Properties

Label 1386.2.a.f
Level $1386$
Weight $2$
Character orbit 1386.a
Self dual yes
Analytic conductor $11.067$
Analytic rank $1$
Dimension $1$
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: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 154)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

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
0
1.00000 0 1.00000 −2.00000 0 −1.00000 1.00000 0 −2.00000
\(n\): 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.f 1
3.b odd 2 1 154.2.a.b 1
7.b odd 2 1 9702.2.a.bz 1
12.b even 2 1 1232.2.a.c 1
15.d odd 2 1 3850.2.a.o 1
15.e even 4 2 3850.2.c.d 2
21.c even 2 1 1078.2.a.b 1
21.g even 6 2 1078.2.e.l 2
21.h odd 6 2 1078.2.e.h 2
24.f even 2 1 4928.2.a.bf 1
24.h odd 2 1 4928.2.a.d 1
33.d even 2 1 1694.2.a.i 1
84.h odd 2 1 8624.2.a.z 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
154.2.a.b 1 3.b odd 2 1
1078.2.a.b 1 21.c even 2 1
1078.2.e.h 2 21.h odd 6 2
1078.2.e.l 2 21.g even 6 2
1232.2.a.c 1 12.b even 2 1
1386.2.a.f 1 1.a even 1 1 trivial
1694.2.a.i 1 33.d even 2 1
3850.2.a.o 1 15.d odd 2 1
3850.2.c.d 2 15.e even 4 2
4928.2.a.d 1 24.h odd 2 1
4928.2.a.bf 1 24.f even 2 1
8624.2.a.z 1 84.h odd 2 1
9702.2.a.bz 1 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(1386))\):

\( T_{5} + 2 \)
\( T_{13} + 4 \)
\( T_{17} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -1 + T \)
$3$ \( T \)
$5$ \( 2 + T \)
$7$ \( 1 + T \)
$11$ \( 1 + T \)
$13$ \( 4 + T \)
$17$ \( T \)
$19$ \( -4 + T \)
$23$ \( 4 + T \)
$29$ \( 2 + T \)
$31$ \( 10 + T \)
$37$ \( 6 + T \)
$41$ \( T \)
$43$ \( 4 + T \)
$47$ \( 10 + T \)
$53$ \( -14 + T \)
$59$ \( 10 + T \)
$61$ \( 8 + T \)
$67$ \( -8 + T \)
$71$ \( -4 + T \)
$73$ \( -4 + T \)
$79$ \( -16 + T \)
$83$ \( 4 + T \)
$89$ \( 10 + T \)
$97$ \( -6 + T \)
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