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: \( x^{4} + 7x^{2} + 49 \) Copy content Toggle raw display
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.

\(f(q)\) \(=\) \( q + ( - \beta_{2} - 1) q^{2} + \beta_{2} q^{4} + \beta_1 q^{5} - \beta_1 q^{7} + q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} - 1) q^{2} + \beta_{2} q^{4} + \beta_1 q^{5} - \beta_1 q^{7} + q^{8} + ( - \beta_{3} - \beta_1) q^{10} - \beta_{2} q^{11} - 4 q^{13} + (\beta_{3} + \beta_1) q^{14} + ( - \beta_{2} - 1) q^{16} + 3 \beta_{2} q^{17} - 2 \beta_1 q^{19} + \beta_{3} q^{20} - q^{22} - \beta_1 q^{23} + 2 \beta_{2} q^{25} + (4 \beta_{2} + 4) q^{26} - \beta_{3} q^{28} - 2 q^{29} - 4 \beta_{2} q^{31} + \beta_{2} q^{32} + 3 q^{34} - 7 \beta_{2} q^{35} + (4 \beta_{2} - 2 \beta_1 + 4) q^{37} + (2 \beta_{3} + 2 \beta_1) q^{38} + \beta_1 q^{40} - 9 q^{41} + ( - 2 \beta_{3} + 4) q^{43} + (\beta_{2} + 1) q^{44} + (\beta_{3} + \beta_1) q^{46} + (4 \beta_{2} - 3 \beta_1 + 4) q^{47} + 7 \beta_{2} q^{49} + 2 q^{50} - 4 \beta_{2} q^{52} - 4 \beta_{2} q^{53} - \beta_{3} q^{55} - \beta_1 q^{56} + (2 \beta_{2} + 2) q^{58} + ( - 4 \beta_{3} + 4 \beta_{2} - 4 \beta_1) q^{59} + ( - 4 \beta_{2} - 3 \beta_1 - 4) q^{61} - 4 q^{62} + q^{64} - 4 \beta_1 q^{65} + (4 \beta_{3} - 3 \beta_{2} + 4 \beta_1) q^{67} + ( - 3 \beta_{2} - 3) q^{68} - 7 q^{70} + (2 \beta_{3} + 8) q^{71} + ( - 2 \beta_{3} - 10 \beta_{2} - 2 \beta_1) q^{73} + (2 \beta_{3} - 4 \beta_{2} + 2 \beta_1) q^{74} - 2 \beta_{3} q^{76} + \beta_{3} q^{77} + ( - 8 \beta_{2} - \beta_1 - 8) q^{79} + ( - \beta_{3} - \beta_1) q^{80} + (9 \beta_{2} + 9) q^{82} + (4 \beta_{3} - 5) q^{83} + 3 \beta_{3} q^{85} + ( - 4 \beta_{2} - 2 \beta_1 - 4) q^{86} - \beta_{2} q^{88} + (8 \beta_{2} - 2 \beta_1 + 8) q^{89} + 4 \beta_1 q^{91} - \beta_{3} q^{92} + (3 \beta_{3} - 4 \beta_{2} + 3 \beta_1) q^{94} - 14 \beta_{2} q^{95} + (4 \beta_{3} - 1) q^{97} + 7 q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{2} - 2 q^{4} + 4 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{2} - 2 q^{4} + 4 q^{8} + 2 q^{11} - 16 q^{13} - 2 q^{16} - 6 q^{17} - 4 q^{22} - 4 q^{25} + 8 q^{26} - 8 q^{29} + 8 q^{31} - 2 q^{32} + 12 q^{34} + 14 q^{35} + 8 q^{37} - 36 q^{41} + 16 q^{43} + 2 q^{44} + 8 q^{47} - 14 q^{49} + 8 q^{50} + 8 q^{52} + 8 q^{53} + 4 q^{58} - 8 q^{59} - 8 q^{61} - 16 q^{62} + 4 q^{64} + 6 q^{67} - 6 q^{68} - 28 q^{70} + 32 q^{71} + 20 q^{73} + 8 q^{74} - 16 q^{79} + 18 q^{82} - 20 q^{83} - 8 q^{86} + 2 q^{88} + 16 q^{89} + 8 q^{94} + 28 q^{95} - 4 q^{97} + 28 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 7 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} ) / 7 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 7\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 7\beta_{3} \) Copy content Toggle raw display

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 \(S_{2}^{\mathrm{new}}(1386, [\chi])\):

\( T_{5}^{4} + 7T_{5}^{2} + 49 \) Copy content Toggle raw display
\( T_{13} + 4 \) Copy content Toggle raw display
\( T_{17}^{2} + 3T_{17} + 9 \) Copy content Toggle raw display
\( T_{23}^{4} + 7T_{23}^{2} + 49 \) Copy content Toggle raw display

Hecke characteristic polynomials

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