Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [612,2,Mod(205,612)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(612, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 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("612.205");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 612 = 2^{2} \cdot 3^{2} \cdot 17 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 612.i (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: | \(4.88684460370\) |
| Analytic rank: | \(0\) |
| Dimension: | \(10\) |
| Relative dimension: | \(5\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | 10.0.1425109843323.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{10} - x^{9} - 5x^{8} + 12x^{7} + 9x^{6} - 54x^{5} + 27x^{4} + 108x^{3} - 135x^{2} - 81x + 243 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 3 \) |
| 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
\( x^{10} - x^{9} - 5x^{8} + 12x^{7} + 9x^{6} - 54x^{5} + 27x^{4} + 108x^{3} - 135x^{2} - 81x + 243 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{9} + 4\nu^{8} - 7\nu^{7} + 9\nu^{6} + 18\nu^{5} - 54\nu^{4} + 27\nu^{3} + 135\nu^{2} - 270\nu + 81 ) / 162 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{9} - \nu^{8} - 5\nu^{7} + 12\nu^{6} + 9\nu^{5} - 54\nu^{4} + 27\nu^{3} + 108\nu^{2} - 135\nu - 81 ) / 81 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{9} + 4\nu^{8} - 7\nu^{7} - 9\nu^{6} + 36\nu^{5} - 18\nu^{4} - 81\nu^{3} + 135\nu^{2} - 81 ) / 54 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -\nu^{9} + 9\nu^{7} - 19\nu^{6} + 72\nu^{4} - 81\nu^{3} - 81\nu^{2} + 270\nu - 189 ) / 54 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 2\nu^{9} - 2\nu^{8} - 10\nu^{7} + 33\nu^{6} - 18\nu^{5} - 72\nu^{4} + 162\nu^{3} + 27\nu^{2} - 351\nu + 405 ) / 81 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -2\nu^{9} + 8\nu^{8} - 5\nu^{7} - 27\nu^{6} + 54\nu^{5} + 18\nu^{4} - 135\nu^{3} + 135\nu^{2} + 27\nu - 81 ) / 81 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( -\nu^{9} + \nu^{8} + 5\nu^{7} - 15\nu^{6} + 3\nu^{5} + 42\nu^{4} - 63\nu^{3} - 18\nu^{2} + 135\nu - 135 ) / 27 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 2\nu^{9} - 11\nu^{8} + 8\nu^{7} + 42\nu^{6} - 90\nu^{5} - 45\nu^{4} + 297\nu^{3} - 216\nu^{2} - 270\nu + 405 ) / 81 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{8} + \beta_{6} + \beta_{3} + \beta _1 + 1 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{9} + 2\beta_{7} + \beta_{6} + \beta_{5} + \beta_{3} - \beta_{2} + 2\beta _1 - 3 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{8} + 4\beta_{6} + 3\beta_{5} + \beta_{4} + \beta_{3} - \beta _1 - 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( 2\beta_{9} - 5\beta_{8} + 4\beta_{7} + 8\beta_{5} + \beta_{4} + 3\beta_{3} + \beta_{2} - 3\beta _1 + 1 \)
|
| \(\nu^{6}\) | \(=\) |
\( -4\beta_{9} - 3\beta_{8} - 8\beta_{7} + 2\beta_{6} + 8\beta_{5} - \beta_{3} + 16\beta_{2} - 2\beta _1 + 6 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 3 \beta_{9} - 4 \beta_{8} + 3 \beta_{7} - 10 \beta_{6} + 3 \beta_{5} - 16 \beta_{4} + 11 \beta_{3} + \cdots + 35 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 20 \beta_{9} + 5 \beta_{8} - 4 \beta_{7} - 9 \beta_{6} - 26 \beta_{5} - 37 \beta_{4} - 12 \beta_{3} + \cdots - 10 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 32 \beta_{9} + 12 \beta_{8} + 17 \beta_{7} - 2 \beta_{6} - 44 \beta_{5} - 72 \beta_{4} + 28 \beta_{3} + \cdots + 30 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/612\mathbb{Z}\right)^\times\).
