Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1106,2,Mod(1,1106)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1106, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1106.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1106 = 2 \cdot 7 \cdot 79 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1106.a (trivial) |
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: | yes |
| Analytic conductor: | \(8.83145446355\) |
| Analytic rank: | \(0\) |
| Dimension: | \(9\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{9} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{9} - 2x^{8} - 18x^{7} + 34x^{6} + 105x^{5} - 184x^{4} - 212x^{3} + 342x^{2} + 72x - 108 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | yes |
| Fricke sign: | \(-1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-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 \(1,\beta_1,\ldots,\beta_{8}\) 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
\( x^{9} - 2x^{8} - 18x^{7} + 34x^{6} + 105x^{5} - 184x^{4} - 212x^{3} + 342x^{2} + 72x - 108 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{8} + \nu^{7} - 15\nu^{6} - 11\nu^{5} + 72\nu^{4} + 23\nu^{3} - 134\nu^{2} + 12\nu + 54 ) / 9 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{8} - 2\nu^{7} + 14\nu^{6} + 26\nu^{5} - 55\nu^{4} - 86\nu^{3} + 60\nu^{2} + 50\nu - 6 ) / 6 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -2\nu^{8} - 2\nu^{7} + 30\nu^{6} + 22\nu^{5} - 135\nu^{4} - 46\nu^{3} + 196\nu^{2} - 24\nu - 45 ) / 9 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -3\nu^{8} - 2\nu^{7} + 46\nu^{6} + 24\nu^{5} - 215\nu^{4} - 66\nu^{3} + 332\nu^{2} + 10\nu - 96 ) / 6 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 4\nu^{8} + 4\nu^{7} - 60\nu^{6} - 53\nu^{5} + 270\nu^{4} + 191\nu^{3} - 392\nu^{2} - 177\nu + 108 ) / 9 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( -7\nu^{8} - 6\nu^{7} + 112\nu^{6} + 74\nu^{5} - 563\nu^{4} - 218\nu^{3} + 976\nu^{2} + 70\nu - 318 ) / 6 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{7} + \beta_{6} - \beta_{5} + \beta_{4} + 7\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{5} + 2\beta_{3} + 8\beta_{2} + 25 \)
|
| \(\nu^{5}\) | \(=\) |
\( 10\beta_{7} + 11\beta_{6} - 13\beta_{5} + 11\beta_{4} + 52\beta _1 + 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( \beta_{8} - 2\beta_{6} + 13\beta_{5} - \beta_{4} + 26\beta_{3} + 62\beta_{2} + 177 \)
|
| \(\nu^{7}\) | \(=\) |
\( -\beta_{8} + 87\beta_{7} + 104\beta_{6} - 128\beta_{5} + 97\beta_{4} - \beta_{3} + 401\beta _1 + 30 \)
|
| \(\nu^{8}\) | \(=\) |
\( 16\beta_{8} - 36\beta_{6} + 131\beta_{5} - 14\beta_{4} + 256\beta_{3} + 488\beta_{2} - 2\beta _1 + 1329 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | |||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
