Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [630,2,Mod(307,630)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(630, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([0, 1, 2]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("630.307");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 630 = 2 \cdot 3^{2} \cdot 5 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 630.p (of order \(4\), degree \(2\), 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: | \(5.03057532734\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Relative dimension: | \(4\) over \(\Q(i)\) |
| Coefficient field: | \(\Q(\zeta_{16})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 70) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-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 primitive root of unity \(\zeta_{16}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/630\mathbb{Z}\right)^\times\).
| \(n\) | \(127\) | \(281\) | \(451\) |
| \(\chi(n)\) | \(\zeta_{16}^{4}\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 307.1 |
|
−0.707107 | − | 0.707107i | 0 | 1.00000i | −2.23044 | + | 0.158513i | 0 | −2.14065 | + | 1.55487i | 0.707107 | − | 0.707107i | 0 | 1.68925 | + | 1.46508i | ||||||||||||||||||||||||||||||||
| 307.2 | −0.707107 | − | 0.707107i | 0 | 1.00000i | 2.23044 | − | 0.158513i | 0 | 1.55487 | − | 2.14065i | 0.707107 | − | 0.707107i | 0 | −1.68925 | − | 1.46508i | |||||||||||||||||||||||||||||||||
| 307.3 | 0.707107 | + | 0.707107i | 0 | 1.00000i | −0.158513 | − | 2.23044i | 0 | −0.941740 | − | 2.47247i | −0.707107 | + | 0.707107i | 0 | 1.46508 | − | 1.68925i | |||||||||||||||||||||||||||||||||
| 307.4 | 0.707107 | + | 0.707107i | 0 | 1.00000i | 0.158513 | + | 2.23044i | 0 | −2.47247 | − | 0.941740i | −0.707107 | + | 0.707107i | 0 | −1.46508 | + | 1.68925i | |||||||||||||||||||||||||||||||||
| 433.1 | −0.707107 | + | 0.707107i | 0 | − | 1.00000i | −2.23044 | − | 0.158513i | 0 | −2.14065 | − | 1.55487i | 0.707107 | + | 0.707107i | 0 | 1.68925 | − | 1.46508i | ||||||||||||||||||||||||||||||||
| 433.2 | −0.707107 | + | 0.707107i | 0 | − | 1.00000i | 2.23044 | + | 0.158513i | 0 | 1.55487 | + | 2.14065i | 0.707107 | + | 0.707107i | 0 | −1.68925 | + | 1.46508i | ||||||||||||||||||||||||||||||||
| 433.3 | 0.707107 | − | 0.707107i | 0 | − | 1.00000i | −0.158513 | + | 2.23044i | 0 | −0.941740 | + | 2.47247i | −0.707107 | − | 0.707107i | 0 | 1.46508 | + | 1.68925i | ||||||||||||||||||||||||||||||||
| 433.4 | 0.707107 | − | 0.707107i | 0 | − | 1.00000i | 0.158513 | − | 2.23044i | 0 | −2.47247 | + | 0.941740i | −0.707107 | − | 0.707107i | 0 | −1.46508 | − | 1.68925i | ||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.c | odd | 4 | 1 | inner |
| 7.b | odd | 2 | 1 | inner |
| 35.f | even | 4 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 630.2.p.a | 8 | |
| 3.b | odd | 2 | 1 | 70.2.g.a | ✓ | 8 | |
| 5.c | odd | 4 | 1 | inner | 630.2.p.a | 8 | |
| 7.b | odd | 2 | 1 | inner | 630.2.p.a | 8 | |
