Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1911,4,Mod(1,1911)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1911, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1911.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1911 = 3 \cdot 7^{2} \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1911.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: | \(112.752650021\) |
| Analytic rank: | \(1\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.4.6295500.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} - 19x^{2} + 19x + 16 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 273) |
| 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
\( x^{4} - x^{3} - 19x^{2} + 19x + 16 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 10 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} - 17\nu + 4 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 10 \)
|
| \(\nu^{3}\) | \(=\) |
\( 2\beta_{3} + 17\beta _1 - 4 \)
|
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 |
|
−5.27680 | 3.00000 | 19.8446 | 4.51607 | −15.8304 | 0 | −62.5017 | 9.00000 | −23.8304 | ||||||||||||||||||||||||||||||
| 1.2 | −1.55157 | 3.00000 | −5.59262 | 8.15605 | −4.65472 | 0 | 21.0900 | 9.00000 | −12.6547 | |||||||||||||||||||||||||||||||
| 1.3 | 0.606564 | 3.00000 | −7.63208 | −10.1890 | 1.81969 | 0 | −9.48186 | 9.00000 | −6.18031 | |||||||||||||||||||||||||||||||
| 1.4 | 3.22181 | 3.00000 | 2.38007 | 0.516925 | 9.66543 | 0 | −18.1064 | 9.00000 | 1.66543 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(7\) | \(-1\) |
| \(13\) | \(-1\) |
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 | 1911.4.a.m | 4 | |
| 7.b | odd | 2 | 1 | 273.4.a.e | ✓ | 4 | |
| 21.c | even | 2 | 1 | 819.4.a.f | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 273.4.a.e | ✓ | 4 | 7.b | odd | 2 | 1 | |
| 819.4.a.f | 4 | 21.c | even | 2 | 1 | ||
| 1911.4.a.m | 4 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1911))\):
|
\( T_{2}^{4} + 3T_{2}^{3} - 16T_{2}^{2} - 18T_{2} + 16 \)
|
|
\( T_{5}^{4} - 3T_{5}^{3} - 91T_{5}^{2} + 423T_{5} - 194 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} + 3 T^{3} + \cdots + 16 \)
|
| $3$ |
\( (T - 3)^{4} \)
|
| $5$ |
\( T^{4} - 3 T^{3} + \cdots - 194 \)
|
| $7$ |
\( T^{4} \)
|
| $11$ |
\( T^{4} + 9 T^{3} + \cdots - 356544 \)
|
| $13$ |
\( (T - 13)^{4} \)
|
| $17$ |
\( T^{4} - 75 T^{3} + \cdots + 590080 \)
|
| $19$ |
\( T^{4} + 87 T^{3} + \cdots - 16852464 \)
|
| $23$ |
\( T^{4} + 105 T^{3} + \cdots + 67639330 \)
|
| $29$ |
\( T^{4} + 219 T^{3} + \cdots - 801390134 \)
|
| $31$ |
\( T^{4} + 62 T^{3} + \cdots - 136356704 \)
|
| $37$ |
\( T^{4} + \cdots - 3284796224 \)
|
| $41$ |
\( T^{4} + \cdots - 1666728704 \)
|
| $43$ |
\( T^{4} + \cdots + 5419905316 \)
|
| $47$ |
\( T^{4} + \cdots - 23998005344 \)
|
| $53$ |
\( T^{4} + \cdots - 1606568084 \)
|
| $59$ |
\( T^{4} + \cdots - 5794181120 \)
|
| $61$ |
\( T^{4} + \cdots - 1122357384 \)
|
| $67$ |
\( T^{4} + \cdots + 20420417376 \)
|
| $71$ |
\( T^{4} + \cdots - 3362829824 \)
|
| $73$ |
\( T^{4} + 613 T^{3} + \cdots + 995273506 \)
|
| $79$ |
\( T^{4} + \cdots - 38487270224 \)
|
| $83$ |
\( T^{4} + \cdots - 158927495364 \)
|
| $89$ |
\( T^{4} + \cdots + 865777562380 \)
|
| $97$ |
\( T^{4} + \cdots + 73983712516 \)
|
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