Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2200,2,Mod(1849,2200)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2200, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 1, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2200.1849");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2200 = 2^{3} \cdot 5^{2} \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2200.b (of order \(2\), degree \(1\), 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: | \(17.5670884447\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.0.44836416.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} + 12x^{4} + 36x^{2} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 1 \) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} + 12x^{4} + 36x^{2} + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} + 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} + 6\nu \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{4} + 6\nu^{2} \)
|
| \(\beta_{5}\) | \(=\) |
\( \nu^{5} + 10\nu^{3} + 24\nu \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} - 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} - 6\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{4} - 6\beta_{2} + 24 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{5} - 10\beta_{3} + 36\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2200\mathbb{Z}\right)^\times\).
| \(n\) | \(177\) | \(551\) | \(1101\) | \(1201\) |
| \(\chi(n)\) | \(-1\) | \(1\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1849.1 |
|
0 | − | 3.36147i | 0 | 0 | 0 | − | 0.576535i | 0 | −8.29947 | 0 | ||||||||||||||||||||||||||||||||||
| 1849.2 | 0 | − | 1.52892i | 0 | 0 | 0 | 1.39543i | 0 | 0.662410 | 0 | ||||||||||||||||||||||||||||||||||||
| 1849.3 | 0 | − | 1.16745i | 0 | 0 | 0 | 4.97196i | 0 | 1.63706 | 0 | ||||||||||||||||||||||||||||||||||||
| 1849.4 | 0 | 1.16745i | 0 | 0 | 0 | − | 4.97196i | 0 | 1.63706 | 0 | ||||||||||||||||||||||||||||||||||||
| 1849.5 | 0 | 1.52892i | 0 | 0 | 0 | − | 1.39543i | 0 | 0.662410 | 0 | ||||||||||||||||||||||||||||||||||||
| 1849.6 | 0 | 3.36147i | 0 | 0 | 0 | 0.576535i | 0 | −8.29947 | 0 | |||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 2200.2.b.l | 6 | |
| 4.b | odd | 2 | 1 | 4400.2.b.bc | 6 | ||
| 5.b | even | 2 | 1 | inner | 2200.2.b.l | 6 | |
| 5.c | odd | 4 | 1 | 2200.2.a.t | ✓ | 3 | |
| 5.c | odd | 4 | 1 | 2200.2.a.w | yes | 3 | |
| 20.d | odd | 2 | 1 | 4400.2.b.bc | 6 | ||
| 20.e | even | 4 | 1 | 4400.2.a.bx | 3 | ||
| 20.e | even | 4 | 1 | 4400.2.a.ca | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2200.2.a.t | ✓ | 3 | 5.c | odd | 4 | 1 | |
| 2200.2.a.w | yes | 3 | 5.c | odd | 4 | 1 | |
| 2200.2.b.l | 6 | 1.a | even | 1 | 1 | trivial | |
| 2200.2.b.l | 6 | 5.b | even | 2 | 1 | inner | |
| 4400.2.a.bx | 3 | 20.e | even | 4 | 1 | ||
| 4400.2.a.ca | 3 | 20.e | even | 4 | 1 | ||
| 4400.2.b.bc | 6 | 4.b | odd | 2 | 1 | ||
| 4400.2.b.bc | 6 | 20.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2200, [\chi])\):
|
\( T_{3}^{6} + 15T_{3}^{4} + 45T_{3}^{2} + 36 \)
|
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\( T_{7}^{6} + 27T_{7}^{4} + 57T_{7}^{2} + 16 \)
|
|
\( T_{13}^{6} + 51T_{13}^{4} + 531T_{13}^{2} + 225 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
|
| $3$ |
\( T^{6} + 15 T^{4} + \cdots + 36 \)
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| $5$ |
\( T^{6} \)
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| $7$ |
\( T^{6} + 27 T^{4} + \cdots + 16 \)
|
| $11$ |
\( (T + 1)^{6} \)
|
| $13$ |
\( T^{6} + 51 T^{4} + \cdots + 225 \)
|
| $17$ |
\( T^{6} + 99 T^{4} + \cdots + 20736 \)
|
| $19$ |
\( (T + 1)^{6} \)
|
| $23$ |
\( T^{6} + 24 T^{4} + \cdots + 25 \)
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| $29$ |
\( (T^{3} + 12 T^{2} + \cdots + 31)^{2} \)
|
| $31$ |
\( (T^{3} + 3 T^{2} - 45 T + 73)^{2} \)
|
| $37$ |
\( T^{6} + 90 T^{4} + \cdots + 16 \)
|
| $41$ |
\( (T^{3} - 6 T^{2} - 3 T + 20)^{2} \)
|
| $43$ |
\( T^{6} + 33 T^{4} + \cdots + 144 \)
|
| $47$ |
\( T^{6} + 162 T^{4} + \cdots + 64 \)
|
| $53$ |
\( T^{6} + 219 T^{4} + \cdots + 293764 \)
|
| $59$ |
\( (T^{3} + 24 T^{2} + \cdots - 100)^{2} \)
|
| $61$ |
\( (T^{3} + 3 T^{2} + \cdots - 346)^{2} \)
|
| $67$ |
\( T^{6} + 132 T^{4} + \cdots + 16384 \)
|
| $71$ |
\( (T^{3} + 15 T^{2} + \cdots - 1500)^{2} \)
|
| $73$ |
\( T^{6} + 387 T^{4} + \cdots + 1401856 \)
|
| $79$ |
\( (T^{3} + 15 T^{2} + \cdots - 172)^{2} \)
|
| $83$ |
\( T^{6} + 240 T^{4} + \cdots + 169 \)
|
| $89$ |
\( (T^{3} + 6 T^{2} + \cdots - 453)^{2} \)
|
| $97$ |
\( T^{6} + 564 T^{4} + \cdots + 6135529 \)
|
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