Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [900,3,Mod(451,900)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(900, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 0, 0]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("900.451");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 900.c (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: | \(24.5232237924\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.0.8405.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 2x^{3} + 8x^{2} - 7x + 2 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | no (minimal twist has level 100) |
| 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
\( x^{4} - 2x^{3} + 8x^{2} - 7x + 2 \)
:
| \(\beta_{1}\) | \(=\) |
\( -\nu^{3} + 2\nu^{2} - 6\nu + 4 \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{3} - 2\nu^{2} + 8\nu - 5 \)
|
| \(\beta_{3}\) | \(=\) |
\( 3\nu^{3} - 4\nu^{2} + 22\nu - 9 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + \beta _1 + 1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} - 2\beta_{2} + \beta _1 - 5 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} - 5\beta_{2} - 3\beta _1 - 4 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/900\mathbb{Z}\right)^\times\).
| \(n\) | \(101\) | \(451\) | \(577\) |
| \(\chi(n)\) | \(1\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 451.1 |
|
−1.35078 | − | 1.47492i | 0 | −0.350781 | + | 3.98459i | 0 | 0 | 10.9190i | 6.35078 | − | 4.86493i | 0 | 0 | ||||||||||||||||||||||||
| 451.2 | −1.35078 | + | 1.47492i | 0 | −0.350781 | − | 3.98459i | 0 | 0 | − | 10.9190i | 6.35078 | + | 4.86493i | 0 | 0 | ||||||||||||||||||||||||
| 451.3 | 1.85078 | − | 0.758030i | 0 | 2.85078 | − | 2.80590i | 0 | 0 | − | 4.09573i | 3.14922 | − | 7.35408i | 0 | 0 | ||||||||||||||||||||||||
| 451.4 | 1.85078 | + | 0.758030i | 0 | 2.85078 | + | 2.80590i | 0 | 0 | 4.09573i | 3.14922 | + | 7.35408i | 0 | 0 | |||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 4.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 900.3.c.m | 4 | |
| 3.b | odd | 2 | 1 | 100.3.b.d | ✓ | 4 | |
| 4.b | odd | 2 | 1 | inner | 900.3.c.m | 4 | |
| 5.b | even | 2 | 1 | 900.3.c.l | 4 | ||
| 5.c | odd | 4 | 2 | 900.3.f.d | 8 | ||
| 12.b | even | 2 | 1 | 100.3.b.d | ✓ | 4 | |
| 15.d | odd | 2 | 1 | 100.3.b.e | yes | 4 | |
| 15.e | even | 4 | 2 | 100.3.d.a | 8 | ||
| 20.d | odd | 2 | 1 | 900.3.c.l | 4 | ||
| 20.e | even | 4 | 2 | 900.3.f.d | 8 | ||
| 24.f | even | 2 | 1 | 1600.3.b.p | 4 | ||
| 24.h | odd | 2 | 1 | 1600.3.b.p | 4 | ||
| 60.h | even | 2 | 1 | 100.3.b.e | yes | 4 | |
| 60.l | odd | 4 | 2 | 100.3.d.a | 8 | ||
| 120.i | odd | 2 | 1 | 1600.3.b.o | 4 | ||
| 120.m | even | 2 | 1 | 1600.3.b.o | 4 | ||
| 120.q | odd | 4 | 2 | 1600.3.h.o | 8 | ||
| 120.w | even | 4 | 2 | 1600.3.h.o | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 100.3.b.d | ✓ | 4 | 3.b | odd | 2 | 1 | |
| 100.3.b.d | ✓ | 4 | 12.b | even | 2 | 1 | |
| 100.3.b.e | yes | 4 | 15.d | odd | 2 | 1 | |
| 100.3.b.e | yes | 4 | 60.h | even | 2 | 1 | |
| 100.3.d.a | 8 | 15.e | even | 4 | 2 | ||
| 100.3.d.a | 8 | 60.l | odd | 4 | 2 | ||
| 900.3.c.l | 4 | 5.b | even | 2 | 1 | ||
| 900.3.c.l | 4 | 20.d | odd | 2 | 1 | ||
| 900.3.c.m | 4 | 1.a | even | 1 | 1 | trivial | |
| 900.3.c.m | 4 | 4.b | odd | 2 | 1 | inner | |
| 900.3.f.d | 8 | 5.c | odd | 4 | 2 | ||
| 900.3.f.d | 8 | 20.e | even | 4 | 2 | ||
| 1600.3.b.o | 4 | 120.i | odd | 2 | 1 | ||
| 1600.3.b.o | 4 | 120.m | even | 2 | 1 | ||
| 1600.3.b.p | 4 | 24.f | even | 2 | 1 | ||
| 1600.3.b.p | 4 | 24.h | odd | 2 | 1 | ||
| 1600.3.h.o | 8 | 120.q | odd | 4 | 2 | ||
| 1600.3.h.o | 8 | 120.w | even | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(900, [\chi])\):
|
\( T_{7}^{4} + 136T_{7}^{2} + 2000 \)
|
|
\( T_{13}^{2} + 4T_{13} - 160 \)
|
|
\( T_{17}^{2} - 6T_{17} - 155 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} - T^{3} + \cdots + 16 \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( T^{4} \)
|
| $7$ |
\( T^{4} + 136T^{2} + 2000 \)
|
| $11$ |
\( T^{4} + 130T^{2} + 125 \)
|
| $13$ |
\( (T^{2} + 4 T - 160)^{2} \)
|
| $17$ |
\( (T^{2} - 6 T - 155)^{2} \)
|
| $19$ |
\( T^{4} + 1370 T^{2} + 465125 \)
|
| $23$ |
\( T^{4} + 776 T^{2} + 147920 \)
|
| $29$ |
\( (T^{2} + 20 T - 64)^{2} \)
|
| $31$ |
\( T^{4} + 520T^{2} + 2000 \)
|
| $37$ |
\( (T^{2} - 64 T + 860)^{2} \)
|
| $41$ |
\( (T + 17)^{4} \)
|
| $43$ |
\( T^{4} + 3056 T^{2} + 128000 \)
|
| $47$ |
\( T^{4} + 4976 T^{2} + 5242880 \)
|
| $53$ |
\( (T^{2} + 76 T - 1180)^{2} \)
|
| $59$ |
\( T^{4} + 13680 T^{2} + 41472000 \)
|
| $61$ |
\( (T^{2} + 56 T - 3316)^{2} \)
|
| $67$ |
\( T^{4} + 3586 T^{2} + 1915805 \)
|
| $71$ |
\( T^{4} + 19120 T^{2} + 67712000 \)
|
| $73$ |
\( (T^{2} + 114 T + 3085)^{2} \)
|
| $79$ |
\( T^{4} + 7720 T^{2} + 6962000 \)
|
| $83$ |
\( T^{4} + 5426 T^{2} + 3621005 \)
|
| $89$ |
\( (T^{2} - 110 T + 2861)^{2} \)
|
| $97$ |
\( (T^{2} + 156 T + 180)^{2} \)
|
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