Properties

Label 387.3.b.a
Level 387
Weight 3
Character orbit 387.b
Self dual yes
Analytic conductor 10.545
Analytic rank 0
Dimension 1
CM discriminant -43
Inner twists 2

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) \(=\) \( 387 = 3^{2} \cdot 43 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 387.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(10.5449862307\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 43)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

\(f(q)\) \(=\) \( q + 4q^{4} + O(q^{10}) \) \( q + 4q^{4} + 21q^{11} - 17q^{13} + 16q^{16} + 9q^{17} - 3q^{23} + 25q^{25} + 19q^{31} - 39q^{41} - 43q^{43} + 84q^{44} + 78q^{47} + 49q^{49} - 68q^{52} - 63q^{53} + 54q^{59} + 64q^{64} + 91q^{67} + 36q^{68} - 14q^{79} - 123q^{83} - 12q^{92} - 193q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/387\mathbb{Z}\right)^\times\).

\(n\) \(46\) \(173\)
\(\chi(n)\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
343.1
0
0 0 4.00000 0 0 0 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
43.b odd 2 1 CM by \(\Q(\sqrt{-43}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 387.3.b.a 1
3.b odd 2 1 43.3.b.a 1
12.b even 2 1 688.3.b.a 1
43.b odd 2 1 CM 387.3.b.a 1
129.d even 2 1 43.3.b.a 1
516.h odd 2 1 688.3.b.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
43.3.b.a 1 3.b odd 2 1
43.3.b.a 1 129.d even 2 1
387.3.b.a 1 1.a even 1 1 trivial
387.3.b.a 1 43.b odd 2 1 CM
688.3.b.a 1 12.b even 2 1
688.3.b.a 1 516.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(387, [\chi])\):

\( T_{2} \)
\( T_{11} - 21 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - 2 T )( 1 + 2 T ) \)
$3$ 1
$5$ \( ( 1 - 5 T )( 1 + 5 T ) \)
$7$ \( ( 1 - 7 T )( 1 + 7 T ) \)
$11$ \( 1 - 21 T + 121 T^{2} \)
$13$ \( 1 + 17 T + 169 T^{2} \)
$17$ \( 1 - 9 T + 289 T^{2} \)
$19$ \( ( 1 - 19 T )( 1 + 19 T ) \)
$23$ \( 1 + 3 T + 529 T^{2} \)
$29$ \( ( 1 - 29 T )( 1 + 29 T ) \)
$31$ \( 1 - 19 T + 961 T^{2} \)
$37$ \( ( 1 - 37 T )( 1 + 37 T ) \)
$41$ \( 1 + 39 T + 1681 T^{2} \)
$43$ \( 1 + 43 T \)
$47$ \( 1 - 78 T + 2209 T^{2} \)
$53$ \( 1 + 63 T + 2809 T^{2} \)
$59$ \( 1 - 54 T + 3481 T^{2} \)
$61$ \( ( 1 - 61 T )( 1 + 61 T ) \)
$67$ \( 1 - 91 T + 4489 T^{2} \)
$71$ \( ( 1 - 71 T )( 1 + 71 T ) \)
$73$ \( ( 1 - 73 T )( 1 + 73 T ) \)
$79$ \( 1 + 14 T + 6241 T^{2} \)
$83$ \( 1 + 123 T + 6889 T^{2} \)
$89$ \( ( 1 - 89 T )( 1 + 89 T ) \)
$97$ \( 1 + 193 T + 9409 T^{2} \)
show more
show less