Embedded newform 483.2.bf.a.19.16

• Label: 483.2.bf.a.19.16
• Level: $483$
• Weight: $2$
• Character: 483.19
• Analytic conductor: $3.857$
• Analytic rank: $0$
• Dimension: $640$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 483 = 3 \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	483.bf (of order \(66\), degree \(20\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.85677441763\)
Analytic rank:	\(0\)
Dimension:	\(640\)
Relative dimension:	\(32\) over \(\Q(\zeta_{66})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{66}]$

Embedding invariants

Embedding label			19.16
Character	\(\chi\)	\(=\)	483.19
Dual form			483.2.bf.a.178.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.0349460 - 0.00333694i) q^{2} +(0.690079 - 0.723734i) q^{3} +(-1.96265 - 0.378269i) q^{4} +(2.03168 - 1.04740i) q^{5} +(-0.0265306 + 0.0229889i) q^{6} +(2.31441 + 1.28200i) q^{7} +(0.134690 + 0.0395486i) q^{8} +(-0.0475819 - 0.998867i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 640 q - 4 q^{2} + 36 q^{4} + 24 q^{8} - 32 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(323\)	\(346\)	\(442\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{15}{22}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.0349460	−	0.00333694i	−0.0247105	−	0.00235957i	0.0826934	−	0.996575i	\(-0.473648\pi\)
							−0.107404	+	0.994215i	\(0.534254\pi\)
\(3\)	0.690079	−	0.723734i	0.398417	−	0.417848i
							\(5\)	2.03168	−	1.04740i	0.908594	−	0.468413i	0.0604458	−	0.998171i	\(-0.480748\pi\)
0.848149	+	0.529759i	\(0.177717\pi\)
\(7\)	2.31441	+	1.28200i	0.874763	+	0.484551i
							\(11\)	0.370720	+	3.88236i	0.111776	+	1.17058i	0.861123	+	0.508397i	\(0.169762\pi\)
−0.749346	+	0.662178i	\(0.769632\pi\)
\(13\)	2.62062	+	0.376788i	0.726828	+	0.104502i	0.495790	−	0.868443i	\(-0.334879\pi\)
							0.231038	+	0.972945i	\(0.425788\pi\)
\(17\)	1.04528	−	3.02013i	0.253517	−	0.732488i	−0.744386	−	0.667750i	\(-0.767258\pi\)
							0.997902	−	0.0647383i	\(-0.0206213\pi\)
\(19\)	−2.37063	−	6.84947i	−0.543859	−	1.57138i	−0.797020	−	0.603953i	\(-0.793592\pi\)
							0.253161	−	0.967424i	\(-0.418530\pi\)
\(23\)	0.436634	−	4.77591i	0.0910445	−	0.995847i
							\(29\)	0.975526	+	1.12582i	0.181151	+	0.209059i	0.839061	−	0.544037i	\(-0.183105\pi\)
−0.657910	+	0.753096i	\(0.728559\pi\)
\(31\)	1.12219	−	0.272241i	0.201552	−	0.0488960i	−0.133712	−	0.991020i	\(-0.542690\pi\)
							0.335264	+	0.942124i	\(0.391175\pi\)
\(37\)	−0.0348868	+	0.00166186i	−0.00573536	+	0.000273209i	−0.0504494	−	0.998727i	\(-0.516065\pi\)
							0.0447140	+	0.999000i	\(0.485762\pi\)
\(41\)	3.74996	−	5.83505i	0.585645	−	0.911282i	−0.414354	−	0.910116i	\(-0.635993\pi\)
							0.999999	−	0.00116586i	\(-0.000371105\pi\)
\(43\)	3.20775	+	10.9246i	0.489177	+	1.66598i	0.720772	+	0.693172i	\(0.243787\pi\)
							−0.231595	+	0.972812i	\(0.574394\pi\)
\(47\)	−10.4537	−	6.03543i	−1.52482	−	0.880358i	−0.999567	−	0.0294148i	\(-0.990636\pi\)
							−0.525258	−	0.850943i	\(-0.676031\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	483.2.bf.a.19.16	✓	640
7.3	odd	6	inner	483.2.bf.a.157.17	yes	640
23.17	odd	22	inner	483.2.bf.a.40.17	yes	640
161.17	even	66	inner	483.2.bf.a.178.16	yes	640

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.bf.a.19.16	✓	640	1.1	even	1	trivial
483.2.bf.a.40.17	yes	640	23.17	odd	22	inner
483.2.bf.a.157.17	yes	640	7.3	odd	6	inner
483.2.bf.a.178.16	yes	640	161.17	even	66	inner