# Properties

 Label 189.2.e.f 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 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.

 $$f(q)$$ $$=$$ $$q + ( 1 - \beta_{4} + \beta_{5} ) q^{2} + ( -\beta_{1} - \beta_{2} - \beta_{3} - 2 \beta_{4} + \beta_{5} ) q^{4} + \beta_{2} q^{5} + ( 1 - \beta_{1} + \beta_{3} - \beta_{4} + \beta_{5} ) q^{7} + ( -4 - \beta_{1} - 2 \beta_{3} ) q^{8} +O(q^{10})$$ $$q + ( 1 - \beta_{4} + \beta_{5} ) q^{2} + ( -\beta_{1} - \beta_{2} - \beta_{3} - 2 \beta_{4} + \beta_{5} ) q^{4} + \beta_{2} q^{5} + ( 1 - \beta_{1} + \beta_{3} - \beta_{4} + \beta_{5} ) q^{7} + ( -4 - \beta_{1} - 2 \beta_{3} ) q^{8} + ( \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} ) q^{10} + ( -\beta_{1} + 2 \beta_{4} + \beta_{5} ) q^{11} + ( 1 + 2 \beta_{1} - \beta_{3} ) q^{13} + ( -2 - \beta_{1} - \beta_{2} - \beta_{3} - 2 \beta_{4} + 2 \beta_{5} ) q^{14} + ( -5 + \beta_{2} + 5 \beta_{4} - 4 \beta_{5} ) q^{16} + ( \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + \beta_{4} - \beta_{5} ) q^{17} + ( -1 - \beta_{2} + \beta_{4} + \beta_{5} ) q^{19} + ( 5 + 2 \beta_{1} ) q^{20} + ( -1 + 2 \beta_{1} - \beta_{3} ) q^{22} + ( 2 - \beta_{2} - 2 \beta_{4} - \beta_{5} ) q^{23} + ( -\beta_{1} + 2 \beta_{2} + 2 \beta_{3} + \beta_{4} + \beta_{5} ) q^{25} + ( 6 - \beta_{2} - 6 \beta_{4} ) q^{26} + ( -6 - 3 \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} ) q^{28} + ( -4 - \beta_{1} + 2 \beta_{3} ) q^{29} + ( 2 \beta_{1} - \beta_{2} - \beta_{3} + 3 \beta_{4} - 2 \beta_{5} ) q^{31} + ( 4 \beta_{1} + \beta_{2} + \beta_{3} + 10 \beta_{4} - 4 \beta_{5} ) q^{32} + ( 6 + 3 \beta_{1} + 3 \beta_{3} ) q^{34} + ( -3 + \beta_{1} - \beta_{2} + \beta_{3} + 4 \beta_{4} - \beta_{5} ) q^{35} + ( 3 - 2 \beta_{2} - 3 \beta_{4} - \beta_{5} ) q^{37} + ( -2 \beta_{2} - 2 \beta_{3} - 3 \beta_{4} ) q^{38} + ( 9 - 9 \beta_{4} + 3 \beta_{5} ) q^{40} + ( -2 - 5 \beta_{1} + \beta_{3} ) q^{41} + ( -4 - 3 \beta_{1} ) q^{43} -\beta_{2} q^{44} + ( -3 \beta_{1} + 3 \beta_{5} ) q^{46} + ( 2 - \beta_{2} - 2 \beta_{4} - 4 \beta_{5} ) q^{47} + ( 1 - 3 \beta_{1} - \beta_{2} - 2 \beta_{3} + 3 \beta_{5} ) q^{49} + ( 3 \beta_{1} + \beta_{3} ) q^{50} + ( -3 \beta_{1} - 3 \beta_{2} - 3 \beta_{3} - 5 \beta_{4} + 3 \beta_{5} ) q^{52} + ( -2 \beta_{1} - \beta_{2} - \beta_{3} + 7 \beta_{4} + 2 \beta_{5} ) q^{53} + ( 1 + \beta_{1} - 2 \beta_{3} ) q^{55} + ( -13 + 2 \beta_{1} + 3 \beta_{2} + \beta_{3} + 9 \beta_{4} - 6 \beta_{5} ) q^{56} + ( -5 - \beta_{2} + 5 \beta_{4} - 2 \beta_{5} ) q^{58} + ( \beta_{1} - \beta_{2} - \beta_{3} - 5 \beta_{4} - \beta_{5} ) q^{59} + ( -1 + 2 \beta_{2} + \beta_{4} - 2 \beta_{5} ) q^{61} + ( 8 + 2 \beta_{1} + \beta_{3} ) q^{62} + ( 13 + 3 \beta_{1} + 3 \beta_{3} ) q^{64} + ( 2 + 3 \beta_{2} - 2 \beta_{4} - \beta_{5} ) q^{65} + ( \beta_{1} + \beta_{2} + \beta_{3} - 4 \beta_{4} - \beta_{5} ) q^{67} + ( 16 - 2 \beta_{2} - 16 \beta_{4} + 7 \beta_{5} ) q^{68} + ( 5 + 3 \beta_{1} - 2 \beta_{2} + \beta_{4} - \beta_{5} ) q^{70} + ( 3 + 3 \beta_{1} + 3 \beta_{3} ) q^{71} + ( 5 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} - 5 \beta_{5} ) q^{73} + ( -5 \beta_{1} - \beta_{2} - \beta_{3} - 2 \beta_{4} + 5 \beta_{5} ) q^{74} + ( -7 - 3 \beta_{1} ) q^{76} + ( -1 - 2 \beta_{1} + 3 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + 4 \beta_{5} ) q^{77} + ( -2 + 3 \beta_{2} + 2 \beta_{4} + 3 \beta_{5} ) q^{79} + ( -5 \beta_{1} - 3 \beta_{2} - 3 \beta_{3} - 8 \beta_{4} + 5 \beta_{5} ) q^{80} + ( -16 + 4 \beta_{2} + 16 \beta_{4} - \beta_{5} ) q^{82} + ( 6 \beta_{1} - 3 \beta_{3} ) q^{83} + ( -9 - 3 \beta_{1} + 3 \beta_{3} ) q^{85} + ( -13 + 3 \beta_{2} + 13 \beta_{4} - 4 \beta_{5} ) q^{86} + ( 3 \beta_{1} - 3 \beta_{2} - 3 \beta_{3} - 3 \beta_{4} - 3 \beta_{5} ) q^{88} + ( -3 + 4 \beta_{2} + 3 \beta_{4} ) q^{89} + ( -1 + 3 \beta_{1} - \beta_{2} + \beta_{3} - 6 \beta_{4} ) q^{91} + ( -5 - 2 \beta_{1} - \beta_{3} ) q^{92} + ( -3 \beta_{1} + 3 \beta_{2} + 3 \beta_{3} + 9 \beta_{4} + 3 \beta_{5} ) q^{94} + ( 2 \beta_{1} - 3 \beta_{2} - 3 \beta_{3} + 5 \beta_{4} - 2 \beta_{5} ) q^{95} + ( -6 - 2 \beta_{1} - 2 \beta_{3} ) q^{97} + ( -10 - \beta_{1} + \beta_{2} - 4 \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 $$\left(\mathbb{Z}/189\mathbb{Z}\right)^\times$$.

