Properties

Label 1386.2.k.s
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: \(\Q(\sqrt{-3}, \sqrt{7})\)
Defining polynomial: \(x^{4} + 7 x^{2} + 49\)
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 7 x^{2} + 49\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} \)\(/7\)
\(\beta_{3}\)\(=\)\( \nu^{3} \)\(/7\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(7 \beta_{2}\)
\(\nu^{3}\)\(=\)\(7 \beta_{3}\)

Character values

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

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

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
−1.32288 + 2.29129i
1.32288 2.29129i
−1.32288 2.29129i
1.32288 + 2.29129i
−0.500000 + 0.866025i 0 −0.500000 0.866025i −0.822876 + 1.42526i 0 −1.32288 + 2.29129i 1.00000 0 −0.822876 1.42526i
793.2 −0.500000 + 0.866025i 0 −0.500000 0.866025i 1.82288 3.15731i 0 1.32288 2.29129i 1.00000 0 1.82288 + 3.15731i
991.1 −0.500000 0.866025i 0 −0.500000 + 0.866025i −0.822876 1.42526i 0 −1.32288 2.29129i 1.00000 0 −0.822876 + 1.42526i
991.2 −0.500000 0.866025i 0 −0.500000 + 0.866025i 1.82288 + 3.15731i 0 1.32288 + 2.29129i 1.00000 0 1.82288 3.15731i
\(n\): 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.s 4
3.b odd 2 1 154.2.e.f 4
7.c even 3 1 inner 1386.2.k.s 4
7.c even 3 1 9702.2.a.cz 2
7.d odd 6 1 9702.2.a.dr 2
12.b even 2 1 1232.2.q.g 4
21.c even 2 1 1078.2.e.v 4
21.g even 6 1 1078.2.a.n 2
21.g even 6 1 1078.2.e.v 4
21.h odd 6 1 154.2.e.f 4
21.h odd 6 1 1078.2.a.s 2
84.j odd 6 1 8624.2.a.bk 2
84.n even 6 1 1232.2.q.g 4
84.n even 6 1 8624.2.a.ca 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
154.2.e.f 4 3.b odd 2 1
154.2.e.f 4 21.h odd 6 1
1078.2.a.n 2 21.g even 6 1
1078.2.a.s 2 21.h odd 6 1
1078.2.e.v 4 21.c even 2 1
1078.2.e.v 4 21.g even 6 1
1232.2.q.g 4 12.b even 2 1
1232.2.q.g 4 84.n even 6 1
1386.2.k.s 4 1.a even 1 1 trivial
1386.2.k.s 4 7.c even 3 1 inner
8624.2.a.bk 2 84.j odd 6 1
8624.2.a.ca 2 84.n even 6 1
9702.2.a.cz 2 7.c even 3 1
9702.2.a.dr 2 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 \(S_{2}^{\mathrm{new}}(1386, [\chi])\):

\( T_{5}^{4} - 2 T_{5}^{3} + 10 T_{5}^{2} + 12 T_{5} + 36 \)
\( T_{13} - 5 \)
\( T_{17}^{2} - 6 T_{17} + 36 \)
\( T_{23}^{4} + 2 T_{23}^{3} + 10 T_{23}^{2} - 12 T_{23} + 36 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T + T^{2} )^{2} \)
$3$ \( T^{4} \)
$5$ \( 36 + 12 T + 10 T^{2} - 2 T^{3} + T^{4} \)
$7$ \( 49 + 7 T^{2} + T^{4} \)
$11$ \( ( 1 - T + T^{2} )^{2} \)
$13$ \( ( -5 + T )^{4} \)
$17$ \( ( 36 - 6 T + T^{2} )^{2} \)
$19$ \( 4 - 12 T + 34 T^{2} - 6 T^{3} + T^{4} \)
$23$ \( 36 - 12 T + 10 T^{2} + 2 T^{3} + T^{4} \)
$29$ \( ( -27 + 2 T + T^{2} )^{2} \)
$31$ \( ( 16 - 4 T + T^{2} )^{2} \)
$37$ \( 36 - 12 T + 10 T^{2} + 2 T^{3} + T^{4} \)
$41$ \( ( -54 + 6 T + T^{2} )^{2} \)
$43$ \( ( 4 + T )^{4} \)
$47$ \( 1296 - 576 T + 220 T^{2} - 16 T^{3} + T^{4} \)
$53$ \( 36 - 12 T + 10 T^{2} + 2 T^{3} + T^{4} \)
$59$ \( 9 + 12 T + 19 T^{2} - 4 T^{3} + T^{4} \)
$61$ \( 2809 + 954 T + 271 T^{2} + 18 T^{3} + T^{4} \)
$67$ \( 2209 + 376 T + 111 T^{2} - 8 T^{3} + T^{4} \)
$71$ \( ( 42 + 14 T + T^{2} )^{2} \)
$73$ \( 4 + 12 T + 34 T^{2} + 6 T^{3} + T^{4} \)
$79$ \( 49 + 7 T^{2} + T^{4} \)
$83$ \( ( 36 + 16 T + T^{2} )^{2} \)
$89$ \( 9216 - 768 T + 160 T^{2} + 8 T^{3} + T^{4} \)
$97$ \( ( 93 + 22 T + T^{2} )^{2} \)
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