Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1911,2,Mod(883,1911)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1911, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1911.883");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1911 = 3 \cdot 7^{2} \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1911.c (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: | \(15.2594118263\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | 8.0.43134305344.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} + 11x^{6} + 36x^{4} + 43x^{2} + 16 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 273) |
| 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
\( x^{8} + 11x^{6} + 36x^{4} + 43x^{2} + 16 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} + 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{7} - 9\nu^{5} - 18\nu^{3} - 7\nu ) / 2 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{7} + 4\nu^{6} + 11\nu^{5} + 36\nu^{4} + 36\nu^{3} + 72\nu^{2} + 35\nu + 28 ) / 8 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -\nu^{7} + 4\nu^{6} - 11\nu^{5} + 36\nu^{4} - 36\nu^{3} + 72\nu^{2} - 35\nu + 28 ) / 8 \)
|
| \(\beta_{6}\) | \(=\) |
\( \nu^{6} + 10\nu^{4} + 26\nu^{2} + 17 \)
|
| \(\beta_{7}\) | \(=\) |
\( -\nu^{7} - 10\nu^{5} - 26\nu^{3} - 17\nu \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} - 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{7} - 2\beta_{5} + 2\beta_{4} - \beta_{3} - 4\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{6} - \beta_{5} - \beta_{4} - 8\beta_{2} + 14 \)
|
| \(\nu^{5}\) | \(=\) |
\( -9\beta_{7} + 16\beta_{5} - 16\beta_{4} + 10\beta_{3} + 22\beta_1 \)
|
| \(\nu^{6}\) | \(=\) |
\( -9\beta_{6} + 10\beta_{5} + 10\beta_{4} + 54\beta_{2} - 79 \)
|
| \(\nu^{7}\) | \(=\) |
\( 63\beta_{7} - 108\beta_{5} + 108\beta_{4} - 74\beta_{3} - 133\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1911\mathbb{Z}\right)^\times\).
| \(n\) | \(638\) | \(1471\) | \(1522\) |
| \(\chi(n)\) | \(1\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 883.1 |
|
− | 2.72457i | −1.00000 | −5.42329 | 2.12682i | 2.72457i | 0 | 9.32701i | 1.00000 | 5.79467 | |||||||||||||||||||||||||||||||||||||||||
| 883.2 | − | 1.95497i | −1.00000 | −1.82190 | 0.150750i | 1.95497i | 0 | − | 0.348171i | 1.00000 | 0.294712 | |||||||||||||||||||||||||||||||||||||||||
| 883.3 | − | 0.644503i | −1.00000 | 1.58462 | 4.28149i | 0.644503i | 0 | − | 2.31030i | 1.00000 | 2.75944 | |||||||||||||||||||||||||||||||||||||||||
| 883.4 | − | 0.582595i | −1.00000 | 1.66058 | − | 1.45696i | 0.582595i | 0 | − | 2.13264i | 1.00000 | −0.848816 | ||||||||||||||||||||||||||||||||||||||||
| 883.5 | 0.582595i | −1.00000 | 1.66058 | 1.45696i | − | 0.582595i | 0 | 2.13264i | 1.00000 | −0.848816 | ||||||||||||||||||||||||||||||||||||||||||
