Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [20,12,Mod(3,20)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(20, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([2, 3]))
N = Newforms(chi, 12, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("20.3");
S:= CuspForms(chi, 12);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 20 = 2^{2} \cdot 5 \) |
| Weight: | \( k \) | \(=\) | \( 12 \) |
| Character orbit: | \([\chi]\) | \(=\) | 20.e (of order \(4\), degree \(2\), 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: | \(15.3668636112\) |
| Analytic rank: | \(0\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{-1}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{U}(1)[D_{4}]$ |
$q$-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 \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/20\mathbb{Z}\right)^\times\).
| \(n\) | \(11\) | \(17\) |
| \(\chi(n)\) | \(-1\) | \(i\) |
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 |
|
32.0000 | − | 32.0000i | 0 | − | 2048.00i | −2642.00 | − | 6469.00i | 0 | 0 | −65536.0 | − | 65536.0i | 177147.i | −291552. | − | 122464.i | |||||||||||||||
| 7.1 | 32.0000 | + | 32.0000i | 0 | 2048.00i | −2642.00 | + | 6469.00i | 0 | 0 | −65536.0 | + | 65536.0i | − | 177147.i | −291552. | + | 122464.i | ||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 4.b | odd | 2 | 1 | CM by \(\Q(\sqrt{-1}) \) |
| 5.c | odd | 4 | 1 | inner |
| 20.e | even | 4 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 20.12.e.a | ✓ | 2 |
| 4.b | odd | 2 | 1 | CM | 20.12.e.a | ✓ | 2 |
| 5.b | even | 2 | 1 | 100.12.e.b | 2 | ||
| 5.c | odd | 4 | 1 | inner | 20.12.e.a | ✓ | 2 |
| 5.c | odd | 4 | 1 | 100.12.e.b | 2 | ||
| 20.d | odd | 2 | 1 | 100.12.e.b | 2 | ||
| 20.e | even | 4 | 1 | inner | 20.12.e.a | ✓ | 2 |
| 20.e | even | 4 | 1 | 100.12.e.b | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 20.12.e.a | ✓ | 2 | 1.a | even | 1 | 1 | trivial |
| 20.12.e.a | ✓ | 2 | 4.b | odd | 2 | 1 | CM |
| 20.12.e.a | ✓ | 2 | 5.c | odd | 4 | 1 | inner |
| 20.12.e.a | ✓ | 2 | 20.e | even | 4 | 1 | inner |
| 100.12.e.b | 2 | 5.b | even | 2 | 1 | ||
| 100.12.e.b | 2 | 5.c | odd | 4 | 1 | ||
| 100.12.e.b | 2 | 20.d | odd | 2 | 1 | ||
| 100.12.e.b | 2 | 20.e | even | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3} \)
acting on \(S_{12}^{\mathrm{new}}(20, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} - 64T + 2048 \)
|
| $3$ |
\( T^{2} \)
|
| $5$ |
\( T^{2} + 5284 T + 48828125 \)
|
| $7$ |
\( T^{2} \)
|
| $11$ |
\( T^{2} \)
|
| $13$ |
\( T^{2} + \cdots + 4879419339698 \)
|
| $17$ |
\( T^{2} + \cdots + 15123031064082 \)
|
| $19$ |
\( T^{2} \)
|
| $23$ |
\( T^{2} \)
|
| $29$ |
\( T^{2} + 36\!\cdots\!16 \)
|
| $31$ |
\( T^{2} \)
|
| $37$ |
\( T^{2} + \cdots + 52\!\cdots\!02 \)
|
| $41$ |
\( (T - 1347482392)^{2} \)
|
| $43$ |
\( T^{2} \)
|
| $47$ |
\( T^{2} \)
|
| $53$ |
\( T^{2} + \cdots + 21\!\cdots\!38 \)
|
| $59$ |
\( T^{2} \)
|
| $61$ |
\( (T - 2774217612)^{2} \)
|
| $67$ |
\( T^{2} \)
|
| $71$ |
\( T^{2} \)
|
| $73$ |
\( T^{2} + \cdots + 47\!\cdots\!58 \)
|
| $79$ |
\( T^{2} \)
|
| $83$ |
\( T^{2} \)
|
| $89$ |
\( T^{2} + 10\!\cdots\!56 \)
|
| $97$ |
\( T^{2} + \cdots + 11\!\cdots\!62 \)
|
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