Properties

Label 819.2.o.d
Level $819$
Weight $2$
Character orbit 819.o
Analytic conductor $6.540$
Analytic rank $0$
Dimension $6$
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: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.771147.1
Defining polynomial: \(x^{6} - x^{5} + 5 x^{4} + 6 x^{3} + 15 x^{2} + 4 x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 273)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 1 - \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} ) q^{2} + ( -2 - \beta_{1} + \beta_{2} - 2 \beta_{4} - \beta_{5} ) q^{4} + ( -1 + \beta_{2} + 2 \beta_{3} ) q^{5} + ( 1 + \beta_{4} ) q^{7} + ( 1 - \beta_{2} + \beta_{3} ) q^{8} +O(q^{10})\) \( q + ( 1 - \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} ) q^{2} + ( -2 - \beta_{1} + \beta_{2} - 2 \beta_{4} - \beta_{5} ) q^{4} + ( -1 + \beta_{2} + 2 \beta_{3} ) q^{5} + ( 1 + \beta_{4} ) q^{7} + ( 1 - \beta_{2} + \beta_{3} ) q^{8} + ( 2 + 3 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + 6 \beta_{4} + 2 \beta_{5} ) q^{10} + ( \beta_{1} + 3 \beta_{4} ) q^{11} + ( -1 - \beta_{1} + \beta_{3} - \beta_{4} - 2 \beta_{5} ) q^{13} + ( -\beta_{2} - \beta_{3} ) q^{14} + ( 2 + \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - \beta_{4} + 2 \beta_{5} ) q^{16} + ( 1 + \beta_{4} - \beta_{5} ) q^{17} + ( 3 + 2 \beta_{1} - 2 \beta_{2} + 3 \beta_{4} ) q^{19} + ( -7 - 3 \beta_{1} + 3 \beta_{2} - 7 \beta_{4} - 5 \beta_{5} ) q^{20} + ( -2 - \beta_{1} + \beta_{2} - 2 \beta_{4} - 4 \beta_{5} ) q^{22} + ( 1 - \beta_{1} - \beta_{2} - \beta_{3} - 3 \beta_{4} + \beta_{5} ) q^{23} + ( 3 - \beta_{2} + 3 \beta_{3} ) q^{25} + ( 7 + 3 \beta_{1} - 4 \beta_{2} - \beta_{3} + 2 \beta_{4} + 3 \beta_{5} ) q^{26} + ( -1 - \beta_{1} + \beta_{2} + \beta_{3} - 2 \beta_{4} - \beta_{5} ) q^{28} + ( 3 + 2 \beta_{1} - 3 \beta_{2} - 3 \beta_{3} + 4 \beta_{4} + 3 \beta_{5} ) q^{29} + ( -4 + \beta_{2} + 4 \beta_{3} ) q^{31} + ( \beta_{1} - \beta_{2} + 2 \beta_{5} ) q^{32} + ( 3 - 2 \beta_{2} - \beta_{3} ) q^{34} + ( 1 + \beta_{1} - \beta_{2} + \beta_{4} + 2 \beta_{5} ) q^{35} + ( 2 + \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + \beta_{4} + 2 \beta_{5} ) q^{37} + ( 4 - 3 \beta_{2} - 5 \beta_{3} ) q^{38} + ( 1 + 4 \beta_{2} + 6 \beta_{3} ) q^{40} + ( -4 + 4 \beta_{2} + 4 \beta_{3} - 2 \beta_{4} - 4 \beta_{5} ) q^{41} + ( 7 + \beta_{1} - \beta_{2} + 7 \beta_{4} + \beta_{5} ) q^{43} + ( 