Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [168,4,Mod(25,168)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(168, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 0, 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("168.25");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 168 = 2^{3} \cdot 3 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 168.q (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: | \(9.91232088096\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Relative dimension: | \(3\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | 6.0.10253065563.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 3x^{5} + 58x^{4} - 111x^{3} + 802x^{2} - 747x + 189 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{4}\cdot 3 \) |
| Twist minimal: | yes |
| 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
\( x^{6} - 3x^{5} + 58x^{4} - 111x^{3} + 802x^{2} - 747x + 189 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -\nu^{5} - 3\nu^{4} - 43\nu^{3} - 81\nu^{2} - 331\nu + 213 ) / 66 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{4} - 2\nu^{3} + 29\nu^{2} - 28\nu + 3 ) / 6 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -2\nu^{5} + 5\nu^{4} - 108\nu^{3} + 157\nu^{2} - 1366\nu + 723 ) / 132 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{4} + 2\nu^{3} - 41\nu^{2} + 40\nu - 219 ) / 12 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 29\nu^{5} - 67\nu^{4} + 1687\nu^{3} - 2249\nu^{2} + 23415\nu - 10203 ) / 132 \)
|
| \(\nu\) | \(=\) |
\( ( -2\beta_{3} + \beta_{2} + 2\beta _1 + 4 ) / 6 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -3\beta_{4} - \beta_{3} - \beta_{2} + \beta _1 - 52 ) / 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 6\beta_{5} - 6\beta_{4} + 142\beta_{3} - 32\beta_{2} - 55\beta _1 - 230 ) / 6 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 6\beta_{5} + 81\beta_{4} + 143\beta_{3} + 29\beta_{2} - 56\beta _1 + 1325 ) / 3 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -294\beta_{5} + 258\beta_{4} - 6140\beta_{3} + 1033\beta_{2} + 1481\beta _1 + 10318 ) / 6 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/168\mathbb{Z}\right)^\times\).
| \(n\) | \(73\) | \(85\) | \(113\) | \(127\) |
| \(\chi(n)\) | \(-\beta_{3}\) | \(1\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 25.1 |
|
0 | −1.50000 | + | 2.59808i | 0 | −5.07832 | − | 8.79590i | 0 | 4.83472 | + | 17.8781i | 0 | −4.50000 | − | 7.79423i | 0 | ||||||||||||||||||||||||||||
| 25.2 | 0 | −1.50000 | + | 2.59808i | 0 | −0.119644 | − | 0.207230i | 0 | 12.9074 | − | 13.2815i | 0 | −4.50000 | − | 7.79423i | 0 | |||||||||||||||||||||||||||||
| 25.3 | 0 | −1.50000 | + | 2.59808i | 0 | 10.6980 | + | 18.5294i | 0 | −18.2421 | + | 3.19770i | 0 | −4.50000 | − | 7.79423i | 0 | |||||||||||||||||||||||||||||
