# Properties

 Label 1170.2.i.o Level $1170$ Weight $2$ Character orbit 1170.i 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.i (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$9.34249703649$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{17})$$ Defining polynomial: $$x^{4} - x^{3} + 5 x^{2} + 4 x + 16$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ 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 basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 1 - \beta_{2} ) q^{2} -\beta_{2} q^{4} - q^{5} + ( -\beta_{1} - \beta_{2} ) q^{7} - q^{8} +O(q^{10})$$ $$q + ( 1 - \beta_{2} ) q^{2} -\beta_{2} q^{4} - q^{5} + ( -\beta_{1} - \beta_{2} ) q^{7} - q^{8} + ( -1 + \beta_{2} ) q^{10} + ( 1 + 2 \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{11} + ( -\beta_{1} + \beta_{2} + \beta_{3} ) q^{13} + ( -1 + \beta_{3} ) q^{14} + ( -1 + \beta_{2} ) q^{16} + 2 \beta_{1} q^{17} + ( \beta_{1} + \beta_{2} ) q^{19} + \beta_{2} q^{20} + ( 2 \beta_{1} - \beta_{2} ) q^{22} + ( 3 \beta_{1} + 3 \beta_{3} ) q^{23} + q^{25} + ( 1 + \beta_{1} + 2 \beta_{3} ) q^{26} + ( -1 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{28} + ( 4 - \beta_{1} - 4 \beta_{2} - \beta_{3} ) q^{29} + ( -2 - 3 \beta_{3} ) q^{31} + \beta_{2} q^{32} -2 \beta_{3} q^{34} + ( \beta_{1} + \beta_{2} ) q^{35} + ( 1 + 2 \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{37} + ( 1 - \beta_{3} ) q^{38} + q^{40} + ( 6 + 4 \beta_{1} - 6 \beta_{2} + 4 \beta_{3} ) q^{41} + ( -\beta_{1} - 2 \beta_{2} ) q^{43} + ( -1 - 2 \beta_{3} ) q^{44} + 3 \beta_{1} q^{46} -7 q^{47} + ( 2 + 3 \beta_{1} - 2 \beta_{2} + 3 \beta_{3} ) q^{49} + ( 1 - \beta_{2} ) q^{50} + ( 1 + 2 \beta_{1} - \beta_{2} + \beta_{3} ) q^{52} + ( -7 - \beta_{3} ) q^{53} + ( -1 - 2 \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{55} + ( \beta_{1} + \beta_{2} ) q^{56} + ( -\beta_{1} - 4 \beta_{2} ) q^{58} + ( \beta_{1} + 8 \beta_{2} ) q^{59} -6 \beta_{2} q^{61} + ( -2 - 3 \beta_{1} + 2 \beta_{2} - 3 \beta_{3} ) q^{62} + q^{64} + ( \beta_{1} - \beta_{2} - \beta_{3} ) q^{65} + ( -4 + 4 \beta_{1} + 4 \beta_{2} + 4 \beta_{3} ) q^{67} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{68} + ( 1 - \beta_{3} ) q^{70} + ( 2 \beta_{1} - 10 \beta_{2} ) q^{71} + 6 \beta_{3} q^{73} + ( 2 \beta_{1} - \beta_{2} ) q^{74} + ( 1 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{76} + ( 7 - 3 \beta_{3} ) q^{77} + ( 10 + \beta_{3} ) q^{79} + ( 1 - \beta_{2} ) q^{80} + ( 4 \beta_{1} - 6 \beta_{2} ) q^{82} + ( 4 + 2 \beta_{3} ) q^{83} -2 \beta_{1} q^{85} + ( -2 + \beta_{3} ) q^{86} + ( -1 - 2 \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{88} + ( -11 - 5 \beta_{1} + 11 \beta_{2} - 5 \beta_{3} ) q^{89} + ( -3 + 3 \beta_{1} + 7 \beta_{2} + \beta_{3} ) q^{91} -3 \beta_{3} q^{92} + ( -7 + 7 \beta_{2} ) q^{94} + ( -\beta_{1} - \beta_{2} ) q^{95} + ( -2 \beta_{1} + 4 \beta_{2} ) q^{97} + ( 3 \beta_{1} - 2 \beta_{2} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 2q^{2} - 2q^{4} - 4q^{5} - 3q^{7} - 4q^{8} + O(q^{10})$$ $$4q + 2q^{2} - 2q^{4} - 4q^{5} - 3q^{7} - 4q^{8} - 2q^{10} - q^{13} - 6q^{14} - 2q^{16} + 2q^{17} + 3q^{19} + 2q^{20} - 3q^{23} + 4q^{25} + q^{26} - 3q^{28} + 9q^{29} - 2q^{31} + 2q^{32} + 4q^{34} + 3q^{35} + 6q^{38} + 4q^{40} + 8q^{41} - 5q^{43} + 3q^{46} - 28q^{47} + q^{49} + 2q^{50} + 2q^{52} - 26q^{53} + 3q^{56} - 9q^{58} + 17q^{59} - 12q^{61} - q^{62} + 4q^{64} + q^{65} - 12q^{67} + 2q^{68} + 6q^{70} - 18q^{71} - 12q^{73} + 3q^{76} + 34q^{77} + 38q^{79} + 2q^{80} - 8q^{82} + 12q^{83} - 2q^{85} - 10q^{86} - 17q^{89} + 3q^{91} + 6q^{92} - 14q^{94} - 3q^{95} + 6q^{97} - q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} + 5 x^{2} + 4 x + 16$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{3} + 5 \nu^{2} - 5 \nu + 16$$$$)/20$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{3} + 4$$$$)/5$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{3} + 4 \beta_{2} + \beta_{1} - 4$$ $$\nu^{3}$$ $$=$$ $$5 \beta_{3} - 4$$

