LMFDB - Newform orbit 1323.2.f.c

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1323,2,Mod(442,1323)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1323, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([2, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1323.442");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$1323 = 3^{3} \cdot 7^{2}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	1323.f (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:	$10.5642081874$
Analytic rank:	$0$
Dimension:	$6$
Relative dimension:	$3$ over $\Q(\zeta_{3})$
Coefficient field:	6.0.309123.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{6} - 3x^{5} + 10x^{4} - 15x^{3} + 19x^{2} - 12x + 3$ x^6 - 3x^5 + 10x^4 - 15x^3 + 19x^2 - 12*x + 3
Coefficient ring:	$\Z[a_1, \ldots, a_{4}]$
Coefficient ring index:	$3$
Twist minimal:	no (minimal twist has level 63)
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 $1,\beta_1,\ldots,\beta_{5}$ for the coefficient ring described below. We also show the integral $q$ -expansion of the trace form.

$f(q)$	$=$	$q + ( - \beta_{5} + \beta_1) q^{2} + (\beta_{5} + \beta_{4} + \beta_{2} - 1) q^{4} + ( - 2 \beta_{4} - \beta_{2} + 2) q^{5} + (\beta_{3} - \beta_1 + 2) q^{8} + (3 \beta_1 + 1) q^{10} + (2 \beta_{5} - \beta_{4} + \cdots - 2 \beta_1) q^{11}+ \cdots + ( - 5 \beta_{5} + 2 \beta_{4} + \cdots + 5 \beta_1) q^{97}+O(q^{100})$ q + (-b5 + b1) * q^2 + (b5 + b4 + b2 - 1) * q^4 + (-2b4 - b2 + 2) q^5 + (b3 - b1 + 2) * q^8 + (3b1 + 1) q^10 + (2b5 - b4 + b3 + b2 - 2b1) * q^11 + (-b4 + 1) * q^13 + (-2b5 + b3 + b2 + 2b1) * q^16 + (b3 - b1 - 4) * q^17 + (b3 - b1 + 1) * q^19 + (2b5 + 5b4 + b3 + b2 - 2b1) q^20 + (-2b5 - 5b4 - 2b2 + 5) q^22 + (b5 - b4 - 2b2 + 1) q^23 + (b5 - 3b4 - 2b3 - 2b2 - b1) q^25 + b1 * q^26 + (b5 - b1) * q^29 + (-2b5 + 2b4 + b2 - 2) * q^31 + (-b5 + 3b4 - 3) q^32 + (2b5 - 2b4 - b3 - b2 - 2b1) q^34 + 3b3 q^37 + (-3b5 - 2b4 - b3 - b2 + 3b1) q^38 + (-2b5 - 7b4 - 2b2 + 7) q^40 + (-7b4 + b2 + 7) q^41 + (4b5 + b3 + b2 - 4b1) * q^43 + (5b1 - 6) q^44 + (b3 + 2b1 + 5) q^46 + (3b5 + 3b4 + 3b3 + 3b2 - 3b1) q^47 + (4b5 - 5b4 - b2 + 5) * q^50 + (b5 + b4 + b3 + b2 - b1) * q^52 + (-2b3 - b1 + 5) q^53 + (-b3 - 5b1) q^55 + (-b5 - 3b4 - b2 + 3) q^58 + (b5 - 4b4 - 2b2 + 4) * q^59 + (-b5 - b4 + 2b3 + 2b2 + b1) * q^61 + (-2b3 - b1 - 7) q^62 + (b3 + 2b1 - 3) q^64 + (-2b4 - b3 - b2) q^65 + (-7b5 + 2b4 - b2 - 2) * q^67 + (-b5 + b4 - 4b2 - 1) q^68 + (2b3 + b1 - 2) q^71 + (5b3 + b1 + 1) q^73 + (-3b5 + 3b4 + 3b1) q^74 + (4b5 + 6b4 + b2 - 6) * q^76 + (-5b5 - 3b4 + b3 + b2 + 5b1) q^79 + (7b1 + 6) q^80 + (6b1 - 1) q^82 + (-b5 + 4b4 - b3 - b2 + b1) q^83 + (-2b5 + 5b4 + 4b2 - 5) q^85 + (-5b5 - 11b4 - 4b2 + 11) q^86 + (7b5 + 5b4 + b3 + b2 - 7b1) q^88 + (-b3 + 6b1 + 1) q^89 + (-2b5 + 5b4 - 2b3 - 2b2 + 2b1) q^92 + (-9b5 - 6b4 - 3b2 + 6) q^94 + (-2b5 - 5b4 - b2 + 5) * q^95 + (-5b5 + 2b4 - 2b3 - 2b2 + 5b1) q^97
$\operatorname{Tr}(f)(q)$	$=$	$6 q - q^{2} - 3 q^{4} + 5 q^{5} + 12 q^{8} - 2 q^{11} + 3 q^{13} - 3 q^{16} - 24 q^{17} + 6 q^{19} + 16 q^{20} + 15 q^{22} - 6 q^{25} - 2 q^{26} + q^{29} - 3 q^{31} - 8 q^{32} - 3 q^{34} - 6 q^{37} - 8 q^{38}+ \cdots + 3 q^{97}+O(q^{100})$ 6 * q - q^2 - 3 * q^4 + 5 * q^5 + 12 * q^8 - 2 * q^11 + 3 * q^13 - 3 * q^16 - 24 * q^17 + 6 * q^19 + 16 * q^20 + 15 * q^22 - 6 * q^25 - 2 * q^26 + q^29 - 3 * q^31 - 8 * q^32 - 3 * q^34 - 6 * q^37 - 8 * q^38 + 21 * q^40 + 22 * q^41 + 3 * q^43 - 46 * q^44 + 24 * q^46 + 9 * q^47 + 10 * q^50 + 3 * q^52 + 36 * q^53 + 12 * q^55 + 9 * q^58 + 9 * q^59 - 6 * q^61 - 36 * q^62 - 24 * q^64 - 5 * q^65 - 6 * q^68 - 18 * q^71 - 6 * q^73 + 6 * q^74 - 21 * q^76 - 15 * q^79 + 22 * q^80 - 18 * q^82 + 12 * q^83 - 9 * q^85 + 34 * q^86 + 21 * q^88 - 4 * q^89 + 15 * q^92 + 24 * q^94 + 16 * q^95 + 3 * q^97

