Properties

Label 208.4.a.e
Level $208$
Weight $4$
Character orbit 208.a
Self dual yes
Analytic conductor $12.272$
Analytic rank $0$
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] = [208,4,Mod(1,208)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(208, 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("208.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 208 = 2^{4} \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 208.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: \(12.2723972812\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 26)
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)
 
\(f(q)\) \(=\) \( q + q^{3} + 17 q^{5} + 35 q^{7} - 26 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{3} + 17 q^{5} + 35 q^{7} - 26 q^{9} - 2 q^{11} + 13 q^{13} + 17 q^{15} - 19 q^{17} - 94 q^{19} + 35 q^{21} + 72 q^{23} + 164 q^{25} - 53 q^{27} + 246 q^{29} + 100 q^{31} - 2 q^{33} + 595 q^{35} - 11 q^{37} + 13 q^{39} - 280 q^{41} - 241 q^{43} - 442 q^{45} - 137 q^{47} + 882 q^{49} - 19 q^{51} - 232 q^{53} - 34 q^{55} - 94 q^{57} + 386 q^{59} + 64 q^{61} - 910 q^{63} + 221 q^{65} + 670 q^{67} + 72 q^{69} - 55 q^{71} - 838 q^{73} + 164 q^{75} - 70 q^{77} - 1016 q^{79} + 649 q^{81} - 420 q^{83} - 323 q^{85} + 246 q^{87} - 934 q^{89} + 455 q^{91} + 100 q^{93} - 1598 q^{95} - 1154 q^{97} + 52 q^{99}+O(q^{100}) \) 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 1.00000 0 17.0000 0 35.0000 0 −26.0000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(13\) \(-1\)

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 208.4.a.e 1
3.b odd 2 1 1872.4.a.b 1
4.b odd 2 1 26.4.a.b 1
8.b even 2 1 832.4.a.g 1
8.d odd 2 1 832.4.a.j 1
12.b even 2 1 234.4.a.a 1
20.d odd 2 1 650.4.a.c 1
20.e even 4 2 650.4.b.d 2
28.d even 2 1 1274.4.a.f 1
52.b odd 2 1 338.4.a.b 1
52.f even 4 2 338.4.b.b 2
52.i odd 6 2 338.4.c.g 2
52.j odd 6 2 338.4.c.c 2
52.l even 12 4 338.4.e.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.4.a.b 1 4.b odd 2 1
208.4.a.e 1 1.a even 1 1 trivial
234.4.a.a 1 12.b even 2 1
338.4.a.b 1 52.b odd 2 1
338.4.b.b 2 52.f even 4 2
338.4.c.c 2 52.j odd 6 2
338.4.c.g 2 52.i odd 6 2
338.4.e.c 4 52.l even 12 4
650.4.a.c 1 20.d odd 2 1
650.4.b.d 2 20.e even 4 2
832.4.a.g 1 8.b even 2 1
832.4.a.j 1 8.d odd 2 1
1274.4.a.f 1 28.d even 2 1
1872.4.a.b 1 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} - 1 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(208))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T - 1 \) Copy content Toggle raw display
$5$ \( T - 17 \) Copy content Toggle raw display
$7$ \( T - 35 \) Copy content Toggle raw display
$11$ \( T + 2 \) Copy content Toggle raw display
$13$ \( T - 13 \) Copy content Toggle raw display
$17$ \( T + 19 \) Copy content Toggle raw display
$19$ \( T + 94 \) Copy content Toggle raw display
$23$ \( T - 72 \) Copy content Toggle raw display
$29$ \( T - 246 \) Copy content Toggle raw display
$31$ \( T - 100 \) Copy content Toggle raw display
$37$ \( T + 11 \) Copy content Toggle raw display
$41$ \( T + 280 \) Copy content Toggle raw display
$43$ \( T + 241 \) Copy content Toggle raw display
$47$ \( T + 137 \) Copy content Toggle raw display
$53$ \( T + 232 \) Copy content Toggle raw display
$59$ \( T - 386 \) Copy content Toggle raw display
$61$ \( T - 64 \) Copy content Toggle raw display
$67$ \( T - 670 \) Copy content Toggle raw display
$71$ \( T + 55 \) Copy content Toggle raw display
$73$ \( T + 838 \) Copy content Toggle raw display
$79$ \( T + 1016 \) Copy content Toggle raw display
$83$ \( T + 420 \) Copy content Toggle raw display
$89$ \( T + 934 \) Copy content Toggle raw display
$97$ \( T + 1154 \) Copy content Toggle raw display
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