Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [475,2,Mod(324,475)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(475, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([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("475.324");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 475 = 5^{2} \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 475.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: | \(3.79289409601\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.0.1827904.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} + 9x^{4} + 14x^{2} + 1 \)
|
| 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
\( x^{6} + 9x^{4} + 14x^{2} + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{4} + 5\nu^{2} - 1 ) / 5 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{4} + 10\nu^{2} + 14 ) / 5 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{5} + 10\nu^{3} + 19\nu ) / 5 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -3\nu^{5} - 25\nu^{3} - 27\nu ) / 5 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} - \beta_{2} - 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{5} + 3\beta_{4} - 6\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -5\beta_{3} + 10\beta_{2} + 16 \)
|
| \(\nu^{5}\) | \(=\) |
\( -10\beta_{5} - 25\beta_{4} + 41\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/475\mathbb{Z}\right)^\times\).
| \(n\) | \(77\) | \(401\) |
| \(\chi(n)\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 324.1 |
|
− | 2.37720i | 1.27389i | −3.65109 | 0 | 3.02830 | − | 0.726109i | 3.92498i | 1.37720 | 0 | ||||||||||||||||||||||||||||||||||
| 324.2 | − | 1.65109i | − | 2.37720i | −0.726109 | 0 | −3.92498 | − | 0.377203i | − | 2.10331i | −2.65109 | 0 | |||||||||||||||||||||||||||||||||
| 324.3 | − | 1.27389i | − | 1.65109i | 0.377203 | 0 | −2.10331 | − | 3.65109i | − | 3.02830i | 0.273891 | 0 | |||||||||||||||||||||||||||||||||
| 324.4 | 1.27389i | 1.65109i | 0.377203 | 0 | −2.10331 | 3.65109i | 3.02830i | 0.273891 | 0 | |||||||||||||||||||||||||||||||||||||
| 324.5 | 1.65109i | 2.37720i | −0.726109 | 0 | −3.92498 | 0.377203i | 2.10331i | −2.65109 | 0 | |||||||||||||||||||||||||||||||||||||
| 324.6 | 2.37720i | − | 1.27389i | −3.65109 | 0 | 3.02830 | 0.726109i | − | 3.92498i | 1.37720 | 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 | 475.2.b.b | 6 | |
| 5.b | even | 2 | 1 | inner | 475.2.b.b | 6 | |
| 5.c | odd | 4 | 1 | 475.2.a.e | ✓ | 3 | |
| 5.c | odd | 4 | 1 | 475.2.a.g | yes | 3 | |
| 15.e | even | 4 | 1 | 4275.2.a.ba | 3 | ||
| 15.e | even | 4 | 1 | 4275.2.a.bm | 3 | ||
| 20.e | even | 4 | 1 | 7600.2.a.bh | 3 | ||
| 20.e | even | 4 | 1 | 7600.2.a.cc | 3 | ||
| 95.g | even | 4 | 1 | 9025.2.a.y | 3 | ||
| 95.g | even | 4 | 1 | 9025.2.a.bc | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 475.2.a.e | ✓ | 3 | 5.c | odd | 4 | 1 | |
| 475.2.a.g | yes | 3 | 5.c | odd | 4 | 1 | |
| 475.2.b.b | 6 | 1.a | even | 1 | 1 | trivial | |
| 475.2.b.b | 6 | 5.b | even | 2 | 1 | inner | |
| 4275.2.a.ba | 3 | 15.e | even | 4 | 1 | ||
| 4275.2.a.bm | 3 | 15.e | even | 4 | 1 | ||
| 7600.2.a.bh | 3 | 20.e | even | 4 | 1 | ||
| 7600.2.a.cc | 3 | 20.e | even | 4 | 1 | ||
| 9025.2.a.y | 3 | 95.g | even | 4 | 1 | ||
| 9025.2.a.bc | 3 | 95.g | even | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{6} + 10T_{2}^{4} + 29T_{2}^{2} + 25 \)
acting on \(S_{2}^{\mathrm{new}}(475, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} + 10 T^{4} + \cdots + 25 \)
|
| $3$ |
\( T^{6} + 10 T^{4} + \cdots + 25 \)
|
| $5$ |
\( T^{6} \)
|
| $7$ |
\( T^{6} + 14 T^{4} + \cdots + 1 \)
|
| $11$ |
\( (T^{3} - T^{2} - 4 T - 1)^{2} \)
|
| $13$ |
\( T^{6} + 81 T^{4} + \cdots + 10609 \)
|
| $17$ |
\( T^{6} + 74 T^{4} + \cdots + 6241 \)
|
| $19$ |
\( (T + 1)^{6} \)
|
| $23$ |
\( T^{6} + 82 T^{4} + \cdots + 15625 \)
|
| $29$ |
\( (T^{3} - 5 T^{2} + 4 T + 5)^{2} \)
|
| $31$ |
\( (T^{3} + T^{2} - 30 T + 53)^{2} \)
|
| $37$ |
\( T^{6} + 173 T^{4} + \cdots + 156025 \)
|
| $41$ |
\( (T^{3} - T^{2} - 82 T + 155)^{2} \)
|
| $43$ |
\( T^{6} + 251 T^{4} + \cdots + 100489 \)
|
| $47$ |
\( T^{6} + 209 T^{4} + \cdots + 96721 \)
|
| $53$ |
\( T^{6} + 355 T^{4} + \cdots + 819025 \)
|
| $59$ |
\( (T^{3} - 6 T^{2} + \cdots + 200)^{2} \)
|
| $61$ |
\( (T^{3} - 3 T^{2} - 10 T - 1)^{2} \)
|
| $67$ |
\( T^{6} + 169 T^{4} + \cdots + 28561 \)
|
| $71$ |
\( (T^{3} - 7 T^{2} - T + 47)^{2} \)
|
| $73$ |
\( T^{6} + 61 T^{4} + \cdots + 2809 \)
|
| $79$ |
\( (T^{3} - 18 T^{2} + \cdots + 395)^{2} \)
|
| $83$ |
\( T^{6} + 549 T^{4} + \cdots + 17161 \)
|
| $89$ |
\( (T^{3} - 20 T^{2} + \cdots + 125)^{2} \)
|
| $97$ |
\( T^{6} + 169 T^{4} + \cdots + 28561 \)
|
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