Embedded newform 177.2.f.a.11.12

• Label: 177.2.f.a.11.12
• Level: $177$
• Weight: $2$
• Character: 177.11
• Analytic conductor: $1.413$
• Analytic rank: $0$
• Dimension: $504$
• 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			11.12
Character	\(\chi\)	\(=\)	177.11
Dual form			177.2.f.a.161.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.845498 - 0.186108i) q^{2} +(-0.0436336 + 1.73150i) q^{3} +(-1.13492 + 0.525070i) q^{4} +(1.97263 + 0.547699i) q^{5} +(0.285354 + 1.47210i) q^{6} +(0.395808 - 0.374930i) q^{7} +(-2.24027 + 1.70301i) q^{8} +(-2.99619 - 0.151103i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 504 q - 27 q^{3} - 70 q^{4} - 29 q^{6} - 58 q^{7} - 19 q^{9}+O(q^{10}) \)

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	0.0942759	−	0.995546i	\(-0.469946\pi\)
							0.503581	+	0.863948i	\(0.332015\pi\)
\(3\)	−0.0436336	+	1.73150i	−0.0251919	+	0.999683i
							\(5\)	1.97263	+	0.547699i	0.882189	+	0.244939i	0.678945	−	0.734190i	\(-0.262438\pi\)
0.203244	+	0.979128i	\(0.434851\pi\)
\(7\)	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	0.988887	−	0.148672i	\(-0.0474998\pi\)
							−0.306697	+	0.951807i	\(0.599224\pi\)
\(13\)	−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	−0.284232	−	0.958756i	\(-0.591739\pi\)
							0.972737	+	0.231909i	\(0.0744972\pi\)
\(19\)	−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	0.422563	−	0.906334i	\(-0.361131\pi\)
							0.607518	+	0.794306i	\(0.292165\pi\)
\(29\)	1.72564	−	7.83968i	0.320444	−	1.45579i	−0.490897	−	0.871218i	\(-0.663331\pi\)
							0.811341	−	0.584574i	\(-0.198738\pi\)
\(31\)	−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	\(0.0964025\pi\)
							−0.996287	+	0.0860892i	\(0.972563\pi\)
\(43\)	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	\(0.0242868\pi\)
							−0.815066	+	0.579368i	\(0.803299\pi\)

(See \(a_n\) instead)

Twists

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

    

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