Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [14,14,Mod(9,14)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(14, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([2]))
N = Newforms(chi, 14, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("14.9");
S:= CuspForms(chi, 14);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 14 = 2 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 14 \) |
| Character orbit: | \([\chi]\) | \(=\) | 14.c (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: | \(15.0123300533\) |
| 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: |
\( x^{8} - 2 x^{7} + 692094 x^{6} + 445371928 x^{5} + 480078817147 x^{4} + 153633603309480 x^{3} + \cdots + 23\!\cdots\!25 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{14}\cdot 3\cdot 7^{4} \) |
| 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
\( x^{8} - 2 x^{7} + 692094 x^{6} + 445371928 x^{5} + 480078817147 x^{4} + 153633603309480 x^{3} + \cdots + 23\!\cdots\!25 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( - 27\!\cdots\!26 \nu^{7} + \cdots - 92\!\cdots\!05 ) / 90\!\cdots\!05 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 14\!\cdots\!70 \nu^{7} + \cdots - 42\!\cdots\!05 ) / 25\!\cdots\!60 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 16\!\cdots\!30 \nu^{7} + \cdots - 84\!\cdots\!45 ) / 25\!\cdots\!60 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 99\!\cdots\!39 \nu^{7} + \cdots - 33\!\cdots\!05 ) / 19\!\cdots\!60 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 17\!\cdots\!37 \nu^{7} + \cdots - 14\!\cdots\!70 ) / 23\!\cdots\!70 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 20\!\cdots\!67 \nu^{7} + \cdots + 21\!\cdots\!15 ) / 23\!\cdots\!70 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 10\!\cdots\!81 \nu^{7} + \cdots - 35\!\cdots\!95 ) / 10\!\cdots\!40 \)
|
| \(\nu\) | \(=\) |
\( ( 3\beta_{7} + \beta_{6} + 2\beta_{5} - 9\beta_{4} - 4\beta_{3} + 8\beta_{2} + 171\beta _1 + 174 ) / 336 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -87\beta_{7} + 386\beta_{6} + 193\beta_{5} - 5751\beta_{4} + 193\beta_{3} + 193\beta_{2} + 29070783\beta_1 ) / 84 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 813235\beta_{6} - 813235\beta_{5} + 2971559\beta_{3} - 16649389\beta_{2} - 56301080754 ) / 336 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( - 26739030 \beta_{7} - 47369614 \beta_{6} - 94739228 \beta_{5} + 1122208434 \beta_{4} + \cdots - 5071555289829 ) / 21 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( - 859127137203 \beta_{7} - 1474522292642 \beta_{6} - 737261146321 \beta_{5} + \cdots - 64\!\cdots\!31 \beta_1 ) / 336 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( - 176934179918233 \beta_{6} + 176934179918233 \beta_{5} - 502838085390899 \beta_{3} + \cdots + 17\!\cdots\!76 ) / 84 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 69\!\cdots\!09 \beta_{7} + \cdots + 63\!\cdots\!34 ) / 336 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/14\mathbb{Z}\right)^\times\).
| \(n\) | \(3\) |
| \(\chi(n)\) | \(-1 - \beta_{1}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 9.1 |
|
−32.0000 | + | 55.4256i | −855.669 | − | 1482.06i | −2048.00 | − | 3547.24i | 12545.8 | − | 21729.9i | 109526. | 48813.9 | − | 307419.i | 262144. | −667176. | + | 1.15558e6i | 802928. | + | 1.39071e6i | ||||||||||||||||||||||||||||
| 9.2 | −32.0000 | + | 55.4256i | −679.656 | − | 1177.20i | −2048.00 | − | 3547.24i | −29011.3 | + | 50249.0i | 86996.0 | −173759. | + | 258257.i | 262144. | −126703. | + | 219456.i | −1.85672e6 | − | 3.21593e6i | |||||||||||||||||||||||||||||
