Embedded newform 177.4.f.a.89.11

• Label: 177.4.f.a.89.11
• 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			89.11
Character	\(\chi\)	\(=\)	177.89
Dual form			177.4.f.a.2.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.212337 - 3.91633i) q^{2} +(4.58888 - 2.43766i) q^{3} +(-7.33945 + 0.798213i) q^{4} +(-0.421405 - 1.25069i) q^{5} +(-10.5211 - 17.4540i) q^{6} +(-5.56806 + 8.21227i) q^{7} +(-0.391673 - 2.38910i) q^{8} +(15.1156 - 22.3723i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1624 q - 21 q^{3} - 278 q^{4} - 29 q^{6} - 42 q^{7} - 25 q^{9}+O(q^{10}) \)

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	\(-0.773383\pi\)
							0.682024	−	0.731329i	\(-0.261100\pi\)
\(3\)	4.58888	−	2.43766i	0.883130	−	0.469128i
							\(5\)	−0.421405	−	1.25069i	−0.0376916	−	0.111865i	0.927155	−	0.374677i	\(-0.122246\pi\)
−0.964847	+	0.262813i	\(0.915350\pi\)
\(7\)	−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	−0.214464	−	0.976732i	\(-0.568801\pi\)
							−0.604762	+	0.796406i	\(0.706732\pi\)
\(13\)	−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	−0.378006	−	0.925803i	\(-0.623390\pi\)
							0.720135	+	0.693834i	\(0.244080\pi\)
\(19\)	−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	0.0403403	−	0.999186i	\(-0.487156\pi\)
							−0.995537	+	0.0943760i	\(0.969914\pi\)
\(29\)	169.586	+	9.19467i	1.08591	+	0.0588761i	0.588389	−	0.808578i	\(-0.299762\pi\)
							0.497517	+	0.867454i	\(0.334245\pi\)
\(31\)	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	0.708583	−	0.705627i	\(-0.249335\pi\)
							−0.742954	+	0.669342i	\(0.766576\pi\)
\(43\)	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	0.247362	−	0.968923i	\(-0.420436\pi\)
							−0.389444	+	0.921050i	\(0.627333\pi\)

(See \(a_n\) instead)

Twists

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

    

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