Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [20,3,Mod(11,20)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(20, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([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("20.11");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 20 = 2^{2} \cdot 5 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 20.b (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: | \(0.544960528721\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\zeta_{10})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} + x^{2} - x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2^{4} \) |
| Twist minimal: | yes |
| 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
| \(\beta_{1}\) | \(=\) |
\( 2\zeta_{10}^{2} \)
|
| \(\beta_{2}\) | \(=\) |
\( 2\zeta_{10}^{3} \)
|
| \(\beta_{3}\) | \(=\) |
\( 2\zeta_{10}^{3} - 2\zeta_{10}^{2} + 4\zeta_{10} - 2 \)
|
| \(\zeta_{10}\) | \(=\) |
\( ( \beta_{3} - \beta_{2} + \beta _1 + 2 ) / 4 \)
|
| \(\zeta_{10}^{2}\) | \(=\) |
\( ( \beta_1 ) / 2 \)
|
| \(\zeta_{10}^{3}\) | \(=\) |
\( ( \beta_{2} ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/20\mathbb{Z}\right)^\times\).
| \(n\) | \(11\) | \(17\) |
| \(\chi(n)\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 11.1 |
|
−1.61803 | − | 1.17557i | − | 3.80423i | 1.23607 | + | 3.80423i | 2.23607 | −4.47214 | + | 6.15537i | 8.50651i | 2.47214 | − | 7.60845i | −5.47214 | −3.61803 | − | 2.62866i | |||||||||||||||||||
| 11.2 | −1.61803 | + | 1.17557i | 3.80423i | 1.23607 | − | 3.80423i | 2.23607 | −4.47214 | − | 6.15537i | − | 8.50651i | 2.47214 | + | 7.60845i | −5.47214 | −3.61803 | + | 2.62866i | ||||||||||||||||||||
| 11.3 | 0.618034 | − | 1.90211i | 2.35114i | −3.23607 | − | 2.35114i | −2.23607 | 4.47214 | + | 1.45309i | 5.25731i | −6.47214 | + | 4.70228i | 3.47214 | −1.38197 | + | 4.25325i | |||||||||||||||||||||
| 11.4 | 0.618034 | + | 1.90211i | − | 2.35114i | −3.23607 | + | 2.35114i | −2.23607 | 4.47214 | − | 1.45309i | − | 5.25731i | −6.47214 | − | 4.70228i | 3.47214 | −1.38197 | − | 4.25325i | |||||||||||||||||||
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 | 20.3.b.a | ✓ | 4 |
| 3.b | odd | 2 | 1 | 180.3.c.a | 4 | ||
| 4.b | odd | 2 | 1 | inner | 20.3.b.a | ✓ | 4 |
| 5.b | even | 2 | 1 | 100.3.b.f | 4 | ||
| 5.c | odd | 4 | 2 | 100.3.d.b | 8 | ||
| 8.b | even | 2 | 1 | 320.3.b.c | 4 | ||
| 8.d | odd | 2 | 1 | 320.3.b.c | 4 | ||
| 12.b | even | 2 | 1 | 180.3.c.a | 4 | ||
| 15.d | odd | 2 | 1 | 900.3.c.k | 4 | ||
| 15.e | even | 4 | 2 | 900.3.f.e | 8 | ||
| 16.e | even | 4 | 2 | 1280.3.g.e | 8 | ||
| 16.f | odd | 4 | 2 | 1280.3.g.e | 8 | ||
