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: | \(6\) |
| Coefficient field: | 6.0.12258833328.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - x^{5} + 29x^{4} - 20x^{3} + 808x^{2} - 672x + 576 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{5}\cdot 3 \) |
| Twist minimal: | no (minimal twist has level 112) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - x^{5} + 29x^{4} - 20x^{3} + 808x^{2} - 672x + 576 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 17\nu^{5} - 493\nu^{4} - 8463\nu^{3} - 13736\nu^{2} + 11424\nu - 285776 ) / 45520 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -29\nu^{5} + 841\nu^{4} - 1629\nu^{3} + 23432\nu^{2} - 19488\nu + 428592 ) / 45520 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -197\nu^{5} + 23\nu^{4} - 667\nu^{3} - 5834\nu^{2} + 117976\nu - 52824 ) / 68280 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 203\nu^{5} - 197\nu^{4} + 5713\nu^{3} + 986\nu^{2} + 159176\nu - 64104 ) / 68280 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 1191\nu^{5} - 399\nu^{4} + 34331\nu^{3} - 17788\nu^{2} + 982432\nu - 404688 ) / 45520 \)
|
| \(\nu\) | \(=\) |
\( ( 3\beta_{4} + 3\beta_{3} + \beta_{2} + \beta _1 + 2 ) / 12 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -12\beta_{5} + 111\beta_{4} + 3\beta_{3} + 11\beta_{2} - \beta _1 - 110 ) / 12 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -17\beta_{2} - 29\beta _1 - 22 ) / 6 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 116\beta_{5} - 1029\beta_{4} - 33\beta_{3} + 105\beta_{2} - 11\beta _1 - 1018 ) / 4 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 396\beta_{5} - 1851\beta_{4} - 2463\beta_{3} + 425\beta_{2} + 821\beta _1 + 1030 ) / 12 \)
|
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 | −9.29424 | 0 | − | 19.8510i | 0 | 0 | 0 | 59.3829 | 0 | |||||||||||||||||||||||||||||||||||
| 783.2 | 0 | −9.29424 | 0 | 19.8510i | 0 | 0 | 0 | 59.3829 | 0 | |||||||||||||||||||||||||||||||||||||
| 783.3 | 0 | −4.76834 | 0 | − | 16.8950i | 0 | 0 | 0 | −4.26295 | 0 | ||||||||||||||||||||||||||||||||||||
| 783.4 | 0 | −4.76834 | 0 | 16.8950i | 0 | 0 | 0 | −4.26295 | 0 | |||||||||||||||||||||||||||||||||||||
| 783.5 | 0 | 7.06258 | 0 | − | 1.22397i | 0 | 0 | 0 | 22.8800 | 0 | ||||||||||||||||||||||||||||||||||||
| 783.6 | 0 | 7.06258 | 0 | 1.22397i | 0 | 0 | 0 | 22.8800 | 0 | |||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 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.g | 6 | |
| 4.b | odd | 2 | 1 | 784.4.f.h | 6 | ||
| 7.b | odd | 2 | 1 | 784.4.f.h | 6 | ||
| 7.c | even | 3 | 1 | 112.4.p.g | yes | 6 | |
| 7.d | odd | 6 | 1 | 112.4.p.f | ✓ | 6 | |
| 28.d | even | 2 | 1 | inner | 784.4.f.g | 6 | |
| 28.f | even | 6 | 1 | 112.4.p.g | yes | 6 | |
| 28.g | odd | 6 | 1 | 112.4.p.f | ✓ | 6 | |
| 56.j | odd | 6 | 1 | 448.4.p.g | 6 | ||
| 56.k | odd | 6 | 1 | 448.4.p.g | 6 | ||
| 56.m | even | 6 | 1 | 448.4.p.f | 6 | ||
| 56.p | even | 6 | 1 | 448.4.p.f | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 112.4.p.f | ✓ | 6 | 7.d | odd | 6 | 1 | |
| 112.4.p.f | ✓ | 6 | 28.g | odd | 6 | 1 | |
| 112.4.p.g | yes | 6 | 7.c | even | 3 | 1 | |
| 112.4.p.g | yes | 6 | 28.f | even | 6 | 1 | |
| 448.4.p.f | 6 | 56.m | even | 6 | 1 | ||
| 448.4.p.f | 6 | 56.p | even | 6 | 1 | ||
| 448.4.p.g | 6 | 56.j | odd | 6 | 1 | ||
| 448.4.p.g | 6 | 56.k | odd | 6 | 1 | ||
| 784.4.f.g | 6 | 1.a | even | 1 | 1 | trivial | |
| 784.4.f.g | 6 | 28.d | even | 2 | 1 | inner | |
| 784.4.f.h | 6 | 4.b | odd | 2 | 1 | ||
| 784.4.f.h | 6 | 7.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{3} + 7T_{3}^{2} - 55T_{3} - 313 \)
acting on \(S_{4}^{\mathrm{new}}(784, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
|
| $3$ |
\( (T^{3} + 7 T^{2} + \cdots - 313)^{2} \)
|
| $5$ |
\( T^{6} + 681 T^{4} + \cdots + 168507 \)
|
| $7$ |
\( T^{6} \)
|
| $11$ |
\( T^{6} + 1821 T^{4} + \cdots + 4876875 \)
|
| $13$ |
\( T^{6} + \cdots + 2167603200 \)
|
| $17$ |
\( T^{6} + 29361 T^{4} + \cdots + 710833347 \)
|
| $19$ |
\( (T^{3} - 143 T^{2} + \cdots - 33271)^{2} \)
|
| $23$ |
\( T^{6} + \cdots + 8915001507 \)
|
| $29$ |
\( (T^{3} + 174 T^{2} + \cdots - 4622400)^{2} \)
|
| $31$ |
\( (T^{3} - 205 T^{2} + \cdots + 1532779)^{2} \)
|
| $37$ |
\( (T^{3} - 249 T^{2} + \cdots + 3813125)^{2} \)
|
| $41$ |
\( T^{6} + \cdots + 38591270836992 \)
|
| $43$ |
\( T^{6} + \cdots + 83474849412288 \)
|
| $47$ |
\( (T^{3} - 75 T^{2} + \cdots + 143829)^{2} \)
|
| $53$ |
\( (T^{3} - 645 T^{2} + \cdots + 61174089)^{2} \)
|
| $59$ |
\( (T^{3} - 321 T^{2} + \cdots - 908793)^{2} \)
|
| $61$ |
\( T^{6} + \cdots + 912934776095523 \)
|
| $67$ |
\( T^{6} + \cdots + 34\!\cdots\!75 \)
|
| $71$ |
\( T^{6} + \cdots + 10\!\cdots\!48 \)
|
| $73$ |
\( T^{6} + \cdots + 48\!\cdots\!03 \)
|
| $79$ |
\( T^{6} + \cdots + 65\!\cdots\!23 \)
|
| $83$ |
\( (T^{3} + 12 T^{2} + \cdots - 308212992)^{2} \)
|
| $89$ |
\( T^{6} + \cdots + 60\!\cdots\!03 \)
|
| $97$ |
\( T^{6} + \cdots + 19\!\cdots\!08 \)
|
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