Embedded newform 177.2.f.a.38.11

• Label: 177.2.f.a.38.11
• Level: $177$
• Weight: $2$
• Character: 177.38
• 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			38.11
Character	\(\chi\)	\(=\)	177.38
Dual form			177.2.f.a.14.11

$q$-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 \(\left(\mathbb{Z}/177\mathbb{Z}\right)^\times\).

\(n\)	\(61\)	\(119\)
\(\chi(n)\)	\(e\left(\frac{39}{58}\right)\)	\(-1\)

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.2.f.a.14.8	✓	504	59.14	odd	58	inner
177.2.f.a.14.11	yes	504	177.14	even	58	inner
177.2.f.a.38.8	yes	504	3.2	odd	2	inner
177.2.f.a.38.11	yes	504	1.1	even	1	trivial