Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [126,8,Mod(37,126)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(126, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([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("126.37");
S:= CuspForms(chi, 8);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 126 = 2 \cdot 3^{2} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| Character orbit: | \([\chi]\) | \(=\) | 126.g (of order \(3\), degree \(2\), 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: | \(39.3605132110\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Relative dimension: | \(3\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{6} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - x^{5} + 2119x^{4} - 65706x^{3} + 4519836x^{2} - 71825616x + 1150023744 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{2}\cdot 3\cdot 7^{3} \) |
| Twist minimal: | no (minimal twist has level 42) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - x^{5} + 2119x^{4} - 65706x^{3} + 4519836x^{2} - 71825616x + 1150023744 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( - 748007 \nu^{5} + 742355 \nu^{4} - 1573050245 \nu^{3} + 23770157970 \nu^{2} + \cdots + 53320105165680 ) / 53726255217936 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 728225 \nu^{5} - 41175703 \nu^{4} + 87251314657 \nu^{3} - 65641237782 \nu^{2} + \cdots - 29\!\cdots\!48 ) / 26863127608968 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 748007 \nu^{5} + 742355 \nu^{4} - 1573050245 \nu^{3} + 23770157970 \nu^{2} + \cdots - 406150052256 ) / 26863127608968 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 18599041 \nu^{5} + 2308123375 \nu^{4} + 83739829295 \nu^{3} + 3344056237766 \nu^{2} + \cdots + 41\!\cdots\!96 ) / 2984791956552 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 710240977 \nu^{5} + 4683229597 \nu^{4} - 969387646387 \nu^{3} + 66817667987142 \nu^{2} + \cdots + 45\!\cdots\!12 ) / 26863127608968 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} - 2\beta _1 + 2 ) / 7 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 5\beta_{5} - \beta_{4} + \beta_{3} - 25\beta_{2} - 9896\beta_1 ) / 7 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( \beta_{5} + 4\beta_{4} - 2093\beta_{3} + 2094\beta_{2} + 223252 ) / 7 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -8476\beta_{5} + 10595\beta_{4} + 82650\beta_{3} + 2119\beta_{2} + 20668652\beta _1 - 20668652 ) / 7 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 148375\beta_{5} - 29675\beta_{4} + 29675\beta_{3} - 5196005\beta_{2} - 787772236\beta_1 ) / 7 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/126\mathbb{Z}\right)^\times\).
| \(n\) | \(29\) | \(73\) |
| \(\chi(n)\) | \(1\) | \(-1 + \beta_{1}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 37.1 |
|
−4.00000 | + | 6.92820i | 0 | −32.0000 | − | 55.4256i | −201.921 | + | 349.738i | 0 | 514.742 | + | 747.384i | 512.000 | 0 | −1615.37 | − | 2797.90i | ||||||||||||||||||||||||||
| 37.2 | −4.00000 | + | 6.92820i | 0 | −32.0000 | − | 55.4256i | 47.5526 | − | 82.3635i | 0 | −906.806 | − | 35.2857i | 512.000 | 0 | 380.420 | + | 658.908i | |||||||||||||||||||||||||||
