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: |
\( x^{6} + 22015x^{4} - 28740x^{3} + 484660225x^{2} - 316355550x + 206496900 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{2}\cdot 3\cdot 7 \) |
| 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
\( x^{6} + 22015x^{4} - 28740x^{3} + 484660225x^{2} - 316355550x + 206496900 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -\nu^{5} - 22015\nu^{3} + 14370\nu^{2} - 484660225\nu + 316355550 ) / 316355550 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{4} - 95\nu^{3} - 22015\nu^{2} + 14370\nu - 321418780 ) / 968660 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 7331 \nu^{5} + 21555 \nu^{4} + 160227995 \nu^{3} + 211009080 \nu^{2} + 3551627119700 \nu + 4662091145100 ) / 6959822100 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -7331\nu^{5} - 160709390\nu^{3} + 263524245\nu^{2} - 3538017220850\nu + 2309393934300 ) / 6959822100 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 14684 \nu^{5} - 7185 \nu^{4} + 324381935 \nu^{3} - 685542405 \nu^{2} + 7142478791900 \nu - 6980570943000 ) / 6959822100 \)
|
| \(\nu\) | \(=\) |
\( ( 4\beta_{5} + 13\beta_{4} + 5\beta_{3} + 11\beta_{2} + 4\beta _1 - 5 ) / 42 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -475\beta_{5} + 803\beta_{4} - 95\beta_{3} + 190\beta_{2} - 616279\beta _1 + 95 ) / 42 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 3145\beta_{5} - 6290\beta_{4} - 12580\beta_{3} - 40885\beta_{2} + 3145\beta _1 + 98800 ) / 6 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 8423180 \beta_{5} - 13308385 \beta_{4} + 10528975 \beta_{3} - 17519975 \beta_{2} + 13565348240 \beta _1 - 13567454035 ) / 42 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( - 2430126875 \beta_{5} - 5319723365 \beta_{4} - 486025375 \beta_{3} + 972050750 \beta_{2} + \cdots + 486025375 ) / 42 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/126\mathbb{Z}\right)^\times\).
| \(n\) | \(29\) | \(73\) |
| \(\chi(n)\) | \(1\) | \(-\beta_{1}\) |
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 | −253.777 | + | 439.555i | 0 | −673.468 | + | 608.263i | −512.000 | 0 | 2030.22 | + | 3516.44i | ||||||||||||||||||||||||||
| 37.2 | 4.00000 | − | 6.92820i | 0 | −32.0000 | − | 55.4256i | 64.3732 | − | 111.498i | 0 | −291.106 | − | 859.535i | −512.000 | 0 | −514.986 | − | 891.982i | |||||||||||||||||||||||||||
| 37.3 | 4.00000 | − | 6.92820i | 0 | −32.0000 | − | 55.4256i | 154.404 | − | 267.436i | 0 | 517.075 | + | 745.773i | −512.000 | 0 | −1235.23 | − | 2139.49i | |||||||||||||||||||||||||||
| 109.1 | 4.00000 | + | 6.92820i | 0 | −32.0000 | + | 55.4256i | −253.777 | − | 439.555i | 0 | −673.468 | − | 608.263i | −512.000 | 0 | 2030.22 | − | 3516.44i | |||||||||||||||||||||||||||
| 109.2 | 4.00000 | + | 6.92820i | 0 | −32.0000 | + | 55.4256i | 64.3732 | + | 111.498i | 0 | −291.106 | + | 859.535i | −512.000 | 0 | −514.986 | + | 891.982i | |||||||||||||||||||||||||||
| 109.3 | 4.00000 | + | 6.92820i | 0 | −32.0000 | + | 55.4256i | 154.404 | + | 267.436i | 0 | 517.075 | − | 745.773i | −512.000 | 0 | −1235.23 | + | 2139.49i | |||||||||||||||||||||||||||
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.k | 6 | |
| 3.b | odd | 2 | 1 | 42.8.e.d | ✓ | 6 | |
| 7.c | even | 3 | 1 | inner | 126.8.g.k | 6 | |
| 21.c | even | 2 | 1 | 294.8.e.z | 6 | ||
| 21.g | even | 6 | 1 | 294.8.a.x | 3 | ||
| 21.g | even | 6 | 1 | 294.8.e.z | 6 | ||
| 21.h | odd | 6 | 1 | 42.8.e.d | ✓ | 6 | |
| 21.h | odd | 6 | 1 | 294.8.a.y | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 42.8.e.d | ✓ | 6 | 3.b | odd | 2 | 1 | |
| 42.8.e.d | ✓ | 6 | 21.h | odd | 6 | 1 | |
| 126.8.g.k | 6 | 1.a | even | 1 | 1 | trivial | |
| 126.8.g.k | 6 | 7.c | even | 3 | 1 | inner | |
| 294.8.a.x | 3 | 21.g | even | 6 | 1 | ||
| 294.8.a.y | 3 | 21.h | odd | 6 | 1 | ||
| 294.8.e.z | 6 | 21.c | even | 2 | 1 | ||
| 294.8.e.z | 6 | 21.g | even | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{6} + 70T_{5}^{5} + 187225T_{5}^{4} - 53121450T_{5}^{3} + 31829851125T_{5}^{2} - 3679199988750T_{5} + 407206166422500 \)
acting on \(S_{8}^{\mathrm{new}}(126, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} - 8 T + 64)^{3} \)
|
| $3$ |
\( T^{6} \)
|
| $5$ |
\( T^{6} + \cdots + 407206166422500 \)
|
| $7$ |
\( T^{6} + \cdots + 55\!\cdots\!07 \)
|
| $11$ |
\( T^{6} + \cdots + 49\!\cdots\!16 \)
|
| $13$ |
\( (T^{3} + 1919 T^{2} + \cdots + 58693162752)^{2} \)
|
| $17$ |
\( T^{6} + \cdots + 54\!\cdots\!16 \)
|
| $19$ |
\( T^{6} + \cdots + 15\!\cdots\!36 \)
|
| $23$ |
\( T^{6} + \cdots + 36\!\cdots\!36 \)
|
| $29$ |
\( (T^{3} + \cdots + 15\!\cdots\!76)^{2} \)
|
| $31$ |
\( T^{6} + \cdots + 21\!\cdots\!25 \)
|
| $37$ |
\( T^{6} + \cdots + 24\!\cdots\!56 \)
|
| $41$ |
\( (T^{3} + \cdots - 11\!\cdots\!32)^{2} \)
|
| $43$ |
\( (T^{3} + \cdots + 95\!\cdots\!84)^{2} \)
|
| $47$ |
\( T^{6} + \cdots + 89\!\cdots\!24 \)
|
| $53$ |
\( T^{6} + \cdots + 16\!\cdots\!56 \)
|
| $59$ |
\( T^{6} + \cdots + 64\!\cdots\!96 \)
|
| $61$ |
\( T^{6} + \cdots + 27\!\cdots\!64 \)
|
| $67$ |
\( T^{6} + \cdots + 14\!\cdots\!96 \)
|
| $71$ |
\( (T^{3} + \cdots + 62\!\cdots\!36)^{2} \)
|
| $73$ |
\( T^{6} + \cdots + 10\!\cdots\!64 \)
|
| $79$ |
\( T^{6} + \cdots + 32\!\cdots\!89 \)
|
| $83$ |
\( (T^{3} + \cdots - 11\!\cdots\!18)^{2} \)
|
| $89$ |
\( T^{6} + \cdots + 21\!\cdots\!04 \)
|
| $97$ |
\( (T^{3} + \cdots - 18\!\cdots\!06)^{2} \)
|
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