# Properties

 Label 1170.2.bs.c 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 130) 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} + (3 \zeta_{12}^{3} - 3 \zeta_{12}) q^{11} + (\zeta_{12}^{2} + 3) q^{13} + 3 q^{14} - \zeta_{12}^{2} q^{16} + ( - 6 \zeta_{12}^{3} - 3 \zeta_{12}^{2} + 3 \zeta_{12} + 3) q^{17} + ( - 2 \zeta_{12}^{2} + 3 \zeta_{12} - 2) q^{19} - \zeta_{12} q^{20} + (3 \zeta_{12}^{2} - 3) q^{22} + (2 \zeta_{12}^{3} + 6 \zeta_{12}^{2} + 2 \zeta_{12}) q^{23} - q^{25} + ( - 3 \zeta_{12}^{3} + 4 \zeta_{12}) q^{26} + ( - 3 \zeta_{12}^{3} + 3 \zeta_{12}) q^{28} + (2 \zeta_{12}^{3} - 6 \zeta_{12}^{2} + 2 \zeta_{12}) q^{29} + ( - 3 \zeta_{12}^{3} - 2 \zeta_{12}^{2} + 1) q^{31} - \zeta_{12} q^{32} + ( - 3 \zeta_{12}^{3} - 6 \zeta_{12}^{2} + 3) q^{34} + ( - 3 \zeta_{12}^{2} + 3) q^{35} + (6 \zeta_{12}^{3} - 3 \zeta_{12}^{2} - 6 \zeta_{12} + 6) q^{37} + (2 \zeta_{12}^{3} - 4 \zeta_{12} + 3) q^{38} - q^{40} + ( - 6 \zeta_{12}^{2} + 12) q^{41} + (2 \zeta_{12}^{2} - 2) q^{43} + 3 \zeta_{12}^{3} q^{44} + (2 \zeta_{12}^{2} + 6 \zeta_{12} + 2) q^{46} + 3 \zeta_{12}^{3} q^{47} + 2 \zeta_{12}^{2} q^{49} + (\zeta_{12}^{3} - \zeta_{12}) q^{50} + ( - 3 \zeta_{12}^{2} + 4) q^{52} + (2 \zeta_{12}^{3} - 4 \zeta_{12} + 3) q^{53} + 3 \zeta_{12}^{2} q^{55} + ( - 3 \zeta_{12}^{2} + 3) q^{56} + (2 \zeta_{12}^{2} - 6 \zeta_{12} + 2) q^{58} + ( - 6 \zeta_{12}^{2} - 6) q^{59} + (6 \zeta_{12}^{3} + \zeta_{12}^{2} - 3 \zeta_{12} - 1) q^{61} + ( - \zeta_{12}^{3} - 3 \zeta_{12}^{2} - \zeta_{12}) q^{62} - q^{64} + ( - 4 \zeta_{12}^{3} + \zeta_{12}) q^{65} + ( - 3 \zeta_{12}^{3} - 3 \zeta_{12}^{2} - 3 \zeta_{12}) q^{68} - 3 \zeta_{12}^{3} q^{70} + 6 \zeta_{12} q^{71} + ( - 3 \zeta_{12}^{3} - 10 \zeta_{12}^{2} + 5) q^{73} + ( - 6 \zeta_{12}^{3} + 6 \zeta_{12}^{2} + 3 \zeta_{12} - 6) q^{74} + ( - 3 \zeta_{12}^{3} + 2 \zeta_{12}^{2} + 3 \zeta_{12} - 4) q^{76} - 9 q^{77} + (3 \zeta_{12}^{3} - 6 \zeta_{12} + 1) q^{79} + (\zeta_{12}^{3} - \zeta_{12}) q^{80} + ( - 12 \zeta_{12}^{3} + 6 \zeta_{12}) q^{82} + ( - 3 \zeta_{12}^{3} - 6 \zeta_{12}^{2} + 3) q^{83} + ( - 3 \zeta_{12}^{2} - 3 \zeta_{12} - 3) q^{85} + 2 \zeta_{12}^{3} q^{86} + 3 \zeta_{12}^{2} q^{88} + (12 \zeta_{12}^{3} - 3 \zeta_{12}^{2} - 12 \zeta_{12} + 6) q^{89} + (3 \zeta_{12}^{3} + 9 \zeta_{12}) q^{91} + ( - 2 \zeta_{12}^{3} + 4 \zeta_{12} + 6) q^{92} + 3 \zeta_{12}^{2} q^{94} + (4 \zeta_{12}^{3} - 3 \zeta_{12}^{2} - 2 \zeta_{12} + 3) q^{95} + (7 \zeta_{12}^{2} - 3 \zeta_{12} + 7) 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 + (3*z^3 - 3*z) * q^11 + (z^2 + 3) * q^13 + 3 * q^14 - z^2 * q^16 + (-6*z^3 - 3*z^2 + 3*z + 3) * q^17 + (-2*z^2 + 3*z - 2) * q^19 - z * q^20 + (3*z^2 - 3) * q^22 + (2*z^3 + 6*z^2 + 2*z) * q^23 - q^25 + (-3*z^3 + 4*z) * q^26 + (-3*z^3 + 3*z) * q^28 + (2*z^3 - 6*z^2 + 2*z) * q^29 + (-3*z^3 - 2*z^2 + 1) * q^31 - z * q^32 + (-3*z^3 - 6*z^2 + 3) * q^34 + (-3*z^2 + 3) * q^35 + (6*z^3 - 3*z^2 - 6*z + 6) * q^37 + (2*z^3 - 4*z + 3) * q^38 - q^40 + (-6*z^2 + 12) * q^41 + (2*z^2 - 2) * q^43 + 3*z^3 * q^44 + (2*z^2 + 6*z + 2) * q^46 + 3*z^3 * q^47 + 2*z^2 * q^49 + (z^3 - z) * q^50 + (-3*z^2 + 4) * q^52 + (2*z^3 - 4*z + 3) * q^53 + 3*z^2 * q^55 + (-3*z^2 + 3) * q^56 + (2*z^2 - 6*z + 2) * q^58 + (-6*z^2 - 6) * q^59 + (6*z^3 + z^2 - 3*z - 1) * q^61 + (-z^3 - 3*z^2 - z) * q^62 - q^64 + (-4*z^3 + z) * q^65 + (-3*z^3 - 3*z^2 - 3*z) * q^68 - 3*z^3 * q^70 + 6*z * q^71 + (-3*z^3 - 10*z^2 + 5) * q^73 + (-6*z^3 + 6*z^2 + 3*z - 6) * q^74 + (-3*z^3 + 2*z^2 + 3*z - 4) * q^76 - 9 * q^77 + (3*z^3 - 6*z + 1) * q^79 + (z^3 - z) * q^80 + (-12*z^3 + 6*z) * q^82 + (-3*z^3 - 6*z^2 + 3) * q^83 + (-3*z^2 - 3*z - 3) * q^85 + 2*z^3 * q^86 + 3*z^2 * q^88 + (12*z^3 - 3*z^2 - 12*z + 6) * q^89 + (3*z^3 + 9*z) * q^91 + (-2*z^3 + 4*z + 6) * q^92 + 3*z^2 * q^94 + (4*z^3 - 3*z^2 - 2*z + 3) * q^95 + (7*z^2 - 3*z + 7) * q^97 + 2*z * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 2 q^{4}+O(q^{10})$$ 4 * q + 2 * q^4 $$4 q + 2 q^{4} - 2 q^{10} + 14 q^{13} + 12 q^{14} - 2 q^{16} + 6 q^{17} - 12 q^{19} - 6 q^{22} + 12 q^{23} - 4 q^{25} - 12 q^{29} + 6 q^{35} + 18 q^{37} + 12 q^{38} - 4 q^{40} + 36 q^{41} - 4 q^{43} + 12 q^{46} + 4 q^{49} + 10 q^{52} + 12 q^{53} + 6 q^{55} + 6 q^{56} + 12 q^{58} - 36 q^{59} - 2 q^{61} - 6 q^{62} - 4 q^{64} - 6 q^{68} - 12 q^{74} - 12 q^{76} - 36 q^{77} + 4 q^{79} - 18 q^{85} + 6 q^{88} + 18 q^{89} + 24 q^{92} + 6 q^{94} + 6 q^{95} + 42 q^{97}+O(q^{100})$$ 4 * q + 2 * q^4 - 2 * q^10 + 14 * q^13 + 12 * q^14 - 2 * q^16 + 6 * q^17 - 12 * q^19 - 6 * q^22 + 12 * q^23 - 4 * q^25 - 12 * q^29 + 6 * q^35 + 18 * q^37 + 12 * q^38 - 4 * q^40 + 36 * q^41 - 4 * q^43 + 12 * q^46 + 4 * q^49 + 10 * q^52 + 12 * q^53 + 6 * q^55 + 6 * q^56 + 12 * q^58 - 36 * q^59 - 2 * q^61 - 6 * q^62 - 4 * q^64 - 6 * q^68 - 12 * q^74 - 12 * q^76 - 36 * q^77 + 4 * q^79 - 18 * q^85 + 6 * q^88 + 18 * q^89 + 24 * q^92 + 6 * q^94 + 6 * q^95 + 42 * 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.c 4
