Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [304,3,Mod(65,304)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(304, base_ring=CyclotomicField(6))
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("304.65");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 304 = 2^{4} \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 304.r (of order \(6\), degree \(2\), not 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: | \(8.28340003655\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{6})\) |
| Coefficient field: | \(\Q(\sqrt{-2}, \sqrt{-3})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 2x^{2} + 4 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 38) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-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
\( x^{4} - 2x^{2} + 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} ) / 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( 2\beta_{2} \)
|
| \(\nu^{3}\) | \(=\) |
\( 2\beta_{3} \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/304\mathbb{Z}\right)^\times\).
| \(n\) | \(97\) | \(191\) | \(229\) |
| \(\chi(n)\) | \(1 - \beta_{2}\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 65.1 |
|
0 | −2.72474 | − | 1.57313i | 0 | 0.500000 | − | 0.866025i | 0 | −6.89898 | 0 | 0.449490 | + | 0.778539i | 0 | ||||||||||||||||||||||||
| 65.2 | 0 | −0.275255 | − | 0.158919i | 0 | 0.500000 | − | 0.866025i | 0 | 2.89898 | 0 | −4.44949 | − | 7.70674i | 0 | |||||||||||||||||||||||||
| 145.1 | 0 | −2.72474 | + | 1.57313i | 0 | 0.500000 | + | 0.866025i | 0 | −6.89898 | 0 | 0.449490 | − | 0.778539i | 0 | |||||||||||||||||||||||||
| 145.2 | 0 | −0.275255 | + | 0.158919i | 0 | 0.500000 | + | 0.866025i | 0 | 2.89898 | 0 | −4.44949 | + | 7.70674i | 0 | |||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 19.d | odd | 6 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 304.3.r.a | 4 | |
| 4.b | odd | 2 | 1 | 38.3.d.a | ✓ | 4 | |
| 12.b | even | 2 | 1 | 342.3.m.a | 4 | ||
| 19.d | odd | 6 | 1 | inner | 304.3.r.a | 4 | |
| 76.f | even | 6 | 1 | 38.3.d.a | ✓ | 4 | |
| 76.f | even | 6 | 1 | 722.3.b.b | 4 | ||
| 76.g | odd | 6 | 1 | 722.3.b.b | 4 | ||
| 228.n | odd | 6 | 1 | 342.3.m.a | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 38.3.d.a | ✓ | 4 | 4.b | odd | 2 | 1 | |
| 38.3.d.a | ✓ | 4 | 76.f | even | 6 | 1 | |
| 304.3.r.a | 4 | 1.a | even | 1 | 1 | trivial | |
| 304.3.r.a | 4 | 19.d | odd | 6 | 1 | inner | |
| 342.3.m.a | 4 | 12.b | even | 2 | 1 | ||
| 342.3.m.a | 4 | 228.n | odd | 6 | 1 | ||
| 722.3.b.b | 4 | 76.f | even | 6 | 1 | ||
| 722.3.b.b | 4 | 76.g | odd | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} + 6T_{3}^{3} + 13T_{3}^{2} + 6T_{3} + 1 \)
acting on \(S_{3}^{\mathrm{new}}(304, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} + 6 T^{3} + \cdots + 1 \)
|
| $5$ |
\( (T^{2} - T + 1)^{2} \)
|
| $7$ |
\( (T^{2} + 4 T - 20)^{2} \)
|
| $11$ |
\( (T^{2} - 20 T + 76)^{2} \)
|
| $13$ |
\( T^{4} + 30 T^{3} + \cdots + 9 \)
|
| $17$ |
\( T^{4} - 14 T^{3} + \cdots + 625 \)
|
| $19$ |
\( T^{4} + 16 T^{3} + \cdots + 130321 \)
|
| $23$ |
\( T^{4} - 10 T^{3} + \cdots + 212521 \)
|
| $29$ |
\( T^{4} - 66 T^{3} + \cdots + 81225 \)
|
| $31$ |
\( (T^{2} + 972)^{2} \)
|
| $37$ |
\( T^{4} + 1320 T^{2} + 404496 \)
|
| $41$ |
\( T^{4} - 18 T^{3} + \cdots + 12257001 \)
|
| $43$ |
\( T^{4} + 38 T^{3} + \cdots + 7912969 \)
|
| $47$ |
\( T^{4} - 70 T^{3} + \cdots + 885481 \)
|
| $53$ |
\( T^{4} + 42 T^{3} + \cdots + 19900521 \)
|
| $59$ |
\( T^{4} + 42 T^{3} + \cdots + 4124961 \)
|
| $61$ |
\( T^{4} - 74 T^{3} + \cdots + 27889 \)
|
| $67$ |
\( T^{4} + 102 T^{3} + \cdots + 49999041 \)
|
| $71$ |
\( T^{4} + 306 T^{3} + \cdots + 58384881 \)
|
| $73$ |
\( T^{4} - 98 T^{3} + \cdots + 14010049 \)
|
| $79$ |
\( T^{4} - 126 T^{3} + \cdots + 194481 \)
|
| $83$ |
\( (T^{2} - 32 T + 40)^{2} \)
|
| $89$ |
\( T^{4} + 6 T^{3} + \cdots + 4761 \)
|
| $97$ |
\( T^{4} + 486 T^{3} + \cdots + 384591321 \)
|
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