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: | \(4\) |
| Coefficient field: | \(\Q(i, \sqrt{33})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} + 17x^{2} + 64 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 33) |
| 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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} + 17x^{2} + 64 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} + 9\nu ) / 8 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{2} + 9 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} - 9 \)
|
| \(\nu^{3}\) | \(=\) |
\( 8\beta_{2} - 9\beta_1 \)
|
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 |
|
− | 3.37228i | 3.00000i | −3.37228 | 0 | 10.1168 | 4.74456i | − | 15.6060i | −9.00000 | 0 | ||||||||||||||||||||||||||||
| 199.2 | − | 2.37228i | − | 3.00000i | 2.37228 | 0 | −7.11684 | 6.74456i | − | 24.6060i | −9.00000 | 0 | ||||||||||||||||||||||||||||
| 199.3 | 2.37228i | 3.00000i | 2.37228 | 0 | −7.11684 | − | 6.74456i | 24.6060i | −9.00000 | 0 | ||||||||||||||||||||||||||||||
| 199.4 | 3.37228i | − | 3.00000i | −3.37228 | 0 | 10.1168 | − | 4.74456i | 15.6060i | −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.i | 4 | |
| 5.b | even | 2 | 1 | inner | 825.4.c.i | 4 | |
| 5.c | odd | 4 | 1 | 33.4.a.d | ✓ | 2 | |
| 5.c | odd | 4 | 1 | 825.4.a.k | 2 | ||
| 15.e | even | 4 | 1 | 99.4.a.e | 2 | ||
| 15.e | even | 4 | 1 | 2475.4.a.o | 2 | ||
| 20.e | even | 4 | 1 | 528.4.a.o | 2 | ||
| 35.f | even | 4 | 1 | 1617.4.a.j | 2 | ||
| 40.i | odd | 4 | 1 | 2112.4.a.ba | 2 | ||
| 40.k | even | 4 | 1 | 2112.4.a.bh | 2 | ||
| 55.e | even | 4 | 1 | 363.4.a.j | 2 | ||
| 60.l | odd | 4 | 1 | 1584.4.a.x | 2 | ||
| 165.l | odd | 4 | 1 | 1089.4.a.t | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 33.4.a.d | ✓ | 2 | 5.c | odd | 4 | 1 | |
| 99.4.a.e | 2 | 15.e | even | 4 | 1 | ||
| 363.4.a.j | 2 | 55.e | even | 4 | 1 | ||
| 528.4.a.o | 2 | 20.e | even | 4 | 1 | ||
| 825.4.a.k | 2 | 5.c | odd | 4 | 1 | ||
| 825.4.c.i | 4 | 1.a | even | 1 | 1 | trivial | |
| 825.4.c.i | 4 | 5.b | even | 2 | 1 | inner | |
| 1089.4.a.t | 2 | 165.l | odd | 4 | 1 | ||
| 1584.4.a.x | 2 | 60.l | odd | 4 | 1 | ||
| 1617.4.a.j | 2 | 35.f | even | 4 | 1 | ||
| 2112.4.a.ba | 2 | 40.i | odd | 4 | 1 | ||
| 2112.4.a.bh | 2 | 40.k | even | 4 | 1 | ||
| 2475.4.a.o | 2 | 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 \(S_{4}^{\mathrm{new}}(825, [\chi])\):
|
\( T_{2}^{4} + 17T_{2}^{2} + 64 \)
|
|
\( T_{7}^{4} + 68T_{7}^{2} + 1024 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} + 17T^{2} + 64 \)
|
| $3$ |
\( (T^{2} + 9)^{2} \)
|
| $5$ |
\( T^{4} \)
|
| $7$ |
\( T^{4} + 68T^{2} + 1024 \)
|
| $11$ |
\( (T - 11)^{4} \)
|
| $13$ |
\( T^{4} + 3944 T^{2} + 839056 \)
|
| $17$ |
\( T^{4} + 15188 T^{2} + 52649536 \)
|
| $19$ |
\( (T^{2} - 54 T - 1944)^{2} \)
|
| $23$ |
\( (T^{2} + 12544)^{2} \)
|
| $29$ |
\( (T^{2} + 222 T - 5136)^{2} \)
|
| $31$ |
\( (T^{2} + 40 T - 88832)^{2} \)
|
| $37$ |
\( T^{4} + 33096 T^{2} + 237036816 \)
|
| $41$ |
\( (T^{2} + 494 T + 60976)^{2} \)
|
| $43$ |
\( T^{4} + \cdots + 3591365184 \)
|
| $47$ |
\( T^{4} + 40064 T^{2} + 323424256 \)
|
| $53$ |
\( T^{4} + \cdots + 17796627216 \)
|
| $59$ |
\( (T + 196)^{4} \)
|
| $61$ |
\( (T^{2} + 1104 T + 282396)^{2} \)
|
| $67$ |
\( T^{4} + 811808 T^{2} + 609497344 \)
|
| $71$ |
\( (T^{2} - 456 T - 227328)^{2} \)
|
| $73$ |
\( T^{4} + \cdots + 190350709264 \)
|
| $79$ |
\( (T^{2} - 230 T - 31952)^{2} \)
|
| $83$ |
\( T^{4} + \cdots + 698521522176 \)
|
| $89$ |
\( (T^{2} + 972 T + 235668)^{2} \)
|
| $97$ |
\( T^{4} + \cdots + 1220662586896 \)
|
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