Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [416,2,Mod(31,416)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(416, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([2, 0, 3]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("416.31");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 416 = 2^{5} \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 416.k (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: | \(3.32177672409\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Relative dimension: | \(4\) over \(\Q(i)\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{8} + \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} + 17x^{6} + 84x^{4} + 100x^{2} + 16 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2 \) |
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} + 17x^{6} + 84x^{4} + 100x^{2} + 16 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{7} + 15\nu^{5} + 62\nu^{3} + 48\nu ) / 16 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{6} + 11\nu^{4} + 26\nu^{2} + 8\nu + 8 ) / 8 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{6} - 11\nu^{4} - 26\nu^{2} + 8\nu - 8 ) / 8 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{7} + 19\nu^{5} + 106\nu^{3} + 136\nu ) / 16 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{4} + 9\nu^{2} + 8 ) / 2 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -\nu^{7} + \nu^{6} - 19\nu^{5} + 11\nu^{4} - 98\nu^{3} + 18\nu^{2} - 80\nu - 24 ) / 16 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -\nu^{7} - \nu^{6} - 19\nu^{5} - 11\nu^{4} - 98\nu^{3} - 18\nu^{2} - 80\nu + 24 ) / 16 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} + \beta_{2} ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 2\beta_{7} - 2\beta_{6} - \beta_{3} + \beta_{2} - 8 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 2\beta_{7} + 2\beta_{6} + 4\beta_{4} - 7\beta_{3} - 7\beta_{2} ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -18\beta_{7} + 18\beta_{6} + 4\beta_{5} + 9\beta_{3} - 9\beta_{2} + 56 ) / 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -22\beta_{7} - 22\beta_{6} - 36\beta_{4} + 55\beta_{3} + 55\beta_{2} - 8\beta_1 ) / 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 146\beta_{7} - 146\beta_{6} - 44\beta_{5} - 81\beta_{3} + 81\beta_{2} - 424 ) / 2 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 206\beta_{7} + 206\beta_{6} + 292\beta_{4} - 439\beta_{3} - 439\beta_{2} + 152\beta_1 ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/416\mathbb{Z}\right)^\times\).
| \(n\) | \(261\) | \(287\) | \(353\) |
| \(\chi(n)\) | \(1\) | \(-1\) | \(-\beta_{1}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 31.1 |
|
0 | − | 3.16816i | 0 | 1.68833 | − | 1.68833i | 0 | 2.12286 | − | 2.12286i | 0 | −7.03721 | 0 | |||||||||||||||||||||||||||||||||||||
| 31.2 | 0 | − | 1.19225i | 0 | −0.720056 | + | 0.720056i | 0 | −3.60545 | + | 3.60545i | 0 | 1.57854 | 0 | ||||||||||||||||||||||||||||||||||||||
| 31.3 | 0 | 1.46091i | 0 | 1.87831 | − | 1.87831i | 0 | 0.675897 | − | 0.675897i | 0 | 0.865736 | 0 | |||||||||||||||||||||||||||||||||||||||
| 31.4 | 0 | 2.89949i | 0 | −2.84658 | + | 2.84658i | 0 | −0.193303 | + | 0.193303i | 0 | −5.40706 | 0 | |||||||||||||||||||||||||||||||||||||||
