Properties

Label 8.20.b.a
Level $8$
Weight $20$
Character orbit 8.b
Analytic conductor $18.305$
Analytic rank $0$
Dimension $18$
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [8,20,Mod(5,8)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 20, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("8.5");
 
S:= CuspForms(chi, 20);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8 = 2^{3} \)
Weight: \( k \) \(=\) \( 20 \)
Character orbit: \([\chi]\) \(=\) 8.b (of order \(2\), degree \(1\), 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: \(18.3053357245\)
Analytic rank: \(0\)
Dimension: \(18\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} - 9 x^{17} + 847482358 x^{16} - 6779858660 x^{15} + \cdots + 11\!\cdots\!76 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: multiple of \( 2^{153}\cdot 3^{16}\cdot 5^{4}\cdot 7 \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{17}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 - 25) q^{2} + ( - \beta_{2} + 5 \beta_1 + 2) q^{3} + (\beta_{3} - 27 \beta_1 - 22907) q^{4} + ( - \beta_{6} - \beta_{3} + 5 \beta_{2} + \cdots - 34) q^{5}+ \cdots + (\beta_{8} + 3 \beta_{7} + \cdots - 344295579) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 - 25) q^{2} + ( - \beta_{2} + 5 \beta_1 + 2) q^{3} + (\beta_{3} - 27 \beta_1 - 22907) q^{4} + ( - \beta_{6} - \beta_{3} + 5 \beta_{2} + \cdots - 34) q^{5}+ \cdots + ( - 49655184 \beta_{17} + \cdots - 30\!\cdots\!62) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q - 458 q^{2} - 412108 q^{4} - 47240948 q^{6} - 80707216 q^{7} - 173313752 q^{8} - 6198727826 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 18 q - 458 q^{2} - 412108 q^{4} - 47240948 q^{6} - 80707216 q^{7} - 173313752 q^{8} - 6198727826 q^{9} + 758800104 q^{10} - 9351642296 q^{12} - 122453668784 q^{14} + 156097960432 q^{15} - 696212072432 q^{16} + 14121426692 q^{17} + 1813497117770 q^{18} + 6517087595632 q^{20} - 11074654117412 q^{22} + 2177121583952 q^{23} - 36473014189168 q^{24} - 44414474211734 q^{25} + 26782030269304 q^{26} - 97002327802784 q^{28} + 327847200544208 q^{30} + 428505770260416 q^{31} + 122449430282912 q^{32} - 185380269683736 q^{33} - 478448330325748 q^{34} - 10\!\cdots\!36 q^{36}+ \cdots + 88\!\cdots\!22 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{18} - 9 x^{17} + 847482358 x^{16} - 6779858660 x^{15} + \cdots + 11\!\cdots\!76 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 10\!\cdots\!25 \nu^{17} + \cdots - 70\!\cdots\!64 ) / 91\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 10\!\cdots\!25 \nu^{17} + \cdots - 70\!\cdots\!64 ) / 18\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 32\!\cdots\!29 \nu^{17} + \cdots + 21\!\cdots\!20 ) / 33\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 57\!\cdots\!61 \nu^{17} + \cdots - 55\!\cdots\!08 ) / 14\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 42\!\cdots\!71 \nu^{17} + \cdots - 13\!\cdots\!60 ) / 43\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 28\!\cdots\!28 \nu^{17} + \cdots - 13\!\cdots\!16 ) / 22\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 70\!\cdots\!91 \nu^{17} + \cdots - 46\!\cdots\!60 ) / 91\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 17\!\cdots\!25 \nu^{17} + \cdots + 17\!\cdots\!72 ) / 91\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 33\!\cdots\!22 \nu^{17} + \cdots + 32\!\cdots\!52 ) / 75\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 22\!\cdots\!07 \nu^{17} + \cdots + 45\!\cdots\!80 ) / 91\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 22\!\cdots\!11 \nu^{17} + \cdots - 16\!\cdots\!52 ) / 27\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 96\!\cdots\!13 \nu^{17} + \cdots + 14\!\cdots\!64 ) / 91\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 17\!\cdots\!37 \nu^{17} + \cdots - 12\!\cdots\!88 ) / 91\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 88\!\cdots\!31 \nu^{17} + \cdots + 48\!\cdots\!40 ) / 41\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 34\!\cdots\!51 \nu^{17} + \cdots + 31\!\cdots\!96 ) / 45\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 66\!