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: | \(6\) |
| Coefficient field: | 6.0.245110336.2 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} + 19x^{4} + 101x^{2} + 100 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | no (minimal twist has level 165) |
| 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
\( x^{6} + 19x^{4} + 101x^{2} + 100 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{5} + 9\nu^{3} + \nu ) / 10 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 3\nu^{5} + 37\nu^{3} + 83\nu ) / 10 \)
|
| \(\beta_{3}\) | \(=\) |
\( -\nu^{4} - 9\nu^{2} - 4 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 2\nu^{5} + 23\nu^{3} + 52\nu ) / 5 \)
|
| \(\beta_{5}\) | \(=\) |
\( \nu^{4} + 11\nu^{2} + 17 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{4} - \beta_{2} - \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{5} + \beta_{3} - 13 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( -4\beta_{4} + 5\beta_{2} + \beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -9\beta_{5} - 11\beta_{3} + 109 ) / 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 71\beta_{4} - 89\beta_{2} + 3\beta_1 ) / 2 \)
|
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 |
|
− | 4.59486i | − | 3.00000i | −13.1127 | 0 | −13.7846 | − | 20.6383i | 23.4921i | −9.00000 | 0 | |||||||||||||||||||||||||||||||||
| 199.2 | − | 4.38835i | 3.00000i | −11.2577 | 0 | 13.1651 | − | 11.7304i | 14.2958i | −9.00000 | 0 | |||||||||||||||||||||||||||||||||||
| 199.3 | − | 0.793499i | − | 3.00000i | 7.37036 | 0 | −2.38050 | 2.90793i | − | 12.1964i | −9.00000 | 0 | ||||||||||||||||||||||||||||||||||
| 199.4 | 0.793499i | 3.00000i | 7.37036 | 0 | −2.38050 | − | 2.90793i | 12.1964i | −9.00000 | 0 | ||||||||||||||||||||||||||||||||||||
| 199.5 | 4.38835i | − | 3.00000i | −11.2577 | 0 | 13.1651 | 11.7304i | − | 14.2958i | −9.00000 | 0 | |||||||||||||||||||||||||||||||||||
| 199.6 | 4.59486i | 3.00000i | −13.1127 | 0 | −13.7846 | 20.6383i | − | 23.4921i | −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.m | 6 | |
| 5.b | even | 2 | 1 | inner | 825.4.c.m | 6 | |
| 5.c | odd | 4 | 1 | 165.4.a.g | ✓ | 3 | |
| 5.c | odd | 4 | 1 | 825.4.a.p | 3 | ||
| 15.e | even | 4 | 1 | 495.4.a.i | 3 | ||
| 15.e | even | 4 | 1 | 2475.4.a.z | 3 | ||
| 55.e | even | 4 | 1 | 1815.4.a.q | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 165.4.a.g | ✓ | 3 | 5.c | odd | 4 | 1 | |
| 495.4.a.i | 3 | 15.e | even | 4 | 1 | ||
| 825.4.a.p | 3 | 5.c | odd | 4 | 1 | ||
| 825.4.c.m | 6 | 1.a | even | 1 | 1 | trivial | |
| 825.4.c.m | 6 | 5.b | even | 2 | 1 | inner | |
| 1815.4.a.q | 3 | 55.e | even | 4 | 1 | ||
| 2475.4.a.z | 3 | 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 \(S_{4}^{\mathrm{new}}(825, [\chi])\):
|
\( T_{2}^{6} + 41T_{2}^{4} + 432T_{2}^{2} + 256 \)
|
|
\( T_{7}^{6} + 572T_{7}^{4} + 63376T_{7}^{2} + 495616 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} + 41 T^{4} + \cdots + 256 \)
|
| $3$ |
\( (T^{2} + 9)^{3} \)
|
| $5$ |
\( T^{6} \)
|
| $7$ |
\( T^{6} + 572 T^{4} + \cdots + 495616 \)
|
| $11$ |
\( (T - 11)^{6} \)
|
| $13$ |
\( T^{6} + \cdots + 6100234816 \)
|
| $17$ |
\( T^{6} + 6384 T^{4} + \cdots + 502835776 \)
|
| $19$ |
\( (T^{3} + 116 T^{2} + \cdots + 80)^{2} \)
|
| $23$ |
\( T^{6} + \cdots + 32480690176 \)
|
| $29$ |
\( (T^{3} + 238 T^{2} + \cdots - 428416)^{2} \)
|
| $31$ |
\( (T^{3} - 92 T^{2} + \cdots + 6769664)^{2} \)
|
| $37$ |
\( T^{6} + \cdots + 40502634332224 \)
|
| $41$ |
\( (T^{3} + 46 T^{2} + \cdots - 245888)^{2} \)
|
| $43$ |
\( T^{6} + \cdots + 5670875449600 \)
|
| $47$ |
\( T^{6} + \cdots + 120565319090176 \)
|
| $53$ |
\( T^{6} + \cdots + 298010064169024 \)
|
| $59$ |
\( (T^{3} + 1236 T^{2} + \cdots + 32923904)^{2} \)
|
| $61$ |
\( (T^{3} - 342 T^{2} + \cdots - 2655176)^{2} \)
|
| $67$ |
\( T^{6} + \cdots + 23\!\cdots\!56 \)
|
| $71$ |
\( (T^{3} - 1816 T^{2} + \cdots - 198158720)^{2} \)
|
| $73$ |
\( T^{6} + \cdots + 26347524744256 \)
|
| $79$ |
\( (T^{3} - 96 T^{2} + \cdots + 167159872)^{2} \)
|
| $83$ |
\( T^{6} + \cdots + 29\!\cdots\!76 \)
|
| $89$ |
\( (T^{3} + 838 T^{2} + \cdots - 693013592)^{2} \)
|
| $97$ |
\( T^{6} + \cdots + 12\!\cdots\!36 \)
|
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