Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [98,10,Mod(1,98)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(98, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 10, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("98.1");
S:= CuspForms(chi, 10);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 98 = 2 \cdot 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 10 \) |
| Character orbit: | \([\chi]\) | \(=\) | 98.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: | \(50.4735119441\) |
| Analytic rank: | \(0\) |
| Dimension: | \(3\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{3} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 4037x + 70980 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{3}\cdot 7 \) |
| Twist minimal: | no (minimal twist has level 14) |
| 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 \(1,\beta_1,\beta_2\) 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
\( x^{3} - x^{2} - 4037x + 70980 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 2\nu^{2} + 20\nu - 5397 ) / 21 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 10\nu^{2} + 688\nu - 27153 ) / 21 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} - 5\beta _1 + 8 ) / 28 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -5\beta_{2} + 172\beta _1 + 37739 ) / 14 \)
|
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 |
|
−16.0000 | −179.327 | 256.000 | 168.288 | 2869.24 | 0 | −4096.00 | 12475.3 | −2692.61 | |||||||||||||||||||||||||||
| 1.2 | −16.0000 | 76.8690 | 256.000 | 1092.26 | −1229.90 | 0 | −4096.00 | −13774.2 | −17476.2 | ||||||||||||||||||||||||||||
| 1.3 | −16.0000 | 173.458 | 256.000 | −2345.55 | −2775.34 | 0 | −4096.00 | 10404.8 | 37528.8 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(7\) | \(-1\) |
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 | 98.10.a.h | 3 | |
| 7.b | odd | 2 | 1 | 98.10.a.g | 3 | ||
| 7.c | even | 3 | 2 | 98.10.c.l | 6 | ||
| 7.d | odd | 6 | 2 | 14.10.c.b | ✓ | 6 | |
| 21.g | even | 6 | 2 | 126.10.g.e | 6 | ||
| 28.f | even | 6 | 2 | 112.10.i.a | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 14.10.c.b | ✓ | 6 | 7.d | odd | 6 | 2 | |
| 98.10.a.g | 3 | 7.b | odd | 2 | 1 | ||
| 98.10.a.h | 3 | 1.a | even | 1 | 1 | trivial | |
| 98.10.c.l | 6 | 7.c | even | 3 | 2 | ||
| 112.10.i.a | 6 | 28.f | even | 6 | 2 | ||
| 126.10.g.e | 6 | 21.g | even | 6 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{3} - 71T_{3}^{2} - 31557T_{3} + 2391075 \)
acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(98))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 16)^{3} \)
|
| $3$ |
\( T^{3} - 71 T^{2} + \cdots + 2391075 \)
|
| $5$ |
\( T^{3} + 1085 T^{2} + \cdots + 431145855 \)
|
| $7$ |
\( T^{3} \)
|
| $11$ |
\( T^{3} + \cdots + 55717735209129 \)
|
| $13$ |
\( T^{3} + \cdots - 368974841338200 \)
|
| $17$ |
\( T^{3} + \cdots - 34\!\cdots\!65 \)
|
| $19$ |
\( T^{3} + \cdots + 19\!\cdots\!25 \)
|
| $23$ |
\( T^{3} + \cdots + 98\!\cdots\!19 \)
|
| $29$ |
\( T^{3} + \cdots - 11\!\cdots\!32 \)
|
| $31$ |
\( T^{3} + \cdots - 19\!\cdots\!53 \)
|
| $37$ |
\( T^{3} + \cdots - 47\!\cdots\!87 \)
|
| $41$ |
\( T^{3} + \cdots - 12\!\cdots\!24 \)
|
| $43$ |
\( T^{3} + \cdots + 85\!\cdots\!92 \)
|
| $47$ |
\( T^{3} + \cdots + 46\!\cdots\!85 \)
|
| $53$ |
\( T^{3} + \cdots + 10\!\cdots\!89 \)
|
| $59$ |
\( T^{3} + \cdots - 94\!\cdots\!25 \)
|
| $61$ |
\( T^{3} + \cdots + 17\!\cdots\!55 \)
|
| $67$ |
\( T^{3} + \cdots - 54\!\cdots\!95 \)
|
| $71$ |
\( T^{3} + \cdots - 12\!\cdots\!40 \)
|
| $73$ |
\( T^{3} + \cdots + 49\!\cdots\!79 \)
|
| $79$ |
\( T^{3} + \cdots - 27\!\cdots\!85 \)
|
| $83$ |
\( T^{3} + \cdots + 87\!\cdots\!00 \)
|
| $89$ |
\( T^{3} + \cdots + 10\!\cdots\!75 \)
|
| $97$ |
\( T^{3} + \cdots - 14\!\cdots\!40 \)
|
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