Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [303,3,Mod(302,303)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(303, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 1]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("303.302");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 303 = 3 \cdot 101 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 303.d (of order \(2\), degree \(1\), 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: | yes |
| Analytic conductor: | \(8.25615201012\) |
| Analytic rank: | \(0\) |
| Dimension: | \(5\) |
| Coefficient field: | 5.5.286903125.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{5} - 20x^{3} + 80x - 37 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{U}(1)[D_{2}]$ |
$q$-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
\( x^{5} - 20x^{3} + 80x - 37 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} + \nu^{2} - 12\nu - 8 ) / 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} - 2\nu^{2} - 12\nu + 16 ) / 3 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{4} - 16\nu^{2} - 5\nu + 32 ) / 3 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{3} + \beta_{2} + 8 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 2\beta_{2} + 12\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( 3\beta_{4} - 16\beta_{3} + 16\beta_{2} + 5\beta _1 + 96 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/303\mathbb{Z}\right)^\times\).
| \(n\) | \(103\) | \(203\) |
| \(\chi(n)\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | |||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 302.1 |
|
−3.92736 | 3.00000 | 11.4241 | 0 | −11.7821 | 0 | −29.1572 | 9.00000 | 0 | |||||||||||||||||||||||||||||||||
| 302.2 | −1.93534 | 3.00000 | −0.254442 | 0 | −5.80603 | 0 | 8.23381 | 9.00000 | 0 | ||||||||||||||||||||||||||||||||||
| 302.3 | −0.491895 | 3.00000 | −3.75804 | 0 | −1.47568 | 0 | 3.81614 | 9.00000 | 0 | ||||||||||||||||||||||||||||||||||
| 302.4 | 2.73125 | 3.00000 | 3.45971 | 0 | 8.19374 | 0 | −1.47567 | 9.00000 | 0 | ||||||||||||||||||||||||||||||||||
| 302.5 | 3.62335 | 3.00000 | 9.12865 | 0 | 10.8700 | 0 | 18.5829 | 9.00000 | 0 | ||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 303.d | odd | 2 | 1 | CM by \(\Q(\sqrt{-303}) \) |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 303.3.d.b | yes | 5 |
| 3.b | odd | 2 | 1 | 303.3.d.a | ✓ | 5 | |
| 101.b | even | 2 | 1 | 303.3.d.a | ✓ | 5 | |
| 303.d | odd | 2 | 1 | CM | 303.3.d.b | yes | 5 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 303.3.d.a | ✓ | 5 | 3.b | odd | 2 | 1 | |
| 303.3.d.a | ✓ | 5 | 101.b | even | 2 | 1 | |
| 303.3.d.b | yes | 5 | 1.a | even | 1 | 1 | trivial |
| 303.3.d.b | yes | 5 | 303.d | odd | 2 | 1 | CM |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{5} - 20T_{2}^{3} + 80T_{2} + 37 \)
acting on \(S_{3}^{\mathrm{new}}(303, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} - 20 T^{3} + \cdots + 37 \)
|
| $3$ |
\( (T - 3)^{5} \)
|
| $5$ |
\( T^{5} \)
|
| $7$ |
\( T^{5} \)
|
| $11$ |
\( T^{5} - 605 T^{3} + \cdots - 296246 \)
|
| $13$ |
\( T^{5} - 845 T^{3} + \cdots + 658486 \)
|
| $17$ |
\( T^{5} \)
|
| $19$ |
\( T^{5} - 1805 T^{3} + \cdots + 1157494 \)
|
| $23$ |
\( T^{5} \)
|
| $29$ |
\( T^{5} - 4205 T^{3} + \cdots - 36396902 \)
|
| $31$ |
\( T^{5} - 4805 T^{3} + \cdots - 1233602 \)
|
| $37$ |
\( T^{5} - 6845 T^{3} + \cdots + 124505158 \)
|
| $41$ |
\( T^{5} - 8405 T^{3} + \cdots + 62241634 \)
|
| $43$ |
\( T^{5} - 9245 T^{3} + \cdots + 17526502 \)
|
| $47$ |
\( T^{5} \)
|
| $53$ |
\( T^{5} - 14045 T^{3} + \cdots - 797315702 \)
|
| $59$ |
\( T^{5} - 17405 T^{3} + \cdots - 972481814 \)
|
| $61$ |
\( T^{5} \)
|
| $67$ |
\( T^{5} \)
|
| $71$ |
\( T^{5} \)
|
| $73$ |
\( T^{5} \)
|
| $79$ |
\( T^{5} + \cdots + 6140687902 \)
|
| $83$ |
\( T^{5} + \cdots + 7752032314 \)
|
| $89$ |
\( T^{5} + \cdots - 7884325694 \)
|
| $97$ |
\( T^{5} + \cdots - 1151439362 \)
|
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