Embedded newform 483.2.y.b.4.7

• Label: 483.2.y.b.4.7
• Level: $483$
• Weight: $2$
• Character: 483.4
• Analytic conductor: $3.857$
• Analytic rank: $0$
• Dimension: $320$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 483 = 3 \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	483.y (of order \(33\), degree \(20\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.85677441763\)
Analytic rank:	\(0\)
Dimension:	\(320\)
Relative dimension:	\(16\) over \(\Q(\zeta_{33})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{33}]$

Embedding invariants

Embedding label			4.7
Character	\(\chi\)	\(=\)	483.4
Dual form			483.2.y.b.121.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.342635 + 0.481163i) q^{2} +(0.235759 - 0.971812i) q^{3} +(0.540016 + 1.56028i) q^{4} +(0.0918733 - 1.92866i) q^{5} +(0.386821 + 0.446415i) q^{6} +(-2.12937 + 1.57028i) q^{7} +(-2.06930 - 0.607603i) q^{8} +(-0.888835 - 0.458227i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 320 q + 2 q^{2} + 16 q^{3} + 18 q^{4} - 2 q^{5} + 18 q^{6} + 2 q^{7} - 12 q^{8} + 16 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(323\)	\(346\)	\(442\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{2}{11}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.342635	+	0.481163i	−0.242279	+	0.340234i	−0.917754	−	0.397149i	\(-0.869999\pi\)
							0.675475	+	0.737383i	\(0.263939\pi\)
\(3\)	0.235759	−	0.971812i	0.136115	−	0.561076i
							\(5\)	0.0918733	−	1.92866i	0.0410870	−	0.862522i	−0.880428	−	0.474179i	\(-0.842745\pi\)
0.921515	−	0.388342i	\(-0.126952\pi\)
\(7\)	−2.12937	+	1.57028i	−0.804826	+	0.593510i
							\(11\)	3.54514	+	4.97845i	1.06890	+	1.50106i	0.851056	+	0.525075i	\(0.175963\pi\)
0.217843	+	0.975984i	\(0.430098\pi\)
\(13\)	−0.534711	+	3.71900i	−0.148302	+	1.03146i	0.770696	+	0.637203i	\(0.219909\pi\)
							−0.918998	+	0.394261i	\(0.871000\pi\)
\(17\)	4.91186	−	0.946684i	1.19130	−	0.229605i	0.445197	−	0.895433i	\(-0.353134\pi\)
							0.746105	+	0.665828i	\(0.231922\pi\)
\(19\)	1.97899	+	0.381419i	0.454012	+	0.0875036i	0.411130	−	0.911577i	\(-0.365134\pi\)
							0.0428815	+	0.999080i	\(0.486346\pi\)
\(23\)	2.80757	+	3.88813i	0.585420	+	0.810730i
							\(29\)	−1.00012	−	1.15420i	−0.185717	−	0.214329i	0.655255	−	0.755408i	\(-0.272561\pi\)
−0.840972	+	0.541079i	\(0.818016\pi\)
\(31\)	2.47476	−	2.35967i	0.444479	−	0.423810i	−0.434499	−	0.900672i	\(-0.643074\pi\)
							0.878978	+	0.476862i	\(0.158226\pi\)
\(37\)	−3.64600	−	1.87965i	−0.599399	−	0.309012i	0.131693	−	0.991291i	\(-0.457959\pi\)
							−0.731092	+	0.682279i	\(0.760989\pi\)
\(41\)	−0.491645	−	0.315961i	−0.0767821	−	0.0493449i	0.501687	−	0.865050i	\(-0.332713\pi\)
							−0.578469	+	0.815705i	\(0.696349\pi\)
\(43\)	−7.39962	+	2.17272i	−1.12843	+	0.331337i	−0.792091	−	0.610403i	\(-0.791007\pi\)
							−0.336340	+	0.941740i	\(0.609189\pi\)
\(47\)	1.57667	+	2.73088i	0.229982	+	0.398340i	0.957802	−	0.287428i	\(-0.0928001\pi\)
							−0.727821	+	0.685767i	\(0.759467\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	483.2.y.b.4.7	✓	320
7.2	even	3	inner	483.2.y.b.142.10	yes	320
23.6	even	11	inner	483.2.y.b.466.10	yes	320
161.121	even	33	inner	483.2.y.b.121.7	yes	320

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.y.b.4.7	✓	320	1.1	even	1	trivial
483.2.y.b.121.7	yes	320	161.121	even	33	inner
483.2.y.b.142.10	yes	320	7.2	even	3	inner
483.2.y.b.466.10	yes	320	23.6	even	11	inner