Properties

Label 208.10.a.b
Level $208$
Weight $10$
Character orbit 208.a
Self dual yes
Analytic conductor $107.127$
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] = [208,10,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, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("208.1");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 208 = 2^{4} \cdot 13 \)
Weight: \( k \) \(=\) \( 10 \)
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: \(107.127453922\)
Analytic rank: \(1\)
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)
 
\(f(q)\) \(=\) \( q - 75 q^{3} - 1979 q^{5} + 10115 q^{7} - 14058 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 75 q^{3} - 1979 q^{5} + 10115 q^{7} - 14058 q^{9} - 18850 q^{11} + 28561 q^{13} + 148425 q^{15} - 142403 q^{17} - 83302 q^{19} - 758625 q^{21} + 536544 q^{23} + 1963316 q^{25} + 2530575 q^{27} - 2600442 q^{29} + 2214004 q^{31} + 1413750 q^{33} - 20017585 q^{35} + 18099241 q^{37} - 2142075 q^{39} + 26812240 q^{41} + 42253475 q^{43} + 27820782 q^{45} - 35914993 q^{47} + 61959618 q^{49} + 10680225 q^{51} - 66514064 q^{53} + 37304150 q^{55} + 6247650 q^{57} + 108164002 q^{59} - 207449912 q^{61} - 142196670 q^{63} - 56522219 q^{65} - 193015514 q^{67} - 40240800 q^{69} + 201833497 q^{71} - 121628110 q^{73} - 147248700 q^{75} - 190667750 q^{77} - 112871912 q^{79} + 86910489 q^{81} - 308254212 q^{83} + 281815537 q^{85} + 195033150 q^{87} - 6374870 q^{89} + 288894515 q^{91} - 166050300 q^{93} + 164854658 q^{95} + 871266886 q^{97} + 264993300 q^{99}+O(q^{100}) \) 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 −75.0000 0 −1979.00 0 10115.0 0 −14058.0 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 208.10.a.b 1
4.b odd 2 1 26.10.a.c 1
12.b even 2 1 234.10.a.a 1
52.b odd 2 1 338.10.a.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.10.a.c 1 4.b odd 2 1
208.10.a.b 1 1.a even 1 1 trivial
234.10.a.a 1 12.b even 2 1
338.10.a.b 1 52.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} + 75 \) acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(208))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T + 75 \) Copy content Toggle raw display
$5$ \( T + 1979 \) Copy content Toggle raw display
$7$ \( T - 10115 \) Copy content Toggle raw display
$11$ \( T + 18850 \) Copy content Toggle raw display
$13$ \( T - 28561 \) Copy content Toggle raw display
$17$ \( T + 142403 \) Copy content Toggle raw display
$19$ \( T + 83302 \) Copy content Toggle raw display
$23$ \( T - 536544 \) Copy content Toggle raw display
$29$ \( T + 2600442 \) Copy content Toggle raw display
$31$ \( T - 2214004 \) Copy content Toggle raw display
$37$ \( T - 18099241 \) Copy content Toggle raw display
$41$ \( T - 26812240 \) Copy content Toggle raw display
$43$ \( T - 42253475 \) Copy content Toggle raw display
$47$ \( T + 35914993 \) Copy content Toggle raw display
$53$ \( T + 66514064 \) Copy content Toggle raw display
$59$ \( T - 108164002 \) Copy content Toggle raw display
$61$ \( T + 207449912 \) Copy content Toggle raw display
$67$ \( T + 193015514 \) Copy content Toggle raw display
$71$ \( T - 201833497 \) Copy content Toggle raw display
$73$ \( T + 121628110 \) Copy content Toggle raw display
$79$ \( T + 112871912 \) Copy content Toggle raw display
$83$ \( T + 308254212 \) Copy content Toggle raw display
$89$ \( T + 6374870 \) Copy content Toggle raw display
$97$ \( T - 871266886 \) Copy content Toggle raw display
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