Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [546,4,Mod(79,546)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(546, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 2, 0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("546.79");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 546 = 2 \cdot 3 \cdot 7 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 546.i (of order \(3\), 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: | \(32.2150428631\) |
| Analytic rank: | \(0\) |
| Dimension: | \(12\) |
| Relative dimension: | \(6\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{12} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{12} - 2 x^{11} + 387 x^{10} - 16 x^{9} + 118610 x^{8} - 17859 x^{7} + 10864988 x^{6} + \cdots + 35073049284 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 3 \) |
| Twist minimal: | yes |
| 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 \(1,\beta_1,\ldots,\beta_{11}\) 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
\( x^{12} - 2 x^{11} + 387 x^{10} - 16 x^{9} + 118610 x^{8} - 17859 x^{7} + 10864988 x^{6} + \cdots + 35073049284 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 97\!\cdots\!57 \nu^{11} + \cdots - 14\!\cdots\!48 ) / 53\!\cdots\!46 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 28\!\cdots\!29 \nu^{11} + \cdots - 12\!\cdots\!70 ) / 69\!\cdots\!06 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 41\!\cdots\!63 \nu^{11} + \cdots + 31\!\cdots\!40 ) / 69\!\cdots\!06 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 86\!\cdots\!06 \nu^{11} + \cdots - 19\!\cdots\!59 ) / 11\!\cdots\!01 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 33\!\cdots\!87 \nu^{11} + \cdots - 10\!\cdots\!76 ) / 17\!\cdots\!42 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 21\!\cdots\!13 \nu^{11} + \cdots + 96\!\cdots\!82 ) / 69\!\cdots\!06 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 13\!\cdots\!86 \nu^{11} + \cdots - 51\!\cdots\!01 ) / 34\!\cdots\!03 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 37\!\cdots\!30 \nu^{11} + \cdots + 22\!\cdots\!76 ) / 69\!\cdots\!06 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 10\!\cdots\!24 \nu^{11} + \cdots - 89\!\cdots\!98 ) / 69\!\cdots\!06 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 22\!\cdots\!97 \nu^{11} + \cdots - 66\!\cdots\!84 ) / 69\!\cdots\!06 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{11} + 2\beta_{10} + 6\beta_{9} + \beta_{8} - \beta_{7} + 3\beta_{6} - 2\beta_{5} - 7\beta_{4} - 126\beta_{2} - 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( -2\beta_{11} + 39\beta_{9} + 22\beta_{8} + 17\beta_{7} + 224\beta_{6} + 39\beta_{4} - 2\beta_{3} - 224\beta _1 - 125 \)
|
| \(\nu^{4}\) | \(=\) |
\( - 1447 \beta_{9} - 1004 \beta_{8} + 1447 \beta_{7} + 915 \beta_{5} + 1919 \beta_{4} - 395 \beta_{3} + \cdots - 29335 \)
|
| \(\nu^{5}\) | \(=\) |
\( - 158 \beta_{11} - 1293 \beta_{10} - 8220 \beta_{9} - 15007 \beta_{8} - 13714 \beta_{7} - 59385 \beta_{6} + \cdots + 1293 \)
|
| \(\nu^{6}\) | \(=\) |
\( 121326 \beta_{11} - 295309 \beta_{10} - 166680 \beta_{9} + 109615 \beta_{8} - 276295 \beta_{7} + \cdots + 8117073 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 1802658 \beta_{9} + 2256630 \beta_{8} + 1802658 \beta_{7} - 695172 \beta_{5} - 2951802 \beta_{4} + \cdots + 14683011 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 35190961 \beta_{11} + 87954869 \beta_{10} + 167073675 \beta_{9} + 53402959 \beta_{8} + \cdots - 87954869 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 3578843 \beta_{11} + 274325772 \beta_{10} + 1429014633 \beta_{9} + 727771297 \beta_{8} + \cdots - 5527428962 \)
|
| \(\nu^{10}\) | \(=\) |
\( - 32029958308 \beta_{9} - 22922575007 \beta_{8} + 32029958308 \beta_{7} + 25590730590 \beta_{5} + \cdots - 620091357583 \)
|
| \(\nu^{11}\) | \(=\) |
\( 7083965437 \beta_{11} - 97178836389 \beta_{10} - 312288354471 \beta_{9} - 422248866934 \beta_{8} + \cdots + 97178836389 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/546\mathbb{Z}\right)^\times\).
