Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [3,19,Mod(2,3)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1]))
N = Newforms(chi, 19, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("3.2");
S:= CuspForms(chi, 19);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 3 \) |
| Weight: | \( k \) | \(=\) | \( 19 \) |
| Character orbit: | \([\chi]\) | \(=\) | 3.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: | \(6.16158413129\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.0.601940665.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} + 123x^{2} - 1744x + 16016 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2^{11}\cdot 3^{11} \) |
| 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
\( x^{4} - x^{3} + 123x^{2} - 1744x + 16016 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -36\nu^{3} + 216\nu^{2} - 11592\nu + 59904 ) / 169 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 882\nu^{3} + 22086\nu^{2} + 201870\nu + 229788 ) / 169 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -4770\nu^{3} + 1242\nu^{2} - 139662\nu + 5911308 ) / 169 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} + \beta_{2} - 108\beta _1 + 1944 ) / 7776 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} + 17\beta_{2} + 284\beta _1 - 158760 ) / 2592 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -38\beta_{3} - 2\beta_{2} + 423\beta _1 + 1181952 ) / 972 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3\mathbb{Z}\right)^\times\).
| \(n\) | \(2\) |
| \(\chi(n)\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 2.1 |
|
− | 932.767i | −4234.02 | + | 19222.2i | −607911. | 1.14247e6i | 1.79299e7 | + | 3.94936e6i | −9.81043e6 | 3.22520e8i | −3.51567e8 | − | 1.62775e8i | 1.06566e9 | |||||||||||||||||||||||
| 2.2 | − | 425.442i | 12172.0 | − | 15468.1i | 81143.0 | − | 787628.i | −6.58078e6 | − | 5.17849e6i | −3.80616e7 | − | 1.46049e8i | −9.11041e7 | − | 3.76556e8i | −3.35090e8 | ||||||||||||||||||||||
| 2.3 | 425.442i | 12172.0 | + | 15468.1i | 81143.0 | 787628.i | −6.58078e6 | + | 5.17849e6i | −3.80616e7 | 1.46049e8i | −9.11041e7 | + | 3.76556e8i | −3.35090e8 | |||||||||||||||||||||||||
| 2.4 | 932.767i | −4234.02 | − | 19222.2i | −607911. | − | 1.14247e6i | 1.79299e7 | − | 3.94936e6i | −9.81043e6 | − | 3.22520e8i | −3.51567e8 | + | 1.62775e8i | 1.06566e9 | |||||||||||||||||||||||
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 | 3.19.b.b | ✓ | 4 |
| 3.b | odd | 2 | 1 | inner | 3.19.b.b | ✓ | 4 |
| 4.b | odd | 2 | 1 | 48.19.e.b | 4 | ||
| 12.b | even | 2 | 1 | 48.19.e.b | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 3.19.b.b | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 3.19.b.b | ✓ | 4 | 3.b | odd | 2 | 1 | inner |
| 48.19.e.b | 4 | 4.b | odd | 2 | 1 | ||
| 48.19.e.b | 4 | 12.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} + 1051056T_{2}^{2} + 157480796160 \)
acting on \(S_{19}^{\mathrm{new}}(3, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} + \cdots + 157480796160 \)
|
| $3$ |
\( T^{4} + \cdots + 15\!\cdots\!21 \)
|
| $5$ |
\( T^{4} + \cdots + 80\!\cdots\!00 \)
|
| $7$ |
\( (T^{2} + \cdots + 373401093446500)^{2} \)
|
| $11$ |
\( T^{4} + \cdots + 44\!\cdots\!00 \)
|
| $13$ |
\( (T^{2} + \cdots - 17\!\cdots\!40)^{2} \)
|
| $17$ |
\( T^{4} + \cdots + 38\!\cdots\!40 \)
|
| $19$ |
\( (T^{2} + \cdots - 79\!\cdots\!64)^{2} \)
|
| $23$ |
\( T^{4} + \cdots + 12\!\cdots\!60 \)
|
| $29$ |
\( T^{4} + \cdots + 21\!\cdots\!00 \)
|
| $31$ |
\( (T^{2} + \cdots - 34\!\cdots\!96)^{2} \)
|
| $37$ |
\( (T^{2} + \cdots + 14\!\cdots\!40)^{2} \)
|
| $41$ |
\( T^{4} + \cdots + 11\!\cdots\!00 \)
|
| $43$ |
\( (T^{2} + \cdots - 34\!\cdots\!00)^{2} \)
|
| $47$ |
\( T^{4} + \cdots + 66\!\cdots\!40 \)
|
| $53$ |
\( T^{4} + \cdots + 14\!\cdots\!40 \)
|
| $59$ |
\( T^{4} + \cdots + 61\!\cdots\!00 \)
|
| $61$ |
\( (T^{2} + \cdots - 11\!\cdots\!96)^{2} \)
|
| $67$ |
\( (T^{2} + \cdots + 84\!\cdots\!80)^{2} \)
|
| $71$ |
\( T^{4} + \cdots + 58\!\cdots\!00 \)
|
| $73$ |
\( (T^{2} + \cdots - 64\!\cdots\!60)^{2} \)
|
| $79$ |
\( (T^{2} + \cdots + 11\!\cdots\!36)^{2} \)
|
| $83$ |
\( T^{4} + \cdots + 24\!\cdots\!40 \)
|
| $89$ |
\( T^{4} + \cdots + 66\!\cdots\!00 \)
|
| $97$ |
\( (T^{2} + \cdots + 74\!\cdots\!20)^{2} \)
|
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