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: | \(3\) |
| Coefficient field: | 3.3.148.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 3x + 1 \)
|
| 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 \(1,\beta_1,\beta_2\) 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
\( x^{3} - x^{2} - 3x + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 2 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | |||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
−2.17009 | −1.70928 | 2.70928 | −1.63090 | 3.70928 | 0.460811 | −1.53919 | −0.0783777 | 3.53919 | |||||||||||||||||||||||||||
| 1.2 | −0.311108 | 2.90321 | −1.90321 | −2.52543 | −0.903212 | 3.21432 | 1.21432 | 5.42864 | 0.785680 | ||||||||||||||||||||||||||||
| 1.3 | 1.48119 | 0.806063 | 0.193937 | 3.15633 | 1.19394 | −0.675131 | −2.67513 | −2.35026 | 4.67513 | ||||||||||||||||||||||||||||
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.c | 3 | |
| 61.b | even | 2 | 1 | 61.2.a.b | ✓ | 3 | |
| 183.d | odd | 2 | 1 | 549.2.a.g | 3 | ||
| 244.c | odd | 2 | 1 | 976.2.a.f | 3 | ||
| 305.d | even | 2 | 1 | 1525.2.a.d | 3 | ||
| 305.h | odd | 4 | 2 | 1525.2.b.b | 6 | ||
| 427.b | odd | 2 | 1 | 2989.2.a.i | 3 | ||
| 488.b | even | 2 | 1 | 3904.2.a.r | 3 | ||
| 488.g | odd | 2 | 1 | 3904.2.a.w | 3 | ||
| 671.d | odd | 2 | 1 | 7381.2.a.f | 3 | ||
| 732.e | even | 2 | 1 | 8784.2.a.bn | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 61.2.a.b | ✓ | 3 | 61.b | even | 2 | 1 | |
| 549.2.a.g | 3 | 183.d | odd | 2 | 1 | ||
| 976.2.a.f | 3 | 244.c | odd | 2 | 1 | ||
| 1525.2.a.d | 3 | 305.d | even | 2 | 1 | ||
| 1525.2.b.b | 6 | 305.h | odd | 4 | 2 | ||
| 2989.2.a.i | 3 | 427.b | odd | 2 | 1 | ||
| 3721.2.a.c | 3 | 1.a | even | 1 | 1 | trivial | |
| 3904.2.a.r | 3 | 488.b | even | 2 | 1 | ||
| 3904.2.a.w | 3 | 488.g | odd | 2 | 1 | ||
| 7381.2.a.f | 3 | 671.d | odd | 2 | 1 | ||
| 8784.2.a.bn | 3 | 732.e | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{3} + T_{2}^{2} - 3T_{2} - 1 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3721))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} + T^{2} - 3T - 1 \)
|
| $3$ |
\( T^{3} - 2 T^{2} + \cdots + 4 \)
|
| $5$ |
\( T^{3} + T^{2} - 9T - 13 \)
|
| $7$ |
\( T^{3} - 3T^{2} - T + 1 \)
|
| $11$ |
\( T^{3} + 13 T^{2} + \cdots + 67 \)
|
| $13$ |
\( T^{3} + 9 T^{2} + \cdots - 37 \)
|
| $17$ |
\( T^{3} - 2 T^{2} + \cdots - 4 \)
|
| $19$ |
\( T^{3} - 48T - 20 \)
|
| $23$ |
\( T^{3} + 5 T^{2} + \cdots - 1 \)
|
| $29$ |
\( T^{3} + 4 T^{2} + \cdots - 20 \)
|
| $31$ |
\( T^{3} - 2 T^{2} + \cdots - 116 \)
|
| $37$ |
\( T^{3} - 6 T^{2} + \cdots + 108 \)
|
| $41$ |
\( T^{3} - 3 T^{2} + \cdots + 191 \)
|
| $43$ |
\( T^{3} - 14 T^{2} + \cdots - 68 \)
|
| $47$ |
\( T^{3} + 4 T^{2} + \cdots + 16 \)
|
| $53$ |
\( T^{3} - 2 T^{2} + \cdots + 8 \)
|
| $59$ |
\( T^{3} + 29 T^{2} + \cdots + 325 \)
|
| $61$ |
\( T^{3} \)
|
| $67$ |
\( T^{3} + 9 T^{2} + \cdots - 559 \)
|
| $71$ |
\( T^{3} + 14 T^{2} + \cdots - 92 \)
|
| $73$ |
\( T^{3} + T^{2} + \cdots - 25 \)
|
| $79$ |
\( T^{3} + 13 T^{2} + \cdots - 625 \)
|
| $83$ |
\( T^{3} + 8 T^{2} + \cdots - 256 \)
|
| $89$ |
\( T^{3} - 4 T^{2} + \cdots - 80 \)
|
| $97$ |
\( T^{3} - 10 T^{2} + \cdots + 1096 \)
|
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