# Properties

 Label 189.3.d.d Level $189$ Weight $3$ Character orbit 189.d Analytic conductor $5.150$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$5.14987699641$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.81622204416.6 Defining polynomial: $$x^{8} + 14 x^{6} + 165 x^{4} + 434 x^{2} + 961$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{4}\cdot 3^{4}$$ 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 $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{2} q^{2} + ( 3 + \beta_{5} ) q^{4} -\beta_{3} q^{5} + ( -2 - \beta_{1} + \beta_{5} - \beta_{7} ) q^{7} + ( -2 \beta_{2} - \beta_{6} ) q^{8} +O(q^{10})$$ $$q -\beta_{2} q^{2} + ( 3 + \beta_{5} ) q^{4} -\beta_{3} q^{5} + ( -2 - \beta_{1} + \beta_{5} - \beta_{7} ) q^{7} + ( -2 \beta_{2} - \beta_{6} ) q^{8} + ( -\beta_{1} - 3 \beta_{7} ) q^{10} -\beta_{2} q^{11} + ( 3 \beta_{1} - 2 \beta_{7} ) q^{13} + ( -\beta_{2} - 2 \beta_{3} + \beta_{4} - \beta_{6} ) q^{14} + ( -1 + 2 \beta_{5} ) q^{16} + ( 3 \beta_{3} - \beta_{4} ) q^{17} -3 \beta_{7} q^{19} + \beta_{4} q^{20} + ( 7 + \beta_{5} ) q^{22} + ( 5 \beta_{2} + 4 \beta_{6} ) q^{23} + ( -2 + 5 \beta_{5} ) q^{25} + ( \beta_{3} - 3 \beta_{4} ) q^{26} + ( 12 - 8 \beta_{1} + \beta_{5} - \beta_{7} ) q^{28} + ( 11 \beta_{2} + \beta_{6} ) q^{29} + ( -4 \beta_{1} - 13 \beta_{7} ) q^{31} + ( 3 \beta_{2} + 2 \beta_{6} ) q^{32} + ( 13 \beta_{1} + 8 \beta_{7} ) q^{34} + ( 6 \beta_{2} + 5 \beta_{3} + \beta_{4} - \beta_{6} ) q^{35} + ( 29 + 2 \beta_{5} ) q^{37} -3 \beta_{3} q^{38} + ( -6 \beta_{1} + 13 \beta_{7} ) q^{40} + ( 5 \beta_{3} + 4 \beta_{4} ) q^{41} + ( -22 + 7 \beta_{5} ) q^{43} + ( -6 \beta_{2} - \beta_{6} ) q^{44} + ( -23 - 21 \beta_{5} ) q^{46} + ( \beta_{3} - 5 \beta_{4} ) q^{47} + ( -5 - 6 \beta_{1} - 8 \beta_{5} + 8 \beta_{7} ) q^{49} + ( -13 \beta_{2} - 5 \beta_{6} ) q^{50} + ( 19 \beta_{1} + 8 \beta_{7} ) q^{52} + ( 8 \beta_{2} - 2 \beta_{6} ) q^{53} + ( -\beta_{1} - 3 \beta_{7} ) q^{55} + ( -11 \beta_{2} - \beta_{3} + 4 \beta_{4} + 3 \beta_{6} ) q^{56} + ( -74 - 15 \beta_{5} ) q^{58} + ( -13 \beta_{3} - \beta_{4} ) q^{59} + ( 2 \beta_{1} + 18 \beta_{7} ) q^{61} + ( -17 \beta_{3} + 4 \beta_{4} ) q^{62} + ( -11 - 19 \beta_{5} ) q^{64} + ( -3 \beta_{2} - 7 \beta_{6} ) q^{65} + ( 44 + 13 \beta_{5} ) q^{67} + ( 9 \beta_{3} - 9 \beta_{4} ) q^{68} + ( -45 - 5 \beta_{1} - 2 \beta_{5} + 16 \beta_{7} ) q^{70} + ( -28 \beta_{2} + 7 \beta_{6} ) q^{71} + ( 11 \beta_{1} - 20 \beta_{7} ) q^{73} + ( -35 \beta_{2} - 2 \beta_{6} ) q^{74} + ( -3 \beta_{1} + 3 \beta_{7} ) q^{76} + ( -\beta_{2} - 2 \beta_{3} + \beta_{4} - \beta_{6} ) q^{77} + ( 14 + \beta_{5} ) q^{79} + ( 7 \beta_{3} + 2 \beta_{4} ) q^{80} + ( -35 \beta_{1} + 19 \beta_{7} ) q^{82} + ( 7 \beta_{3} + 5 \beta_{4} ) q^{83} + ( 72 - 3 \beta_{5} ) q^{85} + ( \beta_{2} - 7 \beta_{6} ) q^{86} + ( 11 + 6 \beta_{5} ) q^{88} -15 \beta_{3} q^{89} + ( 36 + 4 \beta_{1} + 17 \beta_{5} + 18 \beta_{7} ) q^{91} + ( 66 \beta_{2} + 5 \beta_{6} ) q^{92} + ( 51 \beta_{1} - 2 \beta_{7} ) q^{94} + ( 9 \beta_{2} - 6 \beta_{6} ) q^{95} + ( -22 \beta_{1} - 18 \beta_{7} ) q^{97} + ( 29 \beta_{2} + 2 \beta_{3} + 6 \beta_{4} + 8 \beta_{6} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 24q^{4} - 16q^{7} + O(q^{10})$$ $$8q + 24q^{4} - 16q^{7} - 8q^{16} + 56q^{22} - 16q^{25} + 96q^{28} + 232q^{37} - 176q^{43} - 184q^{46} - 40q^{49} - 592q^{58} - 88q^{64} + 352q^{67} - 360q^{70} + 112q^{79} + 576q^{85} + 88q^{88} + 288q^{91} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 14 x^{6} + 165 x^{4} + 434 x^{2} + 961$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-\nu^{6} + 110 \nu^{4} - 1870 \nu^{2} - 2139$$$$)/5115$$ $$\beta_{2}$$ $$=$$ $$($$$$14 \nu^{7} + 165 \nu^{5} + 2310 \nu^{3} + 961 \nu$$$$)/5115$$ $$\beta_{3}$$ $$=$$ $$($$$$26 \nu^{7} + 550 \nu^{5} + 5995 \nu^{3} + 28334 \nu$$$$)/5115$$ $$\beta_{4}$$ $$=$$ $$($$$$14 \nu^{7} + 165 \nu^{5} + 2310 \nu^{3} + 11191 \nu$$$$)/1705$$ $$\beta_{5}$$ $$=$$ $$($$$$\nu^{6} - 721$$$$)/165$$ $$\beta_{6}$$ $$=$$ $$($$$$-28 \nu^{7} - 330 \nu^{5} - 3597 \nu^{3} - 1922 \nu$$$$)/1023$$ $$\beta_{7}$$ $$=$$ $$($$$$83 \nu^{6} + 1100 \nu^{4} + 11990 \nu^{2} + 18972$$$$)/5115$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{4} - 3 \beta_{2}$$$$)/6$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{7} - 3 \beta_{5} - 10 \beta_{1} - 21$$$$)/6$$ $$\nu^{3}$$ $$=$$ $$\beta_{6} + 10 \beta_{2}$$ $$\nu^{4}$$ $$=$$ $$($$$$17 \beta_{7} - 42 \beta_{5} + 109 \beta_{1} - 201$$$$)/6$$ $$\nu^{5}$$ $$=$$ $$($$$$-42 \beta_{6} - 109 \beta_{4} + 126 \beta_{3} - 327 \beta_{2}$$$$)/6$$ $$\nu^{6}$$ $$=$$ $$165 \beta_{5} + 721$$ $$\nu^{7}$$ $$=$$ $$($$$$-495 \beta_{6} + 1216 \beta_{4} - 1485 \beta_{3} - 3648 \beta_{2}$$$$)/6$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
55.1
