Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [17,16,Mod(1,17)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(17, 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("17.1");
S:= CuspForms(chi, 16);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 17 \) |
| Weight: | \( k \) | \(=\) | \( 16 \) |
| Character orbit: | \([\chi]\) | \(=\) | 17.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: | \(24.2578958670\) |
| Analytic rank: | \(1\) |
| Dimension: | \(9\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{9} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{9} - x^{8} - 189946 x^{7} + 135236 x^{6} + 10791723288 x^{5} + 17160527744 x^{4} + \cdots + 15\!\cdots\!32 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{21}\cdot 3^{4}\cdot 5 \) |
| 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 \(1,\beta_1,\ldots,\beta_{8}\) 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
\( x^{9} - x^{8} - 189946 x^{7} + 135236 x^{6} + 10791723288 x^{5} + 17160527744 x^{4} + \cdots + 15\!\cdots\!32 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 216204551700325 \nu^{8} + \cdots + 23\!\cdots\!80 ) / 18\!\cdots\!56 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 216204551700325 \nu^{8} + \cdots - 53\!\cdots\!80 ) / 18\!\cdots\!56 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 97587485477591 \nu^{8} + \cdots - 90\!\cdots\!24 ) / 16\!\cdots\!96 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 25\!\cdots\!83 \nu^{8} + \cdots + 69\!\cdots\!48 ) / 18\!\cdots\!56 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 25\!\cdots\!41 \nu^{8} + \cdots + 72\!\cdots\!56 ) / 22\!\cdots\!32 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 22\!\cdots\!87 \nu^{8} + \cdots - 17\!\cdots\!68 ) / 18\!\cdots\!56 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 23\!\cdots\!07 \nu^{8} + \cdots + 10\!\cdots\!24 ) / 16\!\cdots\!96 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} - \beta_{2} + 42210 \)
|
| \(\nu^{3}\) | \(=\) |
\( 7\beta_{8} - 2\beta_{7} + 12\beta_{6} - 4\beta_{5} + 28\beta_{4} - 18\beta_{3} - 389\beta_{2} + 76377\beta _1 + 9659 \)
|
| \(\nu^{4}\) | \(=\) |
\( - 445 \beta_{8} - 1128 \beta_{7} - 1756 \beta_{6} + 1792 \beta_{5} - 1078 \beta_{4} + 93468 \beta_{3} + \cdots + 3221366031 \)
|
| \(\nu^{5}\) | \(=\) |
\( 894229 \beta_{8} - 25568 \beta_{7} + 1632284 \beta_{6} - 378608 \beta_{5} + 2542574 \beta_{4} + \cdots - 10479340047 \)
|
| \(\nu^{6}\) | \(=\) |
\( - 87624729 \beta_{8} - 178108328 \beta_{7} - 281203692 \beta_{6} + 219161888 \beta_{5} + \cdots + 271281409963515 \)
|
| \(\nu^{7}\) | \(=\) |
\( 96013490677 \beta_{8} + 13925012840 \beta_{7} + 185016379580 \beta_{6} - 25853396192 \beta_{5} + \cdots - 23\!\cdots\!59 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 12074875278033 \beta_{8} - 21893026460776 \beta_{7} - 34306248149132 \beta_{6} + \cdots + 23\!\cdots\!39 \)
|
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 |
|
−321.409 | 3359.94 | 70535.4 | 63959.1 | −1.07991e6 | −1.15696e6 | −1.21388e7 | −3.05968e6 | −2.05570e7 | |||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −305.946 | −5406.74 | 60834.9 | −301966. | 1.65417e6 | −3.95939e6 | −8.58694e6 | 1.48839e7 | 9.23853e7 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −145.770 | 1399.13 | −11519.2 | 66736.6 | −203950. | −521340. | 6.45573e6 | −1.23913e7 | −9.72817e6 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −116.670 | −4782.96 | −19156.2 | −127925. | 558026. | 2.43638e6 | 6.05798e6 | 8.52783e6 | 1.49249e7 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0.512949 | −7189.10 | −32767.7 | 268107. | −3687.64 | −1.34917e6 | −33616.5 | 3.73343e7 | 137525. | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 36.7936 | 3968.81 | −31414.2 | −83963.3 | 146027. | 3.01451e6 | −2.36149e6 | 1.40254e6 | −3.08931e6 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 106.844 | 5068.23 | −21352.5 | 78797.3 | 541508. | −3.75874e6 | −5.78242e6 | 1.13381e7 | 8.41898e6 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 243.626 | −1648.77 | 26585.7 | 150751. | −401684. | −3.14371e6 | −1.50616e6 | −1.16305e7 | 3.67269e7 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 285.017 | 1575.46 | 48466.8 | −337355. | 449035. | −311650. | 4.47444e6 | −1.18668e7 | −9.61519e7 | ||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(17\) | \(-1\) |
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 | 17.16.a.a | ✓ | 9 |
| 3.b | odd | 2 | 1 | 153.16.a.c | 9 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 17.16.a.a | ✓ | 9 | 1.a | even | 1 | 1 | trivial |
| 153.16.a.c | 9 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{9} + 217 T_{2}^{8} - 169018 T_{2}^{7} - 30868820 T_{2}^{6} + 8517240408 T_{2}^{5} + \cdots - 23\!\cdots\!96 \)
acting on \(S_{16}^{\mathrm{new}}(\Gamma_0(17))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{9} + \cdots - 23\!\cdots\!96 \)
|
| $3$ |
\( T^{9} + \cdots - 45\!\cdots\!04 \)
|
| $5$ |
\( T^{9} + \cdots - 14\!\cdots\!00 \)
|
| $7$ |
\( T^{9} + \cdots + 87\!\cdots\!08 \)
|
| $11$ |
\( T^{9} + \cdots - 64\!\cdots\!08 \)
|
| $13$ |
\( T^{9} + \cdots - 81\!\cdots\!96 \)
|
| $17$ |
\( (T - 410338673)^{9} \)
|
| $19$ |
\( T^{9} + \cdots - 19\!\cdots\!68 \)
|
| $23$ |
\( T^{9} + \cdots + 60\!\cdots\!00 \)
|
| $29$ |
\( T^{9} + \cdots + 54\!\cdots\!00 \)
|
| $31$ |
\( T^{9} + \cdots + 13\!\cdots\!52 \)
|
| $37$ |
\( T^{9} + \cdots + 61\!\cdots\!24 \)
|
| $41$ |
\( T^{9} + \cdots - 36\!\cdots\!96 \)
|
| $43$ |
\( T^{9} + \cdots + 63\!\cdots\!04 \)
|
| $47$ |
\( T^{9} + \cdots - 58\!\cdots\!76 \)
|
| $53$ |
\( T^{9} + \cdots - 80\!\cdots\!48 \)
|
| $59$ |
\( T^{9} + \cdots - 38\!\cdots\!68 \)
|
| $61$ |
\( T^{9} + \cdots - 68\!\cdots\!00 \)
|
| $67$ |
\( T^{9} + \cdots - 62\!\cdots\!00 \)
|
| $71$ |
\( T^{9} + \cdots + 75\!\cdots\!48 \)
|
| $73$ |
\( T^{9} + \cdots + 10\!\cdots\!36 \)
|
| $79$ |
\( T^{9} + \cdots + 11\!\cdots\!12 \)
|
| $83$ |
\( T^{9} + \cdots - 22\!\cdots\!72 \)
|
| $89$ |
\( T^{9} + \cdots - 33\!\cdots\!96 \)
|
| $97$ |
\( T^{9} + \cdots - 36\!\cdots\!00 \)
|
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