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: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.152089744.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 3x^{5} - 7x^{4} + 17x^{3} + 16x^{2} - 18x - 2 \)
|
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - 3x^{5} - 7x^{4} + 17x^{3} + 16x^{2} - 18x - 2 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{5} - 4\nu^{4} - \nu^{3} + 12\nu^{2} - 4\nu + 2 ) / 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{4} - 3\nu^{3} - 3\nu^{2} + 8\nu \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{5} - 4\nu^{4} - \nu^{3} + 14\nu^{2} - 6\nu - 4 ) / 2 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -\nu^{5} + 4\nu^{4} + 3\nu^{3} - 16\nu^{2} - 6\nu + 4 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{4} - \beta_{2} + \beta _1 + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{5} + 2\beta_{4} - \beta_{2} + 7\beta _1 + 3 \)
|
| \(\nu^{4}\) | \(=\) |
\( 3\beta_{5} + 9\beta_{4} + \beta_{3} - 6\beta_{2} + 16\beta _1 + 18 \)
|
| \(\nu^{5}\) | \(=\) |
\( 13\beta_{5} + 26\beta_{4} + 4\beta_{3} - 11\beta_{2} + 63\beta _1 + 37 \)
|
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 | −2.23909 | 1.00000 | 2.39864 | −2.23909 | 2.85398 | 1.00000 | 2.01353 | 2.39864 | ||||||||||||||||||||||||||||||||||||
| 1.2 | 1.00000 | −1.43539 | 1.00000 | 4.02642 | −1.43539 | −2.53070 | 1.00000 | −0.939663 | 4.02642 | |||||||||||||||||||||||||||||||||||||
| 1.3 | 1.00000 | 0.153901 | 1.00000 | −1.49208 | 0.153901 | −0.638137 | 1.00000 | −2.97631 | −1.49208 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 1.00000 | 1.10279 | 1.00000 | −0.269279 | 1.10279 | −1.61737 | 1.00000 | −1.78386 | −0.269279 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 1.00000 | 2.63787 | 1.00000 | −1.28201 | 2.63787 | 3.60250 | 1.00000 | 3.95837 | −1.28201 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 1.00000 | 2.77992 | 1.00000 | 3.61831 | 2.77992 | −0.670277 | 1.00000 | 4.72795 | 3.61831 | |||||||||||||||||||||||||||||||||||||
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.ba | 6 | |
| 13.b | even | 2 | 1 | 7774.2.a.x | 6 | ||
| 13.d | odd | 4 | 2 | 598.2.c.c | ✓ | 12 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 598.2.c.c | ✓ | 12 | 13.d | odd | 4 | 2 | |
| 7774.2.a.x | 6 | 13.b | even | 2 | 1 | ||
| 7774.2.a.ba | 6 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7774))\):
|
\( T_{3}^{6} - 3T_{3}^{5} - 7T_{3}^{4} + 21T_{3}^{3} + 10T_{3}^{2} - 28T_{3} + 4 \)
|
|
\( T_{5}^{6} - 7T_{5}^{5} + 5T_{5}^{4} + 39T_{5}^{3} - 24T_{5}^{2} - 76T_{5} - 18 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 1)^{6} \)
|
| $3$ |
\( T^{6} - 3 T^{5} + \cdots + 4 \)
|
| $5$ |
\( T^{6} - 7 T^{5} + \cdots - 18 \)
|
| $7$ |
\( T^{6} - T^{5} + \cdots + 18 \)
|
| $11$ |
\( T^{6} - 6 T^{5} + \cdots + 1392 \)
|
| $13$ |
\( T^{6} \)
|
| $17$ |
\( T^{6} + T^{5} + \cdots - 408 \)
|
| $19$ |
\( T^{6} - 10 T^{5} + \cdots - 12 \)
|
| $23$ |
\( (T - 1)^{6} \)
|
| $29$ |
\( T^{6} + 2 T^{5} + \cdots - 3528 \)
|
| $31$ |
\( T^{6} - 4 T^{5} + \cdots + 576 \)
|
| $37$ |
\( T^{6} - 15 T^{5} + \cdots - 8178 \)
|
| $41$ |
\( T^{6} - 8 T^{5} + \cdots - 112272 \)
|
| $43$ |
\( T^{6} - 17 T^{5} + \cdots - 4148 \)
|
| $47$ |
\( T^{6} - 19 T^{5} + \cdots - 2712 \)
|
| $53$ |
\( T^{6} - 2 T^{5} + \cdots - 27048 \)
|
| $59$ |
\( T^{6} + 4 T^{5} + \cdots - 13008 \)
|
| $61$ |
\( T^{6} - 26 T^{5} + \cdots - 62528 \)
|
| $67$ |
\( T^{6} + 6 T^{5} + \cdots + 4608 \)
|
| $71$ |
\( T^{6} - 17 T^{5} + \cdots - 4968 \)
|
| $73$ |
\( T^{6} - 20 T^{5} + \cdots - 118704 \)
|
| $79$ |
\( T^{6} - 4 T^{5} + \cdots - 25600 \)
|
| $83$ |
\( T^{6} - 16 T^{5} + \cdots - 1572 \)
|
| $89$ |
\( T^{6} - 34 T^{5} + \cdots - 172320 \)
|
| $97$ |
\( T^{6} - 30 T^{5} + \cdots + 3108972 \)
|
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