# Properties

 Label 1170.2.bs.e Level $1170$ Weight $2$ Character orbit 1170.bs 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.bs (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.

 $$f(q)$$ $$=$$ $$q + ( -\zeta_{12} + \zeta_{12}^{3} ) q^{2} + ( 1 - \zeta_{12}^{2} ) q^{4} -\zeta_{12}^{3} q^{5} + 2 \zeta_{12} q^{7} + \zeta_{12}^{3} q^{8} +O(q^{10})$$ $$q + ( -\zeta_{12} + \zeta_{12}^{3} ) q^{2} + ( 1 - \zeta_{12}^{2} ) q^{4} -\zeta_{12}^{3} q^{5} + 2 \zeta_{12} q^{7} + \zeta_{12}^{3} q^{8} + \zeta_{12}^{2} q^{10} + ( -4 + 3 \zeta_{12} + 2 \zeta_{12}^{2} - 3 \zeta_{12}^{3} ) q^{11} + ( -1 + 4 \zeta_{12}^{2} ) q^{13} -2 q^{14} -\zeta_{12}^{2} q^{16} + ( -4 + 4 \zeta_{12}^{2} ) q^{17} + ( 2 - 4 \zeta_{12} + 2 \zeta_{12}^{2} ) q^{19} -\zeta_{12} q^{20} + ( -3 + 2 \zeta_{12} + 3 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{22} + ( -\zeta_{12} + 2 \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{23} - q^{25} + ( -3 \zeta_{12} - \zeta_{12}^{3} ) q^{26} + ( 2 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{28} + ( \zeta_{12} + 2 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{29} + ( -1 + 2 \zeta_{12}^{2} ) q^{31} + \zeta_{12} q^{32} -4 \zeta_{12}^{3} q^{34} + ( 2 - 2 \zeta_{12}^{2} ) q^{35} + ( 6 - 4 \zeta_{12} - 3 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{37} + ( 4 - 4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{38} + q^{40} + ( 2 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{41} + ( -5 + 4 \zeta_{12} + 5 \zeta_{12}^{2} - 8 \zeta_{12}^{3} ) q^{43} + ( -2 + 4 \zeta_{12}^{2} - 3 \zeta_{12}^{3} ) q^{44} + ( 1 - 2 \zeta_{12} + \zeta_{12}^{2} ) q^{46} + ( -2 + 4 \zeta_{12}^{2} + 7 \zeta_{12}^{3} ) q^{47} -3 \zeta_{12}^{2} q^{49} + ( \zeta_{12} - \zeta_{12}^{3} ) q^{50} + ( 3 + \zeta_{12}^{2} ) q^{52} + ( 6 + 8 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{53} + ( 2 \zeta_{12} - 3 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{55} + ( -2 + 2 \zeta_{12}^{2} ) q^{56} + ( -1 - 2 \zeta_{12} - \zeta_{12}^{2} ) q^{58} + ( -2 + 5 \zeta_{12} - 2 \zeta_{12}^{2} ) q^{59} + ( -6 \zeta_{12} + 12 \zeta_{12}^{3} ) q^{61} + ( -\zeta_{12} - \zeta_{12}^{3} ) q^{62} - q^{64} + ( 4 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{65} + ( -4 + 8 \zeta_{12} + 2 \zeta_{12}^{2} - 8 \zeta_{12}^{3} ) q^{67} + 4 \zeta_{12}^{2} q^{68} + 2 \zeta_{12}^{3} q^{70} + ( 6 - 2 \zeta_{12} + 6 \zeta_{12}^{2} ) q^{71} + 2 \zeta_{12}^{3} q^{73} + ( 4 - 3 \zeta_{12} - 4 \zeta_{12}^{2} + 6 \zeta_{12}^{3} ) q^{74} + ( 4 - 4 \zeta_{12} - 2 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{76} + ( 6 - 8 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{77} + ( -7 + 8 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{79} + ( -\zeta_{12} + \zeta_{12}^{3} ) q^{80} + ( -2 + 2 \zeta_{12}^{2} ) q^{82} + ( -4 + 8 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{83} + 4 \zeta_{12} q^{85} + ( -4 + 8 \zeta_{12}^{2} - 5 \zeta_{12}^{3} ) q^{86} + ( -2 \zeta_{12} + 3 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{88} + ( -4 - 4 \zeta_{12} + 2 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{89} + ( -2 \zeta_{12} + 8 \zeta_{12}^{3} ) q^{91} + ( 2 - 2 \zeta_{12} + \zeta_{12}^{3} ) q^{92} + ( -2 \zeta_{12} - 7 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{94} + ( -4 + 2 \zeta_{12} + 4 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{95} + ( -2 - 4 \zeta_{12} - 2 \zeta_{12}^{2} ) q^{97} + 3 \zeta_{12} q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 2q^{4} + O(q^{10})$$ $$4q + 2q^{4} + 2q^{10} - 12q^{11} + 4q^{13} - 8q^{14} - 2q^{16} - 8q^{17} + 12q^{19} - 6q^{22} + 4q^{23} - 4q^{25} + 4q^{29} + 4q^{35} + 18q^{37} + 16q^{38} + 4q^{40} - 10q^{43} + 6q^{46} - 6q^{49} + 14q^{52} + 24q^{53} - 6q^{55} - 4q^{56} - 6q^{58} - 12q^{59} - 4q^{64} - 12q^{67} + 8q^{68} + 36q^{71} + 8q^{74} + 12q^{76} + 24q^{77} - 28q^{79} - 4q^{82} + 6q^{88} - 12q^{89} + 8q^{92} - 14q^{94} - 8q^{95} - 12q^{97} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
361.1
 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 1.00000i 0 1.73205 + 1.00000i 1.00000i 0 0.500000 + 0.866025i
361.2 0.866025 0.500000i 0 0.500000 0.866025i 1.00000i 0 −1.73205 1.00000i 1.00000i 0 0.500000 + 0.866025i
901.1 −0.866025 0.500000i 0 0.500000 + 0.866025i 1.00000i 0 1.73205 1.00000i 1.00000i 0 0.500000 0.866025i
901.2 0.866025 + 0.500000i 0 0.500000 + 0.866025i 1.00000i 0 −1.73205 + 1.00000i 1.00000i 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.e 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.bs.e 4
3.b odd 2 1 390.2.bb.b 4
13.e even 6 1 inner 1170.2.bs.e 4
15.d odd 2 1 1950.2.bc.b 4
15.e even 4 1 1950.2.y.c 4
15.e even 4 1 1950.2.y.f 4
39.h odd 6 1 390.2.bb.b 4
39.h odd 6 1 5070.2.b.o 4
39.i odd 6 1 5070.2.b.o 4
39.k even 12 1 5070.2.a.y 2
39.k even 12 1 5070.2.a.bg 2
195.y odd 6 1 1950.2.bc.b 4
195.bf even 12 1 1950.2.y.c 4
195.bf even 12 1 1950.2.y.f 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.2.bb.b 4 3.b odd 2 1
390.2.bb.b 4 39.h odd 6 1
1170.2.bs.e 4 1.a even 1 1 trivial
1170.2.bs.e 4 13.e even 6 1 inner
1950.2.y.c 4 15.e even 4 1
1950.2.y.c 4 195.bf even 12 1
1950.2.y.f 4 15.e even 4 1
1950.2.y.f 4 195.bf even 12 1
1950.2.bc.b 4 15.d odd 2 1
1950.2.bc.b 4 195.y odd 6 1
5070.2.a.y 2 39.k even 12 1
5070.2.a.bg 2 39.k even 12 1
5070.2.b.o 4 39.h odd 6 1
5070.2.b.o 4 39.i odd 6 1

