# Properties

 Label 189.2.e.e Level $189$ Weight $2$ Character orbit 189.e Analytic conductor $1.509$ Analytic rank $0$ Dimension $6$ 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: $$6$$ Relative dimension: $$3$$ over $$\Q(\zeta_{3})$$ Coefficient field: 6.0.309123.1 Defining polynomial: $$x^{6} - 3 x^{5} + 10 x^{4} - 15 x^{3} + 19 x^{2} - 12 x + 3$$ Coefficient ring: $$\Z[a_1, \ldots, a_{4}]$$ Coefficient ring index: $$3$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -\beta_{1} - \beta_{4} + \beta_{5} ) q^{2} + ( -2 + \beta_{2} + 2 \beta_{4} - \beta_{5} ) q^{4} + ( \beta_{2} + \beta_{3} ) q^{5} + ( \beta_{3} + \beta_{4} - \beta_{5} ) q^{7} + ( 4 + \beta_{1} + 2 \beta_{3} ) q^{8} +O(q^{10})$$ $$q + ( -\beta_{1} - \beta_{4} + \beta_{5} ) q^{2} + ( -2 + \beta_{2} + 2 \beta_{4} - \beta_{5} ) q^{4} + ( \beta_{2} + \beta_{3} ) q^{5} + ( \beta_{3} + \beta_{4} - \beta_{5} ) q^{7} + ( 4 + \beta_{1} + 2 \beta_{3} ) q^{8} + ( 1 - \beta_{2} - \beta_{4} + \beta_{5} ) q^{10} + ( -2 + 2 \beta_{4} + \beta_{5} ) q^{11} + ( 1 + 2 \beta_{1} - \beta_{3} ) q^{13} + ( 4 - \beta_{1} - \beta_{2} - 2 \beta_{4} + 2 \beta_{5} ) q^{14} + ( -4 \beta_{1} - \beta_{2} - \beta_{3} - 5 \beta_{4} + 4 \beta_{5} ) q^{16} + ( -1 + 2 \beta_{2} + \beta_{4} - \beta_{5} ) q^{17} + ( \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} ) q^{19} + ( -5 - 2 \beta_{1} ) q^{20} + ( -1 + 2 \beta_{1} - \beta_{3} ) q^{22} + ( \beta_{1} - \beta_{2} - \beta_{3} - 2 \beta_{4} - \beta_{5} ) q^{23} + ( 1 - 2 \beta_{2} - \beta_{4} - \beta_{5} ) q^{25} + ( -\beta_{2} - \beta_{3} - 6 \beta_{4} ) q^{26} + ( -7 - 3 \beta_{1} - \beta_{2} - 2 \beta_{3} + \beta_{4} ) q^{28} + ( 4 + \beta_{1} - 2 \beta_{3} ) q^{29} + ( 3 + \beta_{2} - 3 \beta_{4} + 2 \beta_{5} ) q^{31} + ( -10 + \beta_{2} + 10 \beta_{4} - 4 \beta_{5} ) q^{32} + ( 6 + 3 \beta_{1} + 3 \beta_{3} ) q^{34} + ( -1 - \beta_{2} - 2 \beta_{3} + 4 \beta_{4} - \beta_{5} ) q^{35} + ( -\beta_{1} + 2 \beta_{2} + 2 \beta_{3} + 3 \beta_{4} + \beta_{5} ) q^{37} + ( 3 - 2 \beta_{2} - 3 \beta_{4} ) q^{38} + ( 3 \beta_{1} + 9 \beta_{4} - 3 \beta_{5} ) q^{40} + ( 2 + 5 \beta_{1} - \beta_{3} ) q^{41} + ( -4 - 3 \beta_{1} ) q^{43} + ( -\beta_{2} - \beta_{3} ) q^{44} -3 \beta_{5} q^{46} + ( 4 \beta_{1} - \beta_{2} - \beta_{3} - 2 \beta_{4} - 4 \beta_{5} ) q^{47} + ( 1 + \beta_{2} - \beta_{3} - 3 \beta_{5} ) q^{49} + ( -3 \beta_{1} - \beta_{3} ) q^{50} + ( -5 + 3 \beta_{2} + 5 \beta_{4} - 3 \beta_{5} ) q^{52} + ( -7 - \beta_{2} + 7 \beta_{4} + 2 \beta_{5} ) q^{53} + ( 1 + \beta_{1} - 2 \beta_{3} ) q^{55} + ( 4 + 4 \beta_{1} + 3 \beta_{2} + 2 \beta_{3} + 9 \beta_{4} - 6 \beta_{5} ) q^{56} + ( -2 \beta_{1} + \beta_{2} + \beta_{3} - 5 \beta_{4} + 2 \beta_{5} ) q^{58} + ( 5 - \beta_{2} - 5 \beta_{4} - \beta_{5} ) q^{59} + ( -2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - \beta_{4} + 2 \beta_{5} ) q^{61} + ( -8 - 2 \beta_{1} - \beta_{3} ) q^{62} + ( 13 + 3 \beta_{1} + 3 \beta_{3} ) q^{64} + ( \beta_{1} + 3 \beta_{2} + 3 \beta_{3} - 2 \beta_{4} - \beta_{5} ) q^{65} + ( -4 - \beta_{2} + 4 \beta_{4} + \beta_{5} ) q^{67} + ( -7 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - 16 \beta_{4} + 7 \beta_{5} ) q^{68} + ( 6 + 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} - \beta_{4} + \beta_{5} ) q^{70} + ( -3 - 3 \beta_{1} - 3 \beta_{3} ) q^{71} + ( -2 \beta_{2} + 5 \beta_{5} ) q^{73} + ( 2 - \beta_{2} - 2 \beta_{4} + 5 \beta_{5} ) q^{74} + ( -7 - 3 \beta_{1} ) q^{76} + ( -1 - 2 \beta_{1} + 3 \beta_{2} + \beta_{3} + 2 \beta_{4} + 4 \beta_{5} ) q^{77} + ( 3 \beta_{1} - 3 \beta_{2} - 3 \beta_{3} - 2 \beta_{4} - 3 \beta_{5} ) q^{79} + ( 8 - 3 \beta_{2} - 8 \beta_{4} + 5 \beta_{5} ) q^{80} + ( -\beta_{1} - 4 \beta_{2} - 4 \beta_{3} - 16 \beta_{4} + \beta_{5} ) q^{82} + ( -6 \beta_{1} + 3 \beta_{3} ) q^{83} + ( -9 - 3 \beta_{1} + 3 \beta_{3} ) q^{85} + ( 4 \beta_{1} + 3 \beta_{2} + 3 \beta_{3} + 13 \beta_{4} - 4 \beta_{5} ) q^{86} + ( -3 + 3 \beta_{2} + 3 \beta_{4} + 3 \beta_{5} ) q^{88} + ( 4 \beta_{2} + 4 \beta_{3} + 3 \beta_{4} ) q^{89} + ( -7 + 3 \beta_{1} + \beta_{2} + 2 \beta_{3} + 6 \beta_{4} ) q^{91} + ( 5 + 2 \beta_{1} + \beta_{3} ) q^{92} + ( 9 - 3 \beta_{2} - 9 \beta_{4} - 3 \beta_{5} ) q^{94} + ( -5 - 3 \beta_{2} + 5 \beta_{4} - 2 \beta_{5} ) q^{95} + ( -6 - 2 \beta_{1} - 2 \beta_{3} ) q^{97} + ( 10 + \beta_{1} + \beta_{2} + 5 \beta_{3} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6q - 2q^{2} - 4q^{4} - q^{5} + 2q^{7} + 18q^{8} + O(q^{10})$$ $$6q - 2q^{2} - 4q^{4} - q^{5} + 2q^{7} + 18q^{8} + q^{10} - 7q^{11} + 4q^{13} + 17q^{14} - 10q^{16} - 5q^{19} - 26q^{20} - 8q^{22} - 6q^{23} + 2q^{25} - 17q^{26} - 30q^{28} + 26q^{29} + 8q^{31} - 25q^{32} + 24q^{34} + 10q^{35} + 8q^{37} + 7q^{38} + 24q^{40} + 4q^{41} - 18q^{43} + q^{44} + 