# Properties

 Label 189.4.a.l Level $189$ Weight $4$ Character orbit 189.a Self dual yes Analytic conductor $11.151$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$189 = 3^{3} \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 189.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$11.1513609911$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{5}, \sqrt{13})$$ Defining polynomial: $$x^{4} - 9 x^{2} + 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{2}\cdot 3^{2}$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -\beta_{1} - \beta_{2} ) q^{2} + ( 6 - \beta_{3} ) q^{4} + ( -4 \beta_{1} - 3 \beta_{2} ) q^{5} -7 q^{7} + ( -5 \beta_{1} - 13 \beta_{2} ) q^{8} +O(q^{10})$$ $$q + ( -\beta_{1} - \beta_{2} ) q^{2} + ( 6 - \beta_{3} ) q^{4} + ( -4 \beta_{1} - 3 \beta_{2} ) q^{5} -7 q^{7} + ( -5 \beta_{1} - 13 \beta_{2} ) q^{8} + ( 50 - 3 \beta_{3} ) q^{10} + ( -6 \beta_{1} + 11 \beta_{2} ) q^{11} + ( -14 + 6 \beta_{3} ) q^{13} + ( 7 \beta_{1} + 7 \beta_{2} ) q^{14} + ( 70 - 5 \beta_{3} ) q^{16} + ( -4 \beta_{1} - 2 \beta_{2} ) q^{17} + ( 45 + 4 \beta_{3} ) q^{19} + ( -39 \beta_{1} - 71 \beta_{2} ) q^{20} + ( -18 + 11 \beta_{3} ) q^{22} + ( -8 \beta_{1} - 43 \beta_{2} ) q^{23} + ( 64 - 8 \beta_{3} ) q^{25} + ( 56 \beta_{1} + 104 \beta_{2} ) q^{26} + ( -42 + 7 \beta_{3} ) q^{28} + ( 22 \beta_{1} + 28 \beta_{2} ) q^{29} + ( 107 + 10 \beta_{3} ) q^{31} + ( -65 \beta_{1} - 41 \beta_{2} ) q^{32} + ( 44 - 2 \beta_{3} ) q^{34} + ( 28 \beta_{1} + 21 \beta_{2} ) q^{35} + ( 327 - 6 \beta_{3} ) q^{37} + ( -17 \beta_{1} + 15 \beta_{2} ) q^{38} + ( 338 - 47 \beta_{3} ) q^{40} + ( 22 \beta_{1} + 111 \beta_{2} ) q^{41} + ( 28 - 2 \beta_{3} ) q^{43} + ( 143 \beta_{1} + 95 \beta_{2} ) q^{44} + ( 322 - 43 \beta_{3} ) q^{46} + ( 114 \beta_{1} - 6 \beta_{2} ) q^{47} + 49 q^{49} + ( -120 \beta_{1} - 184 \beta_{2} ) q^{50} + ( -960 + 56 \beta_{3} ) q^{52} + ( 148 \beta_{1} + 20 \beta_{2} ) q^{53} + ( 113 + 50 \beta_{3} ) q^{55} + ( 35 \beta_{1} + 91 \beta_{2} ) q^{56} + ( -344 + 28 \beta_{3} ) q^{58} + ( -94 \beta_{1} - 74 \beta_{2} ) q^{59} + ( -672 - 8 \beta_{3} ) q^{61} + ( -37 \beta_{1} + 43 \beta_{2} ) q^{62} + ( 206 - \beta_{3} ) q^{64} + ( 146 \beta_{1} + 360 \beta_{2} ) q^{65} + ( 342 - 38 \beta_{3} ) q^{67} + ( -26 \beta_{1} - 58 \beta_{2} ) q^{68} + ( -350 + 21 \beta_{3} ) q^{70} + ( -136 \beta_{1} + 61 \beta_{2} ) q^{71} + ( -104 - 10 \beta_{3} ) q^{73} + ( -369 \beta_{1} - 417 \beta_{2} ) q^{74} + ( -314 - 17 \beta_{3} ) q^{76} + ( 42 \beta_{1} - 77 \beta_{2} ) q^{77} + ( 292 - 26 \beta_{3} ) q^{79} + ( -355 \beta_{1} - 475 \beta_{2} ) q^{80} + ( -842 + 111 \beta_{3} ) q^{82} + ( 158 \beta_{1} + 4 \beta_{2} ) q^{83} + ( 178 - 4 \beta_{3} ) q^{85} + ( -42 \beta_{1} - 58 \beta_{2} ) q^{86} + ( -1570 + 7 \beta_{3} ) q^{88} + ( 110 \beta_{1} + 87 \beta_{2} ) q^{89} + ( 98 - 42 \beta_{3} ) q^{91} + ( -559 \beta_{1} - 623 \beta_{2} ) q^{92} + ( -876 - 6 \beta_{3} ) q^{94} + ( -120 \beta_{1} + 77 \beta_{2} ) q^{95} + ( -152 + 8 \beta_{3} ) q^{97} + ( -49 \beta_{1} - 49 \beta_{2} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 26q^{4} - 28q^{7} + O(q^{10})$$ $$4q + 26q^{4} - 28q^{7} + 206q^{10} - 68q^{13} + 290q^{16} + 172q^{19} - 94q^{22} + 272q^{25} - 182q^{28} + 408q^{31} + 180q^{34} + 1320q^{37} + 1446q^{40} + 116q^{43} + 1374q^{46} + 196q^{49} - 3952q^{52} + 352q^{55} - 1432q^{58} - 2672q^{61} + 826q^{64} + 1444q^{67} - 1442q^{70} - 396q^{73} - 1222q^{76} + 1220q^{79} - 3590q^{82} + 720q^{85} - 6294q^{88} + 476q^{91} - 3492q^{94} - 624q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 9 x^{2} + 4$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{3} - 5 \nu$$$$)/2$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{3} - 11 \nu$$$$)/2$$ $$\beta_{3}$$ $$=$$ $$3 \nu^{2} - 14$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{2} + \beta_{1}$$$$)/3$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{3} + 14$$$$)/3$$ $$\nu^{3}$$ $$=$$ $$($$$$-5 \beta_{2} + 11 \beta_{1}$$$$)/3$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −0.684742 2.92081 −2.92081 0.684742
−5.15688 0 18.5934 −17.0220 0 −7.00000 −54.6288 0 87.7802
1.2 −1.55133 0 −5.59339 −9.81086 0 −7.00000 21.0878 0 15.2198
1.3 1.55133 0 −5.59339 9.81086 0 −7.00000 −21.0878 0 15.2198
1.4 5.15688 0 18.5934 17.0220 0 −7.00000 54.6288 0 87.7802
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$1$$
$$7$$ $$1$$

