Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [112,12,Mod(65,112)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(112, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 0, 2]))
N = Newforms(chi, 12, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("112.65");
S:= CuspForms(chi, 12);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 112 = 2^{4} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 12 \) |
| Character orbit: | \([\chi]\) | \(=\) | 112.i (of order \(3\), degree \(2\), not 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: | \(86.0544362227\) |
| 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: |
\( x^{8} - x^{7} + 149344 x^{6} + 5578711 x^{5} + 20557200983 x^{4} + 408905884576 x^{3} + \cdots + 30\!\cdots\!24 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{19}]\) |
| Coefficient ring index: | \( 2^{12}\cdot 3\cdot 7^{3} \) |
| Twist minimal: | no (minimal twist has level 14) |
| 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
\( x^{8} - x^{7} + 149344 x^{6} + 5578711 x^{5} + 20557200983 x^{4} + 408905884576 x^{3} + \cdots + 30\!\cdots\!24 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 35\!\cdots\!25 \nu^{7} + \cdots - 26\!\cdots\!44 ) / 52\!\cdots\!04 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 170605955991169 \nu^{7} + \cdots - 87\!\cdots\!00 ) / 21\!\cdots\!54 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 76\!\cdots\!75 \nu^{7} + \cdots + 72\!\cdots\!68 ) / 11\!\cdots\!80 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 20\!\cdots\!37 \nu^{7} + \cdots - 17\!\cdots\!60 ) / 11\!\cdots\!80 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 43\!\cdots\!45 \nu^{7} + \cdots + 27\!\cdots\!48 ) / 11\!\cdots\!80 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 10\!\cdots\!81 \nu^{7} + \cdots + 15\!\cdots\!40 ) / 77\!\cdots\!60 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 5\beta_{7} + \beta_{6} + 25\beta_{5} - \beta_{4} + 57\beta_{3} - 298656\beta_{2} - 2 ) / 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 1756 \beta_{7} + 2195 \beta_{6} - 3301 \beta_{5} - 3740 \beta_{4} + 504127 \beta_{3} + 878 \beta_{2} + \cdots - 16931881 ) / 8 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( - 299125 \beta_{7} + 1196500 \beta_{6} + 299125 \beta_{5} + 7762535 \beta_{4} + 75274567413 \beta_{2} + \cdots - 75273969163 ) / 8 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( - 178225175 \beta_{7} - 35645035 \beta_{6} + 207672605 \beta_{5} + 35645035 \beta_{4} - 34326018287 \beta_{3} + \cdots + 71290070 ) / 4 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( - 169793717028 \beta_{7} - 212242146285 \beta_{6} - 1059317406417 \beta_{5} - 1016868977160 \beta_{4} + \cdots + 10\!\cdots\!91 ) / 8 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 10735791588669 \beta_{7} - 42943166354676 \beta_{6} - 10735791588669 \beta_{5} + 23495063646081 \beta_{4} + \cdots + 81\!\cdots\!23 ) / 8 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/112\mathbb{Z}\right)^\times\).
| \(n\) | \(15\) | \(17\) | \(85\) |
| \(\chi(n)\) | \(1\) | \(-\beta_{2}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 65.1 |
|
0 | −347.050 | − | 601.109i | 0 | 2646.67 | − | 4584.17i | 0 | −22119.1 | − | 38575.5i | 0 | −152314. | + | 263816.i | 0 | ||||||||||||||||||||||||||||||||||
| 65.2 | 0 | −69.1868 | − | 119.835i | 0 | 274.515 | − | 475.474i | 0 | 25454.3 | + | 36461.0i | 0 | 78999.9 | − | 136832.i | 0 | |||||||||||||||||||||||||||||||||||
