Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [50,5,Mod(7,50)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(50, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([1]))
N = Newforms(chi, 5, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("50.7");
S:= CuspForms(chi, 5);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 50 = 2 \cdot 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 5 \) |
| Character orbit: | \([\chi]\) | \(=\) | 50.c (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: | \(5.16849815419\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(i)\) |
| Coefficient field: | \(\Q(i, \sqrt{6})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} + 9 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 5^{2} \) |
| 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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} + 9 \)
:
| \(\beta_{1}\) | \(=\) |
\( 5\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} ) / 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 5\nu^{3} ) / 3 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_1 ) / 5 \)
|
| \(\nu^{2}\) | \(=\) |
\( 3\beta_{2} \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 3\beta_{3} ) / 5 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/50\mathbb{Z}\right)^\times\).
| \(n\) | \(27\) |
| \(\chi(n)\) | \(-\beta_{2}\) |
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 | − | 2.00000i | −12.1237 | + | 12.1237i | 8.00000i | 0 | 48.4949 | −11.5051 | − | 11.5051i | 16.0000 | − | 16.0000i | − | 212.969i | 0 | |||||||||||||||||||||
| 7.2 | −2.00000 | − | 2.00000i | 0.123724 | − | 0.123724i | 8.00000i | 0 | −0.494897 | −60.4949 | − | 60.4949i | 16.0000 | − | 16.0000i | 80.9694i | 0 | |||||||||||||||||||||||
| 43.1 | −2.00000 | + | 2.00000i | −12.1237 | − | 12.1237i | − | 8.00000i | 0 | 48.4949 | −11.5051 | + | 11.5051i | 16.0000 | + | 16.0000i | 212.969i | 0 | ||||||||||||||||||||||
| 43.2 | −2.00000 | + | 2.00000i | 0.123724 | + | 0.123724i | − | 8.00000i | 0 | −0.494897 | −60.4949 | + | 60.4949i | 16.0000 | + | 16.0000i | − | 80.9694i | 0 | |||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.c | odd | 4 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 50.5.c.c | ✓ | 4 |
| 3.b | odd | 2 | 1 | 450.5.g.j | 4 | ||
| 4.b | odd | 2 | 1 | 400.5.p.n | 4 | ||
| 5.b | even | 2 | 1 | 50.5.c.d | yes | 4 | |
| 5.c | odd | 4 | 1 | inner | 50.5.c.c | ✓ | 4 |
| 5.c | odd | 4 | 1 | 50.5.c.d | yes | 4 | |
| 15.d | odd | 2 | 1 | 450.5.g.i | 4 | ||
| 15.e | even | 4 | 1 | 450.5.g.i | 4 | ||
| 15.e | even | 4 | 1 | 450.5.g.j | 4 | ||
| 20.d | odd | 2 | 1 | 400.5.p.e | 4 | ||
| 20.e | even | 4 | 1 | 400.5.p.e | 4 | ||
| 20.e | even | 4 | 1 | 400.5.p.n | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 50.5.c.c | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 50.5.c.c | ✓ | 4 | 5.c | odd | 4 | 1 | inner |
| 50.5.c.d | yes | 4 | 5.b | even | 2 | 1 | |
| 50.5.c.d | yes | 4 | 5.c | odd | 4 | 1 | |
| 400.5.p.e | 4 | 20.d | odd | 2 | 1 | ||
| 400.5.p.e | 4 | 20.e | even | 4 | 1 | ||
| 400.5.p.n | 4 | 4.b | odd | 2 | 1 | ||
| 400.5.p.n | 4 | 20.e | even | 4 | 1 | ||
| 450.5.g.i | 4 | 15.d | odd | 2 | 1 | ||
| 450.5.g.i | 4 | 15.e | even | 4 | 1 | ||
| 450.5.g.j | 4 | 3.b | odd | 2 | 1 | ||
| 450.5.g.j | 4 | 15.e | even | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} + 24T_{3}^{3} + 288T_{3}^{2} - 72T_{3} + 9 \)
acting on \(S_{5}^{\mathrm{new}}(50, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 4 T + 8)^{2} \)
|
| $3$ |
\( T^{4} + 24 T^{3} + \cdots + 9 \)
|
| $5$ |
\( T^{4} \)
|
| $7$ |
\( T^{4} + 144 T^{3} + \cdots + 1937664 \)
|
| $11$ |
\( (T^{2} + 126 T - 1431)^{2} \)
|
| $13$ |
\( T^{4} + 144 T^{3} + \cdots + 471237264 \)
|
| $17$ |
\( T^{4} + \cdots + 1814504409 \)
|
| $19$ |
\( T^{4} + \cdots + 2077080625 \)
|
| $23$ |
\( T^{4} + \cdots + 18351579024 \)
|
| $29$ |
\( T^{4} + \cdots + 617324490000 \)
|
| $31$ |
\( (T^{2} + 76 T - 1748156)^{2} \)
|
| $37$ |
\( T^{4} + \cdots + 6434380145664 \)
|
| $41$ |
\( (T^{2} + 3006 T + 1913409)^{2} \)
|
| $43$ |
\( T^{4} + \cdots + 249728073984 \)
|
| $47$ |
\( T^{4} + \cdots + 9862941243024 \)
|
| $53$ |
\( T^{4} + \cdots + 1007590348944 \)
|
| $59$ |
\( T^{4} + \cdots + 14720266890000 \)
|
| $61$ |
\( (T^{2} - 1144 T + 132784)^{2} \)
|
| $67$ |
\( T^{4} + \cdots + 843170207049 \)
|
| $71$ |
\( (T^{2} - 7524 T + 11042244)^{2} \)
|
| $73$ |
\( T^{4} + \cdots + 13\!\cdots\!09 \)
|
| $79$ |
\( T^{4} + \cdots + 239457055360000 \)
|
| $83$ |
\( T^{4} + \cdots + 7286744163609 \)
|
| $89$ |
\( T^{4} + \cdots + 13\!\cdots\!25 \)
|
| $97$ |
\( T^{4} + \cdots + 12\!\cdots\!64 \)
|
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