Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1764,2,Mod(373,1764)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1764, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 2, 2]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1764.373");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1764.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: | \(14.0856109166\) |
| Analytic rank: | \(0\) |
| Dimension: | \(14\) |
| Relative dimension: | \(7\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{14} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{14} - 5x^{12} - 3x^{11} + 7x^{10} + 30x^{9} - 117x^{7} + 270x^{5} + 189x^{4} - 243x^{3} - 1215x^{2} + 2187 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 3^{5} \) |
| Twist minimal: | no (minimal twist has level 252) |
| 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 \(1,\beta_1,\ldots,\beta_{13}\) 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
\( x^{14} - 5x^{12} - 3x^{11} + 7x^{10} + 30x^{9} - 117x^{7} + 270x^{5} + 189x^{4} - 243x^{3} - 1215x^{2} + 2187 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( - 5 \nu^{13} + 105 \nu^{12} - 191 \nu^{11} - 456 \nu^{10} - 971 \nu^{9} + 720 \nu^{8} + \cdots - 189540 ) / 43011 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 35 \nu^{13} + 72 \nu^{12} + 157 \nu^{11} + 312 \nu^{10} - 290 \nu^{9} - 1383 \nu^{8} + \cdots - 3645 ) / 43011 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{12} - 5\nu^{10} - 3\nu^{9} + 7\nu^{8} + 30\nu^{7} - 117\nu^{5} + 270\nu^{3} + 189\nu^{2} - 243\nu - 972 ) / 243 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - \nu^{13} + 5 \nu^{11} + 3 \nu^{10} - 7 \nu^{9} - 30 \nu^{8} + 117 \nu^{6} - 270 \nu^{4} + \cdots + 1215 \nu ) / 729 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 8 \nu^{13} - 2 \nu^{12} + 23 \nu^{11} - 5 \nu^{10} - 37 \nu^{9} + 127 \nu^{8} - 78 \nu^{7} + \cdots + 8505 ) / 4779 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 8 \nu^{13} - 2 \nu^{12} + 23 \nu^{11} - 5 \nu^{10} - 37 \nu^{9} + 127 \nu^{8} - 78 \nu^{7} + \cdots + 8505 ) / 4779 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 8 \nu^{13} - 2 \nu^{12} + 23 \nu^{11} - 5 \nu^{10} - 37 \nu^{9} + 127 \nu^{8} - 78 \nu^{7} + \cdots + 13284 ) / 4779 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 26 \nu^{13} + 7 \nu^{12} - 70 \nu^{11} - 266 \nu^{10} - 301 \nu^{9} + 955 \nu^{8} + 846 \nu^{7} + \cdots - 30618 ) / 14337 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 55 \nu^{13} + 513 \nu^{12} - 193 \nu^{11} - 1104 \nu^{10} - 394 \nu^{9} - 462 \nu^{8} + \cdots - 187353 ) / 43011 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 75 \nu^{13} + 40 \nu^{12} + 276 \nu^{11} + 52 \nu^{10} - 231 \nu^{9} - 755 \nu^{8} + 669 \nu^{7} + \cdots + 9234 ) / 14337 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 151 \nu^{13} - 381 \nu^{12} + 170 \nu^{11} + 1602 \nu^{10} + 2606 \nu^{9} - 3282 \nu^{8} + \cdots + 331695 ) / 43011 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( 506 \nu^{13} - 114 \nu^{12} - 1036 \nu^{11} - 813 \nu^{10} + 1193 \nu^{9} + 8820 \nu^{8} + \cdots + 306909 ) / 