Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [784,4,Mod(783,784)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(784, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 0, 1]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("784.783");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 784 = 2^{4} \cdot 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 784.f (of order \(2\), degree \(1\), minimal) |
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: | no |
| Analytic conductor: | \(46.2574974445\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | 8.0.16928550682624.32 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - 4x^{7} + 62x^{6} - 152x^{5} + 1187x^{4} - 1424x^{3} + 7038x^{2} + 1452x + 5287 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{17}]\) |
| Coefficient ring index: | \( 2^{5}\cdot 7^{4} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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
\( x^{8} - 4x^{7} + 62x^{6} - 152x^{5} + 1187x^{4} - 1424x^{3} + 7038x^{2} + 1452x + 5287 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( - 4164 \nu^{7} - 358345 \nu^{6} - 650409 \nu^{5} - 16098218 \nu^{4} - 15747163 \nu^{3} + \cdots - 788088157 ) / 10361911 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 6093932 \nu^{7} + 25171336 \nu^{6} - 423467291 \nu^{5} + 1234423939 \nu^{4} + \cdots + 2237396676 ) / 10703854063 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 6106 \nu^{7} + 66782 \nu^{6} - 431171 \nu^{5} + 2452981 \nu^{4} - 6194717 \nu^{3} + \cdots - 91499108 ) / 10361911 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 10107994 \nu^{7} + 19374127 \nu^{6} - 457850077 \nu^{5} + 577333526 \nu^{4} + \cdots + 3307731061 ) / 10703854063 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 1630 \nu^{7} + 4476 \nu^{6} + 47706 \nu^{5} + 326529 \nu^{4} + 461416 \nu^{3} + 5257845 \nu^{2} + \cdots + 18437249 ) / 1480273 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 53734552 \nu^{7} - 275479040 \nu^{6} + 2941079496 \nu^{5} - 8802476688 \nu^{4} + \cdots - 52627155866 ) / 10703854063 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 108735962 \nu^{7} - 493132366 \nu^{6} + 6381684922 \nu^{5} - 17094460415 \nu^{4} + \cdots - 27356882655 ) / 10703854063 \)
|
| \(\nu\) | \(=\) |
\( ( 2\beta_{7} - 3\beta_{6} - 2\beta_{5} - 4\beta_{4} + 12\beta_{2} + 14 ) / 28 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -7\beta_{6} + 8\beta_{5} - 8\beta_{4} - 28\beta_{3} - 4\beta_{2} - 378 ) / 28 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -17\beta_{7} + 22\beta_{6} + 49\beta_{5} + 89\beta_{4} - 9\beta_{3} - 155\beta_{2} + 3\beta _1 - 392 ) / 14 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 245\beta_{6} + 12\beta_{5} + 824\beta_{4} + 640\beta_{3} + 160\beta_{2} - 8\beta _1 + 5810 ) / 28 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 642 \beta_{7} - 417 \beta_{6} - 2918 \beta_{5} - 3138 \beta_{4} + 1210 \beta_{3} + 6306 \beta_{2} + \cdots + 22694 ) / 28 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( - 53 \beta_{7} - 3235 \beta_{6} - 2653 \beta_{5} - 16240 \beta_{4} - 5246 \beta_{3} - 2142 \beta_{2} + \cdots - 33894 ) / 14 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( - 11648 \beta_{7} - 4823 \beta_{6} + 72224 \beta_{5} - 3070 \beta_{4} - 39396 \beta_{3} + \cdots - 536830 ) / 28 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/784\mathbb{Z}\right)^\times\).
