Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [208,3,Mod(79,208)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(208, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 0, 0]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("208.79");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 208 = 2^{4} \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 208.d (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: | \(5.66758949869\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | 8.0.29540328129.2 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - x^{7} - 6x^{6} - 23x^{5} + 45x^{4} + 80x^{3} - 20x^{2} + 96x + 144 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{10}\cdot 3 \) |
| Twist minimal: | yes |
| 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
\( x^{8} - x^{7} - 6x^{6} - 23x^{5} + 45x^{4} + 80x^{3} - 20x^{2} + 96x + 144 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -49\nu^{7} + 17\nu^{6} + 154\nu^{5} + 875\nu^{4} + 1187\nu^{3} + 532\nu^{2} + 1680\nu - 33928 ) / 9872 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -411\nu^{7} + 747\nu^{6} + 2702\nu^{5} + 5929\nu^{4} - 24495\nu^{3} - 36436\nu^{2} + 46528\nu - 55912 ) / 19744 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 2225 \nu^{7} - 2837 \nu^{6} - 15102 \nu^{5} - 43963 \nu^{4} + 127977 \nu^{3} + 201136 \nu^{2} + \cdots + 67464 ) / 59232 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 5537 \nu^{7} - 6857 \nu^{6} - 22338 \nu^{5} - 128491 \nu^{4} + 255813 \nu^{3} + 221236 \nu^{2} + \cdots + 507000 ) / 59232 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 10079 \nu^{7} - 23543 \nu^{6} - 32382 \nu^{5} - 184213 \nu^{4} + 710907 \nu^{3} - 69236 \nu^{2} + \cdots + 1262088 ) / 59232 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 5539 \nu^{7} + 12247 \nu^{6} + 18466 \nu^{5} + 100321 \nu^{4} - 380323 \nu^{3} + 39336 \nu^{2} + \cdots - 675000 ) / 19744 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 2913\nu^{7} - 5997\nu^{6} - 10918\nu^{5} - 57659\nu^{4} + 188977\nu^{3} + 29368\nu^{2} - 24424\nu + 345192 ) / 9872 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{7} - \beta_{6} - 3\beta_{5} - 3\beta_{3} - 3\beta_{2} + 3\beta_1 ) / 12 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -2\beta_{7} + 2\beta_{6} + 5\beta_{5} + 3\beta_{4} - 2\beta_{3} - 6\beta_{2} + 6\beta _1 + 12 ) / 8 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -8\beta_{7} - 16\beta_{6} - 15\beta_{5} + 9\beta_{4} + 84\beta _1 + 264 ) / 24 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -5\beta_{7} - 11\beta_{6} - 12\beta_{5} + 5\beta_{4} - 17\beta_{3} - 33\beta_{2} + 17\beta _1 - 4 ) / 4 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -286\beta_{7} + 82\beta_{6} + 465\beta_{5} + 327\beta_{4} - 138\beta_{3} - 246\beta_{2} + 450\beta _1 + 1104 ) / 24 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( -124\beta_{7} - 248\beta_{6} - 257\beta_{5} + 143\beta_{4} + 528\beta _1 + 1924 ) / 8 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( - 877 \beta_{7} - 785 \beta_{6} - 249 \beta_{5} + 1014 \beta_{4} - 1263 \beta_{3} - 2355 \beta_{2} + \cdots - 2394 ) / 12 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/208\mathbb{Z}\right)^\times\).
