Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [78,4,Mod(43,78)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(78, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 1]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("78.43");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 78 = 2 \cdot 3 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 78.i (of order \(6\), 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: | \(4.60214898045\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Relative dimension: | \(4\) over \(\Q(\zeta_{6})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{8} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - 2x^{7} + 2x^{6} + 98x^{5} + 34501x^{4} - 71304x^{3} + 78408x^{2} + 5248584x + 175668516 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-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
\( x^{8} - 2x^{7} + 2x^{6} + 98x^{5} + 34501x^{4} - 71304x^{3} + 78408x^{2} + 5248584x + 175668516 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 7895 \nu^{7} + 1283102 \nu^{6} + 1354444 \nu^{5} + 84502 \nu^{4} + 45079295 \nu^{3} + \cdots + 805226438364 ) / 1586230820616 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 320118769 \nu^{7} - 17460049518 \nu^{6} + 20855063827 \nu^{5} + 10183863441 \nu^{4} + \cdots + 852980598959076 ) / 56\!\cdots\!22 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 320118769 \nu^{7} - 17460049518 \nu^{6} + 20855063827 \nu^{5} + 10183863441 \nu^{4} + \cdots + 852980598959076 ) / 28\!\cdots\!11 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 1494593066071 \nu^{7} + 17753583297236 \nu^{6} - 203136310662636 \nu^{5} + \cdots - 18\!\cdots\!24 ) / 61\!\cdots\!44 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 18528231007489 \nu^{7} - 140843653813386 \nu^{6} + \cdots - 20\!\cdots\!32 ) / 12\!\cdots\!88 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 19248176363307 \nu^{7} - 223879080092014 \nu^{6} + \cdots - 80\!\cdots\!12 ) / 12\!\cdots\!88 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 102708275179027 \nu^{7} + 122065643434058 \nu^{6} - 256841529742980 \nu^{5} + \cdots - 29\!\cdots\!84 ) / 12\!\cdots\!88 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{3} + 2\beta_{2} ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{7} - \beta_{5} - \beta_{4} - 62\beta_{3} - 50\beta _1 + 25 \)
|
| \(\nu^{3}\) | \(=\) |
\( -48\beta_{7} + 125\beta_{6} - 125\beta_{5} + 73\beta_{4} - 62\beta_{3} + 24\beta_{2} + 98\beta _1 - 73 \)
|
| \(\nu^{4}\) | \(=\) |
\( 199\beta_{7} + 150\beta_{6} + 49\beta_{5} - 6077\beta_{4} - 2939\beta_{3} - 150\beta_{2} - 17175 \)
|
| \(\nu^{5}\) | \(=\) |
\( -5927\beta_{6} - 5927\beta_{5} + 5049\beta_{4} + 10661\beta_{3} - 17524\beta_{2} - 21952\beta _1 + 12227 \)
|
| \(\nu^{6}\) | \(=\) |
\( - 33549 \beta_{7} + 18154 \beta_{6} + 15395 \beta_{5} + 33549 \beta_{4} + 1298609 \beta_{3} + \cdots - 1167121 \)
|
| \(\nu^{7}\) | \(=\) |
\( 2303452 \beta_{7} - 2648921 \beta_{6} + 2648921 \beta_{5} - 3249477 \beta_{4} + 137565 \beta_{3} + \cdots + 669985 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/78\mathbb{Z}\right)^\times\).
| \(n\) | \(53\) | \(67\) |
| \(\chi(n)\) | \(1\) | \(1 - \beta_{1}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 43.1 |
|
−1.73205 | − | 1.00000i | 1.50000 | − | 2.59808i | 2.00000 | + | 3.46410i | − | 0.346567i | −5.19615 | + | 3.00000i | 19.8163 | − | 11.4409i | − | 8.00000i | −4.50000 | − | 7.79423i | −0.346567 | + | 0.600271i | ||||||||||||||||||||||||||
| 43.2 | −1.73205 | − | 1.00000i | 1.50000 | − | 2.59808i | 2.00000 | + | 3.46410i | 13.0068i | −5.19615 | + | 3.00000i | −23.3426 | + | 13.4768i | − | 8.00000i | −4.50000 | − | 7.79423i | 13.0068 | − | 22.5285i | ||||||||||||||||||||||||||||
