Embedded newform 483.2.y.b.4.6

• Label: 483.2.y.b.4.6
• 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			4.6
Character	\(\chi\)	\(=\)	483.4
Dual form			483.2.y.b.121.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.765883 + 1.07553i) q^{2} +(0.235759 - 0.971812i) q^{3} +(0.0839437 + 0.242539i) q^{4} +(-0.0355806 + 0.746929i) q^{5} +(0.864651 + 0.997860i) q^{6} +(-2.58720 + 0.553515i) q^{7} +(-2.85890 - 0.839448i) q^{8} +(-0.888835 - 0.458227i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 320 q + 2 q^{2} + 16 q^{3} + 18 q^{4} - 2 q^{5} + 18 q^{6} + 2 q^{7} - 12 q^{8} + 16 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.765883	+	1.07553i	−0.541561	+	0.760516i	−0.991154	−	0.132718i	\(-0.957630\pi\)
							0.449593	+	0.893234i	\(0.351569\pi\)
\(3\)	0.235759	−	0.971812i	0.136115	−	0.561076i
							\(5\)	−0.0355806	+	0.746929i	−0.0159121	+	0.334037i	0.976849	+	0.213930i	\(0.0686266\pi\)
−0.992761	+	0.120106i	\(0.961676\pi\)
\(7\)	−2.58720	+	0.553515i	−0.977871	+	0.209209i
							\(11\)	−2.96822	−	4.16828i	−0.894952	−	1.25678i	−0.965798	−	0.259296i	\(-0.916509\pi\)
0.0708462	−	0.997487i	\(-0.477430\pi\)
\(13\)	0.287414	−	1.99901i	0.0797142	−	0.554424i	−0.910353	−	0.413832i	\(-0.864190\pi\)
							0.990068	−	0.140593i	\(-0.0449008\pi\)
\(17\)	2.11552	−	0.407732i	0.513088	−	0.0988896i	0.0738661	−	0.997268i	\(-0.476466\pi\)
							0.439222	+	0.898379i	\(0.355254\pi\)
\(19\)	−3.04600	−	0.587069i	−0.698801	−	0.134683i	−0.172536	−	0.985003i	\(-0.555196\pi\)
							−0.526265	+	0.850320i	\(0.676408\pi\)
\(23\)	−4.50172	−	1.65364i	−0.938673	−	0.344808i
							\(29\)	−2.40197	−	2.77202i	−0.446035	−	0.514751i	0.487556	−	0.873091i	\(-0.337888\pi\)
−0.933591	+	0.358340i	\(0.883343\pi\)
\(31\)	−3.06362	+	2.92116i	−0.550243	+	0.524656i	−0.913395	−	0.407074i	\(-0.866549\pi\)
							0.363152	+	0.931730i	\(0.381701\pi\)
\(37\)	7.72948	+	3.98483i	1.27072	+	0.655101i	0.955800	−	0.294017i	\(-0.0949923\pi\)
							0.314919	+	0.949119i	\(0.398023\pi\)
\(41\)	−9.28182	−	5.96507i	−1.44958	−	0.931587i	−0.999250	−	0.0387233i	\(-0.987671\pi\)
							−0.450327	−	0.892864i	\(-0.648693\pi\)
\(43\)	−5.97029	+	1.75304i	−0.910461	+	0.267335i	−0.703235	−	0.710958i	\(-0.748262\pi\)
							−0.207226	+	0.978293i	\(0.566444\pi\)
\(47\)	−0.132826	−	0.230061i	−0.0193747	−	0.0335579i	0.856176	−	0.516685i	\(-0.172834\pi\)
							−0.875550	+	0.483127i	\(0.839501\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	483.2.y.b.4.6	✓	320
7.2	even	3	inner	483.2.y.b.142.11	yes	320
23.6	even	11	inner	483.2.y.b.466.11	yes	320
161.121	even	33	inner	483.2.y.b.121.6	yes	320

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.y.b.4.6	✓	320	1.1	even	1	trivial
483.2.y.b.121.6	yes	320	161.121	even	33	inner
483.2.y.b.142.11	yes	320	7.2	even	3	inner
483.2.y.b.466.11	yes	320	23.6	even	11	inner