Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [960,3,Mod(193,960)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(960, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([0, 0, 0, 3]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("960.193");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 960 = 2^{6} \cdot 3 \cdot 5 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 960.bg (of order \(4\), degree \(2\), not 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: | \(26.1581053786\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Relative dimension: | \(4\) over \(\Q(i)\) |
| Coefficient field: | 8.0.443364212736.2 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - x^{4} + 625 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{4} \) |
| Twist minimal: | no (minimal twist has level 480) |
| 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
\( x^{8} - x^{4} + 625 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{5} + 59\nu ) / 35 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{6} + 24\nu^{2} ) / 175 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 2\nu^{4} - 1 ) / 7 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{5} + 11\nu ) / 35 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -2\nu^{7} + 127\nu^{3} ) / 875 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -\nu^{6} + 26\nu^{2} ) / 25 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 12\nu^{7} + 113\nu^{3} ) / 875 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{4} + \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{6} + 7\beta_{2} ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{7} + 6\beta_{5} \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 7\beta_{3} + 1 ) / 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -59\beta_{4} + 11\beta_1 ) / 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( -12\beta_{6} + 91\beta_{2} \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 127\beta_{7} - 113\beta_{5} ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/960\mathbb{Z}\right)^\times\).
| \(n\) | \(511\) | \(577\) | \(641\) | \(901\) |
| \(\chi(n)\) | \(1\) | \(\beta_{2}\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 193.1 |
|
0 | −1.22474 | − | 1.22474i | 0 | −4.91548 | − | 0.915476i | 0 | −7.14143 | + | 7.14143i | 0 | 3.00000i | 0 | ||||||||||||||||||||||||||||||||||||
| 193.2 | 0 | −1.22474 | − | 1.22474i | 0 | 0.915476 | + | 4.91548i | 0 | 7.14143 | − | 7.14143i | 0 | 3.00000i | 0 | |||||||||||||||||||||||||||||||||||||
| 193.3 | 0 | 1.22474 | + | 1.22474i | 0 | −4.91548 | − | 0.915476i | 0 | 7.14143 | − | 7.14143i | 0 | 3.00000i | 0 | |||||||||||||||||||||||||||||||||||||
| 193.4 | 0 | 1.22474 | + | 1.22474i | 0 | 0.915476 | + | 4.91548i | 0 | −7.14143 | + | 7.14143i | 0 | 3.00000i | 0 | |||||||||||||||||||||||||||||||||||||
| 577.1 | 0 | −1.22474 | + | 1.22474i | 0 | −4.91548 | + | 0.915476i | 0 | −7.14143 | − | 7.14143i | 0 | − | 3.00000i | 0 | ||||||||||||||||||||||||||||||||||||
| 577.2 | 0 | −1.22474 | + | 1.22474i | 0 | 0.915476 | − | 4.91548i | 0 | 7.14143 | + | 7.14143i | 0 | − | 3.00000i | 0 | ||||||||||||||||||||||||||||||||||||
| 577.3 | 0 | 1.22474 | − | 1.22474i | 0 | −4.91548 | + | 0.915476i | 0 | 7.14143 | + | 7.14143i | 0 | − | 3.00000i | 0 | ||||||||||||||||||||||||||||||||||||
| 577.4 | 0 | 1.22474 | − | 1.22474i | 0 | 0.915476 | − | 4.91548i | 0 | −7.14143 | − | 7.14143i | 0 | − | 3.00000i | 0 | ||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 4.b | odd | 2 | 1 | inner |
| 5.c | odd | 4 | 1 | inner |
| 20.e | even | 4 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 960.3.bg.j | 8 | |
| 4.b | odd | 2 | 1 | inner | 960.3.bg.j | 8 | |
| 5.c | odd | 4 | 1 | inner | 960.3.bg.j | 8 | |
| 8.b | even | 2 | 1 | 480.3.bg.b | ✓ | 8 | |
| 8.d | odd | 2 | 1 | 480.3.bg.b | ✓ | 8 | |
| 20.e | even | 4 | 1 | inner | 960.3.bg.j | 8 | |
| 40.i | odd | 4 | 1 | 480.3.bg.b | ✓ | 8 | |
| 40.k | even | 4 | 1 | 480.3.bg.b | ✓ | 8 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 480.3.bg.b | ✓ | 8 | 8.b | even | 2 | 1 | |
| 480.3.bg.b | ✓ | 8 | 8.d | odd | 2 | 1 | |
| 480.3.bg.b | ✓ | 8 | 40.i | odd | 4 | 1 | |
| 480.3.bg.b | ✓ | 8 | 40.k | even | 4 | 1 | |
| 960.3.bg.j | 8 | 1.a | even | 1 | 1 | trivial | |
| 960.3.bg.j | 8 | 4.b | odd | 2 | 1 | inner | |
| 960.3.bg.j | 8 | 5.c | odd | 4 | 1 | inner | |
| 960.3.bg.j | 8 | 20.e | even | 4 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{4} + 10404 \)
acting on \(S_{3}^{\mathrm{new}}(960, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} \)
|
| $3$ |
\( (T^{4} + 9)^{2} \)
|
| $5$ |
\( (T^{4} + 8 T^{3} + \cdots + 625)^{2} \)
|
| $7$ |
\( (T^{4} + 10404)^{2} \)
|
| $11$ |
\( (T^{2} - 150)^{4} \)
|
| $13$ |
\( (T^{4} - 24 T^{3} + \cdots + 16)^{2} \)
|
| $17$ |
\( (T^{4} + 48 T^{3} + \cdots + 48400)^{2} \)
|
| $19$ |
\( (T^{4} + 456 T^{2} + 32400)^{2} \)
|
| $23$ |
\( (T^{4} + 11664)^{2} \)
|
| $29$ |
\( (T^{4} + 3460 T^{2} + 2566404)^{2} \)
|
| $31$ |
\( (T^{2} - 384)^{4} \)
|
| $37$ |
\( (T^{4} + 40 T^{3} + \cdots + 169744)^{2} \)
|
| $41$ |
\( (T^{2} - 100 T + 2364)^{4} \)
|
| $43$ |
\( T^{8} + \cdots + 172485565157376 \)
|
| $47$ |
\( T^{8} + \cdots + 101185065216 \)
|
| $53$ |
\( (T^{4} + 48 T^{3} + \cdots + 640000)^{2} \)
|
| $59$ |
\( (T^{4} + 15276 T^{2} + 49702500)^{2} \)
|
| $61$ |
\( (T^{2} - 36 T - 900)^{4} \)
|
| $67$ |
\( T^{8} + \cdots + 79\!\cdots\!76 \)
|
| $71$ |
\( (T^{4} - 9600 T^{2} + 2985984)^{2} \)
|
| $73$ |
\( (T^{4} - 60 T^{3} + \cdots + 40322500)^{2} \)
|
| $79$ |
\( (T^{4} + 29280 T^{2} + 169208064)^{2} \)
|
| $83$ |
\( (T^{4} + 18766224)^{2} \)
|
| $89$ |
\( (T^{4} + 5704 T^{2} + 2250000)^{2} \)
|
| $97$ |
\( (T^{4} + 180 T^{3} + \cdots + 8773444)^{2} \)
|
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