Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [82,2,Mod(23,82)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(82, base_ring=CyclotomicField(10))
chi = DirichletCharacter(H, H._module([9]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("82.23");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 82 = 2 \cdot 41 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 82.f (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: | \(0.654773296574\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{10})\) |
| Coefficient field: | 8.0.1816890625.4 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - 2x^{7} - x^{6} - 8x^{5} + 41x^{4} - 40x^{3} - 25x^{2} - 250x + 625 \)
|
| 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
\( x^{8} - 2x^{7} - x^{6} - 8x^{5} + 41x^{4} - 40x^{3} - 25x^{2} - 250x + 625 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -7\nu^{7} + 4\nu^{6} - 23\nu^{5} + 41\nu^{4} - 157\nu^{3} + 270\nu^{2} - 225\nu + 1250 ) / 1500 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -2\nu^{7} - \nu^{6} + 2\nu^{5} + 16\nu^{4} - 32\nu^{3} + 30\nu^{2} + 15\nu + 550 ) / 300 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -2\nu^{7} - 3\nu^{6} + 6\nu^{5} - 7\nu^{4} - 41\nu^{3} - 77\nu^{2} + 320\nu + 275 ) / 300 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -12\nu^{7} - \nu^{6} - 13\nu^{5} + 146\nu^{4} - 342\nu^{3} - 70\nu^{2} - 275\nu + 4625 ) / 1500 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -\nu^{7} + 10\nu^{4} - 10\nu^{3} - 7\nu^{2} + 10\nu + 250 ) / 60 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 37\nu^{7} - 14\nu^{6} - 32\nu^{5} - 231\nu^{4} + 787\nu^{3} + 230\nu^{2} - 575\nu - 7875 ) / 1500 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{6} - \beta_{5} - \beta_{4} + 4\beta_{3} + \beta_{2} + \beta_1 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{7} + 4\beta_{6} - 5\beta_{5} - \beta_{3} + \beta_{2} - \beta _1 + 5 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{7} + \beta_{6} + 8\beta_{5} - 7\beta_{4} + 7\beta_{3} - 12\beta_{2} + 7\beta _1 - 20 \)
|
| \(\nu^{5}\) | \(=\) |
\( -12\beta_{7} - 12\beta_{6} - 12\beta_{5} - 12\beta_{4} + 40\beta_{3} - 40\beta_{2} - 5 \)
|
| \(\nu^{6}\) | \(=\) |
\( -40\beta_{7} - 40\beta_{5} - 40\beta_{4} - 60\beta_{3} + 23\beta _1 + 60 \)
|
| \(\nu^{7}\) | \(=\) |
\( -83\beta_{6} + 137\beta_{5} - 63\beta_{4} + 52\beta_{3} - 137\beta_{2} + 83\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/82\mathbb{Z}\right)^\times\).
| \(n\) | \(47\) |
| \(\chi(n)\) | \(\beta_{3}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 23.1 |
|
0.309017 | + | 0.951057i | − | 2.67237i | −0.809017 | + | 0.587785i | 0.500000 | − | 0.363271i | 2.54157 | − | 0.825808i | 2.54157 | + | 0.825808i | −0.809017 | − | 0.587785i | −4.14156 | 0.500000 | + | 0.363271i | |||||||||||||||||||||||||||
| 23.2 | 0.309017 | + | 0.951057i | 1.49680i | −0.809017 | + | 0.587785i | 0.500000 | − | 0.363271i | −1.42354 | + | 0.462536i | −1.42354 | − | 0.462536i | −0.809017 | − | 0.587785i | 0.759593 | 0.500000 | + | 0.363271i | |||||||||||||||||||||||||||||
| 25.1 | 0.309017 | − | 0.951057i | − | 1.49680i | −0.809017 | − | 0.587785i | 0.500000 | + | 0.363271i | −1.42354 | − | 0.462536i | −1.42354 | + | 0.462536i | −0.809017 | + | 0.587785i | 0.759593 | 0.500000 | − | 0.363271i | ||||||||||||||||||||||||||||
