Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [342,4,Mod(163,342)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(342, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 4]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("342.163");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 342 = 2 \cdot 3^{2} \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 342.g (of order \(3\), degree \(2\), 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: | \(20.1786532220\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Relative dimension: | \(3\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | 6.0.6967728.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - x^{5} + 8x^{4} + 5x^{3} + 50x^{2} - 7x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{19}]\) |
| Coefficient ring index: | \( 2^{6}\cdot 3^{2} \) |
| Twist minimal: | no (minimal twist has level 114) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-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
\( x^{6} - x^{5} + 8x^{4} + 5x^{3} + 50x^{2} - 7x + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 10\nu^{5} - 80\nu^{4} + 116\nu^{3} - 500\nu^{2} + 70\nu - 2525 ) / 131 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -52\nu^{5} + 23\nu^{4} - 184\nu^{3} - 544\nu^{2} - 1936\nu + 161 ) / 393 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -56\nu^{5} + 55\nu^{4} - 440\nu^{3} - 344\nu^{2} - 2750\nu - 8 ) / 393 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -58\nu^{5} + 71\nu^{4} - 568\nu^{3} - 244\nu^{2} - 1978\nu - 289 ) / 393 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -1036\nu^{5} + 821\nu^{4} - 8140\nu^{3} - 6364\nu^{2} - 52840\nu - 148 ) / 393 \)
|
| \(\nu\) | \(=\) |
\( ( 2\beta_{4} - 3\beta_{3} + \beta_{2} + 1 ) / 6 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 3\beta_{5} + \beta_{4} - 60\beta_{3} + 2\beta_{2} - 3\beta _1 - 58 ) / 12 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -5\beta_{4} + 5\beta_{2} - \beta _1 - 25 ) / 4 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -6\beta_{5} - 10\beta_{4} + 126\beta_{3} - 5\beta_{2} - 5 ) / 3 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -21\beta_{5} - 62\beta_{4} + 537\beta_{3} - 124\beta_{2} + 21\beta _1 + 413 ) / 6 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/342\mathbb{Z}\right)^\times\).
| \(n\) | \(191\) | \(325\) |
| \(\chi(n)\) | \(1\) | \(-1 - \beta_{3}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 163.1 |
|
−1.00000 | + | 1.73205i | 0 | −2.00000 | − | 3.46410i | −6.49420 | + | 11.2483i | 0 | −25.6033 | 8.00000 | 0 | −12.9884 | − | 22.4966i | ||||||||||||||||||||||||||||
| 163.2 | −1.00000 | + | 1.73205i | 0 | −2.00000 | − | 3.46410i | −2.60679 | + | 4.51510i | 0 | 31.4905 | 8.00000 | 0 | −5.21359 | − | 9.03020i | |||||||||||||||||||||||||||||
| 163.3 | −1.00000 | + | 1.73205i | 0 | −2.00000 | − | 3.46410i | 10.1010 | − | 17.4954i | 0 | −22.8872 | 8.00000 | 0 | 20.2020 | + | 34.9909i | |||||||||||||||||||||||||||||
| 235.1 | −1.00000 | − | 1.73205i | 0 | −2.00000 | + | 3.46410i | −6.49420 | − | 11.2483i | 0 | −25.6033 | 8.00000 | 0 | −12.9884 | + | 22.4966i | |||||||||||||||||||||||||||||
| 235.2 | −1.00000 | − | 1.73205i | 0 | −2.00000 | + | 3.46410i | −2.60679 | − | 4.51510i | 0 | 31.4905 | 8.00000 | 0 | −5.21359 | + | 9.03020i | |||||||||||||||||||||||||||||
| 235.3 | −1.00000 | − | 1.73205i | 0 | −2.00000 | + | 3.46410i | 10.1010 | + | 17.4954i | 0 | −22.8872 | 8.00000 | 0 | 20.2020 | − | 34.9909i | |||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 19.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 342.4.g.g | 6 | |
| 3.b | odd | 2 | 1 | 114.4.e.e | ✓ | 6 | |
| 19.c | even | 3 | 1 | inner | 342.4.g.g | 6 | |
| 57.f | even | 6 | 1 | 2166.4.a.w | 3 | ||
| 57.h | odd | 6 | 1 | 114.4.e.e | ✓ | 6 | |
| 57.h | odd | 6 | 1 | 2166.4.a.s | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 114.4.e.e | ✓ | 6 | 3.b | odd | 2 | 1 | |
| 114.4.e.e | ✓ | 6 | 57.h | odd | 6 | 1 | |
| 342.4.g.g | 6 | 1.a | even | 1 | 1 | trivial | |
| 342.4.g.g | 6 | 19.c | even | 3 | 1 | inner | |
| 2166.4.a.s | 3 | 57.h | odd | 6 | 1 | ||
| 2166.4.a.w | 3 | 57.f | even | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{6} - 2T_{5}^{5} + 304T_{5}^{4} + 3336T_{5}^{3} + 87264T_{5}^{2} + 410400T_{5} + 1871424 \)
acting on \(S_{4}^{\mathrm{new}}(342, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 2 T + 4)^{3} \)
|
| $3$ |
\( T^{6} \)
|
| $5$ |
\( T^{6} - 2 T^{5} + \cdots + 1871424 \)
|
| $7$ |
\( (T^{3} + 17 T^{2} + \cdots - 18453)^{2} \)
|
| $11$ |
\( (T^{3} - 52 T^{2} + \cdots + 32688)^{2} \)
|
| $13$ |
\( T^{6} + 75 T^{5} + \cdots + 174424849 \)
|
| $17$ |
\( T^{6} + 48 T^{5} + \cdots + 107495424 \)
|
| $19$ |
\( T^{6} + \cdots + 322687697779 \)
|
| $23$ |
\( T^{6} + \cdots + 37266922552896 \)
|
| $29$ |
\( T^{6} + \cdots + 93900570624 \)
|
| $31$ |
\( (T^{3} - 107 T^{2} + \cdots + 1435247)^{2} \)
|
| $37$ |
\( (T^{3} - 305 T^{2} + \cdots + 28900349)^{2} \)
|
| $41$ |
\( T^{6} + \cdots + 49106682003456 \)
|
| $43$ |
\( T^{6} + \cdots + 9744392803201 \)
|
| $47$ |
\( T^{6} + \cdots + 407898773822016 \)
|
| $53$ |
\( T^{6} + \cdots + 16\!\cdots\!76 \)
|
| $59$ |
\( T^{6} + \cdots + 10\!\cdots\!44 \)
|
| $61$ |
\( T^{6} + \cdots + 61\!\cdots\!69 \)
|
| $67$ |
\( T^{6} + \cdots + 762088088919249 \)
|
| $71$ |
\( T^{6} + \cdots + 288657109767744 \)
|
| $73$ |
\( T^{6} + \cdots + 27\!\cdots\!21 \)
|
| $79$ |
\( T^{6} + \cdots + 40\!\cdots\!29 \)
|
| $83$ |
\( (T^{3} - 2148 T^{2} + \cdots - 211415616)^{2} \)
|
| $89$ |
\( T^{6} + \cdots + 121838150433024 \)
|
| $97$ |
\( T^{6} + \cdots + 62\!\cdots\!44 \)
|
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