# Properties

 Label 27.3.b.a Level 27 Weight 3 Character orbit 27.b Self dual yes Analytic conductor 0.736 Analytic rank 0 Dimension 1 CM discriminant -3 Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$27 = 3^{3}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 27.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: yes Analytic conductor: $$0.735696713773$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + 4q^{4} - 13q^{7} + O(q^{10})$$ $$q + 4q^{4} - 13q^{7} - q^{13} + 16q^{16} + 11q^{19} + 25q^{25} - 52q^{28} - 46q^{31} + 47q^{37} - 22q^{43} + 120q^{49} - 4q^{52} - 121q^{61} + 64q^{64} - 109q^{67} - 97q^{73} + 44q^{76} + 131q^{79} + 13q^{91} + 167q^{97} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/27\mathbb{Z}\right)^\times$$.

 $$n$$ $$2$$ $$\chi(n)$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
26.1
 0
0 0 4.00000 0 0 −13.0000 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by $$\Q(\sqrt{-3})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 27.3.b.a 1
3.b odd 2 1 CM 27.3.b.a 1
4.b odd 2 1 432.3.e.b 1
5.b even 2 1 675.3.c.c 1
5.c odd 4 2 675.3.d.c 2
8.b even 2 1 1728.3.e.a 1
8.d odd 2 1 1728.3.e.d 1
9.c even 3 2 81.3.d.a 2
9.d odd 6 2 81.3.d.a 2
12.b even 2 1 432.3.e.b 1
15.d odd 2 1 675.3.c.c 1
15.e even 4 2 675.3.d.c 2
24.f even 2 1 1728.3.e.d 1
24.h odd 2 1 1728.3.e.a 1
36.f odd 6 2 1296.3.q.a 2
36.h even 6 2 1296.3.q.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
27.3.b.a 1 1.a even 1 1 trivial
27.3.b.a 1 3.b odd 2 1 CM
81.3.d.a 2 9.c even 3 2
81.3.d.a 2 9.d odd 6 2
432.3.e.b 1 4.b odd 2 1
432.3.e.b 1 12.b even 2 1
675.3.c.c 1 5.b even 2 1
675.3.c.c 1 15.d odd 2 1
675.3.d.c 2 5.c odd 4 2
675.3.d.c 2 15.e even 4 2
1296.3.q.a 2 36.f odd 6 2
1296.3.q.a 2 36.h even 6 2
1728.3.e.a 1 8.b even 2 1
1728.3.e.a 1 24.h odd 2 1
1728.3.e.d 1 8.d odd 2 1
1728.3.e.d 1 24.f even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}$$ acting on $$S_{3}^{\mathrm{new}}(27, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 - 2 T )( 1 + 2 T )$$
$3$ 1
$5$ $$( 1 - 5 T )( 1 + 5 T )$$
$7$ $$1 + 13 T + 49 T^{2}$$
$11$ $$( 1 - 11 T )( 1 + 11 T )$$
$13$ $$1 + T + 169 T^{2}$$
$17$ $$( 1 - 17 T )( 1 + 17 T )$$
$19$ $$1 - 11 T + 361 T^{2}$$
$23$ $$( 1 - 23 T )( 1 + 23 T )$$
$29$ $$( 1 - 29 T )( 1 + 29 T )$$
$31$ $$1 + 46 T + 961 T^{2}$$
$37$ $$1 - 47 T + 1369 T^{2}$$
$41$ $$( 1 - 41 T )( 1 + 41 T )$$
$43$ $$1 + 22 T + 1849 T^{2}$$
$47$ $$( 1 - 47 T )( 1 + 47 T )$$
$53$ $$( 1 - 53 T )( 1 + 53 T )$$
$59$ $$( 1 - 59 T )( 1 + 59 T )$$
$61$ $$1 + 121 T + 3721 T^{2}$$
$67$ $$1 + 109 T + 4489 T^{2}$$
$71$ $$( 1 - 71 T )( 1 + 71 T )$$
$73$ $$1 + 97 T + 5329 T^{2}$$
$79$ $$1 - 131 T + 6241 T^{2}$$
$83$ $$( 1 - 83 T )( 1 + 83 T )$$
$89$ $$( 1 - 89 T )( 1 + 89 T )$$
$97$ $$1 - 167 T + 9409 T^{2}$$