Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [930,2,Mod(559,930)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(930, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 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("930.559");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 930 = 2 \cdot 3 \cdot 5 \cdot 31 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 930.d (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: | \(7.42608738798\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.0.3534400.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 2x^{5} - 3x^{4} + 16x^{3} + x^{2} - 12x + 40 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| 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
\( x^{6} - 2x^{5} - 3x^{4} + 16x^{3} + x^{2} - 12x + 40 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 4\nu^{5} - 70\nu^{4} + 183\nu^{3} + 120\nu^{2} - 966\nu + 240 ) / 445 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 13\nu^{5} - 5\nu^{4} - 184\nu^{3} + 390\nu^{2} + 643\nu - 1000 ) / 445 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -3\nu^{5} + 8\nu^{4} - 26\nu^{3} - \nu^{2} + 57\nu - 180 ) / 89 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -9\nu^{5} + 24\nu^{4} + 11\nu^{3} - 92\nu^{2} - 7\nu - 6 ) / 178 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( 2\beta_{5} - \beta_{4} + 2\beta_{3} + \beta_{2} + 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( 4\beta_{5} - 4\beta_{4} + 2\beta_{3} + \beta_{2} + 2\beta _1 - 4 \)
|
| \(\nu^{4}\) | \(=\) |
\( 14\beta_{5} - 14\beta_{4} + 9\beta_{3} - 3\beta_{2} - 10\beta _1 - 6 \)
|
| \(\nu^{5}\) | \(=\) |
\( 2\beta_{5} - 32\beta_{4} + 6\beta_{3} - 17\beta_{2} - 25\beta _1 - 42 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/930\mathbb{Z}\right)^\times\).
| \(n\) | \(187\) | \(311\) | \(871\) |
| \(\chi(n)\) | \(-1\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 559.1 |
|
− | 1.00000i | − | 1.00000i | −1.00000 | −2.14620 | + | 0.627553i | −1.00000 | − | 0.255105i | 1.00000i | −1.00000 | 0.627553 | + | 2.14620i | |||||||||||||||||||||||||||||
| 559.2 | − | 1.00000i | − | 1.00000i | −1.00000 | −1.30135 | − | 1.81837i | −1.00000 | 4.63675i | 1.00000i | −1.00000 | −1.81837 | + | 1.30135i | |||||||||||||||||||||||||||||||
| 559.3 | − | 1.00000i | − | 1.00000i | −1.00000 | 0.447553 | + | 2.19082i | −1.00000 | − | 3.38164i | 1.00000i | −1.00000 | 2.19082 | − | 0.447553i | ||||||||||||||||||||||||||||||
| 559.4 | 1.00000i | 1.00000i | −1.00000 | −2.14620 | − | 0.627553i | −1.00000 | 0.255105i | − | 1.00000i | −1.00000 | 0.627553 | − | 2.14620i | ||||||||||||||||||||||||||||||||
| 559.5 | 1.00000i | 1.00000i | −1.00000 | −1.30135 | + | 1.81837i | −1.00000 | − | 4.63675i | − | 1.00000i | −1.00000 | −1.81837 | − | 1.30135i | |||||||||||||||||||||||||||||||
| 559.6 | 1.00000i | 1.00000i | −1.00000 | 0.447553 | − | 2.19082i | −1.00000 | 3.38164i | − | 1.00000i | −1.00000 | 2.19082 | + | 0.447553i | ||||||||||||||||||||||||||||||||
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 | 930.2.d.h | ✓ | 6 |
| 3.b | odd | 2 | 1 | 2790.2.d.k | 6 | ||
| 5.b | even | 2 | 1 | inner | 930.2.d.h | ✓ | 6 |
| 5.c | odd | 4 | 1 | 4650.2.a.ck | 3 | ||
| 5.c | odd | 4 | 1 | 4650.2.a.cn | 3 | ||
| 15.d | odd | 2 | 1 | 2790.2.d.k | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 930.2.d.h | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 930.2.d.h | ✓ | 6 | 5.b | even | 2 | 1 | inner |
| 2790.2.d.k | 6 | 3.b | odd | 2 | 1 | ||
| 2790.2.d.k | 6 | 15.d | odd | 2 | 1 | ||
| 4650.2.a.ck | 3 | 5.c | odd | 4 | 1 | ||
| 4650.2.a.cn | 3 | 5.c | odd | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(930, [\chi])\):
|
\( T_{7}^{6} + 33T_{7}^{4} + 248T_{7}^{2} + 16 \)
|
|
\( T_{11}^{3} - T_{11}^{2} - 35T_{11} + 67 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 1)^{3} \)
|
| $3$ |
\( (T^{2} + 1)^{3} \)
|
| $5$ |
\( T^{6} + 6 T^{5} + \cdots + 125 \)
|
| $7$ |
\( T^{6} + 33 T^{4} + \cdots + 16 \)
|
| $11$ |
\( (T^{3} - T^{2} - 35 T + 67)^{2} \)
|
| $13$ |
\( T^{6} + 62 T^{4} + \cdots + 1600 \)
|
| $17$ |
\( T^{6} + 42 T^{4} + \cdots + 400 \)
|
| $19$ |
\( (T^{3} + T^{2} - 45 T - 85)^{2} \)
|
| $23$ |
\( T^{6} + 41 T^{4} + \cdots + 1600 \)
|
| $29$ |
\( (T^{3} + 6 T^{2} - 16 T - 16)^{2} \)
|
| $31$ |
\( (T - 1)^{6} \)
|
| $37$ |
\( T^{6} \)
|
| $41$ |
\( (T^{3} - 6 T^{2} + \cdots + 472)^{2} \)
|
| $43$ |
\( T^{6} + 33 T^{4} + \cdots + 16 \)
|
| $47$ |
\( T^{6} + 106 T^{4} + \cdots + 1600 \)
|
| $53$ |
\( T^{6} + 161 T^{4} + \cdots + 8464 \)
|
| $59$ |
\( (T^{3} + 2 T^{2} - 44 T + 40)^{2} \)
|
| $61$ |
\( (T^{3} - 16 T^{2} + \cdots - 10)^{2} \)
|
| $67$ |
\( T^{6} + 114 T^{4} + \cdots + 11236 \)
|
| $71$ |
\( (T^{3} - 9 T^{2} - 75 T + 43)^{2} \)
|
| $73$ |
\( T^{6} + 233 T^{4} + \cdots + 26896 \)
|
| $79$ |
\( (T^{3} + 11 T^{2} + \cdots - 1175)^{2} \)
|
| $83$ |
\( T^{6} + 166 T^{4} + \cdots + 62500 \)
|
| $89$ |
\( (T^{3} + 29 T^{2} + \cdots + 736)^{2} \)
|
| $97$ |
\( T^{6} + 138 T^{4} + \cdots + 100 \)
|
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