4 + 3 \beta_{3} ) q^{44} + ( -1 - \beta_{4} + 4 \beta_{5} ) q^{46} + ( 6 - 3 \beta_{2} + 2 \beta_{3} ) q^{47} + \beta_{4} q^{49} + ( 10 + 7 \beta_{1} - 10 \beta_{2} - 10 \beta_{3} + 11 \beta_{4} + 10 \beta_{5} ) q^{50} + ( -1 + \beta_{1} - 3 \beta_{2} - 5 \beta_{3} + \beta_{4} + 2 \beta_{5} ) q^{52} + ( -3 - 2 \beta_{2} - 2 \beta_{3} ) q^{53} + ( 7 + 2 \beta_{1} - 7 \beta_{2} - 7 \beta_{3} + 3 \beta_{4} + 7 \beta_{5} ) q^{55} + ( 2 + 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{4} + \beta_{5} ) q^{56} + ( -11 - 5 \beta_{1} + 5 \beta_{2} - 11 \beta_{4} - 6 \beta_{5} ) q^{58} + ( -1 + \beta_{1} - \beta_{2} - \beta_{4} - \beta_{5} ) q^{59} + ( -5 - 5 \beta_{4} - 2 \beta_{5} ) q^{61} + ( 3 + 7 \beta_{1} - 3 \beta_{2} - 3 \beta_{3} + 9 \beta_{4} + 3 \beta_{5} ) q^{62} + ( 2 - 5 \beta_{3} ) q^{64} + ( -9 - 4 \beta_{1} + 7 \beta_{2} + 4 \beta_{3} - 11 \beta_{4} - 3 \beta_{5} ) q^{65} + ( 3 - 2 \beta_{1} - 3 \beta_{2} - 3 \beta_{3} + 2 \beta_{4} + 3 \beta_{5} ) q^{67} + ( 1 - \beta_{2} - \beta_{3} + \beta_{5} ) q^{68} + ( -4 + \beta_{2} - 2 \beta_{3} ) q^{70} + ( 4 + 5 \beta_{1} - 5 \beta_{2} + 4 \beta_{4} - \beta_{5} ) q^{71} + ( -1 - 2 \beta_{2} + 3 \beta_{3} ) q^{73} + ( -6 - 3 \beta_{1} + 3 \beta_{2} - 6 \beta_{4} - 2 \beta_{5} ) q^{74} + ( -3 - 3 \beta_{1} + 3 \beta_{2} + 3 \beta_{3} - 8 \beta_{4} - 3 \beta_{5} ) q^{76} + ( -3 + \beta_{2} ) q^{77} + ( -4 + 9 \beta_{2} + 2 \beta_{3} ) q^{79} + ( -1 + 2 \beta_{1} + \beta_{2} + \beta_{3} + 9 \beta_{4} - \beta_{5} ) q^{80} + ( 14 + 4 \beta_{1} - 4 \beta_{2} + 14 \beta_{4} + 2 \beta_{5} ) q^{82} + ( 1 + \beta_{2} - 6 \beta_{3} ) q^{83} + ( -4 - \beta_{1} + \beta_{2} - 4 \beta_{4} + 2 \beta_{5} ) q^{85} + ( -1 - 6 \beta_{2} - 8 \beta_{3} ) q^{86} + ( 2 + 4 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + 9 \beta_{4} + 2 \beta_{5} ) q^{88} + ( -6 - 4 \beta_{1} + 6 \beta_{2} + 6 \beta_{3} + 3 \beta_{4} - 6 \beta_{5} ) q^{89} + ( -1 + \beta_{1} + 2 \beta_{3} - \beta_{5} ) q^{91} + ( -4 + \beta_{2} - \beta_{3} ) q^{92} + ( 13 + 7 \beta_{1} - 13 \beta_{2} - 13 \beta_{3} + 9 \beta_{4} + 13 \beta_{5} ) q^{94} + ( 3 + \beta_{1} - \beta_{2} + 3 \beta_{4} + 8 \beta_{5} ) q^{95} + ( -8 + \beta_{1} - \beta_{2} - 8 \beta_{4} + 2 \beta_{5} ) q^{97} + ( -1 - \beta_{4} - \beta_{5} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q - 2q^{2} - 4q^{4} + 3q^{7} + 6q^{8} + O(q^{10}) \) \( 6q - 2q^{2} - 4q^{4} + 