Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [93,3,Mod(2,93)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(93, base_ring=CyclotomicField(10))
chi = DirichletCharacter(H, H._module([5, 8]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("93.2");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 93 = 3 \cdot 31 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 93.l (of order \(10\), degree \(4\), 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: | \(2.53406645855\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{10})\) |
| Coefficient field: | 8.0.3844000000.4 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - x^{6} + 51x^{4} - 341x^{2} + 961 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{10}]$ |
$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} - x^{6} + 51x^{4} - 341x^{2} + 961 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -48\nu^{6} - 386\nu^{4} - 10229\nu^{2} - 9610 ) / 51491 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -168\nu^{6} - 1351\nu^{4} - 10056\nu^{2} + 17856 ) / 51491 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 168\nu^{7} + 1351\nu^{5} + 10056\nu^{3} - 17856\nu ) / 51491 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 203\nu^{6} - 513\nu^{4} + 12151\nu^{2} - 73067 ) / 51491 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -323\nu^{7} - 452\nu^{5} - 11978\nu^{3} + 49042\nu ) / 51491 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 371\nu^{7} + 838\nu^{5} + 22207\nu^{3} - 90923\nu ) / 51491 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( 2\beta_{3} - 7\beta_{2} - 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( 7\beta_{7} + 7\beta_{6} - 2\beta_{4} + 5\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -24\beta_{5} - 29\beta_{3} - 24 \)
|
| \(\nu^{5}\) | \(=\) |
\( -24\beta_{7} + 53\beta_{4} - 24\beta_1 \)
|
| \(\nu^{6}\) | \(=\) |
\( 193\beta_{5} - 193\beta_{3} + 419\beta_{2} + 419 \)
|
| \(\nu^{7}\) | \(=\) |
\( -226\beta_{7} - 419\beta_{6} \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/93\mathbb{Z}\right)^\times\).
| \(n\) | \(32\) | \(34\) |
| \(\chi(n)\) | \(-1\) | \(\beta_{5}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 2.1 |
|
−1.68686 | + | 2.32176i | −2.42705 | + | 1.76336i | −1.30902 | − | 4.02874i | 9.28705i | − | 8.60957i | −0.354102 | − | 1.08981i | 0.644323 | + | 0.209353i | 2.78115 | − | 8.55951i | −21.5623 | − | 15.6659i | |||||||||||||||||||||||||||
| 2.2 | 1.68686 | − | 2.32176i | −2.42705 | + | 1.76336i | −1.30902 | − | 4.02874i | − | 9.28705i | 8.60957i | −0.354102 | − | 1.08981i | −0.644323 | − | 0.209353i | 2.78115 | − | 8.55951i | −21.5623 | − | 15.6659i | ||||||||||||||||||||||||||||
| 8.1 | −1.84513 | − | 0.599519i | 0.927051 | + | 2.85317i | −0.190983 | − | 0.138757i | − | 2.39808i | − | 5.82026i | 6.35410 | + | 4.61653i | 4.83062 | + | 6.64877i | −7.28115 | + | 5.29007i | −1.43769 | + | 4.42477i | |||||||||||||||||||||||||||
| 8.2 | 1.84513 | + | 0.599519i | 0.927051 | + | 2.85317i | −0.190983 | − | 0.138757i | 2.39808i | 5.82026i | 6.35410 | + | 4.61653i | −4.83062 | − | 6.64877i | −7.28115 | + | 5.29007i | −1.43769 | + | 4.42477i | |||||||||||||||||||||||||||||
