Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1216,2,Mod(255,1216)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1216, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([3, 0, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1216.255");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1216 = 2^{6} \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1216.n (of order \(6\), 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: | \(9.70980888579\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Relative dimension: | \(3\) over \(\Q(\zeta_{6})\) |
| Coefficient field: | 6.0.31726512.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - x^{5} + 10x^{4} + 3x^{3} + 84x^{2} - 27x + 9 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 304) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-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
\( x^{6} - x^{5} + 10x^{4} + 3x^{3} + 84x^{2} - 27x + 9 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{5} + 10\nu^{4} - 100\nu^{3} + 84\nu^{2} - 27\nu + 270 ) / 813 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 30\nu^{5} - 29\nu^{4} + 290\nu^{3} + 190\nu^{2} + 2436\nu - 783 ) / 813 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -161\nu^{5} - 16\nu^{4} - 1466\nu^{3} - 1923\nu^{2} - 14916\nu - 5310 ) / 2439 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 191\nu^{5} - 284\nu^{4} + 2027\nu^{3} - 597\nu^{2} + 15726\nu - 10107 ) / 2439 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -2\beta_{5} + \beta_{4} + 6\beta_{3} - \beta_{2} + \beta_1 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{5} - \beta_{4} - 10\beta_{2} - 3 \)
|
| \(\nu^{4}\) | \(=\) |
\( 10\beta_{5} - 20\beta_{4} - 57\beta_{3} - 16\beta _1 - 57 \)
|
| \(\nu^{5}\) | \(=\) |
\( 32\beta_{5} - 16\beta_{4} - 66\beta_{3} + 103\beta_{2} - 103\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1216\mathbb{Z}\right)^\times\).
| \(n\) | \(191\) | \(705\) | \(837\) |
| \(\chi(n)\) | \(-1\) | \(-\beta_{3}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 255.1 |
|
0 | −1.35887 | + | 2.35363i | 0 | −1.85887 | + | 3.21966i | 0 | 2.36936i | 0 | −2.19306 | − | 3.79849i | 0 | ||||||||||||||||||||||||||||||
| 255.2 | 0 | 0.162698 | − | 0.281802i | 0 | −0.337302 | + | 0.584224i | 0 | − | 3.59084i | 0 | 1.44706 | + | 2.50638i | 0 | ||||||||||||||||||||||||||||||
| 255.3 | 0 | 1.69617 | − | 2.93786i | 0 | 1.19617 | − | 2.07183i | 0 | 1.22147i | 0 | −4.25400 | − | 7.36814i | 0 | |||||||||||||||||||||||||||||||
| 639.1 | 0 | −1.35887 | − | 2.35363i | 0 | −1.85887 | − | 3.21966i | 0 | − | 2.36936i | 0 | −2.19306 | + | 3.79849i | 0 | ||||||||||||||||||||||||||||||
| 639.2 | 0 | 0.162698 | + | 0.281802i | 0 | −0.337302 | − | 0.584224i | 0 | 3.59084i | 0 | 1.44706 | − | 2.50638i | 0 | |||||||||||||||||||||||||||||||
