Embedded newform 304.3.e.e.113.2

• Label: 304.3.e.e.113.2
• Level: $304$
• Weight: $3$
• Character: 304.113
• Self dual: yes
• Analytic conductor: $8.283$
• Analytic rank: $0$
• Dimension: $2$
• CM discriminant: -19
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 304 = 2^{4} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	304.e (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual:	yes
Analytic conductor:	\(8.28340003655\)
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			113.2
Root			\(-3.27492\) of defining polynomial
Character	\(\chi\)	\(=\)	304.113

$q$-expansion

\(f(q)\)	\(=\)	\(q+8.27492 q^{5} +8.82475 q^{7} +9.00000 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 9 q^{5} - 5 q^{7} + 18 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/304\mathbb{Z}\right)^\times\).

\(n\)	\(97\)	\(191\)	\(229\)
\(\chi(n)\)	\(-1\)	\(1\)	\(1\)

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)

\(p\)	\(a_p\)	\(a_p / p^{(k-1)/2}\)	\( \alpha_p\)			\( \theta_p \)
\(2\)	0	0
			\(3\)	0	0	1.00000			\(0\)
−1.00000						\(\pi\)
\(5\)	8.27492	1.65498	0.827492	−	0.561478i	\(-0.189767\pi\)
			0.827492	+	0.561478i	\(0.189767\pi\)
\(7\)	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	\(-0.789792\pi\)
			−0.789754	+	0.613424i	\(0.789792\pi\)
\(13\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(17\)	−33.9244	−1.99555	−0.997777	−	0.0666402i	\(-0.978772\pi\)
			−0.997777	+	0.0666402i	\(0.978772\pi\)
\(19\)	19.0000	1.00000
			\(23\)	30.0000	1.30435	0.652174	−	0.758069i	\(-0.273857\pi\)
0.652174	+	0.758069i				\(0.273857\pi\)
\(29\)	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	\(-0.539292\pi\)
			−0.123127	+	0.992391i	\(0.539292\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	304.3.e.e.113.2		2
3.2	odd	2		2736.3.o.c.721.1		2
4.3	odd	2		76.3.c.b.37.2	✓	2
8.3	odd	2		1216.3.e.f.1025.1		2
8.5	even	2		1216.3.e.e.1025.1		2
12.11	even	2		684.3.h.a.37.1		2
19.18	odd	2	CM	304.3.e.e.113.2		2
20.3	even	4		1900.3.g.a.949.3		4
20.7	even	4		1900.3.g.a.949.2		4
20.19	odd	2		1900.3.e.a.1101.2		2
57.56	even	2		2736.3.o.c.721.1		2
76.75	even	2		76.3.c.b.37.2	✓	2
152.37	odd	2		1216.3.e.e.1025.1		2
152.75	even	2		1216.3.e.f.1025.1		2
228.227	odd	2		684.3.h.a.37.1		2
380.227	odd	4		1900.3.g.a.949.2		4
380.303	odd	4		1900.3.g.a.949.3		4
380.379	even	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	4.3	odd	2
76.3.c.b.37.2	✓	2	76.75	even	2
304.3.e.e.113.2		2	1.1	even	1	trivial
304.3.e.e.113.2		2	19.18	odd	2	CM
684.3.h.a.37.1		2	12.11	even	2
684.3.h.a.37.1		2	228.227	odd	2
1216.3.e.e.1025.1		2	8.5	even	2
1216.3.e.e.1025.1		2	152.37	odd	2
1216.3.e.f.1025.1		2	8.3	odd	2
1216.3.e.f.1025.1		2	152.75	even	2
1900.3.e.a.1101.2		2	20.19	odd	2
1900.3.e.a.1101.2		2	380.379	even	2
1900.3.g.a.949.2		4	20.7	even	4
1900.3.g.a.949.2		4	380.227	odd	4
1900.3.g.a.949.3		4	20.3	even	4
1900.3.g.a.949.3		4	380.303	odd	4
2736.3.o.c.721.1		2	3.2	odd	2
2736.3.o.c.721.1		2	57.56	even	2