Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [42,12,Mod(25,42)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(42, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 4]))
N = Newforms(chi, 12, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("42.25");
S:= CuspForms(chi, 12);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 42 = 2 \cdot 3 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 12 \) |
| Character orbit: | \([\chi]\) | \(=\) | 42.e (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: | \(32.2704135835\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Relative dimension: | \(3\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{6} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - x^{5} + 1516x^{4} + 1461x^{3} + 2295252x^{2} - 40905x + 729 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{8}\cdot 3\cdot 7^{2} \) |
| 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
\( x^{6} - x^{5} + 1516x^{4} + 1461x^{3} + 2295252x^{2} - 40905x + 729 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -71\nu^{5} + 107636\nu^{4} + 2517211\nu^{3} + 162962892\nu^{2} - 2904255\nu + 168770690484 ) / 331386774 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 765580 \nu^{5} + 765571 \nu^{4} - 1160605636 \nu^{3} - 1139196684 \nu^{2} - 1757178368892 \nu - 368388 ) / 31316050143 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 780125804 \nu^{5} + 779789393 \nu^{4} - 1182160719788 \nu^{3} - 1130021145285 \nu^{2} + \cdots + 31897285120665 ) / 62632100286 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 1280769578 \nu^{5} + 1211425091 \nu^{4} - 1940907271766 \nu^{3} - 1958717957451 \nu^{2} + \cdots - 52349083817967 ) / 62632100286 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 1280838689 \nu^{5} + 1316197367 \nu^{4} - 1943161791467 \nu^{3} - 1800090796479 \nu^{2} + \cdots + 104821582929939 ) / 62632100286 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{5} - 2\beta_{4} + 5\beta_{3} - 38\beta_{2} - 5\beta _1 + 2 ) / 112 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 6\beta_{5} + 3\beta_{4} + 13\beta_{3} - 14152\beta_{2} - 14155 ) / 14 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -213\beta_{5} + 213\beta_{4} + 1097\beta _1 - 24179 ) / 16 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -2181\beta_{5} - 4362\beta_{4} - 10319\beta_{3} + 10730561\beta_{2} + 10319\beta _1 + 4362 ) / 7 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 4449234\beta_{5} + 2224617\beta_{4} - 11795981\beta_{3} + 422791574\beta_{2} + 420566957 ) / 112 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/42\mathbb{Z}\right)^\times\).
| \(n\) | \(29\) | \(31\) |
| \(\chi(n)\) | \(1\) | \(-1 - \beta_{2}\) |
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 |
|
−16.0000 | − | 27.7128i | 121.500 | − | 210.444i | −512.000 | + | 886.810i | −1098.21 | − | 1902.16i | −7776.00 | 39703.5 | − | 20023.9i | 32768.0 | −29524.5 | − | 51137.9i | −35142.9 | + | 60869.3i | ||||||||||||||||||||||
