Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [104,3,Mod(33,104)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(104, base_ring=CyclotomicField(12))
chi = DirichletCharacter(H, H._module([0, 0, 11]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("104.33");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 104 = 2^{3} \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 104.v (of order \(12\), 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: | \(2.83379474935\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{12})\) |
| Coefficient field: | 8.0.44991500544.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - 38x^{6} + 555x^{4} - 3674x^{2} + 9409 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{9}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{12}]$ |
$q$-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
\( x^{8} - 38x^{6} + 555x^{4} - 3674x^{2} + 9409 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{4} - 19\nu^{2} + 2\nu + 97 ) / 4 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{4} + 19\nu^{2} + 2\nu - 97 ) / 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{7} - 38\nu^{5} + 458\nu^{3} - 1831\nu - 194 ) / 388 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 11\nu^{7} - 97\nu^{6} - 321\nu^{5} + 2716\nu^{4} + 3389\nu^{3} - 25414\nu^{2} - 12478\nu + 78279 ) / 5044 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -11\nu^{7} - 97\nu^{6} + 321\nu^{5} + 2716\nu^{4} - 3389\nu^{3} - 25414\nu^{2} + 12478\nu + 78279 ) / 5044 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( \nu^{6} - 28\nu^{4} + 288\nu^{2} - 1067 ) / 26 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -84\nu^{7} + 97\nu^{6} + 2222\nu^{5} - 2716\nu^{4} - 19460\nu^{3} + 27936\nu^{2} + 54476\nu - 100977 ) / 5044 \)
|
| \(\nu\) | \(=\) |
\( \beta_{2} + \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{6} + \beta_{5} + \beta_{4} + 10 \)
|
| \(\nu^{3}\) | \(=\) |
\( 2\beta_{7} - \beta_{6} - 10\beta_{5} + 10\beta_{4} - 4\beta_{3} + 9\beta_{2} + 9\beta _1 - 3 \)
|
| \(\nu^{4}\) | \(=\) |
\( 19\beta_{6} + 19\beta_{5} + 19\beta_{4} - 2\beta_{2} + 2\beta _1 + 93 \)
|
| \(\nu^{5}\) | \(=\) |
\( 34\beta_{7} - 17\beta_{6} - 196\beta_{5} + 196\beta_{4} - 112\beta_{3} + 74\beta_{2} + 74\beta _1 - 73 \)
|
| \(\nu^{6}\) | \(=\) |
\( 270\beta_{6} + 244\beta_{5} + 244\beta_{4} - 56\beta_{2} + 56\beta _1 + 791 \)
|
| \(\nu^{7}\) | \(=\) |
\( 376\beta_{7} - 188\beta_{6} - 2868\beta_{5} + 2868\beta_{4} - 2036\beta_{3} + 521\beta_{2} + 521\beta _1 - 1206 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/104\mathbb{Z}\right)^\times\).
| \(n\) | \(41\) | \(53\) | \(79\) |
| \(\chi(n)\) | \(\beta_{5}\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 33.1 |
|
0 | −2.26125 | + | 3.91659i | 0 | 3.07976 | + | 3.07976i | 0 | −11.8234 | − | 3.16808i | 0 | −5.72647 | − | 9.91853i | 0 | ||||||||||||||||||||||||||||||||||
| 33.2 | 0 | 1.12727 | − | 1.95249i | 0 | −6.17784 | − | 6.17784i | 0 | −8.43491 | − | 2.26013i | 0 | 1.95852 | + | 3.39225i | 0 | |||||||||||||||||||||||||||||||||||
