Embedded newform 177.4.f.a.2.19

• Label: 177.4.f.a.2.19
• Level: $177$
• Weight: $4$
• Character: 177.2
• 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			2.19
Character	\(\chi\)	\(=\)	177.2
Dual form			177.4.f.a.89.19

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.123545 + 2.27865i) q^{2} +(-4.04503 - 3.26155i) q^{3} +(2.77611 + 0.301920i) q^{4} +(-0.776377 + 2.30421i) q^{5} +(7.93169 - 8.81427i) q^{6} +(-10.4076 - 15.3501i) q^{7} +(-3.98444 + 24.3040i) q^{8} +(5.72454 + 26.3862i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1624 q - 21 q^{3} - 278 q^{4} - 29 q^{6} - 42 q^{7} - 25 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.123545	+	2.27865i	−0.0436797	+	0.805625i	0.891840	+	0.452351i	\(0.149415\pi\)
							−0.935520	+	0.353274i	\(0.885068\pi\)
\(3\)	−4.04503	−	3.26155i	−0.778466	−	0.627686i
							\(5\)	−0.776377	+	2.30421i	−0.0694413	+	0.206094i	−0.976755	−	0.214361i	\(-0.931233\pi\)
0.907313	+	0.420455i	\(0.138130\pi\)
\(7\)	−10.4076	−	15.3501i	−0.561957	−	0.828825i	0.435493	−	0.900192i	\(-0.356574\pi\)
							−0.997450	+	0.0713670i	\(0.977264\pi\)
\(11\)	2.09406	−	0.460938i	0.0573985	−	0.0126344i	−0.186178	−	0.982516i	\(-0.559610\pi\)
							0.243577	+	0.969882i	\(0.421679\pi\)
\(13\)	59.2134	−	50.2963i	1.26330	−	1.07305i	0.269367	−	0.963038i	\(-0.413186\pi\)
							0.993930	−	0.110016i	\(-0.0350902\pi\)
\(17\)	−68.4059	−	46.3803i	−0.975933	−	0.661699i	−0.0346793	−	0.999398i	\(-0.511041\pi\)
							−0.941254	+	0.337699i	\(0.890351\pi\)
\(19\)	61.8020	−	116.571i	0.746229	−	1.40754i	−0.161024	−	0.986951i	\(-0.551480\pi\)
							0.907253	−	0.420586i	\(-0.138176\pi\)
\(23\)	25.7877	−	24.4274i	0.233787	−	0.221455i	−0.561833	−	0.827250i	\(-0.689904\pi\)
							0.795620	+	0.605796i	\(0.207145\pi\)
\(29\)	152.730	−	8.28078i	0.977974	−	0.0530242i	0.441784	−	0.897121i	\(-0.354346\pi\)
							0.536190	+	0.844097i	\(0.319863\pi\)
\(31\)	194.084	−	102.897i	1.12447	−	0.596156i	0.200909	−	0.979610i	\(-0.435610\pi\)
							0.923560	+	0.383454i	\(0.125266\pi\)
\(37\)	−246.655	+	40.4371i	−1.09594	+	0.179671i	−0.682519	−	0.730868i	\(-0.739116\pi\)
							−0.413424	+	0.910539i	\(0.635667\pi\)
\(41\)	248.110	−	261.926i	0.945079	−	0.997708i	−0.0549206	−	0.998491i	\(-0.517491\pi\)
							1.00000		0.000782822i	\(0.000249180\pi\)
\(43\)	16.9639	−	77.0678i	0.0601621	−	0.273319i	−0.936931	−	0.349515i	\(-0.886346\pi\)
							0.997093	+	0.0761958i	\(0.0242774\pi\)
\(47\)	132.324	−	44.5851i	0.410669	−	0.138370i	−0.106382	−	0.994325i	\(-0.533927\pi\)
							0.517050	+	0.855955i	\(0.327030\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.4.f.a.2.19	✓	1624
3.2	odd	2	inner	177.4.f.a.2.40	yes	1624
59.30	odd	58	inner	177.4.f.a.89.40	yes	1624
177.89	even	58	inner	177.4.f.a.89.19	yes	1624

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.4.f.a.2.19	✓	1624	1.1	even	1	trivial
177.4.f.a.2.40	yes	1624	3.2	odd	2	inner
177.4.f.a.89.19	yes	1624	177.89	even	58	inner
177.4.f.a.89.40	yes	1624	59.30	odd	58	inner