Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [3525,1,Mod(1268,3525)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3525, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([2, 3, 2]))
N = Newforms(chi, 1, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("3525.1268");
S:= CuspForms(chi, 1);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 3525 = 3 \cdot 5^{2} \cdot 47 \) |
| Weight: | \( k \) | \(=\) | \( 1 \) |
| Character orbit: | \([\chi]\) | \(=\) | 3525.l (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: | \(1.75920416953\) |
| Analytic rank: | \(0\) |
| Dimension: | \(16\) |
| Relative dimension: | \(8\) over \(\Q(i)\) |
| Coefficient field: | \(\Q(\zeta_{40})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{16} - x^{12} + x^{8} - x^{4} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Projective image: | \(D_{10}\) |
| Projective field: | Galois closure of 10.0.3705507759375.1 |
$q$-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 \(\left(\mathbb{Z}/3525\mathbb{Z}\right)^\times\).
| \(n\) | \(1552\) | \(2026\) | \(2351\) |
| \(\chi(n)\) | \(-\zeta_{40}^{10}\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1268.1 |
|
−1.14412 | − | 1.14412i | −0.453990 | − | 0.891007i | 1.61803i | 0 | −0.500000 | + | 1.53884i | 0.831254 | − | 0.831254i | 0.707107 | − | 0.707107i | −0.587785 | + | 0.809017i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1268.2 | −1.14412 | − | 1.14412i | 0.891007 | + | 0.453990i | 1.61803i | 0 | −0.500000 | − | 1.53884i | −0.831254 | + | 0.831254i | 0.707107 | − | 0.707107i | 0.587785 | + | 0.809017i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1268.3 | −0.437016 | − | 0.437016i | 0.156434 | − | 0.987688i | − | 0.618034i | 0 | −0.500000 | + | 0.363271i | −1.34500 | + | 1.34500i | −0.707107 | + | 0.707107i | −0.951057 | − | 0.309017i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1268.4 | −0.437016 | − | 0.437016i | 0.987688 | − | 0.156434i | − | 0.618034i | 0 | −0.500000 | − | 0.363271i | 1.34500 | − | 1.34500i | −0.707107 | + | 0.707107i | 0.951057 | − | 0.309017i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1268.5 | 0.437016 | + | 0.437016i | −0.987688 | + | 0.156434i | − | 0.618034i | 0 | −0.500000 | − | 0.363271i | −1.34500 | + | 1.34500i | 0.707107 | − | 0.707107i | 0.951057 | − | 0.309017i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1268.6 | 0.437016 | + | 0.437016i | −0.156434 | + | 0.987688i | − | 0.618034i | 0 | −0.500000 | + | 0.363271i | 1.34500 | − | 1.34500i | 0.707107 | − | 0.707107i | −0.951057 | − | 0.309017i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1268.7 | 1.14412 | + | 1.14412i | −0.891007 | − | 0.453990i | 1.61803i | 0 | −0.500000 | − | 1.53884i | 0.831254 | − | 0.831254i | −0.707107 | + | 0.707107i | 0.587785 | + | 0.809017i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1268.8 | 1.14412 | + | 1.14412i | 0.453990 | + | 0.891007i | 1.61803i | 0 | −0.500000 | + | 1.53884i | −0.831254 | + | 0.831254i | −0.707107 | + | 0.707107i | −0.587785 | + | 0.809017i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1832.1 | −1.14412 | + | 1.14412i | −0.453990 | + | 0.891007i | − | 1.61803i | 0 | −0.500000 | − | 1.53884i | 0.831254 | + | 0.831254i | 0.707107 | + | 0.707107i | −0.587785 | − | 0.809017i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1832.2 | −1.14412 | + | 1.14412i | 0.891007 | − | 0.453990i | − | 1.61803i | 0 | −0.500000 | + | 1.53884i | −0.831254 | − | 0.831254i | 0.707107 | + | 0.707107i | 0.587785 | − | 0.809017i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1832.3 | −0.437016 | + | 0.437016i | 0.156434 | + | 0.987688i | 0.618034i | 0 | −0.500000 | − | 0.363271i | −1.34500 | − | 1.34500i | −0.707107 | − | 0.707107i | −0.951057 | + | 0.309017i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1832.4 | −0.437016 | + | 0.437016i | 0.987688 | + | 0.156434i | 