Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [392,3,Mod(99,392)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(392, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 1, 0]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("392.99");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 392 = 2^{3} \cdot 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 392.g (of order \(2\), degree \(1\), 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: | \(10.6812263629\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.0.700560112.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 2x^{4} - 6x^{3} - 8x^{2} + 64 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | no (minimal twist has level 56) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - 2x^{4} - 6x^{3} - 8x^{2} + 64 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{4} - 2\nu^{2} - 2\nu - 4 ) / 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{5} - 2\nu^{3} + 10\nu^{2} - 8\nu - 16 ) / 16 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{5} - 2\nu^{3} - 6\nu^{2} - 8\nu ) / 16 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{5} + 2\nu^{4} + 6\nu^{3} - 2\nu^{2} - 20\nu - 40 ) / 8 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{4} + \beta_{3} + 1 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{5} - \beta_{4} - \beta_{3} - \beta_{2} + \beta _1 + 3 \)
|
| \(\nu^{4}\) | \(=\) |
\( -2\beta_{4} + 2\beta_{3} + 4\beta_{2} + 2\beta _1 + 6 \)
|
| \(\nu^{5}\) | \(=\) |
\( 2\beta_{5} + 8\beta_{4} + 4\beta_{3} - 2\beta_{2} + 10\beta _1 + 12 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/392\mathbb{Z}\right)^\times\).
| \(n\) | \(197\) | \(295\) | \(297\) |
| \(\chi(n)\) | \(-1\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 99.1 |
|
−1.75274 | − | 0.963276i | 4.50548 | 2.14420 | + | 3.37675i | − | 7.01018i | −7.89694 | − | 4.34002i | 0 | −0.505481 | − | 7.98401i | 11.2994 | −6.75274 | + | 12.2870i | |||||||||||||||||||||||||
| 99.2 | −1.75274 | + | 0.963276i | 4.50548 | 2.14420 | − | 3.37675i | 7.01018i | −7.89694 | + | 4.34002i | 0 | −0.505481 | + | 7.98401i | 11.2994 | −6.75274 | − | 12.2870i | |||||||||||||||||||||||||||
| 99.3 | −0.217214 | − | 1.98817i | 1.43443 | −3.90564 | + | 0.863716i | − | 2.62413i | −0.311578 | − | 2.85189i | 0 | 2.56557 | + | 7.57746i | −6.94242 | −5.21721 | + | 0.569997i | ||||||||||||||||||||||||||
| 99.4 | −0.217214 | + | 1.98817i | 1.43443 | −3.90564 | − | 0.863716i | 2.62413i | −0.311578 | + | 2.85189i | 0 | 2.56557 | − | 7.57746i | −6.94242 | −5.21721 | − | 0.569997i | |||||||||||||||||||||||||||
| 99.5 | 1.96995 | − | 0.345370i | −2.93991 | 3.76144 | − | 1.36073i | − | 8.77333i | −5.79148 | + | 1.01536i | 0 | 6.93991 | − | 3.97966i | −0.356939 | −3.03005 | − | 17.2831i | ||||||||||||||||||||||||||
