Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [60,4,Mod(7,60)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(60, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([2, 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("60.7");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 60 = 2^{2} \cdot 3 \cdot 5 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 60.j (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: | \(3.54011460034\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Relative dimension: | \(4\) over \(\Q(i)\) |
| Coefficient field: | 8.0.157351936.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} + x^{4} + 16 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{4} \) |
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} + x^{4} + 16 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 2\nu^{4} + 1 ) / 3 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{5} + 11\nu ) / 6 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{6} + 5\nu^{2} ) / 12 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{5} + \nu ) / 6 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -\nu^{7} + 7\nu^{3} ) / 24 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -\nu^{6} + 3\nu^{2} ) / 4 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 5\nu^{7} + 13\nu^{3} ) / 24 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{4} + \beta_{2} ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{6} + 3\beta_{3} ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( \beta_{7} + 5\beta_{5} ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 3\beta _1 - 1 ) / 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -11\beta_{4} + \beta_{2} ) / 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( -5\beta_{6} + 9\beta_{3} ) / 2 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 7\beta_{7} - 13\beta_{5} ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/60\mathbb{Z}\right)^\times\).
| \(n\) | \(31\) | \(37\) | \(41\) |
| \(\chi(n)\) | \(-1\) | \(\beta_{3}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 7.1 |
|
−2.57794 | − | 1.16372i | −2.12132 | − | 2.12132i | 5.29150 | + | 6.00000i | 2.61249 | + | 10.8708i | 3.00000 | + | 7.93725i | 17.4714 | − | 17.4714i | −6.65882 | − | 21.6255i | 9.00000i | 5.91580 | − | 31.0645i | ||||||||||||||||||||||||||
| 7.2 | −1.16372 | − | 2.57794i | 2.12132 | + | 2.12132i | −5.29150 | + | 6.00000i | 2.61249 | + | 10.8708i | 3.00000 | − | 7.93725i | −17.4714 | + | 17.4714i | 21.6255 | + | 6.65882i | 9.00000i | 24.9841 | − | 19.3854i | |||||||||||||||||||||||||||
| 7.3 | 1.16372 | + | 2.57794i | −2.12132 | − | 2.12132i | −5.29150 | + | 6.00000i | −8.61249 | + | 7.12917i | 3.00000 | − | 7.93725i | −8.98612 | + | 8.98612i | −21.6255 | − | 6.65882i | 9.00000i | −28.4011 | − | 13.9061i | |||||||||||||||||||||||||||
| 7.4 | 2.57794 | + | 1.16372i | 2.12132 | + | 2.12132i | 5.29150 | + | 6.00000i | −8.61249 | + | 7.12917i | 3.00000 | + | 7.93725i | 8.98612 | − | 8.98612i | 6.65882 | + | 21.6255i | 9.00000i | −30.4988 | + | 8.35600i | |||||||||||||||||||||||||||
| 43.1 | −2.57794 | + | 1.16372i | −2.12132 | + | 2.12132i | 5.29150 | − | 6.00000i | 2.61249 | − | 10.8708i | 3.00000 | − | 7.93725i | 17.4714 | + | 17.4714i | −6.65882 | + | 21.6255i | − | 9.00000i | 5.91580 | + | 31.0645i | ||||||||||||||||||||||||||
