LMFDB - Newform orbit 1323.2.h.c

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1323,2,Mod(226,1323)]

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, 2]))

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

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

chi := DirichletCharacter("1323.226");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$1323 = 3^{3} \cdot 7^{2}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	1323.h (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:	$\Q(\zeta_{18})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{6} - x^{3} + 1$ x^6 - x^3 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
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.

$f(q)$	$=$	$q + (\beta_{4} + \beta_{3} - 1) q^{2} + ( - 2 \beta_{4} - \beta_{3} + 1) q^{4} + ( - \beta_{5} + \beta_{4} + \cdots + \beta_1) q^{5} + (2 \beta_{4} - \beta_{3} - 2) q^{8} + (\beta_{5} - \beta_{4} + \cdots + 2 \beta_{2}) q^{10}+ \cdots + ( - 7 \beta_{5} + 7 \beta_{4} + \cdots + \beta_1) q^{97}+O(q^{100})$ q + (b4 + b3 - 1) * q^2 + (-2b4 - b3 + 1) q^4 + (-b5 + b4 + b3 - b2 + b1) * q^5 + (2b4 - b3 - 2) q^8 + (b5 - b4 - b3 + 2b2) q^10 + (-b5 + b3 - b2 - 2b1 + 2) q^11 + (-2b5 + 4b3 - 4b2 - b1 + 1) q^13 + (-3b4 + 1) q^16 + (-b2 + 2b1) q^17 + (2b5 - b3 + b2 - b1 + 1) q^19 + (-b5 + b4 + b3 - b2 - 2b1) q^20 + (4b5 - 2b3 + 2b2 + 3b1 - 3) * q^22 + (-b5 + b4 + b3 + 2b2 + 4b1) * q^23 + (-b5 + 2b3 - 2b2 - 2b1 + 2) q^25 + (7b5 - 6b3 + 6b2 + b1 - 1) q^26 + (-5b5 + 5b4 + 5b3 - 4b2 + 3b1) q^29 + (3b4 + 6b3 - 1) * q^31 + (3b4 + 3b3) * q^32 + (-b5 + b4 + b3 + b2 - 3b1) q^34 + (-3b5 - 3b3 + 3b2 - b1 + 1) q^37 + (-2b5 + 3b3 - 3b2 - 2b1 + 2) * q^38 + (2b5 - 2b4 - 2b3 - 2b2 + 3b1) q^40 + (b5 + b3 - b2) * q^41 + (-b5 + b4 + b3 + b2 + b1) * q^43 + (-7b5 + 4b3 - 4b2 - 5b1 + 5) * q^44 + (-5b5 + 5b4 + 5b3 - b2) q^46 + (-2b4 - 5b3 - 1) * q^47 + (5b5 - 3b3 + 3b2 + 2b1 - 2) * q^50 + (-10b5 + 5b3 - 5b2 - 7b1 + 7) * q^52 + (3b5 - 3b4 - 3b3 + 2b2 + 2b1) q^53 + (2b4 + b3) q^55 + (6b5 - 6b4 - 6b3 + 9b2 + 3b1) q^58 + (5b4 + 5b3 + 1) * q^59 + (3b4 + 3b3 + 2) * q^61 + (-b4 - b3 + 10) * q^62 + (3b4 + 4) q^64 + (b4 + 6b3 - 5) q^65 + (3b3 - 4) q^67 + (3b5 - 3b4 - 3b3 + 2b2 + 2b1) q^68 + (-3b4 + 3b3 - 3) * q^71 + (4b5 - 4b4 - 4b3 - b2 + 7b1) * q^73 + (b5 + 10b1 - 10) q^74 + (3b5 - 3b3 + 3b2 + 5b1 - 5) * q^76 + (3b3 - 7) q^79 + (-b5 + b4 + b3 + 2b2 - 5b1) * q^80 + (-3b1 + 3) q^82 + (4b5 - 4b4 - 4b3 - b2 - 6b1) * q^83 + (-2b5 + 4b3 - 4b2 + 3b1 - 3) * q^85 + (-b5 + b4 + b3 + 2b1) q^86 + (8b5 - 7b3 + 7b2 + 9b1 - 9) * q^88 + (-3b5 + 7b3 - 7b2 - 4b1 + 4) * q^89 + (8b5 - 8b4 - 8b3 + 2b2 + b1) * q^92 + (-2b4 - b3 - 6) q^94 + (b4 + 4) * q^95 + (-7b5 + 7b4 + 7b3 - 8b2 + b1) * q^97
$\operatorname{Tr}(f)(q)$	$=$	$6 q - 6 q^{2} + 6 q^{4} + 3 q^{5} - 12 q^{8} + 6 q^{11} + 3 q^{13} + 6 q^{16} + 6 q^{17} + 3 q^{19} - 6 q^{20} - 9 q^{22} + 12 q^{23} + 6 q^{25} - 3 q^{26} + 9 q^{29} - 6 q^{31} - 9 q^{34} + 3 q^{37} + 6 q^{38}+ \cdots + 3 q^{97}+O(q^{100})$ 6 * q - 6 * q^2 + 6 * q^4 + 3 * q^5 - 12 * q^8 + 6 * q^11 + 3 * q^13 + 6 * q^16 + 6 * q^17 + 3 * q^19 - 6 * q^20 - 9 * q^22 + 12 * q^23 + 6 * q^25 - 3 * q^26 + 9 * q^29 - 6 * q^31 - 9 * q^34 + 3 * q^37 + 6 * q^38 + 9 * q^40 + 3 * q^43 + 15 * q^44 - 6 * q^47 - 6 * q^50 + 21 * q^52 + 6 * q^53 + 9 * q^58 + 6 * q^59 + 12 * q^61 + 60 * q^62 + 24 * q^64 - 30 * q^65 - 24 * q^67 + 6 * q^68 - 18 * q^71 + 21 * q^73 - 30 * q^74 - 15 * q^76 - 42 * q^79 - 15 * q^80 + 9 * q^82 - 18 * q^83 - 9 * q^85 + 6 * q^86 - 27 * q^88 + 12 * q^89 + 3 * q^92 - 36 * q^94 + 24 * q^95 + 3 * q^97

