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: | \(10\) |
| Relative dimension: | \(5\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{10} + \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{10} + 306x^{8} + 26725x^{6} + 579423x^{4} + 2058904x^{2} + 1651692 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 3^{3} \) |
| 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 \(1,\beta_1,\ldots,\beta_{9}\) 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
\( x^{10} + 306x^{8} + 26725x^{6} + 579423x^{4} + 2058904x^{2} + 1651692 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 5254009 \nu^{8} + 1581630884 \nu^{6} + 133565610437 \nu^{4} + 2575732699969 \nu^{2} + 4364050568682 ) / 201366429312 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -1448081\nu^{8} - 437233060\nu^{6} - 36863457085\nu^{4} - 688840220633\nu^{2} - 1044578936058 ) / 50341607328 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 24025 \nu^{9} - 7466660 \nu^{7} - 677811749 \nu^{5} - 16887829537 \nu^{3} + \cdots + 43021563648 ) / 86043127296 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 2224572631 \nu^{9} + 11005205246 \nu^{8} - 686094858524 \nu^{7} + 3353939312248 \nu^{6} + \cdots + 13\!\cdots\!12 ) / 298827781099008 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 473596456 \nu^{9} - 174287267 \nu^{8} + 143533147680 \nu^{7} - 57959532876 \nu^{6} + \cdots - 14\!\cdots\!22 ) / 49804630183168 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 1420789368 \nu^{9} + 6583868767 \nu^{8} + 430599443040 \nu^{7} + 2006490597692 \nu^{6} + \cdots + 63\!\cdots\!02 ) / 149413890549504 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 1420789368 \nu^{9} + 6583868767 \nu^{8} - 430599443040 \nu^{7} + 2006490597692 \nu^{6} + \cdots + 63\!\cdots\!02 ) / 149413890549504 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 630742687 \nu^{9} + 556924954 \nu^{8} - 192021916220 \nu^{7} + 167652873704 \nu^{6} + \cdots + 462589360280292 ) / 42689683014144 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 630742687 \nu^{9} - 613986344 \nu^{8} + 192021916220 \nu^{7} - 185386817440 \nu^{6} + \cdots - 442901468888592 ) / 42689683014144 \)
|
| \(\nu\) | \(=\) |
\( ( -2\beta_{9} - 2\beta_{8} + \beta_{2} + \beta_1 ) / 3 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -8\beta_{7} + \beta_{6} - 9\beta_{5} - 6\beta_{2} + 4\beta _1 - 180 ) / 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 236 \beta_{9} + 272 \beta_{8} - 17 \beta_{7} + 42 \beta_{6} - 25 \beta_{5} - 50 \beta_{4} + 586 \beta_{3} + \cdots - 293 ) / 3 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 1195\beta_{7} - 98\beta_{6} + 1293\beta_{5} + 1437\beta_{2} - 95\beta _1 + 23538 ) / 3 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( - 34652 \beta_{9} - 39230 \beta_{8} + 1671 \beta_{7} - 6891 \beta_{6} + 5220 \beta_{5} + 10440 \beta_{4} + \cdots + 74955 ) / 3 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( -174686\beta_{7} + 19846\beta_{6} - 194532\beta_{5} - 267009\beta_{2} - 58958\beta _1 - 3439119 ) / 3 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 5364842 \beta_{9} + 5866286 \beta_{8} - 68180 \beta_{7} + 1032483 \beta_{6} - 964303 \beta_{5} + \cdots - 14764271 ) / 3 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( 26129329\beta_{7} - 3973223\beta_{6} + 30102552\beta_{5} + 46789089\beta_{2} + 18317383\beta _1 + 522665496 ) / 3 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( - 848273108 \beta_{9} - 900262766 \beta_{8} - 14004306 \beta_{7} - 155991159 \beta_{6} + \cdots + 2685186714 ) / 3 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/546\mathbb{Z}\right)^\times\).
