Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [810,3,Mod(163,810)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(810, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([0, 3]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("810.163");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 810 = 2 \cdot 3^{4} \cdot 5 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 810.g (of order \(4\), degree \(2\), 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: | \(22.0709014132\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Relative dimension: | \(4\) over \(\Q(i)\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{8} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - 2x^{7} + 11x^{6} - 32x^{5} - 485x^{4} + 3254x^{3} + 8942x^{2} + 68144x + 310792 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2\cdot 5 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-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
\( x^{8} - 2x^{7} + 11x^{6} - 32x^{5} - 485x^{4} + 3254x^{3} + 8942x^{2} + 68144x + 310792 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -9\nu^{7} - 216\nu^{6} - 715\nu^{5} + 6698\nu^{4} + 3513\nu^{3} + 27052\nu^{2} - 497126\nu - 6273572 ) / 625000 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 31\nu^{7} - 256\nu^{6} - 315\nu^{5} + 5818\nu^{4} - 16767\nu^{3} + 136932\nu^{2} - 29566\nu - 3705252 ) / 625000 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 9\nu^{7} + 16\nu^{6} - 285\nu^{5} + 302\nu^{4} - 2113\nu^{3} + 17748\nu^{2} + 181526\nu + 237172 ) / 25000 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 131\nu^{7} + 144\nu^{6} - 1815\nu^{5} - 6382\nu^{4} - 967\nu^{3} + 242132\nu^{2} + 2903334\nu + 8566548 ) / 312500 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -83\nu^{7} + 758\nu^{6} - 1455\nu^{5} - 1424\nu^{4} + 23981\nu^{3} - 355826\nu^{2} + 605588\nu + 3525336 ) / 156250 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -101\nu^{7} + 576\nu^{6} - 385\nu^{5} - 1778\nu^{4} + 18007\nu^{3} - 271472\nu^{2} - 419164\nu + 2096192 ) / 156250 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -97\nu^{7} - 328\nu^{6} + 3405\nu^{5} - 866\nu^{4} + 18029\nu^{3} - 153384\nu^{2} - 3065108\nu - 4661026 ) / 156250 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{7} - 2\beta_{3} + 2\beta_{2} + 1 ) / 5 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{7} + 10\beta_{6} - 5\beta_{5} + 10\beta_{4} - 3\beta_{3} + 28\beta_{2} + 5\beta _1 - 21 ) / 5 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 37\beta_{7} + 10\beta_{6} + 20\beta_{5} + 85\beta_{4} - 6\beta_{3} + 156\beta_{2} + 80\beta _1 - 27 ) / 5 \)
|
| \(\nu^{4}\) | \(=\) |
\( 31\beta_{7} + 35\beta_{5} + 55\beta_{4} - 8\beta_{3} + 343\beta_{2} - 45\beta _1 + 285 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 544\beta_{7} + 25\beta_{6} + 600\beta_{5} + 1175\beta_{4} - 562\beta_{3} + 7012\beta_{2} - 2400\beta _1 - 7044 ) / 5 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 4766 \beta_{7} + 2185 \beta_{6} + 2145 \beta_{5} + 6585 \beta_{4} + 3227 \beta_{3} + 29173 \beta_{2} + \cdots - 72286 ) / 5 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 30437 \beta_{7} - 20465 \beta_{6} + 23870 \beta_{5} + 16510 \beta_{4} + 36544 \beta_{3} + 53706 \beta_{2} + \cdots - 259227 ) / 5 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/810\mathbb{Z}\right)^\times\).