Basis of coefficient ring in terms of a root $\nu$ of $x^{6} - 3x^{5} + 10x^{4} - 15x^{3} + 19x^{2} - 12x + 3$ x^6 - 3*x^5 + 10*x^4 - 15*x^3 + 19*x^2 - 12*x + 3 :

$\beta_{1}$	$=$	$\nu^{2} - \nu + 2$ v^2 - v + 2
$\beta_{2}$	$=$	$( -\nu^{5} + \nu^{4} - 8\nu^{3} + 5\nu^{2} - 18\nu + 6 ) / 3$ (-v^5 + v^4 - 8v^3 + 5v^2 - 18*v + 6) / 3
$\beta_{3}$	$=$	$\nu^{4} - 2\nu^{3} + 6\nu^{2} - 5\nu + 3$ v^4 - 2v^3 + 6v^2 - 5*v + 3
$\beta_{4}$	$=$	$( -2\nu^{5} + 5\nu^{4} - 16\nu^{3} + 19\nu^{2} - 21\nu + 9 ) / 3$ (-2v^5 + 5v^4 - 16v^3 + 19v^2 - 21*v + 9) / 3
$\beta_{5}$	$=$	$( 2\nu^{5} - 5\nu^{4} + 19\nu^{3} - 22\nu^{2} + 30\nu - 9 ) / 3$ (2v^5 - 5v^4 + 19v^3 - 22v^2 + 30*v - 9) / 3

Display $\nu^j$ in terms of $\beta_i$

$\nu$	$=$	$( -2\beta_{5} - \beta_{4} - \beta_{3} - 2\beta_{2} + \beta _1 + 2 ) / 3$ (-2b5 - b4 - b3 - 2b2 + b1 + 2) / 3
$\nu^{2}$	$=$	$( -2\beta_{5} - \beta_{4} - \beta_{3} - 2\beta_{2} + 4\beta _1 - 4 ) / 3$ (-2b5 - b4 - b3 - 2b2 + 4*b1 - 4) / 3
$\nu^{3}$	$=$	$( 7\beta_{5} + 5\beta_{4} + 2\beta_{3} + 4\beta_{2} + \beta _1 - 10 ) / 3$ (7b5 + 5b4 + 2b3 + 4b2 + b1 - 10) / 3
$\nu^{4}$	$=$	$( 16\beta_{5} + 11\beta_{4} + 8\beta_{3} + 10\beta_{2} - 17\beta _1 + 5 ) / 3$ (16b5 + 11b4 + 8b3 + 10b2 - 17*b1 + 5) / 3
$\nu^{5}$	$=$	$( -14\beta_{5} - 16\beta_{4} + 5\beta_{3} - 5\beta_{2} - 23\beta _1 + 47 ) / 3$ (-14b5 - 16b4 + 5b3 - 5b2 - 23*b1 + 47) / 3

Display $\beta_i$ in terms of $\nu^j$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1323\mathbb{Z}\right)^\times$ .