| \(n\) | \(197\) | \(687\) | \(689\) |
| \(\chi(n)\) | \(1\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 783.1 |
|
0 | −9.98491 | 0 | − | 9.37011i | 0 | 0 | 0 | 72.6985 | 0 | |||||||||||||||||||||||||||||||||||||||||
| 783.2 | 0 | −9.98491 | 0 | 9.37011i | 0 | 0 | 0 | 72.6985 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 783.3 | 0 | −6.34835 | 0 | − | 6.94269i | 0 | 0 | 0 | 13.3015 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 783.4 | 0 | −6.34835 | 0 | 6.94269i | 0 | 0 | 0 | 13.3015 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 783.5 | 0 | 6.34835 | 0 | − | 6.94269i | 0 | 0 | 0 | 13.3015 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 783.6 | 0 | 6.34835 | 0 | 6.94269i | 0 | 0 | 0 | 13.3015 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 783.7 | 0 | 9.98491 | 0 | − | 9.37011i | 0 | 0 | 0 | 72.6985 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 783.8 | 0 | 9.98491 | 0 | 9.37011i | 0 | 0 | 0 | 72.6985 | 0 | |||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 4.b | odd | 2 | 1 | inner |
| 7.b | odd | 2 | 1 | inner |
| 28.d | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 784.4.f.i | ✓ | 8 |
| 4.b | odd | 2 | 1 | inner | 784.4.f.i | ✓ | 8 |
| 7.b | odd | 2 | 1 | inner | 784.4.f.i | ✓ | 8 |
| 28.d | even | 2 | 1 | inner | 784.4.f.i | ✓ | 8 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 784.4.f.i | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 784.4.f.i | ✓ | 8 | 4.b | odd | 2 | 1 | inner |
| 784.4.f.i | ✓ | 8 | 7.b | odd | 2 | 1 | inner |
| 784.4.f.i | ✓ | 8 | 28.d | even | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} - 140T_{3}^{2} + 4018 \)
acting on \(S_{4}^{\mathrm{new}}(784, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} \)
|
| $3$ |
\( (T^{4} - 140 T^{2} + 4018)^{2} \)
|
| $5$ |
\( (T^{4} + 136 T^{2} + 4232)^{2} \)
|
| $7$ |
\( T^{8} \)
|
| $11$ |
\( (T^{4} + 1988 T^{2} + 8036)^{2} \)
|
| $13$ |
\( (T^{4} + 5288 T^{2} + 668168)^{2} \)
|
| $17$ |
\( (T^{4} + 7892 T^{2} + 14183138)^{2} \)
|
| $19$ |
\( (T^{4} - 11452 T^{2} + 31826578)^{2} \)
|
| $23$ |
\( (T^{4} + 61208 T^{2} + 896464016)^{2} \)
|
| $29$ |
\( (T^{2} + 224 T + 12152)^{4} \)
|
| $31$ |
\( (T^{4} - 63840 T^{2} + 257152)^{2} \)
|
| $37$ |
\( (T^{2} - 76832)^{4} \)
|
| $41$ |
\( (T^{4} + 150916 T^{2} + 5692658402)^{2} \)
|
| $43$ |
\( (T^{4} + 373828 T^{2} + 34800388196)^{2} \)
|
| $47$ |
\( (T^{4} - 191856 T^{2} + 3196977952)^{2} \)
|
| $53$ |
\( (T^{2} - 52 T - 112612)^{4} \)
|
| $59$ |
\( (T^{4} - 272524 T^{2} + 8136164722)^{2} \)
|
| $61$ |
\( (T^{4} + 35080 T^{2} + 286370312)^{2} \)
|
| $67$ |
\( (T^{4} + 247184 T^{2} + 6980262464)^{2} \)
|
| $71$ |
\( (T^{4} + 439488 T^{2} + 48157369344)^{2} \)
|
| $73$ |
\( (T^{4} + 786404 T^{2} + 16405487522)^{2} \)
|
| $79$ |
\( (T^{4} + 1403024 T^{2} + 223691510336)^{2} \)
|
| $83$ |
\( (T^{4} - 643468 T^{2} + 60208287538)^{2} \)
|
| $89$ |
\( (T^{4} + 3860 T^{2} + 3230882)^{2} \)
|
| $97$ |
\( (T^{4} + 2885284 T^{2} + 796703169602)^{2} \)
|
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