# Properties

 Label 189.3.b.c Level $189$ Weight $3$ Character orbit 189.b Analytic conductor $5.150$ Analytic rank $0$ Dimension $8$ 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: $$8$$ Coefficient field: 8.0.1997017344.2 Defining polynomial: $$x^{8} + 14 x^{6} + 53 x^{4} + 56 x^{2} + 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{4}\cdot 3^{5}$$ 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.

 $$f(q)$$ $$=$$ $$q -\beta_{4} q^{2} + ( -1 - \beta_{2} - \beta_{3} ) q^{4} + ( -\beta_{4} - \beta_{6} + \beta_{7} ) q^{5} -\beta_{2} q^{7} + ( \beta_{1} + 4 \beta_{4} - 2 \beta_{7} ) q^{8} +O(q^{10})$$ $$q -\beta_{4} q^{2} + ( -1 - \beta_{2} - \beta_{3} ) q^{4} + ( -\beta_{4} - \beta_{6} + \beta_{7} ) q^{5} -\beta_{2} q^{7} + ( \beta_{1} + 4 \beta_{4} - 2 \beta_{7} ) q^{8} + ( -6 - 3 \beta_{2} - \beta_{3} + \beta_{5} ) q^{10} + ( \beta_{1} - 3 \beta_{4} + 2 \beta_{6} - 2 \beta_{7} ) q^{11} + ( 4 + 2 \beta_{2} + \beta_{3} + 2 \beta_{5} ) q^{13} + ( \beta_{4} - \beta_{7} ) q^{14} + ( 15 + 7 \beta_{2} + 3 \beta_{3} - 2 \beta_{5} ) q^{16} + ( -\beta_{1} - \beta_{4} - 3 \beta_{6} ) q^{17} + ( 2 - 7 \beta_{2} - \beta_{3} - \beta_{5} ) q^{19} + ( 2 \beta_{1} + 12 \beta_{4} - 2 \beta_{6} - 4 \beta_{7} ) q^{20} + ( -16 - 2 \beta_{3} - 2 \beta_{5} ) q^{22} + ( 3 \beta_{1} + 4 \beta_{4} + 2 \beta_{6} ) q^{23} + ( -13 + 3 \beta_{2} - \beta_{3} + 2 \beta_{5} ) q^{25} + ( \beta_{1} - 10 \beta_{4} + 4 \beta_{6} - 5 \beta_{7} ) q^{26} + ( 6 + \beta_{2} + 2 \beta_{3} - \beta_{5} ) q^{28} + ( -2 \beta_{1} - 5 \beta_{4} - 2 \beta_{6} + 4 \beta_{7} ) q^{29} + ( -2 + 4 \beta_{2} - 3 \beta_{3} - 2 \beta_{5} ) q^{31} + ( -\beta_{1} - 26 \beta_{4} - 4 \beta_{6} + 10 \beta_{7} ) q^{32} + ( -2 + 6 \beta_{2} + \beta_{3} ) q^{34} + ( -\beta_{1} + 4 \beta_{4} - 3 \beta_{6} + \beta_{7} ) q^{35} + ( -2 - 13 \beta_{2} + 3 \beta_{3} ) q^{37} + ( 10 \beta_{4} - 2 \beta_{6} - 4 \beta_{7} ) q^{38} + ( 34 + 18 \beta_{2} + 16 \beta_{3} ) q^{40} + ( -2 \beta_{1} + \beta_{4} + 5 \beta_{6} + 3 \beta_{7} ) q^{41} + ( -15 + 6 \beta_{2} - 8 \beta_{3} ) q^{43} + ( 4 \beta_{1} + 14 \beta_{4} + 4 \beta_{6} - 2 \beta_{7} ) q^{44} + ( 11 - 3 \beta_{2} + 5 \beta_{3} ) q^{46} + ( -6 \beta_{1} - 9 \beta_{4} - 3 \beta_{6} + 7 \beta_{7} ) q^{47} + 