Properties

Label 1386.2.a.r
Level $1386$
Weight $2$
Character orbit 1386.a
Self dual yes
Analytic conductor $11.067$
Analytic rank $0$
Dimension $3$
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: \(3\)
Coefficient field: 3.3.1304.1
Defining polynomial: \(x^{3} - 11 x - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

\(f(q)\) \(=\) \( q + q^{2} + q^{4} + ( 1 + \beta_{2} ) q^{5} + q^{7} + q^{8} +O(q^{10})\) \( q + q^{2} + q^{4} + ( 1 + \beta_{2} ) q^{5} + q^{7} + q^{8} + ( 1 + \beta_{2} ) q^{10} + q^{11} + ( \beta_{1} - \beta_{2} ) q^{13} + q^{14} + q^{16} + ( 1 + \beta_{2} ) q^{17} + ( 3 - \beta_{1} ) q^{19} + ( 1 + \beta_{2} ) q^{20} + q^{22} + ( -\beta_{1} - \beta_{2} ) q^{23} + ( 3 - \beta_{1} + \beta_{2} ) q^{25} + ( \beta_{1} - \beta_{2} ) q^{26} + q^{28} + ( -2 + \beta_{1} - \beta_{2} ) q^{29} + ( -1 - 3 \beta_{2} ) q^{31} + q^{32} + ( 1 + \beta_{2} ) q^{34} + ( 1 + \beta_{2} ) q^{35} + ( 4 + 2 \beta_{2} ) q^{37} + ( 3 - \beta_{1} ) q^{38} + ( 1 + \beta_{2} ) q^{40} + ( -1 - \beta_{2} ) q^{41} + ( 4 + \beta_{1} - 3 \beta_{2} ) q^{43} + q^{44} + ( -\beta_{1} - \beta_{2} ) q^{46} + ( -3 - \beta_{1} + 2 \beta_{2} ) q^{47} + q^{49} + ( 3 - \beta_{1} + \beta_{2} ) q^{50} + ( \beta_{1} - \beta_{2} ) q^{52} + ( 4 - 2 \beta_{2} ) q^{53} + ( 1 + \beta_{2} ) q^{55} + q^{56} + ( -2 + \beta_{1} - \beta_{2} ) q^{58} + ( -2 + 2 \beta_{1} ) q^{59} + ( 4 + \beta_{1} + 3 \beta_{2} ) q^{61} + ( -1 - 3 \beta_{2} ) q^{62} + q^{64} + ( -6 + 2 \beta_{1} - 4 \beta_{2} ) q^{65} + ( 2 + \beta_{1} + \beta_{2} ) q^{67} + ( 1 + \beta_{2} ) q^{68} + ( 1 + \beta_{2} ) q^{70} + ( -2 - \beta_{1} - 3 \beta_{2} ) q^{71} + ( -5 - \beta_{2} ) q^{73} + ( 4 + 2 \beta_{2} ) q^{74} + ( 3 - \beta_{1} ) q^{76} + q^{77} + ( 2 - \beta_{1} - \beta_{2} ) q^{79} + ( 1 + \beta_{2} ) q^{80} + ( -1 - \beta_{2} ) q^{82} + ( -7 - \beta_{2} ) q^{83} + ( 8 - \beta_{1} + \beta_{2} ) q^{85} + ( 4 + \beta_{1} - 3 \beta_{2} ) q^{86} + q^{88} + ( -6 - \beta_{1} - \beta_{2} ) q^{89} + ( \beta_{1} - \beta_{2} ) q^{91} + ( -\beta_{1} - \beta_{2} ) q^{92} + ( -3 - \beta_{1} + 2 \beta_{2} ) q^{94} + ( 2 - \beta_{1} + 7 \beta_{2} ) q^{95} + ( 6 - 2 \beta_{1} + 2 \beta_{2} ) q^{97} + q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + 3q^{2} + 3q^{4} + 2q^{5} + 3q^{7} + 3q^{8} + O(q^{10}) \) \( 3q + 3q^{2} + 3q^{4} + 2q^{5} + 3q^{7} + 3q^{8} + 2q^{10} + 3q^{11} + 3q^{14} + 3q^{16} + 2q^{17} + 10q^{19} + 2q^{20} + 3q^{22} + 2q^{23} + 9q^{25} + 3q^{28} - 6q^{29} + 3q^{32} + 2q^{34} + 2q^{35} + 10q^{37} + 10q^{38} + 2q^{40} - 2q^{41} + 14q^{43} + 3q^{44} + 2q^{46} - 10q^{47} + 3q^{49} + 9q^{50} + 14q^{53} + 2q^{55} + 3q^{56} - 6q^{58} - 8q^{59} + 8q^{61} + 3q^{64} - 16q^{65} + 4q^{67} + 2q^{68} + 2q^{70} - 2q^{71} - 14q^{73} + 10q^{74} + 10q^{76} + 3q^{77} + 8q^{79} + 2q^{80} - 2q^{82} - 20q^{83} + 24q^{85} + 14q^{86} + 3q^{88} - 16q^{89} + 2q^{92} - 10q^{94} + 18q^{97} + 3q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - 11 x - 2\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{2} + 3 \nu - 8 \)\()/2\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{2} - \nu - 8 \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{2} + \beta_{1}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(3 \beta_{2} + \beta_{1} + 16\)\()/2\)

