Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [112,9,Mod(97,112)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(112, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 1]))
N = Newforms(chi, 9, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("112.97");
S:= CuspForms(chi, 9);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 112 = 2^{4} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 9 \) |
| Character orbit: | \([\chi]\) | \(=\) | 112.c (of order \(2\), degree \(1\), 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: | \(45.6264043268\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.0.3520512.3 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} + 120x^{2} + 3438 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{7}\cdot 3 \) |
| Twist minimal: | no (minimal twist has level 14) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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
\( x^{4} + 120x^{2} + 3438 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 2\nu^{3} + 156\nu ) / 9 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 2\nu^{3} + 108\nu ) / 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 4\nu^{2} + 240 ) / 3 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{2} + 3\beta_1 ) / 16 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 3\beta_{3} - 240 ) / 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 39\beta_{2} - 81\beta_1 ) / 8 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/112\mathbb{Z}\right)^\times\).
| \(n\) | \(15\) | \(17\) | \(85\) |
| \(\chi(n)\) | \(1\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 97.1 |
|
0 | − | 126.458i | 0 | − | 143.012i | 0 | 2350.56 | − | 489.571i | 0 | −9430.58 | 0 | ||||||||||||||||||||||||||
| 97.2 | 0 | − | 63.0589i | 0 | 390.317i | 0 | 687.442 | − | 2300.48i | 0 | 2584.58 | 0 | ||||||||||||||||||||||||||||
| 97.3 | 0 | 63.0589i | 0 | − | 390.317i | 0 | 687.442 | + | 2300.48i | 0 | 2584.58 | 0 | ||||||||||||||||||||||||||||
| 97.4 | 0 | 126.458i | 0 | 143.012i | 0 | 2350.56 | + | 489.571i | 0 | −9430.58 | 0 | |||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 112.9.c.c | 4 | |
| 4.b | odd | 2 | 1 | 14.9.b.a | ✓ | 4 | |
| 7.b | odd | 2 | 1 | inner | 112.9.c.c | 4 | |
| 12.b | even | 2 | 1 | 126.9.c.a | 4 | ||
| 20.d | odd | 2 | 1 | 350.9.b.a | 4 | ||
| 20.e | even | 4 | 2 | 350.9.d.a | 8 | ||
| 28.d | even | 2 | 1 | 14.9.b.a | ✓ | 4 | |
| 28.f | even | 6 | 2 | 98.9.d.a | 8 | ||
| 28.g | odd | 6 | 2 | 98.9.d.a | 8 | ||
| 84.h | odd | 2 | 1 | 126.9.c.a | 4 | ||
| 140.c | even | 2 | 1 | 350.9.b.a | 4 | ||
| 140.j | odd | 4 | 2 | 350.9.d.a | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 14.9.b.a | ✓ | 4 | 4.b | odd | 2 | 1 | |
| 14.9.b.a | ✓ | 4 | 28.d | even | 2 | 1 | |
| 98.9.d.a | 8 | 28.f | even | 6 | 2 | ||
| 98.9.d.a | 8 | 28.g | odd | 6 | 2 | ||
| 112.9.c.c | 4 | 1.a | even | 1 | 1 | trivial | |
| 112.9.c.c | 4 | 7.b | odd | 2 | 1 | inner | |
| 126.9.c.a | 4 | 12.b | even | 2 | 1 | ||
| 126.9.c.a | 4 | 84.h | odd | 2 | 1 | ||
| 350.9.b.a | 4 | 20.d | odd | 2 | 1 | ||
| 350.9.b.a | 4 | 140.c | even | 2 | 1 | ||
| 350.9.d.a | 8 | 20.e | even | 4 | 2 | ||
| 350.9.d.a | 8 | 140.j | odd | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} + 19968T_{3}^{2} + 63589248 \)
acting on \(S_{9}^{\mathrm{new}}(112, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} + 19968 T^{2} + 63589248 \)
|
| $5$ |
\( T^{4} + \cdots + 3115873152 \)
|
| $7$ |
\( T^{4} + \cdots + 33232930569601 \)
|
| $11$ |
\( (T^{2} - 6780 T - 43255548)^{2} \)
|
| $13$ |
\( T^{4} + \cdots + 202796647224192 \)
|
| $17$ |
\( T^{4} + \cdots + 23\!\cdots\!92 \)
|
| $19$ |
\( T^{4} + \cdots + 22\!\cdots\!88 \)
|
| $23$ |
\( (T^{2} - 447036 T + 45389086596)^{2} \)
|
| $29$ |
\( (T^{2} - 158532 T - 580343359356)^{2} \)
|
| $31$ |
\( T^{4} + \cdots + 10\!\cdots\!92 \)
|
| $37$ |
\( (T^{2} + 1247548 T + 131756883844)^{2} \)
|
| $41$ |
\( T^{4} + \cdots + 54\!\cdots\!12 \)
|
| $43$ |
\( (T^{2} + \cdots + 2551346607364)^{2} \)
|
| $47$ |
\( T^{4} + \cdots + 25\!\cdots\!88 \)
|
| $53$ |
\( (T^{2} + \cdots + 90844298538372)^{2} \)
|
| $59$ |
\( T^{4} + \cdots + 28\!\cdots\!00 \)
|
| $61$ |
\( T^{4} + \cdots + 38\!\cdots\!48 \)
|
| $67$ |
\( (T^{2} + \cdots - 12\!\cdots\!88)^{2} \)
|
| $71$ |
\( (T^{2} + \cdots - 599142107080188)^{2} \)
|
| $73$ |
\( T^{4} + \cdots + 24\!\cdots\!32 \)
|
| $79$ |
\( (T^{2} + \cdots - 50828841800444)^{2} \)
|
| $83$ |
\( T^{4} + \cdots + 97\!\cdots\!88 \)
|
| $89$ |
\( T^{4} + \cdots + 95\!\cdots\!32 \)
|
| $97$ |
\( T^{4} + \cdots + 34\!\cdots\!12 \)
|
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