Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1008,2,Mod(209,1008)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1008, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 0, 1, 3]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1008.209");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1008.cc (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: | \(8.04892052375\) |
| Analytic rank: | \(0\) |
| Dimension: | \(16\) |
| Relative dimension: | \(8\) over \(\Q(\zeta_{6})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{16} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{16} - 6x^{14} + 9x^{12} + 54x^{10} - 288x^{8} + 486x^{6} + 729x^{4} - 4374x^{2} + 6561 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 3^{4} \) |
| Twist minimal: | no (minimal twist has level 126) |
| 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 \(1,\beta_1,\ldots,\beta_{15}\) 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
\( x^{16} - 6x^{14} + 9x^{12} + 54x^{10} - 288x^{8} + 486x^{6} + 729x^{4} - 4374x^{2} + 6561 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{14} - 6\nu^{12} + 36\nu^{10} - 108\nu^{8} - 288\nu^{6} + 486\nu^{4} - 1215\nu^{2} ) / 5832 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 2\nu^{14} - 21\nu^{12} + 18\nu^{10} + 108\nu^{8} - 576\nu^{6} + 648\nu^{4} + 972\nu^{2} - 3645 ) / 5832 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 2\nu^{15} + 9\nu^{13} - 18\nu^{11} + 396\nu^{7} - 216\nu^{5} + 324\nu^{3} + 9477\nu ) / 5832 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 5 \nu^{15} - 3 \nu^{14} - 12 \nu^{13} - 36 \nu^{12} - 90 \nu^{11} + 216 \nu^{10} + 594 \nu^{9} + \cdots - 8748 ) / 17496 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -5\nu^{15} + 12\nu^{13} + 144\nu^{11} - 432\nu^{9} + 468\nu^{7} + 2754\nu^{5} - 9477\nu^{3} + 13122\nu ) / 17496 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 2 \nu^{15} + 6 \nu^{14} - 12 \nu^{13} + 18 \nu^{12} - 9 \nu^{11} - 27 \nu^{10} + 270 \nu^{9} + \cdots + 26244 ) / 8748 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 5 \nu^{15} + 3 \nu^{14} - 12 \nu^{13} + 36 \nu^{12} - 90 \nu^{11} - 216 \nu^{10} + 594 \nu^{9} + \cdots + 8748 ) / 17496 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 5 \nu^{15} + 9 \nu^{14} + 12 \nu^{13} - 27 \nu^{12} + 90 \nu^{11} - 162 \nu^{10} - 594 \nu^{9} + \cdots - 19683 ) / 17496 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( \nu^{15} - 6\nu^{13} + 9\nu^{11} + 54\nu^{9} - 288\nu^{7} + 486\nu^{5} + 729\nu^{3} - 4374\nu ) / 2187 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( \nu^{15} - 21 \nu^{14} + 12 \nu^{13} + 153 \nu^{12} - 72 \nu^{11} - 108 \nu^{10} + 54 \nu^{9} + \cdots + 111537 ) / 17496 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 13\nu^{14} - 42\nu^{12} - 72\nu^{10} + 864\nu^{8} - 1800\nu^{6} - 1134\nu^{4} + 16281\nu^{2} - 26244 ) / 5832 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( -\nu^{15} + 3\nu^{13} + 9\nu^{11} - 81\nu^{9} + 126\nu^{7} + 135\nu^{5} - 1458\nu^{3} + 2187\nu ) / 1458 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( -\nu^{15} + 6\nu^{11} - 36\nu^{9} + 18\nu^{7} + 108\nu^{5} - 513\nu^{3} ) / 972 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( - 2 \nu^{15} + 27 \nu^{14} + 12 \nu^{13} - 81 \nu^{12} + 9 \nu^{11} - 81 \nu^{10} - 270 \nu^{9} + \cdots - 37179 ) / 8748 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( -3\nu^{15} + 10\nu^{13} + 12\nu^{11} - 180\nu^{9} + 432\nu^{7} + 198\nu^{5} - 3483\nu^{3} + 5832\nu ) / 1944 \)
