Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [260,4,Mod(177,260)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(260, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([0, 1, 1]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("260.177");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 260 = 2^{2} \cdot 5 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 260.r (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: | \(15.3404966015\) |
| 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: | 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 \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/260\mathbb{Z}\right)^\times\).
| \(n\) | \(41\) | \(131\) | \(157\) |
| \(\chi(n)\) | \(i\) | \(1\) | \(i\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 177.1 |
|
0 | −1.00000 | + | 1.00000i | 0 | 10.0000 | − | 5.00000i | 0 | 10.0000 | 0 | 25.0000i | 0 | ||||||||||||||||||||
| 213.1 | 0 | −1.00000 | − | 1.00000i | 0 | 10.0000 | + | 5.00000i | 0 | 10.0000 | 0 | − | 25.0000i | 0 | ||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 65.f | even | 4 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 260.4.r.a | yes | 2 |
| 5.c | odd | 4 | 1 | 260.4.m.a | ✓ | 2 | |
| 13.d | odd | 4 | 1 | 260.4.m.a | ✓ | 2 | |
| 65.f | even | 4 | 1 | inner | 260.4.r.a | yes | 2 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 260.4.m.a | ✓ | 2 | 5.c | odd | 4 | 1 | |
| 260.4.m.a | ✓ | 2 | 13.d | odd | 4 | 1 | |
| 260.4.r.a | yes | 2 | 1.a | even | 1 | 1 | trivial |
| 260.4.r.a | yes | 2 | 65.f | even | 4 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{2} + 2T_{3} + 2 \)
acting on \(S_{4}^{\mathrm{new}}(260, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} \)
|
| $3$ |
\( T^{2} + 2T + 2 \)
|
| $5$ |
\( T^{2} - 20T + 125 \)
|
| $7$ |
\( (T - 10)^{2} \)
|
| $11$ |
\( T^{2} - 2T + 2 \)
|
| $13$ |
\( T^{2} - 52T + 2197 \)
|
| $17$ |
\( T^{2} + 130T + 8450 \)
|
| $19$ |
\( T^{2} - 150T + 11250 \)
|
| $23$ |
\( T^{2} - 118T + 6962 \)
|
| $29$ |
\( T^{2} + 23104 \)
|
| $31$ |
\( T^{2} - 470T + 110450 \)
|
| $37$ |
\( (T - 56)^{2} \)
|
| $41$ |
\( T^{2} - 482T + 116162 \)
|
| $43$ |
\( T^{2} - 398T + 79202 \)
|
| $47$ |
\( (T - 370)^{2} \)
|
| $53$ |
\( T^{2} - 682T + 232562 \)
|
| $59$ |
\( T^{2} + 174T + 15138 \)
|
| $61$ |
\( (T + 646)^{2} \)
|
| $67$ |
\( T^{2} + 646416 \)
|
| $71$ |
\( T^{2} - 1118 T + 624962 \)
|
| $73$ |
\( T^{2} + 396900 \)
|
| $79$ |
\( T^{2} + 925444 \)
|
| $83$ |
\( (T + 558)^{2} \)
|
| $89$ |
\( T^{2} + 1210 T + 732050 \)
|
| $97$ |
\( T^{2} + 42436 \)
|
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