Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [95,2,Mod(11,95)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(95, 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("95.11");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 95 = 5 \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 95.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: | \(0.758578819202\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Relative dimension: | \(3\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | 6.0.3518667.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - x^{5} + 7x^{4} - 8x^{3} + 43x^{2} - 42x + 49 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| 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
\( x^{6} - x^{5} + 7x^{4} - 8x^{3} + 43x^{2} - 42x + 49 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{5} + 7\nu^{4} - 49\nu^{3} + 43\nu^{2} - 42\nu + 294 ) / 259 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -6\nu^{5} + 5\nu^{4} - 35\nu^{3} - \nu^{2} - 215\nu - 49 ) / 259 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -5\nu^{5} + 35\nu^{4} + 14\nu^{3} + 215\nu^{2} - 210\nu + 952 ) / 259 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 18\nu^{5} + 22\nu^{4} + 105\nu^{3} + 3\nu^{2} + 608\nu + 147 ) / 259 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{5} + \beta_{4} - 4\beta_{3} + \beta_{2} - 5 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{4} - 5\beta_{2} + 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( 7\beta_{5} + 21\beta_{3} + \beta_1 \)
|
| \(\nu^{5}\) | \(=\) |
\( 6\beta_{5} - 6\beta_{4} - 25\beta_{3} + 29\beta_{2} - 35\beta _1 - 19 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/95\mathbb{Z}\right)^\times\).
| \(n\) | \(21\) | \(77\) |
| \(\chi(n)\) | \(-1 - \beta_{3}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 11.1 |
|
−1.14257 | + | 1.97899i | 1.25351 | − | 2.17114i | −1.61094 | − | 2.79023i | 0.500000 | − | 0.866025i | 2.86445 | + | 4.96137i | 3.50702 | 2.79216 | −1.64257 | − | 2.84502i | 1.14257 | + | 1.97899i | ||||||||||||||||||||||
| 11.2 | −0.610938 | + | 1.05818i | −1.14257 | + | 1.97899i | 0.253509 | + | 0.439091i | 0.500000 | − | 0.866025i | −1.39608 | − | 2.41808i | −1.28514 | −3.06327 | −1.11094 | − | 1.92420i | 0.610938 | + | 1.05818i | |||||||||||||||||||||||
| 11.3 | 1.25351 | − | 2.17114i | −0.610938 | + | 1.05818i | −2.14257 | − | 3.71104i | 0.500000 | − | 0.866025i | 1.53163 | + | 2.65287i | −0.221876 | −5.72889 | 0.753509 | + | 1.30512i | −1.25351 | − | 2.17114i | |||||||||||||||||||||||
| 26.1 | −1.14257 | − | 1.97899i | 1.25351 | + | 2.17114i | −1.61094 | + | 2.79023i | 0.500000 | + | 0.866025i | 2.86445 | − | 4.96137i | 3.50702 | 2.79216 | −1.64257 | + | 2.84502i | 1.14257 | − | 1.97899i | |||||||||||||||||||||||
| 26.2 | −0.610938 | − | 1.05818i | −1.14257 | − | 1.97899i | 0.253509 | − | 0.439091i | 0.500000 | + | 0.866025i | −1.39608 | + | 2.41808i | −1.28514 | −3.06327 | −1.11094 | + | 1.92420i | 0.610938 | − | 1.05818i | |||||||||||||||||||||||
