Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [847,4,Mod(1,847)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(847, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([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("847.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 847 = 7 \cdot 11^{2} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 847.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: | \(49.9746177749\) |
| 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: |
\( x^{8} - 2x^{7} - 43x^{6} + 53x^{5} + 520x^{4} - 375x^{3} - 1310x^{2} + 1404x - 296 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| 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
\( x^{8} - 2x^{7} - 43x^{6} + 53x^{5} + 520x^{4} - 375x^{3} - 1310x^{2} + 1404x - 296 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -5\nu^{7} + 8\nu^{6} + 215\nu^{5} - 163\nu^{4} - 2454\nu^{3} + 199\nu^{2} + 3836\nu - 1108 ) / 128 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -5\nu^{7} + 8\nu^{6} + 215\nu^{5} - 163\nu^{4} - 2582\nu^{3} + 455\nu^{2} + 6268\nu - 2900 ) / 128 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -5\nu^{7} + 8\nu^{6} + 215\nu^{5} - 163\nu^{4} - 2582\nu^{3} + 583\nu^{2} + 6140\nu - 4308 ) / 128 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -7\nu^{7} + 8\nu^{6} + 301\nu^{5} - 97\nu^{4} - 3490\nu^{3} - 739\nu^{2} + 6756\nu - 2172 ) / 128 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -7\nu^{7} + 8\nu^{6} + 317\nu^{5} - 145\nu^{4} - 4002\nu^{3} + 301\nu^{2} + 10516\nu - 5532 ) / 128 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -11\nu^{7} + 8\nu^{6} + 489\nu^{5} + 51\nu^{4} - 5946\nu^{3} - 3623\nu^{2} + 12836\nu - 620 ) / 128 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{4} - \beta_{3} + \beta _1 + 11 \)
|
| \(\nu^{3}\) | \(=\) |
\( 2\beta_{4} - 3\beta_{3} + \beta_{2} + 21\beta _1 + 8 \)
|
| \(\nu^{4}\) | \(=\) |
\( 2\beta_{7} - 2\beta_{6} - 4\beta_{5} + 28\beta_{4} - 26\beta_{3} + 2\beta_{2} + 45\beta _1 + 226 \)
|
| \(\nu^{5}\) | \(=\) |
\( 6\beta_{7} + 2\beta_{6} - 20\beta_{5} + 83\beta_{4} - 109\beta_{3} + 38\beta_{2} + 507\beta _1 + 429 \)
|
| \(\nu^{6}\) | \(=\) |
\( 82\beta_{7} - 82\beta_{6} - 204\beta_{5} + 796\beta_{4} - 697\beta_{3} + 121\beta_{2} + 1603\beta _1 + 5438 \)
|
| \(\nu^{7}\) | \(=\) |
\( 324\beta_{7} + 20\beta_{6} - 1056\beta_{5} + 2988\beta_{4} - 3522\beta_{3} + 1246\beta_{2} + 13399\beta _1 + 16070 \)
|
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 |
|
−4.60464 | 3.94660 | 13.2027 | −17.6105 | −18.1727 | 7.00000 | −23.9567 | −11.4244 | 81.0901 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −3.22083 | 2.34706 | 2.37377 | 11.1794 | −7.55949 | 7.00000 | 18.1211 | −21.4913 | −36.0069 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −0.367077 | −3.02533 | −7.86525 | −7.89286 | 1.11053 | 7.00000 | 5.82377 | −17.8474 | 2.89729 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0.152961 | −9.20128 | −7.97660 | 19.1558 | −1.40744 | 7.00000 | −2.44379 | 57.6636 | 2.93008 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0.702568 | 8.31370 | −7.50640 | −2.37642 | 5.84094 | 7.00000 | −10.8943 | 42.1177 | −1.66960 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 3.22432 | −3.46519 | 2.39627 | 2.67983 | −11.1729 | 7.00000 | −18.0682 | −14.9925 | 8.64065 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 4.70839 | 6.19073 | 14.1689 | 13.3936 | 29.1484 | 7.00000 | 29.0457 | 11.3252 | 63.0625 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 5.40431 | −3.10629 | 21.2066 | −20.5288 | −16.7874 | 7.00000 | 71.3724 | −17.3509 | −110.944 | |||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(7\) | \( -1 \) |
| \(11\) | \( -1 \) |
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 | 847.4.a.l | yes | 8 |
| 11.b | odd | 2 | 1 | 847.4.a.i | ✓ | 8 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 847.4.a.i | ✓ | 8 | 11.b | odd | 2 | 1 | |
| 847.4.a.l | yes | 8 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{8} - 6T_{2}^{7} - 29T_{2}^{6} + 191T_{2}^{5} + 140T_{2}^{4} - 1361T_{2}^{3} + 556T_{2}^{2} + 260T_{2} - 48 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(847))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} - 6 T^{7} + \cdots - 48 \)
|
| $3$ |
\( T^{8} - 2 T^{7} + \cdots + 142848 \)
|
| $5$ |
\( T^{8} + 2 T^{7} + \cdots + 52121480 \)
|
| $7$ |
\( (T - 7)^{8} \)
|
| $11$ |
\( T^{8} \)
|
| $13$ |
\( T^{8} + \cdots + 13713545074352 \)
|
| $17$ |
\( T^{8} + \cdots - 125260808505416 \)
|
| $19$ |
\( T^{8} + \cdots - 2911754048256 \)
|
| $23$ |
\( T^{8} + \cdots + 203191379919296 \)
|
| $29$ |
\( T^{8} + \cdots - 21\!\cdots\!92 \)
|
| $31$ |
\( T^{8} + \cdots + 17226111324160 \)
|
| $37$ |
\( T^{8} + \cdots + 51\!\cdots\!36 \)
|
| $41$ |
\( T^{8} + \cdots + 70\!\cdots\!76 \)
|
| $43$ |
\( T^{8} + \cdots - 49\!\cdots\!40 \)
|
| $47$ |
\( T^{8} + \cdots - 84\!\cdots\!24 \)
|
| $53$ |
\( T^{8} + \cdots + 41\!\cdots\!52 \)
|
| $59$ |
\( T^{8} + \cdots - 77\!\cdots\!76 \)
|
| $61$ |
\( T^{8} + \cdots - 17\!\cdots\!56 \)
|
| $67$ |
\( T^{8} + \cdots - 23\!\cdots\!56 \)
|
| $71$ |
\( T^{8} + \cdots + 27\!\cdots\!80 \)
|
| $73$ |
\( T^{8} + \cdots - 49\!\cdots\!04 \)
|
| $79$ |
\( T^{8} + \cdots + 17\!\cdots\!40 \)
|
| $83$ |
\( T^{8} + \cdots + 49\!\cdots\!92 \)
|
| $89$ |
\( T^{8} + \cdots - 32\!\cdots\!28 \)
|
| $97$ |
\( T^{8} + \cdots + 13\!\cdots\!00 \)
|
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