Embedded newform 177.3.h.a.104.33

• Label: 177.3.h.a.104.33
• Level: $177$
• Weight: $3$
• Character: 177.104
• 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			104.33
Character	\(\chi\)	\(=\)	177.104
Dual form			177.3.h.a.80.33

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.62065 + 2.69355i) q^{2} +(0.850882 - 2.87680i) q^{3} +(-2.75504 + 5.19655i) q^{4} +(-0.554592 - 5.09939i) q^{5} +(9.12779 - 2.37041i) q^{6} +(11.1862 - 3.76907i) q^{7} +(-5.90649 + 0.320240i) q^{8} +(-7.55200 - 4.89564i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1064 q - 29 q^{3} + 18 q^{4} - 21 q^{6} - 46 q^{7} - 49 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(61\)	\(119\)
\(\chi(n)\)	\(e\left(\frac{24}{29}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.62065	+	2.69355i	0.810326	+	1.34677i	0.935026	+	0.354578i	\(0.115376\pi\)
							−0.124700	+	0.992194i	\(0.539797\pi\)
\(3\)	0.850882	−	2.87680i	0.283627	−	0.958935i
							\(5\)	−0.554592	−	5.09939i	−0.110918	−	1.01988i	−0.908848	−	0.417127i	\(-0.863037\pi\)
0.797930	−	0.602751i	\(-0.205929\pi\)
\(7\)	11.1862	−	3.76907i	1.59803	−	0.538439i	0.627334	−	0.778750i	\(-0.284146\pi\)
							0.970696	+	0.240311i	\(0.0772495\pi\)
\(11\)	−4.73762	−	3.21218i	−0.430692	−	0.292017i	0.326514	−	0.945192i	\(-0.394126\pi\)
							−0.757206	+	0.653176i	\(0.773436\pi\)
\(13\)	−4.17519	−	3.95495i	−0.321168	−	0.304227i	0.510081	−	0.860126i	\(-0.329615\pi\)
							−0.831249	+	0.555900i	\(0.812374\pi\)
\(17\)	−5.52274	+	16.3909i	−0.324867	+	0.964171i	0.653434	+	0.756983i	\(0.273328\pi\)
							−0.978301	+	0.207187i	\(0.933569\pi\)
\(19\)	5.56679	+	33.9559i	0.292989	+	1.78715i	0.565240	+	0.824927i	\(0.308784\pi\)
							−0.272251	+	0.962226i	\(0.587768\pi\)
\(23\)	5.69327	−	1.58073i	0.247533	−	0.0687273i	−0.141545	−	0.989932i	\(-0.545207\pi\)
							0.389079	+	0.921204i	\(0.372793\pi\)
\(29\)	−6.00669	+	9.98320i	−0.207127	+	0.344248i	−0.942955	−	0.332921i	\(-0.891966\pi\)
							0.735827	+	0.677169i	\(0.236793\pi\)
\(31\)	4.85906	−	29.6389i	0.156744	−	0.956095i	−0.784934	−	0.619579i	\(-0.787303\pi\)
							0.941678	−	0.336516i	\(-0.109248\pi\)
\(37\)	2.50735	−	46.2454i	0.0677662	−	1.24987i	−0.744912	−	0.667162i	\(-0.767509\pi\)
							0.812679	−	0.582712i	\(-0.198008\pi\)
\(41\)	−60.9482	−	16.9222i	−1.48654	−	0.412736i	−0.573024	−	0.819538i	\(-0.694230\pi\)
							−0.913516	+	0.406802i	\(0.866644\pi\)
\(43\)	34.2757	+	50.5529i	0.797110	+	1.17565i	0.981248	+	0.192749i	\(0.0617404\pi\)
							−0.184138	+	0.982900i	\(0.558949\pi\)
\(47\)	−3.49040	+	32.0937i	−0.0742638	+	0.682844i	0.896539	+	0.442965i	\(0.146073\pi\)
							−0.970803	+	0.239879i	\(0.922892\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.3.h.a.104.33	yes	1064
3.2	odd	2	inner	177.3.h.a.104.6	yes	1064
59.21	even	29	inner	177.3.h.a.80.6	✓	1064
177.80	odd	58	inner	177.3.h.a.80.33	yes	1064

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.3.h.a.80.6	✓	1064	59.21	even	29	inner
177.3.h.a.80.33	yes	1064	177.80	odd	58	inner
177.3.h.a.104.6	yes	1064	3.2	odd	2	inner
177.3.h.a.104.33	yes	1064	1.1	even	1	trivial