Embedded newform 177.2.e.a.28.3

• Label: 177.2.e.a.28.3
• Level: $177$
• Weight: $2$
• Character: 177.28
• Analytic conductor: $1.413$
• Analytic rank: $0$
• Dimension: $140$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 177 = 3 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	177.e (of order \(29\), degree \(28\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.41335211578\)
Analytic rank:	\(0\)
Dimension:	\(140\)
Relative dimension:	\(5\) over \(\Q(\zeta_{29})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{29}]$

Embedding invariants

Embedding label			28.3
Character	\(\chi\)	\(=\)	177.28
Dual form			177.2.e.a.19.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.0292574 + 0.0551852i) q^{2} +(-0.725995 - 0.687699i) q^{3} +(1.12018 - 1.65215i) q^{4} +(-2.41343 - 0.531236i) q^{5} +(0.0167101 - 0.0601845i) q^{6} +(-2.91820 - 2.21836i) q^{7} +(0.248138 + 0.0269866i) q^{8} +(0.0541389 + 0.998533i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 140 q - q^{2} - 5 q^{3} - 9 q^{4} - 2 q^{5} - q^{6} - 2 q^{7} - 9 q^{8} - 5 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/177\mathbb{Z}\right)^\times\).

\(n\)	\(61\)	\(119\)
\(\chi(n)\)	\(e\left(\frac{10}{29}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.0292574	+	0.0551852i	0.0206881	+	0.0390219i	0.893641	−	0.448783i	\(-0.148142\pi\)
							−0.872953	+	0.487805i	\(0.837798\pi\)
\(3\)	−0.725995	−	0.687699i	−0.419154	−	0.397043i
							\(5\)	−2.41343	−	0.531236i	−1.07932	−	0.237576i	−0.360490	−	0.932763i	\(-0.617391\pi\)
−0.718829	+	0.695187i	\(0.755322\pi\)
\(7\)	−2.91820	−	2.21836i	−1.10298	−	0.838461i	−0.115046	−	0.993360i	\(-0.536702\pi\)
							−0.987930	+	0.154900i	\(0.950495\pi\)
\(11\)	−1.21445	+	3.04803i	−0.366170	+	0.919016i	0.624288	+	0.781195i	\(0.285389\pi\)
							−0.990457	+	0.137822i	\(0.955990\pi\)
\(13\)	0.167630	−	3.09175i	0.0464921	−	0.857497i	−0.878554	−	0.477643i	\(-0.841491\pi\)
							0.925046	−	0.379854i	\(-0.124026\pi\)
\(17\)	5.50074	−	4.18155i	1.33413	−	1.01418i	0.336763	−	0.941589i	\(-0.390668\pi\)
							0.997362	−	0.0725862i	\(-0.0231253\pi\)
\(19\)	4.59260	+	1.54743i	1.05362	+	0.355004i	0.792225	−	0.610229i	\(-0.208923\pi\)
							0.261390	+	0.965233i	\(0.415819\pi\)
\(23\)	−1.26009	−	0.758170i	−0.262747	−	0.158089i	0.378085	−	0.925771i	\(-0.376583\pi\)
							−0.640832	+	0.767682i	\(0.721410\pi\)
\(29\)	0.671518	−	1.26662i	0.124698	−	0.235205i	−0.813301	−	0.581843i	\(-0.802332\pi\)
							0.937999	+	0.346638i	\(0.112677\pi\)
\(31\)	5.47847	−	1.84591i	0.983963	−	0.331536i	0.219081	−	0.975707i	\(-0.429694\pi\)
							0.764882	+	0.644171i	\(0.222797\pi\)
\(37\)	−0.515008	+	0.0560105i	−0.0846668	+	0.00920807i	−0.150354	−	0.988632i	\(-0.548041\pi\)
							0.0656875	+	0.997840i	\(0.479076\pi\)
\(41\)	−4.21757	+	2.53763i	−0.658674	+	0.396311i	−0.805289	−	0.592883i	\(-0.797990\pi\)
							0.146615	+	0.989194i	\(0.453162\pi\)
\(43\)	3.09969	+	7.77963i	0.472698	+	1.18638i	0.951277	+	0.308338i	\(0.0997728\pi\)
							−0.478579	+	0.878045i	\(0.658848\pi\)
\(47\)	0.114449	−	0.0251921i	0.0166941	−	0.00367465i	−0.206616	−	0.978422i	\(-0.566245\pi\)
							0.223310	+	0.974748i	\(0.428314\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.2.e.a.28.3	yes	140
3.2	odd	2		531.2.i.c.28.3		140
59.19	even	29	inner	177.2.e.a.19.3	✓	140
177.137	odd	58		531.2.i.c.19.3		140

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.2.e.a.19.3	✓	140	59.19	even	29	inner
177.2.e.a.28.3	yes	140	1.1	even	1	trivial
531.2.i.c.19.3		140	177.137	odd	58
531.2.i.c.28.3		140	3.2	odd	2