Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [882,2,Mod(227,882)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(882, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([1, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("882.227");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 882 = 2 \cdot 3^{2} \cdot 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 882.l (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: | \(7.04280545828\) |
| 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: |
\( x^{16} - 8 x^{15} + 23 x^{14} - 8 x^{13} - 131 x^{12} + 380 x^{11} - 289 x^{10} - 880 x^{9} + \cdots + 6561 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 3^{2} \) |
| 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
\( x^{16} - 8 x^{15} + 23 x^{14} - 8 x^{13} - 131 x^{12} + 380 x^{11} - 289 x^{10} - 880 x^{9} + \cdots + 6561 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( - 50 \nu^{15} - 1352 \nu^{14} + 6827 \nu^{13} - 7676 \nu^{12} - 27422 \nu^{11} + 107246 \nu^{10} + \cdots - 2825604 ) / 142155 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 1292 \nu^{15} - 10486 \nu^{14} + 25660 \nu^{13} + 10145 \nu^{12} - 192280 \nu^{11} + 408694 \nu^{10} + \cdots - 9270693 ) / 142155 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 2846 \nu^{15} - 22369 \nu^{14} + 55246 \nu^{13} + 17972 \nu^{12} - 402586 \nu^{11} + \cdots - 20783061 ) / 142155 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 2782 \nu^{15} + 15918 \nu^{14} - 26947 \nu^{13} - 42629 \nu^{12} + 270897 \nu^{11} + \cdots + 7405182 ) / 47385 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 3648 \nu^{15} + 23120 \nu^{14} - 45376 \nu^{13} - 47012 \nu^{12} + 401996 \nu^{11} + \cdots + 14281839 ) / 47385 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 16948 \nu^{15} + 107120 \nu^{14} - 210206 \nu^{13} - 216202 \nu^{12} + 1856546 \nu^{11} + \cdots + 66620394 ) / 142155 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 4120 \nu^{15} + 25571 \nu^{14} - 48788 \nu^{13} - 55006 \nu^{12} + 441224 \nu^{11} + \cdots + 14935023 ) / 28431 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 24706 \nu^{15} - 155855 \nu^{14} + 305147 \nu^{13} + 316579 \nu^{12} - 2700647 \nu^{11} + \cdots - 96324228 ) / 142155 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 2015 \nu^{15} - 12538 \nu^{14} + 24088 \nu^{13} + 26576 \nu^{12} - 216643 \nu^{11} + 381184 \nu^{10} + \cdots - 7453296 ) / 10935 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 26777 \nu^{15} + 172141 \nu^{14} - 345205 \nu^{13} - 329150 \nu^{12} + 2992915 \nu^{11} + \cdots + 112005018 ) / 142155 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 32006 \nu^{15} - 201190 \nu^{14} + 390892 \nu^{13} + 415394 \nu^{12} - 3479542 \nu^{11} + \cdots - 121376313 ) / 142155 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( - 10990 \nu^{15} + 68842 \nu^{14} - 133062 \nu^{13} - 143724 \nu^{12} + 1190062 \nu^{11} + \cdots + 41433444 ) / 47385 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( - 35897 \nu^{15} + 219337 \nu^{14} - 409846 \nu^{13} - 492632 \nu^{12} + 3774151 \nu^{11} + \cdots + 122931270 ) / 142155 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( - 14807 \nu^{15} + 92828 \nu^{14} - 179757 \nu^{13} - 193044 \nu^{12} + 1605257 \nu^{11} + \cdots + 55960227 ) / 47385 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( - 68699 \nu^{15} + 433198 \nu^{14} - 844756 \nu^{13} - 889547 \nu^{12} + 7506256 \nu^{11} + \cdots + 264515463 ) / 142155 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{15} - \beta_{9} - \beta_{8} + \beta_{7} + \beta_{6} - 2\beta_{4} + 2\beta_{3} - 2\beta_{2} + \beta _1 + 2 ) / 3 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( - \beta_{15} + 4 \beta_{14} + \beta_{13} - 2 \beta_{12} + 2 \beta_{11} - \beta_{10} - \beta_{9} + \cdots + 3 ) / 