Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [121,3,Mod(40,121)]
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([7]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("121.40");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 121 = 11^{2} \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 121.d (of order \(10\), 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: | \(3.29701119876\) |
| 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: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 11) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{10}]$ |
$q$-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 \(\left(\mathbb{Z}/121\mathbb{Z}\right)^\times\).
| \(n\) | \(2\) |
| \(\chi(n)\) | \(\zeta_{10}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 40.1 |
|
1.80902 | − | 2.48990i | 1.11803 | − | 3.44095i | −1.69098 | − | 5.20431i | −3.23607 | + | 2.35114i | −6.54508 | − | 9.00854i | 0.854102 | − | 0.277515i | −4.30902 | − | 1.40008i | −3.30902 | − | 2.40414i | 12.3107i | ||||||||||||||
| 94.1 | 0.690983 | − | 0.224514i | −1.11803 | − | 0.812299i | −2.80902 | + | 2.04087i | 1.23607 | − | 3.80423i | −0.954915 | − | 0.310271i | −5.85410 | − | 8.05748i | −3.19098 | + | 4.39201i | −2.19098 | − | 6.74315i | − | 2.90617i | ||||||||||||||
| 112.1 | 0.690983 | + | 0.224514i | −1.11803 | + | 0.812299i | −2.80902 | − | 2.04087i | 1.23607 | + | 3.80423i | −0.954915 | + | 0.310271i | −5.85410 | + | 8.05748i | −3.19098 | − | 4.39201i | −2.19098 | + | 6.74315i | 2.90617i | |||||||||||||||
| 118.1 | 1.80902 | + | 2.48990i | 1.11803 | + | 3.44095i | −1.69098 | + | 5.20431i | −3.23607 | − | 2.35114i | −6.54508 | + | 9.00854i | 0.854102 | + | 0.277515i | −4.30902 | + | 1.40008i | −3.30902 | + | 2.40414i | − | 12.3107i | ||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 11.d | odd | 10 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 121.3.d.d | 4 | |
| 11.b | odd | 2 | 1 | 11.3.d.a | ✓ | 4 | |
| 11.c | even | 5 | 1 | 11.3.d.a | ✓ | 4 | |
| 11.c | even | 5 | 1 | 121.3.b.b | 4 | ||
| 11.c | even | 5 | 1 | 121.3.d.a | 4 | ||
| 11.c | even | 5 | 1 | 121.3.d.c | 4 | ||
| 11.d | odd | 10 | 1 | 121.3.b.b | 4 | ||
| 11.d | odd | 10 | 1 | 121.3.d.a | 4 | ||
| 11.d | odd | 10 | 1 | 121.3.d.c | 4 | ||
| 11.d | odd | 10 | 1 | inner | 121.3.d.d | 4 | |
| 33.d | even | 2 | 1 | 99.3.k.a | 4 | ||
| 33.f | even | 10 | 1 | 1089.3.c.e | 4 | ||
| 33.h | odd | 10 | 1 | 99.3.k.a | 4 | ||
| 33.h | odd | 10 | 1 | 1089.3.c.e | 4 | ||
| 44.c | even | 2 | 1 | 176.3.n.a | 4 | ||
| 44.h | odd | 10 | 1 | 176.3.n.a | 4 | ||
| 55.d | odd | 2 | 1 | 275.3.x.e | 4 | ||
| 55.e | even | 4 | 2 | 275.3.q.d | 8 | ||
| 55.j | even | 10 | 1 | 275.3.x.e | 4 | ||
| 55.k | odd | 20 | 2 | 275.3.q.d | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 11.3.d.a | ✓ | 4 | 11.b | odd | 2 | 1 | |
| 11.3.d.a | ✓ | 4 | 11.c | even | 5 | 1 | |
| 99.3.k.a | 4 | 33.d | even | 2 | 1 | ||
| 99.3.k.a | 4 | 33.h | odd | 10 | 1 | ||
| 121.3.b.b | 4 | 11.c | even | 5 | 1 | ||
| 121.3.b.b | 4 | 11.d | odd | 10 | 1 | ||
| 121.3.d.a | 4 | 11.c | even | 5 | 1 | ||
| 121.3.d.a | 4 | 11.d | odd | 10 | 1 | ||
| 121.3.d.c | 4 | 11.c | even | 5 | 1 | ||
| 121.3.d.c | 4 | 11.d | odd | 10 | 1 | ||
| 121.3.d.d | 4 | 1.a | even | 1 | 1 | trivial | |
| 121.3.d.d | 4 | 11.d | odd | 10 | 1 | inner | |
| 176.3.n.a | 4 | 44.c | even | 2 | 1 | ||
| 176.3.n.a | 4 | 44.h | odd | 10 | 1 | ||
| 275.3.q.d | 8 | 55.e | even | 4 | 2 | ||
| 275.3.q.d | 8 | 55.k | odd | 20 | 2 | ||
| 275.3.x.e | 4 | 55.d | odd | 2 | 1 | ||
| 275.3.x.e | 4 | 55.j | even | 10 | 1 | ||
| 1089.3.c.e | 4 | 33.f | even | 10 | 1 | ||
| 1089.3.c.e | 4 | 33.h | odd | 10 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} - 5T_{2}^{3} + 15T_{2}^{2} - 15T_{2} + 5 \)
acting on \(S_{3}^{\mathrm{new}}(121, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} - 5 T^{3} + \cdots + 5 \)
|
| $3$ |
\( T^{4} + 10 T^{2} + \cdots + 25 \)
|
| $5$ |
\( T^{4} + 4 T^{3} + \cdots + 256 \)
|
| $7$ |
\( T^{4} + 10 T^{3} + \cdots + 80 \)
|
| $11$ |
\( T^{4} \)
|
| $13$ |
\( T^{4} - 20 T^{3} + \cdots + 2000 \)
|
| $17$ |
\( T^{4} + 10985 T + 142805 \)
|
| $19$ |
\( T^{4} + 25 T^{3} + \cdots + 605 \)
|
| $23$ |
\( (T^{2} + 10 T + 20)^{2} \)
|
| $29$ |
\( T^{4} - 40 T^{3} + \cdots + 9680 \)
|
| $31$ |
\( T^{4} + 58 T^{3} + \cdots + 55696 \)
|
| $37$ |
\( T^{4} - 90 T^{3} + \cdots + 2624400 \)
|
| $41$ |
\( T^{4} - 80 T^{3} + \cdots + 8405 \)
|
| $43$ |
\( T^{4} + 1625 T^{2} + 581405 \)
|
| $47$ |
\( T^{4} + 30 T^{3} + \cdots + 384400 \)
|
| $53$ |
\( T^{4} - 120 T^{3} + \cdots + 810000 \)
|
| $59$ |
\( T^{4} - 23 T^{3} + \cdots + 5041 \)
|
| $61$ |
\( T^{4} + 10 T^{3} + \cdots + 403280 \)
|
| $67$ |
\( (T^{2} + 115 T + 2945)^{2} \)
|
| $71$ |
\( T^{4} - 148 T^{3} + \cdots + 22619536 \)
|
| $73$ |
\( T^{4} + 300 T^{3} + \cdots + 93787805 \)
|
| $79$ |
\( T^{4} + 70 T^{3} + \cdots + 67280 \)
|
| $83$ |
\( T^{4} + 225 T^{3} + \cdots + 22281605 \)
|
| $89$ |
\( (T^{2} - 61 T - 7681)^{2} \)
|
| $97$ |
\( T^{4} + 165 T^{3} + \cdots + 31416025 \)
|
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