Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [688,2,Mod(97,688)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(688, base_ring=CyclotomicField(14))
chi = DirichletCharacter(H, H._module([0, 0, 10]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("688.97");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 688 = 2^{4} \cdot 43 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 688.u (of order \(7\), degree \(6\), 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: | \(5.49370765906\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | \(\Q(\zeta_{14})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 43) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{7}]$ |
$q$-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 \(\zeta_{14}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/688\mathbb{Z}\right)^\times\).
| \(n\) | \(431\) | \(433\) | \(517\) |
| \(\chi(n)\) | \(1\) | \(-\zeta_{14}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 97.1 |
|
0 | 2.02446 | − | 2.53859i | 0 | 1.80194 | − | 0.867767i | 0 | 1.19806 | 0 | −1.67845 | − | 7.35376i | 0 | ||||||||||||||||||||||||||||||
| 145.1 | 0 | −0.346011 | + | 1.51597i | 0 | −1.24698 | − | 1.56366i | 0 | 4.24698 | 0 | 0.524459 | + | 0.252566i | 0 | |||||||||||||||||||||||||||||||
| 193.1 | 0 | −0.178448 | + | 0.0859360i | 0 | 0.445042 | + | 1.94986i | 0 | 2.55496 | 0 | −1.84601 | + | 2.31482i | 0 | |||||||||||||||||||||||||||||||
| 305.1 | 0 | 2.02446 | + | 2.53859i | 0 | 1.80194 | + | 0.867767i | 0 | 1.19806 | 0 | −1.67845 | + | 7.35376i | 0 | |||||||||||||||||||||||||||||||
| 385.1 | 0 | −0.178448 | − | 0.0859360i | 0 | 0.445042 | − | 1.94986i | 0 | 2.55496 | 0 | −1.84601 | − | 2.31482i | 0 | |||||||||||||||||||||||||||||||
| 465.1 | 0 | −0.346011 | − | 1.51597i | 0 | −1.24698 | + | 1.56366i | 0 | 4.24698 | 0 | 0.524459 | − | 0.252566i | 0 | |||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 43.e | even | 7 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 688.2.u.c | 6 | |
| 4.b | odd | 2 | 1 | 43.2.e.b | ✓ | 6 | |
| 12.b | even | 2 | 1 | 387.2.u.a | 6 | ||
| 43.e | even | 7 | 1 | inner | 688.2.u.c | 6 | |
| 172.j | even | 14 | 1 | 1849.2.a.l | 3 | ||
| 172.k | odd | 14 | 1 | 43.2.e.b | ✓ | 6 | |
| 172.k | odd | 14 | 1 | 1849.2.a.i | 3 | ||
| 516.v | even | 14 | 1 | 387.2.u.a | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 43.2.e.b | ✓ | 6 | 4.b | odd | 2 | 1 | |
| 43.2.e.b | ✓ | 6 | 172.k | odd | 14 | 1 | |
| 387.2.u.a | 6 | 12.b | even | 2 | 1 | ||
| 387.2.u.a | 6 | 516.v | even | 14 | 1 | ||
| 688.2.u.c | 6 | 1.a | even | 1 | 1 | trivial | |
| 688.2.u.c | 6 | 43.e | even | 7 | 1 | inner | |
| 1849.2.a.i | 3 | 172.k | odd | 14 | 1 | ||
| 1849.2.a.l | 3 | 172.j | even | 14 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{6} - 3T_{3}^{5} + 9T_{3}^{4} + T_{3}^{3} + 25T_{3}^{2} + 9T_{3} + 1 \)
acting on \(S_{2}^{\mathrm{new}}(688, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
|
| $3$ |
\( T^{6} - 3 T^{5} + \cdots + 1 \)
|
| $5$ |
\( T^{6} - 2 T^{5} + \cdots + 64 \)
|
| $7$ |
\( (T^{3} - 8 T^{2} + 19 T - 13)^{2} \)
|
| $11$ |
\( T^{6} + 14 T^{5} + \cdots + 49 \)
|
| $13$ |
\( T^{6} + 9 T^{5} + \cdots + 1 \)
|
| $17$ |
\( T^{6} - 8 T^{5} + \cdots + 64 \)
|
| $19$ |
\( T^{6} - 4 T^{5} + \cdots + 841 \)
|
| $23$ |
\( T^{6} + T^{5} + \cdots + 1681 \)
|
| $29$ |
\( T^{6} - 9 T^{5} + \cdots + 169 \)
|
| $31$ |
\( T^{6} - 18 T^{5} + \cdots + 9409 \)
|
| $37$ |
\( (T^{3} + 2 T^{2} + \cdots + 251)^{2} \)
|
| $41$ |
\( T^{6} - 23 T^{5} + \cdots + 32761 \)
|
| $43$ |
\( T^{6} - 29 T^{5} + \cdots + 79507 \)
|
| $47$ |
\( T^{6} + T^{5} + T^{4} + \cdots + 1 \)
|
| $53$ |
\( T^{6} + 10 T^{5} + \cdots + 841 \)
|
| $59$ |
\( T^{6} + 22 T^{5} + \cdots + 113569 \)
|
| $61$ |
\( T^{6} - 19 T^{5} + \cdots + 6889 \)
|
| $67$ |
\( T^{6} - 4 T^{5} + \cdots + 169 \)
|
| $71$ |
\( T^{6} + 28 T^{5} + \cdots + 3136 \)
|
| $73$ |
\( T^{6} - 21 T^{5} + \cdots + 90601 \)
|
| $79$ |
\( (T^{3} + 3 T^{2} - 25 T - 83)^{2} \)
|
| $83$ |
\( T^{6} - 39 T^{5} + \cdots + 851929 \)
|
| $89$ |
\( T^{6} - 11 T^{5} + \cdots + 1413721 \)
|
| $97$ |
\( T^{6} + 19 T^{5} + \cdots + 21077281 \)
|
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