| \(n\) | \(157\) | \(365\) | \(379\) |
| \(\chi(n)\) | \(-\beta_{2}\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 79.1 |
|
1.00000 | − | 1.73205i | −1.50000 | − | 2.59808i | −2.00000 | − | 3.46410i | −9.60453 | + | 16.6355i | −6.00000 | −5.33734 | + | 17.7345i | −8.00000 | −4.50000 | + | 7.79423i | 19.2091 | + | 33.2711i | ||||||||||||||||||||||||||||||||||||||||
| 79.2 | 1.00000 | − | 1.73205i | −1.50000 | − | 2.59808i | −2.00000 | − | 3.46410i | −6.31437 | + | 10.9368i | −6.00000 | −0.858993 | − | 18.5003i | −8.00000 | −4.50000 | + | 7.79423i | 12.6287 | + | 21.8736i | |||||||||||||||||||||||||||||||||||||||||
| 79.3 | 1.00000 | − | 1.73205i | −1.50000 | − | 2.59808i | −2.00000 | − | 3.46410i | −2.15987 | + | 3.74100i | −6.00000 | 16.7072 | − | 7.99187i | −8.00000 | −4.50000 | + | 7.79423i | 4.31974 | + | 7.48201i | |||||||||||||||||||||||||||||||||||||||||
| 79.4 | 1.00000 | − | 1.73205i | −1.50000 | − | 2.59808i | −2.00000 | − | 3.46410i | 0.579850 | − | 1.00433i | −6.00000 | 17.1146 | + | 7.07752i | −8.00000 | −4.50000 | + | 7.79423i | −1.15970 | − | 2.00866i | |||||||||||||||||||||||||||||||||||||||||
| 79.5 | 1.00000 | − | 1.73205i | −1.50000 | − | 2.59808i | −2.00000 | − | 3.46410i | 3.21637 | − | 5.57092i | −6.00000 | 0.538262 | + | 18.5124i | −8.00000 | −4.50000 | + | 7.79423i | −6.43275 | − | 11.1418i | |||||||||||||||||||||||||||||||||||||||||
| 79.6 | 1.00000 | − | 1.73205i | −1.50000 | − | 2.59808i | −2.00000 | − | 3.46410i | 7.28254 | − | 12.6137i | −6.00000 | −13.6637 | − | 12.5021i | −8.00000 | −4.50000 | + | 7.79423i | −14.5651 | − | 25.2275i | |||||||||||||||||||||||||||||||||||||||||
| 235.1 | 1.00000 | + | 1.73205i | −1.50000 | + | 2.59808i | −2.00000 | + | 3.46410i | −9.60453 | − | 16.6355i | −6.00000 | −5.33734 | − | 17.7345i | −8.00000 | −4.50000 | − | 7.79423i | 19.2091 | − | 33.2711i | |||||||||||||||||||||||||||||||||||||||||
| 235.2 | 1.00000 | + | 1.73205i | −1.50000 | + | 2.59808i | −2.00000 | + | 3.46410i | −6.31437 | − | 10.9368i | −6.00000 | −0.858993 | + | 18.5003i | −8.00000 | −4.50000 | − | 7.79423i | 12.6287 | − | 21.8736i | |||||||||||||||||||||||||||||||||||||||||
| 235.3 | 1.00000 | + | 1.73205i | −1.50000 | + | 2.59808i | −2.00000 | + | 3.46410i | −2.15987 | − | 3.74100i | −6.00000 | 16.7072 | + | 7.99187i | −8.00000 | −4.50000 | − | 7.79423i | 4.31974 | − | 7.48201i | |||||||||||||||||||||||||||||||||||||||||
