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: | \(8\) |
| Relative dimension: | \(4\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{8} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - 2 x^{7} + 27351029 x^{6} + 70092626926 x^{5} + 716243548908965 x^{4} + \cdots + 10\!\cdots\!56 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{6}\cdot 3^{3}\cdot 7^{3} \) |
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} - 2 x^{7} + 27351029 x^{6} + 70092626926 x^{5} + 716243548908965 x^{4} + \cdots + 10\!\cdots\!56 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 17\!\cdots\!91 \nu^{7} + \cdots + 12\!\cdots\!00 ) / 31\!\cdots\!24 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 55\!\cdots\!31 \nu^{7} + \cdots - 32\!\cdots\!52 ) / 85\!\cdots\!72 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 10\!\cdots\!55 \nu^{7} + \cdots - 94\!\cdots\!00 ) / 42\!\cdots\!36 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 22\!\cdots\!25 \nu^{7} + \cdots + 52\!\cdots\!68 ) / 57\!\cdots\!14 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 38\!\cdots\!09 \nu^{7} + \cdots + 89\!\cdots\!72 ) / 34\!\cdots\!88 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 72\!\cdots\!83 \nu^{7} + \cdots + 29\!\cdots\!04 ) / 34\!\cdots\!88 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 18\!\cdots\!03 \nu^{7} + \cdots - 16\!\cdots\!28 ) / 34\!\cdots\!88 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{6} + 2\beta_{5} - 2\beta_{4} + 3\beta_{3} + 3\beta_{2} + 21\beta _1 + 1 ) / 42 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( - 121 \beta_{7} + 1327 \beta_{6} - 4102 \beta_{5} + 15605 \beta_{4} + 12830 \beta_{3} + \cdots - 287187774 ) / 21 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( - 12555167 \beta_{7} - 11008901 \beta_{6} - 36119235 \beta_{5} + 135760227 \beta_{4} + \cdots - 552421997762 ) / 21 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( - 36297902707 \beta_{7} - 57138875753 \beta_{6} - 5384043385 \beta_{5} + 41681946092 \beta_{4} + \cdots + 15456929661 ) / 21 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 289501212016223 \beta_{7} - 88834718983290 \beta_{6} + 556005368966093 \beta_{5} + \cdots + 25\!\cdots\!33 ) / 21 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 27\!\cdots\!95 \beta_{7} + \cdots + 29\!\cdots\!18 ) / 3 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 36\!\cdots\!62 \beta_{7} + \cdots + 46\!\cdots\!17 ) / 21 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/42\mathbb{Z}\right)^\times\).
| \(n\) | \(29\) | \(31\) |
| \(\chi(n)\) | \(1\) | \(-\beta_{1}\) |
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 |
|
−16.0000 | − | 27.7128i | −121.500 | + | 210.444i | −512.000 | + | 886.810i | −5651.67 | − | 9788.98i | 7776.00 | 27632.8 | + | 34839.0i | 32768.0 | −29524.5 | − | 51137.9i | −180853. | + | 313247.i | ||||||||||||||||||||||||||||
| 25.2 | −16.0000 | − | 27.7128i | −121.500 | + | 210.444i | −512.000 | + | 886.810i | −4061.85 | − | 7035.33i | 7776.00 | −15634.4 | − | 41628.0i | 32768.0 | −29524.5 | − | 51137.9i | −129979. | + | 225131.i | |||||||||||||||||||||||||||||
| 25.3 | −16.0000 | − | 27.7128i | −121.500 | + | 210.444i | −512.000 | + | 886.810i | 731.917 | + | 1267.72i | 7776.00 | −39593.4 | + | 20240.8i | 32768.0 | −29524.5 | − | 51137.9i | 23421.4 | − | 40567.0i | |||||||||||||||||||||||||||||
