Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [441,2,Mod(68,441)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(441, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([5, 5]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("441.68");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 441 = 3^{2} \cdot 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 441.i (of order \(6\), 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: | \(3.52140272914\) |
| Analytic rank: | \(0\) |
| Dimension: | \(12\) |
| Relative dimension: | \(6\) over \(\Q(\zeta_{6})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{12} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{12} - 7x^{10} + 37x^{8} - 78x^{6} + 123x^{4} - 36x^{2} + 9 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 3 \) |
| Twist minimal: | no (minimal twist has level 63) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-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 \(1,\beta_1,\ldots,\beta_{11}\) 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
\( x^{12} - 7x^{10} + 37x^{8} - 78x^{6} + 123x^{4} - 36x^{2} + 9 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -28\nu^{10} + 148\nu^{8} + 446\nu^{6} - 3807\nu^{4} + 17052\nu^{2} + 4446 ) / 12897 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -175\nu^{11} + 925\nu^{9} - 3661\nu^{7} - 1224\nu^{5} + 16296\nu^{3} - 43146\nu ) / 12897 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -164\nu^{10} + 1481\nu^{8} - 7214\nu^{6} + 17007\nu^{4} - 16197\nu^{2} - 3438 ) / 12897 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -175\nu^{10} + 925\nu^{8} - 3661\nu^{6} - 1224\nu^{4} + 16296\nu^{2} - 30249 ) / 12897 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 148\nu^{10} - 987\nu^{8} + 5217\nu^{6} - 10175\nu^{4} + 17343\nu^{2} - 777 ) / 4299 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 730\nu^{11} - 5701\nu^{9} + 30748\nu^{7} - 76698\nu^{5} + 122898\nu^{3} - 66168\nu ) / 12897 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -877\nu^{10} + 6478\nu^{8} - 34855\nu^{6} + 79281\nu^{4} - 123654\nu^{2} + 31473 ) / 12897 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( -1349\nu^{11} + 9587\nu^{9} - 50060\nu^{7} + 105999\nu^{5} - 158631\nu^{3} + 51147\nu ) / 12897 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( -1496\nu^{11} + 10364\nu^{9} - 54167\nu^{7} + 108582\nu^{5} - 159387\nu^{3} + 3555\nu ) / 12897 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( -1804\nu^{11} + 11992\nu^{9} - 62158\nu^{7} + 118293\nu^{5} - 178167\nu^{3} + 13770\nu ) / 12897 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 2237\nu^{11} - 15509\nu^{9} + 81362\nu^{7} - 167049\nu^{5} + 249792\nu^{3} - 42912\nu ) / 12897 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{11} - \beta_{10} - \beta_{9} + 2\beta_{8} + 2\beta_{6} - \beta_{2} ) / 3 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{7} + 2\beta_{5} + \beta_{4} - \beta_{3} \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -8\beta_{11} - 5\beta_{10} - 5\beta_{9} + \beta_{8} + 4\beta_{6} + \beta_{2} ) / 3 \)
|
| \(\nu^{4}\) | \(=\) |
\( 4\beta_{7} + 6\beta_{5} - 6\beta_{3} + 5\beta _1 - 12 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -16\beta_{11} + 2\beta_{10} - 19\beta_{9} - 19\beta_{8} - 16\beta_{6} + 17\beta_{2} ) / 3 \)
|
| \(\nu^{6}\) | \(=\) |
\( -7\beta_{5} - 16\beta_{4} - 7\beta_{3} + 30\beta _1 - 51 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 67\beta_{11} + 88\beta_{10} - 23\beta_{9} - 65\beta_{8} - 134\beta_{6} + 88\beta_{2} ) / 3 \)
|
| \(\nu^{8}\) | \(=\) |
\( -67\beta_{7} - 118\beta_{5} - 67\beta_{4} + 104\beta_{3} + 37\beta_1 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( 578\beta_{11} + 266\beta_{10} + 266\beta_{9} + 134\beta_{8} - 289\beta_{6} + 134\beta_{2} ) / 3 \)
|
| \(\nu^{10}\) | \(=\) |
\( -289\beta_{7} - 333\beta_{5} + 645\beta_{3} - 467\beta _1 + 978 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( 1267\beta_{11} - 668\beta_{10} + 1801\beta_{9} + 1801\beta_{8} + 1267\beta_{6} - 1133\beta_{2} ) / 3 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/441\mathbb{Z}\right)^\times\).
