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 $4$

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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.

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

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:

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

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