Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [222,2,Mod(7,222)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(222, base_ring=CyclotomicField(18))
chi = DirichletCharacter(H, H._module([0, 16]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("222.7");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 222 = 2 \cdot 3 \cdot 37 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 222.k (of order \(9\), degree \(6\), 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: | \(1.77267892487\) |
| Analytic rank: | \(0\) |
| Dimension: | \(12\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{9})\) |
| Coefficient field: | 12.0.686339028913329.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{12} - 8x^{9} + 37x^{6} - 216x^{3} + 729 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 3 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{9}]$ |
$q$-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
\( x^{12} - 8x^{9} + 37x^{6} - 216x^{3} + 729 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 8\nu^{11} - 37\nu^{8} + 1295\nu^{5} - 729\nu^{2} ) / 17982 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 13\nu^{10} - 185\nu^{7} + 481\nu^{4} - 2808\nu ) / 5994 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -8\nu^{9} + 37\nu^{6} - 296\nu^{3} + 1728 ) / 999 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{11} - 253\nu^{2} ) / 666 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -3\nu^{10} - \nu^{9} - 37\nu^{6} - 37\nu^{3} + 759\nu + 216 ) / 666 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -13\nu^{11} + 9\nu^{9} + 185\nu^{8} + 333\nu^{6} - 481\nu^{5} + 333\nu^{3} + 2808\nu^{2} - 1944 ) / 5994 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 40\nu^{10} - 185\nu^{7} + 481\nu^{4} - 3645\nu ) / 5994 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( -16\nu^{10} - 3\nu^{9} + 74\nu^{7} - 111\nu^{6} - 592\nu^{4} - 111\nu^{3} + 1458\nu + 648 ) / 1998 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 16\nu^{10} - 3\nu^{9} - 74\nu^{7} - 111\nu^{6} + 592\nu^{4} - 111\nu^{3} - 3456\nu + 648 ) / 1998 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 2\nu^{11} - \nu^{9} - 37\nu^{6} - 37\nu^{3} + 160\nu^{2} + 216 ) / 666 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( -35\nu^{10} + 120\nu^{9} + 37\nu^{7} - 555\nu^{6} - 1295\nu^{4} + 1443\nu^{3} + 7560\nu - 10935 ) / 5994 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{9} - \beta_{8} + 2\beta_{7} + 2\beta_{5} - 2\beta_{2} ) / 3 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 3\beta_{10} - \beta_{9} - \beta_{8} - \beta_{7} - \beta_{5} - 6\beta_{4} + \beta_{2} ) / 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -6\beta_{11} - 4\beta_{9} + 2\beta_{8} + 2\beta_{7} + 2\beta_{5} - 15\beta_{3} + 4\beta_{2} + 15 ) / 3 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 5\beta_{9} - 10\beta_{8} - 13\beta_{7} + 5\beta_{5} - 5\beta_{2} ) / 3 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -3\beta_{10} + 2\beta_{9} + 2\beta_{8} + 2\beta_{7} + 3\beta_{6} + 2\beta_{5} - 3\beta_{4} - 2\beta_{2} + 45\beta_1 ) / 3 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( -16\beta_{9} - 16\beta_{8} - 16\beta_{7} - 16\beta_{5} + 9\beta_{3} + 16\beta_{2} ) / 3 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 26\beta_{9} - 13\beta_{8} - 13\beta_{7} - 13\beta_{5} - 131\beta_{2} ) / 3 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( 35\beta_{9} + 35\beta_{8} + 35\beta_{7} + 105\beta_{6} + 35\beta_{5} + 117\beta_{4} - 35\beta_{2} + 117\beta_1 ) / 3 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( 222\beta_{11} + 74\beta_{9} - 148\beta_{8} - 148\beta_{7} - 148\beta_{5} + 222\beta_{3} - 74\beta_{2} + 93 ) / 3 \)
|
| \(\nu^{10}\) | \(=\) |
\( ( -31\beta_{9} - 31\beta_{8} + 728\beta_{7} + 62\beta_{5} - 728\beta_{2} ) / 3 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( 759\beta_{10} - 253\beta_{9} - 253\beta_{8} - 253\beta_{7} - 253\beta_{5} + 480\beta_{4} + 253\beta_{2} ) / 3 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/222\mathbb{Z}\right)^\times\).
