Properties

Label 1386.4.a.g
Level $1386$
Weight $4$
Character orbit 1386.a
Self dual yes
Analytic conductor $81.777$
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: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1386.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(81.7766472680\)
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

\(f(q)\) \(=\) \( q - 2 q^{2} + 4 q^{4} + 14 q^{5} + 7 q^{7} - 8 q^{8} + O(q^{10}) \) \( q - 2 q^{2} + 4 q^{4} + 14 q^{5} + 7 q^{7} - 8 q^{8} - 28 q^{10} + 11 q^{11} - 16 q^{13} - 14 q^{14} + 16 q^{16} - 108 q^{17} + 116 q^{19} + 56 q^{20} - 22 q^{22} - 68 q^{23} + 71 q^{25} + 32 q^{26} + 28 q^{28} - 122 q^{29} - 262 q^{31} - 32 q^{32} + 216 q^{34} + 98 q^{35} + 130 q^{37} - 232 q^{38} - 112 q^{40} - 204 q^{41} - 396 q^{43} + 44 q^{44} + 136 q^{46} - 166 q^{47} + 49 q^{49} - 142 q^{50} - 64 q^{52} - 442 q^{53} + 154 q^{55} - 56 q^{56} + 244 q^{58} - 702 q^{59} + 196 q^{61} + 524 q^{62} + 64 q^{64} - 224 q^{65} - 416 q^{67} - 432 q^{68} - 196 q^{70} - 492 q^{71} + 408 q^{73} - 260 q^{74} + 464 q^{76} + 77 q^{77} + 600 q^{79} + 224 q^{80} + 408 q^{82} + 1212 q^{83} - 1512 q^{85} + 792 q^{86} - 88 q^{88} - 1146 q^{89} - 112 q^{91} - 272 q^{92} + 332 q^{94} + 1624 q^{95} - 482 q^{97} - 98 q^{98} + O(q^{100}) \)

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
−2.00000 0 4.00000 14.0000 0 7.00000 −8.00000 0 −28.0000
\(n\): 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 1386.4.a.g 1
3.b odd 2 1 154.4.a.c 1
12.b even 2 1 1232.4.a.i 1
21.c even 2 1 1078.4.a.h 1
33.d even 2 1 1694.4.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
154.4.a.c 1 3.b odd 2 1
1078.4.a.h 1 21.c even 2 1
1232.4.a.i 1 12.b even 2 1
1386.4.a.g 1 1.a even 1 1 trivial
1694.4.a.a 1 33.d even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1386))\):

\( T_{5} - 14 \)
\( T_{13} + 16 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 2 + T \)
$3$ \( T \)
$5$ \( -14 + T \)
$7$ \( -7 + T \)
$11$ \( -11 + T \)
$13$ \( 16 + T \)
$17$ \( 108 + T \)
$19$ \( -116 + T \)
$23$ \( 68 + T \)
$29$ \( 122 + T \)
$31$ \( 262 + T \)
$37$ \( -130 + T \)
$41$ \( 204 + T \)
$43$ \( 396 + T \)
$47$ \( 166 + T \)
$53$ \( 442 + T \)
$59$ \( 702 + T \)
$61$ \( -196 + T \)
$67$ \( 416 + T \)
$71$ \( 492 + T \)
$73$ \( -408 + T \)
$79$ \( -600 + T \)
$83$ \( -1212 + T \)
$89$ \( 1146 + T \)
$97$ \( 482 + T \)
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