Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [112,6,Mod(65,112)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(112, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 0, 2]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("112.65");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 112 = 2^{4} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 112.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: | \(17.9629878191\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\Q(\sqrt{-3}, \sqrt{37})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} + 10x^{2} + 9x + 81 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{19}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | no (minimal twist has level 7) |
| 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
\( x^{4} - x^{3} + 10x^{2} + 9x + 81 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -\nu^{3} + 10\nu^{2} - 10\nu + 81 ) / 90 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} - 10\nu^{2} + 190\nu - 81 ) / 90 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} + 14 ) / 5 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} + \beta_{2} + 19\beta _1 - 19 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( 5\beta_{3} - 14 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/112\mathbb{Z}\right)^\times\).
| \(n\) | \(15\) | \(17\) | \(85\) |
| \(\chi(n)\) | \(1\) | \(-1 + \beta_{1}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 65.1 |
|
0 | −5.04138 | − | 8.73193i | 0 | 39.9138 | − | 69.1328i | 0 | −43.1587 | − | 122.247i | 0 | 70.6689 | − | 122.402i | 0 | ||||||||||||||||||||||
| 65.2 | 0 | 1.04138 | + | 1.80373i | 0 | −20.9138 | + | 36.2238i | 0 | 127.159 | + | 25.2522i | 0 | 119.331 | − | 206.687i | 0 | |||||||||||||||||||||||
| 81.1 | 0 | −5.04138 | + | 8.73193i | 0 | 39.9138 | + | 69.1328i | 0 | −43.1587 | + | 122.247i | 0 | 70.6689 | + | 122.402i | 0 | |||||||||||||||||||||||
| 81.2 | 0 | 1.04138 | − | 1.80373i | 0 | −20.9138 | − | 36.2238i | 0 | 127.159 | − | 25.2522i | 0 | 119.331 | + | 206.687i | 0 | |||||||||||||||||||||||
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 | 112.6.i.c | 4 | |
| 4.b | odd | 2 | 1 | 7.6.c.a | ✓ | 4 | |
| 7.c | even | 3 | 1 | inner | 112.6.i.c | 4 | |
| 7.c | even | 3 | 1 | 784.6.a.ba | 2 | ||
| 7.d | odd | 6 | 1 | 784.6.a.t | 2 | ||
| 12.b | even | 2 | 1 | 63.6.e.d | 4 | ||
| 28.d | even | 2 | 1 | 49.6.c.f | 4 | ||
| 28.f | even | 6 | 1 | 49.6.a.e | 2 | ||
| 28.f | even | 6 | 1 | 49.6.c.f | 4 | ||
| 28.g | odd | 6 | 1 | 7.6.c.a | ✓ | 4 | |
| 28.g | odd | 6 | 1 | 49.6.a.d | 2 | ||
| 84.j | odd | 6 | 1 | 441.6.a.m | 2 | ||
| 84.n | even | 6 | 1 | 63.6.e.d | 4 | ||
| 84.n | even | 6 | 1 | 441.6.a.n | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 7.6.c.a | ✓ | 4 | 4.b | odd | 2 | 1 | |
| 7.6.c.a | ✓ | 4 | 28.g | odd | 6 | 1 | |
| 49.6.a.d | 2 | 28.g | odd | 6 | 1 | ||
| 49.6.a.e | 2 | 28.f | even | 6 | 1 | ||
| 49.6.c.f | 4 | 28.d | even | 2 | 1 | ||
| 49.6.c.f | 4 | 28.f | even | 6 | 1 | ||
| 63.6.e.d | 4 | 12.b | even | 2 | 1 | ||
| 63.6.e.d | 4 | 84.n | even | 6 | 1 | ||
| 112.6.i.c | 4 | 1.a | even | 1 | 1 | trivial | |
| 112.6.i.c | 4 | 7.c | even | 3 | 1 | inner | |
| 441.6.a.m | 2 | 84.j | odd | 6 | 1 | ||
| 441.6.a.n | 2 | 84.n | even | 6 | 1 | ||
| 784.6.a.t | 2 | 7.d | odd | 6 | 1 | ||
| 784.6.a.ba | 2 | 7.c | even | 3 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} + 8T_{3}^{3} + 85T_{3}^{2} - 168T_{3} + 441 \)
acting on \(S_{6}^{\mathrm{new}}(112, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} + 8 T^{3} + \cdots + 441 \)
|
| $5$ |
\( T^{4} - 38 T^{3} + \cdots + 11148921 \)
|
| $7$ |
\( T^{4} - 168 T^{3} + \cdots + 282475249 \)
|
| $11$ |
\( T^{4} - 424 T^{3} + \cdots + 643687641 \)
|
| $13$ |
\( (T^{2} + 924 T + 184436)^{2} \)
|
| $17$ |
\( T^{4} + \cdots + 534713400081 \)
|
| $19$ |
\( T^{4} + \cdots + 7876852004329 \)
|
| $23$ |
\( T^{4} + \cdots + 31018606641 \)
|
| $29$ |
\( (T^{2} + 7052 T - 5697324)^{2} \)
|
| $31$ |
\( T^{4} + \cdots + 248637676526001 \)
|
| $37$ |
\( T^{4} + \cdots + 58604000855625 \)
|
| $41$ |
\( (T^{2} - 3500 T - 24814188)^{2} \)
|
| $43$ |
\( (T^{2} - 12680 T - 26638832)^{2} \)
|
| $47$ |
\( T^{4} + \cdots + 10\!\cdots\!81 \)
|
| $53$ |
\( T^{4} + \cdots + 14\!\cdots\!09 \)
|
| $59$ |
\( T^{4} + \cdots + 11\!\cdots\!81 \)
|
| $61$ |
\( T^{4} + \cdots + 35\!\cdots\!49 \)
|
| $67$ |
\( T^{4} + \cdots + 17\!\cdots\!41 \)
|
| $71$ |
\( (T^{2} - 2208 T - 175265856)^{2} \)
|
| $73$ |
\( T^{4} + \cdots + 18\!\cdots\!01 \)
|
| $79$ |
\( T^{4} + \cdots + 95\!\cdots\!81 \)
|
| $83$ |
\( (T^{2} - 104328 T + 1959796944)^{2} \)
|
| $89$ |
\( T^{4} + \cdots + 93\!\cdots\!49 \)
|
| $97$ |
\( (T^{2} - 209132 T + 10932626964)^{2} \)
|
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