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( - 3 \beta_{15} - \beta_{14} + 5 \beta_{13} + 2 \beta_{12} + \beta_{11} + \beta_{10} - 3 \beta_{8} + \cdots - 4 ) / 3 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( - \beta_{15} + 2 \beta_{14} + 11 \beta_{13} - 4 \beta_{12} + 7 \beta_{11} + 4 \beta_{10} + 8 \beta_{9} + \cdots + 7 ) / 3 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( - 11 \beta_{15} - 6 \beta_{14} + 9 \beta_{13} + 12 \beta_{11} + 6 \beta_{10} - 2 \beta_{9} + \beta_{8} + \cdots + 10 ) / 3 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( - 5 \beta_{15} + 26 \beta_{14} + 2 \beta_{13} - 31 \beta_{12} + 28 \beta_{11} + 13 \beta_{10} + \cdots + 15 ) / 3 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( - 15 \beta_{15} + 16 \beta_{14} - 20 \beta_{13} + 16 \beta_{12} + 29 \beta_{11} - 13 \beta_{10} + \cdots - 11 ) / 3 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( 43 \beta_{15} + 28 \beta_{14} - 8 \beta_{13} + 10 \beta_{12} + 44 \beta_{11} - 16 \beta_{10} + 25 \beta_{9} + \cdots - 1 ) / 3 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( 50 \beta_{15} - 12 \beta_{14} - 18 \beta_{13} + 102 \beta_{12} + 102 \beta_{11} - 144 \beta_{10} + \cdots + 134 ) / 3 \)
|
| \(\nu^{10}\) | \(=\) |
\( ( 140 \beta_{15} + 139 \beta_{14} + 31 \beta_{13} - 224 \beta_{12} + 95 \beta_{11} - 46 \beta_{10} + \cdots - 109 \beta_1 ) / 3 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( 60 \beta_{15} + 503 \beta_{14} + 23 \beta_{13} - 313 \beta_{12} + 133 \beta_{11} - 218 \beta_{10} + \cdots - 193 ) / 3 \)
|
| \(\nu^{12}\) | \(=\) |
\( ( - 94 \beta_{15} + 722 \beta_{14} + 191 \beta_{13} - 343 \beta_{12} - 182 \beta_{11} + 238 \beta_{10} + \cdots - 1112 ) / 3 \)
|
| \(\nu^{13}\) | \(=\) |
\( ( - 425 \beta_{15} + 1353 \beta_{14} + 255 \beta_{13} + 1086 \beta_{12} + 1068 \beta_{11} - 588 \beta_{10} + \cdots + 727 ) / 3 \)
|
| \(\nu^{14}\) | \(=\) |
\( ( - 953 \beta_{15} + 446 \beta_{14} + 35 \beta_{13} + 2231 \beta_{12} + 1294 \beta_{11} + 82 \beta_{10} + \cdots + 1896 ) / 3 \)
|
| \(\nu^{15}\) | \(=\) |
\( ( 1563 \beta_{15} + 1963 \beta_{14} - 725 \beta_{13} + 2308 \beta_{12} + 3086 \beta_{11} - 1306 \beta_{10} + \cdots + 6694 ) / 3 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/882\mathbb{Z}\right)^\times\).
| \(n\) | \(199\) | \(785\) |
| \(\chi(n)\) | \(1 + \beta_{7}\) | \(1 + \beta_{7}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 227.1 |
|
− | 1.00000i | −1.64774 | + | 0.533822i | −1.00000 | 0.450129 | − | 0.779646i | 0.533822 | + | 1.64774i | 0 | 1.00000i | 2.43007 | − | 1.75919i | −0.779646 | − | 0.450129i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 227.2 | − | 1.00000i | 0.734581 | + | 1.56856i | −1.00000 | −0.483662 | + | 0.837727i | 1.56856 | − | 0.734581i | 0 | 1.00000i | −1.92078 | + | 2.30447i | 0.837727 | + | 0.483662i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 227.3 | − | 1.00000i | 1.08509 | − | 1.35003i | −1.00000 | −1.77612 | + | 3.07634i | −1.35003 | − | 1.08509i | 0 | 1.00000i | −0.645160 | − | 2.92981i | 3.07634 | + | 1.77612i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 227.4 | − | 1.00000i | 1.56012 | − | 0.752355i | −1.00000 | 1.80966 | − | 3.13442i | −0.752355 | − | 1.56012i | 0 | 1.00000i | 1.86792 | − | 2.34752i | −3.13442 | − | 1.80966i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 227.5 | 1.00000i | −1.52765 | − | 0.816261i | −1.00000 | 1.82207 | − | 3.15592i | 0.816261 | − | 1.52765i | 0 | − | 1.00000i | 1.66744 | + | 2.49392i | 3.15592 | + | 1.82207i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 227.6 | 1.00000i | −1.38631 | + | 1.03834i | −1.00000 | −0.714925 | + | 1.23829i | −1.03834 | − | 1.38631i | 0 | − | 1.00000i | 0.843698 | − | 2.87892i | −1.23829 | − | 0.714925i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 227.7 | 1.00000i | 0.290993 | − | 1.70743i | −1.00000 | 0.0338034 | − | 0.0585493i | 1.70743 | + | 0.290993i | 0 | − | 1.00000i | −2.83065 | − | 0.993700i | 0.0585493 | + | 0.0338034i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 227.8 | 1.00000i | 0.890915 | + | 1.48535i | −1.00000 | −1.14095 | + | 1.97618i | −1.48535 | + | 0.890915i | 0 | − | 1.00000i | −1.41254 | + | 2.64665i | −1.97618 | − | 1.14095i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 509.1 | − | 1.00000i | −1.52765 | + | 0.816261i | −1.00000 | 1.82207 | + | 3.15592i | 0.816261 | + | 1.52765i | 0 | 1.00000i | 1.66744 | − | 2.49392i | 3.15592 | − | 1.82207i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 509.2 | − | 1.00000i | −1.38631 | − | 1.03834i | −1.00000 | −0.714925 | − | 1.23829i | −1.03834 | + | 1.38631i | 0 | 1.00000i | 0.843698 | + | 2.87892i | −1.23829 | + | 0.714925i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 509.3 | − | 1.00000i | 0.290993 | + | 1.70743i | −1.00000 | 0.0338034 | + | 0.0585493i | 1.70743 | − | 0.290993i | 0 | 1.00000i | −2.83065 | + | 0.993700i | 0.0585493 | − | 0.0338034i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 509.4 | − | 1.00000i | 0.890915 | − | 1.48535i | −1.00000 | −1.14095 | − | 1.97618i | −1.48535 | − | 0.890915i | 0 | 1.00000i | −1.41254 | − | 2.64665i | −1.97618 | + | 1.14095i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 509.5 | 1.00000i | −1.64774 | − | 0.533822i | −1.00000 | 0.450129 | + | 0.779646i | 0.533822 | − | 1.64774i | 0 | − | 1.00000i | 2.43007 | + | 1.75919i | −0.779646 | + | 0.450129i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 509.6 | 1.00000i | 0.734581 | − | 1.56856i | −1.00000 | −0.483662 | − | 0.837727i | 1.56856 | + | 0.734581i | 0 | − | 1.00000i | −1.92078 | − | 2.30447i | 0.837727 | − | 0.483662i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 509.7 | 1.00000i | 1.08509 | + | 1.35003i | −1.00000 | −1.77612 | − | 3.07634i | −1.35003 | + | 1.08509i | 0 | − | 1.00000i | −0.645160 | + | 2.92981i | 3.07634 | − | 1.77612i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 509.8 | 1.00000i | 1.56012 | + | 0.752355i | −1.00000 | 1.80966 | + | 3.13442i | −0.752355 | + | 1.56012i | 0 | − | 1.00000i | 1.86792 | + | 2.34752i | −3.13442 | + | 1.80966i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 63.i | even | 6 | 1 | inner |
Twists
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{16} + 24 T_{5}^{14} + 24 T_{5}^{13} + 423 T_{5}^{12} + 450 T_{5}^{11} + 3582 T_{5}^{10} + \cdots + 81 \)
acting on \(S_{2}^{\mathrm{new}}(882, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 1)^{8} \)
|
| $3$ |
\( T^{16} + 12 T^{13} + \cdots + 6561 \)
|
| $5$ |
\( T^{16} + 24 T^{14} + \cdots + 81 \)
|
| $7$ |
\( T^{16} \)
|
| $11$ |
\( T^{16} - 12 T^{15} + \cdots + 61732449 \)
|
| $13$ |
\( T^{16} + \cdots + 390971529 \)
|
| $17$ |
\( T^{16} - 18 T^{15} + \cdots + 56070144 \)
|
| $19$ |
\( T^{16} - 72 T^{14} + \cdots + 9199089 \)
|
| $23$ |
\( T^{16} + \cdots + 187388721 \)
|
| $29$ |
\( T^{16} - 6 T^{15} + \cdots + 1108809 \)
|
| $31$ |
\( T^{16} + 204 T^{14} + \cdots + 65610000 \)
|
| $37$ |
\( T^{16} + \cdots + 32746159681 \)
|
| $41$ |
\( T^{16} - 6 T^{15} + \cdots + 81 \)
|
| $43$ |
\( T^{16} + \cdots + 2999643361 \)
|
| $47$ |
\( (T^{8} - 18 T^{7} + \cdots + 766944)^{2} \)
|
| $53$ |
\( T^{16} + \cdots + 36759242529 \)
|
| $59$ |
\( (T^{8} + 30 T^{7} + \cdots + 465300)^{2} \)
|
| $61$ |
\( T^{16} + \cdots + 547560000 \)
|
| $67$ |
\( (T^{8} + 14 T^{7} + \cdots + 51028)^{2} \)
|
| $71$ |
\( T^{16} + 486 T^{14} + \cdots + 65610000 \)
|
| $73$ |
\( T^{16} - 150 T^{14} + \cdots + 71115489 \)
|
| $79$ |
\( (T^{8} - 16 T^{7} + \cdots - 985100)^{2} \)
|
| $83$ |
\( T^{16} + \cdots + 953512641 \)
|
| $89$ |
\( T^{16} + \cdots + 131145120363321 \)
|
| $97$ |
\( T^{16} + \cdots + 9120206721024 \)
|
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