Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [234,3,Mod(73,234)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(234, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([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("234.73");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 234 = 2 \cdot 3^{2} \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 234.i (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: | \(6.37603818603\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Relative dimension: | \(3\) over \(\Q(i)\) |
| Coefficient field: | 6.0.353139264.2 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} + 24x^{4} + 144x^{2} + 169 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{4} \) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} + 24x^{4} + 144x^{2} + 169 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{3} + 12\nu ) / 13 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{5} + \nu^{4} + 20\nu^{3} + 25\nu^{2} + 83\nu + 104 ) / 13 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{5} + \nu^{4} - 20\nu^{3} + 25\nu^{2} - 83\nu + 104 ) / 13 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{5} - 3\nu^{4} - 20\nu^{3} - 49\nu^{2} - 57\nu - 104 ) / 13 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -\nu^{5} + 3\nu^{4} - 20\nu^{3} + 49\nu^{2} - 57\nu + 104 ) / 13 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} ) / 4 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -\beta_{5} + \beta_{4} + 3\beta_{3} + 3\beta_{2} - 32 ) / 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( -3\beta_{5} - 3\beta_{4} + 3\beta_{3} - 3\beta_{2} + 13\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 25\beta_{5} - 25\beta_{4} - 49\beta_{3} - 49\beta_{2} + 384 ) / 4 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 157\beta_{5} + 157\beta_{4} - 183\beta_{3} + 183\beta_{2} - 1040\beta_1 ) / 4 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/234\mathbb{Z}\right)^\times\).
| \(n\) | \(145\) | \(209\) |
| \(\chi(n)\) | \(-\beta_{1}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 73.1 |
|
1.00000 | + | 1.00000i | 0 | 2.00000i | −6.20982 | − | 6.20982i | 0 | −2.48722 | + | 2.48722i | −2.00000 | + | 2.00000i | 0 | − | 12.4196i | |||||||||||||||||||||||||||
| 73.2 | 1.00000 | + | 1.00000i | 0 | 2.00000i | 0.802841 | + | 0.802841i | 0 | −5.34124 | + | 5.34124i | −2.00000 | + | 2.00000i | 0 | 1.60568i | |||||||||||||||||||||||||||||
| 73.3 | 1.00000 | + | 1.00000i | 0 | 2.00000i | 2.40698 | + | 2.40698i | 0 | 7.82846 | − | 7.82846i | −2.00000 | + | 2.00000i | 0 | 4.81396i | |||||||||||||||||||||||||||||
| 109.1 | 1.00000 | − | 1.00000i | 0 | − | 2.00000i | −6.20982 | + | 6.20982i | 0 | −2.48722 | − | 2.48722i | −2.00000 | − | 2.00000i | 0 | 12.4196i | ||||||||||||||||||||||||||||
| 109.2 | 1.00000 | − | 1.00000i | 0 | − | 2.00000i | 0.802841 | − | 0.802841i | 0 | −5.34124 | − | 5.34124i | −2.00000 | − | 2.00000i | 0 | − | 1.60568i | |||||||||||||||||||||||||||
| 109.3 | 1.00000 | − | 1.00000i | 0 | − | 2.00000i | 2.40698 | − | 2.40698i | 0 | 7.82846 | + | 7.82846i | −2.00000 | − | 2.00000i | 0 | − | 4.81396i | |||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 13.d | odd | 4 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 234.3.i.d | yes | 6 |
| 3.b | odd | 2 | 1 | 234.3.i.c | ✓ | 6 | |
| 13.d | odd | 4 | 1 | inner | 234.3.i.d | yes | 6 |
| 39.f | even | 4 | 1 | 234.3.i.c | ✓ | 6 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 234.3.i.c | ✓ | 6 | 3.b | odd | 2 | 1 | |
| 234.3.i.c | ✓ | 6 | 39.f | even | 4 | 1 | |
| 234.3.i.d | yes | 6 | 1.a | even | 1 | 1 | trivial |
| 234.3.i.d | yes | 6 | 13.d | odd | 4 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{6} + 6T_{5}^{5} + 18T_{5}^{4} - 264T_{5}^{3} + 1296T_{5}^{2} - 1728T_{5} + 1152 \)
acting on \(S_{3}^{\mathrm{new}}(234, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} - 2 T + 2)^{3} \)
|
| $3$ |
\( T^{6} \)
|
| $5$ |
\( T^{6} + 6 T^{5} + \cdots + 1152 \)
|
| $7$ |
\( T^{6} + 416 T^{3} + \cdots + 86528 \)
|
| $11$ |
\( T^{6} - 12 T^{5} + \cdots + 14066208 \)
|
| $13$ |
\( T^{6} - 12 T^{5} + \cdots + 4826809 \)
|
| $17$ |
\( T^{6} + 1584 T^{4} + \cdots + 147456 \)
|
| $19$ |
\( T^{6} + 48 T^{5} + \cdots + 270606848 \)
|
| $23$ |
\( T^{6} + 936 T^{4} + \cdots + 6230016 \)
|
| $29$ |
\( (T^{3} + 42 T^{2} + \cdots - 66432)^{2} \)
|
| $31$ |
\( T^{6} - 96 T^{5} + \cdots + 73544192 \)
|
| $37$ |
\( T^{6} - 54 T^{5} + \cdots + 1352 \)
|
| $41$ |
\( T^{6} + \cdots + 1301724288 \)
|
| $43$ |
\( T^{6} + 2304 T^{4} + \cdots + 56070144 \)
|
| $47$ |
\( T^{6} - 36 T^{5} + \cdots + 3548448 \)
|
| $53$ |
\( (T^{3} + 96 T^{2} + \cdots + 7296)^{2} \)
|
| $59$ |
\( T^{6} - 204 T^{5} + \cdots + 389539872 \)
|
| $61$ |
\( (T^{3} - 6 T^{2} + \cdots + 250752)^{2} \)
|
| $67$ |
\( T^{6} + \cdots + 18870804992 \)
|
| $71$ |
\( T^{6} + \cdots + 3092753952 \)
|
| $73$ |
\( T^{6} + \cdots + 23225297288 \)
|
| $79$ |
\( (T^{3} - 204 T^{2} + \cdots - 126528)^{2} \)
|
| $83$ |
\( T^{6} + \cdots + 31564788768 \)
|
| $89$ |
\( T^{6} + \cdots + 7379667072 \)
|
| $97$ |
\( T^{6} + \cdots + 29642229128 \)
|
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