Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [215,2,Mod(6,215)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(215, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 4]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("215.6");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 215 = 5 \cdot 43 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 215.e (of order \(3\), degree \(2\), 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: | \(1.71678364346\) |
| Analytic rank: | \(0\) |
| Dimension: | \(10\) |
| Relative dimension: | \(5\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{10} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{10} - 2x^{9} + 9x^{8} - 10x^{7} + 44x^{6} - 49x^{5} + 99x^{4} - 20x^{3} + 31x^{2} - 3x + 9 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| 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 \(1,\beta_1,\ldots,\beta_{9}\) 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
\( x^{10} - 2x^{9} + 9x^{8} - 10x^{7} + 44x^{6} - 49x^{5} + 99x^{4} - 20x^{3} + 31x^{2} - 3x + 9 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 4212 \nu^{9} + 3675 \nu^{8} - 15575 \nu^{7} - 15678 \nu^{6} - 74025 \nu^{5} + 88200 \nu^{4} + \cdots + 2078070 ) / 773879 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 4749 \nu^{9} + 22333 \nu^{8} - 57798 \nu^{7} + 111303 \nu^{6} - 118188 \nu^{5} + 535992 \nu^{4} + \cdots + 37908 ) / 773879 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 19451 \nu^{9} + 168550 \nu^{8} - 235263 \nu^{7} + 1002431 \nu^{6} - 631225 \nu^{5} + \cdots + 1479609 ) / 2321637 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 28988 \nu^{9} + 23213 \nu^{8} + 85878 \nu^{7} + 451846 \nu^{6} + 416857 \nu^{5} + 2104870 \nu^{4} + \cdots + 188058 ) / 2321637 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 24776 \nu^{9} - 26888 \nu^{8} - 70303 \nu^{7} - 436168 \nu^{6} - 342832 \nu^{5} - 2193070 \nu^{4} + \cdots - 2266128 ) / 773879 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 27063 \nu^{9} + 58338 \nu^{8} - 247242 \nu^{7} + 286205 \nu^{6} - 1175094 \nu^{5} + 1400112 \nu^{4} + \cdots + 86964 ) / 773879 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 108252 \nu^{9} + 233352 \nu^{8} - 988968 \nu^{7} + 1144820 \nu^{6} - 4700376 \nu^{5} + \cdots + 347856 ) / 773879 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 329980 \nu^{9} + 541460 \nu^{8} - 2773827 \nu^{7} + 2469163 \nu^{6} - 13670405 \nu^{5} + \cdots + 681951 ) / 2321637 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{6} + 3\beta_{5} + \beta_{2} \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{8} - 4\beta_{7} - 4\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -5\beta_{6} - 13\beta_{5} - \beta_{4} + \beta_{3} - 13 \)
|
| \(\nu^{5}\) | \(=\) |
\( 2\beta_{9} - 7\beta_{8} + 19\beta_{7} + \beta_{5} + 7\beta_{3} \)
|
| \(\nu^{6}\) | \(=\) |
\( 9\beta_{9} - 9\beta_{8} + \beta_{7} + 9\beta_{4} - 24\beta_{2} + \beta _1 + 60 \)
|
| \(\nu^{7}\) | \(=\) |
\( \beta_{6} - 6\beta_{5} + 18\beta_{4} - 42\beta_{3} + 93\beta _1 - 6 \)
|
| \(\nu^{8}\) | \(=\) |
\( -60\beta_{9} + 61\beta_{8} - 13\beta_{7} + 117\beta_{6} + 285\beta_{5} - 61\beta_{3} + 117\beta_{2} \)
|
| \(\nu^{9}\) | \(=\) |
\( -121\beta_{9} + 238\beta_{8} - 462\beta_{7} - 121\beta_{4} + 14\beta_{2} - 462\beta _1 + 20 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/215\mathbb{Z}\right)^\times\).
