Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [7774,2,Mod(1,7774)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7774, 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("7774.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 7774 = 2 \cdot 13^{2} \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 7774.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: | \(62.0757025313\) |
| 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: |
\( x^{7} - 2x^{6} - 12x^{5} + 18x^{4} + 39x^{3} - 42x^{2} - 27x + 18 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 598) |
| 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
\( x^{7} - 2x^{6} - 12x^{5} + 18x^{4} + 39x^{3} - 42x^{2} - 27x + 18 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{5} - 2\nu^{4} - 9\nu^{3} + 9\nu^{2} + 18\nu ) / 6 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{5} - 2\nu^{4} - 9\nu^{3} + 15\nu^{2} + 12\nu - 24 ) / 6 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{6} - \nu^{5} - 11\nu^{4} + 33\nu^{2} + 24\nu - 24 ) / 6 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{6} - \nu^{5} - 14\nu^{4} + 6\nu^{3} + 54\nu^{2} - 3\nu - 42 ) / 6 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 2\nu^{6} - 3\nu^{5} - 23\nu^{4} + 21\nu^{3} + 66\nu^{2} - 33\nu - 30 ) / 6 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} - \beta_{2} + \beta _1 + 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{6} - \beta_{5} - \beta_{4} + 2\beta_{3} - \beta_{2} + 8\beta _1 + 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( 2\beta_{6} - 4\beta_{5} + 11\beta_{3} - 9\beta_{2} + 14\beta _1 + 26 \)
|
| \(\nu^{5}\) | \(=\) |
\( 13\beta_{6} - 17\beta_{5} - 9\beta_{4} + 31\beta_{3} - 12\beta_{2} + 73\beta _1 + 34 \)
|
| \(\nu^{6}\) | \(=\) |
\( 35\beta_{6} - 61\beta_{5} - 3\beta_{4} + 119\beta_{3} - 78\beta_{2} + 170\beta _1 + 212 \)
|
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 |
|
1.00000 | −3.31190 | 1.00000 | −2.86091 | −3.31190 | 4.18188 | 1.00000 | 7.96867 | −2.86091 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | 1.00000 | −2.16620 | 1.00000 | 1.57371 | −2.16620 | −3.71435 | 1.00000 | 1.69242 | 1.57371 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | 1.00000 | −1.27077 | 1.00000 | −0.183572 | −1.27077 | 0.278618 | 1.00000 | −1.38514 | −0.183572 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | 1.00000 | −0.488639 | 1.00000 | 1.61517 | −0.488639 | 4.62230 | 1.00000 | −2.76123 | 1.61517 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | 1.00000 | 0.854881 | 1.00000 | 2.19969 | 0.854881 | −3.11496 | 1.00000 | −2.26918 | 2.19969 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | 1.00000 | 1.91675 | 1.00000 | −4.10312 | 1.91675 | 1.63489 | 1.00000 | 0.673941 | −4.10312 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | 1.00000 | 2.46587 | 1.00000 | −2.24097 | 2.46587 | −4.88839 | 1.00000 | 3.08053 | −2.24097 | ||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(13\) | \( +1 \) |
| \(23\) | \( -1 \) |
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 | 7774.2.a.bd | 7 | |
| 13.b | even | 2 | 1 | 7774.2.a.bb | 7 | ||
| 13.e | even | 6 | 2 | 598.2.e.d | ✓ | 14 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 598.2.e.d | ✓ | 14 | 13.e | even | 6 | 2 | |
| 7774.2.a.bb | 7 | 13.b | even | 2 | 1 | ||
| 7774.2.a.bd | 7 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(7774))\):
|
\( T_{3}^{7} + 2T_{3}^{6} - 12T_{3}^{5} - 18T_{3}^{4} + 39T_{3}^{3} + 42T_{3}^{2} - 27T_{3} - 18 \)
|
|
\( T_{5}^{7} + 4T_{5}^{6} - 12T_{5}^{5} - 41T_{5}^{4} + 61T_{5}^{3} + 111T_{5}^{2} - 129T_{5} - 27 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 1)^{7} \)
|
| $3$ |
\( T^{7} + 2 T^{6} + \cdots - 18 \)
|
| $5$ |
\( T^{7} + 4 T^{6} + \cdots - 27 \)
|
| $7$ |
\( T^{7} + T^{6} + \cdots + 498 \)
|
| $11$ |
\( T^{7} + 4 T^{6} + \cdots + 408 \)
|
| $13$ |
\( T^{7} \)
|
| $17$ |
\( T^{7} + 3 T^{6} + \cdots - 1677 \)
|
| $19$ |
\( T^{7} + 9 T^{6} + \cdots + 14448 \)
|
| $23$ |
\( (T - 1)^{7} \)
|
| $29$ |
\( T^{7} + 10 T^{6} + \cdots - 35991 \)
|
| $31$ |
\( T^{7} + 9 T^{6} + \cdots - 516 \)
|
| $37$ |
\( T^{7} - 4 T^{6} + \cdots - 201541 \)
|
| $41$ |
\( T^{7} + 20 T^{6} + \cdots + 1186497 \)
|
| $43$ |
\( T^{7} + 6 T^{6} + \cdots + 161808 \)
|
| $47$ |
\( T^{7} + 35 T^{6} + \cdots - 10116 \)
|
| $53$ |
\( T^{7} - 17 T^{6} + \cdots - 1093797 \)
|
| $59$ |
\( T^{7} + 25 T^{6} + \cdots + 962604 \)
|
| $61$ |
\( T^{7} + 15 T^{6} + \cdots - 3700287 \)
|
| $67$ |
\( T^{7} + 6 T^{6} + \cdots - 292738 \)
|
| $71$ |
\( T^{7} + 27 T^{6} + \cdots + 75978 \)
|
| $73$ |
\( T^{7} + 9 T^{6} + \cdots + 2859 \)
|
| $79$ |
\( T^{7} + 5 T^{6} + \cdots + 516 \)
|
| $83$ |
\( T^{7} + 16 T^{6} + \cdots + 536046 \)
|
| $89$ |
\( T^{7} + 25 T^{6} + \cdots + 4255722 \)
|
| $97$ |
\( T^{7} - 20 T^{6} + \cdots + 2094 \)
|
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