Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [343,2,Mod(50,343)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(343, base_ring=CyclotomicField(14))
chi = DirichletCharacter(H, H._module([12]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("343.50");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 343 = 7^{3} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 343.e (of order \(7\), degree \(6\), 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: | \(2.73886878933\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | \(\Q(\zeta_{14})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 49) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{7}]$ |
$q$-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 \(\zeta_{14}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/343\mathbb{Z}\right)^\times\).
| \(n\) | \(3\) |
| \(\chi(n)\) | \(-\zeta_{14}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 50.1 |
|
−0.500000 | − | 0.626980i | 0.500000 | − | 0.240787i | 0.301938 | − | 1.32288i | −2.52446 | + | 1.21572i | −0.400969 | − | 0.193096i | 0 | −2.42543 | + | 1.16802i | −1.67845 | + | 2.10471i | 2.02446 | + | 0.974928i | ||||||||||||||||||||
| 99.1 | −0.500000 | + | 2.19064i | 0.500000 | − | 0.626980i | −2.74698 | − | 1.32288i | −0.153989 | + | 0.193096i | 1.12349 | + | 1.40881i | 0 | 1.46950 | − | 1.84270i | 0.524459 | + | 2.29780i | −0.346011 | − | 0.433884i | |||||||||||||||||||||
| 148.1 | −0.500000 | + | 0.240787i | 0.500000 | − | 2.19064i | −1.05496 | + | 1.32288i | −0.321552 | + | 1.40881i | 0.277479 | + | 1.21572i | 0 | 0.455927 | − | 1.99755i | −1.84601 | − | 0.888992i | −0.178448 | − | 0.781831i | |||||||||||||||||||||
| 197.1 | −0.500000 | − | 0.240787i | 0.500000 | + | 2.19064i | −1.05496 | − | 1.32288i | −0.321552 | − | 1.40881i | 0.277479 | − | 1.21572i | 0 | 0.455927 | + | 1.99755i | −1.84601 | + | 0.888992i | −0.178448 | + | 0.781831i | |||||||||||||||||||||
| 246.1 | −0.500000 | − | 2.19064i | 0.500000 | + | 0.626980i | −2.74698 | + | 1.32288i | −0.153989 | − | 0.193096i | 1.12349 | − | 1.40881i | 0 | 1.46950 | + | 1.84270i | 0.524459 | − | 2.29780i | −0.346011 | + | 0.433884i | |||||||||||||||||||||
| 295.1 | −0.500000 | + | 0.626980i | 0.500000 | + | 0.240787i | 0.301938 | + | 1.32288i | −2.52446 | − | 1.21572i | −0.400969 | + | 0.193096i | 0 | −2.42543 | − | 1.16802i | −1.67845 | − | 2.10471i | 2.02446 | − | 0.974928i | |||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 49.e | even | 7 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 343.2.e.a | 6 | |
| 7.b | odd | 2 | 1 | 49.2.e.a | ✓ | 6 | |
| 7.c | even | 3 | 2 | 343.2.g.e | 12 | ||
