Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [825,4,Mod(199,825)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(825, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1, 0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("825.199");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 825 = 3 \cdot 5^{2} \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 825.c (of order \(2\), degree \(1\), not 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: | \(48.6765757547\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{8} + \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} + 574x^{6} + 121601x^{4} + 11262916x^{2} + 384787456 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{4} \) |
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} + 574x^{6} + 121601x^{4} + 11262916x^{2} + 384787456 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{6} + 431\nu^{4} + 59278\nu^{2} + 2597792 ) / 19320 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 19\nu^{7} + 2324\nu^{5} - 542483\nu^{3} - 63786772\nu ) / 23686320 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -71\nu^{7} - 30946\nu^{5} - 4406423\nu^{3} - 204733372\nu ) / 13535040 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{6} + 546\nu^{4} + 93893\nu^{2} + 5075812 ) / 3220 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 71\nu^{7} + 30946\nu^{5} + 4406423\nu^{3} + 231803452\nu ) / 6767520 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 4483\nu^{7} + 1906298\nu^{5} + 257684419\nu^{3} + 10956545996\nu ) / 94745280 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( \nu^{6} + 431\nu^{4} + 60658\nu^{2} + 2786162 ) / 345 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{5} + 2\beta_{3} ) / 4 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{7} - 56\beta _1 - 546 ) / 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -56\beta_{6} - 151\beta_{5} - 790\beta_{3} + 112\beta_{2} ) / 4 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -301\beta_{7} + 112\beta_{4} + 16184\beta _1 + 78154 ) / 4 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 16184\beta_{6} + 23735\beta_{5} + 186070\beta_{3} - 48272\beta_{2} ) / 4 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 70453\beta_{7} - 48272\beta_{4} - 3578456\beta _1 - 11709754 ) / 4 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( -3578456\beta_{6} - 3857279\beta_{5} - 38600774\beta_{3} + 14088816\beta_{2} ) / 4 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/825\mathbb{Z}\right)^\times\).
| \(n\) | \(376\) | \(551\) | \(727\) |
| \(\chi(n)\) | \(1\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 199.1 |
|
− | 2.56155i | − | 3.00000i | 1.43845 | 0 | −7.68466 | − | 29.1157i | − | 24.1771i | −9.00000 | 0 | ||||||||||||||||||||||||||||||||||||||
| 199.2 | − | 2.56155i | − | 3.00000i | 1.43845 | 0 | −7.68466 | 25.6772i | − | 24.1771i | −9.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 199.3 | − | 1.56155i | 3.00000i | 5.56155 | 0 | 4.68466 | − | 16.7054i | − | 21.1771i | −9.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 199.4 | − | 1.56155i | 3.00000i | 5.56155 | 0 | 4.68466 | 24.2670i | − | 21.1771i | −9.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||
| 199.5 | 1.56155i | − | 3.00000i | 5.56155 | 0 | 4.68466 | − | 24.2670i | 21.1771i | −9.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||
| 199.6 | 1.56155i | − | 3.00000i | 5.56155 | 0 | 4.68466 | 16.7054i | 21.1771i | −9.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 199.7 | 2.56155i | 3.00000i | 1.43845 | 0 | −7.68466 | − | 25.6772i | 24.1771i | −9.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 199.8 | 2.56155i | 3.00000i | 1.43845 | 0 | −7.68466 | 29.1157i | 24.1771i | −9.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||
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 | 825.4.c.q | 8 | |
| 5.b | even | 2 | 1 | inner | 825.4.c.q | 8 | |
| 5.c | odd | 4 | 1 | 825.4.a.u | ✓ | 4 | |
| 5.c | odd | 4 | 1 | 825.4.a.v | yes | 4 | |
| 15.e | even | 4 | 1 | 2475.4.a.bb | 4 | ||
| 15.e | even | 4 | 1 | 2475.4.a.bd | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 825.4.a.u | ✓ | 4 | 5.c | odd | 4 | 1 | |
| 825.4.a.v | yes | 4 | 5.c | odd | 4 | 1 | |
| 825.4.c.q | 8 | 1.a | even | 1 | 1 | trivial | |
| 825.4.c.q | 8 | 5.b | even | 2 | 1 | inner | |
| 2475.4.a.bb | 4 | 15.e | even | 4 | 1 | ||
| 2475.4.a.bd | 4 | 15.e | even | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(825, [\chi])\):
|
\( T_{2}^{4} + 9T_{2}^{2} + 16 \)
|
|
\( T_{7}^{8} + 2375T_{7}^{6} + 2031311T_{7}^{4} + 732789005T_{7}^{2} + 91853849476 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{4} + 9 T^{2} + 16)^{2} \)
|
| $3$ |
\( (T^{2} + 9)^{4} \)
|
| $5$ |
\( T^{8} \)
|
| $7$ |
\( T^{8} + \cdots + 91853849476 \)
|
| $11$ |
\( (T + 11)^{8} \)
|
| $13$ |
\( T^{8} + \cdots + 2913657819136 \)
|
| $17$ |
\( T^{8} + \cdots + 1726301676544 \)
|
| $19$ |
\( (T^{4} + 118 T^{3} + \cdots - 6797601)^{2} \)
|
| $23$ |
\( T^{8} + \cdots + 411778697413561 \)
|
| $29$ |
\( (T^{4} + 251 T^{3} + \cdots - 42323912)^{2} \)
|
| $31$ |
\( (T^{4} + 135 T^{3} + \cdots + 92275200)^{2} \)
|
| $37$ |
\( T^{8} + \cdots + 77\!\cdots\!96 \)
|
| $41$ |
\( (T^{4} + 103 T^{3} + \cdots + 1343813418)^{2} \)
|
| $43$ |
\( T^{8} + \cdots + 63\!\cdots\!84 \)
|
| $47$ |
\( T^{8} + \cdots + 10\!\cdots\!96 \)
|
| $53$ |
\( T^{8} + \cdots + 10\!\cdots\!64 \)
|
| $59$ |
\( (T^{4} + 951 T^{3} + \cdots - 37958294854)^{2} \)
|
| $61$ |
\( (T^{4} + 350 T^{3} + \cdots + 10189762592)^{2} \)
|
| $67$ |
\( T^{8} + \cdots + 15\!\cdots\!04 \)
|
| $71$ |
\( (T^{4} - 1526 T^{3} + \cdots + 6876641459)^{2} \)
|
| $73$ |
\( T^{8} + \cdots + 39\!\cdots\!56 \)
|
| $79$ |
\( (T^{4} + 2567 T^{3} + \cdots - 51952998816)^{2} \)
|
| $83$ |
\( T^{8} + \cdots + 30\!\cdots\!00 \)
|
| $89$ |
\( (T^{4} + 1053 T^{3} + \cdots - 134213353376)^{2} \)
|
| $97$ |
\( T^{8} + \cdots + 72\!\cdots\!41 \)
|
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