| \(n\) | \(199\) | \(344\) |
| \(\chi(n)\) | \(\beta_{5}\) | \(\beta_{5}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 68.1 |
|
− | 2.27639i | −1.12441 | + | 1.31746i | −3.18194 | 0.717144 | + | 1.24213i | 2.99905 | + | 2.55959i | 0 | 2.69056i | −0.471410 | − | 2.96273i | 2.82757 | − | 1.63250i | |||||||||||||||||||||||||||||||||||||||||||
| 68.2 | − | 2.27639i | 1.12441 | − | 1.31746i | −3.18194 | −0.717144 | − | 1.24213i | −2.99905 | − | 2.55959i | 0 | 2.69056i | −0.471410 | − | 2.96273i | −2.82757 | + | 1.63250i | ||||||||||||||||||||||||||||||||||||||||||||
| 68.3 | − | 0.641589i | −1.40434 | − | 1.01381i | 1.58836 | 1.10552 | + | 1.91482i | −0.650451 | + | 0.901012i | 0 | − | 2.30225i | 0.944368 | + | 2.84748i | 1.22853 | − | 0.709292i | |||||||||||||||||||||||||||||||||||||||||||
| 68.4 | − | 0.641589i | 1.40434 | + | 1.01381i | 1.58836 | −1.10552 | − | 1.91482i | 0.650451 | − | 0.901012i | 0 | − | 2.30225i | 0.944368 | + | 2.84748i | −1.22853 | + | 0.709292i | |||||||||||||||||||||||||||||||||||||||||||
| 68.5 | 1.18593i | −1.66238 | + | 0.486291i | 0.593579 | −1.41899 | − | 2.45776i | −0.576705 | − | 1.97146i | 0 | 3.07579i | 2.52704 | − | 1.61680i | 2.91472 | − | 1.68281i | |||||||||||||||||||||||||||||||||||||||||||||
| 68.6 | 1.18593i | 1.66238 | − | 0.486291i | 0.593579 | 1.41899 | + | 2.45776i | 0.576705 | + | 1.97146i | 0 | 3.07579i | 2.52704 | − | 1.61680i | −2.91472 | + | 1.68281i | |||||||||||||||||||||||||||||||||||||||||||||
| 227.1 | − | 1.18593i | −1.66238 | − | 0.486291i | 0.593579 | −1.41899 | + | 2.45776i | −0.576705 | + | 1.97146i | 0 | − | 3.07579i | 2.52704 | + | 1.61680i | 2.91472 | + | 1.68281i | |||||||||||||||||||||||||||||||||||||||||||
| 227.2 | − | 1.18593i | 1.66238 | + | 0.486291i | 0.593579 | 1.41899 | − | 2.45776i | 0.576705 | − | 1.97146i | 0 | − | 3.07579i | 2.52704 | + | 1.61680i | −2.91472 | − | 1.68281i | |||||||||||||||||||||||||||||||||||||||||||
| 227.3 | 0.641589i | −1.40434 | + | 1.01381i | 1.58836 | 1.10552 | − | 1.91482i | −0.650451 | − | 0.901012i | 0 | 2.30225i | 0.944368 | − | 2.84748i | 1.22853 | + | 0.709292i | |||||||||||||||||||||||||||||||||||||||||||||
| 227.4 | 0.641589i | 1.40434 | − | 1.01381i | 1.58836 | −1.10552 | + | 1.91482i | 0.650451 | + | 0.901012i | 0 | 2.30225i | 0.944368 | − | 2.84748i | −1.22853 | − | 0.709292i | |||||||||||||||||||||||||||||||||||||||||||||
| 227.5 | 2.27639i | −1.12441 | − | 1.31746i | −3.18194 | 0.717144 | − | 1.24213i | 2.99905 | − | 2.55959i | 0 | − | 2.69056i | −0.471410 | + | 2.96273i | 2.82757 | + | 1.63250i | ||||||||||||||||||||||||||||||||||||||||||||
| 227.6 | 2.27639i | 1.12441 | + | 1.31746i | −3.18194 | −0.717144 | + | 1.24213i | −2.99905 | + | 2.55959i | 0 | − | 2.69056i | −0.471410 | + | 2.96273i | −2.82757 | − | 1.63250i | ||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.b | odd | 2 | 1 | inner |
| 63.i | even | 6 | 1 | inner |
| 63.j | odd | 6 | 1 | inner |
Twists
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{6} + 7T_{2}^{4} + 10T_{2}^{2} + 3 \)
acting on \(S_{2}^{\mathrm{new}}(441, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{6} + 7 T^{4} + 10 T^{2} + 3)^{2} \)
|
| $3$ |
\( T^{12} - 6 T^{10} + \cdots + 729 \)
|
| $5$ |
\( T^{12} + 15 T^{10} + \cdots + 6561 \)
|
| $7$ |
\( T^{12} \)
|
| $11$ |
\( (T^{6} - 11 T^{4} + 121 T^{2} + \cdots + 3)^{2} \)
|
| $13$ |
\( T^{12} - 45 T^{10} + \cdots + 1750329 \)
|
| $17$ |
\( T^{12} + 54 T^{10} + \cdots + 531441 \)
|
| $19$ |
\( T^{12} - 63 T^{10} + \cdots + 4782969 \)
|
| $23$ |
\( (T^{6} + 12 T^{5} + \cdots + 27)^{2} \)
|
| $29$ |
\( (T^{6} - 15 T^{5} + \cdots + 243)^{2} \)
|
| $31$ |
\( (T^{6} + 129 T^{4} + \cdots + 27)^{2} \)
|
| $37$ |
\( (T^{6} - T^{5} + 5 T^{4} + \cdots + 1)^{2} \)
|
| $41$ |
\( T^{12} + 72 T^{10} + \cdots + 531441 \)
|
| $43$ |
\( (T^{6} + 5 T^{5} + \cdots + 38809)^{2} \)
|
| $47$ |
\( (T^{6} - 93 T^{4} + \cdots - 81)^{2} \)
|
| $53$ |
\( (T^{6} + 6 T^{5} + \cdots + 20667)^{2} \)
|
| $59$ |
\( (T^{6} - 219 T^{4} + \cdots - 136161)^{2} \)
|
| $61$ |
\( (T^{6} + 24 T^{4} + \cdots + 243)^{2} \)
|
| $67$ |
\( (T^{3} + 6 T^{2} - 15 T + 7)^{4} \)
|
| $71$ |
\( (T^{6} + 163 T^{4} + \cdots + 363)^{2} \)
|
| $73$ |
\( T^{12} - 279 T^{10} + \cdots + 4782969 \)
|
| $79$ |
\( (T^{3} - 3 T^{2} - 24 T + 79)^{4} \)
|
| $83$ |
\( T^{12} + \cdots + 132211504881 \)
|
| $89$ |
\( T^{12} + \cdots + 282429536481 \)
|
| $97$ |
\( T^{12} - 69 T^{10} + \cdots + 10673289 \)
|
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