Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [190,2,Mod(39,190)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(190, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("190.39");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 190 = 2 \cdot 5 \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 190.b (of order \(2\), degree \(1\), 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: | \(1.51715763840\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.0.5161984.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 4x^{3} + 25x^{2} - 20x + 8 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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
\( x^{6} - 4x^{3} + 25x^{2} - 20x + 8 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -5\nu^{5} - 2\nu^{4} - 25\nu^{3} + 10\nu^{2} - 121\nu + 100 ) / 121 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 7\nu^{5} + 27\nu^{4} + 35\nu^{3} - 14\nu^{2} + 223 ) / 121 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -25\nu^{5} - 10\nu^{4} - 4\nu^{3} + 50\nu^{2} - 605\nu + 258 ) / 242 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -65\nu^{5} - 26\nu^{4} + 38\nu^{3} + 372\nu^{2} - 1331\nu + 574 ) / 242 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{5} - 3\beta_{4} + \beta_{2} - \beta_1 \)
|
| \(\nu^{3}\) | \(=\) |
\( 2\beta_{4} - 5\beta_{2} + 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( 5\beta_{3} + 7\beta_{2} + 7\beta _1 - 15 \)
|
| \(\nu^{5}\) | \(=\) |
\( 2\beta_{5} - 16\beta_{4} - 2\beta_{3} - 29\beta _1 + 16 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/190\mathbb{Z}\right)^\times\).
| \(n\) | \(21\) | \(77\) |
| \(\chi(n)\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 39.1 |
|
− | 1.00000i | − | 2.12489i | −1.00000 | 1.80487 | + | 1.32001i | −2.12489 | − | 4.12489i | 1.00000i | −1.51514 | 1.32001 | − | 1.80487i | |||||||||||||||||||||||||||||
| 39.2 | − | 1.00000i | 1.36333i | −1.00000 | 1.38900 | − | 1.75233i | 1.36333 | − | 0.636672i | 1.00000i | 1.14134 | −1.75233 | − | 1.38900i | |||||||||||||||||||||||||||||||
| 39.3 | − | 1.00000i | 2.76156i | −1.00000 | −2.19388 | + | 0.432320i | 2.76156 | 0.761557i | 1.00000i | −4.62620 | 0.432320 | + | 2.19388i | ||||||||||||||||||||||||||||||||
| 39.4 | 1.00000i | − | 2.76156i | −1.00000 | −2.19388 | − | 0.432320i | 2.76156 | − | 0.761557i | − | 1.00000i | −4.62620 | 0.432320 | − | 2.19388i | ||||||||||||||||||||||||||||||
| 39.5 | 1.00000i | − | 1.36333i | −1.00000 | 1.38900 | + | 1.75233i | 1.36333 | 0.636672i | − | 1.00000i | 1.14134 | −1.75233 | + | 1.38900i | |||||||||||||||||||||||||||||||
| 39.6 | 1.00000i | 2.12489i | −1.00000 | 1.80487 | − | 1.32001i | −2.12489 | 4.12489i | − | 1.00000i | −1.51514 | 1.32001 | + | 1.80487i | ||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 190.2.b.b | ✓ | 6 |
| 3.b | odd | 2 | 1 | 1710.2.d.d | 6 | ||
| 4.b | odd | 2 | 1 | 1520.2.d.j | 6 | ||
| 5.b | even | 2 | 1 | inner | 190.2.b.b | ✓ | 6 |
| 5.c | odd | 4 | 1 | 950.2.a.i | 3 | ||
| 5.c | odd | 4 | 1 | 950.2.a.n | 3 | ||
| 15.d | odd | 2 | 1 | 1710.2.d.d | 6 | ||
| 15.e | even | 4 | 1 | 8550.2.a.ck | 3 | ||
| 15.e | even | 4 | 1 | 8550.2.a.cl | 3 | ||
| 20.d | odd | 2 | 1 | 1520.2.d.j | 6 | ||
| 20.e | even | 4 | 1 | 7600.2.a.bi | 3 | ||
| 20.e | even | 4 | 1 | 7600.2.a.cd | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 190.2.b.b | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 190.2.b.b | ✓ | 6 | 5.b | even | 2 | 1 | inner |
| 950.2.a.i | 3 | 5.c | odd | 4 | 1 | ||
| 950.2.a.n | 3 | 5.c | odd | 4 | 1 | ||
| 1520.2.d.j | 6 | 4.b | odd | 2 | 1 | ||
| 1520.2.d.j | 6 | 20.d | odd | 2 | 1 | ||
| 1710.2.d.d | 6 | 3.b | odd | 2 | 1 | ||
| 1710.2.d.d | 6 | 15.d | odd | 2 | 1 | ||
| 7600.2.a.bi | 3 | 20.e | even | 4 | 1 | ||
| 7600.2.a.cd | 3 | 20.e | even | 4 | 1 | ||
| 8550.2.a.ck | 3 | 15.e | even | 4 | 1 | ||
| 8550.2.a.cl | 3 | 15.e | even | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{6} + 14T_{3}^{4} + 57T_{3}^{2} + 64 \)
acting on \(S_{2}^{\mathrm{new}}(190, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 1)^{3} \)
|
| $3$ |
\( T^{6} + 14 T^{4} + \cdots + 64 \)
|
| $5$ |
\( T^{6} - 2 T^{5} + \cdots + 125 \)
|
| $7$ |
\( T^{6} + 18 T^{4} + \cdots + 4 \)
|
| $11$ |
\( (T^{3} - 10 T - 8)^{2} \)
|
| $13$ |
\( T^{6} + 38 T^{4} + \cdots + 4 \)
|
| $17$ |
\( T^{6} + 18 T^{4} + \cdots + 16 \)
|
| $19$ |
\( (T + 1)^{6} \)
|
| $23$ |
\( T^{6} + 98 T^{4} + \cdots + 14884 \)
|
| $29$ |
\( (T^{3} - 8 T^{2} + \cdots + 410)^{2} \)
|
| $31$ |
\( (T^{3} - 4 T^{2} + \cdots + 232)^{2} \)
|
| $37$ |
\( T^{6} + 116 T^{4} + \cdots + 256 \)
|
| $41$ |
\( (T^{3} - 2 T^{2} + \cdots - 100)^{2} \)
|
| $43$ |
\( T^{6} + 128 T^{4} + \cdots + 21904 \)
|
| $47$ |
\( T^{6} + 132 T^{4} + \cdots + 4096 \)
|
| $53$ |
\( T^{6} + 102 T^{4} + \cdots + 11236 \)
|
| $59$ |
\( (T^{3} + 2 T^{2} - 29 T - 80)^{2} \)
|
| $61$ |
\( (T^{3} + 30 T^{2} + \cdots + 892)^{2} \)
|
| $67$ |
\( T^{6} + 126 T^{4} + \cdots + 4096 \)
|
| $71$ |
\( (T^{3} + 8 T^{2} + \cdots - 1016)^{2} \)
|
| $73$ |
\( T^{6} + 290 T^{4} + \cdots + 26896 \)
|
| $79$ |
\( (T^{3} - 228 T + 880)^{2} \)
|
| $83$ |
\( T^{6} + 92 T^{4} + \cdots + 64 \)
|
| $89$ |
\( (T^{3} - 14 T^{2} + \cdots + 20)^{2} \)
|
| $97$ |
\( T^{6} + 300 T^{4} + \cdots + 238144 \)
|
| show more | |
| show less |