| 25.2 | 0.309017 | − | 0.951057i | 2.67237i | −0.809017 | − | 0.587785i | 0.500000 | + | 0.363271i | 2.54157 | + | 0.825808i | 2.54157 | − | 0.825808i | −0.809017 | + | 0.587785i | −4.14156 | 0.500000 | − | 0.363271i | |||||||||||||||||||||||||||||
| 31.1 | −0.809017 | − | 0.587785i | − | 3.16567i | 0.309017 | + | 0.951057i | 0.500000 | + | 1.53884i | −1.86073 | + | 2.56108i | −1.86073 | − | 2.56108i | 0.309017 | − | 0.951057i | −7.02146 | 0.500000 | − | 1.53884i | ||||||||||||||||||||||||||||
| 31.2 | −0.809017 | − | 0.587785i | 1.26356i | 0.309017 | + | 0.951057i | 0.500000 | + | 1.53884i | 0.742700 | − | 1.02224i | 0.742700 | + | 1.02224i | 0.309017 | − | 0.951057i | 1.40343 | 0.500000 | − | 1.53884i | |||||||||||||||||||||||||||||
| 45.1 | −0.809017 | + | 0.587785i | − | 1.26356i | 0.309017 | − | 0.951057i | 0.500000 | − | 1.53884i | 0.742700 | + | 1.02224i | 0.742700 | − | 1.02224i | 0.309017 | + | 0.951057i | 1.40343 | 0.500000 | + | 1.53884i | ||||||||||||||||||||||||||||
| 45.2 | −0.809017 | + | 0.587785i | 3.16567i | 0.309017 | − | 0.951057i | 0.500000 | − | 1.53884i | −1.86073 | − | 2.56108i | −1.86073 | + | 2.56108i | 0.309017 | + | 0.951057i | −7.02146 | 0.500000 | + | 1.53884i | |||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 41.f | even | 10 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 82.2.f.a | ✓ | 8 |
| 3.b | odd | 2 | 1 | 738.2.n.d | 8 | ||
| 4.b | odd | 2 | 1 | 656.2.be.c | 8 | ||
| 41.f | even | 10 | 1 | inner | 82.2.f.a | ✓ | 8 |
| 41.g | even | 20 | 2 | 3362.2.a.w | 8 | ||
| 123.l | odd | 10 | 1 | 738.2.n.d | 8 | ||
| 164.l | odd | 10 | 1 | 656.2.be.c | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 82.2.f.a | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 82.2.f.a | ✓ | 8 | 41.f | even | 10 | 1 | inner |
| 656.2.be.c | 8 | 4.b | odd | 2 | 1 | ||
| 656.2.be.c | 8 | 164.l | odd | 10 | 1 | ||
| 738.2.n.d | 8 | 3.b | odd | 2 | 1 | ||
| 738.2.n.d | 8 | 123.l | odd | 10 | 1 | ||
| 3362.2.a.w | 8 | 41.g | even | 20 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{8} + 21T_{3}^{6} + 141T_{3}^{4} + 336T_{3}^{2} + 256 \)
acting on \(S_{2}^{\mathrm{new}}(82, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{4} + T^{3} + T^{2} + \cdots + 1)^{2} \)
|
| $3$ |
\( T^{8} + 21 T^{6} + \cdots + 256 \)
|
| $5$ |
\( (T^{4} - 2 T^{3} + 4 T^{2} + \cdots + 1)^{2} \)
|
| $7$ |
\( T^{8} - 4 T^{6} + \cdots + 256 \)
|
| $11$ |
\( T^{8} - 5 T^{7} + \cdots + 256 \)
|
| $13$ |
\( T^{8} + 5 T^{7} + \cdots + 1 \)
|
| $17$ |
\( T^{8} - 5 T^{7} + \cdots + 255025 \)
|
| $19$ |
\( T^{8} - 5 T^{7} + \cdots + 256 \)
|
| $23$ |
\( T^{8} + 19 T^{7} + \cdots + 256 \)
|
| $29$ |
\( T^{8} + 5 T^{7} + \cdots + 20736 \)
|
| $31$ |
\( T^{8} + 30 T^{6} + \cdots + 6400 \)
|
| $37$ |
\( T^{8} - 4 T^{7} + \cdots + 10201 \)
|
| $41$ |
\( T^{8} - 20 T^{7} + \cdots + 2825761 \)
|
| $43$ |
\( T^{8} - 6 T^{7} + \cdots + 1478656 \)
|
| $47$ |
\( (T^{4} + 20 T^{3} + \cdots + 1280)^{2} \)
|
| $53$ |
\( T^{8} + 45 T^{7} + \cdots + 15046641 \)
|
| $59$ |
\( T^{8} + 2 T^{7} + \cdots + 614656 \)
|
| $61$ |
\( T^{8} + 31 T^{7} + \cdots + 136161 \)
|
| $67$ |
\( T^{8} + 30 T^{7} + \cdots + 495616 \)
|
| $71$ |
\( T^{8} - 15 T^{7} + \cdots + 9834496 \)
|
| $73$ |
\( (T^{4} - 23 T^{3} + \cdots - 164)^{2} \)
|
| $79$ |
\( (T^{4} + 80 T^{2} + 1280)^{2} \)
|
| $83$ |
\( (T^{4} - 27 T^{3} + \cdots - 4464)^{2} \)
|
| $89$ |
\( T^{8} - 5 T^{7} + \cdots + 1048576 \)
|
| $97$ |
\( T^{8} + 10 T^{7} + \cdots + 4626801 \)
|
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