Embedded newform 495.2.bg.b.31.9

• Label: 495.2.bg.b.31.9
• Level: $495$
• Weight: $2$
• Character: 495.31
• 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			31.9
Character	\(\chi\)	\(=\)	495.31
Dual form			495.2.bg.b.16.9

$q$-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 \(\left(\mathbb{Z}/495\mathbb{Z}\right)^\times\).

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

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
495.2.bg.b.16.9	✓	192	99.16	even	15	inner
495.2.bg.b.31.9	yes	192	1.1	even	1	trivial
495.2.bg.b.196.16	yes	192	9.7	even	3	inner
495.2.bg.b.346.16	yes	192	11.5	even	5	inner