Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [10,9,Mod(3,10)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(10, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([3]))
N = Newforms(chi, 9, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("10.3");
S:= CuspForms(chi, 9);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 10 = 2 \cdot 5 \) |
| Weight: | \( k \) | \(=\) | \( 9 \) |
| Character orbit: | \([\chi]\) | \(=\) | 10.c (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: | \(4.07378610061\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(i)\) |
| Coefficient field: | \(\Q(i, \sqrt{249})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} + 125x^{2} + 3844 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2\cdot 5^{2} \) |
| 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
\( x^{4} + 125x^{2} + 3844 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{3} + 63\nu ) / 62 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 3\nu^{3} + 310\nu^{2} + 499\nu + 19406 ) / 62 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} - 155\nu^{2} + 218\nu - 9703 ) / 31 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} + \beta_{2} - 5\beta_1 ) / 10 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -\beta_{3} + \beta_{2} - \beta _1 - 626 ) / 10 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -63\beta_{3} - 63\beta_{2} + 935\beta_1 ) / 10 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/10\mathbb{Z}\right)^\times\).
| \(n\) | \(7\) |
| \(\chi(n)\) | \(-\beta_{1}\) |
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 |
|
−8.00000 | + | 8.00000i | −25.9493 | − | 25.9493i | − | 128.000i | 535.341 | − | 322.544i | 415.189 | 2031.01 | − | 2031.01i | 1024.00 | + | 1024.00i | − | 5214.26i | −1702.38 | + | 6863.08i | ||||||||||||||||
| 3.2 | −8.00000 | + | 8.00000i | 52.9493 | + | 52.9493i | − | 128.000i | −490.341 | + | 387.544i | −847.189 | −2624.01 | + | 2624.01i | 1024.00 | + | 1024.00i | − | 953.736i | 822.379 | − | 7023.08i | |||||||||||||||||
| 7.1 | −8.00000 | − | 8.00000i | −25.9493 | + | 25.9493i | 128.000i | 535.341 | + | 322.544i | 415.189 | 2031.01 | + | 2031.01i | 1024.00 | − | 1024.00i | 5214.26i | −1702.38 | − | 6863.08i | |||||||||||||||||||
| 7.2 | −8.00000 | − | 8.00000i | 52.9493 | − | 52.9493i | 128.000i | −490.341 | − | 387.544i | −847.189 | −2624.01 | − | 2624.01i | 1024.00 | − | 1024.00i | 953.736i | 822.379 | + | 7023.08i | |||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.c | odd | 4 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 10.9.c.a | ✓ | 4 |
| 3.b | odd | 2 | 1 | 90.9.g.b | 4 | ||
| 4.b | odd | 2 | 1 | 80.9.p.b | 4 | ||
| 5.b | even | 2 | 1 | 50.9.c.e | 4 | ||
| 5.c | odd | 4 | 1 | inner | 10.9.c.a | ✓ | 4 |
| 5.c | odd | 4 | 1 | 50.9.c.e | 4 | ||
| 15.e | even | 4 | 1 | 90.9.g.b | 4 | ||
| 20.e | even | 4 | 1 | 80.9.p.b | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 10.9.c.a | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 10.9.c.a | ✓ | 4 | 5.c | odd | 4 | 1 | inner |
| 50.9.c.e | 4 | 5.b | even | 2 | 1 | ||
| 50.9.c.e | 4 | 5.c | odd | 4 | 1 | ||
| 80.9.p.b | 4 | 4.b | odd | 2 | 1 | ||
| 80.9.p.b | 4 | 20.e | even | 4 | 1 | ||
| 90.9.g.b | 4 | 3.b | odd | 2 | 1 | ||
| 90.9.g.b | 4 | 15.e | even | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} - 54T_{3}^{3} + 1458T_{3}^{2} + 148392T_{3} + 7551504 \)
acting on \(S_{9}^{\mathrm{new}}(10, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 16 T + 128)^{2} \)
|
| $3$ |
\( T^{4} - 54 T^{3} + \cdots + 7551504 \)
|
| $5$ |
\( T^{4} + \cdots + 152587890625 \)
|
| $7$ |
\( T^{4} + \cdots + 113609761628944 \)
|
| $11$ |
\( (T^{2} + 9926 T - 82195856)^{2} \)
|
| $13$ |
\( T^{4} + \cdots + 24\!\cdots\!04 \)
|
| $17$ |
\( T^{4} + \cdots + 24\!\cdots\!64 \)
|
| $19$ |
\( T^{4} + \cdots + 11\!\cdots\!00 \)
|
| $23$ |
\( T^{4} + \cdots + 75\!\cdots\!24 \)
|
| $29$ |
\( T^{4} + \cdots + 25\!\cdots\!00 \)
|
| $31$ |
\( (T^{2} + 1164946 T - 344028072296)^{2} \)
|
| $37$ |
\( T^{4} + \cdots + 37\!\cdots\!64 \)
|
| $41$ |
\( (T^{2} + \cdots - 8475379721456)^{2} \)
|
| $43$ |
\( T^{4} + \cdots + 12\!\cdots\!24 \)
|
| $47$ |
\( T^{4} + \cdots + 31\!\cdots\!44 \)
|
| $53$ |
\( T^{4} + \cdots + 42\!\cdots\!44 \)
|
| $59$ |
\( T^{4} + \cdots + 28\!\cdots\!00 \)
|
| $61$ |
\( (T^{2} + \cdots + 454269173871504)^{2} \)
|
| $67$ |
\( T^{4} + \cdots + 16\!\cdots\!24 \)
|
| $71$ |
\( (T^{2} + \cdots + 793510166720824)^{2} \)
|
| $73$ |
\( T^{4} + \cdots + 19\!\cdots\!64 \)
|
| $79$ |
\( T^{4} + \cdots + 60\!\cdots\!00 \)
|
| $83$ |
\( T^{4} + \cdots + 30\!\cdots\!44 \)
|
| $89$ |
\( T^{4} + \cdots + 35\!\cdots\!00 \)
|
| $97$ |
\( T^{4} + \cdots + 55\!\cdots\!04 \)
|
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