Properties

Label 1386.2.k.e
Level $1386$
Weight $2$
Character orbit 1386.k
Analytic conductor $11.067$
Analytic rank $1$
Dimension $2$
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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
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 primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

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

Character values

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

\(n\) \(155\) \(199\) \(1135\)
\(\chi(n)\) \(1\) \(-\zeta_{6}\) \(1\)

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 0.866025i
0.500000 + 0.866025i
−0.500000 + 0.866025i 0 −0.500000 0.866025i 0 0 −0.500000 2.59808i 1.00000 0 0
991.1 −0.500000 0.866025i 0 −0.500000 + 0.866025i 0 0 −0.500000 + 2.59808i 1.00000 0 0
\(n\): 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.e 2
3.b odd 2 1 154.2.e.c 2
7.c even 3 1 inner 1386.2.k.e 2
7.c even 3 1 9702.2.a.br 1
7.d odd 6 1 9702.2.a.bs 1
12.b even 2 1 1232.2.q.d 2
21.c even 2 1 1078.2.e.k 2
21.g even 6 1 1078.2.a.c 1
21.g even 6 1 1078.2.e.k 2
21.h odd 6 1 154.2.e.c 2
21.h odd 6 1 1078.2.a.e 1
84.j odd 6 1 8624.2.a.u 1
84.n even 6 1 1232.2.q.d 2
84.n even 6 1 8624.2.a.k 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
154.2.e.c 2 3.b odd 2 1
154.2.e.c 2 21.h odd 6 1
1078.2.a.c 1 21.g even 6 1
1078.2.a.e 1 21.h odd 6 1
1078.2.e.k 2 21.c even 2 1
1078.2.e.k 2 21.g even 6 1
1232.2.q.d 2 12.b even 2 1
1232.2.q.d 2 84.n even 6 1
1386.2.k.e 2 1.a even 1 1 trivial
1386.2.k.e 2 7.c even 3 1 inner
8624.2.a.k 1 84.n even 6 1
8624.2.a.u 1 84.j odd 6 1
9702.2.a.br 1 7.c even 3 1
9702.2.a.bs 1 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 \(S_{2}^{\mathrm{new}}(1386, [\chi])\):

\( T_{5} \)
\( T_{13} + 1 \)
\( T_{17}^{2} + 6 T_{17} + 36 \)
\( T_{23}^{2} + 6 T_{23} + 36 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T + T^{2} \)
$3$ \( T^{2} \)
$5$ \( T^{2} \)
$7$ \( 7 + T + T^{2} \)
$11$ \( 1 + T + T^{2} \)
$13$ \( ( 1 + T )^{2} \)
$17$ \( 36 + 6 T + T^{2} \)
$19$ \( 4 + 2 T + T^{2} \)
$23$ \( 36 + 6 T + T^{2} \)
$29$ \( ( 9 + T )^{2} \)
$31$ \( 16 - 4 T + T^{2} \)
$37$ \( 4 + 2 T + T^{2} \)
$41$ \( ( -6 + T )^{2} \)
$43$ \( ( 4 + T )^{2} \)
$47$ \( 36 + 6 T + T^{2} \)
$53$ \( T^{2} \)
$59$ \( 9 + 3 T + T^{2} \)
$61$ \( 121 + 11 T + T^{2} \)
$67$ \( 121 + 11 T + T^{2} \)
$71$ \( T^{2} \)
$73$ \( 4 + 2 T + T^{2} \)
$79$ \( 25 + 5 T + T^{2} \)
$83$ \( ( -6 + T )^{2} \)
$89$ \( 324 + 18 T + T^{2} \)
$97$ \( ( 13 + T )^{2} \)
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