Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2600,1,Mod(259,2600)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2600, base_ring=CyclotomicField(10))
chi = DirichletCharacter(H, H._module([5, 5, 7, 5]))
N = Newforms(chi, 1, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2600.259");
S:= CuspForms(chi, 1);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2600 = 2^{3} \cdot 5^{2} \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 1 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2600.cp (of order \(10\), degree \(4\), 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: | \(1.29756903285\) |
| Analytic rank: | \(0\) |
| Dimension: | \(16\) |
| Relative dimension: | \(4\) over \(\Q(\zeta_{10})\) |
| 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: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Projective image: | \(D_{30}\) |
| Projective field: | Galois closure of \(\mathbb{Q}[x]/(x^{30} + \cdots)\) |
$q$-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 \(\left(\mathbb{Z}/2600\mathbb{Z}\right)^\times\).
| \(n\) | \(1301\) | \(1601\) | \(1951\) | \(1977\) |
| \(\chi(n)\) | \(-1\) | \(-1\) | \(-1\) | \(-\zeta_{60}^{24}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 259.1 |
|
−0.587785 | + | 0.809017i | −1.64728 | + | 0.535233i | −0.309017 | − | 0.951057i | 0.866025 | − | 0.500000i | 0.535233 | − | 1.64728i | 0.209057i | 0.951057 | + | 0.309017i | 1.61803 | − | 1.17557i | −0.104528 | + | 0.994522i | ||||||||||||||||||||||||||||||||||||||||||||||||||
| 259.2 | −0.587785 | + | 0.809017i | 1.64728 | − | 0.535233i | −0.309017 | − | 0.951057i | −0.866025 | − | 0.500000i | −0.535233 | + | 1.64728i | − | 1.82709i | 0.951057 | + | 0.309017i | 1.61803 | − | 1.17557i | 0.913545 | − | 0.406737i | ||||||||||||||||||||||||||||||||||||||||||||||||||
| 259.3 | 0.587785 | − | 0.809017i | −1.64728 | + | 0.535233i | −0.309017 | − | 0.951057i | −0.866025 | + | 0.500000i | −0.535233 | + | 1.64728i | − | 0.209057i | −0.951057 | − | 0.309017i | 1.61803 | − | 1.17557i | −0.104528 | + | 0.994522i | ||||||||||||||||||||||||||||||||||||||||||||||||||
| 259.4 | 0.587785 | − | 0.809017i | 1.64728 | − | 0.535233i | −0.309017 | − | 0.951057i | 0.866025 | + | 0.500000i | 0.535233 | − | 1.64728i | 1.82709i | −0.951057 | − | 0.309017i | 1.61803 | − | 1.17557i | 0.913545 | − | 0.406737i | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 779.1 | −0.951057 | − | 0.309017i | −1.01807 | + | 1.40126i | 0.809017 | + | 0.587785i | −0.866025 | − | 0.500000i | 1.40126 | − | 1.01807i | − | 1.33826i | −0.587785 | − | 0.809017i | −0.618034 | − | 1.90211i | 0.669131 | + | 0.743145i | ||||||||||||||||||||||||||||||||||||||||||||||||||
| 779.2 | −0.951057 | − | 0.309017i | 1.01807 | − | 1.40126i | 0.809017 | + | 0.587785i | 0.866025 | − | 0.500000i | −1.40126 | + | 1.01807i | 1.95630i | −0.587785 | − | 0.809017i | −0.618034 | − | 1.90211i | −0.978148 | + | 0.207912i | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 779.3 | 0.951057 | + | 0.309017i | −1.01807 | + | 1.40126i | 0.809017 | + | 0.587785i | 0.866025 | + | 0.500000i | −1.40126 | + | 1.01807i | 1.33826i | 0.587785 | + | 0.809017i | −0.618034 | − | 1.90211i | 0.669131 | + | 0.743145i | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 779.4 | 0.951057 | + | 0.309017i | 1.01807 | − | 1.40126i | 0.809017 | + | 0.587785i | −0.866025 | + | 0.500000i | 1.40126 | − | 1.01807i | − | 1.95630i | 0.587785 | + | 0.809017i | −0.618034 | − | 1.90211i | −0.978148 | + | 0.207912i | ||||||||||||||||||||||||||||||||||||||||||||||||||
| 1819.1 | −0.951057 | + | 0.309017i | −1.01807 | − | 1.40126i | 0.809017 | − | 0.587785i | −0.866025 | + | 0.500000i | 1.40126 | + | 1.01807i | 1.33826i | −0.587785 | + | 0.809017i | −0.618034 | + | 1.90211i | 0.669131 | − | 0.743145i | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 1819.2 | −0.951057 | + | 0.309017i | 1.01807 | + | 1.40126i | 0.809017 | − | 0.587785i | 0.866025 | + | 0.500000i | −1.40126 | − | 1.01807i | − | 1.95630i | −0.587785 | + | 0.809017i | −0.618034 | + | 1.90211i | −0.978148 | − | 0.207912i | ||||||||||||||||||||||||||||||||||||||||||||||||||
