Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [40,3,Mod(17,40)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(40, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([0, 0, 1]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("40.17");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 40 = 2^{3} \cdot 5 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 40.l (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: | \(1.08992105744\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(i)\) |
| Coefficient field: | \(\Q(i, \sqrt{41})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} + 21x^{2} + 100 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2 \) |
| 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
\( x^{4} + 21x^{2} + 100 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{3} + 11\nu ) / 10 \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} + \nu + 11 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} - 10\nu^{2} + 21\nu - 110 ) / 10 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} + \beta_{2} - \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -\beta_{3} + \beta_{2} + \beta _1 - 22 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -11\beta_{3} - 11\beta_{2} + 31\beta_1 ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/40\mathbb{Z}\right)^\times\).
| \(n\) | \(17\) | \(21\) | \(31\) |
| \(\chi(n)\) | \(-\beta_{1}\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 17.1 |
|
0 | −3.70156 | + | 3.70156i | 0 | 1.70156 | + | 4.70156i | 0 | 0.298438 | + | 0.298438i | 0 | − | 18.4031i | 0 | |||||||||||||||||||||||
| 17.2 | 0 | 2.70156 | − | 2.70156i | 0 | −4.70156 | − | 1.70156i | 0 | 6.70156 | + | 6.70156i | 0 | − | 5.59688i | 0 | ||||||||||||||||||||||||
| 33.1 | 0 | −3.70156 | − | 3.70156i | 0 | 1.70156 | − | 4.70156i | 0 | 0.298438 | − | 0.298438i | 0 | 18.4031i | 0 | |||||||||||||||||||||||||
| 33.2 | 0 | 2.70156 | + | 2.70156i | 0 | −4.70156 | + | 1.70156i | 0 | 6.70156 | − | 6.70156i | 0 | 5.59688i | 0 | |||||||||||||||||||||||||
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 | 40.3.l.b | ✓ | 4 |
| 3.b | odd | 2 | 1 | 360.3.v.c | 4 | ||
| 4.b | odd | 2 | 1 | 80.3.p.d | 4 | ||
| 5.b | even | 2 | 1 | 200.3.l.e | 4 | ||
| 5.c | odd | 4 | 1 | inner | 40.3.l.b | ✓ | 4 |
| 5.c | odd | 4 | 1 | 200.3.l.e | 4 | ||
| 8.b | even | 2 | 1 | 320.3.p.l | 4 | ||
| 8.d | odd | 2 | 1 | 320.3.p.i | 4 | ||
| 12.b | even | 2 | 1 | 720.3.bh.l | 4 | ||
| 15.d | odd | 2 | 1 | 1800.3.v.k | 4 | ||
| 15.e | even | 4 | 1 | 360.3.v.c | 4 | ||
| 15.e | even | 4 | 1 | 1800.3.v.k | 4 | ||
| 20.d | odd | 2 | 1 | 400.3.p.i | 4 | ||
| 20.e | even | 4 | 1 | 80.3.p.d | 4 | ||
| 20.e | even | 4 | 1 | 400.3.p.i | 4 | ||
| 40.i | odd | 4 | 1 | 320.3.p.l | 4 | ||
| 40.k | even | 4 | 1 | 320.3.p.i | 4 | ||
| 60.l | odd | 4 | 1 | 720.3.bh.l | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 40.3.l.b | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 40.3.l.b | ✓ | 4 | 5.c | odd | 4 | 1 | inner |
| 80.3.p.d | 4 | 4.b | odd | 2 | 1 | ||
| 80.3.p.d | 4 | 20.e | even | 4 | 1 | ||
| 200.3.l.e | 4 | 5.b | even | 2 | 1 | ||
| 200.3.l.e | 4 | 5.c | odd | 4 | 1 | ||
| 320.3.p.i | 4 | 8.d | odd | 2 | 1 | ||
| 320.3.p.i | 4 | 40.k | even | 4 | 1 | ||
| 320.3.p.l | 4 | 8.b | even | 2 | 1 | ||
| 320.3.p.l | 4 | 40.i | odd | 4 | 1 | ||
| 360.3.v.c | 4 | 3.b | odd | 2 | 1 | ||
| 360.3.v.c | 4 | 15.e | even | 4 | 1 | ||
| 400.3.p.i | 4 | 20.d | odd | 2 | 1 | ||
| 400.3.p.i | 4 | 20.e | even | 4 | 1 | ||
| 720.3.bh.l | 4 | 12.b | even | 2 | 1 | ||
| 720.3.bh.l | 4 | 60.l | odd | 4 | 1 | ||
| 1800.3.v.k | 4 | 15.d | odd | 2 | 1 | ||
| 1800.3.v.k | 4 | 15.e | even | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} + 2T_{3}^{3} + 2T_{3}^{2} - 40T_{3} + 400 \)
acting on \(S_{3}^{\mathrm{new}}(40, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} + 2 T^{3} + \cdots + 400 \)
|
| $5$ |
\( T^{4} + 6 T^{3} + \cdots + 625 \)
|
| $7$ |
\( T^{4} - 14 T^{3} + \cdots + 16 \)
|
| $11$ |
\( (T^{2} - 10 T - 16)^{2} \)
|
| $13$ |
\( T^{4} + 32 T^{3} + \cdots + 2116 \)
|
| $17$ |
\( T^{4} - 16 T^{3} + \cdots + 2500 \)
|
| $19$ |
\( T^{4} + 528T^{2} + 4096 \)
|
| $23$ |
\( T^{4} - 38 T^{3} + \cdots + 16 \)
|
| $29$ |
\( T^{4} + 2112 T^{2} + 65536 \)
|
| $31$ |
\( (T^{2} + 18 T + 40)^{2} \)
|
| $37$ |
\( T^{4} + 60 T^{3} + \cdots + 743044 \)
|
| $41$ |
\( (T^{2} - 26 T - 3152)^{2} \)
|
| $43$ |
\( T^{4} - 94 T^{3} + \cdots + 1175056 \)
|
| $47$ |
\( T^{4} + 106 T^{3} + \cdots + 795664 \)
|
| $53$ |
\( (T^{2} + 6 T + 18)^{2} \)
|
| $59$ |
\( T^{4} + 6160 T^{2} + 7573504 \)
|
| $61$ |
\( (T^{2} + 102 T + 2560)^{2} \)
|
| $67$ |
\( T^{4} - 126 T^{3} + \cdots + 3857296 \)
|
| $71$ |
\( (T^{2} - 78 T + 1480)^{2} \)
|
| $73$ |
\( T^{4} - 148 T^{3} + \cdots + 2033476 \)
|
| $79$ |
\( T^{4} + 21504 T^{2} + 104857600 \)
|
| $83$ |
\( T^{4} + 186 T^{3} + \cdots + 3400336 \)
|
| $89$ |
\( T^{4} + 8448 T^{2} + 1048576 \)
|
| $97$ |
\( (T^{2} - 98 T + 4802)^{2} \)
|
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