Embedded newform 177.4.e.b.7.1

• Label: 177.4.e.b.7.1
• Level: $177$
• Weight: $4$
• Character: 177.7
• 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			7.1
Character	\(\chi\)	\(=\)	177.7
Dual form			177.4.e.b.76.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.86839 - 4.23056i) q^{2} +(0.162417 - 2.99560i) q^{3} +(-6.70886 + 16.8380i) q^{4} +(19.1429 - 8.85647i) q^{5} +(-13.1389 + 7.90544i) q^{6} +(-6.91419 + 24.9026i) q^{7} +(50.5433 - 11.1254i) q^{8} +(-8.94724 - 0.973071i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 420 q + 2 q^{2} - 45 q^{3} - 56 q^{4} - 14 q^{5} + 6 q^{6} + 6 q^{7} + 66 q^{8} - 135 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(61\)	\(119\)
\(\chi(n)\)	\(e\left(\frac{9}{29}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.86839	−	4.23056i	−1.01413	−	1.49573i	−0.861703	−	0.507413i	\(-0.830602\pi\)
							−0.152426	−	0.988315i	\(-0.548709\pi\)
\(3\)	0.162417	−	2.99560i	0.0312571	−	0.576504i
							\(5\)	19.1429	−	8.85647i	1.71220	−	0.792147i	0.716697	−	0.697384i	\(-0.245653\pi\)
0.995500	−	0.0947625i	\(-0.0302092\pi\)
\(7\)	−6.91419	+	24.9026i	−0.373331	+	1.34462i	0.503574	+	0.863952i	\(0.332018\pi\)
							−0.876905	+	0.480664i	\(0.840396\pi\)
\(11\)	25.0461	−	23.7249i	0.686517	−	0.650303i	−0.262718	−	0.964873i	\(-0.584619\pi\)
							0.949234	+	0.314570i	\(0.101860\pi\)
\(13\)	64.9362	−	7.06224i	1.38539	−	0.150670i	0.615143	−	0.788416i	\(-0.289098\pi\)
							0.770247	+	0.637746i	\(0.220133\pi\)
\(17\)	−29.8127	−	107.375i	−0.425331	−	1.53190i	−0.797256	−	0.603642i	\(-0.793716\pi\)
							0.371925	−	0.928263i	\(-0.378698\pi\)
\(19\)	7.36338	−	5.59749i	0.0889092	−	0.0675870i	−0.559775	−	0.828644i	\(-0.689113\pi\)
							0.648685	+	0.761057i	\(0.275319\pi\)
\(23\)	13.6913	−	25.8246i	0.124123	−	0.234121i	−0.813661	−	0.581340i	\(-0.802529\pi\)
							0.937784	+	0.347218i	\(0.112874\pi\)
\(29\)	−118.909	+	175.378i	−0.761410	+	1.12300i	0.227397	+	0.973802i	\(0.426978\pi\)
							−0.988808	+	0.149194i	\(0.952332\pi\)
\(31\)	106.036	+	80.6063i	0.614342	+	0.467010i	0.865482	−	0.500940i	\(-0.167012\pi\)
							−0.251140	+	0.967951i	\(0.580805\pi\)
\(37\)	−193.555	−	42.6046i	−0.860006	−	0.189302i	−0.236991	−	0.971512i	\(-0.576161\pi\)
							−0.623015	+	0.782210i	\(0.714092\pi\)
\(41\)	−33.7223	−	63.6069i	−0.128452	−	0.242286i	0.810941	−	0.585127i	\(-0.198956\pi\)
							−0.939393	+	0.342841i	\(0.888611\pi\)
\(43\)	235.042	+	222.643i	0.833570	+	0.789600i	0.979869	−	0.199641i	\(-0.0639775\pi\)
							−0.146299	+	0.989240i	\(0.546736\pi\)
\(47\)	−168.679	−	78.0393i	−0.523498	−	0.242196i	0.140298	−	0.990109i	\(-0.455194\pi\)
							−0.663796	+	0.747913i	\(0.731056\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.4.e.b.7.1	✓	420
59.17	even	29	inner	177.4.e.b.76.1	yes	420

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.4.e.b.7.1	✓	420	1.1	even	1	trivial
177.4.e.b.76.1	yes	420	59.17	even	29	inner