# Properties

 Label 189.2.c.a Level 189 Weight 2 Character orbit 189.c Analytic conductor 1.509 Analytic rank 0 Dimension 2 CM disc. -3 Inner twists 4

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: No Analytic conductor: $$1.5091725982$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$3$$ Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + 3\sqrt{-3})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q$$ $$+ 2 q^{4}$$ $$+ \beta q^{7}$$ $$+O(q^{10})$$ $$q$$ $$+ 2 q^{4}$$ $$+ \beta q^{7}$$ $$+ ( 1 - 2 \beta ) q^{13}$$ $$+ 4 q^{16}$$ $$+ ( -1 + 2 \beta ) q^{19}$$ $$-5 q^{25}$$ $$+ 2 \beta q^{28}$$ $$+ ( 2 - 4 \beta ) q^{31}$$ $$-11 q^{37}$$ $$-8 q^{43}$$ $$+ ( -7 + \beta ) q^{49}$$ $$+ ( 2 - 4 \beta ) q^{52}$$ $$+ ( -3 + 6 \beta ) q^{61}$$ $$+ 8 q^{64}$$ $$+ 5 q^{67}$$ $$+ ( 3 - 6 \beta ) q^{73}$$ $$+ ( -2 + 4 \beta ) q^{76}$$ $$+ 17 q^{79}$$ $$+ ( 14 - \beta ) q^{91}$$ $$+ ( 1 - 2 \beta ) q^{97}$$ $$+O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q$$ $$\mathstrut +\mathstrut 4q^{4}$$ $$\mathstrut +\mathstrut q^{7}$$ $$\mathstrut +\mathstrut O(q^{10})$$ $$2q$$ $$\mathstrut +\mathstrut 4q^{4}$$ $$\mathstrut +\mathstrut q^{7}$$ $$\mathstrut +\mathstrut 8q^{16}$$ $$\mathstrut -\mathstrut 10q^{25}$$ $$\mathstrut +\mathstrut 2q^{28}$$ $$\mathstrut -\mathstrut 22q^{37}$$ $$\mathstrut -\mathstrut 16q^{43}$$ $$\mathstrut -\mathstrut 13q^{49}$$ $$\mathstrut +\mathstrut 16q^{64}$$ $$\mathstrut +\mathstrut 10q^{67}$$ $$\mathstrut +\mathstrut 34q^{79}$$ $$\mathstrut +\mathstrut 27q^{91}$$ $$\mathstrut +\mathstrut O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
188.1
 0.5 − 0.866025i 0.5 + 0.866025i
0 0 2.00000 0 0 0.500000 2.59808i 0 0 0
188.2 0 0 2.00000 0 0 0.500000 + 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. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
3.b Odd 1 CM by $$\Q(\sqrt{-3})$$ yes
7.b Odd 1 yes
21.c Even 1 yes

## Hecke kernels

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