ( 3 + \zeta_{12} - 3 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{19} + ( 1 + 2 \zeta_{12}^{3} ) q^{20} + ( -1 - \zeta_{12} - \zeta_{12}^{2} ) q^{22} + ( 6 - \zeta_{12} - 3 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{23} + ( 3 - 4 \zeta_{12} - 3 \zeta_{12}^{2} ) q^{25} + ( 3 - 2 \zeta_{12} - 3 \zeta_{12}^{2} ) q^{26} + ( -2 + \zeta_{12} + \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{28} + ( -2 \zeta_{12} + 3 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{29} + 4 q^{31} -\zeta_{12} q^{32} + ( -1 + 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{34} + ( -1 + 4 \zeta_{12} - 2 \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{35} + ( 8 - \zeta_{12} - 4 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{37} + ( 1 - 2 \zeta_{12}^{2} - 3 \zeta_{12}^{3} ) q^{38} + ( \zeta_{12} + 2 \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{40} + ( 3 \zeta_{12} - 4 \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{41} + ( 1 + 5 \zeta_{12} + \zeta_{12}^{2} ) q^{43} + ( -1 - 2 \zeta_{12} + \zeta_{12}^{3} ) q^{44} + ( -1 + 3 \zeta_{12} + \zeta_{12}^{2} - 6 \zeta_{12}^{3} ) q^{46} + ( 5 - 10 \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{47} + ( -2 \zeta_{12} - 3 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{49} + ( -4 - 3 \zeta_{12}^{3} ) q^{50} + ( -2 - 3 \zeta_{12}^{3} ) q^{52} + ( -1 + 2 \zeta_{12}^{2} - 6 \zeta_{12}^{3} ) q^{53} + ( 3 + 3 \zeta_{12} + \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{55} + ( 1 - \zeta_{12} - \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{56} + ( -2 + 3 \zeta_{12} - 2 \zeta_{12}^{2} ) q^{58} + ( 2 - 6 \zeta_{12} - 2 \zeta_{12}^{2} + 12 \zeta_{12}^{3} ) q^{59} + ( -2 - 7 \zeta_{12} + 2 \zeta_{12}^{2} + 14 \zeta_{12}^{3} ) q^{61} + ( 4 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{62} - q^{64} + ( -4 + 7 \zeta_{12} + 4 \zeta_{12}^{2} ) q^{65} + ( 10 + \zeta_{12} - 5 \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{67} + ( 4 - \zeta_{12} - 2 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{68} + ( 4 - 3 \zeta_{12} - \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{70} + ( 3 + \zeta_{12} - 3 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{71} + ( -5 + 10 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{73} + ( -1 + 4 \zeta_{12} + \zeta_{12}^{2} - 8 \zeta_{12}^{3} ) q^{74} + ( -\zeta_{12} - 3 \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{76} + 2 \zeta_{12}^{3} q^{77} -12 q^{79} + ( 1 + 2 \zeta_{12} - \zeta_{12}^{2} ) q^{80} + ( 3 - 4 \zeta_{12} + 3 \zeta_{12}^{2} ) q^{82} + ( 1 - 2 \zeta_{12}^{2} - 7 \zeta_{12}^{3} ) q^{83} + ( -8 \zeta_{12} + 4 \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{85} + ( 5 + 2 \zeta_{12} - \zeta_{12}^{3} ) q^{86} + ( -2 - \zeta_{12} + \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{88} + ( -4 \zeta_{12} - 2 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{89} + ( 1 - 6 \zeta_{12} + 4 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{91} + ( 3 - 6 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{92} + ( -5 \zeta_{12} - \zeta_{12}^{2} - 5 \zeta_{12}^{3} ) q^{94} + ( 1 + 2 \zeta_{12} + 4 \zeta_{12}^{2} + 5 \zeta_{12}^{3} ) q^{95} -10 \zeta_{12} q^{97} + ( -2 - 3 \zeta_{12} - 2 \zeta_{12}^{2} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 2 q^{4} + 2 q^{5} - 6 q^{7} + O(q^{10})$$ $$4 q + 2 q^{4} + 2 q^{5} - 6 q^{7} - 4 q^{10} - 2 q^{11} - 4 q^{13} + 4 q^{14} - 2 q^{16} + 12 q^{17} + 6 q^{19} + 4 q^{20} - 6 q^{22} + 18 q^{23} + 6 q^{25} + 6 q^{26} - 6 q^{28} + 6 q^{29} + 16 q^{31} - 4 q^{34} - 8 q^{35} + 24 q^{37} + 4 q^{40} - 8 q^{41} + 6 q^{43} - 4 q^{44} - 2 q^{46} - 6 q^{49} - 16 q^{50} - 8 q^{52} + 14 q^{55} + 2 q^{56} - 12 q^{58} + 4 q^{59} - 4 q^{61} - 4 q^{64} - 8 q^{65} + 30 q^{67} + 12 q^{68} + 14 q^{70} + 6 q^{71} - 2 q^{74} - 6 q^{76} - 48 q^{79} + 2 q^{80} + 18 q^{82} + 8 q^{85} + 20 q^{86} - 6 q^{88} - 4 q^{89} + 12 q^{91} - 2 q^{94} + 12 q^{95} - 12 q^{98} + O(q^{100})$$

