Embedded newform 171.2.a.b.1.1

• Label: 171.2.a.b.1.1
• Level: $171$
• Weight: $2$
• Character: 171.1
• Self dual: yes
• Analytic conductor: $1.365$
• Analytic rank: $1$
• Dimension: $1$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 171 = 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	171.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(1.36544187456\)
Analytic rank:	\(1\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 19)
Fricke sign:	\(+1\)
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	\(\chi\)	\(=\)	171.1

$q$-expansion

\(f(q)\)

\(=\)

\(q-2.00000 q^{4} -3.00000 q^{5} -1.00000 q^{7} -3.00000 q^{11} -4.00000 q^{13} +4.00000 q^{16} +3.00000 q^{17} +1.00000 q^{19} +6.00000 q^{20} +4.00000 q^{25} +2.00000 q^{28} -6.00000 q^{29} -4.00000 q^{31} +3.00000 q^{35} +2.00000 q^{37} +6.00000 q^{41} -1.00000 q^{43} +6.00000 q^{44} +3.00000 q^{47} -6.00000 q^{49} +8.00000 q^{52} -12.0000 q^{53} +9.00000 q^{55} +6.00000 q^{59} -1.00000 q^{61} -8.00000 q^{64} +12.0000 q^{65} -4.00000 q^{67} -6.00000 q^{68} -6.00000 q^{71} -7.00000 q^{73} -2.00000 q^{76} +3.00000 q^{77} +8.00000 q^{79} -12.0000 q^{80} -12.0000 q^{83} -9.00000 q^{85} -12.0000 q^{89} +4.00000 q^{91} -3.00000 q^{95} +8.00000 q^{97} +O(q^{100})\)

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		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(3\)	0	0
			\(5\)	−3.00000	−1.34164	−0.670820	−	0.741620i	\(-0.734058\pi\)
−0.670820	+	0.741620i				\(0.734058\pi\)
\(7\)	−1.00000	−0.377964	−0.188982	−	0.981981i	\(-0.560519\pi\)
			−0.188982	+	0.981981i	\(0.560519\pi\)
\(11\)	−3.00000	−0.904534	−0.452267	−	0.891883i	\(-0.649385\pi\)
			−0.452267	+	0.891883i	\(0.649385\pi\)
\(13\)	−4.00000	−1.10940	−0.554700	−	0.832050i	\(-0.687167\pi\)
			−0.554700	+	0.832050i	\(0.687167\pi\)
\(17\)	3.00000	0.727607	0.363803	−	0.931476i	\(-0.381478\pi\)
			0.363803	+	0.931476i	\(0.381478\pi\)
\(19\)	1.00000	0.229416
			\(23\)	0	0		−	1.00000i	\(-0.5\pi\)
		1.00000i				\(0.5\pi\)
\(29\)	−6.00000	−1.11417	−0.557086	−	0.830455i	\(-0.688081\pi\)
			−0.557086	+	0.830455i	\(0.688081\pi\)
\(31\)	−4.00000	−0.718421	−0.359211	−	0.933257i	\(-0.616954\pi\)
			−0.359211	+	0.933257i	\(0.616954\pi\)
\(37\)	2.00000	0.328798	0.164399	−	0.986394i	\(-0.447432\pi\)
			0.164399	+	0.986394i	\(0.447432\pi\)
\(41\)	6.00000	0.937043	0.468521	−	0.883452i	\(-0.344787\pi\)
			0.468521	+	0.883452i	\(0.344787\pi\)
\(43\)	−1.00000	−0.152499	−0.0762493	−	0.997089i	\(-0.524294\pi\)
			−0.0762493	+	0.997089i	\(0.524294\pi\)
\(47\)	3.00000	0.437595	0.218797	−	0.975770i	\(-0.429787\pi\)
			0.218797	+	0.975770i	\(0.429787\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	171.2.a.b.1.1		1
3.2	odd	2		19.2.a.a.1.1	✓	1
4.3	odd	2		2736.2.a.c.1.1		1
5.4	even	2		4275.2.a.i.1.1		1
7.6	odd	2		8379.2.a.j.1.1		1
12.11	even	2		304.2.a.f.1.1		1
15.2	even	4		475.2.b.a.324.2		2
15.8	even	4		475.2.b.a.324.1		2
15.14	odd	2		475.2.a.b.1.1		1
19.18	odd	2		3249.2.a.d.1.1		1
21.2	odd	6		931.2.f.c.704.1		2
21.5	even	6		931.2.f.b.704.1		2
21.11	odd	6		931.2.f.c.324.1		2
21.17	even	6		931.2.f.b.324.1		2
21.20	even	2		931.2.a.a.1.1		1
24.5	odd	2		1216.2.a.o.1.1		1
24.11	even	2		1216.2.a.b.1.1		1
33.32	even	2		2299.2.a.b.1.1		1
39.38	odd	2		3211.2.a.a.1.1		1
51.50	odd	2		5491.2.a.b.1.1		1
57.2	even	18		361.2.e.e.99.1		6
57.5	odd	18		361.2.e.d.234.1		6
57.8	even	6		361.2.c.a.292.1		2
57.11	odd	6		361.2.c.c.292.1		2
57.14	even	18		361.2.e.e.234.1		6
57.17	odd	18		361.2.e.d.99.1		6
57.23	odd	18		361.2.e.d.54.1		6
57.26	odd	6		361.2.c.c.68.1		2
57.29	even	18		361.2.e.e.62.1		6
57.32	even	18		361.2.e.e.245.1		6
57.35	odd	18		361.2.e.d.28.1		6
57.41	even	18		361.2.e.e.28.1		6
57.44	odd	18		361.2.e.d.245.1		6
57.47	odd	18		361.2.e.d.62.1		6
57.50	even	6		361.2.c.a.68.1		2
57.53	even	18		361.2.e.e.54.1		6
57.56	even	2		361.2.a.b.1.1		1
60.59	even	2		7600.2.a.c.1.1		1
228.227	odd	2		5776.2.a.c.1.1		1
285.284	even	2		9025.2.a.d.1.1		1

