Embedded newform 483.2.y.a.121.2

• Label: 483.2.y.a.121.2
• Level: $483$
• Weight: $2$
• Character: 483.121
• 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			121.2
Character	\(\chi\)	\(=\)	483.121
Dual form			483.2.y.a.4.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.42823 - 2.00566i) q^{2} +(-0.235759 - 0.971812i) q^{3} +(-1.32872 + 3.83909i) q^{4} +(0.0678834 + 1.42505i) q^{5} +(-1.61241 + 1.86082i) q^{6} +(-0.624376 - 2.57102i) q^{7} +(4.87268 - 1.43075i) q^{8} +(-0.888835 + 0.458227i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 320 q + 2 q^{2} - 16 q^{3} + 18 q^{4} - 2 q^{5} - 18 q^{6} + 2 q^{7} - 12 q^{8} + 16 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(323\)	\(346\)	\(442\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{9}{11}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.42823	−	2.00566i	−1.00991	−	1.41822i	−0.905826	−	0.423651i	\(-0.860748\pi\)
							−0.104084	−	0.994569i	\(-0.533191\pi\)
\(3\)	−0.235759	−	0.971812i	−0.136115	−	0.561076i
							\(5\)	0.0678834	+	1.42505i	0.0303584	+	0.637301i	0.961857	+	0.273554i	\(0.0881994\pi\)
−0.931498	+	0.363746i	\(0.881498\pi\)
\(7\)	−0.624376	−	2.57102i	−0.235992	−	0.971755i
							\(11\)	−3.24233	+	4.55322i	−0.977600	+	1.37285i	−0.0502156	+	0.998738i	\(0.515991\pi\)
−0.927385	+	0.374109i	\(0.877949\pi\)
\(13\)	0.300255	+	2.08832i	0.0832758	+	0.579196i	0.988147	+	0.153511i	\(0.0490579\pi\)
							−0.904871	+	0.425685i	\(0.860033\pi\)
\(17\)	0.748480	+	0.144258i	0.181533	+	0.0349876i	0.279208	−	0.960231i	\(-0.409928\pi\)
							−0.0976744	+	0.995218i	\(0.531140\pi\)
\(19\)	2.77177	−	0.534216i	0.635888	−	0.122557i	0.138889	−	0.990308i	\(-0.455647\pi\)
							0.497000	+	0.867751i	\(0.334435\pi\)
\(23\)	3.45206	+	3.32916i	0.719804	+	0.694177i
							\(29\)	−2.84225	+	3.28013i	−0.527793	+	0.609105i	−0.955565	−	0.294781i	\(-0.904753\pi\)
0.427772	+	0.903887i	\(0.359299\pi\)
\(31\)	3.67362	+	3.50279i	0.659802	+	0.629120i	0.944117	−	0.329611i	\(-0.106918\pi\)
							−0.284315	+	0.958731i	\(0.591766\pi\)
\(37\)	2.37050	−	1.22208i	0.389708	−	0.200909i	−0.252226	−	0.967668i	\(-0.581163\pi\)
							0.641934	+	0.766760i	\(0.278132\pi\)
\(41\)	−6.50294	+	4.17918i	−1.01559	+	0.652679i	−0.938834	−	0.344371i	\(-0.888092\pi\)
							−0.0767547	+	0.997050i	\(0.524456\pi\)
\(43\)	−0.404503	−	0.118773i	−0.0616861	−	0.0181127i	0.250744	−	0.968053i	\(-0.419325\pi\)
							−0.312430	+	0.949941i	\(0.601143\pi\)
\(47\)	0.625055	−	1.08263i	0.0911736	−	0.157917i	−0.816832	−	0.576876i	\(-0.804272\pi\)
							0.908005	+	0.418959i	\(0.137605\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	483.2.y.a.121.2	yes	320
7.4	even	3	inner	483.2.y.a.466.15	yes	320
23.4	even	11	inner	483.2.y.a.142.15	yes	320
161.4	even	33	inner	483.2.y.a.4.2	✓	320

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.y.a.4.2	✓	320	161.4	even	33	inner
483.2.y.a.121.2	yes	320	1.1	even	1	trivial
483.2.y.a.142.15	yes	320	23.4	even	11	inner
483.2.y.a.466.15	yes	320	7.4	even	3	inner