Embedded newform 507.2.t.a.10.5

• Label: 507.2.t.a.10.5
• Level: $507$
• Weight: $2$
• Character: 507.10
• Analytic conductor: $4.048$
• Analytic rank: $0$
• Dimension: $336$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 507 = 3 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	507.t (of order \(78\), degree \(24\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.04841538248\)
Analytic rank:	\(0\)
Dimension:	\(336\)
Relative dimension:	\(14\) over \(\Q(\zeta_{78})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{78}]$

Embedding invariants

Embedding label			10.5
Character	\(\chi\)	\(=\)	507.10
Dual form			507.2.t.a.355.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.886049 - 0.180888i) q^{2} +(0.987050 - 0.160411i) q^{3} +(-1.08760 - 0.463382i) q^{4} +(0.108486 - 0.440146i) q^{5} +(-0.903591 - 0.0364135i) q^{6} +(-1.78657 + 0.847738i) q^{7} +(2.36833 + 1.63474i) q^{8} +(0.948536 - 0.316668i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 336 q + 14 q^{3} - 12 q^{4} + 14 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(170\)	\(340\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{5}{78}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.886049	−	0.180888i	−0.626531	−	0.127907i	−0.123739	−	0.992315i	\(-0.539489\pi\)
							−0.502792	+	0.864408i	\(0.667694\pi\)
\(3\)	0.987050	−	0.160411i	0.569874	−	0.0926135i
							\(5\)	0.108486	−	0.440146i	0.0485165	−	0.196839i	−0.941683	−	0.336503i	\(-0.890756\pi\)
0.990199	+	0.139663i	\(0.0446021\pi\)
\(7\)	−1.78657	+	0.847738i	−0.675260	+	0.320415i	−0.735262	−	0.677783i	\(-0.762941\pi\)
							0.0600016	+	0.998198i	\(0.480889\pi\)
\(11\)	−1.42276	+	4.26169i	−0.428979	+	1.28495i	0.483327	+	0.875440i	\(0.339428\pi\)
							−0.912306	+	0.409509i	\(0.865700\pi\)
\(13\)	−2.79694	+	2.27533i	−0.775732	+	0.631062i
							\(17\)	−0.619538	−	1.30565i	−0.150260	−	0.316666i	0.814338	−	0.580391i	\(-0.197100\pi\)
−0.964598	+	0.263725i	\(0.915049\pi\)
\(19\)	5.00433	+	2.88925i	1.14807	+	0.662840i	0.948417	−	0.317027i	\(-0.102684\pi\)
							0.199655	+	0.979866i	\(0.436018\pi\)
\(23\)	−1.80418	−	3.12494i	−0.376198	−	0.651594i	0.614307	−	0.789067i	\(-0.289435\pi\)
							−0.990506	+	0.137473i	\(0.956102\pi\)
\(29\)	−0.847011	+	4.14893i	−0.157286	+	0.770437i	0.822243	+	0.569136i	\(0.192722\pi\)
							−0.979529	+	0.201301i	\(0.935483\pi\)
\(31\)	4.73107	+	9.01429i	0.849724	+	1.61901i	0.783652	+	0.621200i	\(0.213355\pi\)
							0.0660724	+	0.997815i	\(0.478953\pi\)
\(37\)	−3.04001	+	4.80739i	−0.499774	+	0.790330i	−0.997009	−	0.0772863i	\(-0.975374\pi\)
							0.497235	+	0.867616i	\(0.334349\pi\)
\(41\)	−1.07393	−	6.60818i	−0.167720	−	1.03202i	−0.925287	−	0.379267i	\(-0.876176\pi\)
							0.757567	−	0.652757i	\(-0.226388\pi\)
\(43\)	2.29316	−	1.45011i	0.349704	−	0.221140i	−0.348051	−	0.937475i	\(-0.613157\pi\)
							0.697756	+	0.716336i	\(0.254182\pi\)
\(47\)	3.27002	+	0.397053i	0.476982	+	0.0579161i	0.355498	−	0.934677i	\(-0.384311\pi\)
							0.121485	+	0.992593i	\(0.461235\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	507.2.t.a.10.5	✓	336
169.17	even	78	inner	507.2.t.a.355.5	yes	336

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
507.2.t.a.10.5	✓	336	1.1	even	1	trivial
507.2.t.a.355.5	yes	336	169.17	even	78	inner