Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [195,2,Mod(16,195)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(195, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 0, 2]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("195.16");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 195 = 3 \cdot 5 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 195.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: | \(1.55708283941\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Relative dimension: | \(3\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | 6.0.1714608.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 3x^{5} + 12x^{4} - 19x^{3} + 30x^{2} - 21x + 7 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - 3x^{5} + 12x^{4} - 19x^{3} + 30x^{2} - 21x + 7 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{5} + \nu^{4} + 2\nu^{3} + 10\nu^{2} - 7\nu ) / 7 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{2} - \nu + 3 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -2\nu^{5} + 5\nu^{4} - 18\nu^{3} + 22\nu^{2} - 28\nu + 7 ) / 7 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 2\nu^{5} - 5\nu^{4} + 25\nu^{3} - 29\nu^{2} + 56\nu - 14 ) / 7 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + \beta _1 - 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{5} + \beta_{4} + \beta_{3} - 3\beta _1 - 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( 2\beta_{5} + 3\beta_{4} - 4\beta_{3} + 2\beta_{2} - 6\beta _1 + 13 \)
|
| \(\nu^{5}\) | \(=\) |
\( -4\beta_{5} - 5\beta_{4} - 8\beta_{3} + 5\beta_{2} + 9\beta _1 + 21 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/195\mathbb{Z}\right)^\times\).
| \(n\) | \(106\) | \(131\) | \(157\) |
| \(\chi(n)\) | \(\beta_{4}\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 16.1 |
|
−1.30084 | − | 2.25312i | 0.500000 | + | 0.866025i | −2.38437 | + | 4.12985i | 1.00000 | 1.30084 | − | 2.25312i | −1.80084 | + | 3.11915i | 7.20336 | −0.500000 | + | 0.866025i | −1.30084 | − | 2.25312i | ||||||||||||||||||||||
| 16.2 | 0.169938 | + | 0.294342i | 0.500000 | + | 0.866025i | 0.942242 | − | 1.63201i | 1.00000 | −0.169938 | + | 0.294342i | −0.330062 | + | 0.571683i | 1.32025 | −0.500000 | + | 0.866025i | 0.169938 | + | 0.294342i | |||||||||||||||||||||||
| 16.3 | 1.13090 | + | 1.95878i | 0.500000 | + | 0.866025i | −1.55787 | + | 2.69832i | 1.00000 | −1.13090 | + | 1.95878i | 0.630901 | − | 1.09275i | −2.52360 | −0.500000 | + | 0.866025i | 1.13090 | + | 1.95878i | |||||||||||||||||||||||
| 61.1 | −1.30084 | + | 2.25312i | 0.500000 | − | 0.866025i | −2.38437 | − | 4.12985i | 1.00000 | 1.30084 | + | 2.25312i | −1.80084 | − | 3.11915i | 7.20336 | −0.500000 | − | 0.866025i | −1.30084 | + | 2.25312i | |||||||||||||||||||||||
| 61.2 | 0.169938 | − | 0.294342i | 0.500000 | − | 0.866025i | 0.942242 | + | 1.63201i | 1.00000 | −0.169938 | − | 0.294342i | −0.330062 | − | 0.571683i | 1.32025 | −0.500000 | − | 0.866025i | 0.169938 | − | 0.294342i | |||||||||||||||||||||||