| \(n\) | \(157\) | \(365\) | \(379\) |
| \(\chi(n)\) | \(-1 + \beta_{3}\) | \(1\) | \(1\) |
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 | −6.08880 | + | 10.5461i | 6.00000 | 14.3714 | + | 11.6817i | −8.00000 | −4.50000 | + | 7.79423i | 12.1776 | + | 21.0922i | ||||||||||||||||||||||||||||||||||
| 79.2 | 1.00000 | − | 1.73205i | 1.50000 | + | 2.59808i | −2.00000 | − | 3.46410i | −5.34443 | + | 9.25682i | 6.00000 | 2.97712 | − | 18.2794i | −8.00000 | −4.50000 | + | 7.79423i | 10.6889 | + | 18.5136i | |||||||||||||||||||||||||||||||||||
| 79.3 | 1.00000 | − | 1.73205i | 1.50000 | + | 2.59808i | −2.00000 | − | 3.46410i | −5.28868 | + | 9.16026i | 6.00000 | −17.4236 | − | 6.27828i | −8.00000 | −4.50000 | + | 7.79423i | 10.5774 | + | 18.3205i | |||||||||||||||||||||||||||||||||||
| 79.4 | 1.00000 | − | 1.73205i | 1.50000 | + | 2.59808i | −2.00000 | − | 3.46410i | 4.01556 | − | 6.95515i | 6.00000 | −7.52050 | + | 16.9246i | −8.00000 | −4.50000 | + | 7.79423i | −8.03112 | − | 13.9103i | |||||||||||||||||||||||||||||||||||
| 79.5 | 1.00000 | − | 1.73205i | 1.50000 | + | 2.59808i | −2.00000 | − | 3.46410i | 6.20635 | − | 10.7497i | 6.00000 | 11.5956 | − | 14.4410i | −8.00000 | −4.50000 | + | 7.79423i | −12.4127 | − | 21.4994i | |||||||||||||||||||||||||||||||||||
| 235.1 | 1.00000 | + | 1.73205i | 1.50000 | − | 2.59808i | −2.00000 | + | 3.46410i | −6.08880 | − | 10.5461i | 6.00000 | 14.3714 | − | 11.6817i | −8.00000 | −4.50000 | − | 7.79423i | 12.1776 | − | 21.0922i | |||||||||||||||||||||||||||||||||||
| 235.2 | 1.00000 | + | 1.73205i | 1.50000 | − | 2.59808i | −2.00000 | + | 3.46410i | −5.34443 | − | 9.25682i | 6.00000 | 2.97712 | + | 18.2794i | −8.00000 | −4.50000 | − | 7.79423i | 10.6889 | − | 18.5136i | |||||||||||||||||||||||||||||||||||
| 235.3 | 1.00000 | + | 1.73205i | 1.50000 | − | 2.59808i | −2.00000 | + | 3.46410i | −5.28868 | − | 9.16026i | 6.00000 | −17.4236 | + | 6.27828i | −8.00000 | −4.50000 | − | 7.79423i | 10.5774 | − | 18.3205i | |||||||||||||||||||||||||||||||||||
| 235.4 | 1.00000 | + | 1.73205i | 1.50000 | − | 2.59808i | −2.00000 | + | 3.46410i | 4.01556 | + | 6.95515i | 6.00000 | −7.52050 | − | 16.9246i | −8.00000 | −4.50000 | − | 7.79423i | −8.03112 | + | 13.9103i | |||||||||||||||||||||||||||||||||||
| 235.5 | 1.00000 | + | 1.73205i | 1.50000 | − | 2.59808i | −2.00000 | + | 3.46410i | 6.20635 | + | 10.7497i | 6.00000 | 11.5956 | + | 14.4410i | −8.00000 | −4.50000 | − | 7.79423i | −12.4127 | + | 21.4994i | |||||||||||||||||||||||||||||||||||
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.d | ✓ | 10 |
| 7.c | even | 3 | 1 | inner | 546.4.i.d | ✓ | 10 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 546.4.i.d | ✓ | 10 | 1.a | even | 1 | 1 | trivial |
| 546.4.i.d | ✓ | 10 | 7.c | even | 3 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{10} + 13 T_{5}^{9} + 381 T_{5}^{8} + 3034 T_{5}^{7} + 73639 T_{5}^{6} + 518550 T_{5}^{5} + \cdots + 18837562500 \)
acting on \(S_{4}^{\mathrm{new}}(546, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} - 2 T + 4)^{5} \)
|
| $3$ |
\( (T^{2} - 3 T + 9)^{5} \)
|
| $5$ |
\( T^{10} + \cdots + 18837562500 \)
|
| $7$ |
\( T^{10} + \cdots + 4747561509943 \)
|
| $11$ |
\( T^{10} + \cdots + 698703489 \)
|
| $13$ |
\( (T - 13)^{10} \)
|
| $17$ |
\( T^{10} + \cdots + 69226377017121 \)
|
| $19$ |
\( T^{10} + \cdots + 25\!\cdots\!01 \)
|
| $23$ |
\( T^{10} + \cdots + 714345678744516 \)
|
| $29$ |
\( (T^{5} + 26 T^{4} + \cdots + 32560032573)^{2} \)
|
| $31$ |
\( T^{10} + \cdots + 97\!\cdots\!00 \)
|
| $37$ |
\( T^{10} + \cdots + 81\!\cdots\!76 \)
|
| $41$ |
\( (T^{5} - 26 T^{4} + \cdots + 7142115834)^{2} \)
|
| $43$ |
\( (T^{5} + 184 T^{4} + \cdots + 752717885452)^{2} \)
|
| $47$ |
\( T^{10} + \cdots + 50\!\cdots\!96 \)
|
| $53$ |
\( T^{10} + \cdots + 18\!\cdots\!09 \)
|
| $59$ |
\( T^{10} + \cdots + 37\!\cdots\!25 \)
|
| $61$ |
\( T^{10} + \cdots + 99\!\cdots\!21 \)
|
| $67$ |
\( T^{10} + \cdots + 61\!\cdots\!09 \)
|
| $71$ |
\( (T^{5} + \cdots - 3092007693843)^{2} \)
|
| $73$ |
\( T^{10} + \cdots + 65\!\cdots\!04 \)
|
| $79$ |
\( T^{10} + \cdots + 18\!\cdots\!00 \)
|
| $83$ |
\( (T^{5} + \cdots - 14054061499452)^{2} \)
|
| $89$ |
\( T^{10} + \cdots + 78\!\cdots\!00 \)
|
| $97$ |
\( (T^{5} + \cdots + 78787660010698)^{2} \)
|
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