| 65.3 | 0 | 158.327 | + | 274.230i | 0 | −6009.47 | + | 10408.7i | 0 | −34168.3 | − | 28457.9i | 0 | 38438.9 | − | 66578.1i | 0 | |||||||||||||||||||||||||||||||||||
| 65.4 | 0 | 390.910 | + | 677.077i | 0 | 4992.28 | − | 8646.88i | 0 | −24330.9 | + | 37220.1i | 0 | −217048. | + | 375939.i | 0 | |||||||||||||||||||||||||||||||||||
| 81.1 | 0 | −347.050 | + | 601.109i | 0 | 2646.67 | + | 4584.17i | 0 | −22119.1 | + | 38575.5i | 0 | −152314. | − | 263816.i | 0 | |||||||||||||||||||||||||||||||||||
| 81.2 | 0 | −69.1868 | + | 119.835i | 0 | 274.515 | + | 475.474i | 0 | 25454.3 | − | 36461.0i | 0 | 78999.9 | + | 136832.i | 0 | |||||||||||||||||||||||||||||||||||
| 81.3 | 0 | 158.327 | − | 274.230i | 0 | −6009.47 | − | 10408.7i | 0 | −34168.3 | + | 28457.9i | 0 | 38438.9 | + | 66578.1i | 0 | |||||||||||||||||||||||||||||||||||
| 81.4 | 0 | 390.910 | − | 677.077i | 0 | 4992.28 | + | 8646.88i | 0 | −24330.9 | − | 37220.1i | 0 | −217048. | − | 375939.i | 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 | 112.12.i.b | 8 | |
| 4.b | odd | 2 | 1 | 14.12.c.b | ✓ | 8 | |
| 7.c | even | 3 | 1 | inner | 112.12.i.b | 8 | |
| 12.b | even | 2 | 1 | 126.12.g.c | 8 | ||
| 28.d | even | 2 | 1 | 98.12.c.n | 8 | ||
| 28.f | even | 6 | 1 | 98.12.a.h | 4 | ||
| 28.f | even | 6 | 1 | 98.12.c.n | 8 | ||
| 28.g | odd | 6 | 1 | 14.12.c.b | ✓ | 8 | |
| 28.g | odd | 6 | 1 | 98.12.a.i | 4 | ||
| 84.n | even | 6 | 1 | 126.12.g.c | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 14.12.c.b | ✓ | 8 | 4.b | odd | 2 | 1 | |
| 14.12.c.b | ✓ | 8 | 28.g | odd | 6 | 1 | |
| 98.12.a.h | 4 | 28.f | even | 6 | 1 | ||
| 98.12.a.i | 4 | 28.g | odd | 6 | 1 | ||
| 98.12.c.n | 8 | 28.d | even | 2 | 1 | ||
| 98.12.c.n | 8 | 28.f | even | 6 | 1 | ||
| 112.12.i.b | 8 | 1.a | even | 1 | 1 | trivial | |
| 112.12.i.b | 8 | 7.c | even | 3 | 1 | inner | |
| 126.12.g.c | 8 | 12.b | even | 2 | 1 | ||
| 126.12.g.c | 8 | 84.n | even | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{8} - 266 T_{3}^{7} + 641596 T_{3}^{6} - 49334964 T_{3}^{5} + 328837501557 T_{3}^{4} + \cdots + 56\!\cdots\!25 \)
acting on \(S_{12}^{\mathrm{new}}(112, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} \)
|
| $3$ |
\( T^{8} + \cdots + 56\!\cdots\!25 \)
|
| $5$ |
\( T^{8} + \cdots + 12\!\cdots\!25 \)
|
| $7$ |
\( T^{8} + \cdots + 15\!\cdots\!01 \)
|
| $11$ |
\( T^{8} + \cdots + 87\!\cdots\!25 \)
|
| $13$ |
\( (T^{4} + \cdots + 24\!\cdots\!40)^{2} \)
|
| $17$ |
\( T^{8} + \cdots + 87\!\cdots\!25 \)
|
| $19$ |
\( T^{8} + \cdots + 13\!\cdots\!25 \)
|
| $23$ |
\( T^{8} + \cdots + 40\!\cdots\!81 \)
|
| $29$ |
\( (T^{4} + \cdots - 20\!\cdots\!32)^{2} \)
|
| $31$ |
\( T^{8} + \cdots + 45\!\cdots\!25 \)
|
| $37$ |
\( T^{8} + \cdots + 25\!\cdots\!41 \)
|
| $41$ |
\( (T^{4} + \cdots - 13\!\cdots\!48)^{2} \)
|
| $43$ |
\( (T^{4} + \cdots - 42\!\cdots\!40)^{2} \)
|
| $47$ |
\( T^{8} + \cdots + 13\!\cdots\!25 \)
|
| $53$ |
\( T^{8} + \cdots + 10\!\cdots\!41 \)
|
| $59$ |
\( T^{8} + \cdots + 18\!\cdots\!25 \)
|
| $61$ |
\( T^{8} + \cdots + 51\!\cdots\!25 \)
|
| $67$ |
\( T^{8} + \cdots + 18\!\cdots\!25 \)
|
| $71$ |
\( (T^{4} + \cdots + 23\!\cdots\!40)^{2} \)
|
| $73$ |
\( T^{8} + \cdots + 40\!\cdots\!25 \)
|
| $79$ |
\( T^{8} + \cdots + 28\!\cdots\!25 \)
|
| $83$ |
\( (T^{4} + \cdots + 54\!\cdots\!00)^{2} \)
|
| $89$ |
\( T^{8} + \cdots + 87\!\cdots\!25 \)
|
| $97$ |
\( (T^{4} + \cdots - 56\!\cdots\!00)^{2} \)
|
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