| 235.4 | 1.00000 | + | 1.73205i | −1.50000 | + | 2.59808i | −2.00000 | + | 3.46410i | 0.579850 | + | 1.00433i | −6.00000 | 17.1146 | − | 7.07752i | −8.00000 | −4.50000 | − | 7.79423i | −1.15970 | + | 2.00866i | |||||||||||||||||||||||||||||||||||||||||
| 235.5 | 1.00000 | + | 1.73205i | −1.50000 | + | 2.59808i | −2.00000 | + | 3.46410i | 3.21637 | + | 5.57092i | −6.00000 | 0.538262 | − | 18.5124i | −8.00000 | −4.50000 | − | 7.79423i | −6.43275 | + | 11.1418i | |||||||||||||||||||||||||||||||||||||||||
| 235.6 | 1.00000 | + | 1.73205i | −1.50000 | + | 2.59808i | −2.00000 | + | 3.46410i | 7.28254 | + | 12.6137i | −6.00000 | −13.6637 | + | 12.5021i | −8.00000 | −4.50000 | − | 7.79423i | −14.5651 | + | 25.2275i | |||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 546.4.i.f | ✓ | 12 |
| 7.c | even | 3 | 1 | inner | 546.4.i.f | ✓ | 12 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 546.4.i.f | ✓ | 12 | 1.a | even | 1 | 1 | trivial |
| 546.4.i.f | ✓ | 12 | 7.c | even | 3 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{12} + 14 T_{5}^{11} + 499 T_{5}^{10} + 2188 T_{5}^{9} + 119096 T_{5}^{8} + 560299 T_{5}^{7} + \cdots + 12964555044 \)
acting on \(S_{4}^{\mathrm{new}}(546, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} - 2 T + 4)^{6} \)
|
| $3$ |
\( (T^{2} + 3 T + 9)^{6} \)
|
| $5$ |
\( T^{12} + \cdots + 12964555044 \)
|
| $7$ |
\( T^{12} + \cdots + 16\!\cdots\!49 \)
|
| $11$ |
\( T^{12} + \cdots + 10\!\cdots\!25 \)
|
| $13$ |
\( (T - 13)^{12} \)
|
| $17$ |
\( T^{12} + \cdots + 22\!\cdots\!96 \)
|
| $19$ |
\( T^{12} + \cdots + 29\!\cdots\!00 \)
|
| $23$ |
\( T^{12} + \cdots + 38\!\cdots\!00 \)
|
| $29$ |
\( (T^{6} - 383 T^{5} + \cdots - 3818302011)^{2} \)
|
| $31$ |
\( T^{12} + \cdots + 57\!\cdots\!76 \)
|
| $37$ |
\( T^{12} + \cdots + 82\!\cdots\!84 \)
|
| $41$ |
\( (T^{6} + \cdots - 9981093300360)^{2} \)
|
| $43$ |
\( (T^{6} + \cdots + 16605035309968)^{2} \)
|
| $47$ |
\( T^{12} + \cdots + 30\!\cdots\!44 \)
|
| $53$ |
\( T^{12} + \cdots + 22\!\cdots\!89 \)
|
| $59$ |
\( T^{12} + \cdots + 52\!\cdots\!61 \)
|
| $61$ |
\( T^{12} + \cdots + 23\!\cdots\!00 \)
|
| $67$ |
\( T^{12} + \cdots + 25\!\cdots\!24 \)
|
| $71$ |
\( (T^{6} + \cdots - 65\!\cdots\!70)^{2} \)
|
| $73$ |
\( T^{12} + \cdots + 18\!\cdots\!96 \)
|
| $79$ |
\( T^{12} + \cdots + 17\!\cdots\!96 \)
|
| $83$ |
\( (T^{6} + \cdots - 20\!\cdots\!92)^{2} \)
|
| $89$ |
\( T^{12} + \cdots + 63\!\cdots\!16 \)
|
| $97$ |
\( (T^{6} + \cdots - 11\!\cdots\!06)^{2} \)
|
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