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Properties

Label 208.8.i.b
Level $208$
Weight $8$
Character orbit 208.i
Analytic conductor $64.976$
Analytic rank $0$
Dimension $8$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [208,8,Mod(81,208)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(208, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 2]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("208.81");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 208 = 2^{4} \cdot 13 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 208.i (of order \(3\), degree \(2\), 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: \(64.9760853007\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 4654x^{6} + 7012369x^{4} + 3763719168x^{2} + 637953638400 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{6}\cdot 3 \)
Twist minimal: no (minimal twist has level 26)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{3} q^{3} + ( - \beta_{4} + \beta_{3} - \beta_{2} + 69) q^{5} + (2 \beta_{6} + 7 \beta_{2} + \cdots + 138) q^{7}+ \cdots + ( - 6 \beta_{7} - 3 \beta_{6} + \cdots - 1302) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{3} q^{3} + ( - \beta_{4} + \beta_{3} - \beta_{2} + 69) q^{5} + (2 \beta_{6} + 7 \beta_{2} + \cdots + 138) q^{7}+ \cdots + (1020 \beta_{5} + 7152 \beta_{4} + \cdots + 2592480) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 556 q^{5} + 548 q^{7} - 5214 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 556 q^{5} + 548 q^{7} - 5214 q^{9} + 7392 q^{11} - 25818 q^{13} - 15528 q^{15} + 28316 q^{17} + 99888 q^{19} + 182148 q^{21} + 33388 q^{23} + 173756 q^{25} - 212544 q^{27} + 93140 q^{29} - 622320 q^{31} + 238638 q^{33} - 141544 q^{35} - 9636 q^{37} + 22932 q^{39} + 82892 q^{41} + 569264 q^{43} - 2303394 q^{45} + 1148400 q^{47} - 717798 q^{49} + 5459856 q^{51} + 2470700 q^{53} + 1092512 q^{55} + 7056924 q^{57} - 231504 q^{59} + 685684 q^{61} + 5951712 q^{63} - 6216678 q^{65} - 3271056 q^{67} + 5600034 q^{69} + 175012 q^{71} + 14275780 q^{73} - 22200960 q^{75} - 27830412 q^{77} + 14107904 q^{79} + 3758004 q^{81} - 1314576 q^{83} + 11814998 q^{85} + 7182900 q^{87} - 11452234 q^{89} - 16457168 q^{91} + 2984688 q^{93} + 23334088 q^{95} - 428002 q^{97} + 20715312 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 4654x^{6} + 7012369x^{4} + 3763719168x^{2} + 637953638400 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{7} + 4654\nu^{5} + 6213649\nu^{3} + 1905097728\nu - 3143761920 ) / 6287523840 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{4} + 2327\nu^{2} + 11808\nu + 798720 ) / 7872 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{4} - 2327\nu^{2} + 11808\nu - 798720 ) / 7872 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{6} - 3766\nu^{4} - 3863881\nu^{2} - 854819328 ) / 661248 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{6} + 4018\nu^{4} + 4780909\nu^{2} + 1440612480 ) / 330624 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 381 \nu^{7} - 13312 \nu^{6} - 1613430 \nu^{5} - 53487616 \nu^{4} - 2152864077 \nu^{3} + \cdots - 19177433333760 ) / 8802533376 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 625 \nu^{7} - 6656 \nu^{6} - 2429518 \nu^{5} - 25066496 \nu^{4} - 2689763713 \nu^{3} + \cdots - 5685276180480 ) / 8802533376 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{3} + \beta_{2} ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} + 2\beta_{4} + 3\beta_{3} - 3\beta_{2} - 1163 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 48\beta_{7} - 144\beta_{6} - 72\beta_{5} - 24\beta_{4} - 1655\beta_{3} - 1655\beta_{2} - 17760\beta _1 - 8904 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -2327\beta_{5} - 4654\beta_{4} - 10917\beta_{3} + 10917\beta_{2} + 1907581 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 64464 \beta_{7} + 358704 \beta_{6} + 179352 \beta_{5} + 32232 \beta_{4} + 3087889 \beta_{3} + \cdots + 34452312 ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 4899601\beta_{5} + 9137954\beta_{4} + 29521779\beta_{3} - 29521779\beta_{2} - 3545075771 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 1760304 \beta_{7} - 774642960 \beta_{6} - 387321480 \beta_{5} - 880152 \beta_{4} - 5992544039 \beta_{3} + \cdots - 95583443592 ) / 3 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(53\) \(79\) \(145\)
