Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [10,6,Mod(9,10)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(10, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("10.9");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 10 = 2 \cdot 5 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 10.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: | \(1.60383819813\) |
| Analytic rank: | \(0\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{-1}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2 \) |
| 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 \(\beta = 2i\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/10\mathbb{Z}\right)^\times\).
| \(n\) | \(7\) |
| \(\chi(n)\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 9.1 |
|
− | 4.00000i | − | 14.0000i | −16.0000 | 55.0000 | − | 10.0000i | −56.0000 | 158.000i | 64.0000i | 47.0000 | −40.0000 | − | 220.000i | ||||||||||||||||||
| 9.2 | 4.00000i | 14.0000i | −16.0000 | 55.0000 | + | 10.0000i | −56.0000 | − | 158.000i | − | 64.0000i | 47.0000 | −40.0000 | + | 220.000i | |||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 10.6.b.a | ✓ | 2 |
| 3.b | odd | 2 | 1 | 90.6.c.a | 2 | ||
| 4.b | odd | 2 | 1 | 80.6.c.c | 2 | ||
| 5.b | even | 2 | 1 | inner | 10.6.b.a | ✓ | 2 |
| 5.c | odd | 4 | 1 | 50.6.a.c | 1 | ||
| 5.c | odd | 4 | 1 | 50.6.a.e | 1 | ||
| 8.b | even | 2 | 1 | 320.6.c.b | 2 | ||
| 8.d | odd | 2 | 1 | 320.6.c.a | 2 | ||
| 12.b | even | 2 | 1 | 720.6.f.a | 2 | ||
| 15.d | odd | 2 | 1 | 90.6.c.a | 2 | ||
| 15.e | even | 4 | 1 | 450.6.a.c | 1 | ||
| 15.e | even | 4 | 1 | 450.6.a.w | 1 | ||
| 20.d | odd | 2 | 1 | 80.6.c.c | 2 | ||
| 20.e | even | 4 | 1 | 400.6.a.c | 1 | ||
| 20.e | even | 4 | 1 | 400.6.a.k | 1 | ||
| 40.e | odd | 2 | 1 | 320.6.c.a | 2 | ||
| 40.f | even | 2 | 1 | 320.6.c.b | 2 | ||
| 60.h | even | 2 | 1 | 720.6.f.a | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 10.6.b.a | ✓ | 2 | 1.a | even | 1 | 1 | trivial |
| 10.6.b.a | ✓ | 2 | 5.b | even | 2 | 1 | inner |
| 50.6.a.c | 1 | 5.c | odd | 4 | 1 | ||
| 50.6.a.e | 1 | 5.c | odd | 4 | 1 | ||
| 80.6.c.c | 2 | 4.b | odd | 2 | 1 | ||
| 80.6.c.c | 2 | 20.d | odd | 2 | 1 | ||
| 90.6.c.a | 2 | 3.b | odd | 2 | 1 | ||
| 90.6.c.a | 2 | 15.d | odd | 2 | 1 | ||
| 320.6.c.a | 2 | 8.d | odd | 2 | 1 | ||
| 320.6.c.a | 2 | 40.e | odd | 2 | 1 | ||
| 320.6.c.b | 2 | 8.b | even | 2 | 1 | ||
| 320.6.c.b | 2 | 40.f | even | 2 | 1 | ||
| 400.6.a.c | 1 | 20.e | even | 4 | 1 | ||
| 400.6.a.k | 1 | 20.e | even | 4 | 1 | ||
| 450.6.a.c | 1 | 15.e | even | 4 | 1 | ||
| 450.6.a.w | 1 | 15.e | even | 4 | 1 | ||
| 720.6.f.a | 2 | 12.b | even | 2 | 1 | ||
| 720.6.f.a | 2 | 60.h | even | 2 | 1 | ||
Hecke kernels
This newform subspace is the entire newspace \(S_{6}^{\mathrm{new}}(10, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} + 16 \)
|
| $3$ |
\( T^{2} + 196 \)
|
| $5$ |
\( T^{2} - 110T + 3125 \)
|
| $7$ |
\( T^{2} + 24964 \)
|
| $11$ |
\( (T + 148)^{2} \)
|
| $13$ |
\( T^{2} + 467856 \)
|
| $17$ |
\( T^{2} + 4194304 \)
|
| $19$ |
\( (T + 2220)^{2} \)
|
| $23$ |
\( T^{2} + 1552516 \)
|
| $29$ |
\( (T - 270)^{2} \)
|
| $31$ |
\( (T + 2048)^{2} \)
|
| $37$ |
\( T^{2} + 19114384 \)
|
| $41$ |
\( (T + 2398)^{2} \)
|
| $43$ |
\( T^{2} + 5262436 \)
|
| $47$ |
\( T^{2} + 114105124 \)
|
| $53$ |
\( T^{2} + 8785296 \)
|
| $59$ |
\( (T - 39740)^{2} \)
|
| $61$ |
\( (T + 42298)^{2} \)
|
| $67$ |
\( T^{2} + 1030281604 \)
|
| $71$ |
\( (T + 4248)^{2} \)
|
| $73$ |
\( T^{2} + 906250816 \)
|
| $79$ |
\( (T + 35280)^{2} \)
|
| $83$ |
\( T^{2} + 774286276 \)
|
| $89$ |
\( (T - 85210)^{2} \)
|
| $97$ |
\( T^{2} + 9454061824 \)
|
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