Properties

Label 208.6.f.b
Level $208$
Weight $6$
Character orbit 208.f
Analytic conductor $33.360$
Analytic rank $1$
Dimension $2$
Inner twists $2$

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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: \(1\)
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: \( 3 \)
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 \(\beta = 3i\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 13 q^{3} + 17 \beta q^{5} - 35 \beta q^{7} - 74 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + 13 q^{3} + 17 \beta q^{5} - 35 \beta q^{7} - 74 q^{9} - 40 \beta q^{11} + (39 \beta - 598) q^{13} + 221 \beta q^{15} - 1101 q^{17} + 390 \beta q^{19} - 455 \beta q^{21} - 1050 q^{23} + 524 q^{25} - 4121 q^{27} - 4104 q^{29} - 3208 \beta q^{31} - 520 \beta q^{33} + 5355 q^{35} + 2903 \beta q^{37} + (507 \beta - 7774) q^{39} - 3160 \beta q^{41} - 9995 q^{43} - 1258 \beta q^{45} + 981 \beta q^{47} + 5782 q^{49} - 14313 q^{51} - 750 q^{53} + 6120 q^{55} + 5070 \beta q^{57} + 13646 \beta q^{59} - 57920 q^{61} + 2590 \beta q^{63} + ( - 10166 \beta - 5967) q^{65} - 7604 \beta q^{67} - 13650 q^{69} - 21247 \beta q^{71} + 19622 \beta q^{73} + 6812 q^{75} - 12600 q^{77} - 63202 q^{79} - 35591 q^{81} - 18486 \beta q^{83} - 18717 \beta q^{85} - 53352 q^{87} - 34926 \beta q^{89} + (20930 \beta + 12285) q^{91} - 41704 \beta q^{93} - 59670 q^{95} + 53484 \beta q^{97} + 2960 \beta q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 26 q^{3} - 148 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 26 q^{3} - 148 q^{9} - 1196 q^{13} - 2202 q^{17} - 2100 q^{23} + 1048 q^{25} - 8242 q^{27} - 8208 q^{29} + 10710 q^{35} - 15548 q^{39} - 19990 q^{43} + 11564 q^{49} - 28626 q^{51} - 1500 q^{53} + 12240 q^{55} - 115840 q^{61} - 11934 q^{65} - 27300 q^{69} + 13624 q^{75} - 25200 q^{77} - 126404 q^{79} - 71182 q^{81} - 106704 q^{87} + 24570 q^{91} - 119340 q^{95}+O(q^{100}) \) 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 13.0000 0 51.0000i 0 105.000i 0 −74.0000 0
129.2 0 13.0000 0 51.0000i 0 105.000i 0 −74.0000 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.b 2
4.b odd 2 1 26.6.b.a 2
12.b even 2 1 234.6.b.b 2
13.b even 2 1 inner 208.6.f.b 2
52.b odd 2 1 26.6.b.a 2
52.f even 4 1 338.6.a.a 1
52.f even 4 1 338.6.a.c 1
156.h even 2 1 234.6.b.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.6.b.a 2 4.b odd 2 1
26.6.b.a 2 52.b odd 2 1
208.6.f.b 2 1.a even 1 1 trivial
208.6.f.b 2 13.b even 2 1 inner
234.6.b.b 2 12.b even 2 1
234.6.b.b 2 156.h even 2 1
338.6.a.a 1 52.f even 4 1
338.6.a.c 1 52.f even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} - 13 \) 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
$3$ \( (T - 13)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 2601 \) Copy content Toggle raw display
$7$ \( T^{2} + 11025 \) Copy content Toggle raw display
$11$ \( T^{2} + 14400 \) Copy content Toggle raw display
$13$ \( T^{2} + 1196 T + 371293 \) Copy content Toggle raw display
$17$ \( (T + 1101)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 1368900 \) Copy content Toggle raw display
$23$ \( (T + 1050)^{2} \) Copy content Toggle raw display
$29$ \( (T + 4104)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 92621376 \) Copy content Toggle raw display
$37$ \( T^{2} + 75846681 \) Copy content Toggle raw display
$41$ \( T^{2} + 89870400 \) Copy content Toggle raw display
$43$ \( (T + 9995)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 8661249 \) Copy content Toggle raw display
$53$ \( (T + 750)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 1675919844 \) Copy content Toggle raw display
$61$ \( (T + 57920)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 520387344 \) Copy content Toggle raw display
$71$ \( T^{2} + 4062915081 \) Copy content Toggle raw display
$73$ \( T^{2} + 3465205956 \) Copy content Toggle raw display
$79$ \( (T + 63202)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 3075589764 \) Copy content Toggle raw display
$89$ \( T^{2} + 10978429284 \) Copy content Toggle raw display
$97$ \( T^{2} + 25744844304 \) Copy content Toggle raw display
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