# Properties

 Label 189.3.j.a Level $189$ Weight $3$ Character orbit 189.j Analytic conductor $5.150$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$189 = 3^{3} \cdot 7$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 189.j (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$5.14987699641$$ Analytic rank: $$0$$ Dimension: $$6$$ Relative dimension: $$3$$ over $$\Q(\zeta_{6})$$ Coefficient field: 6.0.63369648.1 Defining polynomial: $$x^{6} - x^{5} + 12 x^{4} + 17 x^{3} + 118 x^{2} + 33 x + 9$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 63) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( -1 - \beta_{4} - \beta_{5} ) q^{2} + ( -4 - \beta_{1} ) q^{4} + ( 1 - 2 \beta_{2} - \beta_{5} ) q^{5} + ( -1 + \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} ) q^{7} + ( 2 + \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + 3 \beta_{4} + 3 \beta_{5} ) q^{8} +O(q^{10})$$ $$q + ( -1 - \beta_{4} - \beta_{5} ) q^{2} + ( -4 - \beta_{1} ) q^{4} + ( 1 - 2 \beta_{2} - \beta_{5} ) q^{5} + ( -1 + \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} ) q^{7} + ( 2 + \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + 3 \beta_{4} + 3 \beta_{5} ) q^{8} + ( -10 - 8 \beta_{2} - \beta_{3} - 4 \beta_{4} - 2 \beta_{5} ) q^{10} + ( 4 - \beta_{1} + 2 \beta_{2} - \beta_{3} + 2 \beta_{4} ) q^{11} + ( -2 - \beta_{1} - 5 \beta_{2} + \beta_{3} - \beta_{4} - 2 \beta_{5} ) q^{13} + ( -6 + 2 \beta_{1} - 15 \beta_{2} - \beta_{3} - 4 \beta_{4} ) q^{14} + ( 15 + 2 \beta_{1} - 4 \beta_{4} + 4 \beta_{5} ) q^{16} + ( 1 - 4 \beta_{1} + 5 \beta_{2} + 2 \beta_{3} + 6 \beta_{5} ) q^{17} + ( 2 \beta_{1} + 7 \beta_{2} - 2 \beta_{3} ) q^{19} + ( -2 \beta_{1} + 7 \beta_{2} + \beta_{3} + 7 \beta_{5} ) q^{20} + ( 2 + 5 \beta_{1} - 19 \beta_{2} - 5 \beta_{3} + \beta_{4} + 2 \beta_{5} ) q^{22} + ( -4 + 2 \beta_{1} + \beta_{2} - \beta_{3} - 3 \beta_{5} ) q^{23} + ( -8 + \beta_{1} + 5 \beta_{2} - \beta_{3} - 4 \beta_{4} - 8 \beta_{5} ) q^{25} + ( -18 - 2 \beta_{1} - 9 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} ) q^{26} + ( 2 + 31 \beta_{2} + 3 \beta_{3} - 11 \beta_{4} - \beta_{5} ) q^{28} + ( 5 - 6 \beta_{2} - \beta_{5} ) q^{29} + ( -14 - 4 \beta_{1} + 2 \beta_{4} - 2 \beta_{5} ) q^{31} + ( 25 - 2 \beta_{1} + 60 \beta_{2} + 4 \beta_{3} - 5 \beta_{4} - 5 \beta_{5} ) q^{32} + ( 53 + 42 \beta_{2} + 22 \beta_{4} + 11 \beta_{5} ) q^{34} + ( -3 + 3 \beta_{1} - 10 \beta_{2} - 3 \beta_{3} - 4 \beta_{4} - 10 \beta_{5} ) q^{35} + ( 14 - 5 \beta_{1} + 3 \beta_{2} + 5 \beta_{3} + 7 \beta_{4} + 14 \beta_{5} ) q^{37} + ( 4 + 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + \beta_{4} ) q^{38} + ( 23 + 21 \beta_{2} + 4 \beta_{4} + 2 \beta_{5} ) q^{40} + ( 10 + 4 \beta_{1} + 5 \beta_{2} + 4 \beta_{3} + 10 \beta_{4} ) q^{41} + ( -42 - 38 \beta_{2} + 3 \beta_{3} - 8 \beta_{4} - 4 \beta_{5} ) q^{43} + ( 42 + 2 \beta_{1} + 21 \beta_{2} + 