# Properties

 Label 189.2.e.a Level 189 Weight 2 Character orbit 189.e Analytic conductor 1.509 Analytic rank 0 Dimension 2 CM no Inner twists 2

# 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: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q -\zeta_{6} q^{2} + ( 1 - \zeta_{6} ) q^{4} + 4 \zeta_{6} q^{5} + ( 1 + 2 \zeta_{6} ) q^{7} -3 q^{8} +O(q^{10})$$ $$q -\zeta_{6} q^{2} + ( 1 - \zeta_{6} ) q^{4} + 4 \zeta_{6} q^{5} + ( 1 + 2 \zeta_{6} ) q^{7} -3 q^{8} + ( 4 - 4 \zeta_{6} ) q^{10} + ( -2 + 2 \zeta_{6} ) q^{11} + q^{13} + ( 2 - 3 \zeta_{6} ) q^{14} + \zeta_{6} q^{16} + ( 6 - 6 \zeta_{6} ) q^{17} -4 \zeta_{6} q^{19} + 4 q^{20} + 2 q^{22} -6 \zeta_{6} q^{23} + ( -11 + 11 \zeta_{6} ) q^{25} -\zeta_{6} q^{26} + ( 3 - \zeta_{6} ) q^{28} + 2 q^{29} + ( -3 + 3 \zeta_{6} ) q^{31} + ( -5 + 5 \zeta_{6} ) q^{32} -6 q^{34} + ( -8 + 12 \zeta_{6} ) q^{35} -3 \zeta_{6} q^{37} + ( -4 + 4 \zeta_{6} ) q^{38} -12 \zeta_{6} q^{40} -2 q^{41} - q^{43} + 2 \zeta_{6} q^{44} + ( -6 + 6 \zeta_{6} ) q^{46} -6 \zeta_{6} q^{47} + ( -3 + 8 \zeta_{6} ) q^{49} + 11 q^{50} + ( 1 - \zeta_{6} ) q^{52} + ( 6 - 6 \zeta_{6} ) q^{53} -8 q^{55} + ( -3 - 6 \zeta_{6} ) q^{56} -2 \zeta_{6} q^{58} + ( -6 + 6 \zeta_{6} ) q^{59} + 5 \zeta_{6} q^{61} + 3 q^{62} + 7 q^{64} + 4 \zeta_{6} q^{65} + ( -7 + 7 \zeta_{6} ) q^{67} -6 \zeta_{6} q^{68} + ( 12 - 4 \zeta_{6} ) q^{70} + ( 6 - 6 \zeta_{6} ) q^{73} + ( -3 + 3 \zeta_{6} ) q^{74} -4 q^{76} + ( -6 + 2 \zeta_{6} ) q^{77} -11 \zeta_{6} q^{79} + ( -4 + 4 \zeta_{6} ) q^{80} + 2 \zeta_{6} q^{82} + 6 q^{83} + 24 q^{85} + \zeta_{6} q^{86} + ( 6 - 6 \zeta_{6} ) q^{88} + 4 \zeta_{6} q^{89} + ( 1 + 2 \zeta_{6} ) q^{91} -6 q^{92} + ( -6 + 6 \zeta_{6} ) q^{94} + ( 16 - 16 \zeta_{6} ) q^{95} + 9 q^{97} + ( 8 - 5 \zeta_{6} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - q^{2} + q^{4} + 4q^{5} + 4q^{7} - 6q^{8} + O(q^{10})$$ $$2q - q^{2} + q^{4} + 4q^{5} + 4q^{7} - 6q^{8} + 4q^{10} - 2q^{11} + 2q^{13} + q^{14} + q^{16} + 6q^{17} - 4q^{19} + 8q^{20} + 4q^{22} - 6q^{23} - 11q^{25} - q^{26} + 5q^{28} + 4q^{29} - 3q^{31} - 5q^{32} - 12q^{34} - 4q^{35} - 3q^{37} - 4q^{38} - 12q^{40} - 4q^{41} - 2q^{43} + 2q^{44} - 6q^{46} - 6q^{47} + 2q^{49} + 22q^{50} + q^{52} + 6q^{53} - 16q^{55} - 12q^{56} - 2q^{58} - 6q^{59} + 5q^{61} + 6q^{62} + 14q^{64} + 4q^{65} - 7q^{67} - 6q^{68} + 20q^{70} + 6q^{73} - 3q^{74} - 8q^{76} - 10q^{77} - 11q^{79} - 4q^{80} + 2q^{82} + 12q^{83} + 48q^{85} + q^{86} + 6q^{88} + 4q^{89} + 4q^{91} - 12q^{92} - 6q^{94} + 16q^{95} + 18q^{97} + 11q^{98} + O(q^{100})$$

## Character values

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

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

## 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
−0.500000 0.866025i 0 0.500000 0.866025i 2.00000 + 3.46410i 0 2.00000 + 1.73205i −3.00000 0 2.00000 3.46410i
163.1 −0.500000 + 0.866025i 0 0.500000 + 0.866025i 2.00000 3.46410i 0 2.00000 1.73205i −3.00000 0 2.00000 + 3.46410i
 $$n$$: 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.c even 3 1 inner

## Twists

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

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

## Hecke kernels

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

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 + T - T^{2} + 2 T^{3} + 4 T^{4}$$
$3$ 1
$5$ $$1 - 4 T + 11 T^{2} - 20 T^{3} + 25 T^{4}$$
$7$ $$1 - 4 T + 7 T^{2}$$
$11$ $$1 + 2 T - 7 T^{2} + 22 T^{3} + 121 T^{4}$$
$13$ $$( 1 - T + 13 T^{2} )^{2}$$
$17$ $$1 - 6 T + 19 T^{2} - 102 T^{3} + 289 T^{4}$$
$19$ $$1 + 4 T - 3 T^{2} + 76 T^{3} + 361 T^{4}$$
$23$ $$1 + 6 T + 13 T^{2} + 138 T^{3} + 529 T^{4}$$
$29$ $$( 1 - 2 T + 29 T^{2} )^{2}$$
$31$ $$1 + 3 T - 22 T^{2} + 93 T^{3} + 961 T^{4}$$
$37$ $$1 + 3 T - 28 T^{2} + 111 T^{3} + 1369 T^{4}$$
$41$ $$( 1 + 2 T + 41 T^{2} )^{2}$$
$43$ $$( 1 + T + 43 T^{2} )^{2}$$
$47$ $$1 + 6 T - 11 T^{2} + 282 T^{3} + 2209 T^{4}$$
$53$ $$1 - 6 T - 17 T^{2} - 318 T^{3} + 2809 T^{4}$$
$59$ $$1 + 6 T - 23 T^{2} + 354 T^{3} + 3481 T^{4}$$
$61$ $$1 - 5 T - 36 T^{2} - 305 T^{3} + 3721 T^{4}$$
$67$ $$1 + 7 T - 18 T^{2} + 469 T^{3} + 4489 T^{4}$$
$71$ $$( 1 + 71 T^{2} )^{2}$$
$73$ $$1 - 6 T - 37 T^{2} - 438 T^{3} + 5329 T^{4}$$
$79$ $$1 + 11 T + 42 T^{2} + 869 T^{3} + 6241 T^{4}$$
$83$ $$( 1 - 6 T + 83 T^{2} )^{2}$$
$89$ $$1 - 4 T - 73 T^{2} - 356 T^{3} + 7921 T^{4}$$
$97$ $$( 1 - 9 T + 97 T^{2} )^{2}$$