Newform orbit 147.2.c.a

• Label: 147.2.c.a
• Level: $147$
• Weight: $2$
• Character orbit: 147.c
• Analytic conductor: $1.174$
• Analytic rank: $0$
• Dimension: $2$
• CM discriminant: -3
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 147 = 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	147.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.17380090971\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{-3}) \)
Defining polynomial:	\( x^{2} - x + 1 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 2 \)
Twist minimal:	no (minimal twist has level 21)
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{-3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	q - b * q^3 + 2 * q^4 - 3 * q^9 - 2*b * q^12 - b * q^13 + 4 * q^16 + 3*b * q^19 - 5 * q^25 + 3*b * q^27 + 5*b * q^31 - 6 * q^36 + q^37 - 3 * q^39 - 5 * q^43 - 4*b * q^48 - 2*b * q^52 + 9 * q^57 - 4*b * q^61 + 8 * q^64 + 11 * q^67 - 9*b * q^73 + 5*b * q^75 + 6*b * q^76 - 13 * q^79 + 9 * q^81 + 15 * q^93 - 8*b * q^97
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{4} - 6 q^{9} + 8 q^{16} - 10 q^{25} - 12 q^{36} + 2 q^{37} - 6 q^{39} - 10 q^{43} + 18 q^{57} + 16 q^{64} + 22 q^{67} - 26 q^{79} + 18 q^{81} + 30 q^{93}+O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/147\mathbb{Z}\right)^\times\).

\(n\)	\(50\)	\(52\)
\(\chi(n)\)	\(-1\)	\(-1\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

146.1

0.500000	+	0.866025i
0.500000	−	0.866025i

0

−

1.73205i

2.00000

0

0

0

0

−3.00000

0

146.2

0

1.73205i

2.00000

0

0

0

0

−3.00000

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	CM by \(\Q(\sqrt{-3}) \)
7.b	odd	2	1	inner
21.c	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	147.2.c.a		2
3.b	odd	2	1	CM	147.2.c.a		2
4.b	odd	2	1		2352.2.k.c		2
7.b	odd	2	1	inner	147.2.c.a		2
7.c	even	3	1		21.2.g.a	✓	2
7.c	even	3	1		147.2.g.a		2
7.d	odd	6	1		21.2.g.a	✓	2
7.d	odd	6	1		147.2.g.a		2
12.b	even	2	1		2352.2.k.c		2
21.c	even	2	1	inner	147.2.c.a		2
21.g	even	6	1		21.2.g.a	✓	2
21.g	even	6	1		147.2.g.a		2
21.h	odd	6	1		21.2.g.a	✓	2
21.h	odd	6	1		147.2.g.a		2
28.d	even	2	1		2352.2.k.c		2
28.f	even	6	1		336.2.bc.c		2
28.g	odd	6	1		336.2.bc.c		2
35.i	odd	6	1		525.2.t.c		2
35.j	even	6	1		525.2.t.c		2
35.k	even	12	2		525.2.q.d		4
35.l	odd	12	2		525.2.q.d		4
63.g	even	3	1		567.2.s.a		2
63.h	even	3	1		567.2.i.b		2
63.i	even	6	1		567.2.i.b		2
63.j	odd	6	1		567.2.i.b		2
63.k	odd	6	1		567.2.s.a		2
63.n	odd	6	1		567.2.s.a		2
63.s	even	6	1		567.2.s.a		2
63.t	odd	6	1		567.2.i.b		2
84.h	odd	2	1		2352.2.k.c		2
84.j	odd	6	1		336.2.bc.c		2
84.n	even	6	1		336.2.bc.c		2
105.o	odd	6	1		525.2.t.c		2
105.p	even	6	1		525.2.t.c		2
105.w	odd	12	2		525.2.q.d		4
105.x	even	12	2		525.2.q.d		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
21.2.g.a	✓	2	7.c	even	3	1
21.2.g.a	✓	2	7.d	odd	6	1
21.2.g.a	✓	2	21.g	even	6	1
21.2.g.a	✓	2	21.h	odd	6	1
147.2.c.a		2	1.a	even	1	1	trivial
147.2.c.a		2	3.b	odd	2	1	CM
147.2.c.a		2	7.b	odd	2	1	inner
147.2.c.a		2	21.c	even	2	1	inner
147.2.g.a		2	7.c	even	3	1
147.2.g.a		2	7.d	odd	6	1
147.2.g.a		2	21.g	even	6	1
147.2.g.a		2	21.h	odd	6	1
336.2.bc.c		2	28.f	even	6	1
336.2.bc.c		2	28.g	odd	6	1
336.2.bc.c		2	84.j	odd	6	1
336.2.bc.c		2	84.n	even	6	1
525.2.q.d		4	35.k	even	12	2
525.2.q.d		4	35.l	odd	12	2
525.2.q.d		4	105.w	odd	12	2
525.2.q.d		4	105.x	even	12	2
525.2.t.c		2	35.i	odd	6	1
525.2.t.c		2	35.j	even	6	1
525.2.t.c		2	105.o	odd	6	1
525.2.t.c		2	105.p	even	6	1
567.2.i.b		2	63.h	even	3	1
567.2.i.b		2	63.i	even	6	1
567.2.i.b		2	63.j	odd	6	1
567.2.i.b		2	63.t	odd	6	1
567.2.s.a		2	63.g	even	3	1
567.2.s.a		2	63.k	odd	6	1
567.2.s.a		2	63.n	odd	6	1
567.2.s.a		2	63.s	even	6	1
2352.2.k.c		2	4.b	odd	2	1
2352.2.k.c		2	12.b	even	2	1
2352.2.k.c		2	28.d	even	2	1
2352.2.k.c		2	84.h	odd	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} \) acting on \(S_{2}^{\mathrm{new}}(147, [\chi])\).

Hecke characteristic polynomials

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