Properties

Label 819.2.o.c
Level $819$
Weight $2$
Character orbit 819.o
Analytic conductor $6.540$
Analytic rank $0$
Dimension $4$
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.o (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.53974792554\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{5})\)
Defining polynomial: \(x^{4} - x^{3} + 2 x^{2} + x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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 basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 1 + \beta_{1} + \beta_{3} ) q^{2} + ( 3 \beta_{1} + 3 \beta_{2} ) q^{4} + ( -1 + \beta_{2} ) q^{5} + \beta_{3} q^{7} + ( -1 + 4 \beta_{2} ) q^{8} +O(q^{10})\) \( q + ( 1 + \beta_{1} + \beta_{3} ) q^{2} + ( 3 \beta_{1} + 3 \beta_{2} ) q^{4} + ( -1 + \beta_{2} ) q^{5} + \beta_{3} q^{7} + ( -1 + 4 \beta_{2} ) q^{8} + ( -2 - 3 \beta_{1} - 2 \beta_{3} ) q^{10} + ( -3 + 3 \beta_{1} - 3 \beta_{3} ) q^{11} + ( -4 - 3 \beta_{3} ) q^{13} + ( -1 + \beta_{2} ) q^{14} + ( -5 - 3 \beta_{1} - 5 \beta_{3} ) q^{16} + ( 4 \beta_{1} + 4 \beta_{2} - 5 \beta_{3} ) q^{17} + ( -3 \beta_{1} - 3 \beta_{2} + 3 \beta_{3} ) q^{19} + ( -6 \beta_{1} - 6 \beta_{2} - 3 \beta_{3} ) q^{20} + ( 3 \beta_{1} + 3 \beta_{2} ) q^{22} + ( 2 - 4 \beta_{1} + 2 \beta_{3} ) q^{23} + ( -3 - 3 \beta_{2} ) q^{25} + ( -1 - 4 \beta_{1} - 3 \beta_{2} - 4 \beta_{3} ) q^{26} -3 \beta_{1} q^{28} + ( -1 + 5 \beta_{1} - \beta_{3} ) q^{29} + ( 5 + 6 \beta_{2} ) q^{31} + ( -3 \beta_{1} - 3 \beta_{2} - 6 \beta_{3} ) q^{32} + ( 1 + 3 \beta_{2} ) q^{34} + ( -\beta_{1} - \beta_{2} - \beta_{3} ) q^{35} + ( -4 - 4 \beta_{3} ) q^{37} -3 \beta_{2} q^{38} + ( 5 - 9 \beta_{2} ) q^{40} + ( 4 - 2 \beta_{1} + 4 \beta_{3} ) q^{41} + ( 9 \beta_{1} + 9 \beta_{2} - 2 \beta_{3} ) q^{43} -9 q^{44} + ( -6 \beta_{1} - 6 \beta_{2} - 2 \beta_{3} ) q^{46} + ( -1 - 2 \beta_{2} ) q^{47} + ( -1 - \beta_{3} ) q^{49} + 3 \beta_{1} q^{50} + ( -3 \beta_{1} - 12 \beta_{2} ) q^{52} + ( -7 - 2 \beta_{2} ) q^{53} -3 \beta_{1} q^{55} + ( -4 \beta_{1} - 4 \beta_{2} - \beta_{3} ) q^{56} + ( 9 \beta_{1} + 9 \beta_{2} + 4 \beta_{3} ) q^{58} + ( 