Properties

Label 1386.4.a.i
Level $1386$
Weight $4$
Character orbit 1386.a
Self dual yes
Analytic conductor $81.777$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1386,4,Mod(1,1386)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1386, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1386.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1386 = 2 \cdot 3^{2} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1386.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(81.7766472680\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 462)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \( q + 2 q^{2} + 4 q^{4} - 11 q^{5} - 7 q^{7} + 8 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 q^{2} + 4 q^{4} - 11 q^{5} - 7 q^{7} + 8 q^{8} - 22 q^{10} - 11 q^{11} - 37 q^{13} - 14 q^{14} + 16 q^{16} + 46 q^{17} + 15 q^{19} - 44 q^{20} - 22 q^{22} + 92 q^{23} - 4 q^{25} - 74 q^{26} - 28 q^{28} - 205 q^{29} + 142 q^{31} + 32 q^{32} + 92 q^{34} + 77 q^{35} - 431 q^{37} + 30 q^{38} - 88 q^{40} + 8 q^{41} + 448 q^{43} - 44 q^{44} + 184 q^{46} - 149 q^{47} + 49 q^{49} - 8 q^{50} - 148 q^{52} + 672 q^{53} + 121 q^{55} - 56 q^{56} - 410 q^{58} + 615 q^{59} + 322 q^{61} + 284 q^{62} + 64 q^{64} + 407 q^{65} - 411 q^{67} + 184 q^{68} + 154 q^{70} + 968 q^{71} - 227 q^{73} - 862 q^{74} + 60 q^{76} + 77 q^{77} - 176 q^{80} + 16 q^{82} + 1302 q^{83} - 506 q^{85} + 896 q^{86} - 88 q^{88} + 870 q^{89} + 259 q^{91} + 368 q^{92} - 298 q^{94} - 165 q^{95} - 1736 q^{97} + 98 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
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 −11.0000 0 −7.00000 8.00000 0 −22.0000
\(n\): 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.i 1
3.b odd 2 1 462.4.a.c 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
462.4.a.c 1 3.b odd 2 1
1386.4.a.i 1 1.a even 1 1 trivial

Hecke kernels

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

\( T_{5} + 11 \) Copy content Toggle raw display
\( T_{13} + 37 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 2 \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T + 11 \) Copy content Toggle raw display
$7$ \( T + 7 \) Copy content Toggle raw display
$11$ \( T + 11 \) Copy content Toggle raw display
$13$ \( T + 37 \) Copy content Toggle raw display
$17$ \( T - 46 \) Copy content Toggle raw display
$19$ \( T - 15 \) Copy content Toggle raw display
$23$ \( T - 92 \) Copy content Toggle raw display
$29$ \( T + 205 \) Copy content Toggle raw display
$31$ \( T - 142 \) Copy content Toggle raw display
$37$ \( T + 431 \) Copy content Toggle raw display
$41$ \( T - 8 \) Copy content Toggle raw display
$43$ \( T - 448 \) Copy content Toggle raw display
$47$ \( T + 149 \) Copy content Toggle raw display
$53$ \( T - 672 \) Copy content Toggle raw display
$59$ \( T - 615 \) Copy content Toggle raw display
$61$ \( T - 322 \) Copy content Toggle raw display
$67$ \( T + 411 \) Copy content Toggle raw display
$71$ \( T - 968 \) Copy content Toggle raw display
$73$ \( T + 227 \) Copy content Toggle raw display
$79$ \( T \) Copy content Toggle raw display
$83$ \( T - 1302 \) Copy content Toggle raw display
$89$ \( T - 870 \) Copy content Toggle raw display
$97$ \( T + 1736 \) Copy content Toggle raw display
show more
show less