3q^{46} - 9q^{47} + 12q^{49} + 8q^{50} - 9q^{52} - 24q^{53} + 8q^{55} + 48q^{56} - 14q^{58} + 15q^{59} + q^{61} - 42q^{62} + 66q^{64} - 10q^{65} - 14q^{67} - 39q^{68} + 26q^{70} - 6q^{71} - 7q^{73} - 36q^{76} + q^{77} - 6q^{79} + 16q^{80} - 43q^{82} + 6q^{83} - 54q^{85} + 32q^{86} - 9q^{88} + 5q^{89} - 33q^{91} + 24q^{92} + 27q^{94} - 16q^{95} - 28q^{97} + 49q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 3 x^{5} + 10 x^{4} - 15 x^{3} + 19 x^{2} - 12 x + 3$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu^{2} - \nu + 2$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{5} + \nu^{4} - 8 \nu^{3} + 5 \nu^{2} - 18 \nu + 6$$$$)/3$$ $$\beta_{3}$$ $$=$$ $$\nu^{4} - 2 \nu^{3} + 6 \nu^{2} - 5 \nu + 3$$ $$\beta_{4}$$ $$=$$ $$($$$$-2 \nu^{5} + 5 \nu^{4} - 16 \nu^{3} + 19 \nu^{2} - 21 \nu + 9$$$$)/3$$ $$\beta_{5}$$ $$=$$ $$($$$$2 \nu^{5} - 5 \nu^{4} + 19 \nu^{3} - 22 \nu^{2} + 30 \nu - 9$$$$)/3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-2 \beta_{5} - \beta_{4} - \beta_{3} - 2 \beta_{2} + \beta_{1} + 2$$$$)/3$$ $$\nu^{2}$$ $$=$$ $$($$$$-2 \beta_{5} - \beta_{4} - \beta_{3} - 2 \beta_{2} + 4 \beta_{1} - 4$$$$)/3$$ $$\nu^{3}$$ $$=$$ $$($$$$7 \beta_{5} + 5 \beta_{4} + 2 \beta_{3} + 4 \beta_{2} + \beta_{1} - 10$$$$)/3$$ $$\nu^{4}$$ $$=$$ $$($$$$16 \beta_{5} + 11 \beta_{4} + 8 \beta_{3} + 10 \beta_{2} - 17 \beta_{1} + 5$$$$)/3$$ $$\nu^{5}$$ $$=$$ $$($$$$-14 \beta_{5} - 16 \beta_{4} + 5 \beta_{3} - 5 \beta_{2} - 23 \beta_{1} + 47$$$$)/3$$

## Character values

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

 $$n$$ $$29$$ $$136$$ $$\chi(n)$$ $$1$$ $$-\beta_{4}$$

## 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 − 0.224437i 0.5 + 1.41036i 0.5 − 2.05195i 0.5 + 0.224437i 0.5 − 1.41036i 0.5 + 2.05195i
−1.34981 2.33795i 0 −2.64400 + 4.57954i 0.794182 + 1.37556i 0 1.23855 + 2.33795i 8.87636 0 2.14400 3.71351i
109.2 −0.380438 0.658939i 0 0.710533 1.23068i −1.59097 2.75564i 0 −2.56238 + 0.658939i −2.60301 0 −1.21053 + 2.09671i
109.3 0.730252 + 1.26483i 0 −0.0665372 + 0.115246i 0.296790 + 0.514055i 0 2.32383 1.26483i 2.72665 0 −0.433463 + 0.750780i
163.1 −1.34981 + 2.33795i 0 −2.64400 4.57954i 0.794182 1.37556i 0 1.23855 2.33795i 8.87636 0 2.14400 + 3.71351i
163.2 −0.380438 + 0.658939i 0 0.710533 + 1.23068i −1.59097 + 2.75564i 0 −2.56238 0.658939i −2.60301 0 −1.21053 2.09671i