| 9.3 | −32.0000 | + | 55.4256i | 301.970 | + | 523.027i | −2048.00 | − | 3547.24i | 2521.96 | − | 4368.16i | −38652.2 | 302038. | + | 75247.5i | 262144. | 614790. | − | 1.06485e6i | 161406. | + | 279563.i | |||||||||||||||||||||||||||||
| 9.4 | −32.0000 | + | 55.4256i | 1142.35 | + | 1978.62i | −2048.00 | − | 3547.24i | −18256.5 | + | 31621.1i | −146221. | −233961. | − | 205308.i | 262144. | −1.81279e6 | + | 3.13984e6i | −1.16841e6 | − | 2.02375e6i | |||||||||||||||||||||||||||||
| 11.1 | −32.0000 | − | 55.4256i | −855.669 | + | 1482.06i | −2048.00 | + | 3547.24i | 12545.8 | + | 21729.9i | 109526. | 48813.9 | + | 307419.i | 262144. | −667176. | − | 1.15558e6i | 802928. | − | 1.39071e6i | |||||||||||||||||||||||||||||
| 11.2 | −32.0000 | − | 55.4256i | −679.656 | + | 1177.20i | −2048.00 | + | 3547.24i | −29011.3 | − | 50249.0i | 86996.0 | −173759. | − | 258257.i | 262144. | −126703. | − | 219456.i | −1.85672e6 | + | 3.21593e6i | |||||||||||||||||||||||||||||
| 11.3 | −32.0000 | − | 55.4256i | 301.970 | − | 523.027i | −2048.00 | + | 3547.24i | 2521.96 | + | 4368.16i | −38652.2 | 302038. | − | 75247.5i | 262144. | 614790. | + | 1.06485e6i | 161406. | − | 279563.i | |||||||||||||||||||||||||||||
| 11.4 | −32.0000 | − | 55.4256i | 1142.35 | − | 1978.62i | −2048.00 | + | 3547.24i | −18256.5 | − | 31621.1i | −146221. | −233961. | + | 205308.i | 262144. | −1.81279e6 | − | 3.13984e6i | −1.16841e6 | + | 2.02375e6i | |||||||||||||||||||||||||||||
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 | 14.14.c.a | ✓ | 8 |
| 3.b | odd | 2 | 1 | 126.14.g.d | 8 | ||
| 7.b | odd | 2 | 1 | 98.14.c.n | 8 | ||
| 7.c | even | 3 | 1 | inner | 14.14.c.a | ✓ | 8 |
| 7.c | even | 3 | 1 | 98.14.a.l | 4 | ||
| 7.d | odd | 6 | 1 | 98.14.a.k | 4 | ||
| 7.d | odd | 6 | 1 | 98.14.c.n | 8 | ||
| 21.h | odd | 6 | 1 | 126.14.g.d | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 14.14.c.a | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 14.14.c.a | ✓ | 8 | 7.c | even | 3 | 1 | inner |
| 98.14.a.k | 4 | 7.d | odd | 6 | 1 | ||
| 98.14.a.l | 4 | 7.c | even | 3 | 1 | ||
| 98.14.c.n | 8 | 7.b | odd | 2 | 1 | ||
| 98.14.c.n | 8 | 7.d | odd | 6 | 1 | ||
| 126.14.g.d | 8 | 3.b | odd | 2 | 1 | ||
| 126.14.g.d | 8 | 21.h | odd | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{8} + 182 T_{3}^{7} + 5197084 T_{3}^{6} + 4025617932 T_{3}^{5} + 23908528667637 T_{3}^{4} + \cdots + 10\!\cdots\!25 \)
acting on \(S_{14}^{\mathrm{new}}(14, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 64 T + 4096)^{4} \)
|
| $3$ |
\( T^{8} + \cdots + 10\!\cdots\!25 \)
|
| $5$ |
\( T^{8} + \cdots + 71\!\cdots\!25 \)
|
| $7$ |
\( T^{8} + \cdots + 88\!\cdots\!01 \)
|
| $11$ |
\( T^{8} + \cdots + 29\!\cdots\!25 \)
|
| $13$ |
\( (T^{4} + \cdots - 94\!\cdots\!80)^{2} \)
|
| $17$ |
\( T^{8} + \cdots + 17\!\cdots\!25 \)
|
| $19$ |
\( T^{8} + \cdots + 13\!\cdots\!25 \)
|
| $23$ |
\( T^{8} + \cdots + 17\!\cdots\!81 \)
|
| $29$ |
\( (T^{4} + \cdots + 47\!\cdots\!28)^{2} \)
|
| $31$ |
\( T^{8} + \cdots + 34\!\cdots\!25 \)
|
| $37$ |
\( T^{8} + \cdots + 55\!\cdots\!61 \)
|
| $41$ |
\( (T^{4} + \cdots - 10\!\cdots\!08)^{2} \)
|
| $43$ |
\( (T^{4} + \cdots - 18\!\cdots\!00)^{2} \)
|
| $47$ |
\( T^{8} + \cdots + 82\!\cdots\!25 \)
|
| $53$ |
\( T^{8} + \cdots + 64\!\cdots\!41 \)
|
| $59$ |
\( T^{8} + \cdots + 14\!\cdots\!25 \)
|
| $61$ |
\( T^{8} + \cdots + 12\!\cdots\!25 \)
|
| $67$ |
\( T^{8} + \cdots + 98\!\cdots\!25 \)
|
| $71$ |
\( (T^{4} + \cdots - 79\!\cdots\!40)^{2} \)
|
| $73$ |
\( T^{8} + \cdots + 21\!\cdots\!25 \)
|
| $79$ |
\( T^{8} + \cdots + 33\!\cdots\!25 \)
|
| $83$ |
\( (T^{4} + \cdots + 13\!\cdots\!00)^{2} \)
|
| $89$ |
\( T^{8} + \cdots + 62\!\cdots\!25 \)
|
| $97$ |
\( (T^{4} + \cdots - 27\!\cdots\!00)^{2} \)
|
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