## 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 189.4.a.l 4
3.b odd 2 1 inner 189.4.a.l 4
7.b odd 2 1 1323.4.a.ba 4
21.c even 2 1 1323.4.a.ba 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
189.4.a.l 4 1.a even 1 1 trivial
189.4.a.l 4 3.b odd 2 1 inner
1323.4.a.ba 4 7.b odd 2 1
1323.4.a.ba 4 21.c even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(189))$$:

 $$T_{2}^{4} - 29 T_{2}^{2} + 64$$ $$T_{5}^{4} - 386 T_{5}^{2} + 27889$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$64 - 29 T^{2} + T^{4}$$
$3$ $$T^{4}$$
$5$ $$27889 - 386 T^{2} + T^{4}$$
$7$ $$( 7 + T )^{4}$$
$11$ $$4592449 - 5906 T^{2} + T^{4}$$
$13$ $$( -4976 + 34 T + T^{2} )^{2}$$
$17$ $$( -180 + T^{2} )^{2}$$
$19$ $$( -491 - 86 T + T^{2} )^{2}$$
$23$ $$363016809 - 40986 T^{2} + T^{4}$$
$29$ $$2849344 - 18404 T^{2} + T^{4}$$
$31$ $$( -4221 - 204 T + T^{2} )^{2}$$
$37$ $$( 103635 - 660 T + T^{2} )^{2}$$
$41$ $$15513948025 - 270890 T^{2} + T^{4}$$
$43$ $$( 256 - 58 T + T^{2} )^{2}$$
$47$ $$8950673664 - 395604 T^{2} + T^{4}$$
$53$ $$43477254144 - 568656 T^{2} + T^{4}$$
$59$ $$8087045184 - 217764 T^{2} + T^{4}$$
$61$ $$( 436864 + 1336 T + T^{2} )^{2}$$
$67$ $$( -80864 - 722 T + T^{2} )^{2}$$
$71$ $$68112009 - 848826 T^{2} + T^{4}$$
$73$ $$( -4824 + 198 T + T^{2} )^{2}$$
$79$ $$( -5840 - 610 T + T^{2} )^{2}$$
$83$ $$43147598400 - 707940 T^{2} + T^{4}$$
$89$ $$15083032969 - 298874 T^{2} + T^{4}$$
$97$ $$( 14976 + 312 T + T^{2} )^{2}$$