Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [102,4,Mod(13,102)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(102, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([0, 1]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("102.13");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 102 = 2 \cdot 3 \cdot 17 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 102.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: | \(6.01819482059\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(i)\) |
| Coefficient field: | \(\Q(\zeta_{8})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 3^{2} \) |
| 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
| \(\beta_{1}\) | \(=\) |
\( 3\zeta_{8} \)
|
| \(\beta_{2}\) | \(=\) |
\( \zeta_{8}^{2} \)
|
| \(\beta_{3}\) | \(=\) |
\( 3\zeta_{8}^{3} \)
|
| \(\zeta_{8}\) | \(=\) |
\( ( \beta_1 ) / 3 \)
|
| \(\zeta_{8}^{2}\) | \(=\) |
\( \beta_{2} \)
|
| \(\zeta_{8}^{3}\) | \(=\) |
\( ( \beta_{3} ) / 3 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/102\mathbb{Z}\right)^\times\).
| \(n\) | \(35\) | \(37\) |
| \(\chi(n)\) | \(1\) | \(\beta_{2}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 13.1 |
|
2.00000i | −2.12132 | − | 2.12132i | −4.00000 | 5.24264 | + | 5.24264i | 4.24264 | − | 4.24264i | −7.75736 | + | 7.75736i | − | 8.00000i | 9.00000i | −10.4853 | + | 10.4853i | |||||||||||||||||||
| 13.2 | 2.00000i | 2.12132 | + | 2.12132i | −4.00000 | −3.24264 | − | 3.24264i | −4.24264 | + | 4.24264i | −16.2426 | + | 16.2426i | − | 8.00000i | 9.00000i | 6.48528 | − | 6.48528i | ||||||||||||||||||||
| 55.1 | − | 2.00000i | −2.12132 | + | 2.12132i | −4.00000 | 5.24264 | − | 5.24264i | 4.24264 | + | 4.24264i | −7.75736 | − | 7.75736i | 8.00000i | − | 9.00000i | −10.4853 | − | 10.4853i | |||||||||||||||||||
| 55.2 | − | 2.00000i | 2.12132 | − | 2.12132i | −4.00000 | −3.24264 | + | 3.24264i | −4.24264 | − | 4.24264i | −16.2426 | − | 16.2426i | 8.00000i | − | 9.00000i | 6.48528 | + | 6.48528i | |||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 17.c | even | 4 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 102.4.f.a | ✓ | 4 |
| 3.b | odd | 2 | 1 | 306.4.g.c | 4 | ||
| 17.c | even | 4 | 1 | inner | 102.4.f.a | ✓ | 4 |
| 17.d | even | 8 | 1 | 1734.4.a.i | 2 | ||
| 17.d | even | 8 | 1 | 1734.4.a.l | 2 | ||
| 51.f | odd | 4 | 1 | 306.4.g.c | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 102.4.f.a | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 102.4.f.a | ✓ | 4 | 17.c | even | 4 | 1 | inner |
| 306.4.g.c | 4 | 3.b | odd | 2 | 1 | ||
| 306.4.g.c | 4 | 51.f | odd | 4 | 1 | ||
| 1734.4.a.i | 2 | 17.d | even | 8 | 1 | ||
| 1734.4.a.l | 2 | 17.d | even | 8 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{4} - 4T_{5}^{3} + 8T_{5}^{2} + 136T_{5} + 1156 \)
acting on \(S_{4}^{\mathrm{new}}(102, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 4)^{2} \)
|
| $3$ |
\( T^{4} + 81 \)
|
| $5$ |
\( T^{4} - 4 T^{3} + \cdots + 1156 \)
|
| $7$ |
\( T^{4} + 48 T^{3} + \cdots + 63504 \)
|
| $11$ |
\( T^{4} + 128 T^{3} + \cdots + 3625216 \)
|
| $13$ |
\( (T^{2} + 88 T + 1288)^{2} \)
|
| $17$ |
\( T^{4} - 4 T^{3} + \cdots + 24137569 \)
|
| $19$ |
\( T^{4} + 27536 T^{2} + 111682624 \)
|
| $23$ |
\( T^{4} + 160 T^{3} + \cdots + 51897616 \)
|
| $29$ |
\( T^{4} - 156 T^{3} + \cdots + 741690756 \)
|
| $31$ |
\( T^{4} - 320 T^{3} + \cdots + 38987536 \)
|
| $37$ |
\( T^{4} + \cdots + 2466314244 \)
|
| $41$ |
\( (T^{2} + 74 T + 2738)^{2} \)
|
| $43$ |
\( T^{4} + 4448 T^{2} + 2715904 \)
|
| $47$ |
\( (T^{2} - 800 T + 154168)^{2} \)
|
| $53$ |
\( T^{4} + \cdots + 1599360064 \)
|
| $59$ |
\( T^{4} + \cdots + 2589588544 \)
|
| $61$ |
\( T^{4} + \cdots + 4442755716 \)
|
| $67$ |
\( (T^{2} + 48 T - 649224)^{2} \)
|
| $71$ |
\( T^{4} + \cdots + 11355886096 \)
|
| $73$ |
\( T^{4} + \cdots + 46036851844 \)
|
| $79$ |
\( T^{4} + \cdots + 958636810000 \)
|
| $83$ |
\( T^{4} + \cdots + 51159201856 \)
|
| $89$ |
\( (T^{2} + 96 T - 435744)^{2} \)
|
| $97$ |
\( T^{4} + \cdots + 22304227716 \)
|
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