Properties

Label 208.4.a.f
Level $208$
Weight $4$
Character orbit 208.a
Self dual yes
Analytic conductor $12.272$
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] = [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: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 52)
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)
 
\(f(q)\) \(=\) \( q + 3 q^{3} - 13 q^{5} + 11 q^{7} - 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 3 q^{3} - 13 q^{5} + 11 q^{7} - 18 q^{9} + 2 q^{11} - 13 q^{13} - 39 q^{15} - 51 q^{17} - 150 q^{19} + 33 q^{21} + 4 q^{23} + 44 q^{25} - 135 q^{27} - 118 q^{29} + 116 q^{31} + 6 q^{33} - 143 q^{35} + 63 q^{37} - 39 q^{39} - 288 q^{41} + 293 q^{43} + 234 q^{45} + 335 q^{47} - 222 q^{49} - 153 q^{51} - 708 q^{53} - 26 q^{55} - 450 q^{57} - 566 q^{59} + 904 q^{61} - 198 q^{63} + 169 q^{65} - 382 q^{67} + 12 q^{69} - 7 q^{71} + 518 q^{73} + 132 q^{75} + 22 q^{77} + 100 q^{79} + 81 q^{81} + 1440 q^{83} + 663 q^{85} - 354 q^{87} + 1254 q^{89} - 143 q^{91} + 348 q^{93} + 1950 q^{95} + 1262 q^{97} - 36 q^{99}+O(q^{100}) \) 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 −13.0000 0 11.0000 0 −18.0000 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.f 1
3.b odd 2 1 1872.4.a.n 1
4.b odd 2 1 52.4.a.a 1
8.b even 2 1 832.4.a.f 1
8.d odd 2 1 832.4.a.n 1
12.b even 2 1 468.4.a.c 1
20.d odd 2 1 1300.4.a.d 1
20.e even 4 2 1300.4.c.b 2
52.b odd 2 1 676.4.a.a 1
52.f even 4 2 676.4.d.a 2
52.i odd 6 2 676.4.e.b 2
52.j odd 6 2 676.4.e.a 2
52.l even 12 4 676.4.h.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
52.4.a.a 1 4.b odd 2 1
208.4.a.f 1 1.a even 1 1 trivial
468.4.a.c 1 12.b even 2 1
676.4.a.a 1 52.b odd 2 1
676.4.d.a 2 52.f even 4 2
676.4.e.a 2 52.j odd 6 2
676.4.e.b 2 52.i odd 6 2
676.4.h.d 4 52.l even 12 4
832.4.a.f 1 8.b even 2 1
832.4.a.n 1 8.d odd 2 1
1300.4.a.d 1 20.d odd 2 1
1300.4.c.b 2 20.e even 4 2
1872.4.a.n 1 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} - 3 \) 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
$3$ \( T - 3 \) Copy content Toggle raw display
$5$ \( T + 13 \) Copy content Toggle raw display
$7$ \( T - 11 \) Copy content Toggle raw display
$11$ \( T - 2 \) Copy content Toggle raw display
$13$ \( T + 13 \) Copy content Toggle raw display
$17$ \( T + 51 \) Copy content Toggle raw display
$19$ \( T + 150 \) Copy content Toggle raw display
$23$ \( T - 4 \) Copy content Toggle raw display
$29$ \( T + 118 \) Copy content Toggle raw display
$31$ \( T - 116 \) Copy content Toggle raw display
$37$ \( T - 63 \) Copy content Toggle raw display
$41$ \( T + 288 \) Copy content Toggle raw display
$43$ \( T - 293 \) Copy content Toggle raw display
$47$ \( T - 335 \) Copy content Toggle raw display
$53$ \( T + 708 \) Copy content Toggle raw display
$59$ \( T + 566 \) Copy content Toggle raw display
$61$ \( T - 904 \) Copy content Toggle raw display
$67$ \( T + 382 \) Copy content Toggle raw display
$71$ \( T + 7 \) Copy content Toggle raw display
$73$ \( T - 518 \) Copy content Toggle raw display
$79$ \( T - 100 \) Copy content Toggle raw display
$83$ \( T - 1440 \) Copy content Toggle raw display
$89$ \( T - 1254 \) Copy content Toggle raw display
$97$ \( T - 1262 \) Copy content Toggle raw display
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