Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1800,1,Mod(211,1800)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1800, base_ring=CyclotomicField(30))
chi = DirichletCharacter(H, H._module([15, 15, 10, 24]))
N = Newforms(chi, 1, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1800.211");
S:= CuspForms(chi, 1);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1800 = 2^{3} \cdot 3^{2} \cdot 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 1 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1800.dg (of order \(30\), 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: | \(0.898317022739\) |
| Analytic rank: | \(0\) |
| Dimension: | \(16\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{30})\) |
| Coefficient field: | \(\Q(\zeta_{60})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{16} + x^{14} - x^{10} - x^{8} - x^{6} + x^{2} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Projective image: | \(A_{5}\) |
| Projective field: | Galois closure of 5.1.2025000000.9 |
$q$-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 \(\left(\mathbb{Z}/1800\mathbb{Z}\right)^\times\).
| \(n\) | \(577\) | \(901\) | \(1001\) | \(1351\) |
| \(\chi(n)\) | \(-\zeta_{60}^{18}\) | \(-1\) | \(-\zeta_{60}^{10}\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 211.1 |
|
−0.406737 | + | 0.913545i | −0.309017 | + | 0.951057i | −0.669131 | − | 0.743145i | −0.743145 | + | 0.669131i | −0.743145 | − | 0.669131i | 0.866025 | − | 0.500000i | 0.951057 | − | 0.309017i | −0.809017 | − | 0.587785i | −0.309017 | − | 0.951057i | ||||||||||||||||||||||||||||||||||||||||||||||||
| 211.2 | 0.406737 | − | 0.913545i | −0.309017 | + | 0.951057i | −0.669131 | − | 0.743145i | 0.743145 | − | 0.669131i | 0.743145 | + | 0.669131i | −0.866025 | + | 0.500000i | −0.951057 | + | 0.309017i | −0.809017 | − | 0.587785i | −0.309017 | − | 0.951057i | |||||||||||||||||||||||||||||||||||||||||||||||||
| 331.1 | −0.207912 | + | 0.978148i | 0.809017 | − | 0.587785i | −0.913545 | − | 0.406737i | 0.406737 | − | 0.913545i | 0.406737 | + | 0.913545i | 0.866025 | + | 0.500000i | 0.587785 | − | 0.809017i | 0.309017 | − | 0.951057i | 0.809017 | + | 0.587785i | |||||||||||||||||||||||||||||||||||||||||||||||||
| 331.2 | 0.207912 | − | 0.978148i | 0.809017 | − | 0.587785i | −0.913545 | − | 0.406737i | −0.406737 | + | 0.913545i | −0.406737 | − | 0.913545i | −0.866025 | − | 0.500000i | −0.587785 | + | 0.809017i | 0.309017 | − | 0.951057i | 0.809017 | + | 0.587785i | |||||||||||||||||||||||||||||||||||||||||||||||||
| 571.1 | −0.207912 | − | 0.978148i | 0.809017 | + | 0.587785i | −0.913545 | + | 0.406737i | 0.406737 | + | 0.913545i | 0.406737 | − | 0.913545i | 0.866025 | − | 0.500000i | 0.587785 | + | 0.809017i | 0.309017 | + | 0.951057i | 0.809017 | − | 0.587785i | |||||||||||||||||||||||||||||||||||||||||||||||||
