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: |
\( x^{18} - 9 x^{17} + 847482358 x^{16} - 6779858660 x^{15} + \cdots + 11\!\cdots\!76 \)
|
| 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.
Basis of coefficient ring in terms of a root \(\nu\) of
\( x^{18} - 9 x^{17} + 847482358 x^{16} - 6779858660 x^{15} + \cdots + 11\!\cdots\!76 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( - 10\!\cdots\!25 \nu^{17} + \cdots - 70\!\cdots\!64 ) / 91\!\cdots\!00 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 10\!\cdots\!25 \nu^{17} + \cdots - 70\!\cdots\!64 ) / 18\!\cdots\!00 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 32\!\cdots\!29 \nu^{17} + \cdots + 21\!\cdots\!20 ) / 33\!\cdots\!00 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 57\!\cdots\!61 \nu^{17} + \cdots - 55\!\cdots\!08 ) / 14\!\cdots\!00 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 42\!\cdots\!71 \nu^{17} + \cdots - 13\!\cdots\!60 ) / 43\!\cdots\!00 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 28\!\cdots\!28 \nu^{17} + \cdots - 13\!\cdots\!16 ) / 22\!\cdots\!00 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 70\!\cdots\!91 \nu^{17} + \cdots - 46\!\cdots\!60 ) / 91\!\cdots\!00 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 17\!\cdots\!25 \nu^{17} + \cdots + 17\!\cdots\!72 ) / 91\!\cdots\!00 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 33\!\cdots\!22 \nu^{17} + \cdots + 32\!\cdots\!52 ) / 75\!\cdots\!00 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 22\!\cdots\!07 \nu^{17} + \cdots + 45\!\cdots\!80 ) / 91\!\cdots\!00 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 22\!\cdots\!11 \nu^{17} + \cdots - 16\!\cdots\!52 ) / 27\!\cdots\!00 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( - 96\!\cdots\!13 \nu^{17} + \cdots + 14\!\cdots\!64 ) / 91\!\cdots\!00 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( 17\!\cdots\!37 \nu^{17} + \cdots - 12\!\cdots\!88 ) / 91\!\cdots\!00 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( 88\!\cdots\!31 \nu^{17} + \cdots + 48\!\cdots\!40 ) / 41\!\cdots\!00 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( - 34\!\cdots\!51 \nu^{17} + \cdots + 31\!\cdots\!96 ) / 45\!\cdots\!00 \)
|
| \(\beta_{16}\) | \(=\) |
\( ( 66\!\cdots\!77 \nu^{17} + \cdots + 81\!\cdots\!72 ) / 75\!\cdots\!00 \)
|
| \(\beta_{17}\) | \(=\) |
\( ( 12\!\cdots\!27 \nu^{17} + \cdots + 51\!\cdots\!20 ) / 82\!\cdots\!00 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} - 5\beta_1 ) / 4 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{8} + 3 \beta_{7} + 4 \beta_{6} + 5 \beta_{5} - 2 \beta_{4} - 232 \beta_{3} + 221 \beta_{2} + \cdots - 1506557050 ) / 16 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 304 \beta_{17} - 280 \beta_{16} + 208 \beta_{15} + 3 \beta_{14} - 280 \beta_{13} - 4720 \beta_{12} + \cdots - 2946572438 ) / 64 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( - 5167052 \beta_{17} + 5169812 \beta_{16} + 468922 \beta_{15} + 5173608 \beta_{14} + \cdots + 20\!