Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [126,10,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, 10, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("126.37");
S:= CuspForms(chi, 10);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 126 = 2 \cdot 3^{2} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 10 \) |
| 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: | \(64.8945153566\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\Q(\sqrt{-3}, \sqrt{7081})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} + 1771x^{2} + 1770x + 3132900 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| 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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - x^{3} + 1771x^{2} + 1770x + 3132900 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{3} + 1771\nu^{2} - 1771\nu + 3132900 ) / 3134670 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} + 3541 ) / 1771 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + 1770\beta_{2} + \beta _1 - 1771 \)
|
| \(\nu^{3}\) | \(=\) |
\( 1771\beta_{3} - 3541 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/126\mathbb{Z}\right)^\times\).
| \(n\) | \(29\) | \(73\) |
| \(\chi(n)\) | \(1\) | \(-1 + \beta_{2}\) |
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 |
|
8.00000 | − | 13.8564i | 0 | −128.000 | − | 221.703i | −884.402 | + | 1531.83i | 0 | 1991.79 | − | 6032.11i | −4096.00 | 0 | 14150.4 | + | 24509.3i | ||||||||||||||||||||
| 37.2 | 8.00000 | − | 13.8564i | 0 | −128.000 | − | 221.703i | 419.902 | − | 727.292i | 0 | −6254.79 | + | 1109.63i | −4096.00 | 0 | −6718.44 | − | 11636.7i | |||||||||||||||||||||
| 109.1 | 8.00000 | + | 13.8564i | 0 | −128.000 | + | 221.703i | −884.402 | − | 1531.83i | 0 | 1991.79 | + | 6032.11i | −4096.00 | 0 | 14150.4 | − | 24509.3i | |||||||||||||||||||||
| 109.2 | 8.00000 | + | 13.8564i | 0 | −128.000 | + | 221.703i | 419.902 | + | 727.292i | 0 | −6254.79 | − | 1109.63i | −4096.00 | 0 | −6718.44 | + | 11636.7i | |||||||||||||||||||||
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 | 126.10.g.c | 4 | |
| 3.b | odd | 2 | 1 | 42.10.e.b | ✓ | 4 | |
| 7.c | even | 3 | 1 | inner | 126.10.g.c | 4 | |
| 21.g | even | 6 | 1 | 294.10.a.n | 2 | ||
| 21.h | odd | 6 | 1 | 42.10.e.b | ✓ | 4 | |
| 21.h | odd | 6 | 1 | 294.10.a.o | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 42.10.e.b | ✓ | 4 | 3.b | odd | 2 | 1 | |
| 42.10.e.b | ✓ | 4 | 21.h | odd | 6 | 1 | |
| 126.10.g.c | 4 | 1.a | even | 1 | 1 | trivial | |
| 126.10.g.c | 4 | 7.c | even | 3 | 1 | inner | |
| 294.10.a.n | 2 | 21.g | even | 6 | 1 | ||
| 294.10.a.o | 2 | 21.h | odd | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{4} + 929T_{5}^{3} + 2348491T_{5}^{2} - 1379983050T_{5} + 2206561702500 \)
acting on \(S_{10}^{\mathrm{new}}(126, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} - 16 T + 256)^{2} \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( T^{4} + \cdots + 2206561702500 \)
|
| $7$ |
\( T^{4} + \cdots + 16\!\cdots\!49 \)
|
| $11$ |
\( T^{4} + \cdots + 218147996387556 \)
|
| $13$ |
\( (T^{2} + 90607 T - 124789728)^{2} \)
|
| $17$ |
\( T^{4} + \cdots + 25\!\cdots\!04 \)
|
| $19$ |
\( T^{4} + \cdots + 10\!\cdots\!16 \)
|
| $23$ |
\( T^{4} + \cdots + 14\!\cdots\!44 \)
|
| $29$ |
\( (T^{2} + \cdots - 15397132707492)^{2} \)
|
| $31$ |
\( T^{4} + \cdots + 21\!\cdots\!25 \)
|
| $37$ |
\( T^{4} + \cdots + 41\!\cdots\!96 \)
|
| $41$ |
\( (T^{2} + \cdots + 98023365428688)^{2} \)
|
| $43$ |
\( (T^{2} + \cdots - 144362880560522)^{2} \)
|
| $47$ |
\( T^{4} + \cdots + 72\!\cdots\!44 \)
|
| $53$ |
\( T^{4} + \cdots + 40\!\cdots\!24 \)
|
| $59$ |
\( T^{4} + \cdots + 20\!\cdots\!84 \)
|
| $61$ |
\( T^{4} + \cdots + 90\!\cdots\!24 \)
|
| $67$ |
\( T^{4} + \cdots + 10\!\cdots\!76 \)
|
| $71$ |
\( (T^{2} + \cdots + 18\!\cdots\!16)^{2} \)
|
| $73$ |
\( T^{4} + \cdots + 49\!\cdots\!76 \)
|
| $79$ |
\( T^{4} + \cdots + 42\!\cdots\!49 \)
|
| $83$ |
\( (T^{2} + \cdots + 17\!\cdots\!78)^{2} \)
|
| $89$ |
\( T^{4} + \cdots + 18\!\cdots\!56 \)
|
| $97$ |
\( (T^{2} + \cdots - 11\!\cdots\!42)^{2} \)
|
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