Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [98,16,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, 16, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("98.1");
S:= CuspForms(chi, 16);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 98 = 2 \cdot 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 16 \) |
| 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: | \(139.839634998\) |
| Analytic rank: | \(0\) |
| Dimension: | \(5\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{5} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{5} - 2x^{4} - 11041035x^{3} - 5114877359x^{2} + 21808168029940x + 10956853436633025 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{9}\cdot 3^{2}\cdot 5\cdot 7^{4} \) |
| 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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) 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
\( x^{5} - 2x^{4} - 11041035x^{3} - 5114877359x^{2} + 21808168029940x + 10956853436633025 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu - 1 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -9896\nu^{4} + 14792592\nu^{3} + 65735919900\nu^{2} - 48153710285246\nu - 25914172944503820 ) / 363478628475 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 69424 \nu^{4} + 156013368 \nu^{3} + 638039040120 \nu^{2} - 833085160943914 \nu - 11\!\cdots\!30 ) / 363478628475 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 127888 \nu^{4} - 70022424 \nu^{3} + 1419043468500 \nu^{2} + 830491893242162 \nu - 20\!\cdots\!35 ) / 363478628475 \)
|
| \(\nu\) | \(=\) |
\( ( \beta _1 + 1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{4} + 5\beta_{3} - 48\beta_{2} + 1408\beta _1 + 17665939 ) / 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -6772\beta_{4} + 21805\beta_{3} - 65454\beta_{2} + 28389097\beta _1 + 24663073015 ) / 8 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 3162531\beta_{4} + 99020985\beta_{3} - 1029376683\beta_{2} + 41678035415\beta _1 + 250596026708216 ) / 8 \)
|
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 |
|
128.000 | −4561.02 | 16384.0 | 303815. | −583811. | 0 | 2.09715e6 | 6.45401e6 | 3.88883e7 | |||||||||||||||||||||||||||||||||
| 1.2 | 128.000 | −3122.66 | 16384.0 | −203646. | −399701. | 0 | 2.09715e6 | −4.59789e6 | −2.60666e7 | ||||||||||||||||||||||||||||||||||
| 1.3 | 128.000 | −618.003 | 16384.0 | 17971.1 | −79104.3 | 0 | 2.09715e6 | −1.39670e7 | 2.30030e6 | ||||||||||||||||||||||||||||||||||
| 1.4 | 128.000 | 3507.94 | 16384.0 | −72330.1 | 449016. | 0 | 2.09715e6 | −2.04326e6 | −9.25825e6 | ||||||||||||||||||||||||||||||||||
| 1.5 | 128.000 | 6772.74 | 16384.0 | 255142. | 866911. | 0 | 2.09715e6 | 3.15212e7 | 3.26582e7 | ||||||||||||||||||||||||||||||||||
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 | 98.16.a.k | 5 | |
| 7.b | odd | 2 | 1 | 98.16.a.j | 5 | ||
| 7.c | even | 3 | 2 | 98.16.c.m | 10 | ||
| 7.d | odd | 6 | 2 | 14.16.c.a | ✓ | 10 | |
| 21.g | even | 6 | 2 | 126.16.g.e | 10 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 14.16.c.a | ✓ | 10 | 7.d | odd | 6 | 2 | |
| 98.16.a.j | 5 | 7.b | odd | 2 | 1 | ||
| 98.16.a.k | 5 | 1.a | even | 1 | 1 | trivial | |
| 98.16.c.m | 10 | 7.c | even | 3 | 2 | ||
| 126.16.g.e | 10 | 21.g | even | 6 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{5} - 1979T_{3}^{4} - 42597570T_{3}^{3} + 10795443678T_{3}^{2} + 360707288638545T_{3} + 209119415354765325 \)
acting on \(S_{16}^{\mathrm{new}}(\Gamma_0(98))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 128)^{5} \)
|
| $3$ |
\( T^{5} + \cdots + 20\!\cdots\!25 \)
|
| $5$ |
\( T^{5} + \cdots - 20\!\cdots\!25 \)
|
| $7$ |
\( T^{5} \)
|
| $11$ |
\( T^{5} + \cdots + 14\!\cdots\!15 \)
|
| $13$ |
\( T^{5} + \cdots - 72\!\cdots\!00 \)
|
| $17$ |
\( T^{5} + \cdots - 65\!\cdots\!25 \)
|
| $19$ |
\( T^{5} + \cdots + 41\!\cdots\!75 \)
|
| $23$ |
\( T^{5} + \cdots + 11\!\cdots\!49 \)
|
| $29$ |
\( T^{5} + \cdots - 36\!\cdots\!72 \)
|
| $31$ |
\( T^{5} + \cdots - 26\!\cdots\!35 \)
|
| $37$ |
\( T^{5} + \cdots - 62\!\cdots\!03 \)
|
| $41$ |
\( T^{5} + \cdots - 58\!\cdots\!44 \)
|
| $43$ |
\( T^{5} + \cdots - 29\!\cdots\!40 \)
|
| $47$ |
\( T^{5} + \cdots + 53\!\cdots\!75 \)
|
| $53$ |
\( T^{5} + \cdots - 15\!\cdots\!23 \)
|
| $59$ |
\( T^{5} + \cdots - 87\!\cdots\!75 \)
|
| $61$ |
\( T^{5} + \cdots + 35\!\cdots\!15 \)
|
| $67$ |
\( T^{5} + \cdots + 24\!\cdots\!35 \)
|
| $71$ |
\( T^{5} + \cdots + 15\!\cdots\!00 \)
|
| $73$ |
\( T^{5} + \cdots - 40\!\cdots\!85 \)
|
| $79$ |
\( T^{5} + \cdots - 42\!\cdots\!75 \)
|
| $83$ |
\( T^{5} + \cdots + 17\!\cdots\!00 \)
|
| $89$ |
\( T^{5} + \cdots + 92\!\cdots\!55 \)
|
| $97$ |
\( T^{5} + \cdots + 10\!\cdots\!00 \)
|
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