Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [180,3,Mod(79,180)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(180, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([3, 4, 3]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("180.79");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 180 = 2^{2} \cdot 3^{2} \cdot 5 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 180.p (of order \(6\), 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: | \(4.90464475849\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{6})\) |
| Coefficient field: | \(\Q(\sqrt{-3}, \sqrt{-5})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 5x^{2} + 25 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{U}(1)[D_{6}]$ |
$q$-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
\( x^{4} - 5x^{2} + 25 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} ) / 5 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} ) / 5 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( 5\beta_{2} \)
|
| \(\nu^{3}\) | \(=\) |
\( 5\beta_{3} \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/180\mathbb{Z}\right)^\times\).
| \(n\) | \(37\) | \(91\) | \(101\) |
| \(\chi(n)\) | \(-1\) | \(-1\) | \(-\beta_{2}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 79.1 |
|
−1.00000 | + | 1.73205i | −0.936492 | + | 2.85008i | −2.00000 | − | 3.46410i | 2.50000 | + | 4.33013i | −4.00000 | − | 4.47214i | −4.80948 | + | 8.33026i | 8.00000 | −7.24597 | − | 5.33816i | −10.0000 | ||||||||||||||||
| 79.2 | −1.00000 | + | 1.73205i | 2.93649 | + | 0.614017i | −2.00000 | − | 3.46410i | 2.50000 | + | 4.33013i | −4.00000 | + | 4.47214i | 6.80948 | − | 11.7944i | 8.00000 | 8.24597 | + | 3.60611i | −10.0000 | |||||||||||||||||
| 139.1 | −1.00000 | − | 1.73205i | −0.936492 | − | 2.85008i | −2.00000 | + | 3.46410i | 2.50000 | − | 4.33013i | −4.00000 | + | 4.47214i | −4.80948 | − | 8.33026i | 8.00000 | −7.24597 | + | 5.33816i | −10.0000 | |||||||||||||||||
| 139.2 | −1.00000 | − | 1.73205i | 2.93649 | − | 0.614017i | −2.00000 | + | 3.46410i | 2.50000 | − | 4.33013i | −4.00000 | − | 4.47214i | 6.80948 | + | 11.7944i | 8.00000 | 8.24597 | − | 3.60611i | −10.0000 | |||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 20.d | odd | 2 | 1 | CM by \(\Q(\sqrt{-5}) \) |
| 9.c | even | 3 | 1 | inner |
| 180.p | odd | 6 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 180.3.p.a | ✓ | 4 |
| 4.b | odd | 2 | 1 | 180.3.p.b | yes | 4 | |
| 5.b | even | 2 | 1 | 180.3.p.b | yes | 4 | |
| 9.c | even | 3 | 1 | inner | 180.3.p.a | ✓ | 4 |
| 20.d | odd | 2 | 1 | CM | 180.3.p.a | ✓ | 4 |
| 36.f | odd | 6 | 1 | 180.3.p.b | yes | 4 | |
| 45.j | even | 6 | 1 | 180.3.p.b | yes | 4 | |
| 180.p | odd | 6 | 1 | inner | 180.3.p.a | ✓ | 4 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 180.3.p.a | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 180.3.p.a | ✓ | 4 | 9.c | even | 3 | 1 | inner |
| 180.3.p.a | ✓ | 4 | 20.d | odd | 2 | 1 | CM |
| 180.3.p.a | ✓ | 4 | 180.p | odd | 6 | 1 | inner |
| 180.3.p.b | yes | 4 | 4.b | odd | 2 | 1 | |
| 180.3.p.b | yes | 4 | 5.b | even | 2 | 1 | |
| 180.3.p.b | yes | 4 | 36.f | odd | 6 | 1 | |
| 180.3.p.b | yes | 4 | 45.j | even | 6 | 1 | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{4} - 4T_{7}^{3} + 147T_{7}^{2} + 524T_{7} + 17161 \)
acting on \(S_{3}^{\mathrm{new}}(180, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 2 T + 4)^{2} \)
|
| $3$ |
\( T^{4} - 4 T^{3} + \cdots + 81 \)
|
| $5$ |
\( (T^{2} - 5 T + 25)^{2} \)
|
| $7$ |
\( T^{4} - 4 T^{3} + \cdots + 17161 \)
|
| $11$ |
\( T^{4} \)
|
| $13$ |
\( T^{4} \)
|
| $17$ |
\( T^{4} \)
|
| $19$ |
\( T^{4} \)
|
| $23$ |
\( T^{4} + 44 T^{3} + \cdots + 121801 \)
|
| $29$ |
\( T^{4} + 22 T^{3} + \cdots + 4157521 \)
|
| $31$ |
\( T^{4} \)
|
| $37$ |
\( T^{4} \)
|
| $41$ |
\( T^{4} - 62 T^{3} + \cdots + 1437601 \)
|
| $43$ |
\( (T^{2} + 76 T + 5776)^{2} \)
|
| $47$ |
\( T^{4} - 4 T^{3} + \cdots + 43705321 \)
|
| $53$ |
\( T^{4} \)
|
| $59$ |
\( T^{4} \)
|
| $61$ |
\( T^{4} + 58 T^{3} + \cdots + 60824401 \)
|
| $67$ |
\( T^{4} + 116 T^{3} + \cdots + 121 \)
|
| $71$ |
\( T^{4} \)
|
| $73$ |
\( T^{4} \)
|
| $79$ |
\( T^{4} \)
|
| $83$ |
\( T^{4} - 76 T^{3} + \cdots + 221741881 \)
|
| $89$ |
\( (T^{2} - 142 T - 3599)^{2} \)
|
| $97$ |
\( T^{4} \)
|
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