Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [378,5,Mod(323,378)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(378, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 0]))
N = Newforms(chi, 5, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("378.323");
S:= CuspForms(chi, 5);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 378 = 2 \cdot 3^{3} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 5 \) |
| Character orbit: | \([\chi]\) | \(=\) | 378.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: | \(39.0738460457\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | 8.0.5443747577856.29 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} + 24x^{6} + 180x^{4} + 488x^{2} + 324 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{9}\cdot 3^{4}\cdot 7^{2} \) |
| 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
\( x^{8} + 24x^{6} + 180x^{4} + 488x^{2} + 324 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -\nu^{7} - 18\nu^{5} - 66\nu^{3} + 4\nu ) / 18 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{6} - 10\nu^{4} + 38\nu^{2} + 186 ) / 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 5\nu^{7} + 102\nu^{5} + 522\nu^{3} + 316\nu ) / 36 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -7\nu^{6} - 133\nu^{4} - 616\nu^{2} - 588 ) / 6 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 13\nu^{6} + 247\nu^{4} + 1072\nu^{2} + 660 ) / 6 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 13\nu^{7} + 207\nu^{5} + 534\nu^{3} - 646\nu ) / 36 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -22\nu^{7} - 441\nu^{5} - 2280\nu^{3} - 2846\nu ) / 36 \)
|
| \(\nu\) | \(=\) |
\( ( -2\beta_{7} - 2\beta_{6} - 12\beta_{3} - 21\beta_1 ) / 84 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -7\beta_{5} - 13\beta_{4} - 504 ) / 84 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 3\beta_{7} + 31\beta_{6} + 81\beta_{3} + 371\beta_1 ) / 84 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 49\beta_{5} + 87\beta_{4} + 14\beta_{2} + 2268 ) / 42 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 4\beta_{7} - 220\beta_{6} - 354\beta_{3} - 2359\beta_1 ) / 42 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( -623\beta_{5} - 1117\beta_{4} - 266\beta_{2} - 24444 ) / 42 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( -25\beta_{7} + 419\beta_{6} + 525\beta_{3} + 4203\beta_1 ) / 6 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/378\mathbb{Z}\right)^\times\).
| \(n\) | \(29\) | \(325\) |
| \(\chi(n)\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 323.1 |
|
− | 2.82843i | 0 | −8.00000 | − | 42.2558i | 0 | −18.5203 | 22.6274i | 0 | −119.518 | ||||||||||||||||||||||||||||||||||||||||
| 323.2 | − | 2.82843i | 0 | −8.00000 | − | 9.19100i | 0 | 18.5203 | 22.6274i | 0 | −25.9961 | |||||||||||||||||||||||||||||||||||||||||
| 323.3 | − | 2.82843i | 0 | −8.00000 | − | 3.12468i | 0 | 18.5203 | 22.6274i | 0 | −8.83793 | |||||||||||||||||||||||||||||||||||||||||
| 323.4 | − | 2.82843i | 0 | −8.00000 | 14.9735i | 0 | −18.5203 | 22.6274i | 0 | 42.3515 | ||||||||||||||||||||||||||||||||||||||||||
| 323.5 | 2.82843i | 0 | −8.00000 | − | 14.9735i | 0 | −18.5203 | − | 22.6274i | 0 | 42.3515 | |||||||||||||||||||||||||||||||||||||||||
| 323.6 | 2.82843i | 0 | −8.00000 | 3.12468i | 0 | 18.5203 | − | 22.6274i | 0 | −8.83793 | ||||||||||||||||||||||||||||||||||||||||||
| 323.7 | 2.82843i | 0 | −8.00000 | 9.19100i | 0 | 18.5203 | − | 22.6274i | 0 | −25.9961 | ||||||||||||||||||||||||||||||||||||||||||
| 323.8 | 2.82843i | 0 | −8.00000 | 42.2558i | 0 | −18.5203 | − | 22.6274i | 0 | −119.518 | ||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 378.5.b.a | ✓ | 8 |
| 3.b | odd | 2 | 1 | inner | 378.5.b.a | ✓ | 8 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 378.5.b.a | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 378.5.b.a | ✓ | 8 | 3.b | odd | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{8} + 2104T_{5}^{6} + 590554T_{5}^{4} + 39384216T_{5}^{2} + 330185241 \)
acting on \(S_{5}^{\mathrm{new}}(378, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 8)^{4} \)
|
| $3$ |
\( T^{8} \)
|
| $5$ |
\( T^{8} + 2104 T^{6} + \cdots + 330185241 \)
|
| $7$ |
\( (T^{2} - 343)^{4} \)
|
| $11$ |
\( T^{8} + \cdots + 16\!\cdots\!81 \)
|
| $13$ |
\( (T^{4} + 188 T^{3} + \cdots - 317096316)^{2} \)
|
| $17$ |
\( T^{8} + \cdots + 12\!\cdots\!56 \)
|
| $19$ |
\( (T^{4} - 560 T^{3} + \cdots + 11482979377)^{2} \)
|
| $23$ |
\( T^{8} + \cdots + 13\!\cdots\!01 \)
|
| $29$ |
\( T^{8} + \cdots + 15\!\cdots\!36 \)
|
| $31$ |
\( (T^{4} + 440 T^{3} + \cdots + 88737781689)^{2} \)
|
| $37$ |
\( (T^{4} + \cdots + 1491560880553)^{2} \)
|
| $41$ |
\( T^{8} + \cdots + 54\!\cdots\!21 \)
|
| $43$ |
\( (T^{4} + \cdots - 16528157070236)^{2} \)
|
| $47$ |
\( T^{8} + \cdots + 18\!\cdots\!76 \)
|
| $53$ |
\( T^{8} + \cdots + 96\!\cdots\!36 \)
|
| $59$ |
\( T^{8} + \cdots + 20\!\cdots\!84 \)
|
| $61$ |
\( (T^{4} + \cdots + 394607747749392)^{2} \)
|
| $67$ |
\( (T^{4} + \cdots - 650727964094204)^{2} \)
|
| $71$ |
\( T^{8} + \cdots + 22\!\cdots\!69 \)
|
| $73$ |
\( (T^{4} + \cdots - 20805309026012)^{2} \)
|
| $79$ |
\( (T^{4} + \cdots + 345776930946468)^{2} \)
|
| $83$ |
\( T^{8} + \cdots + 31\!\cdots\!64 \)
|
| $89$ |
\( T^{8} + \cdots + 21\!\cdots\!81 \)
|
| $97$ |
\( (T^{4} + \cdots + 44\!\cdots\!52)^{2} \)
|
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