Properties

Label 819.2.bm.c
Level $819$
Weight $2$
Character orbit 819.bm
Analytic conductor $6.540$
Analytic rank $0$
Dimension $2$
CM discriminant -3
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 819 = 3^{2} \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 819.bm (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.53974792554\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 q^{4} + ( -3 + 2 \zeta_{6} ) q^{7} +O(q^{10})\) \( q + 2 q^{4} + ( -3 + 2 \zeta_{6} ) q^{7} + ( -3 + 4 \zeta_{6} ) q^{13} + 4 q^{16} + ( 5 + 5 \zeta_{6} ) q^{19} + ( -5 + 5 \zeta_{6} ) q^{25} + ( -6 + 4 \zeta_{6} ) q^{28} + ( 5 + 5 \zeta_{6} ) q^{31} + ( 3 - 6 \zeta_{6} ) q^{37} -8 \zeta_{6} q^{43} + ( 5 - 8 \zeta_{6} ) q^{49} + ( -6 + 8 \zeta_{6} ) q^{52} + ( -1 + \zeta_{6} ) q^{61} + 8 q^{64} + ( -14 + 7 \zeta_{6} ) q^{67} + ( 8 + 8 \zeta_{6} ) q^{73} + ( 10 + 10 \zeta_{6} ) q^{76} -13 \zeta_{6} q^{79} + ( 1 - 10 \zeta_{6} ) q^{91} + ( -6 + 3 \zeta_{6} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{4} - 4 q^{7} + O(q^{10}) \) \( 2 q + 4 q^{4} - 4 q^{7} - 2 q^{13} + 8 q^{16} + 15 q^{19} - 5 q^{25} - 8 q^{28} + 15 q^{31} - 8 q^{43} + 2 q^{49} - 4 q^{52} - q^{61} + 16 q^{64} - 21 q^{67} + 24 q^{73} + 30 q^{76} - 13 q^{79} - 8 q^{91} - 9 q^{97} + O(q^{100}) \)

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
478.1
0.500000 0.866025i
0.500000 + 0.866025i
0 0 2.00000 0 0 −2.00000 1.73205i 0 0 0
550.1 0 0 2.00000 0 0 −2.00000 + 1.73205i 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.k even 6 1 inner
273.bp odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 819.2.bm.c 2
3.b odd 2 1 CM 819.2.bm.c 2
7.c even 3 1 819.2.do.b yes 2
13.e even 6 1 819.2.do.b yes 2
21.h odd 6 1 819.2.do.b yes 2
39.h odd 6 1 819.2.do.b yes 2
91.k even 6 1 inner 819.2.bm.c 2
273.bp odd 6 1 inner 819.2.bm.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
819.2.bm.c 2 1.a even 1 1 trivial
819.2.bm.c 2 3.b odd 2 1 CM
819.2.bm.c 2 91.k even 6 1 inner
819.2.bm.c 2 273.bp odd 6 1 inner
819.2.do.b yes 2 7.c even 3 1
819.2.do.b yes 2 13.e even 6 1
819.2.do.b yes 2 21.h odd 6 1
819.2.do.b yes 2 39.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(819, [\chi])\):

\( T_{2} \)
\( T_{5} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( T^{2} \)
$7$ \( 7 + 4 T + T^{2} \)
$11$ \( T^{2} \)
$13$ \( 13 + 2 T + T^{2} \)
$17$ \( T^{2} \)
$19$ \( 75 - 15 T + T^{2} \)
$23$ \( T^{2} \)
$29$ \( T^{2} \)
$31$ \( 75 - 15 T + T^{2} \)
$37$ \( 27 + T^{2} \)
$41$ \( T^{2} \)
$43$ \( 64 + 8 T + T^{2} \)
$47$ \( T^{2} \)
$53$ \( T^{2} \)
$59$ \( T^{2} \)
$61$ \( 1 + T + T^{2} \)
$67$ \( 147 + 21 T + T^{2} \)
$71$ \( T^{2} \)
$73$ \( 192 - 24 T + T^{2} \)
$79$ \( 169 + 13 T + T^{2} \)
$83$ \( T^{2} \)
$89$ \( T^{2} \)
$97$ \( 27 + 9 T + T^{2} \)
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