Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [768,3,Mod(319,768)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(768, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([2, 1, 0]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("768.319");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 768 = 2^{8} \cdot 3 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 768.l (of order \(4\), degree \(2\), 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: | \(20.9264843029\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Relative dimension: | \(4\) over \(\Q(i)\) |
| Coefficient field: | \(\Q(\zeta_{24})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - x^{4} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{10} \) |
| Twist minimal: | yes |
| 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
| \(\beta_{1}\) | \(=\) |
\( \zeta_{24}^{5} + \zeta_{24} \)
|
| \(\beta_{2}\) | \(=\) |
\( \zeta_{24}^{6} \)
|
| \(\beta_{3}\) | \(=\) |
\( 2\zeta_{24}^{7} - \zeta_{24}^{3} \)
|
| \(\beta_{4}\) | \(=\) |
\( -2\zeta_{24}^{5} + 2\zeta_{24}^{3} + 2\zeta_{24} \)
|
| \(\beta_{5}\) | \(=\) |
\( -2\zeta_{24}^{5} - 2\zeta_{24}^{3} + 2\zeta_{24} \)
|
| \(\beta_{6}\) | \(=\) |
\( -2\zeta_{24}^{6} + 4\zeta_{24}^{4} + 4\zeta_{24}^{2} - 2 \)
|
| \(\beta_{7}\) | \(=\) |
\( -2\zeta_{24}^{6} - 4\zeta_{24}^{4} + 4\zeta_{24}^{2} + 2 \)
|
| \(\zeta_{24}\) | \(=\) |
\( ( \beta_{5} + \beta_{4} + 4\beta_1 ) / 8 \)
|
| \(\zeta_{24}^{2}\) | \(=\) |
\( ( \beta_{7} + \beta_{6} + 4\beta_{2} ) / 8 \)
|
| \(\zeta_{24}^{3}\) | \(=\) |
\( ( -\beta_{5} + \beta_{4} ) / 4 \)
|
| \(\zeta_{24}^{4}\) | \(=\) |
\( ( -\beta_{7} + \beta_{6} + 4 ) / 8 \)
|
| \(\zeta_{24}^{5}\) | \(=\) |
\( ( -\beta_{5} - \beta_{4} + 4\beta_1 ) / 8 \)
|
| \(\zeta_{24}^{6}\) | \(=\) |
\( \beta_{2} \)
|
| \(\zeta_{24}^{7}\) | \(=\) |
\( ( -\beta_{5} + \beta_{4} + 4\beta_{3} ) / 8 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/768\mathbb{Z}\right)^\times\).
| \(n\) | \(257\) | \(511\) | \(517\) |
| \(\chi(n)\) | \(1\) | \(-1\) | \(\beta_{2}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 319.1 |
|
0 | −1.22474 | − | 1.22474i | 0 | 2.00000 | + | 2.00000i | 0 | −5.27792 | 0 | 3.00000i | 0 | ||||||||||||||||||||||||||||||||||||||
| 319.2 | 0 | −1.22474 | − | 1.22474i | 0 | 2.00000 | + | 2.00000i | 0 | 0.378937 | 0 | 3.00000i | 0 | |||||||||||||||||||||||||||||||||||||||
| 319.3 | 0 | 1.22474 | + | 1.22474i | 0 | 2.00000 | + | 2.00000i | 0 | −0.378937 | 0 | 3.00000i | 0 | |||||||||||||||||||||||||||||||||||||||
| 319.4 | 0 | 1.22474 | + | 1.22474i | 0 | 2.00000 | + | 2.00000i | 0 | 5.27792 | 0 | 3.00000i | 0 | |||||||||||||||||||||||||||||||||||||||
| 703.1 | 0 | −1.22474 | + | 1.22474i | 0 | 2.00000 | − | 2.00000i | 0 | −5.27792 | 0 | − | 3.00000i | 0 | ||||||||||||||||||||||||||||||||||||||
