Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [54,8,Mod(19,54)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(54, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([4]))
N = Newforms(chi, 8, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("54.19");
S:= CuspForms(chi, 8);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 54 = 2 \cdot 3^{3} \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| Character orbit: | \([\chi]\) | \(=\) | 54.c (of order \(3\), degree \(2\), not 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: | \(16.8687913761\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Relative dimension: | \(4\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{8} + \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} + 1336x^{6} + 633664x^{4} + 125389995x^{2} + 8783438400 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{4}\cdot 3^{18} \) |
| Twist minimal: | no (minimal twist has level 18) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-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
\( x^{8} + 1336x^{6} + 633664x^{4} + 125389995x^{2} + 8783438400 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{7} + 1336\nu^{5} + 539944\nu^{3} + 62785035\nu + 22961400 ) / 45922800 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 27\nu^{4} + 18981\nu^{2} + 2846070 ) / 245 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -2\nu^{6} - 2081\nu^{4} - 675545\nu^{2} - 66902715 ) / 5145 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 1989 \nu^{7} - 31240 \nu^{6} - 2048124 \nu^{5} - 32505220 \nu^{4} - 644054976 \nu^{3} + \cdots - 1044940043400 ) / 160729800 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -4\nu^{6} - 3595\nu^{4} - 1071559\nu^{2} - 113802195 ) / 5145 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 1503 \nu^{7} + 1586268 \nu^{5} + 2952180 \nu^{4} + 529813512 \nu^{3} + 2075382540 \nu^{2} + \cdots + 311189293800 ) / 53576600 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 37141 \nu^{7} - 124960 \nu^{6} - 39404896 \nu^{5} - 112307800 \nu^{4} - 13321965064 \nu^{3} + \cdots - 3555180571800 ) / 321459600 \)
|
| \(\nu\) | \(=\) |
\( ( 2\beta_{7} + 10\beta_{6} - \beta_{5} + 4\beta_{4} - 2\beta_{3} - 5\beta_{2} + 2\beta _1 - 3 ) / 486 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -7\beta_{5} + 14\beta_{3} + 7\beta_{2} - 54101 ) / 162 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( - 1018 \beta_{7} - 3452 \beta_{6} + 509 \beta_{5} + 1366 \beta_{4} - 683 \beta_{3} + 1726 \beta_{2} + \cdots + 88278 ) / 486 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 4921\beta_{5} - 9842\beta_{3} - 3451\beta_{2} + 20956583 ) / 162 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 502874 \beta_{7} + 1358446 \beta_{6} - 251437 \beta_{5} - 1266788 \beta_{4} + 633394 \beta_{3} + \cdots - 98473359 ) / 486 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( -918631\beta_{5} + 1698347\beta_{3} + 408786\beta_{2} - 2983538168 ) / 54 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( - 247746742 \beta_{7} - 578847518 \beta_{6} + 123873371 \beta_{5} + 703725124 \beta_{4} + \cdots + 72924345897 ) / 486 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/54\mathbb{Z}\right)^\times\).
| \(n\) | \(29\) |
| \(\chi(n)\) | \(-1 + \beta_{1}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 19.1 |
|
−4.00000 | + | 6.92820i | 0 | −32.0000 | − | 55.4256i | −218.032 | − | 377.643i | 0 | 239.730 | − | 415.224i | 512.000 | 0 | 3488.51 | ||||||||||||||||||||||||||||||||||
| 19.2 | −4.00000 | + | 6.92820i | 0 | −32.0000 | − | 55.4256i | −103.252 | − | 178.837i | 0 | −808.604 | + | 1400.54i | 512.000 | 0 | 1652.03 | |||||||||||||||||||||||||||||||||||
