Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [49,12,Mod(18,49)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(49, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([4]))
N = Newforms(chi, 12, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("49.18");
S:= CuspForms(chi, 12);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 49 = 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 12 \) |
| Character orbit: | \([\chi]\) | \(=\) | 49.c (of order \(3\), degree \(2\), 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: | \(37.6488158474\) |
| 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: | \(\Z[a_1, \ldots, a_{9}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 1) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-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 \(\left(\mathbb{Z}/49\mathbb{Z}\right)^\times\).
| \(n\) | \(3\) |
| \(\chi(n)\) | \(-\zeta_{6}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 18.1 |
|
12.0000 | + | 20.7846i | −126.000 | + | 218.238i | 736.000 | − | 1274.79i | −2415.00 | − | 4182.90i | −6048.00 | 0 | 84480.0 | 56821.5 | + | 98417.7i | 57960.0 | − | 100390.i | ||||||||||||
| 30.1 | 12.0000 | − | 20.7846i | −126.000 | − | 218.238i | 736.000 | + | 1274.79i | −2415.00 | + | 4182.90i | −6048.00 | 0 | 84480.0 | 56821.5 | − | 98417.7i | 57960.0 | + | 100390.i | |||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 49.12.c.b | 2 | |
| 7.b | odd | 2 | 1 | 49.12.c.c | 2 | ||
| 7.c | even | 3 | 1 | 1.12.a.a | ✓ | 1 | |
| 7.c | even | 3 | 1 | inner | 49.12.c.b | 2 | |
| 7.d | odd | 6 | 1 | 49.12.a.a | 1 | ||
| 7.d | odd | 6 | 1 | 49.12.c.c | 2 | ||
| 21.h | odd | 6 | 1 | 9.12.a.b | 1 | ||
| 28.g | odd | 6 | 1 | 16.12.a.a | 1 | ||
| 35.j | even | 6 | 1 | 25.12.a.b | 1 | ||
| 35.l | odd | 12 | 2 | 25.12.b.b | 2 | ||
| 56.k | odd | 6 | 1 | 64.12.a.f | 1 | ||
| 56.p | even | 6 | 1 | 64.12.a.b | 1 | ||
| 63.g | even | 3 | 1 | 81.12.c.d | 2 | ||
| 63.h | even | 3 | 1 | 81.12.c.d | 2 | ||
| 63.j | odd | 6 | 1 | 81.12.c.b | 2 | ||
| 63.n | odd | 6 | 1 | 81.12.c.b | 2 | ||
| 77.h | odd | 6 | 1 | 121.12.a.b | 1 | ||
| 84.n | even | 6 | 1 | 144.12.a.d | 1 | ||
| 91.r | even | 6 | 1 | 169.12.a.a | 1 | ||
| 105.o | odd | 6 | 1 | 225.12.a.b | 1 | ||
| 105.x | even | 12 | 2 | 225.12.b.d | 2 | ||
| 112.u | odd | 12 | 2 | 256.12.b.c | 2 | ||
| 112.w | even | 12 | 2 | 256.12.b.e | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1.12.a.a | ✓ | 1 | 7.c | even | 3 | 1 | |
| 9.12.a.b | 1 | 21.h | odd | 6 | 1 | ||
| 16.12.a.a | 1 | 28.g | odd | 6 | 1 | ||
| 25.12.a.b | 1 | 35.j | even | 6 | 1 | ||
| 25.12.b.b | 2 | 35.l | odd | 12 | 2 | ||
| 49.12.a.a | 1 | 7.d | odd | 6 | 1 | ||
| 49.12.c.b | 2 | 1.a | even | 1 | 1 | trivial | |
| 49.12.c.b | 2 | 7.c | even | 3 | 1 | inner | |
| 49.12.c.c | 2 | 7.b | odd | 2 | 1 | ||
| 49.12.c.c | 2 | 7.d | odd | 6 | 1 | ||
| 64.12.a.b | 1 | 56.p | even | 6 | 1 | ||
| 64.12.a.f | 1 | 56.k | odd | 6 | 1 | ||
| 81.12.c.b | 2 | 63.j | odd | 6 | 1 | ||
| 81.12.c.b | 2 | 63.n | odd | 6 | 1 | ||
| 81.12.c.d | 2 | 63.g | even | 3 | 1 | ||
| 81.12.c.d | 2 | 63.h | even | 3 | 1 | ||
| 121.12.a.b | 1 | 77.h | odd | 6 | 1 | ||
| 144.12.a.d | 1 | 84.n | even | 6 | 1 | ||
| 169.12.a.a | 1 | 91.r | even | 6 | 1 | ||
| 225.12.a.b | 1 | 105.o | odd | 6 | 1 | ||
| 225.12.b.d | 2 | 105.x | even | 12 | 2 | ||
| 256.12.b.c | 2 | 112.u | odd | 12 | 2 | ||
| 256.12.b.e | 2 | 112.w | even | 12 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{12}^{\mathrm{new}}(49, [\chi])\):
|
\( T_{2}^{2} - 24T_{2} + 576 \)
|
|
\( T_{3}^{2} + 252T_{3} + 63504 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} - 24T + 576 \)
|
| $3$ |
\( T^{2} + 252T + 63504 \)
|
| $5$ |
\( T^{2} + 4830 T + 23328900 \)
|
| $7$ |
\( T^{2} \)
|
| $11$ |
\( T^{2} + \cdots + 285809990544 \)
|
| $13$ |
\( (T + 577738)^{2} \)
|
| $17$ |
\( T^{2} + \cdots + 47691924412356 \)
|
| $19$ |
\( T^{2} + \cdots + 113665876416400 \)
|
| $23$ |
\( T^{2} + \cdots + 347571590865984 \)
|
| $29$ |
\( (T - 128406630)^{2} \)
|
| $31$ |
\( T^{2} + \cdots + 27\!\cdots\!24 \)
|
| $37$ |
\( T^{2} + \cdots + 33\!\cdots\!96 \)
|
| $41$ |
\( (T - 308120442)^{2} \)
|
| $43$ |
\( (T + 17125708)^{2} \)
|
| $47$ |
\( T^{2} + \cdots + 72\!\cdots\!16 \)
|
| $53$ |
\( T^{2} + \cdots + 25\!\cdots\!04 \)
|
| $59$ |
\( T^{2} + \cdots + 26\!\cdots\!00 \)
|
| $61$ |
\( T^{2} + \cdots + 48\!\cdots\!44 \)
|
| $67$ |
\( T^{2} + \cdots + 23\!\cdots\!56 \)
|
| $71$ |
\( (T - 9791485272)^{2} \)
|
| $73$ |
\( T^{2} + \cdots + 21\!\cdots\!84 \)
|
| $79$ |
\( T^{2} + \cdots + 14\!\cdots\!00 \)
|
| $83$ |
\( (T + 29335099668)^{2} \)
|
| $89$ |
\( T^{2} + \cdots + 62\!\cdots\!00 \)
|
| $97$ |
\( (T - 75013568546)^{2} \)
|
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