Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [38,2,Mod(5,38)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(38, base_ring=CyclotomicField(18))
chi = DirichletCharacter(H, H._module([16]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("38.5");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 38 = 2 \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 38.e (of order \(9\), degree \(6\), 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: | \(0.303431527681\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | \(\Q(\zeta_{18})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - x^{3} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{9}]$ |
$q$-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 primitive root of unity \(\zeta_{18}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/38\mathbb{Z}\right)^\times\).
| \(n\) | \(21\) |
| \(\chi(n)\) | \(\zeta_{18}^{2}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 5.1 |
|
−0.939693 | + | 0.342020i | −0.326352 | + | 1.85083i | 0.766044 | − | 0.642788i | 1.53209 | + | 1.28558i | −0.326352 | − | 1.85083i | −2.53209 | − | 4.38571i | −0.500000 | + | 0.866025i | −0.500000 | − | 0.181985i | −1.87939 | − | 0.684040i | ||||||||||||||||||
| 9.1 | 0.173648 | + | 0.984808i | 0.266044 | − | 0.223238i | −0.939693 | + | 0.342020i | −1.87939 | − | 0.684040i | 0.266044 | + | 0.223238i | 0.879385 | − | 1.52314i | −0.500000 | − | 0.866025i | −0.500000 | + | 2.83564i | 0.347296 | − | 1.96962i | |||||||||||||||||||
| 17.1 | 0.173648 | − | 0.984808i | 0.266044 | + | 0.223238i | −0.939693 | − | 0.342020i | −1.87939 | + | 0.684040i | 0.266044 | − | 0.223238i | 0.879385 | + | 1.52314i | −0.500000 | + | 0.866025i | −0.500000 | − | 2.83564i | 0.347296 | + | 1.96962i | |||||||||||||||||||
| 23.1 | −0.939693 | − | 0.342020i | −0.326352 | − | 1.85083i | 0.766044 | + | 0.642788i | 1.53209 | − | 1.28558i | −0.326352 | + | 1.85083i | −2.53209 | + | 4.38571i | −0.500000 | − | 0.866025i | −0.500000 | + | 0.181985i | −1.87939 | + | 0.684040i | |||||||||||||||||||
| 25.1 | 0.766044 | − | 0.642788i | −1.43969 | − | 0.524005i | 0.173648 | − | 0.984808i | 0.347296 | + | 1.96962i | −1.43969 | + | 0.524005i | −1.34730 | + | 2.33359i | −0.500000 | − | 0.866025i | −0.500000 | − | 0.419550i | 1.53209 | + | 1.28558i | |||||||||||||||||||
| 35.1 | 0.766044 | + | 0.642788i | −1.43969 | + | 0.524005i | 0.173648 | + | 0.984808i | 0.347296 | − | 1.96962i | −1.43969 | − | 0.524005i | −1.34730 | − | 2.33359i | −0.500000 | + | 0.866025i | −0.500000 | + | 0.419550i | 1.53209 | − | 1.28558i | |||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 19.e | even | 9 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 38.2.e.a | ✓ | 6 |
| 3.b | odd | 2 | 1 | 342.2.u.c | 6 | ||
| 4.b | odd | 2 | 1 | 304.2.u.c | 6 | ||
| 5.b | even | 2 | 1 | 950.2.l.d | 6 | ||
| 5.c | odd | 4 | 2 | 950.2.u.b | 12 | ||
| 19.b | odd | 2 | 1 | 722.2.e.k | 6 | ||
| 19.c | even | 3 | 1 | 722.2.e.b | 6 | ||
| 19.c | even | 3 | 1 | 722.2.e.m | 6 | ||
| 19.d | odd | 6 | 1 | 722.2.e.a | 6 | ||
| 19.d | odd | 6 | 1 | 722.2.e.l | 6 | ||
| 19.e | even | 9 | 1 | inner | 38.2.e.a | ✓ | 6 |
| 19.e | even | 9 | 1 | 722.2.a.l | 3 | ||
| 19.e | even | 9 | 2 | 722.2.c.k | 6 | ||
| 19.e | even | 9 | 1 | 722.2.e.b | 6 | ||
| 19.e | even | 9 | 1 | 722.2.e.m | 6 | ||
| 19.f | odd | 18 | 1 | 722.2.a.k | 3 | ||
| 19.f | odd | 18 | 2 | 722.2.c.l | 6 | ||
| 19.f | odd | 18 | 1 | 722.2.e.a | 6 | ||