| 639.3 | 0 | 1.69617 | + | 2.93786i | 0 | 1.19617 | + | 2.07183i | 0 | − | 1.22147i | 0 | −4.25400 | + | 7.36814i | 0 | ||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 76.f | even | 6 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 1216.2.n.e | 6 | |
| 4.b | odd | 2 | 1 | 1216.2.n.d | 6 | ||
| 8.b | even | 2 | 1 | 304.2.n.d | ✓ | 6 | |
| 8.d | odd | 2 | 1 | 304.2.n.e | yes | 6 | |
| 19.d | odd | 6 | 1 | 1216.2.n.d | 6 | ||
| 24.f | even | 2 | 1 | 2736.2.bm.m | 6 | ||
| 24.h | odd | 2 | 1 | 2736.2.bm.l | 6 | ||
| 76.f | even | 6 | 1 | inner | 1216.2.n.e | 6 | |
| 152.l | odd | 6 | 1 | 304.2.n.e | yes | 6 | |
| 152.o | even | 6 | 1 | 304.2.n.d | ✓ | 6 | |
| 456.s | odd | 6 | 1 | 2736.2.bm.l | 6 | ||
| 456.v | even | 6 | 1 | 2736.2.bm.m | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 304.2.n.d | ✓ | 6 | 8.b | even | 2 | 1 | |
| 304.2.n.d | ✓ | 6 | 152.o | even | 6 | 1 | |
| 304.2.n.e | yes | 6 | 8.d | odd | 2 | 1 | |
| 304.2.n.e | yes | 6 | 152.l | odd | 6 | 1 | |
| 1216.2.n.d | 6 | 4.b | odd | 2 | 1 | ||
| 1216.2.n.d | 6 | 19.d | odd | 6 | 1 | ||
| 1216.2.n.e | 6 | 1.a | even | 1 | 1 | trivial | |
| 1216.2.n.e | 6 | 76.f | even | 6 | 1 | inner | |
| 2736.2.bm.l | 6 | 24.h | odd | 2 | 1 | ||
| 2736.2.bm.l | 6 | 456.s | odd | 6 | 1 | ||
| 2736.2.bm.m | 6 | 24.f | even | 2 | 1 | ||
| 2736.2.bm.m | 6 | 456.v | even | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{6} - T_{3}^{5} + 10T_{3}^{4} + 3T_{3}^{3} + 84T_{3}^{2} - 27T_{3} + 9 \)
acting on \(S_{2}^{\mathrm{new}}(1216, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
|
| $3$ |
\( T^{6} - T^{5} + 10 T^{4} + \cdots + 9 \)
|
| $5$ |
\( T^{6} + 2 T^{5} + \cdots + 36 \)
|
| $7$ |
\( T^{6} + 20 T^{4} + \cdots + 108 \)
|
| $11$ |
\( T^{6} + 29 T^{4} + \cdots + 507 \)
|
| $13$ |
\( (T^{2} + 6 T + 12)^{3} \)
|
| $17$ |
\( T^{6} + 2 T^{5} + \cdots + 144 \)
|
| $19$ |
\( T^{6} - 17 T^{5} + \cdots + 6859 \)
|
| $23$ |
\( T^{6} - 10 T^{4} + \cdots + 108 \)
|
| $29$ |
\( T^{6} - 12 T^{5} + \cdots + 2700 \)
|
| $31$ |
\( (T^{3} + 2 T^{2} - 54 T + 66)^{2} \)
|
| $37$ |
\( T^{6} + 72 T^{4} + \cdots + 2700 \)
|
| $41$ |
\( T^{6} - 3 T^{5} + \cdots + 54675 \)
|
| $43$ |
\( (T^{2} - 6 T + 12)^{3} \)
|
| $47$ |
\( T^{6} - 18 T^{5} + \cdots + 12 \)
|
| $53$ |
\( (T^{2} - 12 T + 48)^{3} \)
|
| $59$ |
\( T^{6} - 27 T^{5} + \cdots + 263169 \)
|
| $61$ |
\( T^{6} - 10 T^{5} + \cdots + 36 \)
|
| $67$ |
\( T^{6} + 11 T^{5} + \cdots + 245025 \)
|
| $71$ |
\( (T^{2} + 6 T + 36)^{3} \)
|
| $73$ |
\( T^{6} + 5 T^{5} + \cdots + 289 \)
|
| $79$ |
\( T^{6} - 16 T^{5} + \cdots + 2304 \)
|
| $83$ |
\( T^{6} + 113 T^{4} + \cdots + 3 \)
|
| $89$ |
\( T^{6} + 24 T^{5} + \cdots + 97200 \)
|
| $97$ |
\( T^{6} - 21 T^{5} + \cdots + 151875 \)
|
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