Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1950,2,Mod(49,1950)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1950, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 3, 5]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1950.49");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1950 = 2 \cdot 3 \cdot 5^{2} \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1950.y (of order \(6\), degree \(2\), not 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: | \(15.5708283941\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{6})\) |
| 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: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 78) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-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 \(\left(\mathbb{Z}/1950\mathbb{Z}\right)^\times\).
| \(n\) | \(301\) | \(1301\) | \(1327\) |
| \(\chi(n)\) | \(\zeta_{12}^{2}\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 49.1 |
|
0.500000 | + | 0.866025i | −0.866025 | + | 0.500000i | −0.500000 | + | 0.866025i | 0 | −0.866025 | − | 0.500000i | −0.366025 | + | 0.633975i | −1.00000 | 0.500000 | − | 0.866025i | 0 | ||||||||||||||||||
| 49.2 | 0.500000 | + | 0.866025i | 0.866025 | − | 0.500000i | −0.500000 | + | 0.866025i | 0 | 0.866025 | + | 0.500000i | 1.36603 | − | 2.36603i | −1.00000 | 0.500000 | − | 0.866025i | 0 | |||||||||||||||||||
| 199.1 | 0.500000 | − | 0.866025i | −0.866025 | − | 0.500000i | −0.500000 | − | 0.866025i | 0 | −0.866025 | + | 0.500000i | −0.366025 | − | 0.633975i | −1.00000 | 0.500000 | + | 0.866025i | 0 | |||||||||||||||||||
| 199.2 | 0.500000 | − | 0.866025i | 0.866025 | + | 0.500000i | −0.500000 | − | 0.866025i | 0 | 0.866025 | − | 0.500000i | 1.36603 | + | 2.36603i | −1.00000 | 0.500000 | + | 0.866025i | 0 | |||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 65.l | even | 6 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 1950.2.y.g | 4 | |
| 5.b | even | 2 | 1 | 1950.2.y.b | 4 | ||
| 5.c | odd | 4 | 1 | 78.2.i.a | ✓ | 4 | |
| 5.c | odd | 4 | 1 | 1950.2.bc.d | 4 | ||
| 13.e | even | 6 | 1 | 1950.2.y.b | 4 | ||
| 15.e | even | 4 | 1 | 234.2.l.c | 4 | ||
| 20.e | even | 4 | 1 | 624.2.bv.e | 4 | ||
| 60.l | odd | 4 | 1 | 1872.2.by.h | 4 | ||
| 65.f | even | 4 | 1 | 1014.2.e.g | 4 | ||
| 65.h | odd | 4 | 1 | 1014.2.i.a | 4 | ||
| 65.k | even | 4 | 1 | 1014.2.e.i | 4 | ||
| 65.l | even | 6 | 1 | inner | 1950.2.y.g | 4 | |
| 65.o | even | 12 | 1 | 1014.2.a.i | 2 | ||
| 65.o | even | 12 | 1 | 1014.2.e.i | 4 | ||
| 65.q | odd | 12 | 1 | 1014.2.b.e | 4 | ||
| 65.q | odd | 12 | 1 | 1014.2.i.a | 4 | ||
| 65.r | odd | 12 | 1 | 78.2.i.a | ✓ | 4 | |
| 65.r | odd | 12 | 1 | 1014.2.b.e | 4 | ||
| 65.r | odd | 12 | 1 | 1950.2.bc.d | 4 | ||
| 65.t | even | 12 | 1 | 1014.2.a.k | 2 | ||
| 65.t | even | 12 | 1 | 1014.2.e.g | 4 | ||
| 195.bc | odd | 12 | 1 | 3042.2.a.p | 2 | ||
| 195.bf | even | 12 | 1 | 234.2.l.c | 4 | ||
| 195.bf | even | 12 | 1 | 3042.2.b.i | 4 | ||
