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Properties

Label 208.8.i.d
Level $208$
Weight $8$
Character orbit 208.i
Analytic conductor $64.976$
Analytic rank $0$
Dimension $16$
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: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - x^{15} + 796 x^{14} - 475 x^{13} + 449889 x^{12} - 92038 x^{11} + 116806037 x^{10} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{30}\cdot 3\cdot 13^{2} \)
Twist minimal: no (minimal twist has level 13)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{5} - 4 \beta_1) q^{3} + ( - \beta_{9} + \beta_{6} + 24) q^{5} + (\beta_{12} - 6 \beta_{5} + 6 \beta_{4} + \cdots - 22) q^{7} + ( - \beta_{13} + \beta_{12} + \cdots - 744) q^{9} + (3 \beta_{11} - 4 \beta_{8} + \cdots + 626 \beta_1) q^{11}+ \cdots + (774 \beta_{15} + 551 \beta_{13} + \cdots - 1718835) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 28 q^{3} + 384 q^{5} - 196 q^{7} - 5988 q^{9} - 5052 q^{11} + 17064 q^{13} + 184 q^{15} - 22824 q^{17} - 63692 q^{19} - 271240 q^{21} + 72468 q^{23} + 58488 q^{25} - 358616 q^{27} - 221772 q^{29}+ \cdots - 27386736 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - x^{15} + 796 x^{14} - 475 x^{13} + 449889 x^{12} - 92038 x^{11} + 116806037 x^{10} + \cdots + 11\!\cdots\!00 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 29\!\cdots\!41 \nu^{15} + \cdots + 12\!\cdots\!60 ) / 25\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 29\!\cdots\!41 \nu^{15} + \cdots - 37\!\cdots\!00 ) / 12\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 72\!\cdots\!51 \nu^{15} + \cdots - 27\!\cdots\!60 ) / 12\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 76\!\cdots\!37 \nu^{15} + \cdots - 59\!\cdots\!00 ) / 13\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 65\!\cdots\!33 \nu^{15} + \cdots - 13\!\cdots\!40 ) / 89\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 23\!\cdots\!71 \nu^{15} + \cdots - 21\!\cdots\!20 ) / 27\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 41\!\cdots\!23 \nu^{15} + \cdots + 67\!\cdots\!80 ) / 35\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 57\!\cdots\!41 \nu^{15} + \cdots + 26\!\cdots\!40 ) / 17\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 18\!\cdots\!63 \nu^{15} + \cdots - 27\!\cdots\!00 ) / 35\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 20\!\cdots\!53 \nu^{15} + \cdots - 21\!\cdots\!40 ) / 39\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 44\!\cdots\!41 \nu^{15} + \cdots - 81\!\cdots\!40 ) / 74\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 93\!\cdots\!63 \nu^{15} + \cdots - 66\!\cdots\!00 ) / 74\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 90\!\cdots\!09 \nu^{15} + \cdots - 36\!\cdots\!00 ) / 35\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 22\!\cdots\!93 \nu^{15} + \cdots + 37\!\cdots\!00 ) / 35\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 26\!\cdots\!01 \nu^{15} + \cdots + 28\!\cdots\!00 ) / 39\!\cdots\!20 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{2} + 2\beta _1 + 2 ) / 16 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{8} - \beta_{6} - 2\beta_{5} + 795\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 4 \beta_{15} - 12 \beta_{13} - 4 \beta_{12} - 4 \beta_{11} + 16 \beta_{10} - 20 \beta_{9} + \cdots - 1034 ) / 16 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 6 \beta_{15} + 6 \beta_{14} + 46 \beta_{13} - 6 \beta_{12} - 1018 \beta_{10} + 1172 \beta_{9} + \cdots - 534572 ) / 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 2344 \beta_{14} + 1960 \beta_{11} - 11808 \beta_{8} - 6328 \beta_{7} - 19336 \beta_{6} + \cdots + 1077566 \beta_1 ) / 16 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 1886 \beta_{15} - 16982 \beta_{13} + 4350 \beta_{12} + 4350 \beta_{11} + 237465 \beta_{10} + \cdots + 109984912 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 1165164 \beta_{15} - 1165164 \beta_{14} + 2967556 \beta_{13} + 1030892 \beta_{12} - 7032112 \beta_{10} + \cdots - 1221710758 ) / 16 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 1645986 \beta_{14} - 5618530 \beta_{11} - 221066794 \beta_{8} - 19402746 \beta_{7} + \cdots + 96609992030 \beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 553719504 \beta_{15} - 1384991728 \beta_{13} - 518843856 \beta_{12} - 518843856 \beta_{11} + \cdots + 892132064350 ) / 16 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 285930780 \beta_{15} + 285930780 \beta_{14} + 5119776844 \beta_{13} - 1431801564 \beta_{12} + \cdots - 22120138586500 ) / 4 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 258611348308 \beta_{14} + 248941044820 \beta_{11} - 2078827326032 \beta_{8} + \cdots + 561683494545406 \beta_1 ) / 16 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 140735573502 \beta_{15} - 5230985928966 \beta_{13} + 1315965249150 \beta_{12} + 1315965249150 \beta_{11} + \cdots + 20\!\cdots\!84 ) / 8 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 119920950487288 \beta_{15} - 119920950487288 \beta_{14} + 310667226966440 \beta_{13} + \cdots - 31\!\cdots\!70 ) / 16 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 568908236998 \beta_{14} - 285780617164698 \beta_{11} + \cdots + 47\!\cdots\!11 \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 55\!\cdots\!32 \beta_{15} + \cdots + 16\!\cdots\!86 ) / 16 \) 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
