Embedded newform 177.2.f.a.11.7

• Label: 177.2.f.a.11.7
• Level: $177$
• Weight: $2$
• Character: 177.11
• 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			11.7
Character	\(\chi\)	\(=\)	177.11
Dual form			177.2.f.a.161.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.845498 + 0.186108i) q^{2} +(-0.511521 + 1.65479i) q^{3} +(-1.13492 + 0.525070i) q^{4} +(-1.97263 - 0.547699i) q^{5} +(0.124519 - 1.49432i) q^{6} +(0.395808 - 0.374930i) q^{7} +(2.24027 - 1.70301i) q^{8} +(-2.47669 - 1.69293i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 504 q - 27 q^{3} - 70 q^{4} - 29 q^{6} - 58 q^{7} - 19 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(61\)	\(119\)
\(\chi(n)\)	\(e\left(\frac{25}{58}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.845498	+	0.186108i	−0.597857	+	0.131598i	−0.503581	−	0.863948i	\(-0.667985\pi\)
							−0.0942759	+	0.995546i	\(0.530054\pi\)
\(3\)	−0.511521	+	1.65479i	−0.295327	+	0.955396i
							\(5\)	−1.97263	−	0.547699i	−0.882189	−	0.244939i	−0.203244	−	0.979128i	\(-0.565149\pi\)
−0.678945	+	0.734190i	\(0.737562\pi\)
\(7\)	0.395808	−	0.374930i	0.149602	−	0.141710i	−0.609239	−	0.792987i	\(-0.708525\pi\)
							0.758840	+	0.651277i	\(0.225766\pi\)
\(11\)	−2.26257	−	2.66370i	−0.682189	−	0.803135i	0.306697	−	0.951807i	\(-0.400776\pi\)
							−0.988887	+	0.148672i	\(0.952500\pi\)
\(13\)	−0.338061	−	1.00333i	−0.0937612	−	0.278273i	0.890503	−	0.454978i	\(-0.150353\pi\)
							−0.984264	+	0.176705i	\(0.943456\pi\)
\(17\)	−2.83878	+	2.99686i	−0.688505	+	0.726846i	−0.972737	−	0.231909i	\(-0.925503\pi\)
							0.284232	+	0.958756i	\(0.408261\pi\)
\(19\)	−0.103274	−	0.259199i	−0.0236928	−	0.0594644i	0.916654	−	0.399681i	\(-0.130879\pi\)
							−0.940347	+	0.340216i	\(0.889500\pi\)
\(23\)	−4.94010	+	0.537268i	−1.03008	+	0.112028i	−0.607518	−	0.794306i	\(-0.707835\pi\)
							−0.422563	+	0.906334i	\(0.638869\pi\)
\(29\)	−1.72564	+	7.83968i	−0.320444	+	1.45579i	0.490897	+	0.871218i	\(0.336669\pi\)
							−0.811341	+	0.584574i	\(0.801262\pi\)
\(31\)	−7.50474	−	2.99016i	−1.34789	−	0.537049i	−0.419197	−	0.907895i	\(-0.637688\pi\)
							−0.928694	+	0.370847i	\(0.879068\pi\)
\(37\)	0.225030	−	0.296022i	0.0369947	−	0.0486657i	−0.777224	−	0.629224i	\(-0.783373\pi\)
							0.814218	+	0.580559i	\(0.197166\pi\)
\(41\)	0.267646	−	2.46096i	0.0417993	−	0.384338i	−0.954488	−	0.298249i	\(-0.903597\pi\)
							0.996287	−	0.0860892i	\(-0.0274370\pi\)
\(43\)	5.32906	+	4.52655i	0.812674	+	0.690292i	0.954044	−	0.299665i	\(-0.0968750\pi\)
							−0.141370	+	0.989957i	\(0.545151\pi\)
\(47\)	−1.24790	−	4.49452i	−0.182025	−	0.655594i	−0.997091	−	0.0762254i	\(-0.975713\pi\)
							0.815066	−	0.579368i	\(-0.196701\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.2.f.a.11.7	✓	504
3.2	odd	2	inner	177.2.f.a.11.12	yes	504
59.43	odd	58	inner	177.2.f.a.161.12	yes	504
177.161	even	58	inner	177.2.f.a.161.7	yes	504

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.2.f.a.11.7	✓	504	1.1	even	1	trivial
177.2.f.a.11.12	yes	504	3.2	odd	2	inner
177.2.f.a.161.7	yes	504	177.161	even	58	inner
177.2.f.a.161.12	yes	504	59.43	odd	58	inner