Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [196,3,Mod(99,196)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(196, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 0]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("196.99");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 196 = 2^{2} \cdot 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 196.c (of order \(2\), degree \(1\), 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: | \(5.34061318146\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.0.15582448.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 3x^{5} + 13x^{4} - 21x^{3} + 20x^{2} - 10x + 2 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2^{4} \) |
| Twist minimal: | no (minimal twist has level 28) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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
\( x^{6} - 3x^{5} + 13x^{4} - 21x^{3} + 20x^{2} - 10x + 2 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu^{5} - 2\nu^{4} + 11\nu^{3} - 10\nu^{2} + 10\nu - 2 \)
|
| \(\beta_{2}\) | \(=\) |
\( -2\nu^{5} + 5\nu^{4} - 24\nu^{3} + 31\nu^{2} - 28\nu + 9 \)
|
| \(\beta_{3}\) | \(=\) |
\( 2\nu^{5} - 6\nu^{4} + 26\nu^{3} - 42\nu^{2} + 40\nu - 17 \)
|
| \(\beta_{4}\) | \(=\) |
\( -6\nu^{5} + 14\nu^{4} - 68\nu^{3} + 80\nu^{2} - 60\nu + 17 \)
|
| \(\beta_{5}\) | \(=\) |
\( 6\nu^{5} - 16\nu^{4} + 72\nu^{3} - 100\nu^{2} + 78\nu - 23 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} + 2\beta_{2} + 2\beta _1 + 3 ) / 4 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 2\beta_{5} + 2\beta_{4} - \beta_{3} + 2\beta_{2} + 6\beta _1 - 11 ) / 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 2\beta_{5} + 4\beta_{4} - 9\beta_{3} - 18\beta_{2} - 6\beta _1 - 25 ) / 4 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -18\beta_{5} - 14\beta_{4} + \beta_{3} - 38\beta_{2} - 54\beta _1 + 75 ) / 4 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -38\beta_{5} - 52\beta_{4} + 81\beta_{3} + 122\beta_{2} + 2\beta _1 + 293 ) / 4 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/196\mathbb{Z}\right)^\times\).
| \(n\) | \(99\) | \(101\) |
| \(\chi(n)\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 99.1 |
|
−1.67727 | − | 1.08938i | − | 1.88252i | 1.62649 | + | 3.65439i | 2.25297 | −2.05079 | + | 3.15750i | 0 | 1.25297 | − | 7.90127i | 5.45612 | −3.77885 | − | 2.45435i | |||||||||||||||||||||||||
| 99.2 | −1.67727 | + | 1.08938i | 1.88252i | 1.62649 | − | 3.65439i | 2.25297 | −2.05079 | − | 3.15750i | 0 | 1.25297 | + | 7.90127i | 5.45612 | −3.77885 | + | 2.45435i | |||||||||||||||||||||||||||
| 99.3 | 0.789608 | − | 1.83753i | − | 2.15693i | −2.75304 | − | 2.90186i | −6.50608 | −3.96343 | − | 1.70313i | 0 | −7.50608 | + | 2.76746i | 4.34764 | −5.13725 | + | 11.9551i | ||||||||||||||||||||||||||
| 99.4 | 0.789608 | + | 1.83753i | 2.15693i | −2.75304 | + | 2.90186i | −6.50608 | −3.96343 | + | 1.70313i | 0 | −7.50608 | − | 2.76746i | 4.34764 | −5.13725 | − | 11.9551i | |||||||||||||||||||||||||||
