Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [121,4,Mod(3,121)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(121, base_ring=CyclotomicField(10))
chi = DirichletCharacter(H, H._module([8]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("121.3");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 121 = 11^{2} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 121.c (of order \(5\), degree \(4\), 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: | \(7.13923111069\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\zeta_{10})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} + x^{2} - x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{9}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{U}(1)[D_{5}]$ |
$q$-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 \(\zeta_{10}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/121\mathbb{Z}\right)^\times\).
| \(n\) | \(2\) |
| \(\chi(n)\) | \(-\zeta_{10}^{3}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 3.1 |
|
0 | 2.47214 | + | 7.60845i | −2.47214 | + | 7.60845i | −14.5623 | − | 10.5801i | 0 | 0 | 0 | −29.9336 | + | 21.7481i | 0 | ||||||||||||||||||||||
| 9.1 | 0 | −6.47214 | + | 4.70228i | 6.47214 | + | 4.70228i | 5.56231 | + | 17.1190i | 0 | 0 | 0 | 11.4336 | − | 35.1891i | 0 | |||||||||||||||||||||||
| 27.1 | 0 | −6.47214 | − | 4.70228i | 6.47214 | − | 4.70228i | 5.56231 | − | 17.1190i | 0 | 0 | 0 | 11.4336 | + | 35.1891i | 0 | |||||||||||||||||||||||
| 81.1 | 0 | 2.47214 | − | 7.60845i | −2.47214 | − | 7.60845i | −14.5623 | + | 10.5801i | 0 | 0 | 0 | −29.9336 | − | 21.7481i | 0 | |||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 11.b | odd | 2 | 1 | CM by \(\Q(\sqrt{-11}) \) |
| 11.c | even | 5 | 3 | inner |
| 11.d | odd | 10 | 3 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 121.4.c.a | 4 | |
| 11.b | odd | 2 | 1 | CM | 121.4.c.a | 4 | |
| 11.c | even | 5 | 1 | 121.4.a.a | ✓ | 1 | |
| 11.c | even | 5 | 3 | inner | 121.4.c.a | 4 | |
| 11.d | odd | 10 | 1 | 121.4.a.a | ✓ | 1 | |
| 11.d | odd | 10 | 3 | inner | 121.4.c.a | 4 | |
| 33.f | even | 10 | 1 | 1089.4.a.f | 1 | ||
| 33.h | odd | 10 | 1 | 1089.4.a.f | 1 | ||
| 44.g | even | 10 | 1 | 1936.4.a.a | 1 | ||
| 44.h | odd | 10 | 1 | 1936.4.a.a | 1 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 121.4.a.a | ✓ | 1 | 11.c | even | 5 | 1 | |
| 121.4.a.a | ✓ | 1 | 11.d | odd | 10 | 1 | |
| 121.4.c.a | 4 | 1.a | even | 1 | 1 | trivial | |
| 121.4.c.a | 4 | 11.b | odd | 2 | 1 | CM | |
| 121.4.c.a | 4 | 11.c | even | 5 | 3 | inner | |
| 121.4.c.a | 4 | 11.d | odd | 10 | 3 | inner | |
| 1089.4.a.f | 1 | 33.f | even | 10 | 1 | ||
| 1089.4.a.f | 1 | 33.h | odd | 10 | 1 | ||
| 1936.4.a.a | 1 | 44.g | even | 10 | 1 | ||
| 1936.4.a.a | 1 | 44.h | odd | 10 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2} \)
acting on \(S_{4}^{\mathrm{new}}(121, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} + 8 T^{3} + \cdots + 4096 \)
|
| $5$ |
\( T^{4} + 18 T^{3} + \cdots + 104976 \)
|
| $7$ |
\( T^{4} \)
|
| $11$ |
\( T^{4} \)
|
| $13$ |
\( T^{4} \)
|
| $17$ |
\( T^{4} \)
|
| $19$ |
\( T^{4} \)
|
| $23$ |
\( (T + 108)^{4} \)
|
| $29$ |
\( T^{4} \)
|
| $31$ |
\( T^{4} + \cdots + 13363360000 \)
|
| $37$ |
\( T^{4} + \cdots + 35477982736 \)
|
| $41$ |
\( T^{4} \)
|
| $43$ |
\( T^{4} \)
|
| $47$ |
\( T^{4} - 36 T^{3} + \cdots + 1679616 \)
|
| $53$ |
\( T^{4} + \cdots + 296637086736 \)
|
| $59$ |
\( T^{4} + \cdots + 268738560000 \)
|
| $61$ |
\( T^{4} \)
|
| $67$ |
\( (T + 416)^{4} \)
|
| $71$ |
\( T^{4} + \cdots + 140283207936 \)
|
| $73$ |
\( T^{4} \)
|
| $79$ |
\( T^{4} \)
|
| $83$ |
\( T^{4} \)
|
| $89$ |
\( (T - 1674)^{4} \)
|
| $97$ |
\( T^{4} - 34 T^{3} + \cdots + 1336336 \)
|
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