Embedded newform 177.2.f.a.113.8

• Label: 177.2.f.a.113.8
• Level: $177$
• Weight: $2$
• Character: 177.113
• 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			113.8
Character	\(\chi\)	\(=\)	177.113
Dual form			177.2.f.a.47.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.680494 - 0.229285i) q^{2} +(-1.71998 - 0.204154i) q^{3} +(-1.18169 - 0.898294i) q^{4} +(1.34581 + 0.536219i) q^{5} +(1.12363 + 0.533291i) q^{6} +(-2.55263 + 1.18097i) q^{7} +(1.40412 + 2.07093i) q^{8} +(2.91664 + 0.702280i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 504 q - 27 q^{3} - 70 q^{4} - 29 q^{6} - 58 q^{7} - 19 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(61\)	\(119\)
\(\chi(n)\)	\(e\left(\frac{35}{58}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.680494	−	0.229285i	−0.481182	−	0.162129i	0.0682487	−	0.997668i	\(-0.478259\pi\)
							−0.549431	+	0.835539i	\(0.685155\pi\)
\(3\)	−1.71998	−	0.204154i	−0.993029	−	0.117868i
							\(5\)	1.34581	+	0.536219i	0.601863	+	0.239804i	0.651111	−	0.758982i	\(-0.274303\pi\)
−0.0492480	+	0.998787i	\(0.515682\pi\)
\(7\)	−2.55263	+	1.18097i	−0.964804	+	0.446366i	−0.838065	−	0.545570i	\(-0.816313\pi\)
							−0.126739	+	0.991936i	\(0.540451\pi\)
\(11\)	−1.16129	+	4.18258i	−0.350141	+	1.26109i	0.553517	+	0.832838i	\(0.313286\pi\)
							−0.903658	+	0.428256i	\(0.859128\pi\)
\(13\)	0.962953	+	0.510526i	0.267075	+	0.141594i	0.596565	−	0.802565i	\(-0.296532\pi\)
							−0.329490	+	0.944159i	\(0.606877\pi\)
\(17\)	−1.77047	+	3.82680i	−0.429402	+	0.928137i	0.565257	+	0.824915i	\(0.308777\pi\)
							−0.994658	+	0.103221i	\(0.967085\pi\)
\(19\)	0.143458	−	0.0315776i	0.0329116	−	0.00724439i	−0.198484	−	0.980104i	\(-0.563602\pi\)
							0.231396	+	0.972860i	\(0.425671\pi\)
\(23\)	−1.06461	+	6.49382i	−0.221986	+	1.35405i	0.606963	+	0.794730i	\(0.292388\pi\)
							−0.828949	+	0.559324i	\(0.811061\pi\)
\(29\)	−1.51224	−	4.48817i	−0.280816	−	0.833433i	−0.991521	−	0.129943i	\(-0.958521\pi\)
							0.710705	−	0.703490i	\(-0.248376\pi\)
\(31\)	0.503659	−	2.28815i	0.0904598	−	0.410963i	−0.909529	−	0.415640i	\(-0.863558\pi\)
							0.999989	+	0.00467687i	\(0.00148870\pi\)
\(37\)	2.89842	+	1.96518i	0.476498	+	0.323074i	0.775684	−	0.631121i	\(-0.217405\pi\)
							−0.299186	+	0.954195i	\(0.596715\pi\)
\(41\)	4.23868	−	0.694897i	0.661971	−	0.108525i	0.178575	−	0.983926i	\(-0.442851\pi\)
							0.483396	+	0.875402i	\(0.339403\pi\)
\(43\)	−7.33760	+	2.03728i	−1.11897	+	0.310682i	−0.777296	−	0.629135i	\(-0.783409\pi\)
							−0.341678	+	0.939817i	\(0.610995\pi\)
\(47\)	0.818877	+	2.05523i	0.119446	+	0.299786i	0.976527	−	0.215395i	\(-0.0691040\pi\)
							−0.857081	+	0.515181i	\(0.827725\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.2.f.a.113.8	yes	504
3.2	odd	2	inner	177.2.f.a.113.11	yes	504
59.47	odd	58	inner	177.2.f.a.47.11	yes	504
177.47	even	58	inner	177.2.f.a.47.8	✓	504

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.2.f.a.47.8	✓	504	177.47	even	58	inner
177.2.f.a.47.11	yes	504	59.47	odd	58	inner
177.2.f.a.113.8	yes	504	1.1	even	1	trivial
177.2.f.a.113.11	yes	504	3.2	odd	2	inner