Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1110,2,Mod(889,1110)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1110, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1110.889");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1110 = 2 \cdot 3 \cdot 5 \cdot 37 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1110.d (of order \(2\), degree \(1\), 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: | \(8.86339462436\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | 8.0.57815240704.2 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - 2x^{7} + 2x^{6} + 89x^{4} - 170x^{3} + 162x^{2} - 72x + 16 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} - 2x^{7} + 2x^{6} + 89x^{4} - 170x^{3} + 162x^{2} - 72x + 16 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 198\nu^{7} - 217\nu^{6} + 209\nu^{5} + 88\nu^{4} + 18711\nu^{3} - 17653\nu^{2} + 16897\nu - 7524 ) / 9085 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -1881\nu^{7} + 2970\nu^{6} - 2894\nu^{5} - 836\nu^{4} - 167761\nu^{3} + 244926\nu^{2} - 234110\nu + 67844 ) / 36340 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 1969\nu^{7} - 2057\nu^{6} + 968\nu^{5} + 2894\nu^{4} + 176077\nu^{3} - 166969\nu^{2} + 74052\nu + 56002 ) / 18170 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -358\nu^{7} + 374\nu^{6} - 176\nu^{5} - 361\nu^{4} - 32014\nu^{3} + 30358\nu^{2} - 13464\nu - 2749 ) / 1817 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 9117 \nu^{7} + 17838 \nu^{6} - 14166 \nu^{5} - 4052 \nu^{4} - 811589 \nu^{3} + 1516102 \nu^{2} + \cdots + 328276 ) / 36340 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 9261 \nu^{7} - 16344 \nu^{6} + 14318 \nu^{5} + 4116 \nu^{4} + 825197 \nu^{3} - 1388536 \nu^{2} + \cdots - 333748 ) / 36340 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -2\beta_{7} - \beta_{6} - 5\beta_{3} + \beta_1 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{7} - \beta_{6} + \beta_{5} + \beta_{4} + 2\beta_{3} + 9\beta_{2} + \beta _1 + 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( 11\beta_{5} + 20\beta_{4} - 45 \)
|
| \(\nu^{5}\) | \(=\) |
\( 11\beta_{7} + 11\beta_{6} + 11\beta_{5} + 11\beta_{4} - 18\beta_{3} - 96\beta _1 + 18 \)
|
| \(\nu^{6}\) | \(=\) |
\( 192\beta_{7} + 107\beta_{6} + 425\beta_{3} - 4\beta_{2} - 111\beta_1 \)
|
| \(\nu^{7}\) | \(=\) |
\( 115\beta_{7} + 111\beta_{6} - 111\beta_{5} - 115\beta_{4} - 150\beta_{3} - 809\beta_{2} - 111\beta _1 - 150 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1110\mathbb{Z}\right)^\times\).
| \(n\) | \(371\) | \(631\) | \(667\) |
| \(\chi(n)\) | \(1\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 889.1 |
|
− | 1.00000i | − | 1.00000i | −1.00000 | −2.20793 | − | 0.353624i | −1.00000 | − | 4.51003i | 1.00000i | −1.00000 | −0.353624 | + | 2.20793i | |||||||||||||||||||||||||||||||||||
| 889.2 | − | 1.00000i | − | 1.00000i | −1.00000 | −0.594137 | + | 2.15569i | −1.00000 | 1.17795i | 1.00000i | −1.00000 | 2.15569 | + | 0.594137i | |||||||||||||||||||||||||||||||||||||
| 889.3 | − | 1.00000i | − | 1.00000i | −1.00000 | −0.353624 | − | 2.20793i | −1.00000 | 3.94848i | 1.00000i | −1.00000 | −2.20793 | + | 0.353624i | |||||||||||||||||||||||||||||||||||||