Basis of coefficient ring

$\beta_{1}$	$=$	$\zeta_{18}^{3}$ v^3
$\beta_{2}$	$=$	$\zeta_{18}^{5} + \zeta_{18}$ v^5 + v
$\beta_{3}$	$=$	$-\zeta_{18}^{4} + \zeta_{18}^{2} + \zeta_{18}$ -v^4 + v^2 + v
$\beta_{4}$	$=$	$-\zeta_{18}^{5} + \zeta_{18}^{4}$ -v^5 + v^4
$\beta_{5}$	$=$	$-\zeta_{18}^{5} - \zeta_{18}^{4} + \zeta_{18}$ -v^5 - v^4 + v

Display $\zeta_{18}^j$ in terms of $\beta_i$

$\zeta_{18}$	$=$	$( \beta_{5} + \beta_{4} + 2\beta_{2} ) / 3$ (b5 + b4 + 2*b2) / 3
$\zeta_{18}^{2}$	$=$	$( -2\beta_{5} + \beta_{4} + 3\beta_{3} - \beta_{2} ) / 3$ (-2b5 + b4 + 3b3 - b2) / 3
$\zeta_{18}^{3}$	$=$	$\beta_1$ b1
$\zeta_{18}^{4}$	$=$	$( -\beta_{5} + 2\beta_{4} + \beta_{2} ) / 3$ (-b5 + 2*b4 + b2) / 3
$\zeta_{18}^{5}$	$=$	$( -\beta_{5} - \beta_{4} + \beta_{2} ) / 3$ (-b5 - b4 + b2) / 3

Display $\beta_i$ in terms of $\zeta_{18}^j$

Character values

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

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

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}

226.1

−0.766044	+	0.642788i
−0.173648	−	0.984808i
0.939693	+	0.342020i
−0.766044	−	0.642788i
−0.173648	+	0.984808i
0.939693	−	0.342020i

