Embedded newform 177.2.f.a.14.11

• Label: 177.2.f.a.14.11
• Level: $177$
• Weight: $2$
• Character: 177.14
• Analytic conductor: $1.413$
• Analytic rank: $0$
• Dimension: $504$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 177 = 3 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	177.f (of order \(58\), degree \(28\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.41335211578\)
Analytic rank:	\(0\)
Dimension:	\(504\)
Relative dimension:	\(18\) over \(\Q(\zeta_{58})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{58}]$

Embedding invariants

Embedding label			14.11
Character	\(\chi\)	\(=\)	177.14
Dual form			177.2.f.a.38.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.277115 - 0.166734i) q^{2} +(1.51687 + 0.836130i) q^{3} +(-0.887825 + 1.67461i) q^{4} +(0.0154306 + 0.141882i) q^{5} +(0.559758 - 0.0212100i) q^{6} +(-0.858133 + 0.289139i) q^{7} +(0.0682046 + 1.25796i) q^{8} +(1.60177 + 2.53660i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 504 q - 27 q^{3} - 70 q^{4} - 29 q^{6} - 58 q^{7} - 19 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(61\)	\(119\)
\(\chi(n)\)	\(e\left(\frac{19}{58}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.277115	−	0.166734i	0.195950	−	0.117899i	−0.414198	−	0.910187i	\(-0.635938\pi\)
							0.610147	+	0.792288i	\(0.291110\pi\)
\(3\)	1.51687	+	0.836130i	0.875764	+	0.482740i
							\(5\)	0.0154306	+	0.141882i	0.00690077	+	0.0634516i	0.997082	−	0.0763382i	\(-0.0243229\pi\)
−0.990181	+	0.139790i	\(0.955357\pi\)
\(7\)	−0.858133	+	0.289139i	−0.324344	+	0.109284i	−0.476763	−	0.879032i	\(-0.658190\pi\)
							0.152420	+	0.988316i	\(0.451294\pi\)
\(11\)	1.74490	−	2.57353i	0.526107	−	0.775949i	−0.468072	−	0.883690i	\(-0.655051\pi\)
							0.994179	+	0.107741i	\(0.0343617\pi\)
\(13\)	−0.152764	+	0.161271i	−0.0423691	+	0.0447286i	−0.746836	−	0.665009i	\(-0.768428\pi\)
							0.704466	+	0.709737i	\(0.251186\pi\)
\(17\)	0.541626	−	1.60749i	0.131364	−	0.389873i	−0.861705	−	0.507410i	\(-0.830603\pi\)
							0.993069	+	0.117536i	\(0.0374997\pi\)
\(19\)	0.332767	+	2.02979i	0.0763419	+	0.465665i	0.997137	+	0.0756181i	\(0.0240930\pi\)
							−0.920795	+	0.390047i	\(0.872459\pi\)
\(23\)	−2.08199	−	7.49865i	−0.434125	−	1.56358i	−0.780299	−	0.625406i	\(-0.784933\pi\)
							0.346175	−	0.938170i	\(-0.387480\pi\)
\(29\)	1.29490	−	2.15214i	0.240457	−	0.399643i	−0.713003	−	0.701161i	\(-0.752666\pi\)
							0.953461	+	0.301517i	\(0.0974932\pi\)
\(31\)	3.42768	+	0.561939i	0.615629	+	0.100927i	0.461525	−	0.887127i	\(-0.347303\pi\)
							0.154104	+	0.988055i	\(0.450751\pi\)
\(37\)	−4.70214	−	0.254943i	−0.773028	−	0.0419124i	−0.336595	−	0.941649i	\(-0.609275\pi\)
							−0.436432	+	0.899737i	\(0.643758\pi\)
\(41\)	−7.40832	−	2.05691i	−1.15698	−	0.321235i	−0.364545	−	0.931186i	\(-0.618775\pi\)
							−0.792439	+	0.609951i	\(0.791189\pi\)
\(43\)	1.53414	−	1.04017i	0.233954	−	0.158625i	−0.438577	−	0.898693i	\(-0.644517\pi\)
							0.672531	+	0.740069i	\(0.265207\pi\)
\(47\)	−1.35308	−	0.147156i	−0.197367	−	0.0214650i	0.00890144	−	0.999960i	\(-0.497167\pi\)
							−0.206269	+	0.978495i	\(0.566132\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.2.f.a.14.11	yes	504
3.2	odd	2	inner	177.2.f.a.14.8	✓	504
59.38	odd	58	inner	177.2.f.a.38.8	yes	504
177.38	even	58	inner	177.2.f.a.38.11	yes	504

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.2.f.a.14.8	✓	504	3.2	odd	2	inner
177.2.f.a.14.11	yes	504	1.1	even	1	trivial
177.2.f.a.38.8	yes	504	59.38	odd	58	inner
177.2.f.a.38.11	yes	504	177.38	even	58	inner