Properties

Label 208.6.f.a
Level $208$
Weight $6$
Character orbit 208.f
Analytic conductor $33.360$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [208,6,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, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("208.129");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 208 = 2^{4} \cdot 13 \)
Weight: \( k \) \(=\) \( 6 \)
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: \(33.3598345211\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
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 \(\beta = 2i\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 4 q^{3} + 34 \beta q^{5} - 41 \beta q^{7} - 227 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - 4 q^{3} + 34 \beta q^{5} - 41 \beta q^{7} - 227 q^{9} - 195 \beta q^{11} + ( - 169 \beta + 507) q^{13} - 136 \beta q^{15} + 1738 q^{17} + 537 \beta q^{19} + 164 \beta q^{21} - 2104 q^{23} - 1499 q^{25} + 1880 q^{27} - 1690 q^{29} + 715 \beta q^{31} + 780 \beta q^{33} + 5576 q^{35} + 4426 \beta q^{37} + (676 \beta - 2028) q^{39} + 3380 \beta q^{41} + 16916 q^{43} - 7718 \beta q^{45} + 12579 \beta q^{47} + 10083 q^{49} - 6952 q^{51} + 38214 q^{53} + 26520 q^{55} - 2148 \beta q^{57} - 10643 \beta q^{59} - 5458 q^{61} + 9307 \beta q^{63} + (17238 \beta + 22984) q^{65} - 22271 \beta q^{67} + 8416 q^{69} + 8895 \beta q^{71} + 15532 \beta q^{73} + 5996 q^{75} - 31980 q^{77} + 45360 q^{79} + 47641 q^{81} + 62273 \beta q^{83} + 59092 \beta q^{85} + 6760 q^{87} - 9372 \beta q^{89} + ( - 20787 \beta - 27716) q^{91} - 2860 \beta q^{93} - 73032 q^{95} - 60744 \beta q^{97} + 44265 \beta q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 8 q^{3} - 454 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 8 q^{3} - 454 q^{9} + 1014 q^{13} + 3476 q^{17} - 4208 q^{23} - 2998 q^{25} + 3760 q^{27} - 3380 q^{29} + 11152 q^{35} - 4056 q^{39} + 33832 q^{43} + 20166 q^{49} - 13904 q^{51} + 76428 q^{53} + 53040 q^{55} - 10916 q^{61} + 45968 q^{65} + 16832 q^{69} + 11992 q^{75} - 63960 q^{77} + 90720 q^{79} + 95282 q^{81} + 13520 q^{87} - 55432 q^{91} - 146064 q^{95}+O(q^{100}) \) Copy content Toggle raw display

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
1.00000i
1.00000i
0 −4.00000 0 68.0000i 0 82.0000i 0 −227.000 0
129.2 0 −4.00000 0 68.0000i 0 82.0000i 0 −227.000 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 208.6.f.a 2
4.b odd 2 1 26.6.b.b 2
12.b even 2 1 234.6.b.a 2
13.b even 2 1 inner 208.6.f.a 2
52.b odd 2 1 26.6.b.b 2
52.f even 4 1 338.6.a.b 1
52.f even 4 1 338.6.a.e 1
156.h even 2 1 234.6.b.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.6.b.b 2 4.b odd 2 1
26.6.b.b 2 52.b odd 2 1
208.6.f.a 2 1.a even 1 1 trivial
208.6.f.a 2 13.b even 2 1 inner
234.6.b.a 2 12.b even 2 1
234.6.b.a 2 156.h even 2 1
338.6.a.b 1 52.f even 4 1
338.6.a.e 1 52.f even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} + 4 \) acting on \(S_{6}^{\mathrm{new}}(208, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( (T + 4)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 4624 \) Copy content Toggle raw display
$7$ \( T^{2} + 6724 \) Copy content Toggle raw display
$11$ \( T^{2} + 152100 \) Copy content Toggle raw display
$13$ \( T^{2} - 1014 T + 371293 \) Copy content Toggle raw display
$17$ \( (T - 1738)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 1153476 \) Copy content Toggle raw display
$23$ \( (T + 2104)^{2} \) Copy content Toggle raw display
$29$ \( (T + 1690)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 2044900 \) Copy content Toggle raw display
$37$ \( T^{2} + 78357904 \) Copy content Toggle raw display
$41$ \( T^{2} + 45697600 \) Copy content Toggle raw display
$43$ \( (T - 16916)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 632924964 \) Copy content Toggle raw display
$53$ \( (T - 38214)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 453093796 \) Copy content Toggle raw display
$61$ \( (T + 5458)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 1983989764 \) Copy content Toggle raw display
$71$ \( T^{2} + 316484100 \) Copy content Toggle raw display
$73$ \( T^{2} + 964972096 \) Copy content Toggle raw display
$79$ \( (T - 45360)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 15511706116 \) Copy content Toggle raw display
$89$ \( T^{2} + 351337536 \) Copy content Toggle raw display
$97$ \( T^{2} + 14759334144 \) Copy content Toggle raw display
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