## Character values

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

 $$n$$ $$911$$ $$937$$ $$1081$$ $$\chi(n)$$ $$1$$ $$1$$ $$-\beta_{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
451.1
 1.28078 + 2.21837i −0.780776 − 1.35234i 1.28078 − 2.21837i −0.780776 + 1.35234i
0.500000 0.866025i 0 −0.500000 0.866025i −1.00000 0 −1.78078 3.08440i −1.00000 0 −0.500000 + 0.866025i
451.2 0.500000 0.866025i 0 −0.500000 0.866025i −1.00000 0 0.280776 + 0.486319i −1.00000 0 −0.500000 + 0.866025i
991.1 0.500000 + 0.866025i 0 −0.500000 + 0.866025i −1.00000 0 −1.78078 + 3.08440i −1.00000 0 −0.500000 0.866025i
991.2 0.500000 + 0.866025i 0 −0.500000 + 0.866025i −1.00000 0 0.280776 0.486319i −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.o 4
3.b odd 2 1 390.2.i.g 4
13.c even 3 1 inner 1170.2.i.o 4
15.d odd 2 1 1950.2.i.bi 4
15.e even 4 2 1950.2.z.n 8
39.h odd 6 1 5070.2.a.bb 2
39.i odd 6 1 390.2.i.g 4
39.i odd 6 1 5070.2.a.bi 2
39.k even 12 2 5070.2.b.r 4
195.x odd 6 1 1950.2.i.bi 4
195.bl even 12 2 1950.2.z.n 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.2.i.g 4 3.b odd 2 1
390.2.i.g 4 39.i odd 6 1
1170.2.i.o 4 1.a even 1 1 trivial
1170.2.i.o 4 13.c even 3 1 inner
1950.2.i.bi 4 15.d odd 2 1
1950.2.i.bi 4 195.x odd 6 1
1950.2.z.n 8 15.e even 4 2
1950.2.z.n 8 195.bl even 12 2
5070.2.a.bb 2 39.h odd 6 1
5070.2.a.bi 2 39.i odd 6 1
5070.2.b.r 4 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 $$S_{2}^{\mathrm{new}}(1170, [\chi])$$:

 $$T_{7}^{4} + 3 T_{7}^{3} + 11 T_{7}^{2} - 6 T_{7} + 4$$ $$T_{11}^{4} + 17 T_{11}^{2} + 289$$ $$T_{29}^{4} - 9 T_{29}^{3} + 65 T_{29}^{2} - 144 T_{29} + 256$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 - T + T^{2} )^{2}$$
$3$ $$T^{4}$$
$5$ $$( 1 + T )^{4}$$
$7$ $$4 - 6 T + 11 T^{2} + 3 T^{3} + T^{4}$$
$11$ $$289 + 17 T^{2} + T^{4}$$
$13$ $$169 + 13 T - 12 T^{2} + T^{3} + T^{4}$$
$17$ $$256 + 32 T + 20 T^{2} - 2 T^{3} + T^{4}$$
$19$ $$4 + 6 T + 11 T^{2} - 3 T^{3} + T^{4}$$
$23$ $$1296 - 108 T + 45 T^{2} + 3 T^{3} + T^{4}$$
$29$ $$256 - 144 T + 65 T^{2} - 9 T^{3} + T^{4}$$
$31$ $$( -38 + T + T^{2} )^{2}$$
$37$ $$289 + 17 T^{2} + T^{4}$$
$41$ $$2704 + 416 T + 116 T^{2} - 8 T^{3} + T^{4}$$
$43$ $$4 + 10 T + 23 T^{2} + 5 T^{3} + T^{4}$$
$47$ $$( 7 + T )^{4}$$
$53$ $$( 38 + 13 T + T^{2} )^{2}$$
$59$ $$4624 - 1156 T + 221 T^{2} - 17 T^{3} + T^{4}$$
$61$ $$( 36 + 6 T + T^{2} )^{2}$$
$67$ $$1024 - 384 T + 176 T^{2} + 12 T^{3} + T^{4}$$
$71$ $$4096 + 1152 T + 260 T^{2} + 18 T^{3} + T^{4}$$
$73$ $$( -144 + 6 T + T^{2} )^{2}$$
$79$ $$( 86 - 19 T + T^{2} )^{2}$$
$83$ $$( -8 - 6 T + T^{2} )^{2}$$
$89$ $$1156 - 578 T + 323 T^{2} + 17 T^{3} + T^{4}$$
$97$ $$64 + 48 T + 44 T^{2} - 6 T^{3} + T^{4}$$