Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [216,8,Mod(73,216)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(216, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 0, 4]))
N = Newforms(chi, 8, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("216.73");
S:= CuspForms(chi, 8);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 216 = 2^{3} \cdot 3^{3} \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| Character orbit: | \([\chi]\) | \(=\) | 216.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: | \(67.4751655046\) |
| Analytic rank: | \(0\) |
| Dimension: | \(20\) |
| Relative dimension: | \(10\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{20} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{20} - 10 x^{19} + 332929 x^{18} - 2996076 x^{17} + 44578211685 x^{16} - 356557783716 x^{15} + \cdots + 79\!\cdots\!67 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{17}]\) |
| Coefficient ring index: | \( 2^{48}\cdot 3^{45} \) |
| Twist minimal: | no (minimal twist has level 72) |
| 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 \(1,\beta_1,\ldots,\beta_{19}\) 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
\( x^{20} - 10 x^{19} + 332929 x^{18} - 2996076 x^{17} + 44578211685 x^{16} - 356557783716 x^{15} + \cdots + 79\!\cdots\!67 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 45\!\cdots\!34 \nu^{18} + \cdots - 48\!\cdots\!63 ) / 93\!\cdots\!00 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 12\!\cdots\!47 \nu^{18} + \cdots - 67\!\cdots\!71 ) / 48\!\cdots\!00 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 23\!\cdots\!83 \nu^{18} + \cdots + 49\!\cdots\!19 ) / 48\!\cdots\!00 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 34\!\cdots\!91 \nu^{18} + \cdots - 58\!\cdots\!37 ) / 96\!\cdots\!00 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 94\!\cdots\!47 \nu^{18} + \cdots + 64\!\cdots\!31 ) / 19\!\cdots\!00 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 27\!\cdots\!03 \nu^{18} + \cdots - 54\!\cdots\!79 ) / 48\!\cdots\!00 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 59\!\cdots\!03 \nu^{18} + \cdots - 54\!\cdots\!79 ) / 96\!\cdots\!00 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 10\!\cdots\!63 \nu^{18} + \cdots - 15\!\cdots\!59 ) / 96\!\cdots\!00 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 86\!\cdots\!74 \nu^{18} + \cdots - 33\!\cdots\!93 ) / 60\!\cdots\!00 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 37\!\cdots\!46 \nu^{19} + \cdots - 42\!\cdots\!11 ) / 24\!\cdots\!00 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 74\!\cdots\!92 \nu^{19} + \cdots + 17\!\cdots\!53 ) / 24\!\cdots\!00 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( - 59\!\cdots\!23 \nu^{19} + \cdots + 18\!\cdots\!37 ) / 25\!\cdots\!00 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( - 10\!\cdots\!91 \nu^{19} + \cdots - 10\!\cdots\!26 ) / 25\!\cdots\!00 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( - 28\!\cdots\!27 \nu^{19} + \cdots - 52\!\cdots\!82 ) / 51\!\cdots\!00 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( - 51\!\cdots\!63 \nu^{19} + \cdots - 87\!\cdots\!38 ) / 51\!\cdots\!00 \)
|
| \(\beta_{16}\) | \(=\) |
\( ( 33\!\cdots\!39 \nu^{19} + \cdots - 74\!\cdots\!81 ) / 17\!\cdots\!00 \)
|
| \(\beta_{17}\) | \(=\) |
\( ( - 23\!\cdots\!41 \nu^{19} + \cdots - 27\!\cdots\!66 ) / 85\!\cdots\!00 \)
|
| \(\beta_{18}\) | \(=\) |
\( ( 18\!\cdots\!14 \nu^{19} + \cdots - 27\!\cdots\!71 ) / 63\!\cdots\!00 \)
|
| \(\beta_{19}\) | \(=\) |
