# Properties

 Label 819.1.ff.a Level $819$ Weight $1$ Character orbit 819.ff Analytic conductor $0.409$ Analytic rank $0$ Dimension $4$ Projective image $D_{12}$ CM discriminant -3 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$819 = 3^{2} \cdot 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 819.ff (of order $$12$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.408734245346$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{4}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{12}$$ Projective field: Galois closure of $$\mathbb{Q}[x]/(x^{12} - \cdots)$$

## $q$-expansion

The $$q$$-expansion and trace form are shown below.

 $$f(q)$$ $$=$$ $$q -\zeta_{12} q^{4} + \zeta_{12}^{2} q^{7} +O(q^{10})$$ $$q -\zeta_{12} q^{4} + \zeta_{12}^{2} q^{7} + \zeta_{12} q^{13} + \zeta_{12}^{2} q^{16} + ( -\zeta_{12}^{4} - \zeta_{12}^{5} ) q^{19} -\zeta_{12}^{5} q^{25} -\zeta_{12}^{3} q^{28} + ( -\zeta_{12}^{2} + \zeta_{12}^{5} ) q^{31} + ( \zeta_{12}^{3} + \zeta_{12}^{4} ) q^{37} + ( 1 - \zeta_{12}^{4} ) q^{43} + \zeta_{12}^{4} q^{49} -\zeta_{12}^{2} q^{52} + ( -\zeta_{12} + \zeta_{12}^{5} ) q^{61} -\zeta_{12}^{3} q^{64} + ( -1 + \zeta_{12}^{3} ) q^{67} + ( -1 + \zeta_{12} ) q^{73} + ( -1 + \zeta_{12}^{5} ) q^{76} + \zeta_{12}^{3} q^{91} + ( -\zeta_{12}^{3} + \zeta_{12}^{4} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 2 q^{7} + O(q^{10})$$ $$4 q + 2 q^{7} + 2 q^{16} + 2 q^{19} - 2 q^{31} - 2 q^{37} + 6 q^{43} - 2 q^{49} - 2 q^{52} - 4 q^{67} - 4 q^{73} - 4 q^{76} - 2 q^{97} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/819\mathbb{Z}\right)^\times$$.

 $$n$$ $$92$$ $$379$$ $$703$$ $$\chi(n)$$ $$1$$ $$\zeta_{12}^{5}$$ $$\zeta_{12}^{4}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
163.1
 −0.866025 − 0.500000i 0.866025 + 0.500000i −0.866025 + 0.500000i 0.866025 − 0.500000i
0 0 0.866025 + 0.500000i 0 0 0.500000 + 0.866025i 0 0 0
604.1 0 0 −0.866025 0.500000i 0 0 0.500000 + 0.866025i 0 0 0
613.1 0 0 0.866025 0.500000i 0 0 0.500000 0.866025i 0 0 0
739.1 0 0 −0.866025 + 0.500000i 0 0 0.500000 0.866025i 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 $$\Q(\sqrt{-3})$$
91.bd odd 12 1 inner
273.bw even 12 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 819.1.ff.a yes 4
3.b odd 2 1 CM 819.1.ff.a yes 4
7.c even 3 1 819.1.eu.a 4
13.f odd 12 1 819.1.eu.a 4
21.h odd 6 1 819.1.eu.a 4
39.k even 12 1 819.1.eu.a 4
91.bd odd 12 1 inner 819.1.ff.a yes 4
273.bw even 12 1 inner 819.1.ff.a yes 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
819.1.eu.a 4 7.c even 3 1
819.1.eu.a 4 13.f odd 12 1
819.1.eu.a 4 21.h odd 6 1
819.1.eu.a 4 39.k even 12 1
819.1.ff.a yes 4 1.a even 1 1 trivial
819.1.ff.a yes 4 3.b odd 2 1 CM
819.1.ff.a yes 4 91.bd odd 12 1 inner
819.1.ff.a yes 4 273.bw even 12 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(819, [\chi])$$.

## Hecke characteristic polynomials

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