| 7.d | odd | 6 | 2 | 343.2.g.f | 12 | ||
| 21.c | even | 2 | 1 | 441.2.u.a | 6 | ||
| 28.d | even | 2 | 1 | 784.2.u.a | 6 | ||
| 49.e | even | 7 | 1 | inner | 343.2.e.a | 6 | |
| 49.e | even | 7 | 1 | 2401.2.a.a | 3 | ||
| 49.f | odd | 14 | 1 | 49.2.e.a | ✓ | 6 | |
| 49.f | odd | 14 | 1 | 2401.2.a.b | 3 | ||
| 49.g | even | 21 | 2 | 343.2.g.e | 12 | ||
| 49.h | odd | 42 | 2 | 343.2.g.f | 12 | ||
| 147.k | even | 14 | 1 | 441.2.u.a | 6 | ||
| 196.j | even | 14 | 1 | 784.2.u.a | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 49.2.e.a | ✓ | 6 | 7.b | odd | 2 | 1 | |
| 49.2.e.a | ✓ | 6 | 49.f | odd | 14 | 1 | |
| 343.2.e.a | 6 | 1.a | even | 1 | 1 | trivial | |
| 343.2.e.a | 6 | 49.e | even | 7 | 1 | inner | |
| 343.2.g.e | 12 | 7.c | even | 3 | 2 | ||
| 343.2.g.e | 12 | 49.g | even | 21 | 2 | ||
| 343.2.g.f | 12 | 7.d | odd | 6 | 2 | ||
| 343.2.g.f | 12 | 49.h | odd | 42 | 2 | ||
| 441.2.u.a | 6 | 21.c | even | 2 | 1 | ||
| 441.2.u.a | 6 | 147.k | even | 14 | 1 | ||
| 784.2.u.a | 6 | 28.d | even | 2 | 1 | ||
| 784.2.u.a | 6 | 196.j | even | 14 | 1 | ||
| 2401.2.a.a | 3 | 49.e | even | 7 | 1 | ||
| 2401.2.a.b | 3 | 49.f | odd | 14 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(343, [\chi])\):
|
\( T_{2}^{6} + 3T_{2}^{5} + 9T_{2}^{4} + 13T_{2}^{3} + 11T_{2}^{2} + 5T_{2} + 1 \)
|
|
\( T_{3}^{6} - 3T_{3}^{5} + 9T_{3}^{4} - 13T_{3}^{3} + 11T_{3}^{2} - 5T_{3} + 1 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} + 3 T^{5} + \cdots + 1 \)
|
| $3$ |
\( T^{6} - 3 T^{5} + \cdots + 1 \)
|
| $5$ |
\( T^{6} + 6 T^{5} + \cdots + 1 \)
|
| $7$ |
\( T^{6} \)
|
| $11$ |
\( T^{6} - 2 T^{5} + \cdots + 169 \)
|
| $13$ |
\( T^{6} + 14 T^{4} + \cdots + 41209 \)
|
| $17$ |
\( T^{6} + 4 T^{5} + \cdots + 1681 \)
|
| $19$ |
\( (T^{3} - 4 T^{2} - 11 T + 1)^{2} \)
|
| $23$ |
\( T^{6} + 10 T^{5} + \cdots + 169 \)
|
| $29$ |
\( T^{6} + 16 T^{5} + \cdots + 6889 \)
|
| $31$ |
\( (T^{3} - 5 T^{2} + 6 T - 1)^{2} \)
|
| $37$ |
\( T^{6} - 4 T^{5} + \cdots + 10816 \)
|
| $41$ |
\( T^{6} + 56 T^{4} + \cdots + 3136 \)
|
| $43$ |
\( T^{6} - 12 T^{5} + \cdots + 841 \)
|
| $47$ |
\( T^{6} - 15 T^{5} + \cdots + 9409 \)
|
| $53$ |
\( T^{6} + 26 T^{5} + \cdots + 53824 \)
|
| $59$ |
\( T^{6} + 11 T^{5} + \cdots + 57121 \)
|
| $61$ |
\( T^{6} - 8 T^{5} + \cdots + 1849 \)
|
| $67$ |
\( (T^{3} + 6 T^{2} + 5 T - 13)^{2} \)
|
| $71$ |
\( T^{6} - 5 T^{5} + \cdots + 85849 \)
|
| $73$ |
\( T^{6} + 4 T^{5} + \cdots + 5041 \)
|
| $79$ |
\( (T^{3} - 30 T^{2} + \cdots - 937)^{2} \)
|
| $83$ |
\( T^{6} - 14 T^{5} + \cdots + 625681 \)
|
| $89$ |
\( T^{6} + 13 T^{5} + \cdots + 187489 \)
|
| $97$ |
\( (T^{3} - 7 T + 7)^{2} \)
|
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