| 255.1 | 0 | − | 2.89949i | 0 | −2.84658 | − | 2.84658i | 0 | −0.193303 | − | 0.193303i | 0 | −5.40706 | 0 | ||||||||||||||||||||||||||||||||||||||
| 255.2 | 0 | − | 1.46091i | 0 | 1.87831 | + | 1.87831i | 0 | 0.675897 | + | 0.675897i | 0 | 0.865736 | 0 | ||||||||||||||||||||||||||||||||||||||
| 255.3 | 0 | 1.19225i | 0 | −0.720056 | − | 0.720056i | 0 | −3.60545 | − | 3.60545i | 0 | 1.57854 | 0 | |||||||||||||||||||||||||||||||||||||||
| 255.4 | 0 | 3.16816i | 0 | 1.68833 | + | 1.68833i | 0 | 2.12286 | + | 2.12286i | 0 | −7.03721 | 0 | |||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 52.f | even | 4 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 416.2.k.e | ✓ | 8 |
| 4.b | odd | 2 | 1 | 416.2.k.f | yes | 8 | |
| 8.b | even | 2 | 1 | 832.2.k.g | 8 | ||
| 8.d | odd | 2 | 1 | 832.2.k.i | 8 | ||
| 13.d | odd | 4 | 1 | 416.2.k.f | yes | 8 | |
| 52.f | even | 4 | 1 | inner | 416.2.k.e | ✓ | 8 |
| 104.j | odd | 4 | 1 | 832.2.k.i | 8 | ||
| 104.m | even | 4 | 1 | 832.2.k.g | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 416.2.k.e | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 416.2.k.e | ✓ | 8 | 52.f | even | 4 | 1 | inner |
| 416.2.k.f | yes | 8 | 4.b | odd | 2 | 1 | |
| 416.2.k.f | yes | 8 | 13.d | odd | 4 | 1 | |
| 832.2.k.g | 8 | 8.b | even | 2 | 1 | ||
| 832.2.k.g | 8 | 104.m | even | 4 | 1 | ||
| 832.2.k.i | 8 | 8.d | odd | 2 | 1 | ||
| 832.2.k.i | 8 | 104.j | odd | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(416, [\chi])\):
|
\( T_{3}^{8} + 22T_{3}^{6} + 153T_{3}^{4} + 356T_{3}^{2} + 256 \)
|
|
\( T_{7}^{8} + 2T_{7}^{7} + 2T_{7}^{6} - 48T_{7}^{5} + 281T_{7}^{4} - 246T_{7}^{3} + 98T_{7}^{2} + 56T_{7} + 16 \)
|
|
\( T_{11}^{8} - 6T_{11}^{7} + 18T_{11}^{6} + 24T_{11}^{5} + 84T_{11}^{4} - 312T_{11}^{3} + 648T_{11}^{2} + 288T_{11} + 64 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} \)
|
| $3$ |
\( T^{8} + 22 T^{6} + \cdots + 256 \)
|
| $5$ |
\( T^{8} - 16 T^{5} + \cdots + 676 \)
|
| $7$ |
\( T^{8} + 2 T^{7} + \cdots + 16 \)
|
| $11$ |
\( T^{8} - 6 T^{7} + \cdots + 64 \)
|
| $13$ |
\( T^{8} + 2 T^{7} + \cdots + 28561 \)
|
| $17$ |
\( T^{8} + 38 T^{6} + \cdots + 64 \)
|
| $19$ |
\( T^{8} - 6 T^{7} + \cdots + 256 \)
|
| $23$ |
\( (T^{4} + 8 T^{3} + \cdots - 256)^{2} \)
|
| $29$ |
\( (T^{4} - 2 T^{3} + \cdots + 464)^{2} \)
|
| $31$ |
\( T^{8} - 26 T^{7} + \cdots + 43264 \)
|
| $37$ |
\( T^{8} + 8 T^{5} + \cdots + 669124 \)
|
| $41$ |
\( T^{8} + 24 T^{7} + \cdots + 652864 \)
|
| $43$ |
\( (T^{4} + 22 T^{3} + \cdots - 1424)^{2} \)
|
| $47$ |
\( T^{8} - 14 T^{7} + \cdots + 16 \)
|
| $53$ |
\( (T^{4} - 30 T^{2} + \cdots + 104)^{2} \)
|
| $59$ |
\( T^{8} + 22 T^{7} + \cdots + 1048576 \)
|
| $61$ |
\( (T^{4} + 12 T^{3} + \cdots - 4112)^{2} \)
|
| $67$ |
\( T^{8} - 2 T^{7} + \cdots + 43983424 \)
|
| $71$ |
\( T^{8} + 14 T^{7} + \cdots + 204304 \)
|
| $73$ |
\( T^{8} - 20 T^{7} + \cdots + 43264 \)
|
| $79$ |
\( T^{8} + 208 T^{6} + \cdots + 256 \)
|
| $83$ |
\( T^{8} + 6 T^{7} + \cdots + 10816 \)
|
| $89$ |
\( T^{8} - 8 T^{7} + \cdots + 7744 \)
|
| $97$ |
\( T^{8} + 8 T^{7} + \cdots + 35344 \)
|
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