## Hecke kernels

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

 $$T_{7}^{4} - 4 T_{7}^{2} + 16$$ $$T_{11}^{4} + 12 T_{11}^{3} + 51 T_{11}^{2} + 36 T_{11} + 9$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T^{2} + T^{4}$$
$3$ $$T^{4}$$
$5$ $$( 1 + T^{2} )^{2}$$
$7$ $$16 - 4 T^{2} + T^{4}$$
$11$ $$9 + 36 T + 51 T^{2} + 12 T^{3} + T^{4}$$
$13$ $$( 13 - 2 T + T^{2} )^{2}$$
$17$ $$( 16 + 4 T + T^{2} )^{2}$$
$19$ $$16 + 48 T + 44 T^{2} - 12 T^{3} + T^{4}$$
$23$ $$1 - 4 T + 15 T^{2} - 4 T^{3} + T^{4}$$
$29$ $$1 - 4 T + 15 T^{2} - 4 T^{3} + T^{4}$$
$31$ $$( 3 + T^{2} )^{2}$$
$37$ $$121 - 198 T + 119 T^{2} - 18 T^{3} + T^{4}$$
$41$ $$16 - 4 T^{2} + T^{4}$$
$43$ $$529 - 230 T + 123 T^{2} + 10 T^{3} + T^{4}$$
$47$ $$1369 + 122 T^{2} + T^{4}$$
$53$ $$( -12 - 12 T + T^{2} )^{2}$$
$59$ $$169 - 156 T + 35 T^{2} + 12 T^{3} + T^{4}$$
$61$ $$11664 + 108 T^{2} + T^{4}$$
$67$ $$2704 - 624 T - 4 T^{2} + 12 T^{3} + T^{4}$$
$71$ $$10816 - 3744 T + 536 T^{2} - 36 T^{3} + T^{4}$$
$73$ $$( 4 + T^{2} )^{2}$$
$79$ $$( 1 + 14 T + T^{2} )^{2}$$
$83$ $$1936 + 104 T^{2} + T^{4}$$
$89$ $$16 - 48 T + 44 T^{2} + 12 T^{3} + T^{4}$$
$97$ $$16 - 48 T + 44 T^{2} + 12 T^{3} + T^{4}$$