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

## 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 + 2.05195i 0.5 − 1.41036i 0.5 + 0.224437i 0.5 − 2.05195i 0.5 + 1.41036i 0.5 − 0.224437i
−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
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 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
163.1 −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
163.2 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 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
 $$n$$: 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.f yes 6
3.b odd 2 1 189.2.e.e 6
7.c even 3 1 inner 189.2.e.f yes 6
7.c even 3 1 1323.2.a.x 3
7.d odd 6 1 1323.2.a.y 3
9.c even 3 1 567.2.g.i 6
9.c even 3 1 567.2.h.h 6
9.d odd 6 1 567.2.g.h 6
9.d odd 6 1 567.2.h.i 6
21.g even 6 1 1323.2.a.z 3
21.h odd 6 1 189.2.e.e 6
21.h odd 6 1 1323.2.a.ba 3
63.g even 3 1 567.2.h.h 6
63.h even 3 1 567.2.g.i 6
63.j odd 6 1 567.2.g.h 6
63.n odd 6 1 567.2.h.i 6

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

$p$ $F_p(T)$
$2$ $$1 - 2 T + T^{2} + 4 T^{3} - 7 T^{4} + T^{5} + 7 T^{6} + 2 T^{7} - 28 T^{8} + 32 T^{9} + 16 T^{10} - 64 T^{11} + 64 T^{12}$$
$3$ 1
$5$ $$1 - T - 8 T^{2} + 17 T^{3} + 23 T^{4} - 52 T^{5} - 11 T^{6} - 260 T^{7} + 575 T^{8} + 2125 T^{9} - 5000 T^{10} - 3125 T^{11} + 15625 T^{12}$$
$7$ $$1 - 2 T - 4 T^{2} + 31 T^{3} - 28 T^{4} - 98 T^{5} + 343 T^{6}$$
$11$ $$1 - 7 T + 4 T^{2} - T^{3} + 431 T^{4} - 982 T^{5} - 893 T^{6} - 10802 T^{7} + 52151 T^{8} - 1331 T^{9} + 58564 T^{10} - 1127357 T^{11} + 1771561 T^{12}$$
$13$ $$( 1 - 2 T + 20 T^{2} - 5 T^{3} + 260 T^{4} - 338 T^{5} + 2197 T^{6} )^{2}$$
$17$ $$1 - 18 T^{2} + 18 T^{3} + 18 T^{4} - 162 T^{5} + 4399 T^{6} - 2754 T^{7} + 5202 T^{8} + 88434 T^{9} - 1503378 T^{10} + 24137569 T^{12}$$
$19$ $$1 + 5 T - 28 T^{2} - 57 T^{3} + 997 T^{4} + 268 T^{5} - 22757 T^{6} + 5092 T^{7} + 359917 T^{8} - 390963 T^{9} - 3648988 T^{10} + 12380495 T^{11} + 47045881 T^{12}$$
$23$ $$1 - 6 T - 36 T^{2} + 102 T^{3} + 1926 T^{4} - 2526 T^{5} - 42653 T^{6} - 58098 T^{7} + 1018854 T^{8} + 1241034 T^{9} - 10074276 T^{10} - 38618058 T^{11} + 148035889 T^{12}$$
$29$ $$( 1 + 13 T + 117 T^{2} + 763 T^{3} + 3393 T^{4} + 10933 T^{5} + 24389 T^{6} )^{2}$$