7 q^{49} + ( 3 \beta_{1} + 18 \beta_{4} + 4 \beta_{6} - 6 \beta_{7} ) q^{50} + ( -32 + 9 \beta_{2} - 4 \beta_{3} + 3 \beta_{5} ) q^{52} + ( -5 \beta_{1} + 4 \beta_{4} + 6 \beta_{6} - 2 \beta_{7} ) q^{53} + ( 24 - 23 \beta_{2} + \beta_{3} + \beta_{5} ) q^{55} + ( -3 \beta_{1} - 16 \beta_{4} - 2 \beta_{6} + 3 \beta_{7} ) q^{56} + ( -23 - 15 \beta_{2} - 9 \beta_{3} + 4 \beta_{5} ) q^{58} + ( -7 \beta_{1} + 5 \beta_{4} + 3 \beta_{6} - 6 \beta_{7} ) q^{59} + ( 24 - 3 \beta_{2} - 3 \beta_{3} + \beta_{5} ) q^{61} + ( \beta_{1} + 14 \beta_{4} - 4 \beta_{6} + 9 \beta_{7} ) q^{62} + ( -77 - 29 \beta_{2} - 21 \beta_{3} + 2 \beta_{5} ) q^{64} + ( 5 \beta_{1} + 17 \beta_{4} + 16 \beta_{6} + 8 \beta_{7} ) q^{65} + ( -9 + 3 \beta_{2} - 15 \beta_{3} - 2 \beta_{5} ) q^{67} + ( -5 \beta_{1} - 14 \beta_{4} - 12 \beta_{6} + 7 \beta_{7} ) q^{68} + ( 22 + 7 \beta_{2} + 5 \beta_{3} + \beta_{5} ) q^{70} + ( -\beta_{1} + 14 \beta_{4} + 10 \beta_{7} ) q^{71} + ( 30 + 2 \beta_{2} + 8 \beta_{3} - 6 \beta_{5} ) q^{73} + ( -3 \beta_{1} - 3 \beta_{4} - 10 \beta_{7} ) q^{74} + ( 62 + 2 \beta_{2} + 12 \beta_{3} - 8 \beta_{5} ) q^{76} + ( -\beta_{1} - 3 \beta_{4} + 4 \beta_{6} - 6 \beta_{7} ) q^{77} + ( 38 + 5 \beta_{2} + 3 \beta_{3} + 4 \beta_{5} ) q^{79} + ( -8 \beta_{1} - 100 \beta_{4} - 8 \beta_{6} + 18 \beta_{7} ) q^{80} + ( 8 - 19 \beta_{2} - 9 \beta_{3} + 3 \beta_{5} ) q^{82} + ( 10 \beta_{1} - \beta_{4} + 11 \beta_{6} + \beta_{7} ) q^{83} + ( -32 + 29 \beta_{2} + 7 \beta_{3} + 10 \beta_{5} ) q^{85} + ( 8 \beta_{1} + 57 \beta_{4} - 2 \beta_{7} ) q^{86} + ( -4 + 10 \beta_{2} + 8 \beta_{3} - 10 \beta_{5} ) q^{88} + ( -9 \beta_{1} + 8 \beta_{4} - 10 \beta_{6} + \beta_{7} ) q^{89} + ( -9 - 2 \beta_{2} + 4 \beta_{3} + 5 \beta_{5} ) q^{91} + ( 7 \beta_{1} - 22 \beta_{4} + 8 \beta_{6} + 2 \beta_{7} ) q^{92} + ( -34 - 25 \beta_{2} - 19 \beta_{3} + 7 \beta_{5} ) q^{94} + ( -7 \beta_{1} + 15 \beta_{4} - 30 \beta_{6} + 4 \beta_{7} ) q^{95} + ( -52 + 7 \beta_{2} + 13 \beta_{3} - 5 \beta_{5} ) q^{97} -7 \beta_{4} q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 12q^{4} + O(q^{10})$$ $$8q - 12q^{4} - 52q^{10} + 36q^{13} + 132q^{16} + 12q^{19} - 136q^{22} - 108q^{25} + 56q^{28} - 28q^{31} - 12q^{34} - 4q^{37} + 336q^{40} - 152q^{43} + 108q^{46} + 