Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1075,2,Mod(474,1075)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1075, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([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("1075.474");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1075 = 5^{2} \cdot 43 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1075.b (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: | \(8.58391821729\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.0.6594624.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} + 9x^{4} + 18x^{2} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 215) |
| 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
\( x^{6} + 9x^{4} + 18x^{2} + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{4} + 5\nu^{2} + 1 ) / 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{4} + 8\nu^{2} + 10 ) / 3 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{5} + 8\nu^{3} + 13\nu ) / 3 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 2\nu^{5} + 19\nu^{3} + 41\nu ) / 3 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} - \beta_{2} - 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{5} - 2\beta_{4} - 5\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -5\beta_{3} + 8\beta_{2} + 14 \)
|
| \(\nu^{5}\) | \(=\) |
\( -8\beta_{5} + 19\beta_{4} + 27\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1075\mathbb{Z}\right)^\times\).
| \(n\) | \(302\) | \(476\) |
| \(\chi(n)\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 474.1 |
|
− | 2.69963i | 1.69963i | −5.28799 | 0 | 4.58836 | − | 0.888736i | 8.87636i | 0.111264 | 0 | ||||||||||||||||||||||||||||||||||
| 474.2 | − | 1.46050i | 2.46050i | −0.133074 | 0 | 3.59358 | 4.05408i | − | 2.72665i | −3.05408 | 0 | |||||||||||||||||||||||||||||||||||
| 474.3 | − | 0.760877i | − | 0.239123i | 1.42107 | 0 | −0.181943 | 1.94282i | − | 2.60301i | 2.94282 | 0 | ||||||||||||||||||||||||||||||||||
| 474.4 | 0.760877i | 0.239123i | 1.42107 | 0 | −0.181943 | − | 1.94282i | 2.60301i | 2.94282 | 0 | ||||||||||||||||||||||||||||||||||||
| 474.5 | 1.46050i | − | 2.46050i | −0.133074 | 0 | 3.59358 | − | 4.05408i | 2.72665i | −3.05408 | 0 | |||||||||||||||||||||||||||||||||||
| 474.6 | 2.69963i | − | 1.69963i | −5.28799 | 0 | 4.58836 | 0.888736i | − | 8.87636i | 0.111264 | 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 | 1075.2.b.g | 6 | |
| 5.b | even | 2 | 1 | inner | 1075.2.b.g | 6 | |
| 5.c | odd | 4 | 1 | 215.2.a.b | ✓ | 3 | |
| 5.c | odd | 4 | 1 | 1075.2.a.l | 3 | ||
| 15.e | even | 4 | 1 | 1935.2.a.r | 3 | ||
| 15.e | even | 4 | 1 | 9675.2.a.bp | 3 | ||
| 20.e | even | 4 | 1 | 3440.2.a.l | 3 | ||
| 215.g | even | 4 | 1 | 9245.2.a.h | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 215.2.a.b | ✓ | 3 | 5.c | odd | 4 | 1 | |
| 1075.2.a.l | 3 | 5.c | odd | 4 | 1 | ||
| 1075.2.b.g | 6 | 1.a | even | 1 | 1 | trivial | |
| 1075.2.b.g | 6 | 5.b | even | 2 | 1 | inner | |
| 1935.2.a.r | 3 | 15.e | even | 4 | 1 | ||
| 3440.2.a.l | 3 | 20.e | even | 4 | 1 | ||
| 9245.2.a.h | 3 | 215.g | even | 4 | 1 | ||
| 9675.2.a.bp | 3 | 15.e | even | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1075, [\chi])\):
|
\( T_{2}^{6} + 10T_{2}^{4} + 21T_{2}^{2} + 9 \)
|
|
\( T_{3}^{6} + 9T_{3}^{4} + 18T_{3}^{2} + 1 \)
|
|
\( T_{7}^{6} + 21T_{7}^{4} + 78T_{7}^{2} + 49 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} + 10 T^{4} + \cdots + 9 \)
|
| $3$ |
\( T^{6} + 9 T^{4} + \cdots + 1 \)
|
| $5$ |
\( T^{6} \)
|
| $7$ |
\( T^{6} + 21 T^{4} + \cdots + 49 \)
|
| $11$ |
\( (T^{3} - 9 T^{2} + 107)^{2} \)
|
| $13$ |
\( T^{6} + 36 T^{4} + \cdots + 64 \)
|
| $17$ |
\( T^{6} + 68 T^{4} + \cdots + 576 \)
|
| $19$ |
\( (T^{3} + 6 T^{2} - 24 T - 72)^{2} \)
|
| $23$ |
\( T^{6} + 84 T^{4} + \cdots + 5184 \)
|
| $29$ |
\( (T^{3} + 2 T^{2} - 16 T - 8)^{2} \)
|
| $31$ |
\( (T^{3} - 13 T^{2} + \cdots - 41)^{2} \)
|
| $37$ |
\( T^{6} + 81 T^{4} + \cdots + 1 \)
|
| $41$ |
\( (T^{3} - 15 T^{2} + \cdots + 31)^{2} \)
|
| $43$ |
\( (T^{2} + 1)^{3} \)
|
| $47$ |
\( T^{6} + 260 T^{4} + \cdots + 5184 \)
|
| $53$ |
\( T^{6} + 56 T^{4} + \cdots + 576 \)
|
| $59$ |
\( (T^{3} + 13 T^{2} + \cdots - 579)^{2} \)
|
| $61$ |
\( (T^{3} + 10 T^{2} + \cdots - 648)^{2} \)
|
| $67$ |
\( T^{6} + 84 T^{4} + \cdots + 5184 \)
|
| $71$ |
\( (T^{3} + 6 T^{2} + \cdots - 328)^{2} \)
|
| $73$ |
\( T^{6} + 69 T^{4} + \cdots + 1681 \)
|
| $79$ |
\( (T^{3} - 17 T^{2} + \cdots + 287)^{2} \)
|
| $83$ |
\( T^{6} + 360 T^{4} + \cdots + 419904 \)
|
| $89$ |
\( (T^{3} + 8 T^{2} - 84 T + 72)^{2} \)
|
| $97$ |
\( T^{6} + 300 T^{4} + \cdots + 46656 \)
|
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