Embedded newform 507.2.t.b.4.2

• Label: 507.2.t.b.4.2
• Level: $507$
• Weight: $2$
• Character: 507.4
• Analytic conductor: $4.048$
• Analytic rank: $0$
• Dimension: $360$
• 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:	\(360\)
Relative dimension:	\(15\) over \(\Q(\zeta_{78})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{78}]$

Embedding invariants

Embedding label			4.2
Character	\(\chi\)	\(=\)	507.4
Dual form			507.2.t.b.127.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.53957 - 0.102341i) q^{2} +(-0.278217 + 0.960518i) q^{3} +(4.44542 + 0.358871i) q^{4} +(1.66050 + 0.629744i) q^{5} +(0.804853 - 2.41083i) q^{6} +(-2.05060 - 4.81293i) q^{7} +(-6.20652 - 0.753608i) q^{8} +(-0.845190 - 0.534466i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 360 q - 15 q^{3} - 14 q^{4} + 3 q^{7} + 15 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.53957	−	0.102341i	−1.79575	−	0.0723661i	−0.880206	−	0.474592i	\(-0.842596\pi\)
							−0.915540	+	0.402226i	\(0.868237\pi\)
\(3\)	−0.278217	+	0.960518i	−0.160629	+	0.554555i
							\(5\)	1.66050	+	0.629744i	0.742598	+	0.281630i	0.696746	−	0.717318i	\(-0.254631\pi\)
0.0458519	+	0.998948i	\(0.485400\pi\)
\(7\)	−2.05060	−	4.81293i	−0.775052	−	1.81911i	−0.525611	−	0.850725i	\(-0.676163\pi\)
							−0.249441	−	0.968390i	\(-0.580247\pi\)
\(11\)	0.626521	+	0.990764i	0.188903	+	0.298727i	0.926339	−	0.376690i	\(-0.122938\pi\)
							−0.737436	+	0.675417i	\(0.763964\pi\)
\(13\)	−3.48167	+	0.936989i	−0.965643	+	0.259874i
							\(17\)	−4.86253	+	2.07173i	−1.17934	+	0.502468i	−0.890532	−	0.454920i	\(-0.849668\pi\)
−0.288804	+	0.957388i	\(0.593258\pi\)
\(19\)	0.350140	−	0.202153i	0.0803275	−	0.0463771i	−0.459298	−	0.888282i	\(-0.651899\pi\)
							0.539626	+	0.841905i	\(0.318566\pi\)
\(23\)	−1.21725	+	2.10834i	−0.253814	+	0.439619i	−0.964573	−	0.263817i	\(-0.915018\pi\)
							0.710759	+	0.703436i	\(0.248352\pi\)
\(29\)	−0.322649	+	8.00644i	−0.0599143	+	1.48676i	0.637700	+	0.770285i	\(0.279886\pi\)
							−0.697614	+	0.716474i	\(0.745755\pi\)
\(31\)	3.37219	+	3.80642i	0.605663	+	0.683652i	0.968732	−	0.248110i	\(-0.0798094\pi\)
							−0.363069	+	0.931762i	\(0.618271\pi\)
\(37\)	1.51475	−	0.309239i	0.249023	−	0.0508385i	−0.0738923	−	0.997266i	\(-0.523542\pi\)
							0.322916	+	0.946428i	\(0.395337\pi\)
\(41\)	−9.64780	−	2.79452i	−1.50673	−	0.436431i	−0.580777	−	0.814062i	\(-0.697251\pi\)
							−0.925956	+	0.377632i	\(0.876739\pi\)
\(43\)	−1.80773	+	8.85483i	−0.275676	+	1.35035i	0.572183	+	0.820126i	\(0.306097\pi\)
							−0.847858	+	0.530223i	\(0.822108\pi\)
\(47\)	−7.01712	+	4.84357i	−1.02355	+	0.706507i	−0.956646	−	0.291253i	\(-0.905928\pi\)
							−0.0669054	+	0.997759i	\(0.521313\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	507.2.t.b.4.2	✓	360
169.127	even	78	inner	507.2.t.b.127.2	yes	360

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
507.2.t.b.4.2	✓	360	1.1	even	1	trivial
507.2.t.b.127.2	yes	360	169.127	even	78	inner