Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [17,10,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, 10, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("17.1");
S:= CuspForms(chi, 10);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 17 \) |
| Weight: | \( k \) | \(=\) | \( 10 \) |
| 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: | \(8.75560921479\) |
| Analytic rank: | \(0\) |
| Dimension: | \(7\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{7} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{7} - x^{6} - 2986x^{5} + 8252x^{4} + 2252056x^{3} - 10388768x^{2} - 243559296x - 675998208 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{10}\cdot 3^{4} \) |
| 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 \(1,\beta_1,\ldots,\beta_{6}\) 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
\( x^{7} - x^{6} - 2986x^{5} + 8252x^{4} + 2252056x^{3} - 10388768x^{2} - 243559296x - 675998208 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 8711 \nu^{6} + 479085 \nu^{5} - 21966986 \nu^{4} - 962897524 \nu^{3} + 9962276152 \nu^{2} + \cdots + 1595734267008 ) / 3478080000 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 41053 \nu^{6} + 292545 \nu^{5} + 124232878 \nu^{4} - 1085670148 \nu^{3} + \cdots + 1729601433216 ) / 6260544000 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 41053 \nu^{6} + 292545 \nu^{5} + 124232878 \nu^{4} - 1085670148 \nu^{3} + \cdots + 7069845465216 ) / 6260544000 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 843359 \nu^{6} + 2902635 \nu^{5} + 2465651834 \nu^{4} - 12878937644 \nu^{3} + \cdots + 113709524082048 ) / 31302720000 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 419317 \nu^{6} - 2816505 \nu^{5} - 1224135742 \nu^{4} + 10192644772 \nu^{3} + \cdots - 59384105783424 ) / 10434240000 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{4} + \beta_{3} - 3\beta _1 + 853 \)
|
| \(\nu^{3}\) | \(=\) |
\( 8\beta_{6} + 17\beta_{5} - 16\beta_{4} - \beta_{3} + 10\beta_{2} + 1488\beta _1 - 2467 \)
|
| \(\nu^{4}\) | \(=\) |
\( 336\beta_{6} + 203\beta_{5} - 556\beta_{4} + 1817\beta_{3} + 94\beta_{2} - 6760\beta _1 + 1257619 \)
|
| \(\nu^{5}\) | \(=\) |
\( 14912\beta_{6} + 35837\beta_{5} - 46600\beta_{4} + 1331\beta_{3} + 27714\beta_{2} + 2442620\beta _1 - 5771223 \)
|
| \(\nu^{6}\) | \(=\) |
\( 911488 \beta_{6} + 420111 \beta_{5} + 535832 \beta_{4} + 3254641 \beta_{3} + 217494 \beta_{2} + \cdots + 2057396547 \)
|
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 |
|
−42.3973 | 109.740 | 1285.53 | −2498.37 | −4652.67 | 2872.61 | −32795.8 | −7640.20 | 105924. | |||||||||||||||||||||||||||||||||||||||
| 1.2 | −28.6400 | 243.971 | 308.250 | 1776.79 | −6987.32 | −9598.61 | 5835.40 | 39838.7 | −50887.2 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | −16.8116 | −116.887 | −229.369 | −1103.40 | 1965.06 | −5164.29 | 12463.6 | −6020.47 | 18549.9 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | 4.12962 | −254.074 | −494.946 | 151.544 | −1049.23 | 9407.97 | −4158.31 | 44870.8 | 625.818 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | 5.44491 | 106.475 | −482.353 | 1303.94 | 579.746 | 9199.27 | −5414.17 | −8346.12 | 7099.84 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | 34.1532 | 169.801 | 654.438 | 195.287 | 5799.26 | −356.628 | 4864.71 | 9149.54 | 6669.66 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | 43.1213 | −171.025 | 1347.45 | 1536.21 | −7374.84 | 3027.69 | 36025.5 | 9566.70 | 66243.5 | ||||||||||||||||||||||||||||||||||||||||
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 | 17.10.a.b | ✓ | 7 |
| 3.b | odd | 2 | 1 | 153.10.a.f | 7 | ||
| 4.b | odd | 2 | 1 | 272.10.a.g | 7 | ||
| 17.b | even | 2 | 1 | 289.10.a.b | 7 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 17.10.a.b | ✓ | 7 | 1.a | even | 1 | 1 | trivial |
| 153.10.a.f | 7 | 3.b | odd | 2 | 1 | ||
| 272.10.a.g | 7 | 4.b | odd | 2 | 1 | ||
| 289.10.a.b | 7 | 17.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{7} + T_{2}^{6} - 2986T_{2}^{5} - 8252T_{2}^{4} + 2252056T_{2}^{3} + 10388768T_{2}^{2} - 243559296T_{2} + 675998208 \)
acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(17))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{7} + T^{6} + \cdots + 675998208 \)
|
| $3$ |
\( T^{7} + \cdots + 24\!\cdots\!00 \)
|
| $5$ |
\( T^{7} + \cdots - 29\!\cdots\!00 \)
|
| $7$ |
\( T^{7} + \cdots + 13\!\cdots\!04 \)
|
| $11$ |
\( T^{7} + \cdots - 12\!\cdots\!00 \)
|
| $13$ |
\( T^{7} + \cdots + 51\!\cdots\!32 \)
|
| $17$ |
\( (T - 83521)^{7} \)
|
| $19$ |
\( T^{7} + \cdots + 35\!\cdots\!00 \)
|
| $23$ |
\( T^{7} + \cdots + 31\!\cdots\!72 \)
|
| $29$ |
\( T^{7} + \cdots + 34\!\cdots\!60 \)
|
| $31$ |
\( T^{7} + \cdots + 15\!\cdots\!00 \)
|
| $37$ |
\( T^{7} + \cdots + 26\!\cdots\!04 \)
|
| $41$ |
\( T^{7} + \cdots + 11\!\cdots\!40 \)
|
| $43$ |
\( T^{7} + \cdots + 32\!\cdots\!64 \)
|
| $47$ |
\( T^{7} + \cdots - 18\!\cdots\!00 \)
|
| $53$ |
\( T^{7} + \cdots - 68\!\cdots\!00 \)
|
| $59$ |
\( T^{7} + \cdots + 53\!\cdots\!60 \)
|
| $61$ |
\( T^{7} + \cdots - 16\!\cdots\!00 \)
|
| $67$ |
\( T^{7} + \cdots + 18\!\cdots\!16 \)
|
| $71$ |
\( T^{7} + \cdots - 21\!\cdots\!16 \)
|
| $73$ |
\( T^{7} + \cdots + 86\!\cdots\!04 \)
|
| $79$ |
\( T^{7} + \cdots - 18\!\cdots\!00 \)
|
| $83$ |
\( T^{7} + \cdots - 61\!\cdots\!12 \)
|
| $89$ |
\( T^{7} + \cdots - 45\!\cdots\!00 \)
|
| $97$ |
\( T^{7} + \cdots + 15\!\cdots\!00 \)
|
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