Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [6003,2,Mod(1,6003)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6003, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 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("6003.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 6003 = 3^{2} \cdot 23 \cdot 29 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 6003.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: | \(47.9341963334\) |
| Analytic rank: | \(0\) |
| Dimension: | \(5\) |
| Coefficient field: | 5.5.312617.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{5} - 2x^{4} - 5x^{3} + 11x^{2} - x - 3 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 2001) |
| 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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) 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
\( x^{5} - 2x^{4} - 5x^{3} + 11x^{2} - x - 3 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{3} - 5\nu + 1 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{4} - \nu^{3} - 5\nu^{2} + 5\nu + 1 \)
|
| \(\beta_{4}\) | \(=\) |
\( -\nu^{4} + \nu^{3} + 6\nu^{2} - 5\nu - 3 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{4} + \beta_{3} + 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{2} + 5\beta _1 - 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( 5\beta_{4} + 6\beta_{3} + \beta_{2} + 8 \)
|
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 |
|
−2.10891 | 0 | 2.44748 | −1.70032 | 0 | −4.14780 | −0.943696 | 0 | 3.58582 | |||||||||||||||||||||||||||||||||
| 1.2 | −1.51515 | 0 | 0.295689 | 1.86677 | 0 | 1.57109 | 2.58229 | 0 | −2.82845 | ||||||||||||||||||||||||||||||||||
| 1.3 | 1.31874 | 0 | −0.260930 | −2.51371 | 0 | −2.25278 | −2.98157 | 0 | −3.31493 | ||||||||||||||||||||||||||||||||||
| 1.4 | 1.79920 | 0 | 1.23712 | 2.60753 | 0 | 1.37040 | −1.37257 | 0 | 4.69147 | ||||||||||||||||||||||||||||||||||
| 1.5 | 2.50612 | 0 | 4.28064 | 2.73973 | 0 | −1.54091 | 5.71555 | 0 | 6.86609 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(23\) | \(1\) |
| \(29\) | \(1\) |
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 | 6003.2.a.h | 5 | |
| 3.b | odd | 2 | 1 | 2001.2.a.h | ✓ | 5 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2001.2.a.h | ✓ | 5 | 3.b | odd | 2 | 1 | |
| 6003.2.a.h | 5 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6003))\):
|
\( T_{2}^{5} - 2T_{2}^{4} - 7T_{2}^{3} + 13T_{2}^{2} + 11T_{2} - 19 \)
|
|
\( T_{5}^{5} - 3T_{5}^{4} - 9T_{5}^{3} + 28T_{5}^{2} + 17T_{5} - 57 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} - 2 T^{4} + \cdots - 19 \)
|
| $3$ |
\( T^{5} \)
|
| $5$ |
\( T^{5} - 3 T^{4} + \cdots - 57 \)
|
| $7$ |
\( T^{5} + 5 T^{4} + \cdots + 31 \)
|
| $11$ |
\( T^{5} - 8 T^{4} + \cdots + 1 \)
|
| $13$ |
\( T^{5} - 5 T^{4} + \cdots + 9 \)
|
| $17$ |
\( T^{5} + 2 T^{4} + \cdots + 3 \)
|
| $19$ |
\( T^{5} + 9 T^{4} + \cdots - 563 \)
|
| $23$ |
\( (T + 1)^{5} \)
|
| $29$ |
\( (T + 1)^{5} \)
|
| $31$ |
\( T^{5} + 6 T^{4} + \cdots + 139 \)
|
| $37$ |
\( T^{5} - 10 T^{4} + \cdots - 821 \)
|
| $41$ |
\( T^{5} - 11 T^{4} + \cdots + 921 \)
|
| $43$ |
\( T^{5} + 9 T^{4} + \cdots + 7447 \)
|
| $47$ |
\( T^{5} - 13 T^{4} + \cdots - 49139 \)
|
| $53$ |
\( T^{5} + T^{4} + \cdots - 361 \)
|
| $59$ |
\( T^{5} + 6 T^{4} + \cdots - 1824 \)
|
| $61$ |
\( T^{5} - 23 T^{4} + \cdots - 281 \)
|
| $67$ |
\( T^{5} + 10 T^{4} + \cdots - 513 \)
|
| $71$ |
\( T^{5} - 11 T^{4} + \cdots - 57 \)
|
| $73$ |
\( T^{5} - 31 T^{4} + \cdots + 1317 \)
|
| $79$ |
\( T^{5} - 8 T^{4} + \cdots - 2921 \)
|
| $83$ |
\( T^{5} + 7 T^{4} + \cdots - 40743 \)
|
| $89$ |
\( T^{5} + 3 T^{4} + \cdots + 687 \)
|
| $97$ |
\( T^{5} - 3 T^{4} + \cdots + 4337 \)
|
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