2 \beta_{3} - 26 \beta_{4} ) q^{44} + ( -23 - 21 \beta_{2} - 4 \beta_{4} - 2 \beta_{5} ) q^{46} + ( 6 - 2 \beta_{1} - 24 \beta_{2} + 4 \beta_{3} + 18 \beta_{4} + 18 \beta_{5} ) q^{47} + ( -16 - 3 \beta_{1} - 22 \beta_{2} + 6 \beta_{3} - 12 \beta_{5} ) q^{49} + ( -62 - 3 \beta_{1} - 31 \beta_{2} - 3 \beta_{3} + 2 \beta_{4} ) q^{50} + ( 22 - 10 \beta_{2} + 11 \beta_{4} + 22 \beta_{5} ) q^{52} + ( -16 + 6 \beta_{1} + \beta_{2} - 3 \beta_{3} - 15 \beta_{5} ) q^{53} + ( 28 - \beta_{1} + 3 \beta_{4} - 3 \beta_{5} ) q^{55} + ( -47 - 9 \beta_{1} + 23 \beta_{2} + 9 \beta_{3} + 12 \beta_{4} - 12 \beta_{5} ) q^{56} + ( -14 - 8 \beta_{2} - \beta_{3} - 12 \beta_{4} - 6 \beta_{5} ) q^{58} + ( -23 + 3 \beta_{1} - 34 \beta_{2} - 6 \beta_{3} - 6 \beta_{4} - 6 \beta_{5} ) q^{59} + ( 10 - 8 \beta_{1} + 6 \beta_{4} - 6 \beta_{5} ) q^{61} + ( 4 + 6 \beta_{1} - 40 \beta_{2} - 12 \beta_{3} + 24 \beta_{4} + 24 \beta_{5} ) q^{62} + ( -22 - 3 \beta_{1} + 20 \beta_{4} - 20 \beta_{5} ) q^{64} + ( -27 - 38 \beta_{2} - 8 \beta_{4} - 8 \beta_{5} ) q^{65} + ( -35 + 7 \beta_{1} - 6 \beta_{4} + 6 \beta_{5} ) q^{67} + ( 50 + 6 \beta_{1} - 68 \beta_{2} - 3 \beta_{3} - 18 \beta_{5} ) q^{68} + ( -81 - \beta_{1} - 45 \beta_{2} - 3 \beta_{3} - 26 \beta_{4} - 7 \beta_{5} ) q^{70} + ( 10 - 2 \beta_{1} + 32 \beta_{2} + 4 \beta_{3} - 6 \beta_{4} - 6 \beta_{5} ) q^{71} + ( -18 \beta_{2} + 7 \beta_{3} + 36 \beta_{4} + 18 \beta_{5} ) q^{73} + ( 102 + 2 \beta_{1} + 51 \beta_{2} + 2 \beta_{3} + 18 \beta_{4} ) q^{74} + ( -16 + 3 \beta_{1} + 26 \beta_{2} - 3 \beta_{3} - 8 \beta_{4} - 16 \beta_{5} ) q^{76} + ( -9 + 7 \beta_{1} + 46 \beta_{2} + 4 \beta_{3} - 3 \beta_{4} - 20 \beta_{5} ) q^{77} + ( 31 - 3 \beta_{1} - 2 \beta_{4} + 2 \beta_{5} ) q^{79} + ( -5 - 4 \beta_{1} + 12 \beta_{2} + 2 \beta_{3} + 7 \beta_{5} ) q^{80} + ( -34 - 2 \beta_{1} - 68 \beta_{2} + 2 \beta_{3} - 17 \beta_{4} - 34 \beta_{5} ) q^{82} + ( -8 - 18 \beta_{1} + 8 \beta_{2} + 9 \beta_{3} ) q^{83} + ( 46 - 12 \beta_{1} + 72 \beta_{2} + 12 \beta_{3} + 23 \beta_{4} + 46 \beta_{5} ) q^{85} + ( -6 - 14 \beta_{1} + 35 \beta_{2} + 7 \beta_{3} + 29 \beta_{5} ) q^{86} + ( -46 - 12 \beta_{1} + 138 \beta_{2} + 12 \beta_{3} - 23 \beta_{4} - 46 \beta_{5} ) q^{88} + ( 56 + 5 \beta_{1} + 28 \beta_{2} + 5 \beta_{3} + 8 \beta_{4} ) q^{89} + ( 13 + 6 \beta_{1} - 2 \beta_{2} - 9 \beta_{3} - 18 \beta_{4} - 12 \beta_{5} ) q^{91} + ( -11 + 4 \beta_{1} + 20 \beta_{2} - 2 \beta_{3} + 9 \beta_{5} ) q^{92} + ( 144 + 12 \beta_{1} - 6 \beta_{4} + 6 \beta_{5} ) q^{94} + ( 17 + 2 \beta_{1} + 32 \beta_{2} - 4 \beta_{3} + \beta_{4} + \beta_{5} ) q^{95} + ( -30 - 42 \beta_{2} + \beta_{3} + 24 \beta_{4} + 12 \beta_{5} ) q^{97} + ( -110 - 9 \beta_{1} - 96 \beta_{2} - 12 \beta_{3} - 9 \beta_{4} - 5 \beta_{5} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6q - 26q^{4} + 15q^{5} - 2q^{7} + O(q^{10})$$ $$6q - 26q^{4} + 15q^{5} - 