Embedded newform 177.3.h.a.107.4

• Label: 177.3.h.a.107.4
• 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			107.4
Character	\(\chi\)	\(=\)	177.107
Dual form			177.3.h.a.134.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.712483 - 3.23684i) q^{2} +(-2.95279 + 0.530142i) q^{3} +(-6.33922 + 2.93283i) q^{4} +(9.05162 + 2.51317i) q^{5} +(3.81980 + 9.17999i) q^{6} +(-1.39401 + 1.32047i) q^{7} +(5.98672 + 7.87540i) q^{8} +(8.43790 - 3.13079i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1064 q - 29 q^{3} + 18 q^{4} - 21 q^{6} - 46 q^{7} - 49 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.712483	−	3.23684i	−0.356242	−	1.61842i	−0.724898	−	0.688856i	\(-0.758113\pi\)
							0.368657	−	0.929566i	\(-0.379818\pi\)
\(3\)	−2.95279	+	0.530142i	−0.984262	+	0.176714i
							\(5\)	9.05162	+	2.51317i	1.81032	+	0.502634i	0.996874	−	0.0790016i	\(-0.0251732\pi\)
0.813449	+	0.581636i	\(0.197587\pi\)
\(7\)	−1.39401	+	1.32047i	−0.199144	+	0.188639i	−0.780773	−	0.624815i	\(-0.785174\pi\)
							0.581629	+	0.813454i	\(0.302416\pi\)
\(11\)	9.09749	−	7.72747i	0.827044	−	0.702498i	−0.130292	−	0.991476i	\(-0.541591\pi\)
							0.957336	+	0.288978i	\(0.0933155\pi\)
\(13\)	7.09283	−	2.38985i	0.545602	−	0.183835i	−0.0329842	−	0.999456i	\(-0.510501\pi\)
							0.578586	+	0.815621i	\(0.303605\pi\)
\(17\)	−4.87649	+	5.14805i	−0.286852	+	0.302826i	−0.853457	−	0.521164i	\(-0.825498\pi\)
							0.566604	+	0.823990i	\(0.308257\pi\)
\(19\)	5.67166	+	14.2348i	0.298509	+	0.749200i	0.999356	+	0.0358886i	\(0.0114261\pi\)
							−0.700847	+	0.713311i	\(0.747195\pi\)
\(23\)	−3.05208	−	28.0634i	−0.132699	−	1.22015i	−0.851282	−	0.524708i	\(-0.824175\pi\)
							0.718583	−	0.695441i	\(-0.244791\pi\)
\(29\)	6.69601	−	30.4203i	0.230897	−	1.04898i	−0.708384	−	0.705827i	\(-0.750576\pi\)
							0.939281	−	0.343148i	\(-0.111493\pi\)
\(31\)	22.0998	−	55.4664i	0.712898	−	1.78924i	0.106142	−	0.994351i	\(-0.466150\pi\)
							0.606756	−	0.794888i	\(-0.292470\pi\)
\(37\)	−4.54286	−	3.45339i	−0.122780	−	0.0933350i	0.541982	−	0.840390i	\(-0.317674\pi\)
							−0.664762	+	0.747055i	\(0.731467\pi\)
\(41\)	−0.643688	+	5.91861i	−0.0156997	+	0.144356i	−0.999364	−	0.0356555i	\(-0.988648\pi\)
							0.983664	+	0.180012i	\(0.0576136\pi\)
\(43\)	5.19319	−	6.11390i	0.120772	−	0.142184i	−0.698453	−	0.715656i	\(-0.746128\pi\)
							0.819225	+	0.573472i	\(0.194404\pi\)
\(47\)	−36.2560	+	10.0664i	−0.771404	+	0.214179i	−0.630862	−	0.775895i	\(-0.717298\pi\)
							−0.140542	+	0.990075i	\(0.544885\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.3.h.a.107.4	✓	1064
3.2	odd	2	inner	177.3.h.a.107.35	yes	1064
59.16	even	29	inner	177.3.h.a.134.35	yes	1064
177.134	odd	58	inner	177.3.h.a.134.4	yes	1064

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.3.h.a.107.4	✓	1064	1.1	even	1	trivial
177.3.h.a.107.35	yes	1064	3.2	odd	2	inner
177.3.h.a.134.4	yes	1064	177.134	odd	58	inner
177.3.h.a.134.35	yes	1064	59.16	even	29	inner