Embedded newform 507.2.m.b.40.12

• Label: 507.2.m.b.40.12
• Level: $507$
• Weight: $2$
• Character: 507.40
• Analytic conductor: $4.048$
• Analytic rank: $0$
• Dimension: $204$
• 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:	\(204\)
Relative dimension:	\(17\) over \(\Q(\zeta_{13})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{13}]$

Embedding invariants

Embedding label			40.12
Character	\(\chi\)	\(=\)	507.40
Dual form			507.2.m.b.469.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.31905 + 0.325116i) q^{2} +(0.120537 - 0.992709i) q^{3} +(-0.136731 - 0.0717618i) q^{4} +(0.705120 - 1.02154i) q^{5} +(0.481738 - 1.27024i) q^{6} +(-2.12000 - 1.87816i) q^{7} +(-2.19076 - 1.94084i) q^{8} +(-0.970942 - 0.239316i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 204 q - q^{2} - 17 q^{3} - 21 q^{4} - 6 q^{5} - q^{6} - 8 q^{7} - 9 q^{8} - 17 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.31905	+	0.325116i	0.932706	+	0.229891i	0.676257	−	0.736666i	\(-0.263601\pi\)
							0.256450	+	0.966558i	\(0.417447\pi\)
\(3\)	0.120537	−	0.992709i	0.0695919	−	0.573141i
							\(5\)	0.705120	−	1.02154i	0.315339	−	0.456848i	−0.632979	−	0.774169i	\(-0.718168\pi\)
0.948318	+	0.317321i	\(0.102783\pi\)
\(7\)	−2.12000	−	1.87816i	−0.801286	−	0.709877i	0.159491	−	0.987199i	\(-0.449015\pi\)
							−0.960777	+	0.277322i	\(0.910553\pi\)
\(11\)	−2.35036	+	0.579313i	−0.708661	+	0.174669i	−0.577141	−	0.816644i	\(-0.695832\pi\)
							−0.131520	+	0.991314i	\(0.541986\pi\)
\(13\)	−2.86994	−	2.18254i	−0.795977	−	0.605327i
							\(17\)	0.713067	+	0.631722i	0.172944	+	0.153215i	0.745155	−	0.666892i	\(-0.232376\pi\)
−0.572211	+	0.820107i	\(0.693914\pi\)
\(19\)	7.25578			1.66459			0.832295	−	0.554332i	\(-0.187026\pi\)
							0.832295	+	0.554332i	\(0.187026\pi\)
\(23\)	5.35073			1.11570			0.557852	−	0.829941i	\(-0.311626\pi\)
							0.557852	+	0.829941i	\(0.311626\pi\)
\(29\)	−3.19371	−	0.787179i	−0.593057	−	0.146175i	−0.0686467	−	0.997641i	\(-0.521868\pi\)
							−0.524411	+	0.851466i	\(0.675714\pi\)
\(31\)	0.0533093	−	0.140565i	0.00957463	−	0.0252462i	−0.930144	−	0.367196i	\(-0.880318\pi\)
							0.939718	+	0.341949i	\(0.111087\pi\)
\(37\)	3.25475	−	8.58206i	0.535077	−	1.41088i	−0.346145	−	0.938181i	\(-0.612510\pi\)
							0.881222	−	0.472702i	\(-0.156721\pi\)
\(41\)	1.07940	−	8.88968i	0.168574	−	1.38834i	−0.625362	−	0.780335i	\(-0.715049\pi\)
							0.793937	−	0.608001i	\(-0.208028\pi\)
\(43\)	−0.824290	−	2.17348i	−0.125703	−	0.331452i	0.857361	−	0.514716i	\(-0.172103\pi\)
							−0.983064	+	0.183264i	\(0.941334\pi\)
\(47\)	4.95030	−	2.59812i	0.722075	−	0.378974i	−0.0632997	−	0.997995i	\(-0.520162\pi\)
							0.785375	+	0.619020i	\(0.212470\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	507.2.m.b.40.12	✓	204
169.131	even	13	inner	507.2.m.b.469.12	yes	204

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
507.2.m.b.40.12	✓	204	1.1	even	1	trivial
507.2.m.b.469.12	yes	204	169.131	even	13	inner