Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [60,5,Mod(41,60)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(60, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1, 0]))
N = Newforms(chi, 5, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("60.41");
S:= CuspForms(chi, 5);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 60 = 2^{2} \cdot 3 \cdot 5 \) |
| Weight: | \( k \) | \(=\) | \( 5 \) |
| Character orbit: | \([\chi]\) | \(=\) | 60.g (of order \(2\), degree \(1\), 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: | \(6.20219778503\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\sqrt{-5}, \sqrt{34})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 58x^{2} + 1521 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 3\cdot 5 \) |
| Twist minimal: | yes |
| 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
\( x^{4} - 58x^{2} + 1521 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{3} - 13\nu^{2} - 45\nu + 377 ) / 52 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{3} - 13\nu^{2} + 149\nu + 377 ) / 52 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -5\nu^{3} + 95\nu ) / 78 \)
|
| \(\nu\) | \(=\) |
\( ( -3\beta_{3} + 5\beta_{2} - 5\beta_1 ) / 15 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -12\beta_{3} - 10\beta_{2} - 50\beta _1 + 435 ) / 15 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -291\beta_{3} + 95\beta_{2} - 95\beta_1 ) / 15 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/60\mathbb{Z}\right)^\times\).
| \(n\) | \(31\) | \(37\) | \(41\) |
| \(\chi(n)\) | \(1\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 41.1 |
|
0 | −7.91548 | − | 4.28313i | 0 | 11.1803i | 0 | 7.49286 | 0 | 44.3095 | + | 67.8061i | 0 | ||||||||||||||||||||||||||
| 41.2 | 0 | −7.91548 | + | 4.28313i | 0 | − | 11.1803i | 0 | 7.49286 | 0 | 44.3095 | − | 67.8061i | 0 | ||||||||||||||||||||||||||
| 41.3 | 0 | −2.08452 | − | 8.75527i | 0 | − | 11.1803i | 0 | −27.4929 | 0 | −72.3095 | + | 36.5011i | 0 | ||||||||||||||||||||||||||
| 41.4 | 0 | −2.08452 | + | 8.75527i | 0 | 11.1803i | 0 | −27.4929 | 0 | −72.3095 | − | 36.5011i | 0 | |||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 60.5.g.b | ✓ | 4 |
| 3.b | odd | 2 | 1 | inner | 60.5.g.b | ✓ | 4 |
| 4.b | odd | 2 | 1 | 240.5.l.c | 4 | ||
| 5.b | even | 2 | 1 | 300.5.g.g | 4 | ||
| 5.c | odd | 4 | 2 | 300.5.b.d | 8 | ||
| 12.b | even | 2 | 1 | 240.5.l.c | 4 | ||
| 15.d | odd | 2 | 1 | 300.5.g.g | 4 | ||
| 15.e | even | 4 | 2 | 300.5.b.d | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 60.5.g.b | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 60.5.g.b | ✓ | 4 | 3.b | odd | 2 | 1 | inner |
| 240.5.l.c | 4 | 4.b | odd | 2 | 1 | ||
| 240.5.l.c | 4 | 12.b | even | 2 | 1 | ||
| 300.5.b.d | 8 | 5.c | odd | 4 | 2 | ||
| 300.5.b.d | 8 | 15.e | even | 4 | 2 | ||
| 300.5.g.g | 4 | 5.b | even | 2 | 1 | ||
| 300.5.g.g | 4 | 15.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{2} + 20T_{7} - 206 \)
acting on \(S_{5}^{\mathrm{new}}(60, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} + 20 T^{3} + \cdots + 6561 \)
|
| $5$ |
\( (T^{2} + 125)^{2} \)
|
| $7$ |
\( (T^{2} + 20 T - 206)^{2} \)
|
| $11$ |
\( T^{4} + 35280 T^{2} + 29160000 \)
|
| $13$ |
\( (T^{2} + 20 T - 78236)^{2} \)
|
| $17$ |
\( T^{4} + \cdots + 3014010000 \)
|
| $19$ |
\( (T^{2} + 272 T - 12104)^{2} \)
|
| $23$ |
\( T^{4} + \cdots + 152217022500 \)
|
| $29$ |
\( T^{4} + \cdots + 127020960000 \)
|
| $31$ |
\( (T^{2} - 1168 T - 148544)^{2} \)
|
| $37$ |
\( (T^{2} + 140 T - 39164)^{2} \)
|
| $41$ |
\( T^{4} + \cdots + 58612410000 \)
|
| $43$ |
\( (T^{2} - 3700 T + 3165154)^{2} \)
|
| $47$ |
\( T^{4} + \cdots + 3415658422500 \)
|
| $53$ |
\( T^{4} + \cdots + 106545748410000 \)
|
| $59$ |
\( T^{4} + \cdots + 11\!\cdots\!00 \)
|
| $61$ |
\( (T^{2} + 32 T - 27539744)^{2} \)
|
| $67$ |
\( (T^{2} + 4460 T - 3974846)^{2} \)
|
| $71$ |
\( T^{4} + \cdots + 151009689960000 \)
|
| $73$ |
\( (T^{2} - 9220 T + 5344996)^{2} \)
|
| $79$ |
\( (T^{2} + 3656 T - 43201016)^{2} \)
|
| $83$ |
\( T^{4} + \cdots + 891819673222500 \)
|
| $89$ |
\( T^{4} + \cdots + 10\!\cdots\!00 \)
|
| $97$ |
\( (T^{2} + 4940 T - 65581436)^{2} \)
|
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