Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2700,3,Mod(701,2700)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2700, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1, 0]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2700.701");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2700 = 2^{2} \cdot 3^{3} \cdot 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2700.g (of order \(2\), degree \(1\), 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: | \(73.5696713773\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.0.9292960.2 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} + 41x^{2} + 360 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 5 \) |
| Twist minimal: | no (minimal twist has level 540) |
| 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
\( x^{4} + 41x^{2} + 360 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{3} + 11\nu ) / 6 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} + 26\nu ) / 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{2} + 21 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} - 2\beta_1 ) / 5 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} - 21 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -11\beta_{2} + 52\beta_1 ) / 5 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2700\mathbb{Z}\right)^\times\).
| \(n\) | \(1001\) | \(1351\) | \(2377\) |
| \(\chi(n)\) | \(-1\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 701.1 |
|
0 | 0 | 0 | 0 | 0 | −9.26209 | 0 | 0 | 0 | ||||||||||||||||||||||||||||||
| 701.2 | 0 | 0 | 0 | 0 | 0 | −9.26209 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||
| 701.3 | 0 | 0 | 0 | 0 | 0 | 6.26209 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||
| 701.4 | 0 | 0 | 0 | 0 | 0 | 6.26209 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 2700.3.g.o | 4 | |
| 3.b | odd | 2 | 1 | inner | 2700.3.g.o | 4 | |
| 5.b | even | 2 | 1 | 2700.3.g.p | 4 | ||
| 5.c | odd | 4 | 2 | 540.3.b.c | ✓ | 8 | |
| 15.d | odd | 2 | 1 | 2700.3.g.p | 4 | ||
| 15.e | even | 4 | 2 | 540.3.b.c | ✓ | 8 | |
| 20.e | even | 4 | 2 | 2160.3.c.n | 8 | ||
| 45.k | odd | 12 | 4 | 1620.3.t.e | 16 | ||
| 45.l | even | 12 | 4 | 1620.3.t.e | 16 | ||
| 60.l | odd | 4 | 2 | 2160.3.c.n | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 540.3.b.c | ✓ | 8 | 5.c | odd | 4 | 2 | |
| 540.3.b.c | ✓ | 8 | 15.e | even | 4 | 2 | |
| 1620.3.t.e | 16 | 45.k | odd | 12 | 4 | ||
| 1620.3.t.e | 16 | 45.l | even | 12 | 4 | ||
| 2160.3.c.n | 8 | 20.e | even | 4 | 2 | ||
| 2160.3.c.n | 8 | 60.l | odd | 4 | 2 | ||
| 2700.3.g.o | 4 | 1.a | even | 1 | 1 | trivial | |
| 2700.3.g.o | 4 | 3.b | odd | 2 | 1 | inner | |
| 2700.3.g.p | 4 | 5.b | even | 2 | 1 | ||
| 2700.3.g.p | 4 | 15.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(2700, [\chi])\):
|
\( T_{7}^{2} + 3T_{7} - 58 \)
|
|
\( T_{11}^{4} + 235T_{11}^{2} + 250 \)
|
|
\( T_{13} + 3 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( T^{4} \)
|
| $7$ |
\( (T^{2} + 3 T - 58)^{2} \)
|
| $11$ |
\( T^{4} + 235T^{2} + 250 \)
|
| $13$ |
\( (T + 3)^{4} \)
|
| $17$ |
\( T^{4} + 265T^{2} + 4000 \)
|
| $19$ |
\( (T^{2} - 9 T - 522)^{2} \)
|
| $23$ |
\( T^{4} + 2635 T^{2} + 1722250 \)
|
| $29$ |
\( T^{4} + 2115 T^{2} + 20250 \)
|
| $31$ |
\( (T^{2} - 11 T - 512)^{2} \)
|
| $37$ |
\( (T^{2} + 9 T - 1486)^{2} \)
|
| $41$ |
\( T^{4} + 2890 T^{2} + 1600000 \)
|
| $43$ |
\( (T^{2} + 57 T + 752)^{2} \)
|
| $47$ |
\( T^{4} + 6625 T^{2} + 2500000 \)
|
| $53$ |
\( T^{4} + 5310 T^{2} + 6561000 \)
|
| $59$ |
\( T^{4} + 15460 T^{2} + 59536000 \)
|
| $61$ |
\( (T^{2} + 29 T - 332)^{2} \)
|
| $67$ |
\( (T^{2} + 39 T - 1126)^{2} \)
|
| $71$ |
\( T^{4} + 18990 T^{2} + 50625000 \)
|
| $73$ |
\( (T^{2} + 141 T + 3464)^{2} \)
|
| $79$ |
\( (T + 39)^{4} \)
|
| $83$ |
\( T^{4} + 10090 T^{2} + 24964000 \)
|
| $89$ |
\( T^{4} + 15460 T^{2} + 59536000 \)
|
| $97$ |
\( (T^{2} + 153 T - 1438)^{2} \)
|
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