Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [30,3,Mod(7,30)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(30, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([0, 1]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("30.7");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 30 = 2 \cdot 3 \cdot 5 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 30.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: | \(0.817440793081\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(i)\) |
| Coefficient field: | \(\Q(i, \sqrt{6})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} + 9 \)
|
| 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 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
\( x^{4} + 9 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} ) / 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} ) / 3 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( 3\beta_{2} \)
|
| \(\nu^{3}\) | \(=\) |
\( 3\beta_{3} \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/30\mathbb{Z}\right)^\times\).
| \(n\) | \(7\) | \(11\) |
| \(\chi(n)\) | \(-\beta_{2}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 7.1 |
|
1.00000 | + | 1.00000i | −1.22474 | + | 1.22474i | 2.00000i | 4.89898 | − | 1.00000i | −2.44949 | −8.89898 | − | 8.89898i | −2.00000 | + | 2.00000i | − | 3.00000i | 5.89898 | + | 3.89898i | |||||||||||||||||
| 7.2 | 1.00000 | + | 1.00000i | 1.22474 | − | 1.22474i | 2.00000i | −4.89898 | − | 1.00000i | 2.44949 | 0.898979 | + | 0.898979i | −2.00000 | + | 2.00000i | − | 3.00000i | −3.89898 | − | 5.89898i | ||||||||||||||||||
| 13.1 | 1.00000 | − | 1.00000i | −1.22474 | − | 1.22474i | − | 2.00000i | 4.89898 | + | 1.00000i | −2.44949 | −8.89898 | + | 8.89898i | −2.00000 | − | 2.00000i | 3.00000i | 5.89898 | − | 3.89898i | ||||||||||||||||||
| 13.2 | 1.00000 | − | 1.00000i | 1.22474 | + | 1.22474i | − | 2.00000i | −4.89898 | + | 1.00000i | 2.44949 | 0.898979 | − | 0.898979i | −2.00000 | − | 2.00000i | 3.00000i | −3.89898 | + | 5.89898i | ||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.c | odd | 4 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 30.3.f.a | ✓ | 4 |
| 3.b | odd | 2 | 1 | 90.3.g.d | 4 | ||
| 4.b | odd | 2 | 1 | 240.3.bg.b | 4 | ||
| 5.b | even | 2 | 1 | 150.3.f.b | 4 | ||
| 5.c | odd | 4 | 1 | inner | 30.3.f.a | ✓ | 4 |
| 5.c | odd | 4 | 1 | 150.3.f.b | 4 | ||
| 8.b | even | 2 | 1 | 960.3.bg.e | 4 | ||
| 8.d | odd | 2 | 1 | 960.3.bg.g | 4 | ||
| 12.b | even | 2 | 1 | 720.3.bh.i | 4 | ||
| 15.d | odd | 2 | 1 | 450.3.g.j | 4 | ||
| 15.e | even | 4 | 1 | 90.3.g.d | 4 | ||
| 15.e | even | 4 | 1 | 450.3.g.j | 4 | ||
| 20.d | odd | 2 | 1 | 1200.3.bg.d | 4 | ||
| 20.e | even | 4 | 1 | 240.3.bg.b | 4 | ||
| 20.e | even | 4 | 1 | 1200.3.bg.d | 4 | ||
| 40.i | odd | 4 | 1 | 960.3.bg.e | 4 | ||
| 40.k | even | 4 | 1 | 960.3.bg.g | 4 | ||
| 60.l | odd | 4 | 1 | 720.3.bh.i | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 30.3.f.a | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 30.3.f.a | ✓ | 4 | 5.c | odd | 4 | 1 | inner |
| 90.3.g.d | 4 | 3.b | odd | 2 | 1 | ||
| 90.3.g.d | 4 | 15.e | even | 4 | 1 | ||
| 150.3.f.b | 4 | 5.b | even | 2 | 1 | ||
| 150.3.f.b | 4 | 5.c | odd | 4 | 1 | ||
| 240.3.bg.b | 4 | 4.b | odd | 2 | 1 | ||
| 240.3.bg.b | 4 | 20.e | even | 4 | 1 | ||
| 450.3.g.j | 4 | 15.d | odd | 2 | 1 | ||
| 450.3.g.j | 4 | 15.e | even | 4 | 1 | ||
| 720.3.bh.i | 4 | 12.b | even | 2 | 1 | ||
| 720.3.bh.i | 4 | 60.l | odd | 4 | 1 | ||
| 960.3.bg.e | 4 | 8.b | even | 2 | 1 | ||
| 960.3.bg.e | 4 | 40.i | odd | 4 | 1 | ||
| 960.3.bg.g | 4 | 8.d | odd | 2 | 1 | ||
| 960.3.bg.g | 4 | 40.k | even | 4 | 1 | ||
| 1200.3.bg.d | 4 | 20.d | odd | 2 | 1 | ||
| 1200.3.bg.d | 4 | 20.e | even | 4 | 1 | ||
Hecke kernels
This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(30, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} - 2 T + 2)^{2} \)
|
| $3$ |
\( T^{4} + 9 \)
|
| $5$ |
\( T^{4} - 46T^{2} + 625 \)
|
| $7$ |
\( T^{4} + 16 T^{3} + \cdots + 256 \)
|
| $11$ |
\( (T^{2} + 8 T - 80)^{2} \)
|
| $13$ |
\( T^{4} - 12 T^{3} + \cdots + 30276 \)
|
| $17$ |
\( T^{4} - 44 T^{3} + \cdots + 37636 \)
|
| $19$ |
\( T^{4} + 704 T^{2} + 25600 \)
|
| $23$ |
\( T^{4} + 16 T^{3} + \cdots + 160000 \)
|
| $29$ |
\( T^{4} + 704 T^{2} + 25600 \)
|
| $31$ |
\( (T^{2} + 40 T + 16)^{2} \)
|
| $37$ |
\( (T^{2} + 54 T + 1458)^{2} \)
|
| $41$ |
\( (T^{2} - 16 T - 32)^{2} \)
|
| $43$ |
\( T^{4} - 48 T^{3} + \cdots + 831744 \)
|
| $47$ |
\( T^{4} - 96 T^{3} + \cdots + 518400 \)
|
| $53$ |
\( T^{4} - 100 T^{3} + \cdots + 6959044 \)
|
| $59$ |
\( (T^{2} + 400)^{2} \)
|
| $61$ |
\( (T^{2} + 48 T - 960)^{2} \)
|
| $67$ |
\( T^{4} + 16 T^{3} + \cdots + 14868736 \)
|
| $71$ |
\( (T^{2} - 32 T + 160)^{2} \)
|
| $73$ |
\( T^{4} + 188 T^{3} + \cdots + 17859076 \)
|
| $79$ |
\( T^{4} + 32736 T^{2} + 258566400 \)
|
| $83$ |
\( T^{4} - 192 T^{3} + \cdots + 5089536 \)
|
| $89$ |
\( T^{4} + 17216 T^{2} + 47334400 \)
|
| $97$ |
\( T^{4} - 132 T^{3} + \cdots + 6874884 \)
|
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