| 99.5 | 1.88766 | − | 0.660851i | 4.56111i | 3.12655 | − | 2.49493i | 5.25310 | 3.01422 | + | 8.60985i | 0 | 4.25310 | − | 6.77577i | −11.8038 | 9.91610 | − | 3.47152i | |||||||||||||||||||||||||||
| 99.6 | 1.88766 | + | 0.660851i | − | 4.56111i | 3.12655 | + | 2.49493i | 5.25310 | 3.01422 | − | 8.60985i | 0 | 4.25310 | + | 6.77577i | −11.8038 | 9.91610 | + | 3.47152i | ||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 4.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 196.3.c.i | 6 | |
| 4.b | odd | 2 | 1 | inner | 196.3.c.i | 6 | |
| 7.b | odd | 2 | 1 | 196.3.c.h | 6 | ||
| 7.c | even | 3 | 2 | 28.3.g.a | ✓ | 12 | |
| 7.d | odd | 6 | 2 | 196.3.g.i | 12 | ||
| 21.h | odd | 6 | 2 | 252.3.y.c | 12 | ||
| 28.d | even | 2 | 1 | 196.3.c.h | 6 | ||
| 28.f | even | 6 | 2 | 196.3.g.i | 12 | ||
| 28.g | odd | 6 | 2 | 28.3.g.a | ✓ | 12 | |
| 56.k | odd | 6 | 2 | 448.3.r.h | 12 | ||
| 56.p | even | 6 | 2 | 448.3.r.h | 12 | ||
| 84.n | even | 6 | 2 | 252.3.y.c | 12 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 28.3.g.a | ✓ | 12 | 7.c | even | 3 | 2 | |
| 28.3.g.a | ✓ | 12 | 28.g | odd | 6 | 2 | |
| 196.3.c.h | 6 | 7.b | odd | 2 | 1 | ||
| 196.3.c.h | 6 | 28.d | even | 2 | 1 | ||
| 196.3.c.i | 6 | 1.a | even | 1 | 1 | trivial | |
| 196.3.c.i | 6 | 4.b | odd | 2 | 1 | inner | |
| 196.3.g.i | 12 | 7.d | odd | 6 | 2 | ||
| 196.3.g.i | 12 | 28.f | even | 6 | 2 | ||
| 252.3.y.c | 12 | 21.h | odd | 6 | 2 | ||
| 252.3.y.c | 12 | 84.n | even | 6 | 2 | ||
| 448.3.r.h | 12 | 56.k | odd | 6 | 2 | ||
| 448.3.r.h | 12 | 56.p | even | 6 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(196, [\chi])\):
|
\( T_{3}^{6} + 29T_{3}^{4} + 187T_{3}^{2} + 343 \)
|
|
\( T_{5}^{3} - T_{5}^{2} - 37T_{5} + 77 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} - 2 T^{5} + \cdots + 64 \)
|
| $3$ |
\( T^{6} + 29 T^{4} + \cdots + 343 \)
|
| $5$ |
\( (T^{3} - T^{2} - 37 T + 77)^{2} \)
|
| $7$ |
\( T^{6} \)
|
| $11$ |
\( T^{6} + 101 T^{4} + \cdots + 175 \)
|
| $13$ |
\( (T^{3} + 6 T^{2} + \cdots - 280)^{2} \)
|
| $17$ |
\( (T^{3} - T^{2} - 509 T - 4235)^{2} \)
|
| $19$ |
\( T^{6} + 805 T^{4} + \cdots + 5359375 \)
|
| $23$ |
\( T^{6} + 1645 T^{4} + \cdots + 484183 \)
|
| $29$ |
\( (T^{3} - 18 T^{2} + \cdots - 4408)^{2} \)
|
| $31$ |
\( T^{6} + \cdots + 1257858175 \)
|
| $37$ |
\( (T^{3} + 43 T^{2} + \cdots - 43895)^{2} \)
|
| $41$ |
\( (T^{3} - 2 T^{2} + \cdots + 44296)^{2} \)
|
| $43$ |
\( T^{6} + \cdots + 1101463552 \)
|
| $47$ |
\( T^{6} + \cdots + 2074011583 \)
|
| $53$ |
\( (T^{3} - 37 T^{2} + \cdots + 20425)^{2} \)
|
| $59$ |
\( T^{6} + \cdots + 49928529175 \)
|
| $61$ |
\( (T^{3} + 43 T^{2} + \cdots - 75607)^{2} \)
|
| $67$ |
\( T^{6} + \cdots + 2227609447 \)
|
| $71$ |
\( T^{6} + \cdots + 6884147200 \)
|
| $73$ |
\( (T^{3} - 117 T^{2} + \cdots - 44695)^{2} \)
|
| $79$ |
\( T^{6} + 17709 T^{4} + \cdots + 191104375 \)
|
| $83$ |
\( T^{6} + 2944 T^{4} + \cdots + 275365888 \)
|
| $89$ |
\( (T^{3} + 3 T^{2} + \cdots - 7231)^{2} \)
|
| $97$ |
\( (T^{3} - 186 T^{2} + \cdots + 994280)^{2} \)
|
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