Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [798,2,Mod(113,798)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(798, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 0, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("798.113");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 798 = 2 \cdot 3 \cdot 7 \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 798.b (of order \(2\), degree \(1\), 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: | \(6.37206208130\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | 8.0.573783687384.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - x^{7} - 3x^{6} + x^{5} + 12x^{4} + 3x^{3} - 27x^{2} - 27x + 81 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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
\( x^{8} - x^{7} - 3x^{6} + x^{5} + 12x^{4} + 3x^{3} - 27x^{2} - 27x + 81 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 1 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{7} + 2\nu^{6} + 3\nu^{5} + 10\nu^{4} - 12\nu^{3} - 33\nu^{2} + 36\nu + 81 ) / 54 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{7} + \nu^{6} + 3\nu^{5} - \nu^{4} - 12\nu^{3} - 3\nu^{2} + 27\nu + 27 ) / 27 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -\nu^{7} - 2\nu^{6} + 6\nu^{5} + 8\nu^{4} - 15\nu^{3} - 39\nu^{2} + 18\nu + 81 ) / 27 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -2\nu^{7} - \nu^{6} + 9\nu^{5} + 7\nu^{4} - 42\nu^{2} + 18\nu + 81 ) / 27 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -\nu^{7} + \nu^{5} + 5\nu^{4} - 4\nu^{3} - 9\nu^{2} - 3\nu + 27 ) / 9 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 1 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{6} - \beta_{5} - \beta_{4} + \beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{7} - 2\beta_{4} + 2\beta_{3} + 2\beta_{2} + \beta _1 - 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( -2\beta_{7} + 3\beta_{6} + \beta_{4} + 2\beta_{3} + 4\beta_{2} - 5\beta _1 - 3 \)
|
| \(\nu^{6}\) | \(=\) |
\( \beta_{7} + 2\beta_{6} - 8\beta_{5} + 5\beta_{4} + 8\beta_{3} - 2\beta_{2} - 6\beta _1 - 4 \)
|
| \(\nu^{7}\) | \(=\) |
\( -6\beta_{7} - \beta_{6} + 4\beta_{5} - 5\beta_{4} + 12\beta_{3} + 5\beta_{2} - 7\beta _1 + 1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/798\mathbb{Z}\right)^\times\).
| \(n\) | \(115\) | \(211\) | \(533\) |
| \(\chi(n)\) | \(1\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 113.1 |
|
1.00000 | −1.59977 | − | 0.663878i | 1.00000 | − | 0.571257i | −1.59977 | − | 0.663878i | −1.00000 | 1.00000 | 2.11853 | + | 2.12411i | − | 0.571257i | ||||||||||||||||||||||||||||||||||
| 113.2 | 1.00000 | −1.59977 | + | 0.663878i | 1.00000 | 0.571257i | −1.59977 | + | 0.663878i | −1.00000 | 1.00000 | 2.11853 | − | 2.12411i | 0.571257i | |||||||||||||||||||||||||||||||||||||
| 113.3 | 1.00000 | −0.864095 | − | 1.50111i | 1.00000 | 4.03714i | −0.864095 | − | 1.50111i | −1.00000 | 1.00000 | −1.50668 | + | 2.59421i | 4.03714i | |||||||||||||||||||||||||||||||||||||