$31$ $$1 - 8 T - 30 T^{2} + 102 T^{3} + 2506 T^{4} - 1202 T^{5} - 93509 T^{6} - 37262 T^{7} + 2408266 T^{8} + 3038682 T^{9} - 27705630 T^{10} - 229033208 T^{11} + 887503681 T^{12}$$
$37$ $$1 - 8 T - 42 T^{2} + 150 T^{3} + 3322 T^{4} - 1094 T^{5} - 153041 T^{6} - 40478 T^{7} + 4547818 T^{8} + 7597950 T^{9} - 78714762 T^{10} - 554751656 T^{11} + 2565726409 T^{12}$$
$41$ $$( 1 + 2 T + 18 T^{2} - 223 T^{3} + 738 T^{4} + 3362 T^{5} + 68921 T^{6} )^{2}$$
$43$ $$( 1 + 9 T + 117 T^{2} + 673 T^{3} + 5031 T^{4} + 16641 T^{5} + 79507 T^{6} )^{2}$$
$47$ $$1 - 9 T - 18 T^{2} + 819 T^{3} - 2547 T^{4} - 20772 T^{5} + 299095 T^{6} - 976284 T^{7} - 5626323 T^{8} + 85031037 T^{9} - 87834258 T^{10} - 2064105063 T^{11} + 10779215329 T^{12}$$
$53$ $$1 - 24 T + 252 T^{2} - 2202 T^{3} + 20916 T^{4} - 146148 T^{5} + 883411 T^{6} - 7745844 T^{7} + 58753044 T^{8} - 327827154 T^{9} + 1988401212 T^{10} - 10036691832 T^{11} + 22164361129 T^{12}$$
$59$ $$1 + 15 T - 18 T^{2} - 57 T^{3} + 16947 T^{4} + 71898 T^{5} - 430157 T^{6} + 4241982 T^{7} + 58992507 T^{8} - 11706603 T^{9} - 218112498 T^{10} + 10723864485 T^{11} + 42180533641 T^{12}$$
$61$ $$1 - T - 133 T^{2} - 132 T^{3} + 9781 T^{4} + 12493 T^{5} - 645074 T^{6} + 762073 T^{7} + 36395101 T^{8} - 29961492 T^{9} - 1841496853 T^{10} - 844596301 T^{11} + 51520374361 T^{12}$$
$67$ $$1 + 14 T - 58 T^{2} - 258 T^{3} + 20800 T^{4} + 70720 T^{5} - 964241 T^{6} + 4738240 T^{7} + 93371200 T^{8} - 77596854 T^{9} - 1168765018 T^{10} + 18901751498 T^{11} + 90458382169 T^{12}$$
$71$ $$( 1 - 3 T + 105 T^{2} - 669 T^{3} + 7455 T^{4} - 15123 T^{5} + 357911 T^{6} )^{2}$$
$73$ $$1 + 7 T - 36 T^{2} + 513 T^{3} + 733 T^{4} - 46082 T^{5} - 8831 T^{6} - 3363986 T^{7} + 3906157 T^{8} + 199565721 T^{9} - 1022336676 T^{10} + 14511501151 T^{11} + 151334226289 T^{12}$$
$79$ $$1 + 6 T - 132 T^{2} - 634 T^{3} + 10026 T^{4} + 16110 T^{5} - 794253 T^{6} + 1272690 T^{7} + 62572266 T^{8} - 312586726 T^{9} - 5141410692 T^{10} + 18462338394 T^{11} + 243087455521 T^{12}$$
$83$ $$( 1 + 3 T + 69 T^{2} + 1227 T^{3} + 5727 T^{4} + 20667 T^{5} + 571787 T^{6} )^{2}$$
$89$ $$1 + 5 T - 149 T^{2} + 68 T^{3} + 12785 T^{4} - 45481 T^{5} - 1321850 T^{6} - 4047809 T^{7} + 101269985 T^{8} + 47937892 T^{9} - 9348593909 T^{10} + 27920297245 T^{11} + 496981290961 T^{12}$$
$97$ $$( 1 + 14 T + 307 T^{2} + 2692 T^{3} + 29779 T^{4} + 131726 T^{5} + 912673 T^{6} )^{2}$$