Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [7,7,Mod(6,7)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1]))
N = Newforms(chi, 7, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("7.6");
S:= CuspForms(chi, 7);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 7 \) |
| Weight: | \( k \) | \(=\) | \( 7 \) |
| Character orbit: | \([\chi]\) | \(=\) | 7.b (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: | \(1.61037858534\) |
| Analytic rank: | \(0\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{-510}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} + 510 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2 \) |
| 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 \(\beta = 2\sqrt{-510}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/7\mathbb{Z}\right)^\times\).
| \(n\) | \(3\) |
| \(\chi(n)\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 6.1 |
|
−8.00000 | − | 45.1664i | 0 | 45.1664i | 361.331i | 133.000 | − | 316.165i | 512.000 | −1311.00 | − | 361.331i | ||||||||||||||||||||
| 6.2 | −8.00000 | 45.1664i | 0 | − | 45.1664i | − | 361.331i | 133.000 | + | 316.165i | 512.000 | −1311.00 | 361.331i | |||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 7.7.b.b | ✓ | 2 |
| 3.b | odd | 2 | 1 | 63.7.d.d | 2 | ||
| 4.b | odd | 2 | 1 | 112.7.c.b | 2 | ||
| 5.b | even | 2 | 1 | 175.7.d.e | 2 | ||
| 5.c | odd | 4 | 2 | 175.7.c.c | 4 | ||
| 7.b | odd | 2 | 1 | inner | 7.7.b.b | ✓ | 2 |
| 7.c | even | 3 | 2 | 49.7.d.d | 4 | ||
| 7.d | odd | 6 | 2 | 49.7.d.d | 4 | ||
| 8.b | even | 2 | 1 | 448.7.c.d | 2 | ||
| 8.d | odd | 2 | 1 | 448.7.c.c | 2 | ||
| 21.c | even | 2 | 1 | 63.7.d.d | 2 | ||
| 28.d | even | 2 | 1 | 112.7.c.b | 2 | ||
| 35.c | odd | 2 | 1 | 175.7.d.e | 2 | ||
| 35.f | even | 4 | 2 | 175.7.c.c | 4 | ||
| 56.e | even | 2 | 1 | 448.7.c.c | 2 | ||
| 56.h | odd | 2 | 1 | 448.7.c.d | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 7.7.b.b | ✓ | 2 | 1.a | even | 1 | 1 | trivial |
| 7.7.b.b | ✓ | 2 | 7.b | odd | 2 | 1 | inner |
| 49.7.d.d | 4 | 7.c | even | 3 | 2 | ||
| 49.7.d.d | 4 | 7.d | odd | 6 | 2 | ||
| 63.7.d.d | 2 | 3.b | odd | 2 | 1 | ||
| 63.7.d.d | 2 | 21.c | even | 2 | 1 | ||
| 112.7.c.b | 2 | 4.b | odd | 2 | 1 | ||
| 112.7.c.b | 2 | 28.d | even | 2 | 1 | ||
| 175.7.c.c | 4 | 5.c | odd | 4 | 2 | ||
| 175.7.c.c | 4 | 35.f | even | 4 | 2 | ||
| 175.7.d.e | 2 | 5.b | even | 2 | 1 | ||
| 175.7.d.e | 2 | 35.c | odd | 2 | 1 | ||
| 448.7.c.c | 2 | 8.d | odd | 2 | 1 | ||
| 448.7.c.c | 2 | 56.e | even | 2 | 1 | ||
| 448.7.c.d | 2 | 8.b | even | 2 | 1 | ||
| 448.7.c.d | 2 | 56.h | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2} + 8 \)
acting on \(S_{7}^{\mathrm{new}}(7, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 8)^{2} \)
|
| $3$ |
\( T^{2} + 2040 \)
|
| $5$ |
\( T^{2} + 2040 \)
|
| $7$ |
\( T^{2} - 266T + 117649 \)
|
| $11$ |
\( (T - 874)^{2} \)
|
| $13$ |
\( T^{2} + 4898040 \)
|
| $17$ |
\( T^{2} + 35544960 \)
|
| $19$ |
\( T^{2} + 9712440 \)
|
| $23$ |
\( (T - 4738)^{2} \)
|
| $29$ |
\( (T - 11146)^{2} \)
|
| $31$ |
\( T^{2} + 754114560 \)
|
| $37$ |
\( (T - 3002)^{2} \)
|
| $41$ |
\( T^{2} + 3311075040 \)
|
| $43$ |
\( (T - 31418)^{2} \)
|
| $47$ |
\( T^{2} + 5248544640 \)
|
| $53$ |
\( (T + 76406)^{2} \)
|
| $59$ |
\( T^{2} + 12821499960 \)
|
| $61$ |
\( T^{2} + 75684573240 \)
|
| $67$ |
\( (T - 495242)^{2} \)
|
| $71$ |
\( (T + 184406)^{2} \)
|
| $73$ |
\( T^{2} + 3717900000 \)
|
| $79$ |
\( (T + 534934)^{2} \)
|
| $83$ |
\( T^{2} + 511007615160 \)
|
| $89$ |
\( T^{2} + 396306401760 \)
|
| $97$ |
\( T^{2} + 663312168960 \)
|
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