2q^{7} - 19q^{10} + 9q^{11} + 11q^{13} + 24q^{14} + 94q^{16} - 33q^{17} - 19q^{19} - 45q^{20} + 65q^{22} - 15q^{23} - 26q^{25} - 81q^{26} - 42q^{28} + 51q^{29} - 92q^{31} + 93q^{34} + 57q^{35} + 7q^{37} + 21q^{38} + 57q^{40} + 27q^{41} - 99q^{43} + 273q^{44} - 57q^{46} + 6q^{49} - 294q^{50} + 63q^{52} - 45q^{53} + 166q^{55} - 360q^{56} - 7q^{58} + 44q^{61} - 138q^{64} - 196q^{67} + 567q^{68} - 257q^{70} - 101q^{73} + 411q^{74} - 99q^{76} - 105q^{77} + 180q^{79} - 93q^{80} + 151q^{82} - 99q^{83} - 159q^{85} - 249q^{86} - 495q^{88} + 243q^{89} + 177q^{91} - 147q^{92} + 888q^{94} - 161q^{97} - 360q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - x^{5} + 12 x^{4} + 17 x^{3} + 118 x^{2} + 33 x + 9$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$2 \nu^{5} - 24 \nu^{4} + 288 \nu^{3} - 236 \nu^{2} - 66 \nu + 2241$$$$)/1449$$ $$\beta_{2}$$ $$=$$ $$($$$$44 \nu^{5} - 45 \nu^{4} + 540 \nu^{3} + 604 \nu^{2} + 5310 \nu + 36$$$$)/1449$$ $$\beta_{3}$$ $$=$$ $$($$$$2 \nu^{5} - 3 \nu^{4} + 36 \nu^{3} + 16 \nu^{2} + 354 \nu + 99$$$$)/63$$ $$\beta_{4}$$ $$=$$ $$($$$$83 \nu^{5} - 30 \nu^{4} + 843 \nu^{3} + 2281 \nu^{2} + 9819 \nu + 5337$$$$)/1449$$ $$\beta_{5}$$ $$=$$ $$($$$$32 \nu^{5} - 62 \nu^{4} + 422 \nu^{3} + 249 \nu^{2} + 3130 \nu - 1335$$$$)/483$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3} - \beta_{2} - \beta_{1}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$\beta_{5} + 2 \beta_{4} + \beta_{3} - 7 \beta_{2} - 6$$ $$\nu^{3}$$ $$=$$ $$($$$$-2 \beta_{5} + 2 \beta_{4} + 13 \beta_{1} - 33$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$-24 \beta_{5} - 12 \beta_{4} - 19 \beta_{3} + 94 \beta_{2} + 19 \beta_{1} - 24$$ $$\nu^{5}$$ $$=$$ $$($$$$-52 \beta_{5} - 104 \beta_{4} - 187 \beta_{3} + 571 \beta_{2} + 519$$$$)/2$$

## Character values

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

 $$n$$ $$29$$ $$136$$ $$\chi(n)$$ $$-\beta_{2}$$ $$\beta_{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
44.1
 −0.140998 + 0.244215i 1.98253 − 3.43384i −1.34153 + 2.32360i −1.34153 − 2.32360i 1.98253 + 3.43384i −0.140998 − 0.244215i
3.09257i 0 −5.56399 5.67824 3.27834i 0 6.63848 2.22048i 4.83675i 0 −10.1385 17.5604i
44.2 1.03435i 0 2.93011 2.10422 1.21487i 0 −4.75661 + 5.13563i 7.16819i 0 1.25661 + 2.17651i
44.3 3.79027i 0 −10.3661 −0.282467 + 0.163082i 0 −2.88187 6.37925i 24.1293i 0 −0.618126 1.07063i
116.1 3.79027i 0 −10.3661 −0.282467 0.163082i 0 −2.88187 + 6.37925i 24.1293i 0 −0.618126 + 1.07063i
116.2 1.03435i 0 2.93011 2.10422 + 1.21487i 0 −4.75661 5.13563i 7.16819i 0 1.25661 2.17651i
116.3 3.09257i 0 −5.56399 5.67824 + 3.27834i 0 6.63848 + 2.22048i 4.83675i 0 −10.1385 + 17.5604i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 116.3 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
63.j odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 189.3.j.a 6
3.b odd 2 1 63.3.j.a 6
7.c even 3 1 189.3.n.a 6
