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: | \(\mathbb{Q}[x]/(x^{8} + \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} + 92x^{6} + 2949x^{4} + 37548x^{2} + 142884 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{10}\cdot 3^{4}\cdot 7^{2} \) |
| 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
\( x^{8} + 92x^{6} + 2949x^{4} + 37548x^{2} + 142884 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{7} - 34\nu^{5} - 3351\nu^{3} - 37674\nu ) / 15876 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -25\nu^{7} - 1502\nu^{5} - 23241\nu^{3} + 77490\nu ) / 21168 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -5\nu^{6} - 334\nu^{4} - 6429\nu^{2} - 30366 ) / 72 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{6} - 86\nu^{4} - 2073\nu^{2} - 11862 ) / 24 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -5\nu^{6} - 334\nu^{4} - 6237\nu^{2} - 25950 ) / 8 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 5\nu^{7} + 334\nu^{5} + 6177\nu^{3} + 24066\nu ) / 162 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 29\nu^{7} + 1954\nu^{5} + 37914\nu^{3} + 182385\nu ) / 441 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{6} + 28\beta_{2} + 35\beta_1 ) / 168 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{5} - 9\beta_{3} - 552 ) / 24 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 8\beta_{7} - 21\beta_{6} - 108\beta_{2} - 87\beta_1 ) / 24 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -41\beta_{5} - 30\beta_{4} + 387\beta_{3} + 15396 ) / 24 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -364\beta_{7} + 903\beta_{6} + 3228\beta_{2} - 1929\beta_1 ) / 24 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 1453\beta_{5} + 2004\beta_{4} - 14625\beta_{3} - 464448 ) / 24 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 14432\beta_{7} - 34287\beta_{6} - 101460\beta_{2} + 212271\beta_1 ) / 24 \)
|
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 | − | 30.1368i | 0 | 18.5203 | 22.6274i | 0 | −85.2398 | ||||||||||||||||||||||||||||||||||||||||
| 323.2 | − | 2.82843i | 0 | −8.00000 | − | 28.1791i | 0 | −18.5203 | 22.6274i | 0 | −79.7026 | |||||||||||||||||||||||||||||||||||||||||
| 323.3 | − | 2.82843i | 0 | −8.00000 | 17.8674i | 0 | −18.5203 | 22.6274i | 0 | 50.5366 | ||||||||||||||||||||||||||||||||||||||||||
| 323.4 | − | 2.82843i | 0 | −8.00000 | 34.7917i | 0 | 18.5203 | 22.6274i | 0 | 98.4059 | ||||||||||||||||||||||||||||||||||||||||||
| 323.5 | 2.82843i | 0 | −8.00000 | − | 34.7917i | 0 | 18.5203 | − | 22.6274i | 0 | 98.4059 | |||||||||||||||||||||||||||||||||||||||||
| 323.6 | 2.82843i | 0 | −8.00000 | − | 17.8674i | 0 | −18.5203 | − | 22.6274i | 0 | 50.5366 | |||||||||||||||||||||||||||||||||||||||||
| 323.7 | 2.82843i | 0 | −8.00000 | 28.1791i | 0 | −18.5203 | − | 22.6274i | 0 | −79.7026 | ||||||||||||||||||||||||||||||||||||||||||
| 323.8 | 2.82843i | 0 | −8.00000 | 30.1368i | 0 | 18.5203 | − | 22.6274i | 0 | −85.2398 | ||||||||||||||||||||||||||||||||||||||||||
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.b | ✓ | 8 |
| 3.b | odd | 2 | 1 | inner | 378.5.b.b | ✓ | 8 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 378.5.b.b | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 378.5.b.b | ✓ | 8 | 3.b | odd | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{8} + 3232T_{5}^{6} + 3711634T_{5}^{4} + 1761033888T_{5}^{2} + 278692135569 \)
acting on \(S_{5}^{\mathrm{new}}(378, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 8)^{4} \)
|
| $3$ |
\( T^{8} \)
|
| $5$ |
\( T^{8} + \cdots + 278692135569 \)
|
| $7$ |
\( (T^{2} - 343)^{4} \)
|
| $11$ |
\( T^{8} + \cdots + 7401278039841 \)
|
| $13$ |
\( (T^{4} - 148 T^{3} + \cdots + 1130516676)^{2} \)
|
| $17$ |
\( T^{8} + \cdots + 39\!\cdots\!16 \)
|
| $19$ |
\( (T^{4} + 40 T^{3} + \cdots - 274761143)^{2} \)
|
| $23$ |
\( T^{8} + \cdots + 44\!\cdots\!09 \)
|
| $29$ |
\( T^{8} + \cdots + 40\!\cdots\!96 \)
|
| $31$ |
\( (T^{4} + 512 T^{3} + \cdots + 21940230321)^{2} \)
|
| $37$ |
\( (T^{4} - 452 T^{3} + \cdots + 685159025593)^{2} \)
|
| $41$ |
\( T^{8} + \cdots + 59\!\cdots\!69 \)
|
| $43$ |
\( (T^{4} + \cdots + 19781954641348)^{2} \)
|
| $47$ |
\( T^{8} + \cdots + 10\!\cdots\!44 \)
|
| $53$ |
\( T^{8} + \cdots + 16\!\cdots\!24 \)
|
| $59$ |
\( T^{8} + \cdots + 65\!\cdots\!04 \)
|
| $61$ |
\( (T^{4} + \cdots + 37203857115408)^{2} \)
|
| $67$ |
\( (T^{4} + \cdots - 106304126923004)^{2} \)
|
| $71$ |
\( T^{8} + \cdots + 87\!\cdots\!41 \)
|
| $73$ |
\( (T^{4} + \cdots - 33552318651836)^{2} \)
|
| $79$ |
\( (T^{4} + \cdots + 451461470055876)^{2} \)
|
| $83$ |
\( T^{8} + \cdots + 23\!\cdots\!64 \)
|
| $89$ |
\( T^{8} + \cdots + 55\!\cdots\!61 \)
|
| $97$ |
\( (T^{4} + \cdots + 47506778460304)^{2} \)
|
| show more | |
| show less |