| \(n\) | \(53\) | \(79\) | \(145\) |
| \(\chi(n)\) | \(1\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 79.1 |
|
0 | − | 5.10408i | 0 | −6.73131 | 0 | 13.0811i | 0 | −17.0516 | 0 | |||||||||||||||||||||||||||||||||||||||||
| 79.2 | 0 | − | 3.37203i | 0 | 1.12575 | 0 | − | 4.60502i | 0 | −2.37058 | 0 | |||||||||||||||||||||||||||||||||||||||||
| 79.3 | 0 | − | 2.74722i | 0 | 6.89101 | 0 | − | 1.76573i | 0 | 1.45278 | 0 | |||||||||||||||||||||||||||||||||||||||||
| 79.4 | 0 | − | 1.01517i | 0 | −5.28546 | 0 | 5.52812i | 0 | 7.96943 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 79.5 | 0 | 1.01517i | 0 | −5.28546 | 0 | − | 5.52812i | 0 | 7.96943 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 79.6 | 0 | 2.74722i | 0 | 6.89101 | 0 | 1.76573i | 0 | 1.45278 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 79.7 | 0 | 3.37203i | 0 | 1.12575 | 0 | 4.60502i | 0 | −2.37058 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 79.8 | 0 | 5.10408i | 0 | −6.73131 | 0 | − | 13.0811i | 0 | −17.0516 | 0 | ||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 4.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 208.3.d.b | ✓ | 8 |
| 3.b | odd | 2 | 1 | 1872.3.k.f | 8 | ||
| 4.b | odd | 2 | 1 | inner | 208.3.d.b | ✓ | 8 |
| 8.b | even | 2 | 1 | 832.3.d.b | 8 | ||
| 8.d | odd | 2 | 1 | 832.3.d.b | 8 | ||
| 12.b | even | 2 | 1 | 1872.3.k.f | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 208.3.d.b | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 208.3.d.b | ✓ | 8 | 4.b | odd | 2 | 1 | inner |
| 832.3.d.b | 8 | 8.b | even | 2 | 1 | ||
| 832.3.d.b | 8 | 8.d | odd | 2 | 1 | ||
| 1872.3.k.f | 8 | 3.b | odd | 2 | 1 | ||
| 1872.3.k.f | 8 | 12.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{8} + 46T_{3}^{6} + 625T_{3}^{4} + 2832T_{3}^{2} + 2304 \)
acting on \(S_{3}^{\mathrm{new}}(208, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} \)
|
| $3$ |
\( T^{8} + 46 T^{6} + \cdots + 2304 \)
|
| $5$ |
\( (T^{4} + 4 T^{3} + \cdots + 276)^{2} \)
|
| $7$ |
\( T^{8} + 226 T^{6} + \cdots + 345744 \)
|
| $11$ |
\( T^{8} + 508 T^{6} + \cdots + 5531904 \)
|
| $13$ |
\( (T^{2} - 13)^{4} \)
|
| $17$ |
\( (T^{4} + 6 T^{3} + \cdots - 22932)^{2} \)
|
| $19$ |
\( T^{8} + 1596 T^{6} + \cdots + 756470016 \)
|
| $23$ |
\( T^{8} + 1552 T^{6} + \cdots + 150994944 \)
|
| $29$ |
\( (T^{4} + 48 T^{3} + \cdots + 46224)^{2} \)
|
| $31$ |
\( T^{8} + 724 T^{6} + \cdots + 36864 \)
|
| $37$ |
\( (T^{4} + 4 T^{3} + \cdots + 4202068)^{2} \)
|
| $41$ |
\( (T^{4} - 52 T^{3} + \cdots - 374592)^{2} \)
|
| $43$ |
\( T^{8} + \cdots + 13172370149376 \)
|
| $47$ |
\( T^{8} + \cdots + 9853597233936 \)
|
| $53$ |
\( (T^{4} + 80 T^{3} + \cdots - 13622592)^{2} \)
|
| $59$ |
\( T^{8} + \cdots + 2383001856 \)
|
| $61$ |
\( (T^{4} + 68 T^{3} + \cdots + 399936)^{2} \)
|
| $67$ |
\( T^{8} + \cdots + 19580412600576 \)
|
| $71$ |
\( T^{8} + \cdots + 390170132496 \)
|
| $73$ |
\( (T^{4} - 28 T^{3} + \cdots + 11009008)^{2} \)
|
| $79$ |
\( T^{8} + \cdots + 29091611344896 \)
|
| $83$ |
\( T^{8} + \cdots + 118247952384 \)
|
| $89$ |
\( (T^{4} - 140 T^{3} + \cdots + 17585136)^{2} \)
|
| $97$ |
\( (T^{4} - 32 T^{3} + \cdots - 3081264)^{2} \)
|
| show more | |
| show less |