$n$	$785$	$1081$
$\chi(n)$	$-1 + \beta_{4}$	$1$

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}

442.1

0.500000	−	2.05195i
0.500000	+	1.41036i
0.500000	−	0.224437i
0.500000	+	2.05195i
0.500000	−	1.41036i
0.500000	+	0.224437i

−1.23025

−

2.13086i

−2.02704

3.51094i

1.29679

−

2.24611i

5.05408

−6.38151

442.2

−0.119562

−

0.207087i

0.971410

−

1.68253i

−0.590972

1.02359i

−0.942820

0.282630

442.3

0.849814

1.47192i

−0.444368

0.769668i

1.79418

−

3.10761i

1.88874

6.09888

883.1

−1.23025

2.13086i

−2.02704

−

3.51094i

1.29679

2.24611i

5.05408

−6.38151

883.2

−0.119562

0.207087i

0.971410

1.68253i

−0.590972

−

1.02359i

−0.942820

0.282630

883.3

0.849814

−

1.47192i

−0.444368

−

0.769668i

1.79418

3.10761i

1.88874

6.09888

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	1323.2.f.c		6
3.b	odd	2	1		441.2.f.d		6
7.b	odd	2	1		189.2.f.a		6
7.c	even	3	1		1323.2.g.b		6
7.c	even	3	1		1323.2.h.e		6
7.d	odd	6	1		1323.2.g.c		6
7.d	odd	6	1		1323.2.h.d		6
9.c	even	3	1	inner	1323.2.f.c		6
9.c	even	3	1		3969.2.a.p		3
9.d	odd	6	1		441.2.f.d		6
9.d	odd	6	1		3969.2.a.m		3
21.c	even	2	1		63.2.f.b	✓	6
21.g	even	6	1		441.2.g.e		6
21.g	even	6	1		441.2.h.c		6
21.h	odd	6	1		441.2.g.d		6
21.h	odd	6	1		441.2.h.b		6
28.d	even	2	1		3024.2.r.g		6
63.g	even	3	1		1323.2.h.e		6
63.h	even	3	1		1323.2.g.b		6
63.i	even	6	1		441.2.g.e		6
63.j	odd	6	1		441.2.g.d		6
63.k	odd	6	1		1323.2.h.d		6
63.l	odd	6	1		189.2.f.a		6
63.l	odd	6	1		567.2.a.g		3
63.n	odd	6	1		441.2.h.b		6
63.o	even	6	1		63.2.f.b	✓	6
63.o	even	6	1		567.2.a.d		3
63.s	even	6	1		441.2.h.c		6
63.t	odd	6	1		1323.2.g.c		6
84.h	odd	2	1		1008.2.r.k		6
252.s	odd	6	1		1008.2.r.k		6
252.s	odd	6	1		9072.2.a.bq		3
252.bi	even	6	1		3024.2.r.g		6
252.bi	even	6	1		9072.2.a.cd		3

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
63.2.f.b	✓	6	21.c	even	2	1
63.2.f.b	✓	6	63.o	even	6	1
189.2.f.a		6	7.b	odd	2	1
189.2.f.a		6	63.l	odd	6	1
441.2.f.d		6	3.b	odd	2	1
441.2.f.d		6	9.d	odd	6	1
441.2.g.d		6	21.h	odd	6	1
441.2.g.d		6	63.j	odd	6	1
441.2.g.e		6	21.g	even	6	1
441.2.g.e		6	63.i	even	6	1
441.2.h.b		6	21.h	odd	6	1
441.2.h.b		6	63.n	odd	6	1
441.2.h.c		6	21.g	even	6	1
441.2.h.c		6	63.s	even	6	1
567.2.a.d		3	63.o	even	6	1
567.2.a.g		3	63.l	odd	6	1
1008.2.r.k		6	84.h	odd	2	1
1008.2.r.k		6	252.s	odd	6	1
1323.2.f.c		6	1.a	even	1	1	trivial
1323.2.f.c		6	9.c	even	3	1	inner
1323.2.g.b		6	7.c	even	3	1
1323.2.g.b		6	63.h	even	3	1
1323.2.g.c		6	7.d	odd	6	1
1323.2.g.c		6	63.t	odd	6	1
1323.2.h.d		6	7.d	odd	6	1
1323.2.h.d		6	63.k	odd	6	1
1323.2.h.e		6	7.c	even	3	1
1323.2.h.e		6	63.g	even	3	1
3024.2.r.g		6	28.d	even	2	1
3024.2.r.g		6	252.bi	even	6	1
3969.2.a.m		3	9.d	odd	6	1
3969.2.a.p		3	9.c	even	3	1
9072.2.a.bq		3	252.s	odd	6	1
9072.2.a.cd		3	252.bi	even	6	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(1323, [\chi])$ :

