# Properties

 Label 1170.2.bs.d 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: $$x^{4} - x^{2} + 1$$ 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.

 $$f(q)$$ $$=$$ $$q + (\zeta_{12}^{3} - \zeta_{12}) q^{2} + ( - \zeta_{12}^{2} + 1) q^{4} + \zeta_{12}^{3} q^{5} + 3 \zeta_{12} q^{7} + \zeta_{12}^{3} q^{8}+O(q^{10})$$ q + (z^3 - z) * q^2 + (-z^2 + 1) * q^4 + z^3 * q^5 + 3*z * q^7 + z^3 * q^8 $$q + (\zeta_{12}^{3} - \zeta_{12}) q^{2} + ( - \zeta_{12}^{2} + 1) q^{4} + \zeta_{12}^{3} q^{5} + 3 \zeta_{12} q^{7} + \zeta_{12}^{3} q^{8} - \zeta_{12}^{2} q^{10} + (2 \zeta_{12}^{3} - \zeta_{12}^{2} - 2 \zeta_{12} + 2) q^{11} + ( - 4 \zeta_{12}^{3} + \zeta_{12}) q^{13} - 3 q^{14} - \zeta_{12}^{2} q^{16} + ( - 4 \zeta_{12}^{2} + 4) q^{17} + (\zeta_{12}^{2} + 4 \zeta_{12} + 1) q^{19} + \zeta_{12} q^{20} + (2 \zeta_{12}^{3} - 2 \zeta_{12}^{2} - \zeta_{12} + 2) q^{22} + (2 \zeta_{12}^{3} + 2 \zeta_{12}) q^{23} - q^{25} + (4 \zeta_{12}^{2} - 1) q^{26} + ( - 3 \zeta_{12}^{3} + 3 \zeta_{12}) q^{28} + (2 \zeta_{12}^{3} - 2 \zeta_{12}^{2} + 2 \zeta_{12}) q^{29} + (2 \zeta_{12}^{3} - 8 \zeta_{12}^{2} + 4) q^{31} + \zeta_{12} q^{32} + 4 \zeta_{12}^{3} q^{34} + (3 \zeta_{12}^{2} - 3) q^{35} + ( - \zeta_{12}^{3} + 4 \zeta_{12}^{2} + \zeta_{12} - 8) q^{37} + (\zeta_{12}^{3} - 2 \zeta_{12} - 4) q^{38} - q^{40} + ( - 4 \zeta_{12}^{3} + 4 \zeta_{12}) q^{41} + (6 \zeta_{12}^{2} - 6) q^{43} + (2 \zeta_{12}^{3} - 2 \zeta_{12}^{2} + 1) q^{44} + ( - 2 \zeta_{12}^{2} - 2) q^{46} + ( - 3 \zeta_{12}^{3} - 4 \zeta_{12}^{2} + 2) q^{47} + 2 \zeta_{12}^{2} q^{49} + ( - \zeta_{12}^{3} + \zeta_{12}) q^{50} + ( - \zeta_{12}^{3} - 3 \zeta_{12}) q^{52} + (\zeta_{12}^{3} - 2 \zeta_{12} + 2) q^{53} + (\zeta_{12}^{3} - 2 \zeta_{12}^{2} + \zeta_{12}) q^{55} + (3 \zeta_{12}^{2} - 3) q^{56} + ( - 2 \zeta_{12}^{2} + 2 \zeta_{12} - 2) q^{58} + (2 \zeta_{12}^{2} + 8 \zeta_{12} + 2) q^{59} + (4 \zeta_{12}^{3} - 4 \zeta_{12}^{2} - 2 \zeta_{12} + 4) q^{61} + (4 \zeta_{12}^{3} - 2 \zeta_{12}^{2} + 4 \zeta_{12}) q^{62} - q^{64} + (\zeta_{12}^{2} + 3) q^{65} + (2 \zeta_{12}^{3} - 2 \zeta_{12}^{2} - 2 \zeta_{12} + 4) q^{67} - 4 \zeta_{12}^{2} q^{68} - 3 \zeta_{12}^{3} q^{70} + (4 \zeta_{12}^{2} + 6 \zeta_{12} + 4) q^{71} + (8 \zeta_{12}^{2} - 4) q^{73} + ( - 8 \zeta_{12}^{3} + \zeta_{12}^{2} + 4 \zeta_{12} - 1) q^{74} + ( - 4 \zeta_{12}^{3} - \zeta_{12}^{2} + 4 \zeta_{12} + 2) q^{76} + ( - 3 \zeta_{12}^{3} + 6 \zeta_{12} - 6) q^{77} + (4 \zeta_{12}^{3} - 8 \zeta_{12} + 10) q^{79} + ( - \zeta_{12}^{3} + \zeta_{12}) q^{80} + (4 \zeta_{12}^{2} - 4) q^{82} + (6 \zeta_{12}^{3} + 4 \zeta_{12}^{2} - 2) q^{83} + 4 \zeta_{12} q^{85} - 6 \zeta_{12}^{3} q^{86} + (\zeta_{12}^{3} - 2 \zeta_{12}^{2} + \zeta_{12}) q^{88} + ( - 2 \zeta_{12}^{3} - 7 \zeta_{12}^{2} + 2 \zeta_{12} + 14) q^{89} + ( - 