Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [14,10,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, 10, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("14.9");
S:= CuspForms(chi, 10);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 14 = 2 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 10 \) |
| 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: | \(7.21050170629\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Relative dimension: | \(3\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{6} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - x^{5} + 4038x^{4} - 137923x^{3} + 16368349x^{2} - 286546260x + 5038160400 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{6}\cdot 7 \) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - x^{5} + 4038x^{4} - 137923x^{3} + 16368349x^{2} - 286546260x + 5038160400 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( - 2716901 \nu^{5} + 2705071 \nu^{4} - 10923076698 \nu^{3} + 181829734103 \nu^{2} + \cdots - 3390681949200 ) / 778518660033660 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 4027 \nu^{5} + 16261026 \nu^{4} + 146824014 \nu^{3} + 65915341423 \nu^{2} + \cdots + 166009397495697 ) / 690992893521 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 2081085611 \nu^{5} + 53013142539 \nu^{4} + 8366833294278 \nu^{3} - 139277523654233 \nu^{2} + \cdots + 25\!\cdots\!00 ) / 23\!\cdots\!80 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 694639239 \nu^{5} + 35795315171 \nu^{4} + 3747785917342 \nu^{3} + 26700567042283 \nu^{2} + \cdots + 13\!\cdots\!80 ) / 778518660033660 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 352672159 \nu^{5} + 9199967499 \nu^{4} - 77151616502 \nu^{3} + 96406628067317 \nu^{2} + \cdots + 10\!\cdots\!10 ) / 389259330016830 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{5} - 3\beta_{4} + 4\beta_{3} + \beta_{2} - 7\beta_1 ) / 28 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -15\beta_{5} + 10\beta_{4} - 172\beta_{3} + 167\beta_{2} - 37744\beta _1 - 37744 ) / 14 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 8054\beta_{5} + 4027\beta_{4} + 4027\beta_{3} - 19841\beta_{2} + 1883693 ) / 28 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 107323\beta_{5} - 321969\beta_{4} + 1616464\beta_{3} + 107323\beta_{2} + 302364503\beta_1 ) / 28 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( - 25289214 \beta_{5} + 16859476 \beta_{4} - 51395725 \beta_{3} + 42965987 \beta_{2} + \cdots - 6330121189 ) / 14 \)
|
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 |
|
8.00000 | − | 13.8564i | −89.6637 | − | 155.302i | −128.000 | − | 221.703i | 84.1442 | − | 145.742i | −2869.24 | −3162.16 | + | 5509.48i | −4096.00 | −6237.66 | + | 10803.9i | −1346.31 | − | 2331.87i | ||||||||||||||||||||||
| 9.2 | 8.00000 | − | 13.8564i | 38.4345 | + | 66.5705i | −128.000 | − | 221.703i | 546.130 | − | 945.925i | 1229.90 | 6111.84 | − | 1731.76i | −4096.00 | 6887.08 | − | 11928.8i | −8738.08 | − | 15134.8i | |||||||||||||||||||||||
| 9.3 | 8.00000 | − | 13.8564i | 86.7292 | + | 150.219i | −128.000 | − | 221.703i | −1172.77 | + | 2031.30i | 2775.34 | −6347.68 | − | 246.066i | −4096.00 | −5202.42 | + | 9010.85i | 18764.4 | + | 32500.9i | |||||||||||||||||||||||
