Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [33,3,Mod(10,33)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(33, base_ring=CyclotomicField(2))
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("33.10");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 33 = 3 \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 33.c (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: | \(0.899184872389\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.0.39744.5 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 2x^{3} + 12x^{2} + 4x + 22 \)
|
| 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 a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring in terms of a root \(\nu\) of
\( x^{4} - 2x^{3} + 12x^{2} + 4x + 22 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{3} + \nu^{2} + 2\nu + 23 ) / 13 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} + \nu^{2} + 28\nu + 10 ) / 13 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -3\nu^{3} + 10\nu^{2} - 32\nu + 9 ) / 13 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} - \beta _1 + 1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + \beta_{2} + 2\beta _1 - 5 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{3} - 2\beta_{2} + 12\beta _1 - 19 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/33\mathbb{Z}\right)^\times\).
| \(n\) | \(13\) | \(23\) |
| \(\chi(n)\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 10.1 |
|
− | 3.53045i | −1.73205 | −8.46410 | 6.19615 | 6.11492i | − | 2.58447i | 15.7603i | 3.00000 | − | 21.8752i | |||||||||||||||||||||||||||
| 10.2 | − | 2.35285i | 1.73205 | −1.53590 | −4.19615 | − | 4.07525i | 6.42810i | − | 5.79766i | 3.00000 | 9.87291i | ||||||||||||||||||||||||||||
| 10.3 | 2.35285i | 1.73205 | −1.53590 | −4.19615 | 4.07525i | − | 6.42810i | 5.79766i | 3.00000 | − | 9.87291i | |||||||||||||||||||||||||||||
| 10.4 | 3.53045i | −1.73205 | −8.46410 | 6.19615 | − | 6.11492i | 2.58447i | − | 15.7603i | 3.00000 | 21.8752i | |||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 11.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 33.3.c.a | ✓ | 4 |
| 3.b | odd | 2 | 1 | 99.3.c.b | 4 | ||
| 4.b | odd | 2 | 1 | 528.3.j.c | 4 | ||
| 5.b | even | 2 | 1 | 825.3.b.a | 4 | ||
| 5.c | odd | 4 | 2 | 825.3.h.a | 8 | ||
| 8.b | even | 2 | 1 | 2112.3.j.a | 4 | ||
| 8.d | odd | 2 | 1 | 2112.3.j.d | 4 | ||
| 11.b | odd | 2 | 1 | inner | 33.3.c.a | ✓ | 4 |
| 11.c | even | 5 | 4 | 363.3.g.e | 16 | ||
| 11.d | odd | 10 | 4 | 363.3.g.e | 16 | ||
| 12.b | even | 2 | 1 | 1584.3.j.f | 4 | ||
| 33.d | even | 2 | 1 | 99.3.c.b | 4 | ||
| 44.c | even | 2 | 1 | 528.3.j.c | 4 | ||
| 55.d | odd | 2 | 1 | 825.3.b.a | 4 | ||
| 55.e | even | 4 | 2 | 825.3.h.a | 8 | ||
| 88.b | odd | 2 | 1 | 2112.3.j.a | 4 | ||
| 88.g | even | 2 | 1 | 2112.3.j.d | 4 | ||
| 132.d | odd | 2 | 1 | 1584.3.j.f | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 33.3.c.a | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 33.3.c.a | ✓ | 4 | 11.b | odd | 2 | 1 | inner |
| 99.3.c.b | 4 | 3.b | odd | 2 | 1 | ||
| 99.3.c.b | 4 | 33.d | even | 2 | 1 | ||
| 363.3.g.e | 16 | 11.c | even | 5 | 4 | ||
| 363.3.g.e | 16 | 11.d | odd | 10 | 4 | ||
| 528.3.j.c | 4 | 4.b | odd | 2 | 1 | ||
| 528.3.j.c | 4 | 44.c | even | 2 | 1 | ||
| 825.3.b.a | 4 | 5.b | even | 2 | 1 | ||
| 825.3.b.a | 4 | 55.d | odd | 2 | 1 | ||
| 825.3.h.a | 8 | 5.c | odd | 4 | 2 | ||
| 825.3.h.a | 8 | 55.e | even | 4 | 2 | ||
| 1584.3.j.f | 4 | 12.b | even | 2 | 1 | ||
| 1584.3.j.f | 4 | 132.d | odd | 2 | 1 | ||
| 2112.3.j.a | 4 | 8.b | even | 2 | 1 | ||
| 2112.3.j.a | 4 | 88.b | odd | 2 | 1 | ||
| 2112.3.j.d | 4 | 8.d | odd | 2 | 1 | ||
| 2112.3.j.d | 4 | 88.g | even | 2 | 1 | ||
Hecke kernels
This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(33, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} + 18T^{2} + 69 \)
|
| $3$ |
\( (T^{2} - 3)^{2} \)
|
| $5$ |
\( (T^{2} - 2 T - 26)^{2} \)
|
| $7$ |
\( T^{4} + 48T^{2} + 276 \)
|
| $11$ |
\( T^{4} - 20 T^{3} + \cdots + 14641 \)
|
| $13$ |
\( T^{4} + 624 T^{2} + 33396 \)
|
| $17$ |
\( T^{4} + 216T^{2} + 9936 \)
|
| $19$ |
\( T^{4} + 936T^{2} + 9936 \)
|
| $23$ |
\( (T^{2} + 46 T + 454)^{2} \)
|
| $29$ |
\( T^{4} + 1536 T^{2} + 70656 \)
|
| $31$ |
\( (T^{2} - 40 T - 572)^{2} \)
|
| $37$ |
\( (T^{2} + 32 T + 244)^{2} \)
|
| $41$ |
\( T^{4} + 2304 T^{2} + 4416 \)
|
| $43$ |
\( T^{4} + 7272 T^{2} + 3843024 \)
|
| $47$ |
\( (T^{2} - 50 T + 598)^{2} \)
|
| $53$ |
\( (T^{2} - 14 T - 314)^{2} \)
|
| $59$ |
\( (T^{2} - 20 T - 5192)^{2} \)
|
| $61$ |
\( T^{4} + 6144 T^{2} + 2596884 \)
|
| $67$ |
\( (T + 34)^{4} \)
|
| $71$ |
\( (T^{2} + 142 T + 5038)^{2} \)
|
| $73$ |
\( T^{4} + 4608 T^{2} + 4809024 \)
|
| $79$ |
\( T^{4} + 10128 T^{2} + 5643924 \)
|
| $83$ |
\( T^{4} + 5184 T^{2} + 4416 \)
|
| $89$ |
\( (T^{2} - 152 T + 4804)^{2} \)
|
| $97$ |
\( (T^{2} + 188 T + 3544)^{2} \)
|
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