Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [35,4,Mod(13,35)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(35, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([3, 2]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("35.13");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 35 = 5 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 35.f (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: | \(2.06506685020\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(i)\) |
| Coefficient field: | \(\Q(i, \sqrt{5})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} + 3x^{2} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| 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
\( x^{4} + 3x^{2} + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu^{2} + \nu + 1 \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{3} + 2\nu \)
|
| \(\beta_{3}\) | \(=\) |
\( -\nu^{2} + \nu - 1 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} + \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -\beta_{3} + \beta _1 - 2 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{3} + \beta_{2} - \beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/35\mathbb{Z}\right)^\times\).
| \(n\) | \(22\) | \(31\) |
| \(\chi(n)\) | \(\beta_{2}\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 13.1 |
|
−2.00000 | + | 2.00000i | −2.23607 | + | 2.23607i | 0 | −6.70820 | − | 8.94427i | − | 8.94427i | −18.3262 | + | 2.67376i | −16.0000 | − | 16.0000i | 17.0000i | 31.3050 | + | 4.47214i | |||||||||||||||||
| 13.2 | −2.00000 | + | 2.00000i | 2.23607 | − | 2.23607i | 0 | 6.70820 | + | 8.94427i | 8.94427i | −2.67376 | + | 18.3262i | −16.0000 | − | 16.0000i | 17.0000i | −31.3050 | − | 4.47214i | |||||||||||||||||||
| 27.1 | −2.00000 | − | 2.00000i | −2.23607 | − | 2.23607i | 0 | −6.70820 | + | 8.94427i | 8.94427i | −18.3262 | − | 2.67376i | −16.0000 | + | 16.0000i | − | 17.0000i | 31.3050 | − | 4.47214i | ||||||||||||||||||
| 27.2 | −2.00000 | − | 2.00000i | 2.23607 | + | 2.23607i | 0 | 6.70820 | − | 8.94427i | − | 8.94427i | −2.67376 | − | 18.3262i | −16.0000 | + | 16.0000i | − | 17.0000i | −31.3050 | + | 4.47214i | |||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.c | odd | 4 | 1 | inner |
| 7.b | odd | 2 | 1 | inner |
| 35.f | even | 4 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 35.4.f.a | ✓ | 4 |
| 5.b | even | 2 | 1 | 175.4.f.d | 4 | ||
| 5.c | odd | 4 | 1 | inner | 35.4.f.a | ✓ | 4 |
| 5.c | odd | 4 | 1 | 175.4.f.d | 4 | ||
| 7.b | odd | 2 | 1 | inner | 35.4.f.a | ✓ | 4 |
| 7.c | even | 3 | 2 | 245.4.l.a | 8 | ||
| 7.d | odd | 6 | 2 | 245.4.l.a | 8 | ||
| 35.c | odd | 2 | 1 | 175.4.f.d | 4 | ||
| 35.f | even | 4 | 1 | inner | 35.4.f.a | ✓ | 4 |
| 35.f | even | 4 | 1 | 175.4.f.d | 4 | ||
| 35.k | even | 12 | 2 | 245.4.l.a | 8 | ||
| 35.l | odd | 12 | 2 | 245.4.l.a | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 35.4.f.a | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 35.4.f.a | ✓ | 4 | 5.c | odd | 4 | 1 | inner |
| 35.4.f.a | ✓ | 4 | 7.b | odd | 2 | 1 | inner |
| 35.4.f.a | ✓ | 4 | 35.f | even | 4 | 1 | inner |
| 175.4.f.d | 4 | 5.b | even | 2 | 1 | ||
| 175.4.f.d | 4 | 5.c | odd | 4 | 1 | ||
| 175.4.f.d | 4 | 35.c | odd | 2 | 1 | ||
| 175.4.f.d | 4 | 35.f | even | 4 | 1 | ||
| 245.4.l.a | 8 | 7.c | even | 3 | 2 | ||
| 245.4.l.a | 8 | 7.d | odd | 6 | 2 | ||
| 245.4.l.a | 8 | 35.k | even | 12 | 2 | ||
| 245.4.l.a | 8 | 35.l | odd | 12 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{2} + 4T_{2} + 8 \)
acting on \(S_{4}^{\mathrm{new}}(35, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 4 T + 8)^{2} \)
|
| $3$ |
\( T^{4} + 100 \)
|
| $5$ |
\( T^{4} + 70T^{2} + 15625 \)
|
| $7$ |
\( T^{4} + 42 T^{3} + \cdots + 117649 \)
|
| $11$ |
\( (T - 24)^{4} \)
|
| $13$ |
\( T^{4} + 13032100 \)
|
| $17$ |
\( T^{4} + 282576100 \)
|
| $19$ |
\( (T^{2} - 6480)^{2} \)
|
| $23$ |
\( (T^{2} + 118 T + 6962)^{2} \)
|
| $29$ |
\( (T^{2} + 2704)^{2} \)
|
| $31$ |
\( (T^{2} + 3380)^{2} \)
|
| $37$ |
\( (T^{2} - 186 T + 17298)^{2} \)
|
| $41$ |
\( (T^{2} + 720)^{2} \)
|
| $43$ |
\( (T^{2} - 442 T + 97682)^{2} \)
|
| $47$ |
\( T^{4} + 27984100 \)
|
| $53$ |
\( (T^{2} + 174 T + 15138)^{2} \)
|
| $59$ |
\( (T^{2} - 176720)^{2} \)
|
| $61$ |
\( (T^{2} + 115520)^{2} \)
|
| $67$ |
\( (T^{2} + 554 T + 153458)^{2} \)
|
| $71$ |
\( (T + 916)^{4} \)
|
| $73$ |
\( T^{4} + 150062500 \)
|
| $79$ |
\( (T^{2} + 228484)^{2} \)
|
| $83$ |
\( T^{4} + 183603680100 \)
|
| $89$ |
\( (T^{2} - 20)^{2} \)
|
| $97$ |
\( T^{4} + 3387108968100 \)
|
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