Newform orbit 1872.2.by.f

• Label: 1872.2.by.f
• Level: $1872$
• Weight: $2$
• Character orbit: 1872.by
• Analytic conductor: $14.948$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1872 = 2^{4} \cdot 3^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1872.by (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(14.9479952584\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{-3}) \)
Defining polynomial:	\( x^{2} - x + 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 39)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

$q$-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.

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1872\mathbb{Z}\right)^\times\).

\(n\)	\(145\)	\(209\)	\(469\)	\(703\)
\(\chi(n)\)	\(\zeta_{6}\)	\(1\)	\(1\)	\(1\)

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.

Label  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

433.1

0.500000	+	0.866025i
0.500000	−	0.866025i

0

0

0

−

3.46410i

0

1.50000

−

0.866025i

0

0

0

1297.1

0

0

0

3.46410i

0

1.50000

+

0.866025i

0

0

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
13.e	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1872.2.by.f		2
3.b	odd	2	1		624.2.bv.b		2
4.b	odd	2	1		117.2.q.a		2
12.b	even	2	1		39.2.j.a	✓	2
13.e	even	6	1	inner	1872.2.by.f		2
39.h	odd	6	1		624.2.bv.b		2
39.k	even	12	2		8112.2.a.bu		2
52.i	odd	6	1		117.2.q.a		2
52.i	odd	6	1		1521.2.b.f		2
52.j	odd	6	1		1521.2.b.f		2
52.l	even	12	2		1521.2.a.h		2
60.h	even	2	1		975.2.bc.c		2
60.l	odd	4	2		975.2.w.d		4
156.h	even	2	1		507.2.j.b		2
156.l	odd	4	2		507.2.e.f		4
156.p	even	6	1		507.2.b.c		2
156.p	even	6	1		507.2.j.b		2
156.r	even	6	1		39.2.j.a	✓	2
156.r	even	6	1		507.2.b.c		2
156.v	odd	12	2		507.2.a.e		2
156.v	odd	12	2		507.2.e.f		4
780.cb	even	6	1		975.2.bc.c		2
780.cw	odd	12	2		975.2.w.d		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
39.2.j.a	✓	2	12.b	even	2	1
39.2.j.a	✓	2	156.r	even	6	1
117.2.q.a		2	4.b	odd	2	1
117.2.q.a		2	52.i	odd	6	1
507.2.a.e		2	156.v	odd	12	2
507.2.b.c		2	156.p	even	6	1
507.2.b.c		2	156.r	even	6	1
507.2.e.f		4	156.l	odd	4	2
507.2.e.f		4	156.v	odd	12	2
507.2.j.b		2	156.h	even	2	1
507.2.j.b		2	156.p	even	6	1
624.2.bv.b		2	3.b	odd	2	1
624.2.bv.b		2	39.h	odd	6	1
975.2.w.d		4	60.l	odd	4	2
975.2.w.d		4	780.cw	odd	12	2
975.2.bc.c		2	60.h	even	2	1
975.2.bc.c		2	780.cb	even	6	1
1521.2.a.h		2	52.l	even	12	2
1521.2.b.f		2	52.i	odd	6	1
1521.2.b.f		2	52.j	odd	6	1
1872.2.by.f		2	1.a	even	1	1	trivial
1872.2.by.f		2	13.e	even	6	1	inner
8112.2.a.bu		2	39.k	even	12	2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1872, [\chi])\):

\( T_{5}^{2} + 12 \)

\( T_{7}^{2} - 3T_{7} + 3 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} \)
$3$	\( T^{2} \)
$5$	\( T^{2} + 12 \)
$7$	\( T^{2} - 3T + 3 \)
$11$	\( T^{2} + 6T + 12 \)