| 195.bl | even | 12 | 1 | 3042.2.b.i | 4 | ||
| 195.bn | odd | 12 | 1 | 3042.2.a.y | 2 | ||
| 260.be | odd | 12 | 1 | 8112.2.a.bj | 2 | ||
| 260.bg | even | 12 | 1 | 624.2.bv.e | 4 | ||
| 260.bl | odd | 12 | 1 | 8112.2.a.bp | 2 | ||
| 780.cw | odd | 12 | 1 | 1872.2.by.h | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 78.2.i.a | ✓ | 4 | 5.c | odd | 4 | 1 | |
| 78.2.i.a | ✓ | 4 | 65.r | odd | 12 | 1 | |
| 234.2.l.c | 4 | 15.e | even | 4 | 1 | ||
| 234.2.l.c | 4 | 195.bf | even | 12 | 1 | ||
| 624.2.bv.e | 4 | 20.e | even | 4 | 1 | ||
| 624.2.bv.e | 4 | 260.bg | even | 12 | 1 | ||
| 1014.2.a.i | 2 | 65.o | even | 12 | 1 | ||
| 1014.2.a.k | 2 | 65.t | even | 12 | 1 | ||
| 1014.2.b.e | 4 | 65.q | odd | 12 | 1 | ||
| 1014.2.b.e | 4 | 65.r | odd | 12 | 1 | ||
| 1014.2.e.g | 4 | 65.f | even | 4 | 1 | ||
| 1014.2.e.g | 4 | 65.t | even | 12 | 1 | ||
| 1014.2.e.i | 4 | 65.k | even | 4 | 1 | ||
| 1014.2.e.i | 4 | 65.o | even | 12 | 1 | ||
| 1014.2.i.a | 4 | 65.h | odd | 4 | 1 | ||
| 1014.2.i.a | 4 | 65.q | odd | 12 | 1 | ||
| 1872.2.by.h | 4 | 60.l | odd | 4 | 1 | ||
| 1872.2.by.h | 4 | 780.cw | odd | 12 | 1 | ||
| 1950.2.y.b | 4 | 5.b | even | 2 | 1 | ||
| 1950.2.y.b | 4 | 13.e | even | 6 | 1 | ||
| 1950.2.y.g | 4 | 1.a | even | 1 | 1 | trivial | |
| 1950.2.y.g | 4 | 65.l | even | 6 | 1 | inner | |
| 1950.2.bc.d | 4 | 5.c | odd | 4 | 1 | ||
| 1950.2.bc.d | 4 | 65.r | odd | 12 | 1 | ||
| 3042.2.a.p | 2 | 195.bc | odd | 12 | 1 | ||
| 3042.2.a.y | 2 | 195.bn | odd | 12 | 1 | ||
| 3042.2.b.i | 4 | 195.bf | even | 12 | 1 | ||
| 3042.2.b.i | 4 | 195.bl | even | 12 | 1 | ||
| 8112.2.a.bj | 2 | 260.be | odd | 12 | 1 | ||
| 8112.2.a.bp | 2 | 260.bl | odd | 12 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{4} - 2T_{7}^{3} + 6T_{7}^{2} + 4T_{7} + 4 \)
acting on \(S_{2}^{\mathrm{new}}(1950, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} - T + 1)^{2} \)
|
| $3$ |
\( T^{4} - T^{2} + 1 \)
|
| $5$ |
\( T^{4} \)
|
| $7$ |
\( T^{4} - 2 T^{3} + \cdots + 4 \)
|
| $11$ |
\( T^{4} + 6 T^{3} + \cdots + 36 \)
|
| $13$ |
\( (T^{2} - 5 T + 13)^{2} \)
|
| $17$ |
\( T^{4} - 6 T^{3} + \cdots + 169 \)
|
| $19$ |
\( T^{4} + 6 T^{3} + \cdots + 36 \)
|
| $23$ |
\( T^{4} + 18 T^{3} + \cdots + 676 \)
|
| $29$ |
\( T^{4} + 2 T^{3} + \cdots + 121 \)
|
| $31$ |
\( T^{4} + 32T^{2} + 64 \)
|
| $37$ |
\( T^{4} + 14 T^{3} + \cdots + 1369 \)
|
| $41$ |
\( T^{4} - 36 T^{3} + \cdots + 11449 \)
|
| $43$ |
\( T^{4} + 30 T^{3} + \cdots + 5476 \)
|
| $47$ |
\( (T^{2} + 6 T - 18)^{2} \)
|
| $53$ |
\( T^{4} + 42T^{2} + 9 \)
|
| $59$ |
\( T^{4} - 64T^{2} + 4096 \)
|
| $61$ |
\( T^{4} - 8 T^{3} + \cdots + 121 \)
|
| $67$ |
\( T^{4} - 2 T^{3} + \cdots + 21316 \)
|
| $71$ |
\( T^{4} - 6 T^{3} + \cdots + 36 \)
|
| $73$ |
\( (T^{2} - 16 T + 61)^{2} \)
|
| $79$ |
\( (T^{2} - 12 T + 24)^{2} \)
|
| $83$ |
\( (T^{2} - 10 T - 2)^{2} \)
|
| $89$ |
\( T^{4} - 12 T^{3} + \cdots + 576 \)
|
| $97$ |
\( (T^{2} + 6 T + 36)^{2} \)
|
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