Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [5070,2,Mod(1351,5070)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5070, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("5070.1351");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 5070 = 2 \cdot 3 \cdot 5 \cdot 13^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 5070.b (of order \(2\), degree \(1\), not 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: | \(40.4841538248\) |
| Analytic rank: | \(0\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{-1}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 390) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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 \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/5070\mathbb{Z}\right)^\times\).
| \(n\) | \(1691\) | \(1861\) | \(4057\) |
| \(\chi(n)\) | \(1\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1351.1 |
|
− | 1.00000i | −1.00000 | −1.00000 | 1.00000i | 1.00000i | 2.00000i | 1.00000i | 1.00000 | 1.00000 | |||||||||||||||||||||||
| 1351.2 | 1.00000i | −1.00000 | −1.00000 | − | 1.00000i | − | 1.00000i | − | 2.00000i | − | 1.00000i | 1.00000 | 1.00000 | |||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 13.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 5070.2.b.g | 2 | |
| 13.b | even | 2 | 1 | inner | 5070.2.b.g | 2 | |
| 13.d | odd | 4 | 1 | 5070.2.a.f | 1 | ||
| 13.d | odd | 4 | 1 | 5070.2.a.o | 1 | ||
| 13.f | odd | 12 | 2 | 390.2.i.a | ✓ | 2 | |
| 39.k | even | 12 | 2 | 1170.2.i.k | 2 | ||
| 65.o | even | 12 | 2 | 1950.2.z.h | 4 | ||
| 65.s | odd | 12 | 2 | 1950.2.i.s | 2 | ||
| 65.t | even | 12 | 2 | 1950.2.z.h | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 390.2.i.a | ✓ | 2 | 13.f | odd | 12 | 2 | |
| 1170.2.i.k | 2 | 39.k | even | 12 | 2 | ||
| 1950.2.i.s | 2 | 65.s | odd | 12 | 2 | ||
| 1950.2.z.h | 4 | 65.o | even | 12 | 2 | ||
| 1950.2.z.h | 4 | 65.t | even | 12 | 2 | ||
| 5070.2.a.f | 1 | 13.d | odd | 4 | 1 | ||
| 5070.2.a.o | 1 | 13.d | odd | 4 | 1 | ||
| 5070.2.b.g | 2 | 1.a | even | 1 | 1 | trivial | |
| 5070.2.b.g | 2 | 13.b | even | 2 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(5070, [\chi])\):
|
\( T_{7}^{2} + 4 \)
|
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\( T_{11}^{2} + 1 \)
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|
\( T_{17} + 2 \)
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\( T_{31}^{2} + 9 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} + 1 \)
|
| $3$ |
\( (T + 1)^{2} \)
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| $5$ |
\( T^{2} + 1 \)
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| $7$ |
\( T^{2} + 4 \)
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| $11$ |
\( T^{2} + 1 \)
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| $13$ |
\( T^{2} \)
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| $17$ |
\( (T + 2)^{2} \)
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| $19$ |
\( T^{2} + 36 \)
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| $23$ |
\( (T - 3)^{2} \)
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| $29$ |
\( (T + 1)^{2} \)
|
| $31$ |
\( T^{2} + 9 \)
|
| $37$ |
\( T^{2} + 25 \)
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| $41$ |
\( T^{2} + 100 \)
|
| $43$ |
\( (T + 5)^{2} \)
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| $47$ |
\( T^{2} + 9 \)
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| $53$ |
\( (T - 14)^{2} \)
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| $59$ |
\( T^{2} + 25 \)
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| $61$ |
\( (T + 10)^{2} \)
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| $67$ |
\( T^{2} \)
|
| $71$ |
\( T^{2} + 16 \)
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| $73$ |
\( T^{2} + 4 \)
|
| $79$ |
\( (T - 5)^{2} \)
|
| $83$ |
\( T^{2} + 36 \)
|
| $89$ |
\( T^{2} + 100 \)
|
| $97$ |
\( T^{2} + 100 \)
|
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