Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1617,4,Mod(1,1617)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1617, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1617.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1617 = 3 \cdot 7^{2} \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1617.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: | \(95.4060884793\) |
| Analytic rank: | \(0\) |
| Dimension: | \(10\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{10} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{10} - 4 x^{9} - 45 x^{8} + 168 x^{7} + 651 x^{6} - 2176 x^{5} - 3439 x^{4} + 8716 x^{3} + 7840 x^{2} + \cdots - 4032 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{17}]\) |
| Coefficient ring index: | \( 2^{4} \) |
| Twist minimal: | no (minimal twist has level 231) |
| 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 \(1,\beta_1,\ldots,\beta_{9}\) 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
\( x^{10} - 4 x^{9} - 45 x^{8} + 168 x^{7} + 651 x^{6} - 2176 x^{5} - 3439 x^{4} + 8716 x^{3} + 7840 x^{2} + \cdots - 4032 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 10 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - \nu^{2} - 17\nu + 7 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 13 \nu^{9} + 79 \nu^{8} + 432 \nu^{7} - 3144 \nu^{6} - 2391 \nu^{5} + 35809 \nu^{4} - 24404 \nu^{3} + \cdots + 27840 ) / 384 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 11 \nu^{9} - 65 \nu^{8} - 384 \nu^{7} + 2616 \nu^{6} + 2721 \nu^{5} - 30335 \nu^{4} + 13132 \nu^{3} + \cdots - 29568 ) / 192 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 25 \nu^{9} + 139 \nu^{8} + 888 \nu^{7} - 5544 \nu^{6} - 6747 \nu^{5} + 63397 \nu^{4} - 24044 \nu^{3} + \cdots + 49728 ) / 384 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 9 \nu^{9} + 51 \nu^{8} + 320 \nu^{7} - 2040 \nu^{6} - 2475 \nu^{5} + 23453 \nu^{4} - 7668 \nu^{3} + \cdots + 19136 ) / 128 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 37 \nu^{9} + 211 \nu^{8} + 1308 \nu^{7} - 8448 \nu^{6} - 9807 \nu^{5} + 97405 \nu^{4} + \cdots + 97152 ) / 192 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 121 \nu^{9} + 691 \nu^{8} + 4272 \nu^{7} - 27624 \nu^{6} - 31899 \nu^{5} + 317245 \nu^{4} + \cdots + 280512 ) / 384 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 10 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + \beta_{2} + 18\beta _1 + 3 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{8} - \beta_{7} - \beta_{6} + \beta_{5} + 2\beta_{3} + 26\beta_{2} + 29\beta _1 + 173 \)
|
| \(\nu^{5}\) | \(=\) |
\( 2\beta_{9} - 8\beta_{7} - 2\beta_{4} + 30\beta_{3} + 44\beta_{2} + 377\beta _1 + 110 \)
|
| \(\nu^{6}\) | \(=\) |
\( 12 \beta_{9} + 26 \beta_{8} - 50 \beta_{7} - 42 \beta_{6} + 42 \beta_{5} - 4 \beta_{4} + 82 \beta_{3} + \cdots + 3466 \)
|
| \(\nu^{7}\) | \(=\) |
\( 108 \beta_{9} - 10 \beta_{8} - 358 \beta_{7} - 38 \beta_{6} + 14 \beta_{5} - 108 \beta_{4} + 787 \beta_{3} + \cdots + 3815 \)
|
| \(\nu^{8}\) | \(=\) |
\( 612 \beta_{9} + 543 \beta_{8} - 1775 \beta_{7} - 1407 \beta_{6} + 1271 \beta_{5} - 244 \beta_{4} + \cdots + 74493 \)
|
| \(\nu^{9}\) | \(=\) |
\( 4038 \beta_{9} - 566 \beta_{8} - 11874 \beta_{7} - 2410 \beta_{6} + 786 \beta_{5} - 3766 \beta_{4} + \cdots + 123466 \)
|
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 |
