# Properties

 Label 189.2.e.b Level 189 Weight 2 Character orbit 189.e Analytic conductor 1.509 Analytic rank 0 Dimension 2 CM discriminant -3 Inner twists 4

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$1.50917259820$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{3}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( 2 - 2 \zeta_{6} ) q^{4} + ( 1 - 3 \zeta_{6} ) q^{7} +O(q^{10})$$ $$q + ( 2 - 2 \zeta_{6} ) q^{4} + ( 1 - 3 \zeta_{6} ) q^{7} + 2 q^{13} -4 \zeta_{6} q^{16} + 7 \zeta_{6} q^{19} + ( 5 - 5 \zeta_{6} ) q^{25} + ( -4 - 2 \zeta_{6} ) q^{28} + ( -11 + 11 \zeta_{6} ) q^{31} + 10 \zeta_{6} q^{37} -13 q^{43} + ( -8 + 3 \zeta_{6} ) q^{49} + ( 4 - 4 \zeta_{6} ) q^{52} + 13 \zeta_{6} q^{61} -8 q^{64} + ( 16 - 16 \zeta_{6} ) q^{67} + ( 7 - 7 \zeta_{6} ) q^{73} + 14 q^{76} + 4 \zeta_{6} q^{79} + ( 2 - 6 \zeta_{6} ) q^{91} + 5 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{4} - q^{7} + O(q^{10})$$ $$2q + 2q^{4} - q^{7} + 4q^{13} - 4q^{16} + 7q^{19} + 5q^{25} - 10q^{28} - 11q^{31} + 10q^{37} - 26q^{43} - 13q^{49} + 4q^{52} + 13q^{61} - 16q^{64} + 16q^{67} + 7q^{73} + 28q^{76} + 4q^{79} - 2q^{91} + 10q^{97} + O(q^{100})$$

## Character values

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

 $$n$$ $$29$$ $$136$$ $$\chi(n)$$ $$1$$ $$-\zeta_{6}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
109.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 0 1.00000 1.73205i 0 0 −0.500000 2.59808i 0 0 0
163.1 0 0 1.00000 + 1.73205i 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 Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by $$\Q(\sqrt{-3})$$
7.c even 3 1 inner
21.h odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 189.2.e.b 2
3.b odd 2 1 CM 189.2.e.b 2
7.c even 3 1 inner 189.2.e.b 2
7.c even 3 1 1323.2.a.k 1
7.d odd 6 1 1323.2.a.j 1
9.c even 3 1 567.2.g.c 2
9.c even 3 1 567.2.h.d 2
9.d odd 6 1 567.2.g.c 2
9.d odd 6 1 567.2.h.d 2
21.g even 6 1 1323.2.a.j 1
21.h odd 6 1 inner 189.2.e.b 2
21.h odd 6 1 1323.2.a.k 1
63.g even 3 1 567.2.h.d 2
63.h even 3 1 567.2.g.c 2
63.j odd 6 1 567.2.g.c 2
63.n odd 6 1 567.2.h.d 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
189.2.e.b 2 1.a even 1 1 trivial
189.2.e.b 2 3.b odd 2 1 CM
189.2.e.b 2 7.c even 3 1 inner
189.2.e.b 2 21.h odd 6 1 inner
567.2.g.c 2 9.c even 3 1
567.2.g.c 2 9.d odd 6 1
567.2.g.c 2 63.h even 3 1
567.2.g.c 2 63.j odd 6 1
567.2.h.d 2 9.c even 3 1
567.2.h.d 2 9.d odd 6 1
567.2.h.d 2 63.g even 3 1
567.2.h.d 2 63.n odd 6 1
1323.2.a.j 1 7.d odd 6 1
1323.2.a.j 1 21.g even 6 1
1323.2.a.k 1 7.c even 3 1
1323.2.a.k 1 21.h odd 6 1

## Hecke kernels

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

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 - 2 T^{2} + 4 T^{4}$$
$3$ 1
$5$ $$1 - 5 T^{2} + 25 T^{4}$$
$7$ $$1 + T + 7 T^{2}$$
$11$ $$1 - 11 T^{2} + 121 T^{4}$$
$13$ $$( 1 - 2 T + 13 T^{2} )^{2}$$
$17$ $$1 - 17 T^{2} + 289 T^{4}$$
$19$ $$( 1 - 8 T + 19 T^{2} )( 1 + T + 19 T^{2} )$$
$23$ $$1 - 23 T^{2} + 529 T^{4}$$
$29$ $$( 1 + 29 T^{2} )^{2}$$
$31$ $$( 1 + 4 T + 31 T^{2} )( 1 + 7 T + 31 T^{2} )$$
$37$ $$( 1 - 11 T + 37 T^{2} )( 1 + T + 37 T^{2} )$$
$41$ $$( 1 + 41 T^{2} )^{2}$$
$43$ $$( 1 + 13 T + 43 T^{2} )^{2}$$
$47$ $$1 - 47 T^{2} + 2209 T^{4}$$
$53$ $$1 - 53 T^{2} + 2809 T^{4}$$
$59$ $$1 - 59 T^{2} + 3481 T^{4}$$
$61$ $$( 1 - 14 T + 61 T^{2} )( 1 + T + 61 T^{2} )$$
$67$ $$( 1 - 11 T + 67 T^{2} )( 1 - 5 T + 67 T^{2} )$$
$71$ $$( 1 + 71 T^{2} )^{2}$$
$73$ $$( 1 - 17 T + 73 T^{2} )( 1 + 10 T + 73 T^{2} )$$
$79$ $$( 1 - 17 T + 79 T^{2} )( 1 + 13 T + 79 T^{2} )$$
$83$ $$( 1 + 83 T^{2} )^{2}$$
$89$ $$1 - 89 T^{2} + 7921 T^{4}$$
$97$ $$( 1 - 5 T + 97 T^{2} )^{2}$$