Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [273,2,Mod(44,273)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(273, base_ring=CyclotomicField(12))
chi = DirichletCharacter(H, H._module([6, 4, 9]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("273.44");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 273 = 3 \cdot 7 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 273.cd (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: | \(2.17991597518\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{12})\) |
| Coefficient field: | \(\Q(\zeta_{24})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - x^{4} + 1 \)
|
| 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 \(\zeta_{24}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/273\mathbb{Z}\right)^\times\).
| \(n\) | \(92\) | \(106\) | \(157\) |
| \(\chi(n)\) | \(-1\) | \(-\zeta_{24}^{6}\) | \(-\zeta_{24}^{4}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 44.1 |
|
−0.965926 | + | 0.258819i | 1.62484 | − | 0.599900i | −0.866025 | + | 0.500000i | 0.565826 | − | 0.151613i | −1.41421 | + | 1.00000i | −2.38014 | − | 1.15539i | 2.12132 | − | 2.12132i | 2.28024 | − | 1.94949i | −0.507306 | + | 0.292893i | ||||||||||||||||||||||||
| 44.2 | 0.965926 | − | 0.258819i | 1.10721 | + | 1.33195i | −0.866025 | + | 0.500000i | −3.29788 | + | 0.883663i | 1.41421 | + | 1.00000i | 2.38014 | + | 1.15539i | −2.12132 | + | 2.12132i | −0.548188 | + | 2.94949i | −2.95680 | + | 1.70711i | |||||||||||||||||||||||||
| 86.1 | −0.258819 | − | 0.965926i | 0.599900 | + | 1.62484i | 0.866025 | − | 0.500000i | 0.883663 | + | 3.29788i | 1.41421 | − | 1.00000i | 1.15539 | − | 2.38014i | −2.12132 | − | 2.12132i | −2.28024 | + | 1.94949i | 2.95680 | − | 1.70711i | |||||||||||||||||||||||||
| 86.2 | 0.258819 | + | 0.965926i | −1.33195 | + | 1.10721i | 0.866025 | − | 0.500000i | −0.151613 | − | 0.565826i | −1.41421 | − | 1.00000i | −1.15539 | + | 2.38014i | 2.12132 | + | 2.12132i | 0.548188 | − | 2.94949i | 0.507306 | − | 0.292893i | |||||||||||||||||||||||||
| 200.1 | −0.258819 | + | 0.965926i | 0.599900 | − | 1.62484i | 0.866025 | + | 0.500000i | 0.883663 | − | 3.29788i | 1.41421 | + | 1.00000i | 1.15539 | + | 2.38014i | −2.12132 | + | 2.12132i | −2.28024 | − | 1.94949i | 2.95680 | + | 1.70711i | |||||||||||||||||||||||||
| 200.2 | 0.258819 | − | 0.965926i | −1.33195 | − | 1.10721i | 0.866025 | + | 0.500000i | −0.151613 | + | 0.565826i | −1.41421 | + | 1.00000i | −1.15539 | − | 2.38014i | 2.12132 | − | 2.12132i | 0.548188 | + | 2.94949i | 0.507306 | + | 0.292893i | |||||||||||||||||||||||||
| 242.1 | −0.965926 | − | 0.258819i | 1.62484 | + | 0.599900i | −0.866025 | − | 0.500000i | 0.565826 | + | 0.151613i | −1.41421 | − | 1.00000i | −2.38014 | + | 1.15539i | 2.12132 | + | 2.12132i | 2.28024 | + | 1.94949i | −0.507306 | − | 0.292893i | |||||||||||||||||||||||||
