Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [810,2,Mod(323,810)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(810, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([2, 3]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("810.323");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 810 = 2 \cdot 3^{4} \cdot 5 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 810.f (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: | \(6.46788256372\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Relative dimension: | \(4\) over \(\Q(i)\) |
| Coefficient field: | \(\Q(\zeta_{24})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - x^{4} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{2}\cdot 3^{4} \) |
| 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
| \(\beta_{1}\) | \(=\) |
\( \zeta_{24}^{3} \)
|
| \(\beta_{2}\) | \(=\) |
\( \zeta_{24}^{6} \)
|
| \(\beta_{3}\) | \(=\) |
\( -\zeta_{24}^{5} + \zeta_{24} \)
|
| \(\beta_{4}\) | \(=\) |
\( -\zeta_{24}^{6} + 3\zeta_{24}^{4} + 3\zeta_{24}^{2} - 1 \)
|
| \(\beta_{5}\) | \(=\) |
\( -3\zeta_{24}^{7} + \zeta_{24}^{5} + \zeta_{24}^{3} + 2\zeta_{24} \)
|
| \(\beta_{6}\) | \(=\) |
\( 3\zeta_{24}^{7} + \zeta_{24}^{5} - \zeta_{24}^{3} + 2\zeta_{24} \)
|
| \(\beta_{7}\) | \(=\) |
\( -2\zeta_{24}^{6} - 3\zeta_{24}^{4} + 3\zeta_{24}^{2} + 1 \)
|
| \(\zeta_{24}\) | \(=\) |
\( ( \beta_{6} + \beta_{5} + 2\beta_{3} ) / 6 \)
|
| \(\zeta_{24}^{2}\) | \(=\) |
\( ( \beta_{7} + \beta_{4} + 3\beta_{2} ) / 6 \)
|
| \(\zeta_{24}^{3}\) | \(=\) |
\( \beta_1 \)
|
| \(\zeta_{24}^{4}\) | \(=\) |
\( ( -\beta_{7} + \beta_{4} - \beta_{2} + 2 ) / 6 \)
|
| \(\zeta_{24}^{5}\) | \(=\) |
\( ( \beta_{6} + \beta_{5} - 4\beta_{3} ) / 6 \)
|
| \(\zeta_{24}^{6}\) | \(=\) |
\( \beta_{2} \)
|
| \(\zeta_{24}^{7}\) | \(=\) |
\( ( \beta_{6} - \beta_{5} + 2\beta_1 ) / 6 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/810\mathbb{Z}\right)^\times\).
| \(n\) | \(487\) | \(731\) |
| \(\chi(n)\) | \(\beta_{2}\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 323.1 |
|
−0.707107 | − | 0.707107i | 0 | 1.00000i | 0.707107 | + | 2.12132i | 0 | −2.09808 | + | 2.09808i | 0.707107 | − | 0.707107i | 0 | 1.00000 | − | 2.00000i | ||||||||||||||||||||||||||||||||
| 323.2 | −0.707107 | − | 0.707107i | 0 | 1.00000i | 0.707107 | + | 2.12132i | 0 | 3.09808 | − | 3.09808i | 0.707107 | − | 0.707107i | 0 | 1.00000 | − | 2.00000i | |||||||||||||||||||||||||||||||||
| 323.3 | 0.707107 | + | 0.707107i | 0 | 1.00000i | −0.707107 | − | 2.12132i | 0 | −2.09808 | + | 2.09808i | −0.707107 | + | 0.707107i | 0 | 1.00000 | − | 2.00000i | |||||||||||||||||||||||||||||||||
| 323.4 | 0.707107 | + | 0.707107i | 0 | 1.00000i | −0.707107 | − | 2.12132i | 0 | 3.09808 | − | 3.09808i | −0.707107 | + | 0.707107i | 0 | 1.00000 | − | 2.00000i | |||||||||||||||||||||||||||||||||
| 647.1 | −0.707107 | + | 0.707107i | 0 | − | 1.00000i | 0.707107 | − | 2.12132i | 0 | −2.09808 | − | 2.09808i | 0.707107 | + | 0.707107i | 0 | 1.00000 | + | 2.00000i | ||||||||||||||||||||||||||||||||
