Properties

Label 1386.2.k.t
Level $1386$
Weight $2$
Character orbit 1386.k
Analytic conductor $11.067$
Analytic rank $0$
Dimension $4$
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: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-3})\)
Defining polynomial: \(x^{4} + 2 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 154)
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.

\(f(q)\) \(=\) \( q -\beta_{2} q^{2} + ( -1 - \beta_{2} ) q^{4} + ( -\beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{5} + ( \beta_{1} + \beta_{2} - \beta_{3} ) q^{7} - q^{8} +O(q^{10})\) \( q -\beta_{2} q^{2} + ( -1 - \beta_{2} ) q^{4} + ( -\beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{5} + ( \beta_{1} + \beta_{2} - \beta_{3} ) q^{7} - q^{8} + ( 2 - \beta_{1} + 2 \beta_{2} ) q^{10} + ( 1 + \beta_{2} ) q^{11} + ( -1 + 2 \beta_{3} ) q^{13} + ( 1 - \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{14} + \beta_{2} q^{16} + ( -2 + 4 \beta_{1} - 2 \beta_{2} ) q^{17} + ( \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{19} + ( 2 + \beta_{3} ) q^{20} + q^{22} + ( 3 \beta_{1} + 2 \beta_{2} + 3 \beta_{3} ) q^{23} + ( -1 + 4 \beta_{1} - \beta_{2} ) q^{25} + ( 2 \beta_{1} + \beta_{2} + 2 \beta_{3} ) q^{26} + ( 1 - 2 \beta_{1} - \beta_{3} ) q^{28} + ( 3 + 4 \beta_{3} ) q^{29} + ( 4 + 4 \beta_{2} ) q^{31} + ( 1 + \beta_{2} ) q^{32} + ( -2 - 4 \beta_{3} ) q^{34} + ( 3 \beta_{1} - 4 \beta_{2} + 4 \beta_{3} ) q^{35} + ( -\beta_{1} - 8 \beta_{2} - \beta_{3} ) q^{37} + ( -2 + \beta_{1} - 2 \beta_{2} ) q^{38} + ( \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{40} + ( 4 - \beta_{3} ) q^{41} + 4 \beta_{3} q^{43} -\beta_{2} q^{44} + ( 2 + 3 \beta_{1} + 2 \beta_{2} ) q^{46} + ( 6 \beta_{1} + 2 \beta_{2} + 6 \beta_{3} ) q^{47} + ( 5 + 2 \beta_{1} + 5 \beta_{2} + 4 \beta_{3} ) q^{49} + ( -1 - 4 \beta_{3} ) q^{50} + ( 1 + 2 \beta_{1} + \beta_{2} ) q^{52} + ( -2 + 7 \beta_{1} - 2 \beta_{2} ) q^{53} + ( -2 - \beta_{3} ) q^{55} + ( -\beta_{1} - \beta_{2} + \beta_{3} ) q^{56} + ( 4 \beta_{1} - 3 \beta_{2} + 4 \beta_{3} ) q^{58} + ( -7 + \beta_{1} - 7 \beta_{2} ) q^{59} + ( 2 \beta_{1} + 9 \beta_{2} + 2 \beta_{3} ) q^{61} + 4 q^{62} + q^{64} + ( -3 \beta_{1} + 2 \beta_{2} - 3 \beta_{3} ) q^{65} + ( -7 + 3 \beta_{1} - 7 \beta_{2} ) q^{67} + ( -4 \beta_{1} + 2 \beta_{2} - 4 \beta_{3} ) q^{68} + ( -4 + 4 \beta_{1} - 4 \beta_{2} + \beta_{3} ) q^{70} + ( 4 - 5 \beta_{3} ) q^{71} + ( 8 + \beta_{1} + 8 \beta_{2} ) q^{73} + ( -8 - \beta_{1} - 8 \beta_{2} ) q^{74} + ( -2 - \beta_{3} ) q^{76} + ( -1 + 2 \beta_{1} + \beta_{3} ) q^{77} + ( -3 \beta_{1} - 9 \beta_{2} - 3 \beta_{3} ) q^{79} + ( -2 + \beta_{1} - 2 \beta_{2} ) q^{80} + ( -\beta_{1} - 4 \beta_{2} - \beta_{3} ) q^{82} + ( -2 - 10 \beta_{3} ) q^{83} + ( 12 + 10 \beta_{3} ) q^{85} + ( 4 \beta_{1} + 4 \beta_{3} ) q^{86} + ( -1 - \beta_{2} ) q^{88} + ( -6 \beta_{1} - 4 \beta_{2} - 6 \beta_{3} ) q^{89} + ( -8 - 3 \beta_{1} - 5 \beta_{2} - \beta_{3} ) q^{91} + ( 2 - 3 \beta_{3} ) q^{92} + ( 2 + 6 \beta_{1} + 2 \beta_{2} ) q^{94} + ( 6 - 4 \beta_{1} + 6 \beta_{2} ) q^{95} + ( -1 + 2 \beta_{3} ) q^{97} + ( 5 + 4 \beta_{1} + 2 \beta_{3} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 2q^{2} - 2q^{4} - 4q^{5} - 2q^{7} - 4q^{8} + O(q^{10}) \) \( 4q + 2q^{2} - 2q^{4} - 4q^{5} - 2q^{7} - 4q^{8} + 4q^{10} + 2q^{11} - 4q^{13} + 2q^{14} - 2q^{16} - 4q^{17} + 4q^{19} + 8q^{20} + 4q^{22} - 4q^{23} - 2q^{25} - 2q^{26} + 4q^{28} + 12q^{29} + 8q^{31} + 2q^{32} - 8q^{34} + 8q^{35} + 16q^{37} - 4q^{38} + 4q^{40} + 16q^{41} + 2q^{44} + 4q^{46} - 4q^{47} + 10q^{49} - 4q^{50} + 2q^{52} - 4q^{53} - 8q^{55} + 2q^{56} + 6q^{58} - 14q^{59} - 18q^{61} + 16q^{62} + 4q^{64} - 4q^{65} - 14q^{67} - 4q^{68} - 8q^{70} + 16q^{71} + 16q^{73} - 16q^{74} - 8q^{76} - 4q^{77} + 18q^{79} - 4q^{80} + 8q^{82} - 8q^{83} + 48q^{85} - 2q^{88} + 8q^{89} - 22q^{91} + 8q^{92} + 4q^{94} + 12q^{95} - 4q^{97} + 20q^{98} + O(q^{100}) \)

