Properties

Label 1386.2.k.o
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.

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

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 1.00000 1.73205i 0 −2.50000 + 0.866025i −1.00000 0 −1.00000 1.73205i
991.1 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 1.00000 + 1.73205i 0 −2.50000 0.866025i −1.00000 0 −1.00000 + 1.73205i
\(n\): 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.o 2
3.b odd 2 1 154.2.e.a 2
7.c even 3 1 inner 1386.2.k.o 2
7.c even 3 1 9702.2.a.i 1
7.d odd 6 1 9702.2.a.y 1
12.b even 2 1 1232.2.q.e 2
21.c even 2 1 1078.2.e.f 2
21.g even 6 1 1078.2.a.g 1
21.g even 6 1 1078.2.e.f 2
21.h odd 6 1 154.2.e.a 2
21.h odd 6 1 1078.2.a.m 1
84.j odd 6 1 8624.2.a.be 1
84.n even 6 1 1232.2.q.e 2
84.n even 6 1 8624.2.a.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
154.2.e.a 2 3.b odd 2 1
154.2.e.a 2 21.h odd 6 1
1078.2.a.g 1 21.g even 6 1
1078.2.a.m 1 21.h odd 6 1
1078.2.e.f 2 21.c even 2 1
1078.2.e.f 2 21.g even 6 1
1232.2.q.e 2 12.b even 2 1
1232.2.q.e 2 84.n even 6 1
1386.2.k.o 2 1.a even 1 1 trivial
1386.2.k.o 2 7.c even 3 1 inner
8624.2.a.b 1 84.n even 6 1
8624.2.a.be 1 84.j odd 6 1
9702.2.a.i 1 7.c even 3 1
9702.2.a.y 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 \(S_{2}^{\mathrm{new}}(1386, [\chi])\):

\( T_{5}^{2} - 2 T_{5} + 4 \)
\( T_{13} + 7 \)
\( T_{17}^{2} - 2 T_{17} + 4 \)
\( T_{23}^{2} + 8 T_{23} + 64 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - T + T^{2} \)
$3$ \( T^{2} \)
$5$ \( 4 - 2 T + T^{2} \)
$7$ \( 7 + 5 T + T^{2} \)
$11$ \( 1 + T + T^{2} \)
$13$ \( ( 7 + T )^{2} \)
$17$ \( 4 - 2 T + T^{2} \)
$19$ \( T^{2} \)
$23$ \( 64 + 8 T + T^{2} \)
$29$ \( ( -5 + T )^{2} \)
$31$ \( 16 + 4 T + T^{2} \)
$37$ \( 16 + 4 T + T^{2} \)
$41$ \( ( 4 + T )^{2} \)
$43$ \( ( 8 + T )^{2} \)
$47$ \( 4 - 2 T + T^{2} \)
$53$ \( 36 + 6 T + T^{2} \)
$59$ \( 9 - 3 T + T^{2} \)
$61$ \( 1 + T + T^{2} \)
$67$ \( 81 + 9 T + T^{2} \)
$71$ \( ( -2 + T )^{2} \)
$73$ \( 16 + 4 T + T^{2} \)
$79$ \( 81 + 9 T + T^{2} \)
$83$ \( ( 6 + T )^{2} \)
$89$ \( 36 - 6 T + T^{2} \)
$97$ \( ( -7 + T )^{2} \)
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