Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [140,6,Mod(81,140)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(140, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 0, 4]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("140.81");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 140 = 2^{2} \cdot 5 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 140.i (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: | \(22.4537347738\) |
| Analytic rank: | \(0\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{-3}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| 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 primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/140\mathbb{Z}\right)^\times\).
| \(n\) | \(57\) | \(71\) | \(101\) |
| \(\chi(n)\) | \(1\) | \(1\) | \(-\zeta_{6}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 81.1 |
|
0 | −14.5000 | + | 25.1147i | 0 | 12.5000 | + | 21.6506i | 0 | 129.500 | − | 6.06218i | 0 | −299.000 | − | 517.883i | 0 | ||||||||||||||||
| 121.1 | 0 | −14.5000 | − | 25.1147i | 0 | 12.5000 | − | 21.6506i | 0 | 129.500 | + | 6.06218i | 0 | −299.000 | + | 517.883i | 0 | |||||||||||||||||
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 | 140.6.i.a | ✓ | 2 |
| 7.c | even | 3 | 1 | inner | 140.6.i.a | ✓ | 2 |
| 7.c | even | 3 | 1 | 980.6.a.c | 1 | ||
| 7.d | odd | 6 | 1 | 980.6.a.a | 1 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 140.6.i.a | ✓ | 2 | 1.a | even | 1 | 1 | trivial |
| 140.6.i.a | ✓ | 2 | 7.c | even | 3 | 1 | inner |
| 980.6.a.a | 1 | 7.d | odd | 6 | 1 | ||
| 980.6.a.c | 1 | 7.c | even | 3 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{2} + 29T_{3} + 841 \)
acting on \(S_{6}^{\mathrm{new}}(140, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} \)
|
| $3$ |
\( T^{2} + 29T + 841 \)
|
| $5$ |
\( T^{2} - 25T + 625 \)
|
| $7$ |
\( T^{2} - 259T + 16807 \)
|
| $11$ |
\( T^{2} - 718T + 515524 \)
|
| $13$ |
\( (T - 988)^{2} \)
|
| $17$ |
\( T^{2} - 528T + 278784 \)
|
| $19$ |
\( T^{2} + 2394 T + 5731236 \)
|
| $23$ |
\( T^{2} - 743T + 552049 \)
|
| $29$ |
\( (T - 829)^{2} \)
|
| $31$ |
\( T^{2} - 3272 T + 10705984 \)
|
| $37$ |
\( T^{2} + 4316 T + 18627856 \)
|
| $41$ |
\( (T - 17281)^{2} \)
|
| $43$ |
\( (T + 14251)^{2} \)
|
| $47$ |
\( T^{2} + 10504 T + 110334016 \)
|
| $53$ |
\( T^{2} + 3802 T + 14455204 \)
|
| $59$ |
\( T^{2} + 15874 T + 251983876 \)
|
| $61$ |
\( T^{2} + 27069 T + 732730761 \)
|
| $67$ |
\( T^{2} - 28503 T + 812421009 \)
|
| $71$ |
\( (T + 49230)^{2} \)
|
| $73$ |
\( T^{2} + 20496 T + 420086016 \)
|
| $79$ |
\( T^{2} + \cdots + 1386668644 \)
|
| $83$ |
\( (T + 22177)^{2} \)
|
| $89$ |
\( T^{2} + \cdots + 1851150625 \)
|
| $97$ |
\( (T - 152030)^{2} \)
|
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