Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [3332,2,Mod(2549,3332)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3332, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("3332.2549");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 3332 = 2^{2} \cdot 7^{2} \cdot 17 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 3332.b (of order \(2\), degree \(1\), minimal) |
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: | no |
| Analytic conductor: | \(26.6061539535\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | 8.0.980441344.2 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} + 8x^{6} + 18x^{4} + 9x^{2} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{17}]\) |
| Coefficient ring index: | \( 2^{4} \) |
| Twist minimal: | no (minimal twist has level 476) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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
\( x^{8} + 8x^{6} + 18x^{4} + 9x^{2} + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( -\nu^{4} - 3\nu^{2} + 1 \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{4} + 5\nu^{2} + 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{5} + 6\nu^{3} + 8\nu \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{7} + 8\nu^{5} + 17\nu^{3} + 5\nu \)
|
| \(\beta_{5}\) | \(=\) |
\( 2\nu^{6} + 15\nu^{4} + 29\nu^{2} + 7 \)
|
| \(\beta_{6}\) | \(=\) |
\( -2\nu^{7} - 15\nu^{5} - 30\nu^{3} - 8\nu \)
|
| \(\beta_{7}\) | \(=\) |
\( 2\nu^{7} + 15\nu^{5} + 30\nu^{3} + 10\nu \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{7} + \beta_{6} ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{2} + \beta _1 - 4 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -3\beta_{7} - 4\beta_{6} - 2\beta_{4} + \beta_{3} ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -3\beta_{2} - 5\beta _1 + 14 ) / 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( 5\beta_{7} + 8\beta_{6} + 6\beta_{4} - 2\beta_{3} \)
|
| \(\nu^{6}\) | \(=\) |
\( ( \beta_{5} + 8\beta_{2} + 23\beta _1 - 54 ) / 2 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( -34\beta_{7} - 65\beta_{6} - 60\beta_{4} + 15\beta_{3} ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3332\mathbb{Z}\right)^\times\).
| \(n\) | \(785\) | \(885\) | \(1667\) |
| \(\chi(n)\) | \(-1\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 2549.1 |
|
0 | − | 3.23925i | 0 | − | 2.80694i | 0 | 0 | 0 | −7.49277 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 2549.2 | 0 | − | 1.87576i | 0 | 1.74199i | 0 | 0 | 0 | −0.518489 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 2549.3 | 0 | − | 1.82479i | 0 | 3.70737i | 0 | 0 | 0 | −0.329851 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 2549.4 | 0 | − | 0.811721i | 0 | − | 1.15845i | 0 | 0 | 0 | 2.34111 | 0 | |||||||||||||||||||||||||||||||||||||||||
| 2549.5 | 0 | 0.811721i | 0 | 1.15845i | 0 | 0 | 0 | 2.34111 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 2549.6 | 0 | 1.82479i | 0 | − | 3.70737i | 0 | 0 | 0 | −0.329851 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 2549.7 | 0 | 1.87576i | 0 | − | 1.74199i | 0 | 0 | 0 | −0.518489 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 2549.8 | 0 | 3.23925i | 0 | 2.80694i | 0 | 0 | 0 | −7.49277 | 0 | |||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 17.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 3332.2.b.b | 8 | |
| 7.b | odd | 2 | 1 | 476.2.b.a | ✓ | 8 | |
| 17.b | even | 2 | 1 | inner | 3332.2.b.b | 8 | |
| 21.c | even | 2 | 1 | 4284.2.d.e | 8 | ||
| 28.d | even | 2 | 1 | 1904.2.c.f | 8 | ||
| 119.d | odd | 2 | 1 | 476.2.b.a | ✓ | 8 | |
| 119.f | odd | 4 | 1 | 8092.2.a.q | 4 | ||
| 119.f | odd | 4 | 1 | 8092.2.a.r | 4 | ||
| 357.c | even | 2 | 1 | 4284.2.d.e | 8 | ||
| 476.e | even | 2 | 1 | 1904.2.c.f | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 476.2.b.a | ✓ | 8 | 7.b | odd | 2 | 1 | |
| 476.2.b.a | ✓ | 8 | 119.d | odd | 2 | 1 | |
| 1904.2.c.f | 8 | 28.d | even | 2 | 1 | ||
| 1904.2.c.f | 8 | 476.e | even | 2 | 1 | ||
| 3332.2.b.b | 8 | 1.a | even | 1 | 1 | trivial | |
| 3332.2.b.b | 8 | 17.b | even | 2 | 1 | inner | |
| 4284.2.d.e | 8 | 21.c | even | 2 | 1 | ||
| 4284.2.d.e | 8 | 357.c | even | 2 | 1 | ||
| 8092.2.a.q | 4 | 119.f | odd | 4 | 1 | ||
| 8092.2.a.r | 4 | 119.f | odd | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(3332, [\chi])\):
|
\( T_{3}^{8} + 18T_{3}^{6} + 95T_{3}^{4} + 178T_{3}^{2} + 81 \)
|
|
\( T_{13}^{4} + 10T_{13}^{3} + 20T_{13}^{2} - 24T_{13} - 16 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} \)
|
| $3$ |
\( T^{8} + 18 T^{6} + \cdots + 81 \)
|
| $5$ |
\( T^{8} + 26 T^{6} + \cdots + 441 \)
|
| $7$ |
\( T^{8} \)
|
| $11$ |
\( T^{8} + 48 T^{6} + \cdots + 2304 \)
|
| $13$ |
\( (T^{4} + 10 T^{3} + \cdots - 16)^{2} \)
|
| $17$ |
\( T^{8} + 4 T^{6} + \cdots + 83521 \)
|
| $19$ |
\( (T^{4} + 6 T^{3} - 8 T^{2} + \cdots - 16)^{2} \)
|
| $23$ |
\( T^{8} + 124 T^{6} + \cdots + 256 \)
|
| $29$ |
\( T^{8} + 64 T^{6} + \cdots + 256 \)
|
| $31$ |
\( T^{8} + 34 T^{6} + \cdots + 1 \)
|
| $37$ |
\( T^{8} + 60 T^{6} + \cdots + 256 \)
|
| $41$ |
\( T^{8} + 138 T^{6} + \cdots + 196249 \)
|
| $43$ |
\( (T^{4} + 6 T^{3} + \cdots + 2429)^{2} \)
|
| $47$ |
\( (T^{4} + 12 T^{3} + \cdots - 3664)^{2} \)
|
| $53$ |
\( (T^{4} - 10 T^{3} + \cdots + 557)^{2} \)
|
| $59$ |
\( (T^{4} - 12 T^{3} + \cdots - 1104)^{2} \)
|
| $61$ |
\( T^{8} + 178 T^{6} + \cdots + 47961 \)
|
| $67$ |
\( (T^{4} + 2 T^{3} + \cdots - 787)^{2} \)
|
| $71$ |
\( T^{8} + 244 T^{6} + \cdots + 256 \)
|
| $73$ |
\( T^{8} + 514 T^{6} + \cdots + 154828249 \)
|
| $79$ |
\( T^{8} + 480 T^{6} + \cdots + 16128256 \)
|
| $83$ |
\( (T^{4} + 2 T^{3} + \cdots - 368)^{2} \)
|
| $89$ |
\( (T^{4} - 240 T^{2} + \cdots - 1776)^{2} \)
|
| $97$ |
\( T^{8} + 242 T^{6} + \cdots + 978121 \)
|
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