Embedded newform 495.2.bg.b.16.9

• Label: 495.2.bg.b.16.9
• Level: $495$
• Weight: $2$
• Character: 495.16
• Analytic conductor: $3.953$
• Analytic rank: $0$
• Dimension: $192$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 495 = 3^{2} \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	495.bg (of order \(15\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.95259490005\)
Analytic rank:	\(0\)
Dimension:	\(192\)
Relative dimension:	\(24\) over \(\Q(\zeta_{15})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{15}]$

Embedding invariants

Embedding label			16.9
Character	\(\chi\)	\(=\)	495.16
Dual form			495.2.bg.b.31.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.953180 - 0.202605i) q^{2} +(-1.60122 - 0.660382i) q^{3} +(-0.959588 - 0.427236i) q^{4} +(-0.978148 + 0.207912i) q^{5} +(1.39245 + 0.953877i) q^{6} +(0.137460 + 1.30785i) q^{7} +(2.40483 + 1.74721i) q^{8} +(2.12779 + 2.11483i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 192 q + 2 q^{2} + 8 q^{3} + 24 q^{4} + 24 q^{5} - 15 q^{6} + 8 q^{7} - 12 q^{8} + 12 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/495\mathbb{Z}\right)^\times\).

\(n\)	\(46\)	\(56\)	\(397\)
\(\chi(n)\)	\(e\left(\frac{2}{5}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.953180	−	0.202605i	−0.674000	−	0.143263i	−0.141818	−	0.989893i	\(-0.545295\pi\)
							−0.532182	+	0.846630i	\(0.678628\pi\)
\(3\)	−1.60122	−	0.660382i	−0.924463	−	0.381272i
							\(5\)	−0.978148	+	0.207912i	−0.437441	+	0.0929809i
\(7\)	0.137460	+	1.30785i	0.0519550	+	0.494319i								0.989298	+	0.145911i	\(0.0466112\pi\)
							−0.937343	+	0.348409i	\(0.886722\pi\)
\(11\)	−2.78163	−	1.80626i	−0.838692	−	0.544607i
							\(13\)	−2.12797	+	2.36335i	−0.590192	+	0.655474i	−0.962069	−	0.272808i	\(-0.912048\pi\)
0.371877	+	0.928282i	\(0.378714\pi\)
\(17\)	0.549650	−	1.69165i	0.133310	−	0.410285i	−0.862014	−	0.506885i	\(-0.830797\pi\)
							0.995323	+	0.0966001i	\(0.0307968\pi\)
\(19\)	3.04970	+	2.21574i	0.699650	+	0.508325i	0.879818	−	0.475311i	\(-0.157664\pi\)
							−0.180168	+	0.983636i	\(0.557664\pi\)
\(23\)	1.45647	+	2.52268i	0.303695	+	0.526014i	0.976970	−	0.213378i	\(-0.0684464\pi\)
							−0.673275	+	0.739392i	\(0.735113\pi\)
\(29\)	−0.940139	−	8.94483i	−0.174579	−	1.66101i	−0.634403	−	0.773003i	\(-0.718754\pi\)
							0.459823	−	0.888010i	\(-0.347913\pi\)
\(31\)	3.97741	−	4.41736i	0.714363	−	0.793381i	−0.271230	−	0.962515i	\(-0.587430\pi\)
							0.985593	+	0.169134i	\(0.0540971\pi\)
\(37\)	7.60211	−	5.52325i	1.24978	−	0.908017i	0.251569	−	0.967839i	\(-0.419053\pi\)
							0.998209	+	0.0598221i	\(0.0190533\pi\)
\(41\)	1.19415	−	11.3616i	0.186495	−	1.77438i	−0.356161	−	0.934425i	\(-0.615915\pi\)
							0.542656	−	0.839955i	\(-0.317419\pi\)
\(43\)	−4.49581	+	7.78697i	−0.685605	+	1.18750i	0.287642	+	0.957738i	\(0.407129\pi\)
							−0.973246	+	0.229764i	\(0.926205\pi\)
\(47\)	0.311758	−	0.138803i	0.0454745	−	0.0202466i	−0.383873	−	0.923386i	\(-0.625410\pi\)
							0.429348	+	0.903139i	\(0.358743\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	495.2.bg.b.16.9	✓	192
9.4	even	3	inner	495.2.bg.b.346.16	yes	192
11.9	even	5	inner	495.2.bg.b.196.16	yes	192
99.31	even	15	inner	495.2.bg.b.31.9	yes	192

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
495.2.bg.b.16.9	✓	192	1.1	even	1	trivial
495.2.bg.b.31.9	yes	192	99.31	even	15	inner
495.2.bg.b.196.16	yes	192	11.9	even	5	inner
495.2.bg.b.346.16	yes	192	9.4	even	3	inner