Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [62,3,Mod(61,62)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(62, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("62.61");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 62 = 2 \cdot 31 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 62.b (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: | \(1.68937763903\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.0.48128.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} + 14x^{2} + 47 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2 \) |
| 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
\( x^{4} + 14x^{2} + 47 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{3} + 7\nu \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{2} + 7 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} - 7 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 2\beta_{2} - 7\beta_1 ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/62\mathbb{Z}\right)^\times\).
| \(n\) | \(3\) |
| \(\chi(n)\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 61.1 |
|
−1.41421 | − | 4.10225i | 2.00000 | −5.82843 | 5.80145i | −4.41421 | −2.82843 | −7.82843 | 8.24264 | |||||||||||||||||||||||||||||
| 61.2 | −1.41421 | 4.10225i | 2.00000 | −5.82843 | − | 5.80145i | −4.41421 | −2.82843 | −7.82843 | 8.24264 | ||||||||||||||||||||||||||||||
| 61.3 | 1.41421 | − | 3.34239i | 2.00000 | −0.171573 | − | 4.72685i | −1.58579 | 2.82843 | −2.17157 | −0.242641 | |||||||||||||||||||||||||||||
| 61.4 | 1.41421 | 3.34239i | 2.00000 | −0.171573 | 4.72685i | −1.58579 | 2.82843 | −2.17157 | −0.242641 | |||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 31.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 62.3.b.a | ✓ | 4 |
| 3.b | odd | 2 | 1 | 558.3.d.a | 4 | ||
| 4.b | odd | 2 | 1 | 496.3.e.d | 4 | ||
| 5.b | even | 2 | 1 | 1550.3.c.a | 4 | ||
| 5.c | odd | 4 | 2 | 1550.3.d.a | 8 | ||
| 31.b | odd | 2 | 1 | inner | 62.3.b.a | ✓ | 4 |
| 93.c | even | 2 | 1 | 558.3.d.a | 4 | ||
| 124.d | even | 2 | 1 | 496.3.e.d | 4 | ||
| 155.c | odd | 2 | 1 | 1550.3.c.a | 4 | ||
| 155.f | even | 4 | 2 | 1550.3.d.a | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 62.3.b.a | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 62.3.b.a | ✓ | 4 | 31.b | odd | 2 | 1 | inner |
| 496.3.e.d | 4 | 4.b | odd | 2 | 1 | ||
| 496.3.e.d | 4 | 124.d | even | 2 | 1 | ||
| 558.3.d.a | 4 | 3.b | odd | 2 | 1 | ||
| 558.3.d.a | 4 | 93.c | even | 2 | 1 | ||
| 1550.3.c.a | 4 | 5.b | even | 2 | 1 | ||
| 1550.3.c.a | 4 | 155.c | odd | 2 | 1 | ||
| 1550.3.d.a | 8 | 5.c | odd | 4 | 2 | ||
| 1550.3.d.a | 8 | 155.f | even | 4 | 2 | ||
Hecke kernels
This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(62, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} - 2)^{2} \)
|
| $3$ |
\( T^{4} + 28T^{2} + 188 \)
|
| $5$ |
\( (T^{2} + 6 T + 1)^{2} \)
|
| $7$ |
\( (T^{2} + 6 T + 7)^{2} \)
|
| $11$ |
\( T^{4} + 400T^{2} + 3008 \)
|
| $13$ |
\( T^{4} + 260T^{2} + 9212 \)
|
| $17$ |
\( T^{4} + 484 T^{2} + 54332 \)
|
| $19$ |
\( (T^{2} - 18 T - 369)^{2} \)
|
| $23$ |
\( T^{4} + 860 T^{2} + 180668 \)
|
| $29$ |
\( T^{4} + 260T^{2} + 9212 \)
|
| $31$ |
\( T^{4} - 92 T^{3} + \cdots + 923521 \)
|
| $37$ |
\( T^{4} + 3600 T^{2} + 243648 \)
|
| $41$ |
\( (T^{2} - 22 T + 89)^{2} \)
|
| $43$ |
\( T^{4} + 7900 T^{2} + 15485372 \)
|
| $47$ |
\( (T^{2} + 28 T - 2692)^{2} \)
|
| $53$ |
\( T^{4} + 10976 T^{2} + 20225792 \)
|
| $59$ |
\( (T^{2} - 70 T + 775)^{2} \)
|
| $61$ |
\( T^{4} + 4900 T^{2} + 99452 \)
|
| $67$ |
\( (T^{2} + 88 T - 2296)^{2} \)
|
| $71$ |
\( (T^{2} - 14 T + 31)^{2} \)
|
| $73$ |
\( T^{4} + 6112 T^{2} + 3477248 \)
|
| $79$ |
\( T^{4} + 11900 T^{2} + 13806908 \)
|
| $83$ |
\( T^{4} + 37664 T^{2} + 311881472 \)
|
| $89$ |
\( T^{4} + 23044 T^{2} + 52213052 \)
|
| $97$ |
\( (T^{2} + 86 T - 3151)^{2} \)
|
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