| 703.2 | 0 | −1.22474 | + | 1.22474i | 0 | 2.00000 | − | 2.00000i | 0 | 0.378937 | 0 | − | 3.00000i | 0 | ||||||||||||||||||||||||||||||||||||||
| 703.3 | 0 | 1.22474 | − | 1.22474i | 0 | 2.00000 | − | 2.00000i | 0 | −0.378937 | 0 | − | 3.00000i | 0 | ||||||||||||||||||||||||||||||||||||||
| 703.4 | 0 | 1.22474 | − | 1.22474i | 0 | 2.00000 | − | 2.00000i | 0 | 5.27792 | 0 | − | 3.00000i | 0 | ||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 4.b | odd | 2 | 1 | inner |
| 16.e | even | 4 | 1 | inner |
| 16.f | odd | 4 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 768.3.l.d | yes | 8 |
| 4.b | odd | 2 | 1 | inner | 768.3.l.d | yes | 8 |
| 8.b | even | 2 | 1 | 768.3.l.a | ✓ | 8 | |
| 8.d | odd | 2 | 1 | 768.3.l.a | ✓ | 8 | |
| 16.e | even | 4 | 1 | 768.3.l.a | ✓ | 8 | |
| 16.e | even | 4 | 1 | inner | 768.3.l.d | yes | 8 |
| 16.f | odd | 4 | 1 | 768.3.l.a | ✓ | 8 | |
| 16.f | odd | 4 | 1 | inner | 768.3.l.d | yes | 8 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 768.3.l.a | ✓ | 8 | 8.b | even | 2 | 1 | |
| 768.3.l.a | ✓ | 8 | 8.d | odd | 2 | 1 | |
| 768.3.l.a | ✓ | 8 | 16.e | even | 4 | 1 | |
| 768.3.l.a | ✓ | 8 | 16.f | odd | 4 | 1 | |
| 768.3.l.d | yes | 8 | 1.a | even | 1 | 1 | trivial |
| 768.3.l.d | yes | 8 | 4.b | odd | 2 | 1 | inner |
| 768.3.l.d | yes | 8 | 16.e | even | 4 | 1 | inner |
| 768.3.l.d | yes | 8 | 16.f | odd | 4 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{2} - 4T_{5} + 8 \)
acting on \(S_{3}^{\mathrm{new}}(768, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} \)
|
| $3$ |
\( (T^{4} + 9)^{2} \)
|
| $5$ |
\( (T^{2} - 4 T + 8)^{4} \)
|
| $7$ |
\( (T^{4} - 28 T^{2} + 4)^{2} \)
|
| $11$ |
\( T^{8} + 49664 T^{4} + 65536 \)
|
| $13$ |
\( (T^{4} + 12 T^{3} + \cdots + 36)^{2} \)
|
| $17$ |
\( (T^{2} - 4 T - 188)^{4} \)
|
| $19$ |
\( T^{8} + 486176 T^{4} + 479785216 \)
|
| $23$ |
\( (T^{4} - 624 T^{2} + 69696)^{2} \)
|
| $29$ |
\( (T^{4} + 32 T^{3} + \cdots + 1024)^{2} \)
|
| $31$ |
\( (T^{4} + 3324 T^{2} + 133956)^{2} \)
|
| $37$ |
\( (T^{4} - 100 T^{3} + \cdots + 1069156)^{2} \)
|
| $41$ |
\( (T^{4} + 6344 T^{2} + 80656)^{2} \)
|
| $43$ |
\( T^{8} + \cdots + 704014971136 \)
|
| $47$ |
\( (T^{4} + 1008 T^{2} + 5184)^{2} \)
|
| $53$ |
\( (T^{4} - 192 T^{3} + \cdots + 14017536)^{2} \)
|
| $59$ |
\( T^{8} + \cdots + 62882616180736 \)
|
| $61$ |
\( (T^{4} + 12 T^{3} + \cdots + 74753316)^{2} \)
|
| $67$ |
\( T^{8} + \cdots + 180227832610816 \)
|
| $71$ |
\( (T^{4} - 24304 T^{2} + 119946304)^{2} \)
|
| $73$ |
\( (T^{4} + 8576 T^{2} + 15241216)^{2} \)
|
| $79$ |
\( (T^{4} + 13372 T^{2} + 4251844)^{2} \)
|
| $83$ |
\( T^{8} + \cdots + 21743271936 \)
|
| $89$ |
\( (T^{2} + 5476)^{4} \)
|
| $97$ |
\( (T^{2} + 40 T - 4400)^{4} \)
|
| show more | |
| show less |