2 \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{59} -6 \beta_{3} q^{61} + ( -1 - 7 \beta_{1} - \beta_{3} ) q^{62} + ( -1 - 6 \beta_{2} ) q^{64} + ( 4 + 3 \beta_{1} - \beta_{2} + 3 \beta_{3} ) q^{65} + ( 3 + 6 \beta_{1} + 3 \beta_{3} ) q^{67} + ( -12 + 3 \beta_{1} - 12 \beta_{3} ) q^{68} + ( 2 - 3 \beta_{2} ) q^{70} + ( 10 \beta_{1} + 10 \beta_{2} - 2 \beta_{3} ) q^{71} -2 q^{73} + ( -4 \beta_{1} - 4 \beta_{2} - 4 \beta_{3} ) q^{74} + ( 9 + 9 \beta_{3} ) q^{76} + ( 3 + 3 \beta_{2} ) q^{77} + 4 q^{79} + ( 8 + 11 \beta_{1} + 8 \beta_{3} ) q^{80} + 2 \beta_{3} q^{82} + ( 3 + 6 \beta_{2} ) q^{83} + ( -3 \beta_{1} - 3 \beta_{2} + \beta_{3} ) q^{85} + ( -7 + 16 \beta_{2} ) q^{86} + ( -9 - 3 \beta_{1} - 9 \beta_{3} ) q^{88} + ( 13 - 5 \beta_{1} + 13 \beta_{3} ) q^{89} + ( 3 - \beta_{3} ) q^{91} + ( 12 - 6 \beta_{2} ) q^{92} + ( 1 + 3 \beta_{1} + \beta_{3} ) q^{94} + ( 3 \beta_{1} + 3 \beta_{2} ) q^{95} + ( 3 \beta_{1} + 3 \beta_{2} + 14 \beta_{3} ) q^{97} + ( -\beta_{1} - \beta_{2} - \beta_{3} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 3q^{2} - 3q^{4} - 6q^{5} - 2q^{7} - 12q^{8} + O(q^{10}) \) \( 4q + 3q^{2} - 3q^{4} - 6q^{5} - 2q^{7} - 12q^{8} - 7q^{10} - 3q^{11} - 10q^{13} - 6q^{14} - 13q^{16} + 6q^{17} - 3q^{19} + 12q^{20} - 3q^{22} - 6q^{25} + 6q^{26} - 3q^{28} + 3q^{29} + 8q^{31} + 15q^{32} - 2q^{34} + 3q^{35} - 8q^{37} + 6q^{38} + 38q^{40} + 6q^{41} - 5q^{43} - 36q^{44} + 10q^{46} - 2q^{49} + 3q^{50} + 21q^{52} - 24q^{53} - 3q^{55} + 6q^{56} - 17q^{58} + 12q^{61} - 9q^{62} + 8q^{64} + 15q^{65} + 12q^{67} - 21q^{68} + 14q^{70} - 6q^{71} - 8q^{73} + 12q^{74} + 18q^{76} + 6q^{77} + 16q^{79} + 27q^{80} - 4q^{82} + q^{85} - 60q^{86} - 21q^{88} + 21q^{89} + 14q^{91} + 60q^{92} + 5q^{94} - 3q^{95} - 31q^{97} + 3q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - x^{3} + 2 x^{2} + x + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{3} + 1 \)\()/2\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{3} + 2 \nu^{2} - 2 \nu - 1 \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3} + \beta_{2} + \beta_{1}\)
\(\nu^{3}\)\(=\)\(2 \beta_{2} - 1\)