1.00000 | −2.88266 | 1.00000 | 1.63450 | −2.88266 | 1.00000 | 1.00000 | 5.30976 | 1.63450 | |||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | 1.00000 | −2.29544 | 1.00000 | −3.07254 | −2.29544 | 1.00000 | 1.00000 | 2.26903 | −3.07254 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | 1.00000 | −1.87839 | 1.00000 | 3.85604 | −1.87839 | 1.00000 | 1.00000 | 0.528336 | 3.85604 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | 1.00000 | −0.633348 | 1.00000 | −2.25697 | −0.633348 | 1.00000 | 1.00000 | −2.59887 | −2.25697 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 1.00000 | 0.612322 | 1.00000 | 1.06793 | 0.612322 | 1.00000 | 1.00000 | −2.62506 | 1.06793 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 1.00000 | 1.36919 | 1.00000 | 3.57755 | 1.36919 | 1.00000 | 1.00000 | −1.12533 | 3.57755 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 1.00000 | 2.03031 | 1.00000 | 2.23822 | 2.03031 | 1.00000 | 1.00000 | 1.12214 | 2.23822 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 1.00000 | 2.83497 | 1.00000 | −4.00640 | 2.83497 | 1.00000 | 1.00000 | 5.03705 | −4.00640 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 1.00000 | 2.84305 | 1.00000 | 0.961671 | 2.84305 | 1.00000 | 1.00000 | 5.08295 | 0.961671 | ||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(7\) | \(-1\) |
| \(79\) | \(-1\) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 1106.2.a.l | ✓ | 9 |
| 3.b | odd | 2 | 1 | 9954.2.a.bn | 9 | ||
| 4.b | odd | 2 | 1 | 8848.2.a.t | 9 | ||
| 7.b | odd | 2 | 1 | 7742.2.a.bj | 9 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1106.2.a.l | ✓ | 9 | 1.a | even | 1 | 1 | trivial |
| 7742.2.a.bj | 9 | 7.b | odd | 2 | 1 | ||
| 8848.2.a.t | 9 | 4.b | odd | 2 | 1 | ||
| 9954.2.a.bn | 9 | 3.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(1106))\):
|
\( T_{3}^{9} - 2T_{3}^{8} - 18T_{3}^{7} + 34T_{3}^{6} + 105T_{3}^{5} - 184T_{3}^{4} - 212T_{3}^{3} + 342T_{3}^{2} + 72T_{3} - 108 \)
|
|
\( T_{5}^{9} - 4T_{5}^{8} - 26T_{5}^{7} + 120T_{5}^{6} + 140T_{5}^{5} - 1056T_{5}^{4} + 552T_{5}^{3} + 2512T_{5}^{2} - 3680T_{5} + 1440 \)
|
|
\( T_{11}^{9} - 8 T_{11}^{8} - 22 T_{11}^{7} + 232 T_{11}^{6} + 221 T_{11}^{5} - 2088 T_{11}^{4} + \cdots + 1536 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 1)^{9} \)
|
| $3$ |
\( T^{9} - 2 T^{8} + \cdots - 108 \)
|
| $5$ |
\( T^{9} - 4 T^{8} + \cdots + 1440 \)
|
| $7$ |
\( (T - 1)^{9} \)
|
| $11$ |
\( T^{9} - 8 T^{8} + \cdots + 1536 \)
|
| $13$ |
\( T^{9} - 2 T^{8} + \cdots - 138872 \)
|
| $17$ |
\( T^{9} - 6 T^{8} + \cdots - 384 \)
|
| $19$ |
\( T^{9} - 2 T^{8} + \cdots + 5632 \)
|
| $23$ |
\( T^{9} - 152 T^{7} + \cdots + 746496 \)
|
| $29$ |
\( T^{9} - 10 T^{8} + \cdots + 110592 \)
|
| $31$ |
\( T^{9} + 6 T^{8} + \cdots - 7240 \)
|
| $37$ |
\( T^{9} - 10 T^{8} + \cdots - 3963424 \)
|
| $41$ |
\( T^{9} - 10 T^{8} + \cdots + 31296 \)
|
| $43$ |
\( T^{9} - 22 T^{8} + \cdots - 256 \)
|
| $47$ |
\( T^{9} + 14 T^{8} + \cdots - 153600 \)
|
| $53$ |
\( T^{9} - 12 T^{8} + \cdots + 469800 \)
|
| $59$ |
\( T^{9} + 2 T^{8} + \cdots - 1824468 \)
|
| $61$ |
\( T^{9} - 6 T^{8} + \cdots + 2560 \)
|
| $67$ |
\( T^{9} - 24 T^{8} + \cdots + 56138176 \)
|
| $71$ |
\( T^{9} - 20 T^{8} + \cdots + 16390656 \)
|
| $73$ |
\( T^{9} + 12 T^{8} + \cdots - 185857792 \)
|
| $79$ |
\( (T - 1)^{9} \)
|
| $83$ |
\( T^{9} + 22 T^{8} + \cdots - 460800 \)
|
| $89$ |
\( T^{9} - 20 T^{8} + \cdots + 99702528 \)
|
| $97$ |
\( T^{9} + 6 T^{8} + \cdots + 428032 \)
|
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