Embedded newform 177.3.h.a.107.24

• Label: 177.3.h.a.107.24
• 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			107.24
Character	\(\chi\)	\(=\)	177.107
Dual form			177.3.h.a.134.24

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.255803 + 1.16213i) q^{2} +(-2.31735 + 1.90522i) q^{3} +(2.34520 - 1.08500i) q^{4} +(2.94911 + 0.818817i) q^{5} +(-2.80690 - 2.20570i) q^{6} +(7.54209 - 7.14425i) q^{7} +(4.74132 + 6.23710i) q^{8} +(1.74025 - 8.83015i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1064 q - 29 q^{3} + 18 q^{4} - 21 q^{6} - 46 q^{7} - 49 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.255803	+	1.16213i	0.127902	+	0.581063i	0.996356	+	0.0852939i	\(0.0271829\pi\)
							−0.868454	+	0.495770i	\(0.834886\pi\)
\(3\)	−2.31735	+	1.90522i	−0.772451	+	0.635074i
							\(5\)	2.94911	+	0.818817i	0.589823	+	0.163763i	0.549601	−	0.835427i	\(-0.314780\pi\)
0.0402217	+	0.999191i	\(0.487194\pi\)
\(7\)	7.54209	−	7.14425i	1.07744	−	1.02061i	0.0777206	−	0.996975i	\(-0.475236\pi\)
							0.999721	−	0.0236315i	\(-0.00752283\pi\)
\(11\)	−3.02321	+	2.56794i	−0.274837	+	0.233449i	−0.774195	−	0.632947i	\(-0.781845\pi\)
							0.499358	+	0.866396i	\(0.333569\pi\)
\(13\)	6.78310	−	2.28549i	0.521777	−	0.175807i	−0.0460746	−	0.998938i	\(-0.514671\pi\)
							0.567852	+	0.823131i	\(0.307775\pi\)
\(17\)	−10.1410	+	10.7057i	−0.596527	+	0.629746i	−0.952423	−	0.304780i	\(-0.901417\pi\)
							0.355896	+	0.934526i	\(0.384176\pi\)
\(19\)	12.6885	+	31.8458i	0.667818	+	1.67610i	0.734749	+	0.678339i	\(0.237300\pi\)
							−0.0669305	+	0.997758i	\(0.521321\pi\)
\(23\)	−2.41459	−	22.2018i	−0.104982	−	0.965295i	−0.921607	−	0.388125i	\(-0.873123\pi\)
							0.816624	−	0.577169i	\(-0.195843\pi\)
\(29\)	−7.08656	+	32.1946i	−0.244364	+	1.11016i	0.681357	+	0.731951i	\(0.261390\pi\)
							−0.925721	+	0.378206i	\(0.876541\pi\)
\(31\)	5.89894	−	14.8052i	0.190288	−	0.477588i	−0.802617	−	0.596495i	\(-0.796560\pi\)
							0.992905	+	0.118907i	\(0.0379390\pi\)
\(37\)	−25.1389	−	19.1101i	−0.679431	−	0.516490i	0.207592	−	0.978216i	\(-0.433437\pi\)
							−0.887023	+	0.461726i	\(0.847230\pi\)
\(41\)	7.06887	−	64.9972i	0.172411	−	1.58530i	−0.513810	−	0.857904i	\(-0.671766\pi\)
							0.686221	−	0.727393i	\(-0.259268\pi\)
\(43\)	−30.6194	+	36.0479i	−0.712078	+	0.838323i	−0.992588	−	0.121528i	\(-0.961221\pi\)
							0.280510	+	0.959851i	\(0.409496\pi\)
\(47\)	40.9409	−	11.3672i	0.871082	−	0.241855i	0.196903	−	0.980423i	\(-0.436911\pi\)
							0.674179	+	0.738568i	\(0.264498\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.3.h.a.107.24	yes	1064
3.2	odd	2	inner	177.3.h.a.107.15	✓	1064
59.16	even	29	inner	177.3.h.a.134.15	yes	1064
177.134	odd	58	inner	177.3.h.a.134.24	yes	1064

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.3.h.a.107.15	✓	1064	3.2	odd	2	inner
177.3.h.a.107.24	yes	1064	1.1	even	1	trivial
177.3.h.a.134.15	yes	1064	59.16	even	29	inner
177.3.h.a.134.24	yes	1064	177.134	odd	58	inner