Properties

Label 208.8.a.c
Level $208$
Weight $8$
Character orbit 208.a
Self dual yes
Analytic conductor $64.976$
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] = [208,8,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, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("208.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 208 = 2^{4} \cdot 13 \)
Weight: \( k \) \(=\) \( 8 \)
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: \(64.9760853007\)
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)
 
\(f(q)\) \(=\) \( q + 39 q^{3} + 385 q^{5} + 293 q^{7} - 666 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 39 q^{3} + 385 q^{5} + 293 q^{7} - 666 q^{9} + 5402 q^{11} + 2197 q^{13} + 15015 q^{15} - 21011 q^{17} + 27326 q^{19} + 11427 q^{21} + 63072 q^{23} + 70100 q^{25} - 111267 q^{27} + 122238 q^{29} + 208396 q^{31} + 210678 q^{33} + 112805 q^{35} - 442379 q^{37} + 85683 q^{39} + 58000 q^{41} + 202025 q^{43} - 256410 q^{45} - 588511 q^{47} - 737694 q^{49} - 819429 q^{51} + 1684336 q^{53} + 2079770 q^{55} + 1065714 q^{57} + 442630 q^{59} - 1083608 q^{61} - 195138 q^{63} + 845845 q^{65} - 3443486 q^{67} + 2459808 q^{69} - 2084705 q^{71} + 5937890 q^{73} + 2733900 q^{75} + 1582786 q^{77} + 6609256 q^{79} - 2882871 q^{81} + 142740 q^{83} - 8089235 q^{85} + 4767282 q^{87} - 6985286 q^{89} + 643721 q^{91} + 8127444 q^{93} + 10520510 q^{95} - 200762 q^{97} - 3597732 q^{99}+O(q^{100}) \) 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 39.0000 0 385.000 0 293.000 0 −666.000 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 208.8.a.c 1
4.b odd 2 1 26.8.a.a 1
12.b even 2 1 234.8.a.d 1
52.b odd 2 1 338.8.a.c 1
52.f even 4 2 338.8.b.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.8.a.a 1 4.b odd 2 1
208.8.a.c 1 1.a even 1 1 trivial
234.8.a.d 1 12.b even 2 1
338.8.a.c 1 52.b odd 2 1
338.8.b.b 2 52.f even 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} - 39 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(208))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T - 39 \) Copy content Toggle raw display
$5$ \( T - 385 \) Copy content Toggle raw display
$7$ \( T - 293 \) Copy content Toggle raw display
$11$ \( T - 5402 \) Copy content Toggle raw display
$13$ \( T - 2197 \) Copy content Toggle raw display
$17$ \( T + 21011 \) Copy content Toggle raw display
$19$ \( T - 27326 \) Copy content Toggle raw display
$23$ \( T - 63072 \) Copy content Toggle raw display
$29$ \( T - 122238 \) Copy content Toggle raw display
$31$ \( T - 208396 \) Copy content Toggle raw display
$37$ \( T + 442379 \) Copy content Toggle raw display
$41$ \( T - 58000 \) Copy content Toggle raw display
$43$ \( T - 202025 \) Copy content Toggle raw display
$47$ \( T + 588511 \) Copy content Toggle raw display
$53$ \( T - 1684336 \) Copy content Toggle raw display
$59$ \( T - 442630 \) Copy content Toggle raw display
$61$ \( T + 1083608 \) Copy content Toggle raw display
$67$ \( T + 3443486 \) Copy content Toggle raw display
$71$ \( T + 2084705 \) Copy content Toggle raw display
$73$ \( T - 5937890 \) Copy content Toggle raw display
$79$ \( T - 6609256 \) Copy content Toggle raw display
$83$ \( T - 142740 \) Copy content Toggle raw display
$89$ \( T + 6985286 \) Copy content Toggle raw display
$97$ \( T + 200762 \) Copy content Toggle raw display
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