Embedded newform 507.2.p.a.64.3

• Label: 507.2.p.a.64.3
• Level: $507$
• Weight: $2$
• Character: 507.64
• Analytic conductor: $4.048$
• Analytic rank: $0$
• Dimension: $168$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 507 = 3 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	507.p (of order \(26\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.04841538248\)
Analytic rank:	\(0\)
Dimension:	\(168\)
Relative dimension:	\(14\) over \(\Q(\zeta_{26})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{26}]$

Embedding invariants

Embedding label			64.3
Character	\(\chi\)	\(=\)	507.64
Dual form			507.2.p.a.103.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.27015 - 0.275646i) q^{2} +(-0.748511 + 0.663123i) q^{3} +(3.13571 + 0.772883i) q^{4} +(-1.62219 - 3.09083i) q^{5} +(1.88202 - 1.29906i) q^{6} +(2.36774 + 0.897966i) q^{7} +(-2.62905 - 0.997067i) q^{8} +(0.120537 - 0.992709i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 168 q - 14 q^{3} + 12 q^{4} - 14 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/507\mathbb{Z}\right)^\times\).

\(n\)	\(170\)	\(340\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{1}{26}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.27015	−	0.275646i	−1.60524	−	0.194911i	−0.731685	−	0.681642i	\(-0.761266\pi\)
							−0.873552	+	0.486731i	\(0.838189\pi\)
\(3\)	−0.748511	+	0.663123i	−0.432153	+	0.382854i
							\(5\)	−1.62219	−	3.09083i	−0.725466	−	1.38226i	−0.916242	−	0.400626i	\(-0.868793\pi\)
0.190776	−	0.981634i	\(-0.438900\pi\)
\(7\)	2.36774	+	0.897966i	0.894922	+	0.339399i	0.758824	−	0.651296i	\(-0.225774\pi\)
							0.136098	+	0.990695i	\(0.456544\pi\)
\(11\)	2.74341	−	0.333111i	0.827170	−	0.100437i	0.304008	−	0.952670i	\(-0.401675\pi\)
							0.523163	+	0.852233i	\(0.324752\pi\)
\(13\)	−0.983970	+	3.46869i	−0.272904	+	0.962041i
							\(17\)	−0.725266	+	1.91237i	−0.175903	+	0.463818i	−0.993751	−	0.111616i	\(-0.964397\pi\)
0.817849	+	0.575433i	\(0.195167\pi\)
\(19\)		−	3.50341i		−	0.803737i	−0.915697	−	0.401869i	\(-0.868361\pi\)
							0.915697	−	0.401869i	\(-0.131639\pi\)
\(23\)	−0.539330			−0.112458			−0.0562290	−	0.998418i	\(-0.517908\pi\)
							−0.0562290	+	0.998418i	\(0.517908\pi\)
\(29\)	0.269017	−	2.21555i	0.0499552	−	0.411418i	−0.946184	−	0.323629i	\(-0.895097\pi\)
							0.996139	−	0.0877885i	\(-0.0279800\pi\)
\(31\)	0.902367	−	0.622859i	0.162070	−	0.111869i	−0.484269	−	0.874919i	\(-0.660914\pi\)
							0.646339	+	0.763050i	\(0.276299\pi\)
\(37\)	9.28653	−	6.41003i	1.52670	−	1.05380i	0.551044	−	0.834476i	\(-0.314230\pi\)
							0.975652	−	0.219326i	\(-0.0703859\pi\)
\(41\)	5.99191	+	6.76347i	0.935779	+	1.05628i	0.998303	+	0.0582325i	\(0.0185465\pi\)
							−0.0625242	+	0.998043i	\(0.519915\pi\)
\(43\)	4.88558	−	7.07798i	0.745043	−	1.07938i	−0.248797	−	0.968556i	\(-0.580035\pi\)
							0.993840	−	0.110826i	\(-0.0353496\pi\)
\(47\)	0.194320	+	0.788387i	0.0283445	+	0.114998i	0.983410	−	0.181397i	\(-0.0580618\pi\)
							−0.955066	+	0.296395i	\(0.904216\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	507.2.p.a.64.3	✓	168
169.103	even	26	inner	507.2.p.a.103.3	yes	168

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
507.2.p.a.64.3	✓	168	1.1	even	1	trivial
507.2.p.a.103.3	yes	168	169.103	even	26	inner