| 43.2 | −1.16372 | + | 2.57794i | 2.12132 | − | 2.12132i | −5.29150 | − | 6.00000i | 2.61249 | − | 10.8708i | 3.00000 | + | 7.93725i | −17.4714 | − | 17.4714i | 21.6255 | − | 6.65882i | − | 9.00000i | 24.9841 | + | 19.3854i | ||||||||||||||||||||||||||
| 43.3 | 1.16372 | − | 2.57794i | −2.12132 | + | 2.12132i | −5.29150 | − | 6.00000i | −8.61249 | − | 7.12917i | 3.00000 | + | 7.93725i | −8.98612 | − | 8.98612i | −21.6255 | + | 6.65882i | − | 9.00000i | −28.4011 | + | 13.9061i | ||||||||||||||||||||||||||
| 43.4 | 2.57794 | − | 1.16372i | 2.12132 | − | 2.12132i | 5.29150 | − | 6.00000i | −8.61249 | − | 7.12917i | 3.00000 | − | 7.93725i | 8.98612 | + | 8.98612i | 6.65882 | − | 21.6255i | − | 9.00000i | −30.4988 | − | 8.35600i | ||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 4.b | odd | 2 | 1 | inner |
| 5.c | odd | 4 | 1 | inner |
| 20.e | even | 4 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 60.4.j.a | ✓ | 8 |
| 3.b | odd | 2 | 1 | 180.4.k.d | 8 | ||
| 4.b | odd | 2 | 1 | inner | 60.4.j.a | ✓ | 8 |
| 5.c | odd | 4 | 1 | inner | 60.4.j.a | ✓ | 8 |
| 12.b | even | 2 | 1 | 180.4.k.d | 8 | ||
| 15.e | even | 4 | 1 | 180.4.k.d | 8 | ||
| 20.e | even | 4 | 1 | inner | 60.4.j.a | ✓ | 8 |
| 60.l | odd | 4 | 1 | 180.4.k.d | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 60.4.j.a | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 60.4.j.a | ✓ | 8 | 4.b | odd | 2 | 1 | inner |
| 60.4.j.a | ✓ | 8 | 5.c | odd | 4 | 1 | inner |
| 60.4.j.a | ✓ | 8 | 20.e | even | 4 | 1 | inner |
| 180.4.k.d | 8 | 3.b | odd | 2 | 1 | ||
| 180.4.k.d | 8 | 12.b | even | 2 | 1 | ||
| 180.4.k.d | 8 | 15.e | even | 4 | 1 | ||
| 180.4.k.d | 8 | 60.l | odd | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{8} + 398792T_{7}^{4} + 9721171216 \)
acting on \(S_{4}^{\mathrm{new}}(60, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} + 16T^{4} + 4096 \)
|
| $3$ |
\( (T^{4} + 81)^{2} \)
|
| $5$ |
\( (T^{4} + 12 T^{3} + \cdots + 15625)^{2} \)
|
| $7$ |
\( T^{8} + \cdots + 9721171216 \)
|
| $11$ |
\( (T^{4} + 2116 T^{2} + 917764)^{2} \)
|
| $13$ |
\( (T^{4} - 160 T^{3} + \cdots + 8690704)^{2} \)
|
| $17$ |
\( (T^{4} - 120 T^{3} + \cdots + 1210000)^{2} \)
|
| $19$ |
\( (T^{4} - 10376 T^{2} + 2521744)^{2} \)
|
| $23$ |
\( T^{8} + \cdots + 15\!\cdots\!96 \)
|
| $29$ |
\( (T^{4} + 46468 T^{2} + 122456356)^{2} \)
|
| $31$ |
\( (T^{4} + 92016 T^{2} + 1935296064)^{2} \)
|
| $37$ |
\( (T^{4} + 320 T^{3} + \cdots + 313006864)^{2} \)
|
| $41$ |
\( (T^{2} - 276 T + 6444)^{4} \)
|
| $43$ |
\( T^{8} + \cdots + 58\!\cdots\!36 \)
|
| $47$ |
\( T^{8} + \cdots + 14\!\cdots\!00 \)
|
| $53$ |
\( (T^{4} + 600 T^{3} + \cdots + 40968998464)^{2} \)
|
| $59$ |
\( (T^{4} - 423396 T^{2} + 2342753604)^{2} \)
|
| $61$ |
\( (T^{2} + 68 T - 313844)^{4} \)
|
| $67$ |
\( T^{8} + \cdots + 53\!\cdots\!16 \)
|
| $71$ |
\( (T^{4} + 460000 T^{2} + 32256160000)^{2} \)
|
| $73$ |
\( (T^{4} - 220 T^{3} + \cdots + 101566084)^{2} \)
|
| $79$ |
\( (T^{4} - 449936 T^{2} + 44365839424)^{2} \)
|
| $83$ |
\( T^{8} + \cdots + 42\!\cdots\!76 \)
|
| $89$ |
\( (T^{4} + 547912 T^{2} + 36529936)^{2} \)
|
| $97$ |
\( (T^{4} + 20 T^{3} + \cdots + 258502084)^{2} \)
|
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