Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [75,8,Mod(49,75)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(75, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1]))
N = Newforms(chi, 8, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("75.49");
S:= CuspForms(chi, 8);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 75 = 3 \cdot 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| Character orbit: | \([\chi]\) | \(=\) | 75.b (of order \(2\), degree \(1\), 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: | \(23.4288769113\) |
| 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: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 15) |
| 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 \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/75\mathbb{Z}\right)^\times\).
| \(n\) | \(26\) | \(52\) |
| \(\chi(n)\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 49.1 |
|
− | 22.0000i | − | 27.0000i | −356.000 | 0 | −594.000 | − | 420.000i | 5016.00i | −729.000 | 0 | |||||||||||||||||||||
| 49.2 | 22.0000i | 27.0000i | −356.000 | 0 | −594.000 | 420.000i | − | 5016.00i | −729.000 | 0 | ||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 75.8.b.a | 2 | |
| 3.b | odd | 2 | 1 | 225.8.b.a | 2 | ||
| 5.b | even | 2 | 1 | inner | 75.8.b.a | 2 | |
| 5.c | odd | 4 | 1 | 15.8.a.a | ✓ | 1 | |
| 5.c | odd | 4 | 1 | 75.8.a.c | 1 | ||
| 15.d | odd | 2 | 1 | 225.8.b.a | 2 | ||
| 15.e | even | 4 | 1 | 45.8.a.g | 1 | ||
| 15.e | even | 4 | 1 | 225.8.a.a | 1 | ||
| 20.e | even | 4 | 1 | 240.8.a.c | 1 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 15.8.a.a | ✓ | 1 | 5.c | odd | 4 | 1 | |
| 45.8.a.g | 1 | 15.e | even | 4 | 1 | ||
| 75.8.a.c | 1 | 5.c | odd | 4 | 1 | ||
| 75.8.b.a | 2 | 1.a | even | 1 | 1 | trivial | |
| 75.8.b.a | 2 | 5.b | even | 2 | 1 | inner | |
| 225.8.a.a | 1 | 15.e | even | 4 | 1 | ||
| 225.8.b.a | 2 | 3.b | odd | 2 | 1 | ||
| 225.8.b.a | 2 | 15.d | odd | 2 | 1 | ||
| 240.8.a.c | 1 | 20.e | even | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{2} + 484 \)
acting on \(S_{8}^{\mathrm{new}}(75, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} + 484 \)
|
| $3$ |
\( T^{2} + 729 \)
|
| $5$ |
\( T^{2} \)
|
| $7$ |
\( T^{2} + 176400 \)
|
| $11$ |
\( (T + 2944)^{2} \)
|
| $13$ |
\( T^{2} + 121132036 \)
|
| $17$ |
\( T^{2} + 273770116 \)
|
| $19$ |
\( (T - 25364)^{2} \)
|
| $23$ |
\( T^{2} + 34574400 \)
|
| $29$ |
\( (T + 163042)^{2} \)
|
| $31$ |
\( (T + 201600)^{2} \)
|
| $37$ |
\( T^{2} + 14527480900 \)
|
| $41$ |
\( (T + 115910)^{2} \)
|
| $43$ |
\( T^{2} + 366645904 \)
|
| $47$ |
\( T^{2} + 707307912256 \)
|
| $53$ |
\( T^{2} + 251893572100 \)
|
| $59$ |
\( (T - 1586176)^{2} \)
|
| $61$ |
\( (T + 372962)^{2} \)
|
| $67$ |
\( T^{2} + 20803122369936 \)
|
| $71$ |
\( (T - 1512832)^{2} \)
|
| $73$ |
\( T^{2} + 2319254868100 \)
|
| $79$ |
\( (T + 4231920)^{2} \)
|
| $83$ |
\( T^{2} + 3438072473616 \)
|
| $89$ |
\( (T - 6888174)^{2} \)
|
| $97$ |
\( T^{2} + 13690251601156 \)
|
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