Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [975,2,Mod(268,975)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(975, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([0, 3, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("975.268");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 975 = 3 \cdot 5^{2} \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 975.t (of order \(4\), degree \(2\), 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: | \(8\) |
| Relative dimension: | \(4\) over \(\Q(i)\) |
| Coefficient field: | 8.0.959512576.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} + 7x^{4} + 81 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{4} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} + 7x^{4} + 81 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{5} + \nu ) / 15 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{6} + 16\nu^{2} ) / 45 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 2\nu^{4} + 7 ) / 5 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{5} + 31\nu ) / 15 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -2\nu^{7} + 13\nu^{3} ) / 135 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -\nu^{6} + 2\nu^{2} ) / 9 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 8\nu^{7} + 83\nu^{3} ) / 135 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{4} - \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{6} + 5\beta_{2} ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{7} + 4\beta_{5} \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 5\beta_{3} - 7 ) / 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -\beta_{4} + 31\beta_1 ) / 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( -8\beta_{6} + 5\beta_{2} \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 13\beta_{7} - 83\beta_{5} ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/975\mathbb{Z}\right)^\times\).
| \(n\) | \(301\) | \(326\) | \(352\) |
| \(\chi(n)\) | \(\beta_{2}\) | \(1\) | \(-\beta_{2}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 268.1 |
|
−1.41421 | 0.707107 | + | 0.707107i | 0 | 0 | −1.00000 | − | 1.00000i | − | 1.90241i | 2.82843 | 1.00000i | 0 | |||||||||||||||||||||||||||||||||||||
| 268.2 | −1.41421 | 0.707107 | + | 0.707107i | 0 | 0 | −1.00000 | − | 1.00000i | 4.73084i | 2.82843 | 1.00000i | 0 | |||||||||||||||||||||||||||||||||||||||
| 268.3 | 1.41421 | −0.707107 | − | 0.707107i | 0 | 0 | −1.00000 | − | 1.00000i | − | 4.73084i | −2.82843 | 1.00000i | 0 | ||||||||||||||||||||||||||||||||||||||
| 268.4 | 1.41421 | −0.707107 | − | 0.707107i | 0 | 0 | −1.00000 | − | 1.00000i | 1.90241i | −2.82843 | 1.00000i | 0 | |||||||||||||||||||||||||||||||||||||||
| 382.1 | −1.41421 | 0.707107 | − | 0.707107i | 0 | 0 | −1.00000 | + | 1.00000i | − | 4.73084i | 2.82843 | − | 1.00000i | 0 | |||||||||||||||||||||||||||||||||||||
| 382.2 | −1.41421 | 0.707107 | − | 0.707107i | 0 | 0 | −1.00000 | + | 1.00000i | 1.90241i | 2.82843 | − | 1.00000i | 0 | ||||||||||||||||||||||||||||||||||||||
| 382.3 | 1.41421 | −0.707107 | + | 0.707107i | 0 | 0 | −1.00000 | + | 1.00000i | − | 1.90241i | −2.82843 | − | 1.00000i | 0 | |||||||||||||||||||||||||||||||||||||
| 382.4 | 1.41421 | −0.707107 | + | 0.707107i | 0 | 0 | −1.00000 | + | 1.00000i | 4.73084i | −2.82843 | − | 1.00000i | 0 | ||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.b | even | 2 | 1 | inner |
| 65.f | even | 4 | 1 | inner |
| 65.k | even | 4 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 975.2.t.c | yes | 8 |
| 5.b | even | 2 | 1 | inner | 975.2.t.c | yes | 8 |
| 5.c | odd | 4 | 2 | 975.2.k.c | ✓ | 8 | |
| 13.d | odd | 4 | 1 | 975.2.k.c | ✓ | 8 | |
| 65.f | even | 4 | 1 | inner | 975.2.t.c | yes | 8 |
| 65.g | odd | 4 | 1 | 975.2.k.c | ✓ | 8 | |
| 65.k | even | 4 | 1 | inner | 975.2.t.c | yes | 8 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 975.2.k.c | ✓ | 8 | 5.c | odd | 4 | 2 | |
| 975.2.k.c | ✓ | 8 | 13.d | odd | 4 | 1 | |
| 975.2.k.c | ✓ | 8 | 65.g | odd | 4 | 1 | |
| 975.2.t.c | yes | 8 | 1.a | even | 1 | 1 | trivial |
| 975.2.t.c | yes | 8 | 5.b | even | 2 | 1 | inner |
| 975.2.t.c | yes | 8 | 65.f | even | 4 | 1 | inner |
| 975.2.t.c | yes | 8 | 65.k | even | 4 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{2} - 2 \)
acting on \(S_{2}^{\mathrm{new}}(975, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} - 2)^{4} \)
|
| $3$ |
\( (T^{4} + 1)^{2} \)
|
| $5$ |
\( T^{8} \)
|
| $7$ |
\( (T^{4} + 26 T^{2} + 81)^{2} \)
|
| $11$ |
\( (T^{4} + 121)^{2} \)
|
| $13$ |
\( (T^{4} + 18 T^{2} + 169)^{2} \)
|
| $17$ |
\( T^{8} + 1234 T^{4} + 194481 \)
|
| $19$ |
\( (T^{4} - 4 T^{3} + \cdots + 1764)^{2} \)
|
| $23$ |
\( T^{8} + 3506 T^{4} + 28561 \)
|
| $29$ |
\( (T^{2} + 16)^{4} \)
|
| $31$ |
\( T^{8} \)
|
| $37$ |
\( (T^{4} + 58 T^{2} + 49)^{2} \)
|
| $41$ |
\( (T^{4} + 24 T^{3} + \cdots + 3721)^{2} \)
|
| $43$ |
\( T^{8} + 26056 T^{4} + 3111696 \)
|
| $47$ |
\( (T^{4} + 124 T^{2} + 676)^{2} \)
|
| $53$ |
\( T^{8} + 3506 T^{4} + 28561 \)
|
| $59$ |
\( (T^{2} - 12 T + 72)^{4} \)
|
| $61$ |
\( (T^{2} + 6 T - 79)^{4} \)
|
| $67$ |
\( (T^{4} - 104 T^{2} + 1296)^{2} \)
|
| $71$ |
\( (T^{4} + 8 T^{3} + \cdots + 8281)^{2} \)
|
| $73$ |
\( T^{8} \)
|
| $79$ |
\( (T^{2} + 169)^{4} \)
|
| $83$ |
\( (T^{4} + 124 T^{2} + 676)^{2} \)
|
| $89$ |
\( (T^{4} + 121)^{2} \)
|
| $97$ |
\( (T^{4} - 26 T^{2} + 81)^{2} \)
|
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