Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1183,2,Mod(1,1183)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1183, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1183.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1183 = 7 \cdot 13^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1183.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: | \(9.44630255912\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.4.27004.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} - 6x^{2} + 3x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 91) |
| 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
\( x^{4} - x^{3} - 6x^{2} + 3x + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - \nu^{2} - 6\nu + 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + \beta_{2} + 7\beta _1 + 1 \)
|
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 |
|
−2.22001 | 0.549551 | 2.92843 | 4.22001 | −1.22001 | 1.00000 | −2.06113 | −2.69799 | −9.36845 | ||||||||||||||||||||||||||||||
| 1.2 | −0.231361 | −3.32225 | −1.94647 | 2.23136 | 0.768639 | 1.00000 | 0.913059 | 8.03736 | −0.516249 | |||||||||||||||||||||||||||||||
| 1.3 | 0.710287 | 2.40788 | −1.49549 | 1.28971 | 1.71029 | 1.00000 | −2.48280 | 2.79790 | 0.916066 | |||||||||||||||||||||||||||||||
| 1.4 | 2.74108 | 1.36482 | 5.51353 | −0.741082 | 3.74108 | 1.00000 | 9.63087 | −1.13727 | −2.03137 | |||||||||||||||||||||||||||||||
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 | 1183.2.a.l | 4 | |
| 7.b | odd | 2 | 1 | 8281.2.a.bt | 4 | ||
| 13.b | even | 2 | 1 | 1183.2.a.k | 4 | ||
| 13.d | odd | 4 | 2 | 1183.2.c.g | 8 | ||
| 13.e | even | 6 | 2 | 91.2.f.c | ✓ | 8 | |
| 39.h | odd | 6 | 2 | 819.2.o.h | 8 | ||
| 52.i | odd | 6 | 2 | 1456.2.s.q | 8 | ||
| 91.b | odd | 2 | 1 | 8281.2.a.bp | 4 | ||
| 91.k | even | 6 | 2 | 637.2.h.h | 8 | ||
| 91.l | odd | 6 | 2 | 637.2.h.i | 8 | ||
| 91.p | odd | 6 | 2 | 637.2.g.j | 8 | ||
| 91.t | odd | 6 | 2 | 637.2.f.i | 8 | ||
| 91.u | even | 6 | 2 | 637.2.g.k | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 91.2.f.c | ✓ | 8 | 13.e | even | 6 | 2 | |
| 637.2.f.i | 8 | 91.t | odd | 6 | 2 | ||
| 637.2.g.j | 8 | 91.p | odd | 6 | 2 | ||
| 637.2.g.k | 8 | 91.u | even | 6 | 2 | ||
| 637.2.h.h | 8 | 91.k | even | 6 | 2 | ||
| 637.2.h.i | 8 | 91.l | odd | 6 | 2 | ||
| 819.2.o.h | 8 | 39.h | odd | 6 | 2 | ||
| 1183.2.a.k | 4 | 13.b | even | 2 | 1 | ||
| 1183.2.a.l | 4 | 1.a | even | 1 | 1 | trivial | |
| 1183.2.c.g | 8 | 13.d | odd | 4 | 2 | ||
| 1456.2.s.q | 8 | 52.i | odd | 6 | 2 | ||
| 8281.2.a.bp | 4 | 91.b | odd | 2 | 1 | ||
| 8281.2.a.bt | 4 | 7.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1183))\):
|
\( T_{2}^{4} - T_{2}^{3} - 6T_{2}^{2} + 3T_{2} + 1 \)
|
|
\( T_{11}^{4} - T_{11}^{3} - 9T_{11}^{2} + 16T_{11} - 6 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} - T^{3} - 6 T^{2} + \cdots + 1 \)
|
| $3$ |
\( T^{4} - T^{3} - 9 T^{2} + \cdots - 6 \)
|
| $5$ |
\( T^{4} - 7 T^{3} + \cdots - 9 \)
|
| $7$ |
\( (T - 1)^{4} \)
|
| $11$ |
\( T^{4} - T^{3} - 9 T^{2} + \cdots - 6 \)
|
| $13$ |
\( T^{4} \)
|
| $17$ |
\( T^{4} + 4 T^{3} + \cdots - 53 \)
|
| $19$ |
\( T^{4} + T^{3} + \cdots + 500 \)
|
| $23$ |
\( T^{4} + 2 T^{3} + \cdots + 72 \)
|
| $29$ |
\( T^{4} - T^{3} - 22 T^{2} + \cdots - 5 \)
|
| $31$ |
\( T^{4} - 4 T^{3} + \cdots + 54 \)
|
| $37$ |
\( T^{4} - 10 T^{3} + \cdots - 16 \)
|
| $41$ |
\( T^{4} - 22 T^{3} + \cdots - 564 \)
|
| $43$ |
\( T^{4} - 3 T^{3} + \cdots - 2 \)
|
| $47$ |
\( T^{4} + 2 T^{3} + \cdots + 100 \)
|
| $53$ |
\( T^{4} - 2 T^{3} + \cdots - 1389 \)
|
| $59$ |
\( T^{4} - 8 T^{3} + \cdots - 706 \)
|
| $61$ |
\( T^{4} - 8 T^{3} + \cdots - 100 \)
|
| $67$ |
\( T^{4} - 6 T^{3} + \cdots + 11010 \)
|
| $71$ |
\( T^{4} - 14 T^{3} + \cdots - 2032 \)
|
| $73$ |
\( T^{4} - 8 T^{3} + \cdots + 1772 \)
|
| $79$ |
\( T^{4} + 26 T^{3} + \cdots - 7680 \)
|
| $83$ |
\( T^{4} - 97 T^{2} + \cdots - 426 \)
|
| $89$ |
\( T^{4} - T^{3} + \cdots - 108 \)
|
| $97$ |
\( T^{4} + 3 T^{3} + \cdots + 15692 \)
|
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