Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1014,3,Mod(577,1014)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1014, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([0, 3]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1014.577");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1014 = 2 \cdot 3 \cdot 13^{2} \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1014.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: | \(27.6294988061\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Relative dimension: | \(4\) over \(\Q(i)\) |
| Coefficient field: | 8.0.564373557504.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - 86x^{6} + 2523x^{4} - 28394x^{2} + 113569 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{6} \) |
| Twist minimal: | no (minimal twist has level 78) |
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} - 86x^{6} + 2523x^{4} - 28394x^{2} + 113569 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -\nu^{6} + 64\nu^{4} - 946\nu^{2} + 315 ) / 2366 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 8 \nu^{7} - 5055 \nu^{6} - 3047 \nu^{5} + 323520 \nu^{4} + 114207 \nu^{3} - 6376714 \nu^{2} + \cdots + 35878031 ) / 1594684 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 8 \nu^{7} + 5055 \nu^{6} - 3047 \nu^{5} - 323520 \nu^{4} + 114207 \nu^{3} + 6376714 \nu^{2} + \cdots - 35878031 ) / 1594684 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 7 \nu^{7} + 2359 \nu^{6} - 4453 \nu^{5} - 207929 \nu^{4} + 305859 \nu^{3} + 5477935 \nu^{2} + \cdots - 36680428 ) / 797342 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 7 \nu^{7} - 2696 \nu^{6} - 4453 \nu^{5} + 229497 \nu^{4} + 305859 \nu^{3} - 5796737 \nu^{2} + \cdots + 36786583 ) / 797342 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 23\nu^{7} - 1641\nu^{5} + 36461\nu^{3} - 220354\nu ) / 113906 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -\nu^{7} + 86\nu^{5} - 2186\nu^{3} + 13903\nu ) / 4718 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{7} + \beta_{6} + \beta_{3} + \beta_{2} ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} - \beta_{2} + 15\beta _1 + 43 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 56\beta_{7} + 58\beta_{6} + 7\beta_{5} + 7\beta_{4} + 28\beta_{3} + 28\beta_{2} - 7\beta_1 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 14\beta_{5} - 14\beta_{4} + 57\beta_{3} - 57\beta_{2} + 645\beta _1 + 1189 ) / 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 2645\beta_{7} + 2759\beta_{6} + 287\beta_{5} + 287\beta_{4} + 853\beta_{3} + 853\beta_{2} - 287\beta_1 ) / 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( 448\beta_{5} - 448\beta_{4} + 1351\beta_{3} - 1351\beta_{2} + 11179\beta _1 + 18024 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 109521 \beta_{7} + 124389 \beta_{6} + 9380 \beta_{5} + 9380 \beta_{4} + 26053 \beta_{3} + \cdots - 9380 \beta_1 ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1014\mathbb{Z}\right)^\times\).
| \(n\) | \(677\) | \(847\) |
| \(\chi(n)\) | \(1\) | \(-\beta_{6}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 577.1 |
|
1.00000 | − | 1.00000i | −1.73205 | − | 2.00000i | −3.96855 | + | 3.96855i | −1.73205 | + | 1.73205i | −8.11022 | − | 8.11022i | −2.00000 | − | 2.00000i | 3.00000 | 7.93710i | |||||||||||||||||||||||||||||||
| 577.2 | 1.00000 | − | 1.00000i | −1.73205 | − | 2.00000i | 2.23650 | − | 2.23650i | −1.73205 | + | 1.73205i | 8.84227 | + | 8.84227i | −2.00000 | − | 2.00000i | 3.00000 | − | 4.47299i | |||||||||||||||||||||||||||||||
| 577.3 | 1.00000 | − | 1.00000i | 1.73205 | − | 2.00000i | −4.95414 | + | 4.95414i | 1.73205 | − | 1.73205i | 2.89463 | + | 2.89463i | −2.00000 | − | 2.00000i | 3.00000 | 9.90829i | ||||||||||||||||||||||||||||||||
