Properties

Label 1386.2.k.v
Level $1386$
Weight $2$
Character orbit 1386.k
Analytic conductor $11.067$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1386 = 2 \cdot 3^{2} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1386.k (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(11.0672657201\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.21870000.1
Defining polynomial: \(x^{6} - 3 x^{5} + 24 x^{4} - 43 x^{3} + 138 x^{2} - 117 x + 73\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 462)
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.

\(f(q)\) \(=\) \( q -\beta_{2} q^{2} + ( -1 - \beta_{2} ) q^{4} + ( \beta_{1} - \beta_{4} ) q^{5} + ( 1 + \beta_{2} + \beta_{3} ) q^{7} - q^{8} +O(q^{10})\) \( q -\beta_{2} q^{2} + ( -1 - \beta_{2} ) q^{4} + ( \beta_{1} - \beta_{4} ) q^{5} + ( 1 + \beta_{2} + \beta_{3} ) q^{7} - q^{8} + ( -1 + \beta_{1} - \beta_{3} ) q^{10} + ( -1 - \beta_{2} ) q^{11} + ( 1 + \beta_{3} - \beta_{4} + \beta_{5} ) q^{14} + \beta_{2} q^{16} + ( 1 + \beta_{2} - 2 \beta_{3} - 2 \beta_{5} ) q^{17} + ( \beta_{1} - 2 \beta_{2} + \beta_{4} - \beta_{5} ) q^{19} + ( -1 - \beta_{3} + \beta_{4} ) q^{20} - q^{22} + ( -2 \beta_{2} - 2 \beta_{4} + \beta_{5} ) q^{23} + ( -7 + 2 \beta_{1} - 5 \beta_{2} - \beta_{3} + \beta_{5} ) q^{25} + ( -\beta_{2} - \beta_{4} + \beta_{5} ) q^{28} + ( -1 + \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} - 2 \beta_{5} ) q^{29} + ( 2 + 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{5} ) q^{31} + ( 1 + \beta_{2} ) q^{32} + ( 1 + 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} - 4 \beta_{5} ) q^{34} + ( 4 + 2 \beta_{1} - 2 \beta_{2} - \beta_{3} - 3 \beta_{5} ) q^{35} + ( -\beta_{1} + 6 \beta_{2} - \beta_{4} + \beta_{5} ) q^{37} + ( -3 - 3 \beta_{2} + \beta_{3} + \beta_{5} ) q^{38} + ( -\beta_{1} + \beta_{4} ) q^{40} + ( -6 + \beta_{1} + \beta_{2} - 3 \beta_{3} + 3 \beta_{4} - 2 \beta_{5} ) q^{41} + ( 1 + 3 \beta_{1} + 3 \beta_{2} - 3 \beta_{3} + 3 \beta_{4} - 6 \beta_{5} ) q^{43} + \beta_{2} q^{44} + ( -2 + \beta_{1} - \beta_{2} - 2 \beta_{3} - \beta_{5} ) q^{46} + ( -2 \beta_{1} - 2 \beta_{2} + \beta_{5} ) q^{47} + ( 1 - \beta_{1} + 3 \beta_{2} - \beta_{4} - \beta_{5} ) q^{49} + ( -7 - \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} + 2 \beta_{5} ) q^{50} + ( 8 + 8 \beta_{2} ) q^{53} + ( -1 - \beta_{3} + \beta_{4} ) q^{55} + ( -1 - \beta_{2} - \beta_{3} ) q^{56} + ( 3 \beta_{1} + 4 \beta_{2} - \beta_{4} - \beta_{5} ) q^{58} + ( -5 + 2 \beta_{1} - 3 \beta_{2} - \beta_{3} + \beta_{5} ) q^{59} + ( -3 \beta_{1} - 4 \beta_{2} - \beta_{4} + 2 \beta_{5} ) q^{61} + ( 2 + 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} - 4 \beta_{5} ) q^{62} + q^{64} + ( -4 + 2 \beta_{1} - 2 \beta_{2} - 3 \beta_{3} - \beta_{5} ) q^{67} + ( 2 \beta_{1} + \beta_{2} + 2 \beta_{4} - 2 \beta_{5} ) q^{68} + ( -4 + 3 \beta_{1} - 5 \beta_{2} - \beta_{3} + \beta_{4} - 2 \beta_{5} ) q^{70} + ( -3 + 3 \beta_{1} + 3 \beta_{2} - 3 \beta_{3} + 3 \beta_{4} - 6 \beta_{5} ) q^{71} + ( 2 - 2 \beta_{1} + 4 \beta_{3} + 2 \beta_{5} ) q^{73} + ( 7 + 7 \beta_{2} - \beta_{3} - \beta_{5} ) q^{74} + ( -3 - \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} + 2 \beta_{5} ) q^{76} + ( -\beta_{2} - \beta_{4} + \beta_{5} ) q^{77} + ( \beta_{1} + 4 \beta_{2} - \beta_{4} ) q^{79} + ( 1 - \beta_{1} + \beta_{3} ) q^{80} + ( -\beta_{1} + 5 \beta_{2} + 3 \beta_{4} - \beta_{5} ) q^{82} + ( -4 - \beta_{1} - \beta_{2} + 3 \beta_{3} - 3 \beta_{4} + 2 \beta_{5} ) q^{83} + ( 3 - 4 \beta_{1} - 4 \beta_{2} + 7 \beta_{3} - 7 \beta_{4} + 8 \beta_{5} ) q^{85} + ( 3 \beta_{1} + 2 \beta_{2} + 3 \beta_{4} - 3 \beta_{5} ) q^{86} + ( 1 + \beta_{2} ) q^{88} + ( -2 \beta_{1} + 4 \beta_{2} + 2 \beta_{4} ) q^{89} + ( -2 + \beta_{1} + \beta_{2} - 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} ) q^{92} + ( -\beta_{1} - \beta_{2} - \beta_{5} ) q^{94} + ( -4 + 4 \beta_{1} - 6 \beta_{3} - 2 \beta_{5} ) q^{95} + 7 q^{97} + ( 4 + 2 \beta_{1} + 5 \beta_{2} - \beta_{3} - 3 \beta_{5} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q + 3q^{2} - 3q^{4} - 6q^{8} + O(q^{10}) \) \( 6q + 3q^{2} - 3q^{4} - 6q^{8} - 3q^{11} + 3q^{14} - 3q^{16} + 3q^{17} + 9q^{19} - 6q^{22} + 3q^{23} - 15q^{25} + 3q^{28} - 18q^{29} + 6q^{31} + 3q^{32} + 6q^{34} + 30q^{35} - 21q^{37} - 9q^{38} - 24q^{41} + 6q^{43} - 3q^{44} - 3q^{46} + 3q^{47} - 12q^{49} - 30q^{50} + 24q^{53} - 9q^{58} - 9q^{59} + 6q^{61} + 12q^{62} + 6q^{64} - 6q^{67} + 3q^{68} - 18q^{71} + 21q^{74} - 18q^{76} + 3q^{77} - 12q^{79} - 12q^{82} - 36q^{83} + 3q^{86} + 3q^{88} - 12q^{89} - 6q^{92} - 3q^{94} + 42q^{97} + 9q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - 3 x^{5} + 24 x^{4} - 43 x^{3} + 138 x^{2} - 117 x + 73\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -2 \nu^{5} + 5 \nu^{4} - 30 \nu^{3} + 40 \nu^{2} - 70 \nu + 13 \)\()/31\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{5} + 18 \nu^{4} - 15 \nu^{3} + 144 \nu^{2} + 151 \nu - 164 \)\()/62\)
\(\beta_{4}\)\(=\)\((\)\( -\nu^{5} - 13 \nu^{4} + 47 \nu^{3} - 197 \nu^{2} + 461 \nu - 133 \)\()/62\)
\(\beta_{5}\)\(=\)\((\)\( -\nu^{5} - 13 \nu^{4} + 16 \nu^{3} - 166 \nu^{2} + 151 \nu - 102 \)\()/31\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-2 \beta_{5} + 2 \beta_{4} - 2 \beta_{3} + \beta_{2} + 2 \beta_{1} - 8\)
\(\nu^{3}\)\(=\)\(-3 \beta_{5} + 4 \beta_{4} - 2 \beta_{3} + \beta_{2} - 8 \beta_{1} - 7\)
\(\nu^{4}\)\(=\)\(16 \beta_{5} - 16 \beta_{4} + 20 \beta_{3} - 9 \beta_{2} - 28 \beta_{1} + 75\)
\(\nu^{5}\)\(=\)\(45 \beta_{5} - 60 \beta_{4} + 40 \beta_{3} - 33 \beta_{2} + 55 \beta_{1} + 139\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1386\mathbb{Z}\right)^\times\).

