Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [693,2,Mod(100,693)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(693, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 2, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("693.100");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 693 = 3^{2} \cdot 7 \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 693.i (of order \(3\), degree \(2\), 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: | \(5.53363286007\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Relative dimension: | \(4\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{8} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - 2x^{7} + 8x^{6} + 21x^{4} - 4x^{3} + 28x^{2} + 12x + 9 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{4}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 231) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-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
\( x^{8} - 2x^{7} + 8x^{6} + 21x^{4} - 4x^{3} + 28x^{2} + 12x + 9 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 68\nu^{7} - 215\nu^{6} + 357\nu^{5} + 646\nu^{4} - 1444\nu^{3} + 1156\nu^{2} + 561\nu + 5468 ) / 4243 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 84\nu^{7} - 16\nu^{6} + 441\nu^{5} + 798\nu^{4} + 3208\nu^{3} + 1428\nu^{2} + 693\nu + 2262 ) / 4243 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -754\nu^{7} + 1760\nu^{6} - 6080\nu^{5} + 1323\nu^{4} - 13440\nu^{3} + 12640\nu^{2} - 16828\nu + 5760 ) / 12729 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -815\nu^{7} + 3388\nu^{6} - 11704\nu^{5} + 15594\nu^{4} - 25872\nu^{3} + 24332\nu^{2} - 56579\nu + 11088 ) / 12729 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 1052\nu^{7} - 2827\nu^{6} + 9766\nu^{5} - 6978\nu^{4} + 21588\nu^{3} - 20303\nu^{2} + 17165\nu - 9252 ) / 12729 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -556\nu^{7} + 510\nu^{6} - 2919\nu^{5} - 5282\nu^{4} - 8909\nu^{3} - 9452\nu^{2} - 4587\nu - 13760 ) / 4243 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{6} + 2\beta_{4} + \beta_{3} + \beta_{2} + \beta _1 - 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{7} + 5\beta_{3} + 2\beta_{2} - 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( -8\beta_{6} - 2\beta_{5} - 9\beta_{4} - 10\beta_1 \)
|
| \(\nu^{5}\) | \(=\) |
\( -8\beta_{7} - 20\beta_{6} - 8\beta_{5} - 18\beta_{4} - 33\beta_{3} - 20\beta_{2} - 33\beta _1 + 18 \)
|
| \(\nu^{6}\) | \(=\) |
\( -20\beta_{7} - 83\beta_{3} - 61\beta_{2} + 58 \)
|
| \(\nu^{7}\) | \(=\) |
\( 164\beta_{6} + 61\beta_{5} + 146\beta_{4} + 243\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/693\mathbb{Z}\right)^\times\).
| \(n\) | \(155\) | \(199\) | \(442\) |
| \(\chi(n)\) | \(1\) | \(-\beta_{4}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 100.1 |
|
−0.758290 | + | 1.31340i | 0 | −0.150007 | − | 0.259820i | −1.16659 | + | 2.02059i | 0 | −2.28580 | − | 1.33233i | −2.57816 | 0 | −1.76922 | − | 3.06438i | ||||||||||||||||||||||||||||||||
| 100.2 | −0.276205 | + | 0.478401i | 0 | 0.847422 | + | 1.46778i | 0.795012 | − | 1.37700i | 0 | 0.886763 | + | 2.49272i | −2.04107 | 0 | 0.439172 | + | 0.760669i | |||||||||||||||||||||||||||||||||
| 100.3 | 0.643668 | − | 1.11487i | 0 | 0.171383 | + | 0.296844i | 1.95872 | − | 3.39260i | 0 | −0.234193 | − | 2.63537i | 3.01593 | 0 | −2.52153 | − | 4.36742i | |||||||||||||||||||||||||||||||||
