Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1614,2,Mod(1,1614)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1614, 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("1614.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1614 = 2 \cdot 3 \cdot 269 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1614.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: | \(12.8878548862\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{8} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - 33x^{6} + 352x^{4} - 18x^{3} - 1229x^{2} + 178x + 108 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} - 33x^{6} + 352x^{4} - 18x^{3} - 1229x^{2} + 178x + 108 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 86\nu^{7} + 658\nu^{6} - 1884\nu^{5} - 16251\nu^{4} + 11617\nu^{3} + 91212\nu^{2} - 29475\nu + 18092 ) / 17546 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 300\nu^{7} - 765\nu^{6} - 5756\nu^{5} + 18187\nu^{4} + 10533\nu^{3} - 106391\nu^{2} + 158330\nu - 3808 ) / 17546 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -303\nu^{7} + 334\nu^{6} + 8270\nu^{5} - 7929\nu^{4} - 61434\nu^{3} + 48939\nu^{2} + 95381\nu - 22122 ) / 17546 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 323\nu^{7} - 385\nu^{6} - 7484\nu^{5} + 6802\nu^{4} + 41081\nu^{3} - 19770\nu^{2} + 8753\nu - 74050 ) / 17546 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -377\nu^{7} + 1400\nu^{6} + 8871\nu^{5} - 32710\nu^{4} - 53680\nu^{3} + 196308\nu^{2} + 31177\nu - 111954 ) / 17546 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 466\nu^{7} - 311\nu^{6} - 13269\nu^{5} + 6201\nu^{4} + 116198\nu^{3} - 30098\nu^{2} - 314363\nu - 1938 ) / 17546 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{7} + \beta_{6} + \beta_{4} + \beta_{3} - \beta_{2} + 9 \)
|
| \(\nu^{3}\) | \(=\) |
\( 2\beta_{7} - \beta_{5} + 3\beta_{4} + \beta_{3} + 11\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( 18\beta_{7} + 16\beta_{6} - 9\beta_{5} + 15\beta_{4} + 21\beta_{3} - 14\beta_{2} + 4\beta _1 + 104 \)
|
| \(\nu^{5}\) | \(=\) |
\( 41\beta_{7} + \beta_{6} - 21\beta_{5} + 68\beta_{4} + 28\beta_{3} + 3\beta_{2} + 126\beta _1 + 11 \)
|
| \(\nu^{6}\) | \(=\) |
\( 293\beta_{7} + 253\beta_{6} - 216\beta_{5} + 217\beta_{4} + 367\beta_{3} - 183\beta_{2} + 109\beta _1 + 1277 \)
|
| \(\nu^{7}\) | \(=\) |
\( 727\beta_{7} + 49\beta_{6} - 373\beta_{5} + 1198\beta_{4} + 578\beta_{3} + 85\beta_{2} + 1539\beta _1 + 367 \)
|
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 | 1.00000 | 1.00000 | −3.72348 | −1.00000 | −3.69765 | −1.00000 | 1.00000 | 3.72348 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −1.00000 | 1.00000 | 1.00000 | −3.24183 | −1.00000 | 0.788032 | −1.00000 | 1.00000 | 3.24183 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.00000 | 1.00000 | 1.00000 | −2.96228 | −1.00000 | 4.11619 | −1.00000 | 1.00000 | 2.96228 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.00000 | 1.00000 | 1.00000 | −0.234455 | −1.00000 | 2.69948 | −1.00000 | 1.00000 | 0.234455 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −1.00000 | 1.00000 | 1.00000 | 0.386375 | −1.00000 | 2.17487 | −1.00000 | 1.00000 | −0.386375 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −1.00000 | 1.00000 | 1.00000 | 2.67535 | −1.00000 | 3.81683 | −1.00000 | 1.00000 | −2.67535 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | −1.00000 | 1.00000 | 1.00000 | 3.17461 | −1.00000 | −4.10776 | −1.00000 | 1.00000 | −3.17461 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | −1.00000 | 1.00000 | 1.00000 | 3.92571 | −1.00000 | 3.21002 | −1.00000 | 1.00000 | −3.92571 | |||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(3\) | \( -1 \) |
| \(269\) | \( +1 \) |
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 | 1614.2.a.i | ✓ | 8 |
| 3.b | odd | 2 | 1 | 4842.2.a.q | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1614.2.a.i | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 4842.2.a.q | 8 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{8} - 33T_{5}^{6} + 352T_{5}^{4} - 18T_{5}^{3} - 1229T_{5}^{2} + 178T_{5} + 108 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1614))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 1)^{8} \)
|
| $3$ |
\( (T - 1)^{8} \)
|
| $5$ |
\( T^{8} - 33 T^{6} + \cdots + 108 \)
|
| $7$ |
\( T^{8} - 9 T^{7} + \cdots + 3544 \)
|
| $11$ |
\( T^{8} - 8 T^{7} + \cdots + 5043 \)
|
| $13$ |
\( T^{8} + 3 T^{7} + \cdots + 7136 \)
|
| $17$ |
\( T^{8} - 8 T^{7} + \cdots + 109644 \)
|
| $19$ |
\( T^{8} - 18 T^{7} + \cdots + 220 \)
|
| $23$ |
\( T^{8} + 10 T^{7} + \cdots + 96 \)
|
| $29$ |
\( T^{8} + 9 T^{7} + \cdots - 1944 \)
|
| $31$ |
\( T^{8} - 18 T^{7} + \cdots + 4600 \)
|
| $37$ |
\( T^{8} - 22 T^{7} + \cdots - 5068576 \)
|
| $41$ |
\( T^{8} + T^{7} + \cdots + 1364856 \)
|
| $43$ |
\( T^{8} - 14 T^{7} + \cdots - 258472 \)
|
| $47$ |
\( T^{8} - 6 T^{7} + \cdots + 421152 \)
|
| $53$ |
\( T^{8} + 3 T^{7} + \cdots + 65328 \)
|
| $59$ |
\( T^{8} - T^{7} + \cdots + 11862912 \)
|
| $61$ |
\( T^{8} - 18 T^{7} + \cdots + 511232 \)
|
| $67$ |
\( T^{8} - 30 T^{7} + \cdots + 2448832 \)
|
| $71$ |
\( T^{8} + 5 T^{7} + \cdots + 7318377 \)
|
| $73$ |
\( T^{8} - 15 T^{7} + \cdots - 46343 \)
|
| $79$ |
\( T^{8} - 21 T^{7} + \cdots - 1178668 \)
|
| $83$ |
\( T^{8} + 11 T^{7} + \cdots - 48864 \)
|
| $89$ |
\( T^{8} - 22 T^{7} + \cdots - 691224 \)
|
| $97$ |
\( T^{8} - 13 T^{7} + \cdots + 776603 \)
|
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