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: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{-1}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 10) |
| 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 \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/50\mathbb{Z}\right)^\times\).
| \(n\) | \(27\) |
| \(\chi(n)\) | \(i\) |
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 | −1.00000 | + | 1.00000i | 8.00000i | 0 | 4.00000 | 19.0000 | + | 19.0000i | 16.0000 | − | 16.0000i | 79.0000i | 0 | ||||||||||||||||
| 43.1 | −2.00000 | + | 2.00000i | −1.00000 | − | 1.00000i | − | 8.00000i | 0 | 4.00000 | 19.0000 | − | 19.0000i | 16.0000 | + | 16.0000i | − | 79.0000i | 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.a | 2 | |
| 3.b | odd | 2 | 1 | 450.5.g.b | 2 | ||
| 4.b | odd | 2 | 1 | 400.5.p.b | 2 | ||
| 5.b | even | 2 | 1 | 10.5.c.b | ✓ | 2 | |
| 5.c | odd | 4 | 1 | 10.5.c.b | ✓ | 2 | |
| 5.c | odd | 4 | 1 | inner | 50.5.c.a | 2 | |
| 15.d | odd | 2 | 1 | 90.5.g.a | 2 | ||
| 15.e | even | 4 | 1 | 90.5.g.a | 2 | ||
| 15.e | even | 4 | 1 | 450.5.g.b | 2 | ||
| 20.d | odd | 2 | 1 | 80.5.p.c | 2 | ||
| 20.e | even | 4 | 1 | 80.5.p.c | 2 | ||
| 20.e | even | 4 | 1 | 400.5.p.b | 2 | ||
| 40.e | odd | 2 | 1 | 320.5.p.g | 2 | ||
| 40.f | even | 2 | 1 | 320.5.p.d | 2 | ||
| 40.i | odd | 4 | 1 | 320.5.p.d | 2 | ||
| 40.k | even | 4 | 1 | 320.5.p.g | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 10.5.c.b | ✓ | 2 | 5.b | even | 2 | 1 | |
| 10.5.c.b | ✓ | 2 | 5.c | odd | 4 | 1 | |
| 50.5.c.a | 2 | 1.a | even | 1 | 1 | trivial | |
| 50.5.c.a | 2 | 5.c | odd | 4 | 1 | inner | |
| 80.5.p.c | 2 | 20.d | odd | 2 | 1 | ||
| 80.5.p.c | 2 | 20.e | even | 4 | 1 | ||
| 90.5.g.a | 2 | 15.d | odd | 2 | 1 | ||
| 90.5.g.a | 2 | 15.e | even | 4 | 1 | ||
| 320.5.p.d | 2 | 40.f | even | 2 | 1 | ||
| 320.5.p.d | 2 | 40.i | odd | 4 | 1 | ||
| 320.5.p.g | 2 | 40.e | odd | 2 | 1 | ||
| 320.5.p.g | 2 | 40.k | even | 4 | 1 | ||
| 400.5.p.b | 2 | 4.b | odd | 2 | 1 | ||
| 400.5.p.b | 2 | 20.e | even | 4 | 1 | ||
| 450.5.g.b | 2 | 3.b | odd | 2 | 1 | ||
| 450.5.g.b | 2 | 15.e | even | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{2} + 2T_{3} + 2 \)
acting on \(S_{5}^{\mathrm{new}}(50, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} + 4T + 8 \)
|
| $3$ |
\( T^{2} + 2T + 2 \)
|
| $5$ |
\( T^{2} \)
|
| $7$ |
\( T^{2} - 38T + 722 \)
|
| $11$ |
\( (T - 202)^{2} \)
|
| $13$ |
\( T^{2} - 198T + 19602 \)
|
| $17$ |
\( T^{2} - 478T + 114242 \)
|
| $19$ |
\( T^{2} + 1600 \)
|
| $23$ |
\( T^{2} + 1082 T + 585362 \)
|
| $29$ |
\( T^{2} + 40000 \)
|
| $31$ |
\( (T + 758)^{2} \)
|
| $37$ |
\( T^{2} + 282T + 39762 \)
|
| $41$ |
\( (T - 1042)^{2} \)
|
| $43$ |
\( T^{2} - 1518 T + 1152162 \)
|
| $47$ |
\( T^{2} - 918T + 421362 \)
|
| $53$ |
\( T^{2} - 3638 T + 6617522 \)
|
| $59$ |
\( T^{2} + 21160000 \)
|
| $61$ |
\( (T - 2082)^{2} \)
|
| $67$ |
\( T^{2} + 10162 T + 51633122 \)
|
| $71$ |
\( (T + 3478)^{2} \)
|
| $73$ |
\( T^{2} - 6958 T + 24206882 \)
|
| $79$ |
\( T^{2} + 58982400 \)
|
| $83$ |
\( T^{2} + 12162 T + 73957122 \)
|
| $89$ |
\( T^{2} + 32262400 \)
|
| $97$ |
\( T^{2} + 1122 T + 629442 \)
|
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