| 37.3 | −4.00000 | + | 6.92820i | 0 | −32.0000 | − | 55.4256i | 99.3686 | − | 172.111i | 0 | 709.564 | − | 565.740i | 512.000 | 0 | 794.949 | + | 1376.89i | |||||||||||||||||||||||||||
| 109.1 | −4.00000 | − | 6.92820i | 0 | −32.0000 | + | 55.4256i | −201.921 | − | 349.738i | 0 | 514.742 | − | 747.384i | 512.000 | 0 | −1615.37 | + | 2797.90i | |||||||||||||||||||||||||||
| 109.2 | −4.00000 | − | 6.92820i | 0 | −32.0000 | + | 55.4256i | 47.5526 | + | 82.3635i | 0 | −906.806 | + | 35.2857i | 512.000 | 0 | 380.420 | − | 658.908i | |||||||||||||||||||||||||||
| 109.3 | −4.00000 | − | 6.92820i | 0 | −32.0000 | + | 55.4256i | 99.3686 | + | 172.111i | 0 | 709.564 | + | 565.740i | 512.000 | 0 | 794.949 | − | 1376.89i | |||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 126.8.g.h | 6 | |
| 3.b | odd | 2 | 1 | 42.8.e.e | ✓ | 6 | |
| 7.c | even | 3 | 1 | inner | 126.8.g.h | 6 | |
| 21.c | even | 2 | 1 | 294.8.e.ba | 6 | ||
| 21.g | even | 6 | 1 | 294.8.a.w | 3 | ||
| 21.g | even | 6 | 1 | 294.8.e.ba | 6 | ||
| 21.h | odd | 6 | 1 | 42.8.e.e | ✓ | 6 | |
| 21.h | odd | 6 | 1 | 294.8.a.v | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 42.8.e.e | ✓ | 6 | 3.b | odd | 2 | 1 | |
| 42.8.e.e | ✓ | 6 | 21.h | odd | 6 | 1 | |
| 126.8.g.h | 6 | 1.a | even | 1 | 1 | trivial | |
| 126.8.g.h | 6 | 7.c | even | 3 | 1 | inner | |
| 294.8.a.v | 3 | 21.h | odd | 6 | 1 | ||
| 294.8.a.w | 3 | 21.g | even | 6 | 1 | ||
| 294.8.e.ba | 6 | 21.c | even | 2 | 1 | ||
| 294.8.e.ba | 6 | 21.g | even | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{6} + 110T_{5}^{5} + 111865T_{5}^{4} - 26240130T_{5}^{3} + 9113426325T_{5}^{2} - 761505247350T_{5} + 58262536340100 \)
acting on \(S_{8}^{\mathrm{new}}(126, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 8 T + 64)^{3} \)
|
| $3$ |
\( T^{6} \)
|
| $5$ |
\( T^{6} + \cdots + 58262536340100 \)
|
| $7$ |
\( T^{6} + \cdots + 55\!\cdots\!07 \)
|
| $11$ |
\( T^{6} + \cdots + 10\!\cdots\!36 \)
|
| $13$ |
\( (T^{3} + \cdots + 1384679947008)^{2} \)
|
| $17$ |
\( T^{6} + \cdots + 40\!\cdots\!36 \)
|
| $19$ |
\( T^{6} + \cdots + 14\!\cdots\!16 \)
|
| $23$ |
\( T^{6} + \cdots + 31\!\cdots\!96 \)
|
| $29$ |
\( (T^{3} + \cdots + 82770327433116)^{2} \)
|
| $31$ |
\( T^{6} + \cdots + 91\!\cdots\!25 \)
|
| $37$ |
\( T^{6} + \cdots + 20\!\cdots\!16 \)
|
| $41$ |
\( (T^{3} + \cdots - 78\!\cdots\!72)^{2} \)
|
| $43$ |
\( (T^{3} + \cdots + 23\!\cdots\!76)^{2} \)
|
| $47$ |
\( T^{6} + \cdots + 20\!\cdots\!44 \)
|
| $53$ |
\( T^{6} + \cdots + 89\!\cdots\!16 \)
|
| $59$ |
\( T^{6} + \cdots + 22\!\cdots\!96 \)
|
| $61$ |
\( T^{6} + \cdots + 28\!\cdots\!84 \)
|
| $67$ |
\( T^{6} + \cdots + 10\!\cdots\!96 \)
|
| $71$ |
\( (T^{3} + \cdots - 81\!\cdots\!64)^{2} \)
|
| $73$ |
\( T^{6} + \cdots + 26\!\cdots\!84 \)
|
| $79$ |
\( T^{6} + \cdots + 19\!\cdots\!29 \)
|
| $83$ |
\( (T^{3} + \cdots + 11\!\cdots\!98)^{2} \)
|
| $89$ |
\( T^{6} + \cdots + 12\!\cdots\!24 \)
|
| $97$ |
\( (T^{3} + \cdots + 14\!\cdots\!26)^{2} \)
|
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