| \(n\) | \(46\) | \(87\) |
| \(\chi(n)\) | \(\beta_{5}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 6.1 |
|
−2.32183 | −1.09983 | + | 1.90496i | 3.39089 | −0.500000 | + | 0.866025i | 2.55362 | − | 4.42300i | 1.85636 | + | 3.21531i | −3.22941 | −0.919259 | − | 1.59220i | 1.16091 | − | 2.01076i | ||||||||||||||||||||||||||||||||||||
| 6.2 | −1.78972 | 0.769850 | − | 1.33342i | 1.20311 | −0.500000 | + | 0.866025i | −1.37782 | + | 2.38645i | 0.496416 | + | 0.859818i | 1.42621 | 0.314662 | + | 0.545010i | 0.894862 | − | 1.54995i | |||||||||||||||||||||||||||||||||||||
| 6.3 | −0.609440 | −1.05085 | + | 1.82013i | −1.62858 | −0.500000 | + | 0.866025i | 0.640433 | − | 1.10926i | −1.50957 | − | 2.61465i | 2.21140 | −0.708590 | − | 1.22731i | 0.304720 | − | 0.527790i | |||||||||||||||||||||||||||||||||||||
| 6.4 | 0.544198 | 1.47644 | − | 2.55727i | −1.70385 | −0.500000 | + | 0.866025i | 0.803474 | − | 1.39166i | −2.12402 | − | 3.67892i | −2.01563 | −2.85974 | − | 4.95321i | −0.272099 | + | 0.471289i | |||||||||||||||||||||||||||||||||||||
| 6.5 | 2.17679 | 0.404398 | − | 0.700438i | 2.73843 | −0.500000 | + | 0.866025i | 0.880291 | − | 1.52471i | −0.719181 | − | 1.24566i | 1.60742 | 1.17292 | + | 2.03157i | −1.08840 | + | 1.88516i | |||||||||||||||||||||||||||||||||||||
| 36.1 | −2.32183 | −1.09983 | − | 1.90496i | 3.39089 | −0.500000 | − | 0.866025i | 2.55362 | + | 4.42300i | 1.85636 | − | 3.21531i | −3.22941 | −0.919259 | + | 1.59220i | 1.16091 | + | 2.01076i | |||||||||||||||||||||||||||||||||||||
| 36.2 | −1.78972 | 0.769850 | + | 1.33342i | 1.20311 | −0.500000 | − | 0.866025i | −1.37782 | − | 2.38645i | 0.496416 | − | 0.859818i | 1.42621 | 0.314662 | − | 0.545010i | 0.894862 | + | 1.54995i | |||||||||||||||||||||||||||||||||||||
| 36.3 | −0.609440 | −1.05085 | − | 1.82013i | −1.62858 | −0.500000 | − | 0.866025i | 0.640433 | + | 1.10926i | −1.50957 | + | 2.61465i | 2.21140 | −0.708590 | + | 1.22731i | 0.304720 | + | 0.527790i | |||||||||||||||||||||||||||||||||||||
| 36.4 | 0.544198 | 1.47644 | + | 2.55727i | −1.70385 | −0.500000 | − | 0.866025i | 0.803474 | + | 1.39166i | −2.12402 | + | 3.67892i | −2.01563 | −2.85974 | + | 4.95321i | −0.272099 | − | 0.471289i | |||||||||||||||||||||||||||||||||||||
| 36.5 | 2.17679 | 0.404398 | + | 0.700438i | 2.73843 | −0.500000 | − | 0.866025i | 0.880291 | + | 1.52471i | −0.719181 | + | 1.24566i | 1.60742 | 1.17292 | − | 2.03157i | −1.08840 | − | 1.88516i | |||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 43.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 215.2.e.c | ✓ | 10 |
| 43.c | even | 3 | 1 | inner | 215.2.e.c | ✓ | 10 |
| 43.c | even | 3 | 1 | 9245.2.a.k | 5 | ||
| 43.d | odd | 6 | 1 | 9245.2.a.m | 5 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 215.2.e.c | ✓ | 10 | 1.a | even | 1 | 1 | trivial |
| 215.2.e.c | ✓ | 10 | 43.c | even | 3 | 1 | inner |
| 9245.2.a.k | 5 | 43.c | even | 3 | 1 | ||
| 9245.2.a.m | 5 | 43.d | odd | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{5} + 2T_{2}^{4} - 5T_{2}^{3} - 10T_{2}^{2} + T_{2} + 3 \)
acting on \(S_{2}^{\mathrm{new}}(215, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{5} + 2 T^{4} - 5 T^{3} + \cdots + 3)^{2} \)
|
| $3$ |
\( T^{10} - T^{9} + \cdots + 289 \)
|
| $5$ |
\( (T^{2} + T + 1)^{5} \)
|
| $7$ |
\( T^{10} + 4 T^{9} + \cdots + 4624 \)
|
| $11$ |
\( (T^{5} - 13 T^{4} + \cdots - 16)^{2} \)
|
| $13$ |
\( T^{10} + 7 T^{9} + \cdots + 605284 \)
|
| $17$ |
\( T^{10} + 10 T^{9} + \cdots + 1752976 \)
|
| $19$ |
\( T^{10} + 54 T^{8} + \cdots + 842724 \)
|
| $23$ |
\( T^{10} + 10 T^{9} + \cdots + 5308416 \)
|
| $29$ |
\( T^{10} - 4 T^{9} + \cdots + 316969 \)
|
| $31$ |
\( T^{10} + 46 T^{8} + \cdots + 139876 \)
|
| $37$ |
\( T^{10} + \cdots + 462852196 \)
|
| $41$ |
\( (T^{5} - 4 T^{4} - 53 T^{3} + \cdots - 89)^{2} \)
|
| $43$ |
\( T^{10} + \cdots + 147008443 \)
|
| $47$ |
\( (T^{5} - 6 T^{4} - 46 T^{3} + \cdots - 17)^{2} \)
|
| $53$ |
\( T^{10} + 12 T^{9} + \cdots + 37283236 \)
|
| $59$ |
\( (T^{5} - 192 T^{3} + \cdots - 218)^{2} \)
|
| $61$ |
\( T^{10} + 2 T^{9} + \cdots + 191844 \)
|
| $67$ |
\( T^{10} + 6 T^{9} + \cdots + 5489649 \)
|
| $71$ |
\( T^{10} - 13 T^{9} + \cdots + 605284 \)
|
| $73$ |
\( T^{10} + 13 T^{9} + \cdots + 14273284 \)
|
| $79$ |
\( T^{10} - 19 T^{9} + \cdots + 1763584 \)
|
| $83$ |
\( T^{10} + 8 T^{9} + \cdots + 51796809 \)
|
| $89$ |
\( T^{10} - 2 T^{9} + \cdots + 328329 \)
|
| $97$ |
\( (T^{5} + 8 T^{4} + \cdots - 6276)^{2} \)
|
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