Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [78,5,Mod(31,78)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(78, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([0, 3]))
N = Newforms(chi, 5, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("78.31");
S:= CuspForms(chi, 5);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 78 = 2 \cdot 3 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 5 \) |
| Character orbit: | \([\chi]\) | \(=\) | 78.f (of order \(4\), 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: | \(8.06285712054\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Relative dimension: | \(4\) over \(\Q(i)\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{8} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - 4x^{7} + 8x^{6} + 320x^{5} + 12802x^{4} - 26252x^{3} + 53792x^{2} + 2204488x + 45171841 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{5} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-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
\( x^{8} - 4x^{7} + 8x^{6} + 320x^{5} + 12802x^{4} - 26252x^{3} + 53792x^{2} + 2204488x + 45171841 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{7} - 86 \nu^{6} + 7060 \nu^{5} - 578600 \nu^{4} + 1136363 \nu^{3} - 564740 \nu^{2} + \cdots - 3799518216 ) / 92643278 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 971735 \nu^{7} - 30121165 \nu^{6} + 86800197 \nu^{5} + 152274505 \nu^{4} + \cdots + 1075310271321 ) / 29983255778476 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 558175 \nu^{7} - 1681411 \nu^{6} + 3376185 \nu^{5} + 133377025 \nu^{4} + 11263372975 \nu^{3} + \cdots + 1571879036013 ) / 637941612308 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 83 \nu^{7} - 7138 \nu^{6} + 585980 \nu^{5} - 1702161 \nu^{4} + 1674851 \nu^{3} + \cdots - 3754346375 ) / 92643278 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 43460 \nu^{7} + 174357 \nu^{6} - 392142 \nu^{5} - 10257180 \nu^{4} - 855511120 \nu^{3} + \cdots - 71837650456 ) / 47896574726 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 15 \nu^{7} - 671 \nu^{6} + 1953 \nu^{5} + 1745 \nu^{4} - 6545 \nu^{3} - 4207031 \nu^{2} + \cdots + 12427129 ) / 8414692 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 3606146 \nu^{7} - 14382707 \nu^{6} + 25247746 \nu^{5} + 1449618340 \nu^{4} + \cdots + 6011604270386 ) / 47896574726 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{5} - \beta_{3} - \beta_{2} + 1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{6} - \beta_{5} - 83\beta_{2} \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -2\beta_{7} + 4\beta_{6} - 85\beta_{5} + 85\beta_{3} - 247\beta_{2} - 4\beta _1 - 247 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( -2\beta_{7} + 2\beta_{6} + 2\beta_{4} + 168\beta_{3} + 2\beta_{2} - 170\beta _1 - 7057 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -660\beta_{6} + 7393\beta_{5} + 340\beta_{4} + 7393\beta_{3} + 34629\beta_{2} - 1000\beta _1 - 34629 ) / 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( 500\beta_{7} - 21663\beta_{6} + 21341\beta_{5} + 500\beta_{4} + 613637\beta_{2} - 500 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 44326 \beta_{7} - 168008 \beta_{6} + 656641 \beta_{5} - 656641 \beta_{3} + 4033239 \beta_{2} + \cdots + 4033239 ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/78\mathbb{Z}\right)^\times\).
