Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [196,10,Mod(165,196)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(196, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 4]))
N = Newforms(chi, 10, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("196.165");
S:= CuspForms(chi, 10);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 196 = 2^{2} \cdot 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 10 \) |
| Character orbit: | \([\chi]\) | \(=\) | 196.e (of order \(3\), degree \(2\), not 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: | \(100.947023888\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\Q(\sqrt{-3}, \sqrt{11209})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} + 2803x^{2} + 2802x + 7851204 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{9}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | no (minimal twist has level 28) |
| 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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - x^{3} + 2803x^{2} + 2802x + 7851204 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -\nu^{3} + 2803\nu^{2} - 2803\nu + 7851204 ) / 7854006 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} - 2803\nu^{2} + 15710815\nu - 7851204 ) / 7854006 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 2\nu^{3} + 8407 ) / 2803 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} + \beta_{2} + 5605\beta _1 - 5605 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 2803\beta_{3} - 8407 ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/196\mathbb{Z}\right)^\times\).
| \(n\) | \(99\) | \(101\) |
| \(\chi(n)\) | \(1\) | \(-1 + \beta_{1}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 165.1 |
|
0 | −35.4363 | + | 61.3774i | 0 | −1394.29 | − | 2414.98i | 0 | 0 | 0 | 7330.04 | + | 12696.0i | 0 | ||||||||||||||||||||||||
| 165.2 | 0 | 70.4363 | − | 121.999i | 0 | 617.289 | + | 1069.18i | 0 | 0 | 0 | −81.0398 | − | 140.365i | 0 | |||||||||||||||||||||||||
| 177.1 | 0 | −35.4363 | − | 61.3774i | 0 | −1394.29 | + | 2414.98i | 0 | 0 | 0 | 7330.04 | − | 12696.0i | 0 | |||||||||||||||||||||||||
| 177.2 | 0 | 70.4363 | + | 121.999i | 0 | 617.289 | − | 1069.18i | 0 | 0 | 0 | −81.0398 | + | 140.365i | 0 | |||||||||||||||||||||||||
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 | 196.10.e.e | 4 | |
| 7.b | odd | 2 | 1 | 196.10.e.d | 4 | ||
| 7.c | even | 3 | 1 | 28.10.a.b | ✓ | 2 | |
| 7.c | even | 3 | 1 | inner | 196.10.e.e | 4 | |
| 7.d | odd | 6 | 1 | 196.10.a.b | 2 | ||
| 7.d | odd | 6 | 1 | 196.10.e.d | 4 | ||
| 21.h | odd | 6 | 1 | 252.10.a.b | 2 | ||
| 28.g | odd | 6 | 1 | 112.10.a.d | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 28.10.a.b | ✓ | 2 | 7.c | even | 3 | 1 | |
| 112.10.a.d | 2 | 28.g | odd | 6 | 1 | ||
| 196.10.a.b | 2 | 7.d | odd | 6 | 1 | ||
| 196.10.e.d | 4 | 7.b | odd | 2 | 1 | ||
| 196.10.e.d | 4 | 7.d | odd | 6 | 1 | ||
| 196.10.e.e | 4 | 1.a | even | 1 | 1 | trivial | |
| 196.10.e.e | 4 | 7.c | even | 3 | 1 | inner | |
| 252.10.a.b | 2 | 21.h | odd | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} - 70T_{3}^{3} + 14884T_{3}^{2} + 698880T_{3} + 99680256 \)
acting on \(S_{10}^{\mathrm{new}}(196, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} - 70 T^{3} + \cdots + 99680256 \)
|
| $5$ |
\( T^{4} + \cdots + 11852320998400 \)
|
| $7$ |
\( T^{4} \)
|
| $11$ |
\( T^{4} + \cdots + 36\!\cdots\!00 \)
|
| $13$ |
\( (T^{2} - 122766 T + 3683031768)^{2} \)
|
| $17$ |
\( T^{4} + \cdots + 97\!\cdots\!00 \)
|
| $19$ |
\( T^{4} + \cdots + 88\!\cdots\!04 \)
|
| $23$ |
\( T^{4} + \cdots + 43\!\cdots\!84 \)
|
| $29$ |
\( (T^{2} + \cdots - 6012588512356)^{2} \)
|
| $31$ |
\( T^{4} + \cdots + 47\!\cdots\!44 \)
|
| $37$ |
\( T^{4} + \cdots + 26\!\cdots\!00 \)
|
| $41$ |
\( (T^{2} + \cdots + 328158355074444)^{2} \)
|
| $43$ |
\( (T^{2} + \cdots + 89571104447680)^{2} \)
|
| $47$ |
\( T^{4} + \cdots + 99\!\cdots\!36 \)
|
| $53$ |
\( T^{4} + \cdots + 29\!\cdots\!96 \)
|
| $59$ |
\( T^{4} + \cdots + 13\!\cdots\!00 \)
|
| $61$ |
\( T^{4} + \cdots + 30\!\cdots\!00 \)
|
| $67$ |
\( T^{4} + \cdots + 29\!\cdots\!00 \)
|
| $71$ |
\( (T^{2} + \cdots + 13\!\cdots\!20)^{2} \)
|
| $73$ |
\( T^{4} + \cdots + 37\!\cdots\!00 \)
|
| $79$ |
\( T^{4} + \cdots + 68\!\cdots\!96 \)
|
| $83$ |
\( (T^{2} + \cdots - 72\!\cdots\!92)^{2} \)
|
| $89$ |
\( T^{4} + \cdots + 32\!\cdots\!04 \)
|
| $97$ |
\( (T^{2} + \cdots - 72\!\cdots\!36)^{2} \)
|
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