Properties

Label 819.1.cg.a
Level $819$
Weight $1$
Character orbit 819.cg
Analytic conductor $0.409$
Analytic rank $0$
Dimension $2$
Projective image $D_{6}$
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.cg (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.408734245346\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{6}\)
Projective field: Galois closure of 6.2.12960667629.4

$q$-expansion

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q -\zeta_{6}^{2} q^{4} + \zeta_{6} q^{7} +O(q^{10})\) \( q -\zeta_{6}^{2} q^{4} + \zeta_{6} q^{7} -\zeta_{6}^{2} q^{13} -\zeta_{6} q^{16} + ( \zeta_{6} + \zeta_{6}^{2} ) q^{19} -\zeta_{6} q^{25} + q^{28} -\zeta_{6} q^{37} -\zeta_{6} q^{43} + \zeta_{6}^{2} q^{49} -\zeta_{6} q^{52} + ( \zeta_{6} + \zeta_{6}^{2} ) q^{61} - q^{64} + 2 q^{67} + ( -1 + \zeta_{6}^{2} ) q^{73} + ( 1 + \zeta_{6} ) q^{76} + 2 \zeta_{6}^{2} q^{79} + q^{91} + ( -1 + \zeta_{6}^{2} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{4} + q^{7} + O(q^{10}) \) \( 2 q + q^{4} + q^{7} + q^{13} - q^{16} - q^{25} + 2 q^{28} - q^{37} - q^{43} - q^{49} - q^{52} - 2 q^{64} + 4 q^{67} - 3 q^{73} + 3 q^{76} - 2 q^{79} + 2 q^{91} - 3 q^{97} + O(q^{100}) \)

Character values

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

\(n\) \(92\) \(379\) \(703\)
\(\chi(n)\) \(1\) \(-\zeta_{6}\) \(-\zeta_{6}^{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
334.1
0.500000 + 0.866025i
0.500000 0.866025i
0 0 0.500000 0.866025i 0 0 0.500000 + 0.866025i 0 0 0
640.1 0 0 0.500000 + 0.866025i 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.m odd 6 1 inner
273.bf even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 819.1.cg.a 2
3.b odd 2 1 CM 819.1.cg.a 2
7.d odd 6 1 819.1.ef.a yes 2
13.c even 3 1 819.1.ef.a yes 2
21.g even 6 1 819.1.ef.a yes 2
39.i odd 6 1 819.1.ef.a yes 2
91.m odd 6 1 inner 819.1.cg.a 2
273.bf even 6 1 inner 819.1.cg.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
819.1.cg.a 2 1.a even 1 1 trivial
819.1.cg.a 2 3.b odd 2 1 CM
819.1.cg.a 2 91.m odd 6 1 inner
819.1.cg.a 2 273.bf even 6 1 inner
819.1.ef.a yes 2 7.d odd 6 1
819.1.ef.a yes 2 13.c even 3 1
819.1.ef.a yes 2 21.g even 6 1
819.1.ef.a yes 2 39.i odd 6 1

Hecke kernels

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

Hecke characteristic polynomials

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