LMFDB - Newform orbit 3871.1.m.a

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3871,1,Mod(1500,3871)] 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("3871.1500"); S:= CuspForms(chi, 1); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3871, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([2, 3])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]

Level:	$ N $	$=$	$ 3871 = 7^{2} \cdot 79 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	3871.m (of order $6$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,-2,0,-3,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	no
Analytic conductor:	$1.93188066390$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{4}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{2}$
Projective field:	Galois closure of $\Q(\sqrt{-7}, \sqrt{-79})$
Artin image:	$C_3\times D_4$
Artin field:	Galois closure of $\mathbb{Q}[x]/(x^{12} + \cdots)$

$q$-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 - 2 \zeta_{6} q^{2} + 3 \zeta_{6}^{2} q^{4} + 4 q^{8} - \zeta_{6} q^{9} - 2 \zeta_{6}^{2} q^{11} - 5 \zeta_{6} q^{16} + 2 \zeta_{6}^{2} q^{18} - 4 q^{22} + 2 \zeta_{6} q^{23} - \zeta_{6}^{2} q^{25} + 6 \zeta_{6}^{2} q^{32} + \cdots - 2 q^{99} +O(q^{100}) $ q - 2z q^2 + 3z^2 q^4 + 4 * q^8 - z * q^9 - 2z^2 q^11 - 5z q^16 + 2z^2 q^18 - 4 * q^22 + 2z q^23 - z^2 * q^25 + 6z^2 q^32 + 3 * q^36 + 6z q^44 - 4z^2 q^46 - 2 * q^50 + 7 * q^64 - 2z^2 q^67 - 4z q^72 - z * q^79 + z^2 * q^81 - 8z^2 q^88 - 6 * q^92 - 2 * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} - 3 q^{4} + 8 q^{8} - q^{9} + 2 q^{11} - 5 q^{16} - 2 q^{18} - 8 q^{22} + 2 q^{23} + q^{25} - 6 q^{32} + 6 q^{36} + 6 q^{44} + 4 q^{46} - 4 q^{50} + 14 q^{64} + 2 q^{67} - 4 q^{72} - q^{79}+ \cdots - 4 q^{99}+O(q^{100}) $ 2 * q - 2 * q^2 - 3 * q^4 + 8 * q^8 - q^9 + 2 * q^11 - 5 * q^16 - 2 * q^18 - 8 * q^22 + 2 * q^23 + q^25 - 6 * q^32 + 6 * q^36 + 6 * q^44 + 4 * q^46 - 4 * q^50 + 14 * q^64 + 2 * q^67 - 4 * q^72 - q^79 - q^81 + 8 * q^88 - 12 * q^92 - 4 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3871\mathbb{Z}\right)^\times$.

$n$	$1030$	$2845$
$\chi(n)$	$-1$	$-\zeta_{6}$

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} $

1500.1

0.500000	−	0.866025i
0.500000	+	0.866025i

−1.00000

1.73205i

−1.50000

−

2.59808i

4.00000

−0.500000

0.866025i

3791.1

−1.00000

−

1.73205i

−1.50000

2.59808i

4.00000

−0.500000

−

0.866025i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.b	odd	2	1	CM by $\Q(\sqrt{-7}) $
79.b	odd	2	1	CM by $\Q(\sqrt{-79}) $
553.d	even	2	1	RM by $\Q(\sqrt{553}) $
7.c	even	3	1	inner
7.d	odd	6	1	inner
553.l	even	6	1	inner
553.m	odd	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	3871.1.m.a		2
7.b	odd	2	1	CM	3871.1.m.a		2
7.c	even	3	1		3871.1.c.a	✓	1
7.c	even	3	1	inner	3871.1.m.a		2
7.d	odd	6	1		3871.1.c.a	✓	1
7.d	odd	6	1	inner	3871.1.m.a		2
79.b	odd	2	1	CM	3871.1.m.a		2
553.d	even	2	1	RM	3871.1.m.a		2
553.l	even	6	1		3871.1.c.a	✓	1
553.l	even	6	1	inner	3871.1.m.a		2
553.m	odd	6	1		3871.1.c.a	✓	1
553.m	odd	6	1	inner	3871.1.m.a		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
3871.1.c.a	✓	1	7.c	even	3	1
3871.1.c.a	✓	1	7.d	odd	6	1
3871.1.c.a	✓	1	553.l	even	6	1
3871.1.c.a	✓	1	553.m	odd	6	1
3871.1.m.a		2	1.a	even	1	1	trivial
3871.1.m.a		2	7.b	odd	2	1	CM
3871.1.m.a		2	7.c	even	3	1	inner
3871.1.m.a		2	7.d	odd	6	1	inner
3871.1.m.a		2	79.b	odd	2	1	CM
3871.1.m.a		2	553.d	even	2	1	RM
3871.1.m.a		2	553.l	even	6	1	inner
3871.1.m.a		2	553.m	odd	6	1	inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{1}^{\mathrm{new}}(3871, [\chi])$:

