Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2,42,Mod(1,2)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2, base_ring=CyclotomicField(1))
chi = DirichletCharacter(H, H._module([]))
N = Newforms(chi, 42, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2.1");
S:= CuspForms(chi, 42);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2 \) |
| Weight: | \( k \) | \(=\) | \( 42 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2.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: | \(21.2943340913\) |
| Analytic rank: | \(0\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{2} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - x - 1139917559892 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2^{7}\cdot 3^{2}\cdot 5 \) |
| 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 \(\beta = 2880\sqrt{4559670239569}\). 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 |
|
1.04858e6 | −1.71810e9 | 1.09951e12 | 3.51885e14 | −1.80156e15 | −7.89090e16 | 1.15292e18 | −3.35211e19 | 3.68978e20 | ||||||||||||||||||||||||
| 1.2 | 1.04858e6 | 1.05814e10 | 1.09951e12 | −2.54286e14 | 1.10955e16 | 2.96016e17 | 1.15292e18 | 7.54941e19 | −2.66638e20 | |||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
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 | 2.42.a.b | ✓ | 2 |
| 3.b | odd | 2 | 1 | 18.42.a.c | 2 | ||
| 4.b | odd | 2 | 1 | 16.42.a.b | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2.42.a.b | ✓ | 2 | 1.a | even | 1 | 1 | trivial |
| 16.42.a.b | 2 | 4.b | odd | 2 | 1 | ||
| 18.42.a.c | 2 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{2} - 8863347528T_{3} - 18179996484555185904 \)
acting on \(S_{42}^{\mathrm{new}}(\Gamma_0(2))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 1048576)^{2} \)
|
| $3$ |
\( T^{2} + \cdots - 18\!\cdots\!04 \)
|
| $5$ |
\( T^{2} + \cdots - 89\!\cdots\!00 \)
|
| $7$ |
\( T^{2} + \cdots - 23\!\cdots\!16 \)
|
| $11$ |
\( T^{2} + \cdots - 13\!\cdots\!36 \)
|
| $13$ |
\( T^{2} + \cdots + 74\!\cdots\!56 \)
|
| $17$ |
\( T^{2} + \cdots + 26\!\cdots\!04 \)
|
| $19$ |
\( T^{2} + \cdots - 84\!\cdots\!00 \)
|
| $23$ |
\( T^{2} + \cdots + 55\!\cdots\!56 \)
|
| $29$ |
\( T^{2} + \cdots - 42\!\cdots\!00 \)
|
| $31$ |
\( T^{2} + \cdots - 17\!\cdots\!36 \)
|
| $37$ |
\( T^{2} + \cdots + 34\!\cdots\!24 \)
|
| $41$ |
\( T^{2} + \cdots - 19\!\cdots\!76 \)
|
| $43$ |
\( T^{2} + \cdots + 20\!\cdots\!76 \)
|
| $47$ |
\( T^{2} + \cdots + 12\!\cdots\!24 \)
|
| $53$ |
\( T^{2} + \cdots + 20\!\cdots\!76 \)
|
| $59$ |
\( T^{2} + \cdots - 16\!\cdots\!00 \)
|
| $61$ |
\( T^{2} + \cdots - 13\!\cdots\!76 \)
|
| $67$ |
\( T^{2} + \cdots + 11\!\cdots\!84 \)
|
| $71$ |
\( T^{2} + \cdots - 58\!\cdots\!36 \)
|
| $73$ |
\( T^{2} + \cdots + 57\!\cdots\!36 \)
|
| $79$ |
\( T^{2} + \cdots + 16\!\cdots\!00 \)
|
| $83$ |
\( T^{2} + \cdots - 47\!\cdots\!44 \)
|
| $89$ |
\( T^{2} + \cdots + 11\!\cdots\!00 \)
|
| $97$ |
\( T^{2} + \cdots + 19\!\cdots\!04 \)
|
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