| 12.b | even | 2 | 1 | 560.2.bj.c | 8 | ||
| 15.d | odd | 2 | 1 | 350.2.g.a | 8 | ||
| 15.e | even | 4 | 1 | 70.2.g.a | ✓ | 8 | |
| 15.e | even | 4 | 1 | 350.2.g.a | 8 | ||
| 21.c | even | 2 | 1 | 70.2.g.a | ✓ | 8 | |
| 21.g | even | 6 | 2 | 490.2.l.a | 16 | ||
| 21.h | odd | 6 | 2 | 490.2.l.a | 16 | ||
| 35.f | even | 4 | 1 | inner | 630.2.p.a | 8 | |
| 60.l | odd | 4 | 1 | 560.2.bj.c | 8 | ||
| 84.h | odd | 2 | 1 | 560.2.bj.c | 8 | ||
| 105.g | even | 2 | 1 | 350.2.g.a | 8 | ||
| 105.k | odd | 4 | 1 | 70.2.g.a | ✓ | 8 | |
| 105.k | odd | 4 | 1 | 350.2.g.a | 8 | ||
| 105.w | odd | 12 | 2 | 490.2.l.a | 16 | ||
| 105.x | even | 12 | 2 | 490.2.l.a | 16 | ||
| 420.w | even | 4 | 1 | 560.2.bj.c | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 70.2.g.a | ✓ | 8 | 3.b | odd | 2 | 1 | |
| 70.2.g.a | ✓ | 8 | 15.e | even | 4 | 1 | |
| 70.2.g.a | ✓ | 8 | 21.c | even | 2 | 1 | |
| 70.2.g.a | ✓ | 8 | 105.k | odd | 4 | 1 | |
| 350.2.g.a | 8 | 15.d | odd | 2 | 1 | ||
| 350.2.g.a | 8 | 15.e | even | 4 | 1 | ||
| 350.2.g.a | 8 | 105.g | even | 2 | 1 | ||
| 350.2.g.a | 8 | 105.k | odd | 4 | 1 | ||
| 490.2.l.a | 16 | 21.g | even | 6 | 2 | ||
| 490.2.l.a | 16 | 21.h | odd | 6 | 2 | ||
| 490.2.l.a | 16 | 105.w | odd | 12 | 2 | ||
| 490.2.l.a | 16 | 105.x | even | 12 | 2 | ||
| 560.2.bj.c | 8 | 12.b | even | 2 | 1 | ||
| 560.2.bj.c | 8 | 60.l | odd | 4 | 1 | ||
| 560.2.bj.c | 8 | 84.h | odd | 2 | 1 | ||
| 560.2.bj.c | 8 | 420.w | even | 4 | 1 | ||
| 630.2.p.a | 8 | 1.a | even | 1 | 1 | trivial | |
| 630.2.p.a | 8 | 5.c | odd | 4 | 1 | inner | |
| 630.2.p.a | 8 | 7.b | odd | 2 | 1 | inner | |
| 630.2.p.a | 8 | 35.f | even | 4 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(630, [\chi])\):
|
\( T_{11}^{2} - 8 \)
|
|
\( T_{13}^{8} + 1548T_{13}^{4} + 334084 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{4} + 1)^{2} \)
|
| $3$ |
\( T^{8} \)
|
| $5$ |
\( T^{8} - 48T^{4} + 625 \)
|
| $7$ |
\( T^{8} + 8 T^{7} + \cdots + 2401 \)
|
| $11$ |
\( (T^{2} - 8)^{4} \)
|
| $13$ |
\( T^{8} + 1548 T^{4} + 334084 \)
|
| $17$ |
\( T^{8} + 768 T^{4} + 16384 \)
|
| $19$ |
\( (T^{4} - 52 T^{2} + 98)^{2} \)
|
| $23$ |
\( (T^{4} - 4 T^{3} + 8 T^{2} + \cdots + 4)^{2} \)
|
| $29$ |
\( (T^{4} + 24 T^{2} + 16)^{2} \)
|
| $31$ |
\( (T^{4} + 16 T^{2} + 32)^{2} \)
|
| $37$ |
\( (T^{4} - 16 T^{3} + \cdots + 784)^{2} \)
|
| $41$ |
\( (T^{4} + 16 T^{2} + 32)^{2} \)
|
| $43$ |
\( (T^{2} - 8 T + 32)^{4} \)
|
| $47$ |
\( T^{8} + 192T^{4} + 1024 \)
|
| $53$ |
\( (T^{4} - 16 T^{3} + \cdots + 16)^{2} \)
|
| $59$ |
\( (T^{4} - 100 T^{2} + 1250)^{2} \)
|
| $61$ |
\( (T^{4} + 148 T^{2} + 4418)^{2} \)
|
| $67$ |
\( (T^{4} - 8 T^{3} + \cdots + 18496)^{2} \)
|
| $71$ |
\( (T^{2} - 4 T + 2)^{4} \)
|
| $73$ |
\( T^{8} + 3264 T^{4} + 2458624 \)
|
| $79$ |
\( (T^{4} + 108 T^{2} + 2116)^{2} \)
|
| $83$ |
\( T^{8} + 6732 T^{4} + 11303044 \)
|
| $89$ |
\( (T^{4} - 416 T^{2} + 36992)^{2} \)
|
| $97$ |
\( T^{8} + 62208 T^{4} + 107495424 \)
|
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