Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [4400,2,Mod(4049,4400)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4400, 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("4400.4049");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 4400 = 2^{4} \cdot 5^{2} \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 4400.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: | \(35.1341768894\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.0.96668224.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} + 15x^{4} + 61x^{2} + 36 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 2200) |
| 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
\( x^{6} + 15x^{4} + 61x^{2} + 36 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{5} + 9\nu^{3} + 13\nu ) / 6 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{2} + 5 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{5} + 3\nu^{3} - 29\nu ) / 6 \)
|
| \(\beta_{5}\) | \(=\) |
\( \nu^{4} + 8\nu^{2} + 6 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} - 5 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{4} + \beta_{2} - 7\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{5} - 8\beta_{3} + 34 \)
|
| \(\nu^{5}\) | \(=\) |
\( 9\beta_{4} - 3\beta_{2} + 50\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4400\mathbb{Z}\right)^\times\).
| \(n\) | \(177\) | \(1201\) | \(2751\) | \(3301\) |
| \(\chi(n)\) | \(-1\) | \(1\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 4049.1 |
|
0 | − | 2.75153i | 0 | 0 | 0 | 3.57093i | 0 | −4.57093 | 0 | |||||||||||||||||||||||||||||||||||
| 4049.2 | 0 | − | 2.59261i | 0 | 0 | 0 | − | 2.72165i | 0 | −3.72165 | 0 | |||||||||||||||||||||||||||||||||||
| 4049.3 | 0 | − | 0.841083i | 0 | 0 | 0 | − | 3.29258i | 0 | 2.29258 | 0 | |||||||||||||||||||||||||||||||||||
| 4049.4 | 0 | 0.841083i | 0 | 0 | 0 | 3.29258i | 0 | 2.29258 | 0 | |||||||||||||||||||||||||||||||||||||
| 4049.5 | 0 | 2.59261i | 0 | 0 | 0 | 2.72165i | 0 | −3.72165 | 0 | |||||||||||||||||||||||||||||||||||||
| 4049.6 | 0 | 2.75153i | 0 | 0 | 0 | − | 3.57093i | 0 | −4.57093 | 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 | 4400.2.b.bb | 6 | |
| 4.b | odd | 2 | 1 | 2200.2.b.m | 6 | ||
| 5.b | even | 2 | 1 | inner | 4400.2.b.bb | 6 | |
| 5.c | odd | 4 | 1 | 4400.2.a.by | 3 | ||
| 5.c | odd | 4 | 1 | 4400.2.a.bz | 3 | ||
| 20.d | odd | 2 | 1 | 2200.2.b.m | 6 | ||
| 20.e | even | 4 | 1 | 2200.2.a.u | ✓ | 3 | |
| 20.e | even | 4 | 1 | 2200.2.a.v | yes | 3 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2200.2.a.u | ✓ | 3 | 20.e | even | 4 | 1 | |
| 2200.2.a.v | yes | 3 | 20.e | even | 4 | 1 | |
| 2200.2.b.m | 6 | 4.b | odd | 2 | 1 | ||
| 2200.2.b.m | 6 | 20.d | odd | 2 | 1 | ||
| 4400.2.a.by | 3 | 5.c | odd | 4 | 1 | ||
| 4400.2.a.bz | 3 | 5.c | odd | 4 | 1 | ||
| 4400.2.b.bb | 6 | 1.a | even | 1 | 1 | trivial | |
| 4400.2.b.bb | 6 | 5.b | even | 2 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(4400, [\chi])\):
|
\( T_{3}^{6} + 15T_{3}^{4} + 61T_{3}^{2} + 36 \)
|
|
\( T_{7}^{6} + 31T_{7}^{4} + 313T_{7}^{2} + 1024 \)
|
|
\( T_{13}^{2} + 1 \)
|
|
\( T_{17}^{6} + 23T_{17}^{4} + 41T_{17}^{2} + 16 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
|
| $3$ |
\( T^{6} + 15 T^{4} + \cdots + 36 \)
|
| $5$ |
\( T^{6} \)
|
| $7$ |
\( T^{6} + 31 T^{4} + \cdots + 1024 \)
|
| $11$ |
\( (T + 1)^{6} \)
|
| $13$ |
\( (T^{2} + 1)^{3} \)
|
| $17$ |
\( T^{6} + 23 T^{4} + \cdots + 16 \)
|
| $19$ |
\( (T^{3} - 7 T^{2} - 13 T + 27)^{2} \)
|
| $23$ |
\( T^{6} + 48 T^{4} + \cdots + 81 \)
|
| $29$ |
\( (T^{3} + 10 T^{2} + \cdots - 201)^{2} \)
|
| $31$ |
\( (T^{3} - 3 T^{2} + \cdots + 207)^{2} \)
|
| $37$ |
\( T^{6} + 66 T^{4} + \cdots + 1936 \)
|
| $41$ |
\( (T^{3} - 4 T^{2} - 17 T + 24)^{2} \)
|
| $43$ |
\( T^{6} + 193 T^{4} + \cdots + 258064 \)
|
| $47$ |
\( T^{6} + 154 T^{4} + \cdots + 36864 \)
|
| $53$ |
\( T^{6} + 307 T^{4} + \cdots + 20164 \)
|
| $59$ |
\( (T^{3} - 77 T + 192)^{2} \)
|
| $61$ |
\( (T^{3} + T^{2} - 41 T - 114)^{2} \)
|
| $67$ |
\( T^{6} + 228 T^{4} + \cdots + 9216 \)
|
| $71$ |
\( (T^{3} + 17 T^{2} + \cdots + 52)^{2} \)
|
| $73$ |
\( T^{6} + 399 T^{4} + \cdots + 1106704 \)
|
| $79$ |
\( (T^{3} - 11 T^{2} + \cdots + 144)^{2} \)
|
| $83$ |
\( T^{6} + 388 T^{4} + \cdots + 687241 \)
|
| $89$ |
\( (T^{3} + 30 T^{2} + \cdots + 879)^{2} \)
|
| $97$ |
\( T^{6} + 164 T^{4} + \cdots + 529 \)
|
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