Properties

Label 819.2.n.c
Level $819$
Weight $2$
Character orbit 819.n
Analytic conductor $6.540$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 819 = 3^{2} \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 819.n (of order \(3\), 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: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 91)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + \zeta_{6} q^{2} + ( 1 - \zeta_{6} ) q^{4} + ( 3 - 3 \zeta_{6} ) q^{5} + ( -3 + \zeta_{6} ) q^{7} + 3 q^{8} +O(q^{10})\) \( q + \zeta_{6} q^{2} + ( 1 - \zeta_{6} ) q^{4} + ( 3 - 3 \zeta_{6} ) q^{5} + ( -3 + \zeta_{6} ) q^{7} + 3 q^{8} + 3 q^{10} + 3 q^{11} + ( 1 - 4 \zeta_{6} ) q^{13} + ( -1 - 2 \zeta_{6} ) q^{14} + \zeta_{6} q^{16} + ( -2 + 2 \zeta_{6} ) q^{17} - q^{19} -3 \zeta_{6} q^{20} + 3 \zeta_{6} q^{22} -4 \zeta_{6} q^{25} + ( 4 - 3 \zeta_{6} ) q^{26} + ( -2 + 3 \zeta_{6} ) q^{28} + ( 7 - 7 \zeta_{6} ) q^{29} -3 \zeta_{6} q^{31} + ( 5 - 5 \zeta_{6} ) q^{32} -2 q^{34} + ( -6 + 9 \zeta_{6} ) q^{35} -2 \zeta_{6} q^{37} -\zeta_{6} q^{38} + ( 9 - 9 \zeta_{6} ) q^{40} + ( 3 - 3 \zeta_{6} ) q^{41} + 7 \zeta_{6} q^{43} + ( 3 - 3 \zeta_{6} ) q^{44} + ( 1 - \zeta_{6} ) q^{47} + ( 8 - 5 \zeta_{6} ) q^{49} + ( 4 - 4 \zeta_{6} ) q^{50} + ( -3 - \zeta_{6} ) q^{52} + 3 \zeta_{6} q^{53} + ( 9 - 9 \zeta_{6} ) q^{55} + ( -9 + 3 \zeta_{6} ) q^{56} + 7 q^{58} + ( -4 + 4 \zeta_{6} ) q^{59} -13 q^{61} + ( 3 - 3 \zeta_{6} ) q^{62} + 7 q^{64} + ( -9 - 3 \zeta_{6} ) q^{65} -3 q^{67} + 2 \zeta_{6} q^{68} + ( -9 + 3 \zeta_{6} ) q^{70} + 13 \zeta_{6} q^{71} + 13 \zeta_{6} q^{73} + ( 2 - 2 \zeta_{6} ) q^{74} + ( -1 + \zeta_{6} ) q^{76} + ( -9 + 3 \zeta_{6} ) q^{77} + ( 3 - 3 \zeta_{6} ) q^{79} + 3 q^{80} + 3 q^{82} + 6 \zeta_{6} q^{85} + ( -7 + 7 \zeta_{6} ) q^{86} + 9 q^{88} + 6 \zeta_{6} q^{89} + ( 1 + 9 \zeta_{6} ) q^{91} + q^{94} + ( -3 + 3 \zeta_{6} ) q^{95} + 5 \zeta_{6} q^{97} + ( 5 + 3 \zeta_{6} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + q^{2} + q^{4} + 3q^{5} - 5q^{7} + 6q^{8} + O(q^{10}) \) \( 2q + q^{2} + q^{4} + 3q^{5} - 5q^{7} + 6q^{8} + 6q^{10} + 6q^{11} - 2q^{13} - 4q^{14} + q^{16} - 2q^{17} - 2q^{19} - 3q^{20} + 3q^{22} - 4q^{25} + 5q^{26} - q^{28} + 7q^{29} - 3q^{31} + 5q^{32} - 4q^{34} - 3q^{35} - 2q^{37} - q^{38} + 9q^{40} + 3q^{41} + 7q^{43} + 3q^{44} + q^{47} + 11q^{49} + 4q^{50} - 7q^{52} + 3q^{53} + 9q^{55} - 15q^{56} + 14q^{58} - 4q^{59} - 26q^{61} + 3q^{62} + 14q^{64} - 21q^{65} - 6q^{67} + 2q^{68} - 15q^{70} + 13q^{71} + 13q^{73} + 2q^{74} - q^{76} - 15q^{77} + 3q^{79} + 6q^{80} + 6q^{82} + 6q^{85} - 7q^{86} + 18q^{88} + 6q^{89} + 11q^{91} + 2q^{94} - 3q^{95} + 5q^{97} + 13q^{98} + O(q^{100}) \)