| 20.d | odd | 2 | 1 | 100.3.b.f | 4 | ||
| 20.e | even | 4 | 2 | 100.3.d.b | 8 | ||
| 24.f | even | 2 | 1 | 2880.3.e.e | 4 | ||
| 24.h | odd | 2 | 1 | 2880.3.e.e | 4 | ||
| 40.e | odd | 2 | 1 | 1600.3.b.s | 4 | ||
| 40.f | even | 2 | 1 | 1600.3.b.s | 4 | ||
| 40.i | odd | 4 | 2 | 1600.3.h.n | 8 | ||
| 40.k | even | 4 | 2 | 1600.3.h.n | 8 | ||
| 60.h | even | 2 | 1 | 900.3.c.k | 4 | ||
| 60.l | odd | 4 | 2 | 900.3.f.e | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 20.3.b.a | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 20.3.b.a | ✓ | 4 | 4.b | odd | 2 | 1 | inner |
| 100.3.b.f | 4 | 5.b | even | 2 | 1 | ||
| 100.3.b.f | 4 | 20.d | odd | 2 | 1 | ||
| 100.3.d.b | 8 | 5.c | odd | 4 | 2 | ||
| 100.3.d.b | 8 | 20.e | even | 4 | 2 | ||
| 180.3.c.a | 4 | 3.b | odd | 2 | 1 | ||
| 180.3.c.a | 4 | 12.b | even | 2 | 1 | ||
| 320.3.b.c | 4 | 8.b | even | 2 | 1 | ||
| 320.3.b.c | 4 | 8.d | odd | 2 | 1 | ||
| 900.3.c.k | 4 | 15.d | odd | 2 | 1 | ||
| 900.3.c.k | 4 | 60.h | even | 2 | 1 | ||
| 900.3.f.e | 8 | 15.e | even | 4 | 2 | ||
| 900.3.f.e | 8 | 60.l | odd | 4 | 2 | ||
| 1280.3.g.e | 8 | 16.e | even | 4 | 2 | ||
| 1280.3.g.e | 8 | 16.f | odd | 4 | 2 | ||
| 1600.3.b.s | 4 | 40.e | odd | 2 | 1 | ||
| 1600.3.b.s | 4 | 40.f | even | 2 | 1 | ||
| 1600.3.h.n | 8 | 40.i | odd | 4 | 2 | ||
| 1600.3.h.n | 8 | 40.k | even | 4 | 2 | ||
| 2880.3.e.e | 4 | 24.f | even | 2 | 1 | ||
| 2880.3.e.e | 4 | 24.h | odd | 2 | 1 | ||
Hecke kernels
This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(20, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} + 2 T^{3} + \cdots + 16 \)
|
| $3$ |
\( T^{4} + 20T^{2} + 80 \)
|
| $5$ |
\( (T^{2} - 5)^{2} \)
|
| $7$ |
\( T^{4} + 100T^{2} + 2000 \)
|
| $11$ |
\( T^{4} + 400T^{2} + 1280 \)
|
| $13$ |
\( (T^{2} + 8 T - 4)^{2} \)
|
| $17$ |
\( (T^{2} + 12 T - 284)^{2} \)
|
| $19$ |
\( T^{4} + 320 T^{2} + 20480 \)
|
| $23$ |
\( T^{4} + 260T^{2} + 80 \)
|
| $29$ |
\( (T^{2} + 4 T - 76)^{2} \)
|
| $31$ |
\( T^{4} + 2320 T^{2} + 154880 \)
|
| $37$ |
\( (T^{2} - 8 T - 484)^{2} \)
|
| $41$ |
\( (T^{2} + 56 T + 604)^{2} \)
|
| $43$ |
\( T^{4} + 500T^{2} + 2000 \)
|
| $47$ |
\( T^{4} + 4100 T^{2} + 3561680 \)
|
| $53$ |
\( (T^{2} + 88 T + 1436)^{2} \)
|
| $59$ |
\( T^{4} + 5760 T^{2} + 1658880 \)
|
| $61$ |
\( (T^{2} - 64 T - 2356)^{2} \)
|
| $67$ |
\( T^{4} + 10420 T^{2} + 19920080 \)
|
| $71$ |
\( T^{4} + 8080 T^{2} + 10138880 \)
|
| $73$ |
\( (T^{2} - 132 T - 764)^{2} \)
|
| $79$ |
\( T^{4} + 13120 T^{2} + 2478080 \)
|
| $83$ |
\( T^{4} + 6260 T^{2} + 2620880 \)
|
| $89$ |
\( (T^{2} + 44 T - 7516)^{2} \)
|
| $97$ |
\( (T^{2} + 132 T + 3636)^{2} \)
|
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