Embedded newform 177.4.f.a.2.9

• Label: 177.4.f.a.2.9
• Level: $177$
• Weight: $4$
• Character: 177.2
• Analytic conductor: $10.443$
• Analytic rank: $0$
• Dimension: $1624$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 177 = 3 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	177.f (of order \(58\), degree \(28\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(10.4433380710\)
Analytic rank:	\(0\)
Dimension:	\(1624\)
Relative dimension:	\(58\) over \(\Q(\zeta_{58})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{58}]$

Embedding invariants

Embedding label			2.9
Character	\(\chi\)	\(=\)	177.2
Dual form			177.4.f.a.89.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.216913 + 4.00073i) q^{2} +(4.38754 - 2.78379i) q^{3} +(-8.00566 - 0.870667i) q^{4} +(2.33628 - 6.93383i) q^{5} +(10.1855 + 18.1572i) q^{6} +(-13.7785 - 20.3217i) q^{7} +(0.0342646 - 0.209005i) q^{8} +(11.5010 - 24.4280i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1624 q - 21 q^{3} - 278 q^{4} - 29 q^{6} - 42 q^{7} - 25 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(61\)	\(119\)
\(\chi(n)\)	\(e\left(\frac{1}{58}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.216913	+	4.00073i	−0.0766904	+	1.41447i	0.666557	+	0.745454i	\(0.267767\pi\)
							−0.743248	+	0.669016i	\(0.766716\pi\)
\(3\)	4.38754	−	2.78379i	0.844383	−	0.535741i
							\(5\)	2.33628	−	6.93383i	0.208963	−	0.620180i	−0.791002	−	0.611814i	\(-0.790440\pi\)
0.999965	−	0.00836659i	\(-0.00266320\pi\)
\(7\)	−13.7785	−	20.3217i	−0.743968	−	1.09727i	−0.991716	−	0.128447i	\(-0.959001\pi\)
							0.247748	−	0.968825i	\(-0.420309\pi\)
\(11\)	24.1825	−	5.32296i	0.662844	−	0.145903i	0.129208	−	0.991617i	\(-0.458756\pi\)
							0.533636	+	0.845714i	\(0.320825\pi\)
\(13\)	5.19695	−	4.41433i	0.110875	−	0.0941781i	−0.590234	−	0.807232i	\(-0.700965\pi\)
							0.701109	+	0.713054i	\(0.252689\pi\)
\(17\)	61.1945	+	41.4909i	0.873050	+	0.591943i	0.913318	−	0.407246i	\(-0.133511\pi\)
							−0.0402684	+	0.999189i	\(0.512821\pi\)
\(19\)	71.8255	−	135.477i	0.867258	−	1.63582i	0.0995400	−	0.995034i	\(-0.468263\pi\)
							0.767718	−	0.640788i	\(-0.221392\pi\)
\(23\)	92.8492	−	87.9515i	0.841757	−	0.797354i	−0.139467	−	0.990227i	\(-0.544539\pi\)
							0.981224	+	0.192872i	\(0.0617803\pi\)
\(29\)	−155.267	+	8.41832i	−0.994218	+	0.0539049i	−0.544062	−	0.839045i	\(-0.683114\pi\)
							−0.450156	+	0.892950i	\(0.648632\pi\)
\(31\)	−70.4691	+	37.3603i	−0.408278	+	0.216455i	−0.659873	−	0.751377i	\(-0.729390\pi\)
							0.251595	+	0.967833i	\(0.419045\pi\)
\(37\)	195.991	−	32.1310i	0.870829	−	0.142765i	0.290232	−	0.956956i	\(-0.406268\pi\)
							0.580597	+	0.814191i	\(0.302819\pi\)
\(41\)	−203.452	+	214.781i	−0.774971	+	0.818127i	−0.987266	−	0.159079i	\(-0.949148\pi\)
							0.212295	+	0.977206i	\(0.431906\pi\)
\(43\)	−97.3150	+	442.106i	−0.345126	+	1.56792i	0.409841	+	0.912157i	\(0.365584\pi\)
							−0.754966	+	0.655764i	\(0.772347\pi\)
\(47\)	−325.542	+	109.688i	−1.01032	+	0.340417i	−0.775314	−	0.631577i	\(-0.782408\pi\)
							−0.235008	+	0.971993i	\(0.575512\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.4.f.a.2.9	✓	1624
3.2	odd	2	inner	177.4.f.a.2.50	yes	1624
59.30	odd	58	inner	177.4.f.a.89.50	yes	1624
177.89	even	58	inner	177.4.f.a.89.9	yes	1624

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.4.f.a.2.9	✓	1624	1.1	even	1	trivial
177.4.f.a.2.50	yes	1624	3.2	odd	2	inner
177.4.f.a.89.9	yes	1624	177.89	even	58	inner
177.4.f.a.89.50	yes	1624	59.30	odd	58	inner