Embedded newform 2736.3.o.c.721.1

• Label: 2736.3.o.c.721.1
• Level: $2736$
• Weight: $3$
• Character: 2736.721
• Self dual: yes
• Analytic conductor: $74.551$
• Analytic rank: $0$
• Dimension: $2$
• CM discriminant: -19
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 2736 = 2^{4} \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	2736.o (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual:	yes
Analytic conductor:	\(74.5506003290\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{57}) \)
Defining polynomial:	\( x^{2} - x - 14 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 76)
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			721.1
Root			\(4.27492\) of defining polynomial
Character	\(\chi\)	\(=\)	2736.721

$q$-expansion

\(f(q)\)	\(=\)	\(q-8.27492 q^{5} +8.82475 q^{7} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 9 q^{5} - 5 q^{7}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2736\mathbb{Z}\right)^\times\).

\(n\)	\(1009\)	\(1217\)	\(1711\)	\(2053\)
\(\chi(n)\)	\(-1\)	\(1\)	\(1\)	\(1\)

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)

\(p\)	\(a_p\)	\(a_p / p^{(k-1)/2}\)	\( \alpha_p\)			\( \theta_p \)
\(2\)	0	0
			\(3\)	0	0
\(5\)	−8.27492	−1.65498				−0.827492	−	0.561478i	\(-0.810233\pi\)
			−0.827492	+	0.561478i	\(0.810233\pi\)
\(7\)	8.82475	1.26068	0.630339	−	0.776320i	\(-0.282916\pi\)
			0.630339	+	0.776320i	\(0.282916\pi\)
\(11\)	17.3746	1.57951	0.789754	−	0.613424i	\(-0.210208\pi\)
			0.789754	+	0.613424i	\(0.210208\pi\)
\(13\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(17\)	33.9244	1.99555	0.997777	−	0.0666402i	\(-0.0212280\pi\)
			0.997777	+	0.0666402i	\(0.0212280\pi\)
\(19\)	19.0000	1.00000
			\(23\)	−30.0000	−1.30435	−0.652174	−	0.758069i	\(-0.726143\pi\)
−0.652174	+	0.758069i				\(0.726143\pi\)
\(29\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(31\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(37\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(41\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(43\)	−31.1752	−0.725006	−0.362503	−	0.931983i	\(-0.618078\pi\)
			−0.362503	+	0.931983i	\(0.618078\pi\)
\(47\)	11.5739	0.246254	0.123127	−	0.992391i	\(-0.460708\pi\)
			0.123127	+	0.992391i	\(0.460708\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2736.3.o.c.721.1		2
3.2	odd	2		304.3.e.e.113.2		2
4.3	odd	2		684.3.h.a.37.1		2
12.11	even	2		76.3.c.b.37.2	✓	2
19.18	odd	2	CM	2736.3.o.c.721.1		2
24.5	odd	2		1216.3.e.e.1025.1		2
24.11	even	2		1216.3.e.f.1025.1		2
57.56	even	2		304.3.e.e.113.2		2
60.23	odd	4		1900.3.g.a.949.3		4
60.47	odd	4		1900.3.g.a.949.2		4
60.59	even	2		1900.3.e.a.1101.2		2
76.75	even	2		684.3.h.a.37.1		2
228.227	odd	2		76.3.c.b.37.2	✓	2
456.227	odd	2		1216.3.e.f.1025.1		2
456.341	even	2		1216.3.e.e.1025.1		2
1140.227	even	4		1900.3.g.a.949.2		4
1140.683	even	4		1900.3.g.a.949.3		4
1140.1139	odd	2		1900.3.e.a.1101.2		2

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
76.3.c.b.37.2	✓	2	12.11	even	2
76.3.c.b.37.2	✓	2	228.227	odd	2
304.3.e.e.113.2		2	3.2	odd	2
304.3.e.e.113.2		2	57.56	even	2
684.3.h.a.37.1		2	4.3	odd	2
684.3.h.a.37.1		2	76.75	even	2
1216.3.e.e.1025.1		2	24.5	odd	2
1216.3.e.e.1025.1		2	456.341	even	2
1216.3.e.f.1025.1		2	24.11	even	2
1216.3.e.f.1025.1		2	456.227	odd	2
1900.3.e.a.1101.2		2	60.59	even	2
1900.3.e.a.1101.2		2	1140.1139	odd	2
1900.3.g.a.949.2		4	60.47	odd	4
1900.3.g.a.949.2		4	1140.227	even	4
1900.3.g.a.949.3		4	60.23	odd	4
1900.3.g.a.949.3		4	1140.683	even	4
2736.3.o.c.721.1		2	1.1	even	1	trivial
2736.3.o.c.721.1		2	19.18	odd	2	CM