$ T_{2}^{2} + 2T_{2} + 4 $

T2^2 + 2*T2 + 4

$ T_{5} $

T5

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{2} + 2T + 4 $ T^2 + 2*T + 4
$3$	$ T^{2} $ T^2
$5$	$ T^{2} $ T^2
$7$	$ T^{2} $ T^2
$11$	$ T^{2} - 2T + 4 $ T^2 - 2*T + 4
$13$	$ T^{2} $ T^2
$17$	$ T^{2} $ T^2
$19$	$ T^{2} $ T^2
$23$	$ T^{2} - 2T + 4 $ T^2 - 2*T + 4
$29$	$ T^{2} $ T^2
$31$	$ T^{2} $ T^2
$37$	$ T^{2} $ T^2
$41$	$ T^{2} $ T^2
$43$	$ T^{2} $ T^2
$47$	$ T^{2} $ T^2
$53$	$ T^{2} $ T^2
$59$	$ T^{2} $ T^2
$61$	$ T^{2} $ T^2
$67$	$ T^{2} - 2T + 4 $ T^2 - 2*T + 4
$71$	$ T^{2} $ T^2
$73$	$ T^{2} $ T^2
$79$	$ T^{2} + T + 1 $ T^2 + T + 1
$83$	$ T^{2} $ T^2
$89$	$ T^{2} $ T^2
$97$	$ T^{2} $ T^2
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Level:	\( N \)	\(=\)	\( 3871 = 7^{2} \cdot 79 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	3871.m (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\( q - 2 \zeta_{6} q^{2} + 3 \zeta_{6}^{2} q^{4} + 4 q^{8} - \zeta_{6} q^{9} - 2 \zeta_{6}^{2} q^{11} - 5 \zeta_{6} q^{16} + 2 \zeta_{6}^{2} q^{18} - 4 q^{22} + 2 \zeta_{6} q^{23} - \zeta_{6}^{2} q^{25} + 6 \zeta_{6}^{2} q^{32} + \cdots - 2 q^{99} +O(q^{100}) \) q - 2z q^2 + 3z^2 q^4 + 4 * q^8 - z * q^9 - 2z^2 q^11 - 5z q^16 + 2z^2 q^18 - 4 * q^22 + 2z q^23 - z^2 * q^25 + 6z^2 q^32 + 3 * q^36 + 6z q^44 - 4z^2 q^46 - 2 * q^50 + 7 * q^64 - 2z^2 q^67 - 4z q^72 - z * q^79 + z^2 * q^81 - 8z^2 q^88 - 6 * q^92 - 2 * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} - 3 q^{4} + 8 q^{8} - q^{9} + 2 q^{11} - 5 q^{16} - 2 q^{18} - 8 q^{22} + 2 q^{23} + q^{25} - 6 q^{32} + 6 q^{36} + 6 q^{44} + 4 q^{46} - 4 q^{50} + 14 q^{64} + 2 q^{67} - 4 q^{72} - q^{79}+ \cdots - 4 q^{99}+O(q^{100}) \) 2 * q - 2 * q^2 - 3 * q^4 + 8 * q^8 - q^9 + 2 * q^11 - 5 * q^16 - 2 * q^18 - 8 * q^22 + 2 * q^23 + q^25 - 6 * q^32 + 6 * q^36 + 6 * q^44 + 4 * q^46 - 4 * q^50 + 14 * q^64 + 2 * q^67 - 4 * q^72 - q^79 - q^81 + 8 * q^88 - 12 * q^92 - 4 * q^99

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