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: | \(0\) |
| Dimension: | \(5\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{5} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{5} - 2x^{4} - 809x^{3} + 562x^{2} + 88864x + 320704 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{4} \) |
| 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 a basis \(1,\beta_1,\beta_2,\beta_3,\beta_4\) 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
\( x^{5} - 2x^{4} - 809x^{3} + 562x^{2} + 88864x + 320704 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 2\nu - 324 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 3\nu^{4} - 14\nu^{3} - 2213\nu^{2} + 9248\nu + 146036 ) / 318 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -2\nu^{4} + 27\nu^{3} + 1546\nu^{2} - 16677\nu - 127214 ) / 159 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 2\beta _1 + 324 \)
|
| \(\nu^{3}\) | \(=\) |
\( 9\beta_{4} + 12\beta_{3} - 4\beta_{2} + 587\beta _1 + 394 \)
|
| \(\nu^{4}\) | \(=\) |
\( 42\beta_{4} + 162\beta_{3} + 719\beta_{2} + 1132\beta _1 + 192164 \)
|
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 | −22.3640 | 0 | 25.0000 | 0 | 139.452 | 0 | 257.147 | 0 | |||||||||||||||||||||||||||||||||
| 1.2 | 0 | −9.57436 | 0 | 25.0000 | 0 | −205.693 | 0 | −151.332 | 0 | ||||||||||||||||||||||||||||||||||
| 1.3 | 0 | 8.58904 | 0 | 25.0000 | 0 | 218.249 | 0 | −169.228 | 0 | ||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 11.5771 | 0 | 25.0000 | 0 | −76.8335 | 0 | −108.972 | 0 | ||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 29.7722 | 0 | 25.0000 | 0 | 54.8255 | 0 | 643.385 | 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.e | ✓ | 5 |
| 4.b | odd | 2 | 1 | 880.6.a.p | 5 | ||
| 5.b | even | 2 | 1 | 1100.6.a.g | 5 | ||
| 5.c | odd | 4 | 2 | 1100.6.b.g | 10 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 220.6.a.e | ✓ | 5 | 1.a | even | 1 | 1 | trivial |
| 880.6.a.p | 5 | 4.b | odd | 2 | 1 | ||
| 1100.6.a.g | 5 | 5.b | even | 2 | 1 | ||
| 1100.6.b.g | 10 | 5.c | odd | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{5} - 18T_{3}^{4} - 681T_{3}^{3} + 8698T_{3}^{2} + 55296T_{3} - 633888 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(220))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} \)
|
| $3$ |
\( T^{5} - 18 T^{4} + \cdots - 633888 \)
|
| $5$ |
\( (T - 25)^{5} \)
|
| $7$ |
\( T^{5} + \cdots - 26371065600 \)
|
| $11$ |
\( (T - 121)^{5} \)
|
| $13$ |
\( T^{5} + \cdots - 153867162240 \)
|
| $17$ |
\( T^{5} + \cdots - 23117856213936 \)
|
| $19$ |
\( T^{5} + \cdots + 696671609378624 \)
|
| $23$ |
\( T^{5} + \cdots - 88\!\cdots\!16 \)
|
| $29$ |
\( T^{5} + \cdots + 56\!\cdots\!08 \)
|
| $31$ |
\( T^{5} + \cdots + 58\!\cdots\!60 \)
|
| $37$ |
\( T^{5} + \cdots - 37\!\cdots\!04 \)
|
| $41$ |
\( T^{5} + \cdots - 17\!\cdots\!60 \)
|
| $43$ |
\( T^{5} + \cdots - 19\!\cdots\!24 \)
|
| $47$ |
\( T^{5} + \cdots - 30\!\cdots\!80 \)
|
| $53$ |
\( T^{5} + \cdots - 11\!\cdots\!00 \)
|
| $59$ |
\( T^{5} + \cdots + 80\!\cdots\!04 \)
|
| $61$ |
\( T^{5} + \cdots + 81\!\cdots\!24 \)
|
| $67$ |
\( T^{5} + \cdots - 24\!\cdots\!48 \)
|
| $71$ |
\( T^{5} + \cdots - 13\!\cdots\!96 \)
|
| $73$ |
\( T^{5} + \cdots - 38\!\cdots\!96 \)
|
| $79$ |
\( T^{5} + \cdots + 27\!\cdots\!12 \)
|
| $83$ |
\( T^{5} + \cdots - 14\!\cdots\!32 \)
|
| $89$ |
\( T^{5} + \cdots + 53\!\cdots\!64 \)
|
| $97$ |
\( T^{5} + \cdots + 55\!\cdots\!16 \)
|
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