\( ( 16\!\cdots\!79 \nu^{19} + \cdots + 24\!\cdots\!94 ) / 51\!\cdots\!00 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{11} - 2\beta_{10} + \beta _1 + 1 ) / 3 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{11} - 2 \beta_{10} - \beta_{9} + 5 \beta_{8} + \beta_{7} - 9 \beta_{6} + 7 \beta_{5} + \cdots - 99882 ) / 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 2017 \beta_{19} - 231 \beta_{18} - 237 \beta_{17} - 1889 \beta_{16} + 307 \beta_{15} + 4419 \beta_{14} + \cdots + 687392 ) / 9 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 4034 \beta_{19} - 462 \beta_{18} - 474 \beta_{17} - 3778 \beta_{16} + 614 \beta_{15} + \cdots + 19521785487 ) / 9 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( - 875620105 \beta_{19} + 75686333 \beta_{18} + 18210601 \beta_{17} + 860685675 \beta_{16} + \cdots - 1373897338343 ) / 27 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( - 2626890570 \beta_{19} + 227062464 \beta_{18} + 54635358 \beta_{17} + 2582085360 \beta_{16} + \cdots - 45\!\cdots\!55 ) / 27 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 99957245401476 \beta_{19} - 8462058499050 \beta_{18} + 12337712608668 \beta_{17} + \cdots + 23\!\cdots\!17 ) / 27 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( 399841240456802 \beta_{19} - 33849293624266 \beta_{18} + 49350595466350 \beta_{17} + \cdots + 38\!\cdots\!00 ) / 27 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( - 35\!\cdots\!13 \beta_{19} + \cdots - 99\!\cdots\!10 ) / 9 \)
|
| \(\nu^{10}\) | \(=\) |
\( ( - 17\!\cdots\!70 \beta_{19} + \cdots - 11\!\cdots\!37 ) / 9 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( 10\!\cdots\!05 \beta_{19} + \cdots + 34\!\cdots\!39 ) / 27 \)
|
| \(\nu^{12}\) | \(=\) |
\( ( 65\!\cdots\!96 \beta_{19} + \cdots + 31\!\cdots\!79 ) / 27 \)
|
| \(\nu^{13}\) | \(=\) |
\( ( - 10\!\cdots\!80 \beta_{19} + \cdots - 37\!\cdots\!63 ) / 27 \)
|
| \(\nu^{14}\) | \(=\) |
\( ( - 76\!\cdots\!08 \beta_{19} + \cdots - 29\!\cdots\!54 ) / 27 \)
|
| \(\nu^{15}\) | \(=\) |
\( ( 36\!\cdots\!25 \beta_{19} + \cdots + 13\!\cdots\!08 ) / 9 \)
|
| \(\nu^{16}\) | \(=\) |
\( ( 29\!\cdots\!90 \beta_{19} + \cdots + 95\!\cdots\!91 ) / 9 \)
|
| \(\nu^{17}\) | \(=\) |
\( ( - 10\!\cdots\!09 \beta_{19} + \cdots - 40\!\cdots\!03 ) / 27 \)
|
| \(\nu^{18}\) | \(=\) |
\( ( - 98\!\cdots\!06 \beta_{19} + \cdots - 27\!\cdots\!15 ) / 27 \)
|
| \(\nu^{19}\) | \(=\) |
\( ( 10\!\cdots\!56 \beta_{19} + \cdots + 40\!\cdots\!89 ) / 27 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/216\mathbb{Z}\right)^\times\).
| \(n\) | \(55\) | \(109\) | \(137\) |
| \(\chi(n)\) | \(1\) | \(1\) | \(-\beta_{11}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 73.1 |
|
0 | 0 | 0 | −266.720 | − | 461.973i | 0 | −66.5594 | + | 115.284i | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 73.2 | 0 | 0 | 0 | −156.325 | − | 270.763i | 0 | 626.107 | − | 1084.45i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 73.3 | 0 | 0 | 0 | −130.433 | − | 225.916i | 0 | −113.976 | + | 197.412i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 73.4 | 0 | 0 | 0 | −57.8156 | − | 100.140i | 0 | 783.840 | − | 1357.65i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 73.5 | 0 | 0 | 0 | −3.05772 | − | 5.29612i | 0 | −422.475 | + | 731.748i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 73.6 | 0 | 0 | 0 | 8.05553 | + | 13.9526i | 0 | −629.010 | + | 1089.48i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 73.7 | 0 | 0 | 0 | 46.3771 | + | 80.3275i | 0 | −227.962 | + | 394.841i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 73.8 | 0 | 0 | 0 | 167.971 | + | 290.935i | 0 | 410.729 | − | 711.403i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 73.9 | 0 | 0 | 0 | 212.760 | + | 368.511i | 0 | 612.894 | − | 1061.56i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 73.10 | 0 | 0 | 0 | 241.687 | + | 