Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [726,2,Mod(161,726)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(726, base_ring=CyclotomicField(10))
chi = DirichletCharacter(H, H._module([5, 7]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("726.161");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 726 = 2 \cdot 3 \cdot 11^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 726.h (of order \(10\), degree \(4\), 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: | \(5.79713918674\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{10})\) |
| Coefficient field: | 8.0.64000000.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - 2x^{6} + 4x^{4} - 8x^{2} + 16 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 66) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{10}]$ |
$q$-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 basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring in terms of a root \(\nu\) of
\( x^{8} - 2x^{6} + 4x^{4} - 8x^{2} + 16 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} ) / 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} ) / 2 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{4} ) / 4 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{5} ) / 4 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( \nu^{6} ) / 8 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( \nu^{7} ) / 8 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( 2\beta_{2} \)
|
| \(\nu^{3}\) | \(=\) |
\( 2\beta_{3} \)
|
| \(\nu^{4}\) | \(=\) |
\( 4\beta_{4} \)
|
| \(\nu^{5}\) | \(=\) |
\( 4\beta_{5} \)
|
| \(\nu^{6}\) | \(=\) |
\( 8\beta_{6} \)
|
| \(\nu^{7}\) | \(=\) |
\( 8\beta_{7} \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/726\mathbb{Z}\right)^\times\).
| \(n\) | \(485\) | \(607\) |
| \(\chi(n)\) | \(-1\) | \(1 - \beta_{2} + \beta_{4} - \beta_{6}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 161.1 |
|
0.809017 | + | 0.587785i | −1.65401 | + | 0.514040i | 0.309017 | + | 0.951057i | 0.831254 | + | 1.14412i | −1.64027 | − | 0.556338i | 4.03499 | − | 1.31105i | −0.309017 | + | 0.951057i | 2.47152 | − | 1.70046i | 1.41421i | ||||||||||||||||||||||||||
| 161.2 | 0.809017 | + | 0.587785i | 1.03598 | + | 1.38807i | 0.309017 | + | 0.951057i | −0.831254 | − | 1.14412i | 0.0222369 | + | 1.73191i | −4.03499 | + | 1.31105i | −0.309017 | + | 0.951057i | −0.853491 | + | 2.87603i | − | 1.41421i | ||||||||||||||||||||||||||
| 215.1 | −0.309017 | − | 0.951057i | −0.0222369 | + | 1.73191i | −0.809017 | + | 0.587785i | −1.34500 | − | 0.437016i | 1.65401 | − | 0.514040i | 2.49376 | + | 3.43237i | 0.809017 | + | 0.587785i | −2.99901 | − | 0.0770245i | 1.41421i | |||||||||||||||||||||||||||
| 215.2 | −0.309017 | − | 0.951057i | 1.64027 | − | 0.556338i | −0.809017 | + | 0.587785i | 1.34500 | + | 0.437016i | −1.03598 | − | 1.38807i | −2.49376 | − | 3.43237i | 0.809017 | + | 0.587785i | 2.38098 | − | 1.82509i | − | 1.41421i | ||||||||||||||||||||||||||
| 233.1 | −0.309017 | + | 0.951057i | −0.0222369 | − | 1.73191i | −0.809017 | − | 0.587785i | −1.34500 | + | 0.437016i | 1.65401 | + | 0.514040i | 2.49376 | − | 3.43237i | 0.809017 | − | 0.587785i | −2.99901 | + | 0.0770245i | − | 1.41421i | ||||||||||||||||||||||||||
| 233.2 | −0.309017 | + | 0.951057i | 1.64027 | + | 0.556338i | −0.809017 | − | 0.587785i | 1.34500 | − | 0.437016i | −1.03598 | + | 1.38807i | −2.49376 | + | 3.43237i | 0.809017 | − | 0.587785i | 2.38098 | + | 1.82509i | 1.41421i | |||||||||||||||||||||||||||
| 239.1 | 0.809017 | − | 0.587785i | −1.65401 | − | 0.514040i | 0.309017 | − | 0.951057i | 0.831254 | − | 1.14412i | −1.64027 | + | 0.556338i | 4.03499 | + | 1.31105i | −0.309017 | − | 0.951057i | 2.47152 | + | 1.70046i | − | 1.41421i | ||||||||||||||||||||||||||
| 239.2 | 0.809017 | − | 0.587785i | 1.03598 | − | 1.38807i | 0.309017 | − | 0.951057i | −0.831254 | + | 1.14412i | 0.0222369 | − | 1.73191i | −4.03499 | − | 1.31105i | −0.309017 | − | 0.951057i | −0.853491 | − | 2.87603i | 1.41421i | |||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 11.c | even | 5 | 3 | inner |
| 33.d | even | 2 | 1 | inner |
| 33.f | even | 10 | 3 | inner |
Twists
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(726, [\chi])\):
|
\( T_{5}^{8} - 2T_{5}^{6} + 4T_{5}^{4} - 8T_{5}^{2} + 16 \)
|
|
\( T_{7}^{8} - 18T_{7}^{6} + 324T_{7}^{4} - 5832T_{7}^{2} + 104976 \)
|
|
\( T_{17} \)
|
|
\( T_{29}^{4} + 6T_{29}^{3} + 36T_{29}^{2} + 216T_{29} + 1296 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{4} - T^{3} + T^{2} + \cdots + 1)^{2} \)
|
| $3$ |
\( T^{8} - 2 T^{7} + \cdots + 81 \)
|
| $5$ |
\( T^{8} - 2 T^{6} + \cdots + 16 \)
|
| $7$ |
\( T^{8} - 18 T^{6} + \cdots + 104976 \)
|
| $11$ |
\( T^{8} \)
|
| $13$ |
\( T^{8} - 18 T^{6} + \cdots + 104976 \)
|
| $17$ |
\( T^{8} \)
|
| $19$ |
\( T^{8} \)
|
| $23$ |
\( (T^{2} + 2)^{4} \)
|
| $29$ |
\( (T^{4} + 6 T^{3} + \cdots + 1296)^{2} \)
|
| $31$ |
\( (T^{4} - 4 T^{3} + \cdots + 256)^{2} \)
|
| $37$ |
\( (T^{4} + 2 T^{3} + 4 T^{2} + \cdots + 16)^{2} \)
|
| $41$ |
\( (T^{4} - 6 T^{3} + \cdots + 1296)^{2} \)
|
| $43$ |
\( (T^{2} + 72)^{4} \)
|
| $47$ |
\( T^{8} - 98 T^{6} + \cdots + 92236816 \)
|
| $53$ |
\( T^{8} - 50 T^{6} + \cdots + 6250000 \)
|
| $59$ |
\( T^{8} - 128 T^{6} + \cdots + 268435456 \)
|
| $61$ |
\( T^{8} - 18 T^{6} + \cdots + 104976 \)
|
| $67$ |
\( (T + 4)^{8} \)
|
| $71$ |
\( T^{8} - 50 T^{6} + \cdots + 6250000 \)
|
| $73$ |
\( T^{8} \)
|
| $79$ |
\( T^{8} - 18 T^{6} + \cdots + 104976 \)
|
| $83$ |
\( (T^{4} - 12 T^{3} + \cdots + 20736)^{2} \)
|
| $89$ |
\( (T^{2} + 32)^{4} \)
|
| $97$ |
\( (T^{4} + 8 T^{3} + \cdots + 4096)^{2} \)
|
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