Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [208,4,Mod(81,208)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(208, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 0, 2]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("208.81");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 208 = 2^{4} \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 208.i (of order \(3\), degree \(2\), not 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: | \(12.2723972812\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Relative dimension: | \(3\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | 6.0.6622206867.2 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - x^{5} + 36x^{4} - 61x^{3} + 1273x^{2} - 1680x + 2304 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{9}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | no (minimal twist has level 52) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-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
\( x^{6} - x^{5} + 36x^{4} - 61x^{3} + 1273x^{2} - 1680x + 2304 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 17\nu^{5} - 612\nu^{4} - 42\nu^{3} - 21641\nu^{2} + 28560\nu - 498384 ) / 66222 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -97\nu^{5} - 187\nu^{4} + 6732\nu^{3} - 8963\nu^{2} + 238051\nu - 314160 ) / 176592 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -105\nu^{5} + 101\nu^{4} - 3636\nu^{3} + 1221\nu^{2} - 128573\nu - 6912 ) / 176592 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -105\nu^{5} + 101\nu^{4} - 3636\nu^{3} + 1221\nu^{2} + 224611\nu - 6912 ) / 176592 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 595\nu^{5} + 654\nu^{4} + 20604\nu^{3} - 6919\nu^{2} + 668490\nu + 39168 ) / 132444 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{4} - \beta_{3} ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -6\beta_{5} - 47\beta_{3} - \beta_{2} - 6\beta _1 - 47 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( -17\beta_{4} + 17\beta_{2} - 3\beta _1 + 7 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 216\beta_{5} + 49\beta_{4} + 1583\beta_{3} ) / 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 138\beta_{5} - 1163\beta_{3} - 1189\beta_{2} + 138\beta _1 - 1163 ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/208\mathbb{Z}\right)^\times\).
| \(n\) | \(53\) | \(79\) | \(145\) |
| \(\chi(n)\) | \(1\) | \(1\) | \(\beta_{3}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 81.1 |
|
0 | −3.44428 | − | 5.96566i | 0 | 3.10208 | 0 | 8.23075 | − | 14.2561i | 0 | −10.2261 | + | 17.7121i | 0 | ||||||||||||||||||||||||||||||
| 81.2 | 0 | 1.12823 | + | 1.95415i | 0 | 8.88255 | 0 | −11.2672 | + | 19.5154i | 0 | 10.9542 | − | 18.9732i | 0 | |||||||||||||||||||||||||||||||
| 81.3 | 0 | 2.31605 | + | 4.01151i | 0 | −16.9846 | 0 | 11.0365 | − | 19.1158i | 0 | 2.77187 | − | 4.80102i | 0 | |||||||||||||||||||||||||||||||
| 113.1 | 0 | −3.44428 | + | 5.96566i | 0 | 3.10208 | 0 | 8.23075 | + | 14.2561i | 0 | −10.2261 | − | 17.7121i | 0 | |||||||||||||||||||||||||||||||
