Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [185,2,Mod(43,185)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(185, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([3, 3]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("185.43");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 185 = 5 \cdot 37 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 185.f (of order \(4\), 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: | \(1.47723243739\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Relative dimension: | \(3\) over \(\Q(i)\) |
| Coefficient field: | 6.0.350464.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-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
\( x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -4\nu^{5} + 9\nu^{4} - 16\nu^{3} - 4\nu^{2} - 15\nu + 14 ) / 23 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -5\nu^{5} + 17\nu^{4} - 20\nu^{3} - 5\nu^{2} + 10\nu + 29 ) / 23 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 7\nu^{5} - 10\nu^{4} + 5\nu^{3} + 30\nu^{2} + 32\nu - 13 ) / 23 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -11\nu^{5} + 19\nu^{4} - 21\nu^{3} - 11\nu^{2} - 70\nu + 27 ) / 23 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{5} + \beta_{4} - \beta_{2} + \beta_1 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{5} + \beta_{3} - 4\beta_{2} \)
|
| \(\nu^{4}\) | \(=\) |
\( 4\beta_{3} - 5\beta_{2} - 5\beta _1 - 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( -5\beta_{5} - \beta_{4} + 5\beta_{3} - 16\beta _1 - 1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/185\mathbb{Z}\right)^\times\).
| \(n\) | \(76\) | \(112\) |
| \(\chi(n)\) | \(\beta_{4}\) | \(\beta_{4}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 43.1 |
|
− | 1.48119i | −2.07816 | + | 2.07816i | −0.193937 | −1.48119 | − | 1.67513i | 3.07816 | + | 3.07816i | 1.07816 | − | 1.07816i | − | 2.67513i | − | 5.63752i | −2.48119 | + | 2.19394i | |||||||||||||||||||||||
| 43.2 | 0.311108i | 0.762714 | − | 0.762714i | 1.90321 | 0.311108 | + | 2.21432i | 0.237286 | + | 0.237286i | −1.76271 | + | 1.76271i | 1.21432i | 1.83654i | −0.688892 | + | 0.0967881i | |||||||||||||||||||||||||||
| 43.3 | 2.17009i | 0.315449 | − | 0.315449i | −2.70928 | 2.17009 | − | 0.539189i | 0.684551 | + | 0.684551i | −1.31545 | + | 1.31545i | − | 1.53919i | 2.80098i | 1.17009 | + | 4.70928i | ||||||||||||||||||||||||||
| 142.1 | − | 2.17009i | 0.315449 | + | 0.315449i | −2.70928 | 2.17009 | + | 0.539189i | 0.684551 | − | 0.684551i | −1.31545 | − | 1.31545i | 1.53919i | − | 2.80098i | 1.17009 | − | 4.70928i | |||||||||||||||||||||||||
| 142.2 | − | 0.311108i | 0.762714 | + | 0.762714i | 1.90321 | 0.311108 | − | 2.21432i | 0.237286 | − | 0.237286i | −1.76271 | − | 1.76271i | − | 1.21432i | − | 1.83654i | −0.688892 | − | 0.0967881i | ||||||||||||||||||||||||
| 142.3 | 1.48119i | −2.07816 | − | 2.07816i | −0.193937 | −1.48119 | + | 1.67513i | 3.07816 | − | 3.07816i | 1.07816 | + | 1.07816i | 2.67513i | 5.63752i | −2.48119 | − | 2.19394i | |||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 185.f | even | 4 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 185.2.f.c | ✓ | 6 |
| 5.b | even | 2 | 1 | 925.2.f.c | 6 | ||
| 5.c | odd | 4 | 1 | 185.2.k.c | yes | 6 | |
| 5.c | odd | 4 | 1 | 925.2.k.c | 6 | ||
| 37.d | odd | 4 | 1 | 185.2.k.c | yes | 6 | |
| 185.f | even | 4 | 1 | inner | 185.2.f.c | ✓ | 6 |
| 185.j | odd | 4 | 1 | 925.2.k.c | 6 | ||
| 185.k | even | 4 | 1 | 925.2.f.c | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 185.2.f.c | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 185.2.f.c | ✓ | 6 | 185.f | even | 4 | 1 | inner |
| 185.2.k.c | yes | 6 | 5.c | odd | 4 | 1 | |
| 185.2.k.c | yes | 6 | 37.d | odd | 4 | 1 | |
| 925.2.f.c | 6 | 5.b | even | 2 | 1 | ||
| 925.2.f.c | 6 | 185.k | even | 4 | 1 | ||
| 925.2.k.c | 6 | 5.c | odd | 4 | 1 | ||
| 925.2.k.c | 6 | 185.j | odd | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(185, [\chi])\):
|
\( T_{2}^{6} + 7T_{2}^{4} + 11T_{2}^{2} + 1 \)
|
|
\( T_{3}^{6} + 2T_{3}^{5} + 2T_{3}^{4} - 10T_{3}^{3} + 16T_{3}^{2} - 8T_{3} + 2 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} + 7 T^{4} + \cdots + 1 \)
|
| $3$ |
\( T^{6} + 2 T^{5} + \cdots + 2 \)
|
| $5$ |
\( T^{6} - 2 T^{5} + \cdots + 125 \)
|
| $7$ |
\( T^{6} + 4 T^{5} + \cdots + 50 \)
|
| $11$ |
\( T^{6} + 68 T^{4} + \cdots + 10000 \)
|
| $13$ |
\( T^{6} + 64 T^{4} + \cdots + 5476 \)
|
| $17$ |
\( (T^{3} + 4 T^{2} - 8 T - 34)^{2} \)
|
| $19$ |
\( T^{6} - 14 T^{5} + \cdots + 578 \)
|
| $23$ |
\( T^{6} + 68 T^{4} + \cdots + 16 \)
|
| $29$ |
\( T^{6} + 6 T^{5} + \cdots + 8 \)
|
| $31$ |
\( T^{6} + 134 T^{3} + \cdots + 8978 \)
|
| $37$ |
\( (T^{2} + 2 T + 37)^{3} \)
|
| $41$ |
\( T^{6} + 264 T^{4} + \cdots + 384400 \)
|
| $43$ |
\( T^{6} + 160 T^{4} + \cdots + 71824 \)
|
| $47$ |
\( T^{6} + 8 T^{5} + \cdots + 1250 \)
|
| $53$ |
\( T^{6} + 14 T^{5} + \cdots + 200 \)
|
| $59$ |
\( T^{6} - 24 T^{5} + \cdots + 38642 \)
|
| $61$ |
\( T^{6} - 26 T^{5} + \cdots + 27848 \)
|
| $67$ |
\( T^{6} - 22 T^{5} + \cdots + 2 \)
|
| $71$ |
\( (T^{3} - 2 T^{2} + \cdots + 184)^{2} \)
|
| $73$ |
\( T^{6} + 38 T^{5} + \cdots + 63368 \)
|
| $79$ |
\( T^{6} + 24 T^{5} + \cdots + 8978 \)
|
| $83$ |
\( T^{6} - 18 T^{5} + \cdots + 1250 \)
|
| $89$ |
\( T^{6} + 6 T^{5} + \cdots + 63368 \)
|
| $97$ |
\( (T^{3} + 2 T^{2} - 8 T + 4)^{2} \)
|
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