Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [55,3,Mod(6,55)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(55, base_ring=CyclotomicField(10))
chi = DirichletCharacter(H, H._module([0, 9]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("55.6");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 55 = 5 \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 55.i (of order \(10\), degree \(4\), 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: | \(1.49864145398\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\zeta_{10})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} + x^{2} - x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{10}]$ |
$q$-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 primitive root of unity \(\zeta_{10}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/55\mathbb{Z}\right)^\times\).
| \(n\) | \(12\) | \(46\) |
| \(\chi(n)\) | \(1\) | \(\zeta_{10}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 6.1 |
|
2.92705 | − | 0.951057i | −3.73607 | − | 2.71441i | 4.42705 | − | 3.21644i | 0.690983 | − | 2.12663i | −13.5172 | − | 4.39201i | 6.38197 | + | 8.78402i | 2.66312 | − | 3.66547i | 3.80902 | + | 11.7229i | − | 6.88191i | |||||||||||||
| 41.1 | −0.427051 | − | 0.587785i | 0.736068 | + | 2.26538i | 1.07295 | − | 3.30220i | 1.80902 | + | 1.31433i | 1.01722 | − | 1.40008i | 8.61803 | + | 2.80017i | −5.16312 | + | 1.67760i | 2.69098 | − | 1.95511i | − | 1.62460i | ||||||||||||||
| 46.1 | 2.92705 | + | 0.951057i | −3.73607 | + | 2.71441i | 4.42705 | + | 3.21644i | 0.690983 | + | 2.12663i | −13.5172 | + | 4.39201i | 6.38197 | − | 8.78402i | 2.66312 | + | 3.66547i | 3.80902 | − | 11.7229i | 6.88191i | |||||||||||||||
| 51.1 | −0.427051 | + | 0.587785i | 0.736068 | − | 2.26538i | 1.07295 | + | 3.30220i | 1.80902 | − | 1.31433i | 1.01722 | + | 1.40008i | 8.61803 | − | 2.80017i | −5.16312 | − | 1.67760i | 2.69098 | + | 1.95511i | 1.62460i | |||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 11.d | odd | 10 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 55.3.i.b | ✓ | 4 |
| 5.b | even | 2 | 1 | 275.3.x.a | 4 | ||
| 5.c | odd | 4 | 2 | 275.3.q.a | 8 | ||
| 11.c | even | 5 | 1 | 605.3.c.b | 4 | ||
| 11.d | odd | 10 | 1 | inner | 55.3.i.b | ✓ | 4 |
| 11.d | odd | 10 | 1 | 605.3.c.b | 4 | ||
| 55.h | odd | 10 | 1 | 275.3.x.a | 4 | ||
| 55.l | even | 20 | 2 | 275.3.q.a | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 55.3.i.b | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 55.3.i.b | ✓ | 4 | 11.d | odd | 10 | 1 | inner |
| 275.3.q.a | 8 | 5.c | odd | 4 | 2 | ||
| 275.3.q.a | 8 | 55.l | even | 20 | 2 | ||
| 275.3.x.a | 4 | 5.b | even | 2 | 1 | ||
| 275.3.x.a | 4 | 55.h | odd | 10 | 1 | ||
| 605.3.c.b | 4 | 11.c | even | 5 | 1 | ||
| 605.3.c.b | 4 | 11.d | odd | 10 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} - 5T_{2}^{3} + 5T_{2}^{2} + 5T_{2} + 5 \)
acting on \(S_{3}^{\mathrm{new}}(55, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} - 5 T^{3} + \cdots + 5 \)
|
| $3$ |
\( T^{4} + 6 T^{3} + \cdots + 121 \)
|
| $5$ |
\( T^{4} - 5 T^{3} + \cdots + 25 \)
|
| $7$ |
\( T^{4} - 30 T^{3} + \cdots + 9680 \)
|
| $11$ |
\( T^{4} + T^{3} + \cdots + 14641 \)
|
| $13$ |
\( T^{4} + 30 T^{3} + \cdots + 6480 \)
|
| $17$ |
\( T^{4} + 50 T^{3} + \cdots + 18605 \)
|
| $19$ |
\( T^{4} + 45 T^{3} + \cdots + 59405 \)
|
| $23$ |
\( (T^{2} + 8 T - 4)^{2} \)
|
| $29$ |
\( T^{4} + 70 T^{3} + \cdots + 192080 \)
|
| $31$ |
\( T^{4} - 98 T^{3} + \cdots + 2421136 \)
|
| $37$ |
\( T^{4} - 16 T^{3} + \cdots + 190096 \)
|
| $41$ |
\( T^{4} - 40 T^{3} + \cdots + 15125 \)
|
| $43$ |
\( T^{4} + 4325 T^{2} + 4560125 \)
|
| $47$ |
\( T^{4} + 24 T^{3} + \cdots + 3444736 \)
|
| $53$ |
\( T^{4} + 106 T^{3} + \cdots + 6948496 \)
|
| $59$ |
\( T^{4} - 217 T^{3} + \cdots + 46389721 \)
|
| $61$ |
\( T^{4} + 80 T^{3} + \cdots + 16128080 \)
|
| $67$ |
\( (T^{2} - 33 T - 509)^{2} \)
|
| $71$ |
\( T^{4} - 38 T^{3} + \cdots + 5216656 \)
|
| $73$ |
\( T^{4} + 230 T^{3} + \cdots + 32385125 \)
|
| $79$ |
\( T^{4} + 100 T^{3} + \cdots + 237774080 \)
|
| $83$ |
\( T^{4} - 185 T^{3} + \cdots + 13138205 \)
|
| $89$ |
\( (T^{2} - 201 T + 10039)^{2} \)
|
| $97$ |
\( T^{4} - 101 T^{3} + \cdots + 5755201 \)
|
| show more | |
| show less |