Properties

Label 819.2.s.a
Level $819$
Weight $2$
Character orbit 819.s
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.s (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: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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.

\(f(q)\) \(=\) \( q - q^{2} - q^{4} + 3 \zeta_{6} q^{5} + ( - 2 \zeta_{6} + 3) q^{7} + 3 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} - q^{4} + 3 \zeta_{6} q^{5} + ( - 2 \zeta_{6} + 3) q^{7} + 3 q^{8} - 3 \zeta_{6} q^{10} - 3 \zeta_{6} q^{11} + ( - 4 \zeta_{6} + 1) q^{13} + (2 \zeta_{6} - 3) q^{14} - q^{16} + 2 q^{17} + ( - \zeta_{6} + 1) q^{19} - 3 \zeta_{6} q^{20} + 3 \zeta_{6} q^{22} + (4 \zeta_{6} - 4) q^{25} + (4 \zeta_{6} - 1) q^{26} + (2 \zeta_{6} - 3) q^{28} + ( - 7 \zeta_{6} + 7) q^{29} + (3 \zeta_{6} - 3) q^{31} - 5 q^{32} - 2 q^{34} + (3 \zeta_{6} + 6) q^{35} + 2 q^{37} + (\zeta_{6} - 1) q^{38} + 9 \zeta_{6} q^{40} + ( - 3 \zeta_{6} + 3) q^{41} + 7 \zeta_{6} q^{43} + 3 \zeta_{6} q^{44} + \zeta_{6} q^{47} + ( - 8 \zeta_{6} + 5) q^{49} + ( - 4 \zeta_{6} + 4) q^{50} + (4 \zeta_{6} - 1) q^{52} + ( - 3 \zeta_{6} + 3) q^{53} + ( - 9 \zeta_{6} + 9) q^{55} + ( - 6 \zeta_{6} + 9) q^{56} + (7 \zeta_{6} - 7) q^{58} + 4 q^{59} + ( - 13 \zeta_{6} + 13) q^{61} + ( - 3 \zeta_{6} + 3) q^{62} + 7 q^{64} + ( - 9 \zeta_{6} + 12) q^{65} + 3 \zeta_{6} q^{67} - 2 q^{68} + ( - 3 \zeta_{6} - 6) q^{70} + 13 \zeta_{6} q^{71} + ( - 13 \zeta_{6} + 13) q^{73} - 2 q^{74} + (\zeta_{6} - 1) q^{76} + ( - 3 \zeta_{6} - 6) q^{77} + 3 \zeta_{6} q^{79} - 3 \zeta_{6} q^{80} + (3 \zeta_{6} - 3) q^{82} + 6 \zeta_{6} q^{85} - 7 \zeta_{6} q^{86} - 9 \zeta_{6} q^{88} - 6 q^{89} + ( - 6 \zeta_{6} - 5) q^{91} - \zeta_{6} q^{94} + 3 q^{95} + 5 \zeta_{6} q^{97} + (8 \zeta_{6} - 5) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} - 2 q^{4} + 3 q^{5} + 4 q^{7} + 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} - 2 q^{4} + 3 q^{5} + 4 q^{7} + 6 q^{8} - 3 q^{10} - 3 q^{11} - 2 q^{13} - 4 q^{14} - 2 q^{16} + 4 q^{17} + q^{19} - 3 q^{20} + 3 q^{22} - 4 q^{25} + 2 q^{26} - 4 q^{28} + 7 q^{29} - 3 q^{31} - 10 q^{32} - 4 q^{34} + 15 q^{35} + 4 q^{37} - q^{38} + 9 q^{40} + 3 q^{41} + 7 q^{43} + 3 q^{44} + q^{47} + 2 q^{49} + 4 q^{50} + 2 q^{52} + 3 q^{53} + 9 q^{55} + 12 q^{56} - 7 q^{58} + 8 q^{59} + 13 q^{61} + 3 q^{62} + 14 q^{64} + 15 q^{65} + 3 q^{67} - 4 q^{68} - 15 q^{70} + 13 q^{71} + 13 q^{73} - 4 q^{74} - q^{76} - 15 q^{77} + 3 q^{79} - 3 q^{80} - 3 q^{82} + 6 q^{85} - 7 q^{86} - 9 q^{88} - 12 q^{89} - 16 q^{91} - q^{94} + 6 q^{95} + 5 q^{97} - 2 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
289.1
0.500000 0.866025i
0.500000 + 0.866025i
−1.00000 0 −1.00000 1.50000 2.59808i 0 2.00000 + 1.73205i 3.00000 0 −1.50000 + 2.59808i
802.1 −1.00000 0 −1.00000 1.50000 + 2.59808i 0 2.00000 1.73205i 3.00000 0 −1.50000 2.59808i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Hecke kernels

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

\( T_{2} + 1 \) Copy content Toggle raw display
\( T_{11}^{2} + 3T_{11} + 9 \) Copy content Toggle raw display

Hecke characteristic polynomials

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