Embedded newform 483.2.y.a.4.10

• Label: 483.2.y.a.4.10
• 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			4.10
Character	\(\chi\)	\(=\)	483.4
Dual form			483.2.y.a.121.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.327609 - 0.460062i) q^{2} +(-0.235759 + 0.971812i) q^{3} +(0.549806 + 1.58856i) q^{4} +(-0.198810 + 4.17354i) q^{5} +(0.369857 + 0.426838i) q^{6} +(2.54045 + 0.738979i) q^{7} +(1.99478 + 0.585719i) q^{8} +(-0.888835 - 0.458227i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 320 q + 2 q^{2} - 16 q^{3} + 18 q^{4} - 2 q^{5} - 18 q^{6} + 2 q^{7} - 12 q^{8} + 16 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.327609	−	0.460062i	0.231654	−	0.325313i	−0.682344	−	0.731031i	\(-0.739039\pi\)
							0.913998	+	0.405718i	\(0.132979\pi\)
\(3\)	−0.235759	+	0.971812i	−0.136115	+	0.561076i
							\(5\)	−0.198810	+	4.17354i	−0.0889105	+	1.86646i	0.311657	+	0.950195i	\(0.399116\pi\)
−0.400567	+	0.916267i	\(0.631187\pi\)
\(7\)	2.54045	+	0.738979i	0.960202	+	0.279308i
							\(11\)	−3.13960	−	4.40895i	−0.946624	−	1.32935i	−0.944208	−	0.329350i	\(-0.893170\pi\)
−0.00241619	−	0.999997i	\(-0.500769\pi\)
\(13\)	0.0787052	−	0.547407i	0.0218289	−	0.151823i	−0.975992	−	0.217807i	\(-0.930110\pi\)
							0.997821	+	0.0659840i	\(0.0210186\pi\)
\(17\)	−1.13830	+	0.219389i	−0.276078	+	0.0532097i	−0.325411	−	0.945573i	\(-0.605503\pi\)
							0.0493327	+	0.998782i	\(0.484291\pi\)
\(19\)	3.13924	+	0.605038i	0.720191	+	0.138805i	0.536161	−	0.844115i	\(-0.319874\pi\)
							0.184029	+	0.982921i	\(0.441086\pi\)
\(23\)	3.87686	−	2.82311i	0.808382	−	0.588659i
							\(29\)	−2.55233	−	2.94555i	−0.473956	−	0.546974i	0.467552	−	0.883966i	\(-0.345136\pi\)
−0.941508	+	0.336992i	\(0.890591\pi\)
\(31\)	3.98940	−	3.80388i	0.716517	−	0.683197i	−0.241341	−	0.970440i	\(-0.577587\pi\)
							0.957858	+	0.287243i	\(0.0927388\pi\)
\(37\)	8.01091	+	4.12991i	1.31699	+	0.678953i	0.966271	−	0.257529i	\(-0.0829082\pi\)
							0.350715	+	0.936482i	\(0.385939\pi\)
\(41\)	2.12944	+	1.36851i	0.332563	+	0.213725i	0.696256	−	0.717794i	\(-0.254848\pi\)
							−0.363693	+	0.931519i	\(0.618484\pi\)
\(43\)	0.539191	−	0.158321i	0.0822258	−	0.0241437i	−0.240361	−	0.970684i	\(-0.577266\pi\)
							0.322587	+	0.946540i	\(0.395448\pi\)
\(47\)	3.26385	+	5.65316i	0.476082	+	0.824598i	0.999625	−	0.0274014i	\(-0.00872323\pi\)
							−0.523543	+	0.852000i	\(0.675390\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	483.2.y.a.4.10	✓	320
7.2	even	3	inner	483.2.y.a.142.7	yes	320
23.6	even	11	inner	483.2.y.a.466.7	yes	320
161.121	even	33	inner	483.2.y.a.121.10	yes	320

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.y.a.4.10	✓	320	1.1	even	1	trivial
483.2.y.a.121.10	yes	320	161.121	even	33	inner
483.2.y.a.142.7	yes	320	7.2	even	3	inner
483.2.y.a.466.7	yes	320	23.6	even	11	inner