Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [400,2,Mod(149,400)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(400, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([0, 1, 2]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("400.149");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 400 = 2^{4} \cdot 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 400.q (of order \(4\), degree \(2\), 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: | \(3.19401608085\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(i)\) |
| Coefficient field: | \(\Q(i, \sqrt{11})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 5x^{2} + 9 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-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 a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring in terms of a root \(\nu\) of
\( x^{4} - 5x^{2} + 9 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{3} - 2\nu ) / 3 \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} + \nu - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( -\nu^{2} + \nu + 3 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} + \beta_{2} ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -\beta_{3} + \beta_{2} + 6 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + \beta_{2} + 3\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/400\mathbb{Z}\right)^\times\).
| \(n\) | \(101\) | \(177\) | \(351\) |
| \(\chi(n)\) | \(\beta_{1}\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 149.1 |
|
1.00000 | + | 1.00000i | −2.15831 | − | 2.15831i | 2.00000i | 0 | − | 4.31662i | −2.31662 | −2.00000 | + | 2.00000i | 6.31662i | 0 | |||||||||||||||||||||||
| 149.2 | 1.00000 | + | 1.00000i | 1.15831 | + | 1.15831i | 2.00000i | 0 | 2.31662i | 4.31662 | −2.00000 | + | 2.00000i | − | 0.316625i | 0 | ||||||||||||||||||||||||
| 349.1 | 1.00000 | − | 1.00000i | −2.15831 | + | 2.15831i | − | 2.00000i | 0 | 4.31662i | −2.31662 | −2.00000 | − | 2.00000i | − | 6.31662i | 0 | |||||||||||||||||||||||
| 349.2 | 1.00000 | − | 1.00000i | 1.15831 | − | 1.15831i | − | 2.00000i | 0 | − | 2.31662i | 4.31662 | −2.00000 | − | 2.00000i | 0.316625i | 0 | |||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 80.q | even | 4 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 400.2.q.d | 4 | |
| 4.b | odd | 2 | 1 | 1600.2.q.d | 4 | ||
| 5.b | even | 2 | 1 | 400.2.q.c | 4 | ||
| 5.c | odd | 4 | 1 | 400.2.l.d | ✓ | 4 | |
| 5.c | odd | 4 | 1 | 400.2.l.e | yes | 4 | |
| 16.e | even | 4 | 1 | 400.2.q.c | 4 | ||
| 16.f | odd | 4 | 1 | 1600.2.q.c | 4 | ||
| 20.d | odd | 2 | 1 | 1600.2.q.c | 4 | ||
| 20.e | even | 4 | 1 | 1600.2.l.d | 4 | ||
| 20.e | even | 4 | 1 | 1600.2.l.e | 4 | ||
| 80.i | odd | 4 | 1 | 400.2.l.e | yes | 4 | |
| 80.j | even | 4 | 1 | 1600.2.l.e | 4 | ||
| 80.k | odd | 4 | 1 | 1600.2.q.d | 4 | ||
| 80.q | even | 4 | 1 | inner | 400.2.q.d | 4 | |
| 80.s | even | 4 | 1 | 1600.2.l.d | 4 | ||
| 80.t | odd | 4 | 1 | 400.2.l.d | ✓ | 4 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 400.2.l.d | ✓ | 4 | 5.c | odd | 4 | 1 | |
| 400.2.l.d | ✓ | 4 | 80.t | odd | 4 | 1 | |
| 400.2.l.e | yes | 4 | 5.c | odd | 4 | 1 | |
| 400.2.l.e | yes | 4 | 80.i | odd | 4 | 1 | |
| 400.2.q.c | 4 | 5.b | even | 2 | 1 | ||
| 400.2.q.c | 4 | 16.e | even | 4 | 1 | ||
| 400.2.q.d | 4 | 1.a | even | 1 | 1 | trivial | |
| 400.2.q.d | 4 | 80.q | even | 4 | 1 | inner | |
| 1600.2.l.d | 4 | 20.e | even | 4 | 1 | ||
| 1600.2.l.d | 4 | 80.s | even | 4 | 1 | ||
| 1600.2.l.e | 4 | 20.e | even | 4 | 1 | ||
| 1600.2.l.e | 4 | 80.j | even | 4 | 1 | ||
| 1600.2.q.c | 4 | 16.f | odd | 4 | 1 | ||
| 1600.2.q.c | 4 | 20.d | odd | 2 | 1 | ||
| 1600.2.q.d | 4 | 4.b | odd | 2 | 1 | ||
| 1600.2.q.d | 4 | 80.k | odd | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} + 2T_{3}^{3} + 2T_{3}^{2} - 10T_{3} + 25 \)
acting on \(S_{2}^{\mathrm{new}}(400, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} - 2 T + 2)^{2} \)
|
| $3$ |
\( T^{4} + 2 T^{3} + \cdots + 25 \)
|
| $5$ |
\( T^{4} \)
|
| $7$ |
\( (T^{2} - 2 T - 10)^{2} \)
|
| $11$ |
\( T^{4} - 6 T^{3} + \cdots + 1 \)
|
| $13$ |
\( T^{4} - 4 T^{3} + \cdots + 400 \)
|
| $17$ |
\( T^{4} + 30T^{2} + 49 \)
|
| $19$ |
\( T^{4} + 6 T^{3} + \cdots + 1 \)
|
| $23$ |
\( (T^{2} + 6 T - 2)^{2} \)
|
| $29$ |
\( (T^{2} + 4 T + 8)^{2} \)
|
| $31$ |
\( (T^{2} - 2 T - 10)^{2} \)
|
| $37$ |
\( T^{4} + 16 T^{3} + \cdots + 100 \)
|
| $41$ |
\( (T^{2} + 25)^{2} \)
|
| $43$ |
\( T^{4} + 4 T^{3} + \cdots + 7396 \)
|
| $47$ |
\( (T^{2} + 64)^{2} \)
|
| $53$ |
\( T^{4} + 484 \)
|
| $59$ |
\( T^{4} + 8 T^{3} + \cdots + 196 \)
|
| $61$ |
\( T^{4} - 12 T^{3} + \cdots + 4900 \)
|
| $67$ |
\( T^{4} - 30 T^{3} + \cdots + 11449 \)
|
| $71$ |
\( T^{4} + 96T^{2} + 1600 \)
|
| $73$ |
\( (T^{2} - 20 T + 89)^{2} \)
|
| $79$ |
\( (T^{2} + 2 T - 10)^{2} \)
|
| $83$ |
\( T^{4} - 22 T^{3} + \cdots + 3025 \)
|
| $89$ |
\( T^{4} + 270T^{2} + 3969 \)
|
| $97$ |
\( (T^{2} + 44)^{2} \)
|
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