Properties

Label 819.2.do.d
Level $819$
Weight $2$
Character orbit 819.do
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.do (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.53974792554\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-7})\)
Defining polynomial: \(x^{4} - x^{3} - x^{2} - 2 x + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 273)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( -1 + \beta_{3} ) q^{2} + ( \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{4} + ( 1 + \beta_{2} ) q^{5} + ( -1 + 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{7} + ( -1 + 2 \beta_{2} ) q^{8} +O(q^{10})\) \( q + ( -1 + \beta_{3} ) q^{2} + ( \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{4} + ( 1 + \beta_{2} ) q^{5} + ( -1 + 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{7} + ( -1 + 2 \beta_{2} ) q^{8} + ( -2 + \beta_{1} + \beta_{3} ) q^{10} + ( 2 - 4 \beta_{2} ) q^{11} + ( -3 + 4 \beta_{2} ) q^{13} + ( 1 - 2 \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{14} + ( 1 - 2 \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{16} + ( 1 - \beta_{2} ) q^{17} + ( -2 + 4 \beta_{1} + 4 \beta_{2} - 4 \beta_{3} ) q^{19} + ( 1 + \beta_{2} - 3 \beta_{3} ) q^{20} + ( 2 - 4 \beta_{1} + 2 \beta_{3} ) q^{22} + ( -2 + 4 \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{23} -2 \beta_{2} q^{25} + ( -1 + 4 \beta_{1} - 3 \beta_{3} ) q^{26} + ( -4 + \beta_{1} - 3 \beta_{2} ) q^{28} + ( -7 + 7 \beta_{2} ) q^{29} + ( -4 + \beta_{2} + 2 \beta_{3} ) q^{31} + ( -4 + \beta_{1} - 3 \beta_{2} ) q^{32} + ( -\beta_{1} + \beta_{3} ) q^{34} + ( -1 + 2 \beta_{1} + 3 \beta_{2} - 4 \beta_{3} ) q^{35} + ( -2 - \beta_{2} + 4 \beta_{3} ) q^{37} + ( 2 - 4 \beta_{1} + 4 \beta_{2} + 2 \beta_{3} ) q^{38} + ( -3 + 3 \beta_{2} ) q^{40} + ( 1 - 4 \beta_{1} - 3 \beta_{2} ) q^{41} + ( -2 + 4 \beta_{1} + 3 \beta_{2} - 2 \beta_{3} ) q^{43} + ( -4 + 6 \beta_{1} + 2 \beta_{2} ) q^{44} + ( 9 - 7 \beta_{1} + 2 \beta_{2} ) q^{46} + ( -1 - 2 \beta_{1} - 3 \beta_{2} ) q^{47} -7 q^{49} + ( 2 - 2 \beta_{1} ) q^{50} + ( 4 - 7 \beta_{1} - 3 \beta_{2} + 2 \beta_{3} ) q^{52} + ( -4 + 8 \beta_{1} + 9 \beta_{2} - 4 \beta_{3} ) q^{53} + ( 6 - 6 \beta_{2} ) q^{55} + ( 1 - 2 \beta_{1} - 2 \beta_{3} ) q^{56} + ( 7 \beta_{1} - 7 \beta_{3} ) q^{58} + ( 3 + 2 \beta_{1} + 5 \beta_{2} ) q^{59} + ( 2 + 4 \beta_{1} + 4 \beta_{3} ) q^{61} + ( 5 + 3 \beta_{1} - 2 \beta_{2} - 6 \beta_{3} ) q^{62} + ( 9 - 2 \beta_{1} - 2 \beta_{3} ) q^{64} + ( -7 + 5 \beta_{2} ) q^{65} + ( -2 + 8 \beta_{1} + 4 \beta_{2} - 8 \beta_{3} ) q^{67} + ( -1 + 2 \beta_{1} + \beta_{2} - \beta_{3} ) q^{68} + ( -2 - 3 \beta_{1} + 4 \beta_{2} + 3 \beta_{3} ) q^{70} + ( -4 + 5 \beta_{2} - 6 \beta_{3} ) q^{71} + ( 10 - 5 \beta_{2} ) q^{73} + ( 7 + 3 \beta_{1} - 4 \beta_{2} - 6 \beta_{3} ) q^{74} + ( -8 + 2 \beta_{1} - 6 \beta_{2} ) q^{76} + ( -2 + 4 \beta_{1} + 4 \beta_{3} ) q^{77} + ( 7 - 2 \beta_{1} - 9 \beta_{2} + 4 \beta_{3} ) q^{79} + ( 2 - 3 \beta_{1} - 4 \beta_{2} + 3 \beta_{3} ) q^{80} + ( -6 + \beta_{1} + \beta_{3} ) q^{82} + ( -2 + 4 \beta_{2} ) q^{83} + ( 2 - \beta_{2} ) q^{85} + ( 5 - 3 \beta_{1} + 2 \beta_{2} ) q^{86} + 6 q^{88} + ( 18 - 9 \beta_{2} ) q^{89} + ( 3 - 6 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{91} + ( -15 + 5 \beta_{1} + 5 \beta_{3} ) q^{92} + ( -\beta_{1} - \beta_{3} ) q^{94} + ( -2 + 4 \beta_{1} + 6 \beta_{2} - 8 \beta_{3} ) q^{95} + ( 6 - \beta_{2} - 4 \beta_{3} ) q^{97} + ( 7 - 7 \beta_{3} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 3q^{2} + q^{4} + 6q^{5} + O(q^{10}) \) \( 4q - 3q^{2} + q^{4} + 6q^{5} - 6q^{10} - 4q^{13} + 7q^{14} - q^{16} + 2q^{17} + 3q^{20} + 6q^{22} - 8q^{23} - 4q^{25} - 3q^{26} - 21q^{28} - 14q^{29} - 12q^{31} - 21q^{32} - 6q^{37} + 14q^{38} - 6q^{40} - 6q^{41} - 6q^{44} + 33q^{46} - 12q^{47} - 28q^{49} + 6q^{50} + 5q^{52} + 6q^{53} + 12q^{55} + 24q^{59} + 16q^{61} + 13q^{62} + 32q^{64} - 18q^{65} - q^{68} - 12q^{71} + 30q^{73} + 17q^{74} - 42q^{76} + 12q^{79} - 22q^{82} + 6q^{85} + 21q^{86} + 24q^{88} + 54q^{89} - 50q^{92} - 2q^{94} + 18q^{97} + 21q^{98} + O(q^{100}) \)