| 35.1 | −1.84513 | + | 0.599519i | 0.927051 | − | 2.85317i | −0.190983 | + | 0.138757i | 2.39808i | 5.82026i | 6.35410 | − | 4.61653i | 4.83062 | − | 6.64877i | −7.28115 | − | 5.29007i | −1.43769 | − | 4.42477i | |||||||||||||||||||||||||||||
| 35.2 | 1.84513 | − | 0.599519i | 0.927051 | − | 2.85317i | −0.190983 | + | 0.138757i | − | 2.39808i | − | 5.82026i | 6.35410 | − | 4.61653i | −4.83062 | + | 6.64877i | −7.28115 | − | 5.29007i | −1.43769 | − | 4.42477i | |||||||||||||||||||||||||||
| 47.1 | −1.68686 | − | 2.32176i | −2.42705 | − | 1.76336i | −1.30902 | + | 4.02874i | − | 9.28705i | 8.60957i | −0.354102 | + | 1.08981i | 0.644323 | − | 0.209353i | 2.78115 | + | 8.55951i | −21.5623 | + | 15.6659i | ||||||||||||||||||||||||||||
| 47.2 | 1.68686 | + | 2.32176i | −2.42705 | − | 1.76336i | −1.30902 | + | 4.02874i | 9.28705i | − | 8.60957i | −0.354102 | + | 1.08981i | −0.644323 | + | 0.209353i | 2.78115 | + | 8.55951i | −21.5623 | + | 15.6659i | ||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
| 31.d | even | 5 | 1 | inner |
| 93.l | odd | 10 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 93.3.l.a | ✓ | 8 |
| 3.b | odd | 2 | 1 | inner | 93.3.l.a | ✓ | 8 |
| 31.d | even | 5 | 1 | inner | 93.3.l.a | ✓ | 8 |
| 93.l | odd | 10 | 1 | inner | 93.3.l.a | ✓ | 8 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 93.3.l.a | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 93.3.l.a | ✓ | 8 | 3.b | odd | 2 | 1 | inner |
| 93.3.l.a | ✓ | 8 | 31.d | even | 5 | 1 | inner |
| 93.3.l.a | ✓ | 8 | 93.l | odd | 10 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{8} - T_{2}^{6} + 51T_{2}^{4} - 341T_{2}^{2} + 961 \)
acting on \(S_{3}^{\mathrm{new}}(93, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} - T^{6} + \cdots + 961 \)
|
| $3$ |
\( (T^{4} + 3 T^{3} + 9 T^{2} + \cdots + 81)^{2} \)
|
| $5$ |
\( (T^{4} + 92 T^{2} + 496)^{2} \)
|
| $7$ |
\( (T^{4} - 12 T^{3} + \cdots + 81)^{2} \)
|
| $11$ |
\( T^{8} + 20 T^{6} + \cdots + 153760000 \)
|
| $13$ |
\( (T^{4} + 90 T^{2} + \cdots + 2025)^{2} \)
|
| $17$ |
\( T^{8} + \cdots + 227200942336 \)
|
| $19$ |
\( (T^{4} - 58 T^{3} + \cdots + 271441)^{2} \)
|
| $23$ |
\( T^{8} + 280 T^{6} + \cdots + 153760000 \)
|
| $29$ |
\( T^{8} + \cdots + 3601920256 \)
|
| $31$ |
\( (T^{4} + 89 T^{3} + \cdots + 923521)^{2} \)
|
| $37$ |
\( (T^{2} - 5 T + 5)^{4} \)
|
| $41$ |
\( T^{8} + \cdots + 8207629090816 \)
|
| $43$ |
\( (T^{4} + 59 T^{3} + \cdots + 591361)^{2} \)
|
| $47$ |
\( T^{8} + \cdots + 3601920256 \)
|
| $53$ |
\( T^{8} + \cdots + 23632198799616 \)
|
| $59$ |
\( T^{8} + \cdots + 264044820295936 \)
|
| $61$ |
\( (T^{2} + 65 T - 1255)^{4} \)
|
| $67$ |
\( (T^{2} - 41 T - 241)^{4} \)
|
| $71$ |
\( T^{8} + \cdots + 25600523612416 \)
|
| $73$ |
\( (T^{4} - 102 T^{3} + \cdots + 6765201)^{2} \)
|
| $79$ |
\( (T^{4} + 129 T^{3} + \cdots + 15046641)^{2} \)
|
| $83$ |
\( T^{8} + \cdots + 2981064683776 \)
|
| $89$ |
\( T^{8} + \cdots + 25600523612416 \)
|
| $97$ |
\( (T^{4} - 231 T^{3} + \cdots + 116834481)^{2} \)
|
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