−2.53209

4.41147

−0.439693

−

0.761570i

−6.10607

1.11334

1.92836i

226.2

−1.34730

−0.184793

1.26604

2.19285i

2.94356

−1.70574

−

2.95442i

226.3

0.879385

−1.22668

0.673648

1.16679i

−2.83750

0.592396

1.02606i

802.1

−2.53209

4.41147

−0.439693

0.761570i

−6.10607

1.11334

−

1.92836i

802.2

−1.34730

−0.184793

1.26604

−

2.19285i

2.94356

−1.70574

2.95442i

802.3

0.879385

−1.22668

0.673648

−

1.16679i

−2.83750

0.592396

−

1.02606i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
63.h	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.h.c		6
3.b	odd	2	1		441.2.h.d		6
7.b	odd	2	1		1323.2.h.b		6
7.c	even	3	1		189.2.f.b		6
7.c	even	3	1		1323.2.g.d		6
7.d	odd	6	1		1323.2.f.d		6
7.d	odd	6	1		1323.2.g.e		6
9.c	even	3	1		1323.2.g.d		6
9.d	odd	6	1		441.2.g.c		6
21.c	even	2	1		441.2.h.e		6
21.g	even	6	1		441.2.f.c		6
21.g	even	6	1		441.2.g.b		6
21.h	odd	6	1		63.2.f.a	✓	6
21.h	odd	6	1		441.2.g.c		6
28.g	odd	6	1		3024.2.r.k		6
63.g	even	3	1		189.2.f.b		6
63.h	even	3	1		567.2.a.c		3
63.h	even	3	1	inner	1323.2.h.c		6
63.i	even	6	1		441.2.h.e		6
63.i	even	6	1		3969.2.a.q		3
63.j	odd	6	1		441.2.h.d		6
63.j	odd	6	1		567.2.a.h		3
63.k	odd	6	1		1323.2.f.d		6
63.l	odd	6	1		1323.2.g.e		6
63.n	odd	6	1		63.2.f.a	✓	6
63.o	even	6	1		441.2.g.b		6
63.s	even	6	1		441.2.f.c		6
63.t	odd	6	1		1323.2.h.b		6
63.t	odd	6	1		3969.2.a.l		3
84.n	even	6	1		1008.2.r.h		6
252.o	even	6	1		1008.2.r.h		6
252.u	odd	6	1		9072.2.a.bs		3
252.bb	even	6	1		9072.2.a.ca		3
252.bl	odd	6	1		3024.2.r.k		6

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
63.2.f.a	✓	6	21.h	odd	6	1
63.2.f.a	✓	6	63.n	odd	6	1
189.2.f.b		6	7.c	even	3	1
189.2.f.b		6	63.g	even	3	1
441.2.f.c		6	21.g	even	6	1
441.2.f.c		6	63.s	even	6	1
441.2.g.b		6	21.g	even	6	1
441.2.g.b		6	63.o	even	6	1
441.2.g.c		6	9.d	odd	6	1
441.2.g.c		6	21.h	odd	6	1
441.2.h.d		6	3.b	odd	2	1
441.2.h.d		6	63.j	odd	6	1
441.2.h.e		6	21.c	even	2	1
441.2.h.e		6	63.i	even	6	1
567.2.a.c		3	63.h	even	3	1
567.2.a.h		3	63.j	odd	6	1
1008.2.r.h		6	84.n	even	6	1
1008.2.r.h		6	252.o	even	6	1
1323.2.f.d		6	7.d	odd	6	1
1323.2.f.d		6	63.k	odd	6	1
1323.2.g.d		6	7.c	even	3	1
1323.2.g.d		6	9.c	even	3	1
1323.2.g.e		6	7.d	odd	6	1
1323.2.g.e		6	63.l	odd	6	1
1323.2.h.b		6	7.b	odd	2	1
1323.2.h.b		6	63.t	odd	6	1
1323.2.h.c		6	1.a	even	1	1	trivial
1323.2.h.c		6	63.h	even	3	1	inner
3024.2.r.k		6	28.g	odd	6	1
3024.2.r.k		6	252.bl	odd	6	1
3969.2.a.l		3	63.t	odd	6	1
3969.2.a.q		3	63.i	even	6	1
9072.2.a.bs		3	252.u	odd	6	1
9072.2.a.ca		3	252.bb	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}^{3} + 3T_{2}^{2} - 3

