Embedded newform 177.3.h.a.104.22

• Label: 177.3.h.a.104.22
• 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			104.22
Character	\(\chi\)	\(=\)	177.104
Dual form			177.3.h.a.80.22

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.328667 + 0.546249i) q^{2} +(0.396308 - 2.97371i) q^{3} +(1.68327 - 3.17498i) q^{4} +(0.662994 + 6.09613i) q^{5} +(1.75464 - 0.760877i) q^{6} +(7.88513 - 2.65681i) q^{7} +(4.83383 - 0.262083i) q^{8} +(-8.68588 - 2.35701i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1064 q - 29 q^{3} + 18 q^{4} - 21 q^{6} - 46 q^{7} - 49 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.328667	+	0.546249i	0.164333	+	0.273124i	0.928072	−	0.372402i	\(-0.121466\pi\)
							−0.763738	+	0.645526i	\(0.776638\pi\)
\(3\)	0.396308	−	2.97371i	0.132103	−	0.991236i
							\(5\)	0.662994	+	6.09613i	0.132599	+	1.21923i	0.851591	+	0.524207i	\(0.175638\pi\)
−0.718992	+	0.695019i	\(0.755396\pi\)
\(7\)	7.88513	−	2.65681i	1.12645	−	0.379544i	0.306436	−	0.951891i	\(-0.400864\pi\)
							0.820011	+	0.572347i	\(0.193967\pi\)
\(11\)	2.49591	+	1.69227i	0.226901	+	0.153843i	0.669341	−	0.742955i	\(-0.266577\pi\)
							−0.442440	+	0.896798i	\(0.645887\pi\)
\(13\)	−0.610518	−	0.578314i	−0.0469629	−	0.0444857i	0.663858	−	0.747858i	\(-0.268918\pi\)
							−0.710821	+	0.703373i	\(0.751676\pi\)
\(17\)	5.17053	−	15.3456i	0.304149	−	0.902681i	−0.681195	−	0.732102i	\(-0.738540\pi\)
							0.985344	−	0.170579i	\(-0.0545639\pi\)
\(19\)	−1.70363	−	10.3917i	−0.0896645	−	0.546929i	−0.993011	−	0.118019i	\(-0.962346\pi\)
							0.903347	−	0.428911i	\(-0.141102\pi\)
\(23\)	−4.13895	+	1.14917i	−0.179954	+	0.0499641i	−0.356337	−	0.934358i	\(-0.615974\pi\)
							0.176382	+	0.984322i	\(0.443561\pi\)
\(29\)	−9.80783	+	16.3007i	−0.338201	+	0.562095i	−0.978555	−	0.205985i	\(-0.933960\pi\)
							0.640354	+	0.768080i	\(0.278788\pi\)
\(31\)	−7.81429	+	47.6651i	−0.252074	+	1.53758i	0.492660	+	0.870222i	\(0.336025\pi\)
							−0.744734	+	0.667361i	\(0.767424\pi\)
\(37\)	0.869744	−	16.0415i	0.0235066	−	0.433554i	−0.962966	−	0.269624i	\(-0.913100\pi\)
							0.986472	−	0.163929i	\(-0.0524168\pi\)
\(41\)	13.3053	+	3.69420i	0.324520	+	0.0901024i	0.425968	−	0.904738i	\(-0.359934\pi\)
							−0.101448	+	0.994841i	\(0.532348\pi\)
\(43\)	28.5351	+	42.0861i	0.663607	+	0.978748i	0.999323	+	0.0367843i	\(0.0117114\pi\)
							−0.335716	+	0.941963i	\(0.608978\pi\)
\(47\)	−1.68036	+	15.4507i	−0.0357524	+	0.328738i	0.962579	+	0.271000i	\(0.0873543\pi\)
							−0.998332	+	0.0577379i	\(0.981611\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.3.h.a.104.22	yes	1064
3.2	odd	2	inner	177.3.h.a.104.17	yes	1064
59.21	even	29	inner	177.3.h.a.80.17	✓	1064
177.80	odd	58	inner	177.3.h.a.80.22	yes	1064

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.3.h.a.80.17	✓	1064	59.21	even	29	inner
177.3.h.a.80.22	yes	1064	177.80	odd	58	inner
177.3.h.a.104.17	yes	1064	3.2	odd	2	inner
177.3.h.a.104.22	yes	1064	1.1	even	1	trivial