 −1.67650 − 2.90379i −1.67650 + 2.90379i −0.830265 + 1.43806i −0.830265 − 1.43806i 0.830265 + 1.43806i 0.830265 − 1.43806i 1.67650 − 2.90379i 1.67650 + 2.90379i
−3.35300 0 7.24264 2.40558i 0 2.24264 + 6.63103i −10.8726 0 8.06591i
55.2 −3.35300 0 7.24264 2.40558i 0 2.24264 6.63103i −10.8726 0 8.06591i
55.3 −1.66053 0 −1.24264 6.94357i 0 −6.24264 + 3.16693i 8.70556 0 11.5300i
55.4 −1.66053 0 −1.24264 6.94357i 0 −6.24264 3.16693i 8.70556 0 11.5300i
55.5 1.66053 0 −1.24264 6.94357i 0 −6.24264 3.16693i −8.70556 0 11.5300i
55.6 1.66053 0 −1.24264 6.94357i 0 −6.24264 + 3.16693i −8.70556 0 11.5300i
55.7 3.35300 0 7.24264 2.40558i 0 2.24264 6.63103i 10.8726 0 8.06591i
55.8 3.35300 0 7.24264 2.40558i 0 2.24264 + 6.63103i 10.8726 0 8.06591i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 55.8 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
7.b odd 2 1 inner
21.c even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 189.3.d.d 8
3.b odd 2 1 inner 189.3.d.d 8
7.b odd 2 1 inner 189.3.d.d 8
21.c even 2 1 inner 189.3.d.d 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
189.3.d.d 8 1.a even 1 1 trivial
189.3.d.d 8 3.b odd 2 1 inner
189.3.d.d 8 7.b odd 2 1 inner
189.3.d.d 8 21.c even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} - 14 T_{2}^{2} + 31$$ acting on $$S_{3}^{\mathrm{new}}(189, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 31 - 14 T^{2} + T^{4} )^{2}$$
$3$ $$T^{8}$$
$5$ $$( 279 + 54 T^{2} + T^{4} )^{2}$$
$7$ $$( 2401 + 392 T + 42 T^{2} + 8 T^{3} + T^{4} )^{2}$$
$11$ $$( 31 - 14 T^{2} + T^{4} )^{2}$$
$13$ $$( 34596 + 396 T^{2} + T^{4} )^{2}$$
$17$ $$( 90396 + 756 T^{2} + T^{4} )^{2}$$
$19$ $$( 729 + 162 T^{2} + T^{4} )^{2}$$
$23$ $$( 1770751 - 2702 T^{2} + T^{4} )^{2}$$
$29$ $$( 65596 - 1724 T^{2} + T^{4} )^{2}$$
$31$ $$( 2752281 + 3618 T^{2} + T^{4} )^{2}$$
$37$ $$( 769 - 58 T + T^{2} )^{4}$$
$41$ $$( 15936759 + 8118 T^{2} + T^{4} )^{2}$$
$43$ $$( -398 + 44 T + T^{2} )^{4}$$
$47$ $$( 10500444 + 9324 T^{2} + T^{4} )^{2}$$
$53$ $$( 476656 - 1736 T^{2} + T^{4} )^{2}$$
$59$ $$( 1875996 + 9972 T^{2} + T^{4} )^{2}$$
$61$ $$( 3283344 + 5976 T^{2} + T^{4} )^{2}$$
$67$ $$( -1106 - 88 T + T^{2} )^{4}$$
$71$ $$( 71528191 - 21266 T^{2} + T^{4} )^{2}$$
$73$ $$( 23097636 + 11556 T^{2} + T^{4} )^{2}$$
$79$ $$( 178 - 28 T + T^{2} )^{4}$$
$83$ $$( 41569884 + 13356 T^{2} + T^{4} )^{2}$$
$89$ $$( 14124375 + 12150 T^{2} + T^{4} )^{2}$$
$97$ $$( 57335184 + 23256 T^{2} + T^{4} )^{2}$$