Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1694,4,Mod(1,1694)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1694, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1694.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1694 = 2 \cdot 7 \cdot 11^{2} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1694.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: | \(99.9492355497\) |
| Analytic rank: | \(1\) |
| Dimension: | \(6\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{6} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 2x^{5} - 95x^{4} + 60x^{3} + 2097x^{2} + 2700x - 648 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - 2x^{5} - 95x^{4} + 60x^{3} + 2097x^{2} + 2700x - 648 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{5} - 95\nu^{3} - 20\nu^{2} + 1947\nu + 1314 ) / 990 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{5} - 40\nu^{3} - 75\nu^{2} - 1188\nu + 324 ) / 495 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{5} + 95\nu^{3} + 185\nu^{2} - 2112\nu - 6759 ) / 495 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -\nu^{5} + 5\nu^{4} + 80\nu^{3} - 300\nu^{2} - 1197\nu + 891 ) / 135 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( 3\beta_{4} + 6\beta_{2} + \beta _1 + 33 \)
|
| \(\nu^{3}\) | \(=\) |
\( 3\beta_{4} + 9\beta_{3} - 12\beta_{2} + 58\beta _1 + 51 \)
|
| \(\nu^{4}\) | \(=\) |
\( 27\beta_{5} + 201\beta_{4} + 27\beta_{3} + 546\beta_{2} + 88\beta _1 + 1824 \)
|
| \(\nu^{5}\) | \(=\) |
\( 345\beta_{4} + 855\beta_{3} - 30\beta_{2} + 3583\beta _1 + 4191 \)
|
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 |
|
2.00000 | −9.59770 | 4.00000 | 7.18126 | −19.1954 | −7.00000 | 8.00000 | 65.1159 | 14.3625 | ||||||||||||||||||||||||||||||||||||
| 1.2 | 2.00000 | −6.01968 | 4.00000 | −9.03678 | −12.0394 | −7.00000 | 8.00000 | 9.23654 | −18.0736 | |||||||||||||||||||||||||||||||||||||
| 1.3 | 2.00000 | −3.79520 | 4.00000 | 10.8462 | −7.59039 | −7.00000 | 8.00000 | −12.5965 | 21.6924 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 2.00000 | −1.79331 | 4.00000 | −14.5118 | −3.58662 | −7.00000 | 8.00000 | −23.7840 | −29.0237 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 2.00000 | 4.81488 | 4.00000 | 2.85084 | 9.62975 | −7.00000 | 8.00000 | −3.81697 | 5.70168 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 2.00000 | 6.39101 | 4.00000 | −5.32969 | 12.7820 | −7.00000 | 8.00000 | 13.8451 | −10.6594 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(7\) | \(1\) |
| \(11\) | \(1\) |
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 | 1694.4.a.be | yes | 6 |
| 11.b | odd | 2 | 1 | 1694.4.a.bd | ✓ | 6 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1694.4.a.bd | ✓ | 6 | 11.b | odd | 2 | 1 | |
| 1694.4.a.be | yes | 6 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1694))\):
|
\( T_{3}^{6} + 10T_{3}^{5} - 55T_{3}^{4} - 620T_{3}^{3} + 257T_{3}^{2} + 8800T_{3} + 12100 \)
|
|
\( T_{5}^{6} + 8T_{5}^{5} - 217T_{5}^{4} - 1148T_{5}^{3} + 12175T_{5}^{2} + 33372T_{5} - 155199 \)
|
|
\( T_{13}^{6} + 20T_{13}^{5} - 6363T_{13}^{4} - 213248T_{13}^{3} + 8487516T_{13}^{2} + 457197824T_{13} + 5239963408 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 2)^{6} \)
|
| $3$ |
\( T^{6} + 10 T^{5} + \cdots + 12100 \)
|
| $5$ |
\( T^{6} + 8 T^{5} + \cdots - 155199 \)
|
| $7$ |
\( (T + 7)^{6} \)
|
| $11$ |
\( T^{6} \)
|
| $13$ |
\( T^{6} + \cdots + 5239963408 \)
|
| $17$ |
\( T^{6} + \cdots - 2881416827 \)
|
| $19$ |
\( T^{6} + \cdots - 29056028144 \)
|
| $23$ |
\( T^{6} + \cdots - 2304739184 \)
|
| $29$ |
\( T^{6} + \cdots + 809852039821 \)
|
| $31$ |
\( T^{6} + \cdots + 392809680400 \)
|
| $37$ |
\( T^{6} + \cdots - 198969191216 \)
|
| $41$ |
\( T^{6} + \cdots + 13157780705833 \)
|
| $43$ |
\( T^{6} + \cdots + 790358645328 \)
|
| $47$ |
\( T^{6} + \cdots + 12770728898656 \)
|
| $53$ |
\( T^{6} + \cdots + 221110726462701 \)
|
| $59$ |
\( T^{6} + \cdots - 34\!\cdots\!68 \)
|
| $61$ |
\( T^{6} + \cdots - 268542448673312 \)
|
| $67$ |
\( T^{6} + \cdots + 17\!\cdots\!16 \)
|
| $71$ |
\( T^{6} + \cdots + 38\!\cdots\!44 \)
|
| $73$ |
\( T^{6} + \cdots + 65\!\cdots\!68 \)
|
| $79$ |
\( T^{6} + \cdots + 11\!\cdots\!64 \)
|
| $83$ |
\( T^{6} + \cdots - 21\!\cdots\!56 \)
|
| $89$ |
\( T^{6} + \cdots + 76\!\cdots\!37 \)
|
| $97$ |
\( T^{6} + \cdots + 11\!\cdots\!17 \)
|
| show more | |
| show less |