|
−4.49955 | 3.00000 | 12.2460 | −9.43139 | −13.4987 | 0 | −19.1049 | 9.00000 | 42.4370 | ||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −4.05895 | 3.00000 | 8.47509 | 17.0425 | −12.1769 | 0 | −1.92836 | 9.00000 | −69.1748 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.71637 | 3.00000 | −5.05408 | 14.8426 | −5.14911 | 0 | 22.4056 | 9.00000 | −25.4753 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.70299 | 3.00000 | −5.09984 | −3.64858 | −5.10896 | 0 | 22.3088 | 9.00000 | 6.21348 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −0.373481 | 3.00000 | −7.86051 | −15.6838 | −1.12044 | 0 | 5.92360 | 9.00000 | 5.85760 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 1.11872 | 3.00000 | −6.74846 | −4.70241 | 3.35616 | 0 | −16.4994 | 9.00000 | −5.26068 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 2.38864 | 3.00000 | −2.29442 | 9.60191 | 7.16591 | 0 | −24.5896 | 9.00000 | 22.9355 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 3.53481 | 3.00000 | 4.49490 | −15.9961 | 10.6044 | 0 | −12.3899 | 9.00000 | −56.5431 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 4.14909 | 3.00000 | 9.21491 | 21.1121 | 12.4473 | 0 | 5.04076 | 9.00000 | 87.5957 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 5.16008 | 3.00000 | 18.6265 | 6.86319 | 15.4802 | 0 | 54.8334 | 9.00000 | 35.4146 | |||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(7\) | \(-1\) |
| \(11\) | \(1\) |
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 | 1617.4.a.z | 10 | |
| 7.b | odd | 2 | 1 | 1617.4.a.y | 10 | ||
| 7.d | odd | 6 | 2 | 231.4.i.b | ✓ | 20 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 231.4.i.b | ✓ | 20 | 7.d | odd | 6 | 2 | |
| 1617.4.a.y | 10 | 7.b | odd | 2 | 1 | ||
| 1617.4.a.z | 10 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1617))\):
|
\( T_{2}^{10} - 4 T_{2}^{9} - 45 T_{2}^{8} + 168 T_{2}^{7} + 651 T_{2}^{6} - 2176 T_{2}^{5} - 3439 T_{2}^{4} + \cdots - 4032 \)
|
|
\( T_{5}^{10} - 20 T_{5}^{9} - 661 T_{5}^{8} + 12588 T_{5}^{7} + 151892 T_{5}^{6} - 2602348 T_{5}^{5} + \cdots - 14287026432 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{10} - 4 T^{9} + \cdots - 4032 \)
|
| $3$ |
\( (T - 3)^{10} \)
|
| $5$ |
\( T^{10} + \cdots - 14287026432 \)
|
| $7$ |
\( T^{10} \)
|
| $11$ |
\( (T + 11)^{10} \)
|
| $13$ |
\( T^{10} + \cdots - 41\!\cdots\!24 \)
|
| $17$ |
\( T^{10} + \cdots + 80\!\cdots\!13 \)
|
| $19$ |
\( T^{10} + \cdots - 41\!\cdots\!12 \)
|
| $23$ |
\( T^{10} + \cdots - 16\!\cdots\!33 \)
|
| $29$ |
\( T^{10} + \cdots + 49\!\cdots\!08 \)
|
| $31$ |
\( T^{10} + \cdots - 15\!\cdots\!16 \)
|
| $37$ |
\( T^{10} + \cdots + 12\!\cdots\!96 \)
|
| $41$ |
\( T^{10} + \cdots + 21\!\cdots\!44 \)
|
| $43$ |
\( T^{10} + \cdots + 71\!\cdots\!12 \)
|
| $47$ |
\( T^{10} + \cdots + 52\!\cdots\!51 \)
|
| $53$ |
\( T^{10} + \cdots + 33\!\cdots\!72 \)
|
| $59$ |
\( T^{10} + \cdots + 95\!\cdots\!52 \)
|
| $61$ |
\( T^{10} + \cdots - 16\!\cdots\!44 \)
|
| $67$ |
\( T^{10} + \cdots + 34\!\cdots\!92 \)
|
| $71$ |
\( T^{10} + \cdots - 54\!\cdots\!84 \)
|
| $73$ |
\( T^{10} + \cdots + 18\!\cdots\!16 \)
|
| $79$ |
\( T^{10} + \cdots - 28\!\cdots\!04 \)
|
| $83$ |
\( T^{10} + \cdots + 18\!\cdots\!48 \)
|
| $89$ |
\( T^{10} + \cdots + 30\!\cdots\!64 \)
|
| $97$ |
\( T^{10} + \cdots + 15\!\cdots\!81 \)
|
| show more | |
| show less |