Embedded newform 171.2.h.c.49.12

• Label: 171.2.h.c.49.12
• Level: $171$
• Weight: $2$
• Character: 171.49
• 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.h (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			49.12
Character	\(\chi\)	\(=\)	171.49
Dual form			171.2.h.c.7.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+1.60662 q^{2} +(1.66723 + 0.469417i) q^{3} +0.581222 q^{4} +(-1.87940 - 3.25521i) q^{5} +(2.67860 + 0.754174i) q^{6} +(2.27973 + 3.94861i) q^{7} -2.27943 q^{8} +(2.55930 + 1.56525i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 2 q^{2} + q^{3} + 34 q^{4} + 3 q^{5} - 7 q^{6} + q^{7} - 36 q^{8} + 17 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(20\)	\(154\)
\(\chi(n)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{2}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.60662			1.13605			0.568025	−	0.823011i	\(-0.307708\pi\)
							0.568025	+	0.823011i	\(0.307708\pi\)
\(3\)	1.66723	+	0.469417i	0.962574	+	0.271018i
							\(5\)	−1.87940	−	3.25521i	−0.840492	−	1.45578i	−0.889479	−	0.456976i	\(-0.848932\pi\)
0.0489864	−	0.998799i	\(-0.484401\pi\)
\(7\)	2.27973	+	3.94861i	0.861656	+	1.49243i	0.870329	+	0.492470i	\(0.163906\pi\)
							−0.00867312	+	0.999962i	\(0.502761\pi\)
\(11\)	−1.29616	−	2.24501i	−0.390806	−	0.676896i	0.601750	−	0.798684i	\(-0.294470\pi\)
							−0.992556	+	0.121789i	\(0.961137\pi\)
\(13\)	−0.537714			−0.149135			−0.0745675	−	0.997216i	\(-0.523758\pi\)
							−0.0745675	+	0.997216i	\(0.523758\pi\)
\(17\)	−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
							\(23\)	−0.208923			−0.0435635			−0.0217818	−	0.999763i	\(-0.506934\pi\)
−0.0217818	+	0.999763i	\(0.506934\pi\)
\(29\)	0.426901	−	0.739414i	0.0792735	−	0.137306i	−0.823663	−	0.567079i	\(-0.808073\pi\)
							0.902937	+	0.429774i	\(0.141407\pi\)
\(31\)	3.83524	−	6.64283i	0.688829	−	1.19309i	−0.283388	−	0.959005i	\(-0.591458\pi\)
							0.972217	−	0.234082i	\(-0.0752084\pi\)
\(37\)	4.41513			0.725843			0.362921	−	0.931820i	\(-0.381779\pi\)
							0.362921	+	0.931820i	\(0.381779\pi\)
\(41\)	−0.469658	−	0.813472i	−0.0733483	−	0.127043i	0.827019	−	0.562175i	\(-0.190035\pi\)
							−0.900367	+	0.435132i	\(0.856702\pi\)
\(43\)	3.98834			0.608217			0.304108	−	0.952637i	\(-0.401641\pi\)
							0.304108	+	0.952637i	\(0.401641\pi\)
\(47\)	1.57045	−	2.72010i	0.229073	−	0.396767i	−0.728460	−	0.685088i	\(-0.759764\pi\)
							0.957534	+	0.288321i	\(0.0930971\pi\)

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
171.2.g.c.106.5	✓	32	9.7	even	3
171.2.g.c.121.5	yes	32	19.7	even	3
171.2.h.c.7.12	yes	32	171.7	even	3	inner
171.2.h.c.49.12	yes	32	1.1	even	1	trivial
513.2.g.c.64.12		32	57.26	odd	6
513.2.g.c.505.12		32	9.2	odd	6
513.2.h.c.235.5		32	171.83	odd	6
513.2.h.c.334.5		32	3.2	odd	2