Embedded newform 507.2.t.b.322.14

• Label: 507.2.t.b.322.14
• Level: $507$
• Weight: $2$
• Character: 507.322
• 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			322.14
Character	\(\chi\)	\(=\)	507.322
Dual form			507.2.t.b.433.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.179755 - 2.22666i) q^{2} +(0.845190 + 0.534466i) q^{3} +(-2.95160 - 0.479682i) q^{4} +(0.332790 - 0.375642i) q^{5} +(1.34200 - 1.78588i) q^{6} +(2.16814 + 2.08252i) q^{7} +(-0.529437 + 2.14801i) q^{8} +(0.428693 + 0.903450i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 360 q - 15 q^{3} - 14 q^{4} + 3 q^{7} + 15 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(170\)	\(340\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{41}{78}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.179755	−	2.22666i	0.127106	−	1.57449i	−0.547817	−	0.836598i	\(-0.684541\pi\)
							0.674923	−	0.737888i	\(-0.264177\pi\)
\(3\)	0.845190	+	0.534466i	0.487971	+	0.308574i
							\(5\)	0.332790	−	0.375642i	0.148828	−	0.167992i	−0.669369	−	0.742930i	\(-0.733435\pi\)
0.818197	+	0.574938i	\(0.194974\pi\)
\(7\)	2.16814	+	2.08252i	0.819479	+	0.787120i	0.979807	−	0.199944i	\(-0.0640760\pi\)
							−0.160329	+	0.987064i	\(0.551255\pi\)
\(11\)	3.41636	+	1.62108i	1.03007	+	0.488775i	0.867246	−	0.497880i	\(-0.165888\pi\)
							0.162825	+	0.986655i	\(0.447939\pi\)
\(13\)	0.799644	−	3.51576i	0.221781	−	0.975096i
							\(17\)	2.67268	−	2.78255i	0.648219	−	0.674868i	−0.313284	−	0.949659i	\(-0.601429\pi\)
0.961504	+	0.274791i	\(0.0886088\pi\)
\(19\)	−2.59333	−	1.49726i	−0.594952	−	0.343495i	0.172101	−	0.985079i	\(-0.444944\pi\)
							−0.767053	+	0.641584i	\(0.778278\pi\)
\(23\)	−0.824404	−	1.42791i	−0.171900	−	0.297740i	0.767184	−	0.641427i	\(-0.221657\pi\)
							−0.939084	+	0.343687i	\(0.888324\pi\)
\(29\)	3.80868	+	0.307468i	0.707254	+	0.0570955i	0.428859	−	0.903371i	\(-0.358916\pi\)
							0.278395	+	0.960467i	\(0.410198\pi\)
\(31\)	−5.27452	−	0.640443i	−0.947332	−	0.115027i	−0.367736	−	0.929930i	\(-0.619867\pi\)
							−0.579597	+	0.814903i	\(0.696790\pi\)
\(37\)	0.127725	+	0.299781i	0.0209978	+	0.0492837i	0.930148	−	0.367184i	\(-0.119678\pi\)
							−0.909150	+	0.416468i	\(0.863268\pi\)
\(41\)	−0.142727	+	0.225705i	−0.0222902	+	0.0352492i	−0.856151	−	0.516725i	\(-0.827151\pi\)
							0.833861	+	0.551974i	\(0.186125\pi\)
\(43\)	−9.83741	−	4.19133i	−1.50019	−	0.639172i	−0.523154	−	0.852238i	\(-0.675245\pi\)
							−0.977038	+	0.213066i	\(0.931655\pi\)
\(47\)	1.07895	+	0.409191i	0.157381	+	0.0596866i	0.432037	−	0.901856i	\(-0.357795\pi\)
							−0.274657	+	0.961542i	\(0.588564\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	507.2.t.b.322.14	✓	360
169.95	even	78	inner	507.2.t.b.433.14	yes	360

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
507.2.t.b.322.14	✓	360	1.1	even	1	trivial
507.2.t.b.433.14	yes	360	169.95	even	78	inner