Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [70,3,Mod(23,70)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(70, base_ring=CyclotomicField(12))
chi = DirichletCharacter(H, H._module([9, 4]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("70.23");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 70 = 2 \cdot 5 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 70.l (of order \(12\), 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: | \(1.90736185052\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{12})\) |
| Coefficient field: | 8.0.303595776.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} + 5x^{6} + 16x^{4} + 45x^{2} + 81 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{12}]$ |
$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} + 5x^{6} + 16x^{4} + 45x^{2} + 81 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{6} + 32\nu^{4} + 16\nu^{2} + 45 ) / 144 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{7} + 32\nu^{5} + 16\nu^{3} + 45\nu ) / 432 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -5\nu^{6} - 16\nu^{4} - 80\nu^{2} - 225 ) / 144 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{7} + 13\nu ) / 48 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -\nu^{6} - 13 ) / 16 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 5\nu^{7} + 16\nu^{5} + 80\nu^{3} + 225\nu ) / 144 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{6} - 2\beta_{4} - \beta_{2} - 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( 2\beta_{7} - 3\beta_{5} - 3\beta_{3} - 2\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{4} + 5\beta_{2} \)
|
| \(\nu^{5}\) | \(=\) |
\( -\beta_{7} + 15\beta_{3} \)
|
| \(\nu^{6}\) | \(=\) |
\( -16\beta_{6} - 13 \)
|
| \(\nu^{7}\) | \(=\) |
\( 48\beta_{5} - 13\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/70\mathbb{Z}\right)^\times\).
| \(n\) | \(31\) | \(57\) |
| \(\chi(n)\) | \(-1 - \beta_{4}\) | \(-\beta_{3} - \beta_{5}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 23.1 |
|
−0.366025 | − | 1.36603i | −0.789997 | + | 2.94831i | −1.73205 | + | 1.00000i | −3.57025 | + | 3.50048i | 4.31662 | −3.67639 | + | 5.95686i | 2.00000 | + | 2.00000i | −0.274205 | − | 0.158312i | 6.08854 | + | 3.59579i | ||||||||||||||||||||||||||
| 23.2 | −0.366025 | − | 1.36603i | 0.423972 | − | 1.58228i | −1.73205 | + | 1.00000i | −3.12590 | − | 3.90240i | −2.31662 | −2.78771 | − | 6.42096i | 2.00000 | + | 2.00000i | 5.47036 | + | 3.15831i | −4.18662 | + | 5.69844i | |||||||||||||||||||||||||||
| 37.1 | 1.36603 | − | 0.366025i | −1.58228 | − | 0.423972i | 1.73205 | − | 1.00000i | 4.94253 | + | 0.755913i | −2.31662 | 6.42096 | − | 2.78771i | 2.00000 | − | 2.00000i | −5.47036 | − | 3.15831i | 7.02830 | − | 0.776495i | |||||||||||||||||||||||||||
| 37.2 | 1.36603 | − | 0.366025i | 2.94831 | + | 0.789997i | 1.73205 | − | 1.00000i | −1.24638 | + | 4.84216i | 4.31662 | −5.95686 | − | 3.67639i | 2.00000 | − | 2.00000i | 0.274205 | + | 0.158312i | 0.0697723 | + | 7.07072i | |||||||||||||||||||||||||||
| 53.1 | 1.36603 | + | 0.366025i | −1.58228 | + | 0.423972i | 1.73205 | + | 1.00000i | 4.94253 | − | 0.755913i | −2.31662 | 6.42096 | + | 2.78771i | 2.00000 | + | 2.00000i | −5.47036 | + | 3.15831i | 7.02830 | + | 0.776495i | |||||||||||||||||||||||||||
| 53.2 | 1.36603 | + | 0.366025i | 2.94831 | − | 0.789997i | 1.73205 | + | 1.00000i | −1.24638 | − | 4.84216i | 4.31662 | −5.95686 | + | 3.67639i | 2.00000 | + | 2.00000i | 0.274205 | − | 0.158312i | 0.0697723 | − | 7.07072i | |||||||||||||||||||||||||||