T_{2}^{6} + T_{2}^{5} + 5T_{2}^{4} - 2T_{2}^{3} + 17T_{2}^{2} + 4T_{2} + 1

T_{5}^{6} - 5T_{5}^{5} + 23T_{5}^{4} - 32T_{5}^{3} + 59T_{5}^{2} + 22T_{5} + 121

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{6} + T^{5} + 5 T^{4} + \cdots + 1$ T^6 + T^5 + 5T^4 - 2T^3 + 17T^2 + 4T + 1
$3$	$T^{6}$ T^6
$5$	$T^{6} - 5 T^{5} + \cdots + 121$ T^6 - 5T^5 + 23T^4 - 32T^3 + 59T^2 + 22*T + 121
$7$	$T^{6}$ T^6
$11$	$T^{6} + 2 T^{5} + \cdots + 2209$ T^6 + 2T^5 + 23T^4 + 56T^3 + 455T^2 + 893*T + 2209
$13$	$(T^{2} - T + 1)^{3}$ (T^2 - T + 1)^3
$17$	$(T^{3} + 12 T^{2} + \cdots + 27)^{2}$ (T^3 + 12T^2 + 39T + 27)^2
$19$	$(T^{3} - 3 T^{2} - 6 T + 7)^{2}$ (T^3 - 3T^2 - 6T + 7)^2
$23$	$T^{6} + 33 T^{4} + \cdots + 81$ T^6 + 33T^4 + 18T^3 + 1089T^2 + 297T + 81
$29$	$T^{6} - T^{5} + 5 T^{4} + \cdots + 1$ T^6 - T^5 + 5T^4 + 2T^3 + 17T^2 - 4T + 1
$31$	$T^{6} + 3 T^{5} + \cdots + 729$ T^6 + 3T^5 + 33T^4 - 126T^3 + 495T^2 - 648*T + 729
$37$	$(T^{3} + 3 T^{2} - 54 T + 81)^{2}$ (T^3 + 3T^2 - 54T + 81)^2
$41$	$T^{6} - 22 T^{5} + \cdots + 124609$ T^6 - 22T^5 + 329T^4 - 2704T^3 + 16259T^2 - 54715*T + 124609
$43$	$T^{6} - 3 T^{5} + \cdots + 14641$ T^6 - 3T^5 + 75T^4 + 440T^3 + 3993T^2 + 7986*T + 14641
$47$	$T^{6} - 9 T^{5} + \cdots + 35721$ T^6 - 9T^5 + 135T^4 + 108T^3 + 4617T^2 - 10206*T + 35721
$53$	$(T^{3} - 18 T^{2} + 75 T - 9)^{2}$ (T^3 - 18T^2 + 75T - 9)^2
$59$	$T^{6} - 9 T^{5} + \cdots + 3969$ T^6 - 9T^5 + 87T^4 - 72T^3 + 603T^2 - 378*T + 3969
$61$	$T^{6} + 6 T^{5} + \cdots + 4489$ T^6 + 6T^5 + 57T^4 + 8T^3 + 843T^2 + 1407*T + 4489
$67$	$T^{6} + 207 T^{4} + \cdots + 466489$ T^6 + 207T^4 + 1366T^3 + 42849T^2 + 141381T + 466489
$71$	$(T^{3} + 9 T^{2} - 6 T - 81)^{2}$ (T^3 + 9T^2 - 6T - 81)^2
$73$	$(T^{3} + 3 T^{2} + \cdots + 243)^{2}$ (T^3 + 3T^2 - 168T + 243)^2
$79$	$T^{6} + 15 T^{5} + \cdots + 591361$ T^6 + 15T^5 + 273T^4 + 818T^3 + 13839T^2 + 36912*T + 591361
$83$	$T^{6} - 12 T^{5} + \cdots + 729$ T^6 - 12T^5 + 105T^4 - 414T^3 + 1197T^2 - 1053*T + 729
$89$	$(T^{3} + 2 T^{2} + \cdots + 379)^{2}$ (T^3 + 2T^2 - 151T + 379)^2
$97$	$T^{6} - 3 T^{5} + \cdots + 363609$ T^6 - 3T^5 + 123T^4 - 864T^3 + 14805T^2 - 68742*T + 363609
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Classical	Maass
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Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

$n$ :		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 442.3
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