LMFDB - Newform orbit 2736.2.bm.o

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2736,2,Mod(559,2736)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2736, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 0, 0, 1]))

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

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

chi := DirichletCharacter("2736.559");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2736 = 2^{4} \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2736.bm (of order $6$, 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:	$21.8470699930$
Analytic rank:	$0$
Dimension:	$6$
Relative dimension:	$3$ over $\Q(\zeta_{6})$
Coefficient field:	6.0.954288.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 2x^{4} + 3x^{3} - 6x^{2} - 9x + 27 $ x^6 - x^5 - 2x^4 + 3x^3 - 6x^2 - 9x + 27
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 912)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

$q$-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} - \beta_{2} + 1) q^{5} + ( - \beta_{5} + 2 \beta_{3} + \beta_1 + 1) q^{7}+O(q^{10}) $ q + (b4 + b3 - b2 + 1) * q^5 + (-b5 + 2b3 + b1 + 1) q^7	$ q + (\beta_{4} + \beta_{3} - \beta_{2} + 1) q^{5} + ( - \beta_{5} + 2 \beta_{3} + \beta_1 + 1) q^{7} + (\beta_{5} - \beta_1) q^{11} + ( - \beta_{5} + \beta_{4} - 2 \beta_{2}) q^{13} + ( - 2 \beta_{5} + \beta_{4} + \beta_{3} + \cdots + 1) q^{17}+ \cdots + (\beta_{4} + 3 \beta_{3} + \beta_{2} + \cdots - 3) q^{97}+O(q^{100}) $ q + (b4 + b3 - b2 + 1) * q^5 + (-b5 + 2b3 + b1 + 1) q^7 + (b5 - b1) * q^11 + (-b5 + b4 - 2b2) q^13 + (-2b5 + b4 + b3 - b2 + b1 + 1) q^17 + (-2b5 + b4 + b3 + b1 + 1) q^19 + (2b5 - b4 + b3 + 2b2 + 2) * q^23 + (-b5 + b4 + 2b3 + 2b1) * q^25 + (b5 - b4 - b3 + 2b2 - 2) q^29 - b2 * q^31 + (2b4 + 2b2 + b1) * q^35 + (-2b5 + 4b4 - 2b3 - 2b2 + 2b1 - 1) q^37 + (-2b4 - 2b3 - 2b2 + 2) q^41 + (-b4 - 4b3 - b2 + 4) q^43 + (4b5 - 2b3 - 4) * q^47 + (b5 - b2 + b1 - 1) * q^49 + (-b5 - 4b3 - 8) q^53 + (-b4 + b3 - b2 - b1 - 1) * q^55 + (b4 + b3 - b2 + 1) * q^59 + (-b5 - 3b4 - 2b3 + 2b1) q^61 + (-4b5 + 2b4 + 10b3 - b2 + 4b1 + 5) * q^65 + (-b5 + 2b4 + 7b3 + 2b1) q^67 + (2b5 + b4 - 5b3 - b2 - b1 - 5) * q^71 + (4b5 - 2b4 + 3b3 + 2b2 - 2b1 + 3) q^73 + (b2 + 5) * q^77 + (-3b4 - 4b3 + 3b2 - 4) q^79 + (3b5 - 2b4 - 2b3 + b2 - 3b1 - 1) * q^83 + (-2b5 + 4b4 + 4b3 + 4b1) * q^85 + (b5 + 2b4 - 4b2) * q^89 + (5b4 + 2b3) * q^91 + (4b4 - 2b3 + 3b1 - 6) q^95 + (b4 + 3b3 + b2 - b1 - 3) q^97
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 2 q^{5}+O(q^{10}) $ 6 * q + 2 * q^5	$ 6 q + 2 q^{5} - 3 q^{13} + 2 q^{17} + 4 q^{19} + 12 q^{23} - 5 q^{25} - 6 q^{29} - 2 q^{31} + 6 q^{35} + 12 q^{41} + 33 q^{43} - 18 q^{47} - 8 q^{49} - 36 q^{53} - 12 q^{55} + 2 q^{59} + 3 q^{61} - 19 q^{67} - 16 q^{71} + 11 q^{73} + 32 q^{77} - 9 q^{79} - 8 q^{85} - 6 q^{89} - q^{91} - 26 q^{95} - 24 q^{97}+O(q^{100}) $ 6 * q + 2 * q^5 - 3 * q^13 + 2 * q^17 + 4 * q^19 + 12 * q^23 - 5 * q^25 - 6 * q^29 - 2 * q^31 + 6 * q^35 + 12 * q^41 + 33 * q^43 - 18 * q^47 - 8 * q^49 - 36 * q^53 - 12 * q^55 + 2 * q^59 + 3 * q^61 - 19 * q^67 - 16 * q^71 + 11 * q^73 + 32 * q^77 - 9 * q^79 - 8 * q^85 - 6 * q^89 - q^91 - 26 * q^95 - 24 * q^97

