Embedded newform 177.4.f.a.89.7

• Label: 177.4.f.a.89.7
• Level: $177$
• Weight: $4$
• Character: 177.89
• 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			89.7
Character	\(\chi\)	\(=\)	177.89
Dual form			177.4.f.a.2.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.247929 - 4.57278i) q^{2} +(0.343239 + 5.18480i) q^{3} +(-12.8957 + 1.40250i) q^{4} +(2.38761 + 7.08619i) q^{5} +(23.6239 - 2.85502i) q^{6} +(6.25613 - 9.22711i) q^{7} +(3.68350 + 22.4684i) q^{8} +(-26.7644 + 3.55926i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1624 q - 21 q^{3} - 278 q^{4} - 29 q^{6} - 42 q^{7} - 25 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(61\)	\(119\)
\(\chi(n)\)	\(e\left(\frac{57}{58}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.247929	−	4.57278i	−0.0876561	−	1.61672i	−0.630020	−	0.776579i	\(-0.716953\pi\)
							0.542364	−	0.840143i	\(-0.317529\pi\)
\(3\)	0.343239	+	5.18480i	0.0660564	+	0.997816i
							\(5\)	2.38761	+	7.08619i	0.213555	+	0.633808i	0.999872	+	0.0159853i	\(0.00508850\pi\)
−0.786317	+	0.617823i	\(0.788015\pi\)
\(7\)	6.25613	−	9.22711i	0.337799	−	0.498217i	−0.620413	−	0.784276i	\(-0.713035\pi\)
							0.958212	+	0.286059i	\(0.0923452\pi\)
\(11\)	−53.6150	−	11.8016i	−1.46959	−	0.323482i	−0.593191	−	0.805061i	\(-0.702132\pi\)
							−0.876402	+	0.481579i	\(0.840063\pi\)
\(13\)	−13.5489	−	11.5086i	−0.289062	−	0.245531i	0.491109	−	0.871098i	\(-0.336592\pi\)
							−0.780170	+	0.625567i	\(0.784868\pi\)
\(17\)	−81.8632	+	55.5046i	−1.16793	+	0.791873i	−0.981663	−	0.190626i	\(-0.938948\pi\)
							−0.186263	+	0.982500i	\(0.559638\pi\)
\(19\)	53.5774	+	101.058i	0.646921	+	1.22022i	0.962489	+	0.271320i	\(0.0874602\pi\)
							−0.315568	+	0.948903i	\(0.602195\pi\)
\(23\)	−24.9926	−	23.6743i	−0.226579	−	0.214627i	0.565985	−	0.824415i	\(-0.308496\pi\)
							−0.792564	+	0.609788i	\(0.791255\pi\)
\(29\)	−162.541	−	8.81273i	−1.04080	−	0.0564304i	−0.474194	−	0.880421i	\(-0.657260\pi\)
							−0.566604	+	0.823990i	\(0.691743\pi\)
\(31\)	−209.268	−	110.947i	−1.21244	−	0.642796i	−0.265472	−	0.964119i	\(-0.585528\pi\)
							−0.946970	+	0.321323i	\(0.895873\pi\)
\(37\)	−214.379	−	35.1457i	−0.952534	−	0.156160i	−0.334574	−	0.942370i	\(-0.608592\pi\)
							−0.617960	+	0.786210i	\(0.712041\pi\)
\(41\)	−46.9311	−	49.5445i	−0.178766	−	0.188721i	0.630454	−	0.776226i	\(-0.282869\pi\)
							−0.809220	+	0.587506i	\(0.800110\pi\)
\(43\)	52.2623	+	237.430i	0.185347	+	0.842041i	0.973883	+	0.227052i	\(0.0729086\pi\)
							−0.788536	+	0.614989i	\(0.789160\pi\)
\(47\)	495.188	+	166.848i	1.53682	+	0.517815i	0.955280	−	0.295702i	\(-0.0955535\pi\)
							0.581541	+	0.813517i	\(0.302450\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.4.f.a.89.7	yes	1624
3.2	odd	2	inner	177.4.f.a.89.52	yes	1624
59.2	odd	58	inner	177.4.f.a.2.52	yes	1624
177.2	even	58	inner	177.4.f.a.2.7	✓	1624

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.4.f.a.2.7	✓	1624	177.2	even	58	inner
177.4.f.a.2.52	yes	1624	59.2	odd	58	inner
177.4.f.a.89.7	yes	1624	1.1	even	1	trivial
177.4.f.a.89.52	yes	1624	3.2	odd	2	inner