| 571.2 | 0.207912 | + | 0.978148i | 0.809017 | + | 0.587785i | −0.913545 | + | 0.406737i | −0.406737 | − | 0.913545i | −0.406737 | + | 0.913545i | −0.866025 | + | 0.500000i | −0.587785 | − | 0.809017i | 0.309017 | + | 0.951057i | 0.809017 | − | 0.587785i | |||||||||||||||||||||||||||||||||||||||||||||||||
| 691.1 | −0.406737 | − | 0.913545i | −0.309017 | − | 0.951057i | −0.669131 | + | 0.743145i | −0.743145 | − | 0.669131i | −0.743145 | + | 0.669131i | 0.866025 | + | 0.500000i | 0.951057 | + | 0.309017i | −0.809017 | + | 0.587785i | −0.309017 | + | 0.951057i | |||||||||||||||||||||||||||||||||||||||||||||||||
| 691.2 | 0.406737 | + | 0.913545i | −0.309017 | − | 0.951057i | −0.669131 | + | 0.743145i | 0.743145 | + | 0.669131i | 0.743145 | − | 0.669131i | −0.866025 | − | 0.500000i | −0.951057 | − | 0.309017i | −0.809017 | + | 0.587785i | −0.309017 | + | 0.951057i | |||||||||||||||||||||||||||||||||||||||||||||||||
| 931.1 | −0.743145 | − | 0.669131i | 0.809017 | − | 0.587785i | 0.104528 | + | 0.994522i | −0.994522 | + | 0.104528i | −0.994522 | − | 0.104528i | −0.866025 | + | 0.500000i | 0.587785 | − | 0.809017i | 0.309017 | − | 0.951057i | 0.809017 | + | 0.587785i | |||||||||||||||||||||||||||||||||||||||||||||||||
| 931.2 | 0.743145 | + | 0.669131i | 0.809017 | − | 0.587785i | 0.104528 | + | 0.994522i | 0.994522 | − | 0.104528i | 0.994522 | + | 0.104528i | 0.866025 | − | 0.500000i | −0.587785 | + | 0.809017i | 0.309017 | − | 0.951057i | 0.809017 | + | 0.587785i | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1291.1 | −0.994522 | − | 0.104528i | −0.309017 | − | 0.951057i | 0.978148 | + | 0.207912i | 0.207912 | − | 0.978148i | 0.207912 | + | 0.978148i | 0.866025 | − | 0.500000i | −0.951057 | − | 0.309017i | −0.809017 | + | 0.587785i | −0.309017 | + | 0.951057i | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1291.2 | 0.994522 | + | 0.104528i | −0.309017 | − | 0.951057i | 0.978148 | + | 0.207912i | −0.207912 | + | 0.978148i | −0.207912 | − | 0.978148i | −0.866025 | + | 0.500000i | 0.951057 | + | 0.309017i | −0.809017 | + | 0.587785i | −0.309017 | + | 0.951057i | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1411.1 | −0.994522 | + | 0.104528i | −0.309017 | + | 0.951057i | 0.978148 | − | 0.207912i | 0.207912 | + | 0.978148i | 0.207912 | − | 0.978148i | 0.866025 | + | 0.500000i | −0.951057 | + | 0.309017i | −0.809017 | − | 0.587785i | −0.309017 | − | 0.951057i | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1411.2 | 0.994522 | − | 0.104528i | −0.309017 | + | 0.951057i | 0.978148 | − | 0.207912i | −0.207912 | − | 0.978148i | −0.207912 | + | 0.978148i | −0.866025 | − | 0.500000i | 0.951057 | − | 0.309017i | −0.809017 | − | 0.587785i | −0.309017 | − | 0.951057i | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1771.1 | −0.743145 | + | 0.669131i | 0.809017 | + | 0.587785i | 0.104528 | − | 0.994522i | −0.994522 | − | 0.104528i | −0.994522 | + | 0.104528i | −0.866025 | − | 0.500000i | 0.587785 | + | 0.809017i | 0.309017 | + | 0.951057i | 0.809017 | − | 0.587785i | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1771.2 | 0.743145 | − | 0.669131i | 0.809017 | + | 0.587785i | 0.104528 | − | 0.994522i | 0.994522 | + | 0.104528i | 0.994522 | − | 0.104528i | 0.866025 | + | 0.500000i | −0.587785 | − | 0.809017i | 0.309017 | + | 0.951057i | 0.809017 | − | 0.587785i | |||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 8.d | odd | 2 | 1 | inner |
| 9.c | even | 3 | 1 | inner |
| 25.d | even | 5 | 1 | inner |
| 72.p | odd | 6 | 1 | inner |
| 200.n | odd | 10 | 1 | inner |
| 225.q | even | 15 | 1 | inner |
| 1800.dg | odd | 30 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 1800.1.dg.a | ✓ | 16 |
| 8.d | odd | 2 | 1 | inner | 1800.1.dg.a | ✓ | 16 |
| 9.c | even | 3 | 1 | inner | 1800.1.dg.a | ✓ | 16 |
| 25.d | even | 5 | 1 | inner | 1800.1.dg.a | ✓ | 16 |
| 72.p | odd | 6 | 1 | inner | 1800.1.dg.a | ✓ | 16 |
| 200.n | odd | 10 | 1 | inner | 1800.1.dg.a | ✓ | 16 |
| 225.q | even | 15 | 1 | inner | 1800.1.dg.a | ✓ | 16 |
| 1800.dg | odd | 30 | 1 | inner | 1800.1.dg.a | ✓ | 16 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1800.1.dg.a | ✓ | 16 | 1.a | even | 1 | 1 | trivial |
| 1800.1.dg.a | ✓ | 16 | 8.d | odd | 2 | 1 | inner |
| 1800.1.dg.a | ✓ | 16 | 9.c | even | 3 | 1 | inner |
| 1800.1.dg.a | ✓ | 16 | 25.d | even | 5 | 1 | inner |
| 1800.1.dg.a | ✓ | 16 | 72.p | odd | 6 | 1 | inner |
| 1800.1.dg.a | ✓ | 16 | 200.n | odd | 10 | 1 | inner |
| 1800.1.dg.a | ✓ | 16 | 225.q | even | 15 | 1 | inner |
| 1800.1.dg.a | ✓ | 16 | 1800.dg | odd | 30 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(1800, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{16} + T^{14} + \cdots + 1 \)
|
| $3$ |
\( (T^{4} - T^{3} + T^{2} + \cdots + 1)^{4} \)
|
| $5$ |
\( T^{16} + T^{14} + \cdots + 1 \)
|
| $7$ |
\( (T^{4} - T^{2} + 1)^{4} \)
|
| $11$ |
\( (T^{8} - 3 T^{7} + 5 T^{6} + \cdots + 1)^{2} \)
|
| $13$ |
\( T^{16} + 4 T^{14} + \cdots + 1 \)
|
| $17$ |
\( (T^{4} + T^{3} + T^{2} + \cdots + 1)^{4} \)
|
| $19$ |
\( (T^{4} + T^{3} + T^{2} + \cdots + 1)^{4} \)
|
| $23$ |
\( T^{16} + T^{14} + \cdots + 1 \)
|
| $29$ |
\( T^{16} + 4 T^{14} + \cdots + 1 \)
|
| $31$ |
\( T^{16} + T^{14} + \cdots + 1 \)
|
| $37$ |
\( T^{16} \)
|
| $41$ |
\( (T^{8} - 2 T^{7} - 2 T^{5} + \cdots + 1)^{2} \)
|
| $43$ |
\( (T^{2} + T + 1)^{8} \)
|
| $47$ |
\( T^{16} + 4 T^{14} + \cdots + 1 \)
|
| $53$ |
\( (T^{8} + T^{6} + 6 T^{4} + \cdots + 1)^{2} \)
|
| $59$ |
\( T^{16} \)
|
| $61$ |
\( T^{16} + T^{14} + \cdots + 1 \)
|
| $67$ |
\( (T^{8} - 3 T^{7} + 5 T^{6} + \cdots + 1)^{2} \)
|
| $71$ |
\( T^{16} \)
|
| $73$ |
\( T^{16} \)
|
| $79$ |
\( T^{16} + 4 T^{14} + \cdots + 1 \)
|
| $83$ |
\( (T^{8} - 2 T^{7} - 2 T^{5} + \cdots + 1)^{2} \)
|
| $89$ |
\( (T^{4} + T^{3} + T^{2} + \cdots + 1)^{4} \)
|
| $97$ |
\( T^{16} \)
|
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