Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [975,2,Mod(649,975)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(975, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("975.649");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 975 = 3 \cdot 5^{2} \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 975.h (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: | \(7.78541419707\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(i, \sqrt{17})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} + 9x^{2} + 16 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 195) |
| 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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} + 9x^{2} + 16 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} + 5\nu ) / 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{2} + 5 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} - 5 \)
|
| \(\nu^{3}\) | \(=\) |
\( 4\beta_{2} - 5\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/975\mathbb{Z}\right)^\times\).
| \(n\) | \(301\) | \(326\) | \(352\) |
| \(\chi(n)\) | \(-1\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 649.1 |
|
−2.56155 | − | 1.00000i | 4.56155 | 0 | 2.56155i | −3.56155 | −6.56155 | −1.00000 | 0 | |||||||||||||||||||||||||||||
| 649.2 | −2.56155 | 1.00000i | 4.56155 | 0 | − | 2.56155i | −3.56155 | −6.56155 | −1.00000 | 0 | ||||||||||||||||||||||||||||||
| 649.3 | 1.56155 | − | 1.00000i | 0.438447 | 0 | − | 1.56155i | 0.561553 | −2.43845 | −1.00000 | 0 | |||||||||||||||||||||||||||||
| 649.4 | 1.56155 | 1.00000i | 0.438447 | 0 | 1.56155i | 0.561553 | −2.43845 | −1.00000 | 0 | |||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 65.d | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 975.2.h.e | 4 | |
| 5.b | even | 2 | 1 | 975.2.h.g | 4 | ||
| 5.c | odd | 4 | 1 | 195.2.b.c | ✓ | 4 | |
| 5.c | odd | 4 | 1 | 975.2.b.f | 4 | ||
| 13.b | even | 2 | 1 | 975.2.h.g | 4 | ||
| 15.e | even | 4 | 1 | 585.2.b.e | 4 | ||
| 20.e | even | 4 | 1 | 3120.2.g.n | 4 | ||
| 65.d | even | 2 | 1 | inner | 975.2.h.e | 4 | |
| 65.f | even | 4 | 1 | 2535.2.a.q | 2 | ||
| 65.h | odd | 4 | 1 | 195.2.b.c | ✓ | 4 | |
| 65.h | odd | 4 | 1 | 975.2.b.f | 4 | ||
| 65.k | even | 4 | 1 | 2535.2.a.p | 2 | ||
| 195.j | odd | 4 | 1 | 7605.2.a.bh | 2 | ||
| 195.s | even | 4 | 1 | 585.2.b.e | 4 | ||
| 195.u | odd | 4 | 1 | 7605.2.a.bc | 2 | ||
| 260.p | even | 4 | 1 | 3120.2.g.n | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 195.2.b.c | ✓ | 4 | 5.c | odd | 4 | 1 | |
| 195.2.b.c | ✓ | 4 | 65.h | odd | 4 | 1 | |
| 585.2.b.e | 4 | 15.e | even | 4 | 1 | ||
| 585.2.b.e | 4 | 195.s | even | 4 | 1 | ||
| 975.2.b.f | 4 | 5.c | odd | 4 | 1 | ||
| 975.2.b.f | 4 | 65.h | odd | 4 | 1 | ||
| 975.2.h.e | 4 | 1.a | even | 1 | 1 | trivial | |
| 975.2.h.e | 4 | 65.d | even | 2 | 1 | inner | |
| 975.2.h.g | 4 | 5.b | even | 2 | 1 | ||
| 975.2.h.g | 4 | 13.b | even | 2 | 1 | ||
| 2535.2.a.p | 2 | 65.k | even | 4 | 1 | ||
| 2535.2.a.q | 2 | 65.f | even | 4 | 1 | ||
| 3120.2.g.n | 4 | 20.e | even | 4 | 1 | ||
| 3120.2.g.n | 4 | 260.p | even | 4 | 1 | ||
| 7605.2.a.bc | 2 | 195.u | odd | 4 | 1 | ||
| 7605.2.a.bh | 2 | 195.j | odd | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(975, [\chi])\):
|
\( T_{2}^{2} + T_{2} - 4 \)
|
|
\( T_{7}^{2} + 3T_{7} - 2 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + T - 4)^{2} \)
|
| $3$ |
\( (T^{2} + 1)^{2} \)
|
| $5$ |
\( T^{4} \)
|
| $7$ |
\( (T^{2} + 3 T - 2)^{2} \)
|
| $11$ |
\( T^{4} + 33T^{2} + 64 \)
|
| $13$ |
\( T^{4} - 6 T^{3} + \cdots + 169 \)
|
| $17$ |
\( T^{4} + 21T^{2} + 4 \)
|
| $19$ |
\( T^{4} + 52T^{2} + 64 \)
|
| $23$ |
\( T^{4} + 33T^{2} + 64 \)
|
| $29$ |
\( (T^{2} - 68)^{2} \)
|
| $31$ |
\( (T^{2} + 36)^{2} \)
|
| $37$ |
\( (T^{2} - 7 T + 8)^{2} \)
|
| $41$ |
\( T^{4} + 21T^{2} + 4 \)
|
| $43$ |
\( T^{4} + 52T^{2} + 64 \)
|
| $47$ |
\( (T + 4)^{4} \)
|
| $53$ |
\( T^{4} + 213 T^{2} + 11236 \)
|
| $59$ |
\( (T^{2} + 16)^{2} \)
|
| $61$ |
\( (T^{2} + 21 T + 106)^{2} \)
|
| $67$ |
\( (T^{2} - 14 T + 32)^{2} \)
|
| $71$ |
\( T^{4} + 89T^{2} + 1024 \)
|
| $73$ |
\( (T^{2} + 14 T + 32)^{2} \)
|
| $79$ |
\( (T^{2} - 7 T + 8)^{2} \)
|
| $83$ |
\( (T^{2} + 2 T - 152)^{2} \)
|
| $89$ |
\( T^{4} + 189T^{2} + 324 \)
|
| $97$ |
\( (T^{2} + T - 208)^{2} \)
|
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