\(\chi(n)\) \(1\) \(1\) \(-1 - \beta_{1}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
81.1
41.0833i
23.0850i
− 18.4011i
− 45.7672i
− 41.0833i
− 23.0850i
18.4011i
45.7672i
0 −35.5792 − 61.6250i 0 523.489 0 −208.401 + 360.960i 0 −1438.26 + 2491.14i 0
81.2 0 −19.9922 − 34.6275i 0 −323.700 0 284.296 − 492.415i 0 294.123 − 509.436i 0
81.3 0 15.9358 + 27.6017i 0 −54.4265 0 −556.354 + 963.633i 0 585.599 − 1014.29i 0
81.4 0 39.6356 + 68.6509i 0 132.638 0 754.459 − 1306.76i 0 −2048.46 + 3548.04i 0
113.1 0 −35.5792 + 61.6250i 0 523.489 0 −208.401 − 360.960i 0 −1438.26 − 2491.14i 0
113.2 0 −19.9922 + 34.6275i 0 −323.700 0 284.296 + 492.415i 0 294.123 + 509.436i 0
113.3 0 15.9358 − 27.6017i 0 −54.4265 0 −556.354 − 963.633i 0 585.599 + 1014.29i 0
113.4 0 39.6356 − 68.6509i 0 132.638 0 754.459 + 1306.76i 0 −2048.46 − 3548.04i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 81.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 208.8.i.b 8
4.b odd 2 1 26.8.c.b ✓ 8
12.b even 2 1 234.8.h.b 8
13.c even 3 1 inner 208.8.i.b 8
52.i odd 6 1 338.8.a.j 4
52.j odd 6 1 26.8.c.b ✓ 8
52.j odd 6 1 338.8.a.i 4
52.l even 12 2 338.8.b.h 8
156.p even 6 1 234.8.h.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.8.c.b ✓ 8 4.b odd 2 1
26.8.c.b ✓ 8 52.j odd 6 1
208.8.i.b 8 1.a even 1 1 trivial
208.8.i.b 8 13.c even 3 1 inner
234.8.h.b 8 12.b even 2 1
234.8.h.b 8 156.p even 6 1
338.8.a.i 4 52.j odd 6 1
338.8.a.j 4 52.i odd 6 1
338.8.b.h 8 52.l even 12 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} + 6981 T_{3}^{6} + 70848 T_{3}^{5} + 41545881 T_{3}^{4} + 247294944 T_{3}^{3} + \cdots + 51674244710400 \) acting on \(S_{8}^{\mathrm{new}}(208, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} + \cdots + 51674244710400 \) Copy content Toggle raw display
$5$ \( (T^{4} - 278 T^{3} + \cdots + 1223288100)^{2} \) Copy content Toggle raw display
$7$ \( T^{8} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
$11$ \( T^{8} + \cdots + 28\!\cdots\!36 \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots + 15\!\cdots\!21 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 35\!\cdots\!41 \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 16\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 64\!\cdots\!36 \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots + 65\!\cdots\!69 \) Copy content Toggle raw display
$31$ \( (T^{4} + \cdots + 10\!\cdots\!28)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 27\!\cdots\!41 \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 75\!\cdots\!25 \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots + 19\!\cdots\!96 \) Copy content Toggle raw display
$47$ \( (T^{4} + \cdots + 87\!\cdots\!28)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + \cdots + 10\!\cdots\!32)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 87\!\cdots\!64 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 36\!\cdots\!25 \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 16\!\cdots\!96 \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots + 58\!\cdots\!44 \) Copy content Toggle raw display
$73$ \( (T^{4} + \cdots - 21\!\cdots\!40)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + \cdots + 46\!\cdots\!08)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} + \cdots - 67\!\cdots\!00)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 27\!\cdots\!36 \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 23\!\cdots\!84 \) Copy content Toggle raw display
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