9.c even 3 1 63.3.n.a yes 6
9.d odd 6 1 189.3.n.a 6
21.c even 2 1 441.3.j.c 6
21.g even 6 1 441.3.n.c 6
21.g even 6 1 441.3.r.c 6
21.h odd 6 1 63.3.n.a yes 6
21.h odd 6 1 441.3.r.b 6
63.g even 3 1 441.3.r.b 6
63.h even 3 1 63.3.j.a 6
63.j odd 6 1 inner 189.3.j.a 6
63.k odd 6 1 441.3.r.c 6
63.l odd 6 1 441.3.n.c 6
63.t odd 6 1 441.3.j.c 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.3.j.a 6 3.b odd 2 1
63.3.j.a 6 63.h even 3 1
63.3.n.a yes 6 9.c even 3 1
63.3.n.a yes 6 21.h odd 6 1
189.3.j.a 6 1.a even 1 1 trivial
189.3.j.a 6 63.j odd 6 1 inner
189.3.n.a 6 7.c even 3 1
189.3.n.a 6 9.d odd 6 1
441.3.j.c 6 21.c even 2 1
441.3.j.c 6 63.t odd 6 1
441.3.n.c 6 21.g even 6 1
441.3.n.c 6 63.l odd 6 1
441.3.r.b 6 21.h odd 6 1
441.3.r.b 6 63.g even 3 1
441.3.r.c 6 21.g even 6 1
441.3.r.c 6 63.k odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{6} + 25 T_{2}^{4} + 163 T_{2}^{2} + 147$$ acting on $$S_{3}^{\mathrm{new}}(189, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$147 + 163 T^{2} + 25 T^{4} + T^{6}$$
$3$ $$T^{6}$$
$5$ $$27 + 117 T + 124 T^{2} - 195 T^{3} + 88 T^{4} - 15 T^{5} + T^{6}$$
$7$ $$117649 + 4802 T - 49 T^{2} - 532 T^{3} - T^{4} + 2 T^{5} + T^{6}$$
$11$ $$1982907 - 524385 T + 38908 T^{2} + 1935 T^{3} - 188 T^{4} - 9 T^{5} + T^{6}$$
$13$ $$499849 - 43127 T + 11498 T^{2} - 743 T^{3} + 182 T^{4} - 11 T^{5} + T^{6}$$
$17$ $$31434507 + 5972265 T + 271404 T^{2} - 20295 T^{3} - 252 T^{4} + 33 T^{5} + T^{6}$$
$19$ $$214369 + 28243 T + 12518 T^{2} - 233 T^{3} + 422 T^{4} + 19 T^{5} + T^{6}$$
$23$ $$128547 + 98739 T + 22176 T^{2} - 2385 T^{3} - 84 T^{4} + 15 T^{5} + T^{6}$$
$29$ $$694083 - 399711 T + 101260 T^{2} - 14127 T^{3} + 1144 T^{4} - 51 T^{5} + T^{6}$$
$31$ $$( -4632 - 324 T + 46 T^{2} + T^{3} )^{2}$$
$37$ $$564110001 - 51800931 T + 4923018 T^{2} - 32235 T^{3} + 2230 T^{4} - 7 T^{5} + T^{6}$$
$41$ $$1387137027 - 160949955 T + 5644444 T^{2} + 67365 T^{3} - 2252 T^{4} - 27 T^{5} + T^{6}$$
$43$ $$432265681 - 42476013 T + 6232158 T^{2} + 243839 T^{3} + 7758 T^{4} + 99 T^{5} + T^{6}$$
$47$ $$29274835968 + 31836672 T^{2} + 10128 T^{4} + T^{6}$$
$53$ $$5141134827 + 347610609 T + 5971536 T^{2} - 125955 T^{3} - 2124 T^{4} + 45 T^{5} + T^{6}$$
$59$ $$813189888 + 5757696 T^{2} + 4896 T^{4} + T^{6}$$
$61$ $$( 160696 - 4996 T - 22 T^{2} + T^{3} )^{2}$$
$67$ $$( -210392 - 1156 T + 98 T^{2} + T^{3} )^{2}$$
$71$ $$109734912 + 1654272 T^{2} + 5568 T^{4} + T^{6}$$
$73$ $$653628591729 + 6439487445 T + 145096998 T^{2} + 812481 T^{3} + 18166 T^{4} + 101 T^{5} + T^{6}$$
$79$ $$( -16712 + 2268 T - 90 T^{2} + T^{3} )^{2}$$
$83$ $$165842832483 + 7002078939 T + 75268548 T^{2} - 982773 T^{3} - 6660 T^{4} + 99 T^{5} + T^{6}$$
$89$ $$15857178627 + 696422037 T - 7471580 T^{2} - 775899 T^{3} + 22876 T^{4} - 243 T^{5} + T^{6}$$
$97$ $$36908941689 - 701419167 T + 44260638 T^{2} + 972045 T^{3} + 22270 T^{4} + 161 T^{5} + T^{6}$$