Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [9801,2,Mod(1,9801)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9801, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("9801.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 9801 = 3^{4} \cdot 11^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 9801.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: | \(78.2613790211\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | 8.8.287107358976.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - 15x^{6} + 74x^{4} - 147x^{2} + 100 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} - 15x^{6} + 74x^{4} - 147x^{2} + 100 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 3\nu^{7} - 35\nu^{5} + 102\nu^{3} - 71\nu ) / 10 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -2\nu^{7} + 25\nu^{5} - 88\nu^{3} + 94\nu ) / 5 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{7} - 13\nu^{5} + 48\nu^{3} - 51\nu ) / 2 \)
|
| \(\beta_{6}\) | \(=\) |
\( \nu^{6} - 12\nu^{4} + 38\nu^{2} - 33 \)
|
| \(\beta_{7}\) | \(=\) |
\( -\nu^{6} + 13\nu^{4} - 48\nu^{2} + 51 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{5} - 2\beta_{4} - \beta_{3} + 5\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{7} + \beta_{6} + 10\beta_{2} + 22 \)
|
| \(\nu^{5}\) | \(=\) |
\( -12\beta_{5} - 21\beta_{4} - 8\beta_{3} + 32\beta_1 \)
|
| \(\nu^{6}\) | \(=\) |
\( 12\beta_{7} + 13\beta_{6} + 82\beta_{2} + 145 \)
|
| \(\nu^{7}\) | \(=\) |
\( -106\beta_{5} - 177\beta_{4} - 56\beta_{3} + 227\beta_1 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
−2.81758 | 0 | 5.93878 | −2.21184 | 0 | 3.52835 | −11.0978 | 0 | 6.23205 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −2.81758 | 0 | 5.93878 | 2.21184 | 0 | −3.52835 | −11.0978 | 0 | −6.23205 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −0.220586 | 0 | −1.95134 | −3.98072 | 0 | −0.113960 | 0.871612 | 0 | 0.878093 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −0.220586 | 0 | −1.95134 | 3.98072 | 0 | 0.113960 | 0.871612 | 0 | −0.878093 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0.683225 | 0 | −1.53320 | −2.18081 | 0 | 3.85674 | −2.41397 | 0 | −1.48998 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0.683225 | 0 | −1.53320 | 2.18081 | 0 | −3.85674 | −2.41397 | 0 | 1.48998 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 2.35495 | 0 | 3.54577 | −2.91645 | 0 | −1.28969 | 3.64020 | 0 | −6.86808 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 2.35495 | 0 | 3.54577 | 2.91645 | 0 | 1.28969 | 3.64020 | 0 | 6.86808 | |||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \( -1 \) |
| \(11\) | \( +1 \) |
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 33.d | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 9801.2.a.bx | ✓ | 8 |
| 3.b | odd | 2 | 1 | 9801.2.a.by | yes | 8 | |
| 11.b | odd | 2 | 1 | 9801.2.a.by | yes | 8 | |
| 33.d | even | 2 | 1 | inner | 9801.2.a.bx | ✓ | 8 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 9801.2.a.bx | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 9801.2.a.bx | ✓ | 8 | 33.d | even | 2 | 1 | inner |
| 9801.2.a.by | yes | 8 | 3.b | odd | 2 | 1 | |
| 9801.2.a.by | yes | 8 | 11.b | odd | 2 | 1 | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(9801))\):
|
\( T_{2}^{4} - 7T_{2}^{2} + 3T_{2} + 1 \)
|
|
\( T_{5}^{8} - 34T_{5}^{6} + 393T_{5}^{4} - 1867T_{5}^{2} + 3136 \)
|
|
\( T_{7}^{8} - 29T_{7}^{6} + 231T_{7}^{4} - 311T_{7}^{2} + 4 \)
|
|
\( T_{17}^{4} + 6T_{17}^{3} - 19T_{17}^{2} - 165T_{17} - 242 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{4} - 7 T^{2} + 3 T + 1)^{2} \)
|
| $3$ |
\( T^{8} \)
|
| $5$ |
\( T^{8} - 34 T^{6} + \cdots + 3136 \)
|
| $7$ |
\( T^{8} - 29 T^{6} + \cdots + 4 \)
|
| $11$ |
\( T^{8} \)
|
| $13$ |
\( T^{8} - 81 T^{6} + \cdots + 99225 \)
|
| $17$ |
\( (T^{4} + 6 T^{3} + \cdots - 242)^{2} \)
|
| $19$ |
\( T^{8} - 83 T^{6} + \cdots + 5776 \)
|
| $23$ |
\( T^{8} - 79 T^{6} + \cdots + 4624 \)
|
| $29$ |
\( (T^{4} + 6 T^{3} + \cdots - 413)^{2} \)
|
| $31$ |
\( (T^{4} - 6 T^{3} - 45 T^{2} + \cdots + 36)^{2} \)
|
| $37$ |
\( (T^{4} + 4 T^{3} + \cdots + 895)^{2} \)
|
| $41$ |
\( (T^{4} - 6 T^{3} + \cdots + 379)^{2} \)
|
| $43$ |
\( T^{8} - 386 T^{6} + \cdots + 51984100 \)
|
| $47$ |
\( T^{8} - 271 T^{6} + \cdots + 217156 \)
|
| $53$ |
\( T^{8} - 153 T^{6} + \cdots + 164025 \)
|
| $59$ |
\( T^{8} - 255 T^{6} + \cdots + 54756 \)
|
| $61$ |
\( T^{8} - 266 T^{6} + \cdots + 5313025 \)
|
| $67$ |
\( (T^{4} + 9 T^{3} - 15 T^{2} + \cdots - 18)^{2} \)
|
| $71$ |
\( T^{8} - 322 T^{6} + \cdots + 4990756 \)
|
| $73$ |
\( T^{8} - 278 T^{6} + \cdots + 9333025 \)
|
| $79$ |
\( T^{8} - 219 T^{6} + \cdots + 1296 \)
|
| $83$ |
\( (T^{4} + 3 T^{3} + \cdots + 8854)^{2} \)
|
| $89$ |
\( T^{8} - 319 T^{6} + \cdots + 2062096 \)
|
| $97$ |
\( (T^{4} + T^{3} - 228 T^{2} + \cdots + 9448)^{2} \)
|
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