\cdots\!77 \nu^{17} + \cdots + 81\!\cdots\!72 ) / 75\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 12\!\cdots\!27 \nu^{17} + \cdots + 51\!\cdots\!20 ) / 82\!\cdots\!00 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} - 5\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{8} + 3 \beta_{7} + 4 \beta_{6} + 5 \beta_{5} - 2 \beta_{4} - 232 \beta_{3} + 221 \beta_{2} + \cdots - 1506557050 ) / 16 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 304 \beta_{17} - 280 \beta_{16} + 208 \beta_{15} + 3 \beta_{14} - 280 \beta_{13} - 4720 \beta_{12} + \cdots - 2946572438 ) / 64 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 5167052 \beta_{17} + 5169812 \beta_{16} + 468922 \beta_{15} + 5173608 \beta_{14} + \cdots + 20\!\cdots\!17 ) / 128 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 339623258964 \beta_{17} + 361957444368 \beta_{16} - 273742380294 \beta_{15} - 1008021542328 \beta_{14} + \cdots + 20\!\cdots\!65 ) / 256 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 45\!\cdots\!92 \beta_{17} + \cdots - 79\!\cdots\!23 ) / 256 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 42\!\cdots\!10 \beta_{17} + \cdots - 19\!\cdots\!15 ) / 128 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 73\!\cdots\!16 \beta_{17} + \cdots + 87\!\cdots\!34 ) / 128 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 20\!\cdots\!04 \beta_{17} + \cdots + 11\!\cdots\!43 ) / 256 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 42\!\cdots\!40 \beta_{17} + \cdots - 40\!\cdots\!33 ) / 256 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 26\!\cdots\!22 \beta_{17} + \cdots - 21\!\cdots\!58 ) / 128 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 59\!\cdots\!08 \beta_{17} + \cdots + 49\!\cdots\!49 ) / 128 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 13\!\cdots\!88 \beta_{17} + \cdots + 16\!\cdots\!81 ) / 256 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 31\!\cdots\!48 \beta_{17} + \cdots - 24\!\cdots\!91 ) / 256 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 17\!\cdots\!30 \beta_{17} + \cdots - 30\!\cdots\!09 ) / 128 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( ( - 42\!\cdots\!52 \beta_{17} + \cdots + 31\!\cdots\!36 ) / 128 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( ( - 88\!\cdots\!52 \beta_{17} + \cdots + 20\!\cdots\!67 ) / 256 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/8\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(7\)
\(\chi(n)\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1
0.500000 + 222.384i
0.500000 222.384i
0.500000 + 16156.1i
0.500000 16156.1i
0.500000 12463.1i
0.500000 + 12463.1i
0.500000 520.665i
0.500000 + 520.665i
0.500000 + 7755.46i
0.500000 7755.46i
0.500000 11170.2i
0.500000 + 11170.2i
0.500000 + 5761.07i
0.500000 5761.07i
0.500000 + 12610.3i
0.500000 12610.3i
0.500000 7326.87i
0.500000 + 7326.87i
−696.194 199.004i 889.536i 445083. + 277090.i 5.56401e6i −177021. + 619289.i −4.51327e6 −2.54722e8 2.81482e8i 1.16147e9 1.10726e9 3.87363e9i
5.2 −696.194 + 199.004i 889.536i 445083. 277090.i 5.56401e6i −177021. 619289.i −4.51327e6 −2.54722e8 + 2.81482e8i 1.16147e9 1.10726e9 + 3.87363e9i
5.3 −600.469 404.629i 64624.3i 196838. + 485935.i 4.52012e6i −2.61489e7 + 3.88049e7i −6.45051e7 7.84279e7 3.71435e8i −3.01404e9 −1.82897e9 + 2.71419e9i
5.4 −600.469 + 404.629i 64624.3i 196838. 485935.i 4.52012e6i −2.61489e7 3.88049e7i −6.45051e7 7.84279e7 + 3.71435e8i −3.01404e9 −1.82897e9 2.71419e9i
5.5 −540.302 482.039i 49852.5i 59564.6 + 520893.i 3.24834e6i 2.40309e7 2.69354e7i 1.08353e8 2.18908e8 3.10152e8i −1.32301e9 −1.56583e9 + 1.75508e9i
5.6 −540.302 + 482.039i 49852.5i 59564.6 520893.i 3.24834e6i 2.40309e7 + 2.69354e7i 1.08353e8 2.18908e8 + 3.10152e8i −1.32301e9 −1.56583e9 1.75508e9i
5.7 −238.377 683.714i 2082.66i −410641. + 325963.i 1.39037e6i 1.42394e6 496458.i −1.59434e8 3.20753e8 + 2.03059e8i 1.15792e9 −9.50615e8 + 3.31432e8i
5.8 −238.377 + 683.714i 2082.66i −410641. 325963.i 1.39037e6i 1.42394e6 + 496458.i −1.59434e8 3.20753e8 2.03059e8i 1.15792e9 −9.50615e8 3.31432e8i
