Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [58,3,Mod(17,58)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(58, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([3]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("58.17");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 58 = 2 \cdot 29 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 58.c (of order \(4\), degree \(2\), 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: | \(1.58038553329\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Relative dimension: | \(3\) over \(\Q(i)\) |
| Coefficient field: | 6.0.9296045056.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 2x^{5} + 2x^{4} + 4x^{3} + 225x^{2} - 390x + 338 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - 2x^{5} + 2x^{4} + 4x^{3} + 225x^{2} - 390x + 338 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 2\nu^{5} - 255\nu^{4} + 34\nu^{3} + 4\nu^{2} - 52\nu - 41147 ) / 4263 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 17\nu^{5} - 36\nu^{4} + 289\nu^{3} + 34\nu^{2} + 3821\nu - 6578 ) / 4263 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -253\nu^{5} + 285\nu^{4} - 38\nu^{3} - 4769\nu^{2} - 57367\nu + 48997 ) / 55419 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -2783\nu^{5} + 3135\nu^{4} - 418\nu^{3} + 2960\nu^{2} - 631037\nu + 538967 ) / 55419 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{5} - 11\beta_{4} \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{5} + 2\beta_{4} + 15\beta_{3} - \beta_{2} + 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( 2\beta_{3} - 17\beta_{2} - 2\beta _1 - 161 \)
|
| \(\nu^{5}\) | \(=\) |
\( -19\beta_{5} - 12\beta_{4} - 19\beta_{2} - 229\beta _1 + 12 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/58\mathbb{Z}\right)^\times\).
| \(n\) | \(31\) |
| \(\chi(n)\) | \(-\beta_{4}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 17.1 |
|
−1.00000 | + | 1.00000i | −3.68866 | + | 3.68866i | − | 2.00000i | − | 4.45780i | − | 7.37732i | −7.37732 | 2.00000 | + | 2.00000i | − | 18.2124i | 4.45780 | + | 4.45780i | ||||||||||||||||||||||||
| 17.2 | −1.00000 | + | 1.00000i | −0.147635 | + | 0.147635i | − | 2.00000i | 8.54695i | − | 0.295269i | −0.295269 | 2.00000 | + | 2.00000i | 8.95641i | −8.54695 | − | 8.54695i | |||||||||||||||||||||||||||
| 17.3 | −1.00000 | + | 1.00000i | 1.83630 | − | 1.83630i | − | 2.00000i | − | 6.08915i | 3.67259i | 3.67259 | 2.00000 | + | 2.00000i | 2.25603i | 6.08915 | + | 6.08915i | |||||||||||||||||||||||||||
| 41.1 | −1.00000 | − | 1.00000i | −3.68866 | − | 3.68866i | 2.00000i | 4.45780i | 7.37732i | −7.37732 | 2.00000 | − | 2.00000i | 18.2124i | 4.45780 | − | 4.45780i | |||||||||||||||||||||||||||||
| 41.2 | −1.00000 | − | 1.00000i | −0.147635 | − | 0.147635i | 2.00000i | − | 8.54695i | 0.295269i | −0.295269 | 2.00000 | − | 2.00000i | − | 8.95641i | −8.54695 | + | 8.54695i | |||||||||||||||||||||||||||
| 41.3 | −1.00000 | − | 1.00000i | 1.83630 | + | 1.83630i | 2.00000i | 6.08915i | − | 3.67259i | 3.67259 | 2.00000 | − | 2.00000i | − | 2.25603i | 6.08915 | − | 6.08915i | |||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 29.c | odd | 4 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 58.3.c.b | ✓ | 6 |
| 3.b | odd | 2 | 1 | 522.3.f.e | 6 | ||
| 4.b | odd | 2 | 1 | 464.3.l.b | 6 | ||
| 29.c | odd | 4 | 1 | inner | 58.3.c.b | ✓ | 6 |
| 87.f | even | 4 | 1 | 522.3.f.e | 6 | ||
| 116.e | even | 4 | 1 | 464.3.l.b | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 58.3.c.b | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 58.3.c.b | ✓ | 6 | 29.c | odd | 4 | 1 | inner |
| 464.3.l.b | 6 | 4.b | odd | 2 | 1 | ||
| 464.3.l.b | 6 | 116.e | even | 4 | 1 | ||
| 522.3.f.e | 6 | 3.b | odd | 2 | 1 | ||
| 522.3.f.e | 6 | 87.f | even | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{6} + 4T_{3}^{5} + 8T_{3}^{4} - 48T_{3}^{3} + 169T_{3}^{2} + 52T_{3} + 8 \)
acting on \(S_{3}^{\mathrm{new}}(58, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 2 T + 2)^{3} \)
|
| $3$ |
\( T^{6} + 4 T^{5} + \cdots + 8 \)
|
| $5$ |
\( T^{6} + 130 T^{4} + \cdots + 53824 \)
|
| $7$ |
\( (T^{3} + 4 T^{2} - 26 T - 8)^{2} \)
|
| $11$ |
\( T^{6} + 12 T^{5} + \cdots + 5832 \)
|
| $13$ |
\( T^{6} + 150 T^{4} + \cdots + 116964 \)
|
| $17$ |
\( T^{6} + 14 T^{5} + \cdots + 1083392 \)
|
| $19$ |
\( T^{6} - 28 T^{5} + \cdots + 332201088 \)
|
| $23$ |
\( (T^{3} + 20 T^{2} + \cdots - 18688)^{2} \)
|
| $29$ |
\( T^{6} + 4 T^{5} + \cdots + 594823321 \)
|
| $31$ |
\( T^{6} + 60 T^{5} + \cdots + 6728 \)
|
| $37$ |
\( T^{6} + \cdots + 7660249088 \)
|
| $41$ |
\( T^{6} + \cdots + 24446142728 \)
|
| $43$ |
\( T^{6} - 92 T^{5} + \cdots + 336234312 \)
|
| $47$ |
\( T^{6} + 124 T^{5} + \cdots + 322884872 \)
|
| $53$ |
\( (T^{3} - 50 T^{2} + \cdots + 76576)^{2} \)
|
| $59$ |
\( (T^{3} - 64 T^{2} + \cdots - 1024)^{2} \)
|
| $61$ |
\( T^{6} + 10 T^{5} + \cdots + 19668992 \)
|
| $67$ |
\( T^{6} + \cdots + 6954225664 \)
|
| $71$ |
\( T^{6} + \cdots + 1434288384 \)
|
| $73$ |
\( T^{6} + 146 T^{5} + \cdots + 675942912 \)
|
| $79$ |
\( T^{6} + \cdots + 54485645832 \)
|
| $83$ |
\( (T + 104)^{6} \)
|
| $89$ |
\( T^{6} + \cdots + 47049621768 \)
|
| $97$ |
\( T^{6} + \cdots + 6068934792 \)
|
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