Embedded newform 177.2.f.a.11.13

• Label: 177.2.f.a.11.13
• Level: $177$
• Weight: $2$
• Character: 177.11
• 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			11.13
Character	\(\chi\)	\(=\)	177.11
Dual form			177.2.f.a.161.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.917082 - 0.201865i) q^{2} +(-1.67788 + 0.429798i) q^{3} +(-1.01486 + 0.469524i) q^{4} +(-1.27873 - 0.355038i) q^{5} +(-1.45199 + 0.732865i) q^{6} +(-3.08222 + 2.91963i) q^{7} +(-2.33105 + 1.77202i) q^{8} +(2.63055 - 1.44230i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 504 q - 27 q^{3} - 70 q^{4} - 29 q^{6} - 58 q^{7} - 19 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(61\)	\(119\)
\(\chi(n)\)	\(e\left(\frac{25}{58}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.917082	−	0.201865i	0.648475	−	0.142740i	0.121460	−	0.992596i	\(-0.461242\pi\)
							0.527015	+	0.849856i	\(0.323311\pi\)
\(3\)	−1.67788	+	0.429798i	−0.968723	+	0.248144i
							\(5\)	−1.27873	−	0.355038i	−0.571866	−	0.158778i	−0.0304556	−	0.999536i	\(-0.509696\pi\)
−0.541411	+	0.840758i	\(0.682110\pi\)
\(7\)	−3.08222	+	2.91963i	−1.16497	+	1.10352i	−0.172006	+	0.985096i	\(0.555025\pi\)
							−0.992963	+	0.118421i	\(0.962217\pi\)
\(11\)	0.0607546	+	0.0715258i	0.0183182	+	0.0215659i	0.771245	−	0.636539i	\(-0.219634\pi\)
							−0.752927	+	0.658104i	\(0.771359\pi\)
\(13\)	−0.0319516	−	0.0948290i	−0.00886178	−	0.0263008i	0.943131	−	0.332421i	\(-0.107866\pi\)
							−0.951993	+	0.306120i	\(0.900969\pi\)
\(17\)	1.23975	−	1.30879i	0.300684	−	0.317428i	−0.558093	−	0.829779i	\(-0.688467\pi\)
							0.858776	+	0.512351i	\(0.171225\pi\)
\(19\)	1.14638	+	2.87720i	0.262998	+	0.660076i	0.999873	−	0.0159536i	\(-0.00507842\pi\)
							−0.736874	+	0.676030i	\(0.763699\pi\)
\(23\)	−5.95945	+	0.648129i	−1.24263	+	0.135144i	−0.705718	−	0.708493i	\(-0.749375\pi\)
							−0.536914	+	0.843637i	\(0.680410\pi\)
\(29\)	−1.07393	+	4.87891i	−0.199424	+	0.905992i	0.765495	+	0.643442i	\(0.222494\pi\)
							−0.964919	+	0.262549i	\(0.915437\pi\)
\(31\)	8.35692	+	3.32970i	1.50095	+	0.598032i	0.968629	−	0.248513i	\(-0.0799418\pi\)
							0.532318	+	0.846545i	\(0.321321\pi\)
\(37\)	3.26645	−	4.29694i	0.537001	−	0.706413i	−0.444948	−	0.895556i	\(-0.646778\pi\)
							0.981950	+	0.189143i	\(0.0605710\pi\)
\(41\)	−0.971436	+	8.93220i	−0.151713	+	1.39498i	0.632537	+	0.774530i	\(0.282014\pi\)
							−0.784249	+	0.620446i	\(0.786952\pi\)
\(43\)	−5.25362	−	4.46247i	−0.801170	−	0.680520i	0.150171	−	0.988660i	\(-0.452018\pi\)
							−0.951341	+	0.308140i	\(0.900293\pi\)
\(47\)	2.50015	+	9.00471i	0.364684	+	1.31347i	0.887414	+	0.460974i	\(0.152500\pi\)
							−0.522730	+	0.852498i	\(0.675086\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.2.f.a.11.13	yes	504
3.2	odd	2	inner	177.2.f.a.11.6	✓	504
59.43	odd	58	inner	177.2.f.a.161.6	yes	504
177.161	even	58	inner	177.2.f.a.161.13	yes	504

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.2.f.a.11.6	✓	504	3.2	odd	2	inner
177.2.f.a.11.13	yes	504	1.1	even	1	trivial
177.2.f.a.161.6	yes	504	59.43	odd	58	inner
177.2.f.a.161.13	yes	504	177.161	even	58	inner