\(n\) \(155\) \(199\) \(1135\)
\(\chi(n)\) \(1\) \(\beta_{2}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
793.1
0.500000 0.679547i
0.500000 + 3.05087i
0.500000 3.23735i
0.500000 + 0.679547i
0.500000 3.05087i
0.500000 + 3.23735i
0.500000 0.866025i 0 −0.500000 0.866025i −1.40280 + 2.42972i 0 −1.40280 2.24325i −1.00000 0 1.40280 + 2.42972i
793.2 0.500000 0.866025i 0 −0.500000 0.866025i −0.806615 + 1.39710i 0 −0.806615 + 2.51980i −1.00000 0 0.806615 + 1.39710i
793.3 0.500000 0.866025i 0 −0.500000 0.866025i 2.20942 3.82682i 0 2.20942 + 1.45550i −1.00000 0 −2.20942 3.82682i
991.1 0.500000 + 0.866025i 0 −0.500000 + 0.866025i −1.40280 2.42972i 0 −1.40280 + 2.24325i −1.00000 0 1.40280 2.42972i
991.2 0.500000 + 0.866025i 0 −0.500000 + 0.866025i −0.806615 1.39710i 0 −0.806615 2.51980i −1.00000 0 0.806615 1.39710i
991.3 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 2.20942 + 3.82682i 0 2.20942 1.45550i −1.00000 0 −2.20942 + 3.82682i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 991.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1386.2.k.v 6
3.b odd 2 1 462.2.i.g 6
7.c even 3 1 inner 1386.2.k.v 6
7.c even 3 1 9702.2.a.dv 3
7.d odd 6 1 9702.2.a.dw 3
21.g even 6 1 3234.2.a.bh 3
21.h odd 6 1 462.2.i.g 6
21.h odd 6 1 3234.2.a.bf 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
462.2.i.g 6 3.b odd 2 1
462.2.i.g 6 21.h odd 6 1
1386.2.k.v 6 1.a even 1 1 trivial
1386.2.k.v 6 7.c even 3 1 inner
3234.2.a.bf 3 21.h odd 6 1
3234.2.a.bh 3 21.g even 6 1
9702.2.a.dv 3 7.c even 3 1
9702.2.a.dw 3 7.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1386, [\chi])\):

\( T_{5}^{6} + 15 T_{5}^{4} + 40 T_{5}^{3} + 225 T_{5}^{2} + 300 T_{5} + 400 \)
\( T_{13} \)
\( T_{17}^{6} - 3 T_{17}^{5} + 66 T_{17}^{4} - 107 T_{17}^{3} + 3666 T_{17}^{2} - 7923 T_{17} + 19321 \)
\( T_{23}^{6} - 3 T_{23}^{5} + 36 T_{23}^{4} - 97 T_{23}^{3} + 996 T_{23}^{2} - 2403 T_{23} + 7921 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T + T^{2} )^{3} \)
$3$ \( T^{6} \)
$5$ \( 400 + 300 T + 225 T^{2} + 40 T^{3} + 15 T^{4} + T^{6} \)
$7$ \( 343 + 42 T^{2} - 20 T^{3} + 6 T^{4} + T^{6} \)
$11$ \( ( 1 + T + T^{2} )^{3} \)
$13$ \( T^{6} \)
$17$ \( 19321 - 7923 T + 3666 T^{2} - 107 T^{3} + 66 T^{4} - 3 T^{5} + T^{6} \)
$19$ \( 784 + 336 T + 396 T^{2} - 164 T^{3} + 69 T^{4} - 9 T^{5} + T^{6} \)
$23$ \( 7921 - 2403 T + 996 T^{2} - 97 T^{3} + 36 T^{4} - 3 T^{5} + T^{6} \)
$29$ \( ( -348 - 48 T + 9 T^{2} + T^{3} )^{2} \)
$31$ \( 36864 - 9216 T + 3456 T^{2} - 96 T^{3} + 84 T^{4} - 6 T^{5} + T^{6} \)
$37$ \( 51984 + 30096 T + 12636 T^{2} + 2316 T^{3} + 309 T^{4} + 21 T^{5} + T^{6} \)
$41$ \( ( -306 - 27 T + 12 T^{2} + T^{3} )^{2} \)
$43$ \( ( 404 - 132 T - 3 T^{2} + T^{3} )^{2} \)
$47$ \( 81 - 243 T + 756 T^{2} + 63 T^{3} + 36 T^{4} - 3 T^{5} + T^{6} \)
$53$ \( ( 64 - 8 T + T^{2} )^{3} \)
$59$ \( 2304 + 2304 T + 2736 T^{2} - 336 T^{3} + 129 T^{4} + 9 T^{5} + T^{6} \)
$61$ \( 9604 + 6174 T + 3381 T^{2} + 574 T^{3} + 99 T^{4} - 6 T^{5} + T^{6} \)
$67$ \( 44944 + 13356 T + 5241 T^{2} + 46 T^{3} + 99 T^{4} + 6 T^{5} + T^{6} \)
$71$ \( ( -108 - 108 T + 9 T^{2} + T^{3} )^{2} \)
$73$ \( 230400 - 57600 T + 14400 T^{2} - 960 T^{3} + 120 T^{4} + T^{6} \)
$79$ \( 256 - 528 T + 1281 T^{2} + 428 T^{3} + 111 T^{4} + 12 T^{5} + T^{6} \)
$83$ \( ( -164 + 33 T + 18 T^{2} + T^{3} )^{2} \)
$89$ \( 256 + 192 T + 336 T^{2} - 112 T^{3} + 156 T^{4} + 12 T^{5} + T^{6} \)
$97$ \( ( -7 + T )^{6} \)
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