Embedded newform 177.3.h.a.5.7

• Label: 177.3.h.a.5.7
• Level: $177$
• Weight: $3$
• Character: 177.5
• Analytic conductor: $4.823$
• Analytic rank: $0$
• Dimension: $1064$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 177 = 3 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	177.h (of order \(58\), degree \(28\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.82290067918\)
Analytic rank:	\(0\)
Dimension:	\(1064\)
Relative dimension:	\(38\) over \(\Q(\zeta_{58})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{58}]$

Embedding invariants

Embedding label			5.7
Character	\(\chi\)	\(=\)	177.5
Dual form			177.3.h.a.71.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.912087 + 2.70698i) q^{2} +(2.96240 - 0.473468i) q^{3} +(-3.31146 - 2.51731i) q^{4} +(-6.32194 - 2.51889i) q^{5} +(-1.42030 + 8.45101i) q^{6} +(-6.08665 + 2.81599i) q^{7} +(0.377434 - 0.255907i) q^{8} +(8.55166 - 2.80521i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1064 q - 29 q^{3} + 18 q^{4} - 21 q^{6} - 46 q^{7} - 49 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(61\)	\(119\)
\(\chi(n)\)	\(e\left(\frac{3}{29}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.912087	+	2.70698i	−0.456044	+	1.35349i	0.435351	+	0.900261i	\(0.356624\pi\)
							−0.891394	+	0.453229i	\(0.850272\pi\)
\(3\)	2.96240	−	0.473468i	0.987467	−	0.157823i
							\(5\)	−6.32194	−	2.51889i	−1.26439	−	0.503778i	−0.360986	−	0.932571i	\(-0.617560\pi\)
−0.903403	+	0.428793i	\(0.858939\pi\)
\(7\)	−6.08665	+	2.81599i	−0.869522	+	0.402284i	−0.803338	−	0.595524i	\(-0.796945\pi\)
							−0.0661842	+	0.997807i	\(0.521082\pi\)
\(11\)	−16.9551	−	4.70755i	−1.54137	−	0.427959i	−0.610110	−	0.792317i	\(-0.708875\pi\)
							−0.931259	+	0.364358i	\(0.881288\pi\)
\(13\)	−4.39226	+	8.28469i	−0.337866	+	0.637284i	−0.992985	−	0.118243i	\(-0.962274\pi\)
							0.655118	+	0.755526i	\(0.272619\pi\)
\(17\)	−11.2799	+	24.3812i	−0.663526	+	1.43419i	0.224536	+	0.974466i	\(0.427913\pi\)
							−0.888063	+	0.459723i	\(0.847949\pi\)
\(19\)	2.20764	−	0.485937i	0.116191	−	0.0255756i	−0.156494	−	0.987679i	\(-0.550019\pi\)
							0.272685	+	0.962103i	\(0.412088\pi\)
\(23\)	20.8915	+	3.42499i	0.908327	+	0.148913i	0.597794	−	0.801650i	\(-0.296044\pi\)
							0.310533	+	0.950563i	\(0.399492\pi\)
\(29\)	1.66237	+	4.93373i	0.0573230	+	0.170129i	0.972489	−	0.232948i	\(-0.0748373\pi\)
							−0.915166	+	0.403077i	\(0.867941\pi\)
\(31\)	−20.0376	−	4.41061i	−0.646374	−	0.142278i	−0.120329	−	0.992734i	\(-0.538395\pi\)
							−0.526046	+	0.850456i	\(0.676326\pi\)
\(37\)	19.4432	−	28.6765i	0.525491	−	0.775041i	−0.468621	−	0.883399i	\(-0.655249\pi\)
							0.994112	+	0.108359i	\(0.0345594\pi\)
\(41\)	63.9370	−	10.4819i	1.55944	−	0.255657i	0.680556	−	0.732696i	\(-0.261738\pi\)
							0.878882	+	0.477039i	\(0.158290\pi\)
\(43\)	8.90734	+	32.0814i	0.207148	+	0.746078i	0.991983	+	0.126373i	\(0.0403338\pi\)
							−0.784835	+	0.619705i	\(0.787252\pi\)
\(47\)	−24.4046	+	9.72369i	−0.519247	+	0.206887i	−0.615013	−	0.788517i	\(-0.710849\pi\)
							0.0957659	+	0.995404i	\(0.469470\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.3.h.a.5.7	✓	1064
3.2	odd	2	inner	177.3.h.a.5.32	yes	1064
59.12	even	29	inner	177.3.h.a.71.32	yes	1064
177.71	odd	58	inner	177.3.h.a.71.7	yes	1064

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.3.h.a.5.7	✓	1064	1.1	even	1	trivial
177.3.h.a.5.32	yes	1064	3.2	odd	2	inner
177.3.h.a.71.7	yes	1064	177.71	odd	58	inner
177.3.h.a.71.32	yes	1064	59.12	even	29	inner