Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [70,4,Mod(11,70)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(70, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 4]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("70.11");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 70 = 2 \cdot 5 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 70.e (of order \(3\), 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: | \(4.13013370040\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\Q(\sqrt{-3}, \sqrt{46})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} + 46x^{2} + 2116 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-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 a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring in terms of a root \(\nu\) of
\( x^{4} + 46x^{2} + 2116 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} ) / 46 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} ) / 46 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( 46\beta_{2} \)
|
| \(\nu^{3}\) | \(=\) |
\( 46\beta_{3} \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/70\mathbb{Z}\right)^\times\).
| \(n\) | \(31\) | \(57\) |
| \(\chi(n)\) | \(\beta_{2}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 11.1 |
|
1.00000 | + | 1.73205i | −2.89116 | + | 5.00764i | −2.00000 | + | 3.46410i | −2.50000 | − | 4.33013i | −11.5647 | −15.4558 | + | 10.2038i | −8.00000 | −3.21767 | − | 5.57317i | 5.00000 | − | 8.66025i | ||||||||||||||||
| 11.2 | 1.00000 | + | 1.73205i | 3.89116 | − | 6.73970i | −2.00000 | + | 3.46410i | −2.50000 | − | 4.33013i | 15.5647 | 18.4558 | − | 1.54354i | −8.00000 | −16.7823 | − | 29.0678i | 5.00000 | − | 8.66025i | |||||||||||||||||
| 51.1 | 1.00000 | − | 1.73205i | −2.89116 | − | 5.00764i | −2.00000 | − | 3.46410i | −2.50000 | + | 4.33013i | −11.5647 | −15.4558 | − | 10.2038i | −8.00000 | −3.21767 | + | 5.57317i | 5.00000 | + | 8.66025i | |||||||||||||||||
| 51.2 | 1.00000 | − | 1.73205i | 3.89116 | + | 6.73970i | −2.00000 | − | 3.46410i | −2.50000 | + | 4.33013i | 15.5647 | 18.4558 | + | 1.54354i | −8.00000 | −16.7823 | + | 29.0678i | 5.00000 | + | 8.66025i | |||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 70.4.e.d | ✓ | 4 |
| 3.b | odd | 2 | 1 | 630.4.k.l | 4 | ||
| 4.b | odd | 2 | 1 | 560.4.q.j | 4 | ||
| 5.b | even | 2 | 1 | 350.4.e.h | 4 | ||
| 5.c | odd | 4 | 2 | 350.4.j.g | 8 | ||
| 7.b | odd | 2 | 1 | 490.4.e.u | 4 | ||
| 7.c | even | 3 | 1 | inner | 70.4.e.d | ✓ | 4 |
| 7.c | even | 3 | 1 | 490.4.a.r | 2 | ||
| 7.d | odd | 6 | 1 | 490.4.a.t | 2 | ||
| 7.d | odd | 6 | 1 | 490.4.e.u | 4 | ||
| 21.h | odd | 6 | 1 | 630.4.k.l | 4 | ||
| 28.g | odd | 6 | 1 | 560.4.q.j | 4 | ||
| 35.i | odd | 6 | 1 | 2450.4.a.bv | 2 | ||
| 35.j | even | 6 | 1 | 350.4.e.h | 4 | ||
| 35.j | even | 6 | 1 | 2450.4.a.bz | 2 | ||
| 35.l | odd | 12 | 2 | 350.4.j.g | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 70.4.e.d | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 70.4.e.d | ✓ | 4 | 7.c | even | 3 | 1 | inner |
| 350.4.e.h | 4 | 5.b | even | 2 | 1 | ||
| 350.4.e.h | 4 | 35.j | even | 6 | 1 | ||
| 350.4.j.g | 8 | 5.c | odd | 4 | 2 | ||
| 350.4.j.g | 8 | 35.l | odd | 12 | 2 | ||
| 490.4.a.r | 2 | 7.c | even | 3 | 1 | ||
| 490.4.a.t | 2 | 7.d | odd | 6 | 1 | ||
| 490.4.e.u | 4 | 7.b | odd | 2 | 1 | ||
| 490.4.e.u | 4 | 7.d | odd | 6 | 1 | ||
| 560.4.q.j | 4 | 4.b | odd | 2 | 1 | ||
| 560.4.q.j | 4 | 28.g | odd | 6 | 1 | ||
| 630.4.k.l | 4 | 3.b | odd | 2 | 1 | ||
| 630.4.k.l | 4 | 21.h | odd | 6 | 1 | ||
| 2450.4.a.bv | 2 | 35.i | odd | 6 | 1 | ||
| 2450.4.a.bz | 2 | 35.j | even | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} - 2T_{3}^{3} + 49T_{3}^{2} + 90T_{3} + 2025 \)
acting on \(S_{4}^{\mathrm{new}}(70, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} - 2 T + 4)^{2} \)
|
| $3$ |
\( T^{4} - 2 T^{3} + \cdots + 2025 \)
|
| $5$ |
\( (T^{2} + 5 T + 25)^{2} \)
|
| $7$ |
\( T^{4} - 6 T^{3} + \cdots + 117649 \)
|
| $11$ |
\( T^{4} - 68 T^{3} + \cdots + 944784 \)
|
| $13$ |
\( (T^{2} + 52 T - 60)^{2} \)
|
| $17$ |
\( T^{4} + 164 T^{3} + \cdots + 14288400 \)
|
| $19$ |
\( T^{4} - 232 T^{3} + \cdots + 161798400 \)
|
| $23$ |
\( T^{4} - 198 T^{3} + \cdots + 301401 \)
|
| $29$ |
\( (T^{2} + 18 T - 14823)^{2} \)
|
| $31$ |
\( T^{4} + \cdots + 1010985616 \)
|
| $37$ |
\( T^{4} + 160 T^{3} + \cdots + 50176 \)
|
| $41$ |
\( (T^{2} - 62 T - 65463)^{2} \)
|
| $43$ |
\( (T^{2} - 198 T - 18949)^{2} \)
|
| $47$ |
\( T^{4} + 164 T^{3} + \cdots + 14288400 \)
|
| $53$ |
\( T^{4} + 40 T^{3} + \cdots + 6471936 \)
|
| $59$ |
\( T^{4} + \cdots + 245952516096 \)
|
| $61$ |
\( T^{4} + \cdots + 121560309025 \)
|
| $67$ |
\( T^{4} + \cdots + 5074425225 \)
|
| $71$ |
\( (T^{2} + 832 T + 99456)^{2} \)
|
| $73$ |
\( T^{4} + \cdots + 3858397456 \)
|
| $79$ |
\( T^{4} + \cdots + 11124342784 \)
|
| $83$ |
\( (T^{2} + 298 T - 719733)^{2} \)
|
| $89$ |
\( T^{4} + \cdots + 66262482225 \)
|
| $97$ |
\( (T^{2} - 892 T + 92932)^{2} \)
|
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