Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [30,6,Mod(19,30)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(30, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("30.19");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 30 = 2 \cdot 3 \cdot 5 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 30.c (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: | \(4.81151459439\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(i, \sqrt{1249})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} + 625x^{2} + 97344 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2\cdot 5 \) |
| 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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} + 625x^{2} + 97344 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{3} + 313\nu ) / 312 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{3} + 312\nu^{2} - 1249\nu + 97656 ) / 312 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} + 936\nu^{2} + 625\nu + 292656 ) / 312 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} - 3\beta_{2} - 4\beta _1 + 1 ) / 10 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 3\beta_{3} + \beta_{2} - 2\beta _1 - 3127 ) / 10 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -313\beta_{3} + 939\beta_{2} + 4372\beta _1 - 313 ) / 10 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/30\mathbb{Z}\right)^\times\).
| \(n\) | \(7\) | \(11\) |
| \(\chi(n)\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 19.1 |
|
− | 4.00000i | 9.00000i | −16.0000 | −16.1706 | + | 53.5118i | 36.0000 | 119.706i | 64.0000i | −81.0000 | 214.047 | + | 64.6824i | |||||||||||||||||||||||||
| 19.2 | − | 4.00000i | 9.00000i | −16.0000 | 19.1706 | − | 52.5118i | 36.0000 | − | 233.706i | 64.0000i | −81.0000 | −210.047 | − | 76.6824i | |||||||||||||||||||||||||
| 19.3 | 4.00000i | − | 9.00000i | −16.0000 | −16.1706 | − | 53.5118i | 36.0000 | − | 119.706i | − | 64.0000i | −81.0000 | 214.047 | − | 64.6824i | ||||||||||||||||||||||||
| 19.4 | 4.00000i | − | 9.00000i | −16.0000 | 19.1706 | + | 52.5118i | 36.0000 | 233.706i | − | 64.0000i | −81.0000 | −210.047 | + | 76.6824i | |||||||||||||||||||||||||
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 | 30.6.c.b | ✓ | 4 |
| 3.b | odd | 2 | 1 | 90.6.c.c | 4 | ||
| 4.b | odd | 2 | 1 | 240.6.f.b | 4 | ||
| 5.b | even | 2 | 1 | inner | 30.6.c.b | ✓ | 4 |
| 5.c | odd | 4 | 1 | 150.6.a.n | 2 | ||
| 5.c | odd | 4 | 1 | 150.6.a.o | 2 | ||
| 12.b | even | 2 | 1 | 720.6.f.i | 4 | ||
| 15.d | odd | 2 | 1 | 90.6.c.c | 4 | ||
| 15.e | even | 4 | 1 | 450.6.a.bb | 2 | ||
| 15.e | even | 4 | 1 | 450.6.a.bc | 2 | ||
| 20.d | odd | 2 | 1 | 240.6.f.b | 4 | ||
| 60.h | even | 2 | 1 | 720.6.f.i | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 30.6.c.b | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 30.6.c.b | ✓ | 4 | 5.b | even | 2 | 1 | inner |
| 90.6.c.c | 4 | 3.b | odd | 2 | 1 | ||
| 90.6.c.c | 4 | 15.d | odd | 2 | 1 | ||
| 150.6.a.n | 2 | 5.c | odd | 4 | 1 | ||
| 150.6.a.o | 2 | 5.c | odd | 4 | 1 | ||
| 240.6.f.b | 4 | 4.b | odd | 2 | 1 | ||
| 240.6.f.b | 4 | 20.d | odd | 2 | 1 | ||
| 450.6.a.bb | 2 | 15.e | even | 4 | 1 | ||
| 450.6.a.bc | 2 | 15.e | even | 4 | 1 | ||
| 720.6.f.i | 4 | 12.b | even | 2 | 1 | ||
| 720.6.f.i | 4 | 60.h | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{4} + 68948T_{7}^{2} + 782656576 \)
acting on \(S_{6}^{\mathrm{new}}(30, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 16)^{2} \)
|
| $3$ |
\( (T^{2} + 81)^{2} \)
|
| $5$ |
\( T^{4} - 6 T^{3} + \cdots + 9765625 \)
|
| $7$ |
\( T^{4} + 68948 T^{2} + 782656576 \)
|
| $11$ |
\( (T^{2} - 174 T - 23656)^{2} \)
|
| $13$ |
\( T^{4} + \cdots + 31678304256 \)
|
| $17$ |
\( T^{4} + \cdots + 85278016576 \)
|
| $19$ |
\( (T^{2} - 120 T - 4492800)^{2} \)
|
| $23$ |
\( T^{4} + \cdots + 56918507776 \)
|
| $29$ |
\( (T^{2} - 2430 T - 21286800)^{2} \)
|
| $31$ |
\( (T^{2} - 11684 T + 33004864)^{2} \)
|
| $37$ |
\( T^{4} + \cdots + 43\!\cdots\!76 \)
|
| $41$ |
\( (T^{2} - 24984 T + 119953964)^{2} \)
|
| $43$ |
\( T^{4} + \cdots + 23\!\cdots\!36 \)
|
| $47$ |
\( T^{4} + \cdots + 18\!\cdots\!96 \)
|
| $53$ |
\( T^{4} + \cdots + 11\!\cdots\!76 \)
|
| $59$ |
\( (T^{2} - 23730 T - 927897400)^{2} \)
|
| $61$ |
\( (T^{2} - 57124 T + 528018244)^{2} \)
|
| $67$ |
\( T^{4} + \cdots + 18\!\cdots\!96 \)
|
| $71$ |
\( (T^{2} + 11076 T - 2668294656)^{2} \)
|
| $73$ |
\( T^{4} + \cdots + 27\!\cdots\!76 \)
|
| $79$ |
\( (T^{2} + 14220 T - 1173592800)^{2} \)
|
| $83$ |
\( T^{4} + \cdots + 26\!\cdots\!56 \)
|
| $89$ |
\( (T^{2} - 60420 T - 336355900)^{2} \)
|
| $97$ |
\( T^{4} + \cdots + 53\!\cdots\!96 \)
|
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