56q^{49} - 272q^{52} + 196q^{55} - 220q^{58} + 180q^{61} - 700q^{64} - 132q^{67} + 196q^{70} + 272q^{73} + 544q^{76} + 316q^{79} + 28q^{82} - 228q^{85} - 56q^{91} - 348q^{94} - 364q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 14 x^{6} + 53 x^{4} + 56 x^{2} + 4$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu^{3} + 4 \nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{6} + 11 \nu^{4} + 16 \nu^{2} - 20$$$$)/8$$ $$\beta_{3}$$ $$=$$ $$($$$$3 \nu^{6} + 33 \nu^{4} + 72 \nu^{2} + 28$$$$)/8$$ $$\beta_{4}$$ $$=$$ $$($$$$\nu^{7} + 13 \nu^{5} + 42 \nu^{3} + 36 \nu$$$$)/4$$ $$\beta_{5}$$ $$=$$ $$($$$$\nu^{6} + 14 \nu^{4} + 49 \nu^{2} + 28$$$$)/2$$ $$\beta_{6}$$ $$=$$ $$($$$$3 \nu^{7} + 39 \nu^{5} + 118 \nu^{3} + 40 \nu$$$$)/8$$ $$\beta_{7}$$ $$=$$ $$($$$$5 \nu^{7} + 71 \nu^{5} + 272 \nu^{3} + 260 \nu$$$$)/8$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-2 \beta_{6} + 3 \beta_{4} - 2 \beta_{1}$$$$)/9$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{3} - 3 \beta_{2} - 11$$$$)/3$$ $$\nu^{3}$$ $$=$$ $$($$$$8 \beta_{6} - 12 \beta_{4} + 17 \beta_{1}$$$$)/9$$ $$\nu^{4}$$ $$=$$ $$($$$$2 \beta_{5} - 11 \beta_{3} + 25 \beta_{2} + 73$$$$)/3$$ $$\nu^{5}$$ $$=$$ $$($$$$12 \beta_{7} - 56 \beta_{6} + 54 \beta_{4} - 149 \beta_{1}$$$$)/9$$ $$\nu^{6}$$ $$=$$ $$($$$$-22 \beta_{5} + 105 \beta_{3} - 203 \beta_{2} - 567$$$$)/3$$ $$\nu^{7}$$ $$=$$ $$($$$$-156 \beta_{7} + 464 \beta_{6} - 270 \beta_{4} + 1295 \beta_{1}$$$$)/9$$

## 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
 1.92812i 0.277334i 2.92812i − 1.27733i 1.27733i − 2.92812i − 0.277334i − 1.92812i
3.92812i 0 −11.4301 7.06404i 0 −2.64575 29.1863i 0 −27.7484
134.2 2.27733i 0 −1.18625 4.94527i 0 2.64575 6.40785i 0 11.2620
134.3 0.928117i 0 3.13860 4.06404i 0 −2.64575 6.62545i 0 −3.77190
134.4 0.722666i 0 3.47775 7.94527i 0 2.64575 5.40392i 0 −5.74177
134.5 0.722666i 0 3.47775 7.94527i 0 2.64575 5.40392i 0 −5.74177
134.6 0.928117i 0 3.13860 4.06404i 0 −2.64575 6.62545i 0 −3.77190
134.7 2.27733i 0 −1.18625 4.94527i 0 2.64575 6.40785i 0 11.2620
134.8 3.92812i 0 −11.4301 7.06404i 0 −2.64575 29.1863i 0 −27.7484
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 134.8 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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 189.3.b.c 8
