Embedded newform 177.3.h.a.71.10

• Label: 177.3.h.a.71.10
• Level: $177$
• Weight: $3$
• Character: 177.71
• 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			71.10
Character	\(\chi\)	\(=\)	177.71
Dual form			177.3.h.a.5.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.792761 - 2.35283i) q^{2} +(-2.47356 + 1.69749i) q^{3} +(-1.72297 + 1.30976i) q^{4} +(-7.18639 + 2.86332i) q^{5} +(5.95485 + 4.47417i) q^{6} +(10.9013 + 5.04348i) q^{7} +(-3.77238 - 2.55774i) q^{8} +(3.23704 - 8.39771i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1064 q - 29 q^{3} + 18 q^{4} - 21 q^{6} - 46 q^{7} - 49 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(61\)	\(119\)
\(\chi(n)\)	\(e\left(\frac{26}{29}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.792761	−	2.35283i	−0.396380	−	1.17641i	−0.941197	−	0.337859i	\(-0.890297\pi\)
							0.544816	−	0.838555i	\(-0.316599\pi\)
\(3\)	−2.47356	+	1.69749i	−0.824521	+	0.565831i
							\(5\)	−7.18639	+	2.86332i	−1.43728	+	0.572664i	−0.953195	−	0.302357i	\(-0.902226\pi\)
−0.484084	+	0.875022i	\(0.660847\pi\)
\(7\)	10.9013	+	5.04348i	1.55733	+	0.720497i	0.994368	−	0.105986i	\(-0.0337998\pi\)
							0.562962	+	0.826483i	\(0.309662\pi\)
\(11\)	12.6695	−	3.51766i	1.15177	−	0.319787i	0.361395	−	0.932413i	\(-0.382300\pi\)
							0.790373	+	0.612626i	\(0.209887\pi\)
\(13\)	−8.60974	−	16.2397i	−0.662288	−	1.24921i	−0.956015	−	0.293318i	\(-0.905241\pi\)
							0.293727	−	0.955889i	\(-0.405104\pi\)
\(17\)	−0.124018	−	0.268061i	−0.00729519	−	0.0157683i	0.903894	−	0.427757i	\(-0.140696\pi\)
							−0.911189	+	0.411989i	\(0.864834\pi\)
\(19\)	20.6385	+	4.54288i	1.08624	+	0.239099i	0.721781	−	0.692121i	\(-0.243324\pi\)
							0.364457	+	0.931220i	\(0.381255\pi\)
\(23\)	31.7535	−	5.20572i	1.38059	−	0.226335i	0.574679	−	0.818379i	\(-0.305127\pi\)
							0.805906	+	0.592043i	\(0.201679\pi\)
\(29\)	4.51252	−	13.3927i	0.155604	−	0.461817i	−0.841289	−	0.540586i	\(-0.818203\pi\)
							0.996893	+	0.0787693i	\(0.0250991\pi\)
\(31\)	8.57597	−	1.88771i	0.276644	−	0.0608940i	−0.0744812	−	0.997222i	\(-0.523730\pi\)
							0.351126	+	0.936328i	\(0.385799\pi\)
\(37\)	−15.2154	−	22.4410i	−0.411226	−	0.606513i	0.564520	−	0.825420i	\(-0.309061\pi\)
							−0.975746	+	0.218907i	\(0.929751\pi\)
\(41\)	34.3195	+	5.62640i	0.837061	+	0.137229i	0.565042	−	0.825062i	\(-0.308860\pi\)
							0.272020	+	0.962292i	\(0.412308\pi\)
\(43\)	7.45016	−	26.8331i	0.173260	−	0.624025i	−0.825004	−	0.565126i	\(-0.808827\pi\)
							0.998264	−	0.0588984i	\(-0.0187588\pi\)
\(47\)	40.8575	+	16.2791i	0.869308	+	0.346364i	0.761778	−	0.647838i	\(-0.224327\pi\)
							0.107530	+	0.994202i	\(0.465706\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.3.h.a.71.10	yes	1064
3.2	odd	2	inner	177.3.h.a.71.29	yes	1064
59.5	even	29	inner	177.3.h.a.5.29	yes	1064
177.5	odd	58	inner	177.3.h.a.5.10	✓	1064

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.3.h.a.5.10	✓	1064	177.5	odd	58	inner
177.3.h.a.5.29	yes	1064	59.5	even	29	inner
177.3.h.a.71.10	yes	1064	1.1	even	1	trivial
177.3.h.a.71.29	yes	1064	3.2	odd	2	inner