LMFDB - Newform orbit 1085.1.bh.a

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1085,1,Mod(464,1085)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1085.464"); S:= CuspForms(chi, 1); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1085, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 2, 3])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]

Level:	$ N $	$=$	$ 1085 = 5 \cdot 7 \cdot 31 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	1085.bh (of order $6$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [6,0,0,3,0,0,0,0,3,-6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(10)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$0.541485538707$
Analytic rank:	$0$
Dimension:	$6$
Relative dimension:	$3$ over $\Q(\zeta_{6})$
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:	$\Z[a_1, \ldots, a_{4}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{18}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{18} - \cdots)$

$q$-expansion

comment:q-expansion

sage:f.q_expansion() # note that sage often uses an isomorphic number field

gp:mfcoefs(f, 20)

The $q$-expansion and trace form are shown below.

$f(q)$	$=$	$ q + (\zeta_{18}^{2} + \zeta_{18}) q^{2} + (\zeta_{18}^{4} + \cdots + \zeta_{18}^{2}) q^{4} + \zeta_{18}^{7} q^{5} - \zeta_{18} q^{7} + (\zeta_{18}^{6} + \cdots + \zeta_{18}^{3}) q^{8} - \zeta_{18}^{6} q^{9} + \cdots + (\zeta_{18}^{4} + \zeta_{18}^{3}) q^{98} +O(q^{100}) $ q + (z^2 + z) * q^2 + (z^4 + z^3 + z^2) * q^4 + z^7 * q^5 - z * q^7 + (z^6 + z^5 + z^4 + z^3) * q^8 - z^6 * q^9 + (z^8 - 1) * q^10 + (-z^3 - z^2) * q^14 + (z^8 + z^7 + z^6 + z^5 + z^4) * q^16 + (-z^8 - z^7) * q^18 + (-z^8 - z^4) * q^19 + (-z^2 - z - 1) * q^20 - z^5 * q^25 + (-z^5 - z^4 - z^3) * q^28 + z^3 * q^31 + (z^8 + z^7 + z^6 + z^5 - z - 1) * q^32 - z^8 * q^35 + (-z^8 + z + 1) * q^36 + (-z^6 - z^5 + z + 1) * q^38 + (-z^4 - z^3 - z^2 - z) * q^40 + (-z^5 + z^4) * q^41 + z^4 * q^45 + (-z^3 - 1) * q^47 + z^2 * q^49 + (-z^7 - z^6) * q^50 + (-z^7 - z^6 - z^5 - z^4) * q^56 + (z^8 - z^7) * q^59 + (z^5 + z^4) * q^62 + z^7 * q^63 + (z^8 + z^7 + z^6 - z^3 - z^2 - z - 1) * q^64 + (-z^6 + 1) * q^67 + (z + 1) * q^70 + (z^7 - z^2) * q^71 + (z^3 + z^2 + z + 1) * q^72 + (-z^8 - z^7 - z^6 + z^3 + z^2 + z) * q^76 + (-z^6 - z^5 - z^4 - z^3 - z^2) * q^80 - z^3 * q^81 + (-z^7 + z^5) * q^82 + (z^6 + z^5) * q^90 + (-z^5 - z^4 - z^2 - z) * q^94 + (z^6 + z^2) * q^95 + (z^7 + z^2) * q^97 + (z^4 + z^3) * q^98
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 3 q^{4} + 3 q^{9} - 6 q^{10} - 3 q^{14} - 3 q^{16} - 6 q^{20} - 3 q^{28} + 3 q^{31} - 9 q^{32} + 6 q^{36} + 9 q^{38} - 3 q^{40} - 9 q^{47} + 3 q^{50} + 3 q^{56} - 12 q^{64} + 9 q^{67} + 6 q^{70} + 9 q^{72}+ \cdots + 3 q^{98}+O(q^{100}) $ 6 * q + 3 * q^4 + 3 * q^9 - 6 * q^10 - 3 * q^14 - 3 * q^16 - 6 * q^20 - 3 * q^28 + 3 * q^31 - 9 * q^32 + 6 * q^36 + 9 * q^38 - 3 * q^40 - 9 * q^47 + 3 * q^50 + 3 * q^56 - 12 * q^64 + 9 * q^67 + 6 * q^70 + 9 * q^72 + 6 * q^76 - 3 * q^81 - 3 * q^90 - 3 * q^95 + 3 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1085\mathbb{Z}\right)^\times$.

