Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [49,20,Mod(1,49)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(49, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 20, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("49.1");
S:= CuspForms(chi, 20);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 49 = 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 20 \) |
| Character orbit: | \([\chi]\) | \(=\) | 49.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: | \(112.120181313\) |
| Analytic rank: | \(1\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{4} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} - 330513x^{2} - 30288715x + 14876898628 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2^{5}\cdot 3^{2}\cdot 5\cdot 7 \) |
| Twist minimal: | no (minimal twist has level 7) |
| 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 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
\( x^{4} - x^{3} - 330513x^{2} - 30288715x + 14876898628 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} - 366\nu^{2} - 190203\nu + 37566932 ) / 336 \)
|
| \(\beta_{3}\) | \(=\) |
\( 2\nu^{2} - 278\nu - 330444 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} + 139\beta _1 + 330444 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 366\beta_{3} + 672\beta_{2} + 241077\beta _1 + 45808640 ) / 2 \)
|
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 |
|
−902.343 | −40157.9 | 289935. | 3.86047e6 | 3.62362e7 | 0 | 2.11467e8 | 4.50396e8 | −3.48347e9 | ||||||||||||||||||||||||||||||
| 1.2 | −788.739 | 48874.0 | 97821.5 | −1.27186e6 | −3.85488e7 | 0 | 3.36371e8 | 1.22641e9 | 1.00316e9 | |||||||||||||||||||||||||||||||
| 1.3 | 270.188 | 3480.67 | −451287. | −4.93061e6 | 940434. | 0 | −2.63588e8 | −1.15015e9 | −1.33219e9 | |||||||||||||||||||||||||||||||
| 1.4 | 1078.89 | 17329.2 | 639726. | 4.82861e6 | 1.86964e7 | 0 | 1.24545e8 | −8.61959e8 | 5.20956e9 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(7\) | \(-1\) |
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 | 49.20.a.c | 4 | |
| 7.b | odd | 2 | 1 | 7.20.a.a | ✓ | 4 | |
| 21.c | even | 2 | 1 | 63.20.a.b | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 7.20.a.a | ✓ | 4 | 7.b | odd | 2 | 1 | |
| 49.20.a.c | 4 | 1.a | even | 1 | 1 | trivial | |
| 63.20.a.b | 4 | 21.c | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{20}^{\mathrm{new}}(\Gamma_0(49))\):
|
\( T_{2}^{4} + 342T_{2}^{3} - 1278192T_{2}^{2} - 467202816T_{2} + 207467274240 \)
|
|
\( T_{3}^{4} - 29526T_{3}^{3} - 1720978380T_{3}^{2} + 40317400600344T_{3} - 118383501061125216 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} + \cdots + 207467274240 \)
|
| $3$ |
\( T^{4} + \cdots - 11\!\cdots\!16 \)
|
| $5$ |
\( T^{4} + \cdots + 11\!\cdots\!00 \)
|
| $7$ |
\( T^{4} \)
|
| $11$ |
\( T^{4} + \cdots + 25\!\cdots\!76 \)
|
| $13$ |
\( T^{4} + \cdots + 14\!\cdots\!80 \)
|
| $17$ |
\( T^{4} + \cdots + 20\!\cdots\!96 \)
|
| $19$ |
\( T^{4} + \cdots + 68\!\cdots\!00 \)
|
| $23$ |
\( T^{4} + \cdots - 14\!\cdots\!84 \)
|
| $29$ |
\( T^{4} + \cdots + 55\!\cdots\!00 \)
|
| $31$ |
\( T^{4} + \cdots - 39\!\cdots\!72 \)
|
| $37$ |
\( T^{4} + \cdots + 29\!\cdots\!24 \)
|
| $41$ |
\( T^{4} + \cdots + 28\!\cdots\!24 \)
|
| $43$ |
\( T^{4} + \cdots + 16\!\cdots\!40 \)
|
| $47$ |
\( T^{4} + \cdots + 72\!\cdots\!88 \)
|
| $53$ |
\( T^{4} + \cdots - 28\!\cdots\!12 \)
|
| $59$ |
\( T^{4} + \cdots + 48\!\cdots\!00 \)
|
| $61$ |
\( T^{4} + \cdots + 87\!\cdots\!44 \)
|
| $67$ |
\( T^{4} + \cdots - 26\!\cdots\!88 \)
|
| $71$ |
\( T^{4} + \cdots - 55\!\cdots\!40 \)
|
| $73$ |
\( T^{4} + \cdots + 12\!\cdots\!68 \)
|
| $79$ |
\( T^{4} + \cdots - 11\!\cdots\!00 \)
|
| $83$ |
\( T^{4} + \cdots + 98\!\cdots\!48 \)
|
| $89$ |
\( T^{4} + \cdots + 36\!\cdots\!00 \)
|
| $97$ |
\( T^{4} + \cdots + 44\!\cdots\!68 \)
|
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