| 99.6 | 1.96995 | + | 0.345370i | −2.93991 | 3.76144 | + | 1.36073i | 8.77333i | −5.79148 | − | 1.01536i | 0 | 6.93991 | + | 3.97966i | −0.356939 | −3.03005 | + | 17.2831i | |||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 8.d | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 392.3.g.l | 6 | |
| 4.b | odd | 2 | 1 | 1568.3.g.i | 6 | ||
| 7.b | odd | 2 | 1 | 392.3.g.k | 6 | ||
| 7.c | even | 3 | 2 | 392.3.k.k | 12 | ||
| 7.d | odd | 6 | 2 | 56.3.k.c | ✓ | 12 | |
| 8.b | even | 2 | 1 | 1568.3.g.i | 6 | ||
| 8.d | odd | 2 | 1 | inner | 392.3.g.l | 6 | |
| 28.d | even | 2 | 1 | 1568.3.g.k | 6 | ||
| 28.f | even | 6 | 2 | 224.3.o.c | 12 | ||
| 56.e | even | 2 | 1 | 392.3.g.k | 6 | ||
| 56.h | odd | 2 | 1 | 1568.3.g.k | 6 | ||
| 56.j | odd | 6 | 2 | 224.3.o.c | 12 | ||
| 56.k | odd | 6 | 2 | 392.3.k.k | 12 | ||
| 56.m | even | 6 | 2 | 56.3.k.c | ✓ | 12 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 56.3.k.c | ✓ | 12 | 7.d | odd | 6 | 2 | |
| 56.3.k.c | ✓ | 12 | 56.m | even | 6 | 2 | |
| 224.3.o.c | 12 | 28.f | even | 6 | 2 | ||
| 224.3.o.c | 12 | 56.j | odd | 6 | 2 | ||
| 392.3.g.k | 6 | 7.b | odd | 2 | 1 | ||
| 392.3.g.k | 6 | 56.e | even | 2 | 1 | ||
| 392.3.g.l | 6 | 1.a | even | 1 | 1 | trivial | |
| 392.3.g.l | 6 | 8.d | odd | 2 | 1 | inner | |
| 392.3.k.k | 12 | 7.c | even | 3 | 2 | ||
| 392.3.k.k | 12 | 56.k | odd | 6 | 2 | ||
| 1568.3.g.i | 6 | 4.b | odd | 2 | 1 | ||
| 1568.3.g.i | 6 | 8.b | even | 2 | 1 | ||
| 1568.3.g.k | 6 | 28.d | even | 2 | 1 | ||
| 1568.3.g.k | 6 | 56.h | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{3} - 3T_{3}^{2} - 11T_{3} + 19 \)
acting on \(S_{3}^{\mathrm{new}}(392, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} - 2 T^{4} + \cdots + 64 \)
|
| $3$ |
\( (T^{3} - 3 T^{2} - 11 T + 19)^{2} \)
|
| $5$ |
\( T^{6} + 133 T^{4} + \cdots + 26047 \)
|
| $7$ |
\( T^{6} \)
|
| $11$ |
\( (T^{3} - 7 T^{2} + \cdots - 287)^{2} \)
|
| $13$ |
\( T^{6} + 64 T^{4} + \cdots + 5488 \)
|
| $17$ |
\( (T^{3} + 41 T^{2} + \cdots - 1621)^{2} \)
|
| $19$ |
\( (T^{3} + 47 T^{2} + \cdots + 2267)^{2} \)
|
| $23$ |
\( T^{6} + 793 T^{4} + \cdots + 16795807 \)
|
| $29$ |
\( T^{6} + 3208 T^{4} + \cdots + 747251568 \)
|
| $31$ |
\( T^{6} + 2437 T^{4} + \cdots + 525090447 \)
|
| $37$ |
\( T^{6} + 1813 T^{4} + \cdots + 71366743 \)
|
| $41$ |
\( (T^{3} + 30 T^{2} + \cdots - 364)^{2} \)
|
| $43$ |
\( (T^{3} - 10 T^{2} + \cdots - 16408)^{2} \)
|
| $47$ |
\( T^{6} + 4941 T^{4} + \cdots + 2804823 \)
|
| $53$ |
\( T^{6} + \cdots + 13609348543 \)
|
| $59$ |
\( (T^{3} - 31 T^{2} + \cdots - 4377)^{2} \)
|
| $61$ |
\( T^{6} + \cdots + 176973678063 \)
|
| $67$ |
\( (T^{3} - 89 T^{2} + \cdots + 319881)^{2} \)
|
| $71$ |
\( T^{6} + \cdots + 203138627328 \)
|
| $73$ |
\( (T^{3} - 27 T^{2} + \cdots - 4009)^{2} \)
|
| $79$ |
\( T^{6} + \cdots + 27135024847 \)
|
| $83$ |
\( (T^{3} - 98 T^{2} + \cdots + 823528)^{2} \)
|
| $89$ |
\( (T^{3} + 13 T^{2} + \cdots - 9297)^{2} \)
|
| $97$ |
\( (T^{3} - 46 T^{2} + \cdots + 407276)^{2} \)
|
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