Embedded newform 177.3.h.a.107.26

• Label: 177.3.h.a.107.26
• Level: $177$
• Weight: $3$
• Character: 177.107
• 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			107.26
Character	\(\chi\)	\(=\)	177.107
Dual form			177.3.h.a.134.26

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.290405 + 1.31932i) q^{2} +(-2.99908 + 0.0742855i) q^{3} +(1.97402 - 0.913280i) q^{4} +(-6.07537 - 1.68682i) q^{5} +(-0.968954 - 3.93518i) q^{6} +(0.563809 - 0.534068i) q^{7} +(5.04831 + 6.64094i) q^{8} +(8.98896 - 0.445576i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1064 q - 29 q^{3} + 18 q^{4} - 21 q^{6} - 46 q^{7} - 49 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(61\)	\(119\)
\(\chi(n)\)	\(e\left(\frac{27}{29}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.290405	+	1.31932i	0.145202	+	0.659662i	0.991840	+	0.127491i	\(0.0406924\pi\)
							−0.846637	+	0.532170i	\(0.821377\pi\)
\(3\)	−2.99908	+	0.0742855i	−0.999693	+	0.0247618i
							\(5\)	−6.07537	−	1.68682i	−1.21507	−	0.337363i	−0.399888	−	0.916564i	\(-0.630951\pi\)
−0.815185	+	0.579200i	\(0.803365\pi\)
\(7\)	0.563809	−	0.534068i	0.0805442	−	0.0762955i	−0.646369	−	0.763025i	\(-0.723713\pi\)
							0.726913	+	0.686730i	\(0.240954\pi\)
\(11\)	13.5220	−	11.4857i	1.22927	−	1.04415i	0.231822	−	0.972758i	\(-0.425531\pi\)
							0.997448	−	0.0713928i	\(-0.0227444\pi\)
\(13\)	4.18667	−	1.41065i	0.322052	−	0.108512i	−0.153632	−	0.988128i	\(-0.549097\pi\)
							0.475684	+	0.879616i	\(0.342201\pi\)
\(17\)	17.2784	−	18.2406i	1.01638	−	1.07298i	0.0190572	−	0.999818i	\(-0.493934\pi\)
							0.997320	−	0.0731583i	\(-0.0233078\pi\)
\(19\)	−6.12969	−	15.3844i	−0.322615	−	0.809703i	−0.997425	−	0.0717117i	\(-0.977154\pi\)
							0.674810	−	0.737991i	\(-0.264225\pi\)
\(23\)	3.72802	+	34.2786i	0.162088	+	1.49037i	0.739077	+	0.673621i	\(0.235262\pi\)
							−0.576990	+	0.816751i	\(0.695773\pi\)
\(29\)	0.148521	−	0.674740i	0.00512143	−	0.0232669i	−0.973991	−	0.226589i	\(-0.927243\pi\)
							0.979112	+	0.203322i	\(0.0651737\pi\)
\(31\)	12.1470	−	30.4868i	0.391840	−	0.983444i	−0.592247	−	0.805756i	\(-0.701759\pi\)
							0.984087	−	0.177688i	\(-0.0568616\pi\)
\(37\)	−18.1355	−	13.7862i	−0.490148	−	0.372601i	0.330710	−	0.943732i	\(-0.392712\pi\)
							−0.820859	+	0.571131i	\(0.806505\pi\)
\(41\)	2.30251	−	21.1712i	0.0561588	−	0.516372i	−0.931947	−	0.362595i	\(-0.881891\pi\)
							0.988106	−	0.153777i	\(-0.0491437\pi\)
\(43\)	14.4347	−	16.9939i	0.335692	−	0.395207i	−0.568255	−	0.822853i	\(-0.692381\pi\)
							0.903946	+	0.427646i	\(0.140657\pi\)
\(47\)	−37.0722	+	10.2930i	−0.788769	+	0.219001i	−0.638477	−	0.769641i	\(-0.720435\pi\)
							−0.150292	+	0.988642i	\(0.548021\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.3.h.a.107.26	yes	1064
3.2	odd	2	inner	177.3.h.a.107.13	✓	1064
59.16	even	29	inner	177.3.h.a.134.13	yes	1064
177.134	odd	58	inner	177.3.h.a.134.26	yes	1064

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.3.h.a.107.13	✓	1064	3.2	odd	2	inner
177.3.h.a.107.26	yes	1064	1.1	even	1	trivial
177.3.h.a.134.13	yes	1064	59.16	even	29	inner
177.3.h.a.134.26	yes	1064	177.134	odd	58	inner