# Properties

 Label 189.2.a.d Level 189 Weight 2 Character orbit 189.a Self dual yes Analytic conductor 1.509 Analytic rank 0 Dimension 1 CM no Inner twists 1

# Related objects

## Newspace parameters

 Level: $$N$$ = $$189 = 3^{3} \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 189.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$1.50917259820$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

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

## 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
2.00000 0 2.00000 1.00000 0 −1.00000 0 0 2.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 189.2.a.d yes 1
3.b odd 2 1 189.2.a.a 1
4.b odd 2 1 3024.2.a.u 1
5.b even 2 1 4725.2.a.c 1
7.b odd 2 1 1323.2.a.r 1
9.c even 3 2 567.2.f.a 2
9.d odd 6 2 567.2.f.h 2
12.b even 2 1 3024.2.a.l 1
15.d odd 2 1 4725.2.a.s 1
21.c even 2 1 1323.2.a.b 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
189.2.a.a 1 3.b odd 2 1
189.2.a.d yes 1 1.a even 1 1 trivial
567.2.f.a 2 9.c even 3 2
567.2.f.h 2 9.d odd 6 2
1323.2.a.b 1 21.c even 2 1
1323.2.a.r 1 7.b odd 2 1
3024.2.a.l 1 12.b even 2 1
3024.2.a.u 1 4.b odd 2 1
4725.2.a.c 1 5.b even 2 1
4725.2.a.s 1 15.d odd 2 1

## Atkin-Lehner signs

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

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(189))$$:

 $$T_{2} - 2$$ $$T_{5} - 1$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 - 2 T + 2 T^{2}$$
$3$ 1
$5$ $$1 - T + 5 T^{2}$$
$7$ $$1 + T$$
$11$ $$1 - 4 T + 11 T^{2}$$
$13$ $$1 + 2 T + 13 T^{2}$$
$17$ $$1 + 3 T + 17 T^{2}$$
$19$ $$1 + 8 T + 19 T^{2}$$
$23$ $$1 - 6 T + 23 T^{2}$$
$29$ $$1 - 4 T + 29 T^{2}$$
$31$ $$1 - 6 T + 31 T^{2}$$
$37$ $$1 + 3 T + 37 T^{2}$$
$41$ $$1 + T + 41 T^{2}$$
$43$ $$1 - 11 T + 43 T^{2}$$
$47$ $$1 + 9 T + 47 T^{2}$$
$53$ $$1 + 6 T + 53 T^{2}$$
$59$ $$1 - 15 T + 59 T^{2}$$
$61$ $$1 - 4 T + 61 T^{2}$$
$67$ $$1 + 8 T + 67 T^{2}$$
$71$ $$1 - 12 T + 71 T^{2}$$
$73$ $$1 - 6 T + 73 T^{2}$$
$79$ $$1 + T + 79 T^{2}$$
$83$ $$1 - 9 T + 83 T^{2}$$
$89$ $$1 + 2 T + 89 T^{2}$$
$97$ $$1 - 12 T + 97 T^{2}$$