Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [76,3,Mod(7,76)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(76, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([3, 2]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("76.7");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 76 = 2^{2} \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 76.g (of order \(6\), 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: | \(2.07085000914\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{6})\) |
| Coefficient field: | \(\Q(\sqrt{-3}, \sqrt{-10})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 10x^{2} + 100 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| 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
\( x^{4} - 10x^{2} + 100 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} ) / 10 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} ) / 10 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( 10\beta_{2} \)
|
| \(\nu^{3}\) | \(=\) |
\( 10\beta_{3} \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/76\mathbb{Z}\right)^\times\).
| \(n\) | \(21\) | \(39\) |
| \(\chi(n)\) | \(-\beta_{2}\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 7.1 |
|
2.00000 | −4.23861 | + | 2.44716i | 4.00000 | 1.73861 | + | 3.01137i | −8.47723 | + | 4.89433i | 10.0905i | 8.00000 | 7.47723 | − | 12.9509i | 3.47723 | + | 6.02273i | ||||||||||||||||||||
| 7.2 | 2.00000 | 1.23861 | − | 0.715113i | 4.00000 | −3.73861 | − | 6.47547i | 2.47723 | − | 1.43023i | 3.76593i | 8.00000 | −3.47723 | + | 6.02273i | −7.47723 | − | 12.9509i | |||||||||||||||||||||
| 11.1 | 2.00000 | −4.23861 | − | 2.44716i | 4.00000 | 1.73861 | − | 3.01137i | −8.47723 | − | 4.89433i | − | 10.0905i | 8.00000 | 7.47723 | + | 12.9509i | 3.47723 | − | 6.02273i | ||||||||||||||||||||
| 11.2 | 2.00000 | 1.23861 | + | 0.715113i | 4.00000 | −3.73861 | + | 6.47547i | 2.47723 | + | 1.43023i | − | 3.76593i | 8.00000 | −3.47723 | − | 6.02273i | −7.47723 | + | 12.9509i | ||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 76.g | odd | 6 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 76.3.g.b | yes | 4 |
| 4.b | odd | 2 | 1 | 76.3.g.a | ✓ | 4 | |
| 19.c | even | 3 | 1 | 76.3.g.a | ✓ | 4 | |
| 76.g | odd | 6 | 1 | inner | 76.3.g.b | yes | 4 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 76.3.g.a | ✓ | 4 | 4.b | odd | 2 | 1 | |
| 76.3.g.a | ✓ | 4 | 19.c | even | 3 | 1 | |
| 76.3.g.b | yes | 4 | 1.a | even | 1 | 1 | trivial |
| 76.3.g.b | yes | 4 | 76.g | odd | 6 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} + 6T_{3}^{3} + 5T_{3}^{2} - 42T_{3} + 49 \)
acting on \(S_{3}^{\mathrm{new}}(76, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 2)^{4} \)
|
| $3$ |
\( T^{4} + 6 T^{3} + \cdots + 49 \)
|
| $5$ |
\( T^{4} + 4 T^{3} + \cdots + 676 \)
|
| $7$ |
\( T^{4} + 116T^{2} + 1444 \)
|
| $11$ |
\( T^{4} + 506 T^{2} + 61009 \)
|
| $13$ |
\( (T^{2} + 6 T + 36)^{2} \)
|
| $17$ |
\( T^{4} + 32 T^{3} + \cdots + 18496 \)
|
| $19$ |
\( T^{4} - 42 T^{3} + \cdots + 130321 \)
|
| $23$ |
\( T^{4} - 12 T^{3} + \cdots + 636804 \)
|
| $29$ |
\( T^{4} + 4 T^{3} + \cdots + 676 \)
|
| $31$ |
\( T^{4} + 1580 T^{2} + 36100 \)
|
| $37$ |
\( (T^{2} + 48 T - 174)^{2} \)
|
| $41$ |
\( T^{4} - 26 T^{3} + \cdots + 96721 \)
|
| $43$ |
\( T^{4} + 72 T^{3} + \cdots + 322624 \)
|
| $47$ |
\( T^{4} - 490 T^{2} + 240100 \)
|
| $53$ |
\( T^{4} - 124 T^{3} + \cdots + 11316496 \)
|
| $59$ |
\( T^{4} + 78 T^{3} + \cdots + 247009 \)
|
| $61$ |
\( T^{4} + 44 T^{3} + \cdots + 206116 \)
|
| $67$ |
\( T^{4} + 102 T^{3} + \cdots + 12552849 \)
|
| $71$ |
\( T^{4} - 204 T^{3} + \cdots + 2274064 \)
|
| $73$ |
\( T^{4} + 26 T^{3} + \cdots + 96721 \)
|
| $79$ |
\( T^{4} + 360 T^{3} + \cdots + 103225600 \)
|
| $83$ |
\( T^{4} + 9386 T^{2} + 6385729 \)
|
| $89$ |
\( T^{4} + 16 T^{3} + \cdots + 208975936 \)
|
| $97$ |
\( T^{4} + 234 T^{3} + \cdots + 184117761 \)
|
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