Embedded newform 177.3.h.a.104.25

• Label: 177.3.h.a.104.25
• Level: $177$
• Weight: $3$
• Character: 177.104
• 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			104.25
Character	\(\chi\)	\(=\)	177.104
Dual form			177.3.h.a.80.25

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.816112 + 1.35639i) q^{2} +(-2.70180 - 1.30394i) q^{3} +(0.699882 - 1.32012i) q^{4} +(0.752334 + 6.91759i) q^{5} +(-0.436330 - 4.72886i) q^{6} +(-0.454099 + 0.153004i) q^{7} +(8.68441 - 0.470855i) q^{8} +(5.59950 + 7.04597i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1064 q - 29 q^{3} + 18 q^{4} - 21 q^{6} - 46 q^{7} - 49 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(61\)	\(119\)
\(\chi(n)\)	\(e\left(\frac{24}{29}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.816112	+	1.35639i	0.408056	+	0.678194i	0.990932	−	0.134367i	\(-0.0429001\pi\)
							−0.582876	+	0.812561i	\(0.698073\pi\)
\(3\)	−2.70180	−	1.30394i	−0.900602	−	0.434646i
							\(5\)	0.752334	+	6.91759i	0.150467	+	1.38352i	0.789222	+	0.614109i	\(0.210484\pi\)
−0.638755	+	0.769410i	\(0.720550\pi\)
\(7\)	−0.454099	+	0.153004i	−0.0648712	+	0.0218577i	−0.351550	−	0.936169i	\(-0.614345\pi\)
							0.286679	+	0.958027i	\(0.407449\pi\)
\(11\)	−9.77096	−	6.62488i	−0.888269	−	0.602262i	0.0294169	−	0.999567i	\(-0.490635\pi\)
							−0.917686	+	0.397306i	\(0.869945\pi\)
\(13\)	17.4437	+	16.5236i	1.34183	+	1.27105i	0.933670	+	0.358134i	\(0.116587\pi\)
							0.408157	+	0.912912i	\(0.366172\pi\)
\(17\)	−9.50136	+	28.1990i	−0.558903	+	1.65877i	0.178992	+	0.983851i	\(0.442716\pi\)
							−0.737895	+	0.674915i	\(0.764180\pi\)
\(19\)	3.61446	+	22.0472i	0.190235	+	1.16038i	0.893369	+	0.449323i	\(0.148335\pi\)
							−0.703134	+	0.711057i	\(0.748217\pi\)
\(23\)	15.4278	−	4.28351i	0.670775	−	0.186240i	0.0845884	−	0.996416i	\(-0.473042\pi\)
							0.586186	+	0.810176i	\(0.300629\pi\)
\(29\)	4.85292	−	8.06562i	0.167342	−	0.278125i	−0.761826	−	0.647781i	\(-0.775697\pi\)
							0.929168	+	0.369656i	\(0.120525\pi\)
\(31\)	5.02229	−	30.6346i	0.162009	−	0.988213i	−0.773212	−	0.634148i	\(-0.781351\pi\)
							0.935221	−	0.354065i	\(-0.115201\pi\)
\(37\)	2.48349	−	45.8054i	0.0671215	−	1.23798i	−0.749969	−	0.661473i	\(-0.769932\pi\)
							0.817090	−	0.576509i	\(-0.195586\pi\)
\(41\)	−14.8594	−	4.12569i	−0.362424	−	0.100627i	0.0815426	−	0.996670i	\(-0.474015\pi\)
							−0.443967	+	0.896043i	\(0.646429\pi\)
\(43\)	−19.9416	−	29.4117i	−0.463758	−	0.683992i	0.521850	−	0.853037i	\(-0.325242\pi\)
							−0.985608	+	0.169045i	\(0.945932\pi\)
\(47\)	0.359009	−	3.30103i	0.00763849	−	0.0702347i	−0.989698	−	0.143169i	\(-0.954271\pi\)
							0.997337	+	0.0729342i	\(0.0232363\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.3.h.a.104.25	yes	1064
3.2	odd	2	inner	177.3.h.a.104.14	yes	1064
59.21	even	29	inner	177.3.h.a.80.14	✓	1064
177.80	odd	58	inner	177.3.h.a.80.25	yes	1064

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.3.h.a.80.14	✓	1064	59.21	even	29	inner
177.3.h.a.80.25	yes	1064	177.80	odd	58	inner
177.3.h.a.104.14	yes	1064	3.2	odd	2	inner
177.3.h.a.104.25	yes	1064	1.1	even	1	trivial