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: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{5}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - x - 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{23}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 154) |
| 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 \(\beta = \frac{1}{2}(1 + \sqrt{5})\). We also show the integral \(q\)-expansion of the trace form.
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 | 0.763932 | 4.00000 | 12.1803 | 1.52786 | 7.00000 | 8.00000 | −26.4164 | 24.3607 | ||||||||||||||||||||||||
| 1.2 | 2.00000 | 5.23607 | 4.00000 | −10.1803 | 10.4721 | 7.00000 | 8.00000 | 0.416408 | −20.3607 | |||||||||||||||||||||||||
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.o | 2 | |
| 11.b | odd | 2 | 1 | 1694.4.a.k | 2 | ||
| 11.d | odd | 10 | 2 | 154.4.f.b | ✓ | 4 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 154.4.f.b | ✓ | 4 | 11.d | odd | 10 | 2 | |
| 1694.4.a.k | 2 | 11.b | odd | 2 | 1 | ||
| 1694.4.a.o | 2 | 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(1694))\):
|
\( T_{3}^{2} - 6T_{3} + 4 \)
|
|
\( T_{5}^{2} - 2T_{5} - 124 \)
|
|
\( T_{13}^{2} + 76T_{13} - 976 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 2)^{2} \)
|
| $3$ |
\( T^{2} - 6T + 4 \)
|
| $5$ |
\( T^{2} - 2T - 124 \)
|
| $7$ |
\( (T - 7)^{2} \)
|
| $11$ |
\( T^{2} \)
|
| $13$ |
\( T^{2} + 76T - 976 \)
|
| $17$ |
\( T^{2} - 62T - 5164 \)
|
| $19$ |
\( T^{2} + 130T - 3380 \)
|
| $23$ |
\( T^{2} + 209T - 1831 \)
|
| $29$ |
\( T^{2} + 165T + 2155 \)
|
| $31$ |
\( T^{2} - 244T + 13264 \)
|
| $37$ |
\( T^{2} - 93T - 24849 \)
|
| $41$ |
\( T^{2} + 204T + 2404 \)
|
| $43$ |
\( T^{2} + 201T + 5139 \)
|
| $47$ |
\( T^{2} + 772T + 106676 \)
|
| $53$ |
\( T^{2} + 909T + 169159 \)
|
| $59$ |
\( T^{2} + 570T + 81100 \)
|
| $61$ |
\( T^{2} + 1414 T + 499604 \)
|
| $67$ |
\( T^{2} - 53T - 555409 \)
|
| $71$ |
\( T^{2} - 1269 T + 351079 \)
|
| $73$ |
\( T^{2} - 4T - 109516 \)
|
| $79$ |
\( T^{2} + 1605 T + 638055 \)
|
| $83$ |
\( T^{2} - 774T - 411356 \)
|
| $89$ |
\( T^{2} - 2530 T + 1597100 \)
|
| $97$ |
\( T^{2} + 492 T - 1270764 \)
|
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