## Character values

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

 $$n$$ $$911$$ $$937$$ $$1081$$ $$\chi(n)$$ $$1$$ $$-1$$ $$-1 + \zeta_{12}^{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
289.1
 −0.866025 − 0.500000i 0.866025 + 0.500000i −0.866025 + 0.500000i 0.866025 − 0.500000i
−0.866025 + 0.500000i 0 0.500000 0.866025i 2.23205 0.133975i 0 −2.36603 1.36603i 1.00000i 0 −1.86603 + 1.23205i
289.2 0.866025 0.500000i 0 0.500000 0.866025i −1.23205 + 1.86603i 0 −0.633975 0.366025i 1.00000i 0 −0.133975 + 2.23205i
919.1 −0.866025 0.500000i 0 0.500000 + 0.866025i 2.23205 + 0.133975i 0 −2.36603 + 1.36603i 1.00000i 0 −1.86603 1.23205i
919.2 0.866025 + 0.500000i 0 0.500000 + 0.866025i −1.23205 1.86603i 0 −0.633975 + 0.366025i 1.00000i 0 −0.133975 2.23205i
 $$n$$: 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
65.n even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1170.2.bp.d 4
3.b odd 2 1 390.2.y.b 4
5.b even 2 1 1170.2.bp.e 4
13.c even 3 1 1170.2.bp.e 4
15.d odd 2 1 390.2.y.c yes 4
15.e even 4 1 1950.2.i.y 4
15.e even 4 1 1950.2.i.bh 4
39.i odd 6 1 390.2.y.c yes 4
65.n even 6 1 inner 1170.2.bp.d 4
195.x odd 6 1 390.2.y.b 4
195.bl even 12 1 1950.2.i.y 4
195.bl even 12 1 1950.2.i.bh 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.2.y.b 4 3.b odd 2 1
390.2.y.b 4 195.x odd 6 1
390.2.y.c yes 4 15.d odd 2 1
390.2.y.c yes 4 39.i odd 6 1
1170.2.bp.d 4 1.a even 1 1 trivial
1170.2.bp.d 4 65.n even 6 1 inner
1170.2.bp.e 4 5.b even 2 1
1170.2.bp.e 4 13.c even 3 1
1950.2.i.y 4 15.e even 4 1
1950.2.i.y 4 195.bl even 12 1
1950.2.i.bh 4 15.e even 4 1
1950.2.i.bh 4 195.bl even 12 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{4} + 6 T_{7}^{3} + 14 T_{7}^{2} + 12 T_{7} + 4$$ acting on $$S_{2}^{\mathrm{new}}(1170, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T^{2} + T^{4}$$
$3$ $$T^{4}$$
$5$ $$25 - 10 T - T^{2} - 2 T^{3} + T^{4}$$
$7$ $$4 + 12 T + 14 T^{2} + 6 T^{3} + T^{4}$$
$11$ $$4 - 4 T + 6 T^{2} + 2 T^{3} + T^{4}$$
$13$ $$169 + 52 T + 3 T^{2} + 4 T^{3} + T^{4}$$
$17$ $$121 - 132 T + 59 T^{2} - 12 T^{3} + T^{4}$$
$19$ $$36 - 36 T + 30 T^{2} - 6 T^{3} + T^{4}$$
$23$ $$676 - 468 T + 134 T^{2} - 18 T^{3} + T^{4}$$
$29$ $$9 + 18 T + 39 T^{2} - 6 T^{3} + T^{4}$$
$31$ $$( -4 + T )^{4}$$
$37$ $$2209 - 1128 T + 239 T^{2} - 24 T^{3} + T^{4}$$
$41$ $$121 - 88 T + 75 T^{2} + 8 T^{3} + T^{4}$$
$43$ $$484 + 132 T - 10 T^{2} - 6 T^{3} + T^{4}$$
$47$ $$5476 + 152 T^{2} + T^{4}$$
$53$ $$1089 + 78 T^{2} + T^{4}$$
$59$ $$10816 + 416 T + 120 T^{2} - 4 T^{3} + T^{4}$$
$61$ $$20449 - 572 T + 159 T^{2} + 4 T^{3} + T^{4}$$
$67$ $$5476 - 2220 T + 374 T^{2} - 30 T^{3} + T^{4}$$
$71$ $$36 - 36 T + 30 T^{2} - 6 T^{3} + T^{4}$$
$73$ $$3481 + 182 T^{2} + T^{4}$$
$79$ $$( 12 + T )^{4}$$
$83$ $$2116 + 104 T^{2} + T^{4}$$
$89$ $$1936 - 176 T + 60 T^{2} + 4 T^{3} + T^{4}$$
$97$ $$10000 - 100 T^{2} + T^{4}$$