| 25.4 | −16.0000 | − | 27.7128i | −121.500 | + | 210.444i | −512.000 | + | 886.810i | 3441.60 | + | 5961.03i | 7776.00 | 43474.1 | − | 9345.10i | 32768.0 | −29524.5 | − | 51137.9i | 110131. | − | 190753.i | |||||||||||||||||||||||||||||
| 37.1 | −16.0000 | + | 27.7128i | −121.500 | − | 210.444i | −512.000 | − | 886.810i | −5651.67 | + | 9788.98i | 7776.00 | 27632.8 | − | 34839.0i | 32768.0 | −29524.5 | + | 51137.9i | −180853. | − | 313247.i | |||||||||||||||||||||||||||||
| 37.2 | −16.0000 | + | 27.7128i | −121.500 | − | 210.444i | −512.000 | − | 886.810i | −4061.85 | + | 7035.33i | 7776.00 | −15634.4 | + | 41628.0i | 32768.0 | −29524.5 | + | 51137.9i | −129979. | − | 225131.i | |||||||||||||||||||||||||||||
| 37.3 | −16.0000 | + | 27.7128i | −121.500 | − | 210.444i | −512.000 | − | 886.810i | 731.917 | − | 1267.72i | 7776.00 | −39593.4 | − | 20240.8i | 32768.0 | −29524.5 | + | 51137.9i | 23421.4 | + | 40567.0i | |||||||||||||||||||||||||||||
| 37.4 | −16.0000 | + | 27.7128i | −121.500 | − | 210.444i | −512.000 | − | 886.810i | 3441.60 | − | 5961.03i | 7776.00 | 43474.1 | + | 9345.10i | 32768.0 | −29524.5 | + | 51137.9i | 110131. | + | 190753.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.c | ✓ | 8 |
| 3.b | odd | 2 | 1 | 126.12.g.f | 8 | ||
| 7.c | even | 3 | 1 | inner | 42.12.e.c | ✓ | 8 |
| 21.h | odd | 6 | 1 | 126.12.g.f | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 42.12.e.c | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 42.12.e.c | ✓ | 8 | 7.c | even | 3 | 1 | inner |
| 126.12.g.f | 8 | 3.b | odd | 2 | 1 | ||
| 126.12.g.f | 8 | 21.h | odd | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{8} + 11080 T_{5}^{7} + 183023925 T_{5}^{6} + 473791750360 T_{5}^{5} + \cdots + 85\!\cdots\!00 \)
acting on \(S_{12}^{\mathrm{new}}(42, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 32 T + 1024)^{4} \)
|
| $3$ |
\( (T^{2} + 243 T + 59049)^{4} \)
|
| $5$ |
\( T^{8} + \cdots + 85\!\cdots\!00 \)
|
| $7$ |
\( T^{8} + \cdots + 15\!\cdots\!01 \)
|
| $11$ |
\( T^{8} + \cdots + 25\!\cdots\!56 \)
|
| $13$ |
\( (T^{4} + \cdots + 26\!\cdots\!92)^{2} \)
|
| $17$ |
\( T^{8} + \cdots + 57\!\cdots\!84 \)
|
| $19$ |
\( T^{8} + \cdots + 85\!\cdots\!76 \)
|
| $23$ |
\( T^{8} + \cdots + 98\!\cdots\!44 \)
|
| $29$ |
\( (T^{4} + \cdots - 57\!\cdots\!32)^{2} \)
|
| $31$ |
\( T^{8} + \cdots + 25\!\cdots\!25 \)
|
| $37$ |
\( T^{8} + \cdots + 13\!\cdots\!96 \)
|
| $41$ |
\( (T^{4} + \cdots + 48\!\cdots\!12)^{2} \)
|
| $43$ |
\( (T^{4} + \cdots - 15\!\cdots\!88)^{2} \)
|
| $47$ |
\( T^{8} + \cdots + 29\!\cdots\!24 \)
|
| $53$ |
\( T^{8} + \cdots + 11\!\cdots\!84 \)
|
| $59$ |
\( T^{8} + \cdots + 14\!\cdots\!24 \)
|
| $61$ |
\( T^{8} + \cdots + 16\!\cdots\!84 \)
|
| $67$ |
\( T^{8} + \cdots + 21\!\cdots\!76 \)
|
| $71$ |
\( (T^{4} + \cdots + 26\!\cdots\!56)^{2} \)
|
| $73$ |
\( T^{8} + \cdots + 50\!\cdots\!76 \)
|
| $79$ |
\( T^{8} + \cdots + 25\!\cdots\!09 \)
|
| $83$ |
\( (T^{4} + \cdots + 28\!\cdots\!28)^{2} \)
|
| $89$ |
\( T^{8} + \cdots + 43\!\cdots\!56 \)
|
| $97$ |
\( (T^{4} + \cdots - 14\!\cdots\!48)^{2} \)
|
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