Embedded newform 177.3.h.a.104.29

• Label: 177.3.h.a.104.29
• 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			104.29
Character	\(\chi\)	\(=\)	177.104
Dual form			177.3.h.a.80.29

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.05075 + 1.74636i) q^{2} +(2.53757 - 1.60023i) q^{3} +(-0.0720647 + 0.135928i) q^{4} +(-0.320689 - 2.94868i) q^{5} +(5.46093 + 2.75007i) q^{6} +(-3.10019 + 1.04458i) q^{7} +(7.82734 - 0.424386i) q^{8} +(3.87854 - 8.12139i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1064 q - 29 q^{3} + 18 q^{4} - 21 q^{6} - 46 q^{7} - 49 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.05075	+	1.74636i	0.525375	+	0.873180i	0.999975	−	0.00702129i	\(-0.00223496\pi\)
							−0.474600	+	0.880201i	\(0.657407\pi\)
\(3\)	2.53757	−	1.60023i	0.845857	−	0.533409i
							\(5\)	−0.320689	−	2.94868i	−0.0641377	−	0.589737i	−0.981470	−	0.191615i	\(-0.938628\pi\)
0.917332	−	0.398122i	\(-0.130338\pi\)
\(7\)	−3.10019	+	1.04458i	−0.442885	+	0.149225i	−0.531904	−	0.846805i	\(-0.678523\pi\)
							0.0890193	+	0.996030i	\(0.471627\pi\)
\(11\)	−4.65195	−	3.15410i	−0.422904	−	0.286736i	0.331113	−	0.943591i	\(-0.392576\pi\)
							−0.754017	+	0.656855i	\(0.771886\pi\)
\(13\)	16.6757	+	15.7960i	1.28274	+	1.21508i	0.963593	+	0.267375i	\(0.0861561\pi\)
							0.319150	+	0.947704i	\(0.396603\pi\)
\(17\)	−0.371583	+	1.10282i	−0.0218578	+	0.0648717i	−0.958027	−	0.286679i	\(-0.907449\pi\)
							0.936169	+	0.351550i	\(0.114345\pi\)
\(19\)	−3.91419	−	23.8755i	−0.206010	−	1.25660i	−0.863967	−	0.503549i	\(-0.832027\pi\)
							0.657957	−	0.753056i	\(-0.271421\pi\)
\(23\)	−33.8573	+	9.40044i	−1.47206	+	0.408715i	−0.908683	−	0.417487i	\(-0.862911\pi\)
							−0.563375	+	0.826202i	\(0.690497\pi\)
\(29\)	−17.3004	+	28.7534i	−0.596564	+	0.991497i	0.400601	+	0.916253i	\(0.368801\pi\)
							−0.997165	+	0.0752446i	\(0.976026\pi\)
\(31\)	−9.22122	+	56.2470i	−0.297459	+	1.81442i	0.239599	+	0.970872i	\(0.422984\pi\)
							−0.537057	+	0.843546i	\(0.680464\pi\)
\(37\)	−2.70920	+	49.9683i	−0.0732217	+	1.35050i	0.698991	+	0.715131i	\(0.253633\pi\)
							−0.772212	+	0.635365i	\(0.780850\pi\)
\(41\)	−58.0587	−	16.1199i	−1.41607	−	0.393169i	−0.526578	−	0.850127i	\(-0.676525\pi\)
							−0.889489	+	0.456957i	\(0.848939\pi\)
\(43\)	−0.712545	−	1.05093i	−0.0165708	−	0.0244401i	0.819313	−	0.573346i	\(-0.194355\pi\)
							−0.835884	+	0.548906i	\(0.815045\pi\)
\(47\)	3.92064	−	36.0497i	0.0834178	−	0.767014i	−0.875479	−	0.483257i	\(-0.839454\pi\)
							0.958896	−	0.283757i	\(-0.0915808\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.3.h.a.104.29	yes	1064
3.2	odd	2	inner	177.3.h.a.104.10	yes	1064
59.21	even	29	inner	177.3.h.a.80.10	✓	1064
177.80	odd	58	inner	177.3.h.a.80.29	yes	1064

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.3.h.a.80.10	✓	1064	59.21	even	29	inner
177.3.h.a.80.29	yes	1064	177.80	odd	58	inner
177.3.h.a.104.10	yes	1064	3.2	odd	2	inner
177.3.h.a.104.29	yes	1064	1.1	even	1	trivial