Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [91,4,Mod(1,91)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(91, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([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("91.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 91 = 7 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 91.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: | \(5.36917381052\) |
| Analytic rank: | \(1\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.4.5364412.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 27x^{2} - 24x + 76 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| 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 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
\( x^{4} - 27x^{2} - 24x + 76 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} - 2\nu^{2} - 19\nu + 10 ) / 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{2} - \nu - 14 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + \beta _1 + 14 \)
|
| \(\nu^{3}\) | \(=\) |
\( 2\beta_{3} + 4\beta_{2} + 21\beta _1 + 18 \)
|
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 |
|
−5.05873 | 6.23157 | 17.5908 | −18.8190 | −31.5238 | 7.00000 | −48.5170 | 11.8325 | 95.2001 | ||||||||||||||||||||||||||||||
| 1.2 | −3.63459 | −5.33748 | 5.21021 | 11.0031 | 19.3995 | 7.00000 | 10.1397 | 1.48874 | −39.9917 | |||||||||||||||||||||||||||||||
| 1.3 | 0.323612 | 1.75980 | −7.89528 | −5.91876 | 0.569491 | 7.00000 | −5.14391 | −23.9031 | −1.91538 | |||||||||||||||||||||||||||||||
| 1.4 | 4.36970 | −7.65388 | 11.0943 | −22.2654 | −33.4452 | 7.00000 | 13.5212 | 31.5819 | −97.2930 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(7\) | \(-1\) |
| \(13\) | \(1\) |
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 | 91.4.a.b | ✓ | 4 |
| 3.b | odd | 2 | 1 | 819.4.a.h | 4 | ||
| 4.b | odd | 2 | 1 | 1456.4.a.s | 4 | ||
| 5.b | even | 2 | 1 | 2275.4.a.h | 4 | ||
| 7.b | odd | 2 | 1 | 637.4.a.d | 4 | ||
| 13.b | even | 2 | 1 | 1183.4.a.e | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 91.4.a.b | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 637.4.a.d | 4 | 7.b | odd | 2 | 1 | ||
| 819.4.a.h | 4 | 3.b | odd | 2 | 1 | ||
| 1183.4.a.e | 4 | 13.b | even | 2 | 1 | ||
| 1456.4.a.s | 4 | 4.b | odd | 2 | 1 | ||
| 2275.4.a.h | 4 | 5.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} + 4T_{2}^{3} - 21T_{2}^{2} - 74T_{2} + 26 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(91))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} + 4 T^{3} + \cdots + 26 \)
|
| $3$ |
\( T^{4} + 5 T^{3} + \cdots + 448 \)
|
| $5$ |
\( T^{4} + 36 T^{3} + \cdots - 27288 \)
|
| $7$ |
\( (T - 7)^{4} \)
|
| $11$ |
\( T^{4} + 95 T^{3} + \cdots - 151632 \)
|
| $13$ |
\( (T + 13)^{4} \)
|
| $17$ |
\( T^{4} + 146 T^{3} + \cdots - 1065472 \)
|
| $19$ |
\( T^{4} + 48 T^{3} + \cdots + 317232 \)
|
| $23$ |
\( T^{4} + 121 T^{3} + \cdots - 2384104 \)
|
| $29$ |
\( T^{4} + 440 T^{3} + \cdots - 484339768 \)
|
| $31$ |
\( T^{4} + 283 T^{3} + \cdots - 1026856 \)
|
| $37$ |
\( T^{4} + 209 T^{3} + \cdots + 328158128 \)
|
| $41$ |
\( T^{4} + \cdots + 12096773224 \)
|
| $43$ |
\( T^{4} - 526 T^{3} + \cdots + 18583856 \)
|
| $47$ |
\( T^{4} + \cdots - 1054241384 \)
|
| $53$ |
\( T^{4} + \cdots - 11218230832 \)
|
| $59$ |
\( T^{4} + \cdots + 10047112192 \)
|
| $61$ |
\( T^{4} + 141 T^{3} + \cdots + 3710376 \)
|
| $67$ |
\( T^{4} + 523 T^{3} + \cdots - 951710544 \)
|
| $71$ |
\( T^{4} + \cdots + 2887158784 \)
|
| $73$ |
\( T^{4} + \cdots + 38124898514 \)
|
| $79$ |
\( T^{4} + \cdots - 13183278632 \)
|
| $83$ |
\( T^{4} + \cdots - 11400717312 \)
|
| $89$ |
\( T^{4} + \cdots + 205066944356 \)
|
| $97$ |
\( T^{4} + \cdots - 914822530202 \)
|
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