Newform orbit 847.2.l.g

• Label: 847.2.l.g
• Level: $847$
• Weight: $2$
• Character orbit: 847.l
• Analytic conductor: $6.763$
• Analytic rank: $0$
• Dimension: $16$
• CM: no
• Inner twists: $16$

Newspace parameters

Level:	\( N \)	\(=\)	\( 847 = 7 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	847.l (of order \(10\), degree \(4\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.76332905120\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Relative dimension:	\(4\) over \(\Q(\zeta_{10})\)
Coefficient field:	\(\Q(\zeta_{40})\)
Defining polynomial:	\( x^{16} - x^{12} + x^{8} - x^{4} + 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 5^{4} \)
Twist minimal:	no (minimal twist has level 77)
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q - \beta_{6} q^{2} - \beta_{5} q^{3} + \beta_{13} q^{5} + (\beta_{11} - 2 \beta_{9}) q^{6} + (\beta_{14} + \beta_{3}) q^{7} + 2 \beta_{15} q^{8} - 2 \beta_{2} q^{9}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 8 q^{9}+O(q^{10}) \)

Basis of coefficient ring

\(\beta_{1}\)	\(=\)	\( \zeta_{40}^{4} \)
\(\beta_{2}\)	\(=\)	\( \zeta_{40}^{8} \)
\(\beta_{3}\)	\(=\)	\( \zeta_{40}^{11} + \zeta_{40} \)
\(\beta_{4}\)	\(=\)	\( \zeta_{40}^{12} \)
\(\beta_{5}\)	\(=\)	\( -\zeta_{40}^{14} - 2\zeta_{40}^{6} + 2\zeta_{40}^{2} \)
\(\beta_{6}\)	\(=\)	\( -\zeta_{40}^{13} + \zeta_{40}^{3} \)
\(\beta_{7}\)	\(=\)	\( \zeta_{40}^{9} - \zeta_{40}^{7} + \zeta_{40}^{3} + \zeta_{40} \)
\(\beta_{8}\)	\(=\)	\( -2\zeta_{40}^{10} + \zeta_{40}^{6} - 2\zeta_{40}^{2} \)
\(\beta_{9}\)	\(=\)	\( \zeta_{40}^{13} + \zeta_{40}^{11} - \zeta_{40}^{7} + \zeta_{40}^{3} - \zeta_{40} \)
\(\beta_{10}\)	\(=\)	\( 2\zeta_{40}^{14} - \zeta_{40}^{10} + 2\zeta_{40}^{6} \)
\(\beta_{11}\)	\(=\)	\( \zeta_{40}^{13} - \zeta_{40}^{9} - \zeta_{40}^{7} + \zeta_{40}^{5} - \zeta_{40} \)
\(\beta_{12}\)	\(=\)	\( -\zeta_{40}^{15} + \zeta_{40}^{13} - \zeta_{40}^{9} + \zeta_{40}^{5} + \zeta_{40}^{3} \)
\(\beta_{13}\)	\(=\)	\( -2\zeta_{40}^{14} + 2\zeta_{40}^{10} + \zeta_{40}^{2} \)
\(\beta_{14}\)	\(=\)	\( \zeta_{40}^{15} + \zeta_{40}^{13} - \zeta_{40}^{9} + \zeta_{40}^{7} \)
\(\beta_{15}\)	\(=\)	\( -\zeta_{40}^{15} + \zeta_{40}^{11} - \zeta_{40}^{9} - \zeta_{40}^{7} + \zeta_{40}^{3} \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/847\mathbb{Z}\right)^\times\).

