Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1210,3,Mod(241,1210)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1210, base_ring=CyclotomicField(2))
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("1210.241");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1210 = 2 \cdot 5 \cdot 11^{2} \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1210.d (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: | \(32.9701119876\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\sqrt{-2}, \sqrt{5})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} + 6x^{2} + 4 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2^{2} \) |
| 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
\( x^{4} + 6x^{2} + 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{3} + 4\nu ) / 2 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} + 8\nu ) / 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{2} + 3 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} - \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} - 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( -2\beta_{2} + 4\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1210\mathbb{Z}\right)^\times\).
| \(n\) | \(727\) | \(1091\) |
| \(\chi(n)\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 241.1 |
|
− | 1.41421i | −3.23607 | −2.00000 | 2.23607 | 4.57649i | − | 0.333851i | 2.82843i | 1.47214 | − | 3.16228i | |||||||||||||||||||||||||||
| 241.2 | − | 1.41421i | 1.23607 | −2.00000 | −2.23607 | − | 1.74806i | 5.99070i | 2.82843i | −7.47214 | 3.16228i | |||||||||||||||||||||||||||||
| 241.3 | 1.41421i | −3.23607 | −2.00000 | 2.23607 | − | 4.57649i | 0.333851i | − | 2.82843i | 1.47214 | 3.16228i | |||||||||||||||||||||||||||||
| 241.4 | 1.41421i | 1.23607 | −2.00000 | −2.23607 | 1.74806i | − | 5.99070i | − | 2.82843i | −7.47214 | − | 3.16228i | ||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 11.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 1210.3.d.a | ✓ | 4 |
| 11.b | odd | 2 | 1 | inner | 1210.3.d.a | ✓ | 4 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1210.3.d.a | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 1210.3.d.a | ✓ | 4 | 11.b | odd | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{2} + 2T_{3} - 4 \)
acting on \(S_{3}^{\mathrm{new}}(1210, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 2)^{2} \)
|
| $3$ |
\( (T^{2} + 2 T - 4)^{2} \)
|
| $5$ |
\( (T^{2} - 5)^{2} \)
|
| $7$ |
\( T^{4} + 36T^{2} + 4 \)
|
| $11$ |
\( T^{4} \)
|
| $13$ |
\( T^{4} + 216T^{2} + 7744 \)
|
| $17$ |
\( T^{4} + 216T^{2} + 5184 \)
|
| $19$ |
\( (T^{2} + 72)^{2} \)
|
| $23$ |
\( (T^{2} - 14 T + 44)^{2} \)
|
| $29$ |
\( T^{4} + 916 T^{2} + 68644 \)
|
| $31$ |
\( (T^{2} - 24 T + 64)^{2} \)
|
| $37$ |
\( (T^{2} + 4 T - 1276)^{2} \)
|
| $41$ |
\( T^{4} + 5796 T^{2} + 8282884 \)
|
| $43$ |
\( T^{4} + 1396T^{2} + 484 \)
|
| $47$ |
\( (T^{2} - 46 T - 2116)^{2} \)
|
| $53$ |
\( (T^{2} + 36 T - 3596)^{2} \)
|
| $59$ |
\( (T^{2} - 28 T - 3184)^{2} \)
|
| $61$ |
\( T^{4} + 17604 T^{2} + 64288324 \)
|
| $67$ |
\( (T^{2} + 30 T - 180)^{2} \)
|
| $71$ |
\( (T^{2} - 48 T - 1424)^{2} \)
|
| $73$ |
\( T^{4} + 20120 T^{2} + 83905600 \)
|
| $79$ |
\( T^{4} + 26256 T^{2} + 54051904 \)
|
| $83$ |
\( T^{4} + 9780 T^{2} + 22944100 \)
|
| $89$ |
\( (T^{2} + 240 T + 11520)^{2} \)
|
| $97$ |
\( (T^{2} - 10580)^{2} \)
|
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