163.3 0.730252 1.26483i 0 −0.0665372 0.115246i 0.296790 0.514055i 0 2.32383 + 1.26483i 2.72665 0 −0.433463 0.750780i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 163.3 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.e 6
3.b odd 2 1 189.2.e.f yes 6
7.c even 3 1 inner 189.2.e.e 6
7.c even 3 1 1323.2.a.ba 3
7.d odd 6 1 1323.2.a.z 3
9.c even 3 1 567.2.g.h 6
9.c even 3 1 567.2.h.i 6
9.d odd 6 1 567.2.g.i 6
9.d odd 6 1 567.2.h.h 6
21.g even 6 1 1323.2.a.y 3
21.h odd 6 1 189.2.e.f yes 6
21.h odd 6 1 1323.2.a.x 3
63.g even 3 1 567.2.h.i 6
63.h even 3 1 567.2.g.h 6
63.j odd 6 1 567.2.g.i 6
63.n odd 6 1 567.2.h.h 6

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{6} + 2 T_{2}^{5} + 7 T_{2}^{4} + 15 T_{2}^{2} + 9 T_{2} + 9$$ acting on $$S_{2}^{\mathrm{new}}(189, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$9 + 9 T + 15 T^{2} + 7 T^{4} + 2 T^{5} + T^{6}$$
$3$ $$T^{6}$$
$5$ $$9 - 18 T + 33 T^{2} - 12 T^{3} + 7 T^{4} + T^{5} + T^{6}$$
$7$ $$343 - 98 T - 28 T^{2} + 31 T^{3} - 4 T^{4} - 2 T^{5} + T^{6}$$
$11$ $$9 + 36 T + 123 T^{2} + 78 T^{3} + 37 T^{4} + 7 T^{5} + T^{6}$$
$13$ $$( 47 - 19 T - 2 T^{2} + T^{3} )^{2}$$
$17$ $$81 - 297 T + 1089 T^{2} - 18 T^{3} + 33 T^{4} + T^{6}$$
$19$ $$841 + 116 T + 161 T^{2} + 38 T^{3} + 29 T^{4} + 5 T^{5} + T^{6}$$
$23$ $$81 - 27 T + 63 T^{2} + 36 T^{3} + 33 T^{4} + 6 T^{5} + T^{6}$$
$29$ $$( -9 + 30 T - 13 T^{2} + T^{3} )^{2}$$
$31$ $$4761 + 69 T + 553 T^{2} - 146 T^{3} + 63 T^{4} - 8 T^{5} + T^{6}$$
$37$ $$8649 - 465 T + 769 T^{2} - 146 T^{3} + 69 T^{4} - 8 T^{5} + T^{6}$$
$41$ $$( 387 - 105 T - 2 T^{2} + T^{3} )^{2}$$
$43$ $$( -101 - 12 T + 9 T^{2} + T^{3} )^{2}$$
$47$ $$81 - 378 T + 1683 T^{2} - 396 T^{3} + 123 T^{4} + 9 T^{5} + T^{6}$$
$53$ $$59049 + 40095 T + 21393 T^{2} + 3474 T^{3} + 411 T^{4} + 24 T^{5} + T^{6}$$
$59$ $$6561 - 5346 T + 3141 T^{2} - 828 T^{3} + 159 T^{4} - 15 T^{5} + T^{6}$$
$61$ $$14641 - 5929 T + 2522 T^{2} - 193 T^{3} + 50 T^{4} - T^{5} + T^{6}$$
$67$ $$961 + 1643 T + 2375 T^{2} + 680 T^{3} + 143 T^{4} + 14 T^{5} + T^{6}$$
$71$ $$( 243 - 108 T + 3 T^{2} + T^{3} )^{2}$$
$73$ $$962361 + 131454 T + 24823 T^{2} + 1024 T^{3} + 183 T^{4} + 7 T^{5} + T^{6}$$
$79$ $$16129 + 8763 T + 5523 T^{2} - 160 T^{3} + 105 T^{4} + 6 T^{5} + T^{6}$$
$83$ $$( -729 - 180 T - 3 T^{2} + T^{3} )^{2}$$
$89$ $$239121 - 45477 T + 11094 T^{2} - 513 T^{3} + 118 T^{4} - 5 T^{5} + T^{6}$$
$97$ $$( -24 + 16 T + 14 T^{2} + T^{3} )^{2}$$