| 121.1 | 0 | −1.50000 | − | 2.59808i | 0 | −5.07832 | + | 8.79590i | 0 | 4.83472 | − | 17.8781i | 0 | −4.50000 | + | 7.79423i | 0 | |||||||||||||||||||||||||||||
| 121.2 | 0 | −1.50000 | − | 2.59808i | 0 | −0.119644 | + | 0.207230i | 0 | 12.9074 | + | 13.2815i | 0 | −4.50000 | + | 7.79423i | 0 | |||||||||||||||||||||||||||||
| 121.3 | 0 | −1.50000 | − | 2.59808i | 0 | 10.6980 | − | 18.5294i | 0 | −18.2421 | − | 3.19770i | 0 | −4.50000 | + | 7.79423i | 0 | |||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 168.4.q.e | ✓ | 6 |
| 3.b | odd | 2 | 1 | 504.4.s.g | 6 | ||
| 4.b | odd | 2 | 1 | 336.4.q.l | 6 | ||
| 7.c | even | 3 | 1 | inner | 168.4.q.e | ✓ | 6 |
| 7.c | even | 3 | 1 | 1176.4.a.y | 3 | ||
| 7.d | odd | 6 | 1 | 1176.4.a.x | 3 | ||
| 21.h | odd | 6 | 1 | 504.4.s.g | 6 | ||
| 28.f | even | 6 | 1 | 2352.4.a.cj | 3 | ||
| 28.g | odd | 6 | 1 | 336.4.q.l | 6 | ||
| 28.g | odd | 6 | 1 | 2352.4.a.ch | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 168.4.q.e | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 168.4.q.e | ✓ | 6 | 7.c | even | 3 | 1 | inner |
| 336.4.q.l | 6 | 4.b | odd | 2 | 1 | ||
| 336.4.q.l | 6 | 28.g | odd | 6 | 1 | ||
| 504.4.s.g | 6 | 3.b | odd | 2 | 1 | ||
| 504.4.s.g | 6 | 21.h | odd | 6 | 1 | ||
| 1176.4.a.x | 3 | 7.d | odd | 6 | 1 | ||
| 1176.4.a.y | 3 | 7.c | even | 3 | 1 | ||
| 2352.4.a.ch | 3 | 28.g | odd | 6 | 1 | ||
| 2352.4.a.cj | 3 | 28.f | even | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{6} - 11T_{5}^{5} + 341T_{5}^{4} + 2524T_{5}^{3} + 47828T_{5}^{2} + 11440T_{5} + 2704 \)
acting on \(S_{4}^{\mathrm{new}}(168, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
|
| $3$ |
\( (T^{2} + 3 T + 9)^{3} \)
|
| $5$ |
\( T^{6} - 11 T^{5} + \cdots + 2704 \)
|
| $7$ |
\( T^{6} + T^{5} + \cdots + 40353607 \)
|
| $11$ |
\( T^{6} - 19 T^{5} + \cdots + 10732176 \)
|
| $13$ |
\( (T^{3} + 22 T^{2} + \cdots + 26976)^{2} \)
|
| $17$ |
\( T^{6} - 104 T^{5} + \cdots + 16257024 \)
|
| $19$ |
\( T^{6} + \cdots + 4954933537024 \)
|
| $23$ |
\( T^{6} + \cdots + 2349279645696 \)
|
| $29$ |
\( (T^{3} + 73 T^{2} + \cdots + 970992)^{2} \)
|
| $31$ |
\( T^{6} + \cdots + 17277381752449 \)
|
| $37$ |
\( T^{6} - 326 T^{5} + \cdots + 316697616 \)
|
| $41$ |
\( (T^{3} + 516 T^{2} + \cdots - 15002144)^{2} \)
|
| $43$ |
\( (T^{3} - 36 T^{2} + \cdots + 7204222)^{2} \)
|
| $47$ |
\( T^{6} + \cdots + 136472675351104 \)
|
| $53$ |
\( T^{6} + \cdots + 31729584384 \)
|
| $59$ |
\( T^{6} + \cdots + 3215681005824 \)
|
| $61$ |
\( T^{6} + \cdots + 494624360983104 \)
|
| $67$ |
\( T^{6} + \cdots + 742397156205156 \)
|
| $71$ |
\( (T^{3} - 34 T^{2} + \cdots + 207049704)^{2} \)
|
| $73$ |
\( T^{6} + \cdots + 16\!\cdots\!76 \)
|
| $79$ |
\( T^{6} + \cdots + 24\!\cdots\!21 \)
|
| $83$ |
\( (T^{3} + 115 T^{2} + \cdots + 111709668)^{2} \)
|
| $89$ |
\( T^{6} + \cdots + 12\!\cdots\!96 \)
|
| $97$ |
\( (T^{3} + 2941 T^{2} + \cdots + 866695284)^{2} \)
|
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