Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [825,1,Mod(98,825)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(825, base_ring=CyclotomicField(20))
chi = DirichletCharacter(H, H._module([10, 11, 10]))
N = Newforms(chi, 1, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("825.98");
S:= CuspForms(chi, 1);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 825 = 3 \cdot 5^{2} \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 1 \) |
| Character orbit: | \([\chi]\) | \(=\) | 825.co (of order \(20\), degree \(8\), 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: | \(0.411728635422\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | \(\Q(\zeta_{20})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - x^{6} + x^{4} - x^{2} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Projective image: | \(D_{20}\) |
| Projective field: | Galois closure of \(\mathbb{Q}[x]/(x^{20} + \cdots)\) |
$q$-expansion
comment: q-expansion
sage: f.q_expansion() # note that sage often uses an isomorphic number field
gp: mfcoefs(f, 20)
The \(q\)-expansion and trace form are shown below.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/825\mathbb{Z}\right)^\times\).
| \(n\) | \(376\) | \(551\) | \(727\) |
| \(\chi(n)\) | \(-1\) | \(-1\) | \(\zeta_{20}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 98.1 |
|
0 | 0.587785 | + | 0.809017i | 0.951057 | + | 0.309017i | − | 1.00000i | 0 | 0 | 0 | −0.309017 | + | 0.951057i | 0 | |||||||||||||||||||||||||||||||||||
| 197.1 | 0 | 0.951057 | + | 0.309017i | −0.587785 | + | 0.809017i | 1.00000i | 0 | 0 | 0 | 0.809017 | + | 0.587785i | 0 | |||||||||||||||||||||||||||||||||||||
| 263.1 | 0 | −0.587785 | + | 0.809017i | −0.951057 | + | 0.309017i | − | 1.00000i | 0 | 0 | 0 | −0.309017 | − | 0.951057i | 0 | ||||||||||||||||||||||||||||||||||||
| 362.1 | 0 | 0.587785 | − | 0.809017i | 0.951057 | − | 0.309017i | 1.00000i | 0 | 0 | 0 | −0.309017 | − | 0.951057i | 0 | |||||||||||||||||||||||||||||||||||||
| 428.1 | 0 | −0.951057 | − | 0.309017i | 0.587785 | − | 0.809017i | − | 1.00000i | 0 | 0 | 0 | 0.809017 | + | 0.587785i | 0 | ||||||||||||||||||||||||||||||||||||
| 527.1 | 0 | −0.587785 | − | 0.809017i | −0.951057 | − | 0.309017i | 1.00000i | 0 | 0 | 0 | −0.309017 | + | 0.951057i | 0 | |||||||||||||||||||||||||||||||||||||
| 692.1 | 0 | −0.951057 | + | 0.309017i | 0.587785 | + | 0.809017i | 1.00000i | 0 | 0 | 0 | 0.809017 | − | 0.587785i | 0 | |||||||||||||||||||||||||||||||||||||
| 758.1 | 0 | 0.951057 | − | 0.309017i | −0.587785 | − | 0.809017i | − | 1.00000i | 0 | 0 | 0 | 0.809017 | − | 0.587785i | 0 | ||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 11.b | odd | 2 | 1 | CM by \(\Q(\sqrt{-11}) \) |
| 75.l | even | 20 | 1 | inner |
| 825.co | odd | 20 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 825.1.co.a | ✓ | 8 |
| 3.b | odd | 2 | 1 | 825.1.co.b | yes | 8 | |
| 11.b | odd | 2 | 1 | CM | 825.1.co.a | ✓ | 8 |
| 25.f | odd | 20 | 1 | 825.1.co.b | yes | 8 | |
| 33.d | even | 2 | 1 | 825.1.co.b | yes | 8 | |
| 75.l | even | 20 | 1 | inner | 825.1.co.a | ✓ | 8 |
| 275.bo | even | 20 | 1 | 825.1.co.b | yes | 8 | |
| 825.co | odd | 20 | 1 | inner | 825.1.co.a | ✓ | 8 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 825.1.co.a | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 825.1.co.a | ✓ | 8 | 11.b | odd | 2 | 1 | CM |
| 825.1.co.a | ✓ | 8 | 75.l | even | 20 | 1 | inner |
| 825.1.co.a | ✓ | 8 | 825.co | odd | 20 | 1 | inner |
| 825.1.co.b | yes | 8 | 3.b | odd | 2 | 1 | |
| 825.1.co.b | yes | 8 | 25.f | odd | 20 | 1 | |
| 825.1.co.b | yes | 8 | 33.d | even | 2 | 1 | |
| 825.1.co.b | yes | 8 | 275.bo | even | 20 | 1 | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{23}^{8} - 2T_{23}^{7} + 2T_{23}^{6} - 10T_{23}^{5} + 16T_{23}^{4} - 10T_{23}^{3} + 13T_{23}^{2} - 6T_{23} + 1 \)
acting on \(S_{1}^{\mathrm{new}}(825, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} \)
|
| $3$ |
\( T^{8} - T^{6} + T^{4} + \cdots + 1 \)
|
| $5$ |
\( (T^{2} + 1)^{4} \)
|
| $7$ |
\( T^{8} \)
|
| $11$ |
\( T^{8} - T^{6} + T^{4} + \cdots + 1 \)
|
| $13$ |
\( T^{8} \)
|
| $17$ |
\( T^{8} \)
|
| $19$ |
\( T^{8} \)
|
| $23$ |
\( T^{8} - 2 T^{7} + \cdots + 1 \)
|
| $29$ |
\( T^{8} \)
|
| $31$ |
\( T^{8} + 10 T^{4} + \cdots + 25 \)
|
| $37$ |
\( T^{8} + 2 T^{7} + \cdots + 16 \)
|
| $41$ |
\( T^{8} \)
|
| $43$ |
\( T^{8} \)
|
| $47$ |
\( T^{8} - 2 T^{7} + \cdots + 16 \)
|
| $53$ |
\( T^{8} + 2 T^{7} + \cdots + 1 \)
|
| $59$ |
\( (T^{4} - 3 T^{3} + 4 T^{2} + \cdots + 1)^{2} \)
|
| $61$ |
\( T^{8} \)
|
| $67$ |
\( T^{8} - 2 T^{7} + \cdots + 1 \)
|
| $71$ |
\( (T^{4} + 5 T + 5)^{2} \)
|
| $73$ |
\( T^{8} \)
|
| $79$ |
\( T^{8} \)
|
| $83$ |
\( T^{8} \)
|
| $89$ |
\( T^{8} + 10 T^{4} + \cdots + 25 \)
|
| $97$ |
\( T^{8} - 2 T^{7} + \cdots + 1 \)
|
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