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: | 6.0.9935104.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 2x^{5} + 2x^{4} + 6x^{3} + 16x^{2} - 8x + 2 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2^{3} \) |
| 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
\( x^{6} - 2x^{5} + 2x^{4} + 6x^{3} + 16x^{2} - 8x + 2 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -2\nu^{5} + 5\nu^{4} - 12\nu^{3} - 6\nu^{2} + 4\nu - 19 ) / 33 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -5\nu^{5} + 7\nu^{4} + 3\nu^{3} - 48\nu^{2} - 56\nu + 13 ) / 33 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -7\nu^{5} + 12\nu^{4} - 9\nu^{3} - 54\nu^{2} - 118\nu + 27 ) / 33 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 6\nu^{5} - 26\nu^{4} + 36\nu^{3} + 18\nu^{2} - 12\nu - 185 ) / 33 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -39\nu^{5} + 70\nu^{4} - 69\nu^{3} - 216\nu^{2} - 714\nu + 163 ) / 33 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{3} + \beta_{2} + \beta _1 + 1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{5} - 7\beta_{3} + 2\beta_{2} ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( \beta_{5} - \beta_{4} - 9\beta_{3} + 6\beta_{2} - 6\beta _1 - 9 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( -3\beta_{4} - 9\beta _1 - 22 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -9\beta_{5} - 9\beta_{4} + 73\beta_{3} - 40\beta_{2} - 40\beta _1 - 73 ) / 2 \)
|
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 |
|
− | 3.27945i | − | 3.00000i | −2.75481 | 0 | −9.83836 | 33.3329i | − | 17.2014i | −9.00000 | 0 | |||||||||||||||||||||||||||||||||
| 199.2 | − | 2.82009i | 3.00000i | 0.0470959 | 0 | 8.46027 | − | 7.12434i | − | 22.6935i | −9.00000 | 0 | ||||||||||||||||||||||||||||||||||
| 199.3 | − | 0.540637i | − | 3.00000i | 7.70771 | 0 | −1.62191 | − | 24.4573i | − | 8.49217i | −9.00000 | 0 | |||||||||||||||||||||||||||||||||
| 199.4 | 0.540637i | 3.00000i | 7.70771 | 0 | −1.62191 | 24.4573i | 8.49217i | −9.00000 | 0 | |||||||||||||||||||||||||||||||||||||
| 199.5 | 2.82009i | − | 3.00000i | 0.0470959 | 0 | 8.46027 | 7.12434i | 22.6935i | −9.00000 | 0 | ||||||||||||||||||||||||||||||||||||
| 199.6 | 3.27945i | 3.00000i | −2.75481 | 0 | −9.83836 | − | 33.3329i | 17.2014i | −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.o | 6 | |
| 5.b | even | 2 | 1 | inner | 825.4.c.o | 6 | |
| 5.c | odd | 4 | 1 | 165.4.a.f | ✓ | 3 | |
| 5.c | odd | 4 | 1 | 825.4.a.n | 3 | ||
| 15.e | even | 4 | 1 | 495.4.a.g | 3 | ||
| 15.e | even | 4 | 1 | 2475.4.a.w | 3 | ||
| 55.e | even | 4 | 1 | 1815.4.a.p | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 165.4.a.f | ✓ | 3 | 5.c | odd | 4 | 1 | |
| 495.4.a.g | 3 | 15.e | even | 4 | 1 | ||
| 825.4.a.n | 3 | 5.c | odd | 4 | 1 | ||
| 825.4.c.o | 6 | 1.a | even | 1 | 1 | trivial | |
| 825.4.c.o | 6 | 5.b | even | 2 | 1 | inner | |
| 1815.4.a.p | 3 | 55.e | even | 4 | 1 | ||
| 2475.4.a.w | 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 \(S_{4}^{\mathrm{new}}(825, [\chi])\):
|
\( T_{2}^{6} + 19T_{2}^{4} + 91T_{2}^{2} + 25 \)
|
|
\( T_{7}^{6} + 1760T_{7}^{4} + 751360T_{7}^{2} + 33732864 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} + 19 T^{4} + \cdots + 25 \)
|
| $3$ |
\( (T^{2} + 9)^{3} \)
|
| $5$ |
\( T^{6} \)
|
| $7$ |
\( T^{6} + 1760 T^{4} + \cdots + 33732864 \)
|
| $11$ |
\( (T + 11)^{6} \)
|
| $13$ |
\( T^{6} + \cdots + 18883157056 \)
|
| $17$ |
\( T^{6} + \cdots + 32063916096 \)
|
| $19$ |
\( (T^{3} - 280 T^{2} + \cdots + 97056)^{2} \)
|
| $23$ |
\( T^{6} + \cdots + 3672159698944 \)
|
| $29$ |
\( (T^{3} - 290 T^{2} + \cdots + 9251496)^{2} \)
|
| $31$ |
\( (T^{3} + 392 T^{2} + \cdots - 316800)^{2} \)
|
| $37$ |
\( T^{6} + \cdots + 1080818061376 \)
|
| $41$ |
\( (T^{3} + 662 T^{2} + \cdots - 68561784)^{2} \)
|
| $43$ |
\( T^{6} + \cdots + 419883375995136 \)
|
| $47$ |
\( T^{6} + \cdots + 199419586560000 \)
|
| $53$ |
\( T^{6} + \cdots + 5779177536064 \)
|
| $59$ |
\( (T^{3} - 612 T^{2} + \cdots + 162128320)^{2} \)
|
| $61$ |
\( (T^{3} + 582 T^{2} + \cdots - 21355000)^{2} \)
|
| $67$ |
\( T^{6} + \cdots + 15\!\cdots\!00 \)
|
| $71$ |
\( (T^{3} + 1616 T^{2} + \cdots + 40198784)^{2} \)
|
| $73$ |
\( T^{6} + \cdots + 145112476357696 \)
|
| $79$ |
\( (T^{3} + 124 T^{2} + \cdots - 26871328)^{2} \)
|
| $83$ |
\( T^{6} + \cdots + 245122860960000 \)
|
| $89$ |
\( (T^{3} + 838 T^{2} + \cdots - 831946232)^{2} \)
|
| $97$ |
\( T^{6} + \cdots + 85\!\cdots\!16 \)
|
| show more | |
| show less |