Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [70,6,Mod(11,70)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(70, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([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("70.11");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 70 = 2 \cdot 5 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 70.e (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: | \(11.2268673869\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\Q(\sqrt{-3}, \sqrt{79})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} + 79x^{2} + 6241 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2^{2} \) |
| 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 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
\( x^{4} + 79x^{2} + 6241 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} ) / 79 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 2\nu^{3} ) / 79 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( 79\beta_{2} \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 79\beta_{3} ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/70\mathbb{Z}\right)^\times\).
| \(n\) | \(31\) | \(57\) |
| \(\chi(n)\) | \(\beta_{2}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 11.1 |
|
2.00000 | + | 3.46410i | −10.3882 | + | 17.9929i | −8.00000 | + | 13.8564i | 12.5000 | + | 21.6506i | −83.1056 | −30.7174 | + | 125.950i | −64.0000 | −94.3292 | − | 163.383i | −50.0000 | + | 86.6025i | ||||||||||||||||
| 11.2 | 2.00000 | + | 3.46410i | 7.38819 | − | 12.7967i | −8.00000 | + | 13.8564i | 12.5000 | + | 21.6506i | 59.1056 | 93.7174 | − | 89.5771i | −64.0000 | 12.3292 | + | 21.3547i | −50.0000 | + | 86.6025i | |||||||||||||||||
| 51.1 | 2.00000 | − | 3.46410i | −10.3882 | − | 17.9929i | −8.00000 | − | 13.8564i | 12.5000 | − | 21.6506i | −83.1056 | −30.7174 | − | 125.950i | −64.0000 | −94.3292 | + | 163.383i | −50.0000 | − | 86.6025i | |||||||||||||||||
| 51.2 | 2.00000 | − | 3.46410i | 7.38819 | + | 12.7967i | −8.00000 | − | 13.8564i | 12.5000 | − | 21.6506i | 59.1056 | 93.7174 | + | 89.5771i | −64.0000 | 12.3292 | − | 21.3547i | −50.0000 | − | 86.6025i | |||||||||||||||||
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 | 70.6.e.e | ✓ | 4 |
| 7.c | even | 3 | 1 | inner | 70.6.e.e | ✓ | 4 |
| 7.c | even | 3 | 1 | 490.6.a.q | 2 | ||
| 7.d | odd | 6 | 1 | 490.6.a.p | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 70.6.e.e | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 70.6.e.e | ✓ | 4 | 7.c | even | 3 | 1 | inner |
| 490.6.a.p | 2 | 7.d | odd | 6 | 1 | ||
| 490.6.a.q | 2 | 7.c | even | 3 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} + 6T_{3}^{3} + 343T_{3}^{2} - 1842T_{3} + 94249 \)
acting on \(S_{6}^{\mathrm{new}}(70, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} - 4 T + 16)^{2} \)
|
| $3$ |
\( T^{4} + 6 T^{3} + \cdots + 94249 \)
|
| $5$ |
\( (T^{2} - 25 T + 625)^{2} \)
|
| $7$ |
\( T^{4} - 126 T^{3} + \cdots + 282475249 \)
|
| $11$ |
\( T^{4} - 76 T^{3} + \cdots + 32400 \)
|
| $13$ |
\( (T^{2} - 732 T - 18988)^{2} \)
|
| $17$ |
\( T^{4} + \cdots + 5092514355600 \)
|
| $19$ |
\( T^{4} + \cdots + 3198317331456 \)
|
| $23$ |
\( T^{4} + \cdots + 4868364234969 \)
|
| $29$ |
\( (T^{2} - 3286 T + 1405113)^{2} \)
|
| $31$ |
\( T^{4} + \cdots + 68\!\cdots\!00 \)
|
| $37$ |
\( T^{4} + \cdots + 19\!\cdots\!64 \)
|
| $41$ |
\( (T^{2} + 26174 T + 140113233)^{2} \)
|
| $43$ |
\( (T^{2} + 13298 T - 55242635)^{2} \)
|
| $47$ |
\( T^{4} + \cdots + 38\!\cdots\!00 \)
|
| $53$ |
\( T^{4} + \cdots + 27\!\cdots\!76 \)
|
| $59$ |
\( T^{4} + \cdots + 33\!\cdots\!04 \)
|
| $61$ |
\( T^{4} + \cdots + 31\!\cdots\!25 \)
|
| $67$ |
\( T^{4} + \cdots + 18\!\cdots\!25 \)
|
| $71$ |
\( (T^{2} - 8520 T - 314498016)^{2} \)
|
| $73$ |
\( T^{4} + \cdots + 16\!\cdots\!00 \)
|
| $79$ |
\( T^{4} + \cdots + 64\!\cdots\!00 \)
|
| $83$ |
\( (T^{2} + 54982 T + 639325197)^{2} \)
|
| $89$ |
\( T^{4} + \cdots + 28\!\cdots\!25 \)
|
| $97$ |
\( (T^{2} + 106964 T + 737289700)^{2} \)
|
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