Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [861,2,Mod(379,861)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(861, base_ring=CyclotomicField(10))
chi = DirichletCharacter(H, H._module([0, 0, 2]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("861.379");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 861 = 3 \cdot 7 \cdot 41 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 861.n (of order \(5\), 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: | \(6.87511961403\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{5})\) |
| Coefficient field: | 8.0.511890625.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - 3x^{7} + 7x^{6} - 5x^{5} + 16x^{4} + 15x^{3} + 63x^{2} + 81x + 81 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{5}]$ |
$q$-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
\( x^{8} - 3x^{7} + 7x^{6} - 5x^{5} + 16x^{4} + 15x^{3} + 63x^{2} + 81x + 81 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{7} - 6\nu^{6} + 25\nu^{5} - 80\nu^{4} + 256\nu^{3} - 753\nu^{2} + 594\nu + 27 ) / 1728 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{7} + 30\nu^{6} - 97\nu^{5} + 176\nu^{4} - 160\nu^{3} + 273\nu^{2} + 54\nu + 621 ) / 576 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -13\nu^{7} + 66\nu^{6} - 217\nu^{5} + 416\nu^{4} - 496\nu^{3} - 51\nu^{2} - 270\nu - 459 ) / 1728 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{7} - 2\nu^{6} + 5\nu^{5} + 16\nu^{3} + 31\nu^{2} + 94\nu + 111 ) / 64 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 3\nu^{7} - 14\nu^{6} + 39\nu^{5} - 32\nu^{4} + 16\nu^{3} + 61\nu^{2} + 66\nu + 117 ) / 192 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -37\nu^{7} + 138\nu^{6} - 313\nu^{5} + 320\nu^{4} - 592\nu^{3} - 123\nu^{2} - 1494\nu - 459 ) / 1728 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{6} + \beta_{5} - \beta_{3} - 3\beta_{2} \)
|
| \(\nu^{3}\) | \(=\) |
\( -4\beta_{6} + 5\beta_{5} + 3\beta_{4} - 5\beta_{3} - 3\beta_{2} - 4\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( 3\beta_{7} + 9\beta_{5} + 12\beta_{4} - 8\beta_{3} - 8\beta _1 - 3 \)
|
| \(\nu^{5}\) | \(=\) |
\( 24\beta_{7} + 20\beta_{6} + 12\beta_{5} + 24\beta_{2} - 12\beta _1 - 27 \)
|
| \(\nu^{6}\) | \(=\) |
\( 60\beta_{7} + 56\beta_{6} - 60\beta_{4} + 56\beta_{3} + 96\beta_{2} - 15\beta _1 - 96 \)
|
| \(\nu^{7}\) | \(=\) |
\( 107\beta_{6} - 107\beta_{5} - 168\beta_{4} + 223\beta_{3} + 213\beta_{2} - 168 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/861\mathbb{Z}\right)^\times\).
| \(n\) | \(211\) | \(493\) | \(575\) |
| \(\chi(n)\) | \(-\beta_{7}\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 379.1 |
|
−0.892071 | − | 0.648127i | 1.00000 | −0.242313 | − | 0.745763i | 0.840741 | + | 2.58753i | −0.892071 | − | 0.648127i | −0.809017 | + | 0.587785i | −0.948670 | + | 2.91971i | 1.00000 | 0.927051 | − | 2.85317i | ||||||||||||||||||||||||||||
| 379.2 | 2.20109 | + | 1.59918i | 1.00000 | 1.66936 | + | 5.13777i | −0.340741 | − | 1.04869i | 2.20109 | + | 1.59918i | −0.809017 | + | 0.587785i | −2.86035 | + | 8.80324i | 1.00000 | 0.927051 | − | 2.85317i | |||||||||||||||||||||||||||||
