Embedded newform 177.4.f.a.89.17

• Label: 177.4.f.a.89.17
• 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			89.17
Character	\(\chi\)	\(=\)	177.89
Dual form			177.4.f.a.2.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.150879 - 2.78279i) q^{2} +(-4.15765 - 3.11672i) q^{3} +(0.231945 - 0.0252256i) q^{4} +(4.42533 + 13.1339i) q^{5} +(-8.04587 + 12.0401i) q^{6} +(-0.713636 + 1.05253i) q^{7} +(-3.71213 - 22.6430i) q^{8} +(7.57215 + 25.9165i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1624 q - 21 q^{3} - 278 q^{4} - 29 q^{6} - 42 q^{7} - 25 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.150879	−	2.78279i	−0.0533436	−	0.983865i	−0.895620	−	0.444819i	\(-0.853268\pi\)
							0.842277	−	0.539046i	\(-0.181215\pi\)
\(3\)	−4.15765	−	3.11672i	−0.800141	−	0.599813i
							\(5\)	4.42533	+	13.1339i	0.395814	+	1.17473i	0.941578	+	0.336795i	\(0.109343\pi\)
−0.545764	+	0.837939i	\(0.683761\pi\)
\(7\)	−0.713636	+	1.05253i	−0.0385327	+	0.0568315i	−0.846467	−	0.532440i	\(-0.821275\pi\)
							0.807935	+	0.589272i	\(0.200585\pi\)
\(11\)	30.4374	+	6.69977i	0.834292	+	0.183642i	0.611520	−	0.791229i	\(-0.290558\pi\)
							0.222772	+	0.974871i	\(0.428489\pi\)
\(13\)	39.4789	+	33.5337i	0.842268	+	0.715429i	0.960717	−	0.277531i	\(-0.0895162\pi\)
							−0.118449	+	0.992960i	\(0.537792\pi\)
\(17\)	83.0034	−	56.2777i	1.18419	−	0.802903i	0.199964	−	0.979803i	\(-0.435917\pi\)
							0.984229	+	0.176900i	\(0.0566070\pi\)
\(19\)	48.6265	+	91.7193i	0.587141	+	1.10747i	0.982028	+	0.188736i	\(0.0604390\pi\)
							−0.394887	+	0.918730i	\(0.629216\pi\)
\(23\)	−128.540	−	121.760i	−1.16532	−	1.10385i	−0.992915	−	0.118828i	\(-0.962086\pi\)
							−0.172409	−	0.985025i	\(-0.555155\pi\)
\(29\)	162.239	+	8.79632i	1.03886	+	0.0563253i	0.565667	−	0.824634i	\(-0.308619\pi\)
							0.473193	+	0.880959i	\(0.343101\pi\)
\(31\)	−283.306	−	150.199i	−1.64140	−	0.870213i	−0.994159	−	0.107924i	\(-0.965580\pi\)
							−0.647237	−	0.762289i	\(-0.724076\pi\)
\(37\)	264.192	+	43.3122i	1.17386	+	0.192445i	0.716988	−	0.697085i	\(-0.245520\pi\)
							0.456876	+	0.889530i	\(0.348968\pi\)
\(41\)	−144.681	−	152.738i	−0.551108	−	0.581798i	0.389613	−	0.920979i	\(-0.372609\pi\)
							−0.940721	+	0.339181i	\(0.889850\pi\)
\(43\)	101.490	+	461.075i	0.359933	+	1.63519i	0.714123	+	0.700020i	\(0.246826\pi\)
							−0.354190	+	0.935174i	\(0.615243\pi\)
\(47\)	353.055	+	118.958i	1.09571	+	0.369188i	0.808390	−	0.588647i	\(-0.200339\pi\)
							0.287320	+	0.957835i	\(0.407236\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.4.f.a.89.17	yes	1624
3.2	odd	2	inner	177.4.f.a.89.42	yes	1624
59.2	odd	58	inner	177.4.f.a.2.42	yes	1624
177.2	even	58	inner	177.4.f.a.2.17	✓	1624

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.4.f.a.2.17	✓	1624	177.2	even	58	inner
177.4.f.a.2.42	yes	1624	59.2	odd	58	inner
177.4.f.a.89.17	yes	1624	1.1	even	1	trivial
177.4.f.a.89.42	yes	1624	3.2	odd	2	inner