43011 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( - 568 \nu^{13} - 402 \nu^{12} + 1499 \nu^{11} + 2985 \nu^{10} + 2234 \nu^{9} - 10080 \nu^{8} + \cdots + 447606 ) / 43011 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{6} - \beta_{5} ) / 3 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -3\beta_{7} + \beta_{6} + 2\beta_{5} + 3 ) / 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 3\beta_{13} + 3\beta_{12} - 3\beta_{10} + \beta_{6} - \beta_{5} + 6\beta_{4} + 6\beta_{3} + 3\beta_{2} + 6\beta_1 ) / 3 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -9\beta_{8} - 3\beta_{7} + 4\beta_{6} + 5\beta_{5} - 9\beta_{2} + 9\beta _1 + 12 ) / 3 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 12 \beta_{13} + 12 \beta_{12} - 18 \beta_{11} - 3 \beta_{10} - 3 \beta_{9} - 9 \beta_{8} - 3 \beta_{7} + \cdots - 12 ) / 3 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( - 3 \beta_{13} + 12 \beta_{12} + 15 \beta_{10} - 9 \beta_{9} - 24 \beta_{8} - 12 \beta_{7} - 14 \beta_{6} + \cdots + 15 ) / 3 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 30 \beta_{13} + 21 \beta_{12} - 63 \beta_{11} - 12 \beta_{10} - 24 \beta_{9} - 36 \beta_{8} + 30 \beta_{7} + \cdots + 21 ) / 3 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( - 24 \beta_{13} + 15 \beta_{12} + 93 \beta_{10} - 27 \beta_{9} - 21 \beta_{8} - 48 \beta_{7} + \cdots - 63 ) / 3 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( 57 \beta_{13} + 30 \beta_{12} - 81 \beta_{11} - 48 \beta_{10} - 18 \beta_{9} - 18 \beta_{7} - 71 \beta_{6} + \cdots - 180 ) / 3 \)
|
| \(\nu^{10}\) | \(=\) |
\( ( - 18 \beta_{13} + 45 \beta_{12} + 108 \beta_{11} + 63 \beta_{10} - 54 \beta_{9} - 18 \beta_{8} + \cdots - 240 ) / 3 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( 93 \beta_{13} - 69 \beta_{12} - 207 \beta_{11} - 84 \beta_{10} + 24 \beta_{9} - 144 \beta_{8} + \cdots - 606 ) / 3 \)
|
| \(\nu^{12}\) | \(=\) |
\( ( - 57 \beta_{13} + 174 \beta_{12} + 81 \beta_{11} + 339 \beta_{10} + 234 \beta_{9} + 84 \beta_{8} + \cdots - 984 ) / 3 \)
|
| \(\nu^{13}\) | \(=\) |
\( ( - 186 \beta_{13} - 33 \beta_{12} - 144 \beta_{11} - 363 \beta_{10} - 159 \beta_{9} - 522 \beta_{8} + \cdots - 1356 ) / 3 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1764\mathbb{Z}\right)^\times\).
| \(n\) | \(785\) | \(883\) | \(1081\) |
| \(\chi(n)\) | \(-1 + \beta_{2}\) | \(1\) | \(-1 + \beta_{2}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 373.1 |
|
0 | −1.71886 | − | 0.213318i | 0 | 2.07260 | + | 3.58985i | 0 | 0 | 0 | 2.90899 | + | 0.733330i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 373.2 | 0 | −1.40166 | + | 1.01752i | 0 | −1.26013 | − | 2.18261i | 0 | 0 | 0 | 0.929318 | − | 2.85243i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 373.3 | 0 | −0.799754 | + | 1.53636i | 0 | 0.483929 | + | 0.838189i | 0 | 0 | 0 | −1.72079 | − | 2.45742i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 373.4 | 0 | −0.570327 | − | 1.63546i | 0 | −0.764702 | − | 1.32450i | 0 | 0 | 0 | −2.34945 | + | 1.86549i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 373.5 | 0 | 0.492857 | − | 1.66045i | 0 | 1.80173 | + | 3.12069i | 0 | 0 | 0 | −2.51418 | − | 1.63673i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 373.6 | 0 | 1.20601 | + | 1.24319i | 0 | −0.951504 | − | 1.64805i | 0 | 0 | 0 | −0.0910656 | + | 2.99862i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 373.7 | 0 | 1.29174 | − | 1.15387i | 0 | −0.381918 | − | 0.661502i | 0 | 0 | 0 | 0.337180 | − | 2.98099i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1537.1 | 0 | −1.71886 | + | 0.213318i | 0 | 2.07260 | − | 3.58985i | 0 | 0 | 0 | 2.90899 | − | 0.733330i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1537.2 | 0 | −1.40166 | − | 1.01752i | 0 | −1.26013 | + | 2.18261i | 0 | 0 | 0 | 0.929318 | + | 2.85243i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1537.3 | 0 | −0.799754 | − | 1.53636i | 0 | 0.483929 | − | 0.838189i | 0 | 0 | 0 | −1.72079 | + | 2.45742i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1537.4 | 0 | −0.570327 | + | 1.63546i | 0 | −0.764702 | + | 1.32450i | 0 | 0 | 0 | −2.34945 | − | 1.86549i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1537.5 | 0 | 0.492857 | + | 1.66045i | 0 | 1.80173 | − | 3.12069i | 0 | 0 | 0 | −2.51418 | + | 1.63673i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1537.6 | 0 | 1.20601 | − | 1.24319i | 0 | −0.951504 | + | 1.64805i | 0 | 0 | 0 | −0.0910656 | − | 2.99862i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1537.7 | 0 | 1.29174 | + | 1.15387i | 0 | −0.381918 | + | 0.661502i | 0 | 0 | 0 | 0.337180 | + | 2.98099i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 63.h | even | 3 | 1 | inner |
Twists
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{14} - 2 T_{5}^{13} + 24 T_{5}^{12} + 16 T_{5}^{11} + 295 T_{5}^{10} + 357 T_{5}^{9} + 2670 T_{5}^{8} + \cdots + 6561 \)
acting on \(S_{2}^{\mathrm{new}}(1764, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{14} \)
|
| $3$ |
\( T^{14} + 3 T^{13} + \cdots + 2187 \)
|
| $5$ |
\( T^{14} - 2 T^{13} + \cdots + 6561 \)
|
| $7$ |
\( T^{14} \)
|
| $11$ |
\( T^{14} - 2 T^{13} + \cdots + 6561 \)
|
| $13$ |
\( T^{14} + \cdots + 150626529 \)
|
| $17$ |
\( T^{14} + 2 T^{13} + \cdots + 6561 \)
|
| $19$ |
\( T^{14} + 7 T^{13} + \cdots + 4084441 \)
|
| $23$ |
\( T^{14} + \cdots + 105822369 \)
|
| $29$ |
\( T^{14} + \cdots + 145660761 \)
|
| $31$ |
\( (T^{7} + T^{6} + \cdots + 117504)^{2} \)
|
| $37$ |
\( T^{14} + \cdots + 1566893056 \)
|
| $41$ |
\( T^{14} + \cdots + 1108290681 \)
|
| $43$ |
\( T^{14} - 7 T^{13} + \cdots + 4084441 \)
|
| $47$ |
\( (T^{7} + 3 T^{6} + \cdots + 11664)^{2} \)
|
| $53$ |
\( T^{14} + \cdots + 952401321 \)
|
| $59$ |
\( (T^{7} + 14 T^{6} + \cdots - 26244)^{2} \)
|
| $61$ |
\( (T^{7} + 10 T^{6} + \cdots + 12192)^{2} \)
|
| $67$ |
\( (T^{7} + 6 T^{6} + \cdots + 10816)^{2} \)
|
| $71$ |
\( (T^{7} - T^{6} - 116 T^{5} + \cdots - 972)^{2} \)
|
| $73$ |
\( T^{14} + \cdots + 2748590329 \)
|
| $79$ |
\( (T^{7} - 10 T^{6} + \cdots - 233232)^{2} \)
|
| $83$ |
\( T^{14} + \cdots + 901054679121 \)
|
| $89$ |
\( T^{14} + \cdots + 16524331209 \)
|
| $97$ |
\( T^{14} + \cdots + 767677849 \)
|
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