Embedded newform 483.2.y.a.25.13

• Label: 483.2.y.a.25.13
• Level: $483$
• Weight: $2$
• Character: 483.25
• 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			25.13
Character	\(\chi\)	\(=\)	483.25
Dual form			483.2.y.a.58.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.0853128 - 1.79094i) q^{2} +(-0.928368 - 0.371662i) q^{3} +(-1.20923 - 0.115468i) q^{4} +(-0.898382 - 3.70318i) q^{5} +(-0.744826 + 1.63094i) q^{6} +(2.10874 - 1.59788i) q^{7} +(0.200374 - 1.39363i) q^{8} +(0.723734 + 0.690079i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 320 q + 2 q^{2} - 16 q^{3} + 18 q^{4} - 2 q^{5} - 18 q^{6} + 2 q^{7} - 12 q^{8} + 16 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.0853128	−	1.79094i	0.0603253	−	1.26638i	−0.742368	−	0.669992i	\(-0.766298\pi\)
							0.802694	−	0.596391i	\(-0.203399\pi\)
\(3\)	−0.928368	−	0.371662i	−0.535993	−	0.214579i
							\(5\)	−0.898382	−	3.70318i	−0.401768	−	1.65611i	−0.709550	−	0.704655i	\(-0.751102\pi\)
0.307781	−	0.951457i	\(-0.400413\pi\)
\(7\)	2.10874	−	1.59788i	0.797029	−	0.603941i
							\(11\)	−0.146714	−	3.07991i	−0.0442360	−	0.928627i	−0.906510	−	0.422185i	\(-0.861263\pi\)
0.862274	−	0.506442i	\(-0.169040\pi\)
\(13\)	3.50975	+	4.05047i	0.973430	+	1.12340i	0.992335	+	0.123577i	\(0.0394365\pi\)
							−0.0189052	+	0.999821i	\(0.506018\pi\)
\(17\)	2.66654	+	3.74463i	0.646731	+	0.908207i	0.999623	−	0.0274573i	\(-0.00874103\pi\)
							−0.352892	+	0.935664i	\(0.614802\pi\)
\(19\)	1.42336	−	1.99883i	0.326541	−	0.458563i	−0.618371	−	0.785886i	\(-0.712207\pi\)
							0.944913	+	0.327323i	\(0.106146\pi\)
\(23\)	−4.15932	−	2.38747i	−0.867279	−	0.497823i
							\(29\)	−2.52742	+	5.53428i	−0.469330	+	1.02769i	0.515931	+	0.856630i	\(0.327446\pi\)
−0.985261	+	0.171059i	\(0.945281\pi\)
\(31\)	6.02413	+	4.73743i	1.08197	+	0.850868i	0.989454	−	0.144844i	\(-0.0462681\pi\)
							0.0925116	+	0.995712i	\(0.470510\pi\)
\(37\)	−4.24337	−	4.04605i	−0.697606	−	0.665166i	0.255843	−	0.966718i	\(-0.417647\pi\)
							−0.953450	+	0.301552i	\(0.902495\pi\)
\(41\)	−1.60565	−	0.471460i	−0.250760	−	0.0736297i	0.153937	−	0.988081i	\(-0.450805\pi\)
							−0.404697	+	0.914451i	\(0.632623\pi\)
\(43\)	0.888338	+	6.17852i	0.135470	+	0.942216i	0.938254	+	0.345946i	\(0.112442\pi\)
							−0.802784	+	0.596270i	\(0.796649\pi\)
\(47\)	4.89353	+	8.47585i	0.713795	+	1.23633i	0.963422	+	0.267988i	\(0.0863586\pi\)
							−0.249627	+	0.968342i	\(0.580308\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	483.2.y.a.25.13	✓	320
7.2	even	3	inner	483.2.y.a.163.4	yes	320
23.12	even	11	inner	483.2.y.a.403.4	yes	320
161.58	even	33	inner	483.2.y.a.58.13	yes	320

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.y.a.25.13	✓	320	1.1	even	1	trivial
483.2.y.a.58.13	yes	320	161.58	even	33	inner
483.2.y.a.163.4	yes	320	7.2	even	3	inner
483.2.y.a.403.4	yes	320	23.12	even	11	inner