# Properties

 Label 189.3.b.a 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: 4.0.1166592.2 Defining polynomial: $$x^{4} + 20 x^{2} + 93$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ 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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} + ( -6 + \beta_{2} ) q^{4} + ( -\beta_{1} + \beta_{3} ) q^{5} + \beta_{2} q^{7} + ( -3 \beta_{1} + \beta_{3} ) q^{8} +O(q^{10})$$ $$q + \beta_{1} q^{2} + ( -6 + \beta_{2} ) q^{4} + ( -\beta_{1} + \beta_{3} ) q^{5} + \beta_{2} q^{7} + ( -3 \beta_{1} + \beta_{3} ) q^{8} + ( 7 - 10 \beta_{2} ) q^{10} -\beta_{1} q^{11} + ( 9 - \beta_{2} ) q^{13} + ( -\beta_{1} + \beta_{3} ) q^{14} + ( 3 - 8 \beta_{2} ) q^{16} + ( -6 \beta_{1} + \beta_{3} ) q^{17} + ( -12 - 7 \beta_{2} ) q^{19} + ( 13 \beta_{1} - 6 \beta_{3} ) q^{20} + ( 10 - \beta_{2} ) q^{22} + ( 3 \beta_{1} + 2 \beta_{3} ) q^{23} + ( -45 + 7 \beta_{2} ) q^{25} + ( 10 \beta_{1} - \beta_{3} ) q^{26} + ( 7 - 6 \beta_{2} ) q^{28} + ( -10 \beta_{1} - \beta_{3} ) q^{29} + ( 10 + 13 \beta_{2} ) q^{31} + ( -\beta_{1} - 4 \beta_{3} ) q^{32} + ( 57 - 15 \beta_{2} ) q^{34} + 7 \beta_{1} q^{35} + ( 13 + 22 \beta_{2} ) q^{37} + ( -5 \beta_{1} - 7 \beta_{3} ) q^{38} + ( -84 + 27 \beta_{2} ) q^{40} + ( 7 \beta_{1} + \beta_{3} ) q^{41} + ( 35 - 15 \beta_{2} ) q^{43} + ( 7 \beta_{1} - \beta_{3} ) q^{44} + ( -36 - 15 \beta_{2} ) q^{46} + ( 6 \beta_{1} + 5 \beta_{3} ) q^{47} + 7 q^{49} + ( -52 \beta_{1} + 7 \beta_{3} ) q^{50} + ( -61 + 15 \beta_{2} ) q^{52} + ( -6 \beta_{1} - 2 \beta_{3} ) q^{53} + ( -7 + 10 \beta_{2} ) q^{55} + ( 9 \beta_{1} - 2 \beta_{3} ) q^{56} + ( 103 - \beta_{2} ) q^{58} + ( 12 \beta_{1} + 7 \beta_{3} ) q^{59} + ( 72 - 10 \beta_{2} ) q^{61} + ( -3 \beta_{1} + 13 \beta_{3} ) q^{62} + ( 34 + 3 \beta_{2} ) q^{64} + ( -16 \beta_{1} + 9 \beta_{3} ) q^{65} + ( 33 + 23 \beta_{2} ) q^{67} + ( 48 \beta_{1} - 11 \beta_{3} ) q^{68} + ( -70 + 7 \beta_{2} ) q^{70} + ( 3 \beta_{1} - 9 \beta_{3} ) q^{71} + ( -47 - 5 \beta_{2} ) q^{73} + ( -9 \beta_{1} + 22 \beta_{3} ) q^{74} + ( 23 + 30 \beta_{2} ) q^{76} + ( \beta_{1} - \beta_{3} ) q^{77} + ( -19 + 15 \beta_{2} ) q^{79} + ( -59 \beta_{1} + 3 \beta_{3} ) q^{80} + ( -73 - 2 \beta_{2} ) q^{82} + ( 24 \beta_{1} + 3 \beta_{3} ) q^{83} + ( -105 + 57 \beta_{2} ) q^{85} + ( 50 \beta_{1} - 15 \beta_{3} ) q^{86} + ( -27 + 12 \beta_{2} ) q^{88} + ( -\beta_{1} - 5 \beta_{3} ) q^{89} + ( -7 + 9 \beta_{2} ) q^{91} + ( -9 \beta_{1} - 7 \beta_{3} ) q^{92} + ( -75 - 39 \beta_{2} ) q^{94} + ( -37 \beta_{1} - 12 \beta_{3} ) q^{95} + ( 40 + 22 \beta_{2} ) q^{97} + 7 \beta_{1} q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 24q^{4} + O(q^{10})$$ $$4q - 24q^{4} + 28q^{10} + 36q^{13} + 12q^{16} - 48q^{19} + 40q^{22} - 180q^{25} + 28q^{28} + 40q^{31} + 228q^{34} + 52q^{37} - 336q^{40} + 140q^{43} - 144q^{46} + 28q^{49} - 244q^{52} - 28q^{55} + 412q^{58} + 288q^{61} + 136q^{64} + 132q^{67} - 280q^{70} - 188q^{73} + 92q^{76} - 76q^{79} - 292q^{82} - 420q^{85} - 108q^{88} - 28q^{91} - 300q^{94} + 160q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 20 x^{2} + 93$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} + 10$$ $$\beta_{3}$$ $$=$$ $$\nu^{3} + 11 \nu$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} - 10$$ $$\nu^{3}$$ $$=$$ $$\beta_{3} - 11 \beta_{1}$$