5.9 −97.3835 717.499i 31021.8i −505321. + 139745.i 4.13771e6i −2.22581e7 + 3.02101e6i 1.99482e8 1.49477e8 + 3.48958e8i 1.99908e8 2.96880e9 4.02945e8i
5.10 −97.3835 + 717.499i 31021.8i −505321. 139745.i 4.13771e6i −2.22581e7 3.02101e6i 1.99482e8 1.49477e8 3.48958e8i 1.99908e8 2.96880e9 + 4.02945e8i
5.11 255.450 677.520i 44680.9i −393778. 346145.i 2.26130e6i 3.02722e7 + 1.14138e7i −2.45061e7 −3.35111e8 + 1.78370e8i −8.34122e8 1.53207e9 + 5.77649e8i
5.12 255.450 + 677.520i 44680.9i −393778. + 346145.i 2.26130e6i 3.02722e7 1.14138e7i −2.45061e7 −3.35111e8 1.78370e8i −8.34122e8 1.53207e9 5.77649e8i
5.13 424.854 586.333i 23044.3i −163286. 498212.i 7.37490e6i −1.35116e7 9.79045e6i 5.30965e7 −3.61491e8 1.15928e8i 6.31223e8 −4.32415e9 3.13326e9i
5.14 424.854 + 586.333i 23044.3i −163286. + 498212.i 7.37490e6i −1.35116e7 + 9.79045e6i 5.30965e7 −3.61491e8 + 1.15928e8i 6.31223e8 −4.32415e9 + 3.13326e9i
5.15 564.947 452.905i 50441.2i 114042. 511735.i 7.26212e6i −2.28451e7 2.84966e7i −1.80984e8 −1.67340e8 3.40753e8i −1.38205e9 3.28905e9 + 4.10271e9i
5.16 564.947 + 452.905i 50441.2i 114042. + 511735.i 7.26212e6i −2.28451e7 + 2.84966e7i −1.80984e8 −1.67340e8 + 3.40753e8i −1.38205e9 3.28905e9 4.10271e9i
5.17 698.474 190.846i 29307.5i 451443. 266602.i 795283.i 5.59322e6 + 2.04705e7i 3.26570e7 2.64441e8 2.72371e8i 3.03333e8 1.51777e8 + 5.55484e8i
5.18 698.474 + 190.846i 29307.5i 451443. + 266602.i 795283.i 5.59322e6 2.04705e7i 3.26570e7 2.64441e8 + 2.72371e8i 3.03333e8 1.51777e8 5.55484e8i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 5.18
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 8.20.b.a 18
3.b odd 2 1 72.20.d.b 18
4.b odd 2 1 32.20.b.a 18
8.b even 2 1 inner 8.20.b.a 18
8.d odd 2 1 32.20.b.a 18
24.h odd 2 1 72.20.d.b 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.20.b.a 18 1.a even 1 1 trivial
8.20.b.a 18 8.b even 2 1 inner
32.20.b.a 18 4.b odd 2 1
32.20.b.a 18 8.d odd 2 1
72.20.d.b 18 3.b odd 2 1
72.20.d.b 18 24.h odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{20}^{\mathrm{new}}(8, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{18} + \cdots + 29\!\cdots\!48 \) Copy content Toggle raw display
$3$ \( T^{18} + \cdots + 79\!\cdots\!36 \) Copy content Toggle raw display
$5$ \( T^{18} + \cdots + 20\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( (T^{9} + \cdots + 77\!\cdots\!32)^{2} \) Copy content Toggle raw display
$11$ \( T^{18} + \cdots + 73\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( T^{18} + \cdots + 14\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( (T^{9} + \cdots + 40\!\cdots\!92)^{2} \) Copy content Toggle raw display
$19$ \( T^{18} + \cdots + 13\!\cdots\!16 \) Copy content Toggle raw display
$23$ \( (T^{9} + \cdots + 15\!\cdots\!68)^{2} \) Copy content Toggle raw display
$29$ \( T^{18} + \cdots + 14\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( (T^{9} + \cdots - 11\!\cdots\!96)^{2} \) Copy content Toggle raw display
$37$ \( T^{18} + \cdots + 12\!\cdots\!96 \) Copy content Toggle raw display
$41$ \( (T^{9} + \cdots - 40\!\cdots\!00)^{2} \) Copy content Toggle raw display
$43$ \( T^{18} + \cdots + 67\!\cdots\!84 \) Copy content Toggle raw display
$47$ \( (T^{9} + \cdots + 12\!\cdots\!88)^{2} \) Copy content Toggle raw display
$53$ \( T^{18} + \cdots + 41\!\cdots\!76 \) Copy content Toggle raw display
$59$ \( T^{18} + \cdots + 38\!\cdots\!96 \) Copy content Toggle raw display
$61$ \( T^{18} + \cdots + 56\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{18} + \cdots + 47\!\cdots\!24 \) Copy content Toggle raw display
$71$ \( (T^{9} + \cdots + 54\!\cdots\!24)^{2} \) Copy content Toggle raw display
$73$ \( (T^{9} + \cdots - 11\!\cdots\!32)^{2} \) Copy content Toggle raw display
$79$ \( (T^{9} + \cdots - 32\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( T^{18} + \cdots + 98\!\cdots\!04 \) Copy content Toggle raw display
$89$ \( (T^{9} + \cdots + 58\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( (T^{9} + \cdots - 45\!\cdots\!28)^{2} \) Copy content Toggle raw display
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