3.b odd 2 1 inner 189.3.b.c 8
4.b odd 2 1 3024.3.d.j 8
9.c even 3 2 567.3.r.e 16
9.d odd 6 2 567.3.r.e 16
12.b even 2 1 3024.3.d.j 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
189.3.b.c 8 1.a even 1 1 trivial
189.3.b.c 8 3.b odd 2 1 inner
567.3.r.e 16 9.c even 3 2
567.3.r.e 16 9.d odd 6 2
3024.3.d.j 8 4.b odd 2 1
3024.3.d.j 8 12.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{8} + 22 T_{2}^{6} + 109 T_{2}^{4} + 120 T_{2}^{2} + 36$$ acting on $$S_{3}^{\mathrm{new}}(189, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$36 + 120 T^{2} + 109 T^{4} + 22 T^{6} + T^{8}$$
$3$ $$T^{8}$$
$5$ $$1272384 + 174720 T^{2} + 8185 T^{4} + 154 T^{6} + T^{8}$$
$7$ $$( -7 + T^{2} )^{4}$$
$11$ $$84492864 + 5478816 T^{2} + 97876 T^{4} + 616 T^{6} + T^{8}$$
$13$ $$( -26654 + 8010 T - 473 T^{2} - 18 T^{3} + T^{4} )^{2}$$
$17$ $$57562569 + 4276476 T^{2} + 104094 T^{4} + 876 T^{6} + T^{8}$$
$19$ $$( 42256 - 3432 T - 758 T^{2} - 6 T^{3} + T^{4} )^{2}$$
$23$ $$4882655376 + 96949224 T^{2} + 645849 T^{4} + 1614 T^{6} + T^{8}$$
$29$ $$3638261124 + 116441496 T^{2} + 893605 T^{4} + 2110 T^{6} + T^{8}$$
$31$ $$( 32634 + 8526 T - 1253 T^{2} + 14 T^{3} + T^{4} )^{2}$$
$37$ $$( 814032 + 26592 T - 2891 T^{2} + 2 T^{3} + T^{4} )^{2}$$
$41$ $$4052217312144 + 13218995880 T^{2} + 15257449 T^{4} + 7090 T^{6} + T^{8}$$
$43$ $$( -2939831 - 230756 T - 2082 T^{2} + 76 T^{3} + T^{4} )^{2}$$
$47$ $$3631749696 + 13520768832 T^{2} + 24132177 T^{4} + 9426 T^{6} + T^{8}$$
$53$ $$52968101904 + 2786398056 T^{2} + 36868833 T^{4} + 12198 T^{6} + T^{8}$$
$59$ $$68515632191889 + 123309795636 T^{2} + 76404150 T^{4} + 17940 T^{6} + T^{8}$$
$61$ $$( -526784 + 7344 T + 2146 T^{2} - 90 T^{3} + T^{4} )^{2}$$
$67$ $$( 24597736 - 567516 T - 11771 T^{2} + 66 T^{3} + T^{4} )^{2}$$
$71$ $$259415011584 + 4781796768 T^{2} + 23516649 T^{4} + 14742 T^{6} + T^{8}$$
$73$ $$( 9014688 + 380304 T - 3044 T^{2} - 136 T^{3} + T^{4} )^{2}$$
$79$ $$( -3444668 + 4360 T + 6693 T^{2} - 158 T^{3} + T^{4} )^{2}$$
$83$ $$129353590650384 + 260333953992 T^{2} + 144200169 T^{4} + 23058 T^{6} + T^{8}$$
$89$ $$4824840688704 + 140548148544 T^{2} + 94814185 T^{4} + 19018 T^{6} + T^{8}$$
$97$ $$( -59939928 - 1735608 T - 3386 T^{2} + 182 T^{3} + T^{4} )^{2}$$