Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [4,9,Mod(3,4)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1]))
N = Newforms(chi, 9, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("4.3");
S:= CuspForms(chi, 9);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 4 = 2^{2} \) |
| Weight: | \( k \) | \(=\) | \( 9 \) |
| Character orbit: | \([\chi]\) | \(=\) | 4.b (of order \(2\), degree \(1\), 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: | \(1.62951444024\) |
| Analytic rank: | \(0\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{-39}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - x + 10 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | yes |
| 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 \(\beta = 2\sqrt{-39}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4\mathbb{Z}\right)^\times\).
| \(n\) | \(3\) |
| \(\chi(n)\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 3.1 |
|
−10.0000 | − | 12.4900i | − | 99.9200i | −56.0000 | + | 249.800i | 610.000 | −1248.00 | + | 999.200i | − | 1398.88i | 3680.00 | − | 1798.56i | −3423.00 | −6100.00 | − | 7618.90i | ||||||||||||
| 3.2 | −10.0000 | + | 12.4900i | 99.9200i | −56.0000 | − | 249.800i | 610.000 | −1248.00 | − | 999.200i | 1398.88i | 3680.00 | + | 1798.56i | −3423.00 | −6100.00 | + | 7618.90i | |||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 4.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 4.9.b.b | ✓ | 2 |
| 3.b | odd | 2 | 1 | 36.9.d.b | 2 | ||
| 4.b | odd | 2 | 1 | inner | 4.9.b.b | ✓ | 2 |
| 5.b | even | 2 | 1 | 100.9.b.c | 2 | ||
| 5.c | odd | 4 | 2 | 100.9.d.b | 4 | ||
| 8.b | even | 2 | 1 | 64.9.c.b | 2 | ||
| 8.d | odd | 2 | 1 | 64.9.c.b | 2 | ||
| 12.b | even | 2 | 1 | 36.9.d.b | 2 | ||
| 16.e | even | 4 | 2 | 256.9.d.e | 4 | ||
| 16.f | odd | 4 | 2 | 256.9.d.e | 4 | ||
| 20.d | odd | 2 | 1 | 100.9.b.c | 2 | ||
| 20.e | even | 4 | 2 | 100.9.d.b | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 4.9.b.b | ✓ | 2 | 1.a | even | 1 | 1 | trivial |
| 4.9.b.b | ✓ | 2 | 4.b | odd | 2 | 1 | inner |
| 36.9.d.b | 2 | 3.b | odd | 2 | 1 | ||
| 36.9.d.b | 2 | 12.b | even | 2 | 1 | ||
| 64.9.c.b | 2 | 8.b | even | 2 | 1 | ||
| 64.9.c.b | 2 | 8.d | odd | 2 | 1 | ||
| 100.9.b.c | 2 | 5.b | even | 2 | 1 | ||
| 100.9.b.c | 2 | 20.d | odd | 2 | 1 | ||
| 100.9.d.b | 4 | 5.c | odd | 4 | 2 | ||
| 100.9.d.b | 4 | 20.e | even | 4 | 2 | ||
| 256.9.d.e | 4 | 16.e | even | 4 | 2 | ||
| 256.9.d.e | 4 | 16.f | odd | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{2} + 9984 \)
acting on \(S_{9}^{\mathrm{new}}(4, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} + 20T + 256 \)
|
| $3$ |
\( T^{2} + 9984 \)
|
| $5$ |
\( (T - 610)^{2} \)
|
| $7$ |
\( T^{2} + 1956864 \)
|
| $11$ |
\( T^{2} + 341702400 \)
|
| $13$ |
\( (T + 5470)^{2} \)
|
| $17$ |
\( (T - 73090)^{2} \)
|
| $19$ |
\( T^{2} + 379641600 \)
|
| $23$ |
\( T^{2} + 56268585984 \)
|
| $29$ |
\( (T + 128222)^{2} \)
|
| $31$ |
\( T^{2} + 4616601600 \)
|
| $37$ |
\( (T + 3472030)^{2} \)
|
| $41$ |
\( (T - 2146882)^{2} \)
|
| $43$ |
\( T^{2} + 35142983526144 \)
|
| $47$ |
\( T^{2} + 58160324112384 \)
|
| $53$ |
\( (T - 824290)^{2} \)
|
| $59$ |
\( T^{2} + 13879469510400 \)
|
| $61$ |
\( (T + 14746078)^{2} \)
|
| $67$ |
\( T^{2} + 232766284318464 \)
|
| $71$ |
\( T^{2} + 1430516505600 \)
|
| $73$ |
\( (T + 5725630)^{2} \)
|
| $79$ |
\( T^{2} + 12\!\cdots\!00 \)
|
| $83$ |
\( T^{2} + 26\!\cdots\!24 \)
|
| $89$ |
\( (T + 83324222)^{2} \)
|
| $97$ |
\( (T - 120619010)^{2} \)
|
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