Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2,50,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, 50, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2.1");
S:= CuspForms(chi, 50);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2 \) |
| Weight: | \( k \) | \(=\) | \( 50 \) |
| 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: | \(30.4132410198\) |
| Analytic rank: | \(1\) |
| 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: |
\( x^{2} - x - 22129540960032 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2^{7}\cdot 3^{3}\cdot 5^{2} \) |
| 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 \(\beta = 43200\sqrt{88518163840129}\). 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 |
|
−1.67772e7 | −2.65918e11 | 2.81475e14 | −1.00270e17 | 4.46136e18 | 2.61926e20 | −4.72237e21 | −1.68587e23 | 1.68224e24 | ||||||||||||||||||||||||
| 1.2 | −1.67772e7 | 5.46969e11 | 2.81475e14 | 1.83444e17 | −9.17661e18 | −6.94316e20 | −4.72237e21 | 5.98756e22 | −3.07768e24 | |||||||||||||||||||||||||
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 | 2.50.a.a | ✓ | 2 |
| 4.b | odd | 2 | 1 | 16.50.a.a | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2.50.a.a | ✓ | 2 | 1.a | even | 1 | 1 | trivial |
| 16.50.a.a | 2 | 4.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{2} - 281051075592T_{3} - 145448711312147320422384 \)
acting on \(S_{50}^{\mathrm{new}}(\Gamma_0(2))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 16777216)^{2} \)
|
| $3$ |
\( T^{2} + \cdots - 14\!\cdots\!84 \)
|
| $5$ |
\( T^{2} + \cdots - 18\!\cdots\!00 \)
|
| $7$ |
\( T^{2} + \cdots - 18\!\cdots\!76 \)
|
| $11$ |
\( T^{2} + \cdots - 13\!\cdots\!76 \)
|
| $13$ |
\( T^{2} + \cdots - 32\!\cdots\!04 \)
|
| $17$ |
\( T^{2} + \cdots - 11\!\cdots\!76 \)
|
| $19$ |
\( T^{2} + \cdots + 21\!\cdots\!00 \)
|
| $23$ |
\( T^{2} + \cdots - 19\!\cdots\!04 \)
|
| $29$ |
\( T^{2} + \cdots + 89\!\cdots\!00 \)
|
| $31$ |
\( T^{2} + \cdots + 10\!\cdots\!04 \)
|
| $37$ |
\( T^{2} + \cdots - 91\!\cdots\!36 \)
|
| $41$ |
\( T^{2} + \cdots + 97\!\cdots\!24 \)
|
| $43$ |
\( T^{2} + \cdots + 13\!\cdots\!56 \)
|
| $47$ |
\( T^{2} + \cdots - 61\!\cdots\!96 \)
|
| $53$ |
\( T^{2} + \cdots + 38\!\cdots\!56 \)
|
| $59$ |
\( T^{2} + \cdots - 71\!\cdots\!00 \)
|
| $61$ |
\( T^{2} + \cdots - 33\!\cdots\!96 \)
|
| $67$ |
\( T^{2} + \cdots + 18\!\cdots\!64 \)
|
| $71$ |
\( T^{2} + \cdots - 13\!\cdots\!36 \)
|
| $73$ |
\( T^{2} + \cdots + 17\!\cdots\!36 \)
|
| $79$ |
\( T^{2} + \cdots - 62\!\cdots\!00 \)
|
| $83$ |
\( T^{2} + \cdots - 15\!\cdots\!04 \)
|
| $89$ |
\( T^{2} + \cdots + 14\!\cdots\!00 \)
|
| $97$ |
\( T^{2} + \cdots - 15\!\cdots\!56 \)
|
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