Embedded newform 507.2.m.a.40.15

• Label: 507.2.m.a.40.15
• Level: $507$
• Weight: $2$
• Character: 507.40
• Analytic conductor: $4.048$
• Analytic rank: $0$
• Dimension: $180$
• CM: no
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(4.04841538248\)
Analytic rank:	\(0\)
Dimension:	\(180\)
Relative dimension:	\(15\) over \(\Q(\zeta_{13})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{13}]$

Embedding invariants

Embedding label			40.15
Character	\(\chi\)	\(=\)	507.40
Dual form			507.2.m.a.469.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.53537 + 0.624913i) q^{2} +(-0.120537 + 0.992709i) q^{3} +(4.26668 + 2.23932i) q^{4} +(-2.06622 + 2.99344i) q^{5} +(-0.925962 + 2.44156i) q^{6} +(-3.04788 - 2.70019i) q^{7} +(5.50913 + 4.88066i) q^{8} +(-0.970942 - 0.239316i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 180 q - q^{2} + 15 q^{3} - 15 q^{4} - 2 q^{5} + q^{6} + 4 q^{7} + 3 q^{8} - 15 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(170\)	\(340\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{1}{13}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.53537	+	0.624913i	1.79278	+	0.441880i	0.988354	−	0.152175i	\(-0.0486278\pi\)
							0.804424	+	0.594055i	\(0.202474\pi\)
\(3\)	−0.120537	+	0.992709i	−0.0695919	+	0.573141i
							\(5\)	−2.06622	+	2.99344i	−0.924043	+	1.33871i	0.0167973	+	0.999859i	\(0.494653\pi\)
−0.940841	+	0.338849i	\(0.889962\pi\)
\(7\)	−3.04788	−	2.70019i	−1.15199	−	1.02057i	−0.999479	−	0.0322725i	\(-0.989726\pi\)
							−0.152511	−	0.988302i	\(-0.548736\pi\)
\(11\)	0.242876	−	0.0598635i	0.0732298	−	0.0180495i	−0.202531	−	0.979276i	\(-0.564917\pi\)
							0.275760	+	0.961226i	\(0.411070\pi\)
\(13\)	3.42914	+	1.11400i	0.951073	+	0.308967i
							\(17\)	3.60029	+	3.18958i	0.873199	+	0.773586i	0.975230	−	0.221195i	\(-0.0709957\pi\)
−0.102031	+	0.994781i	\(0.532534\pi\)
\(19\)	7.68422			1.76288			0.881440	−	0.472296i	\(-0.156575\pi\)
							0.881440	+	0.472296i	\(0.156575\pi\)
\(23\)	−1.99289			−0.415547			−0.207773	−	0.978177i	\(-0.566622\pi\)
							−0.207773	+	0.978177i	\(0.566622\pi\)
\(29\)	−6.22050	−	1.53322i	−1.15512	−	0.284711i	−0.385187	−	0.922839i	\(-0.625863\pi\)
							−0.769931	+	0.638128i	\(0.779709\pi\)
\(31\)	3.56465	−	9.39921i	0.640230	−	1.68815i	−0.0821219	−	0.996622i	\(-0.526170\pi\)
							0.722352	−	0.691526i	\(-0.243061\pi\)
\(37\)	−0.996112	+	2.62653i	−0.163760	+	0.431799i	−0.991629	−	0.129123i	\(-0.958784\pi\)
							0.827869	+	0.560922i	\(0.189553\pi\)
\(41\)	0.281998	−	2.32246i	0.0440406	−	0.362707i	−0.953983	−	0.299862i	\(-0.903059\pi\)
							0.998023	−	0.0628455i	\(-0.0200175\pi\)
\(43\)	−2.57867	−	6.79938i	−0.393243	−	1.03690i	−0.974739	−	0.223347i	\(-0.928302\pi\)
							0.581496	−	0.813549i	\(-0.302468\pi\)
\(47\)	−4.00449	+	2.10172i	−0.584116	+	0.306568i	−0.730774	−	0.682620i	\(-0.760841\pi\)
							0.146658	+	0.989187i	\(0.453148\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	507.2.m.a.40.15	✓	180
169.131	even	13	inner	507.2.m.a.469.15	yes	180

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
507.2.m.a.40.15	✓	180	1.1	even	1	trivial
507.2.m.a.469.15	yes	180	169.131	even	13	inner