Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [75,8,Mod(32,75)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(75, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([2, 1]))
N = Newforms(chi, 8, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("75.32");
S:= CuspForms(chi, 8);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 75 = 3 \cdot 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| Character orbit: | \([\chi]\) | \(=\) | 75.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: | \(23.4288769113\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(i)\) |
| Coefficient field: | \(\Q(i, \sqrt{6})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} + 9 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| 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 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
\( x^{4} + 9 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} ) / 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} ) / 3 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( 3\beta_{2} \)
|
| \(\nu^{3}\) | \(=\) |
\( 3\beta_{3} \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/75\mathbb{Z}\right)^\times\).
| \(n\) | \(26\) | \(52\) |
| \(\chi(n)\) | \(-1\) | \(\beta_{2}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 32.1 |
|
0 | −33.0681 | + | 33.0681i | 128.000i | 0 | 0 | −927.132 | − | 927.132i | 0 | − | 2187.00i | 0 | |||||||||||||||||||||||||
| 32.2 | 0 | 33.0681 | − | 33.0681i | 128.000i | 0 | 0 | 927.132 | + | 927.132i | 0 | − | 2187.00i | 0 | ||||||||||||||||||||||||||
| 68.1 | 0 | −33.0681 | − | 33.0681i | − | 128.000i | 0 | 0 | −927.132 | + | 927.132i | 0 | 2187.00i | 0 | ||||||||||||||||||||||||||
| 68.2 | 0 | 33.0681 | + | 33.0681i | − | 128.000i | 0 | 0 | 927.132 | − | 927.132i | 0 | 2187.00i | 0 | ||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | CM by \(\Q(\sqrt{-3}) \) |
| 5.b | even | 2 | 1 | inner |
| 5.c | odd | 4 | 2 | inner |
| 15.d | odd | 2 | 1 | inner |
| 15.e | even | 4 | 2 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 75.8.e.b | ✓ | 4 |
| 3.b | odd | 2 | 1 | CM | 75.8.e.b | ✓ | 4 |
| 5.b | even | 2 | 1 | inner | 75.8.e.b | ✓ | 4 |
| 5.c | odd | 4 | 2 | inner | 75.8.e.b | ✓ | 4 |
| 15.d | odd | 2 | 1 | inner | 75.8.e.b | ✓ | 4 |
| 15.e | even | 4 | 2 | inner | 75.8.e.b | ✓ | 4 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 75.8.e.b | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 75.8.e.b | ✓ | 4 | 3.b | odd | 2 | 1 | CM |
| 75.8.e.b | ✓ | 4 | 5.b | even | 2 | 1 | inner |
| 75.8.e.b | ✓ | 4 | 5.c | odd | 4 | 2 | inner |
| 75.8.e.b | ✓ | 4 | 15.d | odd | 2 | 1 | inner |
| 75.8.e.b | ✓ | 4 | 15.e | even | 4 | 2 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2} \)
acting on \(S_{8}^{\mathrm{new}}(75, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} + 4782969 \)
|
| $5$ |
\( T^{4} \)
|
| $7$ |
\( T^{4} + 2955466407609 \)
|
| $11$ |
\( T^{4} \)
|
| $13$ |
\( T^{4} + 84\!\cdots\!49 \)
|
| $17$ |
\( T^{4} \)
|
| $19$ |
\( (T^{2} + 1856834281)^{2} \)
|
| $23$ |
\( T^{4} \)
|
| $29$ |
\( T^{4} \)
|
| $31$ |
\( (T - 331387)^{4} \)
|
| $37$ |
\( T^{4} + 90\!\cdots\!24 \)
|
| $41$ |
\( T^{4} \)
|
| $43$ |
\( T^{4} + 84\!\cdots\!09 \)
|
| $47$ |
\( T^{4} \)
|
| $53$ |
\( T^{4} \)
|
| $59$ |
\( T^{4} \)
|
| $61$ |
\( (T - 1998347)^{4} \)
|
| $67$ |
\( T^{4} + 60\!\cdots\!89 \)
|
| $71$ |
\( T^{4} \)
|
| $73$ |
\( T^{4} + 23\!\cdots\!44 \)
|
| $79$ |
\( (T^{2} + 76790940145936)^{2} \)
|
| $83$ |
\( T^{4} \)
|
| $89$ |
\( T^{4} \)
|
| $97$ |
\( T^{4} + 26\!\cdots\!29 \)
|
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