Properties

Label 816.4.a.a
Level $816$
Weight $4$
Character orbit 816.a
Self dual yes
Analytic conductor $48.146$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [816,4,Mod(1,816)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(816, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("816.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 816 = 2^{4} \cdot 3 \cdot 17 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 816.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: \(48.1455585647\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 51)
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)
 
\(f(q)\) \(=\) \( q - 3 q^{3} - 20 q^{5} + 2 q^{7} + 9 q^{9} + 48 q^{11} - 14 q^{13} + 60 q^{15} - 17 q^{17} - 92 q^{19} - 6 q^{21} + 122 q^{23} + 275 q^{25} - 27 q^{27} - 36 q^{29} + 182 q^{31} - 144 q^{33} - 40 q^{35}+ \cdots + 432 q^{99}+O(q^{100}) \) 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
0 −3.00000 0 −20.0000 0 2.00000 0 9.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(3\) \( +1 \)
\(17\) \( +1 \)

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 816.4.a.a 1
3.b odd 2 1 2448.4.a.r 1
4.b odd 2 1 51.4.a.b 1
12.b even 2 1 153.4.a.c 1
20.d odd 2 1 1275.4.a.e 1
28.d even 2 1 2499.4.a.d 1
68.d odd 2 1 867.4.a.c 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
51.4.a.b 1 4.b odd 2 1
153.4.a.c 1 12.b even 2 1
816.4.a.a 1 1.a even 1 1 trivial
867.4.a.c 1 68.d odd 2 1
1275.4.a.e 1 20.d odd 2 1
2448.4.a.r 1 3.b odd 2 1
2499.4.a.d 1 28.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 \(S_{4}^{\mathrm{new}}(\Gamma_0(816))\):

\( T_{5} + 20 \) Copy content Toggle raw display
\( T_{7} - 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T + 3 \) Copy content Toggle raw display
$5$ \( T + 20 \) Copy content Toggle raw display
$7$ \( T - 2 \) Copy content Toggle raw display
$11$ \( T - 48 \) Copy content Toggle raw display
$13$ \( T + 14 \) Copy content Toggle raw display
$17$ \( T + 17 \) Copy content Toggle raw display
$19$ \( T + 92 \) Copy content Toggle raw display
$23$ \( T - 122 \) Copy content Toggle raw display
$29$ \( T + 36 \) Copy content Toggle raw display
$31$ \( T - 182 \) Copy content Toggle raw display
$37$ \( T - 76 \) Copy content Toggle raw display
$41$ \( T - 294 \) Copy content Toggle raw display
$43$ \( T - 428 \) Copy content Toggle raw display
$47$ \( T - 12 \) Copy content Toggle raw display
$53$ \( T + 234 \) Copy content Toggle raw display
$59$ \( T - 540 \) Copy content Toggle raw display
$61$ \( T + 820 \) Copy content Toggle raw display
$67$ \( T + 700 \) Copy content Toggle raw display
$71$ \( T + 794 \) Copy content Toggle raw display
$73$ \( T + 1038 \) Copy content Toggle raw display
$79$ \( T + 858 \) Copy content Toggle raw display
$83$ \( T + 1052 \) Copy content Toggle raw display
$89$ \( T - 1102 \) Copy content Toggle raw display
$97$ \( T - 710 \) Copy content Toggle raw display
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