Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [3721,2,Mod(1,3721)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3721, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("3721.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 3721 = 61^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 3721.a (trivial) |
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: | yes |
| Analytic conductor: | \(29.7123345921\) |
| Analytic rank: | \(1\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.4.1957.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 4x^{2} - x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 61) |
| Fricke sign: | \(+1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-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
\( x^{4} - 4x^{2} - x + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{3} - 4\nu - 1 \)
|
| \(\beta_{3}\) | \(=\) |
\( -\nu^{3} + \nu^{2} + 3\nu - 1 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + \beta_{2} + \beta _1 + 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{2} + 4\beta _1 + 1 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
−2.06150 | −0.326637 | 2.24978 | 2.09133 | 0.673363 | 0.485084 | −0.514916 | −2.89331 | −4.31128 | ||||||||||||||||||||||||||||||
| 1.2 | −0.396339 | −0.716159 | −1.84292 | −3.64985 | 0.283841 | 2.52310 | 1.52310 | −2.48712 | 1.44658 | |||||||||||||||||||||||||||||||
| 1.3 | 0.693822 | −3.26608 | −1.51861 | 3.18876 | −2.26608 | −1.44129 | −2.44129 | 7.66727 | 2.21243 | |||||||||||||||||||||||||||||||
| 1.4 | 1.76401 | 1.30887 | 1.11175 | 0.369762 | 2.30887 | −0.566889 | −1.56689 | −1.28685 | 0.652266 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(61\) | \( +1 \) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 3721.2.a.e | 4 | |
| 61.b | even | 2 | 1 | 3721.2.a.d | 4 | ||
| 61.c | even | 3 | 2 | 61.2.c.a | ✓ | 8 | |
| 183.k | odd | 6 | 2 | 549.2.e.g | 8 | ||
| 244.j | odd | 6 | 2 | 976.2.i.g | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 61.2.c.a | ✓ | 8 | 61.c | even | 3 | 2 | |
| 549.2.e.g | 8 | 183.k | odd | 6 | 2 | ||
| 976.2.i.g | 8 | 244.j | odd | 6 | 2 | ||
| 3721.2.a.d | 4 | 61.b | even | 2 | 1 | ||
| 3721.2.a.e | 4 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} - 4T_{2}^{2} + T_{2} + 1 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3721))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} - 4T^{2} + T + 1 \)
|
| $3$ |
\( T^{4} + 3 T^{3} + \cdots - 1 \)
|
| $5$ |
\( T^{4} - 2 T^{3} + \cdots - 9 \)
|
| $7$ |
\( T^{4} - T^{3} - 4 T^{2} + 1 \)
|
| $11$ |
\( T^{4} + T^{3} - 12 T^{2} + \cdots + 1 \)
|
| $13$ |
\( T^{4} + 5 T^{3} + \cdots + 1 \)
|
| $17$ |
\( T^{4} + 4 T^{3} + \cdots - 61 \)
|
| $19$ |
\( T^{4} + T^{3} + \cdots + 259 \)
|
| $23$ |
\( T^{4} + 4 T^{3} + \cdots + 489 \)
|
| $29$ |
\( T^{4} - 16 T^{3} + \cdots - 849 \)
|
| $31$ |
\( T^{4} + 11 T^{3} + \cdots - 267 \)
|
| $37$ |
\( T^{4} - 2 T^{3} + \cdots - 513 \)
|
| $41$ |
\( T^{4} - 2 T^{3} + \cdots - 49 \)
|
| $43$ |
\( T^{4} + 10 T^{3} + \cdots + 43 \)
|
| $47$ |
\( T^{4} - 131 T^{2} + \cdots + 1757 \)
|
| $53$ |
\( T^{4} - 19 T^{3} + \cdots - 27 \)
|
| $59$ |
\( T^{4} + T^{3} + \cdots + 1869 \)
|
| $61$ |
\( T^{4} \)
|
| $67$ |
\( T^{4} - 17 T^{3} + \cdots + 2767 \)
|
| $71$ |
\( T^{4} - 3 T^{3} + \cdots + 367 \)
|
| $73$ |
\( T^{4} - 24 T^{3} + \cdots + 139 \)
|
| $79$ |
\( T^{4} - 16 T^{3} + \cdots - 927 \)
|
| $83$ |
\( T^{4} + 16 T^{3} + \cdots - 23 \)
|
| $89$ |
\( T^{4} - 4 T^{3} + \cdots - 1447 \)
|
| $97$ |
\( T^{4} + 13 T^{3} + \cdots + 10511 \)
|
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