3q^{7} + 6q^{8} - 13q^{10} - 8q^{11} - 4q^{14} + 6q^{16} + 4q^{17} + 7q^{19} - 13q^{20} - q^{22} + 9q^{23} + 22q^{25} + 26q^{26} + 4q^{28} - 7q^{29} - 14q^{31} - 3q^{32} + 12q^{34} + 8q^{38} + 26q^{40} + 2q^{41} + 19q^{43} + 30q^{44} - 7q^{46} + 34q^{47} - 3q^{49} - 16q^{50} - 26q^{52} - 26q^{53} + 3q^{56} - 22q^{58} - 3q^{59} - 13q^{61} - 17q^{62} + 2q^{64} - 5q^{67} + q^{68} - 26q^{70} + 8q^{71} - 4q^{73} - 13q^{74} + 18q^{76} - 16q^{77} - 2q^{79} - 26q^{80} + 36q^{82} - 4q^{83} - 13q^{85} - 34q^{86} - 21q^{88} - 19q^{89} - 24q^{92} - 7q^{94} - 27q^{97} - 2q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - x^{5} + 5 x^{4} + 6 x^{3} + 15 x^{2} + 4 x + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{5} + 5 \nu^{4} - 25 \nu^{3} + 15 \nu^{2} + 4 \nu - 20 \)\()/79\)
\(\beta_{3}\)\(=\)\((\)\( -5 \nu^{5} + 25 \nu^{4} - 46 \nu^{3} + 75 \nu^{2} + 20 \nu + 216 \)\()/79\)
\(\beta_{4}\)\(=\)\((\)\( 20 \nu^{5} - 21 \nu^{4} + 105 \nu^{3} + 95 \nu^{2} + 315 \nu + 5 \)\()/79\)
\(\beta_{5}\)\(=\)\((\)\( 38 \nu^{5} - 32 \nu^{4} + 160 \nu^{3} + 299 \nu^{2} + 480 \nu + 128 \)\()/79\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{5} - 2 \beta_{4} - 2 \beta_{2} + 2 \beta_{1} - 2\)
\(\nu^{3}\)\(=\)\(\beta_{3} - 5 \beta_{2} - 4\)
\(\nu^{4}\)\(=\)\(-5 \beta_{5} + 11 \beta_{4} + 5 \beta_{3} + 5 \beta_{2} - 15 \beta_{1} - 5\)
\(\nu^{5}\)\(=\)\(-10 \beta_{5} + 25 \beta_{4} + 41 \beta_{2} - 41 \beta_{1} + 25\)

Character values

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

\(n\) \(92\) \(379\) \(703\)
\(\chi(n)\) \(1\) \(-1 - \beta_{4}\) \(1\)

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
−0.136945 + 0.237196i
1.32555 2.29591i
−0.688601 + 1.19269i
−0.136945 0.237196i
1.32555 + 2.29591i
−0.688601 1.19269i
−1.18860 + 2.05872i 0 −1.82555 3.16194i 4.02830 0 0.500000 + 0.866025i 3.92498 0 −4.78804 + 8.29313i
568.2 −0.636945 + 1.10322i 0 0.188601 + 0.326667i −1.10331 0 0.500000 + 0.866025i −3.02830 0 0.702750 1.21720i
568.3 0.825547 1.42989i 0 −0.363055 0.628829i −2.92498 0 0.500000 + 0.866025i 2.10331 0 −2.41471 + 4.18240i
757.1 −1.18860 2.05872i 0 −1.82555 + 3.16194i 4.02830 0 0.500000 0.866025i 3.92498 0 −4.78804 8.29313i
757.2 −0.636945 1.10322i 0 0.188601 0.326667i −1.10331 0 0.500000 0.866025i −3.02830 0 0.702750 + 1.21720i
757.3 0.825547 + 1.42989i 0 −0.363055 + 0.628829i −2.92498 0 0.500000 0.866025i 2.10331 0 −2.41471 4.18240i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 757.3
