Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [21,3,Mod(10,21)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(21, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 1]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("21.10");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 21 = 3 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 21.f (of order \(6\), 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: | \(0.572208555157\) |
| Analytic rank: | \(0\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{-3}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-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 \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/21\mathbb{Z}\right)^\times\).
| \(n\) | \(8\) | \(10\) |
| \(\chi(n)\) | \(1\) | \(\zeta_{6}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 10.1 |
|
−0.500000 | − | 0.866025i | 1.50000 | + | 0.866025i | 1.50000 | − | 2.59808i | −4.50000 | + | 2.59808i | − | 1.73205i | −3.50000 | + | 6.06218i | −7.00000 | 1.50000 | + | 2.59808i | 4.50000 | + | 2.59808i | |||||||||
| 19.1 | −0.500000 | + | 0.866025i | 1.50000 | − | 0.866025i | 1.50000 | + | 2.59808i | −4.50000 | − | 2.59808i | 1.73205i | −3.50000 | − | 6.06218i | −7.00000 | 1.50000 | − | 2.59808i | 4.50000 | − | 2.59808i | |||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.d | odd | 6 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 21.3.f.b | ✓ | 2 |
| 3.b | odd | 2 | 1 | 63.3.m.c | 2 | ||
| 4.b | odd | 2 | 1 | 336.3.bh.a | 2 | ||
| 5.b | even | 2 | 1 | 525.3.o.g | 2 | ||
| 5.c | odd | 4 | 2 | 525.3.s.c | 4 | ||
| 7.b | odd | 2 | 1 | 147.3.f.c | 2 | ||
| 7.c | even | 3 | 1 | 147.3.d.b | 2 | ||
| 7.c | even | 3 | 1 | 147.3.f.c | 2 | ||
| 7.d | odd | 6 | 1 | inner | 21.3.f.b | ✓ | 2 |
| 7.d | odd | 6 | 1 | 147.3.d.b | 2 | ||
| 12.b | even | 2 | 1 | 1008.3.cg.g | 2 | ||
| 21.c | even | 2 | 1 | 441.3.m.e | 2 | ||
| 21.g | even | 6 | 1 | 63.3.m.c | 2 | ||
| 21.g | even | 6 | 1 | 441.3.d.b | 2 | ||
| 21.h | odd | 6 | 1 | 441.3.d.b | 2 | ||
| 21.h | odd | 6 | 1 | 441.3.m.e | 2 | ||
| 28.f | even | 6 | 1 | 336.3.bh.a | 2 | ||
| 28.f | even | 6 | 1 | 2352.3.f.d | 2 | ||
| 28.g | odd | 6 | 1 | 2352.3.f.d | 2 | ||
| 35.i | odd | 6 | 1 | 525.3.o.g | 2 | ||
| 35.k | even | 12 | 2 | 525.3.s.c | 4 | ||
| 84.j | odd | 6 | 1 | 1008.3.cg.g | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 21.3.f.b | ✓ | 2 | 1.a | even | 1 | 1 | trivial |
| 21.3.f.b | ✓ | 2 | 7.d | odd | 6 | 1 | inner |
| 63.3.m.c | 2 | 3.b | odd | 2 | 1 | ||
| 63.3.m.c | 2 | 21.g | even | 6 | 1 | ||
| 147.3.d.b | 2 | 7.c | even | 3 | 1 | ||
| 147.3.d.b | 2 | 7.d | odd | 6 | 1 | ||
| 147.3.f.c | 2 | 7.b | odd | 2 | 1 | ||
| 147.3.f.c | 2 | 7.c | even | 3 | 1 | ||
| 336.3.bh.a | 2 | 4.b | odd | 2 | 1 | ||
| 336.3.bh.a | 2 | 28.f | even | 6 | 1 | ||
| 441.3.d.b | 2 | 21.g | even | 6 | 1 | ||
| 441.3.d.b | 2 | 21.h | odd | 6 | 1 | ||
| 441.3.m.e | 2 | 21.c | even | 2 | 1 | ||
| 441.3.m.e | 2 | 21.h | odd | 6 | 1 | ||
| 525.3.o.g | 2 | 5.b | even | 2 | 1 | ||
| 525.3.o.g | 2 | 35.i | odd | 6 | 1 | ||
| 525.3.s.c | 4 | 5.c | odd | 4 | 2 | ||
| 525.3.s.c | 4 | 35.k | even | 12 | 2 | ||
| 1008.3.cg.g | 2 | 12.b | even | 2 | 1 | ||
| 1008.3.cg.g | 2 | 84.j | odd | 6 | 1 | ||
| 2352.3.f.d | 2 | 28.f | even | 6 | 1 | ||
| 2352.3.f.d | 2 | 28.g | odd | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{2} + T_{2} + 1 \)
acting on \(S_{3}^{\mathrm{new}}(21, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} + T + 1 \)
|
| $3$ |
\( T^{2} - 3T + 3 \)
|
| $5$ |
\( T^{2} + 9T + 27 \)
|
| $7$ |
\( T^{2} + 7T + 49 \)
|
| $11$ |
\( T^{2} - 11T + 121 \)
|
| $13$ |
\( T^{2} + 48 \)
|
| $17$ |
\( T^{2} - 42T + 588 \)
|
| $19$ |
\( T^{2} + 6T + 12 \)
|
| $23$ |
\( T^{2} + 28T + 784 \)
|
| $29$ |
\( (T - 25)^{2} \)
|
| $31$ |
\( T^{2} + 57T + 1083 \)
|
| $37$ |
\( T^{2} - 58T + 3364 \)
|
| $41$ |
\( T^{2} + 12 \)
|
| $43$ |
\( (T - 26)^{2} \)
|
| $47$ |
\( T^{2} + 132T + 5808 \)
|
| $53$ |
\( T^{2} + 31T + 961 \)
|
| $59$ |
\( T^{2} + 15T + 75 \)
|
| $61$ |
\( T^{2} - 24T + 192 \)
|
| $67$ |
\( T^{2} - 52T + 2704 \)
|
| $71$ |
\( (T - 64)^{2} \)
|
| $73$ |
\( T^{2} - 12T + 48 \)
|
| $79$ |
\( T^{2} + 17T + 289 \)
|
| $83$ |
\( T^{2} + 2883 \)
|
| $89$ |
\( T^{2} + 138T + 6348 \)
|
| $97$ |
\( T^{2} + 8427 \)
|
| show more | |
| show less |