Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [40,3,Mod(19,40)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(40, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 1, 1]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("40.19");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 40 = 2^{3} \cdot 5 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 40.e (of order \(2\), degree \(1\), 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: | \(1.08992105744\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | 8.0.53824000000.2 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} + 6x^{6} + 36x^{4} + 96x^{2} + 256 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{5} \) |
| Twist minimal: | yes |
| 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 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
\( x^{8} + 6x^{6} + 36x^{4} + 96x^{2} + 256 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{5} + 2\nu^{3} + 4\nu ) / 8 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{7} + 6\nu^{5} + 36\nu^{3} + 96\nu ) / 64 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{2} + 1 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{5} + 2\nu^{3} + 20\nu ) / 8 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -\nu^{6} + 2\nu^{4} - 4\nu^{2} + 48 ) / 16 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( \nu^{6} + 6\nu^{4} + 20\nu^{2} + 48 ) / 16 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -3\nu^{7} - 10\nu^{5} - 28\nu^{3} ) / 64 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{4} - \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} - 1 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{7} - 2\beta_{4} + 3\beta_{2} + \beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( 2\beta_{6} + 2\beta_{5} - 2\beta_{3} - 10 \)
|
| \(\nu^{5}\) | \(=\) |
\( -2\beta_{7} + 2\beta_{4} - 6\beta_{2} + 8\beta_1 \)
|
| \(\nu^{6}\) | \(=\) |
\( 4\beta_{6} - 12\beta_{5} - 8\beta_{3} + 32 \)
|
| \(\nu^{7}\) | \(=\) |
\( -24\beta_{7} + 12\beta_{4} - 8\beta_{2} - 36\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/40\mathbb{Z}\right)^\times\).
| \(n\) | \(17\) | \(21\) | \(31\) |
| \(\chi(n)\) | \(-1\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 19.1 |
|
−1.34500 | − | 1.48020i | 4.79002i | −0.381966 | + | 3.98172i | −4.35250 | + | 2.46084i | 7.09017 | − | 6.44256i | 7.67752 | 6.40747 | − | 4.79002i | −13.9443 | 9.49663 | + | 3.13274i | ||||||||||||||||||||||||||||||
| 19.2 | −1.34500 | + | 1.48020i | − | 4.79002i | −0.381966 | − | 3.98172i | −4.35250 | − | 2.46084i | 7.09017 | + | 6.44256i | 7.67752 | 6.40747 | + | 4.79002i | −13.9443 | 9.49663 | − | 3.13274i | ||||||||||||||||||||||||||||||
| 19.3 | −0.831254 | − | 1.81907i | − | 2.24849i | −2.61803 | + | 3.02422i | 1.02749 | − | 4.89329i | −4.09017 | + | 1.86907i | −6.40747 | 7.67752 | + | 2.24849i | 3.94427 | −9.75534 | + | 2.19850i | ||||||||||||||||||||||||||||||
| 19.4 | −0.831254 | + | 1.81907i | 2.24849i | −2.61803 | − | 3.02422i | 1.02749 | + | 4.89329i | −4.09017 | − | 1.86907i | −6.40747 | 7.67752 | − | 2.24849i | 3.94427 | −9.75534 | − | 2.19850i | |||||||||||||||||||||||||||||||
| 19.5 | 0.831254 | − | 1.81907i | − | 2.24849i | −2.61803 | − | 3.02422i | −1.02749 | + | 4.89329i | −4.09017 | − | 1.86907i | 6.40747 | −7.67752 | + | 2.24849i | 3.94427 | 8.04713 | + | 5.93663i | ||||||||||||||||||||||||||||||
| 19.6 | 0.831254 | + | 1.81907i | 2.24849i | −2.61803 | + | 3.02422i | −1.02749 | − | 4.89329i | −4.09017 | + | 1.86907i | 6.40747 | −7.67752 | − | 2.24849i | 3.94427 | 8.04713 | − | 5.93663i | |||||||||||||||||||||||||||||||
| 19.7 | 1.34500 | − | 1.48020i | 4.79002i | −0.381966 | − | 3.98172i | 4.35250 | − | 2.46084i | 7.09017 | + | 6.44256i | −7.67752 | −6.40747 | − | 4.79002i | −13.9443 | 2.21158 | − | 9.75238i | |||||||||||||||||||||||||||||||
