Embedded newform 177.3.h.a.5.14

• Label: 177.3.h.a.5.14
• 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			5.14
Character	\(\chi\)	\(=\)	177.5
Dual form			177.3.h.a.71.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.386111 + 1.14594i) q^{2} +(1.94700 + 2.28237i) q^{3} +(2.02029 + 1.53578i) q^{4} +(-1.21804 - 0.485310i) q^{5} +(-3.36720 + 1.34988i) q^{6} +(-3.89579 + 1.80238i) q^{7} +(-6.54344 + 4.43656i) q^{8} +(-1.41841 + 8.88753i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1064 q - 29 q^{3} + 18 q^{4} - 21 q^{6} - 46 q^{7} - 49 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.386111	+	1.14594i	−0.193055	+	0.572968i	−0.999840	−	0.0178776i	\(-0.994309\pi\)
							0.806785	+	0.590845i	\(0.201206\pi\)
\(3\)	1.94700	+	2.28237i	0.648999	+	0.760790i
							\(5\)	−1.21804	−	0.485310i	−0.243607	−	0.0970619i	0.245139	−	0.969488i	\(-0.421166\pi\)
−0.488746	+	0.872426i	\(0.662546\pi\)
\(7\)	−3.89579	+	1.80238i	−0.556541	+	0.257483i	−0.677936	−	0.735121i	\(-0.737125\pi\)
							0.121395	+	0.992604i	\(0.461263\pi\)
\(11\)	4.89275	+	1.35846i	0.444795	+	0.123497i	0.482702	−	0.875784i	\(-0.339655\pi\)
							−0.0379071	+	0.999281i	\(0.512069\pi\)
\(13\)	0.583496	−	1.10059i	0.0448843	−	0.0846608i	−0.860055	−	0.510201i	\(-0.829571\pi\)
							0.904940	+	0.425540i	\(0.139916\pi\)
\(17\)	2.39479	−	5.17625i	0.140870	−	0.304485i	−0.824280	−	0.566182i	\(-0.808420\pi\)
							0.965150	+	0.261697i	\(0.0842820\pi\)
\(19\)	7.69961	−	1.69481i	0.405242	−	0.0892006i	−0.00767171	−	0.999971i	\(-0.502442\pi\)
							0.412914	+	0.910770i	\(0.364511\pi\)
\(23\)	14.1932	+	2.32685i	0.617094	+	0.101167i	0.462218	−	0.886767i	\(-0.347054\pi\)
							0.154876	+	0.987934i	\(0.450502\pi\)
\(29\)	13.6363	+	40.4712i	0.470218	+	1.39556i	0.876320	+	0.481730i	\(0.159991\pi\)
							−0.406102	+	0.913828i	\(0.633112\pi\)
\(31\)	10.6397	+	2.34197i	0.343215	+	0.0755474i	0.383233	−	0.923652i	\(-0.374811\pi\)
							−0.0400181	+	0.999199i	\(0.512742\pi\)
\(37\)	18.4262	−	27.1766i	0.498006	−	0.734504i	−0.492749	−	0.870172i	\(-0.664008\pi\)
							0.990754	+	0.135668i	\(0.0433181\pi\)
\(41\)	38.0426	−	6.23676i	0.927868	−	0.152116i	0.321145	−	0.947030i	\(-0.395932\pi\)
							0.606723	+	0.794914i	\(0.292484\pi\)
\(43\)	−6.05782	−	21.8183i	−0.140880	−	0.507402i	0.859111	−	0.511790i	\(-0.171017\pi\)
							−0.999990	+	0.00438736i	\(0.998603\pi\)
\(47\)	28.7305	−	11.4473i	0.611288	−	0.243559i	−0.0438871	−	0.999036i	\(-0.513974\pi\)
							0.655175	+	0.755477i	\(0.272595\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.3.h.a.5.14	✓	1064
3.2	odd	2	inner	177.3.h.a.5.25	yes	1064
59.12	even	29	inner	177.3.h.a.71.25	yes	1064
177.71	odd	58	inner	177.3.h.a.71.14	yes	1064

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.3.h.a.5.14	✓	1064	1.1	even	1	trivial
177.3.h.a.5.25	yes	1064	3.2	odd	2	inner
177.3.h.a.71.14	yes	1064	177.71	odd	58	inner
177.3.h.a.71.25	yes	1064	59.12	even	29	inner