| \(n\) | \(149\) | \(187\) |
| \(\chi(n)\) | \(1\) | \(-\beta_{7}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 7.1 |
|
0.939693 | − | 0.342020i | 0.939693 | + | 0.342020i | 0.766044 | − | 0.642788i | −0.597330 | − | 3.38763i | 1.00000 | 0.176868 | + | 1.00307i | 0.500000 | − | 0.866025i | 0.766044 | + | 0.642788i | −1.71994 | − | 2.97903i | ||||||||||||||||||||||||||||||||||||||
| 7.2 | 0.939693 | − | 0.342020i | 0.939693 | + | 0.342020i | 0.766044 | − | 0.642788i | 0.537023 | + | 3.04561i | 1.00000 | −0.563528 | − | 3.19592i | 0.500000 | − | 0.866025i | 0.766044 | + | 0.642788i | 1.54629 | + | 2.67826i | |||||||||||||||||||||||||||||||||||||||
| 49.1 | −0.766044 | + | 0.642788i | −0.766044 | − | 0.642788i | 0.173648 | − | 0.984808i | −1.94896 | + | 0.709365i | 1.00000 | 1.46639 | − | 0.533723i | 0.500000 | + | 0.866025i | 0.173648 | + | 0.984808i | 1.03702 | − | 1.79618i | |||||||||||||||||||||||||||||||||||||||
| 49.2 | −0.766044 | + | 0.642788i | −0.766044 | − | 0.642788i | 0.173648 | − | 0.984808i | 0.182920 | − | 0.0665776i | 1.00000 | −4.67213 | + | 1.70052i | 0.500000 | + | 0.866025i | 0.173648 | + | 0.984808i | −0.0973299 | + | 0.168580i | |||||||||||||||||||||||||||||||||||||||
| 127.1 | 0.939693 | + | 0.342020i | 0.939693 | − | 0.342020i | 0.766044 | + | 0.642788i | −0.597330 | + | 3.38763i | 1.00000 | 0.176868 | − | 1.00307i | 0.500000 | + | 0.866025i | 0.766044 | − | 0.642788i | −1.71994 | + | 2.97903i | |||||||||||||||||||||||||||||||||||||||
| 127.2 | 0.939693 | + | 0.342020i | 0.939693 | − | 0.342020i | 0.766044 | + | 0.642788i | 0.537023 | − | 3.04561i | 1.00000 | −0.563528 | + | 3.19592i | 0.500000 | + | 0.866025i | 0.766044 | − | 0.642788i | 1.54629 | − | 2.67826i | |||||||||||||||||||||||||||||||||||||||
| 145.1 | −0.766044 | − | 0.642788i | −0.766044 | + | 0.642788i | 0.173648 | + | 0.984808i | −1.94896 | − | 0.709365i | 1.00000 | 1.46639 | + | 0.533723i | 0.500000 | − | 0.866025i | 0.173648 | − | 0.984808i | 1.03702 | + | 1.79618i | |||||||||||||||||||||||||||||||||||||||
| 145.2 | −0.766044 | − | 0.642788i | −0.766044 | + | 0.642788i | 0.173648 | + | 0.984808i | 0.182920 | + | 0.0665776i | 1.00000 | −4.67213 | − | 1.70052i | 0.500000 | − | 0.866025i | 0.173648 | − | 0.984808i | −0.0973299 | − | 0.168580i | |||||||||||||||||||||||||||||||||||||||
| 157.1 | −0.173648 | − | 0.984808i | −0.173648 | + | 0.984808i | −0.939693 | + | 0.342020i | −2.21994 | − | 1.86275i | 1.00000 | −1.32277 | − | 1.10993i | 0.500000 | + | 0.866025i | −0.939693 | − | 0.342020i | −1.44896 | + | 2.50968i | |||||||||||||||||||||||||||||||||||||||
| 157.2 | −0.173648 | − | 0.984808i | −0.173648 | + | 0.984808i | −0.939693 | + | 0.342020i | 1.04629 | + | 0.877946i | 1.00000 | 0.415163 | + | 0.348363i | 0.500000 | + | 0.866025i | −0.939693 | − | 0.342020i | 0.682920 | − | 1.18285i | |||||||||||||||||||||||||||||||||||||||
