Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [315,2,Mod(157,315)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(315, base_ring=CyclotomicField(12))
chi = DirichletCharacter(H, H._module([4, 3, 2]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("315.157");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 315 = 3^{2} \cdot 5 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 315.cg (of order \(12\), degree \(4\), 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: | \(2.51528766367\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\zeta_{12})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{2} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{12}]$ |
$q$-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 \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/315\mathbb{Z}\right)^\times\).
| \(n\) | \(127\) | \(136\) | \(281\) |
| \(\chi(n)\) | \(\zeta_{12}^{3}\) | \(\zeta_{12}^{2}\) | \(-1 + \zeta_{12}^{2}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 157.1 |
|
1.86603 | − | 0.500000i | 1.50000 | + | 0.866025i | 1.50000 | − | 0.866025i | −1.00000 | + | 2.00000i | 3.23205 | + | 0.866025i | 0.500000 | − | 2.59808i | −0.366025 | + | 0.366025i | 1.50000 | + | 2.59808i | −0.866025 | + | 4.23205i | ||||||||||||
| 187.1 | 0.133975 | − | 0.500000i | 1.50000 | − | 0.866025i | 1.50000 | + | 0.866025i | −1.00000 | + | 2.00000i | −0.232051 | − | 0.866025i | 0.500000 | + | 2.59808i | 1.36603 | − | 1.36603i | 1.50000 | − | 2.59808i | 0.866025 | + | 0.767949i | |||||||||||||
| 283.1 | 0.133975 | + | 0.500000i | 1.50000 | + | 0.866025i | 1.50000 | − | 0.866025i | −1.00000 | − | 2.00000i | −0.232051 | + | 0.866025i | 0.500000 | − | 2.59808i | 1.36603 | + | 1.36603i | 1.50000 | + | 2.59808i | 0.866025 | − | 0.767949i | |||||||||||||
| 313.1 | 1.86603 | + | 0.500000i | 1.50000 | − | 0.866025i | 1.50000 | + | 0.866025i | −1.00000 | − | 2.00000i | 3.23205 | − | 0.866025i | 0.500000 | + | 2.59808i | −0.366025 | − | 0.366025i | 1.50000 | − | 2.59808i | −0.866025 | − | 4.23205i | |||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 315.cg | even | 12 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 315.2.cg.d | yes | 4 |
| 3.b | odd | 2 | 1 | 945.2.cj.b | 4 | ||
| 5.c | odd | 4 | 1 | 315.2.cg.b | yes | 4 | |
| 7.d | odd | 6 | 1 | 315.2.bs.d | yes | 4 | |
| 9.c | even | 3 | 1 | 315.2.bs.a | ✓ | 4 | |
| 9.d | odd | 6 | 1 | 945.2.bv.b | 4 | ||
| 15.e | even | 4 | 1 | 945.2.cj.c | 4 | ||
| 21.g | even | 6 | 1 | 945.2.bv.c | 4 | ||
| 35.k | even | 12 | 1 | 315.2.bs.a | ✓ | 4 | |
| 45.k | odd | 12 | 1 | 315.2.bs.d | yes | 4 | |
| 45.l | even | 12 | 1 | 945.2.bv.c | 4 | ||
| 63.k | odd | 6 | 1 | 315.2.cg.b | yes | 4 | |
| 63.s | even | 6 | 1 | 945.2.cj.c | 4 | ||
| 105.w | odd | 12 | 1 | 945.2.bv.b | 4 | ||
| 315.bw | odd | 12 | 1 | 945.2.cj.b | 4 | ||
| 315.cg | even | 12 | 1 | inner | 315.2.cg.d | yes | 4 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 315.2.bs.a | ✓ | 4 | 9.c | even | 3 | 1 | |
| 315.2.bs.a | ✓ | 4 | 35.k | even | 12 | 1 | |
| 315.2.bs.d | yes | 4 | 7.d | odd | 6 | 1 | |
| 315.2.bs.d | yes | 4 | 45.k | odd | 12 | 1 | |
| 315.2.cg.b | yes | 4 | 5.c | odd | 4 | 1 | |
| 315.2.cg.b | yes | 4 | 63.k | odd | 6 | 1 | |
| 315.2.cg.d | yes | 4 | 1.a | even | 1 | 1 | trivial |
| 315.2.cg.d | yes | 4 | 315.cg | even | 12 | 1 | inner |
| 945.2.bv.b | 4 | 9.d | odd | 6 | 1 | ||
| 945.2.bv.b | 4 | 105.w | odd | 12 | 1 | ||
| 945.2.bv.c | 4 | 21.g | even | 6 | 1 | ||
| 945.2.bv.c | 4 | 45.l | even | 12 | 1 | ||
| 945.2.cj.b | 4 | 3.b | odd | 2 | 1 | ||
| 945.2.cj.b | 4 | 315.bw | odd | 12 | 1 | ||
| 945.2.cj.c | 4 | 15.e | even | 4 | 1 | ||
| 945.2.cj.c | 4 | 63.s | even | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(315, [\chi])\):
|
\( T_{2}^{4} - 4T_{2}^{3} + 5T_{2}^{2} - 2T_{2} + 1 \)
|
|
\( T_{11} + 2 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} - 4 T^{3} + \cdots + 1 \)
|
| $3$ |
\( (T^{2} - 3 T + 3)^{2} \)
|
| $5$ |
\( (T^{2} + 2 T + 5)^{2} \)
|
| $7$ |
\( (T^{2} - T + 7)^{2} \)
|
| $11$ |
\( (T + 2)^{4} \)
|
| $13$ |
\( T^{4} - 10 T^{3} + \cdots + 4 \)
|
| $17$ |
\( T^{4} + 8 T^{3} + \cdots + 256 \)
|
| $19$ |
\( T^{4} + 2 T^{3} + \cdots + 4 \)
|
| $23$ |
\( T^{4} - 4 T^{3} + \cdots + 484 \)
|
| $29$ |
\( T^{4} + 24 T^{3} + \cdots + 1521 \)
|
| $31$ |
\( T^{4} + 12 T^{3} + \cdots + 64 \)
|
| $37$ |
\( T^{4} - 18 T^{3} + \cdots + 36 \)
|
| $41$ |
\( T^{4} - 9T^{2} + 81 \)
|
| $43$ |
\( T^{4} - 6 T^{3} + \cdots + 9 \)
|
| $47$ |
\( T^{4} - 6 T^{3} + \cdots + 1521 \)
|
| $53$ |
\( T^{4} + 4 T^{3} + \cdots + 16 \)
|
| $59$ |
\( T^{4} - 6 T^{3} + \cdots + 19044 \)
|
| $61$ |
\( T^{4} - 24 T^{3} + \cdots + 1024 \)
|
| $67$ |
\( T^{4} + 10 T^{3} + \cdots + 484 \)
|
| $71$ |
\( (T^{2} - 6 T + 6)^{2} \)
|
| $73$ |
\( T^{4} + 144 T^{2} + \cdots + 2304 \)
|
| $79$ |
\( T^{4} + 12 T^{3} + \cdots + 2704 \)
|
| $83$ |
\( T^{4} + 26 T^{3} + \cdots + 14641 \)
|
| $89$ |
\( T^{4} + 16 T^{3} + \cdots + 256 \)
|
| $97$ |
\( T^{4} - 34 T^{3} + \cdots + 20164 \)
|
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