# Properties

 Label 189.3.d.b Level $189$ Weight $3$ Character orbit 189.d Analytic conductor $5.150$ Analytic rank $0$ Dimension $2$ CM discriminant -3 Inner twists $4$

# 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: $$3$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q -4 q^{4} + ( 5 + 3 \zeta_{6} ) q^{7} +O(q^{10})$$ $$q -4 q^{4} + ( 5 + 3 \zeta_{6} ) q^{7} + ( -15 + 30 \zeta_{6} ) q^{13} + 16 q^{16} + ( -21 + 42 \zeta_{6} ) q^{19} + 25 q^{25} + ( -20 - 12 \zeta_{6} ) q^{28} + ( 24 - 48 \zeta_{6} ) q^{31} -47 q^{37} + 22 q^{43} + ( 16 + 39 \zeta_{6} ) q^{49} + ( 60 - 120 \zeta_{6} ) q^{52} + ( -9 + 18 \zeta_{6} ) q^{61} -64 q^{64} -109 q^{67} + ( 63 - 126 \zeta_{6} ) q^{73} + ( 84 - 168 \zeta_{6} ) q^{76} + 131 q^{79} + ( -165 + 195 \zeta_{6} ) q^{91} + ( 57 - 114 \zeta_{6} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 8q^{4} + 13q^{7} + O(q^{10})$$ $$2q - 8q^{4} + 13q^{7} + 32q^{16} + 50q^{25} - 52q^{28} - 94q^{37} + 44q^{43} + 71q^{49} - 128q^{64} - 218q^{67} + 262q^{79} - 135q^{91} + O(q^{100})$$

## 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
0 0 −4.00000 0 0 6.50000 2.59808i 0 0 0
55.2 0 0 −4.00000 0 0 6.50000 + 2.59808i 0 0 0
 $$n$$: 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
3.b odd 2 1 CM by $$\Q(\sqrt{-3})$$
7.b odd 2 1 inner
21.c even 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.b 2
3.b odd 2 1 CM 189.3.d.b 2
7.b odd 2 1 inner 189.3.d.b 2
21.c even 2 1 inner 189.3.d.b 2

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

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$49 - 13 T + T^{2}$$
$11$ $$T^{2}$$
$13$ $$675 + T^{2}$$
$17$ $$T^{2}$$
$19$ $$1323 + T^{2}$$
$23$ $$T^{2}$$
$29$ $$T^{2}$$
$31$ $$1728 + T^{2}$$
$37$ $$( 47 + T )^{2}$$
$41$ $$T^{2}$$
$43$ $$( -22 + T )^{2}$$
$47$ $$T^{2}$$
$53$ $$T^{2}$$
$59$ $$T^{2}$$
$61$ $$243 + T^{2}$$
$67$ $$( 109 + T )^{2}$$
$71$ $$T^{2}$$
$73$ $$11907 + T^{2}$$
$79$ $$( -131 + T )^{2}$$
$83$ $$T^{2}$$
$89$ $$T^{2}$$
$97$ $$9747 + T^{2}$$