Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [3,28,Mod(1,3)]
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([0]))
N = Newforms(chi, 28, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("3.1");
S:= CuspForms(chi, 28);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 3 \) |
| Weight: | \( k \) | \(=\) | \( 28 \) |
| Character orbit: | \([\chi]\) | \(=\) | 3.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: | \(13.8556672451\) |
| Analytic rank: | \(1\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{6469}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - x - 1617 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 2^{5}\cdot 3^{2} \) |
| 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 \(\beta = 144\sqrt{6469}\). We also show the integral \(q\)-expansion of the trace form.
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 |
|
−9997.93 | 1.59432e6 | −3.42591e7 | −1.80194e8 | −1.59399e10 | 1.89150e11 | 1.68442e12 | 2.54187e12 | 1.80157e12 | ||||||||||||||||||||||||
| 1.2 | 13165.9 | 1.59432e6 | 3.91241e7 | −4.72587e9 | 2.09908e10 | −3.40807e11 | −1.25200e12 | 2.54187e12 | −6.22205e13 | |||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
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 | 3.28.a.a | ✓ | 2 |
| 3.b | odd | 2 | 1 | 9.28.a.c | 2 | ||
| 4.b | odd | 2 | 1 | 48.28.a.d | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 3.28.a.a | ✓ | 2 | 1.a | even | 1 | 1 | trivial |
| 9.28.a.c | 2 | 3.b | odd | 2 | 1 | ||
| 48.28.a.d | 2 | 4.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{2} - 3168T_{2} - 131632128 \)
acting on \(S_{28}^{\mathrm{new}}(\Gamma_0(3))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} - 3168 T - 131632128 \)
|
| $3$ |
\( (T - 1594323)^{2} \)
|
| $5$ |
\( T^{2} + \cdots + 85\!\cdots\!00 \)
|
| $7$ |
\( T^{2} + \cdots - 64\!\cdots\!20 \)
|
| $11$ |
\( T^{2} + \cdots - 14\!\cdots\!08 \)
|
| $13$ |
\( T^{2} + \cdots - 39\!\cdots\!88 \)
|
| $17$ |
\( T^{2} + \cdots + 13\!\cdots\!16 \)
|
| $19$ |
\( T^{2} + \cdots + 90\!\cdots\!80 \)
|
| $23$ |
\( T^{2} + \cdots - 42\!\cdots\!60 \)
|
| $29$ |
\( T^{2} + \cdots + 31\!\cdots\!20 \)
|
| $31$ |
\( T^{2} + \cdots - 42\!\cdots\!00 \)
|
| $37$ |
\( T^{2} + \cdots - 23\!\cdots\!80 \)
|
| $41$ |
\( T^{2} + \cdots - 34\!\cdots\!40 \)
|
| $43$ |
\( T^{2} + \cdots + 31\!\cdots\!76 \)
|
| $47$ |
\( T^{2} + \cdots - 12\!\cdots\!36 \)
|
| $53$ |
\( T^{2} + \cdots + 30\!\cdots\!00 \)
|
| $59$ |
\( T^{2} + \cdots + 43\!\cdots\!20 \)
|
| $61$ |
\( T^{2} + \cdots - 28\!\cdots\!44 \)
|
| $67$ |
\( T^{2} + \cdots - 20\!\cdots\!64 \)
|
| $71$ |
\( T^{2} + \cdots + 42\!\cdots\!44 \)
|
| $73$ |
\( T^{2} + \cdots - 68\!\cdots\!40 \)
|
| $79$ |
\( T^{2} + \cdots - 13\!\cdots\!00 \)
|
| $83$ |
\( T^{2} + \cdots + 13\!\cdots\!68 \)
|
| $89$ |
\( T^{2} + \cdots + 61\!\cdots\!40 \)
|
| $97$ |
\( T^{2} + \cdots - 18\!\cdots\!64 \)
|
| show more | |
| show less |