0.618034i | 0 | −0.500000 | + | 0.363271i | 1.34500 | + | 1.34500i | −0.707107 | − | 0.707107i | 0.951057 | + | 0.309017i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1832.5 | 0.437016 | − | 0.437016i | −0.987688 | − | 0.156434i | 0.618034i | 0 | −0.500000 | + | 0.363271i | −1.34500 | − | 1.34500i | 0.707107 | + | 0.707107i | 0.951057 | + | 0.309017i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1832.6 | 0.437016 | − | 0.437016i | −0.156434 | − | 0.987688i | 0.618034i | 0 | −0.500000 | − | 0.363271i | 1.34500 | + | 1.34500i | 0.707107 | + | 0.707107i | −0.951057 | + | 0.309017i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1832.7 | 1.14412 | − | 1.14412i | −0.891007 | + | 0.453990i | − | 1.61803i | 0 | −0.500000 | + | 1.53884i | 0.831254 | + | 0.831254i | −0.707107 | − | 0.707107i | 0.587785 | − | 0.809017i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1832.8 | 1.14412 | − | 1.14412i | 0.453990 | − | 0.891007i | − | 1.61803i | 0 | −0.500000 | − | 1.53884i | −0.831254 | − | 0.831254i | −0.707107 | − | 0.707107i | −0.587785 | − | 0.809017i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 47.b | odd | 2 | 1 | CM by \(\Q(\sqrt{-47}) \) |
| 3.b | odd | 2 | 1 | inner |
| 5.b | even | 2 | 1 | inner |
| 5.c | odd | 4 | 2 | inner |
| 15.d | odd | 2 | 1 | inner |
| 15.e | even | 4 | 2 | inner |
| 141.c | even | 2 | 1 | inner |
| 235.b | odd | 2 | 1 | inner |
| 235.e | even | 4 | 2 | inner |
| 705.g | even | 2 | 1 | inner |
| 705.l | odd | 4 | 2 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 3525.1.l.c | ✓ | 16 |
| 3.b | odd | 2 | 1 | inner | 3525.1.l.c | ✓ | 16 |
| 5.b | even | 2 | 1 | inner | 3525.1.l.c | ✓ | 16 |
| 5.c | odd | 4 | 2 | inner | 3525.1.l.c | ✓ | 16 |
| 15.d | odd | 2 | 1 | inner | 3525.1.l.c | ✓ | 16 |
| 15.e | even | 4 | 2 | inner | 3525.1.l.c | ✓ | 16 |
| 47.b | odd | 2 | 1 | CM | 3525.1.l.c | ✓ | 16 |
| 141.c | even | 2 | 1 | inner | 3525.1.l.c | ✓ | 16 |
| 235.b | odd | 2 | 1 | inner | 3525.1.l.c | ✓ | 16 |
| 235.e | even | 4 | 2 | inner | 3525.1.l.c | ✓ | 16 |
| 705.g | even | 2 | 1 | inner | 3525.1.l.c | ✓ | 16 |
| 705.l | odd | 4 | 2 | inner | 3525.1.l.c | ✓ | 16 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 3525.1.l.c | ✓ | 16 | 1.a | even | 1 | 1 | trivial |
| 3525.1.l.c | ✓ | 16 | 3.b | odd | 2 | 1 | inner |
| 3525.1.l.c | ✓ | 16 | 5.b | even | 2 | 1 | inner |
| 3525.1.l.c | ✓ | 16 | 5.c | odd | 4 | 2 | inner |
| 3525.1.l.c | ✓ | 16 | 15.d | odd | 2 | 1 | inner |
| 3525.1.l.c | ✓ | 16 | 15.e | even | 4 | 2 | inner |
| 3525.1.l.c | ✓ | 16 | 47.b | odd | 2 | 1 | CM |
| 3525.1.l.c | ✓ | 16 | 141.c | even | 2 | 1 | inner |
| 3525.1.l.c | ✓ | 16 | 235.b | odd | 2 | 1 | inner |
| 3525.1.l.c | ✓ | 16 | 235.e | even | 4 | 2 | inner |
| 3525.1.l.c | ✓ | 16 | 705.g | even | 2 | 1 | inner |
| 3525.1.l.c | ✓ | 16 | 705.l | odd | 4 | 2 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{8} + 7T_{2}^{4} + 1 \)
acting on \(S_{1}^{\mathrm{new}}(3525, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{8} + 7 T^{4} + 1)^{2} \)
|
| $3$ |
\( T^{16} - T^{12} + \cdots + 1 \)
|
| $5$ |
\( T^{16} \)
|
| $7$ |
\( (T^{8} + 15 T^{4} + 25)^{2} \)
|
| $11$ |
\( T^{16} \)
|
| $13$ |
\( T^{16} \)
|
| $17$ |
\( (T^{8} + 7 T^{4} + 1)^{2} \)
|
| $19$ |
\( T^{16} \)
|
| $23$ |
\( T^{16} \)
|
| $29$ |
\( T^{16} \)
|
| $31$ |
\( T^{16} \)
|
| $37$ |
\( (T^{8} + 15 T^{4} + 25)^{2} \)
|
| $41$ |
\( T^{16} \)
|
| $43$ |
\( T^{16} \)
|
| $47$ |
\( (T^{4} + 1)^{4} \)
|
| $53$ |
\( (T^{8} + 7 T^{4} + 1)^{2} \)
|
| $59$ |
\( (T^{4} - 5 T^{2} + 5)^{4} \)
|
| $61$ |
\( (T^{2} - T - 1)^{8} \)
|
| $67$ |
\( T^{16} \)
|
| $71$ |
\( (T^{4} + 5 T^{2} + 5)^{4} \)
|
| $73$ |
\( T^{16} \)
|
| $79$ |
\( (T^{4} + 3 T^{2} + 1)^{4} \)
|
| $83$ |
\( (T^{4} + 16)^{4} \)
|
| $89$ |
\( (T^{4} - 5 T^{2} + 5)^{4} \)
|
| $97$ |
\( (T^{8} + 15 T^{4} + 25)^{2} \)
|
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