\(n\)	\(122\)	\(365\)
\(\chi(n)\)	\(-1\)	\(1 - \beta_{1} + \beta_{2} - \beta_{4}\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

118.1

−0.891007	+	0.453990i
−0.453990	−	0.891007i
0.891007	−	0.453990i
0.453990	+	0.891007i
0.987688	−	0.156434i
−0.156434	−	0.987688i
−0.987688	+	0.156434i
0.156434	+	0.987688i
−0.891007	−	0.453990i
−0.453990	+	0.891007i
0.891007	+	0.453990i
0.453990	−	0.891007i
0.987688	+	0.156434i
−0.156434	+	0.987688i
−0.987688	−	0.156434i
0.156434	−	0.987688i

−0.831254

−

1.14412i

−2.12663

+

0.690983i

0

−1.31433

+

1.80902i

2.55834

+

1.85874i

−2.50609

+

0.848228i

−2.68999

+

0.874032i

1.61803

−

1.17557i

3.16228

118.2

−0.831254

−

1.14412i

2.12663

−

0.690983i

0

1.31433

−

1.80902i

−2.55834

−

1.85874i

−1.52890

−

2.15928i

−2.68999

+

0.874032i

1.61803

−

1.17557i

−3.16228

118.3

0.831254

+

1.14412i

−2.12663

+

0.690983i

0

−1.31433

+

1.80902i

−2.55834

−

1.85874i

2.50609

−

0.848228i

2.68999

−

0.874032i

1.61803

−

1.17557i

−3.16228

118.4

0.831254

+

1.14412i

2.12663

−

0.690983i

0

1.31433

−

1.80902i

2.55834

+

1.85874i

1.52890

+

2.15928i

2.68999

−

0.874032i

1.61803

−

1.17557i

3.16228

475.1

−1.34500

−

0.437016i

−1.31433

−

1.80902i

0

2.12663

−

0.690983i

0.977198

+

3.00750i

−0.0322874

−

2.64555i

1.66251

+

2.28825i

−0.618034

+

1.90211i

−3.16228

475.2

−1.34500

−

0.437016i

1.31433

+

1.80902i

0

−2.12663

+

0.690983i

−0.977198

−

3.00750i

2.52605

−

0.786814i

1.66251

+

2.28825i

−0.618034

+

1.90211i

3.16228

475.3

1.34500

+

0.437016i

−1.31433

−

1.80902i

0

2.12663

−

0.690983i

−0.977198

−

3.00750i

0.0322874

+

2.64555i

−1.66251

−

2.28825i

−0.618034

+

1.90211i

3.16228

475.4

1.34500

+

0.437016i

1.31433

+

1.80902i

0

−2.12663

+

0.690983i

0.977198

+

3.00750i

−2.52605

+

0.786814i

−1.66251

−

2.28825i

−0.618034

+

1.90211i

−3.16228

524.1

−0.831254

+

1.14412i

−2.12663

−

0.690983i

0

−1.31433

−

1.80902i

2.55834

−

1.85874i

−2.50609

−

0.848228i

−2.68999

−

0.874032i

1.61803

+

1.17557i

3.16228

524.2

−0.831254

+

1.14412i

2.12663

+

0.690983i

0

1.31433

+

1.80902i

−2.55834

+

1.85874i

−1.52890

+

2.15928i

−2.68999

−

0.874032i

1.61803

+

1.17557i

−3.16228

524.3

0.831254

−

1.14412i

−2.12663

−

0.690983i

0

−1.31433

−

1.80902i

−2.55834

+

1.85874i

2.50609

+

0.848228i

2.68999

+

0.874032i

1.61803

+

1.17557i

−3.16228

524.4

0.831254

−

1.14412i

2.12663

+

0.690983i

0

1.31433

+

1.80902i

2.55834

−

1.85874i

1.52890

−

2.15928i

2.68999

+

0.874032i

1.61803

+

1.17557i

3.16228

699.1

−1.34500

+

0.437016i

−1.31433

+

1.80902i

0

2.12663

+

0.690983i

0.977198

−

3.00750i

−0.0322874