| 19.3 | −4.00000 | + | 6.92820i | 0 | −32.0000 | − | 55.4256i | 47.1766 | + | 81.7123i | 0 | −101.366 | + | 175.572i | 512.000 | 0 | −754.826 | |||||||||||||||||||||||||||||||||||
| 19.4 | −4.00000 | + | 6.92820i | 0 | −32.0000 | − | 55.4256i | 247.107 | + | 428.002i | 0 | 648.241 | − | 1122.79i | 512.000 | 0 | −3953.72 | |||||||||||||||||||||||||||||||||||
| 37.1 | −4.00000 | − | 6.92820i | 0 | −32.0000 | + | 55.4256i | −218.032 | + | 377.643i | 0 | 239.730 | + | 415.224i | 512.000 | 0 | 3488.51 | |||||||||||||||||||||||||||||||||||
| 37.2 | −4.00000 | − | 6.92820i | 0 | −32.0000 | + | 55.4256i | −103.252 | + | 178.837i | 0 | −808.604 | − | 1400.54i | 512.000 | 0 | 1652.03 | |||||||||||||||||||||||||||||||||||
| 37.3 | −4.00000 | − | 6.92820i | 0 | −32.0000 | + | 55.4256i | 47.1766 | − | 81.7123i | 0 | −101.366 | − | 175.572i | 512.000 | 0 | −754.826 | |||||||||||||||||||||||||||||||||||
| 37.4 | −4.00000 | − | 6.92820i | 0 | −32.0000 | + | 55.4256i | 247.107 | − | 428.002i | 0 | 648.241 | + | 1122.79i | 512.000 | 0 | −3953.72 | |||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 9.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 54.8.c.b | 8 | |
| 3.b | odd | 2 | 1 | 18.8.c.b | ✓ | 8 | |
| 4.b | odd | 2 | 1 | 432.8.i.b | 8 | ||
| 9.c | even | 3 | 1 | inner | 54.8.c.b | 8 | |
| 9.c | even | 3 | 1 | 162.8.a.i | 4 | ||
| 9.d | odd | 6 | 1 | 18.8.c.b | ✓ | 8 | |
| 9.d | odd | 6 | 1 | 162.8.a.h | 4 | ||
| 12.b | even | 2 | 1 | 144.8.i.b | 8 | ||
| 36.f | odd | 6 | 1 | 432.8.i.b | 8 | ||
| 36.h | even | 6 | 1 | 144.8.i.b | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 18.8.c.b | ✓ | 8 | 3.b | odd | 2 | 1 | |
| 18.8.c.b | ✓ | 8 | 9.d | odd | 6 | 1 | |
| 54.8.c.b | 8 | 1.a | even | 1 | 1 | trivial | |
| 54.8.c.b | 8 | 9.c | even | 3 | 1 | inner | |
| 144.8.i.b | 8 | 12.b | even | 2 | 1 | ||
| 144.8.i.b | 8 | 36.h | even | 6 | 1 | ||
| 162.8.a.h | 4 | 9.d | odd | 6 | 1 | ||
| 162.8.a.i | 4 | 9.c | even | 3 | 1 | ||
| 432.8.i.b | 8 | 4.b | odd | 2 | 1 | ||
| 432.8.i.b | 8 | 36.f | odd | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{8} + 54 T_{5}^{7} + 244431 T_{5}^{6} + 33030990 T_{5}^{5} + 55374420825 T_{5}^{4} + \cdots + 17\!\cdots\!00 \)
acting on \(S_{8}^{\mathrm{new}}(54, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 8 T + 64)^{4} \)
|
| $3$ |
\( T^{8} \)
|
| $5$ |
\( T^{8} + \cdots + 17\!\cdots\!00 \)
|
| $7$ |
\( T^{8} + \cdots + 41\!\cdots\!56 \)
|
| $11$ |
\( T^{8} + \cdots + 12\!\cdots\!25 \)
|
| $13$ |
\( T^{8} + \cdots + 27\!\cdots\!04 \)
|
| $17$ |
\( (T^{4} + \cdots - 21\!\cdots\!28)^{2} \)
|
| $19$ |
\( (T^{4} + \cdots - 57\!\cdots\!48)^{2} \)
|
| $23$ |
\( T^{8} + \cdots + 37\!\cdots\!00 \)
|
| $29$ |
\( T^{8} + \cdots + 18\!\cdots\!76 \)
|
| $31$ |
\( T^{8} + \cdots + 11\!\cdots\!00 \)
|
| $37$ |
\( (T^{4} + \cdots + 14\!\cdots\!48)^{2} \)
|
| $41$ |
\( T^{8} + \cdots + 61\!\cdots\!61 \)
|
| $43$ |
\( T^{8} + \cdots + 14\!\cdots\!81 \)
|
| $47$ |
\( T^{8} + \cdots + 15\!\cdots\!16 \)
|
| $53$ |
\( (T^{4} + \cdots + 32\!\cdots\!80)^{2} \)
|
| $59$ |
\( T^{8} + \cdots + 73\!\cdots\!09 \)
|
| $61$ |
\( T^{8} + \cdots + 40\!\cdots\!24 \)
|
| $67$ |
\( T^{8} + \cdots + 77\!\cdots\!25 \)
|
| $71$ |
\( (T^{4} + \cdots + 10\!\cdots\!28)^{2} \)
|
| $73$ |
\( (T^{4} + \cdots + 17\!\cdots\!44)^{2} \)
|
| $79$ |
\( T^{8} + \cdots + 66\!\cdots\!64 \)
|
| $83$ |
\( T^{8} + \cdots + 72\!\cdots\!64 \)
|
| $89$ |
\( (T^{4} + \cdots - 25\!\cdots\!80)^{2} \)
|
| $97$ |
\( T^{8} + \cdots + 30\!\cdots\!25 \)
|
| show more | |
| show less |