Properties

Label 1386.2.k.w
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: 6.0.1156923.1
Defining polynomial: \(x^{6} - 3 x^{5} + 12 x^{4} - 19 x^{3} + 27 x^{2} - 18 x + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - 3 x^{5} + 12 x^{4} - 19 x^{3} + 27 x^{2} - 18 x + 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{2} - \nu + 3 \)
\(\beta_{2}\)\(=\)\((\)\( 2 \nu^{5} - 5 \nu^{4} + 22 \nu^{3} - 28 \nu^{2} + 43 \nu - 18 \)\()/2\)
\(\beta_{3}\)\(=\)\( -3 \nu^{5} + 7 \nu^{4} - 31 \nu^{3} + 37 \nu^{2} - 56 \nu + 22 \)
\(\beta_{4}\)\(=\)\((\)\( -6 \nu^{5} + 15 \nu^{4} - 64 \nu^{3} + 82 \nu^{2} - 121 \nu + 50 \)\()/2\)
\(\beta_{5}\)\(=\)\( -3 \nu^{5} + 8 \nu^{4} - 33 \nu^{3} + 44 \nu^{2} - 62 \nu + 24 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{5} - 2 \beta_{4} + \beta_{3} + \beta_{1} + 1\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{5} - 2 \beta_{4} + \beta_{3} + 3 \beta_{1} - 5\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(-3 \beta_{5} + 8 \beta_{4} - 3 \beta_{3} + 6 \beta_{2} - \beta_{1} - 5\)\()/2\)
\(\nu^{4}\)\(=\)\((\)\(-5 \beta_{5} + 18 \beta_{4} - 9 \beta_{3} + 12 \beta_{2} - 17 \beta_{1} + 27\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(13 \beta_{5} - 28 \beta_{4} + 3 \beta_{3} - 34 \beta_{2} - 11 \beta_{1} + 49\)\()/2\)

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 2.43956i
0.500000 + 1.51496i
0.500000 + 0.0585812i
0.500000 + 2.43956i
0.500000 1.51496i
0.500000 0.0585812i
0.500000 0.866025i 0 −0.500000 0.866025i −1.60074 + 2.77256i 0 1.60074 2.10657i −1.00000 0 1.60074 + 2.77256i
793.2 0.500000 0.866025i 0 −0.500000 0.866025i 0.227452 0.393958i 0 −0.227452 + 2.63596i −1.00000 0 −0.227452 0.393958i
793.3 0.500000 0.866025i 0 −0.500000 0.866025i 1.37328 2.37860i 0 −1.37328 2.26144i −1.00000 0 −1.37328 2.37860i
991.1 0.500000 + 0.866025i 0 −0.500000 + 0.866025i −1.60074 2.77256i 0 1.60074 + 2.10657i −1.00000 0 1.60074 2.77256i
991.2 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 0.227452 + 0.393958i 0 −0.227452 2.63596i −1.00000 0 −0.227452 + 0.393958i
991.3 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 1.37328 + 2.37860i 0 −1.37328 + 2.26144i −1.00000 0 −1.37328 + 2.37860i
\(n\): 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.w 6
3.b odd 2 1 462.2.i.f 6
7.c even 3 1 inner 1386.2.k.w 6
7.c even 3 1 9702.2.a.dt 3
7.d odd 6 1 9702.2.a.du 3
21.g even 6 1 3234.2.a.bg 3
21.h odd 6 1 462.2.i.f 6
21.h odd 6 1 3234.2.a.bi 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
462.2.i.f 6 3.b odd 2 1
462.2.i.f 6 21.h odd 6 1
1386.2.k.w 6 1.a even 1 1 trivial
1386.2.k.w 6 7.c even 3 1 inner
3234.2.a.bg 3 21.g even 6 1
3234.2.a.bi 3 21.h odd 6 1
9702.2.a.dt 3 7.c even 3 1
9702.2.a.du 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 \(S_{2}^{\mathrm{new}}(1386, [\chi])\):

