Embedded newform 177.4.e.b.4.3

• Label: 177.4.e.b.4.3
• Level: $177$
• Weight: $4$
• Character: 177.4
• Analytic conductor: $10.443$
• Analytic rank: $0$
• Dimension: $420$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 177 = 3 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	177.e (of order \(29\), degree \(28\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(10.4433380710\)
Analytic rank:	\(0\)
Dimension:	\(420\)
Relative dimension:	\(15\) over \(\Q(\zeta_{29})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{29}]$

Embedding invariants

Embedding label			4.3
Character	\(\chi\)	\(=\)	177.4
Dual form			177.4.e.b.133.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.97824 - 0.432660i) q^{2} +(1.94216 - 2.28649i) q^{3} +(7.82627 + 1.72269i) q^{4} +(6.73681 + 5.12119i) q^{5} +(-8.71565 + 8.25591i) q^{6} +(-6.58976 + 16.5390i) q^{7} +(-0.0516573 - 0.0174054i) q^{8} +(-1.45604 - 8.88144i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 420 q + 2 q^{2} - 45 q^{3} - 56 q^{4} - 14 q^{5} + 6 q^{6} + 6 q^{7} + 66 q^{8} - 135 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(61\)	\(119\)
\(\chi(n)\)	\(e\left(\frac{1}{29}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.97824	−	0.432660i	−1.40652	−	0.152968i	−0.626842	−	0.779147i	\(-0.715653\pi\)
							−0.779680	+	0.626178i	\(0.784618\pi\)
\(3\)	1.94216	−	2.28649i	0.373769	−	0.440034i
							\(5\)	6.73681	+	5.12119i	0.602559	+	0.458053i	0.861442	−	0.507855i	\(-0.169561\pi\)
−0.258884	+	0.965909i	\(0.583355\pi\)
\(7\)	−6.58976	+	16.5390i	−0.355814	+	0.893025i	0.636725	+	0.771091i	\(0.280289\pi\)
							−0.992538	+	0.121934i	\(0.961091\pi\)
\(11\)	−42.8086	+	19.8054i	−1.17339	+	0.542868i	−0.907055	−	0.421012i	\(-0.861675\pi\)
							−0.266335	+	0.963880i	\(0.585813\pi\)
\(13\)	6.69383	−	40.8306i	0.142810	−	0.871104i	−0.814101	−	0.580724i	\(-0.802770\pi\)
							0.956911	−	0.290381i	\(-0.0937820\pi\)
\(17\)	20.0414	+	50.3002i	0.285927	+	0.717623i	0.999849	+	0.0173975i	\(0.00553807\pi\)
							−0.713921	+	0.700226i	\(0.753083\pi\)
\(19\)	−45.9031	−	67.7020i	−0.554258	−	0.817469i	0.442597	−	0.896721i	\(-0.354057\pi\)
							−0.996855	+	0.0792516i	\(0.974747\pi\)
\(23\)	−1.84786	+	34.0818i	−0.0167524	+	0.308981i	0.978134	+	0.207977i	\(0.0666880\pi\)
							−0.994886	+	0.101003i	\(0.967795\pi\)
\(29\)	−242.890	+	26.4158i	−1.55529	+	0.169148i	−0.845003	−	0.534762i	\(-0.820401\pi\)
							−0.710289	+	0.703910i	\(0.751436\pi\)
\(31\)	−121.895	+	179.782i	−0.706225	+	1.04160i	0.290187	+	0.956970i	\(0.406282\pi\)
							−0.996412	+	0.0846338i	\(0.973028\pi\)
\(37\)	9.02428	−	3.04063i	0.0400968	−	0.0135102i	−0.299182	−	0.954196i	\(-0.596714\pi\)
							0.339278	+	0.940686i	\(0.389817\pi\)
\(41\)	25.0649	+	462.296i	0.0954753	+	1.76094i	0.518740	+	0.854932i	\(0.326401\pi\)
							−0.423264	+	0.906006i	\(0.639116\pi\)
\(43\)	−371.586	−	171.914i	−1.31782	−	0.609690i	−0.370162	−	0.928967i	\(-0.620698\pi\)
							−0.947661	+	0.319277i	\(0.896560\pi\)
\(47\)	−492.476	+	374.371i	−1.52840	+	1.16186i	−0.594631	+	0.803999i	\(0.702702\pi\)
							−0.933774	+	0.357864i	\(0.883505\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.4.e.b.4.3	✓	420
59.15	even	29	inner	177.4.e.b.133.3	yes	420

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.4.e.b.4.3	✓	420	1.1	even	1	trivial
177.4.e.b.133.3	yes	420	59.15	even	29	inner