Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [3364,2,Mod(1,3364)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3364, 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("3364.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 3364 = 2^{2} \cdot 29^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 3364.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: | \(26.8616752400\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.6456289.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - x^{5} - 12x^{4} + 3x^{3} + 40x^{2} + 6x - 29 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 116) |
| 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
\( x^{6} - x^{5} - 12x^{4} + 3x^{3} + 40x^{2} + 6x - 29 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{5} - 2\nu^{4} - 10\nu^{3} + 13\nu^{2} + 23\nu - 17 ) / 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{5} - 2\nu^{4} - 10\nu^{3} + 17\nu^{2} + 19\nu - 33 ) / 4 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{5} + 3\nu^{4} + 7\nu^{3} - 18\nu^{2} - 13\nu + 22 ) / 2 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 2\nu^{5} - 5\nu^{4} - 15\nu^{3} + 27\nu^{2} + 28\nu - 29 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} - \beta_{2} + \beta _1 + 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{5} + \beta_{4} + 2\beta_{3} - 4\beta_{2} + 6\beta _1 + 3 \)
|
| \(\nu^{4}\) | \(=\) |
\( 3\beta_{5} + 5\beta_{4} + 11\beta_{3} - 13\beta_{2} + 13\beta _1 + 24 \)
|
| \(\nu^{5}\) | \(=\) |
\( 16\beta_{5} + 20\beta_{4} + 29\beta_{3} - 49\beta_{2} + 50\beta _1 + 43 \)
|
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 | −2.27029 | 0 | 1.34652 | 0 | 4.27029 | 0 | 2.15422 | 0 | ||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −1.09189 | 0 | 4.05359 | 0 | 3.09189 | 0 | −1.80777 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | 0.160015 | 0 | −1.76058 | 0 | 1.83999 | 0 | −2.97440 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 2.46835 | 0 | 3.45542 | 0 | −0.468352 | 0 | 3.09276 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 2.64685 | 0 | −0.608550 | 0 | −0.646853 | 0 | 4.00583 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 3.08697 | 0 | 3.51360 | 0 | −1.08697 | 0 | 6.52935 | 0 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(29\) | \( +1 \) |
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 | 3364.2.a.p | 6 | |
| 29.b | even | 2 | 1 | 3364.2.a.m | 6 | ||
| 29.c | odd | 4 | 2 | 3364.2.c.j | 12 | ||
| 29.e | even | 14 | 2 | 116.2.g.b | ✓ | 12 | |
| 87.h | odd | 14 | 2 | 1044.2.u.c | 12 | ||
| 116.h | odd | 14 | 2 | 464.2.u.g | 12 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 116.2.g.b | ✓ | 12 | 29.e | even | 14 | 2 | |
| 464.2.u.g | 12 | 116.h | odd | 14 | 2 | ||
| 1044.2.u.c | 12 | 87.h | odd | 14 | 2 | ||
| 3364.2.a.m | 6 | 29.b | even | 2 | 1 | ||
| 3364.2.a.p | 6 | 1.a | even | 1 | 1 | trivial | |
| 3364.2.c.j | 12 | 29.c | odd | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3364))\):
|
\( T_{3}^{6} - 5T_{3}^{5} - 2T_{3}^{4} + 35T_{3}^{3} - 18T_{3}^{2} - 48T_{3} + 8 \)
|
|
\( T_{5}^{6} - 10T_{5}^{5} + 27T_{5}^{4} + 14T_{5}^{3} - 120T_{5}^{2} + 46T_{5} + 71 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
|
| $3$ |
\( T^{6} - 5 T^{5} + \cdots + 8 \)
|
| $5$ |
\( T^{6} - 10 T^{5} + \cdots + 71 \)
|
| $7$ |
\( T^{6} - 7 T^{5} + \cdots - 8 \)
|
| $11$ |
\( T^{6} - 9 T^{5} + \cdots + 216 \)
|
| $13$ |
\( T^{6} - 14 T^{5} + \cdots - 1 \)
|
| $17$ |
\( T^{6} - 5 T^{5} + \cdots + 344 \)
|
| $19$ |
\( T^{6} - 9 T^{5} + \cdots + 16904 \)
|
| $23$ |
\( T^{6} - 15 T^{5} + \cdots + 1016 \)
|
| $29$ |
\( T^{6} \)
|
| $31$ |
\( T^{6} + 11 T^{5} + \cdots - 232 \)
|
| $37$ |
\( T^{6} + 16 T^{5} + \cdots + 12713 \)
|
| $41$ |
\( T^{6} - T^{5} + \cdots - 1624 \)
|
| $43$ |
\( T^{6} - 15 T^{5} + \cdots - 3352 \)
|
| $47$ |
\( T^{6} + 27 T^{5} + \cdots - 128584 \)
|
| $53$ |
\( (T^{3} - 9 T^{2} + 6 T + 29)^{2} \)
|
| $59$ |
\( T^{6} - 18 T^{5} + \cdots + 19264 \)
|
| $61$ |
\( T^{6} + 22 T^{5} + \cdots - 53033 \)
|
| $67$ |
\( T^{6} - 9 T^{5} + \cdots + 70904 \)
|
| $71$ |
\( T^{6} + 7 T^{5} + \cdots + 9848 \)
|
| $73$ |
\( T^{6} - 32 T^{5} + \cdots + 6119 \)
|
| $79$ |
\( T^{6} + 29 T^{5} + \cdots - 630328 \)
|
| $83$ |
\( T^{6} - 3 T^{5} + \cdots + 3688 \)
|
| $89$ |
\( T^{6} + 12 T^{5} + \cdots - 29609 \)
|
| $97$ |
\( T^{6} - 58 T^{5} + \cdots - 1348999 \)
|
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