Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [286,2,Mod(27,286)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(286, base_ring=CyclotomicField(10))
chi = DirichletCharacter(H, H._module([4, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("286.27");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 286 = 2 \cdot 11 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 286.h (of order \(5\), degree \(4\), 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: | \(2.28372149781\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{5})\) |
| Coefficient field: | 8.0.159390625.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - x^{7} + 6x^{6} - 11x^{5} + 21x^{4} - 5x^{3} + 10x^{2} + 25x + 25 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{5}]$ |
$q$-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
\( x^{8} - x^{7} + 6x^{6} - 11x^{5} + 21x^{4} - 5x^{3} + 10x^{2} + 25x + 25 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 555\nu^{7} - 2159\nu^{6} + 7489\nu^{5} - 18164\nu^{4} + 40069\nu^{3} - 84434\nu^{2} + 43855\nu + 375 ) / 94655 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -970\nu^{7} - 1002\nu^{6} - 6608\nu^{5} + 9063\nu^{4} - 14943\nu^{3} + 27673\nu^{2} - 68120\nu + 35160 ) / 94655 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -1604\nu^{7} + 4159\nu^{6} - 12059\nu^{5} + 28414\nu^{4} - 81659\nu^{3} + 38305\nu^{2} - 13500\nu - 13875 ) / 94655 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -2052\nu^{7} + 2252\nu^{6} - 19912\nu^{5} + 21007\nu^{4} - 82042\nu^{3} + 35785\nu^{2} - 19395\nu - 90925 ) / 94655 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -2667\nu^{7} + 6691\nu^{6} - 17466\nu^{5} + 50856\nu^{4} - 82441\nu^{3} + 72554\nu^{2} - 4035\nu - 12035 ) / 94655 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 4024\nu^{7} - 1464\nu^{6} + 21519\nu^{5} - 26434\nu^{4} + 59219\nu^{3} + 22635\nu^{2} + 54640\nu + 66675 ) / 94655 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{7} - \beta_{6} - \beta_{5} - \beta_{3} - 3\beta_{2} + \beta_1 \)
|
| \(\nu^{3}\) | \(=\) |
\( 2\beta_{6} + \beta_{5} - 4\beta_{4} - \beta_{3} - \beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( 7\beta_{7} + 7\beta_{6} + 2\beta_{5} + 13\beta_{3} + 13\beta_{2} - 7 \)
|
| \(\nu^{5}\) | \(=\) |
\( -8\beta_{7} - 11\beta_{6} - 20\beta_{5} + 20\beta_{4} - 11\beta_{2} + 8\beta _1 - 12 \)
|
| \(\nu^{6}\) | \(=\) |
\( -19\beta_{7} - 19\beta_{4} - 68\beta_{3} - 36\beta_{2} - 24\beta _1 + 36 \)
|
| \(\nu^{7}\) | \(=\) |
\( 111\beta_{7} + 81\beta_{6} + 111\beta_{5} - 55\beta_{4} + 81\beta_{3} + 148\beta_{2} - 56\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/286\mathbb{Z}\right)^\times\).
| \(n\) | \(67\) | \(79\) |
| \(\chi(n)\) | \(1\) | \(-\beta_{2}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 27.1 |
|
0.309017 | + | 0.951057i | 0.0756511 | + | 0.0549637i | −0.809017 | + | 0.587785i | 0.644228 | − | 1.98273i | −0.0288961 | + | 0.0889332i | 2.54238 | − | 1.84715i | −0.809017 | − | 0.587785i | −0.924349 | − | 2.84485i | 2.08477 | ||||||||||||||||||||||||||
| 27.2 | 0.309017 | + | 0.951057i | 2.04238 | + | 1.48388i | −0.809017 | + | 0.587785i | −0.571279 | + | 1.75822i | −0.780121 | + | 2.40097i | 0.575651 | − | 0.418235i | −0.809017 | − | 0.587785i | 1.04238 | + | 3.20812i | −1.84870 | |||||||||||||||||||||||||||
| 53.1 | 0.309017 | − | 0.951057i | 0.0756511 | − | 0.0549637i | −0.809017 | − | 0.587785i | 0.644228 | + | 1.98273i | −0.0288961 | − | 0.0889332i | 2.54238 | + | 1.84715i | −0.809017 | + | 0.587785i | −0.924349 | + | 2.84485i | 2.08477 | |||||||||||||||||||||||||||
