Embedded newform 483.2.bf.a.10.23

• Label: 483.2.bf.a.10.23
• Level: $483$
• Weight: $2$
• Character: 483.10
• Analytic conductor: $3.857$
• Analytic rank: $0$
• Dimension: $640$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 483 = 3 \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	483.bf (of order \(66\), degree \(20\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.85677441763\)
Analytic rank:	\(0\)
Dimension:	\(640\)
Relative dimension:	\(32\) over \(\Q(\zeta_{66})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{66}]$

Embedding invariants

Embedding label			10.23
Character	\(\chi\)	\(=\)	483.10
Dual form			483.2.bf.a.145.23

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.467368 - 1.35037i) q^{2} +(-0.458227 - 0.888835i) q^{3} +(-0.0329606 - 0.0259205i) q^{4} +(-3.87397 + 0.369919i) q^{5} +(-1.41442 + 0.203362i) q^{6} +(-0.818796 + 2.51586i) q^{7} +(2.35383 - 1.51272i) q^{8} +(-0.580057 + 0.814576i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 640 q - 4 q^{2} + 36 q^{4} + 24 q^{8} - 32 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/483\mathbb{Z}\right)^\times\).

\(n\)	\(323\)	\(346\)	\(442\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{3}{22}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.467368	−	1.35037i	0.330479	−	0.954856i	−0.650291	−	0.759686i	\(-0.725353\pi\)
							0.980769	−	0.195170i	\(-0.0625259\pi\)
\(3\)	−0.458227	−	0.888835i	−0.264557	−	0.513169i
							\(5\)	−3.87397	+	0.369919i	−1.73249	+	0.165433i	−0.913082	−	0.407775i	\(-0.866305\pi\)
−0.819410	+	0.573208i	\(0.805699\pi\)
\(7\)	−0.818796	+	2.51586i	−0.309476	+	0.950907i
							\(11\)	−2.90865	+	1.00669i	−0.876992	+	0.303530i	−0.728222	−	0.685342i	\(-0.759653\pi\)
−0.148771	+	0.988872i	\(0.547532\pi\)
\(13\)	1.82413	+	6.21240i	0.505921	+	1.72301i	0.675378	+	0.737472i	\(0.263981\pi\)
							−0.169456	+	0.985538i	\(0.554201\pi\)
\(17\)	−5.39006	−	2.15785i	−1.30728	−	0.523356i	−0.389713	−	0.920936i	\(-0.627426\pi\)
							−0.917567	+	0.397580i	\(0.869850\pi\)
\(19\)	2.04711	−	0.819541i	0.469640	−	0.188016i	−0.124757	−	0.992187i	\(-0.539815\pi\)
							0.594397	+	0.804172i	\(0.297391\pi\)
\(23\)	1.31983	+	4.61065i	0.275203	+	0.961386i
							\(29\)	0.917622	+	6.38220i	0.170398	+	1.18515i	0.878045	+	0.478579i	\(0.158848\pi\)
−0.707647	+	0.706567i	\(0.750243\pi\)
\(31\)	−9.29180	+	0.442623i	−1.66886	+	0.0794974i	−0.860583	−	0.509310i	\(-0.829901\pi\)
							−0.808273	+	0.588807i	\(0.799598\pi\)
\(37\)	−5.57057	−	3.96679i	−0.915796	−	0.652135i	0.0218371	−	0.999762i	\(-0.493048\pi\)
							−0.937633	+	0.347626i	\(0.886988\pi\)
\(41\)	0.0145312	+	0.00663618i	0.00226939	+	0.00103640i	0.416549	−	0.909113i	\(-0.363239\pi\)
							−0.414280	+	0.910149i	\(0.635967\pi\)
\(43\)	1.64739	−	2.56339i	0.251225	−	0.390914i	−0.692620	−	0.721303i	\(-0.743544\pi\)
							0.943845	+	0.330389i	\(0.107180\pi\)
\(47\)	−3.20747	+	1.85183i	−0.467857	+	0.270117i	−0.715342	−	0.698774i	\(-0.753729\pi\)
							0.247485	+	0.968892i	\(0.420396\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	483.2.bf.a.10.23	✓	640
7.5	odd	6	inner	483.2.bf.a.355.10	yes	640
23.7	odd	22	inner	483.2.bf.a.283.10	yes	640
161.145	even	66	inner	483.2.bf.a.145.23	yes	640

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.bf.a.10.23	✓	640	1.1	even	1	trivial
483.2.bf.a.145.23	yes	640	161.145	even	66	inner
483.2.bf.a.283.10	yes	640	23.7	odd	22	inner
483.2.bf.a.355.10	yes	640	7.5	odd	6	inner