Properties

Label 1386.2.a.p
Level $1386$
Weight $2$
Character orbit 1386.a
Self dual yes
Analytic conductor $11.067$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1386 = 2 \cdot 3^{2} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1386.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(11.0672657201\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{3}) \)
Defining polynomial: \(x^{2} - 3\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 462)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 2\sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−1.73205
1.73205
1.00000 0 1.00000 −3.46410 0 1.00000 1.00000 0 −3.46410
1.2 1.00000 0 1.00000 3.46410 0 1.00000 1.00000 0 3.46410
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(7\) \(-1\)
\(11\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1386.2.a.p 2
3.b odd 2 1 462.2.a.h 2
7.b odd 2 1 9702.2.a.dd 2
12.b even 2 1 3696.2.a.bc 2
21.c even 2 1 3234.2.a.x 2
33.d even 2 1 5082.2.a.bu 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
462.2.a.h 2 3.b odd 2 1
1386.2.a.p 2 1.a even 1 1 trivial
3234.2.a.x 2 21.c even 2 1
3696.2.a.bc 2 12.b even 2 1
5082.2.a.bu 2 33.d even 2 1
9702.2.a.dd 2 7.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1386))\):

\( T_{5}^{2} - 12 \)
\( T_{13} - 2 \)
\( T_{17}^{2} - 12 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 + T )^{2} \)
$3$ \( T^{2} \)
$5$ \( -12 + T^{2} \)
$7$ \( ( -1 + T )^{2} \)
$11$ \( ( 1 + T )^{2} \)
$13$ \( ( -2 + T )^{2} \)
$17$ \( -12 + T^{2} \)
$19$ \( -8 - 4 T + T^{2} \)
$23$ \( -48 + T^{2} \)
$29$ \( ( -6 + T )^{2} \)
$31$ \( -8 - 4 T + T^{2} \)
$37$ \( -44 - 4 T + T^{2} \)
$41$ \( -12 + T^{2} \)
$43$ \( -32 + 8 T + T^{2} \)
$47$ \( 24 + 12 T + T^{2} \)
$53$ \( -12 + 12 T + T^{2} \)
$59$ \( -48 + T^{2} \)
$61$ \( ( -2 + T )^{2} \)
$67$ \( 16 - 16 T + T^{2} \)
$71$ \( ( -12 + T )^{2} \)
$73$ \( 4 + 8 T + T^{2} \)
$79$ \( -32 + 8 T + T^{2} \)
$83$ \( -72 - 12 T + T^{2} \)
$89$ \( -12 + 12 T + T^{2} \)
$97$ \( ( -2 + T )^{2} \)
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