# Properties

 Label 189.3.d.a Level $189$ Weight $3$ Character orbit 189.d Analytic conductor $5.150$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$189 = 3^{3} \cdot 7$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 189.d (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$5.14987699641$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{2} -3 q^{4} + ( 1 - 2 \zeta_{6} ) q^{5} + ( -7 + 7 \zeta_{6} ) q^{7} + 7 q^{8} +O(q^{10})$$ $$q - q^{2} -3 q^{4} + ( 1 - 2 \zeta_{6} ) q^{5} + ( -7 + 7 \zeta_{6} ) q^{7} + 7 q^{8} + ( -1 + 2 \zeta_{6} ) q^{10} + 17 q^{11} + ( 10 - 20 \zeta_{6} ) q^{13} + ( 7 - 7 \zeta_{6} ) q^{14} + 5 q^{16} + ( 18 - 36 \zeta_{6} ) q^{17} + ( -12 + 24 \zeta_{6} ) q^{19} + ( -3 + 6 \zeta_{6} ) q^{20} -17 q^{22} + 32 q^{23} + 22 q^{25} + ( -10 + 20 \zeta_{6} ) q^{26} + ( 21 - 21 \zeta_{6} ) q^{28} + 2 q^{29} + ( 13 - 26 \zeta_{6} ) q^{31} -33 q^{32} + ( -18 + 36 \zeta_{6} ) q^{34} + ( 7 + 7 \zeta_{6} ) q^{35} + 8 q^{37} + ( 12 - 24 \zeta_{6} ) q^{38} + ( 7 - 14 \zeta_{6} ) q^{40} + ( -8 + 16 \zeta_{6} ) q^{41} + 26 q^{43} -51 q^{44} -32 q^{46} + ( -22 + 44 \zeta_{6} ) q^{47} -49 \zeta_{6} q^{49} -22 q^{50} + ( -30 + 60 \zeta_{6} ) q^{52} - q^{53} + ( 17 - 34 \zeta_{6} ) q^{55} + ( -49 + 49 \zeta_{6} ) q^{56} -2 q^{58} + ( 52 - 104 \zeta_{6} ) q^{59} + ( -40 + 80 \zeta_{6} ) q^{61} + ( -13 + 26 \zeta_{6} ) q^{62} + 13 q^{64} -30 q^{65} -34 q^{67} + ( -54 + 108 \zeta_{6} ) q^{68} + ( -7 - 7 \zeta_{6} ) q^{70} -10 q^{71} + ( -33 + 66 \zeta_{6} ) q^{73} -8 q^{74} + ( 36 - 72 \zeta_{6} ) q^{76} + ( -119 + 119 \zeta_{6} ) q^{77} -46 q^{79} + ( 5 - 10 \zeta_{6} ) q^{80} + ( 8 - 16 \zeta_{6} ) q^{82} + ( 59 - 118 \zeta_{6} ) q^{83} -54 q^{85} -26 q^{86} + 119 q^{88} + ( -30 + 60 \zeta_{6} ) q^{89} + ( 70 + 70 \zeta_{6} ) q^{91} -96 q^{92} + ( 22 - 44 \zeta_{6} ) q^{94} + 36 q^{95} + ( 53 - 106 \zeta_{6} ) q^{97} + 49 \zeta_{6} q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{2} - 6q^{4} - 7q^{7} + 14q^{8} + O(q^{10})$$ $$2q - 2q^{2} - 6q^{4} - 7q^{7} + 14q^{8} + 34q^{11} + 7q^{14} + 10q^{16} - 34q^{22} + 64q^{23} + 44q^{25} + 21q^{28} + 4q^{29} - 66q^{32} + 21q^{35} + 16q^{37} + 52q^{43} - 102q^{44} - 64q^{46} - 49q^{49} - 44q^{50} - 2q^{53} - 49q^{56} - 4q^{58} + 26q^{64} - 60q^{65} - 68q^{67} - 21q^{70} - 20q^{71} - 16q^{74} - 119q^{77} - 92q^{79} - 108q^{85} - 52q^{86} + 238q^{88} + 210q^{91} - 192q^{92} + 72q^{95} + 49q^{98} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/189\mathbb{Z}\right)^\times$$.

 $$n$$ $$29$$ $$136$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
55.1
 0.5 + 0.866025i 0.5 − 0.866025i
−1.00000 0 −3.00000 1.73205i 0 −3.50000 + 6.06218i 7.00000 0 1.73205i
55.2 −1.00000 0 −3.00000 1.73205i 0 −3.50000 6.06218i 7.00000 0 1.73205i
 $$n$$: 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
7.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 189.3.d.a 2
3.b odd 2 1 189.3.d.c yes 2
7.b odd 2 1 inner 189.3.d.a 2
21.c even 2 1 189.3.d.c yes 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
189.3.d.a 2 1.a even 1 1 trivial
189.3.d.a 2 7.b odd 2 1 inner
189.3.d.c yes 2 3.b odd 2 1
189.3.d.c yes 2 21.c even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2} + 1$$ acting on $$S_{3}^{\mathrm{new}}(189, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T )^{2}$$
$3$ $$T^{2}$$
$5$ $$3 + T^{2}$$
$7$ $$49 + 7 T + T^{2}$$
$11$ $$( -17 + T )^{2}$$
$13$ $$300 + T^{2}$$
$17$ $$972 + T^{2}$$
$19$ $$432 + T^{2}$$
$23$ $$( -32 + T )^{2}$$
$29$ $$( -2 + T )^{2}$$
$31$ $$507 + T^{2}$$
$37$ $$( -8 + T )^{2}$$
$41$ $$192 + T^{2}$$
$43$ $$( -26 + T )^{2}$$
$47$ $$1452 + T^{2}$$
$53$ $$( 1 + T )^{2}$$
$59$ $$8112 + T^{2}$$
$61$ $$4800 + T^{2}$$
$67$ $$( 34 + T )^{2}$$
$71$ $$( 10 + T )^{2}$$
$73$ $$3267 + T^{2}$$
$79$ $$( 46 + T )^{2}$$
$83$ $$10443 + T^{2}$$
$89$ $$2700 + T^{2}$$
$97$ $$8427 + T^{2}$$