| 889.4 | − | 1.00000i | − | 1.00000i | −1.00000 | 2.15569 | − | 0.594137i | −1.00000 | 2.38360i | 1.00000i | −1.00000 | −0.594137 | − | 2.15569i | |||||||||||||||||||||||||||||||||||||
| 889.5 | 1.00000i | 1.00000i | −1.00000 | −2.20793 | + | 0.353624i | −1.00000 | 4.51003i | − | 1.00000i | −1.00000 | −0.353624 | − | 2.20793i | ||||||||||||||||||||||||||||||||||||||
| 889.6 | 1.00000i | 1.00000i | −1.00000 | −0.594137 | − | 2.15569i | −1.00000 | − | 1.17795i | − | 1.00000i | −1.00000 | 2.15569 | − | 0.594137i | |||||||||||||||||||||||||||||||||||||
| 889.7 | 1.00000i | 1.00000i | −1.00000 | −0.353624 | + | 2.20793i | −1.00000 | − | 3.94848i | − | 1.00000i | −1.00000 | −2.20793 | − | 0.353624i | |||||||||||||||||||||||||||||||||||||
| 889.8 | 1.00000i | 1.00000i | −1.00000 | 2.15569 | + | 0.594137i | −1.00000 | − | 2.38360i | − | 1.00000i | −1.00000 | −0.594137 | + | 2.15569i | |||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 1110.2.d.j | ✓ | 8 |
| 3.b | odd | 2 | 1 | 3330.2.d.o | 8 | ||
| 5.b | even | 2 | 1 | inner | 1110.2.d.j | ✓ | 8 |
| 5.c | odd | 4 | 1 | 5550.2.a.cl | 4 | ||
| 5.c | odd | 4 | 1 | 5550.2.a.cm | 4 | ||
| 15.d | odd | 2 | 1 | 3330.2.d.o | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1110.2.d.j | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 1110.2.d.j | ✓ | 8 | 5.b | even | 2 | 1 | inner |
| 3330.2.d.o | 8 | 3.b | odd | 2 | 1 | ||
| 3330.2.d.o | 8 | 15.d | odd | 2 | 1 | ||
| 5550.2.a.cl | 4 | 5.c | odd | 4 | 1 | ||
| 5550.2.a.cm | 4 | 5.c | odd | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1110, [\chi])\):
|
\( T_{7}^{8} + 43T_{7}^{6} + 579T_{7}^{4} + 2525T_{7}^{2} + 2500 \)
|
|
\( T_{11}^{4} + 5T_{11}^{3} - 11T_{11}^{2} - 7T_{11} + 4 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 1)^{4} \)
|
| $3$ |
\( (T^{2} + 1)^{4} \)
|
| $5$ |
\( T^{8} + 2 T^{7} + \cdots + 625 \)
|
| $7$ |
\( T^{8} + 43 T^{6} + \cdots + 2500 \)
|
| $11$ |
\( (T^{4} + 5 T^{3} - 11 T^{2} + \cdots + 4)^{2} \)
|
| $13$ |
\( T^{8} + 26 T^{6} + \cdots + 256 \)
|
| $17$ |
\( T^{8} + 75 T^{6} + \cdots + 11236 \)
|
| $19$ |
\( (T^{4} - 12 T^{3} + \cdots + 128)^{2} \)
|
| $23$ |
\( (T^{4} + 81 T^{2} + 1296)^{2} \)
|
| $29$ |
\( (T^{4} + 5 T^{3} - 34 T^{2} + \cdots - 16)^{2} \)
|
| $31$ |
\( (T^{4} + 19 T^{3} + \cdots - 3328)^{2} \)
|
| $37$ |
\( (T^{2} + 1)^{4} \)
|
| $41$ |
\( (T^{4} - 13 T^{3} + \cdots + 172)^{2} \)
|
| $43$ |
\( T^{8} + 25 T^{6} + \cdots + 256 \)
|
| $47$ |
\( T^{8} + 208 T^{6} + \cdots + 719104 \)
|
| $53$ |
\( T^{8} + 335 T^{6} + \cdots + 1747684 \)
|
| $59$ |
\( (T^{4} + 10 T^{3} + \cdots - 128)^{2} \)
|
| $61$ |
\( (T^{4} + 3 T^{3} + \cdots + 1296)^{2} \)
|
| $67$ |
\( T^{8} + 568 T^{6} + \cdots + 11505664 \)
|
| $71$ |
\( (T^{4} + 6 T^{3} - 250 T^{2} + \cdots - 64)^{2} \)
|
| $73$ |
\( T^{8} + 278 T^{6} + \cdots + 11075584 \)
|
| $79$ |
\( (T^{4} - 8 T^{3} + \cdots - 1024)^{2} \)
|
| $83$ |
\( T^{8} + 518 T^{6} + \cdots + 163840000 \)
|
| $89$ |
\( (T^{4} + 32 T^{3} + \cdots + 1616)^{2} \)
|
| $97$ |
\( T^{8} + 337 T^{6} + \cdots + 839056 \)
|
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