Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [4,13,Mod(3,4)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1]))
N = Newforms(chi, 13, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("4.3");
S:= CuspForms(chi, 13);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 4 = 2^{2} \) |
| Weight: | \( k \) | \(=\) | \( 13 \) |
| Character orbit: | \([\chi]\) | \(=\) | 4.b (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: | \(3.65597526911\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.0.8546467905.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} + 139x^{2} + 3741x + 30480 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2^{12} \) |
| 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
\( x^{4} - x^{3} + 139x^{2} + 3741x + 30480 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{3} + 6\nu^{2} + 181\nu + 3280 ) / 64 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -3\nu^{3} - 18\nu^{2} + 1505\nu - 10352 ) / 64 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -27\nu^{3} + 350\nu^{2} - 5399\nu - 52976 ) / 64 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + 3\beta _1 + 8 ) / 32 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 4\beta_{3} + \beta_{2} + 111\beta _1 - 2216 ) / 32 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -24\beta_{3} - 187\beta_{2} + 839\beta _1 - 93112 ) / 32 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4\mathbb{Z}\right)^\times\).
| \(n\) | \(3\) |
| \(\chi(n)\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 3.1 |
|
−6.60060 | − | 63.6587i | − | 292.868i | −4008.86 | + | 840.371i | −15342.2 | −18643.6 | + | 1933.10i | − | 129787.i | 79957.8 | + | 249652.i | 445669. | 101268. | + | 976664.i | ||||||||||||||||||
| 3.2 | −6.60060 | + | 63.6587i | 292.868i | −4008.86 | − | 840.371i | −15342.2 | −18643.6 | − | 1933.10i | 129787.i | 79957.8 | − | 249652.i | 445669. | 101268. | − | 976664.i | |||||||||||||||||||||
| 3.3 | 60.6006 | − | 20.5808i | 1288.37i | 3248.86 | − | 2494.41i | 6162.19 | 26515.6 | + | 78075.9i | − | 44726.1i | 145546. | − | 218027.i | −1.12845e6 | 373432. | − | 126823.i | ||||||||||||||||||||
| 3.4 | 60.6006 | + | 20.5808i | − | 1288.37i | 3248.86 | + | 2494.41i | 6162.19 | 26515.6 | − | 78075.9i | 44726.1i | 145546. | + | 218027.i | −1.12845e6 | 373432. | + | 126823.i | ||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 4.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 4.13.b.b | ✓ | 4 |
| 3.b | odd | 2 | 1 | 36.13.d.b | 4 | ||
| 4.b | odd | 2 | 1 | inner | 4.13.b.b | ✓ | 4 |
| 8.b | even | 2 | 1 | 64.13.c.f | 4 | ||
| 8.d | odd | 2 | 1 | 64.13.c.f | 4 | ||
| 12.b | even | 2 | 1 | 36.13.d.b | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 4.13.b.b | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 4.13.b.b | ✓ | 4 | 4.b | odd | 2 | 1 | inner |
| 36.13.d.b | 4 | 3.b | odd | 2 | 1 | ||
| 36.13.d.b | 4 | 12.b | even | 2 | 1 | ||
| 64.13.c.f | 4 | 8.b | even | 2 | 1 | ||
| 64.13.c.f | 4 | 8.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} + 1745664T_{3}^{2} + 142371717120 \)
acting on \(S_{13}^{\mathrm{new}}(4, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} - 108 T^{3} + \cdots + 16777216 \)
|
| $3$ |
\( T^{4} + \cdots + 142371717120 \)
|
| $5$ |
\( (T^{2} + 9180 T - 94541500)^{2} \)
|
| $7$ |
\( T^{4} + \cdots + 33\!\cdots\!00 \)
|
| $11$ |
\( T^{4} + \cdots + 13\!\cdots\!00 \)
|
| $13$ |
\( (T^{2} + \cdots + 2132952328900)^{2} \)
|
| $17$ |
\( (T^{2} + \cdots + 215318135101700)^{2} \)
|
| $19$ |
\( T^{4} + \cdots + 34\!\cdots\!00 \)
|
| $23$ |
\( T^{4} + \cdots + 50\!\cdots\!80 \)
|
| $29$ |
\( (T^{2} + \cdots - 27\!\cdots\!44)^{2} \)
|
| $31$ |
\( T^{4} + \cdots + 48\!\cdots\!00 \)
|
| $37$ |
\( (T^{2} + \cdots - 27\!\cdots\!00)^{2} \)
|
| $41$ |
\( (T^{2} + \cdots - 28\!\cdots\!36)^{2} \)
|
| $43$ |
\( T^{4} + \cdots + 31\!\cdots\!00 \)
|
| $47$ |
\( T^{4} + \cdots + 11\!\cdots\!20 \)
|
| $53$ |
\( (T^{2} + \cdots - 15\!\cdots\!00)^{2} \)
|
| $59$ |
\( T^{4} + \cdots + 41\!\cdots\!00 \)
|
| $61$ |
\( (T^{2} + \cdots - 14\!\cdots\!96)^{2} \)
|
| $67$ |
\( T^{4} + \cdots + 11\!\cdots\!20 \)
|
| $71$ |
\( T^{4} + \cdots + 28\!\cdots\!00 \)
|
| $73$ |
\( (T^{2} + \cdots + 39\!\cdots\!00)^{2} \)
|
| $79$ |
\( T^{4} + \cdots + 29\!\cdots\!00 \)
|
| $83$ |
\( T^{4} + \cdots + 41\!\cdots\!20 \)
|
| $89$ |
\( (T^{2} + \cdots - 12\!\cdots\!24)^{2} \)
|
| $97$ |
\( (T^{2} + \cdots + 15\!\cdots\!00)^{2} \)
|
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