## 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
 − 3.55609i − 2.71187i 2.71187i 3.55609i
3.55609i 0 −8.64575 9.40852i 0 −2.64575 16.5207i 0 33.4575
134.2 2.71187i 0 −3.35425 7.17494i 0 2.64575 1.75119i 0 −19.4575
134.3 2.71187i 0 −3.35425 7.17494i 0 2.64575 1.75119i 0 −19.4575
134.4 3.55609i 0 −8.64575 9.40852i 0 −2.64575 16.5207i 0 33.4575
 $$n$$: 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.a 4
3.b odd 2 1 inner 189.3.b.a 4
4.b odd 2 1 3024.3.d.f 4
9.c even 3 2 567.3.r.d 8
9.d odd 6 2 567.3.r.d 8
12.b even 2 1 3024.3.d.f 4

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} + 20 T_{2}^{2} + 93$$ acting on $$S_{3}^{\mathrm{new}}(189, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$93 + 20 T^{2} + T^{4}$$
$3$ $$T^{4}$$
$5$ $$4557 + 140 T^{2} + T^{4}$$
$7$ $$( -7 + T^{2} )^{2}$$
$11$ $$93 + 20 T^{2} + T^{4}$$
$13$ $$( 74 - 18 T + T^{2} )^{2}$$
$17$ $$30132 + 780 T^{2} + T^{4}$$
$19$ $$( -199 + 24 T + T^{2} )^{2}$$
$23$ $$837 + 780 T^{2} + T^{4}$$
$29$ $$1208628 + 2252 T^{2} + T^{4}$$
$31$ $$( -1083 - 20 T + T^{2} )^{2}$$
$37$ $$( -3219 - 26 T + T^{2} )^{2}$$
$41$ $$302157 + 1196 T^{2} + T^{4}$$
$43$ $$( -350 - 70 T + T^{2} )^{2}$$
$47$ $$271188 + 4380 T^{2} + T^{4}$$
$53$ $$120528 + 1392 T^{2} + T^{4}$$
$59$ $$30132 + 10356 T^{2} + T^{4}$$
$61$ $$( 4484 - 144 T + T^{2} )^{2}$$
$67$ $$( -2614 - 66 T + T^{2} )^{2}$$
$71$ $$26222373 + 10548 T^{2} + T^{4}$$
$73$ $$( 2034 + 94 T + T^{2} )^{2}$$
$79$ $$( -1214 + 38 T + T^{2} )^{2}$$
$83$ $$41250708 + 13572 T^{2} + T^{4}$$
$89$ $$1796853 + 3380 T^{2} + T^{4}$$
$97$ $$( -1788 - 80 T + T^{2} )^{2}$$