\cdots\!17 ) / 128 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( - 339623258964 \beta_{17} + 361957444368 \beta_{16} - 273742380294 \beta_{15} - 1008021542328 \beta_{14} + \cdots + 20\!\cdots\!65 ) / 256 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 45\!\cdots\!92 \beta_{17} + \cdots - 79\!\cdots\!23 ) / 256 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 42\!\cdots\!10 \beta_{17} + \cdots - 19\!\cdots\!15 ) / 128 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( - 73\!\cdots\!16 \beta_{17} + \cdots + 87\!\cdots\!34 ) / 128 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( - 20\!\cdots\!04 \beta_{17} + \cdots + 11\!\cdots\!43 ) / 256 \)
|
| \(\nu^{10}\) | \(=\) |
\( ( 42\!\cdots\!40 \beta_{17} + \cdots - 40\!\cdots\!33 ) / 256 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( 26\!\cdots\!22 \beta_{17} + \cdots - 21\!\cdots\!58 ) / 128 \)
|
| \(\nu^{12}\) | \(=\) |
\( ( - 59\!\cdots\!08 \beta_{17} + \cdots + 49\!\cdots\!49 ) / 128 \)
|
| \(\nu^{13}\) | \(=\) |
\( ( - 13\!\cdots\!88 \beta_{17} + \cdots + 16\!\cdots\!81 ) / 256 \)
|
| \(\nu^{14}\) | \(=\) |
\( ( 31\!\cdots\!48 \beta_{17} + \cdots - 24\!\cdots\!91 ) / 256 \)
|
| \(\nu^{15}\) | \(=\) |
\( ( 17\!\cdots\!30 \beta_{17} + \cdots - 30\!\cdots\!09 ) / 128 \)
|
| \(\nu^{16}\) | \(=\) |
\( ( - 42\!\cdots\!52 \beta_{17} + \cdots + 31\!\cdots\!36 ) / 128 \)
|
| \(\nu^{17}\) | \(=\) |
\( ( - 88\!\cdots\!52 \beta_{17} + \cdots + 20\!\cdots\!67 ) / 256 \)
|
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 |
|
−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 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
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
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{18} + \cdots + 29\!\cdots\!48 \)
|
| $3$ |
\( T^{18} + \cdots + 79\!\cdots\!36 \)
|
| $5$ |
\( T^{18} + \cdots + 20\!\cdots\!00 \)
|
| $7$ |
\( (T^{9} + \cdots + 77\!\cdots\!32)^{2} \)
|
| $11$ |
\( T^{18} + \cdots + 73\!\cdots\!00 \)
|
| $13$ |
\( T^{18} + \cdots + 14\!\cdots\!00 \)
|
| $17$ |
\( (T^{9} + \cdots + 40\!\cdots\!92)^{2} \)
|
| $19$ |
\( T^{18} + \cdots + 13\!\cdots\!16 \)
|
| $23$ |
\( (T^{9} + \cdots + 15\!\cdots\!68)^{2} \)
|
| $29$ |
\( T^{18} + \cdots + 14\!\cdots\!00 \)
|
| $31$ |
\( (T^{9} + \cdots - 11\!\cdots\!96)^{2} \)
|
| $37$ |
\( T^{18} + \cdots + 12\!\cdots\!96 \)
|
| $41$ |
\( (T^{9} + \cdots - 40\!\cdots\!00)^{2} \)
|
| $43$ |
\( T^{18} + \cdots + 67\!\cdots\!84 \)
|
| $47$ |
\( (T^{9} + \cdots + 12\!\cdots\!88)^{2} \)
|
| $53$ |
\( T^{18} + \cdots + 41\!\cdots\!76 \)
|
| $59$ |
\( T^{18} + \cdots + 38\!\cdots\!96 \)
|
| $61$ |
\( T^{18} + \cdots + 56\!\cdots\!00 \)
|
| $67$ |
\( T^{18} + \cdots + 47\!\cdots\!24 \)
|
| $71$ |
\( (T^{9} + \cdots + 54\!\cdots\!24)^{2} \)
|
| $73$ |
\( (T^{9} + \cdots - 11\!\cdots\!32)^{2} \)
|
| $79$ |
\( (T^{9} + \cdots - 32\!\cdots\!00)^{2} \)
|
| $83$ |
\( T^{18} + \cdots + 98\!\cdots\!04 \)
|
| $89$ |
\( (T^{9} + \cdots + 58\!\cdots\!00)^{2} \)
|
| $97$ |
\( (T^{9} + \cdots - 45\!\cdots\!28)^{2} \)
|
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