| 26.3 | 1.25351 | + | 2.17114i | −0.610938 | − | 1.05818i | −2.14257 | + | 3.71104i | 0.500000 | + | 0.866025i | 1.53163 | − | 2.65287i | −0.221876 | −5.72889 | 0.753509 | − | 1.30512i | −1.25351 | + | 2.17114i | |||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 19.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 95.2.e.b | ✓ | 6 |
| 3.b | odd | 2 | 1 | 855.2.k.g | 6 | ||
| 4.b | odd | 2 | 1 | 1520.2.q.j | 6 | ||
| 5.b | even | 2 | 1 | 475.2.e.d | 6 | ||
| 5.c | odd | 4 | 2 | 475.2.j.b | 12 | ||
| 19.c | even | 3 | 1 | inner | 95.2.e.b | ✓ | 6 |
| 19.c | even | 3 | 1 | 1805.2.a.h | 3 | ||
| 19.d | odd | 6 | 1 | 1805.2.a.g | 3 | ||
| 57.h | odd | 6 | 1 | 855.2.k.g | 6 | ||
| 76.g | odd | 6 | 1 | 1520.2.q.j | 6 | ||
| 95.h | odd | 6 | 1 | 9025.2.a.ba | 3 | ||
| 95.i | even | 6 | 1 | 475.2.e.d | 6 | ||
| 95.i | even | 6 | 1 | 9025.2.a.z | 3 | ||
| 95.m | odd | 12 | 2 | 475.2.j.b | 12 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 95.2.e.b | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 95.2.e.b | ✓ | 6 | 19.c | even | 3 | 1 | inner |
| 475.2.e.d | 6 | 5.b | even | 2 | 1 | ||
| 475.2.e.d | 6 | 95.i | even | 6 | 1 | ||
| 475.2.j.b | 12 | 5.c | odd | 4 | 2 | ||
| 475.2.j.b | 12 | 95.m | odd | 12 | 2 | ||
| 855.2.k.g | 6 | 3.b | odd | 2 | 1 | ||
| 855.2.k.g | 6 | 57.h | odd | 6 | 1 | ||
| 1520.2.q.j | 6 | 4.b | odd | 2 | 1 | ||
| 1520.2.q.j | 6 | 76.g | odd | 6 | 1 | ||
| 1805.2.a.g | 3 | 19.d | odd | 6 | 1 | ||
| 1805.2.a.h | 3 | 19.c | even | 3 | 1 | ||
| 9025.2.a.z | 3 | 95.i | even | 6 | 1 | ||
| 9025.2.a.ba | 3 | 95.h | odd | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{6} + T_{2}^{5} + 7T_{2}^{4} + 8T_{2}^{3} + 43T_{2}^{2} + 42T_{2} + 49 \)
acting on \(S_{2}^{\mathrm{new}}(95, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} + T^{5} + \cdots + 49 \)
|
| $3$ |
\( T^{6} + T^{5} + \cdots + 49 \)
|
| $5$ |
\( (T^{2} - T + 1)^{3} \)
|
| $7$ |
\( (T^{3} - 2 T^{2} - 5 T - 1)^{2} \)
|
| $11$ |
\( (T^{3} + 5 T^{2} + 2 T - 1)^{2} \)
|
| $13$ |
\( (T^{2} - 5 T + 25)^{3} \)
|
| $17$ |
\( T^{6} + T^{5} + \cdots + 49 \)
|
| $19$ |
\( T^{6} + 133T^{3} + 6859 \)
|
| $23$ |
\( T^{6} + 4 T^{5} + \cdots + 2401 \)
|
| $29$ |
\( T^{6} - 2 T^{5} + \cdots + 1 \)
|
| $31$ |
\( (T^{3} + T^{2} - 6 T - 7)^{2} \)
|
| $37$ |
\( (T^{3} + 2 T^{2} + \cdots - 227)^{2} \)
|
| $41$ |
\( T^{6} - 2 T^{5} + \cdots + 1369 \)
|
| $43$ |
\( T^{6} - T^{5} + \cdots + 14641 \)
|
| $47$ |
\( T^{6} + 6 T^{5} + \cdots + 2401 \)
|
| $53$ |
\( T^{6} + 11 T^{5} + \cdots + 96721 \)
|
| $59$ |
\( T^{6} + 6 T^{5} + \cdots + 2401 \)
|
| $61$ |
\( T^{6} - 9 T^{5} + \cdots + 2401 \)
|
| $67$ |
\( T^{6} - 20 T^{5} + \cdots + 7744 \)
|
| $71$ |
\( T^{6} - 29 T^{5} + \cdots + 218089 \)
|
| $73$ |
\( T^{6} - 22 T^{5} + \cdots + 5929 \)
|
| $79$ |
\( T^{6} - 24 T^{5} + \cdots + 61504 \)
|
| $83$ |
\( (T^{3} + 3 T^{2} - 54 T + 77)^{2} \)
|
| $89$ |
\( T^{6} - 14 T^{5} + \cdots + 3136 \)
|
| $97$ |
\( T^{6} + 7 T^{5} + \cdots + 14641 \)
|
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