Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [210,8,Mod(121,210)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(210, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 0, 2]))
N = Newforms(chi, 8, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("210.121");
S:= CuspForms(chi, 8);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 210 = 2 \cdot 3 \cdot 5 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| Character orbit: | \([\chi]\) | \(=\) | 210.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: | \(65.6008553517\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{4} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} + 5593x^{2} + 5592x + 31270464 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 3 \) |
| 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
\( x^{4} - x^{3} + 5593x^{2} + 5592x + 31270464 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -\nu^{3} + 5593\nu^{2} - 5593\nu + 31270464 ) / 31276056 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} - \nu^{2} + 11185\nu + 5592 ) / 5592 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} - 5593\nu + 11185 ) / 5593 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{3} + \beta_{2} + \beta_1 ) / 3 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} + 2\beta_{2} + 16778\beta _1 - 16779 ) / 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 11186\beta_{3} + 5593\beta_{2} + 5593\beta _1 - 33555 ) / 3 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/210\mathbb{Z}\right)^\times\).
| \(n\) | \(31\) | \(71\) | \(127\) |
| \(\chi(n)\) | \(-1 + \beta_{1}\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 121.1 |
|
−4.00000 | + | 6.92820i | −13.5000 | − | 23.3827i | −32.0000 | − | 55.4256i | 62.5000 | − | 108.253i | 216.000 | −804.454 | + | 419.996i | 512.000 | −364.500 | + | 631.333i | 500.000 | + | 866.025i | ||||||||||||||||
| 121.2 | −4.00000 | + | 6.92820i | −13.5000 | − | 23.3827i | −32.0000 | − | 55.4256i | 62.5000 | − | 108.253i | 216.000 | 765.954 | − | 486.680i | 512.000 | −364.500 | + | 631.333i | 500.000 | + | 866.025i | |||||||||||||||||
| 151.1 | −4.00000 | − | 6.92820i | −13.5000 | + | 23.3827i | −32.0000 | + | 55.4256i | 62.5000 | + | 108.253i | 216.000 | −804.454 | − | 419.996i | 512.000 | −364.500 | − | 631.333i | 500.000 | − | 866.025i | |||||||||||||||||
| 151.2 | −4.00000 | − | 6.92820i | −13.5000 | + | 23.3827i | −32.0000 | + | 55.4256i | 62.5000 | + | 108.253i | 216.000 | 765.954 | + | 486.680i | 512.000 | −364.500 | − | 631.333i | 500.000 | − | 866.025i | |||||||||||||||||
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 | 210.8.i.b | ✓ | 4 |
| 7.c | even | 3 | 1 | inner | 210.8.i.b | ✓ | 4 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 210.8.i.b | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 210.8.i.b | ✓ | 4 | 7.c | even | 3 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{11}^{4} - 5325T_{11}^{3} + 47891421T_{11}^{2} + 104028113700T_{11} + 381647325353616 \)
acting on \(S_{8}^{\mathrm{new}}(210, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 8 T + 64)^{2} \)
|
| $3$ |
\( (T^{2} + 27 T + 729)^{2} \)
|
| $5$ |
\( (T^{2} - 125 T + 15625)^{2} \)
|
| $7$ |
\( T^{4} + \cdots + 678223072849 \)
|
| $11$ |
\( T^{4} + \cdots + 381647325353616 \)
|
| $13$ |
\( (T^{2} + 6860 T - 94733909)^{2} \)
|
| $17$ |
\( T^{4} + \cdots + 10\!\cdots\!00 \)
|
| $19$ |
\( T^{4} + \cdots + 29\!\cdots\!29 \)
|
| $23$ |
\( T^{4} + \cdots + 14\!\cdots\!36 \)
|
| $29$ |
\( (T^{2} - 240654 T + 12424911408)^{2} \)
|
| $31$ |
\( T^{4} + \cdots + 82\!\cdots\!00 \)
|
| $37$ |
\( T^{4} + \cdots + 23\!\cdots\!09 \)
|
| $41$ |
\( (T^{2} + 320979 T + 883217088)^{2} \)
|
| $43$ |
\( (T^{2} + 990095 T + 167309324824)^{2} \)
|
| $47$ |
\( T^{4} + \cdots + 52\!\cdots\!00 \)
|
| $53$ |
\( T^{4} + \cdots + 49\!\cdots\!24 \)
|
| $59$ |
\( T^{4} + \cdots + 23\!\cdots\!16 \)
|
| $61$ |
\( T^{4} + \cdots + 13\!\cdots\!00 \)
|
| $67$ |
\( T^{4} + \cdots + 16\!\cdots\!00 \)
|
| $71$ |
\( (T^{2} + \cdots + 6278281186464)^{2} \)
|
| $73$ |
\( T^{4} + \cdots + 12\!\cdots\!36 \)
|
| $79$ |
\( T^{4} + \cdots + 17\!\cdots\!76 \)
|
| $83$ |
\( (T^{2} + \cdots + 59899934002752)^{2} \)
|
| $89$ |
\( T^{4} + \cdots + 36\!\cdots\!00 \)
|
| $97$ |
\( (T^{2} + \cdots - 10295270196512)^{2} \)
|
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