3.b odd 2 1 130.2.l.a 4
12.b even 2 1 1040.2.da.a 4
13.e even 6 1 inner 1170.2.bs.c 4
15.d odd 2 1 650.2.m.a 4
15.e even 4 1 650.2.n.a 4
15.e even 4 1 650.2.n.b 4
39.d odd 2 1 1690.2.l.g 4
39.f even 4 1 1690.2.e.l 4
39.f even 4 1 1690.2.e.n 4
39.h odd 6 1 130.2.l.a 4
39.h odd 6 1 1690.2.d.f 4
39.i odd 6 1 1690.2.d.f 4
39.i odd 6 1 1690.2.l.g 4
39.k even 12 1 1690.2.a.j 2
39.k even 12 1 1690.2.a.m 2
39.k even 12 1 1690.2.e.l 4
39.k even 12 1 1690.2.e.n 4
156.r even 6 1 1040.2.da.a 4
195.y odd 6 1 650.2.m.a 4
195.bf even 12 1 650.2.n.a 4
195.bf even 12 1 650.2.n.b 4
195.bh even 12 1 8450.2.a.bf 2
195.bh even 12 1 8450.2.a.bm 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
130.2.l.a 4 3.b odd 2 1
130.2.l.a 4 39.h odd 6 1
650.2.m.a 4 15.d odd 2 1
650.2.m.a 4 195.y odd 6 1
650.2.n.a 4 15.e even 4 1
650.2.n.a 4 195.bf even 12 1
650.2.n.b 4 15.e even 4 1
650.2.n.b 4 195.bf even 12 1
1040.2.da.a 4 12.b even 2 1
1040.2.da.a 4 156.r even 6 1
1170.2.bs.c 4 1.a even 1 1 trivial
1170.2.bs.c 4 13.e even 6 1 inner
1690.2.a.j 2 39.k even 12 1
1690.2.a.m 2 39.k even 12 1
1690.2.d.f 4 39.h odd 6 1
1690.2.d.f 4 39.i odd 6 1
1690.2.e.l 4 39.f even 4 1
1690.2.e.l 4 39.k even 12 1
1690.2.e.n 4 39.f even 4 1
1690.2.e.n 4 39.k even 12 1
1690.2.l.g 4 39.d odd 2 1
1690.2.l.g 4 39.i odd 6 1
8450.2.a.bf 2 195.bh even 12 1
8450.2.a.bm 2 195.bh even 12 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 $$T_{11}^{4} - 9T_{11}^{2} + 81$$ T11^4 - 9*T11^2 + 81

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - T^{2} + 1$$
$3$ $$T^{4}$$
$5$ $$(T^{2} + 1)^{2}$$
$7$ $$T^{4} - 9T^{2} + 81$$
$11$ $$T^{4} - 9T^{2} + 81$$
$13$ $$(T^{2} - 7 T + 13)^{2}$$
$17$ $$T^{4} - 6 T^{3} + 54 T^{2} + 108 T + 324$$
$19$ $$T^{4} + 12 T^{3} + 51 T^{2} + 36 T + 9$$
$23$ $$T^{4} - 12 T^{3} + 120 T^{2} + \cdots + 576$$
$29$ $$T^{4} + 12 T^{3} + 120 T^{2} + \cdots + 576$$
$31$ $$T^{4} + 24T^{2} + 36$$
$37$ $$T^{4} - 18 T^{3} + 99 T^{2} + 162 T + 81$$
$41$ $$(T^{2} - 18 T + 108)^{2}$$
$43$ $$(T^{2} + 2 T + 4)^{2}$$
$47$ $$(T^{2} + 9)^{2}$$
$53$ $$(T^{2} - 6 T - 3)^{2}$$
$59$ $$(T^{2} + 18 T + 108)^{2}$$
$61$ $$T^{4} + 2 T^{3} + 30 T^{2} - 52 T + 676$$
$67$ $$T^{4}$$
$71$ $$T^{4} - 36T^{2} + 1296$$
$73$ $$T^{4} + 168T^{2} + 4356$$
$79$ $$(T^{2} - 2 T - 26)^{2}$$
$83$ $$T^{4} + 72T^{2} + 324$$
$89$ $$T^{4} - 18 T^{3} - 9 T^{2} + \cdots + 13689$$
$97$ $$T^{4} - 42 T^{3} + 726 T^{2} + \cdots + 19044$$