Embedded newform 171.2.g.c.106.5

• Label: 171.2.g.c.106.5
• Level: $171$
• Weight: $2$
• Character: 171.106
• Analytic conductor: $1.365$
• Analytic rank: $0$
• Dimension: $32$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 171 = 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	171.g (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.36544187456\)
Analytic rank:	\(0\)
Dimension:	\(32\)
Relative dimension:	\(16\) over \(\Q(\zeta_{3})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			106.5
Character	\(\chi\)	\(=\)	171.106
Dual form			171.2.g.c.121.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.803309 - 1.39137i) q^{2} +(-1.24014 - 1.20915i) q^{3} +(-0.290611 + 0.503353i) q^{4} +3.75880 q^{5} +(-0.686165 + 2.69682i) q^{6} +(2.27973 - 3.94861i) q^{7} -2.27943 q^{8} +(0.0758986 + 2.99904i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + q^{2} - 2 q^{3} - 17 q^{4} - 6 q^{5} + 2 q^{6} + q^{7} - 36 q^{8} - 10 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/171\mathbb{Z}\right)^\times\).

\(n\)	\(20\)	\(154\)
\(\chi(n)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{2}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.803309	−	1.39137i	−0.568025	−	0.983849i	−0.996761	−	0.0804188i	\(-0.974374\pi\)
							0.428736	−	0.903430i	\(-0.358959\pi\)
\(3\)	−1.24014	−	1.20915i	−0.715996	−	0.698105i
							\(5\)	3.75880			1.68098			0.840492	−	0.541823i	\(-0.182266\pi\)
0.840492	+	0.541823i	\(0.182266\pi\)
\(7\)	2.27973	−	3.94861i	0.861656	−	1.49243i	−0.00867312	−	0.999962i	\(-0.502761\pi\)
							0.870329	−	0.492470i	\(-0.163906\pi\)
\(11\)	−1.29616	+	2.24501i	−0.390806	+	0.676896i	−0.992556	−	0.121789i	\(-0.961137\pi\)
							0.601750	+	0.798684i	\(0.294470\pi\)
\(13\)	0.268857	−	0.465674i	0.0745675	−	0.129155i	−0.826331	−	0.563185i	\(-0.809576\pi\)
							0.900898	+	0.434031i	\(0.142909\pi\)
\(17\)	−1.73001	+	2.99646i	−0.419588	+	0.726748i	−0.995898	−	0.0904832i	\(-0.971159\pi\)
							0.576310	+	0.817231i	\(0.304492\pi\)
\(19\)	−4.01417	−	1.69894i	−0.920915	−	0.389764i
							\(23\)	0.104462	−	0.180933i	0.0217818	−	0.0377271i	−0.854929	−	0.518745i	\(-0.826399\pi\)
0.876711	+	0.481018i	\(0.159733\pi\)
\(29\)	−0.853801			−0.158547			−0.0792735	−	0.996853i	\(-0.525260\pi\)
							−0.0792735	+	0.996853i	\(0.525260\pi\)
\(31\)	3.83524	+	6.64283i	0.688829	+	1.19309i	0.972217	+	0.234082i	\(0.0752084\pi\)
							−0.283388	+	0.959005i	\(0.591458\pi\)
\(37\)	4.41513			0.725843			0.362921	−	0.931820i	\(-0.381779\pi\)
							0.362921	+	0.931820i	\(0.381779\pi\)
\(41\)	0.939316			0.146697			0.0733483	−	0.997306i	\(-0.476632\pi\)
							0.0733483	+	0.997306i	\(0.476632\pi\)
\(43\)	−1.99417	−	3.45401i	−0.304108	−	0.526731i	0.672954	−	0.739684i	\(-0.265025\pi\)
							−0.977062	+	0.212953i	\(0.931692\pi\)
\(47\)	−3.14090			−0.458147			−0.229073	−	0.973409i	\(-0.573570\pi\)
							−0.229073	+	0.973409i	\(0.573570\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	171.2.g.c.106.5	✓	32
3.2	odd	2		513.2.g.c.505.12		32
9.4	even	3		171.2.h.c.49.12	yes	32
9.5	odd	6		513.2.h.c.334.5		32
19.7	even	3		171.2.h.c.7.12	yes	32
57.26	odd	6		513.2.h.c.235.5		32
171.121	even	3	inner	171.2.g.c.121.5	yes	32
171.140	odd	6		513.2.g.c.64.12		32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
171.2.g.c.106.5	✓	32	1.1	even	1	trivial
171.2.g.c.121.5	yes	32	171.121	even	3	inner
171.2.h.c.7.12	yes	32	19.7	even	3
171.2.h.c.49.12	yes	32	9.4	even	3
513.2.g.c.64.12		32	171.140	odd	6
513.2.g.c.505.12		32	3.2	odd	2
513.2.h.c.235.5		32	57.26	odd	6
513.2.h.c.334.5		32	9.5	odd	6