418.615i | 0 | −351.089 | + | 608.105i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 145.1 | 0 | 0 | 0 | −266.720 | + | 461.973i | 0 | −66.5594 | − | 115.284i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 145.2 | 0 | 0 | 0 | −156.325 | + | 270.763i | 0 | 626.107 | + | 1084.45i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 145.3 | 0 | 0 | 0 | −130.433 | + | 225.916i | 0 | −113.976 | − | 197.412i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 145.4 | 0 | 0 | 0 | −57.8156 | + | 100.140i | 0 | 783.840 | + | 1357.65i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 145.5 | 0 | 0 | 0 | −3.05772 | + | 5.29612i | 0 | −422.475 | − | 731.748i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 145.6 | 0 | 0 | 0 | 8.05553 | − | 13.9526i | 0 | −629.010 | − | 1089.48i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 145.7 | 0 | 0 | 0 | 46.3771 | − | 80.3275i | 0 | −227.962 | − | 394.841i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 145.8 | 0 | 0 | 0 | 167.971 | − | 290.935i | 0 | 410.729 | + | 711.403i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 145.9 | 0 | 0 | 0 | 212.760 | − | 368.511i | 0 | 612.894 | + | 1061.56i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 145.10 | 0 | 0 | 0 | 241.687 | − | 418.615i | 0 | −351.089 | − | 608.105i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 9.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 216.8.i.a | 20 | |
| 3.b | odd | 2 | 1 | 72.8.i.a | ✓ | 20 | |
| 4.b | odd | 2 | 1 | 432.8.i.e | 20 | ||
| 9.c | even | 3 | 1 | inner | 216.8.i.a | 20 | |
| 9.d | odd | 6 | 1 | 72.8.i.a | ✓ | 20 | |
| 12.b | even | 2 | 1 | 144.8.i.e | 20 | ||
| 36.f | odd | 6 | 1 | 432.8.i.e | 20 | ||
| 36.h | even | 6 | 1 | 144.8.i.e | 20 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 72.8.i.a | ✓ | 20 | 3.b | odd | 2 | 1 | |
| 72.8.i.a | ✓ | 20 | 9.d | odd | 6 | 1 | |
| 144.8.i.e | 20 | 12.b | even | 2 | 1 | ||
| 144.8.i.e | 20 | 36.h | even | 6 | 1 | ||
| 216.8.i.a | 20 | 1.a | even | 1 | 1 | trivial | |
| 216.8.i.a | 20 | 9.c | even | 3 | 1 | inner | |
| 432.8.i.e | 20 | 4.b | odd | 2 | 1 | ||
| 432.8.i.e | 20 | 36.f | odd | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{20} - 125 T_{5}^{19} + 507916 T_{5}^{18} - 44805399 T_{5}^{17} + 175596788865 T_{5}^{16} + \cdots + 10\!\cdots\!00 \)
acting on \(S_{8}^{\mathrm{new}}(216, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{20} \)
|
| $3$ |
\( T^{20} \)
|
| $5$ |
\( T^{20} + \cdots + 10\!\cdots\!00 \)
|
| $7$ |
\( T^{20} + \cdots + 41\!\cdots\!96 \)
|
| $11$ |
\( T^{20} + \cdots + 76\!\cdots\!25 \)
|
| $13$ |
\( T^{20} + \cdots + 37\!\cdots\!00 \)
|
| $17$ |
\( (T^{10} + \cdots - 99\!\cdots\!24)^{2} \)
|
| $19$ |
\( (T^{10} + \cdots + 23\!\cdots\!24)^{2} \)
|
| $23$ |
\( T^{20} + \cdots + 23\!\cdots\!64 \)
|
| $29$ |
\( T^{20} + \cdots + 13\!\cdots\!44 \)
|
| $31$ |
\( T^{20} + \cdots + 15\!\cdots\!76 \)
|
| $37$ |
\( (T^{10} + \cdots + 49\!\cdots\!04)^{2} \)
|
| $41$ |
\( T^{20} + \cdots + 65\!\cdots\!25 \)
|
| $43$ |
\( T^{20} + \cdots + 20\!\cdots\!49 \)
|
| $47$ |
\( T^{20} + \cdots + 60\!\cdots\!96 \)
|
| $53$ |
\( (T^{10} + \cdots - 15\!\cdots\!92)^{2} \)
|
| $59$ |
\( T^{20} + \cdots + 15\!\cdots\!29 \)
|
| $61$ |
\( T^{20} + \cdots + 26\!\cdots\!96 \)
|
| $67$ |
\( T^{20} + \cdots + 14\!\cdots\!25 \)
|
| $71$ |
\( (T^{10} + \cdots + 12\!\cdots\!00)^{2} \)
|
| $73$ |
\( (T^{10} + \cdots + 18\!\cdots\!92)^{2} \)
|
| $79$ |
\( T^{20} + \cdots + 31\!\cdots\!96 \)
|
| $83$ |
\( T^{20} + \cdots + 29\!\cdots\!00 \)
|
| $89$ |
\( (T^{10} + \cdots + 27\!\cdots\!00)^{2} \)
|
| $97$ |
\( T^{20} + \cdots + 57\!\cdots\!41 \)
|
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