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

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

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
−0.707107 1.22474i
0.707107 + 1.22474i
−0.707107 + 1.22474i
0.707107 1.22474i
0.500000 0.866025i 0 −0.500000 0.866025i −1.70711 + 2.95680i 0 −2.62132 0.358719i −1.00000 0 1.70711 + 2.95680i
793.2 0.500000 0.866025i 0 −0.500000 0.866025i −0.292893 + 0.507306i 0 1.62132 + 2.09077i −1.00000 0 0.292893 + 0.507306i
991.1 0.500000 + 0.866025i 0 −0.500000 + 0.866025i −1.70711 2.95680i 0 −2.62132 + 0.358719i −1.00000 0 1.70711 2.95680i
991.2 0.500000 + 0.866025i 0 −0.500000 + 0.866025i −0.292893 0.507306i 0 1.62132 2.09077i −1.00000 0 0.292893 0.507306i
\(n\): e.g. 2-40 or 990-1000
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.t 4
3.b odd 2 1 154.2.e.e 4
7.c even 3 1 inner 1386.2.k.t 4
7.c even 3 1 9702.2.a.cx 2
7.d odd 6 1 9702.2.a.ch 2
12.b even 2 1 1232.2.q.f 4
21.c even 2 1 1078.2.e.m 4
21.g even 6 1 1078.2.a.x 2
21.g even 6 1 1078.2.e.m 4
21.h odd 6 1 154.2.e.e 4
21.h odd 6 1 1078.2.a.t 2
84.j odd 6 1 8624.2.a.bh 2
84.n even 6 1 1232.2.q.f 4
84.n even 6 1 8624.2.a.cc 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
154.2.e.e 4 3.b odd 2 1
154.2.e.e 4 21.h odd 6 1
1078.2.a.t 2 21.h odd 6 1
1078.2.a.x 2 21.g even 6 1
1078.2.e.m 4 21.c even 2 1
1078.2.e.m 4 21.g even 6 1
1232.2.q.f 4 12.b even 2 1
1232.2.q.f 4 84.n even 6 1
1386.2.k.t 4 1.a even 1 1 trivial
1386.2.k.t 4 7.c even 3 1 inner
8624.2.a.bh 2 84.j odd 6 1
8624.2.a.cc 2 84.n even 6 1
9702.2.a.ch 2 7.d odd 6 1
9702.2.a.cx 2 7.c even 3 1

