Properties

Label 1386.2.k.r
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 462)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

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 −1.32288 + 2.29129i 0 1.32288 2.29129i 1.00000 0 −1.32288 2.29129i
793.2 −0.500000 + 0.866025i 0 −0.500000 0.866025i 1.32288 2.29129i 0 −1.32288 + 2.29129i 1.00000 0 1.32288 + 2.29129i
991.1 −0.500000 0.866025i 0 −0.500000 + 0.866025i −1.32288 2.29129i 0 1.32288 + 2.29129i 1.00000 0 −1.32288 + 2.29129i
991.2 −0.500000 0.866025i 0 −0.500000 + 0.866025i 1.32288 + 2.29129i 0 −1.32288 2.29129i 1.00000 0 1.32288 2.29129i
\(n\): 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.r 4
3.b odd 2 1 462.2.i.e 4
7.c even 3 1 inner 1386.2.k.r 4
7.c even 3 1 9702.2.a.db 2
7.d odd 6 1 9702.2.a.dm 2
21.g even 6 1 3234.2.a.ba 2
21.h odd 6 1 462.2.i.e 4
21.h odd 6 1 3234.2.a.w 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
462.2.i.e 4 3.b odd 2 1
462.2.i.e 4 21.h odd 6 1
1386.2.k.r 4 1.a even 1 1 trivial
1386.2.k.r 4 7.c even 3 1 inner
3234.2.a.w 2 21.h odd 6 1
3234.2.a.ba 2 21.g even 6 1
9702.2.a.db 2 7.c even 3 1
9702.2.a.dm 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 \(S_{2}^{\mathrm{new}}(1386, [\chi])\):

\( T_{5}^{4} + 7 T_{5}^{2} + 49 \)
\( T_{13} + 4 \)
\( T_{17}^{2} + 3 T_{17} + 9 \)
\( T_{23}^{4} + 7 T_{23}^{2} + 49 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T + T^{2} )^{2} \)
$3$ \( T^{4} \)
$5$ \( 49 + 7 T^{2} + T^{4} \)
$7$ \( 49 + 7 T^{2} + T^{4} \)
$11$ \( ( 1 - T + T^{2} )^{2} \)
$13$ \( ( 4 + T )^{4} \)
$17$ \( ( 9 + 3 T + T^{2} )^{2} \)
$19$ \( 784 + 28 T^{2} + T^{4} \)
$23$ \( 49 + 7 T^{2} + T^{4} \)
$29$ \( ( 2 + T )^{4} \)
$31$ \( ( 16 - 4 T + T^{2} )^{2} \)
$37$ \( 144 + 96 T + 76 T^{2} - 8 T^{3} + T^{4} \)
$41$ \( ( 9 + T )^{4} \)
$43$ \( ( -12 - 8 T + T^{2} )^{2} \)
$47$ \( 2209 + 376 T + 111 T^{2} - 8 T^{3} + T^{4} \)
$53$ \( ( 16 - 4 T + T^{2} )^{2} \)
$59$ \( 9216 - 768 T + 160 T^{2} + 8 T^{3} + T^{4} \)
$61$ \( 2209 - 376 T + 111 T^{2} + 8 T^{3} + T^{4} \)
$67$ \( 10609 + 618 T + 139 T^{2} - 6 T^{3} + T^{4} \)
$71$ \( ( 36 - 16 T + T^{2} )^{2} \)
$73$ \( 5184 - 1440 T + 328 T^{2} - 20 T^{3} + T^{4} \)
$79$ \( 3249 + 912 T + 199 T^{2} + 16 T^{3} + T^{4} \)
$83$ \( ( -87 + 10 T + T^{2} )^{2} \)
$89$ \( 1296 - 576 T + 220 T^{2} - 16 T^{3} + T^{4} \)
$97$ \( ( -111 + 2 T + T^{2} )^{2} \)
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