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.d 6
3.b odd 2 1 273.2.k.d 6
13.c even 3 1 inner 819.2.o.d 6
39.h odd 6 1 3549.2.a.s 3
39.i odd 6 1 273.2.k.d 6
39.i odd 6 1 3549.2.a.h 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.2.k.d 6 3.b odd 2 1
273.2.k.d 6 39.i odd 6 1
819.2.o.d 6 1.a even 1 1 trivial
819.2.o.d 6 13.c even 3 1 inner
3549.2.a.h 3 39.i odd 6 1
3549.2.a.s 3 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 \(S_{2}^{\mathrm{new}}(819, [\chi])\):

\( T_{2}^{6} + 2 T_{2}^{5} + 7 T_{2}^{4} + 4 T_{2}^{3} + 19 T_{2}^{2} + 15 T_{2} + 25 \)
\( T_{11}^{6} + 8 T_{11}^{5} + 47 T_{11}^{4} + 126 T_{11}^{3} + 249 T_{11}^{2} + 85 T_{11} + 25 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 25 + 15 T + 19 T^{2} + 4 T^{3} + 7 T^{4} + 2 T^{5} + T^{6} \)
$3$ \( T^{6} \)
$5$ \( ( -13 - 13 T + T^{3} )^{2} \)
$7$ \( ( 1 - T + T^{2} )^{3} \)
$11$ \( 25 + 85 T + 249 T^{2} + 126 T^{3} + 47 T^{4} + 8 T^{5} + T^{6} \)
$13$ \( 2197 + 65 T^{3} + T^{6} \)
$17$ \( 1 + T + 5 T^{2} - 6 T^{3} + 15 T^{4} - 4 T^{5} + T^{6} \)
$19$ \( 2209 - 47 T + 330 T^{2} - 87 T^{3} + 50 T^{4} - 7 T^{5} + T^{6} \)
$23$ \( 1 - 14 T + 187 T^{2} - 124 T^{3} + 67 T^{4} - 9 T^{5} + T^{6} \)
$29$ \( 25 - 70 T + 161 T^{2} - 108 T^{3} + 63 T^{4} + 7 T^{5} + T^{6} \)
$31$ \( ( -281 - 40 T + 7 T^{2} + T^{3} )^{2} \)
$37$ \( 169 - 169 T + 169 T^{2} - 26 T^{3} + 13 T^{4} + T^{6} \)
$41$ \( 40000 - 13600 T + 5024 T^{2} - 264 T^{3} + 72 T^{4} - 2 T^{5} + T^{6} \)
$43$ \( 52441 - 26564 T + 9105 T^{2} - 1746 T^{3} + 245 T^{4} - 19 T^{5} + T^{6} \)
$47$ \( ( 547 + 14 T - 17 T^{2} + T^{3} )^{2} \)
$53$ \( ( -13 + 39 T + 13 T^{2} + T^{3} )^{2} \)
$59$ \( 625 + 250 T + 175 T^{2} + 20 T^{3} + 19 T^{4} + 3 T^{5} + T^{6} \)
$61$ \( 169 - 507 T + 1690 T^{2} + 533 T^{3} + 130 T^{4} + 13 T^{5} + T^{6} \)
$67$ \( 156025 + 29230 T + 7451 T^{2} + 420 T^{3} + 99 T^{4} + 5 T^{5} + T^{6} \)
$71$ \( 265225 - 58195 T + 16889 T^{2} - 126 T^{3} + 177 T^{4} - 8 T^{5} + T^{6} \)
$73$ \( ( 73 - 81 T + 2 T^{2} + T^{3} )^{2} \)
$79$ \( ( -337 - 290 T + T^{2} + T^{3} )^{2} \)
$83$ \( ( 229 - 185 T + 2 T^{2} + T^{3} )^{2} \)
$89$ \( 1038361 + 1019 T + 19362 T^{2} + 2019 T^{3} + 362 T^{4} + 19 T^{5} + T^{6} \)
$97$ \( 358801 + 137770 T + 36727 T^{2} + 5012 T^{3} + 499 T^{4} + 27 T^{5} + T^{6} \)
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