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{6} - x^{5} - 2x^{4} + 3x^{3} - 6x^{2} - 9x + 27 $ x^6 - x^5 - 2*x^4 + 3*x^3 - 6*x^2 - 9*x + 27 :

$\beta_{1}$	$=$	$ ( \nu^{5} + 5\nu^{4} + \nu^{3} + 9\nu^{2} + 21\nu - 45 ) / 27 $ (v^5 + 5v^4 + v^3 + 9v^2 + 21*v - 45) / 27
$\beta_{2}$	$=$	$ ( \nu^{5} - 4\nu^{4} + \nu^{3} + 18\nu^{2} - 33\nu + 9 ) / 27 $ (v^5 - 4v^4 + v^3 + 18v^2 - 33*v + 9) / 27
$\beta_{3}$	$=$	$ ( -2\nu^{5} - \nu^{4} - 2\nu^{3} + 12\nu + 9 ) / 27 $ (-2v^5 - v^4 - 2v^3 + 12*v + 9) / 27
$\beta_{4}$	$=$	$ ( 2\nu^{5} - 8\nu^{4} + 2\nu^{3} + 9\nu^{2} - 12\nu + 18 ) / 27 $ (2v^5 - 8v^4 + 2v^3 + 9v^2 - 12*v + 18) / 27
$\beta_{5}$	$=$	$ ( -2\nu^{5} - \nu^{4} + 4\nu^{3} - 3\nu^{2} + 12\nu + 27 ) / 9 $ (-2v^5 - v^4 + 4v^3 - 3v^2 + 12v + 27) / 9

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

$\nu$	$=$	$ ( \beta_{4} + \beta_{3} - \beta_{2} + \beta _1 + 1 ) / 2 $ (b4 + b3 - b2 + b1 + 1) / 2
$\nu^{2}$	$=$	$ \beta_{3} + \beta_{2} + \beta _1 + 1 $ b3 + b2 + b1 + 1
$\nu^{3}$	$=$	$ ( 3\beta_{5} - 8\beta_{3} + \beta_{2} + \beta _1 - 5 ) / 2 $ (3b5 - 8b3 + b2 + b1 - 5) / 2
$\nu^{4}$	$=$	$ -3\beta_{4} - 2\beta_{3} + \beta_{2} + \beta _1 + 4 $ -3b4 - 2b3 + b2 + b1 + 4
$\nu^{5}$	$=$	$ ( -3\beta_{5} + 9\beta_{4} - 11\beta_{3} - 8\beta_{2} + 4\beta _1 + 16 ) / 2 $ (-3b5 + 9b4 - 11b3 - 8b2 + 4*b1 + 16) / 2

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2736\mathbb{Z}\right)^\times$.

$n$	$1009$	$1217$	$1711$	$2053$
$\chi(n)$	$-\beta_{3}$	$1$	$-1$	$1$

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} $

559.1

−1.62241	+	0.606458i
1.71903	+	0.211943i
0.403374	−	1.68443i
−1.62241	−	0.606458i
1.71903	−	0.211943i
0.403374	+	1.68443i

−1.33641

2.31473i

−

3.93569i

559.2

0.675970

−

1.17081i

1.45735i

559.3

1.66044

−

2.87597i

−

2.71781i

1855.1

−1.33641

−

2.31473i

3.93569i

1855.2

0.675970

1.17081i

−

1.45735i

1855.3

1.66044

2.87597i

2.71781i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
76.f	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	2736.2.bm.o		6
3.b	odd	2	1		912.2.bb.f	yes	6
4.b	odd	2	1		2736.2.bm.n		6
12.b	even	2	1		912.2.bb.e	✓	6
19.d	odd	6	1		2736.2.bm.n		6
57.f	even	6	1		912.2.bb.e	✓	6
76.f	even	6	1	inner	2736.2.bm.o		6
228.n	odd	6	1		912.2.bb.f	yes	6

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
912.2.bb.e	✓	6	12.b	even	2	1
912.2.bb.e	✓	6	57.f	even	6	1
912.2.bb.f	yes	6	3.b	odd	2	1
912.2.bb.f	yes	6	228.n	odd	6	1
2736.2.bm.n		6	4.b	odd	2	1
2736.2.bm.n		6	19.d	odd	6	1
2736.2.bm.o		6	1.a	even	1	1	trivial
2736.2.bm.o		6	76.f	even	6	1	inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(2736, [\chi])$:

$ T_{5}^{6} - 2T_{5}^{5} + 12T_{5}^{4} - 8T_{5}^{3} + 88T_{5}^{2} - 96T_{5} + 144 $

$ T_{7}^{6} + 25T_{7}^{4} + 163T_{7}^{2} + 243 $

$ T_{11}^{6} + 16T_{11}^{4} + 64T_{11}^{2} + 48 $

$ T_{23}^{6} - 12T_{23}^{5} + 28T_{23}^{4} + 240T_{23}^{3} - 416T_{23}^{2} - 4080T_{23} + 13872 $

$ T_{31}^{3} + T_{31}^{2} - 9T_{31} + 3 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{6} $ T^6
$3$	$ T^{6} $ T^6
$5$	$ T^{6} - 2 T^{5} + \cdots + 144 $ T^6 - 2T^5 + 12T^4 - 8T^3 + 88T^2 - 96*T + 144
$7$	$ T^{6} + 25 T^{4} + \cdots + 243 $ T^6 + 25T^4 + 163T^2 + 243
$11$	$ T^{6} + 16 T^{4} + \cdots + 48 $ T^6 + 16T^4 + 64T^2 + 48
$13$	$ T^{6} + 3 T^{5} + \cdots + 2883 $ T^6 + 3T^5 - 20T^4 - 69T^3 + 436T^2 + 2139*T + 2883
$17$	$ T^{6} - 2 T^{5} + \cdots + 576 $ T^6 - 2T^5 + 24T^4 - 8T^3 + 448T^2 - 480*T + 576
$19$	$ T^{6} - 4 T^{5} + \cdots + 6859 $ T^6 - 4T^5 + 17T^4 - 136T^3 + 323T^2 - 1444*T + 6859
$23$	$ T^{6} - 12 T^{5} + \cdots + 13872 $ T^6 - 12T^5 + 28T^4 + 240T^3 - 416T^2 - 4080*T + 13872
$29$	$ T^{6} + 6 T^{5} + \cdots + 192 $ T^6 + 6T^5 - 8T^4 - 120T^3 + 448T^2 - 480*T + 192
$31$	$ (T^{3} + T^{2} - 9 T + 3)^{2} $ (T^3 + T^2 - 9*T + 3)^2
$37$	$ T^{6} + 217 T^{4} + \cdots + 328683 $ T^6 + 217T^4 + 14971T^2 + 328683
$41$	$ T^{6} - 12 T^{5} + \cdots + 248832 $ T^6 - 12T^5 - 48T^4 + 1152T^3 + 5760T^2 - 82944*T + 248832
$43$	$ T^{6} - 33 T^{5} + \cdots + 2187 $ T^6 - 33T^5 + 456T^4 - 3069T^3 + 9540T^2 - 7533*T + 2187
$47$	$ T^{6} + 18 T^{5} + \cdots + 714432 $ T^6 + 18T^5 + 16T^4 - 1656T^3 - 320T^2 + 134688*T + 714432
$53$	$ T^{6} + 36 T^{5} + \cdots + 80688 $ T^6 + 36T^5 + 568T^4 + 4896T^3 + 24400T^2 + 66912*T + 80688
$59$	$ T^{6} - 2 T^{5} + \cdots + 144 $ T^6 - 2T^5 + 12T^4 - 8T^3 + 88T^2 - 96*T + 144
$61$	$ T^{6} - 3 T^{5} + \cdots + 638401 $ T^6 - 3T^5 + 150T^4 - 1175T^3 + 22278T^2 - 112659*T + 638401
$67$	$ T^{6} + 19 T^{5} + \cdots + 10201 $ T^6 + 19T^5 + 278T^4 + 1375T^3 + 4970T^2 + 8383*T + 10201
$71$	$ T^{6} + 16 T^{5} + \cdots + 2304 $ T^6 + 16T^5 + 216T^4 + 736T^3 + 2368T^2 - 1920*T + 2304
$73$	$ T^{6} - 11 T^{5} + \cdots + 33489 $ T^6 - 11T^5 + 166T^4 + 129T^3 + 4038T^2 - 8235*T + 33489
$79$	$ T^{6} + 9 T^{5} + \cdots + 151321 $ T^6 + 9T^5 + 138T^4 + 265T^3 + 6750T^2 + 22173*T + 151321
$83$	$ T^{6} + 132 T^{4} + \cdots + 15552 $ T^6 + 132T^4 + 4464T^2 + 15552
$89$	$ T^{6} + 6 T^{5} + \cdots + 215472 $ T^6 + 6T^5 - 128T^4 - 840T^3 + 21208T^2 - 112560*T + 215472
$97$	$ T^{6} + 24 T^{5} + \cdots + 62208 $ T^6 + 24T^5 + 208T^4 + 384T^3 - 3200T^2 - 6912*T + 62208