| 242.2 | 0.965926 | + | 0.258819i | 1.10721 | − | 1.33195i | −0.866025 | − | 0.500000i | −3.29788 | − | 0.883663i | 1.41421 | − | 1.00000i | 2.38014 | − | 1.15539i | −2.12132 | − | 2.12132i | −0.548188 | − | 2.94949i | −2.95680 | − | 1.70711i | |||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.c | even | 3 | 1 | inner |
| 39.f | even | 4 | 1 | inner |
| 273.cd | even | 12 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 273.2.cd.c | ✓ | 8 |
| 3.b | odd | 2 | 1 | 273.2.cd.d | yes | 8 | |
| 7.c | even | 3 | 1 | inner | 273.2.cd.c | ✓ | 8 |
| 13.d | odd | 4 | 1 | 273.2.cd.d | yes | 8 | |
| 21.h | odd | 6 | 1 | 273.2.cd.d | yes | 8 | |
| 39.f | even | 4 | 1 | inner | 273.2.cd.c | ✓ | 8 |
| 91.z | odd | 12 | 1 | 273.2.cd.d | yes | 8 | |
| 273.cd | even | 12 | 1 | inner | 273.2.cd.c | ✓ | 8 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 273.2.cd.c | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 273.2.cd.c | ✓ | 8 | 7.c | even | 3 | 1 | inner |
| 273.2.cd.c | ✓ | 8 | 39.f | even | 4 | 1 | inner |
| 273.2.cd.c | ✓ | 8 | 273.cd | even | 12 | 1 | inner |
| 273.2.cd.d | yes | 8 | 3.b | odd | 2 | 1 | |
| 273.2.cd.d | yes | 8 | 13.d | odd | 4 | 1 | |
| 273.2.cd.d | yes | 8 | 21.h | odd | 6 | 1 | |
| 273.2.cd.d | yes | 8 | 91.z | odd | 12 | 1 | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(273, [\chi])\):
|
\( T_{2}^{8} - T_{2}^{4} + 1 \)
|
|
\( T_{5}^{8} + 4T_{5}^{7} + 8T_{5}^{6} + 48T_{5}^{5} + 92T_{5}^{4} - 96T_{5}^{3} + 32T_{5}^{2} - 32T_{5} + 16 \)
|
|
\( T_{19}^{8} + 16 T_{19}^{7} + 128 T_{19}^{6} + 1056 T_{19}^{5} + 7487 T_{19}^{4} + 32736 T_{19}^{3} + \cdots + 923521 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} - T^{4} + 1 \)
|
| $3$ |
\( T^{8} - 4 T^{7} + \cdots + 81 \)
|
| $5$ |
\( T^{8} + 4 T^{7} + \cdots + 16 \)
|
| $7$ |
\( T^{8} + 23T^{4} + 2401 \)
|
| $11$ |
\( T^{8} - 8 T^{7} + \cdots + 1 \)
|
| $13$ |
\( (T^{2} - 4 T + 13)^{4} \)
|
| $17$ |
\( (T^{4} - 2 T^{3} + 11 T^{2} + \cdots + 49)^{2} \)
|
| $19$ |
\( T^{8} + 16 T^{7} + \cdots + 923521 \)
|
| $23$ |
\( (T^{4} - 16 T^{3} + \cdots + 3844)^{2} \)
|
| $29$ |
\( (T^{2} + 1)^{4} \)
|
| $31$ |
\( T^{8} + 8 T^{7} + \cdots + 256 \)
|
| $37$ |
\( T^{8} - 256 T^{4} + 65536 \)
|
| $41$ |
\( (T^{2} - 8 T + 32)^{4} \)
|
| $43$ |
\( (T^{2} + 36)^{4} \)
|
| $47$ |
\( T^{8} + 16 T^{7} + \cdots + 2401 \)
|
| $53$ |
\( T^{8} - 82 T^{6} + \cdots + 279841 \)
|
| $59$ |
\( T^{8} + 24 T^{7} + \cdots + 25411681 \)
|
| $61$ |
\( (T^{2} - T + 1)^{4} \)
|
| $67$ |
\( T^{8} + 24 T^{7} + \cdots + 4879681 \)
|
| $71$ |
\( (T^{4} - 16 T^{3} + \cdots + 49)^{2} \)
|
| $73$ |
\( T^{8} - 8 T^{7} + \cdots + 71639296 \)
|
| $79$ |
\( (T^{4} + 2 T^{2} + 4)^{2} \)
|
| $83$ |
\( (T^{4} + 16 T^{3} + \cdots + 4624)^{2} \)
|
| $89$ |
\( T^{8} + 24 T^{7} + \cdots + 9834496 \)
|
| $97$ |
\( (T^{2} + 14 T + 98)^{4} \)
|
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