9 \zeta_{12}^{2} + 12) q^{91} + ( - 2 \zeta_{12}^{3} + 4 \zeta_{12}) q^{92} + (2 \zeta_{12}^{3} + 3 \zeta_{12}^{2} + 2 \zeta_{12}) q^{94} + (2 \zeta_{12}^{3} + 4 \zeta_{12}^{2} - \zeta_{12} - 4) q^{95} + (6 \zeta_{12}^{2} - 2 \zeta_{12} + 6) q^{97} - 2 \zeta_{12} q^{98} +O(q^{100})$$ q + (z^3 - z) * q^2 + (-z^2 + 1) * q^4 + z^3 * q^5 + 3*z * q^7 + z^3 * q^8 - z^2 * q^10 + (2*z^3 - z^2 - 2*z + 2) * q^11 + (-4*z^3 + z) * q^13 - 3 * q^14 - z^2 * q^16 + (-4*z^2 + 4) * q^17 + (z^2 + 4*z + 1) * q^19 + z * q^20 + (2*z^3 - 2*z^2 - z + 2) * q^22 + (2*z^3 + 2*z) * q^23 - q^25 + (4*z^2 - 1) * q^26 + (-3*z^3 + 3*z) * q^28 + (2*z^3 - 2*z^2 + 2*z) * q^29 + (2*z^3 - 8*z^2 + 4) * q^31 + z * q^32 + 4*z^3 * q^34 + (3*z^2 - 3) * q^35 + (-z^3 + 4*z^2 + z - 8) * q^37 + (z^3 - 2*z - 4) * q^38 - q^40 + (-4*z^3 + 4*z) * q^41 + (6*z^2 - 6) * q^43 + (2*z^3 - 2*z^2 + 1) * q^44 + (-2*z^2 - 2) * q^46 + (-3*z^3 - 4*z^2 + 2) * q^47 + 2*z^2 * q^49 + (-z^3 + z) * q^50 + (-z^3 - 3*z) * q^52 + (z^3 - 2*z + 2) * q^53 + (z^3 - 2*z^2 + z) * q^55 + (3*z^2 - 3) * q^56 + (-2*z^2 + 2*z - 2) * q^58 + (2*z^2 + 8*z + 2) * q^59 + (4*z^3 - 4*z^2 - 2*z + 4) * q^61 + (4*z^3 - 2*z^2 + 4*z) * q^62 - q^64 + (z^2 + 3) * q^65 + (2*z^3 - 2*z^2 - 2*z + 4) * q^67 - 4*z^2 * q^68 - 3*z^3 * q^70 + (4*z^2 + 6*z + 4) * q^71 + (8*z^2 - 4) * q^73 + (-8*z^3 + z^2 + 4*z - 1) * q^74 + (-4*z^3 - z^2 + 4*z + 2) * q^76 + (-3*z^3 + 6*z - 6) * q^77 + (4*z^3 - 8*z + 10) * q^79 + (-z^3 + z) * q^80 + (4*z^2 - 4) * q^82 + (6*z^3 + 4*z^2 - 2) * q^83 + 4*z * q^85 - 6*z^3 * q^86 + (z^3 - 2*z^2 + z) * q^88 + (-2*z^3 - 7*z^2 + 2*z + 14) * q^89 + (-9*z^2 + 12) * q^91 + (-2*z^3 + 4*z) * q^92 + (2*z^3 + 3*z^2 + 2*z) * q^94 + (2*z^3 + 4*z^2 - z - 4) * q^95 + (6*z^2 - 2*z + 6) * q^97 - 2*z * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 2 q^{4}+O(q^{10})$$ 4 * q + 2 * q^4 $$4 q + 2 q^{4} - 2 q^{10} + 6 q^{11} - 12 q^{14} - 2 q^{16} + 8 q^{17} + 6 q^{19} + 4 q^{22} - 4 q^{25} + 4 q^{26} - 4 q^{29} - 6 q^{35} - 24 q^{37} - 16 q^{38} - 4 q^{40} - 12 q^{43} - 12 q^{46} + 4 q^{49} + 8 q^{53} - 4 q^{55} - 6 q^{56} - 12 q^{58} + 12 q^{59} + 8 q^{61} - 4 q^{62} - 4 q^{64} + 14 q^{65} + 12 q^{67} - 8 q^{68} + 24 q^{71} - 2 q^{74} + 6 q^{76} - 24 q^{77} + 40 q^{79} - 8 q^{82} - 4 q^{88} + 42 q^{89} + 30 q^{91} + 6 q^{94} - 8 q^{95} + 36 q^{97}+O(q^{100})$$ 4 * q + 2 * q^4 - 2 * q^10 + 6 * q^11 - 12 * q^14 - 2 * q^16 + 8 * q^17 + 6 * q^19 + 4 * q^22 - 4 * q^25 + 4 * q^26 - 4 * q^29 - 6 * q^35 - 24 * q^37 - 16 * q^38 - 4 * q^40 - 12 * q^43 - 12 * q^46 + 4 * q^49 + 8 * q^53 - 4 * q^55 - 6 * q^56 - 12 * q^58 + 12 * q^59 + 8 * q^61 - 4 * q^62 - 4 * q^64 + 14 * q^65 + 12 * q^67 - 8 * q^68 + 24 * q^71 - 2 * q^74 + 6 * q^76 - 24 * q^77 + 40 * q^79 - 8 * q^82 - 4 * q^88 + 42 * q^89 + 30 * q^91 + 6 * q^94 - 8 * q^95 + 36 * q^97