| 61.3 | 1.13090 | − | 1.95878i | 0.500000 | − | 0.866025i | −1.55787 | − | 2.69832i | 1.00000 | −1.13090 | − | 1.95878i | 0.630901 | + | 1.09275i | −2.52360 | −0.500000 | − | 0.866025i | 1.13090 | − | 1.95878i | |||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 13.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 195.2.i.d | ✓ | 6 |
| 3.b | odd | 2 | 1 | 585.2.j.f | 6 | ||
| 5.b | even | 2 | 1 | 975.2.i.l | 6 | ||
| 5.c | odd | 4 | 2 | 975.2.bb.k | 12 | ||
| 13.c | even | 3 | 1 | inner | 195.2.i.d | ✓ | 6 |
| 13.c | even | 3 | 1 | 2535.2.a.bb | 3 | ||
| 13.e | even | 6 | 1 | 2535.2.a.ba | 3 | ||
| 39.h | odd | 6 | 1 | 7605.2.a.bw | 3 | ||
| 39.i | odd | 6 | 1 | 585.2.j.f | 6 | ||
| 39.i | odd | 6 | 1 | 7605.2.a.bv | 3 | ||
| 65.n | even | 6 | 1 | 975.2.i.l | 6 | ||
| 65.q | odd | 12 | 2 | 975.2.bb.k | 12 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 195.2.i.d | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 195.2.i.d | ✓ | 6 | 13.c | even | 3 | 1 | inner |
| 585.2.j.f | 6 | 3.b | odd | 2 | 1 | ||
| 585.2.j.f | 6 | 39.i | odd | 6 | 1 | ||
| 975.2.i.l | 6 | 5.b | even | 2 | 1 | ||
| 975.2.i.l | 6 | 65.n | even | 6 | 1 | ||
| 975.2.bb.k | 12 | 5.c | odd | 4 | 2 | ||
| 975.2.bb.k | 12 | 65.q | odd | 12 | 2 | ||
| 2535.2.a.ba | 3 | 13.e | even | 6 | 1 | ||
| 2535.2.a.bb | 3 | 13.c | even | 3 | 1 | ||
| 7605.2.a.bv | 3 | 39.i | odd | 6 | 1 | ||
| 7605.2.a.bw | 3 | 39.h | odd | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{6} + 6T_{2}^{4} - 4T_{2}^{3} + 36T_{2}^{2} - 12T_{2} + 4 \)
acting on \(S_{2}^{\mathrm{new}}(195, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} + 6 T^{4} + \cdots + 4 \)
|
| $3$ |
\( (T^{2} - T + 1)^{3} \)
|
| $5$ |
\( (T - 1)^{6} \)
|
| $7$ |
\( T^{6} + 3 T^{5} + \cdots + 9 \)
|
| $11$ |
\( T^{6} + 24 T^{4} + \cdots + 256 \)
|
| $13$ |
\( T^{6} - 3 T^{5} + \cdots + 2197 \)
|
| $17$ |
\( T^{6} + 42 T^{4} + \cdots + 9604 \)
|
| $19$ |
\( T^{6} + 12 T^{5} + \cdots + 16 \)
|
| $23$ |
\( T^{6} + 48 T^{4} + \cdots + 9216 \)
|
| $29$ |
\( T^{6} + 6 T^{5} + \cdots + 196 \)
|
| $31$ |
\( (T^{3} - 3 T^{2} + \cdots + 363)^{2} \)
|
| $37$ |
\( T^{6} + 6 T^{5} + \cdots + 64 \)
|
| $41$ |
\( T^{6} + 24 T^{4} + \cdots + 676 \)
|
| $43$ |
\( T^{6} + 9 T^{5} + \cdots + 121 \)
|
| $47$ |
\( (T^{3} + 12 T^{2} + \cdots - 206)^{2} \)
|
| $53$ |
\( (T^{3} - 24 T + 16)^{2} \)
|
| $59$ |
\( T^{6} - 6 T^{5} + \cdots + 196 \)
|
| $61$ |
\( T^{6} - 3 T^{5} + \cdots + 4489 \)
|
| $67$ |
\( T^{6} + 9 T^{5} + \cdots + 123201 \)
|
| $71$ |
\( T^{6} - 12 T^{5} + \cdots + 465124 \)
|
| $73$ |
\( (T^{3} - 21 T^{2} + \cdots + 89)^{2} \)
|
| $79$ |
\( (T^{3} + 3 T^{2} + \cdots - 363)^{2} \)
|
| $83$ |
\( (T^{3} + 18 T^{2} + \cdots - 168)^{2} \)
|
| $89$ |
\( T^{6} + 30 T^{5} + \cdots + 777924 \)
|
| $97$ |
\( T^{6} + 15 T^{5} + \cdots + 346921 \)
|
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