−5.55707 + 9.62513i
10.3972 − 18.0085i
2.89297 − 5.01078i
−11.0459 + 19.1321i
4.62264 − 8.00665i
−3.54173 + 6.13445i
−5.36984 + 9.30084i
8.10170 − 14.0326i
−5.55707 − 9.62513i
10.3972 + 18.0085i
2.89297 + 5.01078i
−11.0459 − 19.1321i
4.62264 + 8.00665i
−3.54173 − 6.13445i
−5.36984 − 9.30084i
8.10170 + 14.0326i
0 −35.2053 − 60.9774i 0 −163.930 0 123.469 − 213.855i 0 −1385.33 + 2399.46i 0
81.2 0 −33.9160 − 58.7442i 0 157.804 0 421.054 − 729.287i 0 −1207.09 + 2090.74i 0
81.3 0 −15.0885 − 26.1340i 0 93.2351 0 −809.227 + 1401.62i 0 638.177 − 1105.35i 0
81.4 0 2.77530 + 4.80695i 0 126.804 0 −14.6580 + 25.3884i 0 1078.10 − 1867.32i 0
81.5 0 4.84859 + 8.39800i 0 −294.418 0 715.408 − 1239.12i 0 1046.48 − 1812.56i 0
81.6 0 12.8141 + 22.1946i 0 442.310 0 380.773 − 659.518i 0 765.099 − 1325.19i 0
81.7 0 34.7632 + 60.2115i 0 −459.709 0 −430.545 + 745.725i 0 −1323.45 + 2292.29i 0
81.8 0 43.0086 + 74.4931i 0 289.904 0 −484.274 + 838.787i 0 −2605.99 + 4513.70i 0
113.1 0 −35.2053 + 60.9774i 0 −163.930 0 123.469 + 213.855i 0 −1385.33 − 2399.46i 0
113.2 0 −33.9160 + 58.7442i 0 157.804 0 421.054 + 729.287i 0 −1207.09 − 2090.74i 0
113.3 0 −15.0885 + 26.1340i 0 93.2351 0 −809.227 − 1401.62i 0 638.177 + 1105.35i 0
113.4 0 2.77530 − 4.80695i 0 126.804 0 −14.6580 − 25.3884i 0 1078.10 + 1867.32i 0
113.5 0 4.84859 − 8.39800i 0 −294.418 0 715.408 + 1239.12i 0 1046.48 + 1812.56i 0
113.6 0 12.8141 − 22.1946i 0 442.310 0 380.773 + 659.518i 0 765.099 + 1325.19i 0
113.7 0 34.7632 − 60.2115i 0 −459.709 0 −430.545 − 745.725i 0 −1323.45 − 2292.29i 0
113.8 0 43.0086 − 74.4931i 0 289.904 0 −484.274 − 838.787i 0 −2605.99 − 4513.70i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 81.8
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.d 16
4.b odd 2 1 13.8.c.a ✓ 16
12.b even 2 1 117.8.g.d 16
13.c even 3 1 inner 208.8.i.d 16
52.i odd 6 1 169.8.a.e 8
52.j odd 6 1 13.8.c.a ✓ 16
52.j odd 6 1 169.8.a.f 8
52.l even 12 2 169.8.b.e 16
156.p even 6 1 117.8.g.d 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.8.c.a ✓ 16 4.b odd 2 1
13.8.c.a ✓ 16 52.j odd 6 1
117.8.g.d 16 12.b even 2 1
117.8.g.d 16 156.p even 6 1
169.8.a.e 8 52.i odd 6 1
169.8.a.f 8 52.j odd 6 1
169.8.b.e 16 52.l even 12 2
208.8.i.d 16 1.a even 1 1 trivial
208.8.i.d 16 13.c even 3 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{16} - 28 T_{3}^{15} + 12134 T_{3}^{14} - 131248 T_{3}^{13} + 99215291 T_{3}^{12} + \cdots + 14\!\cdots\!44 \) acting on \(S_{8}^{\mathrm{new}}(208, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} + \cdots + 14\!\cdots\!44 \) Copy content Toggle raw display
$5$ \( (T^{8} + \cdots - 53\!\cdots\!00)^{2} \) Copy content Toggle raw display
$7$ \( T^{16} + \cdots + 80\!\cdots\!00 \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 10\!\cdots\!56 \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 24\!\cdots\!41 \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 33\!\cdots\!89 \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots + 50\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 44\!\cdots\!24 \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 25\!\cdots\!49 \) Copy content Toggle raw display
$31$ \( (T^{8} + \cdots + 18\!\cdots\!00)^{2} \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 85\!\cdots\!69 \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 50\!\cdots\!25 \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 66\!\cdots\!84 \) Copy content Toggle raw display
$47$ \( (T^{8} + \cdots - 26\!\cdots\!00)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} + \cdots + 57\!\cdots\!00)^{2} \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 62\!\cdots\!44 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 15\!\cdots\!25 \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 45\!\cdots\!44 \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 29\!\cdots\!16 \) Copy content Toggle raw display
$73$ \( (T^{8} + \cdots - 72\!\cdots\!00)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} + \cdots + 20\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} + \cdots + 92\!\cdots\!00)^{2} \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 16\!\cdots\!44 \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 11\!\cdots\!44 \) Copy content Toggle raw display
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