Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [930,2,Mod(497,930)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(930, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([2, 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("930.497");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 930 = 2 \cdot 3 \cdot 5 \cdot 31 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 930.j (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: | \(7.42608738798\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Relative dimension: | \(4\) over \(\Q(i)\) |
| Coefficient field: | 8.0.1698758656.6 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} + 18x^{6} + 97x^{4} + 176x^{2} + 64 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| 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
\( x^{8} + 18x^{6} + 97x^{4} + 176x^{2} + 64 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{7} + 2\nu^{6} + 18\nu^{5} + 28\nu^{4} + 89\nu^{3} + 74\nu^{2} + 104\nu - 16 ) / 64 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{7} - 2\nu^{6} + 18\nu^{5} - 28\nu^{4} + 89\nu^{3} - 74\nu^{2} + 104\nu + 16 ) / 64 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{7} - 2\nu^{6} - 10\nu^{5} - 20\nu^{4} + 15\nu^{3} - 2\nu^{2} + 120\nu + 80 ) / 64 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{7} - 2\nu^{6} - 10\nu^{5} - 20\nu^{4} + 15\nu^{3} - 2\nu^{2} + 184\nu + 80 ) / 64 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{7} - 2\nu^{6} + 10\nu^{5} - 20\nu^{4} - 15\nu^{3} - 2\nu^{2} - 120\nu + 80 ) / 64 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( \nu^{6} + 14\nu^{4} + 45\nu^{2} + 32 ) / 8 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 3\nu^{7} + 46\nu^{5} + 179\nu^{3} + 168\nu ) / 64 \)
|
| \(\nu\) | \(=\) |
\( \beta_{4} - \beta_{3} \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{6} + 2\beta_{2} - 2\beta _1 - 5 \)
|
| \(\nu^{3}\) | \(=\) |
\( 4\beta_{7} - 2\beta_{5} - 5\beta_{4} + 7\beta_{3} - 4\beta_{2} - 4\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -9\beta_{6} + 4\beta_{5} + 4\beta_{3} - 22\beta_{2} + 22\beta _1 + 37 \)
|
| \(\nu^{5}\) | \(=\) |
\( -52\beta_{7} + 22\beta_{5} + 37\beta_{4} - 59\beta_{3} + 56\beta_{2} + 56\beta_1 \)
|
| \(\nu^{6}\) | \(=\) |
\( 89\beta_{6} - 56\beta_{5} - 56\beta_{3} + 218\beta_{2} - 218\beta _1 - 325 \)
|
| \(\nu^{7}\) | \(=\) |
\( 580\beta_{7} - 218\beta_{5} - 325\beta_{4} + 543\beta_{3} - 620\beta_{2} - 620\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/930\mathbb{Z}\right)^\times\).
| \(n\) | \(187\) | \(311\) | \(871\) |
| \(\chi(n)\) | \(\beta_{7}\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 497.1 |
|
−0.707107 | + | 0.707107i | −1.70711 | − | 0.292893i | − | 1.00000i | −2.23483 | + | 0.0743018i | 1.41421 | − | 1.00000i | 0.218591 | + | 0.218591i | 0.707107 | + | 0.707107i | 2.82843 | + | 1.00000i | 1.52773 | − | 1.63280i | |||||||||||||||||||||||||
| 497.2 | −0.707107 | + | 0.707107i | −1.70711 | − | 0.292893i | − | 1.00000i | 1.52773 | + | 1.63280i | 1.41421 | − | 1.00000i | −1.33991 | − | 1.33991i | 0.707107 | + | 0.707107i | 2.82843 | + | 1.00000i | −2.23483 | − | 0.0743018i | ||||||||||||||||||||||||||
| 497.3 | 0.707107 | − | 0.707107i | −0.292893 | − | 1.70711i | − | 1.00000i | −0.489528 | + | 2.18183i | −1.41421 | − | 1.00000i | −0.474719 | − | 0.474719i | −0.707107 | − | 0.707107i | −2.82843 | + | 1.00000i | 1.19663 | + | 1.88893i | ||||||||||||||||||||||||||
