Embedded newform 177.4.f.a.89.48

• Label: 177.4.f.a.89.48
• Level: $177$
• Weight: $4$
• Character: 177.89
• Analytic conductor: $10.443$
• Analytic rank: $0$
• Dimension: $1624$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 177 = 3 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	177.f (of order \(58\), degree \(28\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(10.4433380710\)
Analytic rank:	\(0\)
Dimension:	\(1624\)
Relative dimension:	\(58\) over \(\Q(\zeta_{58})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{58}]$

Embedding invariants

Embedding label			89.48
Character	\(\chi\)	\(=\)	177.89
Dual form			177.4.f.a.2.48

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.212337 + 3.91633i) q^{2} +(1.11288 + 5.07558i) q^{3} +(-7.33945 + 0.798213i) q^{4} +(0.421405 + 1.25069i) q^{5} +(-19.6413 + 5.43615i) q^{6} +(-5.56806 + 8.21227i) q^{7} +(0.391673 + 2.38910i) q^{8} +(-24.5230 + 11.2970i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1624 q - 21 q^{3} - 278 q^{4} - 29 q^{6} - 42 q^{7} - 25 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(61\)	\(119\)
\(\chi(n)\)	\(e\left(\frac{57}{58}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.212337	+	3.91633i	0.0750726	+	1.38463i	0.757097	+	0.653302i	\(0.226617\pi\)
							−0.682024	+	0.731329i	\(0.738900\pi\)
\(3\)	1.11288	+	5.07558i	0.214174	+	0.976795i
							\(5\)	0.421405	+	1.25069i	0.0376916	+	0.111865i	0.964847	−	0.262813i	\(-0.0846501\pi\)
−0.927155	+	0.374677i	\(0.877754\pi\)
\(7\)	−5.56806	+	8.21227i	−0.300647	+	0.443421i	−0.947765	−	0.318969i	\(-0.896664\pi\)
							0.647118	+	0.762389i	\(0.275974\pi\)
\(11\)	29.8877	+	6.57879i	0.819226	+	0.180325i	0.604762	−	0.796406i	\(-0.293268\pi\)
							0.214464	+	0.976732i	\(0.431199\pi\)
\(13\)	−45.8339	−	38.9317i	−0.977849	−	0.830592i	0.00770232	−	0.999970i	\(-0.497548\pi\)
							−0.985551	+	0.169378i	\(0.945824\pi\)
\(17\)	−23.9808	+	16.2594i	−0.342129	+	0.231969i	−0.720135	−	0.693834i	\(-0.755920\pi\)
							0.378006	+	0.925803i	\(0.376610\pi\)
\(19\)	−4.96284	−	9.36091i	−0.0599239	−	0.113028i	0.851741	−	0.523963i	\(-0.175547\pi\)
							−0.911665	+	0.410935i	\(0.865202\pi\)
\(23\)	105.362	+	99.8043i	0.955196	+	0.904810i	0.995537	−	0.0943760i	\(-0.0300856\pi\)
							−0.0403403	+	0.999186i	\(0.512844\pi\)
\(29\)	−169.586	−	9.19467i	−1.08591	−	0.0588761i	−0.497517	−	0.867454i	\(-0.665755\pi\)
							−0.588389	+	0.808578i	\(0.700238\pi\)
\(31\)	250.261	+	132.680i	1.44994	+	0.768710i	0.992697	−	0.120632i	\(-0.0384921\pi\)
							0.457243	+	0.889342i	\(0.348837\pi\)
\(37\)	48.5555	+	7.96027i	0.215742	+	0.0353692i	0.268684	−	0.963228i	\(-0.413411\pi\)
							−0.0529413	+	0.998598i	\(0.516860\pi\)
\(41\)	9.02334	+	9.52583i	0.0343710	+	0.0362850i	0.742954	−	0.669342i	\(-0.233424\pi\)
							−0.708583	+	0.705627i	\(0.750665\pi\)
\(43\)	31.5889	+	143.510i	0.112029	+	0.508954i	0.998898	+	0.0469327i	\(0.0149446\pi\)
							−0.886869	+	0.462021i	\(0.847124\pi\)
\(47\)	45.7811	+	15.4254i	0.142082	+	0.0478730i	0.389444	−	0.921050i	\(-0.372667\pi\)
							−0.247362	+	0.968923i	\(0.579564\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.4.f.a.89.48	yes	1624
3.2	odd	2	inner	177.4.f.a.89.11	yes	1624
59.2	odd	58	inner	177.4.f.a.2.11	✓	1624
177.2	even	58	inner	177.4.f.a.2.48	yes	1624

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.4.f.a.2.11	✓	1624	59.2	odd	58	inner
177.4.f.a.2.48	yes	1624	177.2	even	58	inner
177.4.f.a.89.11	yes	1624	3.2	odd	2	inner
177.4.f.a.89.48	yes	1624	1.1	even	1	trivial