Character values

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

\(n\) \(92\) \(379\) \(703\)
\(\chi(n)\) \(1\) \(\beta_{3}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
568.1
−0.309017 + 0.535233i
0.809017 1.40126i
−0.309017 0.535233i
0.809017 + 1.40126i
0.190983 0.330792i 0 0.927051 + 1.60570i −0.381966 0 −0.500000 0.866025i 1.47214 0 −0.0729490 + 0.126351i
568.2 1.30902 2.26728i 0 −2.42705 4.20378i −2.61803 0 −0.500000 0.866025i −7.47214 0 −3.42705 + 5.93583i
757.1 0.190983 + 0.330792i 0 0.927051 1.60570i −0.381966 0 −0.500000 + 0.866025i 1.47214 0 −0.0729490 0.126351i
757.2 1.30902 + 2.26728i 0 −2.42705 + 4.20378i −2.61803 0 −0.500000 + 0.866025i −7.47214 0 −3.42705 5.93583i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.c 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.o.c 4
3.b odd 2 1 91.2.f.a 4
12.b even 2 1 1456.2.s.h 4
13.c even 3 1 inner 819.2.o.c 4
21.c even 2 1 637.2.f.c 4
21.g even 6 1 637.2.g.c 4
21.g even 6 1 637.2.h.f 4
21.h odd 6 1 637.2.g.b 4
21.h odd 6 1 637.2.h.g 4
39.h odd 6 1 1183.2.a.c 2
39.i odd 6 1 91.2.f.a 4
39.i odd 6 1 1183.2.a.g 2
39.k even 12 2 1183.2.c.c 4
156.p even 6 1 1456.2.s.h 4
273.r even 6 1 637.2.g.c 4
273.s odd 6 1 637.2.g.b 4
273.u even 6 1 8281.2.a.n 2
273.bf even 6 1 637.2.h.f 4
273.bm odd 6 1 637.2.h.g 4
273.bn even 6 1 637.2.f.c 4
273.bn even 6 1 8281.2.a.bb 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.f.a 4 3.b odd 2 1
91.2.f.a 4 39.i odd 6 1
637.2.f.c 4 21.c even 2 1
637.2.f.c 4 273.bn even 6 1
637.2.g.b 4 21.h odd 6 1
637.2.g.b 4 273.s odd 6 1
637.2.g.c 4 21.g even 6 1
637.2.g.c 4 273.r even 6 1
637.2.h.f 4 21.g even 6 1
637.2.h.f 4 273.bf even 6 1
637.2.h.g 4 21.h odd 6 1
637.2.h.g 4 273.bm odd 6 1
819.2.o.c 4 1.a even 1 1 trivial
819.2.o.c 4 13.c even 3 1 inner
1183.2.a.c 2 39.h odd 6 1
1183.2.a.g 2 39.i odd 6 1
1183.2.c.c 4 39.k even 12 2
1456.2.s.h 4 12.b even 2 1
1456.2.s.h 4 156.p even 6 1
8281.2.a.n 2 273.u even 6 1
8281.2.a.bb 2 273.bn even 6 1

Hecke kernels

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

\( T_{2}^{4} - 3 T_{2}^{3} + 8 T_{2}^{2} - 3 T_{2} + 1 \)
\( T_{11}^{4} + 3 T_{11}^{3} + 18 T_{11}^{2} - 27 T_{11} + 81 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 3 T + 8 T^{2} - 3 T^{3} + T^{4} \)
$3$ \( T^{4} \)
$5$ \( ( 1 + 3 T + T^{2} )^{2} \)
$7$ \( ( 1 + T + T^{2} )^{2} \)
$11$ \( 81 - 27 T + 18 T^{2} + 3 T^{3} + T^{4} \)
$13$ \( ( 13 + 5 T + T^{2} )^{2} \)
$17$ \( 121 + 66 T + 47 T^{2} - 6 T^{3} + T^{4} \)
$19$ \( 81 - 27 T + 18 T^{2} + 3 T^{3} + T^{4} \)
$23$ \( 400 + 20 T^{2} + T^{4} \)
$29$ \( 841 + 87 T + 38 T^{2} - 3 T^{3} + T^{4} \)
$31$ \( ( -41 - 4 T + T^{2} )^{2} \)
$37$ \( ( 16 + 4 T + T^{2} )^{2} \)
$41$ \( 16 - 24 T + 32 T^{2} - 6 T^{3} + T^{4} \)
$43$ \( 9025 - 475 T + 120 T^{2} + 5 T^{3} + T^{4} \)
$47$ \( ( -5 + T^{2} )^{2} \)
$53$ \( ( 31 + 12 T + T^{2} )^{2} \)
$59$ \( 25 + 5 T^{2} + T^{4} \)
$61$ \( ( 36 - 6 T + T^{2} )^{2} \)
$67$ \( 81 + 108 T + 153 T^{2} - 12 T^{3} + T^{4} \)
$71$ \( 13456 - 696 T + 152 T^{2} + 6 T^{3} + T^{4} \)
$73$ \( ( 2 + T )^{4} \)
$79$ \( ( -4 + T )^{4} \)
$83$ \( ( -45 + T^{2} )^{2} \)
$89$ \( 6241 - 1659 T + 362 T^{2} - 21 T^{3} + T^{4} \)
$97$ \( 52441 + 7099 T + 732 T^{2} + 31 T^{3} + T^{4} \)
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