| 113.2 | 0 | 1.12823 | − | 1.95415i | 0 | 8.88255 | 0 | −11.2672 | − | 19.5154i | 0 | 10.9542 | + | 18.9732i | 0 | |||||||||||||||||||||||||||||||
| 113.3 | 0 | 2.31605 | − | 4.01151i | 0 | −16.9846 | 0 | 11.0365 | + | 19.1158i | 0 | 2.77187 | + | 4.80102i | 0 | |||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 13.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 208.4.i.f | 6 | |
| 4.b | odd | 2 | 1 | 52.4.e.a | ✓ | 6 | |
| 12.b | even | 2 | 1 | 468.4.l.c | 6 | ||
| 13.c | even | 3 | 1 | inner | 208.4.i.f | 6 | |
| 52.b | odd | 2 | 1 | 676.4.e.g | 6 | ||
| 52.f | even | 4 | 2 | 676.4.h.g | 12 | ||
| 52.i | odd | 6 | 1 | 676.4.a.f | 3 | ||
| 52.i | odd | 6 | 1 | 676.4.e.g | 6 | ||
| 52.j | odd | 6 | 1 | 52.4.e.a | ✓ | 6 | |
| 52.j | odd | 6 | 1 | 676.4.a.e | 3 | ||
| 52.l | even | 12 | 2 | 676.4.d.c | 6 | ||
| 52.l | even | 12 | 2 | 676.4.h.g | 12 | ||
| 156.p | even | 6 | 1 | 468.4.l.c | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 52.4.e.a | ✓ | 6 | 4.b | odd | 2 | 1 | |
| 52.4.e.a | ✓ | 6 | 52.j | odd | 6 | 1 | |
| 208.4.i.f | 6 | 1.a | even | 1 | 1 | trivial | |
| 208.4.i.f | 6 | 13.c | even | 3 | 1 | inner | |
| 468.4.l.c | 6 | 12.b | even | 2 | 1 | ||
| 468.4.l.c | 6 | 156.p | even | 6 | 1 | ||
| 676.4.a.e | 3 | 52.j | odd | 6 | 1 | ||
| 676.4.a.f | 3 | 52.i | odd | 6 | 1 | ||
| 676.4.d.c | 6 | 52.l | even | 12 | 2 | ||
| 676.4.e.g | 6 | 52.b | odd | 2 | 1 | ||
| 676.4.e.g | 6 | 52.i | odd | 6 | 1 | ||
| 676.4.h.g | 12 | 52.f | even | 4 | 2 | ||
| 676.4.h.g | 12 | 52.l | even | 12 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{6} + 37T_{3}^{4} - 144T_{3}^{3} + 1369T_{3}^{2} - 2664T_{3} + 5184 \)
acting on \(S_{4}^{\mathrm{new}}(208, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
|
| $3$ |
\( T^{6} + 37 T^{4} + \cdots + 5184 \)
|
| $5$ |
\( (T^{3} + 5 T^{2} + \cdots + 468)^{2} \)
|
| $7$ |
\( T^{6} - 16 T^{5} + \cdots + 67043344 \)
|
| $11$ |
\( T^{6} - 12 T^{5} + \cdots + 159668496 \)
|
| $13$ |
\( T^{6} + \cdots + 10604499373 \)
|
| $17$ |
\( T^{6} + \cdots + 18938989161 \)
|
| $19$ |
\( T^{6} + \cdots + 9183772224 \)
|
| $23$ |
\( T^{6} + \cdots + 7862014224 \)
|
| $29$ |
\( T^{6} + \cdots + 19953258146649 \)
|
| $31$ |
\( (T^{3} + 164 T^{2} + \cdots + 1678336)^{2} \)
|
| $37$ |
\( T^{6} + \cdots + 10551849186321 \)
|
| $41$ |
\( T^{6} + \cdots + 1993492671921 \)
|
| $43$ |
\( T^{6} + \cdots + 319975678482496 \)
|
| $47$ |
\( (T^{3} - 552 T^{2} + \cdots - 893952)^{2} \)
|
| $53$ |
\( (T^{3} + 1069 T^{2} + \cdots + 20613204)^{2} \)
|
| $59$ |
\( T^{6} + \cdots + 58\!\cdots\!16 \)
|
| $61$ |
\( T^{6} + \cdots + 25\!\cdots\!69 \)
|
| $67$ |
\( T^{6} + \cdots + 11\!\cdots\!96 \)
|
| $71$ |
\( T^{6} + \cdots + 23\!\cdots\!84 \)
|
| $73$ |
\( (T^{3} + 1857 T^{2} + \cdots + 59964292)^{2} \)
|
| $79$ |
\( (T^{3} - 1668 T^{2} + \cdots + 345851648)^{2} \)
|
| $83$ |
\( (T^{3} + 312 T^{2} + \cdots - 776293632)^{2} \)
|
| $89$ |
\( T^{6} + \cdots + 809953044186276 \)
|
| $97$ |
\( T^{6} + \cdots + 44\!\cdots\!84 \)
|
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