Embedded newform 756.1.h.c.755.1

• Label: 756.1.h.c.755.1
• Level: $756$
• Weight: $1$
• Character: 756.755
• Self dual: yes
• Analytic conductor: $0.377$
• Analytic rank: $0$
• Dimension: $1$
• Projective image: $D_{3}$
• CM discriminant: -84
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 756 = 2^{2} \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	756.h (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	yes
Analytic conductor:	\(0.377293149551\)
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 3.1.756.1
Artin image:	$S_3$
Artin field:	Galois closure of 3.1.756.1
Stark unit:	Root of $x^{3} - 1221x^{2} - 51x - 1$

Embedding invariants

Embedding label			755.1
Character	\(\chi\)	\(=\)	756.755

$q$-expansion

\(f(q)\)

\(=\)

\(q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{8} -1.00000 q^{10} -1.00000 q^{11} +1.00000 q^{14} +1.00000 q^{16} +2.00000 q^{17} -1.00000 q^{19} -1.00000 q^{20} -1.00000 q^{22} -1.00000 q^{23} +1.00000 q^{28} -1.00000 q^{31} +1.00000 q^{32} +2.00000 q^{34} -1.00000 q^{35} -1.00000 q^{37} -1.00000 q^{38} -1.00000 q^{40} -1.00000 q^{41} -1.00000 q^{44} -1.00000 q^{46} +1.00000 q^{49} +1.00000 q^{55} +1.00000 q^{56} -1.00000 q^{62} +1.00000 q^{64} +2.00000 q^{68} -1.00000 q^{70} -1.00000 q^{71} -1.00000 q^{74} -1.00000 q^{76} -1.00000 q^{77} -1.00000 q^{80} -1.00000 q^{82} -2.00000 q^{85} -1.00000 q^{88} -1.00000 q^{89} -1.00000 q^{92} +1.00000 q^{95} +1.00000 q^{98} +O(q^{100})\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/756\mathbb{Z}\right)^\times\).

\(n\)	\(29\)	\(325\)	\(379\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(-1\)

Coefficient data

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

(See \(a_n\) instead)

\(p\)	\(a_p\)	\(a_p / p^{(k-1)/2}\)	\( \alpha_p\)			\( \theta_p \)
\(2\)	1.00000	1.00000
			\(3\)	0	0
\(5\)	−1.00000	−1.00000				−0.500000	−	0.866025i	\(-0.666667\pi\)
			−0.500000	+	0.866025i	\(0.666667\pi\)
\(7\)	1.00000	1.00000
			\(11\)	−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			\(\pi\)
\(17\)	2.00000	2.00000	1.00000			\(0\)
			1.00000			\(0\)
\(19\)	−1.00000	−1.00000	−0.500000	−	0.866025i	\(-0.666667\pi\)
			−0.500000	+	0.866025i	\(0.666667\pi\)
\(23\)	−1.00000	−1.00000	−0.500000	−	0.866025i	\(-0.666667\pi\)
			−0.500000	+	0.866025i	\(0.666667\pi\)
\(29\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(31\)	−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	\(0.666667\pi\)
\(41\)	−1.00000	−1.00000	−0.500000	−	0.866025i	\(-0.666667\pi\)
			−0.500000	+	0.866025i	\(0.666667\pi\)
\(43\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(47\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	756.1.h.c.755.1	yes	1
3.2	odd	2		756.1.h.b.755.1	yes	1
4.3	odd	2		756.1.h.a.755.1	✓	1
7.6	odd	2		756.1.h.d.755.1	yes	1
9.2	odd	6		2268.1.s.c.1511.1		2
9.4	even	3		2268.1.s.b.755.1		2
9.5	odd	6		2268.1.s.c.755.1		2
9.7	even	3		2268.1.s.b.1511.1		2
12.11	even	2		756.1.h.d.755.1	yes	1
21.20	even	2		756.1.h.a.755.1	✓	1
28.27	even	2		756.1.h.b.755.1	yes	1
36.7	odd	6		2268.1.s.d.1511.1		2
36.11	even	6		2268.1.s.a.1511.1		2
36.23	even	6		2268.1.s.a.755.1		2
36.31	odd	6		2268.1.s.d.755.1		2
63.13	odd	6		2268.1.s.a.755.1		2
63.20	even	6		2268.1.s.d.1511.1		2
63.34	odd	6		2268.1.s.a.1511.1		2
63.41	even	6		2268.1.s.d.755.1		2
84.83	odd	2	CM	756.1.h.c.755.1	yes	1
252.83	odd	6		2268.1.s.b.1511.1		2
252.139	even	6		2268.1.s.c.755.1		2
252.167	odd	6		2268.1.s.b.755.1		2
252.223	even	6		2268.1.s.c.1511.1		2

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
756.1.h.a.755.1	✓	1	4.3	odd	2
756.1.h.a.755.1	✓	1	21.20	even	2
756.1.h.b.755.1	yes	1	3.2	odd	2
756.1.h.b.755.1	yes	1	28.27	even	2
756.1.h.c.755.1	yes	1	1.1	even	1	trivial
756.1.h.c.755.1	yes	1	84.83	odd	2	CM
756.1.h.d.755.1	yes	1	7.6	odd	2
756.1.h.d.755.1	yes	1	12.11	even	2
2268.1.s.a.755.1		2	36.23	even	6
2268.1.s.a.755.1		2	63.13	odd	6
2268.1.s.a.1511.1		2	36.11	even	6
2268.1.s.a.1511.1		2	63.34	odd	6
2268.1.s.b.755.1		2	9.4	even	3
2268.1.s.b.755.1		2	252.167	odd	6
2268.1.s.b.1511.1		2	9.7	even	3
2268.1.s.b.1511.1		2	252.83	odd	6
2268.1.s.c.755.1		2	9.5	odd	6
2268.1.s.c.755.1		2	252.139	even	6
2268.1.s.c.1511.1		2	9.2	odd	6
2268.1.s.c.1511.1		2	252.223	even	6
2268.1.s.d.755.1		2	36.31	odd	6
2268.1.s.d.755.1		2	63.41	even	6
2268.1.s.d.1511.1		2	36.7	odd	6
2268.1.s.d.1511.1		2	63.20	even	6