| 113.4 | 1.00000 | −0.864095 | + | 1.50111i | 1.00000 | − | 4.03714i | −0.864095 | + | 1.50111i | −1.00000 | 1.00000 | −1.50668 | − | 2.59421i | − | 4.03714i | |||||||||||||||||||||||||||||||||||
| 113.5 | 1.00000 | 1.32096 | − | 1.12030i | 1.00000 | − | 3.56124i | 1.32096 | − | 1.12030i | −1.00000 | 1.00000 | 0.489863 | − | 2.95974i | − | 3.56124i | |||||||||||||||||||||||||||||||||||
| 113.6 | 1.00000 | 1.32096 | + | 1.12030i | 1.00000 | 3.56124i | 1.32096 | + | 1.12030i | −1.00000 | 1.00000 | 0.489863 | + | 2.95974i | 3.56124i | |||||||||||||||||||||||||||||||||||||
| 113.7 | 1.00000 | 1.64291 | − | 0.548504i | 1.00000 | 2.38593i | 1.64291 | − | 0.548504i | −1.00000 | 1.00000 | 2.39829 | − | 1.80228i | 2.38593i | |||||||||||||||||||||||||||||||||||||
| 113.8 | 1.00000 | 1.64291 | + | 0.548504i | 1.00000 | − | 2.38593i | 1.64291 | + | 0.548504i | −1.00000 | 1.00000 | 2.39829 | + | 1.80228i | − | 2.38593i | |||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 57.d | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 798.2.b.h | yes | 8 |
| 3.b | odd | 2 | 1 | 798.2.b.e | ✓ | 8 | |
| 19.b | odd | 2 | 1 | 798.2.b.e | ✓ | 8 | |
| 57.d | even | 2 | 1 | inner | 798.2.b.h | yes | 8 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 798.2.b.e | ✓ | 8 | 3.b | odd | 2 | 1 | |
| 798.2.b.e | ✓ | 8 | 19.b | odd | 2 | 1 | |
| 798.2.b.h | yes | 8 | 1.a | even | 1 | 1 | trivial |
| 798.2.b.h | yes | 8 | 57.d | even | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(798, [\chi])\):
|
\( T_{5}^{8} + 35T_{5}^{6} + 383T_{5}^{4} + 1298T_{5}^{2} + 384 \)
|
|
\( T_{29}^{4} + 5T_{29}^{3} - 55T_{29}^{2} + 56T_{29} + 36 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 1)^{8} \)
|
| $3$ |
\( T^{8} - T^{7} + \cdots + 81 \)
|
| $5$ |
\( T^{8} + 35 T^{6} + \cdots + 384 \)
|
| $7$ |
\( (T + 1)^{8} \)
|
| $11$ |
\( T^{8} + 17 T^{6} + \cdots + 96 \)
|
| $13$ |
\( T^{8} + 66 T^{6} + \cdots + 6144 \)
|
| $17$ |
\( T^{8} + 68 T^{6} + \cdots + 24576 \)
|
| $19$ |
\( T^{8} - 6 T^{7} + \cdots + 130321 \)
|
| $23$ |
\( T^{8} + 134 T^{6} + \cdots + 497664 \)
|
| $29$ |
\( (T^{4} + 5 T^{3} - 55 T^{2} + \cdots + 36)^{2} \)
|
| $31$ |
\( T^{8} + 238 T^{6} + \cdots + 5809536 \)
|
| $37$ |
\( T^{8} + 35 T^{6} + \cdots + 384 \)
|
| $41$ |
\( (T^{4} - 3 T^{3} + \cdots - 216)^{2} \)
|
| $43$ |
\( (T^{4} - 11 T^{3} + \cdots - 1080)^{2} \)
|
| $47$ |
\( T^{8} + 207 T^{6} + \cdots + 1109400 \)
|
| $53$ |
\( (T^{4} + 15 T^{3} + \cdots + 396)^{2} \)
|
| $59$ |
\( (T^{4} - 3 T^{3} + \cdots + 2592)^{2} \)
|
| $61$ |
\( (T^{4} + 17 T^{3} + \cdots - 1492)^{2} \)
|
| $67$ |
\( T^{8} + 136 T^{6} + \cdots + 1536 \)
|
| $71$ |
\( (T^{4} - 25 T^{3} + \cdots - 12288)^{2} \)
|
| $73$ |
\( (T^{4} - 10 T^{3} + \cdots - 1024)^{2} \)
|
| $79$ |
\( T^{8} + 315 T^{6} + \cdots + 254616 \)
|
| $83$ |
\( T^{8} + 376 T^{6} + \cdots + 124416 \)
|
| $89$ |
\( (T^{4} + T^{3} - 149 T^{2} + \cdots + 2148)^{2} \)
|
| $97$ |
\( T^{8} + 381 T^{6} + \cdots + 38400 \)
|
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