| 67.1 | −0.366025 | + | 1.36603i | −0.789997 | − | 2.94831i | −1.73205 | − | 1.00000i | −3.57025 | − | 3.50048i | 4.31662 | −3.67639 | − | 5.95686i | 2.00000 | − | 2.00000i | −0.274205 | + | 0.158312i | 6.08854 | − | 3.59579i | |||||||||||||||||||||||||||
| 67.2 | −0.366025 | + | 1.36603i | 0.423972 | + | 1.58228i | −1.73205 | − | 1.00000i | −3.12590 | + | 3.90240i | −2.31662 | −2.78771 | + | 6.42096i | 2.00000 | − | 2.00000i | 5.47036 | − | 3.15831i | −4.18662 | − | 5.69844i | |||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.c | odd | 4 | 1 | inner |
| 7.c | even | 3 | 1 | inner |
| 35.l | odd | 12 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 70.3.l.b | ✓ | 8 |
| 5.b | even | 2 | 1 | 350.3.p.b | 8 | ||
| 5.c | odd | 4 | 1 | inner | 70.3.l.b | ✓ | 8 |
| 5.c | odd | 4 | 1 | 350.3.p.b | 8 | ||
| 7.c | even | 3 | 1 | inner | 70.3.l.b | ✓ | 8 |
| 7.c | even | 3 | 1 | 490.3.f.f | 4 | ||
| 7.d | odd | 6 | 1 | 490.3.f.k | 4 | ||
| 35.j | even | 6 | 1 | 350.3.p.b | 8 | ||
| 35.k | even | 12 | 1 | 490.3.f.k | 4 | ||
| 35.l | odd | 12 | 1 | inner | 70.3.l.b | ✓ | 8 |
| 35.l | odd | 12 | 1 | 350.3.p.b | 8 | ||
| 35.l | odd | 12 | 1 | 490.3.f.f | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 70.3.l.b | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 70.3.l.b | ✓ | 8 | 5.c | odd | 4 | 1 | inner |
| 70.3.l.b | ✓ | 8 | 7.c | even | 3 | 1 | inner |
| 70.3.l.b | ✓ | 8 | 35.l | odd | 12 | 1 | inner |
| 350.3.p.b | 8 | 5.b | even | 2 | 1 | ||
| 350.3.p.b | 8 | 5.c | odd | 4 | 1 | ||
| 350.3.p.b | 8 | 35.j | even | 6 | 1 | ||
| 350.3.p.b | 8 | 35.l | odd | 12 | 1 | ||
| 490.3.f.f | 4 | 7.c | even | 3 | 1 | ||
| 490.3.f.f | 4 | 35.l | odd | 12 | 1 | ||
| 490.3.f.k | 4 | 7.d | odd | 6 | 1 | ||
| 490.3.f.k | 4 | 35.k | even | 12 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{8} - 2T_{3}^{7} + 2T_{3}^{6} - 24T_{3}^{5} - T_{3}^{4} + 120T_{3}^{3} + 50T_{3}^{2} + 250T_{3} + 625 \)
acting on \(S_{3}^{\mathrm{new}}(70, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{4} - 2 T^{3} + 2 T^{2} + \cdots + 4)^{2} \)
|
| $3$ |
\( T^{8} - 2 T^{7} + \cdots + 625 \)
|
| $5$ |
\( T^{8} + 6 T^{7} + \cdots + 390625 \)
|
| $7$ |
\( T^{8} + 12 T^{7} + \cdots + 5764801 \)
|
| $11$ |
\( (T^{4} + 8 T^{3} + 59 T^{2} + \cdots + 25)^{2} \)
|
| $13$ |
\( (T^{4} + 8 T^{3} + \cdots + 36100)^{2} \)
|
| $17$ |
\( T^{8} + \cdots + 13841287201 \)
|
| $19$ |
\( T^{8} - 106 T^{6} + \cdots + 1500625 \)
|
| $23$ |
\( T^{8} + \cdots + 1908029761 \)
|
| $29$ |
\( (T^{4} + 1608 T^{2} + 364816)^{2} \)
|
| $31$ |
\( (T^{4} - 12 T^{3} + \cdots + 731025)^{2} \)
|
| $37$ |
\( T^{8} + \cdots + 10474291432801 \)
|
| $41$ |
\( (T^{2} - 80 T + 16)^{4} \)
|
| $43$ |
\( (T^{4} - 8 T^{3} + \cdots + 13764100)^{2} \)
|
| $47$ |
\( T^{8} + \cdots + 1829442899761 \)
|
| $53$ |
\( T^{8} + \cdots + 25041707380561 \)
|
| $59$ |
\( T^{8} + \cdots + 4457203441681 \)
|
| $61$ |
\( (T^{4} + 42 T^{3} + \cdots + 434281)^{2} \)
|
| $67$ |
\( T^{8} + 130 T^{7} + \cdots + 260144641 \)
|
| $71$ |
\( (T^{2} - 100 T + 2324)^{4} \)
|
| $73$ |
\( T^{8} + \cdots + 12\!\cdots\!25 \)
|
| $79$ |
\( T^{8} + \cdots + 85\!\cdots\!25 \)
|
| $83$ |
\( (T^{4} + 80 T^{3} + \cdots + 8514724)^{2} \)
|
| $89$ |
\( T^{8} + \cdots + 6702304832161 \)
|
| $97$ |
\( (T^{4} + 216 T^{3} + \cdots + 33756100)^{2} \)
|
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