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
100.1
0.500000 + 0.866025i
0.500000 0.866025i
0.500000 + 0.866025i 0 0.500000 0.866025i 1.50000 2.59808i 0 −2.50000 + 0.866025i 3.00000 0 3.00000
172.1 0.500000 0.866025i 0 0.500000 + 0.866025i 1.50000 + 2.59808i 0 −2.50000 0.866025i 3.00000 0 3.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
91.g even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 819.2.n.c 2
3.b odd 2 1 91.2.g.a 2
7.c even 3 1 819.2.s.a 2
13.c even 3 1 819.2.s.a 2
21.c even 2 1 637.2.g.a 2
21.g even 6 1 637.2.f.a 2
21.g even 6 1 637.2.h.a 2
21.h odd 6 1 91.2.h.a yes 2
21.h odd 6 1 637.2.f.b 2
39.h odd 6 1 1183.2.e.c 2
39.i odd 6 1 91.2.h.a yes 2
39.i odd 6 1 1183.2.e.a 2
91.g even 3 1 inner 819.2.n.c 2
273.r even 6 1 637.2.f.a 2
273.s odd 6 1 637.2.f.b 2
273.s odd 6 1 1183.2.e.a 2
273.x odd 6 1 8281.2.a.c 1
273.y even 6 1 8281.2.a.g 1
273.bf even 6 1 637.2.g.a 2
273.bf even 6 1 8281.2.a.j 1
273.bm odd 6 1 91.2.g.a 2
273.bm odd 6 1 8281.2.a.i 1
273.bn even 6 1 637.2.h.a 2
273.bp odd 6 1 1183.2.e.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.g.a 2 3.b odd 2 1
91.2.g.a 2 273.bm odd 6 1
91.2.h.a yes 2 21.h odd 6 1
91.2.h.a yes 2 39.i odd 6 1
637.2.f.a 2 21.g even 6 1
637.2.f.a 2 273.r even 6 1
637.2.f.b 2 21.h odd 6 1
637.2.f.b 2 273.s odd 6 1
637.2.g.a 2 21.c even 2 1
637.2.g.a 2 273.bf even 6 1
637.2.h.a 2 21.g even 6 1
637.2.h.a 2 273.bn even 6 1
819.2.n.c 2 1.a even 1 1 trivial
819.2.n.c 2 91.g even 3 1 inner
819.2.s.a 2 7.c even 3 1
819.2.s.a 2 13.c even 3 1
1183.2.e.a 2 39.i odd 6 1
1183.2.e.a 2 273.s odd 6 1
1183.2.e.c 2 39.h odd 6 1
1183.2.e.c 2 273.bp odd 6 1
8281.2.a.c 1 273.x odd 6 1
8281.2.a.g 1 273.y even 6 1
8281.2.a.i 1 273.bm odd 6 1
8281.2.a.j 1 273.bf even 6 1

Hecke kernels

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

\( T_{2}^{2} - T_{2} + 1 \)
\( T_{11} - 3 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - T + T^{2} \)
$3$ \( T^{2} \)
$5$ \( 9 - 3 T + T^{2} \)
$7$ \( 7 + 5 T + T^{2} \)
$11$ \( ( -3 + T )^{2} \)
$13$ \( 13 + 2 T + T^{2} \)
$17$ \( 4 + 2 T + T^{2} \)
$19$ \( ( 1 + T )^{2} \)
$23$ \( T^{2} \)
$29$ \( 49 - 7 T + T^{2} \)
$31$ \( 9 + 3 T + T^{2} \)
$37$ \( 4 + 2 T + T^{2} \)
$41$ \( 9 - 3 T + T^{2} \)
$43$ \( 49 - 7 T + T^{2} \)
$47$ \( 1 - T + T^{2} \)
$53$ \( 9 - 3 T + T^{2} \)
$59$ \( 16 + 4 T + T^{2} \)
$61$ \( ( 13 + T )^{2} \)
$67$ \( ( 3 + T )^{2} \)
$71$ \( 169 - 13 T + T^{2} \)
$73$ \( 169 - 13 T + T^{2} \)
$79$ \( 9 - 3 T + T^{2} \)
$83$ \( T^{2} \)
$89$ \( 36 - 6 T + T^{2} \)
$97$ \( 25 - 5 T + T^{2} \)
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