| 181.1 | −0.173648 | + | 0.984808i | −0.173648 | − | 0.984808i | −0.939693 | − | 0.342020i | −2.21994 | + | 1.86275i | 1.00000 | −1.32277 | + | 1.10993i | 0.500000 | − | 0.866025i | −0.939693 | + | 0.342020i | −1.44896 | − | 2.50968i | |||||||||||||||||||||||||||||||||||||||
| 181.2 | −0.173648 | + | 0.984808i | −0.173648 | − | 0.984808i | −0.939693 | − | 0.342020i | 1.04629 | − | 0.877946i | 1.00000 | 0.415163 | − | 0.348363i | 0.500000 | − | 0.866025i | −0.939693 | + | 0.342020i | 0.682920 | + | 1.18285i | |||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 37.f | even | 9 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 222.2.k.e | ✓ | 12 |
| 3.b | odd | 2 | 1 | 666.2.x.f | 12 | ||
| 37.f | even | 9 | 1 | inner | 222.2.k.e | ✓ | 12 |
| 37.f | even | 9 | 1 | 8214.2.a.bb | 6 | ||
| 37.h | even | 18 | 1 | 8214.2.a.bc | 6 | ||
| 111.p | odd | 18 | 1 | 666.2.x.f | 12 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 222.2.k.e | ✓ | 12 | 1.a | even | 1 | 1 | trivial |
| 222.2.k.e | ✓ | 12 | 37.f | even | 9 | 1 | inner |
| 666.2.x.f | 12 | 3.b | odd | 2 | 1 | ||
| 666.2.x.f | 12 | 111.p | odd | 18 | 1 | ||
| 8214.2.a.bb | 6 | 37.f | even | 9 | 1 | ||
| 8214.2.a.bc | 6 | 37.h | even | 18 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{12} + 6 T_{5}^{11} + 33 T_{5}^{10} + 118 T_{5}^{9} + 333 T_{5}^{8} + 666 T_{5}^{7} + 1089 T_{5}^{6} + \cdots + 289 \)
acting on \(S_{2}^{\mathrm{new}}(222, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{6} - T^{3} + 1)^{2} \)
|
| $3$ |
\( (T^{6} - T^{3} + 1)^{2} \)
|
| $5$ |
\( T^{12} + 6 T^{11} + \cdots + 289 \)
|
| $7$ |
\( T^{12} + 9 T^{11} + \cdots + 576 \)
|
| $11$ |
\( T^{12} + 51 T^{10} + \cdots + 262144 \)
|
| $13$ |
\( T^{12} + 12 T^{11} + \cdots + 12321 \)
|
| $17$ |
\( T^{12} - 9 T^{11} + \cdots + 14722569 \)
|
| $19$ |
\( T^{12} + 21 T^{11} + \cdots + 3474496 \)
|
| $23$ |
\( T^{12} + 63 T^{10} + \cdots + 46656 \)
|
| $29$ |
\( T^{12} + \cdots + 172108161 \)
|
| $31$ |
\( (T^{6} - 3 T^{5} + \cdots + 136)^{2} \)
|
| $37$ |
\( T^{12} + \cdots + 2565726409 \)
|
| $41$ |
\( T^{12} + \cdots + 7386371136 \)
|
| $43$ |
\( (T^{6} + 3 T^{5} - 78 T^{4} + \cdots + 64)^{2} \)
|
| $47$ |
\( T^{12} + \cdots + 5061468736 \)
|
| $53$ |
\( T^{12} + \cdots + 509359761 \)
|
| $59$ |
\( T^{12} + \cdots + 2109381184 \)
|
| $61$ |
\( T^{12} + \cdots + 16360712281 \)
|
| $67$ |
\( T^{12} - 6 T^{11} + \cdots + 7096896 \)
|
| $71$ |
\( T^{12} + 6 T^{11} + \cdots + 36864 \)
|
| $73$ |
\( (T^{6} + 27 T^{5} + \cdots - 90008)^{2} \)
|
| $79$ |
\( T^{12} + 12 T^{11} + \cdots + 788544 \)
|
| $83$ |
\( T^{12} + \cdots + 14424970816 \)
|
| $89$ |
\( T^{12} + \cdots + 37603275886801 \)
|
| $97$ |
\( T^{12} + \cdots + 8606087361 \)
|
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