| 53.2 | 0.309017 | − | 0.951057i | 2.04238 | − | 1.48388i | −0.809017 | − | 0.587785i | −0.571279 | − | 1.75822i | −0.780121 | − | 2.40097i | 0.575651 | + | 0.418235i | −0.809017 | + | 0.587785i | 1.04238 | − | 3.20812i | −1.84870 | |||||||||||||||||||||||||||
| 157.1 | −0.809017 | + | 0.587785i | −0.697759 | − | 2.14748i | 0.309017 | − | 0.951057i | 0.680019 | + | 0.494063i | 1.82676 | + | 1.32722i | 1.07973 | − | 3.32305i | 0.309017 | + | 0.951057i | −1.69776 | + | 1.23349i | −0.840550 | |||||||||||||||||||||||||||
| 157.2 | −0.809017 | + | 0.587785i | 0.579725 | + | 1.78421i | 0.309017 | − | 0.951057i | 2.74703 | + | 1.99584i | −1.51774 | − | 1.10270i | −0.197759 | + | 0.608640i | 0.309017 | + | 0.951057i | −0.420275 | + | 0.305348i | −3.39552 | |||||||||||||||||||||||||||
| 235.1 | −0.809017 | − | 0.587785i | −0.697759 | + | 2.14748i | 0.309017 | + | 0.951057i | 0.680019 | − | 0.494063i | 1.82676 | − | 1.32722i | 1.07973 | + | 3.32305i | 0.309017 | − | 0.951057i | −1.69776 | − | 1.23349i | −0.840550 | |||||||||||||||||||||||||||
| 235.2 | −0.809017 | − | 0.587785i | 0.579725 | − | 1.78421i | 0.309017 | + | 0.951057i | 2.74703 | − | 1.99584i | −1.51774 | + | 1.10270i | −0.197759 | − | 0.608640i | 0.309017 | − | 0.951057i | −0.420275 | − | 0.305348i | −3.39552 | |||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 11.c | even | 5 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 286.2.h.b | ✓ | 8 |
| 11.c | even | 5 | 1 | inner | 286.2.h.b | ✓ | 8 |
| 11.c | even | 5 | 1 | 3146.2.a.ba | 4 | ||
| 11.d | odd | 10 | 1 | 3146.2.a.x | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 286.2.h.b | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 286.2.h.b | ✓ | 8 | 11.c | even | 5 | 1 | inner |
| 3146.2.a.x | 4 | 11.d | odd | 10 | 1 | ||
| 3146.2.a.ba | 4 | 11.c | even | 5 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{8} - 4T_{3}^{7} + 13T_{3}^{6} - 30T_{3}^{5} + 71T_{3}^{4} - 90T_{3}^{3} + 127T_{3}^{2} - 18T_{3} + 1 \)
acting on \(S_{2}^{\mathrm{new}}(286, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{4} + T^{3} + T^{2} + \cdots + 1)^{2} \)
|
| $3$ |
\( T^{8} - 4 T^{7} + \cdots + 1 \)
|
| $5$ |
\( T^{8} - 7 T^{7} + \cdots + 121 \)
|
| $7$ |
\( T^{8} - 8 T^{7} + \cdots + 25 \)
|
| $11$ |
\( T^{8} - 5 T^{7} + \cdots + 14641 \)
|
| $13$ |
\( (T^{4} + T^{3} + T^{2} + \cdots + 1)^{2} \)
|
| $17$ |
\( T^{8} - 4 T^{7} + \cdots + 841 \)
|
| $19$ |
\( (T^{4} + 5 T^{3} + 40 T^{2} + \cdots + 25)^{2} \)
|
| $23$ |
\( (T^{2} + 8 T + 11)^{4} \)
|
| $29$ |
\( T^{8} - 16 T^{7} + \cdots + 1 \)
|
| $31$ |
\( T^{8} + 5 T^{7} + \cdots + 15625 \)
|
| $37$ |
\( T^{8} + 14 T^{7} + \cdots + 841 \)
|
| $41$ |
\( T^{8} + 4 T^{7} + \cdots + 5340721 \)
|
| $43$ |
\( (T^{4} + 2 T^{3} + \cdots + 2255)^{2} \)
|
| $47$ |
\( T^{8} + 3 T^{7} + \cdots + 167281 \)
|
| $53$ |
\( T^{8} - 27 T^{7} + \cdots + 32761 \)
|
| $59$ |
\( T^{8} - 4 T^{7} + \cdots + 15625 \)
|
| $61$ |
\( T^{8} - 8 T^{7} + \cdots + 1 \)
|
| $67$ |
\( (T^{4} + 12 T^{3} + \cdots + 25)^{2} \)
|
| $71$ |
\( T^{8} - 9 T^{7} + \cdots + 29757025 \)
|
| $73$ |
\( T^{8} - 46 T^{7} + \cdots + 47554816 \)
|
| $79$ |
\( T^{8} - 42 T^{7} + \cdots + 37319881 \)
|
| $83$ |
\( T^{8} + 5 T^{7} + \cdots + 22477081 \)
|
| $89$ |
\( (T^{4} + 26 T^{3} + \cdots - 2305)^{2} \)
|
| $97$ |
\( T^{8} - 11 T^{7} + \cdots + 42025 \)
|
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