Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [220,6,Mod(1,220)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(220, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("220.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 220 = 2^{2} \cdot 5 \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 220.a (trivial) |
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: | yes |
| Analytic conductor: | \(35.2844403589\) |
| Analytic rank: | \(1\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{1761}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - x - 440 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Fricke sign: | \(+1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-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 \(\beta = \frac{1}{2}(1 + \sqrt{1761})\). We also show the integral \(q\)-expansion of the trace form.
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
0 | −30.4821 | 0 | 25.0000 | 0 | −170.411 | 0 | 686.161 | 0 | ||||||||||||||||||||||||
| 1.2 | 0 | 11.4821 | 0 | 25.0000 | 0 | 39.4107 | 0 | −111.161 | 0 | |||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(5\) | \( -1 \) |
| \(11\) | \( +1 \) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 220.6.a.a | ✓ | 2 |
| 4.b | odd | 2 | 1 | 880.6.a.h | 2 | ||
| 5.b | even | 2 | 1 | 1100.6.a.c | 2 | ||
| 5.c | odd | 4 | 2 | 1100.6.b.b | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 220.6.a.a | ✓ | 2 | 1.a | even | 1 | 1 | trivial |
| 880.6.a.h | 2 | 4.b | odd | 2 | 1 | ||
| 1100.6.a.c | 2 | 5.b | even | 2 | 1 | ||
| 1100.6.b.b | 4 | 5.c | odd | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{2} + 19T_{3} - 350 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(220))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} \)
|
| $3$ |
\( T^{2} + 19T - 350 \)
|
| $5$ |
\( (T - 25)^{2} \)
|
| $7$ |
\( T^{2} + 131T - 6716 \)
|
| $11$ |
\( (T + 121)^{2} \)
|
| $13$ |
\( T^{2} + 1610 T + 505384 \)
|
| $17$ |
\( T^{2} + 321 T - 4116552 \)
|
| $19$ |
\( T^{2} - 3439 T + 2935108 \)
|
| $23$ |
\( T^{2} - 4536 T + 1417548 \)
|
| $29$ |
\( T^{2} + 5031 T - 2178330 \)
|
| $31$ |
\( T^{2} + 4463 T - 641960 \)
|
| $37$ |
\( T^{2} + 2423 T - 49724978 \)
|
| $41$ |
\( T^{2} - 19668 T + 69630420 \)
|
| $43$ |
\( T^{2} + 14900 T + 15880000 \)
|
| $47$ |
\( T^{2} + 23052 T - 90333420 \)
|
| $53$ |
\( T^{2} + 7203 T - 72649458 \)
|
| $59$ |
\( T^{2} - 50838 T + 507075240 \)
|
| $61$ |
\( T^{2} + 44177 T + 339290602 \)
|
| $67$ |
\( T^{2} + 10610 T + 6445744 \)
|
| $71$ |
\( T^{2} + 1089 T - 373141620 \)
|
| $73$ |
\( T^{2} + \cdots + 2117787760 \)
|
| $79$ |
\( T^{2} + \cdots - 3429379952 \)
|
| $83$ |
\( T^{2} - 88410 T - 837245616 \)
|
| $89$ |
\( T^{2} + \cdots + 1523224182 \)
|
| $97$ |
\( T^{2} + \cdots + 2630384368 \)
|
| show more | |
| show less |