Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1425,2,Mod(799,1425)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1425, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([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("1425.799");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1425 = 3 \cdot 5^{2} \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1425.c (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: | \(11.3786822880\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.0.24681024.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} + 12x^{4} + 36x^{2} + 9 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| 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
\( x^{6} + 12x^{4} + 36x^{2} + 9 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} + 6\nu ) / 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{4} + 6\nu^{2} ) / 3 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{4} + 9\nu^{2} + 12 ) / 3 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{5} + 10\nu^{3} + 21\nu ) / 3 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{4} - \beta_{3} - 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( 3\beta_{2} - 6\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -6\beta_{4} + 9\beta_{3} + 24 \)
|
| \(\nu^{5}\) | \(=\) |
\( 3\beta_{5} - 30\beta_{2} + 39\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1425\mathbb{Z}\right)^\times\).
| \(n\) | \(476\) | \(1027\) | \(1351\) |
| \(\chi(n)\) | \(1\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 799.1 |
|
− | 2.66908i | − | 1.00000i | −5.12398 | 0 | −2.66908 | 0.454904i | 8.33816i | −1.00000 | 0 | ||||||||||||||||||||||||||||||||||
| 799.2 | − | 2.14510i | 1.00000i | −2.60147 | 0 | 2.14510 | − | 2.74657i | 1.29021i | −1.00000 | 0 | |||||||||||||||||||||||||||||||||||
| 799.3 | − | 0.523976i | 1.00000i | 1.72545 | 0 | 0.523976 | 3.20147i | − | 1.95205i | −1.00000 | 0 | |||||||||||||||||||||||||||||||||||
| 799.4 | 0.523976i | − | 1.00000i | 1.72545 | 0 | 0.523976 | − | 3.20147i | 1.95205i | −1.00000 | 0 | |||||||||||||||||||||||||||||||||||
| 799.5 | 2.14510i | − | 1.00000i | −2.60147 | 0 | 2.14510 | 2.74657i | − | 1.29021i | −1.00000 | 0 | |||||||||||||||||||||||||||||||||||
| 799.6 | 2.66908i | 1.00000i | −5.12398 | 0 | −2.66908 | − | 0.454904i | − | 8.33816i | −1.00000 | 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 | 1425.2.c.p | 6 | |
| 5.b | even | 2 | 1 | inner | 1425.2.c.p | 6 | |
| 5.c | odd | 4 | 1 | 1425.2.a.u | ✓ | 3 | |
| 5.c | odd | 4 | 1 | 1425.2.a.v | yes | 3 | |
| 15.e | even | 4 | 1 | 4275.2.a.be | 3 | ||
| 15.e | even | 4 | 1 | 4275.2.a.bh | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1425.2.a.u | ✓ | 3 | 5.c | odd | 4 | 1 | |
| 1425.2.a.v | yes | 3 | 5.c | odd | 4 | 1 | |
| 1425.2.c.p | 6 | 1.a | even | 1 | 1 | trivial | |
| 1425.2.c.p | 6 | 5.b | even | 2 | 1 | inner | |
| 4275.2.a.be | 3 | 15.e | even | 4 | 1 | ||
| 4275.2.a.bh | 3 | 15.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 \(S_{2}^{\mathrm{new}}(1425, [\chi])\):
|
\( T_{2}^{6} + 12T_{2}^{4} + 36T_{2}^{2} + 9 \)
|
|
\( T_{7}^{6} + 18T_{7}^{4} + 81T_{7}^{2} + 16 \)
|
|
\( T_{11}^{3} + 3T_{11}^{2} - 6T_{11} - 12 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} + 12 T^{4} + \cdots + 9 \)
|
| $3$ |
\( (T^{2} + 1)^{3} \)
|
| $5$ |
\( T^{6} \)
|
| $7$ |
\( T^{6} + 18 T^{4} + \cdots + 16 \)
|
| $11$ |
\( (T^{3} + 3 T^{2} - 6 T - 12)^{2} \)
|
| $13$ |
\( T^{6} + 60 T^{4} + \cdots + 4096 \)
|
| $17$ |
\( T^{6} \)
|
| $19$ |
\( (T + 1)^{6} \)
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| $23$ |
\( T^{6} + 57 T^{4} + \cdots + 144 \)
|
| $29$ |
\( (T + 3)^{6} \)
|
| $31$ |
\( (T^{3} - 9 T^{2} + 16)^{2} \)
|
| $37$ |
\( T^{6} + 144 T^{4} + \cdots + 33856 \)
|
| $41$ |
\( (T^{3} - 123 T + 426)^{2} \)
|
| $43$ |
\( (T^{2} + 64)^{3} \)
|
| $47$ |
\( T^{6} + 156 T^{4} + \cdots + 9216 \)
|
| $53$ |
\( T^{6} + 219 T^{4} + \cdots + 4761 \)
|
| $59$ |
\( (T^{3} + 24 T^{2} + \cdots + 444)^{2} \)
|
| $61$ |
\( (T^{3} + 9 T^{2} + \cdots - 113)^{2} \)
|
| $67$ |
\( T^{6} + 213 T^{4} + \cdots + 8464 \)
|
| $71$ |
\( (T^{3} - 6 T^{2} + \cdots + 522)^{2} \)
|
| $73$ |
\( T^{6} + 195 T^{4} + \cdots + 9409 \)
|
| $79$ |
\( (T^{3} + 15 T^{2} + \cdots - 916)^{2} \)
|
| $83$ |
\( T^{6} + 93 T^{4} + \cdots + 7056 \)
|
| $89$ |
\( (T^{3} - 3 T^{2} - 33 T + 3)^{2} \)
|
| $97$ |
\( T^{6} + 480 T^{4} + \cdots + 2096704 \)
|
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