Hecke kernels

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

\( T_{5}^{4} + 4 T_{5}^{3} + 14 T_{5}^{2} + 8 T_{5} + 4 \)
\( T_{13}^{2} + 2 T_{13} - 7 \)
\( T_{17}^{4} + 4 T_{17}^{3} + 44 T_{17}^{2} - 112 T_{17} + 784 \)
\( T_{23}^{4} + 4 T_{23}^{3} + 30 T_{23}^{2} - 56 T_{23} + 196 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T + T^{2} )^{2} \)
$3$ \( T^{4} \)
$5$ \( 4 + 8 T + 14 T^{2} + 4 T^{3} + T^{4} \)
$7$ \( 49 + 14 T - 3 T^{2} + 2 T^{3} + T^{4} \)
$11$ \( ( 1 - T + T^{2} )^{2} \)
$13$ \( ( -7 + 2 T + T^{2} )^{2} \)
$17$ \( 784 - 112 T + 44 T^{2} + 4 T^{3} + T^{4} \)
$19$ \( 4 - 8 T + 14 T^{2} - 4 T^{3} + T^{4} \)
$23$ \( 196 - 56 T + 30 T^{2} + 4 T^{3} + T^{4} \)
$29$ \( ( -23 - 6 T + T^{2} )^{2} \)
$31$ \( ( 16 - 4 T + T^{2} )^{2} \)
$37$ \( 3844 - 992 T + 194 T^{2} - 16 T^{3} + T^{4} \)
$41$ \( ( 14 - 8 T + T^{2} )^{2} \)
$43$ \( ( -32 + T^{2} )^{2} \)
$47$ \( 4624 - 272 T + 84 T^{2} + 4 T^{3} + T^{4} \)
$53$ \( 8836 - 376 T + 110 T^{2} + 4 T^{3} + T^{4} \)
$59$ \( 2209 + 658 T + 149 T^{2} + 14 T^{3} + T^{4} \)
$61$ \( 5329 + 1314 T + 251 T^{2} + 18 T^{3} + T^{4} \)
$67$ \( 961 + 434 T + 165 T^{2} + 14 T^{3} + T^{4} \)
$71$ \( ( -34 - 8 T + T^{2} )^{2} \)
$73$ \( 3844 - 992 T + 194 T^{2} - 16 T^{3} + T^{4} \)
$79$ \( 3969 - 1134 T + 261 T^{2} - 18 T^{3} + T^{4} \)
$83$ \( ( -196 + 4 T + T^{2} )^{2} \)
$89$ \( 3136 + 448 T + 120 T^{2} - 8 T^{3} + T^{4} \)
$97$ \( ( -7 + 2 T + T^{2} )^{2} \)
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