Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [112,7,Mod(17,112)]
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, 1]))
N = Newforms(chi, 7, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("112.17");
S:= CuspForms(chi, 7);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 112 = 2^{4} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 7 \) |
| Character orbit: | \([\chi]\) | \(=\) | 112.s (of order \(6\), 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: | \(25.7660573654\) |
| Analytic rank: | \(0\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{-3}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{9}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 7) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-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 primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/112\mathbb{Z}\right)^\times\).
| \(n\) | \(15\) | \(17\) | \(85\) |
| \(\chi(n)\) | \(1\) | \(\zeta_{6}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 17.1 |
|
0 | 10.5000 | + | 6.06218i | 0 | 157.500 | − | 90.9327i | 0 | 343.000 | 0 | −291.000 | − | 504.027i | 0 | ||||||||||||||||||
| 33.1 | 0 | 10.5000 | − | 6.06218i | 0 | 157.500 | + | 90.9327i | 0 | 343.000 | 0 | −291.000 | + | 504.027i | 0 | |||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.d | odd | 6 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 112.7.s.a | 2 | |
| 4.b | odd | 2 | 1 | 7.7.d.a | ✓ | 2 | |
| 7.d | odd | 6 | 1 | inner | 112.7.s.a | 2 | |
| 12.b | even | 2 | 1 | 63.7.m.a | 2 | ||
| 28.d | even | 2 | 1 | 49.7.d.b | 2 | ||
| 28.f | even | 6 | 1 | 7.7.d.a | ✓ | 2 | |
| 28.f | even | 6 | 1 | 49.7.b.a | 2 | ||
| 28.g | odd | 6 | 1 | 49.7.b.a | 2 | ||
| 28.g | odd | 6 | 1 | 49.7.d.b | 2 | ||
| 84.j | odd | 6 | 1 | 63.7.m.a | 2 | ||
| 84.j | odd | 6 | 1 | 441.7.d.a | 2 | ||
| 84.n | even | 6 | 1 | 441.7.d.a | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 7.7.d.a | ✓ | 2 | 4.b | odd | 2 | 1 | |
| 7.7.d.a | ✓ | 2 | 28.f | even | 6 | 1 | |
| 49.7.b.a | 2 | 28.f | even | 6 | 1 | ||
| 49.7.b.a | 2 | 28.g | odd | 6 | 1 | ||
| 49.7.d.b | 2 | 28.d | even | 2 | 1 | ||
| 49.7.d.b | 2 | 28.g | odd | 6 | 1 | ||
| 63.7.m.a | 2 | 12.b | even | 2 | 1 | ||
| 63.7.m.a | 2 | 84.j | odd | 6 | 1 | ||
| 112.7.s.a | 2 | 1.a | even | 1 | 1 | trivial | |
| 112.7.s.a | 2 | 7.d | odd | 6 | 1 | inner | |
| 441.7.d.a | 2 | 84.j | odd | 6 | 1 | ||
| 441.7.d.a | 2 | 84.n | even | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{2} - 21T_{3} + 147 \)
acting on \(S_{7}^{\mathrm{new}}(112, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} \)
|
| $3$ |
\( T^{2} - 21T + 147 \)
|
| $5$ |
\( T^{2} - 315T + 33075 \)
|
| $7$ |
\( (T - 343)^{2} \)
|
| $11$ |
\( T^{2} - 1479 T + 2187441 \)
|
| $13$ |
\( T^{2} + 235200 \)
|
| $17$ |
\( T^{2} + 5229 T + 9114147 \)
|
| $19$ |
\( T^{2} + 11907 T + 47258883 \)
|
| $23$ |
\( T^{2} + 5913 T + 34963569 \)
|
| $29$ |
\( (T - 3978)^{2} \)
|
| $31$ |
\( T^{2} - 22197 T + 164235603 \)
|
| $37$ |
\( T^{2} + \cdots + 3791726929 \)
|
| $41$ |
\( T^{2} + 12226636800 \)
|
| $43$ |
\( (T - 17414)^{2} \)
|
| $47$ |
\( T^{2} - 53109 T + 940188627 \)
|
| $53$ |
\( T^{2} + \cdots + 3661823169 \)
|
| $59$ |
\( T^{2} + \cdots + 46538854803 \)
|
| $61$ |
\( T^{2} + \cdots + 26486008563 \)
|
| $67$ |
\( T^{2} + \cdots + 72241075729 \)
|
| $71$ |
\( (T + 101922)^{2} \)
|
| $73$ |
\( T^{2} + \cdots + 100898977347 \)
|
| $79$ |
\( T^{2} + \cdots + 131211297361 \)
|
| $83$ |
\( T^{2} + 46995076800 \)
|
| $89$ |
\( T^{2} + \cdots + 1781061603363 \)
|
| $97$ |
\( T^{2} + 2292467116800 \)
|
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