$n$	$311$	$561$	$652$
$\chi(n)$	$\zeta_{18}^{6}$	$-1$	$-1$

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} $

464.1

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

−1.11334

−

0.642788i

0.326352

0.565258i

0.939693

0.342020i

0.173648

0.984808i

0.446476i

0.500000

−

0.866025i

−0.826352

−

0.984808i

464.2

−0.592396

−

0.342020i

−0.266044

−

0.460802i

−0.173648

−

0.984808i

0.766044

−

0.642788i

1.04801i

0.500000

−

0.866025i

−0.233956

0.642788i

464.3

1.70574

0.984808i

1.43969

2.49362i

−0.766044

0.642788i

−0.939693

−

0.342020i

3.70167i

0.500000

−

0.866025i

−1.93969

0.342020i

774.1

−1.11334

0.642788i

0.326352

−

0.565258i

0.939693

−

0.342020i

0.173648

−

0.984808i

−

0.446476i

0.500000

0.866025i

−0.826352

0.984808i

774.2

−0.592396

0.342020i

−0.266044

0.460802i

−0.173648

0.984808i

0.766044

0.642788i

−

1.04801i

0.500000

0.866025i

−0.233956

−

0.642788i

774.3

1.70574

−

0.984808i

1.43969

−

2.49362i

−0.766044

−

0.642788i

−0.939693

0.342020i

−

3.70167i

0.500000

0.866025i

−1.93969

−

0.342020i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
31.b	odd	2	1	CM by $\Q(\sqrt{-31}) $
35.j	even	6	1	inner
1085.bh	odd	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1085.1.bh.a	✓	6
5.b	even	2	1		1085.1.bh.b	yes	6
7.c	even	3	1		1085.1.bh.b	yes	6
31.b	odd	2	1	CM	1085.1.bh.a	✓	6
35.j	even	6	1	inner	1085.1.bh.a	✓	6
155.c	odd	2	1		1085.1.bh.b	yes	6
217.n	odd	6	1		1085.1.bh.b	yes	6
1085.bh	odd	6	1	inner	1085.1.bh.a	✓	6

