Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [570,4,Mod(229,570)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(570, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1, 0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("570.229");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 570 = 2 \cdot 3 \cdot 5 \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 570.d (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: | \(33.6310887033\) |
| 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: |
\( x^{8} + 45x^{6} + 548x^{4} + 1824x^{2} + 400 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| 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
\( x^{8} + 45x^{6} + 548x^{4} + 1824x^{2} + 400 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -\nu^{6} + 7\nu^{4} + 628\nu^{2} - 1540\nu - 600 ) / 1540 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{6} + 7\nu^{4} + 628\nu^{2} + 1540\nu - 600 ) / 1540 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -3\nu^{7} - 133\nu^{5} - 1658\nu^{3} - 6728\nu ) / 3080 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -3\nu^{6} + 21\nu^{4} + 3424\nu^{2} + 16680 ) / 1540 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 8\nu^{6} + 329\nu^{4} + 3061\nu^{2} + 3260 ) / 770 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 25\nu^{7} + 1057\nu^{5} + 11096\nu^{3} + 23624\nu ) / 3080 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -17\nu^{7} - 805\nu^{5} - 10576\nu^{3} - 35148\nu ) / 1540 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} - \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 2\beta_{4} - 3\beta_{2} - 3\beta _1 - 24 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 2\beta_{7} - 4\beta_{6} - 56\beta_{3} - 23\beta_{2} + 23\beta_1 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 4\beta_{5} - 42\beta_{4} + 95\beta_{2} + 95\beta _1 + 512 ) / 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -106\beta_{7} + 92\beta_{6} + 1968\beta_{3} + 587\beta_{2} - 587\beta_1 ) / 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 28\beta_{5} + 962\beta_{4} - 2759\beta_{2} - 2759\beta _1 - 12688 ) / 2 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 3594\beta_{7} - 1868\beta_{6} - 58352\beta_{3} - 15555\beta_{2} + 15555\beta_1 ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/570\mathbb{Z}\right)^\times\).
| \(n\) | \(191\) | \(211\) | \(457\) |
| \(\chi(n)\) | \(1\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 229.1 |
|
− | 2.00000i | 3.00000i | −4.00000 | −11.0288 | − | 1.83443i | 6.00000 | − | 12.2372i | 8.00000i | −9.00000 | −3.66887 | + | 22.0576i | ||||||||||||||||||||||||||||||||||||
| 229.2 | − | 2.00000i | 3.00000i | −4.00000 | 7.24764 | + | 8.51303i | 6.00000 | 24.3035i | 8.00000i | −9.00000 | 17.0261 | − | 14.4953i | ||||||||||||||||||||||||||||||||||||||
| 229.3 | − | 2.00000i | 3.00000i | −4.00000 | 8.27917 | − | 7.51367i | 6.00000 | 5.11592i | 8.00000i | −9.00000 | −15.0273 | − | 16.5583i | ||||||||||||||||||||||||||||||||||||||
| 229.4 | − | 2.00000i | 3.00000i | −4.00000 | 10.5020 | + | 3.83507i | 6.00000 | − | 23.1823i | 8.00000i | −9.00000 | 7.67015 | − | 21.0040i | |||||||||||||||||||||||||||||||||||||
| 229.5 | 2.00000i | − | 3.00000i | −4.00000 | −11.0288 | + | 1.83443i | 6.00000 | 12.2372i | − | 8.00000i | −9.00000 | −3.66887 | − | 22.0576i | |||||||||||||||||||||||||||||||||||||
| 229.6 | 2.00000i | − | 3.00000i | −4.00000 | 7.24764 | − | 8.51303i | 6.00000 | − | 24.3035i | − | 8.00000i | −9.00000 | 17.0261 | + | 14.4953i | ||||||||||||||||||||||||||||||||||||
| 229.7 | 2.00000i | − | 3.00000i | −4.00000 | 8.27917 | + | 7.51367i | 6.00000 | − | 5.11592i | − | 8.00000i | −9.00000 | −15.0273 | + | 16.5583i | ||||||||||||||||||||||||||||||||||||
| 229.8 | 2.00000i | − | 3.00000i | −4.00000 | 10.5020 | − | 3.83507i | 6.00000 | 23.1823i | − | 8.00000i | −9.00000 | 7.67015 | + | 21.0040i | |||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 570.4.d.c | ✓ | 8 |
| 5.b | even | 2 | 1 | inner | 570.4.d.c | ✓ | 8 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 570.4.d.c | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 570.4.d.c | ✓ | 8 | 5.b | even | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{8} + 1304T_{7}^{6} + 519804T_{7}^{4} + 60264260T_{7}^{2} + 1244113984 \)
acting on \(S_{4}^{\mathrm{new}}(570, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 4)^{4} \)
|
| $3$ |
\( (T^{2} + 9)^{4} \)
|
| $5$ |
\( T^{8} - 30 T^{7} + \cdots + 244140625 \)
|
| $7$ |
\( T^{8} + \cdots + 1244113984 \)
|
| $11$ |
\( (T^{4} + 120 T^{3} + \cdots - 105512)^{2} \)
|
| $13$ |
\( T^{8} + \cdots + 2199466963600 \)
|
| $17$ |
\( T^{8} + \cdots + 6105234496 \)
|
| $19$ |
\( (T - 19)^{8} \)
|
| $23$ |
\( T^{8} + \cdots + 35\!\cdots\!84 \)
|
| $29$ |
\( (T^{4} + 198 T^{3} + \cdots - 20960420)^{2} \)
|
| $31$ |
\( (T^{4} + 574 T^{3} + \cdots - 64392824)^{2} \)
|
| $37$ |
\( T^{8} + \cdots + 20\!\cdots\!64 \)
|
| $41$ |
\( (T^{4} + 84 T^{3} + \cdots + 2106700776)^{2} \)
|
| $43$ |
\( T^{8} + \cdots + 27\!\cdots\!64 \)
|
| $47$ |
\( T^{8} + \cdots + 36\!\cdots\!44 \)
|
| $53$ |
\( T^{8} + \cdots + 90\!\cdots\!04 \)
|
| $59$ |
\( (T^{4} + 420 T^{3} + \cdots + 773806960)^{2} \)
|
| $61$ |
\( (T^{4} + 692 T^{3} + \cdots - 101103650984)^{2} \)
|
| $67$ |
\( T^{8} + \cdots + 45\!\cdots\!96 \)
|
| $71$ |
\( (T^{4} + 414 T^{3} + \cdots + 79207110880)^{2} \)
|
| $73$ |
\( T^{8} + \cdots + 66\!\cdots\!96 \)
|
| $79$ |
\( (T^{4} - 370 T^{3} + \cdots - 40506586400)^{2} \)
|
| $83$ |
\( T^{8} + \cdots + 29\!\cdots\!96 \)
|
| $89$ |
\( (T^{4} - 90 T^{3} + \cdots + 624269648320)^{2} \)
|
| $97$ |
\( T^{8} + \cdots + 26\!\cdots\!44 \)
|
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