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: | 6.0.36142572544.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} + 53x^{4} + 632x^{2} + 484 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2 \) |
| 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
\( x^{6} + 53x^{4} + 632x^{2} + 484 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{4} + 27\nu^{2} - 22 ) / 48 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{5} + 75\nu^{3} + 1226\nu ) / 1056 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{4} + 51\nu^{2} + 386 ) / 24 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -13\nu^{5} - 623\nu^{3} - 6082\nu ) / 352 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{4} - 2\beta_{2} - 17 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{5} + 39\beta_{3} - 28\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -27\beta_{4} + 102\beta_{2} + 481 \)
|
| \(\nu^{5}\) | \(=\) |
\( -75\beta_{5} - 1869\beta_{3} + 874\beta_1 \)
|
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 |
|
− | 5.06484i | − | 3.00000i | −17.6526 | 0 | −15.1945 | 27.4348i | 48.8887i | −9.00000 | 0 | ||||||||||||||||||||||||||||||||||
| 199.2 | − | 4.97123i | 3.00000i | −16.7131 | 0 | 14.9137 | − | 5.48376i | 43.3148i | −9.00000 | 0 | |||||||||||||||||||||||||||||||||||
| 199.3 | − | 1.90639i | − | 3.00000i | 4.36567 | 0 | −5.71918 | − | 22.9186i | − | 23.5738i | −9.00000 | 0 | |||||||||||||||||||||||||||||||||
| 199.4 | 1.90639i | 3.00000i | 4.36567 | 0 | −5.71918 | 22.9186i | 23.5738i | −9.00000 | 0 | |||||||||||||||||||||||||||||||||||||
| 199.5 | 4.97123i | − | 3.00000i | −16.7131 | 0 | 14.9137 | 5.48376i | − | 43.3148i | −9.00000 | 0 | |||||||||||||||||||||||||||||||||||
| 199.6 | 5.06484i | 3.00000i | −17.6526 | 0 | −15.1945 | − | 27.4348i | − | 48.8887i | −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.k | 6 | |
| 5.b | even | 2 | 1 | inner | 825.4.c.k | 6 | |
| 5.c | odd | 4 | 1 | 165.4.a.e | ✓ | 3 | |
| 5.c | odd | 4 | 1 | 825.4.a.r | 3 | ||
| 15.e | even | 4 | 1 | 495.4.a.k | 3 | ||
| 15.e | even | 4 | 1 | 2475.4.a.t | 3 | ||
| 55.e | even | 4 | 1 | 1815.4.a.r | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 165.4.a.e | ✓ | 3 | 5.c | odd | 4 | 1 | |
| 495.4.a.k | 3 | 15.e | even | 4 | 1 | ||
| 825.4.a.r | 3 | 5.c | odd | 4 | 1 | ||
| 825.4.c.k | 6 | 1.a | even | 1 | 1 | trivial | |
| 825.4.c.k | 6 | 5.b | even | 2 | 1 | inner | |
| 1815.4.a.r | 3 | 55.e | even | 4 | 1 | ||
| 2475.4.a.t | 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 \(S_{4}^{\mathrm{new}}(825, [\chi])\):
|
\( T_{2}^{6} + 54T_{2}^{4} + 817T_{2}^{2} + 2304 \)
|
|
\( T_{7}^{6} + 1308T_{7}^{4} + 433776T_{7}^{2} + 11888704 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} + 54 T^{4} + \cdots + 2304 \)
|
| $3$ |
\( (T^{2} + 9)^{3} \)
|
| $5$ |
\( T^{6} \)
|
| $7$ |
\( T^{6} + 1308 T^{4} + \cdots + 11888704 \)
|
| $11$ |
\( (T - 11)^{6} \)
|
| $13$ |
\( T^{6} + \cdots + 1385030656 \)
|
| $17$ |
\( T^{6} + 5232 T^{4} + \cdots + 71368704 \)
|
| $19$ |
\( (T^{3} - 58 T^{2} + \cdots - 65520)^{2} \)
|
| $23$ |
\( T^{6} + \cdots + 21970354176 \)
|
| $29$ |
\( (T^{3} - 220 T^{2} + \cdots + 629760)^{2} \)
|
| $31$ |
\( (T^{3} - 248 T^{2} + \cdots + 9589248)^{2} \)
|
| $37$ |
\( T^{6} + \cdots + 346232953016896 \)
|
| $41$ |
\( (T^{3} - 156 T^{2} + \cdots - 3013632)^{2} \)
|
| $43$ |
\( T^{6} + \cdots + 2089181160000 \)
|
| $47$ |
\( T^{6} + \cdots + 19116342706176 \)
|
| $53$ |
\( T^{6} + \cdots + 13353645381696 \)
|
| $59$ |
\( (T^{3} + 548 T^{2} + \cdots + 1206720)^{2} \)
|
| $61$ |
\( (T^{3} - 414 T^{2} + \cdots + 342344792)^{2} \)
|
| $67$ |
\( T^{6} + \cdots + 66187206344704 \)
|
| $71$ |
\( (T^{3} + 912 T^{2} + \cdots - 2867712)^{2} \)
|
| $73$ |
\( T^{6} + \cdots + 700057090622464 \)
|
| $79$ |
\( (T^{3} - 542 T^{2} + \cdots + 88503440)^{2} \)
|
| $83$ |
\( T^{6} + \cdots + 18\!\cdots\!04 \)
|
| $89$ |
\( (T^{3} + 790 T^{2} + \cdots - 1941629400)^{2} \)
|
| $97$ |
\( T^{6} + \cdots + 96\!\cdots\!96 \)
|
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