| 631.1 | −0.448193 | + | 1.37940i | 1.00000 | −0.0838250 | − | 0.0609025i | 1.67339 | + | 1.21579i | −0.448193 | + | 1.37940i | 0.309017 | + | 0.951057i | −2.22519 | + | 1.61670i | 1.00000 | −2.42705 | + | 1.76336i | |||||||||||||||||||||||||||||
| 631.2 | 0.639176 | − | 1.96718i | 1.00000 | −1.84323 | − | 1.33918i | −1.17339 | − | 0.852514i | 0.639176 | − | 1.96718i | 0.309017 | + | 0.951057i | −0.465791 | + | 0.338417i | 1.00000 | −2.42705 | + | 1.76336i | |||||||||||||||||||||||||||||
| 652.1 | −0.892071 | + | 0.648127i | 1.00000 | −0.242313 | + | 0.745763i | 0.840741 | − | 2.58753i | −0.892071 | + | 0.648127i | −0.809017 | − | 0.587785i | −0.948670 | − | 2.91971i | 1.00000 | 0.927051 | + | 2.85317i | |||||||||||||||||||||||||||||
| 652.2 | 2.20109 | − | 1.59918i | 1.00000 | 1.66936 | − | 5.13777i | −0.340741 | + | 1.04869i | 2.20109 | − | 1.59918i | −0.809017 | − | 0.587785i | −2.86035 | − | 8.80324i | 1.00000 | 0.927051 | + | 2.85317i | |||||||||||||||||||||||||||||
| 715.1 | −0.448193 | − | 1.37940i | 1.00000 | −0.0838250 | + | 0.0609025i | 1.67339 | − | 1.21579i | −0.448193 | − | 1.37940i | 0.309017 | − | 0.951057i | −2.22519 | − | 1.61670i | 1.00000 | −2.42705 | − | 1.76336i | |||||||||||||||||||||||||||||
| 715.2 | 0.639176 | + | 1.96718i | 1.00000 | −1.84323 | + | 1.33918i | −1.17339 | + | 0.852514i | 0.639176 | + | 1.96718i | 0.309017 | − | 0.951057i | −0.465791 | − | 0.338417i | 1.00000 | −2.42705 | − | 1.76336i | |||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 41.d | even | 5 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 861.2.n.b | ✓ | 8 |
| 41.d | even | 5 | 1 | inner | 861.2.n.b | ✓ | 8 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 861.2.n.b | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 861.2.n.b | ✓ | 8 | 41.d | even | 5 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{8} - 3T_{2}^{7} + 7T_{2}^{6} - 5T_{2}^{5} + 16T_{2}^{4} + 15T_{2}^{3} + 63T_{2}^{2} + 81T_{2} + 81 \)
acting on \(S_{2}^{\mathrm{new}}(861, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} - 3 T^{7} + \cdots + 81 \)
|
| $3$ |
\( (T - 1)^{8} \)
|
| $5$ |
\( T^{8} - 2 T^{7} + \cdots + 81 \)
|
| $7$ |
\( (T^{4} + T^{3} + T^{2} + \cdots + 1)^{2} \)
|
| $11$ |
\( T^{8} - 10 T^{7} + \cdots + 361 \)
|
| $13$ |
\( T^{8} - T^{7} + \cdots + 25 \)
|
| $17$ |
\( (T^{4} + 3 T^{3} + 4 T^{2} + \cdots + 1)^{2} \)
|
| $19$ |
\( T^{8} - 17 T^{7} + \cdots + 1946025 \)
|
| $23$ |
\( T^{8} + T^{7} + \cdots + 10201 \)
|
| $29$ |
\( T^{8} + 11 T^{7} + \cdots + 625 \)
|
| $31$ |
\( T^{8} - 16 T^{7} + \cdots + 52441 \)
|
| $37$ |
\( T^{8} - 11 T^{7} + \cdots + 81 \)
|
| $41$ |
\( T^{8} - 15 T^{7} + \cdots + 2825761 \)
|
| $43$ |
\( T^{8} - 11 T^{7} + \cdots + 2025 \)
|
| $47$ |
\( T^{8} - 6 T^{7} + \cdots + 20736 \)
|
| $53$ |
\( T^{8} - 17 T^{7} + \cdots + 1846881 \)
|
| $59$ |
\( T^{8} - 9 T^{7} + \cdots + 390625 \)
|
| $61$ |
\( T^{8} + 19 T^{7} + \cdots + 77841 \)
|
| $67$ |
\( T^{8} - 7 T^{7} + \cdots + 600625 \)
|
| $71$ |
\( T^{8} - 9 T^{7} + \cdots + 229441 \)
|
| $73$ |
\( (T^{4} - T^{3} - 49 T^{2} + \cdots + 81)^{2} \)
|
| $79$ |
\( (T^{4} + 28 T^{3} + \cdots - 3280)^{2} \)
|
| $83$ |
\( (T^{4} - 11 T^{3} + \cdots - 5839)^{2} \)
|
| $89$ |
\( T^{8} - 7 T^{7} + \cdots + 16443025 \)
|
| $97$ |
\( T^{8} + 4 T^{7} + \cdots + 123454321 \)
|
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