| 25.2 | −16.0000 | − | 27.7128i | 121.500 | − | 210.444i | −512.000 | + | 886.810i | −405.223 | − | 701.867i | −7776.00 | 24393.5 | + | 37179.1i | 32768.0 | −29524.5 | − | 51137.9i | −12967.1 | + | 22459.7i | |||||||||||||||||||||||
| 25.3 | −16.0000 | − | 27.7128i | 121.500 | − | 210.444i | −512.000 | + | 886.810i | 2025.94 | + | 3509.03i | −7776.00 | −41231.5 | − | 16652.0i | 32768.0 | −29524.5 | − | 51137.9i | 64830.0 | − | 112289.i | |||||||||||||||||||||||
| 37.1 | −16.0000 | + | 27.7128i | 121.500 | + | 210.444i | −512.000 | − | 886.810i | −1098.21 | + | 1902.16i | −7776.00 | 39703.5 | + | 20023.9i | 32768.0 | −29524.5 | + | 51137.9i | −35142.9 | − | 60869.3i | |||||||||||||||||||||||
| 37.2 | −16.0000 | + | 27.7128i | 121.500 | + | 210.444i | −512.000 | − | 886.810i | −405.223 | + | 701.867i | −7776.00 | 24393.5 | − | 37179.1i | 32768.0 | −29524.5 | + | 51137.9i | −12967.1 | − | 22459.7i | |||||||||||||||||||||||
| 37.3 | −16.0000 | + | 27.7128i | 121.500 | + | 210.444i | −512.000 | − | 886.810i | 2025.94 | − | 3509.03i | −7776.00 | −41231.5 | + | 16652.0i | 32768.0 | −29524.5 | + | 51137.9i | 64830.0 | + | 112289.i | |||||||||||||||||||||||
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 | 42.12.e.a | ✓ | 6 |
| 3.b | odd | 2 | 1 | 126.12.g.b | 6 | ||
| 7.c | even | 3 | 1 | inner | 42.12.e.a | ✓ | 6 |
| 21.h | odd | 6 | 1 | 126.12.g.b | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 42.12.e.a | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 42.12.e.a | ✓ | 6 | 7.c | even | 3 | 1 | inner |
| 126.12.g.b | 6 | 3.b | odd | 2 | 1 | ||
| 126.12.g.b | 6 | 21.h | odd | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{6} - 1045 T_{5}^{5} + 11495425 T_{5}^{4} + 25296942000 T_{5}^{3} + 100693465807500 T_{5}^{2} + \cdots + 52\!\cdots\!00 \)
acting on \(S_{12}^{\mathrm{new}}(42, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 32 T + 1024)^{3} \)
|
| $3$ |
\( (T^{2} - 243 T + 59049)^{3} \)
|
| $5$ |
\( T^{6} + \cdots + 52\!\cdots\!00 \)
|
| $7$ |
\( T^{6} + \cdots + 77\!\cdots\!07 \)
|
| $11$ |
\( T^{6} + \cdots + 45\!\cdots\!00 \)
|
| $13$ |
\( (T^{3} + \cdots + 39\!\cdots\!80)^{2} \)
|
| $17$ |
\( T^{6} + \cdots + 30\!\cdots\!16 \)
|
| $19$ |
\( T^{6} + \cdots + 25\!\cdots\!24 \)
|
| $23$ |
\( T^{6} + \cdots + 74\!\cdots\!56 \)
|
| $29$ |
\( (T^{3} + \cdots + 50\!\cdots\!00)^{2} \)
|
| $31$ |
\( T^{6} + \cdots + 13\!\cdots\!81 \)
|
| $37$ |
\( T^{6} + \cdots + 80\!\cdots\!00 \)
|
| $41$ |
\( (T^{3} + \cdots + 38\!\cdots\!60)^{2} \)
|
| $43$ |
\( (T^{3} + \cdots - 41\!\cdots\!70)^{2} \)
|
| $47$ |
\( T^{6} + \cdots + 60\!\cdots\!96 \)
|
| $53$ |
\( T^{6} + \cdots + 18\!\cdots\!44 \)
|
| $59$ |
\( T^{6} + \cdots + 15\!\cdots\!36 \)
|
| $61$ |
\( T^{6} + \cdots + 25\!\cdots\!24 \)
|
| $67$ |
\( T^{6} + \cdots + 79\!\cdots\!44 \)
|
| $71$ |
\( (T^{3} + \cdots - 30\!\cdots\!52)^{2} \)
|
| $73$ |
\( T^{6} + \cdots + 79\!\cdots\!00 \)
|
| $79$ |
\( T^{6} + \cdots + 69\!\cdots\!41 \)
|
| $83$ |
\( (T^{3} + \cdots - 51\!\cdots\!80)^{2} \)
|
| $89$ |
\( T^{6} + \cdots + 75\!\cdots\!00 \)
|
| $97$ |
\( (T^{3} + \cdots - 35\!\cdots\!04)^{2} \)
|
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