Embedded newform 177.4.f.a.89.13

• Label: 177.4.f.a.89.13
• Level: $177$
• Weight: $4$
• Character: 177.89
• Analytic conductor: $10.443$
• Analytic rank: $0$
• Dimension: $1624$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 177 = 3 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	177.f (of order \(58\), degree \(28\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(10.4433380710\)
Analytic rank:	\(0\)
Dimension:	\(1624\)
Relative dimension:	\(58\) over \(\Q(\zeta_{58})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{58}]$

Embedding invariants

Embedding label			89.13
Character	\(\chi\)	\(=\)	177.89
Dual form			177.4.f.a.2.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.194915 - 3.59500i) q^{2} +(3.72957 + 3.61806i) q^{3} +(-4.93296 + 0.536492i) q^{4} +(-1.29349 - 3.83895i) q^{5} +(12.2800 - 14.1130i) q^{6} +(12.2031 - 17.9982i) q^{7} +(-1.76949 - 10.7934i) q^{8} +(0.819319 + 26.9876i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1624 q - 21 q^{3} - 278 q^{4} - 29 q^{6} - 42 q^{7} - 25 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(61\)	\(119\)
\(\chi(n)\)	\(e\left(\frac{57}{58}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.194915	−	3.59500i	−0.0689130	−	1.27103i	−0.804646	−	0.593755i	\(-0.797645\pi\)
							0.735733	−	0.677272i	\(-0.236838\pi\)
\(3\)	3.72957	+	3.61806i	0.717755	+	0.696295i
							\(5\)	−1.29349	−	3.83895i	−0.115693	−	0.343366i	0.874125	−	0.485701i	\(-0.161436\pi\)
−0.989819	+	0.142335i	\(0.954539\pi\)
\(7\)	12.2031	−	17.9982i	0.658903	−	0.971809i	−0.340598	−	0.940209i	\(-0.610629\pi\)
							0.999501	−	0.0315999i	\(-0.0100602\pi\)
\(11\)	0.0389942	+	0.00858328i	0.00106884	+	0.000235269i	0.215505	−	0.976503i	\(-0.430860\pi\)
							−0.214436	+	0.976738i	\(0.568791\pi\)
\(13\)	48.9858	+	41.6089i	1.04509	+	0.887710i	0.993823	−	0.110975i	\(-0.0353973\pi\)
							0.0512698	+	0.998685i	\(0.483673\pi\)
\(17\)	69.5135	−	47.1313i	0.991736	−	0.672414i	0.0465396	−	0.998916i	\(-0.485181\pi\)
							0.945196	+	0.326503i	\(0.105870\pi\)
\(19\)	−58.6727	−	110.668i	−0.708444	−	1.33627i	−0.932315	−	0.361648i	\(-0.882214\pi\)
							0.223871	−	0.974619i	\(-0.428131\pi\)
\(23\)	−117.553	−	111.353i	−1.06572	−	1.00951i	−0.999932	−	0.0116716i	\(-0.996285\pi\)
							−0.0657897	−	0.997834i	\(-0.520957\pi\)
\(29\)	112.692	+	6.11001i	0.721602	+	0.0391241i	0.411282	−	0.911508i	\(-0.365081\pi\)
							0.310320	+	0.950632i	\(0.399564\pi\)
\(31\)	64.4505	+	34.1695i	0.373408	+	0.197968i	0.644534	−	0.764575i	\(-0.277051\pi\)
							−0.271126	+	0.962544i	\(0.587396\pi\)
\(37\)	−419.849	−	68.8307i	−1.86548	−	0.305830i	−0.879915	−	0.475132i	\(-0.842400\pi\)
							−0.985564	+	0.169302i	\(0.945849\pi\)
\(41\)	357.050	+	376.933i	1.36004	+	1.43578i	0.788203	+	0.615415i	\(0.211012\pi\)
							0.571840	+	0.820365i	\(0.306230\pi\)
\(43\)	−0.0374982	−	0.170356i	−0.000132986	−	0.000604163i	0.976554	−	0.215273i	\(-0.0690640\pi\)
							−0.976687	+	0.214668i	\(0.931133\pi\)
\(47\)	209.966	+	70.7458i	0.651632	+	0.219560i	0.625652	−	0.780102i	\(-0.284833\pi\)
							0.0259798	+	0.999662i	\(0.491729\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.4.f.a.89.13	yes	1624
3.2	odd	2	inner	177.4.f.a.89.46	yes	1624
59.2	odd	58	inner	177.4.f.a.2.46	yes	1624
177.2	even	58	inner	177.4.f.a.2.13	✓	1624

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.4.f.a.2.13	✓	1624	177.2	even	58	inner
177.4.f.a.2.46	yes	1624	59.2	odd	58	inner
177.4.f.a.89.13	yes	1624	1.1	even	1	trivial
177.4.f.a.89.46	yes	1624	3.2	odd	2	inner