Embedded newform 177.3.h.a.104.15

• Label: 177.3.h.a.104.15
• 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			104.15
Character	\(\chi\)	\(=\)	177.104
Dual form			177.3.h.a.80.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.720339 - 1.19721i) q^{2} +(-1.59637 - 2.54000i) q^{3} +(0.959203 - 1.80925i) q^{4} +(0.482899 + 4.44019i) q^{5} +(-1.89099 + 3.74086i) q^{6} +(-8.85491 + 2.98357i) q^{7} +(-8.43767 + 0.457477i) q^{8} +(-3.90320 + 8.10956i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1064 q - 29 q^{3} + 18 q^{4} - 21 q^{6} - 46 q^{7} - 49 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.720339	−	1.19721i	−0.360170	−	0.598607i	0.622799	−	0.782382i	\(-0.285996\pi\)
							−0.982968	+	0.183776i	\(0.941168\pi\)
\(3\)	−1.59637	−	2.54000i	−0.532124	−	0.846667i
							\(5\)	0.482899	+	4.44019i	0.0965799	+	0.888037i	0.937760	+	0.347283i	\(0.112896\pi\)
−0.841181	+	0.540754i	\(0.818139\pi\)
\(7\)	−8.85491	+	2.98357i	−1.26499	+	0.426224i	−0.870268	−	0.492578i	\(-0.836055\pi\)
							−0.394719	+	0.918802i	\(0.629158\pi\)
\(11\)	5.00097	+	3.39074i	0.454634	+	0.308249i	0.766925	−	0.641737i	\(-0.221786\pi\)
							−0.312291	+	0.949987i	\(0.601096\pi\)
\(13\)	−5.60890	−	5.31303i	−0.431454	−	0.408695i	0.440910	−	0.897551i	\(-0.354656\pi\)
							−0.872364	+	0.488856i	\(0.837414\pi\)
\(17\)	−2.44500	+	7.25651i	−0.143824	+	0.426854i	−0.995219	−	0.0976715i	\(-0.968861\pi\)
							0.851395	+	0.524525i	\(0.175757\pi\)
\(19\)	0.261408	+	1.59452i	0.0137583	+	0.0839221i	0.992813	−	0.119675i	\(-0.0381851\pi\)
							−0.979055	+	0.203597i	\(0.934737\pi\)
\(23\)	−28.2447	+	7.84212i	−1.22803	+	0.340962i	−0.820190	−	0.572091i	\(-0.806132\pi\)
							−0.407842	+	0.913052i	\(0.633719\pi\)
\(29\)	−16.3989	+	27.2552i	−0.565480	+	0.939835i	0.433767	+	0.901025i	\(0.357184\pi\)
							−0.999247	+	0.0388095i	\(0.987643\pi\)
\(31\)	0.850103	−	5.18540i	0.0274227	−	0.167271i	−0.969565	−	0.244835i	\(-0.921266\pi\)
							0.996987	+	0.0775644i	\(0.0247143\pi\)
\(37\)	0.436841	−	8.05705i	0.0118065	−	0.217758i	−0.986676	−	0.162695i	\(-0.947981\pi\)
							0.998483	−	0.0550629i	\(-0.0175360\pi\)
\(41\)	−34.1691	−	9.48700i	−0.833392	−	0.231390i	−0.175478	−	0.984483i	\(-0.556147\pi\)
							−0.657914	+	0.753093i	\(0.728561\pi\)
\(43\)	−29.2102	−	43.0818i	−0.679307	−	1.00190i	−0.998529	−	0.0542233i	\(-0.982732\pi\)
							0.319221	−	0.947680i	\(-0.396579\pi\)
\(47\)	9.28805	−	85.4023i	0.197618	−	1.81707i	−0.304769	−	0.952426i	\(-0.598579\pi\)
							0.502387	−	0.864643i	\(-0.332455\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.3.h.a.104.15	yes	1064
3.2	odd	2	inner	177.3.h.a.104.24	yes	1064
59.21	even	29	inner	177.3.h.a.80.24	yes	1064
177.80	odd	58	inner	177.3.h.a.80.15	✓	1064

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.3.h.a.80.15	✓	1064	177.80	odd	58	inner
177.3.h.a.80.24	yes	1064	59.21	even	29	inner
177.3.h.a.104.15	yes	1064	1.1	even	1	trivial
177.3.h.a.104.24	yes	1064	3.2	odd	2	inner