Properties

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

Related objects

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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

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

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

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T \)
$3$ \( T \)
$5$ \( 2 + T \)
$7$ \( 1 + T \)
$11$ \( -1 + T \)
$13$ \( -2 + T \)
$17$ \( 2 + T \)
$19$ \( T \)
$23$ \( -8 + T \)
$29$ \( -2 + T \)
$31$ \( 8 + T \)
$37$ \( 2 + T \)
$41$ \( 10 + T \)
$43$ \( -4 + T \)
$47$ \( 8 + T \)
$53$ \( 6 + T \)
$59$ \( T \)
$61$ \( -10 + T \)
$67$ \( 12 + T \)
$71$ \( 16 + T \)
$73$ \( 14 + T \)
$79$ \( T \)
$83$ \( T \)
$89$ \( -6 + T \)
$97$ \( -10 + T \)
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