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: |
\( x^{8} - 2x^{7} - 45x^{6} + 73x^{5} + 584x^{4} - 787x^{3} - 2076x^{2} + 1820x + 1488 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2 \) |
| 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
\( x^{8} - 2x^{7} - 45x^{6} + 73x^{5} + 584x^{4} - 787x^{3} - 2076x^{2} + 1820x + 1488 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 12 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -3\nu^{7} - 65\nu^{6} + 232\nu^{5} + 2447\nu^{4} - 3901\nu^{3} - 22022\nu^{2} + 15632\nu + 30592 ) / 3568 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -25\nu^{7} + 53\nu^{6} + 744\nu^{5} - 1611\nu^{4} - 991\nu^{3} + 15102\nu^{2} - 52296\nu - 17424 ) / 3568 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -23\nu^{7} + 22\nu^{6} + 961\nu^{5} - 269\nu^{4} - 11696\nu^{3} - 1065\nu^{2} + 42390\nu + 240 ) / 1784 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -49\nu^{7} - 21\nu^{6} + 2154\nu^{5} + 1909\nu^{4} - 23725\nu^{3} - 31288\nu^{2} + 36188\nu + 102432 ) / 3568 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -16\nu^{7} + 25\nu^{6} + 717\nu^{5} - 924\nu^{4} - 8689\nu^{3} + 9585\nu^{2} + 21674\nu - 10188 ) / 892 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 12 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{6} - \beta_{5} - \beta_{3} + 2\beta_{2} + 18\beta _1 + 4 \)
|
| \(\nu^{4}\) | \(=\) |
\( -2\beta_{7} + 4\beta_{6} - 2\beta_{4} - 6\beta_{3} + 28\beta_{2} + 5\beta _1 + 240 \)
|
| \(\nu^{5}\) | \(=\) |
\( 2\beta_{7} + 38\beta_{6} - 34\beta_{5} - 12\beta_{4} - 42\beta_{3} + 83\beta_{2} + 382\beta _1 + 234 \)
|
| \(\nu^{6}\) | \(=\) |
\( -70\beta_{7} + 171\beta_{6} - 15\beta_{5} - 92\beta_{4} - 303\beta_{3} + 762\beta_{2} + 302\beta _1 + 5586 \)
|
| \(\nu^{7}\) | \(=\) |
\( 40\beta_{7} + 1196\beta_{6} - 1004\beta_{5} - 566\beta_{4} - 1466\beta_{3} + 2806\beta_{2} + 8881\beta _1 + 9734 \)
|
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 |
|
−4.67671 | −6.60806 | 13.8716 | 11.8609 | 30.9040 | 7.00000 | −27.4597 | 16.6665 | −55.4702 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −4.00760 | 6.42839 | 8.06082 | −10.5476 | −25.7624 | 7.00000 | −0.243753 | 14.3241 | 42.2704 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.90659 | −9.91818 | −4.36491 | −19.7809 | 18.9099 | 7.00000 | 23.5748 | 71.3704 | 37.7142 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −0.562643 | 3.85167 | −7.68343 | 19.5413 | −2.16712 | 7.00000 | 8.82417 | −12.1646 | −10.9948 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 1.44396 | 3.39708 | −5.91498 | −15.8594 | 4.90524 | 7.00000 | −20.0927 | −15.4599 | −22.9003 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 2.48954 | −5.42566 | −1.80220 | 4.00309 | −13.5074 | 7.00000 | −24.4030 | 2.43778 | 9.96584 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 3.79499 | 9.82995 | 6.40195 | 2.41156 | 37.3045 | 7.00000 | −6.06460 | 69.6278 | 9.15184 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 5.42505 | 0.444824 | 21.4312 | 10.3710 | 2.41319 | 7.00000 | 72.8647 | −26.8021 | 56.2631 | |||||||||||||||||||||||||||||||||||||||||||
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.k | yes | 8 |
| 11.b | odd | 2 | 1 | 847.4.a.j | ✓ | 8 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 847.4.a.j | ✓ | 8 | 11.b | odd | 2 | 1 | |
| 847.4.a.k | yes | 8 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{8} - 2T_{2}^{7} - 45T_{2}^{6} + 73T_{2}^{5} + 584T_{2}^{4} - 787T_{2}^{3} - 2076T_{2}^{2} + 1820T_{2} + 1488 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(847))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} - 2 T^{7} + \cdots + 1488 \)
|
| $3$ |
\( T^{8} - 2 T^{7} + \cdots - 130784 \)
|
| $5$ |
\( T^{8} - 2 T^{7} + \cdots - 76784220 \)
|
| $7$ |
\( (T - 7)^{8} \)
|
| $11$ |
\( T^{8} \)
|
| $13$ |
\( T^{8} + \cdots - 35644092144 \)
|
| $17$ |
\( T^{8} + \cdots - 16139110559904 \)
|
| $19$ |
\( T^{8} + \cdots + 134898129615744 \)
|
| $23$ |
\( T^{8} + \cdots - 830693323383552 \)
|
| $29$ |
\( T^{8} + \cdots + 16\!\cdots\!64 \)
|
| $31$ |
\( T^{8} + \cdots - 603630047457280 \)
|
| $37$ |
\( T^{8} + \cdots + 13\!\cdots\!36 \)
|
| $41$ |
\( T^{8} + \cdots + 17\!\cdots\!24 \)
|
| $43$ |
\( T^{8} + \cdots - 10\!\cdots\!60 \)
|
| $47$ |
\( T^{8} + \cdots + 18\!\cdots\!44 \)
|
| $53$ |
\( T^{8} + \cdots + 61\!\cdots\!16 \)
|
| $59$ |
\( T^{8} + \cdots + 13\!\cdots\!52 \)
|
| $61$ |
\( T^{8} + \cdots + 25\!\cdots\!96 \)
|
| $67$ |
\( T^{8} + \cdots + 24\!\cdots\!52 \)
|
| $71$ |
\( T^{8} + \cdots - 21\!\cdots\!80 \)
|
| $73$ |
\( T^{8} + \cdots + 89\!\cdots\!88 \)
|
| $79$ |
\( T^{8} + \cdots + 15\!\cdots\!00 \)
|
| $83$ |
\( T^{8} + \cdots + 10\!\cdots\!84 \)
|
| $89$ |
\( T^{8} + \cdots + 49\!\cdots\!28 \)
|
| $97$ |
\( T^{8} + \cdots + 16\!\cdots\!80 \)
|
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