Embedded newform 507.2.x.b.2.9

• Label: 507.2.x.b.2.9
• Level: $507$
• Weight: $2$
• Character: 507.2
• Analytic conductor: $4.048$
• Analytic rank: $0$
• Dimension: $2784$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 507 = 3 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	507.x (of order \(156\), degree \(48\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.04841538248\)
Analytic rank:	\(0\)
Dimension:	\(2784\)
Relative dimension:	\(58\) over \(\Q(\zeta_{156})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{156}]$

Embedding invariants

Embedding label			2.9
Character	\(\chi\)	\(=\)	507.2
Dual form			507.2.x.b.254.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.0419920 + 2.08489i) q^{2} +(-1.62925 - 0.587818i) q^{3} +(-2.34662 - 0.0945657i) q^{4} +(0.137218 - 0.748772i) q^{5} +(1.29395 - 3.37213i) q^{6} +(-1.58590 + 2.39953i) q^{7} +(0.0438824 - 0.725462i) q^{8} +(2.30894 + 1.91541i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2784 q - 50 q^{3} - 92 q^{4} - 50 q^{6} - 100 q^{7} - 56 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(170\)	\(340\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{1}{156}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.0419920	+	2.08489i	−0.0296929	+	1.47424i	0.660616	+	0.750724i	\(0.270295\pi\)
							−0.690308	+	0.723515i	\(0.742525\pi\)
\(3\)	−1.62925	−	0.587818i	−0.940650	−	0.339377i
							\(5\)	0.137218	−	0.748772i	0.0613655	−	0.334861i	−0.938583	−	0.345055i	\(-0.887860\pi\)
0.999948	+	0.0101940i	\(0.00324491\pi\)
\(7\)	−1.58590	+	2.39953i	−0.599413	+	0.906937i	−0.999956	−	0.00934094i	\(-0.997027\pi\)
							0.400543	+	0.916278i	\(0.368822\pi\)
\(11\)	−1.70635	−	0.939867i	−0.514485	−	0.283380i	0.203942	−	0.978983i	\(-0.434625\pi\)
							−0.718427	+	0.695603i	\(0.755137\pi\)
\(13\)	−2.81348	−	2.25485i	−0.780318	−	0.625383i
							\(17\)	−1.34444	−	6.58548i	−0.326073	−	1.59721i	−0.730678	−	0.682722i	\(-0.760796\pi\)
0.404605	−	0.914492i	\(-0.367409\pi\)
\(19\)	1.23653	+	4.61480i	0.283680	+	1.05871i	0.949798	+	0.312863i	\(0.101288\pi\)
							−0.666118	+	0.745846i	\(0.732045\pi\)
\(23\)	−2.76916	−	4.79633i	−0.577410	−	1.00010i	−0.995775	−	0.0918246i	\(-0.970730\pi\)
							0.418365	−	0.908279i	\(-0.362603\pi\)
\(29\)	4.74555	−	4.55817i	0.881227	−	0.846430i	−0.107799	−	0.994173i	\(-0.534380\pi\)
							0.989026	+	0.147743i	\(0.0472007\pi\)
\(31\)	−5.97908	−	2.69097i	−1.07387	−	0.483312i	−0.205213	−	0.978717i	\(-0.565789\pi\)
							−0.868662	+	0.495406i	\(0.835019\pi\)
\(37\)	−5.57368	+	0.563129i	−0.916307	+	0.0925779i	−0.547383	−	0.836882i	\(-0.684376\pi\)
							−0.368923	+	0.929460i	\(0.620273\pi\)
\(41\)	−6.93080	−	0.983551i	−1.08241	−	0.153605i	−0.423551	−	0.905872i	\(-0.639216\pi\)
							−0.658858	+	0.752268i	\(0.728960\pi\)
\(43\)	4.29249	−	3.50471i	0.654599	−	0.534463i	−0.245915	−	0.969291i	\(-0.579088\pi\)
							0.900513	+	0.434828i	\(0.143191\pi\)
\(47\)	−4.42545	+	1.37903i	−0.645519	+	0.201152i	−0.602746	−	0.797933i	\(-0.705927\pi\)
							−0.0427726	+	0.999085i	\(0.513619\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	507.2.x.b.2.9	✓	2784
3.2	odd	2	inner	507.2.x.b.2.50	yes	2784
169.85	odd	156	inner	507.2.x.b.254.50	yes	2784
507.254	even	156	inner	507.2.x.b.254.9	yes	2784

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
507.2.x.b.2.9	✓	2784	1.1	even	1	trivial
507.2.x.b.2.50	yes	2784	3.2	odd	2	inner
507.2.x.b.254.9	yes	2784	507.254	even	156	inner
507.2.x.b.254.50	yes	2784	169.85	odd	156	inner