Embedded newform 177.4.f.a.2.6

• Label: 177.4.f.a.2.6
• 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			2.6
Character	\(\chi\)	\(=\)	177.2
Dual form			177.4.f.a.89.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.254048 + 4.68564i) q^{2} +(1.63609 + 4.93186i) q^{3} +(-13.9375 - 1.51580i) q^{4} +(5.44329 - 16.1551i) q^{5} +(-23.5245 + 6.41321i) q^{6} +(-18.4501 - 27.2119i) q^{7} +(4.56997 - 27.8756i) q^{8} +(-21.6464 + 16.1380i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1624 q - 21 q^{3} - 278 q^{4} - 29 q^{6} - 42 q^{7} - 25 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.254048	+	4.68564i	−0.0898195	+	1.65662i	0.512736	+	0.858546i	\(0.328632\pi\)
							−0.602556	+	0.798077i	\(0.705851\pi\)
\(3\)	1.63609	+	4.93186i	0.314866	+	0.949136i
							\(5\)	5.44329	−	16.1551i	0.486862	−	1.44496i	−0.369816	−	0.929105i	\(-0.620579\pi\)
0.856678	−	0.515851i	\(-0.172524\pi\)
\(7\)	−18.4501	−	27.2119i	−0.996213	−	1.46930i	−0.879362	−	0.476154i	\(-0.842030\pi\)
							−0.116851	−	0.993149i	\(-0.537280\pi\)
\(11\)	7.76280	−	1.70872i	0.212779	−	0.0468362i	−0.107301	−	0.994227i	\(-0.534221\pi\)
							0.320081	+	0.947390i	\(0.396290\pi\)
\(13\)	54.8811	−	46.6164i	1.17087	−	0.994543i	0.170910	−	0.985287i	\(-0.445329\pi\)
							0.999957	−	0.00925643i	\(-0.00294646\pi\)
\(17\)	−106.595	−	72.2733i	−1.52077	−	1.03111i	−0.982049	−	0.188625i	\(-0.939597\pi\)
							−0.538722	−	0.842484i	\(-0.681093\pi\)
\(19\)	−34.0425	+	64.2111i	−0.411047	+	0.775317i	−0.999447	−	0.0332529i	\(-0.989413\pi\)
							0.588400	+	0.808570i	\(0.299758\pi\)
\(23\)	10.2625	−	9.72113i	0.0930380	−	0.0881302i	−0.639767	−	0.768569i	\(-0.720969\pi\)
							0.732805	+	0.680438i	\(0.238211\pi\)
\(29\)	124.184	−	6.73309i	0.795189	−	0.0431139i	0.347932	−	0.937520i	\(-0.386884\pi\)
							0.447257	+	0.894406i	\(0.352401\pi\)
\(31\)	−146.250	+	77.5367i	−0.847330	+	0.449226i	−0.834707	−	0.550694i	\(-0.814363\pi\)
							−0.0126230	+	0.999920i	\(0.504018\pi\)
\(37\)	−245.247	+	40.2061i	−1.08968	+	0.178645i	−0.679730	−	0.733462i	\(-0.737903\pi\)
							−0.409953	+	0.912107i	\(0.634455\pi\)
\(41\)	105.013	−	110.861i	0.400006	−	0.422282i	−0.494611	−	0.869114i	\(-0.664690\pi\)
							0.894617	+	0.446833i	\(0.147448\pi\)
\(43\)	9.13165	−	41.4855i	0.0323852	−	0.147127i	−0.957653	−	0.287923i	\(-0.907035\pi\)
							0.990039	+	0.140796i	\(0.0449662\pi\)
\(47\)	61.0250	−	20.5617i	0.189392	−	0.0638135i	−0.223007	−	0.974817i	\(-0.571587\pi\)
							0.412399	+	0.911003i	\(0.364691\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.4.f.a.2.6	✓	1624
3.2	odd	2	inner	177.4.f.a.2.53	yes	1624
59.30	odd	58	inner	177.4.f.a.89.53	yes	1624
177.89	even	58	inner	177.4.f.a.89.6	yes	1624

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.4.f.a.2.6	✓	1624	1.1	even	1	trivial
177.4.f.a.2.53	yes	1624	3.2	odd	2	inner
177.4.f.a.89.6	yes	1624	177.89	even	58	inner
177.4.f.a.89.53	yes	1624	59.30	odd	58	inner