# Properties

 Label 189.3.b.b Level $189$ Weight $3$ Character orbit 189.b Analytic conductor $5.150$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$5.14987699641$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{-2}, \sqrt{7})$$ Defining polynomial: $$x^{4} + 8 x^{2} + 9$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$3$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} + \beta_{2} q^{4} -\beta_{3} q^{5} + \beta_{2} q^{7} + ( 2 \beta_{1} + \beta_{3} ) q^{8} +O(q^{10})$$ $$q + \beta_{1} q^{2} + \beta_{2} q^{4} -\beta_{3} q^{5} + \beta_{2} q^{7} + ( 2 \beta_{1} + \beta_{3} ) q^{8} + ( 1 + 2 \beta_{2} ) q^{10} + ( 3 \beta_{1} - 4 \beta_{3} ) q^{11} + ( -3 + 5 \beta_{2} ) q^{13} + ( -2 \beta_{1} + \beta_{3} ) q^{14} + ( -9 + 4 \beta_{2} ) q^{16} + ( \beta_{1} - \beta_{3} ) q^{17} + ( 6 - \beta_{2} ) q^{19} + ( -3 \beta_{1} - 2 \beta_{3} ) q^{20} + ( -8 + 11 \beta_{2} ) q^{22} + ( -3 \beta_{1} + 6 \beta_{3} ) q^{23} + ( 9 - 5 \beta_{2} ) q^{25} + ( -13 \beta_{1} + 5 \beta_{3} ) q^{26} + 7 q^{28} + ( 17 \beta_{1} + 3 \beta_{3} ) q^{29} + ( 16 + 7 \beta_{2} ) q^{31} + ( -9 \beta_{1} + 8 \beta_{3} ) q^{32} + ( -3 + 3 \beta_{2} ) q^{34} + ( -3 \beta_{1} - 2 \beta_{3} ) q^{35} + ( -23 - 14 \beta_{2} ) q^{37} + ( 8 \beta_{1} - \beta_{3} ) q^{38} + ( 18 + 9 \beta_{2} ) q^{40} + ( -20 \beta_{1} - 9 \beta_{3} ) q^{41} + ( -7 - 15 \beta_{2} ) q^{43} + ( -18 \beta_{1} - 5 \beta_{3} ) q^{44} + ( 6 - 15 \beta_{2} ) q^{46} + ( -11 \beta_{1} - \beta_{3} ) q^{47} + 7 q^{49} + ( 19 \beta_{1} - 5 \beta_{3} ) q^{50} + ( 35 - 3 \beta_{2} ) q^{52} + ( 28 \beta_{1} + 2 \beta_{3} ) q^{53} + ( -61 - 14 \beta_{2} ) q^{55} + ( -\beta_{1} + 4 \beta_{3} ) q^{56} + ( -71 + 11 \beta_{2} ) q^{58} + ( 41 \beta_{1} + \beta_{3} ) q^{59} + ( 24 - 34 \beta_{2} ) q^{61} + ( 2 \beta_{1} + 7 \beta_{3} ) q^{62} + ( -8 - 9 \beta_{2} ) q^{64} + ( -15 \beta_{1} - 7 \beta_{3} ) q^{65} + ( 39 - \beta_{2} ) q^{67} + ( -5 \beta_{1} - \beta_{3} ) q^{68} + ( 14 + \beta_{2} ) q^{70} + ( 42 \beta_{1} - 9 \beta_{3} ) q^{71} + ( 49 - 23 \beta_{2} ) q^{73} + ( 5 \beta_{1} - 14 \beta_{3} ) q^{74} + ( -7 + 6 \beta_{2} ) q^{76} + ( -18 \beta_{1} - 5 \beta_{3} ) q^{77} + ( -121 - 9 \beta_{2} ) q^{79} + ( -12 \beta_{1} + \beta_{3} ) q^{80} + ( 89 - 2 \beta_{2} ) q^{82} + ( -31 \beta_{1} + 25 \beta_{3} ) q^{83} + ( -15 - 3 \beta_{2} ) q^{85} + ( 23 \beta_{1} - 15 \beta_{3} ) q^{86} + ( 45 + 36 \beta_{2} ) q^{88} + ( -10 \beta_{1} - 15 \beta_{3} ) q^{89} + ( 35 - 3 \beta_{2} ) q^{91} + ( 24 \beta_{1} + 9 \beta_{3} ) q^{92} + ( 45 - 9 \beta_{2} ) q^{94} + ( 3 \beta_{1} - 4 \beta_{3} ) q^{95} + ( 4 - 38 \beta_{2} ) q^{97} + 7 \beta_{1} q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + O(q^{10})$$ $$4q + 4q^{10} - 12q^{13} - 36q^{16} + 24q^{19} - 32q^{22} + 36q^{25} + 28q^{28} + 64q^{31} - 12q^{34} - 92q^{37} + 72q^{40} - 28q^{43} + 24q^{46} + 28q^{49} + 140q^{52} - 244q^{55} - 284q^{58} + 96q^{61} - 32q^{64} + 156q^{67} + 56q^{70} + 196q^{73} - 28q^{76} - 484q^{79} + 356q^{82} - 60q^{85} + 180q^{88} + 140q^{91} + 180q^{94} + 16q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 8 x^{2} + 9$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} + 4$$ $$\beta_{3}$$ $$=$$ $$\nu^{3} + 6 \nu$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} - 4$$ $$\nu^{3}$$ $$=$$ $$\beta_{3} - 6 \beta_{1}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
134.1
 − 2.57794i − 1.16372i 1.16372i 2.57794i
2.57794i 0 −2.64575 1.66471i 0 −2.64575 3.49117i 0 −4.29150
134.2 1.16372i 0 2.64575 5.40636i 0 2.64575 7.73381i 0 6.29150
134.3 1.16372i 0 2.64575 5.40636i 0 2.64575 7.73381i 0 6.29150
134.4 2.57794i 0 −2.64575 1.66471i 0 −2.64575 3.49117i 0 −4.29150
 $$n$$: 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 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 189.3.b.b 4
3.b odd 2 1 inner 189.3.b.b 4
4.b odd 2 1 3024.3.d.c 4
9.c even 3 2 567.3.r.b 8
9.d odd 6 2 567.3.r.b 8
12.b even 2 1 3024.3.d.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
189.3.b.b 4 1.a even 1 1 trivial
189.3.b.b 4 3.b odd 2 1 inner
567.3.r.b 8 9.c even 3 2
567.3.r.b 8 9.d odd 6 2
3024.3.d.c 4 4.b odd 2 1
3024.3.d.c 4 12.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} + 8 T_{2}^{2} + 9$$ acting on $$S_{3}^{\mathrm{new}}(189, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$9 + 8 T^{2} + T^{4}$$
$3$ $$T^{4}$$
$5$ $$81 + 32 T^{2} + T^{4}$$
$7$ $$( -7 + T^{2} )^{2}$$
$11$ $$68121 + 536 T^{2} + T^{4}$$
$13$ $$( -166 + 6 T + T^{2} )^{2}$$
$17$ $$( 18 + T^{2} )^{2}$$
$19$ $$( 29 - 12 T + T^{2} )^{2}$$
$23$ $$263169 + 1152 T^{2} + T^{4}$$
$29$ $$1954404 + 2804 T^{2} + T^{4}$$
$31$ $$( -87 - 32 T + T^{2} )^{2}$$
$37$ $$( -843 + 46 T + T^{2} )^{2}$$
$41$ $$6922161 + 6512 T^{2} + T^{4}$$
$43$ $$( -1526 + 14 T + T^{2} )^{2}$$
$47$ $$236196 + 1044 T^{2} + T^{4}$$
$53$ $$8928144 + 6624 T^{2} + T^{4}$$
$59$ $$30536676 + 13644 T^{2} + T^{4}$$
$61$ $$( -7516 - 48 T + T^{2} )^{2}$$
$67$ $$( 1514 - 78 T + T^{2} )^{2}$$
$71$ $$729 + 15192 T^{2} + T^{4}$$
$73$ $$( -1302 - 98 T + T^{2} )^{2}$$
$79$ $$( 14074 + 242 T + T^{2} )^{2}$$
$83$ $$145009764 + 24588 T^{2} + T^{4}$$
$89$ $$5625 + 8600 T^{2} + T^{4}$$
$97$ $$( -10092 - 8 T + T^{2} )^{2}$$