| \(n\) | \(487\) | \(731\) |
| \(\chi(n)\) | \(-\beta_{2}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 163.1 |
|
1.00000 | − | 1.00000i | 0 | − | 2.00000i | −4.98164 | + | 0.428082i | 0 | −9.35386 | + | 9.35386i | −2.00000 | − | 2.00000i | 0 | −4.55356 | + | 5.40972i | |||||||||||||||||||||||||||||||
| 163.2 | 1.00000 | − | 1.00000i | 0 | − | 2.00000i | −1.27826 | − | 4.83385i | 0 | 7.85532 | − | 7.85532i | −2.00000 | − | 2.00000i | 0 | −6.11210 | − | 3.55559i | ||||||||||||||||||||||||||||||||
| 163.3 | 1.00000 | − | 1.00000i | 0 | − | 2.00000i | −1.08444 | + | 4.88098i | 0 | 4.92233 | − | 4.92233i | −2.00000 | − | 2.00000i | 0 | 3.79654 | + | 5.96543i | ||||||||||||||||||||||||||||||||
| 163.4 | 1.00000 | − | 1.00000i | 0 | − | 2.00000i | 4.34434 | − | 2.47522i | 0 | −4.42379 | + | 4.42379i | −2.00000 | − | 2.00000i | 0 | 1.86912 | − | 6.81956i | ||||||||||||||||||||||||||||||||
| 487.1 | 1.00000 | + | 1.00000i | 0 | 2.00000i | −4.98164 | − | 0.428082i | 0 | −9.35386 | − | 9.35386i | −2.00000 | + | 2.00000i | 0 | −4.55356 | − | 5.40972i | |||||||||||||||||||||||||||||||||
| 487.2 | 1.00000 | + | 1.00000i | 0 | 2.00000i | −1.27826 | + | 4.83385i | 0 | 7.85532 | + | 7.85532i | −2.00000 | + | 2.00000i | 0 | −6.11210 | + | 3.55559i | |||||||||||||||||||||||||||||||||
| 487.3 | 1.00000 | + | 1.00000i | 0 | 2.00000i | −1.08444 | − | 4.88098i | 0 | 4.92233 | + | 4.92233i | −2.00000 | + | 2.00000i | 0 | 3.79654 | − | 5.96543i | |||||||||||||||||||||||||||||||||
| 487.4 | 1.00000 | + | 1.00000i | 0 | 2.00000i | 4.34434 | + | 2.47522i | 0 | −4.42379 | − | 4.42379i | −2.00000 | + | 2.00000i | 0 | 1.86912 | + | 6.81956i | |||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.c | odd | 4 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 810.3.g.f | yes | 8 |
| 3.b | odd | 2 | 1 | 810.3.g.e | ✓ | 8 | |
| 5.c | odd | 4 | 1 | inner | 810.3.g.f | yes | 8 |
| 15.e | even | 4 | 1 | 810.3.g.e | ✓ | 8 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 810.3.g.e | ✓ | 8 | 3.b | odd | 2 | 1 | |
| 810.3.g.e | ✓ | 8 | 15.e | even | 4 | 1 | |
| 810.3.g.f | yes | 8 | 1.a | even | 1 | 1 | trivial |
| 810.3.g.f | yes | 8 | 5.c | odd | 4 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(810, [\chi])\):
|
\( T_{7}^{8} + 2T_{7}^{7} + 2T_{7}^{6} - 400T_{7}^{5} + 24064T_{7}^{4} - 15872T_{7}^{3} + 128T_{7}^{2} + 102400T_{7} + 40960000 \)
|
|
\( T_{11}^{4} + 4T_{11}^{3} - 345T_{11}^{2} - 500T_{11} + 25000 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} - 2 T + 2)^{4} \)
|
| $3$ |
\( T^{8} \)
|
| $5$ |
\( T^{8} + 6 T^{7} + \cdots + 390625 \)
|
| $7$ |
\( T^{8} + 2 T^{7} + \cdots + 40960000 \)
|
| $11$ |
\( (T^{4} + 4 T^{3} + \cdots + 25000)^{2} \)
|
| $13$ |
\( T^{8} - 6 T^{7} + \cdots + 2304 \)
|
| $17$ |
\( T^{8} - 4 T^{7} + \cdots + 880427584 \)
|
| $19$ |
\( T^{8} + \cdots + 1142440000 \)
|
| $23$ |
\( T^{8} - 4 T^{7} + \cdots + 100000000 \)
|
| $29$ |
\( T^{8} + \cdots + 13929900625 \)
|
| $31$ |
\( (T^{4} - 16 T^{3} + \cdots + 1461808)^{2} \)
|
| $37$ |
\( T^{8} + \cdots + 944434112400 \)
|
| $41$ |
\( (T^{4} - 92 T^{3} + \cdots - 5205824)^{2} \)
|
| $43$ |
\( T^{8} + \cdots + 22177318118400 \)
|
| $47$ |
\( T^{8} + \cdots + 130015051776 \)
|
| $53$ |
\( T^{8} + \cdots + 523564689719296 \)
|
| $59$ |
\( T^{8} + \cdots + 59845696000000 \)
|
| $61$ |
\( (T^{4} + 84 T^{3} + \cdots - 1649280)^{2} \)
|
| $67$ |
\( T^{8} + \cdots + 4647715092736 \)
|
| $71$ |
\( (T^{4} - 214 T^{3} + \cdots + 7620568)^{2} \)
|
| $73$ |
\( T^{8} + \cdots + 65816485405696 \)
|
| $79$ |
\( T^{8} + 13860 T^{6} + \cdots + 513294336 \)
|
| $83$ |
\( T^{8} + \cdots + 3845866143744 \)
|
| $89$ |
\( T^{8} + 29800 T^{6} + \cdots + 98029801 \)
|
| $97$ |
\( T^{8} + \cdots + 22045954777344 \)
|
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