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1085.1.bh.a	✓	6	1.a	even	1	1	trivial
1085.1.bh.a	✓	6	31.b	odd	2	1	CM
1085.1.bh.a	✓	6	35.j	even	6	1	inner
1085.1.bh.a	✓	6	1085.bh	odd	6	1	inner
1085.1.bh.b	yes	6	5.b	even	2	1
1085.1.bh.b	yes	6	7.c	even	3	1
1085.1.bh.b	yes	6	155.c	odd	2	1
1085.1.bh.b	yes	6	217.n	odd	6	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{2}^{6} - 3T_{2}^{4} + 9T_{2}^{2} + 9T_{2} + 3 $ T2^6 - 3*T2^4 + 9*T2^2 + 9*T2 + 3 acting on $S_{1}^{\mathrm{new}}(1085, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{6} - 3 T^{4} + \cdots + 3 $ T^6 - 3T^4 + 9T^2 + 9*T + 3
$3$	$ T^{6} $ T^6
$5$	$ T^{6} - T^{3} + 1 $ T^6 - T^3 + 1
$7$	$ T^{6} + T^{3} + 1 $ T^6 + T^3 + 1
$11$	$ T^{6} $ T^6
$13$	$ T^{6} $ T^6
$17$	$ T^{6} $ T^6
$19$	$ T^{6} + 3 T^{4} + \cdots + 1 $ T^6 + 3T^4 - 2T^3 + 9T^2 - 3T + 1
$23$	$ T^{6} $ T^6
$29$	$ T^{6} $ T^6
$31$	$ (T^{2} - T + 1)^{3} $ (T^2 - T + 1)^3
$37$	$ T^{6} $ T^6
$41$	$ (T^{3} - 3 T + 1)^{2} $ (T^3 - 3*T + 1)^2
$43$	$ T^{6} $ T^6
$47$	$ (T^{2} + 3 T + 3)^{3} $ (T^2 + 3*T + 3)^3
$53$	$ T^{6} $ T^6
$59$	$ T^{6} + 3 T^{4} + \cdots + 1 $ T^6 + 3T^4 + 2T^3 + 9T^2 + 3T + 1
$61$	$ T^{6} $ T^6
$67$	$ (T^{2} - 3 T + 3)^{3} $ (T^2 - 3*T + 3)^3
$71$	$ (T^{3} - 3 T - 1)^{2} $ (T^3 - 3*T - 1)^2
$73$	$ T^{6} $ T^6
$79$	$ T^{6} $ T^6
$83$	$ T^{6} $ T^6
$89$	$ T^{6} $ T^6
$97$	$ T^{6} + 6 T^{4} + \cdots + 3 $ T^6 + 6T^4 + 9T^2 + 3
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Level:	\( N \)	\(=\)	\( 1085 = 5 \cdot 7 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1085.bh (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\( q + (\zeta_{18}^{2} + \zeta_{18}) q^{2} + (\zeta_{18}^{4} + \cdots + \zeta_{18}^{2}) q^{4} + \zeta_{18}^{7} q^{5} - \zeta_{18} q^{7} + (\zeta_{18}^{6} + \cdots + \zeta_{18}^{3}) q^{8} - \zeta_{18}^{6} q^{9} + \cdots + (\zeta_{18}^{4} + \zeta_{18}^{3}) q^{98} +O(q^{100}) \) q + (z^2 + z) * q^2 + (z^4 + z^3 + z^2) * q^4 + z^7 * q^5 - z * q^7 + (z^6 + z^5 + z^4 + z^3) * q^8 - z^6 * q^9 + (z^8 - 1) * q^10 + (-z^3 - z^2) * q^14 + (z^8 + z^7 + z^6 + z^5 + z^4) * q^16 + (-z^8 - z^7) * q^18 + (-z^8 - z^4) * q^19 + (-z^2 - z - 1) * q^20 - z^5 * q^25 + (-z^5 - z^4 - z^3) * q^28 + z^3 * q^31 + (z^8 + z^7 + z^6 + z^5 - z - 1) * q^32 - z^8 * q^35 + (-z^8 + z + 1) * q^36 + (-z^6 - z^5 + z + 1) * q^38 + (-z^4 - z^3 - z^2 - z) * q^40 + (-z^5 + z^4) * q^41 + z^4 * q^45 + (-z^3 - 1) * q^47 + z^2 * q^49 + (-z^7 - z^6) * q^50 + (-z^7 - z^6 - z^5 - z^4) * q^56 + (z^8 - z^7) * q^59 + (z^5 + z^4) * q^62 + z^7 * q^63 + (z^8 + z^7 + z^6 - z^3 - z^2 - z - 1) * q^64 + (-z^6 + 1) * q^67 + (z + 1) * q^70 + (z^7 - z^2) * q^71 + (z^3 + z^2 + z + 1) * q^72 + (-z^8 - z^7 - z^6 + z^3 + z^2 + z) * q^76 + (-z^6 - z^5 - z^4 - z^3 - z^2) * q^80 - z^3 * q^81 + (-z^7 + z^5) * q^82 + (z^6 + z^5) * q^90 + (-z^5 - z^4 - z^2 - z) * q^94 + (z^6 + z^2) * q^95 + (z^7 + z^2) * q^97 + (z^4 + z^3) * q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 3 q^{4} + 3 q^{9} - 6 q^{10} - 3 q^{14} - 3 q^{16} - 6 q^{20} - 3 q^{28} + 3 q^{31} - 9 q^{32} + 6 q^{36} + 9 q^{38} - 3 q^{40} - 9 q^{47} + 3 q^{50} + 3 q^{56} - 12 q^{64} + 9 q^{67} + 6 q^{70} + 9 q^{72}+ \cdots + 3 q^{98}+O(q^{100}) \) 6 * q + 3 * q^4 + 3 * q^9 - 6 * q^10 - 3 * q^14 - 3 * q^16 - 6 * q^20 - 3 * q^28 + 3 * q^31 - 9 * q^32 + 6 * q^36 + 9 * q^38 - 3 * q^40 - 9 * q^47 + 3 * q^50 + 3 * q^56 - 12 * q^64 + 9 * q^67 + 6 * q^70 + 9 * q^72 + 6 * q^76 - 3 * q^81 - 3 * q^90 - 3 * q^95 + 3 * q^98

