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Properties

Label 208.4.f.d
Level $208$
Weight $4$
Character orbit 208.f
Analytic conductor $12.272$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [208,4,Mod(129,208)]
 
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, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("208.129");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 208 = 2^{4} \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 208.f (of order \(2\), degree \(1\), not minimal)

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: no
Analytic conductor: \(12.2723972812\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{217})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 109x^{2} + 2916 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2\cdot 3 \)
Twist minimal: no (minimal twist has level 26)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{3} + 1) q^{3} + (2 \beta_{2} - \beta_1) q^{5} + (\beta_{2} - \beta_1) q^{7} + (3 \beta_{3} + 28) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{3} + 1) q^{3} + (2 \beta_{2} - \beta_1) q^{5} + (\beta_{2} - \beta_1) q^{7} + (3 \beta_{3} + 28) q^{9} + ( - \beta_{2} + 6 \beta_1) q^{11} + (\beta_{3} + 3 \beta_{2} - 5 \beta_1 - 28) q^{13} + (16 \beta_{2} - 11 \beta_1) q^{15} + ( - 5 \beta_{3} + 9) q^{17} + ( - 5 \beta_{2} - 12 \beta_1) q^{19} + (12 \beta_{2} - 5 \beta_1) q^{21} + ( - 10 \beta_{3} + 54) q^{23} + ( - 19 \beta_{3} - 10) q^{25} + (7 \beta_{3} + 163) q^{27} + (14 \beta_{3} + 180) q^{29} + ( - 35 \beta_{2} + 2 \beta_1) q^{31} - 52 \beta_{2} q^{33} + ( - 13 \beta_{3} - 81) q^{35} + ( - 6 \beta_{2} + 17 \beta_1) q^{37} + ( - 26 \beta_{3} + 52 \beta_{2} + \cdots + 26) q^{39}+ \cdots + ( - 181 \beta_{2} + 150 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 6 q^{3} + 118 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 6 q^{3} + 118 q^{9} - 110 q^{13} + 26 q^{17} + 196 q^{23} - 78 q^{25} + 666 q^{27} + 748 q^{29} - 350 q^{35} + 52 q^{39} + 438 q^{43} + 1106 q^{49} - 1046 q^{51} + 48 q^{53} + 960 q^{55} - 564 q^{61} - 1294 q^{65} - 1876 q^{69} - 4240 q^{75} + 1176 q^{77} + 1444 q^{79} - 668 q^{81} + 4160 q^{87} - 1162 q^{91} - 324 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 109x^{2} + 2916 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{3} + 28\nu ) / 27 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + 55\nu ) / 9 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{2} + 55 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( \beta_{2} - 3\beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} - 55 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -28\beta_{2} + 165\beta_1 ) / 3 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/208\mathbb{Z}\right)^\times\).

\(n\) \(53\) \(79\) \(145\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
129.1
7.86546i
− 7.86546i
− 6.86546i
6.86546i
0 −5.86546 0 − 2.13454i 0 3.86546i 0 7.40362 0
129.2 0 −5.86546 0 2.13454i 0 − 3.86546i 0 7.40362 0
129.3 0 8.86546 0 − 16.8655i 0 − 10.8655i 0 51.5964 0
129.4 0 8.86546 0 16.8655i 0 10.8655i 0 51.5964 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 208.4.f.d 4
4.b odd 2 1 26.4.b.a ✓ 4
8.b even 2 1 832.4.f.h 4
8.d odd 2 1 832.4.f.j 4
12.b even 2 1 234.4.b.b 4
13.b even 2 1 inner 208.4.f.d 4
20.d odd 2 1 650.4.d.d 4
20.e even 4 1 650.4.c.e 4
20.e even 4 1 650.4.c.f 4
52.b odd 2 1 26.4.b.a ✓ 4
52.f even 4 1 338.4.a.f 2
52.f even 4 1 338.4.a.i 2
52.i odd 6 2 338.4.e.g 8
52.j odd 6 2 338.4.e.g 8
52.l even 12 2 338.4.c.h 4
52.l even 12 2 338.4.c.i 4
104.e even 2 1 832.4.f.h 4
104.h odd 2 1 832.4.f.j 4
156.h even 2 1 234.4.b.b 4
260.g odd 2 1 650.4.d.d 4
260.p even 4 1 650.4.c.e 4
260.p even 4 1 650.4.c.f 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.4.b.a ✓ 4 4.b odd 2 1
26.4.b.a ✓ 4 52.b odd 2 1
208.4.f.d 4 1.a even 1 1 trivial
208.4.f.d 4 13.b even 2 1 inner
234.4.b.b 4 12.b even 2 1
234.4.b.b 4 156.h even 2 1
338.4.a.f 2 52.f even 4 1
338.4.a.i 2 52.f even 4 1
338.4.c.h 4 52.l even 12 2
338.4.c.i 4 52.l even 12 2
338.4.e.g 8 52.i odd 6 2
338.4.e.g 8 52.j odd 6 2
650.4.c.e 4 20.e even 4 1
650.4.c.e 4 260.p even 4 1
650.4.c.f 4 20.e even 4 1
650.4.c.f 4 260.p even 4 1
650.4.d.d 4 20.d odd 2 1
650.4.d.d 4 260.g odd 2 1
832.4.f.h 4 8.b even 2 1
832.4.f.h 4 104.e even 2 1
832.4.f.j 4 8.d odd 2 1
832.4.f.j 4 104.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - 3T_{3} - 52 \) acting on \(S_{4}^{\mathrm{new}}(208, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} - 3 T - 52)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} + 289T^{2} + 1296 \) Copy content Toggle raw display
$7$ \( T^{4} + 133T^{2} + 1764 \) Copy content Toggle raw display
$11$ \( T^{4} + 4068 T^{2} + 3504384 \) Copy content Toggle raw display
$13$ \( T^{4} + 110 T^{3} + \cdots + 4826809 \) Copy content Toggle raw display
$17$ \( (T^{2} - 13 T - 1314)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} + 22824 T^{2} + 17740944 \) Copy content Toggle raw display
$23$ \( (T^{2} - 98 T - 3024)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} - 374 T + 24336)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} + \cdots + 1747908864 \) Copy content Toggle raw display
$37$ \( T^{4} + 31441 T^{2} + 244484496 \) Copy content Toggle raw display
$41$ \( T^{4} + \cdots + 1052872704 \) Copy content Toggle raw display
$43$ \( (T^{2} - 219 T - 21916)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + \cdots + 2466314244 \) Copy content Toggle raw display
$53$ \( (T^{2} - 24 T - 7668)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + \cdots + 7836498576 \) Copy content Toggle raw display
$61$ \( (T^{2} + 282 T - 94912)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + \cdots + 17799829056 \) Copy content Toggle raw display
$71$ \( T^{4} + \cdots + 11761402500 \) Copy content Toggle raw display
$73$ \( T^{4} + \cdots + 8100000000 \) Copy content Toggle raw display
$79$ \( (T^{2} - 722 T - 166752)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 1623976 T^{2} + 797271696 \) Copy content Toggle raw display
$89$ \( T^{4} + \cdots + 827983524096 \) Copy content Toggle raw display
$97$ \( T^{4} + \cdots + 749484969984 \) Copy content Toggle raw display
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