| 43.3 | 1.73205 | + | 1.00000i | 1.50000 | − | 2.59808i | 2.00000 | + | 3.46410i | − | 15.0170i | 5.19615 | − | 3.00000i | 11.7885 | − | 6.80612i | 8.00000i | −4.50000 | − | 7.79423i | 15.0170 | − | 26.0102i | ||||||||||||||||||||||||||||
| 43.4 | 1.73205 | + | 1.00000i | 1.50000 | − | 2.59808i | 2.00000 | + | 3.46410i | 19.6772i | 5.19615 | − | 3.00000i | 3.73773 | − | 2.15798i | 8.00000i | −4.50000 | − | 7.79423i | −19.6772 | + | 34.0820i | |||||||||||||||||||||||||||||
| 49.1 | −1.73205 | + | 1.00000i | 1.50000 | + | 2.59808i | 2.00000 | − | 3.46410i | − | 13.0068i | −5.19615 | − | 3.00000i | −23.3426 | − | 13.4768i | 8.00000i | −4.50000 | + | 7.79423i | 13.0068 | + | 22.5285i | ||||||||||||||||||||||||||||
| 49.2 | −1.73205 | + | 1.00000i | 1.50000 | + | 2.59808i | 2.00000 | − | 3.46410i | 0.346567i | −5.19615 | − | 3.00000i | 19.8163 | + | 11.4409i | 8.00000i | −4.50000 | + | 7.79423i | −0.346567 | − | 0.600271i | |||||||||||||||||||||||||||||
| 49.3 | 1.73205 | − | 1.00000i | 1.50000 | + | 2.59808i | 2.00000 | − | 3.46410i | − | 19.6772i | 5.19615 | + | 3.00000i | 3.73773 | + | 2.15798i | − | 8.00000i | −4.50000 | + | 7.79423i | −19.6772 | − | 34.0820i | |||||||||||||||||||||||||||
| 49.4 | 1.73205 | − | 1.00000i | 1.50000 | + | 2.59808i | 2.00000 | − | 3.46410i | 15.0170i | 5.19615 | + | 3.00000i | 11.7885 | + | 6.80612i | − | 8.00000i | −4.50000 | + | 7.79423i | 15.0170 | + | 26.0102i | ||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 13.e | even | 6 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 78.4.i.b | ✓ | 8 |
| 3.b | odd | 2 | 1 | 234.4.l.c | 8 | ||
| 4.b | odd | 2 | 1 | 624.4.bv.g | 8 | ||
| 13.c | even | 3 | 1 | 1014.4.b.o | 8 | ||
| 13.e | even | 6 | 1 | inner | 78.4.i.b | ✓ | 8 |
| 13.e | even | 6 | 1 | 1014.4.b.o | 8 | ||
| 13.f | odd | 12 | 1 | 1014.4.a.z | 4 | ||
| 13.f | odd | 12 | 1 | 1014.4.a.ba | 4 | ||
| 39.h | odd | 6 | 1 | 234.4.l.c | 8 | ||
| 52.i | odd | 6 | 1 | 624.4.bv.g | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 78.4.i.b | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 78.4.i.b | ✓ | 8 | 13.e | even | 6 | 1 | inner |
| 234.4.l.c | 8 | 3.b | odd | 2 | 1 | ||
| 234.4.l.c | 8 | 39.h | odd | 6 | 1 | ||
| 624.4.bv.g | 8 | 4.b | odd | 2 | 1 | ||
| 624.4.bv.g | 8 | 52.i | odd | 6 | 1 | ||
| 1014.4.a.z | 4 | 13.f | odd | 12 | 1 | ||
| 1014.4.a.ba | 4 | 13.f | odd | 12 | 1 | ||
| 1014.4.b.o | 8 | 13.c | even | 3 | 1 | ||
| 1014.4.b.o | 8 | 13.e | even | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{8} + 782T_{5}^{6} + 191065T_{5}^{4} + 14794776T_{5}^{2} + 1774224 \)
acting on \(S_{4}^{\mathrm{new}}(78, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{4} - 4 T^{2} + 16)^{2} \)
|
| $3$ |
\( (T^{2} - 3 T + 9)^{4} \)
|
| $5$ |
\( T^{8} + 782 T^{6} + \cdots + 1774224 \)
|
| $7$ |
\( T^{8} + \cdots + 1312902756 \)
|
| $11$ |
\( T^{8} + \cdots + 1461685824 \)
|
| $13$ |
\( T^{8} + \cdots + 23298085122481 \)
|
| $17$ |
\( T^{8} + \cdots + 218270130465024 \)
|
| $19$ |
\( T^{8} + \cdots + 50391484900416 \)
|
| $23$ |
\( T^{8} + \cdots + 49\!\cdots\!16 \)
|
| $29$ |
\( T^{8} + \cdots + 43\!\cdots\!64 \)
|
| $31$ |
\( T^{8} + \cdots + 14\!\cdots\!76 \)
|
| $37$ |
\( T^{8} + \cdots + 22\!\cdots\!56 \)
|
| $41$ |
\( T^{8} + \cdots + 10\!\cdots\!64 \)
|
| $43$ |
\( T^{8} + \cdots + 95\!\cdots\!96 \)
|
| $47$ |
\( T^{8} + \cdots + 44\!\cdots\!36 \)
|
| $53$ |
\( (T^{4} - 606 T^{3} + \cdots - 6144065136)^{2} \)
|
| $59$ |
\( T^{8} + \cdots + 82\!\cdots\!96 \)
|
| $61$ |
\( T^{8} + \cdots + 33\!\cdots\!09 \)
|
| $67$ |
\( T^{8} + \cdots + 47\!\cdots\!56 \)
|
| $71$ |
\( T^{8} + \cdots + 37\!\cdots\!04 \)
|
| $73$ |
\( T^{8} + \cdots + 10\!\cdots\!41 \)
|
| $79$ |
\( (T^{4} - 962 T^{3} + \cdots + 368623918104)^{2} \)
|
| $83$ |
\( T^{8} + \cdots + 13\!\cdots\!36 \)
|
| $89$ |
\( T^{8} + \cdots + 16\!\cdots\!76 \)
|
| $97$ |
\( T^{8} + \cdots + 50\!\cdots\!84 \)
|
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