Embedded newform 672.3.e.a.335.1

• Label: 672.3.e.a.335.1
• Level: $672$
• Weight: $3$
• Character: 672.335
• Self dual: yes
• Analytic conductor: $18.311$
• Analytic rank: $0$
• Dimension: $1$
• CM discriminant: -168
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 672 = 2^{5} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	672.e (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual:	yes
Analytic conductor:	\(18.3106737650\)
Analytic rank:	\(0\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 168)
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			335.1
Character	\(\chi\)	\(=\)	672.335

$q$-expansion

\(f(q)\)

\(=\)

\(q-3.00000 q^{3} -7.00000 q^{7} +9.00000 q^{9} +2.00000 q^{13} -22.0000 q^{17} +21.0000 q^{21} -38.0000 q^{23} +25.0000 q^{25} -27.0000 q^{27} +26.0000 q^{29} +34.0000 q^{31} -6.00000 q^{39} +26.0000 q^{41} +82.0000 q^{43} +49.0000 q^{49} +66.0000 q^{51} -22.0000 q^{53} +106.000 q^{59} -94.0000 q^{61} -63.0000 q^{63} +34.0000 q^{67} +114.000 q^{69} +58.0000 q^{71} -75.0000 q^{75} +81.0000 q^{81} +58.0000 q^{83} -78.0000 q^{87} +122.000 q^{89} -14.0000 q^{91} -102.000 q^{93} +O(q^{100})\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/672\mathbb{Z}\right)^\times\).

\(n\)	\(127\)	\(421\)	\(449\)	\(577\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(-1\)	\(-1\)

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
			\(3\)	−3.00000	−1.00000
\(5\)	0	0				1.00000			\(0\)
			−1.00000			\(\pi\)
\(7\)	−7.00000	−1.00000
			\(11\)	0	0	1.00000			\(0\)
−1.00000						\(\pi\)
\(13\)	2.00000	0.153846	0.0769231	−	0.997037i	\(-0.475490\pi\)
			0.0769231	+	0.997037i	\(0.475490\pi\)
\(17\)	−22.0000	−1.29412	−0.647059	−	0.762440i	\(-0.724001\pi\)
			−0.647059	+	0.762440i	\(0.724001\pi\)
\(19\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(23\)	−38.0000	−1.65217	−0.826087	−	0.563543i	\(-0.809438\pi\)
			−0.826087	+	0.563543i	\(0.809438\pi\)
\(29\)	26.0000	0.896552	0.448276	−	0.893895i	\(-0.352038\pi\)
			0.448276	+	0.893895i	\(0.352038\pi\)
\(31\)	34.0000	1.09677	0.548387	−	0.836225i	\(-0.315242\pi\)
			0.548387	+	0.836225i	\(0.315242\pi\)
\(37\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(41\)	26.0000	0.634146	0.317073	−	0.948401i	\(-0.397300\pi\)
			0.317073	+	0.948401i	\(0.397300\pi\)
\(43\)	82.0000	1.90698	0.953488	−	0.301430i	\(-0.0974639\pi\)
			0.953488	+	0.301430i	\(0.0974639\pi\)
\(47\)	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	672.3.e.a.335.1		1
3.2	odd	2		672.3.e.c.335.1		1
4.3	odd	2		168.3.e.b.83.1	yes	1
7.6	odd	2		672.3.e.d.335.1		1
8.3	odd	2		672.3.e.b.335.1		1
8.5	even	2		168.3.e.d.83.1	yes	1
12.11	even	2		168.3.e.c.83.1	yes	1
21.20	even	2		672.3.e.b.335.1		1
24.5	odd	2		168.3.e.a.83.1	✓	1
24.11	even	2		672.3.e.d.335.1		1
28.27	even	2		168.3.e.a.83.1	✓	1
56.13	odd	2		168.3.e.c.83.1	yes	1
56.27	even	2		672.3.e.c.335.1		1
84.83	odd	2		168.3.e.d.83.1	yes	1
168.83	odd	2	CM	672.3.e.a.335.1		1
168.125	even	2		168.3.e.b.83.1	yes	1

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
168.3.e.a.83.1	✓	1	24.5	odd	2
168.3.e.a.83.1	✓	1	28.27	even	2
168.3.e.b.83.1	yes	1	4.3	odd	2
168.3.e.b.83.1	yes	1	168.125	even	2
168.3.e.c.83.1	yes	1	12.11	even	2
168.3.e.c.83.1	yes	1	56.13	odd	2
168.3.e.d.83.1	yes	1	8.5	even	2
168.3.e.d.83.1	yes	1	84.83	odd	2
672.3.e.a.335.1		1	1.1	even	1	trivial
672.3.e.a.335.1		1	168.83	odd	2	CM
672.3.e.b.335.1		1	8.3	odd	2
672.3.e.b.335.1		1	21.20	even	2
672.3.e.c.335.1		1	3.2	odd	2
672.3.e.c.335.1		1	56.27	even	2
672.3.e.d.335.1		1	7.6	odd	2
672.3.e.d.335.1		1	24.11	even	2