Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [3800,2,Mod(3649,3800)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3800, 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("3800.3649");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 3800 = 2^{3} \cdot 5^{2} \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 3800.d (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: | \(30.3431527681\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.0.59105344.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} + 21x^{4} + 116x^{2} + 64 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | no (minimal twist has level 152) |
| 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
\( x^{6} + 21x^{4} + 116x^{2} + 64 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{5} + 5\nu^{3} - 60\nu ) / 16 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{4} - 9\nu^{2} + 8 ) / 4 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{5} + 13\nu^{3} + 28\nu ) / 8 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{4} + 13\nu^{2} + 20 ) / 4 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{5} + \beta_{3} - 7 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{4} - 2\beta_{2} - 11\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -9\beta_{5} - 13\beta_{3} + 71 \)
|
| \(\nu^{5}\) | \(=\) |
\( -5\beta_{4} + 26\beta_{2} + 115\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3800\mathbb{Z}\right)^\times\).
| \(n\) | \(401\) | \(951\) | \(1901\) | \(1977\) |
| \(\chi(n)\) | \(1\) | \(1\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 3649.1 |
|
0 | − | 3.29707i | 0 | 0 | 0 | 1.78680i | 0 | −7.87067 | 0 | |||||||||||||||||||||||||||||||||||
| 3649.2 | 0 | − | 3.08387i | 0 | 0 | 0 | − | 4.29707i | 0 | −6.51027 | 0 | |||||||||||||||||||||||||||||||||||
| 3649.3 | 0 | − | 0.786802i | 0 | 0 | 0 | − | 2.08387i | 0 | 2.38094 | 0 | |||||||||||||||||||||||||||||||||||
| 3649.4 | 0 | 0.786802i | 0 | 0 | 0 | 2.08387i | 0 | 2.38094 | 0 | |||||||||||||||||||||||||||||||||||||
| 3649.5 | 0 | 3.08387i | 0 | 0 | 0 | 4.29707i | 0 | −6.51027 | 0 | |||||||||||||||||||||||||||||||||||||
| 3649.6 | 0 | 3.29707i | 0 | 0 | 0 | − | 1.78680i | 0 | −7.87067 | 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 | 3800.2.d.j | 6 | |
| 5.b | even | 2 | 1 | inner | 3800.2.d.j | 6 | |
| 5.c | odd | 4 | 1 | 152.2.a.c | ✓ | 3 | |
| 5.c | odd | 4 | 1 | 3800.2.a.r | 3 | ||
| 15.e | even | 4 | 1 | 1368.2.a.n | 3 | ||
| 20.e | even | 4 | 1 | 304.2.a.g | 3 | ||
| 20.e | even | 4 | 1 | 7600.2.a.bv | 3 | ||
| 35.f | even | 4 | 1 | 7448.2.a.bf | 3 | ||
| 40.i | odd | 4 | 1 | 1216.2.a.u | 3 | ||
| 40.k | even | 4 | 1 | 1216.2.a.v | 3 | ||
| 60.l | odd | 4 | 1 | 2736.2.a.bd | 3 | ||
| 95.g | even | 4 | 1 | 2888.2.a.o | 3 | ||
| 380.j | odd | 4 | 1 | 5776.2.a.bp | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 152.2.a.c | ✓ | 3 | 5.c | odd | 4 | 1 | |
| 304.2.a.g | 3 | 20.e | even | 4 | 1 | ||
| 1216.2.a.u | 3 | 40.i | odd | 4 | 1 | ||
| 1216.2.a.v | 3 | 40.k | even | 4 | 1 | ||
| 1368.2.a.n | 3 | 15.e | even | 4 | 1 | ||
| 2736.2.a.bd | 3 | 60.l | odd | 4 | 1 | ||
| 2888.2.a.o | 3 | 95.g | even | 4 | 1 | ||
| 3800.2.a.r | 3 | 5.c | odd | 4 | 1 | ||
| 3800.2.d.j | 6 | 1.a | even | 1 | 1 | trivial | |
| 3800.2.d.j | 6 | 5.b | even | 2 | 1 | inner | |
| 5776.2.a.bp | 3 | 380.j | odd | 4 | 1 | ||
| 7448.2.a.bf | 3 | 35.f | even | 4 | 1 | ||
| 7600.2.a.bv | 3 | 20.e | even | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(3800, [\chi])\):
|
\( T_{3}^{6} + 21T_{3}^{4} + 116T_{3}^{2} + 64 \)
|
|
\( T_{7}^{6} + 26T_{7}^{4} + 153T_{7}^{2} + 256 \)
|
|
\( T_{11}^{3} + 5T_{11}^{2} - 2T_{11} - 8 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
|
| $3$ |
\( T^{6} + 21 T^{4} + \cdots + 64 \)
|
| $5$ |
\( T^{6} \)
|
| $7$ |
\( T^{6} + 26 T^{4} + \cdots + 256 \)
|
| $11$ |
\( (T^{3} + 5 T^{2} - 2 T - 8)^{2} \)
|
| $13$ |
\( T^{6} + 29 T^{4} + \cdots + 64 \)
|
| $17$ |
\( T^{6} + 22 T^{4} + \cdots + 4 \)
|
| $19$ |
\( (T - 1)^{6} \)
|
| $23$ |
\( T^{6} + 153 T^{4} + \cdots + 65536 \)
|
| $29$ |
\( (T^{3} - 9 T^{2} - 4 T + 4)^{2} \)
|
| $31$ |
\( T^{6} \)
|
| $37$ |
\( (T^{2} + 4)^{3} \)
|
| $41$ |
\( (T^{3} - 8 T^{2} + \cdots + 128)^{2} \)
|
| $43$ |
\( T^{6} + 241 T^{4} + \cdots + 135424 \)
|
| $47$ |
\( T^{6} + 145 T^{4} + \cdots + 65536 \)
|
| $53$ |
\( T^{6} + 269 T^{4} + \cdots + 65536 \)
|
| $59$ |
\( (T^{3} - 23 T^{2} + \cdots - 376)^{2} \)
|
| $61$ |
\( (T^{3} - 3 T^{2} - 28 T + 92)^{2} \)
|
| $67$ |
\( T^{6} + 137 T^{4} + \cdots + 1024 \)
|
| $71$ |
\( (T^{3} + 12 T^{2} + \cdots - 928)^{2} \)
|
| $73$ |
\( T^{6} + 150 T^{4} + \cdots + 106276 \)
|
| $79$ |
\( (T^{3} + 26 T^{2} + \cdots + 256)^{2} \)
|
| $83$ |
\( T^{6} + 260 T^{4} + \cdots + 541696 \)
|
| $89$ |
\( (T^{3} + 18 T^{2} + \cdots - 1024)^{2} \)
|
| $97$ |
\( T^{6} + 104 T^{4} + \cdots + 16384 \)
|
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