| 41.1 | 0 | −2.26125 | − | 3.91659i | 0 | 3.07976 | − | 3.07976i | 0 | −11.8234 | + | 3.16808i | 0 | −5.72647 | + | 9.91853i | 0 | |||||||||||||||||||||||||||||||||||
| 41.2 | 0 | 1.12727 | + | 1.95249i | 0 | −6.17784 | + | 6.17784i | 0 | −8.43491 | + | 2.26013i | 0 | 1.95852 | − | 3.39225i | 0 | |||||||||||||||||||||||||||||||||||
| 89.1 | 0 | −2.84881 | − | 4.93429i | 0 | 0.0126012 | + | 0.0126012i | 0 | −0.286634 | − | 1.06973i | 0 | −11.7315 | + | 20.3195i | 0 | |||||||||||||||||||||||||||||||||||
| 89.2 | 0 | −0.0172135 | − | 0.0298147i | 0 | 2.08548 | + | 2.08548i | 0 | 2.54496 | + | 9.49794i | 0 | 4.49941 | − | 7.79320i | 0 | |||||||||||||||||||||||||||||||||||
| 97.1 | 0 | −2.84881 | + | 4.93429i | 0 | 0.0126012 | − | 0.0126012i | 0 | −0.286634 | + | 1.06973i | 0 | −11.7315 | − | 20.3195i | 0 | |||||||||||||||||||||||||||||||||||
| 97.2 | 0 | −0.0172135 | + | 0.0298147i | 0 | 2.08548 | − | 2.08548i | 0 | 2.54496 | − | 9.49794i | 0 | 4.49941 | + | 7.79320i | 0 | |||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 13.f | odd | 12 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 104.3.v.c | ✓ | 8 |
| 4.b | odd | 2 | 1 | 208.3.bd.g | 8 | ||
| 13.f | odd | 12 | 1 | inner | 104.3.v.c | ✓ | 8 |
| 52.l | even | 12 | 1 | 208.3.bd.g | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 104.3.v.c | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 104.3.v.c | ✓ | 8 | 13.f | odd | 12 | 1 | inner |
| 208.3.bd.g | 8 | 4.b | odd | 2 | 1 | ||
| 208.3.bd.g | 8 | 52.l | even | 12 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{8} + 8T_{3}^{7} + 61T_{3}^{6} + 140T_{3}^{5} + 475T_{3}^{4} - 142T_{3}^{3} + 3370T_{3}^{2} + 116T_{3} + 4 \)
acting on \(S_{3}^{\mathrm{new}}(104, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} \)
|
| $3$ |
\( T^{8} + 8 T^{7} + \cdots + 4 \)
|
| $5$ |
\( T^{8} + 2 T^{7} + \cdots + 4 \)
|
| $7$ |
\( T^{8} + 36 T^{7} + \cdots + 1354896 \)
|
| $11$ |
\( T^{8} + 24 T^{7} + \cdots + 5184 \)
|
| $13$ |
\( T^{8} - 26 T^{7} + \cdots + 815730721 \)
|
| $17$ |
\( T^{8} - 42 T^{7} + \cdots + 387263041 \)
|
| $19$ |
\( T^{8} + \cdots + 9825963876 \)
|
| $23$ |
\( T^{8} - 48 T^{7} + \cdots + 58564 \)
|
| $29$ |
\( T^{8} + \cdots + 2031916255209 \)
|
| $31$ |
\( T^{8} + \cdots + 11540775184 \)
|
| $37$ |
\( T^{8} + \cdots + 4927116923521 \)
|
| $41$ |
\( T^{8} + \cdots + 2158810369 \)
|
| $43$ |
\( T^{8} + \cdots + 112912811089936 \)
|
| $47$ |
\( T^{8} + \cdots + 892427859856 \)
|
| $53$ |
\( (T^{4} - 28 T^{3} + \cdots - 20396)^{2} \)
|
| $59$ |
\( T^{8} + \cdots + 74723502736 \)
|
| $61$ |
\( T^{8} + \cdots + 14512450249441 \)
|
| $67$ |
\( T^{8} + \cdots + 12588971032836 \)
|
| $71$ |
\( T^{8} + \cdots + 60\!\cdots\!36 \)
|
| $73$ |
\( T^{8} + \cdots + 4265538441124 \)
|
| $79$ |
\( (T^{4} - 80 T^{3} + \cdots + 5880928)^{2} \)
|
| $83$ |
\( T^{8} + \cdots + 637514424906304 \)
|
| $89$ |
\( T^{8} + \cdots + 449916294564 \)
|
| $97$ |
\( T^{8} + \cdots + 498433397686884 \)
|
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