Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [960,4,Mod(481,960)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(960, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1, 0, 0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("960.481");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 960 = 2^{6} \cdot 3 \cdot 5 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 960.k (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: | \(56.6418336055\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{6} + \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} + 385x^{4} + 36024x^{2} + 176400 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{6} \) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} + 385x^{4} + 36024x^{2} + 176400 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{5} + 805\nu^{3} + 117084\nu ) / 257040 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{5} + 805\nu^{3} + 631164\nu ) / 257040 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{4} + 193\nu^{2} - 726 ) / 306 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{4} + 397\nu^{2} + 25692 ) / 408 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -137\nu^{5} - 46025\nu^{3} - 3509808\nu ) / 128520 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} - \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 4\beta_{4} - 3\beta_{3} - 259 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 4\beta_{5} - 195\beta_{2} + 1291\beta_1 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -772\beta_{4} + 1191\beta_{3} + 51439 ) / 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -3220\beta_{5} + 39891\beta_{2} - 408091\beta_1 ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/960\mathbb{Z}\right)^\times\).
| \(n\) | \(511\) | \(577\) | \(641\) | \(901\) |
| \(\chi(n)\) | \(1\) | \(1\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 481.1 |
|
0 | − | 3.00000i | 0 | − | 5.00000i | 0 | −36.6421 | 0 | −9.00000 | 0 | ||||||||||||||||||||||||||||||||||
| 481.2 | 0 | − | 3.00000i | 0 | − | 5.00000i | 0 | −1.44825 | 0 | −9.00000 | 0 | |||||||||||||||||||||||||||||||||||
| 481.3 | 0 | − | 3.00000i | 0 | − | 5.00000i | 0 | 18.0903 | 0 | −9.00000 | 0 | |||||||||||||||||||||||||||||||||||
| 481.4 | 0 | 3.00000i | 0 | 5.00000i | 0 | −36.6421 | 0 | −9.00000 | 0 | |||||||||||||||||||||||||||||||||||||
| 481.5 | 0 | 3.00000i | 0 | 5.00000i | 0 | −1.44825 | 0 | −9.00000 | 0 | |||||||||||||||||||||||||||||||||||||
| 481.6 | 0 | 3.00000i | 0 | 5.00000i | 0 | 18.0903 | 0 | −9.00000 | 0 | |||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 8.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 960.4.k.c | ✓ | 6 |
| 4.b | odd | 2 | 1 | 960.4.k.d | yes | 6 | |
| 8.b | even | 2 | 1 | inner | 960.4.k.c | ✓ | 6 |
| 8.d | odd | 2 | 1 | 960.4.k.d | yes | 6 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 960.4.k.c | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 960.4.k.c | ✓ | 6 | 8.b | even | 2 | 1 | inner |
| 960.4.k.d | yes | 6 | 4.b | odd | 2 | 1 | |
| 960.4.k.d | yes | 6 | 8.d | odd | 2 | 1 | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{3} + 20T_{7}^{2} - 636T_{7} - 960 \)
acting on \(S_{4}^{\mathrm{new}}(960, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
|
| $3$ |
\( (T^{2} + 9)^{3} \)
|
| $5$ |
\( (T^{2} + 25)^{3} \)
|
| $7$ |
\( (T^{3} + 20 T^{2} + \cdots - 960)^{2} \)
|
| $11$ |
\( T^{6} + 7648 T^{4} + \cdots + 180633600 \)
|
| $13$ |
\( T^{6} + \cdots + 5979619584 \)
|
| $17$ |
\( (T^{3} + 96 T^{2} + \cdots - 783360)^{2} \)
|
| $19$ |
\( T^{6} + \cdots + 70051267584 \)
|
| $23$ |
\( (T^{3} + 2 T^{2} + \cdots - 1316448)^{2} \)
|
| $29$ |
\( T^{6} + \cdots + 42202485366336 \)
|
| $31$ |
\( (T^{3} + 40 T^{2} + \cdots + 4490880)^{2} \)
|
| $37$ |
\( T^{6} + \cdots + 19766916000000 \)
|
| $41$ |
\( (T^{3} + 526 T^{2} + \cdots - 15656376)^{2} \)
|
| $43$ |
\( T^{6} + \cdots + 721894582665216 \)
|
| $47$ |
\( (T^{3} - 62 T^{2} + \cdots - 7228512)^{2} \)
|
| $53$ |
\( T^{6} + \cdots + 318130030440000 \)
|
| $59$ |
\( T^{6} + \cdots + 29\!\cdots\!24 \)
|
| $61$ |
\( T^{6} + \cdots + 15\!\cdots\!24 \)
|
| $67$ |
\( T^{6} + \cdots + 190832342667264 \)
|
| $71$ |
\( (T^{3} - 1452 T^{2} + \cdots - 48600000)^{2} \)
|
| $73$ |
\( (T^{3} - 806 T^{2} + \cdots + 62295800)^{2} \)
|
| $79$ |
\( (T^{3} - 920 T^{2} + \cdots + 807757440)^{2} \)
|
| $83$ |
\( T^{6} + \cdots + 19\!\cdots\!16 \)
|
| $89$ |
\( (T^{3} + 354 T^{2} + \cdots - 7605000)^{2} \)
|
| $97$ |
\( (T^{3} - 926 T^{2} + \cdots + 380592120)^{2} \)
|
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