\( T_{5}^{6} + 9 T_{5}^{4} - 8 T_{5}^{3} + 81 T_{5}^{2} - 36 T_{5} + 16 \)
\( T_{13}^{3} - 36 T_{13} + 32 \)
\( T_{17}^{2} - T_{17} + 1 \)
\( T_{23}^{6} - 9 T_{23}^{5} + 78 T_{23}^{4} - 69 T_{23}^{3} + 198 T_{23}^{2} + 63 T_{23} + 441 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T + T^{2} )^{3} \)
$3$ 1
$5$ \( 1 - 6 T^{2} - 8 T^{3} + 6 T^{4} + 24 T^{5} + 86 T^{6} + 120 T^{7} + 150 T^{8} - 1000 T^{9} - 3750 T^{10} + 15625 T^{12} \)
$7$ \( 1 + 12 T^{2} - 4 T^{3} + 84 T^{4} + 343 T^{6} \)
$11$ \( ( 1 - T + T^{2} )^{3} \)
$13$ \( ( 1 + 3 T^{2} + 32 T^{3} + 39 T^{4} + 2197 T^{6} )^{2} \)
$17$ \( ( 1 - T - 16 T^{2} - 17 T^{3} + 289 T^{4} )^{3} \)
$19$ \( 1 - 3 T - 12 T^{2} + 237 T^{3} - 408 T^{4} - 1839 T^{5} + 24590 T^{6} - 34941 T^{7} - 147288 T^{8} + 1625583 T^{9} - 1563852 T^{10} - 7428297 T^{11} + 47045881 T^{12} \)
$23$ \( 1 - 9 T + 9 T^{2} + 138 T^{3} - 9 T^{4} - 3249 T^{5} + 16702 T^{6} - 74727 T^{7} - 4761 T^{8} + 1679046 T^{9} + 2518569 T^{10} - 57927087 T^{11} + 148035889 T^{12} \)
$29$ \( ( 1 + 9 T + 87 T^{2} + 430 T^{3} + 2523 T^{4} + 7569 T^{5} + 24389 T^{6} )^{2} \)
$31$ \( 1 + 6 T + 39 T^{2} + 262 T^{3} + 942 T^{4} + 7302 T^{5} + 44307 T^{6} + 226362 T^{7} + 905262 T^{8} + 7805242 T^{9} + 36017319 T^{10} + 171774906 T^{11} + 887503681 T^{12} \)
$37$ \( 1 + 9 T - 18 T^{2} - 113 T^{3} + 1620 T^{4} - 3915 T^{5} - 120216 T^{6} - 144855 T^{7} + 2217780 T^{8} - 5723789 T^{9} - 33734898 T^{10} + 624095613 T^{11} + 2565726409 T^{12} \)
$41$ \( ( 1 + 12 T + 144 T^{2} + 902 T^{3} + 5904 T^{4} + 20172 T^{5} + 68921 T^{6} )^{2} \)
$43$ \( ( 1 - 9 T + 117 T^{2} - 610 T^{3} + 5031 T^{4} - 16641 T^{5} + 79507 T^{6} )^{2} \)
$47$ \( 1 + 15 T + 33 T^{2} + 98 T^{3} + 6975 T^{4} + 29487 T^{5} - 87106 T^{6} + 1385889 T^{7} + 15407775 T^{8} + 10174654 T^{9} + 161029473 T^{10} + 3440175105 T^{11} + 10779215329 T^{12} \)
$53$ \( 1 - 123 T^{2} - 64 T^{3} + 8610 T^{4} + 3936 T^{5} - 503059 T^{6} + 208608 T^{7} + 24185490 T^{8} - 9528128 T^{9} - 970529163 T^{10} + 22164361129 T^{12} \)
$59$ \( 1 - 9 T - 141 T^{3} - 864 T^{4} + 30879 T^{5} - 171866 T^{6} + 1821861 T^{7} - 3007584 T^{8} - 28958439 T^{9} - 6434318691 T^{11} + 42180533641 T^{12} \)
$61$ \( 1 - 6 T - 18 T^{2} + 688 T^{3} - 3414 T^{4} - 10122 T^{5} + 398166 T^{6} - 617442 T^{7} - 12703494 T^{8} + 156162928 T^{9} - 249225138 T^{10} - 5067577806 T^{11} + 51520374361 T^{12} \)
$67$ \( 1 + 6 T - 150 T^{2} - 324 T^{3} + 17814 T^{4} + 11310 T^{5} - 1359610 T^{6} + 757770 T^{7} + 79967046 T^{8} - 97447212 T^{9} - 3022668150 T^{10} + 8100750642 T^{11} + 90458382169 T^{12} \)
$71$ \( ( 1 + 3 T + 189 T^{2} + 362 T^{3} + 13419 T^{4} + 15123 T^{5} + 357911 T^{6} )^{2} \)
$73$ \( 1 - 123 T^{2} - 384 T^{3} + 6150 T^{4} + 23616 T^{5} - 289519 T^{6} + 1723968 T^{7} + 32773350 T^{8} - 149382528 T^{9} - 3492983643 T^{10} + 151334226289 T^{12} \)
$79$ \( 1 + 12 T - 60 T^{2} - 608 T^{3} + 7164 T^{4} - 180 T^{5} - 865506 T^{6} - 14220 T^{7} + 44710524 T^{8} - 299767712 T^{9} - 2337004860 T^{10} + 36924676788 T^{11} + 243087455521 T^{12} \)
$83$ \( ( 1 - 6 T + 138 T^{2} - 1184 T^{3} + 11454 T^{4} - 41334 T^{5} + 571787 T^{6} )^{2} \)
$89$ \( 1 + 36 T + 633 T^{2} + 8396 T^{3} + 102066 T^{4} + 1165620 T^{5} + 11868377 T^{6} + 103740180 T^{7} + 808464786 T^{8} + 5918919724 T^{9} + 39715838553 T^{10} + 201026140164 T^{11} + 496981290961 T^{12} \)
$97$ \( ( 1 - 9 T + 210 T^{2} - 1753 T^{3} + 20370 T^{4} - 84681 T^{5} + 912673 T^{6} )^{2} \)
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