| 100.4 | 1.39083 | − | 2.40898i | 0 | −2.86880 | − | 4.96890i | 0.412855 | − | 0.715087i | 0 | 2.63323 | − | 0.257073i | −10.3967 | 0 | −1.14842 | − | 1.98912i | |||||||||||||||||||||||||||||||||
| 298.1 | −0.758290 | − | 1.31340i | 0 | −0.150007 | + | 0.259820i | −1.16659 | − | 2.02059i | 0 | −2.28580 | + | 1.33233i | −2.57816 | 0 | −1.76922 | + | 3.06438i | |||||||||||||||||||||||||||||||||
| 298.2 | −0.276205 | − | 0.478401i | 0 | 0.847422 | − | 1.46778i | 0.795012 | + | 1.37700i | 0 | 0.886763 | − | 2.49272i | −2.04107 | 0 | 0.439172 | − | 0.760669i | |||||||||||||||||||||||||||||||||
| 298.3 | 0.643668 | + | 1.11487i | 0 | 0.171383 | − | 0.296844i | 1.95872 | + | 3.39260i | 0 | −0.234193 | + | 2.63537i | 3.01593 | 0 | −2.52153 | + | 4.36742i | |||||||||||||||||||||||||||||||||
| 298.4 | 1.39083 | + | 2.40898i | 0 | −2.86880 | + | 4.96890i | 0.412855 | + | 0.715087i | 0 | 2.63323 | + | 0.257073i | −10.3967 | 0 | −1.14842 | + | 1.98912i | |||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 693.2.i.i | 8 | |
| 3.b | odd | 2 | 1 | 231.2.i.e | ✓ | 8 | |
| 7.c | even | 3 | 1 | inner | 693.2.i.i | 8 | |
| 7.c | even | 3 | 1 | 4851.2.a.bt | 4 | ||
| 7.d | odd | 6 | 1 | 4851.2.a.bu | 4 | ||
| 21.g | even | 6 | 1 | 1617.2.a.x | 4 | ||
| 21.h | odd | 6 | 1 | 231.2.i.e | ✓ | 8 | |
| 21.h | odd | 6 | 1 | 1617.2.a.z | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 231.2.i.e | ✓ | 8 | 3.b | odd | 2 | 1 | |
| 231.2.i.e | ✓ | 8 | 21.h | odd | 6 | 1 | |
| 693.2.i.i | 8 | 1.a | even | 1 | 1 | trivial | |
| 693.2.i.i | 8 | 7.c | even | 3 | 1 | inner | |
| 1617.2.a.x | 4 | 21.g | even | 6 | 1 | ||
| 1617.2.a.z | 4 | 21.h | odd | 6 | 1 | ||
| 4851.2.a.bt | 4 | 7.c | even | 3 | 1 | ||
| 4851.2.a.bu | 4 | 7.d | odd | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(693, [\chi])\):
|
\( T_{2}^{8} - 2T_{2}^{7} + 8T_{2}^{6} + 21T_{2}^{4} - 4T_{2}^{3} + 28T_{2}^{2} + 12T_{2} + 9 \)
|
|
\( T_{5}^{8} - 4T_{5}^{7} + 20T_{5}^{6} - 24T_{5}^{5} + 108T_{5}^{4} - 176T_{5}^{3} + 352T_{5}^{2} - 240T_{5} + 144 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} - 2 T^{7} + \cdots + 9 \)
|
| $3$ |
\( T^{8} \)
|
| $5$ |
\( T^{8} - 4 T^{7} + \cdots + 144 \)
|
| $7$ |
\( T^{8} - 2 T^{7} + \cdots + 2401 \)
|
| $11$ |
\( (T^{2} + T + 1)^{4} \)
|
| $13$ |
\( (T^{4} + 2 T^{3} - 8 T^{2} + \cdots - 4)^{2} \)
|
| $17$ |
\( T^{8} - 2 T^{7} + \cdots + 225 \)
|
| $19$ |
\( T^{8} + 40 T^{6} + \cdots + 7921 \)
|
| $23$ |
\( T^{8} - 4 T^{7} + \cdots + 216225 \)
|
| $29$ |
\( (T^{4} + 8 T^{3} + 12 T^{2} + \cdots + 3)^{2} \)
|
| $31$ |
\( T^{8} - 12 T^{7} + \cdots + 1948816 \)
|
| $37$ |
\( T^{8} - 4 T^{7} + \cdots + 1 \)
|
| $41$ |
\( (T^{4} + 2 T^{3} - 44 T^{2} + \cdots + 60)^{2} \)
|
| $43$ |
\( (T^{4} - 18 T^{3} + \cdots - 1385)^{2} \)
|
| $47$ |
\( T^{8} - 12 T^{7} + \cdots + 81 \)
|
| $53$ |
\( T^{8} + 12 T^{7} + \cdots + 39087504 \)
|
| $59$ |
\( T^{8} - 12 T^{7} + \cdots + 62001 \)
|
| $61$ |
\( T^{8} + 2 T^{7} + \cdots + 400 \)
|
| $67$ |
\( T^{8} + 28 T^{7} + \cdots + 2131600 \)
|
| $71$ |
\( (T^{4} + 12 T^{3} + \cdots - 699)^{2} \)
|
| $73$ |
\( T^{8} + 6 T^{7} + \cdots + 294328336 \)
|
| $79$ |
\( T^{8} + 2 T^{7} + \cdots + 150544 \)
|
| $83$ |
\( (T^{4} - 12 T^{3} + \cdots + 2592)^{2} \)
|
| $89$ |
\( T^{8} - 8 T^{7} + \cdots + 145154304 \)
|
| $97$ |
\( (T^{4} + 44 T^{3} + \cdots + 8501)^{2} \)
|
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