| \(n\) | \(53\) | \(67\) |
| \(\chi(n)\) | \(1\) | \(-\beta_{2}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 31.1 |
|
−2.00000 | + | 2.00000i | −5.19615 | − | 8.00000i | −28.2923 | + | 28.2923i | 10.3923 | − | 10.3923i | 25.8221 | + | 25.8221i | 16.0000 | + | 16.0000i | 27.0000 | − | 113.169i | ||||||||||||||||||||||||||||||
| 31.2 | −2.00000 | + | 2.00000i | −5.19615 | − | 8.00000i | 7.16797 | − | 7.16797i | 10.3923 | − | 10.3923i | 6.81895 | + | 6.81895i | 16.0000 | + | 16.0000i | 27.0000 | 28.6719i | ||||||||||||||||||||||||||||||||
| 31.3 | −2.00000 | + | 2.00000i | 5.19615 | − | 8.00000i | −3.08989 | + | 3.08989i | −10.3923 | + | 10.3923i | 16.4030 | + | 16.4030i | 16.0000 | + | 16.0000i | 27.0000 | − | 12.3595i | |||||||||||||||||||||||||||||||
| 31.4 | −2.00000 | + | 2.00000i | 5.19615 | − | 8.00000i | 6.21424 | − | 6.21424i | −10.3923 | + | 10.3923i | −53.0440 | − | 53.0440i | 16.0000 | + | 16.0000i | 27.0000 | 24.8570i | ||||||||||||||||||||||||||||||||
| 73.1 | −2.00000 | − | 2.00000i | −5.19615 | 8.00000i | −28.2923 | − | 28.2923i | 10.3923 | + | 10.3923i | 25.8221 | − | 25.8221i | 16.0000 | − | 16.0000i | 27.0000 | 113.169i | |||||||||||||||||||||||||||||||||
| 73.2 | −2.00000 | − | 2.00000i | −5.19615 | 8.00000i | 7.16797 | + | 7.16797i | 10.3923 | + | 10.3923i | 6.81895 | − | 6.81895i | 16.0000 | − | 16.0000i | 27.0000 | − | 28.6719i | ||||||||||||||||||||||||||||||||
| 73.3 | −2.00000 | − | 2.00000i | 5.19615 | 8.00000i | −3.08989 | − | 3.08989i | −10.3923 | − | 10.3923i | 16.4030 | − | 16.4030i | 16.0000 | − | 16.0000i | 27.0000 | 12.3595i | |||||||||||||||||||||||||||||||||
| 73.4 | −2.00000 | − | 2.00000i | 5.19615 | 8.00000i | 6.21424 | + | 6.21424i | −10.3923 | − | 10.3923i | −53.0440 | + | 53.0440i | 16.0000 | − | 16.0000i | 27.0000 | − | 24.8570i | ||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 13.d | odd | 4 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 78.5.f.a | ✓ | 8 |
| 3.b | odd | 2 | 1 | 234.5.i.f | 8 | ||
| 13.d | odd | 4 | 1 | inner | 78.5.f.a | ✓ | 8 |
| 39.f | even | 4 | 1 | 234.5.i.f | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 78.5.f.a | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 78.5.f.a | ✓ | 8 | 13.d | odd | 4 | 1 | inner |
| 234.5.i.f | 8 | 3.b | odd | 2 | 1 | ||
| 234.5.i.f | 8 | 39.f | even | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{8} + 36 T_{5}^{7} + 648 T_{5}^{6} - 21648 T_{5}^{5} + 300624 T_{5}^{4} - 1086048 T_{5}^{3} + \cdots + 242611776 \)
acting on \(S_{5}^{\mathrm{new}}(78, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 4 T + 8)^{4} \)
|
| $3$ |
\( (T^{2} - 27)^{4} \)
|
| $5$ |
\( T^{8} + 36 T^{7} + \cdots + 242611776 \)
|
| $7$ |
\( T^{8} + \cdots + 375538547344 \)
|
| $11$ |
\( T^{8} + \cdots + 14\!\cdots\!96 \)
|
| $13$ |
\( T^{8} + \cdots + 66\!\cdots\!41 \)
|
| $17$ |
\( T^{8} + \cdots + 17\!\cdots\!76 \)
|
| $19$ |
\( T^{8} + \cdots + 62\!\cdots\!36 \)
|
| $23$ |
\( T^{8} + \cdots + 61\!\cdots\!84 \)
|
| $29$ |
\( (T^{4} - 3036 T^{3} + \cdots + 231630219024)^{2} \)
|
| $31$ |
\( T^{8} + \cdots + 29\!\cdots\!96 \)
|
| $37$ |
\( T^{8} + \cdots + 10\!\cdots\!24 \)
|
| $41$ |
\( T^{8} + \cdots + 14\!\cdots\!16 \)
|
| $43$ |
\( T^{8} + \cdots + 74\!\cdots\!56 \)
|
| $47$ |
\( T^{8} + \cdots + 71\!\cdots\!24 \)
|
| $53$ |
\( (T^{4} + 2484 T^{3} + \cdots + 365411917392)^{2} \)
|
| $59$ |
\( T^{8} + \cdots + 44\!\cdots\!64 \)
|
| $61$ |
\( (T^{4} + \cdots + 53787020315712)^{2} \)
|
| $67$ |
\( T^{8} + \cdots + 32\!\cdots\!64 \)
|
| $71$ |
\( T^{8} + \cdots + 50\!\cdots\!24 \)
|
| $73$ |
\( T^{8} + \cdots + 76\!\cdots\!44 \)
|
| $79$ |
\( (T^{4} + \cdots - 338606607600768)^{2} \)
|
| $83$ |
\( T^{8} + \cdots + 22\!\cdots\!04 \)
|
| $89$ |
\( T^{8} + \cdots + 94\!\cdots\!44 \)
|
| $97$ |
\( T^{8} + \cdots + 81\!\cdots\!36 \)
|
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