|
| \(\nu\) | \(=\) |
\( ( - \beta_{15} + \beta_{13} + 2 \beta_{12} - 2 \beta_{11} - 2 \beta_{10} - \beta_{9} + 2 \beta_{8} + \cdots - 2 \beta_1 ) / 3 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{8} + \beta_{4} - \beta_{2} + 1 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{15} + 2\beta_{13} + \beta_{11} + \beta_{10} - \beta_{8} - \beta_{6} + \beta_{5} + 2\beta_{3} + \beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( 3\beta_{14} - 5\beta_{11} - 2\beta_{10} + 3\beta_{8} + \beta_{7} + \beta_{6} + 4\beta_{4} - 3\beta_{2} - 7\beta _1 + 3 \)
|
| \(\nu^{5}\) | \(=\) |
\( 3\beta_{15} + 3\beta_{13} - 3\beta_{12} + 6\beta_{9} - 3\beta_{5} + 6\beta_{3} \)
|
| \(\nu^{6}\) | \(=\) |
\( 3\beta_{10} + 3\beta_{8} - 3\beta_{7} + 3\beta_{6} + 3\beta_{4} + 6\beta_{2} - 6\beta _1 - 18 \)
|
| \(\nu^{7}\) | \(=\) |
\( 6 \beta_{15} + 6 \beta_{13} - 33 \beta_{12} + 18 \beta_{11} + 18 \beta_{10} + 6 \beta_{9} + \cdots + 18 \beta_1 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 6 \beta_{14} + 18 \beta_{11} + 3 \beta_{10} - 30 \beta_{8} - 15 \beta_{7} - 3 \beta_{6} - 18 \beta_{4} + \cdots + 12 \)
|
| \(\nu^{9}\) | \(=\) |
\( 30 \beta_{15} - 21 \beta_{13} - 69 \beta_{12} - 21 \beta_{11} - 21 \beta_{10} - 15 \beta_{9} + \cdots - 21 \beta_1 \)
|
| \(\nu^{10}\) | \(=\) |
\( - 63 \beta_{14} + 135 \beta_{11} + 45 \beta_{10} - 108 \beta_{8} - 90 \beta_{7} - 18 \beta_{6} + \cdots - 18 \)
|
| \(\nu^{11}\) | \(=\) |
\( - 54 \beta_{15} - 36 \beta_{13} - 90 \beta_{12} + 54 \beta_{11} + 54 \beta_{10} - 117 \beta_{9} + \cdots + 54 \beta_1 \)
|
| \(\nu^{12}\) | \(=\) |
\( 54 \beta_{14} + 18 \beta_{11} - 90 \beta_{10} - 162 \beta_{8} - 36 \beta_{7} - 36 \beta_{6} - 36 \beta_{4} + \cdots + 405 \)
|
| \(\nu^{13}\) | \(=\) |
\( 189 \beta_{15} - 459 \beta_{13} + 270 \beta_{12} - 216 \beta_{11} - 216 \beta_{10} - 135 \beta_{9} + \cdots - 216 \beta_1 \)
|
| \(\nu^{14}\) | \(=\) |
\( 486 \beta_{14} - 378 \beta_{11} + 297 \beta_{8} + 54 \beta_{7} + 486 \beta_{6} + 243 \beta_{4} + \cdots - 1053 \)
|
| \(\nu^{15}\) | \(=\) |
\( - 459 \beta_{15} - 1026 \beta_{13} + 1026 \beta_{12} + 891 \beta_{11} + 891 \beta_{10} + \cdots + 891 \beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1008\mathbb{Z}\right)^\times\).
| \(n\) | \(127\) | \(577\) | \(757\) | \(785\) |
| \(\chi(n)\) | \(1\) | \(-1\) | \(1\) | \(1 - \beta_{2}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 209.1 |
|
0 | −1.69547 | + | 0.354107i | 0 | −0.895175 | − | 1.55049i | 0 | 2.30191 | − | 1.30430i | 0 | 2.74922 | − | 1.20075i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 209.2 | 0 | −1.62181 | − | 0.608059i | 0 | −1.94556 | − | 3.36980i | 0 | −2.09985 | + | 1.60954i | 0 | 2.26053 | + | 1.97231i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 209.3 | 0 | −1.40917 | + | 1.00709i | 0 | 1.17468 | + | 2.03460i | 0 | −1.55364 | − | 2.14154i | 0 | 0.971521 | − | 2.83834i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 209.4 | 0 | −0.0967785 | − | 1.72934i | 0 | 0.183299 | + | 0.317483i | 0 | 0.624224 | + | 2.57106i | 0 | −2.98127 | + | 0.334727i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 209.5 | 0 | 0.0967785 | + | 1.72934i | 0 | −0.183299 | − | 0.317483i | 0 | −2.53871 | + | 0.744936i | 0 | −2.98127 | + | 0.334727i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 209.6 | 0 | 1.40917 | − | 1.00709i | 0 | −1.17468 | − | 2.03460i | 0 | 2.63145 | + | 0.274725i | 0 | 0.971521 | − | 2.83834i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 209.7 | 0 | 1.62181 | + | 0.608059i | 0 | 1.94556 | + | 3.36980i | 0 | −0.343982 | + | 2.62329i | 0 | 2.26053 | + | 1.97231i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 209.8 | 0 | 1.69547 | − | 0.354107i | 0 | 0.895175 | + | 1.55049i | 0 | −0.0213944 | − | 2.64566i | 0 | 2.74922 | − | 1.20075i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 545.1 | 0 | −1.69547 | − | 0.354107i | 0 | −0.895175 | + | 1.55049i | 0 | 2.30191 | + | 1.30430i | 0 | 2.74922 | + | 1.20075i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 545.2 | 0 | −1.62181 | + | 0.608059i | 0 | −1.94556 | + | 3.36980i | 0 | −2.09985 | − | 1.60954i | 0 | 2.26053 | − | 1.97231i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 545.3 | 0 | −1.40917 | − | 1.00709i | 0 | 1.17468 | − | 2.03460i | 0 | −1.55364 | + | 2.14154i | 0 | 0.971521 | + | 2.83834i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 545.4 | 0 | −0.0967785 | + | 1.72934i | 0 | 0.183299 | − | 0.317483i | 0 | 0.624224 | − | 2.57106i | 0 | −2.98127 | − | 0.334727i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 545.5 | 0 | 0.0967785 | − | 1.72934i | 0 | −0.183299 | + | 0.317483i | 0 | −2.53871 | − | 0.744936i | 0 | −2.98127 | − | 0.334727i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 545.6 | 0 | 1.40917 | + | 1.00709i | 0 | −1.17468 | + | 2.03460i | 0 | 2.63145 | − | 0.274725i | 0 | 0.971521 | + | 2.83834i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 545.7 | 0 | 1.62181 | − | 0.608059i | 0 | 1.94556 | − | 3.36980i | 0 | −0.343982 | − | 2.62329i | 0 | 2.26053 | − | 1.97231i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 545.8 | 0 | 1.69547 | + | 0.354107i | 0 | 0.895175 | − | 1.55049i | 0 | −0.0213944 | + | 2.64566i | 0 | 2.74922 | + | 1.20075i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.b | odd | 2 | 1 | inner |
| 9.d | odd | 6 | 1 | inner |
| 63.o | even | 6 | 1 | inner |
Twists
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{16} + 24T_{5}^{14} + 423T_{5}^{12} + 3096T_{5}^{10} + 16461T_{5}^{8} + 42336T_{5}^{6} + 77436T_{5}^{4} + 10368T_{5}^{2} + 1296 \)
acting on \(S_{2}^{\mathrm{new}}(1008, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{16} \)
|
| $3$ |
\( T^{16} - 6 T^{14} + \cdots + 6561 \)
|
| $5$ |
\( T^{16} + 24 T^{14} + \cdots + 1296 \)
|
| $7$ |
\( T^{16} + 2 T^{15} + \cdots + 5764801 \)
|
| $11$ |
\( (T^{8} - 6 T^{7} + \cdots + 1296)^{2} \)
|
| $13$ |
\( T^{16} - 36 T^{14} + \cdots + 331776 \)
|
| $17$ |
\( (T^{8} - 42 T^{6} + \cdots + 576)^{2} \)
|
| $19$ |
\( (T^{8} + 54 T^{6} + \cdots + 1521)^{2} \)
|
| $23$ |
\( (T^{8} - 24 T^{7} + \cdots + 443556)^{2} \)
|
| $29$ |
\( (T^{8} + 6 T^{7} + \cdots + 20736)^{2} \)
|
| $31$ |
\( T^{16} + \cdots + 557256278016 \)
|
| $37$ |
\( (T^{4} + 2 T^{3} + \cdots + 1336)^{4} \)
|
| $41$ |
\( T^{16} + \cdots + 73499483897856 \)
|
| $43$ |
\( (T^{8} + 2 T^{7} + \cdots + 10816)^{2} \)
|
| $47$ |
\( T^{16} + \cdots + 1485512441856 \)
|
| $53$ |
\( T^{16} \)
|
| $59$ |
\( T^{16} + 294 T^{14} + \cdots + 1296 \)
|
| $61$ |
\( T^{16} + \cdots + 2425818710016 \)
|
| $67$ |
\( (T^{8} - 14 T^{7} + \cdots + 824464)^{2} \)
|
| $71$ |
\( (T^{8} + 90 T^{6} + \cdots + 82944)^{2} \)
|
| $73$ |
\( (T^{8} + 222 T^{6} + \cdots + 1710864)^{2} \)
|
| $79$ |
\( (T^{8} - 2 T^{7} + \cdots + 1444804)^{2} \)
|
| $83$ |
\( T^{16} + \cdots + 33\!\cdots\!56 \)
|
| $89$ |
\( (T^{8} - 216 T^{6} + \cdots + 186624)^{2} \)
|
| $97$ |
\( T^{16} + \cdots + 45\!\cdots\!56 \)
|
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