LMFDB - Newform orbit 4212.2.i.g

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4212,2,Mod(1405,4212)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4212, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 2, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("4212.1405"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 4212 = 2^{2} \cdot 3^{4} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4212.i (of order $3$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,0,0,-2,0,0,0,0,0,-1,0,0,0,12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

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

Self dual:	no
Analytic conductor:	$33.6329893314$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-3}) $
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:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 156)
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 primitive root of unity $\zeta_{6}$. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + (2 \zeta_{6} - 2) q^{7} - \zeta_{6} q^{13} + 6 q^{17} + 2 q^{19} + ( - 5 \zeta_{6} + 5) q^{25} + (6 \zeta_{6} - 6) q^{29} - 2 \zeta_{6} q^{31} + 2 q^{37} - 12 \zeta_{6} q^{41} + ( - 4 \zeta_{6} + 4) q^{43} + \cdots + ( - 10 \zeta_{6} + 10) q^{97} +O(q^{100}) $ q + (2z - 2) q^7 - z * q^13 + 6 * q^17 + 2 * q^19 + (-5z + 5) q^25 + (6z - 6) q^29 - 2z q^31 + 2 * q^37 - 12z q^41 + (-4z + 4) q^43 + 3z q^49 - 6 * q^53 + 12z q^59 + (2z - 2) q^61 + 10z q^67 - 12 * q^71 + 14 * q^73 + (8z - 8) q^79 + (-12z + 12) q^83 + 2 * q^91 + (-10z + 10) q^97
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{7} - q^{13} + 12 q^{17} + 4 q^{19} + 5 q^{25} - 6 q^{29} - 2 q^{31} + 4 q^{37} - 12 q^{41} + 4 q^{43} + 3 q^{49} - 12 q^{53} + 12 q^{59} - 2 q^{61} + 10 q^{67} - 24 q^{71} + 28 q^{73} - 8 q^{79}+ \cdots + 10 q^{97}+O(q^{100}) $ 2 * q - 2 * q^7 - q^13 + 12 * q^17 + 4 * q^19 + 5 * q^25 - 6 * q^29 - 2 * q^31 + 4 * q^37 - 12 * q^41 + 4 * q^43 + 3 * q^49 - 12 * q^53 + 12 * q^59 - 2 * q^61 + 10 * q^67 - 24 * q^71 + 28 * q^73 - 8 * q^79 + 12 * q^83 + 4 * q^91 + 10 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4212\mathbb{Z}\right)^\times$.

$n$	$2107$	$3485$	$3889$
$\chi(n)$	$1$	$-\zeta_{6}$	$1$

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} $

1405.1

0.500000	−	0.866025i
0.500000	+	0.866025i

−1.00000

−

1.73205i

2809.1

−1.00000

1.73205i

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	4212.2.i.g		2
3.b	odd	2	1		4212.2.i.f		2
9.c	even	3	1		468.2.a.c		1
9.c	even	3	1	inner	4212.2.i.g		2
9.d	odd	6	1		156.2.a.b	✓	1
9.d	odd	6	1		4212.2.i.f		2
36.f	odd	6	1		1872.2.a.i		1
36.h	even	6	1		624.2.a.b		1
45.h	odd	6	1		3900.2.a.a		1
45.l	even	12	2		3900.2.h.e		2
63.o	even	6	1		7644.2.a.a		1
72.j	odd	6	1		2496.2.a.h		1
72.l	even	6	1		2496.2.a.v		1
72.n	even	6	1		7488.2.a.bf		1
72.p	odd	6	1		7488.2.a.bb		1
117.k	odd	6	1		2028.2.i.b		2
117.m	odd	6	1		2028.2.i.c		2
117.n	odd	6	1		2028.2.a.e		1
117.t	even	6	1		6084.2.a.h		1
117.u	odd	6	1		2028.2.i.b		2
117.v	odd	6	1		2028.2.i.c		2
117.x	even	12	2		2028.2.q.d		4
117.y	odd	12	2		6084.2.b.a		2
117.z	even	12	2		2028.2.b.d		2
117.bc	even	12	2		2028.2.q.d		4
468.x	even	6	1		8112.2.a.i		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
156.2.a.b	✓	1	9.d	odd	6	1
468.2.a.c		1	9.c	even	3	1
624.2.a.b		1	36.h	even	6	1
1872.2.a.i		1	36.f	odd	6	1
2028.2.a.e		1	117.n	odd	6	1
2028.2.b.d		2	117.z	even	12	2
2028.2.i.b		2	117.k	odd	6	1
2028.2.i.b		2	117.u	odd	6	1
2028.2.i.c		2	117.m	odd	6	1
2028.2.i.c		2	117.v	odd	6	1
2028.2.q.d		4	117.x	even	12	2
2028.2.q.d		4	117.bc	even	12	2
2496.2.a.h		1	72.j	odd	6	1
2496.2.a.v		1	72.l	even	6	1
3900.2.a.a		1	45.h	odd	6	1
3900.2.h.e		2	45.l	even	12	2
4212.2.i.f		2	3.b	odd	2	1
4212.2.i.f		2	9.d	odd	6	1
4212.2.i.g		2	1.a	even	1	1	trivial
4212.2.i.g		2	9.c	even	3	1	inner
6084.2.a.h		1	117.t	even	6	1
6084.2.b.a		2	117.y	odd	12	2
7488.2.a.bb		1	72.p	odd	6	1
7488.2.a.bf		1	72.n	even	6	1
7644.2.a.a		1	63.o	even	6	1
8112.2.a.i		1	468.x	even	6	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(4212, [\chi])$:

$ T_{5} $

T5

$ T_{7}^{2} + 2T_{7} + 4 $

T7^2 + 2*T7 + 4

$ T_{17} - 6 $

T17 - 6

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{2} $ T^2
$3$	$ T^{2} $ T^2
$5$	$ T^{2} $ T^2
$7$	$ T^{2} + 2T + 4 $ T^2 + 2*T + 4
$11$	$ T^{2} $ T^2
$13$	$ T^{2} + T + 1 $ T^2 + T + 1
$17$	$ (T - 6)^{2} $ (T - 6)^2
$19$	$ (T - 2)^{2} $ (T - 2)^2
$23$	$ T^{2} $ T^2
$29$	$ T^{2} + 6T + 36 $ T^2 + 6*T + 36
$31$	$ T^{2} + 2T + 4 $ T^2 + 2*T + 4
$37$	$ (T - 2)^{2} $ (T - 2)^2
$41$	$ T^{2} + 12T + 144 $ T^2 + 12*T + 144
$43$	$ T^{2} - 4T + 16 $ T^2 - 4*T + 16
$47$	$ T^{2} $ T^2
$53$	$ (T + 6)^{2} $ (T + 6)^2
$59$	$ T^{2} - 12T + 144 $ T^2 - 12*T + 144
$61$	$ T^{2} + 2T + 4 $ T^2 + 2*T + 4
$67$	$ T^{2} - 10T + 100 $ T^2 - 10*T + 100
$71$	$ (T + 12)^{2} $ (T + 12)^2
$73$	$ (T - 14)^{2} $ (T - 14)^2
$79$	$ T^{2} + 8T + 64 $ T^2 + 8*T + 64
$83$	$ T^{2} - 12T + 144 $ T^2 - 12*T + 144
$89$	$ T^{2} $ T^2
$97$	$ T^{2} - 10T + 100 $ T^2 - 10*T + 100
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Elliptic curves over $\Q$
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Level:	\( N \)	\(=\)	\( 4212 = 2^{2} \cdot 3^{4} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4212.i (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\( q + (2 \zeta_{6} - 2) q^{7} - \zeta_{6} q^{13} + 6 q^{17} + 2 q^{19} + ( - 5 \zeta_{6} + 5) q^{25} + (6 \zeta_{6} - 6) q^{29} - 2 \zeta_{6} q^{31} + 2 q^{37} - 12 \zeta_{6} q^{41} + ( - 4 \zeta_{6} + 4) q^{43} + \cdots + ( - 10 \zeta_{6} + 10) q^{97} +O(q^{100}) \) q + (2z - 2) q^7 - z * q^13 + 6 * q^17 + 2 * q^19 + (-5z + 5) q^25 + (6z - 6) q^29 - 2z q^31 + 2 * q^37 - 12z q^41 + (-4z + 4) q^43 + 3z q^49 - 6 * q^53 + 12z q^59 + (2z - 2) q^61 + 10z q^67 - 12 * q^71 + 14 * q^73 + (8z - 8) q^79 + (-12z + 12) q^83 + 2 * q^91 + (-10z + 10) q^97
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{7} - q^{13} + 12 q^{17} + 4 q^{19} + 5 q^{25} - 6 q^{29} - 2 q^{31} + 4 q^{37} - 12 q^{41} + 4 q^{43} + 3 q^{49} - 12 q^{53} + 12 q^{59} - 2 q^{61} + 10 q^{67} - 24 q^{71} + 28 q^{73} - 8 q^{79}+ \cdots + 10 q^{97}+O(q^{100}) \) 2 * q - 2 * q^7 - q^13 + 12 * q^17 + 4 * q^19 + 5 * q^25 - 6 * q^29 - 2 * q^31 + 4 * q^37 - 12 * q^41 + 4 * q^43 + 3 * q^49 - 12 * q^53 + 12 * q^59 - 2 * q^61 + 10 * q^67 - 24 * q^71 + 28 * q^73 - 8 * q^79 + 12 * q^83 + 4 * q^91 + 10 * q^97

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