Introduction

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Newspace parameters

Newform invariants

$q$-expansion

Character values

Embeddings

Inner twists

Twists

Hecke kernels

Hecke characteristic polynomials

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	1085.1.bh.a

Level	$1085$
Weight	$1$
Character orbit	1085.bh
Analytic conductor	$0.541$
Analytic rank	$0$
Dimension	$6$
Projective image	$D_{18}$
CM discriminant	-31
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(0.541485538707\)
Analytic rank:	\(0\)
Dimension:	\(6\)
Relative dimension:	\(3\) over \(\Q(\zeta_{6})\)
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:	\(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Projective image:	\(D_{18}\)
Projective field:	Galois closure of \(\mathbb{Q}[x]/(x^{18} - \cdots)\)

\(n\)	\(311\)	\(561\)	\(652\)
\(\chi(n)\)	\(\zeta_{18}^{6}\)	\(-1\)	\(-1\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 464.3
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{6} - 3 T^{4} + \cdots + 3 \) T^6 - 3T^4 + 9T^2 + 9*T + 3
$3$	\( T^{6} \) T^6
$5$	\( T^{6} - T^{3} + 1 \) T^6 - T^3 + 1
$7$	\( T^{6} + T^{3} + 1 \) T^6 + T^3 + 1
$11$	\( T^{6} \) T^6
$13$	\( T^{6} \) T^6
$17$	\( T^{6} \) T^6
$19$	\( T^{6} + 3 T^{4} + \cdots + 1 \) T^6 + 3T^4 - 2T^3 + 9T^2 - 3T + 1
$23$	\( T^{6} \) T^6
$29$	\( T^{6} \) T^6
$31$	\( (T^{2} - T + 1)^{3} \) (T^2 - T + 1)^3
$37$	\( T^{6} \) T^6
$41$	\( (T^{3} - 3 T + 1)^{2} \) (T^3 - 3*T + 1)^2
$43$	\( T^{6} \) T^6
$47$	\( (T^{2} + 3 T + 3)^{3} \) (T^2 + 3*T + 3)^3
$53$	\( T^{6} \) T^6
$59$	\( T^{6} + 3 T^{4} + \cdots + 1 \) T^6 + 3T^4 + 2T^3 + 9T^2 + 3T + 1
$61$	\( T^{6} \) T^6
$67$	\( (T^{2} - 3 T + 3)^{3} \) (T^2 - 3*T + 3)^3
$71$	\( (T^{3} - 3 T - 1)^{2} \) (T^3 - 3*T - 1)^2
$73$	\( T^{6} \) T^6
$79$	\( T^{6} \) T^6
$83$	\( T^{6} \) T^6
$89$	\( T^{6} \) T^6
$97$	\( T^{6} + 6 T^{4} + \cdots + 3 \) T^6 + 6T^4 + 9T^2 + 3
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