Embedded newform 507.2.t.b.127.1

• Label: 507.2.t.b.127.1
• Level: $507$
• Weight: $2$
• Character: 507.127
• 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			127.1
Character	\(\chi\)	\(=\)	507.127
Dual form			507.2.t.b.4.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.60202 + 0.104858i) q^{2} +(-0.278217 - 0.960518i) q^{3} +(4.76600 - 0.384751i) q^{4} +(-0.705070 + 0.267398i) q^{5} +(0.824645 + 2.47011i) q^{6} +(0.861409 - 2.02180i) q^{7} +(-7.19059 + 0.873096i) q^{8} +(-0.845190 + 0.534466i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 360 q - 15 q^{3} - 14 q^{4} + 3 q^{7} + 15 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(170\)	\(340\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{77}{78}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.60202	+	0.104858i	−1.83991	+	0.0741457i	−0.935668	−	0.352882i	\(-0.885202\pi\)
							−0.904238	+	0.427028i	\(0.859561\pi\)
\(3\)	−0.278217	−	0.960518i	−0.160629	−	0.554555i
							\(5\)	−0.705070	+	0.267398i	−0.315317	+	0.119584i	−0.507186	−	0.861837i	\(-0.669314\pi\)
0.191869	+	0.981421i	\(0.438545\pi\)
\(7\)	0.861409	−	2.02180i	0.325582	−	0.764169i	−0.674092	−	0.738647i	\(-0.735465\pi\)
							0.999674	−	0.0255218i	\(-0.00812474\pi\)
\(11\)	−2.17645	+	3.44179i	−0.656226	+	1.03774i	0.339061	+	0.940764i	\(0.389891\pi\)
							−0.995287	+	0.0969736i	\(0.969084\pi\)
\(13\)	0.939825	+	3.48091i	0.260661	+	0.965431i
							\(17\)	3.69987	+	1.57637i	0.897351	+	0.382326i	0.790866	−	0.611989i	\(-0.209630\pi\)
0.106485	+	0.994314i	\(0.466040\pi\)
\(19\)	−6.05425	−	3.49542i	−1.38894	−	0.801905i	−0.395745	−	0.918361i	\(-0.629513\pi\)
							−0.993196	+	0.116455i	\(0.962847\pi\)
\(23\)	0.272907	+	0.472689i	0.0569050	+	0.0985624i	0.893075	−	0.449909i	\(-0.148543\pi\)
							−0.836170	+	0.548471i	\(0.815210\pi\)
\(29\)	−0.236814	−	5.87648i	−0.0439752	−	1.09123i	−0.859054	−	0.511884i	\(-0.828948\pi\)
							0.815079	−	0.579350i	\(-0.196693\pi\)
\(31\)	−4.18685	+	4.72598i	−0.751981	+	0.848811i	−0.992552	−	0.121820i	\(-0.961127\pi\)
							0.240571	+	0.970631i	\(0.422665\pi\)
\(37\)	9.47866	+	1.93508i	1.55828	+	0.318126i	0.900423	−	0.435015i	\(-0.143257\pi\)
							0.657860	+	0.753141i	\(0.271462\pi\)
\(41\)	−1.15890	+	0.335681i	−0.180990	+	0.0524245i	−0.367475	−	0.930033i	\(-0.619778\pi\)
							0.186485	+	0.982458i	\(0.440291\pi\)
\(43\)	2.26922	+	11.1154i	0.346052	+	1.69508i	0.664624	+	0.747178i	\(0.268592\pi\)
							−0.318572	+	0.947899i	\(0.603203\pi\)
\(47\)	−1.43072	−	0.987558i	−0.208693	−	0.144050i	0.459134	−	0.888367i	\(-0.348160\pi\)
							−0.667827	+	0.744317i	\(0.732775\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	507.2.t.b.127.1	yes	360
169.4	even	78	inner	507.2.t.b.4.1	✓	360

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
507.2.t.b.4.1	✓	360	169.4	even	78	inner
507.2.t.b.127.1	yes	360	1.1	even	1	trivial