Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [301,2,Mod(1,301)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(301, 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("301.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 301 = 7 \cdot 43 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 301.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: | \(2.40349710084\) |
| Analytic rank: | \(0\) |
| Dimension: | \(7\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{7} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{7} - 2x^{6} - 8x^{5} + 13x^{4} + 11x^{3} - 15x^{2} - 5x + 4 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | yes |
| 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 \(1,\beta_1,\ldots,\beta_{6}\) 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
\( x^{7} - 2x^{6} - 8x^{5} + 13x^{4} + 11x^{3} - 15x^{2} - 5x + 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu^{6} - 2\nu^{5} - 7\nu^{4} + 11\nu^{3} + 4\nu^{2} - 5\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{5} - 2\nu^{4} - 7\nu^{3} + 11\nu^{2} + 4\nu - 5 \)
|
| \(\beta_{3}\) | \(=\) |
\( 2\nu^{6} - 4\nu^{5} - 14\nu^{4} + 23\nu^{3} + 7\nu^{2} - 15\nu + 2 \)
|
| \(\beta_{4}\) | \(=\) |
\( -\nu^{6} + 4\nu^{5} + 4\nu^{4} - 27\nu^{3} + 12\nu^{2} + 22\nu - 9 \)
|
| \(\beta_{5}\) | \(=\) |
\( 2\nu^{6} - 5\nu^{5} - 12\nu^{4} + 30\nu^{3} - 5\nu^{2} - 18\nu + 10 \)
|
| \(\beta_{6}\) | \(=\) |
\( -2\nu^{6} + 6\nu^{5} + 11\nu^{4} - 38\nu^{3} + 8\nu^{2} + 25\nu - 9 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{6} + \beta_{4} - \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -\beta_{6} - 2\beta_{5} + \beta_{4} + 2\beta_{3} - 2\beta_{2} - \beta _1 + 6 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( -3\beta_{6} - \beta_{5} + 3\beta_{4} + 2\beta_{3} - \beta_{2} - 5\beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -9\beta_{6} - 16\beta_{5} + 11\beta_{4} + 20\beta_{3} - 20\beta_{2} - 15\beta _1 + 38 ) / 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -45\beta_{6} - 24\beta_{5} + 49\beta_{4} + 46\beta_{3} - 30\beta_{2} - 85\beta _1 + 34 ) / 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( -44\beta_{6} - 65\beta_{5} + 55\beta_{4} + 90\beta_{3} - 85\beta_{2} - 82\beta _1 + 144 \)
|
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 |
|
−2.26642 | −0.859305 | 3.13666 | 2.90597 | 1.94755 | 1.00000 | −2.57616 | −2.26159 | −6.58615 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | −0.931333 | 0.912051 | −1.13262 | −2.28866 | −0.849423 | 1.00000 | 2.91751 | −2.16816 | 2.13150 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | −0.143321 | 3.02374 | −1.97946 | 0.907302 | −0.433364 | 1.00000 | 0.570338 | 6.14298 | −0.130035 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0.628471 | −3.20707 | −1.60502 | −0.530891 | −2.01555 | 1.00000 | −2.26565 | 7.28530 | −0.333650 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | 1.62795 | 1.51574 | 0.650213 | 1.89579 | 2.46754 | 1.00000 | −2.19738 | −0.702533 | 3.08625 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | 2.50133 | 1.48520 | 4.25665 | −3.61774 | 3.71497 | 1.00000 | 5.64461 | −0.794188 | −9.04915 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | 2.58333 | −1.87035 | 4.67358 | 0.728222 | −4.83172 | 1.00000 | 6.90673 | 0.498203 | 1.88124 | ||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(7\) | \(-1\) |
| \(43\) | \(1\) |
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 | 301.2.a.d | ✓ | 7 |
| 3.b | odd | 2 | 1 | 2709.2.a.o | 7 | ||
| 4.b | odd | 2 | 1 | 4816.2.a.x | 7 | ||
| 5.b | even | 2 | 1 | 7525.2.a.h | 7 | ||
| 7.b | odd | 2 | 1 | 2107.2.a.h | 7 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 301.2.a.d | ✓ | 7 | 1.a | even | 1 | 1 | trivial |
| 2107.2.a.h | 7 | 7.b | odd | 2 | 1 | ||
| 2709.2.a.o | 7 | 3.b | odd | 2 | 1 | ||
| 4816.2.a.x | 7 | 4.b | odd | 2 | 1 | ||
| 7525.2.a.h | 7 | 5.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{7} - 4T_{2}^{6} - 3T_{2}^{5} + 25T_{2}^{4} - 13T_{2}^{3} - 23T_{2}^{2} + 11T_{2} + 2 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(301))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{7} - 4 T^{6} + \cdots + 2 \)
|
| $3$ |
\( T^{7} - T^{6} + \cdots + 32 \)
|
| $5$ |
\( T^{7} - 16 T^{5} + \cdots + 16 \)
|
| $7$ |
\( (T - 1)^{7} \)
|
| $11$ |
\( T^{7} - 16 T^{6} + \cdots - 688 \)
|
| $13$ |
\( T^{7} + 2 T^{6} + \cdots + 52 \)
|
| $17$ |
\( T^{7} - 4 T^{6} + \cdots - 292 \)
|
| $19$ |
\( T^{7} - 48 T^{5} + \cdots - 32 \)
|
| $23$ |
\( T^{7} - 6 T^{6} + \cdots + 22336 \)
|
| $29$ |
\( T^{7} - 12 T^{6} + \cdots + 162968 \)
|
| $31$ |
\( T^{7} - 8 T^{6} + \cdots + 2708 \)
|
| $37$ |
\( T^{7} + 7 T^{6} + \cdots + 38336 \)
|
| $41$ |
\( T^{7} - 14 T^{6} + \cdots - 136684 \)
|
| $43$ |
\( (T + 1)^{7} \)
|
| $47$ |
\( T^{7} - T^{6} + \cdots + 2416 \)
|
| $53$ |
\( T^{7} - 15 T^{6} + \cdots + 4748 \)
|
| $59$ |
\( T^{7} - 45 T^{6} + \cdots - 10832 \)
|
| $61$ |
\( T^{7} + 20 T^{6} + \cdots + 236648 \)
|
| $67$ |
\( T^{7} - 7 T^{6} + \cdots + 15152 \)
|
| $71$ |
\( T^{7} - 13 T^{6} + \cdots + 224768 \)
|
| $73$ |
\( T^{7} + 19 T^{6} + \cdots - 107608 \)
|
| $79$ |
\( T^{7} - 12 T^{6} + \cdots + 5237888 \)
|
| $83$ |
\( T^{7} + 6 T^{6} + \cdots - 40636 \)
|
| $89$ |
\( T^{7} + 17 T^{6} + \cdots - 95888 \)
|
| $97$ |
\( T^{7} + 15 T^{6} + \cdots - 636788 \)
|
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