Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [3549,2,Mod(1,3549)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3549, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 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("3549.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 3549 = 3 \cdot 7 \cdot 13^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 3549.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: | \(28.3389076774\) |
| Analytic rank: | \(1\) |
| Dimension: | \(8\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{8} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - 2x^{7} - 13x^{6} + 22x^{5} + 57x^{4} - 72x^{3} - 96x^{2} + 64x + 61 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 273) |
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} - 2x^{7} - 13x^{6} + 22x^{5} + 57x^{4} - 72x^{3} - 96x^{2} + 64x + 61 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -2\nu^{7} + 3\nu^{6} + 21\nu^{5} - 27\nu^{4} - 56\nu^{3} + 64\nu^{2} + 16\nu - 29 ) / 13 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{4} - \nu^{3} - 7\nu^{2} + 4\nu + 8 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 2\nu^{7} - 3\nu^{6} - 21\nu^{5} + 27\nu^{4} + 69\nu^{3} - 64\nu^{2} - 81\nu + 16 ) / 13 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -2\nu^{7} + 3\nu^{6} + 21\nu^{5} - 27\nu^{4} - 69\nu^{3} + 77\nu^{2} + 68\nu - 55 ) / 13 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -2\nu^{7} + 3\nu^{6} + 34\nu^{5} - 40\nu^{4} - 160\nu^{3} + 129\nu^{2} + 172\nu - 55 ) / 13 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( \nu^{7} + 5\nu^{6} - 17\nu^{5} - 58\nu^{4} + 80\nu^{3} + 202\nu^{2} - 112\nu - 200 ) / 13 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{5} + \beta_{4} + \beta _1 + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{4} + \beta_{2} + 5\beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( 7\beta_{5} + 8\beta_{4} + \beta_{3} + \beta_{2} + 8\beta _1 + 14 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{6} + 2\beta_{5} + 11\beta_{4} + \beta_{3} + 8\beta_{2} + 31\beta _1 + 9 \)
|
| \(\nu^{6}\) | \(=\) |
\( 2\beta_{7} + \beta_{6} + 43\beta_{5} + 55\beta_{4} + 12\beta_{3} + 12\beta_{2} + 59\beta _1 + 80 \)
|
| \(\nu^{7}\) | \(=\) |
\( 3\beta_{7} + 12\beta_{6} + 23\beta_{5} + 94\beta_{4} + 15\beta_{3} + 54\beta_{2} + 206\beta _1 + 79 \)
|
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 |
|
−2.72116 | −1.00000 | 5.40472 | −1.58278 | 2.72116 | 1.00000 | −9.26480 | 1.00000 | 4.30699 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −2.54225 | −1.00000 | 4.46304 | 4.07309 | 2.54225 | 1.00000 | −6.26168 | 1.00000 | −10.3548 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.67113 | −1.00000 | 0.792659 | −2.68351 | 1.67113 | 1.00000 | 2.01762 | 1.00000 | 4.48449 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.32829 | −1.00000 | −0.235645 | −1.55828 | 1.32829 | 1.00000 | 2.96959 | 1.00000 | 2.06984 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0.756173 | −1.00000 | −1.42820 | −4.01537 | −0.756173 | 1.00000 | −2.59231 | 1.00000 | −3.03631 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 1.06752 | −1.00000 | −0.860399 | 0.994065 | −1.06752 | 1.00000 | −3.05354 | 1.00000 | 1.06118 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 2.14586 | −1.00000 | 2.60470 | −1.91954 | −2.14586 | 1.00000 | 1.29759 | 1.00000 | −4.11906 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 2.29328 | −1.00000 | 3.25913 | 0.692320 | −2.29328 | 1.00000 | 2.88753 | 1.00000 | 1.58768 | |||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(1\) |
| \(7\) | \(-1\) |
| \(13\) | \(-1\) |
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 | 3549.2.a.ba | 8 | |
| 13.b | even | 2 | 1 | 3549.2.a.bc | 8 | ||
| 13.f | odd | 12 | 2 | 273.2.bd.b | ✓ | 16 | |
| 39.k | even | 12 | 2 | 819.2.ct.c | 16 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 273.2.bd.b | ✓ | 16 | 13.f | odd | 12 | 2 | |
| 819.2.ct.c | 16 | 39.k | even | 12 | 2 | ||
| 3549.2.a.ba | 8 | 1.a | even | 1 | 1 | trivial | |
| 3549.2.a.bc | 8 | 13.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3549))\):
|
\( T_{2}^{8} + 2T_{2}^{7} - 13T_{2}^{6} - 22T_{2}^{5} + 57T_{2}^{4} + 72T_{2}^{3} - 96T_{2}^{2} - 64T_{2} + 61 \)
|
|
\( T_{5}^{8} + 6T_{5}^{7} - 7T_{5}^{6} - 104T_{5}^{5} - 177T_{5}^{4} + 70T_{5}^{3} + 312T_{5}^{2} + 40T_{5} - 143 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} + 2 T^{7} + \cdots + 61 \)
|
| $3$ |
\( (T + 1)^{8} \)
|
| $5$ |
\( T^{8} + 6 T^{7} + \cdots - 143 \)
|
| $7$ |
\( (T - 1)^{8} \)
|
| $11$ |
\( T^{8} + 16 T^{7} + \cdots - 764 \)
|
| $13$ |
\( T^{8} \)
|
| $17$ |
\( T^{8} + 2 T^{7} + \cdots + 15433 \)
|
| $19$ |
\( T^{8} + 14 T^{7} + \cdots - 3068 \)
|
| $23$ |
\( T^{8} + 6 T^{7} + \cdots - 4784 \)
|
| $29$ |
\( T^{8} - 12 T^{7} + \cdots - 285503 \)
|
| $31$ |
\( T^{8} + 16 T^{7} + \cdots + 39232 \)
|
| $37$ |
\( T^{8} - 8 T^{7} + \cdots - 472811 \)
|
| $41$ |
\( T^{8} - 4 T^{7} + \cdots - 85184 \)
|
| $43$ |
\( T^{8} - 14 T^{7} + \cdots + 774292 \)
|
| $47$ |
\( T^{8} + 6 T^{7} + \cdots - 41036 \)
|
| $53$ |
\( T^{8} - 14 T^{7} + \cdots - 8807 \)
|
| $59$ |
\( T^{8} + 50 T^{7} + \cdots - 1521692 \)
|
| $61$ |
\( T^{8} + 2 T^{7} + \cdots + 338917 \)
|
| $67$ |
\( T^{8} - 16 T^{7} + \cdots - 345392 \)
|
| $71$ |
\( T^{8} + 34 T^{7} + \cdots + 25924 \)
|
| $73$ |
\( T^{8} - 8 T^{7} + \cdots + 97837 \)
|
| $79$ |
\( T^{8} - 46 T^{7} + \cdots - 7916 \)
|
| $83$ |
\( T^{8} + 42 T^{7} + \cdots - 1196 \)
|
| $89$ |
\( T^{8} + 28 T^{7} + \cdots - 5457824 \)
|
| $97$ |
\( T^{8} + 16 T^{7} + \cdots + 208 \)
|
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