T_{5}^{6} - 3T_{5}^{5} + 9T_{5}^{4} - 6T_{5}^{3} + 9T_{5}^{2} + 9

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$(T^{3} + 3 T^{2} - 3)^{2}$ (T^3 + 3*T^2 - 3)^2
$3$	$T^{6}$ T^6
$5$	$T^{6} - 3 T^{5} + \cdots + 9$ T^6 - 3T^5 + 9T^4 - 6T^3 + 9T^2 + 9
$7$	$T^{6}$ T^6
$11$	$T^{6} - 6 T^{5} + \cdots + 9$ T^6 - 6T^5 + 27T^4 - 48T^3 + 63T^2 - 27*T + 9
$13$	$T^{6} - 3 T^{5} + \cdots + 11449$ T^6 - 3T^5 + 42T^4 - 115T^3 + 1410T^2 - 3531*T + 11449
$17$	$T^{6} - 6 T^{5} + \cdots + 9$ T^6 - 6T^5 + 27T^4 - 48T^3 + 63T^2 - 27*T + 9
$19$	$T^{6} - 3 T^{5} + \cdots + 289$ T^6 - 3T^5 + 15T^4 - 16T^3 + 87T^2 - 102*T + 289
$23$	$T^{6} - 12 T^{5} + \cdots + 9$ T^6 - 12T^5 + 117T^4 - 330T^3 + 765T^2 + 81*T + 9
$29$	$T^{6} - 9 T^{5} + \cdots + 110889$ T^6 - 9T^5 + 117T^4 - 342T^3 + 4293T^2 - 11988*T + 110889
$31$	$(T^{3} + 3 T^{2} + \cdots - 323)^{2}$ (T^3 + 3T^2 - 78T - 323)^2
$37$	$T^{6} - 3 T^{5} + \cdots + 104329$ T^6 - 3T^5 + 87T^4 - 412T^3 + 7053T^2 - 25194*T + 104329
$41$	$T^{6} + 9 T^{4} + \cdots + 81$ T^6 + 9T^4 + 18T^3 + 81T^2 + 81T + 81
$43$	$T^{6} - 3 T^{5} + \cdots + 1$ T^6 - 3T^5 + 15T^4 + 20T^3 + 33T^2 + 6*T + 1
$47$	$(T^{3} + 3 T^{2} - 54 T + 51)^{2}$ (T^3 + 3T^2 - 54T + 51)^2
$53$	$T^{6} - 6 T^{5} + \cdots + 9$ T^6 - 6T^5 + 45T^4 + 60T^3 + 63T^2 + 27*T + 9
$59$	$(T^{3} - 3 T^{2} - 72 T - 51)^{2}$ (T^3 - 3T^2 - 72T - 51)^2
$61$	$(T^{3} - 6 T^{2} - 15 T + 19)^{2}$ (T^3 - 6T^2 - 15T + 19)^2
$67$	$(T^{3} + 12 T^{2} + \cdots - 17)^{2}$ (T^3 + 12T^2 + 21T - 17)^2
$71$	$(T^{3} + 9 T^{2} - 54 T + 27)^{2}$ (T^3 + 9T^2 - 54T + 27)^2
$73$	$T^{6} - 21 T^{5} + \cdots + 72361$ T^6 - 21T^5 + 357T^4 - 2302T^3 + 12705T^2 + 22596*T + 72361
$79$	$(T^{3} + 21 T^{2} + \cdots + 181)^{2}$ (T^3 + 21T^2 + 120T + 181)^2
$83$	$T^{6} + 18 T^{5} + \cdots + 81$ T^6 + 18T^5 + 279T^4 + 792T^3 + 1863T^2 + 405*T + 81
$89$	$T^{6} - 12 T^{5} + \cdots + 660969$ T^6 - 12T^5 + 207T^4 - 870T^3 + 13725T^2 - 51219*T + 660969
$97$	$T^{6} - 3 T^{5} + \cdots + 104329$ T^6 - 3T^5 + 177T^4 - 142T^3 + 29193T^2 - 54264*T + 104329
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Elliptic curves over $\Q$
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Genus 2 curves over $\Q$
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$n$ :		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 226.3
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