Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [668,2,Mod(1,668)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(668, 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("668.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 668 = 2^{2} \cdot 167 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 668.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: | \(5.33400685502\) |
| Analytic rank: | \(1\) |
| 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: |
\( x^{7} - 11x^{5} - 7x^{4} + 21x^{3} + 17x^{2} - 4x - 4 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| 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
\( x^{7} - 11x^{5} - 7x^{4} + 21x^{3} + 17x^{2} - 4x - 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 7\nu^{6} - 6\nu^{5} - 73\nu^{4} + 13\nu^{3} + 149\nu^{2} + 5\nu - 50 ) / 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 9\nu^{6} - 6\nu^{5} - 95\nu^{4} - \nu^{3} + 191\nu^{2} + 35\nu - 58 ) / 4 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -13\nu^{6} + 10\nu^{5} + 135\nu^{4} - 11\nu^{3} - 263\nu^{2} - 31\nu + 70 ) / 4 \)
|
| \(\beta_{5}\) | \(=\) |
\( -6\nu^{6} + 4\nu^{5} + 63\nu^{4} - 123\nu^{2} - 18\nu + 33 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 27\nu^{6} - 18\nu^{5} - 285\nu^{4} + \nu^{3} + 565\nu^{2} + 81\nu - 154 ) / 4 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{6} - \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} + 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{6} - 2\beta_{5} + 2\beta_{4} - \beta_{3} + 2\beta_{2} + 6\beta _1 + 3 \)
|
| \(\nu^{4}\) | \(=\) |
\( -11\beta_{6} - 12\beta_{5} + 9\beta_{4} + 7\beta_{3} + 9\beta_{2} + 4\beta _1 + 29 \)
|
| \(\nu^{5}\) | \(=\) |
\( -18\beta_{6} - 29\beta_{5} + 27\beta_{4} - 3\beta_{3} + 24\beta_{2} + 48\beta _1 + 48 \)
|
| \(\nu^{6}\) | \(=\) |
\( -107\beta_{6} - 125\beta_{5} + 92\beta_{4} + 51\beta_{3} + 90\beta_{2} + 71\beta _1 + 260 \)
|
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 |
|
0 | −3.06430 | 0 | −0.836956 | 0 | 1.84506 | 0 | 6.38996 | 0 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −3.04732 | 0 | 1.35854 | 0 | −3.73540 | 0 | 6.28618 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −1.07866 | 0 | 4.03761 | 0 | −3.68648 | 0 | −1.83648 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | −0.685166 | 0 | 0.610181 | 0 | −0.184306 | 0 | −2.53055 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 0.568565 | 0 | −1.35423 | 0 | −2.86706 | 0 | −2.67673 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 0.678805 | 0 | −2.77086 | 0 | 1.71364 | 0 | −2.53922 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0 | 2.62809 | 0 | −3.04428 | 0 | −5.08545 | 0 | 3.90685 | 0 | ||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(167\) | \(-1\) |
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 | 668.2.a.c | ✓ | 7 |
| 3.b | odd | 2 | 1 | 6012.2.a.g | 7 | ||
| 4.b | odd | 2 | 1 | 2672.2.a.k | 7 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 668.2.a.c | ✓ | 7 | 1.a | even | 1 | 1 | trivial |
| 2672.2.a.k | 7 | 4.b | odd | 2 | 1 | ||
| 6012.2.a.g | 7 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{7} + 4T_{3}^{6} - 6T_{3}^{5} - 32T_{3}^{4} - 6T_{3}^{3} + 29T_{3}^{2} + 4T_{3} - 7 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(668))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{7} \)
|
| $3$ |
\( T^{7} + 4 T^{6} + \cdots - 7 \)
|
| $5$ |
\( T^{7} + 2 T^{6} + \cdots - 32 \)
|
| $7$ |
\( T^{7} + 12 T^{6} + \cdots + 117 \)
|
| $11$ |
\( T^{7} + 7 T^{6} + \cdots - 1744 \)
|
| $13$ |
\( T^{7} + 9 T^{6} + \cdots + 1184 \)
|
| $17$ |
\( T^{7} + T^{6} + \cdots - 32 \)
|
| $19$ |
\( T^{7} + 11 T^{6} + \cdots + 1996 \)
|
| $23$ |
\( T^{7} + 19 T^{6} + \cdots + 56288 \)
|
| $29$ |
\( T^{7} + 5 T^{6} + \cdots - 2237 \)
|
| $31$ |
\( T^{7} + 13 T^{6} + \cdots - 48068 \)
|
| $37$ |
\( T^{7} + 26 T^{6} + \cdots + 547488 \)
|
| $41$ |
\( T^{7} + 2 T^{6} + \cdots - 36896 \)
|
| $43$ |
\( T^{7} + 24 T^{6} + \cdots - 32 \)
|
| $47$ |
\( T^{7} + 11 T^{6} + \cdots - 482877 \)
|
| $53$ |
\( T^{7} - 4 T^{6} + \cdots - 11392 \)
|
| $59$ |
\( T^{7} + 4 T^{6} + \cdots - 4736 \)
|
| $61$ |
\( T^{7} + 5 T^{6} + \cdots + 36603 \)
|
| $67$ |
\( T^{7} + 42 T^{6} + \cdots - 162784 \)
|
| $71$ |
\( T^{7} - 9 T^{6} + \cdots + 64512 \)
|
| $73$ |
\( T^{7} - 27 T^{6} + \cdots + 71072 \)
|
| $79$ |
\( T^{7} + 8 T^{6} + \cdots - 380448 \)
|
| $83$ |
\( T^{7} - 16 T^{6} + \cdots - 147296 \)
|
| $89$ |
\( T^{7} - 9 T^{6} + \cdots - 369764 \)
|
| $97$ |
\( T^{7} + 8 T^{6} + \cdots + 1674729 \)
|
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