Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1040,2,Mod(209,1040)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1040, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 1, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1040.209");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1040 = 2^{4} \cdot 5 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1040.d (of order \(2\), degree \(1\), not minimal) |
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: | no |
| Analytic conductor: | \(8.30444181021\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.0.350464.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | no (minimal twist has level 520) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -\nu^{5} + 8\nu^{4} - 4\nu^{3} - \nu^{2} + 2\nu + 38 ) / 23 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -5\nu^{5} + 17\nu^{4} - 20\nu^{3} - 5\nu^{2} + 10\nu + 29 ) / 23 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 7\nu^{5} - 10\nu^{4} + 5\nu^{3} + 30\nu^{2} + 32\nu - 13 ) / 23 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -11\nu^{5} + 19\nu^{4} - 21\nu^{3} - 11\nu^{2} - 70\nu + 27 ) / 23 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -14\nu^{5} + 20\nu^{4} - 10\nu^{3} - 37\nu^{2} - 64\nu + 26 ) / 23 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} - \beta _1 + 1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{5} + 2\beta_{3} \)
|
| \(\nu^{3}\) | \(=\) |
\( 2\beta_{5} - \beta_{4} + 2\beta_{3} - \beta_{2} + 2\beta _1 - 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( -\beta_{2} + 5\beta _1 - 7 \)
|
| \(\nu^{5}\) | \(=\) |
\( -8\beta_{5} + 3\beta_{4} - 9\beta_{3} - 3\beta_{2} + 8\beta _1 - 9 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1040\mathbb{Z}\right)^\times\).
| \(n\) | \(261\) | \(417\) | \(561\) | \(911\) |
| \(\chi(n)\) | \(1\) | \(-1\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 209.1 |
|
0 | − | 2.48119i | 0 | 1.67513 | + | 1.48119i | 0 | − | 4.15633i | 0 | −3.15633 | 0 | ||||||||||||||||||||||||||||||||
| 209.2 | 0 | − | 1.17009i | 0 | 0.539189 | + | 2.17009i | 0 | − | 0.630898i | 0 | 1.63090 | 0 | |||||||||||||||||||||||||||||||||
| 209.3 | 0 | − | 0.688892i | 0 | −2.21432 | − | 0.311108i | 0 | 1.52543i | 0 | 2.52543 | 0 | ||||||||||||||||||||||||||||||||||
| 209.4 | 0 | 0.688892i | 0 | −2.21432 | + | 0.311108i | 0 | − | 1.52543i | 0 | 2.52543 | 0 | ||||||||||||||||||||||||||||||||||
| 209.5 | 0 | 1.17009i | 0 | 0.539189 | − | 2.17009i | 0 | 0.630898i | 0 | 1.63090 | 0 | |||||||||||||||||||||||||||||||||||
| 209.6 | 0 | 2.48119i | 0 | 1.67513 | − | 1.48119i | 0 | 4.15633i | 0 | −3.15633 | 0 | |||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 1040.2.d.e | 6 | |
| 4.b | odd | 2 | 1 | 520.2.d.b | ✓ | 6 | |
| 5.b | even | 2 | 1 | inner | 1040.2.d.e | 6 | |
| 5.c | odd | 4 | 1 | 5200.2.a.cc | 3 | ||
| 5.c | odd | 4 | 1 | 5200.2.a.ch | 3 | ||
| 12.b | even | 2 | 1 | 4680.2.l.d | 6 | ||
| 20.d | odd | 2 | 1 | 520.2.d.b | ✓ | 6 | |
| 20.e | even | 4 | 1 | 2600.2.a.x | 3 | ||
| 20.e | even | 4 | 1 | 2600.2.a.y | 3 | ||
| 60.h | even | 2 | 1 | 4680.2.l.d | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 520.2.d.b | ✓ | 6 | 4.b | odd | 2 | 1 | |
| 520.2.d.b | ✓ | 6 | 20.d | odd | 2 | 1 | |
| 1040.2.d.e | 6 | 1.a | even | 1 | 1 | trivial | |
| 1040.2.d.e | 6 | 5.b | even | 2 | 1 | inner | |
| 2600.2.a.x | 3 | 20.e | even | 4 | 1 | ||
| 2600.2.a.y | 3 | 20.e | even | 4 | 1 | ||
| 4680.2.l.d | 6 | 12.b | even | 2 | 1 | ||
| 4680.2.l.d | 6 | 60.h | even | 2 | 1 | ||
| 5200.2.a.cc | 3 | 5.c | odd | 4 | 1 | ||
| 5200.2.a.ch | 3 | 5.c | odd | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{6} + 8T_{3}^{4} + 12T_{3}^{2} + 4 \)
acting on \(S_{2}^{\mathrm{new}}(1040, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
|
| $3$ |
\( T^{6} + 8 T^{4} + \cdots + 4 \)
|
| $5$ |
\( T^{6} - T^{4} + \cdots + 125 \)
|
| $7$ |
\( T^{6} + 20 T^{4} + \cdots + 16 \)
|
| $11$ |
\( (T^{3} - 4 T + 2)^{2} \)
|
| $13$ |
\( (T^{2} + 1)^{3} \)
|
| $17$ |
\( T^{6} + 44 T^{4} + \cdots + 1600 \)
|
| $19$ |
\( (T^{3} + 10 T^{2} + \cdots - 26)^{2} \)
|
| $23$ |
\( T^{6} + 116 T^{4} + \cdots + 17956 \)
|
| $29$ |
\( (T^{3} - 14 T^{2} + \cdots - 76)^{2} \)
|
| $31$ |
\( (T^{3} - 8 T^{2} + \cdots + 130)^{2} \)
|
| $37$ |
\( T^{6} + 72 T^{4} + \cdots + 13456 \)
|
| $41$ |
\( (T^{3} + 4 T^{2} - 16 T - 32)^{2} \)
|
| $43$ |
\( T^{6} + 12 T^{4} + \cdots + 4 \)
|
| $47$ |
\( T^{6} + 404 T^{4} + \cdots + 2085136 \)
|
| $53$ |
\( T^{6} + 208 T^{4} + \cdots + 256 \)
|
| $59$ |
\( (T^{3} - 2 T^{2} + \cdots + 178)^{2} \)
|
| $61$ |
\( (T^{3} + 6 T^{2} + \cdots - 100)^{2} \)
|
| $67$ |
\( T^{6} + 84 T^{4} + \cdots + 4624 \)
|
| $71$ |
\( (T^{3} - 10 T^{2} + \cdots + 802)^{2} \)
|
| $73$ |
\( T^{6} + 248 T^{4} + \cdots + 364816 \)
|
| $79$ |
\( (T^{3} + 28 T^{2} + \cdots + 688)^{2} \)
|
| $83$ |
\( T^{6} + 188 T^{4} + \cdots + 2704 \)
|
| $89$ |
\( (T^{3} - 22 T^{2} + \cdots + 184)^{2} \)
|
| $97$ |
\( T^{6} + 300 T^{4} + \cdots + 506944 \)
|
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