Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [17,12,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, 12, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("17.1");
S:= CuspForms(chi, 12);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 17 \) |
| Weight: | \( k \) | \(=\) | \( 12 \) |
| 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: | \(13.0618340695\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{8} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - x^{7} - 13381 x^{6} - 182353 x^{5} + 49101741 x^{4} + 1188560917 x^{3} - 22633823135 x^{2} + \cdots + 2663203205942 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{8}\cdot 3 \) |
| 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
\( x^{8} - x^{7} - 13381 x^{6} - 182353 x^{5} + 49101741 x^{4} + 1188560917 x^{3} - 22633823135 x^{2} + \cdots + 2663203205942 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 931929083 \nu^{7} - 20518519750 \nu^{6} - 13973604268129 \nu^{5} - 103613033669818 \nu^{4} + \cdots - 88\!\cdots\!98 ) / 30\!\cdots\!24 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 35503553 \nu^{7} - 545037442 \nu^{6} - 446940197443 \nu^{5} - 840392849038 \nu^{4} + \cdots - 34\!\cdots\!58 ) / 82\!\cdots\!76 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 35503553 \nu^{7} + 545037442 \nu^{6} + 446940197443 \nu^{5} + 840392849038 \nu^{4} + \cdots - 10\!\cdots\!26 ) / 41\!\cdots\!88 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 5616807583 \nu^{7} + 254717708030 \nu^{6} + 56016003192221 \nu^{5} + \cdots + 47\!\cdots\!86 ) / 60\!\cdots\!48 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 475278779 \nu^{7} + 1423172102 \nu^{6} - 5816485512157 \nu^{5} - 143912068051258 \nu^{4} + \cdots - 13\!\cdots\!78 ) / 25\!\cdots\!52 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 1933879615 \nu^{7} + 57902991610 \nu^{6} - 27502847914805 \nu^{5} + \cdots - 94\!\cdots\!18 ) / 75\!\cdots\!56 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{4} + 2\beta_{3} + 21\beta _1 + 3343 \)
|
| \(\nu^{3}\) | \(=\) |
\( -9\beta_{7} + 16\beta_{6} + 18\beta_{5} + 26\beta_{4} + 70\beta_{3} + 6\beta_{2} + 6049\beta _1 + 72645 \)
|
| \(\nu^{4}\) | \(=\) |
\( - 209 \beta_{7} + 152 \beta_{6} + 626 \beta_{5} + 7089 \beta_{4} + 16644 \beta_{3} - 738 \beta_{2} + \cdots + 20279848 \)
|
| \(\nu^{5}\) | \(=\) |
\( - 84712 \beta_{7} + 160384 \beta_{6} + 152176 \beta_{5} + 341070 \beta_{4} + 820812 \beta_{3} + \cdots + 858439596 \)
|
| \(\nu^{6}\) | \(=\) |
\( - 3339926 \beta_{7} + 5288672 \beta_{6} + 10174028 \beta_{5} + 52990861 \beta_{4} + 131740070 \beta_{3} + \cdots + 138291828781 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 730271167 \beta_{7} + 1406278112 \beta_{6} + 1301978878 \beta_{5} + 3448122842 \beta_{4} + \cdots + 8021513874987 \)
|
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 |
|
−85.3600 | 403.021 | 5238.33 | 11654.5 | −34401.9 | 59759.3 | −272327. | −14721.1 | −994831. | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −73.4849 | −107.536 | 3352.03 | −2672.40 | 7902.29 | −72464.9 | −95826.5 | −165583. | 196381. | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −8.67209 | −94.4314 | −1972.79 | −10913.0 | 818.918 | 68029.8 | 34868.7 | −168230. | 94638.1 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −0.484371 | −652.991 | −2047.77 | −5436.11 | 316.290 | −59647.9 | 1983.87 | 249250. | 2633.09 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 22.2215 | 818.803 | −1554.21 | 6987.51 | 18195.0 | 31303.5 | −80046.4 | 493291. | 155273. | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 49.8963 | −593.995 | 441.639 | 6656.68 | −29638.1 | 31706.0 | −80151.5 | 175683. | 332143. | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 72.9081 | 470.639 | 3267.59 | −7122.79 | 34313.4 | 82392.3 | 88918.0 | 44353.7 | −519309. | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 77.9755 | 252.491 | 4032.18 | 9437.53 | 19688.1 | −45790.1 | 154717. | −113395. | 735896. | |||||||||||||||||||||||||||||||||||||||||||
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 | 17.12.a.b | ✓ | 8 |
| 3.b | odd | 2 | 1 | 153.12.a.d | 8 | ||
| 4.b | odd | 2 | 1 | 272.12.a.h | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 17.12.a.b | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 153.12.a.d | 8 | 3.b | odd | 2 | 1 | ||
| 272.12.a.h | 8 | 4.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{8} - 55 T_{2}^{7} - 12058 T_{2}^{6} + 726176 T_{2}^{5} + 33040416 T_{2}^{4} - 2383120192 T_{2}^{3} + \cdots + 166084534272 \)
acting on \(S_{12}^{\mathrm{new}}(\Gamma_0(17))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} + \cdots + 166084534272 \)
|
| $3$ |
\( T^{8} + \cdots + 15\!\cdots\!08 \)
|
| $5$ |
\( T^{8} + \cdots + 57\!\cdots\!00 \)
|
| $7$ |
\( T^{8} + \cdots - 65\!\cdots\!16 \)
|
| $11$ |
\( T^{8} + \cdots + 71\!\cdots\!68 \)
|
| $13$ |
\( T^{8} + \cdots + 50\!\cdots\!12 \)
|
| $17$ |
\( (T + 1419857)^{8} \)
|
| $19$ |
\( T^{8} + \cdots + 61\!\cdots\!68 \)
|
| $23$ |
\( T^{8} + \cdots + 12\!\cdots\!60 \)
|
| $29$ |
\( T^{8} + \cdots + 22\!\cdots\!60 \)
|
| $31$ |
\( T^{8} + \cdots + 28\!\cdots\!96 \)
|
| $37$ |
\( T^{8} + \cdots - 55\!\cdots\!08 \)
|
| $41$ |
\( T^{8} + \cdots - 23\!\cdots\!08 \)
|
| $43$ |
\( T^{8} + \cdots + 27\!\cdots\!52 \)
|
| $47$ |
\( T^{8} + \cdots - 15\!\cdots\!92 \)
|
| $53$ |
\( T^{8} + \cdots - 46\!\cdots\!36 \)
|
| $59$ |
\( T^{8} + \cdots + 27\!\cdots\!44 \)
|
| $61$ |
\( T^{8} + \cdots - 20\!\cdots\!40 \)
|
| $67$ |
\( T^{8} + \cdots - 41\!\cdots\!00 \)
|
| $71$ |
\( T^{8} + \cdots + 13\!\cdots\!16 \)
|
| $73$ |
\( T^{8} + \cdots - 43\!\cdots\!12 \)
|
| $79$ |
\( T^{8} + \cdots + 38\!\cdots\!48 \)
|
| $83$ |
\( T^{8} + \cdots + 78\!\cdots\!88 \)
|
| $89$ |
\( T^{8} + \cdots + 54\!\cdots\!56 \)
|
| $97$ |
\( T^{8} + \cdots - 49\!\cdots\!00 \)
|
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