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
19.2.a.a.1.1	✓	1	3.2	odd	2
171.2.a.b.1.1		1	1.1	even	1	trivial
304.2.a.f.1.1		1	12.11	even	2
361.2.a.b.1.1		1	57.56	even	2
361.2.c.a.68.1		2	57.50	even	6
361.2.c.a.292.1		2	57.8	even	6
361.2.c.c.68.1		2	57.26	odd	6
361.2.c.c.292.1		2	57.11	odd	6
361.2.e.d.28.1		6	57.35	odd	18
361.2.e.d.54.1		6	57.23	odd	18
361.2.e.d.62.1		6	57.47	odd	18
361.2.e.d.99.1		6	57.17	odd	18
361.2.e.d.234.1		6	57.5	odd	18
361.2.e.d.245.1		6	57.44	odd	18
361.2.e.e.28.1		6	57.41	even	18
361.2.e.e.54.1		6	57.53	even	18
361.2.e.e.62.1		6	57.29	even	18
361.2.e.e.99.1		6	57.2	even	18
361.2.e.e.234.1		6	57.14	even	18
361.2.e.e.245.1		6	57.32	even	18
475.2.a.b.1.1		1	15.14	odd	2
475.2.b.a.324.1		2	15.8	even	4
475.2.b.a.324.2		2	15.2	even	4
931.2.a.a.1.1		1	21.20	even	2
931.2.f.b.324.1		2	21.17	even	6
931.2.f.b.704.1		2	21.5	even	6
931.2.f.c.324.1		2	21.11	odd	6
931.2.f.c.704.1		2	21.2	odd	6
1216.2.a.b.1.1		1	24.11	even	2
1216.2.a.o.1.1		1	24.5	odd	2
2299.2.a.b.1.1		1	33.32	even	2
2736.2.a.c.1.1		1	4.3	odd	2
3211.2.a.a.1.1		1	39.38	odd	2
3249.2.a.d.1.1		1	19.18	odd	2
4275.2.a.i.1.1		1	5.4	even	2
5491.2.a.b.1.1		1	51.50	odd	2
5776.2.a.c.1.1		1	228.227	odd	2
7600.2.a.c.1.1		1	60.59	even	2
8379.2.a.j.1.1		1	7.6	odd	2
9025.2.a.d.1.1		1	285.284	even	2