Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [70,3,Mod(43,70)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(70, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([3, 0]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("70.43");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 70 = 2 \cdot 5 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 70.f (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: | \(1.90736185052\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(i)\) |
| Coefficient field: | \(\Q(i, \sqrt{14})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} + 49 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-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
\( x^{4} + 49 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} ) / 7 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} ) / 7 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( 7\beta_{2} \)
|
| \(\nu^{3}\) | \(=\) |
\( 7\beta_{3} \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/70\mathbb{Z}\right)^\times\).
| \(n\) | \(31\) | \(57\) |
| \(\chi(n)\) | \(1\) | \(-\beta_{2}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 43.1 |
|
−1.00000 | + | 1.00000i | −0.870829 | − | 0.870829i | − | 2.00000i | 1.12917 | − | 4.87083i | 1.74166 | 1.87083 | − | 1.87083i | 2.00000 | + | 2.00000i | − | 7.48331i | 3.74166 | + | 6.00000i | ||||||||||||||||
| 43.2 | −1.00000 | + | 1.00000i | 2.87083 | + | 2.87083i | − | 2.00000i | 4.87083 | − | 1.12917i | −5.74166 | −1.87083 | + | 1.87083i | 2.00000 | + | 2.00000i | 7.48331i | −3.74166 | + | 6.00000i | ||||||||||||||||||
| 57.1 | −1.00000 | − | 1.00000i | −0.870829 | + | 0.870829i | 2.00000i | 1.12917 | + | 4.87083i | 1.74166 | 1.87083 | + | 1.87083i | 2.00000 | − | 2.00000i | 7.48331i | 3.74166 | − | 6.00000i | |||||||||||||||||||
| 57.2 | −1.00000 | − | 1.00000i | 2.87083 | − | 2.87083i | 2.00000i | 4.87083 | + | 1.12917i | −5.74166 | −1.87083 | − | 1.87083i | 2.00000 | − | 2.00000i | − | 7.48331i | −3.74166 | − | 6.00000i | ||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.c | odd | 4 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 70.3.f.a | ✓ | 4 |
| 3.b | odd | 2 | 1 | 630.3.o.a | 4 | ||
| 4.b | odd | 2 | 1 | 560.3.bh.b | 4 | ||
| 5.b | even | 2 | 1 | 350.3.f.b | 4 | ||
| 5.c | odd | 4 | 1 | inner | 70.3.f.a | ✓ | 4 |
| 5.c | odd | 4 | 1 | 350.3.f.b | 4 | ||
| 7.b | odd | 2 | 1 | 490.3.f.d | 4 | ||
| 15.e | even | 4 | 1 | 630.3.o.a | 4 | ||
| 20.e | even | 4 | 1 | 560.3.bh.b | 4 | ||
| 35.f | even | 4 | 1 | 490.3.f.d | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 70.3.f.a | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 70.3.f.a | ✓ | 4 | 5.c | odd | 4 | 1 | inner |
| 350.3.f.b | 4 | 5.b | even | 2 | 1 | ||
| 350.3.f.b | 4 | 5.c | odd | 4 | 1 | ||
| 490.3.f.d | 4 | 7.b | odd | 2 | 1 | ||
| 490.3.f.d | 4 | 35.f | even | 4 | 1 | ||
| 560.3.bh.b | 4 | 4.b | odd | 2 | 1 | ||
| 560.3.bh.b | 4 | 20.e | even | 4 | 1 | ||
| 630.3.o.a | 4 | 3.b | odd | 2 | 1 | ||
| 630.3.o.a | 4 | 15.e | even | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} - 4T_{3}^{3} + 8T_{3}^{2} + 20T_{3} + 25 \)
acting on \(S_{3}^{\mathrm{new}}(70, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 2 T + 2)^{2} \)
|
| $3$ |
\( T^{4} - 4 T^{3} + \cdots + 25 \)
|
| $5$ |
\( T^{4} - 12 T^{3} + \cdots + 625 \)
|
| $7$ |
\( T^{4} + 49 \)
|
| $11$ |
\( (T^{2} - 2 T - 223)^{2} \)
|
| $13$ |
\( T^{4} - 4 T^{3} + \cdots + 3721 \)
|
| $17$ |
\( T^{4} + 20 T^{3} + \cdots + 85849 \)
|
| $19$ |
\( T^{4} + 676 T^{2} + 96100 \)
|
| $23$ |
\( T^{4} + 40 T^{3} + \cdots + 250000 \)
|
| $29$ |
\( T^{4} + 2898 T^{2} + 1002001 \)
|
| $31$ |
\( (T^{2} - 24 T + 18)^{2} \)
|
| $37$ |
\( T^{4} - 104 T^{3} + \cdots + 1210000 \)
|
| $41$ |
\( (T^{2} - 8 T - 1118)^{2} \)
|
| $43$ |
\( T^{4} - 188 T^{3} + \cdots + 18541636 \)
|
| $47$ |
\( T^{4} + 84 T^{3} + \cdots + 7958041 \)
|
| $53$ |
\( T^{4} - 4 T^{3} + \cdots + 487204 \)
|
| $59$ |
\( T^{4} + 6332 T^{2} + 9821956 \)
|
| $61$ |
\( (T^{2} + 144 T + 4960)^{2} \)
|
| $67$ |
\( T^{4} + 92 T^{3} + \cdots + 128164 \)
|
| $71$ |
\( (T^{2} + 80 T + 1544)^{2} \)
|
| $73$ |
\( T^{4} + 72 T^{3} + \cdots + 23425600 \)
|
| $79$ |
\( T^{4} + 16690 T^{2} + 1385329 \)
|
| $83$ |
\( T^{4} - 256 T^{3} + \cdots + 59969536 \)
|
| $89$ |
\( (T^{2} + 6400)^{2} \)
|
| $97$ |
\( T^{4} + 84 T^{3} + \cdots + 2706025 \)
|
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