| 647.2 | −0.707107 | + | 0.707107i | 0 | − | 1.00000i | 0.707107 | − | 2.12132i | 0 | 3.09808 | + | 3.09808i | 0.707107 | + | 0.707107i | 0 | 1.00000 | + | 2.00000i | ||||||||||||||||||||||||||||||||
| 647.3 | 0.707107 | − | 0.707107i | 0 | − | 1.00000i | −0.707107 | + | 2.12132i | 0 | −2.09808 | − | 2.09808i | −0.707107 | − | 0.707107i | 0 | 1.00000 | + | 2.00000i | ||||||||||||||||||||||||||||||||
| 647.4 | 0.707107 | − | 0.707107i | 0 | − | 1.00000i | −0.707107 | + | 2.12132i | 0 | 3.09808 | + | 3.09808i | −0.707107 | − | 0.707107i | 0 | 1.00000 | + | 2.00000i | ||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
| 5.c | odd | 4 | 1 | inner |
| 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 | 810.2.f.a | ✓ | 8 |
| 3.b | odd | 2 | 1 | inner | 810.2.f.a | ✓ | 8 |
| 5.c | odd | 4 | 1 | inner | 810.2.f.a | ✓ | 8 |
| 9.c | even | 3 | 1 | 810.2.m.b | 8 | ||
| 9.c | even | 3 | 1 | 810.2.m.g | 8 | ||
| 9.d | odd | 6 | 1 | 810.2.m.b | 8 | ||
| 9.d | odd | 6 | 1 | 810.2.m.g | 8 | ||
| 15.e | even | 4 | 1 | inner | 810.2.f.a | ✓ | 8 |
| 45.k | odd | 12 | 1 | 810.2.m.b | 8 | ||
| 45.k | odd | 12 | 1 | 810.2.m.g | 8 | ||
| 45.l | even | 12 | 1 | 810.2.m.b | 8 | ||
| 45.l | even | 12 | 1 | 810.2.m.g | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 810.2.f.a | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 810.2.f.a | ✓ | 8 | 3.b | odd | 2 | 1 | inner |
| 810.2.f.a | ✓ | 8 | 5.c | odd | 4 | 1 | inner |
| 810.2.f.a | ✓ | 8 | 15.e | even | 4 | 1 | inner |
| 810.2.m.b | 8 | 9.c | even | 3 | 1 | ||
| 810.2.m.b | 8 | 9.d | odd | 6 | 1 | ||
| 810.2.m.b | 8 | 45.k | odd | 12 | 1 | ||
| 810.2.m.b | 8 | 45.l | even | 12 | 1 | ||
| 810.2.m.g | 8 | 9.c | even | 3 | 1 | ||
| 810.2.m.g | 8 | 9.d | odd | 6 | 1 | ||
| 810.2.m.g | 8 | 45.k | odd | 12 | 1 | ||
| 810.2.m.g | 8 | 45.l | even | 12 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{4} - 2T_{7}^{3} + 2T_{7}^{2} + 26T_{7} + 169 \)
acting on \(S_{2}^{\mathrm{new}}(810, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{4} + 1)^{2} \)
|
| $3$ |
\( T^{8} \)
|
| $5$ |
\( (T^{4} + 8 T^{2} + 25)^{2} \)
|
| $7$ |
\( (T^{4} - 2 T^{3} + \cdots + 169)^{2} \)
|
| $11$ |
\( (T^{2} + 2)^{4} \)
|
| $13$ |
\( (T^{4} + 6 T^{3} + 18 T^{2} + \cdots + 81)^{2} \)
|
| $17$ |
\( T^{8} + 1784 T^{4} + 456976 \)
|
| $19$ |
\( (T^{4} + 62 T^{2} + 529)^{2} \)
|
| $23$ |
\( (T^{4} + 625)^{2} \)
|
| $29$ |
\( (T^{2} - 32)^{4} \)
|
| $31$ |
\( (T^{2} - 2 T - 26)^{4} \)
|
| $37$ |
\( (T^{4} + 2916)^{2} \)
|
| $41$ |
\( (T^{4} + 76 T^{2} + 121)^{2} \)
|
| $43$ |
\( (T^{4} + 2916)^{2} \)
|
| $47$ |
\( T^{8} + 15714 T^{4} + 6561 \)
|
| $53$ |
\( T^{8} + 2786 T^{4} + 279841 \)
|
| $59$ |
\( (T^{4} - 28 T^{2} + 169)^{2} \)
|
| $61$ |
\( (T^{2} - 18 T + 54)^{4} \)
|
| $67$ |
\( (T^{4} + 4 T^{3} + \cdots + 2704)^{2} \)
|
| $71$ |
\( (T^{4} + 124 T^{2} + 2116)^{2} \)
|
| $73$ |
\( (T^{2} - 8 T + 32)^{4} \)
|
| $79$ |
\( (T^{4} + 72 T^{2} + 324)^{2} \)
|
| $83$ |
\( T^{8} + 40824 T^{4} + 8503056 \)
|
| $89$ |
\( (T^{4} - 172 T^{2} + 484)^{2} \)
|
| $97$ |
\( (T^{2} + 12 T + 72)^{4} \)
|
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