Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [338,3,Mod(19,338)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(338, base_ring=CyclotomicField(12))
chi = DirichletCharacter(H, H._module([5]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("338.19");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 338 = 2 \cdot 13^{2} \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 338.f (of order \(12\), 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: | \(9.20983293538\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\zeta_{12})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{2} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 26) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{12}]$ |
$q$-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 \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/338\mathbb{Z}\right)^\times\).
| \(n\) | \(171\) |
| \(\chi(n)\) | \(\zeta_{12}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 19.1 |
|
0.366025 | + | 1.36603i | 0.866025 | − | 1.50000i | −1.73205 | + | 1.00000i | −1.73205 | + | 1.73205i | 2.36603 | + | 0.633975i | 2.03590 | − | 7.59808i | −2.00000 | − | 2.00000i | 3.00000 | + | 5.19615i | −3.00000 | − | 1.73205i | ||||||||||||
| 89.1 | 0.366025 | − | 1.36603i | 0.866025 | + | 1.50000i | −1.73205 | − | 1.00000i | −1.73205 | − | 1.73205i | 2.36603 | − | 0.633975i | 2.03590 | + | 7.59808i | −2.00000 | + | 2.00000i | 3.00000 | − | 5.19615i | −3.00000 | + | 1.73205i | |||||||||||||
| 249.1 | −1.36603 | − | 0.366025i | −0.866025 | − | 1.50000i | 1.73205 | + | 1.00000i | 1.73205 | − | 1.73205i | 0.633975 | + | 2.36603i | 8.96410 | − | 2.40192i | −2.00000 | − | 2.00000i | 3.00000 | − | 5.19615i | −3.00000 | + | 1.73205i | |||||||||||||
| 319.1 | −1.36603 | + | 0.366025i | −0.866025 | + | 1.50000i | 1.73205 | − | 1.00000i | 1.73205 | + | 1.73205i | 0.633975 | − | 2.36603i | 8.96410 | + | 2.40192i | −2.00000 | + | 2.00000i | 3.00000 | + | 5.19615i | −3.00000 | − | 1.73205i | |||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 13.f | odd | 12 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 338.3.f.d | 4 | |
| 13.b | even | 2 | 1 | 26.3.f.a | ✓ | 4 | |
| 13.c | even | 3 | 1 | 338.3.d.e | 4 | ||
| 13.c | even | 3 | 1 | 338.3.f.c | 4 | ||
| 13.d | odd | 4 | 1 | 338.3.f.c | 4 | ||
| 13.d | odd | 4 | 1 | 338.3.f.f | 4 | ||
| 13.e | even | 6 | 1 | 338.3.d.d | 4 | ||
| 13.e | even | 6 | 1 | 338.3.f.f | 4 | ||
| 13.f | odd | 12 | 1 | 26.3.f.a | ✓ | 4 | |
| 13.f | odd | 12 | 1 | 338.3.d.d | 4 | ||
| 13.f | odd | 12 | 1 | 338.3.d.e | 4 | ||
| 13.f | odd | 12 | 1 | inner | 338.3.f.d | 4 | |
| 39.d | odd | 2 | 1 | 234.3.bb.b | 4 | ||
| 39.k | even | 12 | 1 | 234.3.bb.b | 4 | ||
| 52.b | odd | 2 | 1 | 208.3.bd.c | 4 | ||
| 52.l | even | 12 | 1 | 208.3.bd.c | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 26.3.f.a | ✓ | 4 | 13.b | even | 2 | 1 | |
| 26.3.f.a | ✓ | 4 | 13.f | odd | 12 | 1 | |
| 208.3.bd.c | 4 | 52.b | odd | 2 | 1 | ||
| 208.3.bd.c | 4 | 52.l | even | 12 | 1 | ||
| 234.3.bb.b | 4 | 39.d | odd | 2 | 1 | ||
| 234.3.bb.b | 4 | 39.k | even | 12 | 1 | ||
| 338.3.d.d | 4 | 13.e | even | 6 | 1 | ||
| 338.3.d.d | 4 | 13.f | odd | 12 | 1 | ||
| 338.3.d.e | 4 | 13.c | even | 3 | 1 | ||
| 338.3.d.e | 4 | 13.f | odd | 12 | 1 | ||
| 338.3.f.c | 4 | 13.c | even | 3 | 1 | ||
| 338.3.f.c | 4 | 13.d | odd | 4 | 1 | ||
| 338.3.f.d | 4 | 1.a | even | 1 | 1 | trivial | |
| 338.3.f.d | 4 | 13.f | odd | 12 | 1 | inner | |
| 338.3.f.f | 4 | 13.d | odd | 4 | 1 | ||
| 338.3.f.f | 4 | 13.e | even | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(338, [\chi])\):
|
\( T_{3}^{4} + 3T_{3}^{2} + 9 \)
|
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\( T_{5}^{4} + 36 \)
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\( T_{7}^{4} - 22T_{7}^{3} + 221T_{7}^{2} - 1460T_{7} + 5329 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} + 2 T^{3} + \cdots + 4 \)
|
| $3$ |
\( T^{4} + 3T^{2} + 9 \)
|
| $5$ |
\( T^{4} + 36 \)
|
| $7$ |
\( T^{4} - 22 T^{3} + \cdots + 5329 \)
|
| $11$ |
\( T^{4} - 6 T^{3} + \cdots + 1521 \)
|
| $13$ |
\( T^{4} \)
|
| $17$ |
\( T^{4} - 78 T^{3} + \cdots + 221841 \)
|
| $19$ |
\( T^{4} - 58 T^{3} + \cdots + 167281 \)
|
| $23$ |
\( T^{4} - 12 T^{3} + \cdots + 184041 \)
|
| $29$ |
\( T^{4} + 54 T^{3} + \cdots + 1521 \)
|
| $31$ |
\( T^{4} - 128 T^{3} + \cdots + 3602404 \)
|
| $37$ |
\( T^{4} + 40 T^{3} + \cdots + 11449 \)
|
| $41$ |
\( T^{4} + 1521 T^{2} + \cdots + 257049 \)
|
| $43$ |
\( T^{4} - 120 T^{3} + \cdots + 103041 \)
|
| $47$ |
\( (T^{2} + 66 T + 2178)^{2} \)
|
| $53$ |
\( (T^{2} + 84 T + 312)^{2} \)
|
| $59$ |
\( T^{4} + 6 T^{3} + \cdots + 84681 \)
|
| $61$ |
\( T^{4} - 78 T^{3} + \cdots + 1996569 \)
|
| $67$ |
\( T^{4} + 86 T^{3} + \cdots + 2399401 \)
|
| $71$ |
\( T^{4} + 42 T^{3} + \cdots + 154449 \)
|
| $73$ |
\( T^{4} - 136 T^{3} + \cdots + 4251844 \)
|
| $79$ |
\( (T^{2} - 96 T - 1584)^{2} \)
|
| $83$ |
\( T^{4} + 192 T^{3} + \cdots + 2056356 \)
|
| $89$ |
\( T^{4} - 60 T^{3} + \cdots + 21594609 \)
|
| $97$ |
\( T^{4} + 280 T^{3} + \cdots + 20043529 \)
|
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