Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [242,4,Mod(3,242)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(242, 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("242.3");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 242 = 2 \cdot 11^{2} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 242.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: | \(14.2784622214\) |
| 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: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{5}]$ |
$q$-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 \(\left(\mathbb{Z}/242\mathbb{Z}\right)^\times\).
| \(n\) | \(123\) |
| \(\chi(n)\) | \(-\zeta_{10}^{3}\) |
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 |
|
−1.61803 | + | 1.17557i | 1.23607 | + | 3.80423i | 1.23607 | − | 3.80423i | −2.42705 | − | 1.76336i | −6.47214 | − | 4.70228i | 2.47214 | − | 7.60845i | 2.47214 | + | 7.60845i | 8.89919 | − | 6.46564i | 6.00000 | ||||||||||||||
| 9.1 | 0.618034 | − | 1.90211i | −3.23607 | + | 2.35114i | −3.23607 | − | 2.35114i | 0.927051 | + | 2.85317i | 2.47214 | + | 7.60845i | −6.47214 | − | 4.70228i | −6.47214 | + | 4.70228i | −3.39919 | + | 10.4616i | 6.00000 | |||||||||||||||
| 27.1 | 0.618034 | + | 1.90211i | −3.23607 | − | 2.35114i | −3.23607 | + | 2.35114i | 0.927051 | − | 2.85317i | 2.47214 | − | 7.60845i | −6.47214 | + | 4.70228i | −6.47214 | − | 4.70228i | −3.39919 | − | 10.4616i | 6.00000 | |||||||||||||||
| 81.1 | −1.61803 | − | 1.17557i | 1.23607 | − | 3.80423i | 1.23607 | + | 3.80423i | −2.42705 | + | 1.76336i | −6.47214 | + | 4.70228i | 2.47214 | + | 7.60845i | 2.47214 | − | 7.60845i | 8.89919 | + | 6.46564i | 6.00000 | |||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 11.c | even | 5 | 3 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 242.4.c.c | 4 | |
| 11.b | odd | 2 | 1 | 242.4.c.i | 4 | ||
| 11.c | even | 5 | 1 | 242.4.a.e | yes | 1 | |
| 11.c | even | 5 | 3 | inner | 242.4.c.c | 4 | |
| 11.d | odd | 10 | 1 | 242.4.a.b | ✓ | 1 | |
| 11.d | odd | 10 | 3 | 242.4.c.i | 4 | ||
| 33.f | even | 10 | 1 | 2178.4.a.p | 1 | ||
| 33.h | odd | 10 | 1 | 2178.4.a.f | 1 | ||
| 44.g | even | 10 | 1 | 1936.4.a.f | 1 | ||
| 44.h | odd | 10 | 1 | 1936.4.a.e | 1 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 242.4.a.b | ✓ | 1 | 11.d | odd | 10 | 1 | |
| 242.4.a.e | yes | 1 | 11.c | even | 5 | 1 | |
| 242.4.c.c | 4 | 1.a | even | 1 | 1 | trivial | |
| 242.4.c.c | 4 | 11.c | even | 5 | 3 | inner | |
| 242.4.c.i | 4 | 11.b | odd | 2 | 1 | ||
| 242.4.c.i | 4 | 11.d | odd | 10 | 3 | ||
| 1936.4.a.e | 1 | 44.h | odd | 10 | 1 | ||
| 1936.4.a.f | 1 | 44.g | even | 10 | 1 | ||
| 2178.4.a.f | 1 | 33.h | odd | 10 | 1 | ||
| 2178.4.a.p | 1 | 33.f | even | 10 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(242, [\chi])\):
|
\( T_{3}^{4} + 4T_{3}^{3} + 16T_{3}^{2} + 64T_{3} + 256 \)
|
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\( T_{5}^{4} + 3T_{5}^{3} + 9T_{5}^{2} + 27T_{5} + 81 \)
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\( T_{7}^{4} + 8T_{7}^{3} + 64T_{7}^{2} + 512T_{7} + 4096 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} + 2 T^{3} + \cdots + 16 \)
|
| $3$ |
\( T^{4} + 4 T^{3} + \cdots + 256 \)
|
| $5$ |
\( T^{4} + 3 T^{3} + \cdots + 81 \)
|
| $7$ |
\( T^{4} + 8 T^{3} + \cdots + 4096 \)
|
| $11$ |
\( T^{4} \)
|
| $13$ |
\( T^{4} + 83 T^{3} + \cdots + 47458321 \)
|
| $17$ |
\( T^{4} + 123 T^{3} + \cdots + 228886641 \)
|
| $19$ |
\( T^{4} - 112 T^{3} + \cdots + 157351936 \)
|
| $23$ |
\( (T - 36)^{4} \)
|
| $29$ |
\( T^{4} - 21 T^{3} + \cdots + 194481 \)
|
| $31$ |
\( T^{4} + 128 T^{3} + \cdots + 268435456 \)
|
| $37$ |
\( T^{4} + 107 T^{3} + \cdots + 131079601 \)
|
| $41$ |
\( T^{4} + \cdots + 1632240801 \)
|
| $43$ |
\( (T - 308)^{4} \)
|
| $47$ |
\( T^{4} + \cdots + 58594980096 \)
|
| $53$ |
\( T^{4} + \cdots + 14166950625 \)
|
| $59$ |
\( T^{4} + \cdots + 1731891456 \)
|
| $61$ |
\( T^{4} + \cdots + 48796810000 \)
|
| $67$ |
\( (T + 760)^{4} \)
|
| $71$ |
\( T^{4} + \cdots + 656100000000 \)
|
| $73$ |
\( T^{4} + \cdots + 303120718096 \)
|
| $79$ |
\( T^{4} + 92 T^{3} + \cdots + 71639296 \)
|
| $83$ |
\( T^{4} + \cdots + 557256278016 \)
|
| $89$ |
\( (T + 645)^{4} \)
|
| $97$ |
\( T^{4} + \cdots + 7992538801 \)
|
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