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Elliptic curves over $\Q$
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Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 2736 = 2^{4} \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2736.bm (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\( q + (\beta_{4} + \beta_{3} - \beta_{2} + 1) q^{5} + ( - \beta_{5} + 2 \beta_{3} + \beta_1 + 1) q^{7}+O(q^{10}) \) q + (b4 + b3 - b2 + 1) * q^5 + (-b5 + 2b3 + b1 + 1) q^7	\( q + (\beta_{4} + \beta_{3} - \beta_{2} + 1) q^{5} + ( - \beta_{5} + 2 \beta_{3} + \beta_1 + 1) q^{7} + (\beta_{5} - \beta_1) q^{11} + ( - \beta_{5} + \beta_{4} - 2 \beta_{2}) q^{13} + ( - 2 \beta_{5} + \beta_{4} + \beta_{3} + \cdots + 1) q^{17}+ \cdots + (\beta_{4} + 3 \beta_{3} + \beta_{2} + \cdots - 3) q^{97}+O(q^{100}) \) q + (b4 + b3 - b2 + 1) * q^5 + (-b5 + 2b3 + b1 + 1) q^7 + (b5 - b1) * q^11 + (-b5 + b4 - 2b2) q^13 + (-2b5 + b4 + b3 - b2 + b1 + 1) q^17 + (-2b5 + b4 + b3 + b1 + 1) q^19 + (2b5 - b4 + b3 + 2b2 + 2) * q^23 + (-b5 + b4 + 2b3 + 2b1) * q^25 + (b5 - b4 - b3 + 2b2 - 2) q^29 - b2 * q^31 + (2b4 + 2b2 + b1) * q^35 + (-2b5 + 4b4 - 2b3 - 2b2 + 2b1 - 1) q^37 + (-2b4 - 2b3 - 2b2 + 2) q^41 + (-b4 - 4b3 - b2 + 4) q^43 + (4b5 - 2b3 - 4) * q^47 + (b5 - b2 + b1 - 1) * q^49 + (-b5 - 4b3 - 8) q^53 + (-b4 + b3 - b2 - b1 - 1) * q^55 + (b4 + b3 - b2 + 1) * q^59 + (-b5 - 3b4 - 2b3 + 2b1) q^61 + (-4b5 + 2b4 + 10b3 - b2 + 4b1 + 5) * q^65 + (-b5 + 2b4 + 7b3 + 2b1) q^67 + (2b5 + b4 - 5b3 - b2 - b1 - 5) * q^71 + (4b5 - 2b4 + 3b3 + 2b2 - 2b1 + 3) q^73 + (b2 + 5) * q^77 + (-3b4 - 4b3 + 3b2 - 4) q^79 + (3b5 - 2b4 - 2b3 + b2 - 3b1 - 1) * q^83 + (-2b5 + 4b4 + 4b3 + 4b1) * q^85 + (b5 + 2b4 - 4b2) * q^89 + (5b4 + 2b3) * q^91 + (4b4 - 2b3 + 3b1 - 6) q^95 + (b4 + 3b3 + b2 - b1 - 3) q^97
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 2 q^{5}+O(q^{10}) \) 6 * q + 2 * q^5	\( 6 q + 2 q^{5} - 3 q^{13} + 2 q^{17} + 4 q^{19} + 12 q^{23} - 5 q^{25} - 6 q^{29} - 2 q^{31} + 6 q^{35} + 12 q^{41} + 33 q^{43} - 18 q^{47} - 8 q^{49} - 36 q^{53} - 12 q^{55} + 2 q^{59} + 3 q^{61} - 19 q^{67} - 16 q^{71} + 11 q^{73} + 32 q^{77} - 9 q^{79} - 8 q^{85} - 6 q^{89} - q^{91} - 26 q^{95} - 24 q^{97}+O(q^{100}) \) 6 * q + 2 * q^5 - 3 * q^13 + 2 * q^17 + 4 * q^19 + 12 * q^23 - 5 * q^25 - 6 * q^29 - 2 * q^31 + 6 * q^35 + 12 * q^41 + 33 * q^43 - 18 * q^47 - 8 * q^49 - 36 * q^53 - 12 * q^55 + 2 * q^59 + 3 * q^61 - 19 * q^67 - 16 * q^71 + 11 * q^73 + 32 * q^77 - 9 * q^79 - 8 * q^85 - 6 * q^89 - q^91 - 26 * q^95 - 24 * q^97

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