| \(n\) | \(37\) | \(137\) | \(307\) |
| \(\chi(n)\) | \(1\) | \(-1 - \beta_{2}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 205.1 |
|
0 | −1.66673 | + | 0.471168i | 0 | −0.0966301 | − | 0.167368i | 0 | 0.0625819 | − | 0.108395i | 0 | 2.55600 | − | 1.57062i | 0 | ||||||||||||||||||||||||||||||||||||||||
| 205.2 | 0 | −1.24119 | − | 1.20808i | 0 | 0.791229 | + | 1.37045i | 0 | 2.00327 | − | 3.46976i | 0 | 0.0810929 | + | 2.99890i | 0 | |||||||||||||||||||||||||||||||||||||||||
| 205.3 | 0 | −0.829999 | + | 1.52023i | 0 | 1.66189 | + | 2.87848i | 0 | −0.603137 | + | 1.04466i | 0 | −1.62220 | − | 2.52358i | 0 | |||||||||||||||||||||||||||||||||||||||||
| 205.4 | 0 | 1.60328 | + | 0.655366i | 0 | −1.05551 | − | 1.82820i | 0 | 0.600832 | − | 1.04067i | 0 | 2.14099 | + | 2.10147i | 0 | |||||||||||||||||||||||||||||||||||||||||
| 205.5 | 0 | 1.63464 | − | 0.572661i | 0 | 0.699022 | + | 1.21074i | 0 | −2.06355 | + | 3.57417i | 0 | 2.34412 | − | 1.87219i | 0 | |||||||||||||||||||||||||||||||||||||||||
| 409.1 | 0 | −1.66673 | − | 0.471168i | 0 | −0.0966301 | + | 0.167368i | 0 | 0.0625819 | + | 0.108395i | 0 | 2.55600 | + | 1.57062i | 0 | |||||||||||||||||||||||||||||||||||||||||
| 409.2 | 0 | −1.24119 | + | 1.20808i | 0 | 0.791229 | − | 1.37045i | 0 | 2.00327 | + | 3.46976i | 0 | 0.0810929 | − | 2.99890i | 0 | |||||||||||||||||||||||||||||||||||||||||
| 409.3 | 0 | −0.829999 | − | 1.52023i | 0 | 1.66189 | − | 2.87848i | 0 | −0.603137 | − | 1.04466i | 0 | −1.62220 | + | 2.52358i | 0 | |||||||||||||||||||||||||||||||||||||||||
| 409.4 | 0 | 1.60328 | − | 0.655366i | 0 | −1.05551 | + | 1.82820i | 0 | 0.600832 | + | 1.04067i | 0 | 2.14099 | − | 2.10147i | 0 | |||||||||||||||||||||||||||||||||||||||||
| 409.5 | 0 | 1.63464 | + | 0.572661i | 0 | 0.699022 | − | 1.21074i | 0 | −2.06355 | − | 3.57417i | 0 | 2.34412 | + | 1.87219i | 0 | |||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 9.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 612.2.i.d | ✓ | 10 |
| 3.b | odd | 2 | 1 | 1836.2.i.d | 10 | ||
| 9.c | even | 3 | 1 | inner | 612.2.i.d | ✓ | 10 |
| 9.c | even | 3 | 1 | 5508.2.a.k | 5 | ||
| 9.d | odd | 6 | 1 | 1836.2.i.d | 10 | ||
| 9.d | odd | 6 | 1 | 5508.2.a.l | 5 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 612.2.i.d | ✓ | 10 | 1.a | even | 1 | 1 | trivial |
| 612.2.i.d | ✓ | 10 | 9.c | even | 3 | 1 | inner |
| 1836.2.i.d | 10 | 3.b | odd | 2 | 1 | ||
| 1836.2.i.d | 10 | 9.d | odd | 6 | 1 | ||
| 5508.2.a.k | 5 | 9.c | even | 3 | 1 | ||
| 5508.2.a.l | 5 | 9.d | odd | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{10} - 4T_{5}^{9} + 18T_{5}^{8} - 28T_{5}^{7} + 88T_{5}^{6} - 135T_{5}^{5} + 288T_{5}^{4} - 228T_{5}^{3} + 198T_{5}^{2} + 36T_{5} + 9 \)
acting on \(S_{2}^{\mathrm{new}}(612, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{10} \)
|
| $3$ |
\( T^{10} + T^{9} + \cdots + 243 \)
|
| $5$ |
\( T^{10} - 4 T^{9} + \cdots + 9 \)
|
| $7$ |
\( T^{10} + 18 T^{8} + \cdots + 9 \)
|
| $11$ |
\( T^{10} + 4 T^{9} + \cdots + 9 \)
|
| $13$ |
\( T^{10} - 10 T^{9} + \cdots + 1 \)
|
| $17$ |
\( (T - 1)^{10} \)
|
| $19$ |
\( (T^{5} - 5 T^{4} - 8 T^{3} + \cdots - 8)^{2} \)
|
| $23$ |
\( T^{10} + 3 T^{9} + \cdots + 136161 \)
|
| $29$ |
\( T^{10} - 8 T^{9} + \cdots + 40401 \)
|
| $31$ |
\( T^{10} - 8 T^{9} + \cdots + 889249 \)
|
| $37$ |
\( (T^{5} + 10 T^{4} + \cdots - 146)^{2} \)
|
| $41$ |
\( T^{10} + 5 T^{9} + \cdots + 700569 \)
|
| $43$ |
\( T^{10} - 15 T^{9} + \cdots + 15116544 \)
|
| $47$ |
\( T^{10} + \cdots + 319801689 \)
|
| $53$ |
\( (T^{5} - 18 T^{4} + \cdots - 486)^{2} \)
|
| $59$ |
\( T^{10} - 15 T^{9} + \cdots + 363609 \)
|
| $61$ |
\( T^{10} + 4 T^{9} + \cdots + 822649 \)
|
| $67$ |
\( T^{10} - 27 T^{9} + \cdots + 25431849 \)
|
| $71$ |
\( (T^{5} + 3 T^{4} + \cdots - 1296)^{2} \)
|
| $73$ |
\( (T^{5} + 16 T^{4} + \cdots - 14606)^{2} \)
|
| $79$ |
\( T^{10} + \cdots + 28757037241 \)
|
| $83$ |
\( T^{10} + \cdots + 286387929 \)
|
| $89$ |
\( (T^{5} + 12 T^{4} + \cdots + 15426)^{2} \)
|
| $97$ |
\( T^{10} + \cdots + 869247289 \)
|
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