| 19.8 | 1.34500 | + | 1.48020i | − | 4.79002i | −0.381966 | + | 3.98172i | 4.35250 | + | 2.46084i | 7.09017 | − | 6.44256i | −7.67752 | −6.40747 | + | 4.79002i | −13.9443 | 2.21158 | + | 9.75238i | ||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.b | even | 2 | 1 | inner |
| 8.d | odd | 2 | 1 | inner |
| 40.e | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 40.3.e.c | ✓ | 8 |
| 3.b | odd | 2 | 1 | 360.3.p.g | 8 | ||
| 4.b | odd | 2 | 1 | 160.3.e.c | 8 | ||
| 5.b | even | 2 | 1 | inner | 40.3.e.c | ✓ | 8 |
| 5.c | odd | 4 | 2 | 200.3.g.h | 8 | ||
| 8.b | even | 2 | 1 | 160.3.e.c | 8 | ||
| 8.d | odd | 2 | 1 | inner | 40.3.e.c | ✓ | 8 |
| 12.b | even | 2 | 1 | 1440.3.p.g | 8 | ||
| 15.d | odd | 2 | 1 | 360.3.p.g | 8 | ||
| 16.e | even | 4 | 2 | 1280.3.h.m | 16 | ||
| 16.f | odd | 4 | 2 | 1280.3.h.m | 16 | ||
| 20.d | odd | 2 | 1 | 160.3.e.c | 8 | ||
| 20.e | even | 4 | 2 | 800.3.g.h | 8 | ||
| 24.f | even | 2 | 1 | 360.3.p.g | 8 | ||
| 24.h | odd | 2 | 1 | 1440.3.p.g | 8 | ||
| 40.e | odd | 2 | 1 | inner | 40.3.e.c | ✓ | 8 |
| 40.f | even | 2 | 1 | 160.3.e.c | 8 | ||
| 40.i | odd | 4 | 2 | 800.3.g.h | 8 | ||
| 40.k | even | 4 | 2 | 200.3.g.h | 8 | ||
| 60.h | even | 2 | 1 | 1440.3.p.g | 8 | ||
| 80.k | odd | 4 | 2 | 1280.3.h.m | 16 | ||
| 80.q | even | 4 | 2 | 1280.3.h.m | 16 | ||
| 120.i | odd | 2 | 1 | 1440.3.p.g | 8 | ||
| 120.m | even | 2 | 1 | 360.3.p.g | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 40.3.e.c | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 40.3.e.c | ✓ | 8 | 5.b | even | 2 | 1 | inner |
| 40.3.e.c | ✓ | 8 | 8.d | odd | 2 | 1 | inner |
| 40.3.e.c | ✓ | 8 | 40.e | odd | 2 | 1 | inner |
| 160.3.e.c | 8 | 4.b | odd | 2 | 1 | ||
| 160.3.e.c | 8 | 8.b | even | 2 | 1 | ||
| 160.3.e.c | 8 | 20.d | odd | 2 | 1 | ||
| 160.3.e.c | 8 | 40.f | even | 2 | 1 | ||
| 200.3.g.h | 8 | 5.c | odd | 4 | 2 | ||
| 200.3.g.h | 8 | 40.k | even | 4 | 2 | ||
| 360.3.p.g | 8 | 3.b | odd | 2 | 1 | ||
| 360.3.p.g | 8 | 15.d | odd | 2 | 1 | ||
| 360.3.p.g | 8 | 24.f | even | 2 | 1 | ||
| 360.3.p.g | 8 | 120.m | even | 2 | 1 | ||
| 800.3.g.h | 8 | 20.e | even | 4 | 2 | ||
| 800.3.g.h | 8 | 40.i | odd | 4 | 2 | ||
| 1280.3.h.m | 16 | 16.e | even | 4 | 2 | ||
| 1280.3.h.m | 16 | 16.f | odd | 4 | 2 | ||
| 1280.3.h.m | 16 | 80.k | odd | 4 | 2 | ||
| 1280.3.h.m | 16 | 80.q | even | 4 | 2 | ||
| 1440.3.p.g | 8 | 12.b | even | 2 | 1 | ||
| 1440.3.p.g | 8 | 24.h | odd | 2 | 1 | ||
| 1440.3.p.g | 8 | 60.h | even | 2 | 1 | ||
| 1440.3.p.g | 8 | 120.i | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(40, [\chi])\):
|
\( T_{3}^{4} + 28T_{3}^{2} + 116 \)
|
|
\( T_{7}^{4} - 100T_{7}^{2} + 2420 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} + 6 T^{6} + \cdots + 256 \)
|
| $3$ |
\( (T^{4} + 28 T^{2} + 116)^{2} \)
|
| $5$ |
\( T^{8} + 20 T^{6} + \cdots + 390625 \)
|
| $7$ |
\( (T^{4} - 100 T^{2} + 2420)^{2} \)
|
| $11$ |
\( (T^{2} - 8 T - 4)^{4} \)
|
| $13$ |
\( (T^{4} - 320 T^{2} + 5120)^{2} \)
|
| $17$ |
\( (T^{4} + 368 T^{2} + 1856)^{2} \)
|
| $19$ |
\( (T^{2} - 8 T - 484)^{4} \)
|
| $23$ |
\( (T^{4} - 100 T^{2} + 500)^{2} \)
|
| $29$ |
\( (T^{4} + 1760 T^{2} + 37120)^{2} \)
|
| $31$ |
\( (T^{4} + 2400 T^{2} + 928000)^{2} \)
|
| $37$ |
\( (T^{4} - 3280 T^{2} + 1613120)^{2} \)
|
| $41$ |
\( (T^{2} - 48 T + 496)^{4} \)
|
| $43$ |
\( (T^{4} + 1372 T^{2} + 111476)^{2} \)
|
| $47$ |
\( (T^{4} - 5860 T^{2} + 444020)^{2} \)
|
| $53$ |
\( (T^{4} - 5760 T^{2} + 6635520)^{2} \)
|
| $59$ |
\( (T^{2} + 56 T - 1636)^{4} \)
|
| $61$ |
\( (T^{4} + 1080 T^{2} + 187920)^{2} \)
|
| $67$ |
\( (T^{4} + 12988 T^{2} + 41620916)^{2} \)
|
| $71$ |
\( (T^{4} + 12000 T^{2} + 35672320)^{2} \)
|
| $73$ |
\( (T^{4} + 7408 T^{2} + 3119936)^{2} \)
|
| $79$ |
\( (T^{4} + 22880 T^{2} + 129214720)^{2} \)
|
| $83$ |
\( (T^{4} + 1628 T^{2} + 116)^{2} \)
|
| $89$ |
\( (T^{2} - 28 T - 4924)^{4} \)
|
| $97$ |
\( (T^{4} + 44912 T^{2} + 344772416)^{2} \)
|
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