| 497.4 | 0.707107 | − | 0.707107i | −0.292893 | − | 1.70711i | − | 1.00000i | 1.19663 | − | 1.88893i | −1.41421 | − | 1.00000i | 3.59604 | + | 3.59604i | −0.707107 | − | 0.707107i | −2.82843 | + | 1.00000i | −0.489528 | − | 2.18183i | ||||||||||||||||||||||||||
| 683.1 | −0.707107 | − | 0.707107i | −1.70711 | + | 0.292893i | 1.00000i | −2.23483 | − | 0.0743018i | 1.41421 | + | 1.00000i | 0.218591 | − | 0.218591i | 0.707107 | − | 0.707107i | 2.82843 | − | 1.00000i | 1.52773 | + | 1.63280i | |||||||||||||||||||||||||||
| 683.2 | −0.707107 | − | 0.707107i | −1.70711 | + | 0.292893i | 1.00000i | 1.52773 | − | 1.63280i | 1.41421 | + | 1.00000i | −1.33991 | + | 1.33991i | 0.707107 | − | 0.707107i | 2.82843 | − | 1.00000i | −2.23483 | + | 0.0743018i | |||||||||||||||||||||||||||
| 683.3 | 0.707107 | + | 0.707107i | −0.292893 | + | 1.70711i | 1.00000i | −0.489528 | − | 2.18183i | −1.41421 | + | 1.00000i | −0.474719 | + | 0.474719i | −0.707107 | + | 0.707107i | −2.82843 | − | 1.00000i | 1.19663 | − | 1.88893i | |||||||||||||||||||||||||||
| 683.4 | 0.707107 | + | 0.707107i | −0.292893 | + | 1.70711i | 1.00000i | 1.19663 | + | 1.88893i | −1.41421 | + | 1.00000i | 3.59604 | − | 3.59604i | −0.707107 | + | 0.707107i | −2.82843 | − | 1.00000i | −0.489528 | + | 2.18183i | |||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 15.e | even | 4 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 930.2.j.c | ✓ | 8 |
| 3.b | odd | 2 | 1 | 930.2.j.f | yes | 8 | |
| 5.c | odd | 4 | 1 | 930.2.j.f | yes | 8 | |
| 15.e | even | 4 | 1 | inner | 930.2.j.c | ✓ | 8 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 930.2.j.c | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 930.2.j.c | ✓ | 8 | 15.e | even | 4 | 1 | inner |
| 930.2.j.f | yes | 8 | 3.b | odd | 2 | 1 | |
| 930.2.j.f | yes | 8 | 5.c | odd | 4 | 1 | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(930, [\chi])\):
|
\( T_{7}^{8} - 4T_{7}^{7} + 8T_{7}^{6} + 48T_{7}^{5} + 117T_{7}^{4} + 52T_{7}^{3} + 8T_{7}^{2} - 8T_{7} + 4 \)
|
|
\( T_{11}^{8} + 22T_{11}^{6} + 97T_{11}^{4} + 144T_{11}^{2} + 64 \)
|
|
\( T_{17}^{8} - 4T_{17}^{7} + 8T_{17}^{6} + 168T_{17}^{5} + 3540T_{17}^{4} - 7424T_{17}^{3} + 15488T_{17}^{2} + 339328T_{17} + 3717184 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{4} + 1)^{2} \)
|
| $3$ |
\( (T^{4} + 4 T^{3} + 8 T^{2} + \cdots + 9)^{2} \)
|
| $5$ |
\( T^{8} + 2 T^{6} + \cdots + 625 \)
|
| $7$ |
\( T^{8} - 4 T^{7} + \cdots + 4 \)
|
| $11$ |
\( T^{8} + 22 T^{6} + \cdots + 64 \)
|
| $13$ |
\( T^{8} + 4 T^{7} + \cdots + 3136 \)
|
| $17$ |
\( T^{8} - 4 T^{7} + \cdots + 3717184 \)
|
| $19$ |
\( T^{8} + 118 T^{6} + \cdots + 107584 \)
|
| $23$ |
\( T^{8} + 12 T^{7} + \cdots + 37636 \)
|
| $29$ |
\( (T^{4} + 4 T^{3} + \cdots + 248)^{2} \)
|
| $31$ |
\( (T - 1)^{8} \)
|
| $37$ |
\( (T^{2} + 4 T + 8)^{4} \)
|
| $41$ |
\( T^{8} + 196 T^{6} + \cdots + 1048576 \)
|
| $43$ |
\( T^{8} + 8 T^{7} + \cdots + 85264 \)
|
| $47$ |
\( T^{8} - 28 T^{7} + \cdots + 16384 \)
|
| $53$ |
\( T^{8} + 12 T^{7} + \cdots + 15376 \)
|
| $59$ |
\( (T^{4} - 12 T^{3} + \cdots - 1024)^{2} \)
|
| $61$ |
\( (T^{4} - 28 T^{3} + \cdots - 7792)^{2} \)
|
| $67$ |
\( T^{8} + 16 T^{7} + \cdots + 2408704 \)
|
| $71$ |
\( T^{8} + 286 T^{6} + \cdots + 583696 \)
|
| $73$ |
\( T^{8} + 36 T^{7} + \cdots + 38416 \)
|
| $79$ |
\( T^{8} + 278 T^{6} + \cdots + 602176 \)
|
| $83$ |
\( T^{8} + 12 T^{7} + \cdots + 18800896 \)
|
| $89$ |
\( (T^{4} - 18 T^{3} + \cdots + 248)^{2} \)
|
| $97$ |
\( T^{8} - 12 T^{7} + \cdots + 4129024 \)
|
| show more | |
| show less |