| 11.1 | 8.00000 | + | 13.8564i | −89.6637 | + | 155.302i | −128.000 | + | 221.703i | 84.1442 | + | 145.742i | −2869.24 | −3162.16 | − | 5509.48i | −4096.00 | −6237.66 | − | 10803.9i | −1346.31 | + | 2331.87i | |||||||||||||||||||||||
| 11.2 | 8.00000 | + | 13.8564i | 38.4345 | − | 66.5705i | −128.000 | + | 221.703i | 546.130 | + | 945.925i | 1229.90 | 6111.84 | + | 1731.76i | −4096.00 | 6887.08 | + | 11928.8i | −8738.08 | + | 15134.8i | |||||||||||||||||||||||
| 11.3 | 8.00000 | + | 13.8564i | 86.7292 | − | 150.219i | −128.000 | + | 221.703i | −1172.77 | − | 2031.30i | 2775.34 | −6347.68 | + | 246.066i | −4096.00 | −5202.42 | − | 9010.85i | 18764.4 | − | 32500.9i | |||||||||||||||||||||||
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 | 14.10.c.b | ✓ | 6 |
| 3.b | odd | 2 | 1 | 126.10.g.e | 6 | ||
| 4.b | odd | 2 | 1 | 112.10.i.a | 6 | ||
| 7.b | odd | 2 | 1 | 98.10.c.l | 6 | ||
| 7.c | even | 3 | 1 | inner | 14.10.c.b | ✓ | 6 |
| 7.c | even | 3 | 1 | 98.10.a.g | 3 | ||
| 7.d | odd | 6 | 1 | 98.10.a.h | 3 | ||
| 7.d | odd | 6 | 1 | 98.10.c.l | 6 | ||
| 21.h | odd | 6 | 1 | 126.10.g.e | 6 | ||
| 28.g | odd | 6 | 1 | 112.10.i.a | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 14.10.c.b | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 14.10.c.b | ✓ | 6 | 7.c | even | 3 | 1 | inner |
| 98.10.a.g | 3 | 7.c | even | 3 | 1 | ||
| 98.10.a.h | 3 | 7.d | odd | 6 | 1 | ||
| 98.10.c.l | 6 | 7.b | odd | 2 | 1 | ||
| 98.10.c.l | 6 | 7.d | odd | 6 | 1 | ||
| 112.10.i.a | 6 | 4.b | odd | 2 | 1 | ||
| 112.10.i.a | 6 | 28.g | odd | 6 | 1 | ||
| 126.10.g.e | 6 | 3.b | odd | 2 | 1 | ||
| 126.10.g.e | 6 | 21.h | odd | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{6} - 71T_{3}^{5} + 36598T_{3}^{4} - 2541603T_{3}^{3} + 1165610574T_{3}^{2} - 75455153775T_{3} + 5717239655625 \)
acting on \(S_{10}^{\mathrm{new}}(14, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} - 16 T + 256)^{3} \)
|
| $3$ |
\( T^{6} + \cdots + 5717239655625 \)
|
| $5$ |
\( T^{6} + \cdots + 18\!\cdots\!25 \)
|
| $7$ |
\( T^{6} + \cdots + 65\!\cdots\!43 \)
|
| $11$ |
\( T^{6} + \cdots + 31\!\cdots\!41 \)
|
| $13$ |
\( (T^{3} + \cdots + 368974841338200)^{2} \)
|
| $17$ |
\( T^{6} + \cdots + 11\!\cdots\!25 \)
|
| $19$ |
\( T^{6} + \cdots + 37\!\cdots\!25 \)
|
| $23$ |
\( T^{6} + \cdots + 97\!\cdots\!61 \)
|
| $29$ |
\( (T^{3} + \cdots - 11\!\cdots\!32)^{2} \)
|
| $31$ |
\( T^{6} + \cdots + 38\!\cdots\!09 \)
|
| $37$ |
\( T^{6} + \cdots + 22\!\cdots\!69 \)
|
| $41$ |
\( (T^{3} + \cdots + 12\!\cdots\!24)^{2} \)
|
| $43$ |
\( (T^{3} + \cdots + 85\!\cdots\!92)^{2} \)
|
| $47$ |
\( T^{6} + \cdots + 21\!\cdots\!25 \)
|
| $53$ |
\( T^{6} + \cdots + 10\!\cdots\!21 \)
|
| $59$ |
\( T^{6} + \cdots + 88\!\cdots\!25 \)
|
| $61$ |
\( T^{6} + \cdots + 31\!\cdots\!25 \)
|
| $67$ |
\( T^{6} + \cdots + 29\!\cdots\!25 \)
|
| $71$ |
\( (T^{3} + \cdots - 12\!\cdots\!40)^{2} \)
|
| $73$ |
\( T^{6} + \cdots + 24\!\cdots\!41 \)
|
| $79$ |
\( T^{6} + \cdots + 77\!\cdots\!25 \)
|
| $83$ |
\( (T^{3} + \cdots - 87\!\cdots\!00)^{2} \)
|
| $89$ |
\( T^{6} + \cdots + 11\!\cdots\!25 \)
|
| $97$ |
\( (T^{3} + \cdots + 14\!\cdots\!40)^{2} \)
|
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