| 577.4 | 1.00000 | − | 1.00000i | 1.73205 | − | 2.00000i | 6.68619 | − | 6.68619i | 1.73205 | − | 1.73205i | −5.62668 | − | 5.62668i | −2.00000 | − | 2.00000i | 3.00000 | − | 13.3724i | |||||||||||||||||||||||||||||||
| 775.1 | 1.00000 | + | 1.00000i | −1.73205 | 2.00000i | −3.96855 | − | 3.96855i | −1.73205 | − | 1.73205i | −8.11022 | + | 8.11022i | −2.00000 | + | 2.00000i | 3.00000 | − | 7.93710i | ||||||||||||||||||||||||||||||||
| 775.2 | 1.00000 | + | 1.00000i | −1.73205 | 2.00000i | 2.23650 | + | 2.23650i | −1.73205 | − | 1.73205i | 8.84227 | − | 8.84227i | −2.00000 | + | 2.00000i | 3.00000 | 4.47299i | |||||||||||||||||||||||||||||||||
| 775.3 | 1.00000 | + | 1.00000i | 1.73205 | 2.00000i | −4.95414 | − | 4.95414i | 1.73205 | + | 1.73205i | 2.89463 | − | 2.89463i | −2.00000 | + | 2.00000i | 3.00000 | − | 9.90829i | ||||||||||||||||||||||||||||||||
| 775.4 | 1.00000 | + | 1.00000i | 1.73205 | 2.00000i | 6.68619 | + | 6.68619i | 1.73205 | + | 1.73205i | −5.62668 | + | 5.62668i | −2.00000 | + | 2.00000i | 3.00000 | 13.3724i | |||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 13.d | odd | 4 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 1014.3.f.i | 8 | |
| 13.b | even | 2 | 1 | 78.3.f.b | ✓ | 8 | |
| 13.d | odd | 4 | 1 | 78.3.f.b | ✓ | 8 | |
| 13.d | odd | 4 | 1 | inner | 1014.3.f.i | 8 | |
| 39.d | odd | 2 | 1 | 234.3.i.e | 8 | ||
| 39.f | even | 4 | 1 | 234.3.i.e | 8 | ||
| 52.b | odd | 2 | 1 | 624.3.ba.c | 8 | ||
| 52.f | even | 4 | 1 | 624.3.ba.c | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 78.3.f.b | ✓ | 8 | 13.b | even | 2 | 1 | |
| 78.3.f.b | ✓ | 8 | 13.d | odd | 4 | 1 | |
| 234.3.i.e | 8 | 39.d | odd | 2 | 1 | ||
| 234.3.i.e | 8 | 39.f | even | 4 | 1 | ||
| 624.3.ba.c | 8 | 52.b | odd | 2 | 1 | ||
| 624.3.ba.c | 8 | 52.f | even | 4 | 1 | ||
| 1014.3.f.i | 8 | 1.a | even | 1 | 1 | trivial | |
| 1014.3.f.i | 8 | 13.d | odd | 4 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(1014, [\chi])\):
|
\( T_{5}^{8} + 168T_{5}^{5} + 5748T_{5}^{4} + 15120T_{5}^{3} + 14112T_{5}^{2} - 197568T_{5} + 1382976 \)
|
|
\( T_{7}^{8} + 4T_{7}^{7} + 8T_{7}^{6} + 16T_{7}^{5} + 23056T_{7}^{4} + 113792T_{7}^{3} + 270848T_{7}^{2} - 3438592T_{7} + 21827584 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} - 2 T + 2)^{4} \)
|
| $3$ |
\( (T^{2} - 3)^{4} \)
|
| $5$ |
\( T^{8} + 168 T^{5} + \cdots + 1382976 \)
|
| $7$ |
\( T^{8} + 4 T^{7} + \cdots + 21827584 \)
|
| $11$ |
\( T^{8} + 24 T^{7} + \cdots + 2304 \)
|
| $13$ |
\( T^{8} \)
|
| $17$ |
\( T^{8} + 864 T^{6} + \cdots + 222845184 \)
|
| $19$ |
\( T^{8} + \cdots + 12386799616 \)
|
| $23$ |
\( T^{8} + 1944 T^{6} + \cdots + 422055936 \)
|
| $29$ |
\( (T^{4} + 84 T^{3} + \cdots - 2225856)^{2} \)
|
| $31$ |
\( T^{8} - 52 T^{7} + \cdots + 214798336 \)
|
| $37$ |
\( T^{8} - 16 T^{7} + \cdots + 553002256 \)
|
| $41$ |
\( T^{8} + \cdots + 17404997184 \)
|
| $43$ |
\( T^{8} + \cdots + 1157707137024 \)
|
| $47$ |
\( T^{8} + \cdots + 36580045844736 \)
|
| $53$ |
\( (T^{4} - 72 T^{3} + \cdots - 1571952)^{2} \)
|
| $59$ |
\( T^{8} + \cdots + 5631470717184 \)
|
| $61$ |
\( (T^{4} + 36 T^{3} + \cdots + 28866048)^{2} \)
|
| $67$ |
\( T^{8} + \cdots + 18141329821696 \)
|
| $71$ |
\( T^{8} + \cdots + 306027720347904 \)
|
| $73$ |
\( T^{8} + \cdots + 23289928529296 \)
|
| $79$ |
\( (T^{4} - 120 T^{3} + \cdots - 11500032)^{2} \)
|
| $83$ |
\( T^{8} + \cdots + 22303630819584 \)
|
| $89$ |
\( T^{8} + \cdots + 58096425024 \)
|
| $97$ |
\( T^{8} + \cdots + 28822531984 \)
|
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