Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [847,6,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, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("847.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 847 = 7 \cdot 11^{2} \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| 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: | \(135.845095382\) |
| 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: |
\( x^{8} - 4x^{7} - 203x^{6} + 554x^{5} + 12647x^{4} - 17272x^{3} - 249337x^{2} - 8670x + 225324 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{5}\cdot 3 \) |
| Twist minimal: | no (minimal twist has level 77) |
| 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
\( x^{8} - 4x^{7} - 203x^{6} + 554x^{5} + 12647x^{4} - 17272x^{3} - 249337x^{2} - 8670x + 225324 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 52 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 68 \nu^{7} + 611 \nu^{6} + 12002 \nu^{5} - 98964 \nu^{4} - 631392 \nu^{3} + 4097905 \nu^{2} + \cdots - 23224908 ) / 980112 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 459 \nu^{7} - 3395 \nu^{6} - 64970 \nu^{5} + 495904 \nu^{4} + 531053 \nu^{3} - 17443337 \nu^{2} + \cdots + 129578772 ) / 3920448 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 1441 \nu^{7} - 8701 \nu^{6} - 276214 \nu^{5} + 1210224 \nu^{4} + 16213719 \nu^{3} + \cdots + 15576972 ) / 3920448 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 243 \nu^{7} - 1111 \nu^{6} - 38514 \nu^{5} + 119776 \nu^{4} + 1516581 \nu^{3} - 1866709 \nu^{2} + \cdots - 13951260 ) / 653408 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 244 \nu^{7} + 553 \nu^{6} - 54734 \nu^{5} - 98230 \nu^{4} + 3570340 \nu^{3} + 3446013 \nu^{2} + \cdots - 6959472 ) / 245028 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 52 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{6} - \beta_{5} - 3\beta_{4} - 5\beta_{3} + \beta_{2} + 88\beta _1 + 58 \)
|
| \(\nu^{4}\) | \(=\) |
\( 2\beta_{7} - 20\beta_{5} - 4\beta_{4} - 84\beta_{3} + 117\beta_{2} + 227\beta _1 + 4362 \)
|
| \(\nu^{5}\) | \(=\) |
\( 2\beta_{7} + 213\beta_{6} - 267\beta_{5} - 445\beta_{4} - 995\beta_{3} + 247\beta_{2} + 8902\beta _1 + 9640 \)
|
| \(\nu^{6}\) | \(=\) |
\( 428 \beta_{7} + 430 \beta_{6} - 3962 \beta_{5} - 1126 \beta_{4} - 14442 \beta_{3} + 13283 \beta_{2} + \cdots + 422792 \)
|
| \(\nu^{7}\) | \(=\) |
\( 1288 \beta_{7} + 32173 \beta_{6} - 44333 \beta_{5} - 54983 \beta_{4} - 151121 \beta_{3} + 43649 \beta_{2} + \cdots + 1405730 \)
|
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 |
|
−10.9164 | 10.9440 | 87.1687 | −34.5060 | −119.470 | −49.0000 | −602.246 | −123.228 | 376.682 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −8.22711 | 9.20357 | 35.6854 | 95.7957 | −75.7188 | −49.0000 | −30.3201 | −158.294 | −788.122 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −6.15403 | 28.5192 | 5.87203 | 8.09265 | −175.508 | −49.0000 | 160.792 | 570.348 | −49.8024 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −2.03491 | −23.8080 | −27.8592 | −99.8321 | 48.4472 | −49.0000 | 121.808 | 323.823 | 203.149 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −0.0754222 | −4.71317 | −31.9943 | 27.2752 | 0.355478 | −49.0000 | 4.82659 | −220.786 | −2.05716 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 5.68207 | 24.9902 | 0.285912 | 87.8732 | 141.996 | −49.0000 | −180.202 | 381.511 | 499.302 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 7.49569 | 9.51604 | 24.1853 | −100.869 | 71.3293 | −49.0000 | −58.5763 | −152.445 | −756.080 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 10.2302 | −12.6519 | 72.6561 | 66.1700 | −129.431 | −49.0000 | 415.918 | −82.9291 | 676.929 | |||||||||||||||||||||||||||||||||||||||||||
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 | 847.6.a.g | 8 | |
| 11.b | odd | 2 | 1 | 77.6.a.e | ✓ | 8 | |
| 33.d | even | 2 | 1 | 693.6.a.k | 8 | ||
| 77.b | even | 2 | 1 | 539.6.a.i | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 77.6.a.e | ✓ | 8 | 11.b | odd | 2 | 1 | |
| 539.6.a.i | 8 | 77.b | even | 2 | 1 | ||
| 693.6.a.k | 8 | 33.d | even | 2 | 1 | ||
| 847.6.a.g | 8 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{8} + 4T_{2}^{7} - 203T_{2}^{6} - 692T_{2}^{5} + 12302T_{2}^{4} + 34712T_{2}^{3} - 222832T_{2}^{2} - 507040T_{2} - 36960 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(847))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} + 4 T^{7} + \cdots - 36960 \)
|
| $3$ |
\( T^{8} - 42 T^{7} + \cdots - 969823296 \)
|
| $5$ |
\( T^{8} + \cdots - 42721153061184 \)
|
| $7$ |
\( (T + 49)^{8} \)
|
| $11$ |
\( T^{8} \)
|
| $13$ |
\( T^{8} + \cdots - 62\!\cdots\!32 \)
|
| $17$ |
\( T^{8} + \cdots - 15\!\cdots\!76 \)
|
| $19$ |
\( T^{8} + \cdots + 33\!\cdots\!96 \)
|
| $23$ |
\( T^{8} + \cdots - 64\!\cdots\!60 \)
|
| $29$ |
\( T^{8} + \cdots - 98\!\cdots\!40 \)
|
| $31$ |
\( T^{8} + \cdots - 52\!\cdots\!76 \)
|
| $37$ |
\( T^{8} + \cdots - 89\!\cdots\!20 \)
|
| $41$ |
\( T^{8} + \cdots - 37\!\cdots\!56 \)
|
| $43$ |
\( T^{8} + \cdots - 41\!\cdots\!48 \)
|
| $47$ |
\( T^{8} + \cdots - 45\!\cdots\!08 \)
|
| $53$ |
\( T^{8} + \cdots - 16\!\cdots\!28 \)
|
| $59$ |
\( T^{8} + \cdots + 27\!\cdots\!36 \)
|
| $61$ |
\( T^{8} + \cdots + 16\!\cdots\!56 \)
|
| $67$ |
\( T^{8} + \cdots + 38\!\cdots\!72 \)
|
| $71$ |
\( T^{8} + \cdots - 11\!\cdots\!00 \)
|
| $73$ |
\( T^{8} + \cdots - 39\!\cdots\!60 \)
|
| $79$ |
\( T^{8} + \cdots - 51\!\cdots\!44 \)
|
| $83$ |
\( T^{8} + \cdots - 32\!\cdots\!84 \)
|
| $89$ |
\( T^{8} + \cdots + 29\!\cdots\!00 \)
|
| $97$ |
\( T^{8} + \cdots + 25\!\cdots\!00 \)
|
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