| 19.f | odd | 18 | 1 | 722.2.e.k | 6 | ||
| 19.f | odd | 18 | 1 | 722.2.e.l | 6 | ||
| 57.j | even | 18 | 1 | 6498.2.a.bq | 3 | ||
| 57.l | odd | 18 | 1 | 342.2.u.c | 6 | ||
| 57.l | odd | 18 | 1 | 6498.2.a.bl | 3 | ||
| 76.k | even | 18 | 1 | 5776.2.a.bo | 3 | ||
| 76.l | odd | 18 | 1 | 304.2.u.c | 6 | ||
| 76.l | odd | 18 | 1 | 5776.2.a.bn | 3 | ||
| 95.p | even | 18 | 1 | 950.2.l.d | 6 | ||
| 95.q | odd | 36 | 2 | 950.2.u.b | 12 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 38.2.e.a | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 38.2.e.a | ✓ | 6 | 19.e | even | 9 | 1 | inner |
| 304.2.u.c | 6 | 4.b | odd | 2 | 1 | ||
| 304.2.u.c | 6 | 76.l | odd | 18 | 1 | ||
| 342.2.u.c | 6 | 3.b | odd | 2 | 1 | ||
| 342.2.u.c | 6 | 57.l | odd | 18 | 1 | ||
| 722.2.a.k | 3 | 19.f | odd | 18 | 1 | ||
| 722.2.a.l | 3 | 19.e | even | 9 | 1 | ||
| 722.2.c.k | 6 | 19.e | even | 9 | 2 | ||
| 722.2.c.l | 6 | 19.f | odd | 18 | 2 | ||
| 722.2.e.a | 6 | 19.d | odd | 6 | 1 | ||
| 722.2.e.a | 6 | 19.f | odd | 18 | 1 | ||
| 722.2.e.b | 6 | 19.c | even | 3 | 1 | ||
| 722.2.e.b | 6 | 19.e | even | 9 | 1 | ||
| 722.2.e.k | 6 | 19.b | odd | 2 | 1 | ||
| 722.2.e.k | 6 | 19.f | odd | 18 | 1 | ||
| 722.2.e.l | 6 | 19.d | odd | 6 | 1 | ||
| 722.2.e.l | 6 | 19.f | odd | 18 | 1 | ||
| 722.2.e.m | 6 | 19.c | even | 3 | 1 | ||
| 722.2.e.m | 6 | 19.e | even | 9 | 1 | ||
| 950.2.l.d | 6 | 5.b | even | 2 | 1 | ||
| 950.2.l.d | 6 | 95.p | even | 18 | 1 | ||
| 950.2.u.b | 12 | 5.c | odd | 4 | 2 | ||
| 950.2.u.b | 12 | 95.q | odd | 36 | 2 | ||
| 5776.2.a.bn | 3 | 76.l | odd | 18 | 1 | ||
| 5776.2.a.bo | 3 | 76.k | even | 18 | 1 | ||
| 6498.2.a.bl | 3 | 57.l | odd | 18 | 1 | ||
| 6498.2.a.bq | 3 | 57.j | even | 18 | 1 | ||
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(38, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} + T^{3} + 1 \)
|
| $3$ |
\( T^{6} + 3 T^{5} + 6 T^{4} + 8 T^{3} + \cdots + 1 \)
|
| $5$ |
\( T^{6} + 8T^{3} + 64 \)
|
| $7$ |
\( T^{6} + 6 T^{5} + 36 T^{4} + 48 T^{3} + \cdots + 576 \)
|
| $11$ |
\( T^{6} + 6 T^{5} + 33 T^{4} + 56 T^{3} + \cdots + 361 \)
|
| $13$ |
\( T^{6} - 12 T^{5} + 48 T^{4} - 64 T^{3} + \cdots + 64 \)
|
| $17$ |
\( T^{6} + 12 T^{5} + 108 T^{4} + \cdots + 12321 \)
|
| $19$ |
\( T^{6} - 18 T^{5} + 162 T^{4} + \cdots + 6859 \)
|
| $23$ |
\( T^{6} + 12 T^{5} + 60 T^{4} + 152 T^{3} + \cdots + 64 \)
|
| $29$ |
\( T^{6} + 18 T^{5} + 144 T^{4} + \cdots + 23104 \)
|
| $31$ |
\( T^{6} - 6 T^{5} + 60 T^{4} + 160 T^{3} + \cdots + 64 \)
|
| $37$ |
\( (T^{3} + 6 T^{2} - 24 T - 136)^{2} \)
|
| $41$ |
\( T^{6} - 3 T^{5} + 6 T^{4} - 8 T^{3} + \cdots + 1 \)
|
| $43$ |
\( T^{6} + 6 T^{5} + 42 T^{4} + 271 T^{3} + \cdots + 289 \)
|
| $47$ |
\( T^{6} - 30 T^{5} + 372 T^{4} + \cdots + 87616 \)
|
| $53$ |
\( T^{6} - 24 T^{5} + 276 T^{4} + \cdots + 18496 \)
|
| $59$ |
\( T^{6} + 3 T^{5} + 54 T^{4} + 24 T^{3} + \cdots + 9 \)
|
| $61$ |
\( T^{6} - 6 T^{5} - 12 T^{4} + \cdots + 23104 \)
|
| $67$ |
\( T^{6} + 9 T^{5} + 162 T^{4} + \cdots + 6561 \)
|
| $71$ |
\( T^{6} + 18 T^{5} + 216 T^{4} + \cdots + 23104 \)
|
| $73$ |
\( T^{6} + 30 T^{5} + 279 T^{4} + \cdots + 3249 \)
|
| $79$ |
\( T^{6} - 6 T^{5} - 48 T^{4} + \cdots + 18496 \)
|
| $83$ |
\( T^{6} + 6 T^{5} + 63 T^{4} + \cdots + 2601 \)
|
| $89$ |
\( T^{6} + 36 T^{4} + 315 T^{3} + \cdots + 962361 \)
|
| $97$ |
\( T^{6} - 3 T^{5} + 6 T^{4} - 8 T^{3} + \cdots + 1 \)
|
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