| 883.6 | 0.644503i | −1.00000 | 1.58462 | − | 4.28149i | − | 0.644503i | 0 | 2.31030i | 1.00000 | 2.75944 | |||||||||||||||||||||||||||||||||||||||||
| 883.7 | 1.95497i | −1.00000 | −1.82190 | − | 0.150750i | − | 1.95497i | 0 | 0.348171i | 1.00000 | 0.294712 | |||||||||||||||||||||||||||||||||||||||||
| 883.8 | 2.72457i | −1.00000 | −5.42329 | − | 2.12682i | − | 2.72457i | 0 | − | 9.32701i | 1.00000 | 5.79467 | ||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 13.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 1911.2.c.j | 8 | |
| 7.b | odd | 2 | 1 | 1911.2.c.m | 8 | ||
| 7.c | even | 3 | 2 | 273.2.bj.d | ✓ | 16 | |
| 13.b | even | 2 | 1 | inner | 1911.2.c.j | 8 | |
| 21.h | odd | 6 | 2 | 819.2.dl.g | 16 | ||
| 91.b | odd | 2 | 1 | 1911.2.c.m | 8 | ||
| 91.r | even | 6 | 2 | 273.2.bj.d | ✓ | 16 | |
| 273.w | odd | 6 | 2 | 819.2.dl.g | 16 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 273.2.bj.d | ✓ | 16 | 7.c | even | 3 | 2 | |
| 273.2.bj.d | ✓ | 16 | 91.r | even | 6 | 2 | |
| 819.2.dl.g | 16 | 21.h | odd | 6 | 2 | ||
| 819.2.dl.g | 16 | 273.w | odd | 6 | 2 | ||
| 1911.2.c.j | 8 | 1.a | even | 1 | 1 | trivial | |
| 1911.2.c.j | 8 | 13.b | even | 2 | 1 | inner | |
| 1911.2.c.m | 8 | 7.b | odd | 2 | 1 | ||
| 1911.2.c.m | 8 | 91.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1911, [\chi])\):
|
\( T_{2}^{8} + 12T_{2}^{6} + 37T_{2}^{4} + 23T_{2}^{2} + 4 \)
|
|
\( T_{5}^{8} + 25T_{5}^{6} + 132T_{5}^{4} + 179T_{5}^{2} + 4 \)
|
|
\( T_{17}^{4} - T_{17}^{3} - 22T_{17}^{2} - 23T_{17} - 4 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} + 12 T^{6} + \cdots + 4 \)
|
| $3$ |
\( (T + 1)^{8} \)
|
| $5$ |
\( T^{8} + 25 T^{6} + \cdots + 4 \)
|
| $7$ |
\( T^{8} \)
|
| $11$ |
\( T^{8} + 29 T^{6} + \cdots + 4 \)
|
| $13$ |
\( T^{8} - 3 T^{7} + \cdots + 28561 \)
|
| $17$ |
\( (T^{4} - T^{3} - 22 T^{2} + \cdots - 4)^{2} \)
|
| $19$ |
\( T^{8} + 112 T^{6} + \cdots + 247009 \)
|
| $23$ |
\( (T^{4} + 12 T^{3} + 27 T^{2} + \cdots + 8)^{2} \)
|
| $29$ |
\( (T^{4} - 6 T^{3} + \cdots - 104)^{2} \)
|
| $31$ |
\( T^{8} + 131 T^{6} + \cdots + 5929 \)
|
| $37$ |
\( T^{8} + 206 T^{6} + \cdots + 2307361 \)
|
| $41$ |
\( T^{8} + 212 T^{6} + \cdots + 649636 \)
|
| $43$ |
\( (T^{4} + 19 T^{3} + \cdots + 169)^{2} \)
|
| $47$ |
\( T^{8} + 160 T^{6} + \cdots + 30976 \)
|
| $53$ |
\( (T^{4} + 5 T^{3} + \cdots + 6652)^{2} \)
|
| $59$ |
\( T^{8} + 113 T^{6} + \cdots + 78400 \)
|
| $61$ |
\( (T^{4} + 13 T^{3} + \cdots - 6646)^{2} \)
|
| $67$ |
\( T^{8} + 452 T^{6} + \cdots + 34351321 \)
|
| $71$ |
\( T^{8} + 173 T^{6} + \cdots + 33124 \)
|
| $73$ |
\( T^{8} + 83 T^{6} + \cdots + 5041 \)
|
| $79$ |
\( (T^{4} - 5 T^{3} + \cdots + 10771)^{2} \)
|
| $83$ |
\( T^{8} + 232 T^{6} + \cdots + 4218916 \)
|
| $89$ |
\( T^{8} + 493 T^{6} + \cdots + 3136 \)
|
| $97$ |
\( T^{8} + 67 T^{6} + \cdots + 400 \)
|
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