Embedded newform 483.2.y.a.340.16

• Label: 483.2.y.a.340.16
• Level: $483$
• Weight: $2$
• Character: 483.340
• 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			340.16
Character	\(\chi\)	\(=\)	483.340
Dual form			483.2.y.a.331.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.632789 + 2.60839i) q^{2} +(0.327068 + 0.945001i) q^{3} +(-4.62562 + 2.38467i) q^{4} +(1.95956 - 0.784488i) q^{5} +(-2.25797 + 1.45111i) q^{6} +(0.712373 + 2.54804i) q^{7} +(-5.63184 - 6.49949i) q^{8} +(-0.786053 + 0.618159i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 320 q + 2 q^{2} - 16 q^{3} + 18 q^{4} - 2 q^{5} - 18 q^{6} + 2 q^{7} - 12 q^{8} + 16 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(323\)	\(346\)	\(442\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{6}{11}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.632789	+	2.60839i	0.447450	+	1.84441i	0.530244	+	0.847845i	\(0.322100\pi\)
							−0.0827941	+	0.996567i	\(0.526384\pi\)
\(3\)	0.327068	+	0.945001i	0.188833	+	0.545596i
							\(5\)	1.95956	−	0.784488i	0.876341	−	0.350834i	0.110509	−	0.993875i	\(-0.464752\pi\)
0.765832	+	0.643041i	\(0.222328\pi\)
\(7\)	0.712373	+	2.54804i	0.269252	+	0.963070i
							\(11\)	−1.30113	+	5.36334i	−0.392306	+	1.61711i	0.342973	+	0.939345i	\(0.388566\pi\)
−0.735279	+	0.677764i	\(0.762949\pi\)
\(13\)	0.558523	−	1.22299i	0.154906	−	0.339198i	−0.816228	−	0.577730i	\(-0.803939\pi\)
							0.971135	+	0.238532i	\(0.0766661\pi\)
\(17\)	−0.291393	−	6.11709i	−0.0706732	−	1.48361i	−0.704261	−	0.709942i	\(-0.748721\pi\)
							0.633587	−	0.773671i	\(-0.281582\pi\)
\(19\)	0.188631	−	3.95985i	0.0432749	−	0.908452i	−0.867977	−	0.496604i	\(-0.834580\pi\)
							0.911252	−	0.411848i	\(-0.135117\pi\)
\(23\)	1.46378	+	4.56698i	0.305220	+	0.952282i
							\(29\)	2.23130	−	1.43397i	0.414342	−	0.266282i	−0.316819	−	0.948486i	\(-0.602615\pi\)
0.731162	+	0.682204i	\(0.238979\pi\)
\(31\)	5.25031	−	1.01191i	0.942983	−	0.181745i	0.305485	−	0.952197i	\(-0.401181\pi\)
							0.637498	+	0.770452i	\(0.279969\pi\)
\(37\)	−6.68175	+	5.25458i	−1.09847	+	0.863848i	−0.991460	−	0.130409i	\(-0.958371\pi\)
							−0.107012	+	0.994258i	\(0.534128\pi\)
\(41\)	0.829006	−	5.76586i	0.129469	−	0.900476i	−0.816760	−	0.576978i	\(-0.804232\pi\)
							0.946229	−	0.323498i	\(-0.104859\pi\)
\(43\)	5.84189	−	6.74190i	0.890880	−	1.02813i	−0.108541	−	0.994092i	\(-0.534618\pi\)
							0.999421	−	0.0340382i	\(-0.0108368\pi\)
\(47\)	5.81748	+	10.0762i	0.848567	+	1.46976i	0.882487	+	0.470337i	\(0.155868\pi\)
							−0.0339196	+	0.999425i	\(0.510799\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	483.2.y.a.340.16	yes	320
7.2	even	3	inner	483.2.y.a.478.16	yes	320
23.9	even	11	inner	483.2.y.a.193.16	✓	320
161.9	even	33	inner	483.2.y.a.331.16	yes	320

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.y.a.193.16	✓	320	23.9	even	11	inner
483.2.y.a.331.16	yes	320	161.9	even	33	inner
483.2.y.a.340.16	yes	320	1.1	even	1	trivial
483.2.y.a.478.16	yes	320	7.2	even	3	inner