Embedded newform 177.3.h.a.5.3

• Label: 177.3.h.a.5.3
• Level: $177$
• Weight: $3$
• Character: 177.5
• 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			5.3
Character	\(\chi\)	\(=\)	177.5
Dual form			177.3.h.a.71.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.15362 + 3.42383i) q^{2} +(0.846084 + 2.87822i) q^{3} +(-7.20737 - 5.47890i) q^{4} +(-6.99239 - 2.78602i) q^{5} +(-10.8306 - 0.423531i) q^{6} +(0.665705 - 0.307988i) q^{7} +(15.1117 - 10.2460i) q^{8} +(-7.56829 + 4.87043i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1064 q - 29 q^{3} + 18 q^{4} - 21 q^{6} - 46 q^{7} - 49 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(61\)	\(119\)
\(\chi(n)\)	\(e\left(\frac{3}{29}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.15362	+	3.42383i	−0.576811	+	1.71191i	0.118271	+	0.992981i	\(0.462265\pi\)
							−0.695081	+	0.718931i	\(0.744632\pi\)
\(3\)	0.846084	+	2.87822i	0.282028	+	0.959406i
							\(5\)	−6.99239	−	2.78602i	−1.39848	−	0.557204i	−0.455552	−	0.890209i	\(-0.650558\pi\)
−0.942925	+	0.333005i	\(0.891937\pi\)
\(7\)	0.665705	−	0.307988i	0.0951007	−	0.0439983i	−0.371762	−	0.928328i	\(-0.621246\pi\)
							0.466863	+	0.884330i	\(0.345384\pi\)
\(11\)	9.87482	+	2.74173i	0.897711	+	0.249248i	0.685584	−	0.727994i	\(-0.259547\pi\)
							0.212127	+	0.977242i	\(0.431961\pi\)
\(13\)	−1.92710	+	3.63490i	−0.148239	+	0.279608i	−0.946500	−	0.322703i	\(-0.895408\pi\)
							0.798262	+	0.602311i	\(0.205753\pi\)
\(17\)	9.81341	−	21.2113i	0.577259	−	1.24773i	−0.370527	−	0.928822i	\(-0.620823\pi\)
							0.947787	−	0.318904i	\(-0.103315\pi\)
\(19\)	−25.8306	+	5.68574i	−1.35950	+	0.299249i	−0.834104	−	0.551607i	\(-0.814015\pi\)
							−0.525399	+	0.850856i	\(0.676084\pi\)
\(23\)	−31.1235	−	5.10245i	−1.35320	−	0.221845i	−0.558830	−	0.829282i	\(-0.688749\pi\)
							−0.794368	+	0.607437i	\(0.792198\pi\)
\(29\)	−5.80528	−	17.2295i	−0.200182	−	0.594119i	0.799799	−	0.600268i	\(-0.204939\pi\)
							−0.999981	+	0.00614853i	\(0.998043\pi\)
\(31\)	−13.3422	−	2.93683i	−0.430392	−	0.0947365i	−0.00550947	−	0.999985i	\(-0.501754\pi\)
							−0.424883	+	0.905248i	\(0.639685\pi\)
\(37\)	7.39359	−	10.9047i	0.199827	−	0.294722i	−0.714552	−	0.699582i	\(-0.753369\pi\)
							0.914379	+	0.404860i	\(0.132680\pi\)
\(41\)	56.2137	−	9.21578i	1.37107	−	0.224775i	0.569161	−	0.822226i	\(-0.307268\pi\)
							0.801906	+	0.597451i	\(0.203820\pi\)
\(43\)	−13.6656	−	49.2189i	−0.317804	−	1.14463i	−0.934071	−	0.357088i	\(-0.883770\pi\)
							0.616267	−	0.787537i	\(-0.288644\pi\)
\(47\)	−73.1414	+	29.1422i	−1.55620	+	0.620046i	−0.980307	−	0.197479i	\(-0.936724\pi\)
							−0.575892	+	0.817526i	\(0.695345\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.3.h.a.5.3	✓	1064
3.2	odd	2	inner	177.3.h.a.5.36	yes	1064
59.12	even	29	inner	177.3.h.a.71.36	yes	1064
177.71	odd	58	inner	177.3.h.a.71.3	yes	1064

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.3.h.a.5.3	✓	1064	1.1	even	1	trivial
177.3.h.a.5.36	yes	1064	3.2	odd	2	inner
177.3.h.a.71.3	yes	1064	177.71	odd	58	inner
177.3.h.a.71.36	yes	1064	59.12	even	29	inner