Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [21,8,Mod(1,21)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(21, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 8, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("21.1");
S:= CuspForms(chi, 8);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 21 = 3 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| Character orbit: | \([\chi]\) | \(=\) | 21.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: | \(6.56008553517\) |
| Analytic rank: | \(1\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{1065}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - x - 266 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| 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 \(\beta = \frac{1}{2}(1 + \sqrt{1065})\). 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 |
|
−20.8172 | −27.0000 | 305.355 | 146.343 | 562.064 | 343.000 | −3692.02 | 729.000 | −3046.45 | ||||||||||||||||||||||||
| 1.2 | 11.8172 | −27.0000 | 11.6455 | −506.343 | −319.064 | 343.000 | −1374.98 | 729.000 | −5983.55 | |||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(1\) |
| \(7\) | \(-1\) |
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 | 21.8.a.b | ✓ | 2 |
| 3.b | odd | 2 | 1 | 63.8.a.f | 2 | ||
| 4.b | odd | 2 | 1 | 336.8.a.n | 2 | ||
| 5.b | even | 2 | 1 | 525.8.a.e | 2 | ||
| 7.b | odd | 2 | 1 | 147.8.a.c | 2 | ||
| 7.c | even | 3 | 2 | 147.8.e.h | 4 | ||
| 7.d | odd | 6 | 2 | 147.8.e.g | 4 | ||
| 21.c | even | 2 | 1 | 441.8.a.m | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 21.8.a.b | ✓ | 2 | 1.a | even | 1 | 1 | trivial |
| 63.8.a.f | 2 | 3.b | odd | 2 | 1 | ||
| 147.8.a.c | 2 | 7.b | odd | 2 | 1 | ||
| 147.8.e.g | 4 | 7.d | odd | 6 | 2 | ||
| 147.8.e.h | 4 | 7.c | even | 3 | 2 | ||
| 336.8.a.n | 2 | 4.b | odd | 2 | 1 | ||
| 441.8.a.m | 2 | 21.c | even | 2 | 1 | ||
| 525.8.a.e | 2 | 5.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{2} + 9T_{2} - 246 \)
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(21))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} + 9T - 246 \)
|
| $3$ |
\( (T + 27)^{2} \)
|
| $5$ |
\( T^{2} + 360T - 74100 \)
|
| $7$ |
\( (T - 343)^{2} \)
|
| $11$ |
\( T^{2} + 4932 T - 19176384 \)
|
| $13$ |
\( T^{2} - 7708 T - 7230524 \)
|
| $17$ |
\( T^{2} + 28584 T + 121953804 \)
|
| $19$ |
\( T^{2} + 63728 T + 782053936 \)
|
| $23$ |
\( T^{2} + \cdots + 1392518400 \)
|
| $29$ |
\( T^{2} + \cdots + 46362346164 \)
|
| $31$ |
\( T^{2} + \cdots - 27226807040 \)
|
| $37$ |
\( T^{2} + \cdots + 123432685924 \)
|
| $41$ |
\( T^{2} + \cdots - 416101387716 \)
|
| $43$ |
\( T^{2} + \cdots + 23523686224 \)
|
| $47$ |
\( T^{2} + \cdots + 551244428160 \)
|
| $53$ |
\( T^{2} + \cdots + 800144528964 \)
|
| $59$ |
\( T^{2} + \cdots - 455709488016 \)
|
| $61$ |
\( T^{2} + \cdots - 397808236556 \)
|
| $67$ |
\( T^{2} + \cdots + 4656119933104 \)
|
| $71$ |
\( T^{2} + \cdots - 387209643840 \)
|
| $73$ |
\( T^{2} + \cdots - 22405484001836 \)
|
| $79$ |
\( T^{2} + \cdots + 578840156416 \)
|
| $83$ |
\( T^{2} + \cdots + 6817674434256 \)
|
| $89$ |
\( T^{2} + \cdots - 61835691772164 \)
|
| $97$ |
\( T^{2} + \cdots + 62174212264276 \)
|
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