Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [714,2,Mod(319,714)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(714, base_ring=CyclotomicField(12))
chi = DirichletCharacter(H, H._module([0, 8, 3]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("714.319");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 714 = 2 \cdot 3 \cdot 7 \cdot 17 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 714.ba (of order \(12\), degree \(4\), 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: | \(5.70131870432\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{12})\) |
| 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: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{12}]$ |
$q$-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 primitive root of unity \(\zeta_{24}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/714\mathbb{Z}\right)^\times\).
| \(n\) | \(239\) | \(409\) | \(547\) |
| \(\chi(n)\) | \(1\) | \(-1 + \zeta_{24}^{4}\) | \(-\zeta_{24}^{6}\) |
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 |
|
0.866025 | − | 0.500000i | −0.965926 | + | 0.258819i | 0.500000 | − | 0.866025i | 0.258819 | − | 0.965926i | −0.707107 | + | 0.707107i | 1.02494 | + | 2.43916i | − | 1.00000i | 0.866025 | − | 0.500000i | −0.258819 | − | 0.965926i | |||||||||||||||||||||||||
| 319.2 | 0.866025 | − | 0.500000i | 0.965926 | − | 0.258819i | 0.500000 | − | 0.866025i | −0.258819 | + | 0.965926i | 0.707107 | − | 0.707107i | 2.43916 | + | 1.02494i | − | 1.00000i | 0.866025 | − | 0.500000i | 0.258819 | + | 0.965926i | ||||||||||||||||||||||||||
| 361.1 | −0.866025 | + | 0.500000i | −0.258819 | − | 0.965926i | 0.500000 | − | 0.866025i | 0.965926 | + | 0.258819i | 0.707107 | + | 0.707107i | −1.02494 | + | 2.43916i | 1.00000i | −0.866025 | + | 0.500000i | −0.965926 | + | 0.258819i | |||||||||||||||||||||||||||
| 361.2 | −0.866025 | + | 0.500000i | 0.258819 | + | 0.965926i | 0.500000 | − | 0.866025i | −0.965926 | − | 0.258819i | −0.707107 | − | 0.707107i | −2.43916 | + | 1.02494i | 1.00000i | −0.866025 | + | 0.500000i | 0.965926 | − | 0.258819i | |||||||||||||||||||||||||||
| 625.1 | −0.866025 | − | 0.500000i | −0.258819 | + | 0.965926i | 0.500000 | + | 0.866025i | 0.965926 | − | 0.258819i | 0.707107 | − | 0.707107i | −1.02494 | − | 2.43916i | − | 1.00000i | −0.866025 | − | 0.500000i | −0.965926 | − | 0.258819i | ||||||||||||||||||||||||||
| 625.2 | −0.866025 | − | 0.500000i | 0.258819 | − | 0.965926i | 0.500000 | + | 0.866025i | −0.965926 | + | 0.258819i | −0.707107 | + | 0.707107i | −2.43916 | − | 1.02494i | − | 1.00000i | −0.866025 | − | 0.500000i | 0.965926 | + | 0.258819i | ||||||||||||||||||||||||||
| 667.1 | 0.866025 | + | 0.500000i | −0.965926 | − | 0.258819i | 0.500000 | + | 0.866025i | 0.258819 | + | 0.965926i | −0.707107 | − | 0.707107i | 1.02494 | − | 2.43916i | 1.00000i | 0.866025 | + | 0.500000i | −0.258819 | + | 0.965926i | |||||||||||||||||||||||||||
| 667.2 | 0.866025 | + | 0.500000i | 0.965926 | + | 0.258819i | 0.500000 | + | 0.866025i | −0.258819 | − | 0.965926i | 0.707107 | + | 0.707107i | 2.43916 | − | 1.02494i | 1.00000i | 0.866025 | + | 0.500000i | 0.258819 | − | 0.965926i | |||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.c | even | 3 | 1 | inner |
| 17.c | even | 4 | 1 | inner |
| 119.n | even | 12 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 714.2.ba.c | ✓ | 8 |
| 7.c | even | 3 | 1 | inner | 714.2.ba.c | ✓ | 8 |
| 17.c | even | 4 | 1 | inner | 714.2.ba.c | ✓ | 8 |
| 119.n | even | 12 | 1 | inner | 714.2.ba.c | ✓ | 8 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 714.2.ba.c | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 714.2.ba.c | ✓ | 8 | 7.c | even | 3 | 1 | inner |
| 714.2.ba.c | ✓ | 8 | 17.c | even | 4 | 1 | inner |
| 714.2.ba.c | ✓ | 8 | 119.n | even | 12 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{8} - T_{5}^{4} + 1 \)
acting on \(S_{2}^{\mathrm{new}}(714, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{4} - T^{2} + 1)^{2} \)
|
| $3$ |
\( T^{8} - T^{4} + 1 \)
|
| $5$ |
\( T^{8} - T^{4} + 1 \)
|
| $7$ |
\( T^{8} + 2T^{4} + 2401 \)
|
| $11$ |
\( T^{8} - 4 T^{7} + \cdots + 38416 \)
|
| $13$ |
\( (T^{2} - 2 T - 7)^{4} \)
|
| $17$ |
\( T^{8} - 16 T^{6} + \cdots + 83521 \)
|
| $19$ |
\( (T^{4} - 16 T^{2} + 256)^{2} \)
|
| $23$ |
\( T^{8} - 4 T^{7} + \cdots + 1336336 \)
|
| $29$ |
\( (T^{4} + 16 T^{3} + \cdots + 529)^{2} \)
|
| $31$ |
\( T^{8} + 4 T^{7} + \cdots + 2401 \)
|
| $37$ |
\( T^{8} + 20 T^{7} + \cdots + 4477456 \)
|
| $41$ |
\( (T^{4} + 8 T^{3} + \cdots + 1681)^{2} \)
|
| $43$ |
\( (T^{4} + 164 T^{2} + 2116)^{2} \)
|
| $47$ |
\( (T^{4} - 10 T^{3} + \cdots + 49)^{2} \)
|
| $53$ |
\( T^{8} - 76 T^{6} + \cdots + 1336336 \)
|
| $59$ |
\( T^{8} - 54 T^{6} + \cdots + 279841 \)
|
| $61$ |
\( T^{8} + 4 T^{7} + \cdots + 38416 \)
|
| $67$ |
\( (T^{4} + 12 T^{3} + \cdots + 784)^{2} \)
|
| $71$ |
\( (T^{4} + 20 T^{3} + \cdots + 2116)^{2} \)
|
| $73$ |
\( T^{8} + 20 T^{7} + \cdots + 4477456 \)
|
| $79$ |
\( T^{8} - 1296 T^{4} + 1679616 \)
|
| $83$ |
\( (T^{4} + 246 T^{2} + 14161)^{2} \)
|
| $89$ |
\( (T^{4} + 8 T^{3} + \cdots + 256)^{2} \)
|
| $97$ |
\( (T^{4} + 20736)^{2} \)
|
| show more | |
| show less |