| 1819.3 | 0.951057 | − | 0.309017i | −1.01807 | − | 1.40126i | 0.809017 | − | 0.587785i | 0.866025 | − | 0.500000i | −1.40126 | − | 1.01807i | − | 1.33826i | 0.587785 | − | 0.809017i | −0.618034 | + | 1.90211i | 0.669131 | − | 0.743145i | ||||||||||||||||||||||||||||||||||||||||||||||||||
| 1819.4 | 0.951057 | − | 0.309017i | 1.01807 | + | 1.40126i | 0.809017 | − | 0.587785i | −0.866025 | − | 0.500000i | 1.40126 | + | 1.01807i | 1.95630i | 0.587785 | − | 0.809017i | −0.618034 | + | 1.90211i | −0.978148 | − | 0.207912i | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 2339.1 | −0.587785 | − | 0.809017i | −1.64728 | − | 0.535233i | −0.309017 | + | 0.951057i | 0.866025 | + | 0.500000i | 0.535233 | + | 1.64728i | − | 0.209057i | 0.951057 | − | 0.309017i | 1.61803 | + | 1.17557i | −0.104528 | − | 0.994522i | ||||||||||||||||||||||||||||||||||||||||||||||||||
| 2339.2 | −0.587785 | − | 0.809017i | 1.64728 | + | 0.535233i | −0.309017 | + | 0.951057i | −0.866025 | + | 0.500000i | −0.535233 | − | 1.64728i | 1.82709i | 0.951057 | − | 0.309017i | 1.61803 | + | 1.17557i | 0.913545 | + | 0.406737i | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 2339.3 | 0.587785 | + | 0.809017i | −1.64728 | − | 0.535233i | −0.309017 | + | 0.951057i | −0.866025 | − | 0.500000i | −0.535233 | − | 1.64728i | 0.209057i | −0.951057 | + | 0.309017i | 1.61803 | + | 1.17557i | −0.104528 | − | 0.994522i | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 2339.4 | 0.587785 | + | 0.809017i | 1.64728 | + | 0.535233i | −0.309017 | + | 0.951057i | 0.866025 | − | 0.500000i | 0.535233 | + | 1.64728i | − | 1.82709i | −0.951057 | + | 0.309017i | 1.61803 | + | 1.17557i | 0.913545 | + | 0.406737i | ||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 104.h | odd | 2 | 1 | CM by \(\Q(\sqrt{-26}) \) |
| 8.d | odd | 2 | 1 | inner |
| 13.b | even | 2 | 1 | inner |
| 25.e | even | 10 | 1 | inner |
| 200.s | odd | 10 | 1 | inner |
| 325.p | even | 10 | 1 | inner |
| 2600.cp | odd | 10 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 2600.1.cp.b | ✓ | 16 |
| 8.d | odd | 2 | 1 | inner | 2600.1.cp.b | ✓ | 16 |
| 13.b | even | 2 | 1 | inner | 2600.1.cp.b | ✓ | 16 |
| 25.e | even | 10 | 1 | inner | 2600.1.cp.b | ✓ | 16 |
| 104.h | odd | 2 | 1 | CM | 2600.1.cp.b | ✓ | 16 |
| 200.s | odd | 10 | 1 | inner | 2600.1.cp.b | ✓ | 16 |
| 325.p | even | 10 | 1 | inner | 2600.1.cp.b | ✓ | 16 |
| 2600.cp | odd | 10 | 1 | inner | 2600.1.cp.b | ✓ | 16 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2600.1.cp.b | ✓ | 16 | 1.a | even | 1 | 1 | trivial |
| 2600.1.cp.b | ✓ | 16 | 8.d | odd | 2 | 1 | inner |
| 2600.1.cp.b | ✓ | 16 | 13.b | even | 2 | 1 | inner |
| 2600.1.cp.b | ✓ | 16 | 25.e | even | 10 | 1 | inner |
| 2600.1.cp.b | ✓ | 16 | 104.h | odd | 2 | 1 | CM |
| 2600.1.cp.b | ✓ | 16 | 200.s | odd | 10 | 1 | inner |
| 2600.1.cp.b | ✓ | 16 | 325.p | even | 10 | 1 | inner |
| 2600.1.cp.b | ✓ | 16 | 2600.cp | odd | 10 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{8} - 3T_{3}^{6} + 9T_{3}^{4} - 27T_{3}^{2} + 81 \)
acting on \(S_{1}^{\mathrm{new}}(2600, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{8} - T^{6} + T^{4} + \cdots + 1)^{2} \)
|
| $3$ |
\( (T^{8} - 3 T^{6} + 9 T^{4} + \cdots + 81)^{2} \)
|
| $5$ |
\( (T^{4} - T^{2} + 1)^{4} \)
|
| $7$ |
\( (T^{8} + 9 T^{6} + 26 T^{4} + \cdots + 1)^{2} \)
|
| $11$ |
\( T^{16} \)
|
| $13$ |
\( (T^{8} - T^{6} + T^{4} + \cdots + 1)^{2} \)
|
| $17$ |
\( (T^{8} + 5 T^{7} + 12 T^{6} + \cdots + 1)^{2} \)
|
| $19$ |
\( T^{16} \)
|
| $23$ |
\( T^{16} \)
|
| $29$ |
\( T^{16} \)
|
| $31$ |
\( (T^{8} + 10 T^{4} + \cdots + 25)^{2} \)
|
| $37$ |
\( T^{16} - 2 T^{14} + \cdots + 1 \)
|
| $41$ |
\( T^{16} \)
|
| $43$ |
\( (T^{8} + 7 T^{6} + 14 T^{4} + \cdots + 1)^{2} \)
|
| $47$ |
\( T^{16} - 7 T^{14} + \cdots + 1 \)
|
| $53$ |
\( T^{16} \)
|
| $59$ |
\( T^{16} \)
|
| $61$ |
\( T^{16} \)
|
| $67$ |
\( T^{16} \)
|
| $71$ |
\( T^{16} + T^{14} + \cdots + 1 \)
|
| $73$ |
\( T^{16} \)
|
| $79$ |
\( T^{16} \)
|
| $83$ |
\( T^{16} \)
|
| $89$ |
\( T^{16} \)
|
| $97$ |
\( T^{16} \)
|
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