Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [210,4,Mod(169,210)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(210, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1, 0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("210.169");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 210 = 2 \cdot 3 \cdot 5 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 210.g (of order \(2\), degree \(1\), 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: | \(12.3904011012\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| 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: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{3} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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
\( x^{4} + 9 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{2} ) / 3 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 2\nu^{3} + 6\nu ) / 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -2\nu^{3} + 6\nu ) / 3 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} + \beta_{2} ) / 4 \)
|
| \(\nu^{2}\) | \(=\) |
\( 3\beta_1 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -3\beta_{3} + 3\beta_{2} ) / 4 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/210\mathbb{Z}\right)^\times\).
| \(n\) | \(31\) | \(71\) | \(127\) |
| \(\chi(n)\) | \(1\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 169.1 |
|
− | 2.00000i | − | 3.00000i | −4.00000 | −10.7980 | − | 2.89898i | −6.00000 | − | 7.00000i | 8.00000i | −9.00000 | −5.79796 | + | 21.5959i | |||||||||||||||||||||||
| 169.2 | − | 2.00000i | − | 3.00000i | −4.00000 | 8.79796 | + | 6.89898i | −6.00000 | − | 7.00000i | 8.00000i | −9.00000 | 13.7980 | − | 17.5959i | ||||||||||||||||||||||||
| 169.3 | 2.00000i | 3.00000i | −4.00000 | −10.7980 | + | 2.89898i | −6.00000 | 7.00000i | − | 8.00000i | −9.00000 | −5.79796 | − | 21.5959i | ||||||||||||||||||||||||||
| 169.4 | 2.00000i | 3.00000i | −4.00000 | 8.79796 | − | 6.89898i | −6.00000 | 7.00000i | − | 8.00000i | −9.00000 | 13.7980 | + | 17.5959i | ||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 210.4.g.a | ✓ | 4 |
| 3.b | odd | 2 | 1 | 630.4.g.e | 4 | ||
| 5.b | even | 2 | 1 | inner | 210.4.g.a | ✓ | 4 |
| 5.c | odd | 4 | 1 | 1050.4.a.bc | 2 | ||
| 5.c | odd | 4 | 1 | 1050.4.a.bg | 2 | ||
| 15.d | odd | 2 | 1 | 630.4.g.e | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 210.4.g.a | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 210.4.g.a | ✓ | 4 | 5.b | even | 2 | 1 | inner |
| 630.4.g.e | 4 | 3.b | odd | 2 | 1 | ||
| 630.4.g.e | 4 | 15.d | odd | 2 | 1 | ||
| 1050.4.a.bc | 2 | 5.c | odd | 4 | 1 | ||
| 1050.4.a.bg | 2 | 5.c | odd | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{11}^{2} - 12T_{11} - 348 \)
acting on \(S_{4}^{\mathrm{new}}(210, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 4)^{2} \)
|
| $3$ |
\( (T^{2} + 9)^{2} \)
|
| $5$ |
\( T^{4} + 4 T^{3} + \cdots + 15625 \)
|
| $7$ |
\( (T^{2} + 49)^{2} \)
|
| $11$ |
\( (T^{2} - 12 T - 348)^{2} \)
|
| $13$ |
\( T^{4} + 8072 T^{2} + 929296 \)
|
| $17$ |
\( T^{4} + 10112 T^{2} + 3444736 \)
|
| $19$ |
\( (T^{2} - 4 T - 92)^{2} \)
|
| $23$ |
\( T^{4} + 34496 T^{2} + 231040000 \)
|
| $29$ |
\( (T^{2} + 20 T + 4)^{2} \)
|
| $31$ |
\( (T^{2} + 292 T + 20932)^{2} \)
|
| $37$ |
\( T^{4} + 80192 T^{2} + 9634816 \)
|
| $41$ |
\( (T^{2} + 132 T - 1788)^{2} \)
|
| $43$ |
\( T^{4} + \cdots + 13769614336 \)
|
| $47$ |
\( T^{4} + \cdots + 2799679744 \)
|
| $53$ |
\( T^{4} + \cdots + 33046876944 \)
|
| $59$ |
\( (T^{2} + 224 T - 380672)^{2} \)
|
| $61$ |
\( (T^{2} + 12 T - 161340)^{2} \)
|
| $67$ |
\( T^{4} + \cdots + 4301261056 \)
|
| $71$ |
\( (T^{2} + 156 T - 224412)^{2} \)
|
| $73$ |
\( T^{4} + 92744 T^{2} + 76387600 \)
|
| $79$ |
\( (T^{2} + 376 T + 29200)^{2} \)
|
| $83$ |
\( T^{4} + \cdots + 904720564224 \)
|
| $89$ |
\( (T^{2} + 548 T - 1172540)^{2} \)
|
| $97$ |
\( T^{4} + \cdots + 252506250000 \)
|
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