Embedded newform 177.2.f.a.101.9

• Label: 177.2.f.a.101.9
• Level: $177$
• Weight: $2$
• Character: 177.101
• Analytic conductor: $1.413$
• Analytic rank: $0$
• Dimension: $504$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 177 = 3 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	177.f (of order \(58\), degree \(28\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.41335211578\)
Analytic rank:	\(0\)
Dimension:	\(504\)
Relative dimension:	\(18\) over \(\Q(\zeta_{58})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{58}]$

Embedding invariants

Embedding label			101.9
Character	\(\chi\)	\(=\)	177.101
Dual form			177.2.f.a.170.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.154453 + 0.227801i) q^{2} +(-0.819531 + 1.52590i) q^{3} +(0.712239 + 1.78758i) q^{4} +(-1.09722 + 2.37161i) q^{5} +(-0.221022 - 0.422369i) q^{6} +(-0.787392 - 2.83593i) q^{7} +(-1.05480 - 0.232179i) q^{8} +(-1.65674 - 2.50104i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 504 q - 27 q^{3} - 70 q^{4} - 29 q^{6} - 58 q^{7} - 19 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(61\)	\(119\)
\(\chi(n)\)	\(e\left(\frac{11}{58}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.154453	+	0.227801i	−0.109214	+	0.161079i	−0.878368	−	0.477984i	\(-0.841368\pi\)
							0.769154	+	0.639064i	\(0.220678\pi\)
\(3\)	−0.819531	+	1.52590i	−0.473156	+	0.880979i
							\(5\)	−1.09722	+	2.37161i	−0.490692	+	1.06061i	0.491178	+	0.871059i	\(0.336567\pi\)
−0.981870	+	0.189555i	\(0.939296\pi\)
\(7\)	−0.787392	−	2.83593i	−0.297606	−	1.07188i	−0.949551	−	0.313614i	\(-0.898460\pi\)
							0.651944	−	0.758267i	\(-0.273954\pi\)
\(11\)	−1.07982	−	1.02286i	−0.325577	−	0.308403i	0.507400	−	0.861711i	\(-0.330607\pi\)
							−0.832977	+	0.553308i	\(0.813365\pi\)
\(13\)	−0.324215	+	2.98110i	−0.0899209	+	0.826809i	0.859145	+	0.511732i	\(0.170996\pi\)
							−0.949066	+	0.315077i	\(0.897970\pi\)
\(17\)	6.60144	+	1.83288i	1.60108	+	0.444539i	0.949430	−	0.313977i	\(-0.101662\pi\)
							0.651654	+	0.758516i	\(0.274075\pi\)
\(19\)	1.91308	+	1.45428i	0.438890	+	0.333636i	0.801173	−	0.598432i	\(-0.204209\pi\)
							−0.362283	+	0.932068i	\(0.618003\pi\)
\(23\)	3.50747	+	6.61578i	0.731357	+	1.37949i	0.917809	+	0.397022i	\(0.129956\pi\)
							−0.186452	+	0.982464i	\(0.559699\pi\)
\(29\)	−6.12820	+	4.15502i	−1.13798	+	0.771569i	−0.976545	−	0.215315i	\(-0.930922\pi\)
							−0.161434	+	0.986884i	\(0.551612\pi\)
\(31\)	−4.49967	−	5.91921i	−0.808164	−	1.06312i	−0.996709	−	0.0810573i	\(-0.974170\pi\)
							0.188545	−	0.982065i	\(-0.439623\pi\)
\(37\)	−0.431482	−	1.96024i	−0.0709352	−	0.322262i	0.927769	−	0.373155i	\(-0.121724\pi\)
							−0.998704	+	0.0508935i	\(0.983793\pi\)
\(41\)	−2.48096	−	1.31532i	−0.387461	−	0.205419i	0.263278	−	0.964720i	\(-0.415196\pi\)
							−0.650740	+	0.759301i	\(0.725541\pi\)
\(43\)	4.78961	+	5.05633i	0.730409	+	0.771083i	0.980369	−	0.197173i	\(-0.0631760\pi\)
							−0.249960	+	0.968256i	\(0.580417\pi\)
\(47\)	7.93353	−	3.67044i	1.15722	−	0.535389i	0.255121	−	0.966909i	\(-0.417885\pi\)
							0.902103	+	0.431520i	\(0.142023\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.2.f.a.101.9	✓	504
3.2	odd	2	inner	177.2.f.a.101.10	yes	504
59.52	odd	58	inner	177.2.f.a.170.10	yes	504
177.170	even	58	inner	177.2.f.a.170.9	yes	504

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.2.f.a.101.9	✓	504	1.1	even	1	trivial
177.2.f.a.101.10	yes	504	3.2	odd	2	inner
177.2.f.a.170.9	yes	504	177.170	even	58	inner
177.2.f.a.170.10	yes	504	59.52	odd	58	inner