## 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 2.59808 + 1.50000i 1.00000i 0 −0.500000 0.866025i
361.2 0.866025 0.500000i 0 0.500000 0.866025i 1.00000i 0 −2.59808 1.50000i 1.00000i 0 −0.500000 0.866025i
901.1 −0.866025 0.500000i 0 0.500000 + 0.866025i 1.00000i 0 2.59808 1.50000i 1.00000i 0 −0.500000 + 0.866025i
901.2 0.866025 + 0.500000i 0 0.500000 + 0.866025i 1.00000i 0 −2.59808 + 1.50000i 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.d 4
3.b odd 2 1 390.2.bb.a 4
13.e even 6 1 inner 1170.2.bs.d 4
15.d odd 2 1 1950.2.bc.a 4
15.e even 4 1 1950.2.y.d 4
15.e even 4 1 1950.2.y.e 4
39.h odd 6 1 390.2.bb.a 4
39.h odd 6 1 5070.2.b.p 4
39.i odd 6 1 5070.2.b.p 4
39.k even 12 1 5070.2.a.ba 2
39.k even 12 1 5070.2.a.be 2
195.y odd 6 1 1950.2.bc.a 4
195.bf even 12 1 1950.2.y.d 4
195.bf even 12 1 1950.2.y.e 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.2.bb.a 4 3.b odd 2 1
390.2.bb.a 4 39.h odd 6 1
1170.2.bs.d 4 1.a even 1 1 trivial
1170.2.bs.d 4 13.e even 6 1 inner
1950.2.y.d 4 15.e even 4 1
1950.2.y.d 4 195.bf even 12 1
1950.2.y.e 4 15.e even 4 1
1950.2.y.e 4 195.bf even 12 1
1950.2.bc.a 4 15.d odd 2 1
1950.2.bc.a 4 195.y odd 6 1
5070.2.a.ba 2 39.k even 12 1
5070.2.a.be 2 39.k even 12 1
5070.2.b.p 4 39.h odd 6 1
5070.2.b.p 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 $$S_{2}^{\mathrm{new}}(1170, [\chi])$$:

 $$T_{7}^{4} - 9T_{7}^{2} + 81$$ T7^4 - 9*T7^2 + 81 $$T_{11}^{4} - 6T_{11}^{3} + 11T_{11}^{2} + 6T_{11} + 1$$ T11^4 - 6*T11^3 + 11*T11^2 + 6*T11 + 1

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - T^{2} + 1$$
$3$ $$T^{4}$$
$5$ $$(T^{2} + 1)^{2}$$
$7$ $$T^{4} - 9T^{2} + 81$$
$11$ $$T^{4} - 6 T^{3} + 11 T^{2} + 6 T + 1$$
$13$ $$T^{4} + 23T^{2} + 169$$
$17$ $$(T^{2} - 4 T + 16)^{2}$$
$19$ $$T^{4} - 6 T^{3} - T^{2} + 78 T + 169$$
$23$ $$T^{4} + 12T^{2} + 144$$
$29$ $$T^{4} + 4 T^{3} + 24 T^{2} - 32 T + 64$$
$31$ $$T^{4} + 104T^{2} + 1936$$
$37$ $$T^{4} + 24 T^{3} + 239 T^{2} + \cdots + 2209$$
$41$ $$T^{4} - 16T^{2} + 256$$
$43$ $$(T^{2} + 6 T + 36)^{2}$$
$47$ $$T^{4} + 42T^{2} + 9$$
$53$ $$(T^{2} - 4 T + 1)^{2}$$
$59$ $$T^{4} - 12 T^{3} - 4 T^{2} + \cdots + 2704$$
$61$ $$T^{4} - 8 T^{3} + 60 T^{2} - 32 T + 16$$
$67$ $$T^{4} - 12 T^{3} + 56 T^{2} - 96 T + 64$$
$71$ $$T^{4} - 24 T^{3} + 204 T^{2} + \cdots + 144$$
$73$ $$(T^{2} + 48)^{2}$$
$79$ $$(T^{2} - 20 T + 52)^{2}$$
$83$ $$T^{4} + 96T^{2} + 576$$
$89$ $$T^{4} - 42 T^{3} + 731 T^{2} + \cdots + 20449$$
$97$ $$T^{4} - 36 T^{3} + 536 T^{2} + \cdots + 10816$$