+

2.64555i

1.66251

−

2.28825i

−0.618034

−

1.90211i

−3.16228

699.2

−1.34500

+

0.437016i

1.31433

−

1.80902i

0

−2.12663

−

0.690983i

−0.977198

+

3.00750i

2.52605

+

0.786814i

1.66251

−

2.28825i

−0.618034

−

1.90211i

3.16228

699.3

1.34500

−

0.437016i

−1.31433

+

1.80902i

0

2.12663

+

0.690983i

−0.977198

+

3.00750i

0.0322874

−

2.64555i

−1.66251

+

2.28825i

−0.618034

−

1.90211i

3.16228

699.4

1.34500

−

0.437016i

1.31433

−

1.80902i

0

−2.12663

−

0.690983i

0.977198

−

3.00750i

−2.52605

−

0.786814i

−1.66251

+

2.28825i

−0.618034

−

1.90211i

−3.16228

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.b	odd	2	1	inner
11.b	odd	2	1	inner
11.c	even	5	3	inner
11.d	odd	10	3	inner
77.b	even	2	1	inner
77.j	odd	10	3	inner
77.l	even	10	3	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	847.2.l.g		16
7.b	odd	2	1	inner	847.2.l.g		16
11.b	odd	2	1	inner	847.2.l.g		16
11.c	even	5	1		77.2.b.b	✓	4
11.c	even	5	3	inner	847.2.l.g		16
11.d	odd	10	1		77.2.b.b	✓	4
11.d	odd	10	3	inner	847.2.l.g		16
33.f	even	10	1		693.2.c.b		4
33.h	odd	10	1		693.2.c.b		4
44.g	even	10	1		1232.2.e.c		4
44.h	odd	10	1		1232.2.e.c		4
77.b	even	2	1	inner	847.2.l.g		16
77.j	odd	10	1		77.2.b.b	✓	4
77.j	odd	10	3	inner	847.2.l.g		16
77.l	even	10	1		77.2.b.b	✓	4
77.l	even	10	3	inner	847.2.l.g		16
77.m	even	15	2		539.2.i.b		8
77.n	even	30	2		539.2.i.b		8
77.o	odd	30	2		539.2.i.b		8
77.p	odd	30	2		539.2.i.b		8
231.r	odd	10	1		693.2.c.b		4
231.u	even	10	1		693.2.c.b		4
308.s	odd	10	1		1232.2.e.c		4
308.t	even	10	1		1232.2.e.c		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
77.2.b.b	✓	4	11.c	even	5	1
77.2.b.b	✓	4	11.d	odd	10	1
77.2.b.b	✓	4	77.j	odd	10	1
77.2.b.b	✓	4	77.l	even	10	1
539.2.i.b		8	77.m	even	15	2
539.2.i.b		8	77.n	even	30	2
539.2.i.b		8	77.o	odd	30	2
539.2.i.b		8	77.p	odd	30	2
693.2.c.b		4	33.f	even	10	1
693.2.c.b		4	33.h	odd	10	1
693.2.c.b		4	231.r	odd	10	1
693.2.c.b		4	231.u	even	10	1
847.2.l.g		16	1.a	even	1	1	trivial
847.2.l.g		16	7.b	odd	2	1	inner
847.2.l.g		16	11.b	odd	2	1	inner
847.2.l.g		16	11.c	even	5	3	inner
847.2.l.g		16	11.d	odd	10	3	inner
847.2.l.g		16	77.b	even	2	1	inner
847.2.l.g		16	77.j	odd	10	3	inner
847.2.l.g		16	77.l	even	10	3	inner
1232.2.e.c		4	44.g	even	10	1
1232.2.e.c		4	44.h	odd	10	1
1232.2.e.c		4	308.s	odd	10	1
1232.2.e.c		4	308.t	even	10	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} - 2T_{2}^{6} + 4T_{2}^{4} - 8T_{2}^{2} + 16 \) acting on \(S_{2}^{\mathrm{new}}(847, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	(T^8 - 2*T^6 + 4*T^4 - 8*T^2 + 16)^2
$3$	(T^8 - 5*T^6 + 25*T^4 - 125*T^2 + 625)^2
$5$	(T^8 - 5*T^6 + 25*T^4 - 125*T^2 + 625)^2
$7$	T^16 - 4*T^14 - 33*T^12 + 328*T^10 + 305*T^8 + 16072*T^6 - 79233*T^4 - 470596*T^2 + 5764801
$11$	\( T^{16} \)