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

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

Character values

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

\(n\) \(92\) \(379\) \(703\)
\(\chi(n)\) \(1\) \(\beta_{2}\) \(-1 + \beta_{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
361.1
−0.895644 + 1.09445i
1.39564 0.228425i
−0.895644 1.09445i
1.39564 + 0.228425i
−1.89564 1.09445i 0 1.39564 + 2.41733i 1.50000 0.866025i 0 2.64575i 1.73205i 0 −3.79129
361.2 0.395644 + 0.228425i 0 −0.895644 1.55130i 1.50000 0.866025i 0 2.64575i 1.73205i 0 0.791288
667.1 −1.89564 + 1.09445i 0 1.39564 2.41733i 1.50000 + 0.866025i 0 2.64575i 1.73205i 0 −3.79129
667.2 0.395644 0.228425i 0 −0.895644 + 1.55130i 1.50000 + 0.866025i 0 2.64575i 1.73205i 0 0.791288
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
91.u even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 819.2.do.d 4
3.b odd 2 1 273.2.bl.b yes 4
7.c even 3 1 819.2.bm.d 4
13.e even 6 1 819.2.bm.d 4
21.h odd 6 1 273.2.t.b 4
39.h odd 6 1 273.2.t.b 4
91.u even 6 1 inner 819.2.do.d 4
273.x odd 6 1 273.2.bl.b yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.2.t.b 4 21.h odd 6 1
273.2.t.b 4 39.h odd 6 1
273.2.bl.b yes 4 3.b odd 2 1
273.2.bl.b yes 4 273.x odd 6 1
819.2.bm.d 4 7.c even 3 1
819.2.bm.d 4 13.e even 6 1
819.2.do.d 4 1.a even 1 1 trivial
819.2.do.d 4 91.u even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} + 3 T_{2}^{3} + 2 T_{2}^{2} - 3 T_{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(819, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 3 T + 2 T^{2} + 3 T^{3} + T^{4} \)
$3$ \( T^{4} \)
$5$ \( ( 3 - 3 T + T^{2} )^{2} \)
$7$ \( ( 7 + T^{2} )^{2} \)
$11$ \( ( 12 + T^{2} )^{2} \)
$13$ \( ( 13 + 2 T + T^{2} )^{2} \)
$17$ \( ( 1 - T + T^{2} )^{2} \)
$19$ \( ( 28 + T^{2} )^{2} \)
$23$ \( 25 - 40 T + 69 T^{2} + 8 T^{3} + T^{4} \)
$29$ \( ( 49 + 7 T + T^{2} )^{2} \)
$31$ \( 25 + 60 T + 53 T^{2} + 12 T^{3} + T^{4} \)
$37$ \( 625 - 150 T - 13 T^{2} + 6 T^{3} + T^{4} \)
$41$ \( 625 - 150 T - 13 T^{2} + 6 T^{3} + T^{4} \)
$43$ \( 441 + 21 T^{2} + T^{4} \)
$47$ \( 25 + 60 T + 53 T^{2} + 12 T^{3} + T^{4} \)
$53$ \( 5625 + 450 T + 111 T^{2} - 6 T^{3} + T^{4} \)
$59$ \( 1681 - 984 T + 233 T^{2} - 24 T^{3} + T^{4} \)
$61$ \( ( -68 - 8 T + T^{2} )^{2} \)
$67$ \( 10000 + 248 T^{2} + T^{4} \)
$71$ \( 2601 - 612 T - 3 T^{2} + 12 T^{3} + T^{4} \)
$73$ \( ( 75 - 15 T + T^{2} )^{2} \)
$79$ \( 225 - 180 T + 129 T^{2} - 12 T^{3} + T^{4} \)
$83$ \( ( 12 + T^{2} )^{2} \)
$89$ \( ( 243 - 27 T + T^{2} )^{2} \)
$97$ \( 1 + 18 T + 107 T^{2} - 18 T^{3} + T^{4} \)
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