Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1210,4,Mod(1,1210)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1210, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1210.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1210 = 2 \cdot 5 \cdot 11^{2} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1210.a (trivial) |
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: | yes |
| Analytic conductor: | \(71.3923111069\) |
| Analytic rank: | \(0\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{177}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - x - 44 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 110) |
| Fricke sign: | \(+1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-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 \(\beta = \frac{1}{2}(1 + \sqrt{177})\). We also show the integral \(q\)-expansion of the trace form.
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
2.00000 | −3.15207 | 4.00000 | −5.00000 | −6.30413 | −6.84793 | 8.00000 | −17.0645 | −10.0000 | ||||||||||||||||||||||||
| 1.2 | 2.00000 | 10.1521 | 4.00000 | −5.00000 | 20.3041 | −20.1521 | 8.00000 | 76.0645 | −10.0000 | |||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(5\) | \( +1 \) |
| \(11\) | \( -1 \) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 1210.4.a.r | 2 | |
| 11.b | odd | 2 | 1 | 110.4.a.i | ✓ | 2 | |
| 33.d | even | 2 | 1 | 990.4.a.bf | 2 | ||
| 44.c | even | 2 | 1 | 880.4.a.q | 2 | ||
| 55.d | odd | 2 | 1 | 550.4.a.t | 2 | ||
| 55.e | even | 4 | 2 | 550.4.b.m | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 110.4.a.i | ✓ | 2 | 11.b | odd | 2 | 1 | |
| 550.4.a.t | 2 | 55.d | odd | 2 | 1 | ||
| 550.4.b.m | 4 | 55.e | even | 4 | 2 | ||
| 880.4.a.q | 2 | 44.c | even | 2 | 1 | ||
| 990.4.a.bf | 2 | 33.d | even | 2 | 1 | ||
| 1210.4.a.r | 2 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1210))\):
|
\( T_{3}^{2} - 7T_{3} - 32 \)
|
|
\( T_{7}^{2} + 27T_{7} + 138 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 2)^{2} \)
|
| $3$ |
\( T^{2} - 7T - 32 \)
|
| $5$ |
\( (T + 5)^{2} \)
|
| $7$ |
\( T^{2} + 27T + 138 \)
|
| $11$ |
\( T^{2} \)
|
| $13$ |
\( T^{2} + 42T - 3984 \)
|
| $17$ |
\( T^{2} + 49T - 2984 \)
|
| $19$ |
\( T^{2} + 133T + 4024 \)
|
| $23$ |
\( T^{2} - 162T + 6384 \)
|
| $29$ |
\( T^{2} - 407T + 33934 \)
|
| $31$ |
\( T^{2} - 351T + 28632 \)
|
| $37$ |
\( T^{2} + 9T - 15954 \)
|
| $41$ |
\( T^{2} + 40T - 5972 \)
|
| $43$ |
\( T^{2} - 514T + 26224 \)
|
| $47$ |
\( T^{2} - 98T - 167696 \)
|
| $53$ |
\( T^{2} + 53T - 334226 \)
|
| $59$ |
\( T^{2} + 986T + 234376 \)
|
| $61$ |
\( T^{2} - 25T - 669302 \)
|
| $67$ |
\( T^{2} - 172T - 10304 \)
|
| $71$ |
\( T^{2} + 427T - 119072 \)
|
| $73$ |
\( T^{2} + 646T - 222944 \)
|
| $79$ |
\( T^{2} - 706T + 123016 \)
|
| $83$ |
\( T^{2} - 492T - 961836 \)
|
| $89$ |
\( T^{2} + 273T - 13626 \)
|
| $97$ |
\( T^{2} - 3222 T + 2531424 \)
|
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