Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [240,5,Mod(209,240)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(240, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 1, 1]))
N = Newforms(chi, 5, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("240.209");
S:= CuspForms(chi, 5);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 240 = 2^{4} \cdot 3 \cdot 5 \) |
| Weight: | \( k \) | \(=\) | \( 5 \) |
| Character orbit: | \([\chi]\) | \(=\) | 240.c (of order \(2\), degree \(1\), not 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: | \(24.8087911401\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\sqrt{10}, \sqrt{-26})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} + 8x^{2} + 81 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 3^{2} \) |
| Twist minimal: | no (minimal twist has level 15) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} + 8x^{2} + 81 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{3} + 26\nu ) / 9 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{3} + \nu ) / 9 \)
|
| \(\beta_{3}\) | \(=\) |
\( 3\nu^{2} + 12 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + \beta_1 ) / 3 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} - 12 ) / 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -26\beta_{2} + \beta_1 ) / 3 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/240\mathbb{Z}\right)^\times\).
| \(n\) | \(31\) | \(97\) | \(161\) | \(181\) |
| \(\chi(n)\) | \(1\) | \(-1\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 209.1 |
|
0 | −4.74342 | − | 7.64853i | 0 | 6.32456 | − | 24.1868i | 0 | 15.2971i | 0 | −36.0000 | + | 72.5603i | 0 | ||||||||||||||||||||||||
| 209.2 | 0 | −4.74342 | + | 7.64853i | 0 | 6.32456 | + | 24.1868i | 0 | − | 15.2971i | 0 | −36.0000 | − | 72.5603i | 0 | ||||||||||||||||||||||||
| 209.3 | 0 | 4.74342 | − | 7.64853i | 0 | −6.32456 | + | 24.1868i | 0 | 15.2971i | 0 | −36.0000 | − | 72.5603i | 0 | |||||||||||||||||||||||||
| 209.4 | 0 | 4.74342 | + | 7.64853i | 0 | −6.32456 | − | 24.1868i | 0 | − | 15.2971i | 0 | −36.0000 | + | 72.5603i | 0 | ||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
| 5.b | even | 2 | 1 | inner |
| 15.d | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 240.5.c.c | 4 | |
| 3.b | odd | 2 | 1 | inner | 240.5.c.c | 4 | |
| 4.b | odd | 2 | 1 | 15.5.d.c | ✓ | 4 | |
| 5.b | even | 2 | 1 | inner | 240.5.c.c | 4 | |
| 12.b | even | 2 | 1 | 15.5.d.c | ✓ | 4 | |
| 15.d | odd | 2 | 1 | inner | 240.5.c.c | 4 | |
| 20.d | odd | 2 | 1 | 15.5.d.c | ✓ | 4 | |
| 20.e | even | 4 | 2 | 75.5.c.h | 4 | ||
| 60.h | even | 2 | 1 | 15.5.d.c | ✓ | 4 | |
| 60.l | odd | 4 | 2 | 75.5.c.h | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 15.5.d.c | ✓ | 4 | 4.b | odd | 2 | 1 | |
| 15.5.d.c | ✓ | 4 | 12.b | even | 2 | 1 | |
| 15.5.d.c | ✓ | 4 | 20.d | odd | 2 | 1 | |
| 15.5.d.c | ✓ | 4 | 60.h | even | 2 | 1 | |
| 75.5.c.h | 4 | 20.e | even | 4 | 2 | ||
| 75.5.c.h | 4 | 60.l | odd | 4 | 2 | ||
| 240.5.c.c | 4 | 1.a | even | 1 | 1 | trivial | |
| 240.5.c.c | 4 | 3.b | odd | 2 | 1 | inner | |
| 240.5.c.c | 4 | 5.b | even | 2 | 1 | inner | |
| 240.5.c.c | 4 | 15.d | odd | 2 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{5}^{\mathrm{new}}(240, [\chi])\):
|
\( T_{7}^{2} + 234 \)
|
|
\( T_{17}^{2} - 77440 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} + 72T^{2} + 6561 \)
|
| $5$ |
\( T^{4} + 1090 T^{2} + 390625 \)
|
| $7$ |
\( (T^{2} + 234)^{2} \)
|
| $11$ |
\( (T^{2} + 9360)^{2} \)
|
| $13$ |
\( (T^{2} + 59904)^{2} \)
|
| $17$ |
\( (T^{2} - 77440)^{2} \)
|
| $19$ |
\( (T + 308)^{4} \)
|
| $23$ |
\( (T^{2} - 171610)^{2} \)
|
| $29$ |
\( (T^{2} + 37440)^{2} \)
|
| $31$ |
\( (T + 32)^{4} \)
|
| $37$ |
\( (T^{2} + 1651104)^{2} \)
|
| $41$ |
\( (T^{2} + 4326660)^{2} \)
|
| $43$ |
\( (T^{2} + 6683274)^{2} \)
|
| $47$ |
\( (T^{2} - 5975290)^{2} \)
|
| $53$ |
\( (T^{2} - 2007040)^{2} \)
|
| $59$ |
\( (T^{2} + 15734160)^{2} \)
|
| $61$ |
\( (T + 928)^{4} \)
|
| $67$ |
\( (T^{2} + 6683274)^{2} \)
|
| $71$ |
\( (T^{2} + 21565440)^{2} \)
|
| $73$ |
\( (T^{2} + 17825184)^{2} \)
|
| $79$ |
\( (T + 8)^{4} \)
|
| $83$ |
\( (T^{2} - 19684090)^{2} \)
|
| $89$ |
\( (T^{2} + 86261760)^{2} \)
|
| $97$ |
\( (T^{2} + 6293664)^{2} \)
|
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