Embedded newform 83.1.b.a.82.1

• Label: 83.1.b.a.82.1
• Level: $83$
• Weight: $1$
• Character: 83.82
• Self dual: yes
• Analytic conductor: $0.041$
• Analytic rank: $0$
• Dimension: $1$
• Projective image: $D_{3}$
• CM discriminant: -83
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 83 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	83.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	yes
Analytic conductor:	\(0.0414223960485\)
Analytic rank:	\(0\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Projective image:	\(D_{3}\)
Projective field:	Galois closure of 3.1.83.1
Artin image:	$S_3$
Artin field:	Galois closure of 3.1.83.1
Stark unit:	Root of $x^{3} - 2x^{2} - 2x - 1$

Embedding invariants

Embedding label			82.1
Character	\(\chi\)	\(=\)	83.82

$q$-expansion

\(f(q)\)

\(=\)

\(q-1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{7} -1.00000 q^{11} -1.00000 q^{12} +1.00000 q^{16} -1.00000 q^{17} +1.00000 q^{21} +2.00000 q^{23} +1.00000 q^{25} +1.00000 q^{27} -1.00000 q^{28} -1.00000 q^{29} -1.00000 q^{31} +1.00000 q^{33} -1.00000 q^{37} +2.00000 q^{41} -1.00000 q^{44} -1.00000 q^{48} +1.00000 q^{51} -1.00000 q^{59} -1.00000 q^{61} +1.00000 q^{64} -1.00000 q^{68} -2.00000 q^{69} -1.00000 q^{75} +1.00000 q^{77} -1.00000 q^{81} +1.00000 q^{83} +1.00000 q^{84} +1.00000 q^{87} +2.00000 q^{92} +1.00000 q^{93} +O(q^{100})\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/83\mathbb{Z}\right)^\times\).

\(n\)	\(2\)
\(\chi(n)\)	\(-1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.

Currently showing only \(a_p\); display all \(a_n\)

\(p\)	\(a_p\)	\(a_p / p^{(k-1)/2}\)	\( \alpha_p\)			\( \theta_p \)
\(2\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(3\)	−1.00000	−1.00000	−0.500000	−	0.866025i	\(-0.666667\pi\)
			−0.500000	+	0.866025i	\(0.666667\pi\)
\(5\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(7\)	−1.00000	−1.00000	−0.500000	−	0.866025i	\(-0.666667\pi\)
			−0.500000	+	0.866025i	\(0.666667\pi\)
\(11\)	−1.00000	−1.00000	−0.500000	−	0.866025i	\(-0.666667\pi\)
			−0.500000	+	0.866025i	\(0.666667\pi\)
\(13\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(17\)	−1.00000	−1.00000	−0.500000	−	0.866025i	\(-0.666667\pi\)
			−0.500000	+	0.866025i	\(0.666667\pi\)
\(19\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(23\)	2.00000	2.00000	1.00000			\(0\)
			1.00000			\(0\)
\(29\)	−1.00000	−1.00000	−0.500000	−	0.866025i	\(-0.666667\pi\)
			−0.500000	+	0.866025i	\(0.666667\pi\)
\(31\)	−1.00000	−1.00000	−0.500000	−	0.866025i	\(-0.666667\pi\)
			−0.500000	+	0.866025i	\(0.666667\pi\)
\(37\)	−1.00000	−1.00000	−0.500000	−	0.866025i	\(-0.666667\pi\)
			−0.500000	+	0.866025i	\(0.666667\pi\)
\(41\)	2.00000	2.00000	1.00000			\(0\)
			1.00000			\(0\)
\(43\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(47\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(53\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(59\)	−1.00000	−1.00000	−0.500000	−	0.866025i	\(-0.666667\pi\)
			−0.500000	+	0.866025i	\(0.666667\pi\)
\(61\)	−1.00000	−1.00000	−0.500000	−	0.866025i	\(-0.666667\pi\)
			−0.500000	+	0.866025i	\(0.666667\pi\)
\(67\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(71\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(73\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(79\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(83\)	1.00000	1.00000
			\(89\)	0	0	1.00000			\(0\)
−1.00000						\(\pi\)
\(97\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)

Currently showing only \(a_p\); display all \(a_n\)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	83.1.b.a.82.1	✓	1
3.2	odd	2		747.1.c.a.82.1		1
4.3	odd	2		1328.1.g.a.497.1		1
5.2	odd	4		2075.1.d.c.2074.2		2
5.3	odd	4		2075.1.d.c.2074.1		2
5.4	even	2		2075.1.c.f.1576.1		1
83.82	odd	2	CM	83.1.b.a.82.1	✓	1
249.248	even	2		747.1.c.a.82.1		1
332.331	even	2		1328.1.g.a.497.1		1
415.82	even	4		2075.1.d.c.2074.2		2
415.248	even	4		2075.1.d.c.2074.1		2
415.414	odd	2		2075.1.c.f.1576.1		1

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
83.1.b.a.82.1	✓	1	1.1	even	1	trivial
83.1.b.a.82.1	✓	1	83.82	odd	2	CM
747.1.c.a.82.1		1	3.2	odd	2
747.1.c.a.82.1		1	249.248	even	2
1328.1.g.a.497.1		1	4.3	odd	2
1328.1.g.a.497.1		1	332.331	even	2
2075.1.c.f.1576.1		1	5.4	even	2
2075.1.c.f.1576.1		1	415.414	odd	2
2075.1.d.c.2074.1		2	5.3	odd	4
2075.1.d.c.2074.1		2	415.248	even	4
2075.1.d.c.2074.2		2	5.2	odd	4
2075.1.d.c.2074.2		2	415.82	even	4