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
−0.182370
3.40405
−3.22168
1.00000 0 1.00000 −2.89219 0 1.00000 1.00000 0 −2.89219
1.2 1.00000 0 1.00000 1.09174 0 1.00000 1.00000 0 1.09174
1.3 1.00000 0 1.00000 3.80044 0 1.00000 1.00000 0 3.80044
\(n\): 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.r yes 3
3.b odd 2 1 1386.2.a.q 3
7.b odd 2 1 9702.2.a.dy 3
21.c even 2 1 9702.2.a.dx 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1386.2.a.q 3 3.b odd 2 1
1386.2.a.r yes 3 1.a even 1 1 trivial
9702.2.a.dx 3 21.c even 2 1
9702.2.a.dy 3 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(1386))\):

\( T_{5}^{3} - 2 T_{5}^{2} - 10 T_{5} + 12 \)
\( T_{13}^{3} - 44 T_{13} - 16 \)
\( T_{17}^{3} - 2 T_{17}^{2} - 10 T_{17} + 12 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 + T )^{3} \)
$3$ \( T^{3} \)
$5$ \( 12 - 10 T - 2 T^{2} + T^{3} \)
$7$ \( ( -1 + T )^{3} \)
$11$ \( ( -1 + T )^{3} \)
$13$ \( -16 - 44 T + T^{3} \)
$17$ \( 12 - 10 T - 2 T^{2} + T^{3} \)
$19$ \( 188 - 6 T - 10 T^{2} + T^{3} \)
$23$ \( 48 - 56 T - 2 T^{2} + T^{3} \)
$29$ \( -96 - 32 T + 6 T^{2} + T^{3} \)
$31$ \( -128 - 102 T + T^{3} \)
$37$ \( 152 - 12 T - 10 T^{2} + T^{3} \)
$41$ \( -12 - 10 T + 2 T^{2} + T^{3} \)
$43$ \( 976 - 56 T - 14 T^{2} + T^{3} \)
$47$ \( -396 - 38 T + 10 T^{2} + T^{3} \)
$53$ \( 72 + 20 T - 14 T^{2} + T^{3} \)
$59$ \( -1152 - 136 T + 8 T^{2} + T^{3} \)
$61$ \( 1168 - 140 T - 8 T^{2} + T^{3} \)
$67$ \( 64 - 52 T - 4 T^{2} + T^{3} \)
$71$ \( -864 - 160 T + 2 T^{2} + T^{3} \)
$73$ \( 44 + 54 T + 14 T^{2} + T^{3} \)
$79$ \( 144 - 36 T - 8 T^{2} + T^{3} \)
$83$ \( 216 + 122 T + 20 T^{2} + T^{3} \)
$89$ \( -144 + 28 T + 16 T^{2} + T^{3} \)
$97$ \( 968 - 68 T - 18 T^{2} + T^{3} \)
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