Newform orbit 2106.2.e.b

• Label: 2106.2.e.b
• Level: $2106$
• Weight: $2$
• Character orbit: 2106.e
• Analytic conductor: $16.816$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 2106 = 2 \cdot 3^{4} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2106.e (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(16.8164946657\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{-3}) \)
Defining polynomial:	\( x^{2} - x + 1 \)
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 26)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

$q$-expansion

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.

\(f(q)\)	\(=\)	\( q + (\zeta_{6} - 1) q^{2} - \zeta_{6} q^{4} - 3 \zeta_{6} q^{5} + ( - \zeta_{6} + 1) q^{7} + q^{8} +O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{2} - q^{4} - 3 q^{5} + q^{7} + 2 q^{8}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2106\mathbb{Z}\right)^\times\).

\(n\)	\(1379\)	\(1783\)
\(\chi(n)\)	\(-\zeta_{6}\)	\(1\)

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.

Label  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

703.1

0.500000	+	0.866025i
0.500000	−	0.866025i

−0.500000

+

0.866025i

0

−0.500000

−

0.866025i

−1.50000

−

2.59808i

0

0.500000

−

0.866025i

1.00000

0

3.00000

1405.1

−0.500000

−

0.866025i

0

−0.500000

+

0.866025i

−1.50000

+

2.59808i

0

0.500000

+

0.866025i

1.00000

0

3.00000

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	2106.2.e.b		2
3.b	odd	2	1		2106.2.e.ba		2
9.c	even	3	1		234.2.a.e		1
9.c	even	3	1	inner	2106.2.e.b		2
9.d	odd	6	1		26.2.a.a	✓	1
9.d	odd	6	1		2106.2.e.ba		2
36.f	odd	6	1		1872.2.a.q		1
36.h	even	6	1		208.2.a.a		1
45.h	odd	6	1		650.2.a.j		1
45.j	even	6	1		5850.2.a.p		1
45.k	odd	12	2		5850.2.e.a		2
45.l	even	12	2		650.2.b.d		2
63.i	even	6	1		1274.2.f.r		2
63.j	odd	6	1		1274.2.f.p		2
63.n	odd	6	1		1274.2.f.p		2
63.o	even	6	1		1274.2.a.d		1
63.s	even	6	1		1274.2.f.r		2
72.j	odd	6	1		832.2.a.d		1
72.l	even	6	1		832.2.a.i		1
72.n	even	6	1		7488.2.a.g		1
72.p	odd	6	1		7488.2.a.h		1
99.g	even	6	1		3146.2.a.n		1
117.k	odd	6	1		338.2.c.d		2
117.m	odd	6	1		338.2.c.a		2
117.n	odd	6	1		338.2.a.f		1
117.t	even	6	1		3042.2.a.a		1
117.u	odd	6	1		338.2.c.d		2
117.v	odd	6	1		338.2.c.a		2
117.x	even	12	2		338.2.e.a		4
117.y	odd	12	2		3042.2.b.a		2
117.z	even	12	2		338.2.b.c		2
117.bc	even	12	2		338.2.e.a		4
144.u	even	12	2		3328.2.b.j		2
144.w	odd	12	2		3328.2.b.m		2
153.i	odd	6	1		7514.2.a.c		1
171.l	even	6	1		9386.2.a.j		1
180.n	even	6	1		5200.2.a.x		1
468.x	even	6	1		2704.2.a.f		1
468.ch	odd	12	2		2704.2.f.d		2
585.bo	odd	6	1		8450.2.a.c		1

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
26.2.a.a	✓	1	9.d	odd	6	1
208.2.a.a		1	36.h	even	6	1
234.2.a.e		1	9.c	even	3	1
338.2.a.f		1	117.n	odd	6	1
338.2.b.c		2	117.z	even	12	2
338.2.c.a		2	117.m	odd	6	1
338.2.c.a		2	117.v	odd	6	1
338.2.c.d		2	117.k	odd	6	1
338.2.c.d		2	117.u	odd	6	1
338.2.e.a		4	117.x	even	12	2
338.2.e.a		4	117.bc	even	12	2
650.2.a.j		1	45.h	odd	6	1
650.2.b.d		2	45.l	even	12	2
832.2.a.d		1	72.j	odd	6	1
832.2.a.i		1	72.l	even	6	1
1274.2.a.d		1	63.o	even	6	1
1274.2.f.p		2	63.j	odd	6	1
1274.2.f.p		2	63.n	odd	6	1
1274.2.f.r		2	63.i	even	6	1
1274.2.f.r		2	63.s	even	6	1
1872.2.a.q		1	36.f	odd	6	1
2106.2.e.b		2	1.a	even	1	1	trivial
2106.2.e.b		2	9.c	even	3	1	inner
2106.2.e.ba		2	3.b	odd	2	1
2106.2.e.ba		2	9.d	odd	6	1
2704.2.a.f		1	468.x	even	6	1
2704.2.f.d		2	468.ch	odd	12	2
3042.2.a.a		1	117.t	even	6	1
3042.2.b.a		2	117.y	odd	12	2
3146.2.a.n		1	99.g	even	6	1
3328.2.b.j		2	144.u	even	12	2
3328.2.b.m		2	144.w	odd	12	2
5200.2.a.x		1	180.n	even	6	1
5850.2.a.p		1	45.j	even	6	1
5850.2.e.a		2	45.k	odd	12	2
7488.2.a.g		1	72.n	even	6	1
7488.2.a.h		1	72.p	odd	6	1
7514.2.a.c		1	153.i	odd	6	1
8450.2.a.c		1	585.bo	odd	6	1
9386.2.a.j		1	171.l	even	6	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2106, [\chi])\):

\( T_{5}^{2} + 3T_{5} + 9 \)

\( T_{7}^{2} - T_{7} + 1 \)

\( T_{11}^{2} - 6T_{11} + 36 \)

\( T_{19} - 2 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} + T + 1 \)
$3$	\( T^{2} \)
$5$	\( T^{2} + 3T + 9 \)
$7$	\( T^{2} - T + 1 \)
$11$	\( T^{2} - 6T + 36 \)